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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 raretodense 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 denseto 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 rangeof 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 LummerBrodhun “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 46cm 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 PLANEPARALLEL 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. 359364,
McGrawHill 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 righthand 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 movingpicture ﬁlmeditor 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
ﬁlmeditor gate. A small eightsided 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 1D 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 DIRECTVISION PRISMS As an illustration of ray tracing through several prisms, consider the design of an
important optical device known as a directvision 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 crownglass 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 directvision 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 directvision prism in Fig. 2L is in principle two prisms of the type shown in Fig. 2K placed back
to back. White light FIGURE 2L Directvision 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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