Missing Page 
 
Cornell University Library 
 
 QC 385.F41 
 
 Notes on light. 
 
 3 1924 002 932 345 
 
CHAPTER I. 
 THE NATURE OF LIGHT. 
 
 1. Light is a Wave Motion. — That which is capable of 
 effecting the sensation of siglit is called light. That part of 
 physics which deals with the phenomena and laws of light is 
 called optics. 
 
 At the opening of the nineteenth century Thomas Young 
 performed an experimerit-that is of fundamental importance in 
 the theory of optics. This experiment can be easily repeated in 
 the following manner : A narrow slit S, in a diaphragm A, is 
 
 Mg. 1 
 
 illumined by light from the sun or some other source L. In 
 the diaphragm B there are two narrow slits, 5*1 and 5"2, close 
 together, parallel to >S", and equally distant from it. Light which 
 traverses the slits 5"i and S^ is received on the screen C. 
 
 When the apparatus is arranged as indicated, it is found 
 that the illumination of the screen C is not uniform or contin- 
 uous, but that it consists of a series of narrow bright bands 
 alternating with narrow dark streaks, with their lengths 
 parallel to the slits in the diaphragms. It is observed that if 
 either S^ or S^ is covered, the other remaining open, the illumi- 
 nation of the screen becomes continuous, that is, without bands. 
 When both slits, 5'i and S2, are open, the intensity of the illumi- 
 nation at a bright band is more than double the intensity at 
 the same place when but a single slit is open; but in the space 
 between two adjacent bright bands there is little or no illumina- 
 tion when light comes from both slits, whereas there is consider- 
 able illumination when light comes from a single slit. 
 
 The fact that the illumination at a given point due. to light 
 from two sources can be of zero value, whereas when light 
 comes from but one of the sources the illumination at the same 
 point is not zero, indicates that light is a phenomenon capable 
 of interference effects. Since interference effects can be pro- 
 duced only by wave motions, it follows that light is a wave 
 motion. 
 
NATURE OF LIGHT 
 
 Fig. 2 is a reproduction of a photograph of a set of inter- 
 ference fringes obtained by using a photographic plate for the 
 screen C, Fig. 1, and at B a diaphragm provided witlj two slits, 
 one of which is partly closed by a narrow tongue as shown in 
 
 Fig. 2 
 
 Fig. 3 
 
 Fig. 3. It will be noted that in the photograph, Fig. 2, the 
 regions illumined by light from two slits are crossed by bright 
 and dark fri,nges, whereas the region illumined by light from a 
 single slit contains lio fringes. 
 
 Interference of water waves can be obtained in a manner 
 quite similar to Young's method for obtaining interference of 
 light. Let the plane of the page represent the surface of water 
 
 Fig. 4 
 
 in a large tank or pond. Let A and B^, Fig. 4, represent the 
 edges of two long boards which project through the surface. 
 
LIGHT 3 
 
 In A there is a narrow slit S, and in B there are two similar 
 narrow slits, close together, parallel to and equally distant 
 from S. If the water surface be slightly tapped with the finger 
 at regular intervals at some point X, a circular wave will pro- 
 ceed from this point as a center. A narrow portion of the 
 wave will proceed through the aperture 5" and portions of this 
 wave will emerge from S^ and S^- Since S-^ and 5*2 are 
 arranged symmetrically with respect to S, the waves at 5"i and S^ 
 will at any given instant be in the same phase and have the 
 same amplitude. The portions of the water surface to the right 
 of B will now be traversed by waves that proceed as though 
 they had been set up by two periodic disturbances at S^ and S"j 
 of the same period, phase and amplitude. 
 
 At some particular instant the crests of the waves pro- 
 ceeding from S^ and S^ will be in the positions marked by the 
 circular arcs drawn in full lines (Fig. 4), and the troughs of 
 the waves by the circular arcs drawn in dotted lines. The eleva- 
 tion of the water above the undisturbed surface, at any pioint, 
 equals the sum of the effects of the two waves at the given point. 
 At a point where two crests coincide the elevation is greater 
 than that which would be produced by a wave from either of 
 the sources ; at a point where two troughs coincide the depres- 
 sion below the undisturbed level is greater than that which would 
 be produced by a wave from either of the sources ; at a point 
 where a crest and a trough coincide the elevation is small — ^that 
 is, the waves here tend to neutralize one another's effect and 
 produce destructive interference. Since each particle of water 
 moves up and down with the same frequency, it follows that 
 the particles now displaced the maximum distance above the 
 undisturbed level will at an instant one-half period later be 
 displaced the maximum distance below the undisturbed level. 
 The particles at one time undisturbed will remain undisturbed. 
 Thus along the lines marked mm' the particles are in a state of 
 maximum disturbance, and along the lines mo the particles are 
 undisturbed. 
 
 The condition for maximum disturbance at any g^ven point 
 is that at this point the two waves shall be in the same phase. 
 Now the two sources of disturbance, S^ and ^"2, vibrate in the 
 same period and same phase, consequently there will be maxi- 
 mum disturbance at any point whose distance from one source 
 differs from its distance to the other source by an amount equal 
 to any even number of half wave lengths. Similarly, there will 
 
NATURE OF LIGHT 
 
 Fig. 5 
 
 be minimum disturbance at any 
 point whose distance from one 
 source differs from its distance to 
 the other source by an amount equal 
 to any odd number of half wave 
 lengths. 
 
 Fig. S was made from a photo- 
 graph of the waves on the surface of 
 mercury produced by the vibration up 
 and down of two iron wires.- The two 
 vibrations were of the same period 
 and in the same phase. Note the lines 
 along which complete interference oc- 
 curs. 
 
 2. The Determination of the Wave Length of Light. — 
 
 In Fig. 6 let L represent a point source that emits light waves of 
 uniform length. Light from this source, after traversing the 
 three slits of Young's apparatus, strikes the screen XXj. At a 
 point X light from the slits S-^ and S^ arrives in the same phase. 
 Consequently, at this point there is reinforcement. At some 
 point Fi light from the two slits will arrive in opposite phases 
 if the distance S-^ F^ differs by one-half wave length from the dis- 
 tance S^ Fj. There will here be destructive interference. At 
 some point X^, so situated that its distance from one slit is two 
 half wave lengths less than its distance from the other slit, there 
 
 « 
 
 X. 
 
 Fig. 6 
 
 will again be reinforcement, and consequently a bright band. In 
 the same way it is seen that there will be other dark bands on 
 the screen at places Y^, Fg, etc., such that 
 
LIGHT 
 
 
 
 ^x 
 
 y^ 
 
 - -^3 
 
 Y. 
 
 -h 
 
 
 
 
 
 ^x 
 
 Y, 
 
 - s. 
 
 Y, 
 
 'h 
 
 
 
 etc., 
 and that there will be other- bright bands at 
 such that 
 
 places 
 
 X: 
 
 
 
 ■S-x 
 
 X, 
 
 - s. 
 
 X, 
 
 =i^ 
 
 
 
 
 
 Sx 
 
 ^3 
 
 - s. 
 
 X, 
 
 -h 
 
 
 
 Xs, etc., 
 
 etc. 
 where A represents the wave length of the given disturbance. 
 
 This experiment affords a method for determining the wave 
 length of the light that illumines the slits. The rationale of 
 the method is rendered clear by the following scheme : 
 
 From Xi lay off on X-^S^ a distance X^P equal to Xj^S^, and 
 draw S^P Then, since PX^ = S^X^, and since the angle 
 PX^S^ is very small, SJ' is very nearly perpendicular to the 
 lines ^1^1, AX.^ and S^X^. 
 
 Since AX is perpendicular to S-^ S2, and i", P is very nearly 
 perpendicular to AX^, the angles S^S^P and XAX^ are very 
 nearly equal. Moreover, AXX^ is a right angle, and SJPS^ very 
 nearly a right angle. Consequently, the triangle SJPS^ and 
 X^XA are very nearly similar. 
 
 Consequently, 
 
 ^ = ^ (nearly). (1) 
 
 Since Xj is at the middle of the first bright band, and since 
 PX^ is equal to ^2^x; S\P is one wave length. Letting A denote 
 the wave length of the light, and substituting in (1) the symbols- 
 represented in Fig. 6, we have 
 
 Or, since x is very small compared with b, 
 
 f — = -; — (nearly). 
 a 
 
 Consequently, to the above degree of approximation, the wave 
 
 length of the light emitted by the source is 
 
 A=-'^ (2) 
 
( 
 
 6 NATURE OF LIGHT 
 
 It is found that the wave length of light from a given source 
 depends upon the medium through which the light is traveling. 
 For example, the wave length of light in water is about two- 
 thirds of the wave length in air. 
 
 3. Velocity of Light. — Consider light proceeding along the 
 
 -M 
 
 Fig. 7 
 
 path SM. If a mirror M be placed normal to SM, the incident 
 light will be reflected back along the path MS and will enter an 
 eye placed behind ^. Let a toothed wheel having teeth and spaces 
 of equal width be placed with the axle parallel to the direction 
 of the light. When the wheel is at rest, light traversing the 
 space between two teeth will return through the same space. But 
 the wheel can be rotated at such a speed that while light goes 
 from A to M and back, a tooth will have advanced just enough 
 to intercept the returning light. If the time occupied by a tooth 
 to move into the position occupied by the adjacent space be de- 
 noted by f. then the speed of the light is 
 
 2(AM) 
 
 ' = —r- 
 
 If the angular speed of the wheel be doubled, the reflected 
 light will pass through the next space. If the angular speed be 
 trebled, the reflected light will be intercepted by the second tooth. 
 
 In Fizeau's experiment, the wheel had 720 teeth and 720 spaces, 
 all of the same width. The distance between the wheel and the mirror 
 was 8663 meters. The first eclipse of the light occurred when the 
 wheel was making 12.6 revolutions per second. Hence the time occu- 
 pied by a tooth to move its own width was 
 
 1 
 t=- 
 
 2 (720) (12.6) 
 This would give for the speed of light 
 
 2 (a m) "I _^ 
 
 t 
 
 1 = 2(8663) [2(720) (12.6)] 
 
 = 314,362,944 meters per second. 
 
 = 195,200 miles per second. 
 
LIGHT 7 
 
 More tecent determinations of the speed of light in vacuo 
 give 300,574,000 meters per second, or 186,700 miles per second. 
 The speed depends upon the medium being traversed, being \ 
 greatest in vacuo. (In water, the speed of light is about two- 
 thirds the speed in vacuo.) I n vacu o, the SEgeji of light of all I 
 colors is the same ; but in matter, the speed" of light of different 1 
 colors is not the same. 
 
 4. Color and Frequency. — The speed, s, of any wave 
 motion of wave length, \, is 
 
 s = N\ (3) 
 
 where N represents the frequency, or number of vibrations per 
 unit of time. It is found that when a beam of light traverses 
 different media, the speed and the wave length change in such ai' 
 manner that the ratio of speed to wave length is always constant. I 
 L Therefore the frequency remain s c onstant. Kp^>*t'° e^o-.. t ^^s^ <^ 
 
 In traversing various colorless media, the speed and the 
 wave length of light will change, without any change of the color 
 of the light. Whence, we conclude that color is not a matter 
 of either speed or wave length, but is due to the frequency of the 
 disturbance. A color can be specified by the frequency of the 
 wave that produces it. 
 
 To determine the frequency of a wave we must first know 
 ' the speed and the wave length in some particular medium. But 
 since for any assigned medium the wave length is constant when 
 the period is constant, we can also specify color by the wave 
 length in some assigned medium of the disturbance which pro- 
 duces it. This has led to the custom of specifying color in terms 
 of the wave length in air._ 
 
 It is found that in air the wave length of the light that we 
 call yellow is about 0.000059 cm. An object emitting or reflect- 
 ing light of this particular color would be said to be of the color 
 corresponding to wave length 0.000059 cm. 
 
 The human eye is sensitive to waves in air of lengths from 
 0.000039 cm. to 0.000077 cm. Waves of the former length pro- 
 duce the sensation called violet, and waves of the latter the sen- 
 sation called red. All of the other colors have wave lengths be- 
 tween these limits. Waves in air shorter than the violet cannot 
 be detected by the eye, but ean be observed by photography. 
 Waves in air longer than the red cannot be detected by the eye, 
 but can be observed by means of a thermometer. 
 
 If the two slits S-^ and S^ of Young's experiment, Fig. 6, be 
 illumined by a mixture of two colors, there will be formed^ on 
 the screen two systems of bands, one for each color. If the slits 
 
8 NATURE OF LIGHT 
 
 are illumined by sunlight, the screen will be crossed by a series 
 of rainbow colored bands. This indicates that sunlight is a mix- 
 ture of lights of many wave lengths. Light that contains all 
 visible wave lenghts in about the same proportion as in sunlight 
 is called white' light. Light consisting of waves all of which have 
 the same wave length is called monochromatic or homogeneous 
 light. 
 
 5. Color and Visibility. — For light of certain colors the 
 eye is less sensitive than for light of other colors. In order that 
 deep blue light (wave length in air 0.000046 cm.) and deep red 
 light (wave length in air 0.000066 cm.) may produce equally 
 intense sensations of brightness, the energy of the two waves 
 must be equal. But to produce an equally intense sensation of 
 brightness with yellow-green light (wave length 0.000054 cm.), 
 the energy of the latter must be but one-tenth as great as the 
 energy of either of the other two colors. This is expressed by 
 the statement that the visibility of the yellow-green light is ten 
 
 Fig. 8 
 
 times as great as that of either the deep blue or the deep red 
 light of the specified wave lengths. The relative visibility of the 
 light of different wave lengths is represented graphically in Fig. 8. 
 
 6. Light Waves are Transverse. — Soon after Young's 
 interference experiment, Malus made an observation which 
 showed that the vibrations constituting light waves are transverse 
 to the direction of their propagation through the medium. 
 
 The observation of Malus can be repeated in the following 
 manner: Each of the glass mirrors A and B, Fig. 9, is capable 
 of rotation about a horizontal axis, and the mirror B is in addi- 
 tion capable of rotation about a vertical axis. If sunlight is 
 
LIGHT 
 
 Fig. 9 
 
 Pig. 10 
 
 reflected from the mirror A to the mirror 
 B, it is found that when the horizontal 
 axes of the mirrors are parallel (as in 
 Fig. 9), light is copiously reflected from 
 B, whereas when the horizontal axes of 
 the mirrors are at right angles (as in 
 Fig. 10), a much less amount of light is 
 reflected from B. If the mirrors are so 
 arranged that their horizontal axes are 
 at right angles to one another, and if in 
 addition the plane of each mirror makes 
 an angle of about 33° to the direction of the path of the light 
 incident upon it, the light reflected from B will be minimum. 
 If now the upper mirror is rotated about its vertical axis, this 
 light reflected from B will increase, becoming a maximum when 
 the horizontal axes of the mirrors are parallel; if the rotation 
 is continued the light reflected from B will diminish, becoming 
 again almost zero when the horizontal axes are again at right 
 angles to one another ; again increasing and becoming a maxi- 
 mum when the axes are parallel. Thus in two positions 180° 
 apart the light reflected from B is almost zero, while at two in- 
 termediate positions the light reflected from B is a maximum. 
 Fresnel pointed out that these phenomena show , that light 
 waves cannot be longitudinal. If the vibrations constituting light 
 were longitudinal they would meet a mirror in exactly the same 
 way whether the disturbance had or had not been previously 
 reflected from another mirroi- ; but if the vibrations are trans- 
 verse to the direction of propagation of the disturbance no such 
 symmetry exists. For instance, imagine two transverse wave 
 
 motions a and b to be incident on 
 a mirror M, Fig. 11, perpendicu- 
 lar to the plane of the paper. Let 
 the parallel lines across a indicate 
 that in this wave the vibrations 
 are parallel to the plane of the 
 paper, and let the dots on h in- 
 dicate that in this wave the 
 vibrations are perpendicular to the plane of the paper. These 
 two transverse wave motions meet the mirror in different. ways, 
 and we should expect they would not be similarly reflected. 
 
 Reflection would occur as observed by Malus if light waves 
 consist of vibrations that are successively in a great many direc- 
 tions transverse to the line of propagation, and if in addition the 
 vibrations that are reflected consist of those that are nearly 
 
 Pig. 11 
 
10 NATURE OF LIGHT 
 
 parallel to the surface of the mirror. Thus in Fig. 9, if only 
 those vibrations of the light incident on A that are nearly parallel 
 to the surface of the mirror are reflected, then the vibrations con- 
 stituting the light between A and B are. for the most part parallel 
 to the horizontal axis of A. And since the surface of B is 
 parallel to this direction, the light that is reflected from A will 
 also be reflected from B. But when the mirror B is turned as 
 in Fig. 10, its plane has no Hne parallel to the plane in which 
 occurs the greater part of the vibrations constituting the light 
 between A and B, and consequently there is little light reflected 
 from B. 
 
 It thus appears that a glass mirror can serve as a polarizer 
 of light waves. The degree of polarization depends upon the 
 angle between the direction of the incident light and the plane 
 of the mirror. 
 
 7. The Luminiferous Ether. — For the propagation of 
 waves a medium is necessary. Since light is transmitted through 
 interplanetary space and through the most nearly perfect vacuum 
 that can be produced, we must conclude that ordinary matter is 
 not necessary for its propagation. This compels us to conclude 
 that there is a medium other than matter by which light is propa- 
 gated. This medium is not perceived by our senses, but the 
 above facts convince us of its actual existence. This medium is 
 called the luminiferous ether (i. e., light bearing spirit), or 
 briefly, the ether. The student should be warned against con- 
 fusing the luminiferous ether wifh the various volatile liquids 
 that in Chemistry are called ethers. 
 
 Since ether is the medium by which light is propagated, and 
 light is transmitted by ordinary matter, we conclude that the 
 ether permeates all space by filling the interstices between the 
 molecules and atoms of matter. 
 
 Young's experiment shows that light is propagated by waves 
 in the ether. It does not tell, however, whether these waves are 
 due to a periodic to-and-fro motion of particles of the ether, or 
 whether these waves are due to a periodic change of some one of 
 the properties of ether. By means of waves, energy can be trans- 
 ferred from one region to another not only by a vibration of 
 particles of the intervening medium, but also by means of a 
 periodic electric or magnetic disturbance handed on successively 
 from one portion of the medium to another. 
 
 8. Effects of Light.— Wave motion implies energy. The 
 energy associated with light waves exhibits itself in several 
 effects. First, this energy may be transformed into heat energy. 
 
LIGHT 11 
 
 When light waves go from one medium to a different medium, i 
 at 'the second medium the light is usually in part reflected back 
 into the first medium, in part transmitted through the second 
 medium, and in part absorbed by the second medium. Polished 
 silver is a good reflector; clear polished glass is a good trans- 
 mitter; lamp-black is a good absorber of light. When light is 
 incident on lamp-black, about 98 per cent, of the energy is trans- 
 formed into heat, and about 2 per cent, is reflected back into the 
 first medium. By coating a thermometer with lamp-black the 
 energy of the light incident upon it can be determined. 
 
 Second, this energy of light may be transformed into poten- 
 tial energy. When light waves are incident upon a substance, 
 instead of being transformed into heat, the energy may break 
 down the molecules composing the substance and cause the atoms 
 to unite in a different combination. In other words, a different 
 chemical substance may be formed. The action of light waves 
 on a photographic plate is an example of this effect. 
 
 Third, light may excite the sensation of vision. 
 
CHAPTER II. 
 THE PROPAGATION OF LIGHT. 
 
 §1. Quantity of Light. 
 
 9. Isotropic and Anisotropic Substances. — Any portion 
 of a uniform specimeii of glass or of quartz is similar in all 
 respects to any other portion of the same specimen so long as 
 the portion considered is larger than molecular dimensions. 
 That is, uniform glass is homogeneous and uniform quartz is 
 homogeneous. But glass differs from quartz in that light, heat 
 and mechanical vibrations are transmitted with the same speed 
 in all directions through uniform glass, whereas light, heat and 
 mechanical vibrations are transmitted with unequal speed in dif- 
 ferent -directions through uniform quartz. 
 
 A medium which at any point has the same properties in 
 all directions is said to be isotropic. A medium which at any 
 given point has different properties in different directions is said 
 to be anisotropic or aeolotropic. Under ordinary conditions 
 liquids and gases are isotropic, while crystals are anisotropic. 
 Annealed glass is isotropic, but unannealed strained glass is 
 anisotropic. 
 
 10. Measurement of Solid Angles. — A plane angle <^ is 
 measured by the ratio of the length x of the arc of any circle 
 drawn with O as a center, to the 
 length of the radius of this circle ; 
 that is, 
 
 (^ = — — radians. 
 
 Fig. 12 
 
 A solid angle O is measured in an analogous manner. ' With 
 C as a center. Fig. 13, construct a sphere of radius r. The ele- 
 ments of the pyramidal faces enclosing the solid angle n will cut 
 out of the sphere a surface of area a. Whatever the magnitude 
 of the radius r, the ratio of the area of the surface a to the 
 
 square of the radius r is a constant 
 quantity. Consequently, the meas- 
 ure of the solid angle O is taken as 
 
 " = -^ (4) 
 
 Fig. 13 
 
 For example, since the area of a sphere is A-irf', a sphere sub- 
 tends at its center a solid angle of 4 tt units. This unit is called 
 the space radian, or steradian. 
 
LIGHT 13 
 
 11. Light Units. — The quantity of water, heat, etc., that 
 passes a given surface in unit time is called the rate of flow or 
 flux of the water, heat, etc. The total visible energy emitted by 
 a source per second is called the total flux of light. The flux of 
 light of any given wave length from a source is measured by the 
 total energy emitted per second and by the sensitiveness of the 
 eye for radiant energy of the given wave length. Or, in symbols, 
 
 F^Kx P (5) 
 
 where F represents flux of light of wave length A, P represents 
 the total radiant energy emitted per second, and K\ is the retinal 
 stimulus coefficient for light of wave length A,. 
 
 For purposes of comparison, the light source used as a 
 standard is a lamp burning amyl acetate devised- by Hefner. The 
 unit of light-flux is the light-flux emitted in one space radian by 
 a Hefner lamp and is called the lumen. If the flame of a Hefner 
 lamp is .'It C, Fig. 13, then the light-flux in the solid angle fi is one 
 lumen when this angle is one space radian. Since there are 4 ir 
 space radians in a complete sphere, the total light-flux from the 
 Hefner lamp is 4 tt lumens. 
 
 The intensity of- a point light source is measured by the light- 
 flux emitted per unit solid angle. Thus, if the total flux emitted 
 isF, 
 
 J -I. (6) 
 
 4,r 
 
 This equation shows that the intensity of a point light source will 
 be unity when F ^Av lumens. This is the light-flux of a stand- 
 ard Hefner lamp. Consequently, the unit intensity of a point 
 light source is called the mean spherical hefner. 
 
 Light-flux per unit area of cross section of the beam is 
 called the light-flux density, or the illumination, of the stream 
 of light. 
 
 F 
 £=4- (7) 
 
 a 
 
 From this equation we derive the unit of illumination — one lumen 
 of light-flux per square meter of area. This unit is called the lux. 
 If we have a point source, then at a distance r the illumina- 
 tion would be 
 
 [ a i Ur' 
 Whence, from (6) 
 
14 PROPAGATION OF LIGHT 
 
 From this equation we see that at a distance of one meter 
 from a point source of luminous intensity one hefner, the illu- 
 mination is unity. Consequently, the unit of illumination is also 
 called the hefner-meter. 
 
 The ratio of the luminous intensity to the area of the lumi- 
 nous source is called the intrinsic luminosity or intrinsic brilliancy 
 of the source. The unit of intrinsic luminosity is one hefner per 
 square millimeter. 
 
 Before the Hefner standard lamp was devised, candles were 
 commonly used in Great Britain, France and America as luminous 
 standards. The standard British candle was one that burned 
 120 grains of spermacetti per hour. On account of the lack of 
 uniformity of even the most carefully made candles, candles are 
 now seldom used 'in actual photometric measurements. But as 
 actual candles were employed for a long time, in these countries 
 li^ht quantities are still usually expressed in terms of candles. 
 Thus, luminous intensities are' expressed in candles even though 
 actual candles are not used in making the measurement. By 
 agreement, ten-ninths hefners are taken as the international candle. 
 Illuminations due to point sources are expressed in candle-feet. 
 One candle-foot is the illumination at a distance of one foot from 
 a point source of luminous intensity one candle. In this system, 
 intrinsic brilliancy would be expressed in candles per square inch 
 of radiating surface. For example, the intrinsic brilliancy of the 
 tungsten filament of a certain incandescent lamp was 1000 candles 
 per square inch. After the globe was frosted, the intrinsic bril- 
 liancy of the globe was found to be approximately 10 candles per 
 square inch. 
 
 Problem. — Find the value of the candle-foot in (a), candle-meters, 
 (b), hefners per sq. ft., (c) lux. 
 Solution. — From (8) 
 
 1 (candle) 
 
 1 candle-foot ^=- 
 and since 1 m. = 3.28 ft. 
 
 1 candle-meter = 
 Whence, 1 candle-foot 
 
 (1)^ (ft.) 
 
 1 (candle) 
 (3.28)2 (ft.) 
 _ r 3.28 ] 2 _ 
 
 10.76 
 
 1 candle-meter 
 or, 1 candle-foot = 10.76 cand'e-meters. 
 
 Again, since the luminous intensity of a standard candle is ten- 
 ninths that of a hefner lamp, we have from (7) 
 
LIGHT IS 
 
 10 
 
 „ (hefners) . ., , , , 
 
 1 candle-foot = -^, ^^ — = 1.11 hefners per sq. ft. 
 
 1 (sq. ft.) 
 
 and since 1 sq. ft = 0.092 sq. m. 
 
 -^ (hefners) 
 1 candle-foot = nnno ( — ' 'A ^^ ^^ hefners per sq. m. or lux. 
 
 12. Illumination at Different Distances from a Point 
 Source. — Consider a luminous point of intensity I situated in a 
 transparent isotropic medium. - From (8) the illuminations at 
 
 distances r^ and r^ will be 
 Whence, 
 
 £i = J- and £2 
 
 E, - r,' (9) 
 
 Therefore, in the case of light emitted from a luminous 
 point in a transparent isotropic medium, the illumination of a 
 surface at any given distance is inversely proportional to the 
 square of that distance from the source. If the luminous source 
 is an extended surface, the front of the emitted wave will not be 
 spherical and the above "inverse square" law will not apply. 
 
 13. Photometry. — The art of comparing luminous inten- 
 sities is called photometry. Two beams of light of the same color 
 and different light-flux densities will produce retinal sensations 
 of the same kind, but different in magnitude. The magnitude of 
 the sensation, however, is not proportional to the rate with which 
 energy enters the eye. 
 
 Although the eye cannot accurately compare illuminations 
 of different color (Art. 5), or of different magnitude, even 
 though they are of the same color, still the eye can judge of the 
 equality of illuminations of the same color with a satisfactory 
 degree of precision. This fact is the basis of a simple method 
 of comparing the luminous intensities of two small sources. 
 
 Consider two point sources 
 emitting light of the same color 
 uniformly in all directions. Let 
 a small white screen A, Fig. 14, 
 r-^ +£^ Ijg placed between the two 
 sources and normal to the line 
 connecting them. Let the two 
 Fig. 14 sources be of luminous intensi- 
 
 r- — ■"^- 
 
16 
 
 PROPAGATION OF LIGHT 
 
 ties /i and 4, and let the distances of the screen from the two 
 sources be r^ and r^, respectively. Then the illuminations on the 
 two sides of the screen due to the two point sources are (8) 
 
 A and £. = A 
 
 If the screen be moved back and forth until the two sides 
 are equally illumined, that is, until E^ =£2, then 
 
 £, =. 
 
 Whence, 
 
 
 h ~ U' (10) 
 
 It should be kept in mind that this equation is true only for 
 point sources. But it may be used to a close degree of approxi- 
 mation fbr sources that are so small compared with their dis- 
 tances from the screen that at the screen the wave fronts are 
 nearly spherical. When the wave fronts are nearly plane, as in 
 the case of a search light, this method is entirely inapplicable. 
 
 If in the experiment a Hefner lamp is used for one of the 
 sources, then the equation gives the luminous intensity of the 
 other source in "hefners.". If one of the sources is a standard 
 candle, then the luminous intensity of the other will be expressed 
 in "candles." 
 
 Fig. 15 
 
 In Fig. IS is represented the actual apparatus for the comparison 
 of luminous intensities. In this case the intensity of the gas flame L 
 is to be compared with the intensity of the flame of the Hefner lamp 1. 
 M is a gas meter, R is a pressure regulator, and m is a manometer. 
 
LIGHT 
 
 17 
 
 The screen is in the box P. This "photometer box" is moved back 
 and forth along the scale S till the two sides of the screen are equally 
 illumined. The distances of the screen from the two light sources are 
 then observed. The luminuos intensity of the gas flame can now be 
 computed by means of (10), 
 
 §2. Change of the Form of a Wave Front by Reflection. 
 
 14. Reflection. — Consider the passage of a disturbance 
 from one medium to another in which the speed of the disturb- 
 ance 'is different. To fix the ideas, take the case of a long spiral 
 spring loaded for a part of its length with balls of large inertia 
 and loaded for the remainder of its length with balls of smaller 
 inertia. A disturbance will travel more slowly along the first 
 part of the spring than along the second part. Let the ball a, 
 
 ^frrm^ma^fmu^prwmiimnamramssvunmm 
 
 Pig. 16 
 
 Fig. 16, be displaced to the right a distance represented by the 
 heavy arrow below a. The portion of the spring between a and h 
 being now compressed more than that between b and c, the ball b 
 will be displaced to the right. The ball c will, in turn, be dis- 
 placed an equal amount. The motion of the ball d, however, is 
 opposed by the lesser inertia of the ball e and consequently will 
 be displaced a greater distance than any of the preceding balls. 
 The succeeding balls will be displaced through distances repre- 
 sented by the heavy arrows below them. 
 
 When the large displacement of d has occurred, the portion 
 of the spring between c and d is more extended than that between 
 b and c. Hence the ball c will be displaced to the right through 
 a distance represented by the light arrow below it. Thus, a pulse 
 of compression from a to d is succeeded by a pulse of rarefaction 
 from d to a. Similarly, if a pulse of rarefaction be sent from a, 
 then at d a pulse of rarefaction will continue into the second por- 
 tion of the spring, while a pulse of compression will be reflected 
 back through the first portion of the spring. It follows that if a 
 longitudinal wave originates in the left portion of the spring, then 
 at the junction of the two parts of the spring a compression will 
 be reflected as a rarefaction, and a rarefaction will be reflected 
 as a compression. In fact, whenever a wave of any type travels 
 from one medium to another in which the speed is greater, then 
 
18 
 
 PROPAGATION OF LIGHT 
 
 at the boundary separating the two media part of the. energy 
 will be transmitted into the second medium and the remainder 
 will be reflected with change of phase back through the first 
 medium. 
 
 The case is somewhat different when the wave proceeds from 
 one medium to another in which the speed is less. To fix the 
 
 ^fmn^rmn^pnm^mninmhmmmfnrmmh^^ 
 
 Fig. 17 
 
 ideas, let the ball h, Fig 17, be displaced to the left a distance 
 represented by the heavy arrow below h. The balls g, f and e 
 will, in turn, be displaced almost equal distances. The inertia 
 of d, however, is so great that d is displaced but slightly. The 
 displacements oi d, c,b and a are represented by the heavy arrows 
 below them. 
 
 ' Wlien the small displacement of d has occurred, the portion 
 of the spring between d and e is more compressed than the portion 
 between e and /. Hence, the ball e will be moved to the right 
 through a distance represented by the light arrow below it. Thus, 
 a pulse of compression from /j to d is succeeded by a pulse of 
 compression from d to h. Similarly, if a pulse of rarefaction be 
 sent from h, then at d a pulse of rarefaction will continue into the 
 second portion of the spring, while another pulse of rarefaction 
 will be reflected from d to h. In fact, whenever a wave of any 
 type travels from one medium to another in which the speed iS 
 less, then at the boundary separating the two media part of the' 
 energy will be transmitted into the second medium and the re- 
 mainder will be reflected without change of phase back through 
 the first medium. 
 
 Whenever a wave proceeds from .one medium to another in 
 which the speed is different, part of the energy will he reflected 
 at the interface separating the two media and part ■mil- be trans- 
 mitted into the second medium. 
 
 From 0.92 to 0.95 of all the luminous energy incident on a 
 polished silver surface will be 'reflected. The "reflection coeffi- 
 cients" of some other familiar substances are approximately as 
 follows: Silvered glass mirror, 0.85; mercury coated glass 
 mirror, 0.75 ; white writing paper, 0.70 ; black paper, 0.05 ; black 
 velvet, 0.004. 
 
LIGHT 
 
 19 
 
 15. Construction of Wave Fronts. — Consider the wave 
 motion due to a periodic disturbance at the point 5" in an isotropic 
 medium of indefinite extent. Since the medium is isotropic, the 
 disturbance will have traveled during any given interval of time 
 equal distances from the source 5" in all directions. Consequently 
 the advancing front of the wave is a spherical surface. If either 
 the wave had not originated at a point source or had not traversed 
 an isotropic medium, the advancing wave froijt would not be a 
 spherical surface. The term "wave front" is not limited to the 
 surface containing the particles of the medium that are just on 
 the point of being disturbed. The surface passing through all 
 adjacent particles which are in the same phase of vibration' is 
 called a wave front. A wave front may be plane or curved, con- 
 verging toward a point, diverging from a point, or advancing 
 without either convergence or divergence. 
 
 If a wave front meets a screen AB, Fig. 18, containing an 
 opening, a portion of the wave will be prevented from advancing 
 beyond the screen. A portion abc of the advancing wave front 
 will traverse the aperture and at some later instant will be in 
 the position a'b'c'. The form a'b'c' is 
 determined solely by the form and posi- 
 tion of the part abc, and would be ths 
 same even though the periodic disturb- 
 ance at 5" had ceased after originating 
 the wave. Consequently, the new wave 
 front a'b'c' is due solely to the position 
 abc of the previous wave front, and the 
 energy in abc. 
 
 These considerations lead to the con- 
 clusion that every point in a wave front 
 traversing an isotropic medium is a center of disturbance from 
 which spreads a spherical wave. This result is called Huyghens' 
 Principle. 
 
 For example, in the above figure consider the disturbance 
 sent out by each point of the wave front abc. At some given 
 instant the spherical wave fronts produced by the disturbances 
 at these points will have reached the positions shown. On the 
 surface a'b'c' tangent to all of these wave fronts the disturbance 
 is very great, and it can be shown that at points back of this 
 enveloping surface the waves from the different points of abc 
 produce interference. Consequently, at any instant the wave front 
 of a disturbance is the envelope of all the secondary wave sur- 
 faces which are due to the action as separate sources of all the 
 
 Fig. 18 
 
20 
 
 PROPAGATION OF LIGHT 
 
 points that at some previous instant constituted the wave front. 
 This method of determining the form of a wave front at 
 any instant after some previous position is called Huyghens' Con- 
 struction. For example, wave fronts of different forms, abc, 
 Figs. 19, 20, and 21, advancing to the right in an isotropic 
 
 Fig. 20 
 
 CCXC 
 Fig. 21 
 
 medium, would at some later instant have the form a'b'c' as 
 shown. 
 
 A wave which has a front of continually increasing radius, 
 as in Fig. 19, is called a diverging wave. A wave which has a' 
 front of decreasing radius, as in Fig. 20, is called a converging 
 wave. A plane wave, Fig. 21, may be considered to have a wave 
 front of infinite radius. 
 
 The point from which a wave diverges (Fj, Fig. 19), or to 
 which it converges {F^, Fig. 20), is called a focus of the dis- 
 turbance, or the center of the wave. 
 
 In the above discussion waves in isotropic media only have 
 been considered. But Huyghens' Construction can be applied 
 equally well to waves in anisotropic media. Since in this case 
 the disturbance travels in different directions through the medium 
 with different speeds, a wave originating at a point source will 
 not have a spherical front. Examples of the construction of 
 wave fronts in anisotropic media will be considered in a later 
 article. 
 
 16. Reflection from a Plane Surface. — If light is incident 
 on a rough metallic surface, the incident wave is. broken up and 
 scattered in all directions. If, however, the distance between 
 the elevations of the surface be less than a quarter wave length 
 of light (about 0.000005 inch), the incident wave will be reflected 
 without scattering. Such a surface is said to be polished. A body 
 which has a polished surface capable of reflecting light is called 
 an optical mirror. 
 
LIGHT 
 
 21 
 
 Consider a wave diverging from a point source 5" and inci- 
 dent on a plane mirror M M', Fig. 22. It will be assumed that 
 
 ^s' 
 
 Fig. 22 
 
 the medium surrounding the mirror is air or some other isotropic 
 substance. In this case the wave front of the wave from 5" will 
 be spherical. If the mirror had been absent, the wave front 
 would at some instant occupy the position M Q M'. With the 
 mirror in place, each element of the' mirror struck by the wave 
 becomes a center of disturbance from which energy is propa- 
 gated in every direction. Energy that is propagated back into 
 the first medium is said to be reflected; the energy that is propa- 
 gated through the second medium, — in this case the substance 
 constituting the mirror,^ — is said to be transmitted; while the 
 luminous energy that at the mirror is transformed into heat is 
 said to be absorbed. In general, when light is incident upon a 
 mirror some of the energy is reflected, some is transmitted, and 
 the remainder is absorbed. 
 
 If the mirror were absent, at a certain time after the wave 
 reached the point c the disturbance would have progressed a 
 distance c Q. But the mirror being present, energy is turned 
 back into the first medium, and at the same speed that it came 
 toward the mirror. Consequently, the reflected wave front is 
 somewhere on a sphere of radius c Q, which has c as a center. 
 It will also be seen that the reflected wave front touches every 
 sphere tangent to the surface M Q M' that can be described 
 about centers on the mirror surface. From Huyghens' Con- 
 struction the envelope of all these secondary spherical surfaces 
 constitutes the reflected wave from M P M'. 
 
 The above construction shows that the incident wave front 
 M Q M' and the reflected wave front M P M' have equal radii 
 
22 PROPAGATION OF LIGHT 
 
 of curvature, and that the line S S' joining the centers of curva- 
 ture of the incident and reflected waves is normal to the mirror. 
 Consequently, a plane mirror reverses the direction of the curva- 
 ture of the incident wave without altering the amount of the 
 curvature. 
 
 Since the reflected wave appears to originate at S'j this 
 point is called thevirtual source or virtual focus of the wave. 
 
 17. The Laws of Reflection. — In Fig. 22 consider the 
 light that travels from the source 5" to the mirror along any 
 line 3" b. The path along which light is propagated is called a 
 ray. In an isotropic homogeneous medium a ray is normal to 
 the wave front. From b draw the line b, E normal to the reflected 
 wave front. This line being the path of the light reflected at b, 
 is called the reflected ray at the point b. The angle S b R between 
 the incident ray and the normal b R to the mirror is called the 
 angle of incidence. Similarly, the angle Eb R between the re- 
 flected ray and the normal to the mirror is called the angle of 
 reflection. 
 
 The relation between the angle of incidence and the angle 
 of reflection will now be determined. In Fig. 22 the triangles 
 S c b and S'c b are equal, for c & is common, and in Art. 16 it 
 has been proved that S' c ^S c and that 5" S' is perpendicular to 
 M M'. Since these triangles are equal, 
 
 bSc = bS'c 
 Moreover b S c = S b R (alt. int. angles) 
 
 and Eh R = b S'c (text. int. angles) 
 
 Therefore EbR = SbR 
 
 That is, the angle of reflection equals the angle of incidence. 
 
 It will now be shown that the reflected ray, the incident ray, 
 and the normal to the mirror at the point where reflection occurs, 
 lie in the same plane. Since 5" was chosen in the plane of the 
 paper, and since in an isotropic medium the wave front is nor- 
 mal to the ray, it follows that along an intersection, M Q M' , 
 of the paper, and the incident wave front, the wave front is per- 
 pendicular to the plane of the paper. Again, since (Art. i6) 
 S S' is in the plane of the paper when the plane of the mirror 
 is normal to the plane of the paper, it follows that the point S' 
 is in the plane of the paper. By reasoning similar to that just 
 used for the incident wave front, it follows that the reflected 
 wave front where it intersects the page is also perpendicular to 
 the plane of the paper. Whence, the reflected ray lies in the 
 plane of the paper. Consequently, the reflected ray, the incident 
 ray, and the normal to the mirror at_the^_point where reflection 
 occurs, lie in the same plane. 
 
LIGHT 
 
 23 
 
 Fig. 23 
 
 18. Reflection from a Curved Surface. — Optical mirrors 
 usually have either plane or spherical surfaces. The center C 
 of the spherical surface M P M-^, Fig. 23, is called the center of 
 curvature of the mirror. The middle point 
 P of the reflecting surface is called the pole 
 of the mirror. The right line C P joining 
 the center of curvature and the pole is 
 called the principal axis of the mirror. The 
 diameter M M^ of the circular boundary « 
 of the mirror is called the linear aperture. U 
 
 First, consider the reflection of a plane 
 wave from a concave spherical mirror. 
 By' Huyghens' Construction (Art. 15), the 
 reflected wave front is determined by con- 
 sidering the disturbance set up at each 
 point of the mirror that is struck by the 
 incident wave. If the mirror were absent, 
 then at a given instant the wave front would 
 have reached the plane A^ P B^. But 
 the mirror being present, the original wave is interrupted, and 
 each point of the mirror that is struck by the wave becomes a 
 center of disturbance and originates a secondary spherical wave 
 which goes back into the first medium with the same speed that 
 the original wave approached the mirror. Consequently, if with 
 each point of the mirror as a center, spheres be constructed tan- 
 gent to the plane A^P B^, the tangent surface A J' B^ enveloping 
 these spheres will be the reflected wave front at the ^ven instant. 
 In general, when a spherical wave is incident oh a spherical 
 surface (a plane is a sphere of infinite radius), the reflected wave 
 will not be' spherical. When a spherical wave is incident on a 
 spherical mirror, the deviation from a spherical form of the 
 reflected wave is called spherical aberration. 
 
 The portion of the reflected wave reflected from the mirror 
 near the pole is nearly spherical. Thus, if the aperture of the 
 mirror is small compared with the radius of curvature of the 
 mirror, the reflected wave front is nearly spherical and advances 
 with diminishing radius of curvature until the entire wave front 
 shrinks to nearly point dimensions at F. In all the following 
 consideration of curved mirrors, small apertures will be assumed. 
 A converging spherical wave will shrink to point dimensions. 
 A point at which a converging wave coalesces is called a focus. 
 The point to which a plane wave that is advancing toward a con- 
 cave mirror parallel to the principal axis converges after reflec- 
 
24 
 
 PROPAGATION OF LIGHT 
 
 tion is called the principal focus of the mirror. The distance 
 of this point from the pole of the mirror is called the principal 
 focal length of the mirror. 
 
 19. — The position of the principal focus of a spherical mirror 
 
 will now be determined. Let A, Fig. 24, be 
 a point source on the principal axis of a 
 concave mirror having the center of curva- 
 ture at C. Light traveling along the ray 
 A X will be reflected along the path X A'. 
 Since C X A' = A X C, 
 
 Fig. 24 
 
 AC _ AX^ 
 
 CA'~ A'X 
 If the angle Z C F is small, then AX ^ AY (nearly), and 
 A' X =-A' Y (nearly). In this case, the above equation may be 
 written, 
 
 AC ._ AY^ 
 
 CA' T A'Y (11) 
 
 Putting this equation into the form 
 
 CA' ._ A£ 
 
 A'Y ~. AY 
 we see that so long as the angular aperture W A Y is small, the 
 position of the point A' depends only upon the position of the 
 source and the curvature of the mirror. That is, under this con- 
 dition, all the light from A that strikes the mirror will be reflected 
 to the point A'. Consequently, A' is the focus of the light 
 from A. 
 
 It will be convenient to represent the radius of curvature of 
 the mirror by r, the distance of the source A from the pole Y by u, 
 and the distance of the focus A' from the pole by v. Using this 
 notation, (11-) assumes the form 
 
 u — r ■ u 
 
 V 
 
 V 
 
 or 
 
 uv — vr = ur — uv 
 2 uv = vr -{- ur 
 
 On dividing each term by uvr, 
 
 r • u V 
 
 (12) 
 
LIGHT- 
 
 25 
 
 This equation gives the relation between the distances of the 
 , source and of the focus from a concave mirror of small angular 
 aperture in terms of the radius of curvature of the mirror. If 
 the source be at infinity, that is, ii u {= A Y) == x , the above 
 equation becomes 
 
 2 . 1 
 
 That is, the principal focus of a concave mirror of small angular 
 aperture is midway between the center' of curvature and the pole 
 of the mirror. The principal focal length is one-half the radius 
 of curvature. 
 
 §3. Change of the Form of a Wave Front by Refraction. 
 
 20. Refraction at a Plane Surface. — Imagine a wave 
 originating at a point S, Fig. 25, to pass from one isotropic trans- 
 
 mg. 25 
 
 parent medium to another in which the velocity is greater than 
 in the first. Every point of the plane surface N N' separating 
 the two media that is struck by the wave will be a new center of 
 disturbance. Suppose that in the second medium the velocity of 
 light is l.S times so great as in the first medium. If with the 
 second medium absent the wave front at some given instant were 
 N d N', the actual wave front in the second medium could be 
 constructed as follows : With various points a, c, e, etc., of the 
 interface as centers, construct a number of spheres of radii 
 1.5 {ah), l.S {cd), 1.5 {cf), etc. The surface N P N' envelop- 
 ing these spheres is the wave front in the second medium. If 
 the distance N N' is small compared with S N, it can be proved 
 that the new wave front is nearly spherical in form, and that the 
 wave in the second medium advances as though it originated 
 at a point S'. The real source 6" and the virtual source ^S*' are on 
 a line normal to the surface separating the two media. Conse- 
 
26 
 
 PROPAGATION OF LIGHT 
 
 quently, light traveling from the first medium to the second in 
 any direction except the one normal to the interface will be bent 
 out of its original direction at the interface separating the two 
 media. The phenomenon of the breaking or bending of a ray at 
 the surface separating two media in which light travels with dif- 
 ferent speeds is called refraction. 
 
 When light passes obliquely from a medium in which the 
 speed is less to a medium in which the speed is greater, the 
 ray is bent away from the normal to the surface separating the 
 two media. 
 
 By proceeding in exactly the same manner as above, it can 
 be shown by means of Fig. 26 that when light passes from a 
 point source in a transparent isotropic medium to another in 
 which the speed is, less, the two media being separated by a 
 plane surface, the wave front in the second medium is nearly 
 spherical in form and advances as though it originated at a 
 point S' at a greater distance from the interface than the real 
 source 5'. Also, when the light passes obliquely from a medium 
 in which the speed is greater, to a mediuni in which the speed is 
 less, the ray is bent toward the normal to the surface separating 
 the two media. 
 
 21. The Laws of Refraction. — The amdunt that light is 
 deviated out of its course when it passes from one medium to 
 -'^K another will now be determined. 
 
 Let D G, Fig. 27, be the front of 
 a plane wave advancing toward 
 the plane N N^ separating two 
 transparent isotropic media. As 
 each point of the advancing 
 wave strikes the interface sep- 
 arating the two media a dis-- 
 turbance is there set up which 
 is transmitted back through the 
 first medium and also one that 
 is transmitted forward through 
 ^^' " the second medium. The re- 
 
 flected wave has been considered in Art. 17. The wave trans- 
 mitted through the second medium will now be considered. 
 
 On entering' the sec6nd medium, the light that has 
 traveled along A D will be bent out of this ray. If the velocity 
 of light is less in the second than in the first medium, the ray 
 in the second medium will be bent toward the normal to the 
 surface separating the two media. If the interface between the 
 
LIGHT 27 
 
 two media is plane, all rays in the first medium parallel to A D 
 will, at this interface, be bent toward the normal to the same 
 amount. That is, rays that are parallel before incidence on the 
 interface will be parallel after refraction. Since the interface 
 is plane and the velocity of light in the second medium is uniform, 
 the wave front in the second medium will be plane and normal 
 to the refracted rays. 
 
 The angle between the incident ray A D and the normal 
 to the surface' separating the two media is called the angle of 
 incidence. It will be seen that the angle of incidente equals the 
 angle between the incident wave front, D G, and the surface 
 separating the two media. The angle between the refracted 
 ray D,X and the normal to the surface separating the two media 
 is called the angle of refraction. It will be seen that the angle 
 of refraction equals the angle between the refracted wave front 
 d F and the surface separating the two media. 
 
 The ratio of the speeds of light in two media is called the I 
 relative index of refraction of the two media. The absolute 
 index of refraction of a medium is the ratio of the speed of light 
 in a vacuum to the speed in the given medium. If the speed of 
 light in the first medium be represented by s^, and the speed in 
 the second medium by s^, then from definition, the index of re- 
 fraction of the second "medium relative to the first is 
 
 ^ = f- (13) 
 
 •'2 
 
 Now since light travels in the first medium the distance 
 G F {^D a) during the time t, it follows that 
 
 D a = s-J: 
 During this same time light travels from D into the second 
 medium a distance D d such that 
 
 Dd ^ sj 
 Substituting these values of s^ and s^in (13) we obtain 
 
 Da 
 
 But from the figure D a^ D F s'mi and D d = D F sin r. There- 
 fore, the index of refraction 
 
 /* = — — (14) 
 
 sm r ^ ' 
 
 Consequently, the sine of the angle of incidence hears a con- 
 stant ratio to the sine of the angle of refraction. This is called 
 Snell's Law of Refraction. 
 
 It can also be shown that ivhen both substances are isotropic, 
 
28 
 
 PROPAGATION OF LIGHT 
 
 the refracted ray, the incident ray, and the normal to the refract- 
 ing surface at the point of incidence, lie in the same plane. 
 
 The index of rafraction of a substance is different for light of 
 different wave lengths. 
 
 The indices of • refraction of a few famihar substances for 
 Ught of wave length 0.000058 cm. are as follows: Ice, 1.31; 
 water, 1.33; crown glass, 1.5 to 
 lT6; flint glass, 1.6 to 1.9; Canada 
 balsam, 1^54^; carbon bisulphate, 
 1.63. 
 
 22. As an illustration of the law 
 of refraction consider the passage of 
 light through a specimen bounded by 
 parallel faces. Let the index of re- 
 fraction of the specimen relative to 
 the surrounding medium be fi. Then 
 the index of refraction of the sur- 
 rounding -mediuim— relative- to -th»t-ef~ 
 rounding medium relative to that of the specimen is l/|u,. In Fig. 28 
 
 sin i 1 sin i' 
 
 u. = — :-^ and = — ^-^ , 
 
 sm R yn sm R 
 
 Equating these two values of /j,, we have 
 
 sin i sin R' 
 
 (( 
 
 or, since i' =R| 
 Therefore 
 
 sin R sin i' 
 sin i = sin R'. 
 R' = i. 
 
 Consequently light emerges from a transparent body bounded by 
 parallel sides in a direction parallel to the direction in which it en- 
 tered the body. 
 
 23. The Change of Wave Length on Refraction. — The 
 
 speed of propagation of any wave motion is 
 
 s = NK (15) 
 
 where N represents the number of vibrations per second, and 
 A, represents the distance traveled during the time of one vibra- 
 tion. Since the frequency is constant, (Art. 4), the wave length 
 must change whenever the speed changes. 
 
 The amount of change in the wave length of light on pass- 
 ing from a' medium in which the speed is s^ to a medium in which 
 the speed is s^ is easily determined. Denote the wave lengths of 
 the light in the two media by K-^ and X^, respectively. Then 
 
 ji = AfA.1 and s^ = N^X^ 
 
LIGHT 
 
 Whence, 
 
 
 -= [i 
 
 where /t is the index of refraction of the second medium relative 
 to the first for the particular color. 
 Consequently, 
 
 X, = ^ ■ (16) 
 
 /*. 
 
 24. Total Reflection. — Consider the passage of light from 
 
 a medium in which the speed is less to another in which the speed 
 
 is greater. To fix the ideas, let S, Fig. 29, be a point source of 
 
 light in water. At the surface separating the water from the 
 
 air above there will be both reflec- 
 tion and refraction. Light arriving 
 at a point D will there be partly re- 
 flected, and the remainder will be 
 transmitted into the second "me- 
 dium. At any point on the inter- 
 face separating the air and water 
 the angle of reflection will equal the 
 angle of incidence. And if the ita- 
 dex of refraction of water relative to air is /i, then, for any angle 
 of incidence i the angle of refraction R will be given by the equa- 
 
 Pig. 29 
 
 tion 
 
 1 
 
 sin J 
 sini? 
 
 Since fi is greater than unity, this equation shows that if i is 
 increased by a certain amount the angle R will be increased to a 
 greater amount. At some point of the surface, E, the angle of 
 incidence will be such that i?-=90°. Beyond E light will be re- 
 flected but there will be no light transmitted into the second me- 
 dium. The value of i for which R = 90°, that is, the smallest 
 value of the angle of incidence at which waves of a given length 1 
 will be totally reflected, is called the critical angle for the two 
 media . Denoting this angle by c, we have for the case where 
 R = 90°, sin R = l, and 
 
 1 
 
 sm c 
 
 Whence, the critical angle of incidence is 
 
 c = sm^ 
 
 (17) 
 
30 
 
 PROPAGATION OF LIGHT 
 
 Consider an eye in water at E, Fig. 
 30. Let c be the critical angle of inci- 
 dence for water with respect to air. Now 
 light from any point above the water will 
 enter the eye at E, but this light must 
 traverse the area included between A and 
 B. Thus, to an eye at E the surface of 
 thd water appears to be an opaque re- 
 flecting ceiling pierced by a round win- 
 j-ig 30 dow immediately overhead. Objects 
 
 above the area A B will appear undistorted, but objects near the hori- 
 zon will appear considerably distorted. 
 
 Taking the index of refraction of water to be 1.33 we obtain 
 
 c = sin -1 , : =: sin -1 0.75 = 48° 30' 
 
 1-33 cz. ^^^.jn , 
 
 Whence A E B = 97° 
 
 Fig. 31 shows the view that would be seen by an eye in water 
 directed upward. It was taken by a camera submerged in water. 
 
 Fig. 31 
 
 25. The Total Reflection Prism. — Reflection occurs when- 
 ever Hght is incident on the surface separating two media in 
 which the speed of Hght is different. Usually a part of the 
 energy of the incident wave is transmitted by the second medium. 
 But when light travels from a mediurri in which the speed is less 
 to a medium in which it is greater, there will be no transmitted 
 wave if the angle of incidence at the interface is greater than the 
 
LIGHT 
 
 31 
 
 critical angle for the given media. In many optical instruments 
 application is made of this fact for changing the direction of a 
 beam of light by 90° without sensibly changing its intesity. 
 
 The total reflection prism consists of 
 a right angled glass prism ABC, Fig. 32. 
 The index of refraction of glass relative to 
 air is about 1.5. For a substance of this 
 index of refraction, surrounded by air, the 
 critical angle is 42° (17). Consequently, 
 ^B. if light traverses the face A C normally, it 
 will be totally reflecte(;i at the face A B and 
 emerge from the prism normally to the 
 face B C. 
 
 Fig. 32 
 
 ^sf 
 
 26. The Erecting Prism.- — If a light wave proceeding 
 parallel to the base of an isoceles glass prism be incident on the 
 ^ face A C, Fig 33, it will be refracted and 
 
 strike the base at an angle greater than thfe 
 critical angle of incidence. After total re- 
 flection at the base, the light will proceed to 
 Fig. S3 the face B C and emerge parallel to the in- 
 
 cident direction. The diagram shows that on emergence the 
 relative positions, of the rays .*■ and y are reversed. This device 
 is commonly used to invert an image. When so used, it is called 
 an "erecting prism." 
 
 If the angle between the isoceles faces is smaller than a cer- 
 a part of the light incident on the first face will not 
 strike the base of the prism, Fig. 34, and this 
 light will not emerge in the direction of inci- 
 dence. If the part o'f the prism above the 
 dotted line be removed, then all the light inci- 
 dent on the first isoceles face parallel to the 
 base will emerge from the second isoceles face 
 in the same direction. 
 
 If the angle between the isoceles faces 
 
 ,j, is larger than a certain value, all the light 
 
 ^ incident on the first face will strike the 
 
 ; base and emerge from the second isoceles 
 
 face in the direction of incidence, , Fig. 35, 
 
 tain value, 
 
 a 
 
 part 
 
 
 
 
 
 10 / \ 
 
 
 
 T r--^ 
 
 \ 
 
 z' 
 
 y /y<^ 
 
 ^ 
 
 V 
 
 ^l/>^/ 
 
 \^' 
 
 34 
 
 ^^5<^ 
 
 Fig. 35 
 
 but the prism will be needlessly 'long. 
 
 Problem, — Find the smallest angle between the isoceles faces pf 
 an erecting pris^m that will cause all the light incident on the first face 
 parallel to the base to emerge in the parallel direction from the other 
 isoceles face. , 
 
32 PROPAGATION OF LIGHT 
 
 Solution.— In order that all the light incident on the first isoceles 
 face parallel to the base may emerge from the second isoceles face in 
 the same direction, all the light after the first refraction must strike 
 the base. For minimum length of prism, light incident at the edge, C, 
 Fig. 36, must strike the base at the edge B. That is, after the first re- 
 fraction the rays within the prism must be parallel to the second iso- 
 cles face. Thus, the problem consists in 
 finding an expression for the angle that 
 will cause K L to be parallel to C B. 
 Since the prism is isoceles 
 ® = 180° — 2,^ 
 
 From the geometry of the figure, ij = 90° — <^ and r^ = 90° — 2^. 
 Whence, (14) 
 
 r sin i^ 1 sin (90 — </>) _ cos <!> ^ cos ^ 
 
 '* [ ^ sinr^ J ^ sin (90 — 2^) ~ cos 2,/, ~ 2 cos^ <j, -- 1 
 
 or, 2 |u, cos^ ^ — cos <^ — /<.^0 
 
 Extracting the square root, 
 
 ^>^. 
 
 cos (b = i 
 
 Consequently, 
 
 ® [= 180° —2^]— 180° — 2 cos -1 f. 
 
 1± yi+S/t' 
 4^ 
 
 27. Change of Wave Front Produced by a Convex Lens. 
 
 — Lenses are usually made of glass and are usually bounded by 
 two spherical surfaces which have a common axis. A lens that 
 is thicker at the center than at the edges is called a convex lens, 
 while one that is thinner at the center than at the edges is called 
 a concave lens. The line joining the centers of curvature of the 
 two faces of a lens is called the principal axis of the lens. The 
 points where the principal axis intersects the faces of a lens are 
 called the poles of the lens. 
 
 Consider the change in the form of a wave front produced 
 by passage through a convex lens. In Fig. 37 is represented a 
 plane wave A B proceeding toward a convex lens parallel to the . 
 principal axis. To simplify the present construction, the first 
 face of the lens is taken to be plane. On entering the first sur- 
 face the speed of the wave is changed, but not the form of the 
 wave front. If the glass extended indefinitely to the right, then 
 at some instant the wave front in the glass would be A-^B-i. But 
 when the glass is bounded by the surface represented in the 
 
LIGHT 
 
 33 
 
 figure, we find by Huyghens' construction (assuming that the 
 speed of light in glass is two-thirds the speed in air) , that at this 
 instant the wave front is A,P B~. 
 
 When a spherical wave is incident on a lens bounded by 
 spherical surfaces (a plane is a sphere of infinite radius), the 
 emergent wave will in general not be spherical. If, when a 
 spherical wave is incident on a lens, the emergent wave front is 
 not spherical, the deviation from the spherical form of the 
 emergent wave is called spherical aberration. 
 
 If the incident wave is spherical and proceeds parallel to the 
 principal axis of the lens, and if the lens is covered, except a 
 small area about the pole, the emergent wave will be practically 
 spherical. That is, the spherical aberration will be negligible. 
 Henceforward, as far as Art. 44, we shall assume that the 
 axis of the incident beam of light is so nearly coincident with 
 the principal axis of the lens, and that the aperture of the lens 
 is so small, that the emergent wave is approximately spherical. 
 
 The effect of a convex lens on a wave traversing it may be 
 illustrated by the four following diagrams. In Fig. 38 a plane 
 
 Fig. 38 
 
 wave advancing parallel to the principal axis converges after 
 transmission to the focus F. The point to which a plane wave 
 
34 
 
 PROPAGATION OF LIGHT 
 
 advancing parallel to the principal axis converges after emergence 
 is called the principal focus (i. e., fireplace) of the lens or system 
 of lenses. A lens has a principal focus on each side. 
 
 Fig. 39 
 
 Fig. 39 illustrates the fact that if the wave originates at a 
 point 5" farther from the lens than the principal focus, the 
 emergent yirave will converge to a focus Fi beyond the other 
 principal focus. 
 
 Fig. 40 
 
 Fig. 40 illustrates the converse of Fig. 38, namely, that a 
 wave originating at a principal focus will after transmission by 
 a convex lens be a plane wave. 
 
 Fig. 41 
 
 A wave originating at a point between a convex lens and a 
 principal focus. Fig. 41, will after transmission advance as 
 though it had originated' at a point S' beyond the principal focus. 
 
 A lens, as well as a mirror, imprints on a wave a new curva- 
 ture. A wave emerging from a concave lens is more convergent 
 
LIGHT 
 
 35 
 
 than when it entered the lens. For this reason a convex lens is 
 often called a converging lens. 
 
 A lens or lens system that causes an incident plane wave to 
 erfierge as a convergent wave is called a positive lens or lens sys- 
 tem. A single convex lens is positive for light entering from 
 either side. 
 
 28. Change of Wave Front Produced by a Concave Lens, 
 
 — It is left as an exercise for the student to construct by Huy- 
 ghens' method the wave front of light emerging from a concave 
 lens and to show that the emergent wave is more divergent than 
 the incident wave. A concave lens is often called a diverging lens. 
 
 A plane wave advancing 
 along the principal' axis of a con- 
 cave lens will after transmission 
 become a diverging wave that 
 appears to have originated at a 
 point on the principal axis. This 
 virtual source F, Fig 42, is called 
 the principal focus of the con- 
 Pig. 42 cave lens. 
 A lens or lens system that causes an incident plane wave to 
 emerge as a divergent wave is called a negative lens or lens system. 
 A single concave lens is negative for light entering from either 
 side. 
 
 A lens system consisting of two or more lenses may be posi- 
 tive for light entering from one side, and negative for light en- 
 tering from the other side. 
 
 §4. Diffraction. 
 
 29. Half-Period Elements. — Consider the illumination at 
 a point M, Fig. 43, due to a spherical wave advancing from a 
 point source 5'. The effect at M is the resultant of the individual 
 effects produced by all points of the advancing wave front. Draw 
 a line from .S" to M. The point of . intersection, P, of this line 
 
 Fig. 43 
 
 Pig. 44 
 
36 PROPAGATION OF LIGHT 
 
 with the wave front is called the pole of the wave surface with 
 respect to M. Denote the distance P M hy the symbol a 
 and the wave length by A. With M as center and with radii 
 
 12 3 
 
 o-j- — \ a-\- — \ a-\ \ etc., describe a series of spheres. 
 
 ^ ^ ^ 
 
 These spheres divide the advancing wave front into a series of 
 zones. Looking at the advancing wave front in the direction 
 MP, the appearance of these zones, if they could be rendered 
 visible, would be as represented in Fig. 44. The areas of the 
 concentric zones are called half ^period elements. It can be shown 
 tbat these half-period elements are approximately equal. If 
 the distance of these half -period elements from M were exactly 
 equal, and if their inclinations to MP were equal, the illumina- 
 tion at M produced by any half-period element would equal that 
 produced by any other. But as the radius r of a half-period 
 element increases, there is a gradual increase both in the distance 
 of the element from M and in its inclination to M P- It follows 
 that the illumination at M due to the various elements of the 
 wave is not quite the same, and that the illumination due to any 
 element is slightly greater than the illumination at the same point 
 due to the next outer element. 
 
 The disturbance that reaches M from any point of one half- 
 period element is exactly one-half wave length behind the dis- 
 turbance from some point in the next inner element. Hence, 
 the disturbance at M, at any distant, due to an entire half-period 
 element, is, on an average, one-half wave length behind the dis- 
 turbance due to the next inner element. Therefore, the illumina^ 
 tion due to any two adjacent half-period elements is considerably 
 less than that due to the inner one alone. In fact, it can be 
 shown, though the proof will not here be given, that the illumina- 
 tion at M due to the entire wave very nearly equals one-fourth 
 that which would be produced by the central half-period element 
 if this alone had been effective. Thus, if all the wave had been 
 screened from M except a certain part of the first half-period 
 element, the illumination at M would be the same as if no part 
 of the wave were screened. Consequently, the illumination at M 
 may be regarded as due to a very small portion of the advanc- 
 ing wave front about the pole. All of the remainder of the wave 
 is ineffective so far as the illumination of M is concerned. 
 
 If an opaque circular disc of the diameter of the first half- 
 period element be placed at P, the illumination at M will be ap- 
 proximately that which would be produced by the second half- 
 period element if that alone were sending light to M. If the disc 
 
LIGHT 
 
 37 
 
 cover two half-period elements, the illumination at M will be 
 approximately that which would be produced by the third half- 
 period element if that alone were sending light to M. As the 
 illumination due to each half-period element is practically the 
 same, it follows that the illumination at M will be but slightly 
 affected bv placing a small opaque disc in the path of the light 
 wave. 
 
 If at P there be placed a diaphragm with transparent annular 
 spaces, so arranged that the alternate half-period elements are 
 uncovered, the illumination at M will be many times greater than 
 if the entire wave front were uncovered. Such a diaphragm 
 would have the appearance of Fig. 44, except that the diameter 
 of the rings would be much smaller. It is called a "zone plate." 
 
 30., The Rectilinear Propagatioh of Light. — The approxi- 
 mate radius of any half -period element will now be determined. 
 Let A, Fig. 45, be on the outer edge of the nth. half-period ele- 
 
 Flg. 45 
 
 ment, with respect to M, of a wave front advancing from a point 
 source 5". From the figure, the radius r of this element is given 
 by the equation 
 
 r' = {4My— (BMy 
 
 But AM = a+-^^ and BM=t + a 
 
 where A is the wave length of light. 
 Hence, 
 
 2 a n X 
 
 + 
 
 2at ■ 
 
 2 ' 4 
 
 For any case that would arise in experiment, t and A, are so small 
 compared with r that for the degree of precision here sought it 
 will be justifiable to neglect the square of either of them. Con- 
 sequently, 
 
 r^ = a(MA, — 2 *) nearly. (18) 
 
38 
 
 PROPAGATION OF LIGHT 
 
 Again, from the>figure, 
 
 V2 = b^— (b — ty 
 = 2bt—t^ 
 ^ 2b t nearly. 
 
 Eliminating t between (18) and (19), 
 
 , nab X. • , 
 r^ = ; — r~ nearly. 
 
 (19) 
 
 (20) 
 
 a + b 
 
 For example, the radius of the circular aperture placed 10 m. froin 
 a point source that will permit only the' first half-period element of 
 light of wave-length 0.00004 cm. to reach a screen 10 m. from the aper- 
 ture is 
 
 ,2_ 1(1000) (1000) (0-00004) _„„, 
 2000 
 r = 0.15 cm. nearly. 
 The aperture necessary for the production on the screen of il- 
 lumination equal to that which would be produced by the entire 
 unobstructed wave is even smaller than this value. That is, the 
 illumination at a point is equal to that which would be produced 
 by a very small area of the wave front normal to the direction of 
 propagation. For this reason light is said to travel in a straight 
 line from the source to the point under consideration. 
 
 31. Diffraction around the Edges of an Aperture. — In 
 the model illustrated in Fig. 46 the spherical surface F represents 
 
 Fig. 46 
 
 a portion of the front of a wave originating at 5'. This spherical 
 surface is marked off into half-period elements with respect to 
 the point M. (In constructing the boundaries of these half-period 
 elements a wave length of two inches was assumed.) The board 
 A represents a diaphragm pierced by a circular aperture of an 
 area equal to one-half of the central half-period element. 
 
 Consider the illumination on a vertical screen through C and 
 
LIGHT 
 
 39 
 
 C . Fig. 47 represents a cross-section of the model through the 
 line 5" M, and Fig. 48 represents the wave front advancing through 
 the aperture as seen from the point M. (The diagrams in this 
 
 A 
 
 B 
 
 -p— 
 
 B' 
 
 Fig. 47 
 
 M 
 
 
 
 Fig. 48 
 
 article are not drawn to scale.) Since all of the wave front not 
 covered by the diaphragm belongs to the same half-period ele- 
 ment, light from no two points will arrive, at M in opposite phase. 
 Whence, there is no destructive interference, and the illumination 
 at M is great. 
 
 On raising the end K of the rod 5 K, Fig. 45, the appearance 
 at the diaphragm of the half-period elements with respect to 
 some point M^ will be as represented in Fig. 50. In this case 
 the light that comes from the second half-period element inter- 
 feres with part of that from the first half-period element, so that 
 
 Fig. 49 Fig. 50 
 
 the illumination at M^ is considerably weaker than at M. Since 
 the source 5' is symmetrically placed with respect to the aperture, 
 the illumination on the screen will be the same at all points equally 
 distant from M. Hence the bright central spot at M is sur- 
 rounded by a dark circular band of radius M M-^. , 
 
 At some place above M^ is a poinf M^ for which the half- 
 period elements would be as shown in Figs. 51 and 52. In this 
 case the light from the exposed part of the third half-period ele- 
 ment interferes with a considerable part of that from the, second 
 half-period elerhent. Thus, only a part of the light from the 
 
40 
 
 PROPAGATION OF LIGHT 
 
 Fig. 51 Fig. 52 
 
 second half-period element is left to interfere with that from the 
 first. The illumination at M^ is, therefore, greater than at M-^, but 
 is not nearly so great as that at M. 
 
 Proceeding in this manner, it is found that the central bright 
 
 Fig. 53 
 
 spot is surrounded by a series of bands, alternately dark and 
 bright, that extend far beyond the geometric shadow of the 
 edges of the aperture. 
 
 The phenomenon of the bending of light around the edges 
 of objects is called diffraction. The alternate dark and bright 
 bands that accompany diffraction are termed diffraction fringes 
 or diffraction bands. A central bright spot that is bordered by 
 diffraction bands is called a diffraction disc. 
 
LIGHT 
 
 41 
 
 The distribution of illumiilation on a screen produced by 
 light from a luminous point, traversing apertures of different 
 sizes, is shown in Fig. 53. In this diagram the curve drawn with 
 a dashed line represents the distribution of illumination when the 
 aperture uncovers one-quarter of the first half-period element 
 with respect to M; the curve drawn with a full line represents 
 the distribution when the aperture uncovers one-half of the first 
 half-period element; and the dotted curve represents the dis- 
 tribution when the aperture uncovers the first eight half-period 
 elements. 
 
 §5. Optical Images. 
 
 32. The Formation of Images. — In the preceding article 
 it was shown that when light from a luminous point traverses 
 a small aperture of any form and falls on a screen there is formed 
 on the screen a small bright spot of light' surrounded by a series 
 of dark and bright fringes of less intensity. When the aperture 
 is not small there is formed on the screen a large bright spot hav- 
 ing the form of the aperture. 
 
 If, instead of a single point, the luminous object consists 
 of a number of points, each will produce on the screen its own 
 effect. If the aperture is small, there will be produced on the 
 screen a small bright spot surrounded by a series of faint dif- 
 fraction fringes for each luminous point of the object. For 
 instance. Fig. 54 shows the figure produced on the screen by 
 light from an object consisting of three luminous points. Ex- 
 cept at a', b' and c' the screen is blank because light waves 
 from the aperture meet at all other points in the condition to 
 
 produce total interference. The 
 figure formed on the screen is 
 similar to the object, except that 
 it is reversed ; that is, the right 
 hand side of the figure corre- 
 sponds to the left hand side of 
 the object, and the upper side 
 of the figure corresponds to the 
 Pig. 54 lower side of the object. An 
 
 optical figure resembling a given object and formed by light 
 coming from the object is called an image of the object. An 
 image is due to the interference of light at all points except in 
 a region having a form similar to that of the object. Unless 
 the aperture is so small that it contains but very few half-period 
 elements, there will be formed on the screen a large bright spot 
 for each luminous point of the object, and these spots by over- 
 lapping will constitute a figure that does not resemble the object. 
 
42 
 
 PROPAGATION OF LIGHT 
 
 The effect of the size of the aperture on the sharpness of 
 the image is shown by the series of pictures constituting Fig. 55. 
 The photographs from which these engravings were made were 
 produced as follows. The object consisted of a square luminous 
 spot 1.3 cm. on a side. This object was placed 557 cm. from 
 a diaphragm perforated by a round aperture of known diameter. 
 A photographic plate was placed 532 cm. from the diaphragm. 
 Four sizes of aperture were used. Their diameters were 3.000 cm., 
 1.530 cm., 0.360 cm., and 0.114 cm., respectively. The wave 
 length of the light used was taken as A ^ 0.00004 cm. The num- 
 ber of half -period elements uncovered by the various apertures 
 is found by means of (20) to be 206, 54, 3, and 0.3 respectively. 
 
 :206 
 
 n = 0.3 
 
 Fig. 55 
 
 Fig. 55a, shows that when the aperture contains a large 
 number of half-period elements the bright spot produced on the 
 screen by light from each point of the object is so large that 
 
 :122 
 
 : 5.75 
 Fig. 50 
 
 n = 0.52 
 
 n = 0.08 
 
 the resultant figure has not the form of the object, but has the 
 forip of the aperture. As this sign of the aperture is diminished 
 
LIGHT 
 
 .43 
 
 the figure on the screen becomes more and more like the object 
 until when the aperture is too small the outline of the figure 
 becomes nebulous. In the case now being considered the aperture 
 that would give the best image would have a size between that 
 iised for r and d, Fig. 55. 
 
 Fig. 56 shows a similar progression. Here the object was 
 a luminous circular disc 1.56 cm. in diameter, and the aperture 
 was a square of diflFerent sizes. For the above photographs the 
 side of the square was 2.30 cm., 0.50 cm,. 0.15 cm., and 0.06 cm., 
 respectively. 
 
 Fig. 57 Fig. 58 
 
 For Fig. 57 was used a circular aperture of such a diameter 
 that one half-period element was uncovered. The distances from 
 the diaphragm to the object, and from the diaphragm to the re- 
 ceiving screen, were such that the diameter of the diffraction 
 disc formed on the screen of a luminous point of the object, was 
 0.22 mm. For Fig. 58 was used a circular aperture that un- 
 
 Fig. -59 
 
 covered fifty half-period elements, and the diameter of the image 
 of each lumnious point of the object was 0.03 mm. 
 
44. 
 
 PROPAGATION OF LIGHT 
 
 M 
 
 N 
 
 Fig. 60 
 
 Fig. 59 shows the image of an object produced by light that 
 has traversed an aperture of 0.9 half-period element when the 
 distances are such that the diameter of the image of an object 
 point is 0.142 mm. 
 
 33. The Functions of a Lens. — In the two preceding arti- 
 cles it has been shown that at an image the wavelets from the 
 different points of the aperture reinforce one another, and that 
 this bright- spot is marked off from the surrounding region 
 by a dark space produced by interference of wavelets from the 
 different points of the aperture. A satisfactory image of a 
 
 luminous point should be 
 bright, small, and sharply dif- 
 ferentiated from the surround- 
 ing -'Space. 
 
 In order that it may be 
 sharp, the diffraction fringes 
 must be weak. In order that it 
 may be bright, much light must 
 reach it. In Figs. S3, 55 and 56 
 the sharpest image is that pro- 
 duced when the aperture uncovers about a half of one half- 
 period element; This is on account of the interference of light 
 from different half-period elements. If by any means light from 
 all parts of an aperture could be made to reach a given point in 
 the same phase, then the larger the aperture the brighter would 
 be the image withoilt diminishing the sharpness. Exactly this 
 result is obtained by the use of a zone plate, a converging lens, or 
 a converging mirror. 
 
 Consider the illumination on a screen due to light from a 
 point source traversing a large aperture. If the aperture con- 
 tains a large number of half-period elements the distribution of 
 illumination on the screen will be about as represented in Fig. 60. 
 Now let there be placed in the aperture a lens of such curva- 
 ture that the emergent wave will converge to a focus on the 
 screen. Since all parts of the emergent wave. Fig. 61, are 
 
 at the same distance from the 
 focus M, all the light from the 
 advancing wave that arrives at 
 M will be in the same phase. 
 Hence, if the lens is without 
 aberration-, the spot of light at 
 M will be very small and very 
 bright. The distribution of 
 Fig. 61 light on the screen when a lens 
 
 M 
 
LIGHT 45 
 
 is used will be something as represented in Fig. 61. When, a 
 lens is used the interference ' effects occur just as if the lens 
 were not present and the aperture were not larger than one half- 
 period element. When the receiving screen M is placed at the 
 focus of the lens, the diffraction disc produced by light from the 
 point source will be small, sharp and bright, however great the 
 diarneter of the lens. If the lens be removed, the diffraction disc 
 will be sharp only when the aperture in the diaphragm is small, 
 and the diffraction disc cannot be at the same time bright and 
 small. 
 
 When a lens is used, the diaphragm with the aperture can 
 be omitted. In this case the edge of the lens plays the same role 
 as the edge of the aperture in producing interference effects at 
 the screen. If a lens is broken into pieces and only one part is 
 used, or if a lens is partly covered by an opaque object, the 
 image will be the same as when the entire lens is used, except 
 that the brightness will be less. This corresponds to the fact 
 that without a lens, if the aperture is smaller than one half-period 
 element, the shape of the bright spot is that of the object and is 
 independent of the shape and size of the aperture. 
 
 Optical images are formed only by the reinforcement and 
 interference of light waves. The purpose of the lens is to im- 
 print on the wave front traversing it' such a curvature that the 
 light which arrives at the image from the various parts of the 
 emergent wave front shall be in the same phase, whatever the 
 area of the lens may be. When a converging lens or mirror 
 without aberration is used, the image of each point of the object 
 will be sharp, whatever the size of the aperture. 
 
 34. Size of the Image of a Point Source. — The size of the 
 image of a luminous point is determined by the destructive inter- 
 ference of wavelets from the point. When a converging lens or 
 mirror is used, the image is smaller and brighter than when a 
 simple aperture is used. 
 
 To be useful, the image of an extended object must have 
 a distinct image of each point of the given object. This means 
 that the central diffraction disc produced by light from any given 
 point of the object must be very small. The conditions for pro- 
 ducing small images of point sources will now be studied. 
 
 It has been shown in Arts. 31 and 32 that if all of the wave! 
 front emerging from an aperture belongs to a single half-period 
 element with respect to some point B. then at B there will be 
 a distinct image of the luminous point. In order that the emergent 
 wave front may belong to a single half-period element, (20) shows 
 that for either a plane or diverging wave the aperture must be 
 
46 PROPAGATION OF LIGHT 
 
 small. For a wave converging toward the point B, however, the 
 entire wave front belongs to the same half-period element, what- 
 ever the diameter of the aperture may be. Consequently, when 
 a wave emerging from a large aperture converges itoward a point, 
 the image will be at the same time small, distinct and very bright. 
 Such convergence can be produced by a convex lens or a concave 
 mirror. 
 
 fi 
 
 In Fig. 62, let b represent the point source, X Z the aper- 
 ture, and B the point toward which the wave emerging from the 
 aperture converges. Since every portion of the advancing wave 
 front is at the same distance from B, all the light that reaches 
 this point will be the same phase. Consequently, at this point 
 there will be a strong reinforcement of light. Light from differ- 
 ent portions of the advancing wave front incident on some point 
 slightly above or below B will meet in different phase and pro- 
 duce interference. That is, the bright spot B is surrounded by 
 a region of partial or complete interference. At a point A, so 
 situated that {Z A — X A) equals one wave length of light, the 
 path difference {Y A — X A) is one-half wave length, and con- 
 sequently, waves from X and waves from Y will interfere de- 
 structively. Similarly, at A, waves from points directly below X 
 and waves from points directly below Y will interfere destruc- 
 tively. In fact, waves from any point between X and Y will 
 completely interfere at A with waves from some point between 
 Y and Z. Therefore, the bright spot is marked off from the sur- 
 rounding region by a dark band of radius B A. 
 
 The approximate radius of the image of a luminous point 
 will now be determined. With Z as a center, draw the arc B E, 
 and with X as a center, draw the arc A F. In any actual case 
 where light waves are employed, A B will be so short- compared 
 with the radii of these arcs that each of these arcs may be taken 
 as a right line. 
 
 Denote the radius B A of the image by the symbol r, and 
 the wave length of the light employed by the symbol X. Then, 
 
LIGHT 47 
 
 since there is complete interference at A. 
 
 ZA — XA = \ (21) 
 
 But ZA = ZE + EA = ZB + r sin p' 
 
 Similarly X A = X B —F B = X B — rsin p' 
 
 Since B is on the line through the center of the aperture 
 and normal to it, fi = p' and Z B = X B. 
 Substituting the above values in (21) 
 
 Z A~XA = \ = 2rsmB 
 Whence, the radius of the image of a point source is 
 
 \ 
 
 r = „ ■ - (nearly) (22) 
 
 2 sm j8 ^ ' ^ ' 
 
 This equation shows that to produce a small image of a 
 point source, the diameter of the aperture must be large. 
 
 In the case of a thin lens of small diameter compared with the 
 distance of the lens from the image, 
 
 WZ I WZ 
 
 (approx.) 
 
 sin^ r= 
 
 Z B J ~ Y B 
 
 R 
 
 = (approx.) 
 
 V 
 
 Whence, in this special case (22) becomes 
 
 _ _ (approx.) (23) 
 
 Z K. 
 
 35. The Resolution of Point Sources. — In the preceding 
 article it has been shown that the radius of the image of a point 
 source is small only when the diameter of the aperture is large. 
 Thus, two point sources may be so close together that for a 
 certain aperture the two images will overlap, whereas for a 
 greater aperture the two images will be separate. When the two 
 images can be distinguished as separate images the two point 
 sources are said to be resolved. 
 
 When the linear distance between the centers of two images 
 is greater than the radius of the diffraction disc of one of them, 
 the two images can be distinguished as two separate images. It 
 is commonly assumed that the smallest distance between two 
 point sources that can be resolved is that which will produce 
 images whose centers are separated by a space equal to the 
 radius of the diffraction disc of one of them. 
 
 Fig. 63i is from a photograph of the diffraction disc due to 
 one half-period element of the wave from a luminous point. 
 
48 
 
 PROPAGATION OF LIGHT 
 
 Fig. 682 shows the image of two luminous points very close to- 
 gether. Fig. 683 shows the image of two luminous points so 
 close together that the distance between the centers of the dif- 
 fraction discs of the two points equals the radius of the diiifrac- 
 tion disc of one of them. Fig. 63^ shows the image of two widely 
 
 separated luminous points. In 
 Fig. 632 the points are not re- 
 solved; in Fig. 633 they are 
 barely resolved ; in Fig. 63^ they 
 are widely resolved. 
 
 Let the point sources a and b. 
 Fig. 63 Fig. 64, be separated by such 
 
 a distance D that the center of the image of one coincides with 
 the middle of the first diffraction fringe of the other. Then the 
 distance D is the smallest for which resolution of the two sources 
 is possible, and the distance between the images A and B of 
 these two sources is the value of r given by (22). The value 
 of D will now be determined. 
 
 1 
 
 1 
 1 / 
 
 
 .-,==W\~^~— 
 
 
 1,,/ 
 
 ^ 
 
 Y W 
 
 
 >' 
 
 1 ' ~~~~~~~~- 
 
 -i 
 
 \\l 
 
 ^^^ 
 
 u 
 
 U 
 
 ^^^ 
 
 =^V 
 
 --^'^ 
 
 
 Fig. 64 
 
 In Fig. 64', a and h are the point sources and A and B are 
 their images. Light proceeding from a point source to its image, 
 along whatever path, will reach the image in the same time. 
 Thus, for the light from a that grazes the edge of the aperture 
 and reaches A, 
 
 aX + XA = aZ + ZA or aX — aZ = ZA — XA 
 So that, from (21), 
 
 aX — aZ = 'K (24) 
 
 With X as center and & X as radius, draw an arc from 
 b as far as the line a X. In any actual case in which light waves 
 are concerned, this arc will be so short compared with its radius 
 that it may be taken to be a right line. Similarly, with Z as 
 center and a Z as radius, draw an arc from o as far as the 
 
LIGHT 
 
 49 
 
 line b Z. This arc is so short compared with its radius that we 
 shall treat it as a right line. 
 
 Then aX ^ D ^va.aL -\- h X 
 
 and aZ = h Z — Z)sinoc' 
 
 If h is on the line through the center of the aperture and 
 normal to it, oc = oc ' and b X = b Z. 
 
 Substituting the above values in (24) 
 
 aX — a Z ^=1^ = 2 D sino:. 
 
 Whence, the smallest distance that can be resolved by an 
 aperture or lens in air, is 
 
 ^ '"^'' - (25) 
 
 D = 
 
 2 sinoc 
 
 (nearly) = y^— (nearly) 
 
 2R 
 
 This equation shows that', other things remaining constant, 
 shorter distances can be resolved by a short focus lens than by a 
 long focus lens and by a lens of large aperture than by a lens of 
 smaller aperture. Also, that by using light of short wave length, 
 shorter distances can be resolved than by using light of longer 
 wave length. In astronomical telescapes, points apparently coin-, 
 cident are resolved by the use of lenses of large aperture. Irt 
 microscopes, resolution is increased by the use of short focus 
 lenses and by illuminating the object with light of short wave 
 . length. 
 
 36. The Resolving Power of a Lens or Aperture. — When 
 the images of two point sources can be just distinguished as 
 separate points, the lens or aperture is said to be at its limit of 
 resolution. The ability of an aperture or lens to separate the 
 images of two adjacent point sources is called the resolving 
 power of the aperture or lens. 
 
 Consider the case of two point sources. Fig. 65, separated 
 
 Fig. 65 
 
 by a linear distance D. If the distance between the images is d, 
 then by the geometry of the figure, 
 
 D . d 
 
50 PROPAGATION OF LIGHT 
 
 Denoting by ® the angular distance at the lens between the 
 two sources, then when ® is small, 
 
 r. D d ^ , ' 
 
 = = ( approx. ) 
 
 U V 
 
 When the linear distance between the centers of the images 
 is less than the radius of the central diffraction disc of one of 
 them, the two images cannot be distinguished as two separate 
 images. 
 
 Whence, putting d^r in the above equation, it follows that 
 
 two distinct images of two point sources cannot be obtained if 
 
 tlie angular distance, at the aperture or lens, between the sources, 
 
 is less than 
 
 D r 
 
 © = — =■ — (approx.) (26) 
 
 Therefore from (23) 
 
 = -j^ (approx.) (27) 
 
 The angle ® determines the smallest distance that can exist 
 between two point sources and still permit the formation of dis- 
 tinct images of the sources, and consequently measures the limit 
 of resolution of the aperture or lens. The limit of resolution of 
 the human eye is about one minute. 
 
 The angular resolving power of an aperture or lens is 
 
 — = -^(approx.) (28; 
 
 The resolving power can also be measured in terms of the 
 
 smallest linear distance that can be resolved. From (25), the 
 
 linear resolving power of a lens or aperture is 
 
 1 2 since 
 — p- = — ^; (approx.) (29) 
 
 where D represents the linear distance between the two point 
 sources measured parallel to the plane of the aperture or lens, 
 and cc represents half the angle at one of the sources subtended 
 by the aperture or lens. 
 
CHAPTER III. 
 
 SPHERICAL LENSES AND LENS SYSTEMS. 
 
 §L The Cardinal Points. 
 
 37. Principal Points. — For purposes of graphical repre- 
 sentation it is often convenient to represent the paths along which 
 light travels instead of the fronts of the advancing wave. The 
 path along which light travels is called a ray. In isotropic media, 
 rays are normal to the wave fronts. 
 
 Throughout the present section of this chapter we shall con- 
 sider the effect of a lens bounded by spherical surfaces upon light 
 that traverses the lens near the center and in a direction nearly 
 parallel to the principal axis. For the purpose of legibility, how- 
 ever, the rays in the diagrams illustrating -these articles will not 
 be confined to this narrow region. 
 
 In Fig. 66, the ray starting from A parallel to the principal 
 axis of the lens will he A B C I. Produce A B and I C till they 
 intersect at D. Through D draw a line D P perpendicular to the 
 principal axis. In so far as the portions of the ray outside of the 
 lens are concerned, the light along the ray A B proceeds as though 
 
 Fig. 66 
 
 it had traversed the straight line A D and had then been bent 
 into the path D F. Similarly, in so far as the portions of the 
 ray outside the lens are concerned, light proceeding from the 
 right to the left along the ray A'B' emerges from the lens as 
 though it had traversed the straight line '14' D' and had then been 
 bent into the path D'F'. Through D' draw a line D'P' perpen- 
 dicular to the principal axis. 
 
 Again, light along some ray A J, not parallel to the principal 
 axis, will emerge from the lens in some direction K I as though 
 it had traversed the straight line AIL, thence the line L M 
 parallel to the principal axis, and thence the straight line M K I. 
 That is, there are two planes perpendicular to the principal axis 
 of a lens or lens system which possess the property that any inci- 
 
S2 
 
 PROPAGATION OF LIGHT 
 
 dent ray meets the first, and the corresponding emergent ray 
 meets the second, in points equally distant from the principal 
 axis. They are called the "principal planes" of the lens or lens 
 system. 
 
 The points P and P' where the principal planes are cut by 
 the principal axis are called "principal points." There are a pair 
 of principal points for light of each color. In the case of most 
 lenses, however, the principal points for all colors ai^e so nearly 
 coincident that in the present discussion the departure will be 
 neglected. ^ 
 
 38. Parallel Rays meet in the Principal Focal Plane. — 
 
 Let P, P' represent the principal points, and F, F' the principal 
 foci of a lens. From the point A oi z small object perpendicular 
 to the principal axis, consider light traversing the path A D 
 parallel to the principal axis ; also light along the path A P' 
 toward the first principal point ; and also light along some inter- 
 mediate path A K. After emergence from the lens, the light will 
 proceed along the paths d A.^, P A.^ and k Ay. 
 
 Fig. 67 
 
 Also consider light from some point C such that AC = 
 K P'. That is, the rays A K and C P' are parallel. From the 
 relation between the positions of points of an undistorted image 
 and the points of the object from which they are derived, 
 AC r ^kP_'\ _ AyCy 
 
 AB 
 
 or. 
 
 kP 
 
 (30) 
 
 A,C, 
 
 From the similar triangles kH P and A^ H C^, 
 kP P H 
 
 A, C, H C, 
 and from the triangles d P F and F By Ay 
 
 dP _ P F 
 AyBy ~'FB^ 
 
LIGHT 53 
 
 Therefore, (30) becomes 
 
 P H P F 
 
 Whence, F H is parallel to Si C^. Since the object was 
 taken at right angles to the principal axis, the image is at right 
 angles to the principal axis. Consequently, F H is at right angles 
 to the principal axis. 
 
 The plane through the principal focus normal to the princi- 
 pal axis i^ called the principal focal plane. 
 
 It has now been shown that incident parallel rays {A K and 
 C P' in the diagram), after refraction by a lens, meet at a point 
 in the principal focal plane. 
 
 If the light be supposed to originate at a point H in the 
 principal focal plane we will have the converse proposition : Two 
 rays from a point in the principal focal plane of a lens will after 
 emergence be parallel. 
 
 The distance between a principal focus and the correspond- 
 ing principal point of emergence is called the principal focal dis- 
 tance. In the diagram, for light traversing the lens from left'to 
 right, the principal focal distance is P F; for light traversing 
 the lens from right to left, the principal focal distance is P' F'. 
 It can be shown, though the proof will not be here given, that if 
 the media on the two sides of the lens are the same, then the two 
 principal focal distances are equal. 
 
 The distance between the image and the nearer lens surface 
 is called the back facus of the lens. In the case of a system of 
 lenses or a single thick lens, the principal focal distance may be 
 considerably greater than the back focus. 
 
 In Art. 47 it is shown that for an extended object at a finite 
 distance from the lens, the surface containing the image is spheri- 
 cal, and its diameter is coincident with the axis of the lens. This 
 surface is called the "focal sphere." But since in the present 
 discussion we are considering rays which make small angles with 
 the axis of the lens, the approximations made are justifiable, and 
 we may speak of this surface as the focal plane. 
 
 39. Nodal Points. — Consider a lens with principal points 
 situated at P and P' , and principal foci at F and F' , Fig. 68. 
 Light parallel to the principal axis and incident at B will after 
 emergence pass through the principal focus F. From A. on the 
 focal plane through F' , let light pass parallel to C F to the point 
 /. From the property of rays from a point in the focal plane just 
 proven (Art. 38), the emergent ray K L is parallel to C F. 
 
54 
 
 PROPAGATION OF LIGHT 
 
 Produce A J and L K. The intersections, N' and N, of 
 these lines with the principal axis have the property that incident 
 light directed toward the nearer one will on emergence proceed 
 
 Fig. 
 
 in the parallel line that passes through the other. The two points, 
 N and A''', having the property that if incident light is directed 
 toward one, the emergent light will proceed in a parallel direc- 
 tion from the second, are called nodal points. 
 
 The positions of the nodal points of lenses of several shapes 
 when made of glass of index of refraction 1.5, are given in 
 Fig. 69. It will be noted that, depending upon the curvature of 
 the bounding surface, the nodal points may be within or outside 
 the lens. If the curvature of the left face of {b) were somewhat 
 greater the nodal point N' would be within the lens. Also, if the 
 curvature of the right face of (/) were somewhat greater, the 
 nodal point N would be within the glass. 
 
 The nodal points of a sphere coalesce at the center. Also, a 
 lens with sides of ' different radii of curvature, struck from the 
 same center, has a single nodal point. It is possible to construct 
 a lens of glass such that the nodal points are at infinity. This 
 lens has no principal focal length. 
 
 The principle underlying the ordinary method of locating 
 experimentally the nodal points of a lens or lens combination will 
 now be considered. Let a beam of parallel light be incident on 
 a lens having nodal points A'', N'. After traversing the lens, 
 light incident in the direction A N.' will emerge in the direc- 
 iton A'' B. If the lens be rotated through an angle ® about an axis 
 through A'', light incident in the direction A N/ will emerge in 
 
LIGHT 
 
 55 
 
 the same direction as before. That is, if parallel light be incident 
 on a lens, the direction of the emergent light will be unaffected 
 by a rotation of the lens about an axis through the nodal point 
 of emergence. 
 
 Fig. 70 
 The fact that if a lens or lens system be rotated about a line 
 through the nodal point of emergence, the image will remain 
 stationary is utilied in the "panoramic" camera. In this caniera the 
 lens system is turned about such an axis while the film remains sta- 
 tionary". In order that each part of the image may be equally sharp, 
 this film may be bent into the arc of a circle having a radius equal to 
 the focal length of the lens. By this device, the various parts of an 
 object extending through any angle can be successively brought to 
 sharp focus in proper relation on a long sensitive film. 
 
 The principal points, the nodal points and the principal foci 
 constitute 51 system called the cardinal points of a lens. Knowing 
 the cardinal points of a given lens or lens systerii, the emergent 
 ray corresponding to any assigned incident ray can be constructed. 
 
 40. — It will now be shown that in the usual case in which 
 the media on the two sides of the lens are the same, the nodal 
 points coincide with the principal points. Since in Fig. 68, the 
 lines A N', D F and R L are parallel, and Q P' and R P are 
 parallel, and Q R and F' F are parallel, the triangles Q P' N' and 
 R P N are equal, and the triangles A F' N' and D P F are equal. 
 
 Whence, 
 
 P' ]Sf' ^ PN {= PF — NF) (31) 
 
 and F' P' -\-P' N' = PN + NF 
 
 Or, since P' N' = P N, 
 
 F'P' ^ NF 
 Substituting in (31), 
 
 P' N' = PN = PF — F'P' 
 When the media on the two sides of the lens are the same, 
 the two principal focal lengths are equal ; that is, P' F' =P F 
 Whence, in this case, 
 
 P' TV' = P AT = 
 Consequently, when the media on the two sides of the leiis 
 are the same, the nodal points coincide' with the principal points. 
 
56 PROPAGATION OF LIGHT 
 
 41. Equivalent Points and Planes. — It has been just shown 
 that for a lens or lens system having the same medium on 
 both sides, the principal points .coincide with the nodal points. 
 The term equivalent points is used to denote the superimposed 
 principal and nodal points of a lens or lens system. The two 
 planes through the equivalent points normal to the principal axis 
 are called the equivalent planes of the lens or lens system. , 
 
 The properties of equivalent points and equivalent planes 
 of greatest utility in the study of lenses are as follows : 
 
 (1). — An incident ray parallel to the principal axis of a 
 lens will emerge as though it had proceeded to the second equiva- 
 lent plane and had there been changed in direction so as to pass 
 through the second principal focus. 
 
 (2). — An incident ray directed toward any point on the first 
 equivalent plane, will emerge from the lens as though it came 
 from a point at an equal distance from the principal axi,s on the 
 second equivalent plane. 
 
 (3). — An incident ray directed toward the first equivalent 
 point will emerge from the lens as a parallel ray from the second, 
 equivalent point. 
 
 (4) .—Incident parallel rays meet after emergence in a prin- 
 cipal focal plane. 
 
 (5). — A lens can be rotated about an axis in the equivalent 
 plane of emergence without motion of the image being produced. 
 §2. The Position of the Image. 
 
 42. The Location of the Image of an Object. — An optical 
 figure resembling an object and due to light from the object is 
 called a real image of the object. In the following diagrams, the 
 object will be represented by a heavy arrow, and its real image 
 by a light arrow. After transmission by a lens, light, without 
 forming a real image, may diverge as though it came from a 
 second object similar to the actual object. The region from which 
 the light appears to diverge is called a virtual image. A virtual 
 image will be represented by a light dotted arrow. An image in 
 space used as an object for another lens is called a virtual or 
 aerial object, and will be represented by a heavy dotted arrow. 
 
 When a lens is used, the image of an object is situated at 
 the focus conjugate to the object. The position of the image of 
 an object situated at various distances from a lens will now be 
 determined ; first, graphically, and then, analytically. Only lenses 
 will be considered that are bounded on the two sides by the same 
 medium. 
 
 For uniformity of representation, the light will be always' 
 represented as traveling from the left to the rights 
 
LIGHT 
 
 57 
 
 (a)_. CONVERGING LENS. OBJECT FARTHER FROM THE LENS 
 
 THAN THE PRINCIPAL FOCUS. — In Fig. 71, let F, F' and E, E' rep- 
 resent the principal foci and the equivalent points, respectively, of 
 
 +lens 
 
 Fig, 71 
 
 a converging lens. These four points being given, the actual 
 curvature of the surfaces and the index of refraction of the glass 
 may be dismissed from consideration. 
 
 From the end A oi the object draw two lines, — one parallel 
 to the principal axis as far as the second equivalent plane, and 
 another through the first equivalent point E'- Then, in accord- 
 ance with Property 1, Art. 41, draw a line from D through 
 the principal. focus F. And in accordance with Property 3, draw 
 a line from E parallel to A E' . The image of the point A will 
 be on each of these lines and consequently will be at their inter- 
 section A^. In the same manner, the position of the image of any 
 other point of the object can be found. The image A^ B^ is 
 real and inverted. 
 
 An analytical expression for the position of the image will 
 now be determined. In the figure, the triangles ABE' and 
 A^B^E are similar, and also the triangles D E F and A^ B^ F. 
 Whence, 
 
 ' E' b' AB f DE ) EF 
 
 E B, A^ B^ 
 
 A^BJ ^ FB^ 
 
 It is customary to reprersent the principal focal distance E F 
 by the symbol /, the distance E^ B of an object from the first 
 equivalent plane by u, and the distance E B^ of the image from 
 the second equivalent plane by v. Using this notation, the above 
 equation becomes 
 
 u f 
 
 or, 
 
 ■V V / 
 
 uv — uf = vf 
 
58 LENSES AND LENS SYSTEMS 
 
 Dividing by uvf, 
 
 -^ = - ' + ^ 
 
 V u ' f 
 
 (32) 
 
 (b). CONVERGING LENS. OBJECT NEARER THE LENS THAN 
 
 THE PRINCIPAL FOCUS. — From the end A of the object, Fig. 72, 
 draw two lines,- — one parallel to the principal axis as far 
 as the second equivalent plane, and another through the 
 
 ■ilens 
 
 Fig. Y2 
 
 first equivalent point E' . Then, as per Property 1, Art. 41, 
 draw a line from D through the principal focUs F. And 
 as per Property 3, draw a line from E parallel to A E'. The 
 image of the point A will be on the intersection of these two 
 lines, or of these lines produced. Whence, A^ is the image of A. 
 In the same manner, the position of the image of any other point 
 of the object can be found. The image A^ B^ is virtual and erect. 
 That is, no image is actually formed, but the light emerging from 
 the lens proceeds as though it came from an object similar to 
 A B but situated at A^ B^. 
 
 An analytical expression for the position of the image may 
 be determined as in the precedirig case. From the construction 
 of the figure, the triangles ABE' and A.^ B^ E are similar, and 
 the triangles D E F and A-^ B^ F are similar. Consequently, 
 E' B AB r DE ] EF 
 
 EB, A,B, L A,B,J ~ B,F 
 
 Using the customary notation, indicated in the figure, 
 u f 
 
 V z' + / 
 
 Whence, 
 
 J__ J_ 1_ 
 
 V u f 
 
 (33) 
 
LIGHT 
 
 59 
 
 (c). — CONVERGING LENS. AERIAL OBJECT TO THE RIGHT OF 
 
 THE LENS. — Before striking a lens or mirror, light from a point 
 source is divergent. After reflection from a mirror or refraction 
 by a lens, light proceeds along rays that may be divergent, parallel 
 or convergent. We will now consider the effect of a convex lens 
 on convergent light. 
 
 Let light diverging from a point A, not shown in the figure, 
 and rendered . convergent by a lens not shown, proceed toward 
 
 the point y4j. Fig. 73. Let a converging lens having equivalent 
 points £'', E, and principal foci F', F, be interposed in the path 
 of the light. The ray G E' incident at the first equivalent point 
 will emerge from the lens in the parallel direction E E/ from 
 the second equivalent point (Property 3, Art. 41). An incident 
 ray N O, parallel to the principal axis, will on emergence be bent 
 into the line D D^ passing through the principal focus F (Prop- 
 erty 1, Art. 41). ,The intersection of E E.^' and D D^ is the posi- 
 tion of the image of, the poftit A-^^ when the given lens is used. 
 
 The relation will now be determined between the distances 
 of the actual image A^ B^, the aerial object A^B^ and the principal 
 focus F from the corresponding equivalent planes. 
 
 From the construction of the figure, the triangles A^ B^ E' 
 and A^ B^ E are similar, and. also the triangles D E F and A^ B.^ F 
 Whence, ~ 
 
 £'5i A^bJc £)£! EF 
 
 '■h 
 
 eb, A,B, l A,BJ b,f 
 
 Using the customary notation,' this' becomes 
 u f 
 
 ■f-v 
 
 Whence, 
 
 
 (34) 
 
60 
 
 LENSES- AND LENS SYSTEMS 
 
 (d). — DIVERGING LENS. REAL OBJECT. — The triatigels A B E' 
 and A-i^ B^ E, Fig. 74, are similar, as well as the triangles D E F' 
 and A^ B^ F'. Whence, 
 
 —lens 
 
 Fig. 74 
 
 E' B 
 
 EF' 
 
 — = ABf DE ^__ 
 EB, 'A^B,L~ A^bJ F'B, 
 
 Using the customary notation, this becomes 
 
 u ^ / 
 
 Whence, 
 
 1 
 
 V 
 
 1 
 
 / 
 
 
 1 
 
 — = 
 
 M 
 
 
 
 + 
 
 / 
 
 .(35J 
 
 V 
 
 (e). — ^DIVERGING LENS. AERIAL OBJECT TO THE RIGHT OF THE 
 LENS AND OUTSIDE THE PRINCIPAL FOCAL DISTANCE. Consjider, 
 
 the effect of a diverging lens on light already rendered converg- 
 ent by a preceding lens. Without the diverging lens, Fig. 75, 
 light following the paths G E' and N D would converge at the 
 point /^i. On the introduction of the diverging lens having equiv- 
 
 Fig. 75 
 
 aient points E', E, and principal foci F', F, the emergent light will 
 diverge as though it had come from the point A^. 
 
 The triangles A^ B^ E' and A^ B^ E aresimilar, and also the 
 triangles D E F' and A^ B^ F'. Whence, 
 
LIGHT 
 
 61 
 
 £'J5, A^B:, 
 
 
 EF' 
 
 eb; -a,b, [-'a^b;] =1P^, 
 
 Using the customary notation, this becomes 
 M _ f 
 
 or, 
 
 V 
 
 v-f 
 
 1 
 
 V 
 
 1 
 
 u 
 
 
 1 
 
 (36) 
 
 (f). DIVERGING LENS. AERIAL OBJECT' TO THE RIGHT OF 
 
 THE LENS AND WITHIN THE PRINCIPAL FOCAL DISTANCE. V\)^ith- 
 
 out the diverging lens. Fig. 76, light following the paths G E' 
 and TV D would converge to the point A.^. On the introduction of 
 
 Fig. 76 
 
 the diverging lens having equivalent points £', E, and principal 
 foci F' , F, the emergent light will converge at the point A^. 
 
 The triangles A.^ B^ E' and A^ B^ E are similar, and also the 
 
 triangles DEE' and A^ B^ F'. 
 E' B, A^ B 
 
 [= 
 
 Whence, 
 DE 
 
 EF' 
 
 (37) 
 
 EB, A,B,V A,B,i F'B, 
 Using the customary notation, this becomes 
 u f 
 
 V f -\-v 
 
 1 _ J_ ]_ 
 
 or, V u f 
 
 It should be' noted that when but one lens is employed a real 
 image is always inverted. By means of a second lens this image 
 may be again inyerted ; that is, caused to be erect. 
 
 Before we can use Equations 32-37 in a numerical problem, 
 we must know whether the given problem comes under Case a, .b, 
 c, d, e or /. For example, let it be required to find the principal 
 
62 LENSES AND LENS SYSTEMS 
 
 focal length of a positive lens such that when m = 5 in., v^ IS in. 
 These data could be obtained from lenses of two different prin- 
 cipal focal lengths. Thus, if m > /, we have from (a), 
 / = 3M in., whereas, ii u <i f, we have from (b), / = 7^2 in. 
 
 43. The Standard Lens Formula.^ — ^An inspection of Equa- 
 tions 32-37 shows that the relation between u, v and / for the 
 various cases is the same, except as to the signs of these quan- 
 tities. This suggests tlie possibility of adopting a convention 
 such that by giving proper signs to the quantities in a single equa- 
 tion, one equation could be made to apply to all cases. 
 
 It should be noted that the object distance u is the distance 
 from the incident equivalent plane to the object, whether real or 
 aerial; the image distance v is the distance from the emergent 
 equivalent plane to the image, whether real or virtual; the prin- 
 cipal focal distance / is the distance from the emergent equivalent 
 plane to the focus, either real or virtual, formed by incident 
 parallel rays. 
 
 Measuring u, v and f from the corresponding equivalent 
 planes, these quantities are considered positive when they extend 
 in the direction light is being propagated, and negative when they 
 extend in the opposite direction. With this convention, by giving 
 the proper signs to the quantities u, v and f, the equation 
 
 -=— + ' 
 V u , f (38) 
 
 will apply to either a converging or diverging lens in any assigned 
 
 position relative to either a real object, a real image, or an aerial 
 
 object. 
 
 §3. The Aberrations of Lenses and Lens Systems. 
 
 44. The Spherical Aberrations. — Spherical surfaces being 
 the only ones that can be easily ground with considerable pre- 
 cision, lenses are almost always bounded by such surfaces. In 
 the present Section, as well as in the previous Articles, only such 
 lenses are considered. In Art. 27, it has been shown that a 
 spherical wave, after traversing a simple lens of wide aperture, 
 will, in general, emerge with a front that is not spherical. When 
 a spherical wave is incident on a lens, the departure of the form 
 of the emergent wave front from a spherical form is called 
 spherical aberration. 
 
 By filling with smoke or dust particles the region through 
 which light passes, it is possible to render visible the path of 
 light: By this device, photographs were taken of a beam of 
 parallel light incident on a plano-convex lens, when the plane face 
 
LIGHT 
 
 63 
 
 was toward the light source, Fig. jj, and also when the convex 
 face was toward the light source, Fig. 78. In the first case the 
 
 Fig. 77 
 
 Fig. 78 
 
 lack of a point focus, that is, the departure of the emergent wave 
 from the spherical form, is very marked. Light traversing the 
 lens in this direction suffers great spherical aberration. In 
 Fig. 78, the light converges to a focus of small dimensions. That 
 is, the form of the emergent wave front is nearly spherical. Con- 
 sequently, in this case, the spherical aberration is not great. 
 
 If a screen perpendicular to the principal axis of the lens 
 be placed in the narrow part of the emergent beam of light, one 
 will observe a small bright spot in the center of a larger disc of 
 light. At a certain position the diameter of the disc will be the 
 minimum. The bright disc of smallest diameter is called the 
 circle of least confusion. 
 
 There is no definite principal focus of a lens having spherical 
 aberration, but it is customa,ry to assume the principal focus to 
 be situated at the circle of least confusion, produced by an inci- 
 dent plane wave proceeding toward the center of the lens in the 
 direction of the principal axis. The principal focal length of 
 such a lens is usually taken to be the distance from this circle of 
 least confusion to the emergent equivalent plane. 
 
 Due to spherical aberration, the image of a point source will 
 never be a point, but will be a nebulous spot. The image of an 
 extended object will, in general, be indistinct, curved and dis- 
 torted. 
 
 Five different spherical aberrations are distinguished as fol- 
 lows : axial spherical aberration, astigmatism, curvature of field, 
 distortion and coma. 
 
 45. Axial or Longitudinal Spherical Aberration. — A nar- 
 row cone of light proceeding from a point, or advancing toward 
 a point, is called a light pencil. After refraction by a lens, or 
 reflection from a mirror, a pencil may be cylindrical. A pencil 
 parallel to the principal axis of a lens is called an axial pencil; 
 
64 
 
 LENSES AND LENS SYSTEMS 
 
 one coinciding with the principal axis is called a direct axial 
 pencil; one oblique to the axis and traversing the center of the 
 lens is called an oblique centric pencil; one incident at a point 
 remote from the pole is Called an excentric pencil. A wide pencil 
 is called a beam. 
 
 Light advancing parallel to the principal axis and incident 
 near the edge of a convex lens bounded by spherical faces will 
 be considerably refra(;ted, whereas light advancing parallel. to the 
 principal axis and incident near the center will be but slightly 
 refracted. Thus, light parallel to the principal axis and incident 
 near the edge of a convex lens will come to a focus nearer the 
 lens than will light incident near the center. In the case of a 
 concave lens, the virtual focus for light parallel to the principal 
 axis and incident near the edge is nearer the lens than is the 
 
 Fig. 79 
 
 virtual focus for light incident near the center. The longitudinal 
 distribution of the focus of an axial pencil is called axial or longi- 
 tudinal aberration. 
 
 The axial aberration for the lens illustrated in the figure is 
 measured by the distance (/i f^) between the focus for the mar- 
 ginal pencils and the focus for the central pencil. 
 
 With a convex lens, f^ is to the right of /i, whereas with a 
 concave lens, fi is to the left of f^. The longitudinal aberration 
 of a convex lens is taken to be positive. Then that of a concave 
 lens is negative. , 
 
 Associated with the axial spreading of the focus is a trans- 
 verse spreading. Due to this cause, the image of a point source 
 will have an appreciable magnitude. 
 
 The axial and transverse aberration of a convex lens can 
 be counterbalanced by the addition of a concave lens of greater 
 focal length. 
 
 46. Astigmatism. — In Fig. 80 is shown the effect of a con- 
 vex lens of large aperture on a plane wave incident at a great 
 angle to the principal axis of the lens. It will be observed that 
 
LIGHT 63 
 
 the emergent wave, instead of converging to a point or to a small 
 circle of least confusion, converges to two "focal lines" separated 
 from one another by a certain distance, and that they are at right 
 angles to one another and to the axis of the incident wave. The 
 
 
 Fig. 80 
 
 line focus farther from the lens is in the plane of incidence and 
 nearly coincides with the focal plane of the lens. When a wave 
 which on incidence is spherical, converges on emergence to two , 
 focal lines, the phenomenon of departing from a point focus 
 is called astigmatism (i. e., without a point). 
 
 Imagine a large diagram on a blackboard, in which there are 
 white lines in various directions, to be placed normal to the prin- 
 cipal axis of a converging lens. Let a white screen be placed in 
 the focal plane of the lens. The image formed on this screen will 
 have the following characteristics : the image of all lines 
 directed toward the intersection of the principal axis of the lens 
 and the plane of the diagram will be sharp to the edge of the 
 field. This is due to the fact that although the images of all 
 points near the edge of the object field will be lines, these image 
 lines are radial, and for any radial line of the object the image 
 lines will be superposed. The images of all lines of the object 
 perpendicular to the radial ones just considered will be broad and 
 blurred except near the center of the image field. This is due to 
 the fact that the image of each point of one of these object lines 
 is a line perpendicular to the object line. The length of the image 
 line of a point near the axis of the lens is small, but is appreciable 
 for points farther from the axis. 
 
 If the receiving screen be brought nearer the lens than the 
 principal focus, a position will be reached such that the images of 
 the ends of the radial lines of the object will be broad and 
 nebulous, and the image of lines perpendicular to them will be 
 distinct. 
 
 If with the receiving screen in the focal plane of the lens 
 we use a diagram consisting of black lines on a white background, 
 the images of the radial lines will be distinct and of the same 
 width throughout,, whereas images of the lines perpendicular to 
 
66 
 
 LENSES AND LENS SYSTEMS 
 
 them will be narrower at the edge of the image field than at the 
 center. They will be nebulous at the center and sharp at the 
 edge of the field. 
 
 By means of simple lenses of different thickness and refrac- 
 tive indices bounded by spherical surfaces of different curvatures, 
 a compound lens can be constructed that will produce no astig- 
 matism. 
 
 Lenses are sometimes used which are bounded by surfaces 
 that are cylinders with parallel axes. After traversing a convex 
 cylindrical lens, a wave which on incidence was plane will con- 
 verge to a single focal line. This line focus is the principal focus 
 of the lens. After refraction by a convex cylindrical lens, light 
 from a point source will produce a line focus that may be either 
 real or virtual, depending upon the distance between the source 
 and the lens. A compound lens consisting of a spherical and a 
 cylindrical component produces astigmatism in all pencils whether 
 they are incident axially or obliquely, centricly or excentricly. 
 47. Curvature of Field. — A narrow pencil of homogeneous 
 light A Bj advancing toward the principal axis and incident at 
 any point of a lens, will be brought to a focus at a point nearer 
 the leijs than will a pencil C B, parallel to the principal kxis and 
 incident at the same point of the lens. In 
 other words, the focal length of a lens for 
 the narrow pencil A B is less than the focal 
 length for light advancing parallel to the 
 principal axis and incident on the same point 
 of the lens. The greater the inclination 
 of ^ S to the principal axis, the shorter will be the, focal length. 
 For this reason the image of an object plane normal to the prin- 
 cipal axis will be curved. The error which causes the marginal 
 portions of the image of a plane surface normal to the principal 
 axis to be in a different plane from the central portions, is called 
 curvature of field. 
 
 rig. 81 
 
 Fig. 82 
 
LIGHT 
 
 67 
 
 In Figs. 82 and 83 is represented a convex lens provided 
 with a diaphragm or, "stop" in which there is a small opening 
 opposite the center of the lens. In Fig. 82 the object A B is 
 farther from the lens than the principal focus, whereas in Fig. 83 
 the ohipct is nearer the lens than the principal focus. In both 
 cases the object is so large that it subtends a large angle at the 
 center of the lens. 
 
 An inspection of Fig.\ 71 shows that when an object is be- 
 yond the principal focus of a convex lens, the shorter the focal 
 length of the lens, the nearer to the lens will be the image. Then, 
 since the focal lengths of a lens are smaller for oblique than for 
 axial pencils, the image will be curved, with the concavity toward 
 the- lens. Fig. 82. The image of a plane normal to the principal 
 axis will be saucer shaped. 
 
 Pig. 83 
 
 Fig, 84 
 
 An inspection of Fig. 72 shows that when an object is be- 
 tween a convex lens and the principal focus, the shorter the 
 focal length, the farther from the lens will be the image. Then, 
 since the focal lengths of a lens are smaller for oblique than for 
 axial pencils, the image (virtual, in this case) will be curved, 
 with the concavity away from the lens. Fig. 83. 
 
 An inspection of Fig. 74 shows that the shorter the focal 
 length of a concave lens, the nearer to the lens will be the image. 
 Then, since the focal lenghts of a lens are smaller for oblique 
 than for axial pencils, the image will be curved, with the con- 
 cavity toward the lens. Fig. 84. 
 
 By adding a convex lens to a concave lens, a compound 
 converging lens can be produced that will give zero curvature 
 of field. 
 
 48. Distortion. — An ununiform magnification of an image 
 is called distortion. Consider oblique pencils from points' of a 
 distant object incident on a simple converging lens. In Fig. 85 
 there is a diaphragm between the object and the lens, and in 
 Fig. 86 there is a diaphragm between the lens and the image. 
 Draw through the first equivalent point a ray parallel to the 
 incident pencil. The ray from the second equivalent point will 
 
68 
 
 LENSES AND LENS SYSTEMS 
 
 interseict the principal focal plane at A'. But since a pencil inci- 
 dent near the edge of a simple lens is refracted more than a 
 pencil in a parallel direction incident near the pole of the lens, 
 the pencils in the above figures will converge, not at the point A', 
 
 Fig. 85 
 
 Fig. 86 
 
 but at some point a'. Thus, with a diaphragm between the 
 object and the lens, Fig. 85, the edges of 'the image of an ex- 
 tended object will be less magnified than the central region. 
 When the diaphragm is between the lens and the image, Fig. 86, 
 the edges of the image of an extended object will be more 
 magnified than the central region. All lines of an extended 
 object that do not pass through the principal axis of the lens will 
 appear curved in the image. 
 
 , The field will also be curved. The image of a plane will not 
 be plane, but will be spherical. For this reason the image formed 
 on a plane screen will not be sharp throughout its extent. If the 
 screen be placed at such a position that the central portion of the 
 image is sharp, then the peripheral portions will be nebulous. 
 
 It should be remarked that although for the purpose of 
 simplifying the construction of the diagrams in this Article the 
 
 4 
 
 Fig. 87 
 
 Fig. 88 , 
 
LIGHT 
 
 69 
 
 object has been considered to be at a great distance from the 
 lens, there would be distortion of the image and curvature of 
 field whatever the distance. 
 
 Figs. 87 and 88 are from photographs of images of an object 
 consisting of a set of parallel straight lines crossed at right' angles 
 by another set of parallel straight lines. With the diaphragm 
 between the object and a converging lens, Fig. 87 was produced; 
 w;ith the diaphragm between the lens and the image, Fig. 88 was 
 produced. 
 
 49. Coma. — Even though a lens were corrected for astig- 
 matism, rays AB and AB', Fig. 89, equally inclined .to the line 
 A E', would converge to a point a, while rays A C and A C, 
 equally inclined to the line A E', would converge to a point a'. 
 
 Fig. I 
 
 If the lens were also corrected for curvature of field, the images 
 of a and a' would be in one plane. But even though the lens 
 were corrected for astigmatism and curvature of field, the image 
 of a point source at a distance from the principal axis would be 
 a nebulous volume instead of a sharp point. The aberration that 
 exists when light from a lliminous point incident obliquely on a 
 lens of large aperture forms a volume image instead of a point 
 image is called coma. 
 
 The outline of the volume image can be studied by plac- 
 ing a white receiving screen in the image. and normal to the axis 
 of the pencil. In Fig. 90 is shown the appearance on a screen 
 placed at a a', Fig. 89. The lower end of the bright pear-shaped 
 spot corresponds to the position a', and the middle of the upper 
 curved edge to the position a. The right hand side was out of 
 the plane of Fig. 89, and toward the reader. The cross-section 
 
70 
 
 LENSES AND LENS SYSTEMS 
 
 ^^^^p^ n^wppm^'- 
 
 Fig. 90 
 
 Fig. 91 
 
 Fig. 92 
 
 of the volume image near the farther end is a broad, nebulous, 
 nearly straight line, Fig. 92. The cross-section at a mid position 
 is shown in Fig. 91. 
 
 50. Methods of Reducing the Effects of Spherical Aber- 
 ration. — In the Articles immediately preceding, it has been shown 
 that the spherical aberration of a simple lens depends upon the 
 distance from the center of the lens to the point of incidence, the 
 curvatures of the lens surfaces and the inclination of the incident 
 ray to the principal axis. The effects of spherical aberration can 
 be reduced by several methods. The obvious rnethod is to de- 
 crease the effective aperture of the lens by stopping out either 
 the marginal part or the central part. In cheap cameras this 
 method is commonly employed. The method has the fault that 
 by cutting off a large amount of light the brightness of the image 
 is- considerably diminished. A camera in which this method is 
 used requires a long exposure. 
 
 From an examination of Fig. 79 it appears that a lens which 
 considerably refracts light has a large spherical aberration, 
 whereas one that refracts to but a small extent has a small 
 spherical aberration. The magnitude of spherical aberration in- 
 
 Fig. 93 
 
 Fig. 94 
 
LIGHT 71 
 
 creases rapidly with increase of refraction. To produce small 
 spherical aberration the shape of the lens must be such that at 
 neither face shall there be a large deviation. For a given total 
 deviation, the spherical aberration will be a minimum when the 
 deviation at the two surfaces of the lens are equal, — that is, when 
 the angles of emergence and incidence are equal. Fig. 93 shows 
 light parallel to the principal axis and incident on the plane 
 surface of a plano-convex lens. In this case all of the refraction 
 will occur at one surface and the spherical aberration will be 
 great. In Fig. 94 the light is incident on the convex face of the 
 same lens. Thedight is here refracted almost equally at the two 
 lens surfaces, and the spherical aberration is small. 
 
 Fig 95 shows a double convex lens 
 having the two surfaces of radii of curv- 
 ature in the ratio of 6 to 1, and made of 
 glass of refractive index 1.5. In this 
 case the angle of emergence equals the 
 angle of incidence, and the spherical 
 aberration is the smallest possible for 
 pjg 95 a single lens. This is called a "crossed 
 
 lens." 
 
 When moderate refraction is required, spherical aberration 
 can be reduced to zero by combining a lens of negative spherical 
 aberration with one of positive. 
 
 When a large refraction is required, spherical aberration 
 can be made as small as desired by combining several lenses of 
 such shapes that the refraction produced at each surface shall be 
 small. • Other things being constant, spherical aberration will be 
 least when the whole bending of a ray is equally divided amongst 
 the lens surfaces. It can be shown that for a system of two 
 lenses of given curvatures, and principal focal lengths /^ and f^, 
 the conditions of equal deviation for rays parallel to the principal 
 axis is that the lenses shall be separated by the distance 
 
 ^-h-h (39) 
 
 51. Light of all JPrequencies Traverses the Ether with 
 the Same Speed. — There are certain distant stars, Algol for 
 example, that change greatly in brilliancy within a very short 
 time. During these changes in brilliancy the color remains con- 
 stant. Now if red light traveled through the ether faster than 
 the violet, then when the star suddenly increases in brilliancy the 
 star would first appear red. As no change of color appears, we 
 conclude that in the ether the speed of light of all frequencies 
 is the same. 
 
72 
 
 LENSES AND LENS SYSTEMS 
 
 52. Dispersion. — It is found that although light of all fre- 
 quencies has the same velocity in the ether, light of different fre- 
 quencies traverses transparent matter with different speeds. This 
 fact is described by the stateriient that the index of refraction of 
 a substance is different for light of different frequencies. Since 
 the amount that light is deviated from its course in going from 
 one medium to another depends upon the relative speeds in the 
 two media, it follows that light ol different frequencies will be 
 refracted by different amounts in going from one transparent 
 
 medium to another. Thus, sup- 
 pose the luminous point S, Fig. 96, 
 is a source of waves having the 
 frequency that produces the sen- 
 sation of red, and also of waves 
 that produce the sensation of blue. 
 Fig. 96 Let a small portion of the wave 
 
 front pass through a hole in the diaphragm D and traverse the 
 glass prism P. It is found that a screen placed so as to receive 
 the light after emerging from the prism will exhibit two spots 
 of light, one red and one blue. Whatever may be the mixture 
 of light emitted by the source, the unequal refraction of light 
 of different frequencies effected by traversing the prism will 
 separate the composite light into its constituent colors. The 
 separation of light consisting of a mixture of different frequen- 
 cies into its components is called dispersion. 
 
 If one of two prisms of equal angle deviates light through 
 twice the angle that the other does, it might be supposed that 
 it would also disperse it twice as much. This is not, ho'wever, 
 found in general to be true. Two prisms of different substances 
 placed as in Fig. 97 may give a resultant deviation of zero and a 
 resultant dispersion of finite value. 
 
 Again, one substance may have a higher index of refraction 
 than another and have a smaller dispersion for any two given 
 colors. Thus, if a prism, Fig. 98, of the first substance be placed 
 
 Fig. 97 
 
 Fig. 98 
 
LIGHT 
 
 73 
 
 adjacent to a prism of the second substance, and of such an 
 angle as exactly to neutralize the dispersion produced by the 
 first prism, the light emerging ' from the system will be deviated 
 from its original direction, but will be undispersed. 
 
 53. Chromatic Aberration. — It has been shown (Art. 4) 
 that white light consists of a mixture of a great number of dif- 
 ferent colors. The same fact is more easily shown by means of 
 a prism. Since light, of different frequencies is refracted by dif- 
 
 Fig. 99 
 
 ferent amounts, a lens will not bring the components of white or 
 other unhomogeneous light to a single focus. There will be a 
 separate focus for each: color. The failure of a lens to form a 
 single focus for the various components of unhomogeneous light 
 is called chromatic aberration. 
 
 When white light from an object traverses a simple convex 
 lens. Fig. 99, there will be formed a series of colored images of 
 different sizes at different distances from the lens, — the blue 
 image being nearest the lens and consequently smallest, while 
 the red will be the farthest from the lens and consequently the 
 largest. The addition of a concave lens would cause all these 
 images to retreat from the lens, the violet being more affected 
 than the red. If the concave lens have greater dispersive power 
 aijd smaller refractive power than the convex lens, the images 
 of some two different colors will be brought into coincidence. 
 By cementing together two lenses of proper principal focal 
 lengths and dispersive powers, it is possible to construct a com- 
 pound lens that will cause the images of any two predetermined 
 colors to be coincident and also of the same size. Thus, in Fig. 
 
 Fig, 100 
 
74 
 
 LENSES AND LENS SYSTEMS 
 
 100, if the converging crown glass lens alone were used the red 
 light would be focalized at R, the blue would be focalized at B, 
 and light of other colors at intermediate points. The addition of 
 a diverging lens made of glass having greater dispersive power 
 than crown glass will cause light of two colors, — say red and 
 blue, to come to a common focus F. A lens cannot be made that 
 will bring all wave lengths to the same focus, but by causing the 
 brightest two colors to focalize at a single point the colored border 
 about the central bright spot is practically eliminated. A com- 
 pound lens that focalizes at a common point light of two wave 
 lengths is said to be achromatic for the two selected wave lengths. 
 By means of a lens of three components of different focal 
 lengths and dispersive powers, it is possible to cause images of 
 three predetermined colors to coincide. 
 Such a lens is said to be "apochromatic." I 
 An apochromatic lens used in three-color 
 photography is illustrated in Fig. 101. 
 This system consists of two compound j 
 lenses each of three elements. 
 
 By means of a combination of two I 
 lenses, of the same material, separated by | 
 an interval, light of various wave lengths 
 from a luminous point may be caused 
 to emerge in rays that are parallel but not coincident. Fig. 102. 
 This light on entering the eye will produce the same sensation as 
 
 though the rays were coincident. 
 Thus, by means of two lenses of 
 the same material, separated by 
 a distance, incomplete achro- 
 matism may be produced that 
 for the eye is satisfying. ' Since 
 both lenses have the same dis- 
 persive power, the system is equally achromatic for all colors. 
 It can be shown, though the proof will not be here given, that the 
 conditions for this partial achromatism is that the distances be- 
 tween the component lenses shall be 
 
 Fig. 101 
 
 Jik 
 
 
 KA 
 
 \SL 
 
 ^^""^l 
 
 ■ ^7' 
 
 Sir 
 
 1 
 
 F 
 
 
 E 
 
 Fig. 102 
 
 ^^y^ih+u) 
 
 (40) 
 
CHAPTER IV. 
 
 OPTICAL INSTRUMENTS DEPENDING UPON RE- 
 FRACTION. 
 
 §1. Some Properties of Lens Combinations. 
 
 54. The Angular Field of View of a Lens Combination. — 
 
 Imagine an object of indefinite extent XX' in front of a con- 
 
 Fig. 103 
 
 verging lens E^' E^. The lens through which light from an object 
 first passes is called the objective.'- The image of some point 0, 
 Fig. 103, will be formed at /. It is required to determine the 
 limit of the field of view that is visible through a second lens 
 E/E^. As this second lens is placed near the observing eye, it 
 is called the "eyelens." 
 
 Draw lines from the center and from the extremities of the 
 objective lens to one edge of the eyelens. Denote the intersec- 
 tions of these lines with the focal plane of the objective by 
 p, q and r. A consideration of the diagram shows that when the 
 image /of the point O is between j and p, light from the point 
 O will traverse the eyelens, and that when / is beyond p, no light 
 from the corresponding object point O will traverse the eyelens. 
 When / is between .? and r, all the light from the image will 
 traverse the eyelens, whereas when / is between r and p, only 
 part of the light will be incident on the eyelens. That is, on 
 looking through the eyelens, all points of the object between 
 A and R will -appear bright ; points from R to P will appear less 
 and less bright as their distance from A is greater, while points 
 beyond P will be invisible. 
 
 The field of view of the lens combination is limited by the axes 
 of the extreme pencils transmitted. In the diagram, the angle 
 2(g£iJ)^2®is the angular field of view of the combination. 
 It is usually expressed as the ratio of the radius of the second lens 
 to the distance between the two lenses. 
 
 The angle 2 (r E^ s) ^2 ^ limits the bright part of the field 
 of view. In optical instruments the visible field is usually limited 
 
It 
 
 OPTICAL INSTRUMENTS 
 
 to the bright field by means of a diaphragm containing a circular 
 aperture of radius sr placed in the focal plane of the objective. 
 
 In order that the whole of the bright field may be included 
 by the eye, the pupil must be placed at the point Z, where the 
 pencils emerging from the eyelens cross the principal axis of the 
 system. To locate this point, most optical instruments are pro- 
 vided with a stop, close to which the' eye must be placed when 
 looking through the instrument. 
 
 The field of view of a lens combination depends upon the 
 apertures and the principal focal lengths of the component lenses. 
 A system of lenses may be replaced by a single lens having a 
 principal focal length equal to the equivalent focal length of the 
 system without change of the field of view. 
 
 The visual angle under which an object can be sharply 
 seen through a lens by an eye situated on the principal axis of 
 the lens is limited by the diameter of the pupil and by the area 
 of the sensitive part of the retina. The visual angle may be 
 much less than the angular field of view of the lens to which 
 the eye is applied. For distinct vision, the retinal image must 
 not subtend at the principal point of emergence of the eye an 
 angle much greater than one degree. 
 
 55. The Supplementary Collective Lens. — When a single 
 lens is used, the illumination of an image depend^ upon the 
 brightness of the object, the transparency of the lens, the relation 
 between the size of the object to the size of the image, and upon 
 the angle at the object subtended by the lens. When a train of 
 lenses is employed, the lenses should be so selected that all the 
 light which traverses any lens should also traverse the succeeding 
 lens. 
 
 In Fig. 104, light from a distant object, after being brought 
 
 Fi^. 104 
 
 to a focus in the plane d^, proceeds to the lens (2), from which 
 it converges to a focus in the plane d^. 
 
 , In order that all the light that traverses the objective (i) 
 may appear in the second image, the lens (2) would need to have 
 the radius A B. By employing a lens (2) of shorter focal length. 
 
LIGHT 
 
 n 
 
 the diameter can be somewhat reduced. But the diameter of the 
 apparatus can be still farther reduced by the use of an additional 
 lens. By placing in the plane of d-^ a supplementary lens of proper 
 focal length (lo), Fig. 105, the image in the plane oi d^ will be 
 
 Fig. 105 - 
 
 unaltered in size and position. In order that the image shall now 
 contain all the light transmitted by the objective, the lens (2) 
 need have a radius of only A D. 
 
 The use of a supplementary lens is a common device to main- 
 tain the illumination of an- image without the necessity of lenses 
 of large diameter. Although any convex lens is a collective lens, 
 this term is commonly restricted to a lens employed to increase 
 the illumination of an image. 
 
 56. Magnification. — When light from a luminous object 
 traverses a converging lens and forms a real image, the ratio of 
 the size of the image to that of the object is called the linear mag- 
 nification of the object, or the linear magnifying power of the 
 lens. 
 
 Now the eye can be accommodated to rays that are either 
 parallel or slightly divergent. Vision is most easy when the rays 
 to the eye from a given point are parallel ; that is, when the point 
 is at a great distance. Vision is most distinct when the point is 
 about 10 inches (25 cm.) from the eye. Observers usually 
 focalize a microscope for most distinct vision. When the virtual 
 image of the object seen in a microscope is 10 inches from the 
 eye, the ratio of the size of this virtual image to the size of the 
 object is taken to be the linear magnification of the microscope. 
 
 When the observing eye is not focalized for most distinct 
 vision, magnification is measured in terms of the angles sub- 
 tended at the eye by the image and by the object. The ratio of 
 the angle subtended at the eye by the image to the angle sub- 
 tended at the eye by the object is called the angular magnifica- 
 tion of the object, or the angular magnifying power of the lens 
 or lens system. 
 
 Observers usually focalize telescopes for most easy vision. 
 The term "linear magnifying power" is not applied to telescopes. 
 
78 
 
 OPTICAL INSTRUMENTS 
 
 To be useful, the image must exhibit structure that cannot 
 be distinguished in the object. Magnification without increased 
 resolution is without value. The lens must resolve details that 
 cannot be resolved by the unaided eye. To produce resolution, 
 the aperture of the lens must not be less than that given by (25). 
 The proper magnification is determined by the limits of 
 comfortable vision for the details of the image that are to be 
 observed. The magnification produced depends upon the curva- 
 tures and material of the lens. Experience has shown that most 
 people read most comfortably type of the size used in this book 
 at a distance of about twenty-five centimeters. When the page 
 is held at this distance from the eyes, the angle at the eyes sub- 
 tended by the individual letters has an average value of about 
 0.008 radian. From this fact we conclude that for most com- 
 fortable vision the size of the details should be such as to sub- 
 tend an angle at the eye of about 0.008 radian. However, for 
 short spaces of time we can study details, without undue strain, 
 that are from one-tenth to ten times this size. 
 
 Problem. — A certain optical system consists of two positive lenses 
 having principal focal lengths in the ratio 3 : 1. The distance between 
 the equivalent points of each lens being given, and the distance between 
 the adjacent equivalent points of the two lenses being one-half the 
 suiin of the focal lengths of the two lenses, locate graphically the 
 equivalent points and the principal foci of the system. On the dia- 
 gram indicate the principal focal length of the system. From the 
 diagram show whether the system is positive or negative. 
 
 Solution. — 
 
 Fig. 106 
 
 In the figure, quantities with the subscript "1" irefer to the lens 
 marked "1," and quantities with the subscript "2" refer to the lens 
 marked "2." Quantities with the subscript "s" refer to the lens system 
 taken as a unit. 
 
 An inspection of the diagram shows that for this particular lens 
 system, (a), one equivalent point lies between the lenses at a distance 
 from the short-focus lens equal to about one-third the principal focal 
 length of that lens; (b), the other equivalent point of the system lies 
 beyond the short-focus lens and is almost coincident with the principal 
 
LIGHT 79 
 
 focus of the long-focus lens; (c), for light traversing the combination 
 in one direction the system produces convergence, whereas for light 
 in the opposite direction the system produces divergence; that is, the 
 combination may be either a positive or negative system, depending 
 upon the direction of passage of the light. 
 
 It can be shown that when the principal focus of the lens upon 
 which the light is incident is between the first principal focus and the 
 first equivalent point of the other lens, then the system is negative. 
 Otherwise, it is not. 
 
 §2. The Camera. 
 
 57. The Photographic Camera. — The camera consists of 
 a box having in one end an aperture covered with a converg- 
 ing lens, and in the opposite end a 
 device for holding a screen sensitive to 
 light. The distance between the lens 
 and the sensitive plate is so adjusted 
 that after traversing the aperture and 
 lens the light from an object in front 
 of the camera produces an image on 
 
 the screen. In making this adjustment a plate of grqund glass 
 is first used in place of the photographic plate. When the 
 camera is "focused," that is, when the -distance between the lens 
 ■and the ground glass is so adjusted that the image on the ground 
 glass plate is sharp, the lens is covered and a sensitive photo- 
 graphic plate is substituted for the ground glass. This plate is 
 now "exposed" by uncovering the lens for the proper interval of 
 time, and then "developed" by immersion in certain solutions. 
 
 58. The Photographic Objective. — Most luminous bodies 
 emit waves of different frequencies. The frequency that pro- 
 duces the sensation called yellow affects most strongly the sensa- 
 tion of sight, whilst that which produces the sensation called violet 
 affects most strongly the photographic plate. When the image has 
 been focalized by the eye for the yellow waves, it should be in 
 focus for the violet waves. Consequently, a camera lens should 
 have the same focal length for yellow and for violet; that is, 
 should be without chromatic aberration for waves of these two 
 frequencies. 
 
 ' For landscape photography, a single crown-flint glass com- 
 pound lens is commonly used. Though the various spherical 
 aberrations are present, they will not be obtrusive if the subject 
 does not contain long straight lines. These aberrations can be 
 greatly reduced by diminishing the lens aperture by means of a 
 "stop." The use of a small aperture is attended by the necessity 
 of a longer exposure. 
 
80 ; OPTICAL INSTRUMENTS 
 
 For a short exposure the stop must be large. With a large 
 stop the image will be distorted and unequally sharp in different 
 parts unless the photographic objective is corrected for spherical 
 aberration. The various spherical aberrations can be reduced to 
 any desired degree by the use of two compound lenses separated 
 by a distance. If each compound lens is corrected for chromatic 
 aberration, a system can be constructed that will be corrected for 
 both chromatic and spherical aberration. Three modern corrected 
 photographic objectives are illustrated in Figs. lOSj 109 and 110. 
 
 Fig. 108 
 
 Fig. 109 
 
 Fig. 110 
 
 Various names are given to photographic objectives, depend- 
 ing upon the particular property which it is desired to emphasize. 
 The term "achromatic" is applied to a lens in which chromatic 
 aberration has been corrected for two colors, and the term 
 "apochromatic" to a lens corrected for three colors. The terms 
 "rectilinear" and "orthoscopic" signify freedom from distortion. 
 A lens free from astigmatism is called a "stigmat" (i. e., a point), 
 an "anastigmat" (i. e., back again to a point), an "orthostigmat" 
 (i. e., same point), a "verastigmat" (i. e., true point), etc. 
 
 A lens system nearly free of both chromatic and spherical 
 aberration is termed "aplanatic." A lens that includes on the 
 photographic plate a wide angle of view is called a "wide angle" 
 lens. 
 
 59. The Teleobjective. — The size of the real image of a 
 given object is proportional to the distance between the image 
 and the emergent equivalent point of the lens. With a converg- 
 ing lens or lens system at a fixed distance from the object, this 
 distance increases with increase in the principal focal length of 
 the lens or lens system. 
 
 Fig. Ill represents parallel light from a distant point A inci- 
 dent on a positive l?ns of principal focal length f^. In construct- 
 
LIGHT 
 
 81 
 
 ing the figure, f^ was taken to be 10 in. The image A' B' will 
 be nearly 10 in. from the back face of the lens; that is, the back 
 focus of the lens is nearly 10 in. 
 
 By replacing this lens by one 
 of four tinges the principal focal 
 length, the image will be nearly 
 four times as large, and the back 
 focus will be increased in the same 
 proportion. A bellows extension 
 of 40 inches is possible, but not convenient. 
 
 By placing a negative. lens between the lens in Fig. Ill and 
 the ground glass, the light from the distant point A will proceed 
 
 Fig. Ill 
 
 Fig. 112 
 
 as indicated in Fig. 112. It will be observed that the emergent 
 equivalent plane of the combination Es is at a considerable dis- 
 tance from either lens. The principal focal length /' is now much 
 greater than the back focus of the system. In fact, by using a 
 negative lens of one-third the principal focal length of the posi- 
 tive lens, and a distance between the lenses of 7.66 in., the prin- 
 cipal focal length of the system will be 40 in., and the back focus 
 10 in. 
 
 By means of this device the "back focus" can be kept small 
 and still the image of the distant object may be large. By alter- 
 ing the distance between the two lenses, the principal focal length 
 of the system can be altered (and consequently the size of the 
 image), within wide limits. This property is applied in the con- 
 struction of telephotographic objectives for taking pictures of 
 distant objects. A modern telephotographic objective is illus- 
 trated in Fig. 113. This consists of a corrected objective, shown 
 to the left, and a diverging lens combination, shown to the right. 
 
 By using the converging component of the teleobjective 
 alone, a photographic negative could be taken and an enlarged 
 copy of this negative could be made. But even though the final 
 picture be of the same size as the picture taken at once by means 
 of the teleobjective, it would be decidedly inferior to the latter. 
 In enlarging a picture, the grain of the negative as well as the 
 
82 
 
 OPTICfAL INSTRUMENTS 
 
 minute irregularities of texture become more and more obtrusive 
 as the magnification is increased. So that though the resolving 
 
 Fig. 113 
 
 power of the teleobjective is no greater than that of the converg- 
 ing component, a sharper picture of a distant object can be pro- 
 duced by its use. 
 
 60. Depth of Field. — Light from two points at different 
 distances from a lens will form images that are at different dis- 
 tances from the lens. That is, sharp images of points at different 
 distances from a lens will not be formed on a plane perpendicu- 
 lar to the principal axis of the lens. But in most photographic 
 work the object consists of parts at different distances from the 
 lens, and the plate receiving the image is plane. This condition 
 can be met only by a sacrifice of sharpness. But for most pur- 
 poses maximum sharpness is not required. 
 
 The image of a luminous point at A^, Fig. 114, will be formed 
 at A^'. On a screen placed at A^' there will appear a small and 
 
 distinct diffraction disc. On 
 moving the screen either to 
 the right or. to the left, the 
 diffraction disc will fade into 
 a larger circle of light, bound- 
 ed by the cone of rays con- 
 Fig. 114 verging at A^^'. This circle is 
 called the "circle of confusion." 
 
 If the diameter of the circle of confusion due to each point 
 of an extended object is not greater than about 0.01 inch, the 
 image of the object will be sufficiently sharp for most purposes. 
 
LIGHT 
 
 83 
 
 Hence, to produce a satisfactory picture, the screen may be out 
 of the focal plane by such a distance that the diameter of the 
 circle of confusion due to any object point does not exceed about 
 0.01 inch. 
 
 If the luminous object point be moved to A^, the image will 
 move to A/. If the distance xy is not more than about 0.01 inch, 
 the image formed on a screen at xy, of objects situated anywhere 
 between ^^ and A^, will be satisfactorily sharp. The quality 
 which renders sufficiently sharp the images of objects situated at 
 different distances from the lens is called depth of field, or depth 
 of focus. It depends upon the principal focal length and upon 
 the aperture of the lens. With a short focus lens the distance 
 A^' A/ corresponds to a greater range of object distance A^ A^ 
 than with a long focus lens. Again, since with a small aperture 
 the distance A/ A/ corresponding to a fixed diameter of circle 
 of confusion xy^ is greater than for the same lens and a wider 
 aperture, it follows that with a small aperture a satisfactory image 
 will b,e formed of objects extending through a greater range of 
 distances from the lens than if the same lens were used with a 
 larger aperture. 
 
 Any lens will give depth of field if used with a small aperture. 
 Great speed and great depth of field cannot be obtained at the 
 same time. 
 
 61. The Fixed Focus Camera. — In the preceding Article 
 it has been shown that by means of a stopped lens of short focal 
 length a satisfactory image can be obtained of a landscape or other 
 object that extends through a considerable range along the axis 
 of the lens. This quality is utilized in the design of "universal" 
 or "fixed focus" cameras ; that is, in cameras in which the back 
 focus is essentially the same for objects extending from a few 
 feet from the lens to infinity. 
 
 -+-ff ' In Fig. 115, ^' is the im- 
 
 age of the luminous point A, 
 and X y is the diameter of the 
 blurred image of a luminous 
 point at infinite distance from 
 the lens. If ;ir y is not more 
 than about 0.01 inch, then the 
 ^*s- 113 images of all points from A to 
 
 infinity will be satisfactory. We will now find the distance yl E' , 
 beyond which all points are in satisfactory focus on the screen 
 through A'. This distance is called the "'hyperfocal distance," 
 
84 OPTiqAL INSTRUMENTS 
 
 Since the triangles H K F and x y F are similar, 
 HK _ EF 
 xy ~ FA' 
 or, using the notation indicated in the figure, 
 
 D __ f 
 d v — f 
 
 But since, (32) 
 
 V u ' f 
 
 we can eliminate v and find the distance beyond which all points 
 are in satisfactory focus, 
 
 a 
 In this equation, d is the maximum allowable diameter of the 
 blurred image of an object point at infinity, D is the diameter of 
 the aperture of the lens, and / is the principal focal length of the 
 lens. 
 
 62. Brightness of an Image. — If a point source of light 
 is distant u from a lens having an aperture of diameter D, there 
 will be incident upon the lens a quantity of light which varies 
 directly with the area of the aperture and inversely with the 
 square of the distance between the lens and the source. Conse- 
 quently, the quantity of light in the image of a point source is 
 
 in which fe is a constant of proportionality. 
 
 If the source be an object of diameter x, the quantity of 
 light incident upon the lens is 
 
 kP^x^ 
 
 Since this light is spread over an image of diameter y, the bright- 
 ness of the image of the extended object is 
 
 kP^x^ 
 u^ y^ 
 Now for narrow pencils making small angles with the. prin- 
 cipal axis of a corrected lens, Fig. 116, the ratio of the size of the 
 object to the size of the image equals the ratio of the object dis- 
 tance to the image distance. 
 
That 
 
 IS, 
 
 LIGHT 
 
 85 
 
 mg. 116 
 
 Whence, the brightness of the image of an extended source is 
 
 K^ y^ 
 
 u' V' 
 
 V 
 
 (42) 
 
 If the object be at a great distance from the lens, the image 
 will be near the principal focus, and v nearly equal to /. Under 
 this condition, the brightness of the image of an extended object 
 
 tf] 
 
 (43) 
 
 63. Diameter of Stop and Duration of Exposure. — The 
 
 quantity which expresses the brightness of the image (43) is 
 used in the computation of the duration of the exposure required 
 with stops of different diameters. In taking a photograph, a stop 
 of such a diameter is selected that the image will show the re- 
 quired, resolution and depth of field. After the stop has been 
 selected, the required duration of exposure must be determined. 
 The proper duration of exposure varies inversely with the bright- 
 ness of the image, and (43) shows that this is measured by the 
 ratio of the square of the diameter of the stop employed to the 
 square of the principal focal length of the lens. It is customary 
 to have for each photographic objective a series of stops marked 
 with numbers which express the relative durations of exposure 
 required when the various stops are used. If it be desired to 
 consti-uct a series of stops with which the exposures shall be in 
 the ratio 1:2, 1:4, 1:8, etc., that is, with which the brightness 
 of the image shall be in the ratio 1 : J^, 1 : J4. 1 : >i> etc., the 
 procedure will be as follows : 
 
 We shall assume that the diameter of the largest stop which 
 can be used is one-fourth the principal focal length of the lens. 
 
 For this stop, -j^ = — . Consequently, the brightness of the 
 I 4 
 
 D 2 1 
 
 image, which is measured by —~ , is represented by -j^ 
 
 D ^ 
 the next stop which is to give an image half so bright,^|- 
 
 For 
 
 must 
 
86 
 
 OPTICAL INSTRUMENTS 
 
 equal ^ . 
 
 That 
 
 is, 
 
 f 
 
 1 
 
 5.65' 
 
 or, D^ = 
 
 5.65 
 
 Proceeding : 
 structed. 
 
 in 
 
 like manner, 
 
 the following table was con- 
 
 Diameter of 
 
 Stop. 
 
 
 D. 
 
 f 
 
 
 (fl- 
 
 Relative Exposure 
 
 f 
 
 4 
 
 
 
 1 
 
 4 
 
 
 1 
 
 16 
 
 1 
 
 f 
 5.65 
 
 
 
 1 
 5.65 
 
 
 1 
 32 
 
 2 
 
 f 
 
 8 
 
 
 
 1 
 8 
 
 
 1 
 64 
 
 4 
 
 f 
 11.3 
 
 
 
 1 
 11.3 
 
 
 1 
 128 
 
 8 
 
 f 
 16 
 
 
 
 1 
 16 
 
 
 1 
 256 
 
 16 
 
 f 
 22.6 
 
 
 
 1 
 22.6 
 
 
 1 
 512 
 
 32 
 
 f 
 32 
 
 
 
 1 
 32 
 
 
 1 
 1024 
 
 64 
 
 f 
 45.2 
 
 
 
 1 
 45.2 
 
 
 1 
 2048 
 
 128 
 
 f 
 
 64 
 
 
 
 1 
 64 
 
 
 1 
 4096 
 
 256 
 
 Most makers have adopted the series of diameters given in 
 the table, and mark the individual stops with the numbers given 
 in the first column. Few lenses can be used with such a large 
 aperture as the first one in the table. But in using several lenses 
 it is a great convenience to have the numbering of the stops start 
 with a diameter which, for each lens, is the same fraction of the 
 focal length. In this case, though the larger apertures in the 
 table are unavailable for use, the stops that can be used are num- 
 bered as in the first column. It should be remarked, however, that 
 some makers construct series that start with apertures other than 
 Vn f . The numbering of a series of stops in terms of diameters 
 expressed as a fraction of the focal length of the lens is called 
 the / system of numbering. Thus, the symbol "/ : 32" repre- 
 sents a stop of a diameter one thirty-second of the principal focal 
 length of the lens. 
 
 Stops are also numbered so as to indicate the relative ex- 
 posure required ; that is, according to the numbers given in the 
 last column of the table. This is called the "uniform system" of 
 numbering. Thus the symbol "u. s. 64" represents a stop that 
 requires 64 times the duration of exposure that would be required 
 with the first stop of the series. 
 
LIGHT 
 
 87 
 
 §3. The Human Eye. 
 64. Structure and Function of the Eye. — Optically, the 
 human eye consists of an aperture P (called the pupil) in a dia- 
 phragm / (called the iris), a lens L, and a screen R (called the 
 retina). These parts are enclosed in a nearly spherical opaque 
 
 Fig. UT 
 
 envelope 5" provided with a curved transparent round window C. 
 The cornea,' C, and also the transparent gelatinous liquids which 
 fill the space between the cornea and the lens, and the space be- 
 ' tween the lens and the retina, have an index of refraction nearly 
 equal to that of water. The lens has an index of refraction of 
 about 1.44. 
 
 The two principal points of the human eye are between the 
 cornea and the lens, and the two nodal points are within the lens. 
 The front principal focal length is about 1.5 cm., and the rear 
 principal focal length is about 2.0 cm. The front principal focus 
 lies about 1.28 cm. in front of the cornea. 
 
 Functionally, the eye is equivalent to a simple camera. The 
 image of an object in front of the eye is formed on the retina 
 in exactly the same way that the image is formed on the ground 
 glass of a camera. The means for adjusting the eye so as to 
 focalize on the retina images of objects at different distances is, 
 however, different from the means employed in the case of the 
 camera. With the camera this adjustment is made by altering 
 the distance between the lens and the screen, while in the eye 
 this adjustment is effected principally by an involuntary altera- 
 tion of the curvature of the lens. 
 
 One sees a point source with least effort when the rays 
 to the eye from the point are ^ parallel. The point appears 
 to be most distinct, however, when distant from the eye about 
 10 in. (25 cm.). In using a telescope, the instrument is focalized 
 for most easy vision ; that is, the distance between the lenses are 
 adjusted till the emergent rays are parallel. In using a micro- 
 scope, however, the instrument is focalized for most distinct 
 
88 OPTICAL INSTRUMENTS 
 
 vision ; that is, till the rays entering the eye appear to come from 
 an object about 10 in. distant. 
 
 Different parts of the retina are unequally sensitive. ■ The 
 most sensitive part is a small spot of a diameter that subtends 
 at the principal point of emergence of the lens an angle of less 
 than one degree. The remainder of the retina is so much less 
 sensitive than this spot that a person always involuntarily moves 
 the eye into such a position that the retinal image is formed on 
 this area. 
 
 The eye has chromatic and all the spherical aberrations to 
 a high degree. As an optical instrument it is poor. But the 
 power of accommodation makes it an admirable sense organ. 
 
 Since the eye comprises a single lens, and real images are 
 formed on the retina, these images are inverted. 
 
 65. Ocular Defects and their Correction. — Light from a 
 distant object after traversing a normal eye at rest will forin 
 an image on the retina. In order that the image of a near object 
 may be on the retina the lens of the eye must be made more con- 
 verging. This is accoinplished by the contraction of a miiscle 
 about the edge of the lens automatically increasing the curvature 
 of the lens. This process of changing the focal length of the 
 lens of the eye is called accommodation. 
 
 The accommodative power of the eye decreases with old age.' 
 Loss of accommodative power is called presbyopia. Good images 
 of distant objects are produced on the retina of a presbyopic 
 eye; but since the focus of light from an object near such an 
 eye will be behind the retina, it follows that the image on the 
 retina of a near object will be indistinct. To produce a distinct 
 image of a near object the focus must be brought forward to 
 the retina. This is accomplished by spectacles which effect the 
 proper convergence of the light entering the eye. Since the 
 presbyopic eye is devoid of accommodative power, spectacles of 
 different focal length are required to view objects at different dis- 
 tances from the eye. 
 
 An eye that when ^t rest converges light from a distant 
 point to a focus in front of the retina is said to be myopic. Since 
 sharp images are produced on the retina of a myopic eye by 
 light from near objects, myopia is popularly described by the 
 name near-sightedness. Since the convexity of the lens of the 
 eye cannot be diminished, there is no internal means of neutraliz- 
 ing myopia. Myopia is corrected by spectacles having concave 
 lenses of such curvature that parallel light after traversing the 
 spectacle lens and the eye is brought to a focus on the retina. 
 The appearance of the retinal image of a certain object when 
 
LIGHT 89 
 
 the eye is moderately myopic is shown in Fig. 1 18. The addition 
 of a negative lens of focal length 33 cm. would cause the image 
 of the same object to have the appearance of Fig. 119. 
 
 NXTP iWtJP 
 Z O B Z O B 
 ETG 1 STa 
 
 CDV iC-DV 
 
 i _ 
 
 Fig., lis Fig. 119 
 
 An eye which when at rest converges light from a distant 
 point to a focus behind the retina is said to be hypermetropic. or 
 far-sighted. By accommodation such an eye will focalize light 
 from distant objects on the retina, but in order to bring light 
 from a near object to a focus on the retina convex spectacle lenses 
 must be employed. The appearance of the retinal image of a 
 certain object when the eye is moderately hypermetropic is shown 
 
 HXTP WUP 
 
 Z O B Z O B , 
 ETG ETG 
 
 CDV CDV 
 
 * 
 
 Fig. 120 Fig. 121 
 
 in Fig. 120. When a positive lens of focal length 33 cm. is placed 
 before the eye, the retinal image of the same object would have 
 the appearance of Fig. 121. If with spectacles the images of dis- 
 tant objects are focalized on the retina without accommodation, 
 then by the aid of accommodation objects nearer the eye can be 
 seen. 
 
 In the normal eye the faces of both the cornea and the lens 
 are spherical surfaces. In many eyes, however, the curvature 
 of either the cornea or the lens is not the same in different 
 meridians. With these eyes, the image of a point source is a line 
 
90 OPTICAL INSTRUMENTS 
 
 and not a point. That is, these eyes are astigmatic (Art. 46). 
 The astigmatism is said to be "regular" when the meridians of 
 greatest and of least curvature are at right angles to one another. 
 An eye that is regularly astigmatic cannot focalize at the same 
 time lines of an object that are at right angles to one another. 
 Regular astigmatism may be corrected either by a positive cylin- 
 drical lens that will increase the refraction along the meridian 
 which has the least curvature, or by a negative cylindrical lens 
 that will diminish the refraction along the meridian of greatest 
 curvature. 
 
 For an eye of moderate astigmatism and no other defects, 
 the appearance of a retinal image of a certain object would be 
 as shown in Fig. 122. It will be noticed that horizontal black 
 lines are narrow and nebulous as described on page 65. On 
 placing in front of the eye a negative cylindrical lens of focal 
 length 33 cm., with the axis of the cylindrical faces horizontal, 
 the retinal image would be as shown in Fig. 123. 
 
 If the meridians of greatest and of least curvature are not 
 approximately at right angles, or if there is an irregularity in 
 the curvature along some one meridian, the astigmatism is said 
 to be "irregular." Usually this form of astigmatism cannot be 
 corrected by lenses. 
 
 When an astigmatic eye is also either near-sighted or far- 
 sighted, one face of the spectacle lens is a cylindrical surface and 
 the other face is a spherical surface. 
 
 ff u F 
 
 If U P 
 
 :z O B !Z O B 
 
 T« J^F ii'"*' Ts !»T^ rw 
 
 ^•'■*. '-If-', =5 5- ^■g =?^, ^S- 
 
 Fig. 122 Fig. 123 
 
 Sometimes when the lens of tlie eye is excessively convex or 
 opaque, it is removed. It is then necessary to use spectacles 
 of convex lenses. As such an eye is incapable of any accommo- 
 dation, spectacles of a different convexity will be required for 
 near and for distant vision. 
 
LIGHT 91 
 
 66. The Numbering of Spectacle Lenses. — Since a convex 
 lens of small aperture converges a plane wave to a principal focus, 
 
 the lens changes the curvature of the wave from zero to -j- 
 
 where / denotes the principal focal length of the lens. Similarly, 
 a concave lens of small aperture imprints on the wave traversing 
 
 it a curvature of ~z- . Thus, in general, the effect of a lens is 
 
 / 1 
 
 to imprmt on a wave traversmg it a curvature of -j- . The 
 
 cause of this change in curvature is the greater retardation of the 
 speed of light where the glass is thick than where it is thin. Since 
 this retardation is independent of the curvature of the incident 
 wave, it follows that if the curvature of the wave remains con- 
 stant while traversing the lens, the curvature of the emergent wave 
 always differs from the curvature of the entrant wave by the con- 
 
 1 
 stant amount -p- 
 
 Spectacle lenses are so tliin that no appreciable error is made 
 in assuming that the "power" of a spectacle lens to alter the 
 curvature of a wave, front equals the reciprocal of the principal 
 focal length. It is customary to denote the "power" of a spectacle 
 lens by the reciprocal of the principal focal length expressed in 
 meters. A spectacle lens of one meter focal length is said to have 
 a "power" of one diopter. A lens of two meters focal length has 
 a "power" of 0.5 diopter. One of 0.25 meter focal length has a 
 "power" of four diopters. 
 
 Converging lenses, whether cylindrical or spherical, are called 
 positive; and diverging lenses, whether cylindrical or spherical, 
 are called negative. 
 
 In the case of an astigmatic eye, the positions of the merid- 
 ians of greatest and least curvature are described in terms of the 
 angles they make with the horizontal. Angles are measured 
 counter-clockwise from the right hand end of the horizontal line 
 through the eye, as seen by a person looking toward the eye. 
 
 The notation used to describe spectacle lenses will be illus- 
 trated in the following oculists' prescriptions : 
 
 O. D. 4- 0.75 D. sph. 
 
 O. S. + 2.00 D. sph. 
 means, "for the right eye (oculus dexter) a positive spherical lens 
 of 0.75 diopters, and for the left eye (oculus sinister) a positive 
 spherical lens of 2.00 diopters." Again, 
 
 O.D. + 3.50D.cy.ax. 45°. 
 
 O. S. + 2.25 D.cy. ax. 13S° 
 
92 
 
 OPTICAL INSTRUMENTS 
 
 reads, "for the right eye a positive cylindrical lens of 3.50 diopters 
 with the axis at 45° from the horizontal, and for the left eye a 
 positive cylindrical lens of 2.25 diopters v/ith the axis at 135° 
 irom the horizontal." Again, 
 
 O. D. — 0.50 D. sph. + 1.25 D. cy. ax. 90°. 
 
 0. 5". + 0.75 D. sph.— 1.00 D. cy. ax. 165°. 
 means, "for the right eye a lens that is equivalent to a negative 
 spherical lens of 0.50 diopter in combination with a positive 
 cylindrical lens of 1.25 diopters with the axis vertical and for the 
 left eye a lens that is equivalent to a positive spherical lens of 
 0.75 diopter- in combination with a negative cylindrical lens of 
 1.00 diopter with the axis at 165° from the horizontal." • 
 §4. The Stereopticon or Projection Lantern. 
 
 67. The Elements of the Lantern. — If a positive lens 0, 
 Fig. 124, be placed in front of an object at a distance greater 
 than the principal focal length of the lens, a real image / will 
 
 Fig. 126 
 
 be formed. If this image is much larger than the object, it will 
 be dim. For a magnification of a few diameters, the image will 
 be scarcely visible. 
 
 By placing behind the object a bright point source of light S, 
 Fig. 125, the clear space about the object and the transparent 
 
LIGHT 
 
 93 
 
 parts of the object will be traversed by light. The light which 
 traverses the object and the lens O will form a bright image of 
 the part of the object traversed. Unless the object is considerably 
 smaller than the aperture of the lens, only the central part of the 
 object will be traversed by the light from the source that goes 
 through the lens. The image will then be dim except for the cen- 
 tral part that is due to light from the source that has traversed 
 both the object and the lens. 
 
 By introducing between the object and the light source a 
 positive lens of greater diameter than the object CI Fig. 126, 
 light that traverses the edge of the object will be caused to also 
 traverse the lens O. By this device a bright image of the entire 
 object can be produced. 
 
 The projection lantern consists essentially of an intense 
 small light source, a lens to collect a large .amount of light and 
 condense it on the object, and another lens to focalize light from 
 the object on a screen. The lens or lens system used to illumine 
 the object is called the condenser. The lens or lens system used 
 to focalize light from the object on the screen is called the 
 objective, the projecting lens or the focusing lens. 
 
 These elements are mounted in various ways, according to 
 the purpose for which the instrument is designed. The simplest 
 form is the familiar "magic lantern" for producing on a screen 
 large images of drawings or photographs made on glass "lantern 
 
 Fig. 127 
 
94 
 
 OPTICAL INSTRUMENTS 
 
 slides." In conjunction with a microscope it is used to produce 
 on a screen large images of microscopic preparations. Special 
 forms are used in physics lecture demonstration and in moving 
 picture projection. 
 
 A projection lantern designed especially for physics lecture dem- 
 onstration is illustrated in Fig. 127. The illuminant consists of an 
 electric arc lamp the carbons of which may be either in line or at right 
 angles to one another. The lenses are easily exchanged so as to pro- 
 duce images of different size, and the positions of the lenses are easily 
 altered. The optical system is far enough above the table to permit 
 apparatus of considerable height to be used as the "object." 
 
 For use with objects that must be horizontal, the entire optical, 
 system can be rotated as a unit into the position shown in Fig. 128. 
 When in this position the emergent beam of light is reflected to the. 
 screen by the mirror M. The entire lantern can also be rotated about 
 a vertical axis. 
 
 Fig. 128 
 
 68. The Lantern Objective. — For the projection of images 
 of objects that consist of long straight lines, the objective should 
 be nearly free of spherical aberration. This necessitates a com- 
 bination of lenses similar to a camera objective. For the projec- 
 
LIGHT 
 
 9S 
 
 tion of images of most small objects, however, the aberrations 
 produced by a simple lens are unobtrusive. 
 
 The principal focal length of the objective depends upon the 
 magnification required and upon the distance from the objective 
 
 It 
 
 
 
 + 
 
 
 
 1 
 
 
 -...^^ 
 
 
 \ 
 
 
 
 
 ^^^^^ 
 
 
 
 "x^ 
 
 K 
 
 
 B 
 
 /■ 
 
 
 
 ^>^\ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 A' 
 
 rig. 129 
 
 to the screen. In the case of an objective without spherical aber- 
 ration, the linear magnification M, Fig. 129, is 
 
 A' B' ^ V 
 
 M 
 
 [- 
 
 (44) 
 
 AB ) u 
 Since the object is beyond the principal focus, we have from (32) 
 
 f ~ V ^ u (45) 
 
 In this equation, substituting the value of u from (44) 
 1 1 1 :^ _ 1 + M 
 
 / 
 
 or. 
 
 / 
 
 (46) 
 
 1+M 
 The relation between the object distance and the prmcipal 
 focal length of the objective will now be determined, 
 tuting in (45) the value of v from (44), we obtain 
 
 1 _ 1 \_^ 1 + M 
 
 f Mu^ u 
 
 Mu 
 or, / 
 
 Mu 
 
 Substi- 
 
 (47) 
 
 1 + M 
 
 An inspection of this equation shows that as M increases, 
 / becomes more nearly equal to u. In the case of a projection 
 lantern, M is so large that the distance between the object and 
 the objective is but little greater than the principal focal length of 
 the objective. 
 
 69. The Lantern Condenser. — The function of the con- 
 denser is to illumine the object evenly and intensely. Even illu- 
 mination can be secured, even though the condenser produces 
 
96 
 
 OPTICAL INSTRUMENTS 
 
 some spherical aberration, by placing the object close to the con- 
 denser. To produce a strong concentration of light on the object, 
 the condenser must be near the light source ; that is, it must be 
 of short principal focal length. In order that th^ spherical aber- 
 ration of a short focus condenser may not be excessive, it must 
 consist of either two or three lenses depending upon the focal 
 length of the combination. 
 
 In order that the objective may have a moderate aperture, 
 all the light from the source that traverses the condenser and 
 the object should pass through the first equivalent point of the 
 objective. Consequently the principal focal length of the con- 
 denser is determined by the distance it is to be used from the 
 illuminant, and the distance the objective is to be used from the 
 object. 
 
 Fig. 130 shows the passage of light through the optical sys- 
 tem of a lantern in which there is an extended light source 5", 
 a two-lens condenser C, and a corrected objective 0. 
 
 Fig. 131 shows the passage of light through a lantern pro- 
 vided with a three-lens condenser. 
 
 Fig. 131 
 
 70. Moving Picture Projection. — For the representation 
 on a screen of the motions of bodies there are required a projec- 
 tion lantern and a series of instantaneous pictures of the moving 
 body takpn at regular and short intervals. If the film with the 
 series of pictures be rapidly drawn at a uniform speed through 
 the object plane of the lantern, the screen will show a confused 
 band of light. But if, when a picture is in front of the condenser, 
 it be caused to remain at rest for a brief period, and while another 
 picture is taking its place the light be eclipsed by a shutter, there 
 will be formed on the screen a series of separate pictures each last- 
 ing for a short time. Due to the persistence of retinal impressions 
 
LIGHT 
 
 97 
 
 after the cessation of a retinal image, these i)ictures of brief dura- 
 tion fuse into a picture that appears to be uninterrupted. 
 
 71. The Moving Picture Lantern. — Fig. 132 represents a 
 lantern arranged for the projection of moving pictures. By means 
 
 ffig. 132 
 
 of the sprocket wheels a and b the film F is wound from the reel 
 Ri to the take-up reel R^ at a uniform rate of about one foot per 
 second. The intermediate sprocket wheel c moves intermittently, 
 jerking a picture into the beam of light, pausing, and then jerking 
 the next picture into its place. A picture is at rest for about five 
 times the time required to change pictures. While the picture is 
 being changed the light to the screen is cut off by an opaque wing 
 of the rotating sectored disc D. 
 
 For mechanical and economical reasons the pictures on the 
 film are small, being, in fact, less than one inch in either dimen- 
 sion. In order that the large screen image may be sufficiently 
 bright, the film picture must be more strongly illumined than a 
 lantern slide. To secure the required degree of illumination, the 
 film is placed in the circle of least confusion of the beam from an 
 extended light source. If the source were of nearly point dimen- 
 sions, and the condenser were not fully corrected for longitudinal 
 spherical aberration, there would be a circle of least confusion of 
 rather weak illumination with a brighter central spot (Art. 44). 
 But if the source have a diameter of not less than three milli- 
 meters, the circle of least confusion will be bright and practically 
 uniform. For moving picture projection the source of light is 
 the crater of the positive carbon of an arc lamp operated by a 
 sufficient current to produce a source of the required diameter. 
 A short focus condenser is employed that has the proper amount 
 of longitudinal spherical aberration to produce a uniformly bright 
 circle of least confusion of sufficient size to cover one of the film 
 pictures. The objective should be practically free of chromatic 
 and spherical aberration. 
 
 72. Prevention of Flicker. — The pictures on the screen 
 should appear to fuse into one another without any flicker. The 
 fusion of successive images is due to the persistence of the retinal 
 sensation after the cessation of the retinal image. Lack of flicker 
 
98 
 
 OPTICAL INSTRUMENTS 
 
 requires that the intensity of the retinal sensation produced by one 
 picture shall not appreciably diminish during the time that picture 
 is being- changed for the next. The duration of a retinal sensation 
 after the cessation of the retinal image depends upon the bright- 
 ness of the retinal image. For a bright image the duration is less 
 than for a faint image. 
 
 These considerations lead to the conclusion that flicker can 
 be prevented by diminishing the time the picture is on the screen 
 and the duration of the eclipse ; that is, by sufficiently increasing 
 the number of flashes of the picture on the screen per second. 
 
 It is found that with most films there will be no flicker if 
 there are not less than 48 flashes per second. Moving picture 
 films usually have 16 pictures per foot, and are moved through the 
 beam of light at the rate of one foot per second. It is a common 
 practice to flash each picture on the screen three times during the 
 interval it is at rest. 
 
 §5. The Telescope and the Microscope. 
 
 73. Galileo's Telescope. — This instrument consists of a 
 converging lens, a diverging lens, and means for altering their 
 distance apart. The converging lens is directed toward the object 
 under observation and is called the objective; the diverging lens 
 is toward the eye and is called the ocular. In Fig. 133, if the 
 ocular were not present, light from the object A B would form 
 an inverted real image at A^ B^. If, however, a diverging lens (2) 
 be introduced between this image and the objective (1), the light 
 pencils from points of the object after traversing the ocular will 
 diverge from one another. That is, the image will now be virtual. 
 The diagram shows that this image will be erect. 
 
 For most easy vision, the pencils emerging from the ocular 
 should be cylindrical. This result is accomplished by making the 
 distance from the ocular to the aerial object A.^^ B^ equal to the 
 principal focal length of the ocular. The virtual image will now 
 be at infinity, ' 
 
 Fig. 133 
 
 Assuming that the lenses are without spherical and chromatic 
 aberrations, the magnifying power of the Galilean telescope can be 
 readily obtained. From definition, the angular magnifying power 
 of an optical system is the ratio of the angle at the eye subtended 
 
LIGHT 
 
 99 
 
 by the image to the angle at the eye subtended by the object. In 
 the case of a telescope, the distance from the object to the eye 
 is so nearly equal to the distance from the object to the objective 
 that it is customary to call the magnifying power of a telescope the 
 ratio of the angle subtended at the eye by the image to the angle 
 subtended at the objective by the object. Thus, from Fig. 133, 
 the magnifying power of the Galilean telescope focalized for most 
 easy vision is 
 
 1^ 
 
 The ordinary opera glass, field glass, and marine glass con- 
 sist of two telescopes of this type, one for -each eye. Since 
 (jalileo's telescope gives erect images, is compact and cheap, it 
 is well suited to such use. The serious disadvantage of this in- 
 strument is its very, small field- of view. This is made evident in 
 Fig. 134. Since the cylindrical pencils emerging from the ocular 
 are divergent, the pupil of an eye in front of the ocular will ex- 
 clude all the light except that included within the small angle (8. 
 
 Fig. 134 
 
 That is, light from any point outside of this angle, after emerging 
 from the ocular, will not enter the pupil of the eye. The greater 
 the magnifying power of the ocular, the smaller the field of view. 
 Since the light from different points of the object on emerging 
 from the ocular is strongly divergent, if the eye is not placed, 
 close to the ocular the field of view will be farther diminished. 
 It is unnecessary to have the ocular much larger in diameter than 
 the pupil of the eye. 
 
 74. The Simple Astronomical Telescope. — In its simplest 
 form, this telescope consists of two converging lenses mounted 
 in opposite ends of a tube of adjustable length. The lens directed 
 toward the object under observation is called the objective; the 
 lens toward the eye is called the ocular. In Fig. 135 light from 
 a distant object A B, after traversing the objective (1), forms an 
 inverted real image A^ B^. If the positive ocular (2) be placed 
 at a distance equal to its principal focal lenght to the right of this 
 
100 
 
 OPTICAL INSTRUMENTS 
 
 image, the pencils emerging from the ocular will be cylindrical 
 and convergent. An eye placed in front of the ocular will see 
 an inverted image at infinity. 
 
 Fig. 135 
 
 If the objective and ocular be without spherical and chro- 
 matic aberrations, the magnifying power of a simple astronomical 
 telescope focalized for most easy vision will be 
 
 ^ = T = 
 
 
 That is, the magnifying power of, this instrument varies directly 
 with the principal focal length of the objective, and inversely 
 with the principal focal length of the eyepiece. The principal 
 focal length of the objective of the Yerkes telescope is 62 feet. 
 By the use of eyepieces of various principal focal lengths, the 
 magnifying power of a telescope may be changed within wide 
 limits. 
 
 For the semi-angle A between the axes of the extreme pencils 
 that enter the eye. Fig. 135, the total semi-angular field of view 
 of the simple astronomical telescope is represeiited by the angle 8. 
 This angle depends upon the diameter of the second lens, the dis- 
 tance between the two lenses, and the angle between the axis 
 of the extreme pencils which entering the eye will produce distinct 
 vision. 
 
 In iising the telescope, the pupil of the eye should be placed 
 in front of the ocular where the pencils emerging from the ocular 
 cross, the axis of the instrument. This place is usually indicated 
 by a diaphragm containing a small aperture not much larger than 
 the pupil of the eye. 
 
 75. The Use of a Telescope for Sighting.— In surveying and in 
 a great variety of astronomical and physical determinations angles are 
 measured h-jf the aid of telescopes. Suppose the angle X C Y, Fig. 136, 
 is required. Pointing a telescope placed at C toward the object X, an 
 image will be formed at some point in the focal plane of the objective. 
 If the telescope be rotated about an axis through C till light from Y 
 
LIGHT 
 
 101 
 
 forms an image at the same point, the angle through which the tele- 
 scope has been turned equals the angle X C Y. This method requires 
 a suitable circular scale attached to the telescope, and also a fixed 
 point in the fotal plane which can be caused to coincide ifl succession 
 
 with the images X and Y. This fixed 
 point of reference usually consists of 
 the point of intersection of two very 
 fine wires or fibers placed in the focal 
 plane of the objective. When the eye- 
 piece is focalized on the cross-wires 
 the eye will see the cross-wires and 
 the image of the distant object coinci- 
 dent. This condition is attained by 
 moving the eyepiece back and forth 
 till a position is found such that when the eye is moved slightly from 
 one side to the other there will be no displacement of the image rela- 
 tive to the cross-wires. ^ 
 
 Small angular displacements of a body are frequently determined 
 by means of the "Telescope and Scale Method." Suppose it is' re- 
 quired to measure the deflection' or angular .displacement of a small 
 
 Fig. 136 
 
 Fig. 137 
 
 magnetic, needle produced by an electric current in a neighboring wire. 
 The angle © between the first position n s. Fig. 137, and the second 
 position n' s' is the angle required. A small mirror is attached to 
 the magnetic needle, 'and the image of a stationary scale O' O" re- 
 flected by the moving mirror is observed with a telescope. If the 
 mirror is also perpendicular to the axis of the telescope, an observer 
 looking into the telescope sees the image of the point O of the scale 
 on the cross-wires; and when the mirror is deflected through an 
 angle ®, the image of the point O' of the scale is on the cross-wires. 
 When the mirror is turned through the angle ®, the normal to the 
 mirror has also moved through an angle ®. And since the angle of 
 reflection equals the angle of incidence (Art. 17), the angle O' CO 
 
102 OPTICAL INSTRUMENTS 
 
 equals 2 @. Denoting the distance of the scale from the mirror by L 
 
 and the linear deflection O O' by s, we haye 
 
 s 
 tan 2 ® = J — 
 
 whence the deflection © can be obtained. 
 
 It should be kept in mind that cross-wires coiricide in position 
 with a real image. If the instrument has no real image, cross-wires 
 are useless. For this reason, Galileo's telescope cannot be used for 
 •sighting. 
 
 76. Telescope Objectives. — The function of the objective 
 of an optical instrument is to collect a large amount of light 
 emanating from the object under observation and concentrate 
 it into either a real image (as in the astronomical telescope), or 
 into an aerial object (as in the Galilean telescope). By means of 
 an objective of large aperture, stars can be perceived that would 
 be invisible to the naked eye. Since the resolving power of a lens 
 increases when the aperture increases, the minimum diameter of 
 the objective is determined by the resolving power required of the 
 instrument. 
 
 By combining a converging lens of crown glass and a diverg- 
 ing lens of flint glass, a compound lens may be produced that is 
 achromatic. The great light gathering power and the great re- 
 solving power of the Yerkes telescope are due to the great diame- 
 ter of the objective. This is an achromatic doublet 40 inches in 
 diameter. It cost $66,000. The remainder of the telescope cost 
 $55,000. 
 
 In the case of a telescope, the pencils from a point under 
 observation are incident on the objective so nearly centricly that 
 the distortion produced by the objective is very small. The re- 
 maining spherical aberrations are kept low by giving the outer 
 faces of the objective proper curvatures. The chromatic aber- 
 ration is reduced by forming the objective of two lenses of dif- 
 ferent refractive indices and curvatures. An objective usually 
 consists of a convex lens of crown glass and a concave lens of 
 flint glass. In order that the light incident centricly on the first 
 component lens may be incident centricly on the second compo- 
 nent also, the two components must be close together. They are 
 usually cemented together so as to constitute a single compound 
 lens. 
 
 77. Oculars or Eyepieces. — The purpose of the ocular or 
 eyepiece of an optical instrument is to magnify the image formed 
 by the light that has traversed the objective. Since the pencils 
 incident on the ocular are oblique and exc^ntric, the errors due to 
 
LIGHT 103 
 
 spherical aberration are much greater than they are for the 
 objective. The defects are reduced to a minimum by employing 
 such lenses that the bending shall be as small as possible at each 
 surface. The bending at each refracting surface is reduced to a 
 minimum, (a) by increasing the number of refracting surfaces ; 
 (b) by causing the bending at each surface to be the same ; (c) by 
 a proper selection of the curvatures of the refracting surfaces. 
 
 . For most purposes, two lenses in the ocular are sufficient to 
 make the bending at each surface small enough to reduce the 
 spherical aberrations of the ocular to a proper amount. In the 
 case of two lenses of given curvatures, and of principal focal 
 lengths, /i and f^, respectively, it can be shown that the condition 
 for equal deviation of incident rays parallel to the principal axis 
 is that the lenses shall be separated by the distance 
 
 x-h-h (48) 
 
 For a system of two given lenses made of the same kind of 
 glass, and of principal focal lengths /^ and f^, respectively, it can 
 be shown that there will be minimum chromatic aberration when 
 the combination is convergent, and the component lenses are 
 separated by the distance 
 
 ^ = ^ (A +/.) _ (49) 
 
 The lens of the ocular toward the eye is called the eye lens; 
 the one toward the objettive is called the field lens. 
 
 78. The Huyghens Eyepiece. — This eyepiece is a com- 
 bination of two lenses of the same kind of glass designed to re- 
 duce to a minimum the effects of spherical and chromatic aberra- 
 tions. If the conditixDn of minimum spherical aberration (48) 
 be combined with the condition of minimum chromatic aberra- 
 tion (49), then, for a system of two lenses made of the same kind 
 of glass, we have 
 
 h-h = y-ih + h) 
 
 Whence, fi^^fi 
 
 Consequently, for axial light pencils, the principal focal 
 length of the field lens should be three times that of the eye lens, 
 and the two lenses should be separated by a distance equal to the 
 mean of their principal focal lengths. 
 
 This particular two-lens combination was devised by Huy- 
 ghens and is called the Huyghens Eyepiece or Huyghens Ocular. 
 The component lenses are usually convexo-plane ; that is, the con- 
 vex surfaces are toward the incident light, Figs. 138 and 139. 
 
 Besides this combination of two lenses of principal focal 
 lengths in the ratio 3 to 1, Huyghens also used a combination 
 
104 
 
 OPTICAL INSTRUMENTS 
 
 of two lenses of principal focal lengths in the ratio of 2 to 1. 
 He also devised a three-lens eyepiece that is considerably used. 
 The distance the field lens of a Huyghens 3 ; 1 eyepiece must be 
 from an object or a real image, in order that after traversing the 
 combination the emergent fays may be parallel, will now be deter- 
 mined. In order that light emerging from the eye lens may be 
 parallel, there must be a real image to the feft at a distance equal to . 
 the principal focal length of the eye lens. And in order that a real 
 image may be formed at this place, the field lens must be in a certain 
 position relative to the aerial object due the objective (objective not 
 shown in the figure) that will now be determined. 
 For the Huyghenien eyepiece of this type 
 
 Fig. 138 
 
 Fig. 139 
 
 andiassuming that the distance between the lenses equals the mean 
 of their principal focal length, (49), 
 
 = ^(f^ + f^y]=i_[,^+^^^|_,^ 
 
 Whence, f^ _ ^^ ^ 
 
 Therefore, F^ must be midway between the field lens and the eye 
 lens. Since the real image is formed at this point, ' 
 
 , V =f r=iL 
 
 ' ' [ , 3 
 
 Substituting this value in (34), 
 
 — [=—--! = — 
 
 2 
 
 17 
 
 Consequently, the field lens must be placed between the objective 
 and the aerial object due to light that has traversed the objective, and 
 at a distance from this aerial object equal to half the principal focal 
 length of the field lens. 
 
 The aerial object will be curved and the peripheral portions 
 less magnified than the central portion (Art. 48).. The virtual 
 
LIGHT 105 
 
 image will be curved in the opposite direction (Art. 47), and the 
 peripheral portions more magnified than the central portion. 
 Consequently, the ilnage seen by the eye will be nearly free of 
 distortion and curvature. The spherical aberration can be far- 
 ther reduced by using for the eyelens a "crossed lens" (Art. 50), 
 and for the field lens a convexo-concave lens having radii of 
 curvature in the ratio 4:11. 
 
 If cross- wires were placed in the image within the Huyghens 
 eyepiece, light from them would traverse but one lens. The 
 image of the cross-wires seen by the eye would be distorted, and 
 the greater the magnification of the eye lens, the greater the dis- 
 tortion. Since the final image of the object and the image of the 
 cross-wires are unequally distorted, cross-wires are not employed 
 in the Huyghens eyepiece except when the magnification is low. 
 
 79. The Ramsden Eyepiece. — For measuring angles or 
 distances, the cross-wires or scale must be placed in the real 
 image of the object under observation, and any slight distortion 
 produced by an observing eyepiece must affect in the same way 
 any subsequent images of both the cross-wires and the original 
 image of the object. Ramsden's eyepiece was designed with 
 especial regard to these requirements. It consists of two con- 
 verging lenses of equal focal length placed beyond the image 
 formed by light that has traversed an objective lens. 
 
 For minimum chromatic aberration of two simple converg- 
 ing leijses of the same material, the distance between them should 
 be, (40) , x=y2 (/1-I-/2) . - Or, in the case of the Ramsden combina- 
 tion, the distance between the two equal components should be 
 equal to, the focal length of one of them. If this separation were 
 made, then the field lens would be in the principal focal plane 
 of the eye lens and the field lens would coincide with the image 
 to be rnagnified. Such an arrangement would have the fault 
 that dust or spots on the field lens would show in the field of 
 view. To obviate this fault, the distance between the two lenses 
 is made less than required to make the chromatic aberration 
 minimum. The separation is usually made two-thirds the prin- 
 cipal focal length of one of the lenses. With simple lenses the 
 departure from achromatism will be slight. If better achroma- 
 tism is required, each lens may be composed of a flint-crown- 
 ■ glass combination as described in Art. S3. The other defects of 
 spherical aberration are remedied through the selection of the 
 curvatures of the lens faces. 
 
 Usually, the lenses are simple plano-convex, with the curved 
 surfaces toward one another. Light from some point of an 
 object, after traversing an objective not shown in the figure, will 
 
106 
 
 OPTICAL INSTRUMENTS 
 
 converge to a real image A, Fig 140. After traversing the field 
 lens (1), the light will diverge as though it came from a virtual 
 image A'. 
 
 Fig. 140 ^ Fig. 141 
 
 For most easy vision, light from a' point source should emerge 
 from the eye lens in a parallel pencil. In order that light from a point 
 source shall emerge from the eye lens in a parallel pencil, the first 
 equivalent plane of the field lens must be at a definite distance from 
 the image A, which will now be determined. 
 
 With reference to the lens (1), Fig. 140, A is the source and 
 A' is the itnage. Then, from (33), 
 
 1 1 1 
 
 + 
 
 BE', 
 
 B' E, 
 
 f. 
 
 (SO) 
 
 Now from the figure. 
 
 B' E^ = B' Ej- 
 
 •E,E„ 
 
 But when light from a point source emerges from the eye lens in a 
 parallel pencil, 
 
 B' E, = f. 
 And since the distance between the lenses, 
 
 2 f. 
 
 2 f„ 
 
 it follows that 
 
 B'EJ=B'E,-E^EJ =f^ ^ ^ 
 
 On substituting this value in (SO), and remembering that f^^ = f^. 
 obtain 
 
 1 
 
 1 
 
 BE' 
 
 Whence, 
 
 BE', = 
 
 Consequently, the field lens of a Ramsden eyepiece is placed at 
 a distance equal to one-fourth of its own focal length beyond the ima^e 
 of the object formed by light that has traversed the objective. 
 
 80. The Erecting Eyepiece. — The image produced by light 
 after traversing an objective and either a simple converging, a 
 
LIGHT 
 
 107 
 
 Hiiyghens, or a Ramsden eyepiece, is inverted. For an astro- 
 nomical telescope or a microscope this leads to no inconvenience. 
 But for a telescope used to view terrestrial objects we require an 
 eyepiece that will give an erect image. 
 
 A single converging lens placed between the objective and 
 any one of the eyepieces mentioned above will erect the image. 
 By using two lenses, however, the refraction at any surface may 
 be kept low, and consequently the spherical aberration (Art. SO). 
 A common form of erecting eyepiece consists of either a Huy- 
 ghens or a Ramsden eyepiece to which has been added two con- 
 verging lenses of equal focal length separated by any convenient 
 distance. The four lenses are fitted into a tube and constitiite a 
 single unit. The arrangement of the lenses is shown in Fig. 142. 
 A' B' represents the inverted image of the object produced by 
 light that has traversed the objective, not shown in the figure. 
 
 Fig. 142 
 
 (2) and (3) are the two lenses of equal principal focal length 
 added to a Huyghens eyepiece (4)-(5). The lens (2) is placed 
 at its principal focal length from A' B' . 
 
 Light diverging from A', after traversing (2), will proceed 
 in parallel rays to (3). After traversing (3), the light will be 
 converged toward A". Before reaching A", however, the light 
 strikes (4) and is focalized at A"' . . Traversing this point and 
 lens (5), the light emerges in a parallel pencil. The final image 
 A"' B"' is erect. 
 
 Fig. 143 
 
 An erecting eyepiece of this type is shown in Fig. 143, and 
 a telescope with an eyepiece of this type is shown in Fig. 144. 
 
 Fig. 144 
 
108 
 
 OPTICAL INSTRUMENTS 
 
 81. Ray Diagrams of the Ordinary Types of Telescopes. 
 
 —The paths of light through telescopes with the eyepieces already 
 described are indicated in the following diagrams. In these dia- 
 grams, the principal focal length of the objective is the same in 
 all cases. ' Each instrument is focaUzed for most easy vision for 
 objects at a great distance. It is left as an exercise to the student 
 to compute the lengfths of the various telescopes in terms of the 
 principal focal lengths of the lenses employed. ,, 
 
 Fig. 145. Galileo's Telescope. 
 
 Fig. 146. Simple TTro-Iiens Astronomical Telescope. 
 
 Fig. 147. Telescope with Huyghens' Eyepiece. 
 
 Fig. 148. Telescope with Eamsden's Eyepeice. 
 
 Fig. 149. Telescope with Huyghens' Eyepiece and Erecting Lenses. 
 
LIGHT 
 
 109 
 
 82. 
 
 in their 
 has the 
 
 multiple 
 ject. 
 
 The Prism Binocular. — For viewing distant objects 
 proper relations, the ordinary "field glass," Fig. 145, 
 advantage of compactness, but the serious disadvantage 
 of a limited field of view. The "spy 
 glass," Fig. 149, has a satisfactory 
 field view,, but it too long for con- 
 venient use. 
 
 In the "prism binocular" we have 
 the same objective and eyepiece that 
 give the "spy glass" its wide field of 
 view, and by bending the path of light 
 twice back upon itself the length of 
 the instrument is brought within con- 
 venient limits. In Fig. ISO is shown 
 the arrangement of the parts of a 
 prism binocular. 
 
 The reflections are produced by 
 two double totally reflecting prisms P^ 
 and Fj. The prism P^ inverts the 
 image, i. e., reverses top and bottom, 
 leaving the right and left aspects un- 
 changed. The prism P^ perverts the 
 image, i. e., interchanges the. right and 
 Fig. 150a left sides. Thus, by means of these 
 
 reeflctions, the image has the same aspect as the ob- 
 
 83. The Hyperscope and the Periscope. — The hyperscope 
 and the periscope are instruments designed to view objects from 
 a station which cannot be seen from the objects sighted upon. 
 The hyperscope consists of two mirrors in a tube as shown in 
 Fig. 151. By means of this simple device, soldiers in a trench can 
 view the ground in front of them without showing their heads 
 above the surface. 
 
 The periscope is a telescope bent in two right angles, with the 
 objective end capable of rotation so that all parts of the horizon 
 can be viewed without change of position of the observer. By 
 means of a periscope an officer in a submarine can view objects 
 above the surface of the sea when all of the vessel is submerged 
 except the end of an inconspicuous tube. Also, by its aid an 
 officer on a gun platform of a hidden battery can view distant 
 objects on the other side of a hill or protecting shield. One form 
 of periscope is diagrammed in Fig. 152. This consists of an ob- 
 
110 
 
 OPTICAL INSTRUMENTS 
 
 Fig. 151 
 
 Fig. 152 
 
 jective and eyepiece in combination with two totally reflecting 
 prisms^ Pi and P^, for bending the light beams into the required 
 direction, and two erecting prisms, A-^ and A^, for the purpose 
 of rendering the image erect and Imperverted. The upper totally 
 reflecting prism F^ is capable of rotation about a vertical axis 
 for the purpose of bringing objects at di-fferent azimuths into 
 the field of yiew. 
 
 84. The Compound Microscope. — In principle, the com- 
 pound microscope is the same as the astronomical telescope. 
 But in the case of the microscope the distance of the objective 
 .from the object under examination is but slightly greater than 
 the focal length of the objective. After traversing the objective, 
 the light from the object forms a real image, inverted and larger 
 than the o^bject. This image is farther magnified by means of an 
 eyepiece. To produce great resolution a very short focus objec- 
 tive is used (25). This requires the objective to be so close to 
 the object that the light entering the objective from aiiy point 
 of the object is strongly divergent. Excessive spherical aberra- 
 tion would thereby result if special precautions were not taken to 
 correct it. This is accpmplished by building up the objective 
 of several lenses. 
 
LIGHT 
 
 111 
 
 .As the image is so much larger than the object, in order 
 that the image may be sufficiently bright to be distinct, the object 
 must be brilliantly illumined? When the objective is close to th,e 
 object it is so difficult to properly illumine the object from above 
 that usually light is supplied from below by means of a mirror. 
 
 The arrangernent of lenses and the path of light in a modern 
 compound microscope are shown in Fig. 153. The object under 
 
 Fig. 153 
 
 investigation is placed on the stage S, and is illumined by light 
 reflected from the mirror M. Light from the illumined object 
 after traversing the compound objective passes through the 
 field lens F and forms a real image at the place shown in the 
 figure. This image is highly magnified by means of the eye 
 lens E. The distance from the object to the objective is adjusted 
 by means of the pinion X and the screw x. 
 
 The smallest distance that can be resolved by a lens is, (25), 
 
 X 
 
 D = 
 
 - (nearly) 
 
 2 since 
 
 where X is the wave length of the light entering the lens, and 
 cc' is half the angle subtended at the object by the lens. 
 
112 OPTICAL INSTRUMENTS 
 
 If the space between the object and the objective of the 
 microscope be filled with a liquid of index of refraction fi com- 
 pared with air, light entering the objective will not have the 
 wave length Ai it had in air, but will have a difJFerent value Aa, 
 (16), such that 
 
 x^ \_ 
 
 Whence, for an "immersion" lens, we have 
 
 D =^r^ — (nearly) 
 2 fi since 
 
 Consequently, by filling the space between object and objec-' 
 five with a liquid of index of refraction relative to air greater 
 than unity, smaller distances can be resolved than when the 
 objective is not immersed. By using a liquid of about the same 
 index of refraction as glass, an added advantage is secured — 
 the object can be illumined by convergent light without loss by 
 reflection at the surface of the objective. For this reason, cedar 
 oil of /u, = 1.5 is commonly employed in microscopic work. 
 
 The resolving power of the objective is also increased by 
 increasing oz. ^ The greatest possible value of oc is 90°. The 
 object would then be in contact with the objective. ' 
 
 With oc=90°, and /x,==1.5, resolution would be effected 
 of a distance so short as 
 
 where Aj is the wave length in air of the light used to illumine 
 ' the object. This . represents about the ultimate possible limit 
 of resolution. We cannot expect to distinguish visually objects 
 much smaller than one-third to one-fifth the wave length of 
 violet light. This limit is set, not by the degree of perfection of 
 our instruments, but by the properties of light. The image may 
 be magnified by the eyepiece, however much, but no finer detail 
 can be distinguished than is resolved by the objective. 
 
 In any optical system, each succeeding lens should have a 
 resolving power not less than that of the objective. Greater re- 
 solving power is useless. 
 
LIGHT 
 
 113 
 
 §6. The Spectroscope and Spectrum Analysis. 
 85. The Prism Spectroscope. — The spectroscope is an in- 
 strument for separating the radiant energy emitted by a luminous 
 body into waves of the various frequencies of which it is com- 
 posed. The dispersing action of a 
 prism is often employed for this 
 purpose. In the ordinary prism 
 spectroscope, a very narrow slit sy 
 Figs. 154 and ISS, is illumined by 
 light from the source S, under in- 
 vestigation. The spherical waves 
 from the various points Of this slit, 
 after being rendered plane by a 
 lens C, are bent out of their cource by a prism P, and brought to a 
 focus F by means of a lens 0. If the luminous source emits waves 
 of a single frequency, there will be formed at F a. single image 
 of the slit. If, however, the luminous source emits waves of 
 several different frequencies, these waves of different frequen- 
 cies will be unequally refracted by the prism, and will form at F 
 a separate image of the slit for each dififerent frequency. This 
 series of parallel images of the slit is called a spectrum. The 
 
 Fig. 15i 
 
 Fig. 155 
 
 spectrum is viewed by means of an eyepiece E. The tube C 
 with the slit and the lens, is called the collimator. The tube with 
 the lens and eyepiece E constitutes an ordinary reading tele- 
 scope. The telescope is usually provided with a set of cross- 
 wires. 
 
 86. The Direct Vision Spectroscope. — A combination of 
 prisms made of glasses of different indices of refraction and 
 dififerent dispersive powers can be made that will produce dis- 
 persion without deviation (Art. 52). This fact is utilized in 
 the production of a simple spectroscope. In Fig. 156, CCC 
 represent three crown glass prisms, and F F represent two flint 
 
114 
 
 OPTICAL INSTRUMENTS 
 
 glass prisms of such angles that light traversing the combina- 
 tion will be dispersed, but will emerge in the same direction 
 
 Fig. 156 
 
 it entered. Light from a source S after traversing a narrow 
 slit and (:he compound prism is focalized by the eyepiece E. 
 In the focal plane of this eyepiece there is formed an image of 
 the slit for light of each frequency emitted by the source. That 
 is, a spectrum of the source is formed in the focal plane of the 
 eyepiece. 
 
 By covering one-half of the slit by a totally reflecting prism, 
 the spectrum of a second source S' can be produced by the side 
 of the spectrum from the other source. By this means the 
 spectra of two bodies can be readily compared. 
 
 The instrument is often provided with a scale ruled on 
 glass, K, so placed that light after traversing the scale and lens, 
 and being reflected at the air-glass surface of the last prism, 
 forms an image of the scale beside the spectra. A particular 
 spectrum, line can then be indicated by reference to its position 
 on this scale. 
 
 87. Spectrum Analysis. — The spectrum produced by an 
 incandescent solid or liquid consists of a continuous ribbon of 
 color, blue .at one end, red at the other, and a whole series of 
 other colors in between. Self-luminous bodies, whether solid or 
 melted, e. g. the incandescent particles of carbon-forming flames, the 
 filament of an incandescent electric lamp, the hot carbon points 
 of an electric arc lamp, give the continuous^ spectrum. If light 
 from an unknown source gives a continuous spectrum we infer 
 that the luminous source is either an incandescent solid or liquid. 
 
 When a gas free from solid particles is heated to incan- 
 descence, the spectrum consists of a series of bright images of 
 the slit separated by dark spaces. This is called the bright line 
 spectrum. If the spectrum of a star is of the bright line type, 
 we infer that the star consists of a mass of incandescent gas. 
 When raised to a sufficiently high temperature, any element 
 becomes an incandescent gas. The bright line spectrum of an 
 element consists of a particular grouping of lines- that dis- 
 tinguishes that element from all others. When raised to a suf- 
 
LIGHT lis 
 
 ficiently high temperature any mixture or compound becomes 
 dissociated into the elements composing it, and each com- 
 ponent element gives its own spectrum independently of all 
 the others. In this manner different substances can be iden- 
 tified by their spectra. This is the basis of Spectrum Analysis. 
 
 1 
 
 
 ft ' =i If 
 
 . ■ 
 
 1 
 
 Fig. 1.17 
 
 In Fig. 157 are shown three bright line spectra. The upper spec- 
 trum is of copper, the lower one is of zinc, and the .middle one is of 
 brass. The fact that for each line of the spectra of copper and of 
 zinc there is in the spectrum of brass a similar line in the sam'e rela- 
 tive position shows that brass is Composed of copper and zinc. Any 
 lines occurring in the spectrum of brass that do not occur in the 
 spectra of copper or of zinc are due to some other element. 
 
 If the waves emitted by an incandescent solid or liquid 
 fall upon any nonluminous body, certain of the incident waves 
 are absorbed and the remainder are transmitted. The spec- 
 trum of the transmitted waves has the appearance of a contin- 
 uous spectrum from which certain regions have been absorbed 
 or blotted out. A spectrum of this sort is called the absorption 
 spectrum of the body which produced the absorption. In the 
 absorption spectra of solids and liquids the dark spaces are broad 
 with nebulous edges. In the absorption spectra of gases the dark 
 spaces consist of narrow lines with sharply defined edges. 
 
 From theoretical consideratiotis it can be shown that the 
 waves absorbed by any gas are of the same frequencies as those 
 that would be emitted if the gas were heated to incandescence. 
 In agreement with this conclusion we find that the dark lines of 
 
116 OPTICAL INSTRUMENTS 
 
 the absorption spectrum* of any gas occupy the same relative 
 position as do the bright lines in the emission spectrum of the 
 same gas. 
 
 Since the spectrum of the sun is a continuous ribbon of 
 color crossed by narrow dark lines, we infer that the sun con- 
 sists of an incandescent solid or liquid nucleus surrounded by 
 a layer of cooler gas. The dark lines of the solar spectrum 
 permit the identification of the elements which enter into the 
 composition of the gaseous envelope of the sun. 
 
 Since when sunlight is incident on the moon, we get from 
 the moon the same spectrum that we do from the sun, whereas, 
 when sunlight is not incident on the moon we get nothing, we 
 infer that the moon is not self-luminous, but reflects sunlight. 
 
 88. Quantitative Spectrum Analysis.— The degree of 
 blackness of an absorption band depends upon the amount of 
 absorbing material through which the light travels. .For instance, 
 the fraction of the light incident upon a cell of absorbing solution 
 that ernerges from the cell depends upon the concentration of the 
 solution. Thus, by using cells of the same thickness we can 
 compare the concentration of two absorbing solutions of the same 
 substance from determinations of the fraction of the incident light 
 transmitted by each. 
 
 For such a determination a spectroscope is arranged so that 
 part of the light from a luminous source traverses the cell in front 
 of the slit while another part enters a different portion of the slit 
 without traversing the cell. In the eyepiece now appears a con- 
 tinuous spectrum of the luminous source beside the absorption 
 spectrum of the solution. By means of a diaphragm in the focal 
 plain of the eyepiece all of the two spectra can be obscured except' 
 a selected portion of one absorption band, and the corresponding 
 portion of the continuous spectrum. The observer now sees side 
 by side two rectangular patches of light of the same wave length, 
 one due to light that has traversed the solution, and one due to 
 light from the same source that has not traversed the solution. 
 A device is provided which permits the observer to diminish the 
 brightness of the continuous spectrum by kno^ivn amounts till 
 the two parts of .the field of view are equally bright. A spectro- 
 scope provided with & device to compare the brightness of cor- 
 responding parts of two spectra is called a spectrophotometer. 
 
 By means of a spectrophotometer, the concentration of an 
 absorbing solution can be quickly determined with considerable 
 accuracy. In the case of certain classes of substance the degree 
 of accuracy possible is as great as by chemical analysis. The 
 method is available for the determination of the speed of chem- 
 ical reactions in a considerable number of cases. 
 
CHAPTER V. 
 
 -O 
 
 DOUBLE REFRACTION. 
 
 89. The Phenomena of IDouble Refraction. — In the case 
 of glass and other isotropic media, to which our attention has 
 been hmited up to the present time, light is transmitted with 
 the same speed in all directions. There is, however, a large 
 class, of crystals in which light, heat, and mechanical vibrations 
 are transmitted with different speeds in different directions. 
 Such substances are called anisotropic media. Calcite, some- 
 times called Iceland spar, is a striking example, while quartz 
 and tourmaline have the same property to a less degree. 
 
 On looking through a crystal of calcite placed in front of 
 
 a brightly illumined aperture 
 one will see two spots of light 
 instead of one. If the direction 
 of the inicdent light, 5" A, Fig. 
 158, is normal to the base .of the 
 crystal^ one portion of the light 
 will be transmitted undeviated 
 as in the case of glass, while an- 
 'f*^- ^°^ other portion will be deviated at 
 
 A, again at B, and will emerge parallel to the original direction. 
 On rotating the crystal about an axis through A normal to the 
 base of the crystal, the spot of light O will remain stationary 
 while the spot E will rotate about it. ' The light propagated in 
 the directibn A obeys the ordinary laws of refraction (Art. 
 21) but the light propagated along the path ABE does not 
 obey these laws. If the direction of the incident light is 
 oblique, the ratio of the sine of the angle of incidence to the 
 sine of the angle of refraction for one of the refracted rays will 
 be a constant quantity whatever the angle of incidence. For 
 the other refracted ray this ration will not be constant for all 
 angles of incidence. The ray that obeys the ordinary laws of 
 refraction is called the ordinary ray ; the ray that does not obey 
 these laws is called the extraordinary ray. 
 
 Since -the light along the ordinary and along the extraor- 
 dinary rays has been deviated from its original direction by 
 different amounts, the speed along these two rays must be dif- 
 ferent. On looking through the crystal toward the illumined 
 aperture in the diaphragm one will observe that the bright 
 spot corresponding to the ordinary ray appears to be nearer 
 the observer than the other. Therefore the light propagated 
 along the ordinary ray suffers the greater change of speed on 
 
118 ■ DOUBLE REFRACTION 
 
 emerging into the air. Consequently, in the crystal the speed 
 along the ordinary ray is less than the speed along the extra- 
 ordinary ray. 
 
 There is one direction in which light can be propagated 
 through a crystal of calcite without suffering double refraction. 
 When light traverses a crystal in this direction, the ordinary ray 
 and the extraordinary become coincident. The direction in which 
 light can be propagated in an anisotropic substance without the 
 occurrence of double refraction is called the optic axis of the 
 substance. Some substances have two optic axps. 
 
 If the light emerging from the crystal be examined by the 
 aid of a mirror placed at the angle of maximum polarization 
 (Art. 6), or by the aid of any other detector of plane polarized 
 light, it will be found that the light along the ordinary ray and 
 that along the extraordinary ray are plane , polarized, — the 
 planes of polarization of the light in the two rays being at right 
 angles to one another. ' , 
 
 Glass or other isotropic substance is rendered doubly re- 
 fracting by the application of mechanical stress. Ordinary 
 light becomes polarized on traversing a piece of glass under 
 stress. By examining the transmitted light with the aid of a 
 detector of polarized light the direction of the stress can be 
 determined and also the approximate magnitude of the stress. 
 
 When a mass of hot glass cools too suddenly stresses are 
 set up in it which diminish its ability to withstand shocks. The 
 presence of these stresses can be easily detected by examining 
 the specimen with the aid of some detector of polarized light. 
 To prevent these internal stresses glass must be annealed by 
 cooling so slowly that the internal strains have time to be re- 
 lieved before the molecules acquire fixed positions. 
 
 90. Fresnel's Theory of Double Refraction. — A theory 
 of double refraction must coordinate the following facts. Light 
 traverses certain substances with different speeds in different 
 directions. There is one, — and in some substances two, — direc- 
 tions in which light can be transmitted without double refrac- 
 tion. In all other directions the incident wave is divided into 
 two which traverse the substances with different speeds. The 
 light composing these two waves is plane polarized and the 
 planes of polarization of the two waves are at right angles to 
 one another. 
 
 A crude and imperfect analogy suggests itself. Consider 
 a long steel rod of rectangular cross section with one end 
 fastened in a vise and the other end free. If the free end. Fig. 
 
LIGHT 119 
 
 159, be displaced in the direction 
 A A' and then released, a re- 
 storing force will be developed 
 in the direction A' A which will 
 cause the free end to vibrate 
 along this line. This vibration 
 will be propagated along the 
 length of the rod with a certain 
 speed. If the free end had been 
 ^'^^- ^^® displaced in the direction B B' 
 
 a greater restoring force would have been developed in the 
 direction B' B and tl^e vibration would have been transmitted 
 down the rod with greater speed. If, however, the free end is 
 displaced, in the direction C C the restoring force thereby de- 
 veloped will not be in this direction and consequently the free 
 end will not vibrate in the line C C. At any instant the restor- 
 ing force will be the resultant of forces in the directions A' A 
 and B' B. This produces a vibration of the free end of the rod 
 which is the resultant of a vibration in the line A A' and an- 
 other in the direction B B' . Since these vibrations are of dif- 
 ferent periods two waves of different speeds will be sent sim- 
 ultaneously down the rod. Consequently, the disturbance pro- 
 duced by the original displacement in the direction C C has 
 been resolved into two waves polarized a right angles to one 
 another and traveling with unequal speeds. 
 
 Ordinary or unpolarized light consists of vibrations in sll 
 directions normal to the line of propagation. Plane polarized 
 light consists of vibrations in a single direction normal to the 
 line of propagation. These facts suggest that on entering a ' 
 doubly refracting substance the vibrations of a wave of ordin- 
 ary light are quickly altered in direction till all the vibrations 
 are limited to two directions at right angles to one another. 
 That is, the incident wave has been polarized in two planes at 
 right angles to one another. This change has been effected 
 without absorption of energy and the energy in the incident 
 wave has been equally divided between the two transmitted 
 waves. 
 
 The phenomena of double refraction suggest that light vi- 
 brations in a doubly refracting crystal, like mechanical vibrations 
 in a flat spring, have different speeds for different directions of 
 vibration. The facts of experiment agree with the hypothesis 
 that in the case of substances of which calcite is an example, the 
 speed parallel to the optic axis is least, and the speed perpendicu- 
 lar to the optic axis is greatest. In terms of the previously con- 
 sidered analogy, the light vibrations in a specimen of calcite C, 
 
120 
 
 DOUBLE REFRACTION 
 
 Fig. 160 
 
 Fig. 160, in which- the optic axis 
 ^ Z is in the plane of the paper, 
 correspond to the mechanical vi- 
 brations of a flat spring 6" P that 
 is wider in the plane of the paper 
 than that in the direction normal 
 to the plane of the paper. When 
 the free end of the spring is dis- 
 placed in any direction inclined 
 to the long axis of its cross sec- 
 tion, and then released, two plane polarized waves will be pro- 
 duced. These move along the length of the spring with different 
 speeds, and their planes of polarization are perpendicular to one 
 another. Assuming the hypothesis that light vibrations have dif- 
 ferent speeds in different directions of a doubly refracting subi- 
 stance, we would expect that if a wave consisting of transverse 
 vibrations in all directions is incident on such a substance, the 
 wave will, in general, be resolved- into two components polarized 
 at right angles to one another. In the section represented in 
 • Fig. 160, one set of vibrations will be in the plane of the paper 
 and the other in a plane perpendicular to the plane of the paper. 
 If, however, the incident wave advances in the direction of the 
 optic axis, all the vibrations will be perpendicular to the axis and 
 no polarization will be produced. 
 
 A plane that includes the optic axis and the normal to a face 
 of a crystal is called a principal plane of the specimen. Fig. 161 
 represents a section of a doubly refract- 
 ing substance cut parallel to a principal 
 plane. Each point of the surface B D 
 on which the incident wave impinges will 
 be the center from which two waves will 
 be propagated in all directions. For the 
 sake of clearness only the wave consist- 
 ing of vibrations in the principal plane 
 is represented in Fig. 161. The wave 
 originating at some point C will in a 
 certain interval of time have traveled a 
 distance C E parallel to the optic axis and 
 a greater distance C F perpendicular to the optic axis. The trace 
 of this elementary wave front on the principal plane of the crystal 
 is not circular but is elliptical. Traces of the waves originating- 
 at points B and D are also shown in the diagram. The envelope 
 of these elementary wave fronts, G H, is the wave front of the 
 refracted wave. Light from' C meets thfe new wave front at the 
 poiri't of tangency of the elementary wave and the envelope. In 
 
LIGHT 
 
 121 
 
 the crystal the ray is not normal to the wave front. On emerg- 
 ing from the crystal, each point of the surface becomes a center 
 of disturbance from which spherjcal waves are sent into the air. 
 Consequently the emergent ray is normal to the wave front in 
 the air. 
 
 By the aid of Fig. 162 we shall consider the wave in the 
 crystal due to vibrations normal to the principal plane. All 
 of these vibrations, being perpendicular 
 to the optic axis, the disturbance orig- 
 inating at some point C will in a certain 
 interval of time have traveled a certain 
 distance C E parallel to the optic axis 
 and an equal distance C F perpendicular 
 to the optic axis. Consequently the 
 trace on the principal .plane of the ele- 
 mentary wave from C is circular. The 
 envelope of the elementary wave fronts 
 from all the points from 5 to D is the 
 refracted wave front G H due to the 
 component under consideration. A ray from C will be normal to 
 this wave front and will obey the ordinary laws of refrection. 
 
 91. The Planes of Polarization. — When light from, any 
 source is copiously reflected by a mirror set in one position, but 
 is not reflected when the mirror is rotated 90° about an axis co- 
 incident with the incident ray, the incident light is said to be 
 plane polarized in a plane parallel to the plane >of incidence of 
 the light reflected from the mirror. The plane of polarisation 
 is defined as that p3.rticular plane of incidence in which light 
 is most copiously reflected. 
 
 A plane parallel to the optic axis of a crystal and perpen- 
 dicular to the face, on which light is incident is called a prin- 
 cipal plane of the crystal. It is found that when plane polarized 
 light is incident upon a crystal of ca.lcite, only the ordinary ray 
 is transmitted if the principal plane of the crystal is parallel 
 to the plane of polarization of the incident light, whereas only 
 the extraordinary ray is transmitted if the principal plane of 
 the crystal is perpendicular to the plane of polarization of the 
 incident light. Consequently, the ordinary ray is polarized in 
 the principal plane of calcite, while the extraordinary ray is 
 polarized perpendicularly to the principal plane. 
 
 According to the generally accepted theory of light, the 
 vibrations of plane polarized light are perpendicular to the plane 
 of polarization. 
 
122 DOUBLE REFRACTION 
 
 92. Polarizing Prisms. — The most efficient means of 
 producing polarized light is by the use of doubly refracting 
 crystals. When transmitted by a doubly refracting substance, 
 ordinary light is separated into two parts each consisting of 
 plane polarized light. On emerging from the crystals these 
 two parts usually recombine and form ordinary light. But if 
 one of these 'parts can be eliminated by. being absorbed or by 
 being reflected to one side, the emergen't light will be plane 
 polarized. 
 
 The polarization produced when light traverses tourma- 
 line is due to the absorption of one of the components into which 
 the incident light is resolved. A tourmaline polariscope. consists 
 of simply two plates of tourmaline cut parallel to the axis of the 
 crystal. But as tourmaline crystals are small and usually strongly 
 colored, this type of polariscope is unavailable for most pur- 
 poses? 
 
 The most effective means of producing plane polarized 
 light is by means of a piece of calcite in which the second face 
 makes 'such an angle with the ordinary and extraordinary rays 
 that one of these components is transmitted while the other is 
 totally reflected to one side. 
 
 Foucault's prism consists. Fig. 163, of an equilateral rhomb 
 of calcite cut' by a plane B C. The two sides of the cut are 
 polished and separated by a thin film of air. On entering the 
 calcite light is resolved into an ordinary and an extraordinary 
 ray which meet the face"5 C at different angles. The light in 
 the ordinary ray is incident on the air film at an angle greater 
 than the critical angle (Art. 24) and is totally reflected. The 
 light in the extraordinary ray is incident on the air film at an 
 angle less than the critical angle and is not totally reflected. 
 But as the index of refraction of air is considerably different 
 from the index of calcite there will be considerable reflection of 
 the light in the extraordinary ray. The emergent light E is 
 plane polarized. The second half B C D oi the prism prevents 
 dispersion and deviation. 
 
 » M N 
 
 Pig. 164 
 
 To prevent the excessive loss of light in the emergent ray 
 due to reflection at the air film, Nicol substituted for the air film 
 a thin layer of Canada balsam. As the index of refraction of 
 
LIGHT 123 
 
 Canada balsam is nearly that of calcite, there will be little re- 
 flection at the surface of the Canada balsam unless the light is 
 incident on it at an angle not less than the critical angle. The 
 critical angle from calcite to Canada balsam being greater than 
 that from calcite to air, light must be incident on the Canada 
 balsam film, Fig. 164, at a larger angle than in the case of the 
 air film. Fig. 163. This requires that the Nicol prism shall be 
 about three times as long as a Foucault prism. But although 
 large clear crystals of calcite are expensive, there is so much 
 more light transmitted by a Nicol prism than by a Foucault prism 
 that the Nicol prism is much more commonly used. 
 
 According to the generally accepted relation between the 
 direction of vibration and the plane of polarization, the direction 
 of vibration of the light emerging from either a Foucault or a 
 Nicol prism is parallel to the shorter diagonal of the end face. 
 The plane which contains the shorter diagonals of the two end 
 faces of a polarizing prism is called the principal section of the 
 prism. 
 
 Either a Foucault or a Nicol prism can be used as an ana- 
 lyzer or as a polarizer. When two polarizing prisms are placed 
 end to end, with their principal sections parallel, the light trans- 
 mitted by the first is transmitt-ed by the second without diminu- 
 tion. If one of the prisms is rotated about an axis coinciding 
 with the incident light, the intensity of the light emerging from 
 the second prism gradually diminishes, until when the principal 
 sections of the two prisms are "crossed," that is, are at right 
 angles to one another, — the intensity of the emergent light is 
 zero. If the rotation is continued, the intensity of the emergent 
 light increases until it attains a maximum value when the princi- 
 pal sections of the two prisms are again parallel. 
 
 93. Rotation of the Plane of Polarization. — If a plate of 
 quartz cut perpeiidicular to the optic axis be interposed in the 
 path of light that traverses a polarizer and analyzer set for ex- 
 tinction, the field of view of the analyzer becomes bright. If 
 the light be monochromatic, the light emerging from the quartz 
 can be quenched by rotating the analyzer through a certain angle. 
 Thus, the light transmitted by the quartz plate is plane polarized 
 in a plane inclined to the plane of polarization of the incident 
 light. This fact is expressed by the statement that in traversing 
 the quartz plate, the plane of polarization of light is rotated 
 through a certain angle. The ability to rotate the plane of polar- 
 ization of light is possessed by many substances, solid, Hquid, 
 and gaseous. Some produce a rotation in the clockwise direction, 
 while others produce a rotation in the counterclockwise direction.- 
 
124 DOUBLE REFRACTION 
 
 Biot found, (a), that the amount of rotation produced by any 
 substance is proportional to the thickness; (b), that when light 
 traverses more than one substance, the rotation equals the alge- 
 braic sum of the rotation due to the separate substances; (c), the 
 rotation depends upon the wave length of the light transmitted; 
 (d), in the case of solutions of active substances in inactive solv- 
 ents, the rotation is proportional to the concentration. 
 
 A plate of quartz one millimeter thick, cut perpendicular to 
 the optic axis, rotates the plane of polarization of red light about 
 18?, and of yellow light about 22°. A column of fifty per cent 
 aqueous 'cane sugar solution, ten centimeters long, produces a 
 rotation of the plane of polarization of yellow light of about 
 21.7°. , 
 
 94. Elementary Explanation of the Rotation of the Plane 
 of Polarization. — It can be shown, although the proof will not 
 here be given, that any simple harmonic motion can be resolved 
 into two uniform circular motions of equal period in opposite 
 directions. This fact can be illustrated by the following device.: 
 Let a bead M be so mounted on a rod O M that it can rotate 
 clockwise in its own plane at a uniform rate about an axis 
 through 0; and let this axis through O simultaneously rotate 
 with the same angular speed in the counterclockwise direction 
 about another parallel axis through C. In Fig. 165, Oi, O^, O^ 
 and O4 represent positions of the end O of the rod, and Mj, M^, 
 
 a>- .4. 
 
 
 v: \ 
 
 X 
 
 
 \ 
 
 
 > 
 
 . \^ / 
 
 Fig. 165 Fig. 166 
 
 Mg and M^ represent the corresponding positions of the other 
 end M. With this understanding, an inspection of the annexed 
 figure shows that under these two component circular motions, 
 the bead travels back and forth along the line XX'. At intervals 
 of one quarter of the period of rotation, the bead occupies the 
 positions Ml, M^_, M^, M^, M^, etc. 
 
LIGHT 125 
 
 It will now be shown that if the speed of one of the cir- 
 cular components be retarded for a time, and then be allowed 
 to resume its former value, the path of the resultant simple 
 harmonic motion will be rotated from the axis X X' to some new 
 position A A', Fig. 166. To 'fix the ideas, suppose that while 
 the counterclockwise component has traveled 360°, the clockwise 
 component has been so retarded that it has traveled only 330°. 
 At this instant the bead M is at the position M', Fig. 166. If 
 from this instant the retardation ceases and both components 
 become of the same speed, the resultant motion will be back and 
 forth along the axis A C A'. 
 
 The explanation of the rotation of .the plane of polarization 
 is based on the principles above illustrated and has been fully 
 verified by Fresnel both theoretically and experimentally. Ac- 
 cording to the work of Fresnel it appears that plane polarized 
 light may be regarded as being composed of two portions, cir- 
 cularly polarized in opposite directions. In traversing any sub- 
 stance plane polarized light will be resolved into its two circularly 
 polarized components, and these components may traverse the 
 substance with different speeds. If the two components traverse 
 the given substance in any direction with equal speed, they will 
 combine on emergence into plane polarized light polarized in the 
 same plane as before entering the given substance. If, however, 
 the two components traverse the given substance in the direction 
 of the optic axis with unequal speeds, the emergent light will be 
 plane polarized in a plane inclined to the plane of polarization 
 of the entrant light. That is, in traversing a substance of this 
 latter sort, the plane of polarization of light is rotated. 
 
 If plane polarized light is incident normally upon a plate of 
 doubly refracting substance cut parallel to the optic axis, the 
 erriergent light may be plane polarized, circularly polarized, or 
 elliptical polarized, depending upon the relative retardation- 
 of the two components produced by the substance. When the 
 emergent light is plane polarized, the plane of polarization will 
 be inclined to the plane of polarization of the incident light if, in 
 traversing the substance, the light in the ordinary ray and the 
 light in the extraordinary ray are unequally retarded. 
 
 95. Laurent's Half Shade Analyzer. — The obvious meth- 
 od of determining the amount of rotation of the plane of polar- 
 ization produced by any substance would be to set two Nicol 
 prisms for extinction ; place the substance under investigation 
 between the prisms ; and rotate one prism until the field of view 
 again becomes dark. The trouble with this method is that the 
 eye is not very sensitive to small changes of brightness, and the 
 mind cannot accurately compare the brightness of two things 
 
126 DOUBLE REFRACTION 
 
 unless seen simultaneously and in juxtaposition. To overcome 
 this difficulty several methods have been devised in which the 
 plane polarized light entering the analyzing Nicol is divided into 
 two plane polarized portions with the. planes of polarization in- 
 clined to one another at a small angle. Thus, suppose that, in 
 advancing toward the observer, monochromatic plane polarized 
 light vibrating parallel to B, Fig. 167, is divided into two parts, 
 — one which illumines the right half of the field of view and re- 
 tains its original plane of vibration, and an- 
 other ,portion of equal brightness which il- 
 lumines the left half of the field of view and 
 has its plane of vibration parallel to O A. 
 Without an analyzing Nicol, both halves of the 
 field of view are of the same brightness, but 
 with an analyzing Nicol the two halves are 
 unequally bright except when the principal 
 plane of the analyzer is parallel to Y Y' . 
 The device used by Laurent to produce this separation of 
 plane polarized light into two portions consists of a plate of 
 quartz Y X Y' and a plate of glass Y X' Y', Fig. 167, joined to- 
 gether along one edge. The plate of quartz is cut with the 
 optic axisparallel to the joint Y Y', and is of such a thickness 
 that during the passage of light through it, light in the extra- 
 ordinary ray is retarded more than light in the ordinary ray by 
 an amount equal to one-half wave length of the monochromatic 
 light used. The glass plate is of such a thickness that the light 
 which traverses it is reduced in brightness, through absorption 
 and reflection, by the same amount as the light that traverses 
 the quartz plate. 
 
 Suppose the monochromatic plane polarized light incident 
 on the compound quartz-glass plate vibrates parallel to some line 
 O B. The part of the light incident on the glass plate emerges 
 vibrating in the same plane B, but the portion incident on the 
 quartz emerges as plane polarized light vibrating in some other 
 plane A. It can be shown, though the proof will not be here 
 given, that the planes of vibration of the emergent light, O B 
 and A, are equally inclined to the joint Y Y'. Consequently, 
 when a Nicol prism is placed in front of the quartz-glass plate 
 with the principal plane parallel to the joint Y Y', the field of 
 view is uniformly bright. With the Nicol turned out of this 
 position, even very slightly, the two halves of the field of view 
 are of very unequal brightness. This quartz-glass plate is called 
 Laurent's half shade analyzer. 
 
 As usually employed, the half shade analyzer is placed be- 
 tween two Nic6l prisms with the joint between the quartz and 
 
LIGHT 
 
 127 
 
 glass plates slightly inclined to the principal plane of the polar- 
 izing Nicol. The polarizing prism is illumined with monochro- 
 matic light and the analyzing prism is turned till the field of view 
 is uniform. The specimen under investigation is then placed 
 between the half shade analyzer and the analyzing prism. If a 
 rotation of the plane of polarization has been produced, one-half 
 the field of view is now dark and the other is' bright. The angle 
 the » analyzing Nicol must be turned to bring the two halves to 
 equal brightness is the amount of rotation produced by the 
 specimen. 
 
 96. The Laurent Saccharimeter. — In custom houses and 
 sugar refineries it is necessary to have an accurate method for 
 quickly determining , the percentage of pure sugar in a given 
 specimen of syrup or solid sugar. The most convenient means, 
 and, the one usually employed, is afforded by the fact that sugar 
 rotates the plane of polarization of light passing through it. 
 A tube with glass ends is first filled with a solution of pure 
 sugar of known concentration, and the amount of rotation of 
 the plane of polarization produced by it is observed. The same 
 tube is then filled with a solution of the given specimen and the 
 amount of rotation produced by it is observed. Since for a 
 layer of constant thickness and temperature, the rotation de- 
 pends directly upon the concentration, the per cent of sugar in 
 the given specimen can be readily computed. An instrument for 
 determining the sugar content of a solution is called a sacchari- 
 meter. 
 
 E A 
 
 20S[ 
 
 Laurent's saccharimeter consists of a lens 0, Fig. 168, for 
 parallelizing the light emitted by some source not shown in the 
 engraving, a polarizing prism P, a half shade analyzer D, a 
 specimen ,tube S, an analyzing Nicol A, an eyepiece E, and a 
 
128 DOUBLE REFRACTION 
 
 divided circle V for reading the angle through which the ana- 
 lyzing Nicol is turned. The half shade analyzer is usually made 
 for yellow light. To produce light of the proper color, either 
 the luminous source is a gas flame supplied with common salt, 
 or there is interposed between the polarizer and the light source 
 a plate of potassium bichromate. 
 
 There are sevferal sorts of sugar, and some produce right 
 handed rotation while others produce left handed rotations^ 
 Oftentimes a specimen is a mixture of right handed and left 
 handed sugar. The concentration of such specimens can also be 
 determined, but the methods employed in such cases belong par- 
 ticularly to the laboratory and will not here be described. 
 
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