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Chapter2 - 2 PLANE SURFACES AND PRISMS The behavior of a...

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Unformatted text preview: 2 PLANE SURFACES AND PRISMS The behavior of a beam of light upon reflection or refraction at a plane surface is of basic importance in geometrical optics. Its study will reveal several of the features that will later have to be considered in the more diflicult case of a curved surface. Plane surfaces often occur in nature, e.g., as the cleavage surfaces of crystals or as the surfaces of liquids. Artificial plane surfaces are used in optical instruments to bring about deviations or lateral displacements of rays as well as to break light into its colors. The most important devices of this type are prisms, but before taking up this case of two surfaces inclined to each other, we must examine rather thoroughly what happens at a single plane surface. 2. 1 PARALLEL BEAM In a beam or pencil of parallel light, each ray meets the surface traveling in the same direction. Therefore any one ray may be taken as representative of all the others. The parallel beam remains parallel after reflection or refraction at a plane surface, as shown in Fig. 2A(a). Refraction causes a change in width of the beam which is easily seen to be in the ratio (cos ¢')/(cos 4)), whereas the reflected beam remains of the same f“ PLANE SURFACES AND PRISMS 25 n<n' n>n' n>n' FIGURE 2A Reflection and refraction of a parallel beam: (a) external reflection; (b) internal reflection at an angle smaller than the critical angle; (c) total reflection at or greater than the critical angle. width. There is also chromatic dispersion of the refracted beam but not of the reflected one. Reflection at a surface where n increases, as in Fig. 2A(a), is called external reflection. It is also frequently termed rare-to-dense reflection because the relative magnitudes of n correspond roughly (though not exactly) to those of the actual densities of materials. In Fig. 2A(b) is shown a case of internal reflection or dense-to- rare reflection. In this particular case the refracted beam is narrow because 41’ is close to 90°. 2.2 THE CRITICAL ANGLE AND TOTAL REFLECTION We have already seen in Fig. 2A(a) that as light passes from one medium like air into another medium like glass or water the angle of refraction is always less than the angle of incidence. While a decrease in angle occurs for all angles of incidence, there exists a range-of refracted angles for which no refracted light is possible. A diagram illustrating this principle is shown in Fig. 2B, where for several angles of incidence, from 0 to 90°, the corresponding angles of refraction are shown from 0° to 4),, respectively. It will be seen that in the limiting case, where the incident rays approach an angle of 90° with the normal, the refracted rays approach a fixed angle ()5, beyond which no refracted light is possible. This particular angle (1),, for which 4) = 90°, is called the critical angle. A formula for calculating the critical angle is obtained by substituting ¢ = 90°, or sin (15 = 1, in Snell’s law [Eq. (1m)], nx 1=n’sin4>c 0 so that sin ¢c = 2, (2a) n 26 FUNDAMENTALS or OPTICS M ,/r\ . ll, >//// ,r/4 //d FIGURE 2B Refraction and total reflection: (a) the critical angle is the limiting angle of refraction; (b) total reflection beyond the critical angle. a quantity which is always less than unity. For a common crown glass of index 1.520 surrounded by air sin die = 0.6579, and the = 41°8’. If we apply the principle of reversibility of light rays to Fig. 23(a), all incident rays will lie within a cone subtending an angle of 2¢c, while the corresponding refracted rays will lie within a cone of 180°. For angles of incidence greater than 4% there can be no refracted light and every ray undergoes total reflection as shown in Fig. 2B(b). The critical angle for the boundary separating two optical media is defined as the smallest angle of incidence, in the medium of greater index, for which light is totally reflected. Total reflection is really total in the sense that no energy is lost upon reflection. In any device intended to utilize this property there will, however, be small losses due to absorption in the medium and to reflections at the surfaces where the light enters and leaves the medium. The commonest devices of this kind are called total reflection prisms, which are glass prisms with two angles of 45° and one of 90°. As shown in ' Fig. 2C(a), the light usually enters perpendicular to one of the shorter faces, is totally reflected from the hypotenuse, and leaves at right angles to the other short face. This deviates the rays through a right angle. Such a prism may also be used in two other ways which are illustrated in (b) and (c) of the figure. The Dove prism (c) inter- changes the two rays, and if the prism is rotated about the direction of the light, they rotate around each other with twice the angular velocity of the prism. Many other forms of prisms which use total reflection have been devised for special purposes. Two common ones are illustrated in Fig. 2C(d) and (e). The roof , prism accomplishes the same purpose as the total reflection prism (a) except that it introduces an extra inversion. The triple mirror (e) is made by cutting off the corner of a cube by a plane which makes equal angles with the three faces intersecting at that PLANE SURFACES AND PRISMS 27 Tofa/ reflect/hr; Porro Dave or inverf/‘ng 1 2 Amici or roof Trip/e mirror Lummer—Brodhun FIGURE 2C Reflecting prisms utilizing the principle of total reflection. corner.* It has the useful property that any ray striking it will, after being internally reflected at each of the three faces, be sent back parallel to its original direction. The Lummer-Brodhun “cube” shown in (f) is used in photometry to compare the illumination of two surfaces, one of which is viewed by rays (2) coming directly through the circular region where the prisms are in contact, the other by rays (1) which are totally reflected in the area around this region. Since, in the examples shown, the angles of incidence can be as small as 45°, it is essential that this exceed the critical angle in order that the reflection be total. Supposing the second medium to be air (n’ = 1), this requirement sets a lower limit on the value of the index n of the prism. By Eq. (2a) we must have I: = —1- 2 sin 45° n n so that n 2 \/ 2 = 1.414. This condition always holds for glass and is even fulfilled for optical materials having low refractive indices such as Lucite (n = 1.49) and fused quartz (n 1.46). The principle of most accurate refractometers (instruments for the determination of refractive index) is based on the measurement of the critical angle 4),. In both the Pulfrich and Abbe types a convergent beam strikes the surface between the unknown sample, of index n, and a prism of known index n’. Now n’ is greater than n, so the ‘ A 46-cm array of 100 of these prisms is located on the moon’s surface, 3.84 x 10‘ m from the earth. This retrodirector, placed there during the Apollo 11 moon flight, is used to return light from a laser beam from the earth to a point on the earth close to the source. Such a marker can be used to accurately determine the distance to the moon at difi‘erent times. See J. E. Foller and E. J. Wampler, The Lunar Reflector, Sci. Am., March 1970, p. 38. For more details see Sec. 30.13. 28 FUNDAMENTALS or OPTICS FIGURE 2D Refraction by the prism in a Pulfrich refractometer. two must be interchanged in Eq. (2a). The beam is so oriented that some of its rays just graze the surface (Fig. 2D) so that one observes in the transmitted light a sharp boundary between light and dark. Measurement of the angle at which this boundary occurs allows one to compute the value of 4% and hence of n. There are important precautions that must be observed if the results are to be at all accurate.* 2.3 PLANE-PARALLEL PLATE When a single ray traverses a glass plate with plane surfaces that are parallel to each other, it emerges parallel to its original direction but with a lateral displacement d which increases with the angle of incidence (1). Using the notation shown in Fig. 2E, we may apply the law of refraction and some simple trigonometry to find the dis- placement d. Starting with the right triangle ABE, we can write d = [sin (95 — ¢’) (2b) which, by the trigonometric relation for the sine of the 'difirerence between two angles, can be written d = I(sin ()5 cos ¢’ — sin (15’ cos 4)) (20) From the right triangle ABC we can write t l = . cos d)’ which, substituted in Eq. (2c), gives d = t<sm¢cos¢ _ srn¢ cos 4)) (2d) cos ¢’ cos 4)’ From Snell’s law [Eq. (lm)] we obtain . n. s1n¢’=——,sm¢ n ‘ For a valuable description of this and other methods of determining indices of refraction see A. C. Hardy and F. H. Perrin, “Principles of Optics," pp. 359-364, McGraw-Hill Book Company, New York, 1932. PLANE SURFACES AND PRISMS 29 WWW/W W WWWW ////7//.. / FIGURE 2E Refraction by a plane—parallel plate. which upon substitution in Eq. (2d), gives d= t<sin¢ — ”54’ ” —sin¢> cos¢’ n’ d— = tsin¢(— n cos¢) (26) n’ cosd)’ From 0° up to appreciably large angles, d is nearly proportional to d), for as the ratio of the cosines becomes appreciably less than 1, causing the right-hand factor to increase, the sine factor drops below the angle itself in almost the same proportion.* -2.4 REFRACTION BY A PRISM In a prism the two surfaces are inclined at some angle a so that the deviation produced by the first surface is not annulled by the second but is further increased. The chromatic dispersion (Sec. 1.10) is also increased, and this is usually the main function of a prism. First let us consider, however, the geometrical optics of the prism for light of a single color, i.e., for monochromatic light such as is obtained from a sodium arc. ‘ This principle is made use of in most of the home moving-picture film-editor devices in common use today. Instead of starting and stopping intermittently, as it does in the normal film projector, the film moves smoothly and continuously through the film-editor gate. A small eight-sided prism, immediately behind the film, produces a stationary image of each picture on the viewing screen of the editor. See Prob. 2.2 at the end of this chapter. 30 FUNDAMENTALS or OPTICS FIGURE 2F The geometry associated with refraction by a prism. The solid ray in Fig. 2F shows the path of a ray incident on the first surface at_ the angle (15,. Its‘ refraction at the second surface, as well as at the first surface, obeys Snell’s law, so that in terms of the angles shown I sin 4), _ _ _ sin 422 (21‘) sin 42’, n sin qb'z The angle of deviation produced by the first surface is [i = 451 — (121, and that produced by the second surface is y = (i); — 453. The total angle of deviation 5 between the incident and emergent rays is given by 5=l3+7 (23) Since NN’ and MN ' are perpendicular to the two prism faces, a is also the angle at N’. From triangle ABN' and the exterior angle a, we obtain = 051 + ¢'2 (2h) Combining the above equations, we obtain 5=5+Y=¢1—¢'1+¢2—¢'2=¢1+¢2—(¢'1+¢'2) or 6=¢1+¢2—a (2i) 2.5 MINIMUM DEVIATION When the total angle of deviation 5 for any given prism is calculated by the use of the above equations, it is found to vary considerably with the angle of incidence. The angles thus calculated are in exact agreement with the experimental measurements. If during the time a ray of light is refracted by a prism the prism is rotated con- tinuously in one direction about an axis (A in Fig. 2F) parallel to the refracting edge, the angle of deviation 6 will be observed to decrease, reach a minimum, and then increase again, as shown 1n Fig. 2G. The smallest deviation angle, called the angle of minimum deviation 6",, occurs at that particular angle of incidence where the refracted ray inside the prism makes equal angles with the two prism faces (see Fig. 2H). In this special case ¢1=¢2 4>’1=¢’2 fi=7 (21') To prove these angles equal, assume ¢1 does not equal 4;, when minimum deviation occurs. By the principle of the reversibility of light rays (see Sec. 1.8), PLANE SURFACES AND PRISMS 31 20 'l'lllll 0 40 50 60 7O 80 9 «p 60 50 3O 20 '5 O 1 FIGURE 2G A graph of the deviation produced by a 60° glass prism of index n’ = 1.50. At minimum deviation 6... = 37.2°, m = 48.6°, and ¢,’ = 30.0°. there would be two diflerent angles of incidence capable of giving minimum deviation. Since experimentally we find only one, there must be symmetry and the above equalities must hold. In the triangle ABC in Fig. 2H the exterior angle 6," equals the sum of the opposite interior angles [3 + y. Similarly, for the triangle ABN’, the exterior angle 0: equals the sum d)’, + 42'2. Consequently a=2¢i 5m=213 ¢1=¢l+fi Solving these three equations for ¢’1 and (1), gives ¢1= id 4’1 = flat + 5...) Since by Snell’s law n’/n = (sin ¢l)/(sin 43’1), 7_1_' = sin ‘Hd + (5,.) n sin in (2k) FIGURE 2H The geometry of a light ray traversing a prism at minimum deviation. 32 FUNDAMENTALS or OPTICS The most accurate measurements of refractive index are made by placing the sample in the form of a prism on the table of a spectrometer and measuring the angles 6,, and a, the former for each color desired. When prisms are used in spectroscopes and spectrographs, they are always set as nearly as possible at minimum deviation because otherwise any slight divergence or convergence of the incident light would cause astigmatism in the image. 2.6 THIN PRISMS The equations for the prism become much simpler when the refracting angle or becomes small enough to ensure that its sine and the sine of the angle of deviation 6 may be set equal to the angles themselves. Even at an angle of 0.1 rad, or 5.7°, the difference between the angle and its sine is less than 0.2 percent. For prisms having a refracting angle of only a few degrees, we can therefore simplify 'Eq. (2k) by writing n sm aka a O and 5 = (n' —— l)oc (21) Thin prism in air The subscript on 6 has been dropped because such prisms are always used at or near minimum deviation, and n has been dropped because it will be assumed that the surrounding medium is air, n = 1. It is customary to measure the power of a prism by the deflection of the ray in centimeters at a distance of 1 m, in which case the unit of power is called the prism diopter (D). A prism having a power of 1 prism diopter therefore displaces the ray on a screen 1 m away by 1 cm. In Fig. 21(a) the deflection on the screen is x cm and is numerically equal to the power of the prism. For small values of 6 it will be seen that the power in prism diopters is essentially the angle of deviation 6 measured in units of 0.01 rad, or O.573°. For the dense flint glass of Table 1A, 125 = 1.67050, and Eq. (21) shows that the refracting angle of a 1-D prism should be 0.57300 a = = 0.85459° 0.67050 2.7 COMBINATIONS OF THIN PRISMS In measuring binocular accommodation, ophthalmologists make use of a com- bination of two thin prisms of equal power which can be rotated in opposite directions in their own plane [Fig. 21(b)]. Such a device, known as the Risley or Herschel prism, is equivalent to a single prism of variable power. When the prisms are parallel, the power is twice that of either one; when they are opposed, the power is zero. To find how the power and direction of deviation depend on the angle between the PLANE SURFACES AND PRISMS 33 FIGURE 21 Thin prisms: (a) the displacement x in centimeters at a distance of 1 m gives the power of the prism in diopters; (b) Risley prism of variable power; (c) vector addition of prism deviations. components, we use the fact that the deviations add vectorially. In Fig. 21(c) it will be seen that the resultant deviation 6 will in general be, from the law of cosines, 6 = V612 + 622 + 26152 cos [3 (2m) where B is the angle between the two prisms. To find the angle y between the resultant deviation and that due to prism 1 alone (or, we may say, between the “equivalent” prism and prism 1) we have the relation 62 sin [3 tany=——— 61 +52cosfi (2n) Since almost always 51 = 62, we may call the deviation by either component 6,, and the equations simplify to 5 = x/26f(l + cos [3) = J45} cosza = 25,- cos’g (20) and tan'y = fl— = tang l + cos B 2 so that _ E 7 — 2 (2p) 2.8 GRAPHICAL METHOD OF RAYVTRACING It is often desirable in the process of designing optical instruments to be able to trace rays of light through the system quickly. For prism instruments the principles pre- sented below are extremely useful. Consider first a 60° prism of index n’ = 1.50 surrounded by air of index n = 1.00. After the prism has been drawn to scale, as in Fig. 2J, and the angle of incidence d), has been selected, the construction begins as in Fig. 1G. Line OR is drawn parallel to JA, and, with an origin at 0, the two circular arcs are drawn with radii proportional to n and 71’. Line RP is drawn parallel to NN’, 34 FUNDAMENTALS or opncs FIGURE 2] A graphical method for ray tracing through a prism. and OP is drawn to give the direction of the refracted ray AB. Carrying on from the point P, a line is drawn parallel to MN’ to intersect the are n at Q. The line 0Q then gives the correct direction of the final refracted ray BT. In the construction diagram at the left the angle RPQ is equal to the prism angle at, and the angle ROQ is equal to the total angle of deviation 6. 2.9 DIRECT-VISION PRISMS As an illustration of ray tracing through several prisms, consider the design of an important optical device known as a direct-vision prism. The primary function of such an instrument is to produce a visible spectrum the central color of which emerges from the prism parallel to the incident light. The simplest type of such a combination usually consists of a crown-glass prism of index n' and angle a’ opposed to a flint- glass prism of index n” and angle a”, as shown in Fig. 2K. The indices 11’ and n” chosen for the prisms are those for the central color of the spectrum, namely, for the sodium yellow D lines. Let us assume that the angle a” of the flint prism is selected and the construction proceeds with the light emerging perpendicular to the last surface and the angle a’ of the crown prism as the unknown. The flint prism is first drawn with its second face vertical. The horizontal line OP is next drawn, and, with a center at 0, three arcs are drawn with radii proportional to n, n', and n”. Through the intersection at P a line is drawn perpendicular to AC intersecting n’ at Q. The line RQ is next drawn, and normal to it the side AB of the crown prism. All directions and angles are now known. 0R gives the direction of the incident ray, 0Q the direction of the refracted ray inside the crown prism, 0P the direction of the refracted ray inside the flint prism, and finally OP the direction of the emergent ray on the right. The angle ot’ of the crown prism is the supplement of angle RQP. If more accurate determinations of angles are required, the construction diagram will be found useful in keeping track of the trigonometric calculations. If the dispersion of white light by the prism combination is desired, the indices 71’ and n” for the red and violet light can be drawn in and new ray diagrams constructed proceeding now PLANE SURFACES AND PRISMS 35 FIGURE 2K Graphical ray tracing applied to the design of a direct-vision prism. from left to right in Fig. 2K(b). These rays, however, will not emerge perpendicular to the last prism face. The principles just outlined are readily extended to additional prism com- binations like those shown in Fig. 2L. It should be noted that the upper direct-vision prism in Fig. 2L is in principle two prisms of the type shown in Fig. 2K placed back to back. White light FIGURE 2L Direct-vision prisms used for producing a spectrum with its central color in line with the incident white light. 36 FUNDAMENTALS or OPTICS FIGURE 2M The reflection of divergent rays of light from a plane surface. 2.10 REFLECTION OF DIVERGENT RAYS When a divergent pencil of light is reflected at a plane surface, it remains divergent. All rays originating from a point Q (Fig. 2M) will after reflection a...
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