1. JENKINS! WHITE inn t jT- QBQ" and QCQ" > QBQ". Therefore
the real path QBQ" is a minimum. A graph of hypothetical paths
close to the real path QBQ", as shown in the lower right of the
diagram, indicates the meaning of a minimum, LIGHT RAYS Fig. H.
Geometry of a refracted ray used in illustrating Fermat's
principle. 11. and the flatness of the curve between -4 and C
illustrates that to a first approximation adjacent paths are equal
to the real optical path. Consider finally the optical properties
of an ellipsoidal reflector as shown in Fig. IF. All rays emanating
from a point source Q at one focus are reflected according to the
law of reflection and come together at the other focus Q'.
Furthermore all paths are equal in length. (It will be recalled
that an ellipse can be drawn with a string of fixed length with its
ends fastened at the foci.) Because all optical paths are equal,
this is a stationary case as mentioned above. On the graph in Fig.
1G(6) equal path lengths are represented by a straight horizontal
line. Some attention will here be devoted to other reflecting
surfaces like (a) and (c) in Fig. IF. If these surfaces are tangent
to the ellipsoid at the point B, the line NB is normal to all three
surfaces and QBQ' is a real path for all three. Adjacent paths from
Q to points along these mirrors, however, will give a minimum
condition for the real path to and from reflector c, and a maximum
condition for the real path to and from reflector a (see Fig. IG).
It is readily shown mathematically that both the laws of reflection
and refraction follow Fermat's principle. Figure H, which
represents the refraction of a ray at a plane surface, may be used
to prove the law of refraction (Eq. lc). The length of the optical
path between a point Q in the upper medium of index n, and another
point Q' in the lower medium of index n' , passing through any
point .4 on the surface, is [d] = nd+ n'd' {IK) where d and d'
represent the distances QA and AQ', respectively. Now if we let h
and h' represent perpendicular distances to the surface and p the
total length of the x axis intercepted by these perpendiculars, we
can invoke the Pythagorean theorem concerning right triangles and
write d 2 = h 2 + (p - x) 2 d' 2 = h' 2 + x 2 When these values of
d and d' are substituted in Eq. h, we obtain [d] = n[h 2 + (p -
a:)*]i + n'(h' 2 + as*)* (It) According to Fermat's principle [d]
must be a minimum or a maximum (or in general stationary) for the
actual path. One method for finding 10 GEOMETRICAL OPTICS a minimum
or maximum for the optical path is to plot a graph of [d] against x
and find at what value of x a tangent to the curve is parallel to
the x axis (see Fig. IG). The mathematical means for doing the same
thing is, first, to differentiate Eq. i with respect to the
variable x, thus obtaining an equation for the slope of the graph,
and, second, to set this resultant equation equal to zero, thus
finding the value of x for which the slope of the curve is zero. By
differentiating Eq. It with respect to x and setting the result
equal to zero, we obtain 12. This gives or, more simply, p x , = n
[h 2 + (p - a;) 2 ]* (h' 2 + x 2 ) p x , x By reference to Fig. 1/7
it will be seen that the multipliers of n and n' are just the sines
of the corresponding angles, so that we have now proved Eq. lc,
namely, n sin = n' sin ' (lj) A diagram for reflected light,
similar to Fig. IH, can be drawn and the same mathematics applied
to prove the law of reflection. 1.7. Color Dispersion. It is well
known to those who have studied elementary physics that refraction
causes a separation of white light into its component colors. Thus,
as is shown in Fig. II, the incident ray of white light gives rise
to refracted rays of different colors (really a con- tinuous
spectrum) each of which has a different value of '. By Eq. lc the
value of n' must therefore vary with color. It is customary in the
exact specification of indices of refraction to use the particular
colors corresponding to certain dark lines in the spectrum of the
sun. Tables of these so-called Fraunhofer* lines, which are
designated by the letters A, B, C, . . . , starting at the extreme
red end, are given later in Tables 21-11 and 23-1. The ones most
commonly used are those in Fig. II. The angular divergence of rays
F and C is a measure of the dispersion produced, and has been
greatly exaggerated in the figure relative to the * Joseph
Fraunhofer (1787-1826). Son of a poor Bavarian glazier, Fraunhofer
learned glass grinding, and entered the field of optics from the
practical side. His rare experimental skill enabled him to produce
much better spectra than those of his predecessors and led to his
study of the solar lines with which his name is now associ- ated.
Fraunhofer was one of the first to produce diffraction gratings
(Chap. 17). LIGHT RAYS 1 I 13. average deviation of the spectrum,
which is measured by the angle through which ray D is bent. To take
a typical case of crown glass, the refractive indices as given in
Table 23-1 are n F = 1.53303 n D = 1.52704 n c = 1.52441 Now it is
readily shown from Eq. d that for a given small angle $ the
dispersion of the F and C rays (4>' F ' c ) is proportional to n
F - n c = 0.00862 while the deviation of the D ray ( 4>' D )
depends on n D 1 = 0.52704 1.5 n^p^. 5p F' n C n D - LO i i Fig.
1/. Upon refraction white light is spread out into a spectrum. This
is called dispersion. 14. F DC Violet Blue Green Yellow Red Fio.
1/. A graph showing the variation of refractive index with color.
and is thus more than sixty times as great. The ratio of these two
quan- tities varies greatly for different kinds of glass and is an
important char- acteristic of any optical substance. It is called
the dispersive power and is defined by the equation 1 _ n F nc v nD
1 (Ik) The reciprocal of the dispersive power, designated by the
Greek letter v, lies between 30 and 60 for most optical glasses.
Figure J illustrates schematically the type of variation of n with
color that is usually encountered for optical materials. The
numerator of Eq. Ik, which is a measure of the dispersion, is
determined by the difference in the index at two points near the
ends of the spectrum. The denominator, which measures the average
deviation, represents the mag- nitude in excess of unity of an
intermediate index of refraction. It is customary in most
treatments of geometrical optics to neglect chromatic effects and
assume, as we have in the next seven chapters, 12 GEOMETRICAL
OPTICS that the refractive index of each specific element of an
optical instrument is that determined for yellow sodium D light.
PROBLEMS 1. A ray of light in air is incident on the polished
surface of a piece of glass at an angle of 15. What percentage
error in the angle of refraction is made by assuming that the sines
of angles in Snell's law can be replaced by the angles themselves?
Assume n' = 1.520. 2. A ray of light in air is incident at an angle
of 45 on glass of index 1.560. Find the angle of refraction (a)
graphically, and (6) by calculation using Snell's law. (c) What is
the angle of deviation? Ans. (a) 27. (6) 2657'. (c) 183'. 3. A
straight hollow pipe exactly 1 m long is closed at either end with
quartz plates 10 mm thick. The pipe is evacuated, and the index of
quartz is 1.460. (a) What is 15. the optical path between the two
outer quartz surfaces? (6) By how much is the optical path
increased if the pipe is filled with a gas at 1 atm pressure if the
index is 1.000250? 4. The points Q and Q' in Fig. H arc at a
distance h = 10 cm and h' = 10 cm, respectively, from the surface
separating water of index n = 1.333 from glasss of index n' =
1.500. If the distance x is 4 cm, find the optical path [d] from Q
to Q'. Ans. 30.83 cm. 6. An approximate law of refraction was given
by Kepler in the form 4> = ), where k = (n' l)/n', n' being the
relative index of refraction. Calculate the angle of incidence for
glass of index n' = 1 .600, if the angle of refrac- tion ' 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 A ft fft // /// d Fig. 2B. Refraction and
total reflection, (a) The critical angle is the limiting angle of
refraction, (b) Total reflection beyond the critical angle. 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 18. light is possible. A diagram illustrating this
principle is shown in Fig. 2B, where for several angles of
incidence, from to 90, the corresponding angles of refraction are
shown from 0 to c , 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 c beyond
which no refracted light is possible. This particular angle c , for
which = 90, is called the critical angle. A formula for calculating
the critical angle is obtained by substituting = 90, or sin = 1, in
SnelPs law (Eq. lc), so that n X 1 = n' sin 4> c sin 4> c = n
(2a) 16 GEOMETRICAL OPTICS I quantity which is always less than
unity. For a common crown glass of index 1.520 surrounded by air
sin 4> c = 0.6579, and c = 418'. If we apply the principle of
reversibility of light rays to Fig. 26(a), all incident rays will
lie within a cone subtending an angle of 20 c , while the
corresponding refracted rays will lie within a cone of 180. For
angles of incidence greater than c 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, Total reflection 1^ (a) Porro Dove or inverting 19. c) 1 2
Amici or roof Triple mirror Lummer-Brodhun Fig. 2C. Reflecting
prisms utilizing the principle of total reflection. however, be
small losses due to absorption in the medium and to reflec- tions
at the surfaces where the light enters and leaves the medium. The
commonest device of this kind is the total reflection prism, which
is a glass prism 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 (6) and (c) of the figure. The Dove
prism (c) interchanges 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. PLANE SURFACES 17
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 corner. It has the useful property that any
ray striking it will, after being inter- nally reflected at each of
the three faces, be sent back parallel to its original direction.
The Lummer-Brodhun "cube" shown in (/) is used in photometry to
compare the illumina- tion 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 shall exceed .,..., . , ,, ,, Fig. 2D.
Refraction in the prism of a the critical angle in order that the p
u lfrich refractometer. 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 Vl =
I ^ sin 450 n n 20. so that n ^ y/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 deter- mination of refractive index) is based
on the measurement of the critical angle 4> c - 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 two must be inter- changed 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 < c and hence of n. There are important precautions that must
be observed if the results are to be at all accurate.* * 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," 1st ed., pp. 359- 364, McGraw-Hill Book Company, Inc.,
New York, 1932. 18 GEOMETRICAL OPTICS 2.3. 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. 22?) will after reflection appear to come from another point
Q' symmetrically placed behind the mirror. The proof of this
proposition follows at once from the application of the law of
reflection (Eq. la), according to which all the angles labeled <
in the figure must be equal. Under these conditions the distances Q
A and A Q' along the line QAQ' drawn perpendicular to the surface
must be equal: i.e., s' = s The point Q' is said to be a virtual
image of Q, since when the eye receives the reflected rays they
appear to come from a source at Q' but do not _a I S-lZlyj^ i j
Fig. 2E. The reflection of a divergent pencil of light. 21. Fig.
IF. The refraction of a divergent pencil of light. actually pass
through Q' as would be the case if it were a real image. In order
to produce a real image a surface other than a plane one is
required. 2.4. Refraction of Divergent Rays. Referring to Fig. 2F,
let us find the position of the point Q' where the lower refracted
ray, when produced backward, crosses the perpendicular to the
surface drawn through Q. Let QA = s, Q'A = s', and AB = h. Then so
that h = s tan (f> = s' tan sin ' s = s ^7 = s (26) tan 0' " sin
0' cos Now according to the law of refraction (Eq. lc) the ratio
sin We therefore have , = = const, sin n , n' cos does not equal 2
when mini- mum deviation occurs. By the principle of the
reversibility of light rays (see Sec. 1.4), there would be two
different angles of incidence capable of giving minimum deviation.
Since experimentally we find 60 50 t 40 30 26. 20- - Om , 1 27. 1
20 30 40 70 80 90 50 60 Fig. 2J. A graph of the deviation produced
by a 60 glass prism of index n' = 1.50. At minimum deviation 5 m =
37.2, 0, = 48.6, and n. If in Fig. 3F the 36 GEOMETRICAL OPTICS
medium on the left were to have the greater index, so that ri <
n, the surface would have a diverging effect and each of the focal
points F and F' would he on the opposite side of the vertex from
that shown, just as they do in Fig. 3G. Similarly, if we made n'
< n in Fig. 3G, the surface would have a converging effect and
the focal points would he as they do in Fig. 3F. Since any ray
through the center of curvature is undeviated and has all the
properties of the principal axis it may be called an auxiliary
axis. 3.7. Oblique-ray Methods. Method 1. In more complicated
optical systems that are treated in the following chapters it is
convenient to be able graphically to trace a ray across a spherical
boundary for any given W Fig. ZH. Illustrating the oblique-ray
method for graphically locating images formed by a single spherical
surface. angle of incidence. The oblique- ray methods permit this
to be done with considerable ease. In these constructions one is
free to choose any two 42. rays coming from a common object point
and, after tracing them through the system, find where they finally
intersect. This intersection is then the image point. Let MT in
Fig. 3H represent any ray incident on the surface from the left.
Through the center of curvature C a dashed line RC is drawn,
parallel to MT, and extended to the point where it crosses the
secondary focal plane. The line TX is then drawn as the refracted
ray and extended to the point where it crosses the axis at M '.
Since the axis may here be considered as a second ray of light, M
represents an axial object point and M' its conjugate image point.
The principle involved in this construction is the following: If M
T and RA were parallel incident rays of light, they would, after
refraction, and by the definition of focal planes, intersect the
secondary focal plane WF' at X. Since RA is directed toward C, the
refracted ray ACX remains undeviated from its original direction.
Method 2. This method is shown in Fig. 31. After drawing the axis
MM' and the arc representing the spherical surface with a center C,
any line such as 1 is drawn to represent any oblique ray of light.
Next, an auxiliary diagram is started by drawing XZ parallel to the
axis. SPHERICAL SURFACES 37 With an origin at 0, line intervals OK
and OL are laid off proportional to n and n' , respectively, and
perpendiculars are drawn through K, L, and A. From here the
construction proceeds in the order of the numbers 1, 2, 3, 4, 5,
and 6. Line 2 is drawn through parallel to line 1, line 4 is drawn
through J parallel to line 3, and line 6 is drawn through T
parallel to line 5. Fig. 3/. Illustrating the auxiliary-diagram
method for graphically locating images formed by paraxial rays. A
proof for this construction is readily obtained by writing down
proportionalities from three pairs of similar triangles in the two
diagrams. These proportionalities are h i+3 n' n We now transpose n
and n' to the left in all three equations. 43. hn = i hn' = J h(n'
n) = i + j We finally add the first two equations and for the
right-hand side substi- tute the third equality. hn , hn' = i + j
and 4- _ s 7" n n It should be noted that to employ method 1 the
secondary focal length /' must be known or it must first be
calculated from the known radius of curvature and the refractive
indices n and n'. Method 2 can be applied without knowing either of
the focal lengths. 3.8. Magnification. In any optical system the
ratio between the trans- verse dimension of the final image and the
corresponding dimension of the original object is called the
lateral magnification. To determine the relative size of the image
formed by a single spherical surface, reference 44. 38 GEOMETRICAL
OPTICS is made to the geometry of Fig. 3F. Here the undeviated ray
5 forms two similar right triangles QMC and Q'M'C. The theorem of
the proportionality of corresponding sides requires that M'Q' CM'
or II s r MQ CM y s + r We will now define y'/y as the lateral
magnification m and obtain U m = = y 8 r s + r (30) If m is
positive, the image will be virtual and erect, while if it is nega-
tive, the image is real and inverted. 3.9. Reduced Vergence. In the
formulas for a single spherical refract- ing surface (Eqs. 36 to
3/), the distance s, s', r, f, and /' appear in the 45. n' % ?/ //
A Fig. 3J. Illustrating the refraction of light waves at a single
spherical surface. denominators. The reciprocals 1/s, 1/s', 1/r,
1//, and 1//' actually represent curvatures of which s, s', r, f,
and /' are the radii. Reference to Fig. 3J will show that if we
think of M in the left-hand diagram as a point source of waves
their refraction by the spherical boundary causes them to converge
toward the image point M ' . In the right-hand diagram plane waves
are refracted so as to converge toward the secondary focal point F'
. Note that these curved lines representing the crests of light
waves are everywhere perpendicular to the correspond- ing light
rays that could have been drawn from object point to image point.
As the waves from M strike the vertex A, they have a radius s and a
curvature 1/s, and as they leave A converging toward M' , they have
a radius s' and a curvature 1/s'. Similarly the incident waves
arriving at A in the second diagram have an infinite radius c and a
curvature of 1/co, or zero. At the vertex where they leave the
surface the radius of the refracted waves is equal to/' and their
curvature is equal to 1//'. The Gaussian formulas may therefore be
considered as involving the addition and subtraction of quantities
proportional to the curvatures SPHERICAL SURFACES 39 of spherical
surfaces. When these curvatures, rather than radii are used, the
formulas become simpler in form and for some purposes more con-
venient. We therefore introduce at this point the following
quantities: V = - V = -, K = - P = - f P = %r (3/0 s s r f f The
first two of these, V and V, are called reduced vergences because
they are direct measures of the convergence and divergence of the
object and image wave fronts, respectively. For a divergent wave
from the object s is positive, and so is the vergence V. For a
convergent wave, on the other hand, s is negative, and so is its
vergence. For a converging wave front toward the image V is
positive, and for a diverging wave front V is negative. Note that
in each case the refractive index involved is that of the medium in
which the wave front is located. 46. The third quantity K is the
curvature of the refracting surface (recip- rocal of its radius),
while the fourth and fifth quantities are, according to Eq. 3e,
equal and define the refracting power. When all distances are
measured in meters, the reduced vergences V and V, the curvature K,
and the power P are in units called diopters. We may think of V as
the power of the object wave front just as it touches the
refracting surface and V as the power of the corresponding image
wave front which is tangent to the refracting surface. In these new
terms, Eq. 36 becomes V + V = P (Si) where P = ^-^ or P = (n' - n)K
(3j) Example: One end of a glass rod of refractive index 1.50 is
ground and polished with a convex spherical surface of radius 10
cm. An object is placed in the air on the axis 40 cm to the left of
the vertex. Find (a) the power of the surface, and (b) the position
of the image. Solution: To find the solution to (a), we make use of
Eq. 3j, substitute the given distance in meters, and obtain For the
answer to part (6), we first use Eq. 3/i to find the vergence V. _
1.00 v ~ cuo = +2 - 5 D Direct substitution in Eq. 3t gives 2.5 + V
= 5 from which V = +2.5 D y 40 GEOMETRICAL OPTICS To find the image
distance, we have V = n'/s', so that n' 1.50 8 or V 2.5 s' = GO cm
= 0.60 m This answer should be verified by the student, using one
of the graphical 47. methods of construction drawn to a convenient
scale. 3.10. Derivation of the Gaussian Formula. The basic equation
36 is of sufficient importance to warrant its derivation in some
detail. While there are many ways of performing a derivation, a
method involving oblique rays will be given here. In Fig. 3K an
oblique ray from an Fig. 3K. Geometry for the derivation of the
paraxial formula used in locating images. axial object point M is
shown incident on the surface at an angle 0, and refracted at an
angle '. The refracted ray crosses the axis at the image point M' .
If the incident and refracted rays MT and TM' are paraxial, the
angles and ' will be small enough so that we may put the sines of
the two angles equal to the angles themselves and for Snell's law
write A = VL ' n (3*0 Since is an exterior angle of the triangle
MTC and equals the sum of the opposite interior angles, = a + P
(3Z) Similarly /3 is an exterior angle of the triangle TCM', so
that /3 = ' + y, and *' - - y (3m) Substituting these values of
and
