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Unformatted text preview: 1 PROPERTIES OF LIGHT All the known properties of light are described in terms of the experiments by which
they were discovered and the many and varied demonstrations by which they are
frequently illustrated. Numerous though these properties are, their demonstrations
can be grouped together and classiﬁed under one of three heads: geometrical optics,
wave optics, and quantum optics, each of which may be subdivided as follows: Geometrical optics
Rectilinear propagation
Finite speed
Reﬂection
Refraction
Dispersion Wave optics
Interference
Diffraction
Electromagnetic character
Polarization
Double refraction Quantum optics
Atomic orbits
Probability densities
Energy levels
Quanta
Lasers 4 FUNDAMENTALS or OPTICS Screen Screen FIGURE 1A
A demonstration experiment illustrating the principle that light rays travel in
straight lines. The rectilinear propagation of light. The ﬁrst group of phenomena classiﬁed as geometrical optics are treated in the ﬁrst 10
chapters of this text and are most easily described in terms of straight lines and plane
geometry. The second group, wave optics, deals with the wave nature of light, and is
treated in Chaps. 11 to 28. The third group, quantum optics, deals with light as made up of tiny bundles of energy called quanta, and is treated from the optical standpoint
in Chaps. 29 to 33. 1.1 THE RECTILINEAR PROPAGATION OF LIGHT The rectilinear propagation of light is the technical terminology applied to the
principle that “light travels in straight lines.” The fact that objects can be made to
cast fairly sharp shadows may be considered a good demonstration of this principle.
Another illustration is found in the pinhole camera. In this simple and inexpensive
device the image of a stationary object is formed on a photographic ﬁlm or plate by
light passing through a small opening, as diagramed in Fig. 1A. In this ﬁgure the
object is an ornamental light bulb emitting white light. To see how an image is
formed, consider the rays of light emanating from a single point a near the top of the.
bulb. Of the many rays of light radiating in many directions the ray that travels in the
exact direction of the hole passes through to the point a’ near the bottom of the image
screen. Similarly, a ray leaving b near the bottom of the bulb and passing through the hole will arrive at b’, near the top of the image screen. Thus it can be seen how an
inverted image of the entire bulb is formed. If the image screen is moved closer to the pinhole screen, the image will be
proportionately smaller, whereas if it is moved farther away, the image will
be proportionately larger. Excellent sharp photographs of stationary objects can be
made with this arrangement. By making a pinhole in one end of a small box and
placing a photographic ﬁlm or plate at the other end, taking several time exposures
as trial runs, good pictures are attainable. For good, sharp photographs the hole .EE mmd H 20: 25:3 WEE 9m 2:898
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«m ‘J 6 FUNDAMENTALS OF OPTICS must be very small, because its size determines the amount of blurring in the image.
A small square hole is quite satisfactory. A piece of household aluminum foil is
folded twice and the corner fold cut oﬁ‘ with a razor blade, leaving good clean edges.
After several such trials, and examination with a magnifying glass, a good square
hole can be selected. The photograph reproduced in Fig. 1B was taken with such a
pinhole camera. Note the undistorted perspective lines as well as thedepth of focus
in the picture. 1.2 THE SPEED OF LIGHT The ancient astronomers believed that light traveled with an inﬁnite speed. Any major
event that occurred among the distant stars was believed to be observable instantly
at all other points in the universe. It is said that around 1600 Galileo tried to measure the speed of light but was
not successful. He stationed himself on a hilltop with a lamp and his assistant on a
distant hilltop with another lamp. The plan was for Galileo to uncover his lamp at
an agreed signal, thereby sending a ﬂash of light toward his assistant. Upon seeing
the light the assistant was to uncover his lamp, sending a ﬂash of light back to Galileo,
who observed the total elapsed time. Many repetitions of this experiment, performed
at greater and greater distances between the two observers, convinced Galileo that
light must travel at an inﬁnite speed. We now know that the speed of light is ﬁnite and that it has an approximate
value of v = 300,000 km/s = 186,400 mi/s In 1849 the French physicist Fizeau* became the ﬁrst man to measure the speed
of light here on earth. His apparatus is believed to have looked like Fig. 1C. His
account of this experiment is quite detailed, but no diagram of his apparatus is given
in his notes. An intense beam of light from a source S is ﬁrst reﬂected from a halfsilvered
mirror G and then brought to a focus at the point 0 by means of lens L1. The diverg
ing beam from 0 is made into a parallel beam by lens L2. After traveling a distance
of 8.67 km to a distant lens L3 and mirror M, the light is reﬂected back toward the
source. This returning beam retraces its path through L2, 0, and L1, half of it
passing through G and entering the observer’s eye at E. The function of the toothed wheel is to cut the light beam into short pulses and
to measure the time required for these pulses to travel to the distant mirror and back.
When the wheel is at rest, light is permitted to pass through one of the openings at 0. “ Armand H. L. Fizeau (1819—1896), French physicist, was born of a wealthy French
family that enabled him to be ﬁnancially independent. Instead of shunning work,
however, he devoted his life to diligent scientiﬁc experiment. His most important
achievement was the measurement of the speed of light in 1849, carried on in Paris
between Montmartre and Suresnes. He also gave the correct explanation of the
Doppler principle as applied to light coming from the stars and showed how the
effect could be used to measure stellar velocities. He carried out his experiments
on the velocity oflight in a moving medium in 1851 and showed that light is dragged
along by a moving stream of water. PROPERTIES or LIGHT 7 FIGURE 1C Experimental arrangement described by the French physicist Fizeau, with which
be determined the speed of light in air in 1849. In this position all lenses and the distant mirror are aligned so that an image of the
light source S can be seen by the observer at E. The wheel is then set rotating with slowly increasing speed. At some point the
light passing through 0 will return just in timel'to be stopped by tooth a. At this
same speed light passing through opening 1 will return in time to be stopped by the
next tooth b. Under these circumstances the light S is completely eclipsed from the
observer. At twice this speed the light will reappear and reach a maximum intensity.
This condition occurs when the light pulses getting through openings 1, 2, 3, 4, . . .
return just in time to get through openings 2, 3, 4, 5, . . . , respectively. Since the wheel contained 720 teeth, Fizeau found the maximum intensity to
occur when its speed was 25 rev/s. The time required for each light pulse to travel over and back could be calculated by (ﬁx21;) = l / 18,000 5. From the measured
distance over and back of 17.34 km, this gave a speed of v = 4 = m = 312,000 km/s
z 1/18,000 s In the years that followed Fizeau’s ﬁrst experiments on the speed of light, a
number of experimenters improved on his apparatus and obtained more and more
accurate values for this, universal constant. About threequarters of a century passed,
however, before A. A. Michelson, and others following him, applied new and improved 8 FUNDAMENTALS or OPTICS methods to visible light, radio waves, and microwaves and obtained the speed of light
accurate to approximately six signiﬁcant ﬁgures. Electromagnetic waves of all wavelengths, from X rays at one end of the
spectrum to the longest radio waves, are believed to travel with exactly the same speed
in a vacuum. These more recent experiments will be treated in detail in Chap. 19,
but we give here the most generally accepted value of this universal constant, . c = 299,792.5 km/s = 2.997925 x 108 m/s (la) For practical purposes where calculations are to be made to four signiﬁcant
ﬁgures, the speed of light in air or in a vacuum may be taken to be c=3.0x108m/s (lb) One is often justiﬁed in using this rounded value since it differs from the more
accurate value in Eq. (la) by less than 0.1 percent. 1.3 THE SPEED OF LIGHT IN STATIONARY MATTER In 1850, the French physicist Foucault* completed and published the results of an
experiment in which he had measured the speed of light in water. Foucault’s experi
ment was of great importance for it settled a long controversy over the nature of light.
Newton and his followers in England and on the Continent believed light to be made
up of small particles emitted by every light source. The Dutch physicist Huygens,
on the other hand, believed light to be composed of waves, similar to water or sound
waves. According to Newton’s corpuscular theory, light should travel faster in an
optically dense medium like water than in a less dense medium like air. Huygens’
wave theory required light to travel slower in the more optically dense medium.
Upon sending a beam of light back and forth through a long tube containing water,
Foucault found the speed of light to be less than in air. This result was considered
by many to be a strong conﬁrmation of the wave theory. Foucault’s apparatus for this experiment is shown in Fig. 1D. Light coming
through a slit S is reﬂected from a plane rotating mirror R to the equidistant concave
mirrors M1 and M 2. When R is in the position 1, the light travels to M1, back along
the same path to R, through the lens L, and by reﬂection to the eye at E. When R
is in position 2, the light travels the lower path through an auxiliary lens L’ and tube * Jean Bernard Leon Foucault (1819—1868), French physicist. After studying
medicine he became interested in experimental physics and with A. H. L. Fizeau
carried out experiments on the speed of light. After working together for some time,
they quarreled over the best method to use for “chopping” up a light beam, and
thereafter went their respective ways. Fizeau (using a toothed wheel) and Foucault
(using a rotating mirror) did admirable work, each supplementing the work of the
other. With a rotating mirror Foucault in 1850 was able to measure the speed of
light in a number of different media. In 1851 he demonstrated the earth‘s rotation
by the rotation of the plane of oscillation of a long, freely suspended, heavy
pendulum. For the development of this device, known .today as a Foucault pendulum,
and his invention of the gyroscope, he received the Copley medal of the Royal
Society of London, in 1855. He also discovered the eddy currents induced in a copper disk moving in a strong magnetic ﬁeld and invented the optical polarizer
which bears his name. PROPERTIES or LIGHT 9 FIGURE lD
Foucault’s apparatus for determining the speed of light in water. T to M 2, back to R, through L to G, and then to the eye E. If now the tube T is
ﬁlled with water and the mirror is set into rotation, there will be a displacement of the
images from E to El and E2. Foucault observed that the light ray through the tube
was more displaced than the other. This means that it takes the light longer to travel
the lower path through water than it does the upper path through the air. The image observed was due to a ﬁne wire parallel to, and stretched across, the
slit. Since sharp images were desired at E1 and E2, the auxiliary lens L’ was necessary
to avoid bending the light rays at the ends of the tube T. Over 40 years later the American physicist Michelson (ﬁrst American Nobel
laureate 1907) measured the speed of light in air and water. For water he found the
value of 225,000 km/s, which is just threefourths the speed in a vacuum. In ordinary
optical glass, the speed was still lower, about twothirds the speed in a vacuum. The speed of light in air at normal temperature and pressure is about 87 km/s less
than in a vacuum, or v = 299,706 km/s. For many practical purposes this difference may be neglected and the speed of light in air taken to be the same as in a vacuum,
v = 3.0 x 108 m/s. 1.4 THE REFRACTIVE INDEX The index of refraction, or refractive index, of any optical medium is deﬁned as the
ratio between the speed of light in a vacuum and the speed of light in the medium: . . s eed in vacuum
Refractlve 1ndex = i____ (10) speed in medium 10 FUNDAMENTALS OF OPTICS In algebraic symbols 0 n = (1d) CI?) The letter n is customarily used to represent this ratio. Using the speeds given in
Sec. 1.3, we obtain the following values for the refractive indices: For glass: n = 1.520 (1e)
For water: n = 1.333 (1f)
For air: 71 = 1.000 (1g) Accurate determination of the refractive index of air at standard temperature (0°C)
and pressure (760 mmHg) give n = 1.000292 for air (1h) Different kinds of glass and plastics have diﬂerent refractive indices. The most
commonly used optical glasses range from 1.52 to 1.72 (see Table 1A). The optical density of any transparent medium is a measure of its refractive
index. A medium with a relatively high refractive index is said to have a high optical
density, while one with a low index is said to have a low optical density. 1.5 OPTICAL PATH To derive one of the most fundamental principles in geometric optics, it is appropriate
to deﬁne a quantity called the optical path. The path d of a ray of light in any medium
is given by the product velocity times time: d=vt Since by deﬁnition n = c/v, which gives 12 = c/n, we can write c
d=—t or nd=ct The product nd is called the optical path A:
A = nd The optical path represents the distance light travels in a vacuum in the same
time it travels a distance d in the medium. If a light ray travels through a series of
optical media of thickness d, d’, d”, . . . and refractive indices 11, n’, n”, . . . , the total
optical path is just the sum of theseparate values: 0 A = nd + n’d' + n”d” + (1i) A diagram illustrating the meaning of optical path is shown in Fig. 1E. Three
media of length d, d’, and d”, with refractive indices n, n’, and n", respectively, are
shown touching each other. Line AB shows the length of the actual light path through these media, while the line CD shows the distance A, the distance light would travel
in a vacuum in the same amount of time t. PROPERTIES or LIGHT 11 Equivalent optical path in a vacuum
A FIGURE 1E
The optical path through a series of optical media. 1.6 LAWS OF REFLECTION AND REFRACTION Whenever a ray of light is incident on the boundary separating two different media,
part of the ray is reﬂected back into the ﬁrst medium and the remainder is refracted
(bent in its path) as it enters the second medium (see Fig. 1F). The directions taken
by these rays can best be described by two wellestablished laws of nature. According to the simplest of these laws, the angle at which the incident ray
strikes the interface MM’ is exactly equal to the angle the reﬂected ray makes with the
same interface. Instead of measuring the angle of incidence and the angle of reﬂection
from the interface MM’, it is customary to measure both from a common line
perpendicular to this surface. This line NN’ in the diagram is called the normal. As
the angle of incidence ¢ increases, the angle of reﬂection also increases by exactly the
same amount, so that for all angles of incidence 0 angle of incidence = angle of reﬂection (lj) A second and equally important part of this law stipulates that the reﬂected
ray lies in the plane of incidence and on the opposite side of the normal, the plane of
incidence being deﬁned as the plane containing the incident ray and the normal.
In other words, the incident ray, the normal, and the reected ray all lie in the same
plane, which is perpendicular to the interface separating the two media. The second law is concerned with the incident and refracted rays of light, and
states that the sine of the angle of incidence and the sine of the angle of refraction
bear a constant ratio one to the other, for all angles of incidence: = const (1k) 12 FUNDAMENTALS or omcs FIGURE 1F 1 ~, , _//
Reﬂection and refraction at the bound I ~ / if, ' f' " ‘ "’ /  a ;/
ary separating two media With refractive / ' 7274/ (1.  W4 ,7
indices n and n’, respectively. Furthermore, the refracted ray also lies in the plane of incidence and on the opposite
side of the normal. This relationship, experimentally established by Snell,* is known
as Snell’s law. In addition the constant is found to have exactly the ratio of the
refractive indices of the two media n and 11’. Hence we can write 5?“ ¢ = ’i (11)
s1n d)’ n which can be written in the symmetrical form 0 n sin ¢ = n’ sin (15’ (lm) By Eqs. (1c) and (1d) the refractive indices of different optical media are deﬁned
as t and n = (1n)
v 3
II
elo where c is the speed of light in a vacuum (c = 2.997925 + 108 m/s) and v and v’
are the speeds of light in the two media. ‘ Willebrord Snell (1591—1626), Dutch astronomer and mathematician, was born at
Leyden. At twentyone he succeeded his father as professor of mathematics at the
University of Leyden. In 1617, he determined the size of the earth from measure
ments of its curvature between Alkmaar and BergenopZoom. He announced
what is essentially the law of refraction in an unpublished paper in 1621. His
geometrical construction requires that the ratios of the cosecants of 96 and ¢’ be
constant. Descartes was the ﬁrst to use the ratio of the sines, and the law is known
as Descartes’ law in France. PROPERTIES or LIGHT 13 By the substitution of Eqs. (1c) in Eq. (11), we obtain,
sin ¢
’ sin ¢’
If one or both indices are different from unity, the ratio n’/n is often called the
relative index n’ and Snell’s law can be written . sin (1) sm d)’ If the ﬁrst medium is a vacuum, n = 1.0, the relative index has just the value of the second index and Eq. (1p) is again valid. If the ﬁrst index is air at normal temperature and pressure (n 1.000292), and if threeﬁgure accuracy is satisfactory,
Eq. (1p) is again used. Wherever practical, we shall use unprimed symbols to refer to the ﬁrst medium, primed symbols for the second medium, double primed symbols for the third medium, etc. When the angles of incidence and refraction are very small, a good approximation
is obtained by setting the sines of angles equal to the angles themselves, obtaining = 3, (lo)
I) = 11’ (1p) (1(1) 1.7 GRAPHICAL CONSTRUCTION FOR REFRACTION A simple method for tracing a ray of light across a boundary separating two optically
transparent media is shown in Fig. 1G. Because the principles involved in this
construction are readily extended to complicated optical systems, the method is
useful in the preliminary design of many different kinds of optical instruments. After the line GH is drawn, representing the boundary separating the two media
of index n and n’, the angle of incidence 4) of the incident ray JA is selected and the
construction proceeds as follows. At one side of the drawing, and as reasonably close
as possible, a line OR is drawn parallel to JA. With a point of origin 0, two circular
arcs are drawn with their radii proportional to the two indices n and 11’, respectively.
Through the point of intersection R a line is drawn parallel to the boundary normal
NN’, intersecting the are n’ at P. The line OP is next drawn in; parallel to it, through
A, the refracted ray AB is drawn. The angle [3 between the incident and refracted
ray, called the angle of deviation, is given by B = 4> — ¢’ (1r)
To prove that this construction follows Snell’s law exactly, we apply the law of sines to the triangle ORP:
0R 0P sin 45’ = sin (1: —— ()5)
Since sin (7: — 4)) = sin ([5, OR = n, and GP = n’, substitution gives directly I n 7: sin d)’ sin (I) (18)
which is Snell’s law [Eq. (11)]. 14 FUNDAMENTALS or orrrcs J 7
l
¢
,, s.
B
/
FIGURElG Graphical construction for refraction at a smooth surface separating two media
of index n and n’. 1.8 THE PRINCIPLE OF REVERSIBILITY The symmetry of Eqs. (1 j) and (1m) with respect to the symbols used shows at once
that if a reﬂected or refracted ray is reversed in direction, it will retrace its original
path. For any given pair of media with indices n and n' any one value of d} is cor
related with a corresponding value of n’. This will be equally true when the ray is
reversed and (12’ becomes the angle of incidence in the medium of n’; the angle of
refraction will then be 4). Since reversibility holds at each reﬂecting and refracting
surface, it holds also for even the most complicated light paths. This useful principle
has more than a purely geometrical foundation, and later it will be shown to follow
from the application of wave motion to a principle in mechanics. 1.9 FERMAT’S PRINCIPLE . The term optical path was introduced in Sec. 1.5, where it was deﬁned as the distance
a light ray would travel in a vacuum in the same time it travels from one point to
another, a speciﬁed distance, through one or more optical media. The real path of a
ray of light through a prism, with media of different refractive index on either side,
is shown in Fig. 1H. The optical path from the point Q in medium n, through medium
71’, and to the point Q" in medium n" is given by O A = nd + n’d’ + n”a'” (1t) One can also deﬁne an optical path in a medium of continuously varying
refractive index by replacing the summation by an integral. The paths of the rays
are then curved, and Snell’s law of refraction loses its meaning. PROPERTIES or LIGHT 15 o
0000 0'. o
o
to
O '9‘.
o o o
'0 o
5.0:
’9‘. O
i O V
O O. H j:\ \
// 5555:2532:5;:;:;:;:;:;:;.;::.;.:.; ‘ \\\
\ V ‘\
. \ FIGURE 1H
The refraction of light by a prism and the meaning of optical path A. We shall now consider Fermat’s* principle, which is applicable to any type of
variation of n and hence contains within it the laws of reﬂection and refraction as well: The path taken by a light ray in going from one point to another through
any set of media is such as to render its optical path equal, in the ﬁrst
approximation, to other paths closely adjacent to the actual one. The other paths must be possible ones in the sense that they may undergo devi
ations only where there are reﬂecting or refracting surfaces. Fermat’s principle will
hold for a ray whose optical path is a minimum with respect to adjacent hypothetical
paths. Fermat himself stated that the time required by the light to traverse the path
is a minimum and the optical path is a measure of this time. But there are plenty of
cases in which the optical path is a maximum or neither a maximum nor a minimum
but merely stationary (at a point of inﬂection) at the position of the true ray.
Consider a ray of light that must pass through a point Q and then, after reﬂection
from a plane surface, pass through a second point Q” (see Fig. 11). To ﬁnd the real
path, we ﬁrst drop a perpendicular to GH and extend it an equal distance on the other
side to Q’. The straight line Q’Q” is drawn in, and from its intersection B the line QB ‘ Pierre de Fermat (1601—1665), French mathematician, born at Beaumontde
Lomagne. In his youth, with Pascal, he made discoveries about the properties of
numbers, on which he later built his method of calculating probabilities. His
brilliant researches in the theory of numbers rank him as the founder of modern,
theory. He also studied the reﬂection of light and enunciated his principle of least
time. His justiﬁcation for this principle was that nature is economical, but he was
unaware of circumstances where exactly the opposite is true. Fermat was a
counselor for the parliament of Toulouse, distinguished for both legal knowledge
and for strict integrity of conduct. He was also an accomplished general scholar
and linguist. 16 FUNDAMENTALS or OPTICS FIGURE 11
Fermat’s principle applied to reﬂection
at a plane surface. is drawn. The real light path is therefore QBQ”, and, as can be seen from the sym
metry relations in the diagram, it obeys the law of reﬂection. Consider now adjacent paths to points like A and C on the mirror surface close
to B. Since a straight line is the shortest path between two points, both the paths
Q'AQ” and Q’CQ” are greater than Q’BQ". By the above construction and equivalent
triangles, QA = Q'A, and QC = Q’C, so that QAQ” > 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, and the ﬂatness of the curve between A 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. II. 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 remembered 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. 1K(b)
equal path lengths are represented by a straight horizontal line. Some attention will be devoted here to other reﬂecting surfaces like a and c
shown dotted in Fig. II. If these surfaces are tangent to the ellipsoid at the point B, FIGURE 11 Fermat’s principle applied to an ellipti
cal reﬂector. PROPERTIES or LIGHT 17 FIGURE 1K 2
Graphs of optical paths involving re
ﬂection illustrating conditions for (a)
maximum, (b) stationary, and (c) min
imum light paths. Fermat’s principle. 9—» 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. 1K). _ It is readily shown mathematically that both the laws of reﬂection and refraction
follow from Fermat’s principle. Figure 1L, which represents the refraction of a ray
at a plane surface, can be used to prove the law of refraction [Eq. (lm)]. The length
of the optical path between the point Q in the upper medium of index n and another
point Q’ in the lower medium of index n' passing through any point A on the surface is A nd + n’d’ (lu) where d and (1’ 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 alz=h2+(p—'x)2 d’2=h'2+x2
When these values of d and d’ are substituted in Eq. (1i), we obtain
A =_n[h2 + (p — x)2]”2 + n’(h’2 + 3:2)"2 (1v) According to Fermat’s principle, A must be a minimum or a maximum (or in
general stationary) for the actual path. One method of ﬁnding a minimum or max
imum for the optical path is to plot a graph of A against x and ﬁnd at what value of x FIGURE 1L Geometry of a refracted ray used in
illustrating Fermat’s principle. 18 FUNDAMENTALS or orrrcs a tangent to the curve is parallel to the x axis (see Fig. 1K). The mathematical means
for doing the same thing is, ﬁrst, to differentiate Eq. (IV) 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. (IV) with respect to x and setting the result equal to zero,
we obtain
dA in in'
=__——— —2 +2 +——_2 =0
dx [h2+(p—x)2]1/2( p x) x' (hIZ + x2)1/2 which ives n —R—_x___ : n’ ___x___.
g [I12 + _ x)2]l/2 (hi2 + x2)1/2 . P — x I x or sun 1 n = n —
py d (1’ By reference to Fig. 1L 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. (1m), namely ‘ n sin (I) = n’ sin d)’ (1w) A diagram for reﬂected light, similar to Fig. 1L, can be drawn and the same
mathematics applied to prove the law of reﬂection. 1.10 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. 1M,
the incident ray of white light gives rise to refracted rays of different colors (really a
continuous spectrum) each of which has a diﬁ‘erent value of ¢’. By Eq. (1m) the value
of 11’ 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. These Fraunhofer* lines, which are designated by the letters A, B, C, . . . , starting at the extreme red end, are given in Table 1A. The ones
most commonly used are those in Fig. 1M. 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 average deviation of the " Joseph von Fraunhofer (1787—1826) was the son of a Bavarian glazier. He learned
glass grinding from his father and entered the ﬁeld of optics from the practical side.
Fraunhofer gained great skill in the manufacture of achromatic lenses and optical
instruments. While measuring the refractive index of diﬁ‘erent kinds of glass and
its variation with color or wavelength, he noticed and made use of the yellow
D lines of the sodium spectrum. He was one of the ﬁrst to produce diffraction
gratings, and his rare skill with these devices enabled him to produce better spectra
than his predecessors. Although the dark lines of the solar spectrum were ﬁrst
observed by W. H. Wollaston, they were carefully observed by Fraunhofer, under
high dispersion and resolution, and the wavelengths of the most prominent lines
were measured with precision. He mapped 576 of these lines, the principal ones,
denoted by the letters A through K, being known by his name. PROPERTIES or LIGHT 19 FIGURE 1M Upon refraction, white light is spread
out into a spectrum. This is called dis
persion. 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 1A are nF = 1.52933 nD = 1.52300 nC = 1.52042 Now it is readily shown from Eq. (lq) that for a given small angle 45 the dispersion
of the F and C rays (4)} — (be) is proportional to up —— nC = 0.00891 while the deviation of the D ray (4) — (#3,) depends on nD — l which is equal to
0.52300. Thus it is nearly 60 times as great. The ratio of these two quantities varies
greatly for different kinds of glass and is an important characteristic of any optical
substance. It is called the dispersive power and is deﬁned by the equation = —"F ‘ "C (1x)
nD — 1
The reciprocal of the dispersive power is called the dispersive index v:
o v = "ID—"1. (1y) "F—nc For most optical glasses v lies between 20 and 60 (see Table 1B and Appendix 111). Table 1A FRAUNHOFER’S DESIGNATIONS, ELENIENT SOURCE, WAVELENGTH,
AND REFRACI'IVE INDEX FOR FOUR OPTICAL GLASSES‘ Designa Chemical Wavelength, Spectacle Light Dense Extra dense
tion element AT crown ﬂint ﬂint ﬂint C H 6563 1.52042 1.57208 1.66650 1.71303 D Na 5892 1.52300 1.57600 1.67050 1.72000 F H 4861 1.52933 [1.58606 1.68059 1.73780 G’ H 4340 1.53435 1.59441 1.68882 1.75324 ‘ For other glasses and crystals see Appendices III and IV. T To change wavelengths in'angstroms (A) to nanometers (nm), move decimal point one place to the left
(see Appendix VI). 20 FUNDAMENTALS or OPTICS FIGURE IN
The variation of refractive index with F D C
color. Violet Blue Green Yel/ow Red Figure 1N illustrates schematically the type of variation of n with color that is
usually encountered for optical materials. The denominator of Eq. (1y), 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 numerator, which measures the average deviation,
represents the magnitude 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, that the refractive index of each speciﬁc element of an optical instrument is that determined for yellow sodium
D light. Table 18 DISPERSION INDEX FOR FOUR OPTICAL GLASSES‘
Spectacle Light Dense Extra dense
Glass crown ﬂint ﬂint ﬂint
v 58.7 41.2 47.6 29.08 _
“ See Table 1A. ' PROBLEMS* 1.1 A boy makes a pinhole camera out of a cardboard box with the dimensions 10.0 cm x
10.0cm x 16.0 cm. A pinhole is located in one end, and a ﬁlm 8.0 cm x 8.0 cm is
placed in the other end. How far away from a tree 25.0 m high should the boy place
his camera if the image of the tree is to be 6.0 cm high on the ﬁlm? Ans. 66.7 m 1.2 A physics student wishes to repeat Fizeau’s experiment for measuring the speed of
light. If he uses a toothed wheel containing 1440 teeth and his distant mirror is located
in a laboratory window across the college campus 412.60 m away, how fast must his
wheel be rotated if the returning light pulses show the ﬁrst maximum intensity? 1.3 If the mirror R in Foucault’s experiment were to rotate at 12,000 rev/min, ﬁnd (a) the
rotational speed of the mirror R in revolutions per second and (b) the rotational speed
of the sweeping beam RM 1 in radians per second. Find the time it takes the light to
traverse the path (c) RM IR and ((1) RM 2R. What is the obserVed slit deﬂection (e) EEI ,
and (f) EEZ? Assume the distances RM 1 = RM; = 6.0 m, RS = RE = 6.0 m, the " Before solving any problems in this text, read Appendix VI. 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 PROPERTIES or LIGHT 21 length of the water tube T = 5.0 m, the refractive index of water is 1.3330, and the
speed of light in air is 3.0 x 108 m/s. If the refractive index for a piece of optical glass is 1.5250, calculate the speed of light
in the glass. Ans. 1.9659 x 108 m/s
Calculate the difference between the speed of light in kilometers per second in a vacuum and the speed of light in air if the refractive index of air is 1.0002340. Use velocity
values to seven signiﬁcant ﬁgures. If the moon’s distance from the earth is 3.840 x 105 km, how long will it take micro
waves to travel from the earth to the moon and back again?
How long does it take light from the sun to reach the earth? Assume the earth’s
distance from the sun to be 1.50 x 108 km. Ans. 500 s, or 8 min 20 s
A beam of light passes through a block of glass 10.0 cm thick, then through water for
a distance of 30.5 cm, and ﬁnally through another block of glass 5.0 cm thick. If the
refractive index of both pieces of glass is 1.5250 and of water is 1.3330, ﬁnd the total
optical path.
A water tank is 62.0 cm long inside and has glass ends which are each 2.50 cm thick.
If the refractive index of water is 1.3330 and of glass is 1.6240, ﬁnd the overall optical
path.
A beam of light passes through 285.60 cm of water of index 1.3330, then through
15.40 cm of glass of index 1.6360, and ﬁnally through 174.20 cm of oil of index 1.3870.
Find to three signiﬁcant ﬁgures (0) each of the separate optical paths and (b) the total
optical path. Ans. (a) 380.7, 25.19, and 241.6 cm, (b) 647 cm
A ray of light in air is incident on the polished surface of a block of glass at an angle
of 10°. ((1) If the refractive index of the glass is 1.5258, ﬁnd the angle of refraction to
four signiﬁcant ﬁgures. (b) Assuming the sines of the angles in Snell’s law can be
replaced by the angles themselves, what would be the angle of refraction? (c) Find
the percentage error.
Find the answers to Prob. 1.11, if the angle of incidence is 45.0° and the refractive
index is 1.4265.
A ray of light in air is incident at an angle of 54.0° on the smooth surface of a piece
of glass. (a) If the refractive index is 1.5152, ﬁnd the angle of refraction to four
signiﬁcant ﬁgures. (b) Find the angle of refraction graphically.‘ (See Fig. Pl.13).
Ans. (a) 32.272°, (b) 32.3° FIGURE Pl.13
Graph for part (b) of Prob. 1.13. 22 FUNDAMENTALS or orncs 1.14 1.15 1.16
1.17 1.18 1.19 1.20 1 .21 1.22 ' 1.23 1.24 1.25 A straight hollow pipe exactly 1.250 m long, with glass plates 8.50 mm thick to close
the two ends, is thoroughly evacuated. (a) If the glass plates have a refractive index
of 1.5250, ﬁnd the overall optical path between the two outer glass surfaces. (b) By
how much is the optical path increased if the pipe is ﬁlled with water of refractive index
1.33300. Give answers to ﬁve signiﬁcant ﬁgures.
Referring to Fig. 1L, the distancex = 6.0 cm, h = 12.0 cm, h’ = 15.0 cm, n =
and n’ = 1.5250. Find ¢’, «)5, d, d’, p, and A, to three signiﬁcant ﬁgures.
Ans. ¢’ = 21.80°, ¢ = 25.14°, d = 13.26 cm, d’ = 16.16 cm,
p = 11.63 cm, A = 42.3 cm 1.3330, Solve Prob. 1.15 graphically.
In studying the refraction of light Kepler arrived at a refraction formula =——é——— where k=n—1
1—ksec¢’ n’ #5 n’ being the relative index of refraction. Calculate the angle of incidence of for a piece
of glass for which n’ = 1.7320 and the angle of refraction ¢’ = 32.0° according to
(a) Kepler’s formula and (b) Snell’s law. Note that see ¢’ = l/(cos 43’).
White light is incident at an angle of 55.0° on the polished surface of a piece of glass.
If the refractive indices for red C light and blue F light are nc = 1.53828, and n1: =
1.54735, respectively, what is the angular dispersion between these two colors? (a)
Find the two angles to ﬁve signiﬁcant ﬁgures and (b) the dispersion to three signiﬁcant
ﬁgures. Ans. (a) ¢'C = 32.1753°, ¢’F = 31.9643°, (b) 0.2110°
A piece of dense ﬂint glass is to be made into a prism. If the refractive indices for red,
yellow, and blue light are speciﬁed as no = 1.64357, no = 1.64900, and n; = 1.66270,
ﬁnd (a) the dispersive power and (b) the dispersion constant for this glass.
A block of spectacle crown glass is to be made into a lens. The refractive indices fur
nished by the glass manufacturer are speciﬁed as nc = 1.52042, nD = 1.52300, and
n; = 1.52933. Determine the value of (a) the dispersion constant and (b) the dispersive
power.
A piece of extra dense ﬂint glass is to be made into a prism. The refractive indices
furnished by the glass manufacturer are those given in Table 1A. Find the value of
(a) the dispersive power and (b) the dispersion constant. Ans. (a) 0.034403, (b) 29.067
Two plane mirrors are inclined to each other at an angle 0:. Applying the law of reﬂec
tion show that any ray whose plane of incidence is perpendicular to the line of inter
section of the two mirrors is deviated by two reﬂections by an angle 5 which is
independent of the angle of incidence. Express this deviation in terms of a.
An ellipsoidal mirror has a major axis of 10.0 cm, a minor axis of 8.0 cm, and foci
6.0 cm apart. If there is a point source of light at one focus Q, there are only two rays
of light that pass through the point C, midway between B and Q’, as can, be drawn in
Fig. I]. Draw such an ellipse and graphically determine whether these two paths
QBC and QDC are maxima, minima, or stationary.
A ray of light in air enters the center of one face of a prism at an angle making 55.0°
with the normal. Traveling through the glass, the ray is again refracted into the air
beyond. Assume the angle between the two prism faces to be 60.0° and the glass to
have a refractive index of 1.650. Find the deviation of the ray (a) at the ﬁrst surface
and (b) the second surface. Find the total deviation (c) by calculation and (d)
graphically.
One end of a glass rod is ground and polished to the shape of a hemisphere with a
diameter of 10.0 em. Five parallel rays of light 2.0 cm apart and in the same plane are 1.26 PROPERTIES or LIGHT 23 incident on this curved end, with one ray traversing the center of the hemisphere
parallel to the rod axis. If the refractive index is 1.5360, calculate the distances from
the front surface to the point where the refracted rays cross the axis. Crystals of clear strontium titanate are made into semiprecious gems. The refractive
indices for different colors of light are as follows: ____________________———————————— Red Yellow Blue Violet
____________________——————
2, A 6563 5892 4861 4340
n 2.37287 2.41208 2.49242 2.57168 ______________________————————— Calculate the value of (a) the dispersion constant and (b) the dispersive power. Plot a
graph of the wavelength A against the refractive index n. Use the blue, yellow, and red
indices. ' ...
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