Unit I (Wave Optics)
Contents
1 Wavefront
1
2 Huygens’s Principle
2
3 Reflection
3
3.1
Reflection of a plane wave at a plane surface using wavefront . . . . . . . . . . . .
4 Refraction
4.1
5
Refraction of a plane wave at a plane surface using wavefront . . . . . . . . . . . .
5 Total Internal Reflection
5.1
5
6
7
Total internal reflection of a plane wave at a plane surface using wavefront . . . .
6 Youngs double slit experiment
9
9
7 Interference
10
7.1
Conditions for observing sustained interference . . . . . . . . . . . . . . . . . . . .
13
7.2
Geometrical path and optical path . . . . . . . . . . . . . . . . . . . . . . . . . .
13
7.3
Techniques of obtaining interference . . . . . . . . . . . . . . . . . . . . . . . . . .
16
7.4
Interference by division of wave front . . . . . . . . . . . . . . . . . . . . . . . . .
16
7.4.1
Optical Path Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Interference by division of wave amplitude . . . . . . . . . . . . . . . . . . . . . .
20
7.5.1
Thin film interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
7.5.2
Interference due to thin film of uniform thickness . . . . . . . . . . . . . .
21
7.5.3
Conditions for Maxima (Brightness) and Minima (Darkness) . . . . . . . .
24
7.5.4
Interference at wedge-shaped film . . . . . . . . . . . . . . . . . . . . . . .
25
7.5
8 Newton’s rings
26
8.1
Condition for Bright and Dark Rings . . . . . . . . . . . . . . . . . . . . . . . . .
28
8.2
Circular Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
8.3
Radii of Dark Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
9 Anti-Reflecting Coatings
30
i
10 Ultrasonic interferometer
33
11 DIFFRACTION
34
11.1 Dependence of the phenomenon on wavelength . . . . . . . . . . . . . . . . . . . .
34
11.2 Distinction between interference and diffraction . . . . . . . . . . . . . . . . . . .
35
11.3 Types of diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
11.4 Condition of Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
11.5 Intensity Distribution in Diffraction Pattern Due to a Single Slit . . . . . . . . . .
40
11.6 Diffraction grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
A Refractive index
44
ii
1
Wavefront
A wavefront is a surface that comprises particles in the same state of vibration. It is usually
perpendicular to the direction of wave propagation. The wavefronts can be classified into three
types based on the shape of the light source.
• Spherical wavefront: The wavefronts that arise from a point source are spherical in shape.
Figure 1: Spherical wavefront
• Plane wavefront: A plane wavefront is formed if the source of the light is infinity. As a
result, wavefronts generated by the sun or stars are considered planar. The waves having
plane wave fronts are called plane waves. In the plane waves, the rays are parallel to one
another.
Figure 2: Plane wavefront.
• Cylindrical wavefront: The wavefronts that arise from a line source are cylindrical in shape.
All the points are equidistant from the source.
Figure 3: Cylindrical wavefront.
2
Huygens’s Principle
In the late 17th century, scientists were debating whether light was a wave or a particle. Newton
was the first who proposed a corpuscular theory which states that light is made up of microscopic
particles called corpuscles. Later, Christian Huygens proposed that light is made up of waves that
vibrate up and down perpendicular to the direction of wave propagation, i.e. a wave propagating
through space like ripples in water or sound in air. Hence, light spreads out like a wave in all
directions from a source. A wavefront is a collection of points that moved some distance over a
fixed time interval.
Huygens’s Principle states that ”Every point on a wavefront (called as primary wavefront) is
the source of a secondary disturbance (called spherical wavelets) that spread out at the speed of
light in the forward direction. A surface touching these secondary wavelets, tangentially in the
forward direction at any instant gives the new wavefront at that instant. This is called secondary
wavefront.”
2
Figure 4: Propagation of wavefront.
3
Reflection
The reflection refers to the phenomenon when a wave travelling in any medium hits the boundary
and returns back in the medium after striking.
• Smooth surface: If the surface irregularities are small compared to the wavelength of the
incident light, the boundary is said to be smooth surface. If surface is smooth, it reflects
light specularly.
• Rough surface: If the surface irregularities are larger than the wavelength of the incident
light, the boundary is said to be rough surface.
• Specular reflection: It refers to the case when the angle of reflection of light equals the angle
of incidence. Polished metals, mirrors, liquid surfaces etc., smooth surfaces reflect light
specularly. Figure 5 shows the specular relection at smooth surfaces.
3
Figure 5: Specular reflection at smooth surface. The reflected rays are parallel.
• Diffuse reflection: The rough surfaces have microscopic irregularities. The reflection of light
at many angles is called the diffuse reflection. This is why glare does not occur when light
is reflected from a wall, whereas glare occurs when light is specularly reflected from a water
surface. Figure
Figure 6: Diffuse reflection at rough surface. The reflected rays are in random directions.
In Physics, the term ”reflection” refers to specular reflection.
• Laws of reflection: The specular reflection obeys the following two laws known as laws of
reflection:
– The incident ray, reflected ray and the normal to the surface all lie in the same plane
(see Figure 5)
4
– The angle of reflection θ2 is equal to the angle of incidence θ1 , i.e.
θ1 = θ2
3.1
Reflection of a plane wave at a plane surface using wavefront
As the wave front strikes the mirror, wavelets are first emitted from the left part of the mirror and
then from the right. The wavelets closer to the left have had time to travel farther, producing a
wave front traveling in the direction shown. Figure 7 shows the reflection of a plane wave at plane
surface.
Figure 7: Huygenss principle applied to a plane wave front striking a mirror. The wavelets shown
were emitted as each point on the wave front struck the mirror. The tangent to these wavelets
shows that the new wave front has been reflected at an angle equal to the incident angle. The
direction of propagation is perpendicular to the wave front, as shown by the downward-pointing
arrows.
4
Refraction
The bending of a wave as it travels from one medium to another is known as refraction of the wave.
When waves strike a different medium’s surface, some are reflected, while others bend and change
direction. The boundary is known as a refracting surface. The angle of refraction r depends on
the properties of the two media and on the angle of incidence i.
sin i
v1
c/µ1
µ2
=
=
=
= µ21
sin r
v2
c/µ2
µ1
5
(1)
where v1 is the velocity of light in medium 1 having a refractive index µ1 (c.f. Appendix A) and
v2 is the velocity of light in medium 2 having a refractive index µ2 . The ratio
v1
v2
or
µ1
µ2
is known as
the relative refractive index. Eq. 1 is known as Snell’s law. An increase in the angle of incidence
leads to an increase in the angle of refraction. Further, for a given angle of incidence (i ), the angle
of refraction (r ) depends on the wavelength of the incident light and varies from colour to colour.
The laws of refraction states that the incident ray, refracted ray and the normal to the refracting
surface all lie in the same plane. The process of refraction is shown in figure 8.
Figure 8: Phenomenon of refraction- A ray obliquely incident on air-glass interface bends toward
the normal in glass.
4.1
Refraction of a plane wave at a plane surface using wavefront
Figure 11 shows the refraction of a plane wave at a plane interface. The position of the refracted
wave is formed using the idea of secondary wavelets.
Figure 9: Refraction of a plane wave at a plane surface.
6
One side of the wave moves from A to C (a distance vg t) in glass in the same time that the
other side of the wavefront moves from B to D (a distance va t) in the same time in air. The wave
front recombines at CD.
5
Total Internal Reflection
A medium having a lower refractive index is said to be an optically rarer medium while a medium
having a higher refractive index is known as an optically denser medium. When a ray of light
passes from a denser medium to a rarer medium, it is bent away from the normal in the rarer
medium (Figure 10(a)).
(a)
Figure 10:
(b)
(c)
(a) When light rays travel from a denser medium into a rarer medium, they bend
away from the normal in the rarer medium. As the angle of incidence θ1 increases, the angle of
refraction θ2 increases (b) The angle of incidence θc which produces an angle of refraction 90 is
called the critical angle.
From Snell’s law
sin θ2 = (
µ1
) sin θ1
µ2
(2)
where θ1 is the angle of incidence of light ray in the denser medium and θ2 is the angle of refraction
in the rarer medium. Also µ1 > µ2 . When the angle of incidence, θ1 in the denser medium is
increased, the transmission angle, θ2 increases and the refracted rays bend more and more away
from the normal. At some particular angle θc the refracted ray glides along the boundary surface
so that θc = 90◦ , as seen in Figure 10(b). At angles greater than θc there are no refracted rays at
7
all. The rays are reflected back into the denser medium as though they encountered a specular
reflecting surface (Figure 10(c)). Thus,
• If θ1 < θc , the ray refracts into the rarer medium
• If θ1 = θc , the ray just touches the interface of rarer-to-denser media
• If θ1 > θc , the ray is reflected back into the denser medium
The phenomenon in which light is totally reflected from a denser-to-rarer medium boundary is
known as total internal reflection. The rays that experience total internal reflection obey the laws
of reflection. Therefore, the critical angle can be determined from Snells law.
When θ1 = θc , θ2 = 90◦ . Therefore,
µ1 sin θc = µ2 sin 90◦ = µ2
Or,
sin θc =
µ2
µ1
(3)
If the rarer medium is air, µ2 = 1 and writing µ1 = µ, we obtain
sin θc =
1
µ
(4)
Total internal reflection does not take place when light propagates from a rarer to a denser medium.
The critical angle is small for substances having high refractive index. For example, θc = 24◦
for diamond - air interface, while it is about 42◦ for glass-to-air interface. This phenomenon is
exploited in obtaining sparkle in crystal glass and diamonds. The phenomenon of total internal
reflection has made possible to guide light through optical fibres.
Example: Determine the critical angle for a light ray traveling from water (µ = 1.333) to
air.
Solution:
sin θc =
1
1
=
µ
1.333
(5)
Or
θc = sin−1 (0.75) = 48.61◦
8
(6)
5.1
Total internal reflection of a plane wave at a plane surface using
wavefront
Figure 11 shows the total internal reflection of a plane wave at a plane surface. The position of
the reflected wave is formed using the idea of secondary wavelets.
Figure 11: Total internal reflection of a plane wave at a plane surface.
6
Youngs double slit experiment
Young gave the first demonstration of the interference of light waves in 1801. Fig. 12 shows a
plan view of the basic arrangement of his double slit experiment. The primary light source at S
is a monochromatic source; it is generally a sodium lamp, which emits yellow light of wavelength
at around 5893 Å. The expanding wavefront from the primary light source S falls on two narrow
closely spaced slits, S1 and S2 as shown in Fig. 12. The slits at S1 and S2 are very narrow and
partition the incident wavefront. If the slits are equidistant from S, the phase of the wave at S1
will be the same as the phase at S2 because parts of the same wavefront emerging from S pass
through S1 and S2 . Further, waves leaving S1 and S2 have the same frequency as the primary
source. Hence, sources S1 and S2 act as secondary coherent sources. The waves leaving from S1
and S2 interfere and produce alternate bright and dark bands on the screen at T.
9
Figure 12:
Youngs double slit arrangement - The narrow slit S acts as a source of cylindrical
waves which illuminate the slits S1 and S2. S1 and S2 behave as coherent sources and produce
interference.
Thomas Young used this experiment to make the first measurement of the wavelength of the
light.
Now, if light is allowed to illuminate the slits S1 and S2 directly, instead of through slit S,
interference will not be produced and the observation screen will be uniformly illuminated.
7
Interference
Interference is an important consequence of superposition of waves. Let us consider two (or more)
light waves of same frequency and having a constant phase difference travel in the same region of
a medium simultaneously and cross each other (see Fig. 13). Waves having the same frequency
and a constant phase difference are known as coherent waves. Waves having fluctuating phase
difference are known as incoherent waves. At the point of crossing, waves overlap or superpose on
each other.
10
Figure 13: Two light waves meet at point P.
According to the principle of superposition, the combined effect of coherent waves at each
point of the region of superposition is obtained by adding algebraically the disturbances due to
individual waves. The resultant intensity at any point in the region of superposition depends upon
the amplitudes and the phase relationships of the component waves. Let us assume here that the
component waves are of the same amplitude, say A.
Figure 14: Two possibilities of resultant wave at point P.
• If the two waves, meeting at P, reach their maxima, zeros and minima at the same instant
of time, then their phase difference is zero (or an integral multiple of 2π). Such waves will
have a crest-to-crest and trough-to-trough correspondence, as shown in Fig. 14 (a). Then
the waves are said to be in phase. The amplitude of the resultant wave at the point will
then be equal to the sum of the amplitudes of the two waves, as shown in Fig. 14 (a).
The amplitude of the resultant wave = A + A = 2A. Hence, the intensity of the resultant
wave is IR ∝ 22 A2 = 22 I. It is obvious that the resultant intensity is greater than the
11
sum of the intensities (I + I = 2I) due to individual waves. The interference produced at
such points is known as constructive interference. When two waves are not displaced with
respect to each other or when they are displaced through an integral number of wavelengths,
constructive interference takes place. Bright bands of light are observed at those points. As
S1 and S2 are coherent sources, the bright bands are stationary.
• If one of the waves reaches its maximum at the same time when the other reaches its
minimum, then their phase difference is π radians. Such waves have a crest-to- trough
correspondence, as shown in Fig. 14(b). The waves are said to be in opposite phase or 180◦
out-of-phase. The amplitude of the resultant wave at the point will also be equal to the sum
of the amplitudes of the two waves, as shown in Fig. 14(b). As the amplitude of one of the
waves is negative, the amplitude of the resultant wave = A - A = 0. Hence, the intensity of
the resultant wave is IR ∝ 02 = 0. It is obvious that the resultant intensity is less than the
sum of the intensities (I + I = 2I) due to individual waves. The interference produced at
such points is known as destructive interference. When two waves are displaced with respect
to each other by an odd number of half-wavelengths, destructive interference results. Dark
bands of light are observed at those points. As S1 and S2 are coherent sources, the dark
bands are stationary.
Thus, when two or more coherent waves of light are superposed, the resultant effect is that at
certain points brightness is produced while at other points darkness is produced in the medium.
The phenomenon of redistribution of light energy due to the superposition of light waves from
two or more coherent sources is known as interference. The stationary bands of alternate darkness
and brightness are known as fringes. The fringe pattern is obtained only when the interfering
waves are coherent.
When incoherent waves overlap on each other, the resultant intensity is a simple addition of the
intensities in the region of overlap. Therefore, the resultant intensity due to two incoherent waves
of equal intensity is I + I = 2I and the region of overlap is uniformly bright without the formation
of fringes. It means that interference does not take place when incoherent waves superpose on
each other.
12
7.1
Conditions for observing sustained interference
• The waves from the two light sources must be of the same frequency, i.e. light source must
be monochromatic.
• The waves from the two light sources must maintain a constant phase difference, i.e., light
source must be coherent.
7.2
Geometrical path and optical path
• Geometrical Path: Light travels along a straight line path from a point A to another point B
and it is known as the path of the light. The shortest path between any two points A and B
is called the geometrical path length (GPL). GPL remains the same whether it is measured
in a vacuum or in any medium.
• Optical Path: Light travels µ times slower in a medium. Therefore, it takes µ times more
time to cover the distance AB in the medium than it takes to cover the same distance in
a vacuum. This time delay is accounted for by introducing another distance called optical
path length (OPL). It is defined as
O.P.L. = µ × G.P.L.
(7)
∆ = µL
(8)
Or,
The optical path length, ∆ signifies the number of wavelengths that are accommodated in
a given medium over the corresponding geometrical path length.
• Path Difference: Light rays travel along different paths, which may lie in the same medium
or in different media. The difference between optical paths of two rays travelling in different
directions is known as the optical path difference.
• Phase Difference: The phase of a wave arriving at a point depends on the optical path length
it traversed. We know that if a wave covers in air a distance of one wavelength, 1λ, its phase
13
changes by 2π radians. Therefore, we compute that if a wave travels a distance L in air, its
phase change is given by
δ=
2πL
λ
(9)
When the wave travels the distance L in a medium, then
δ=
2πL∆
2πµL
=
λ
λ
(10)
Comparing Eq. 9 and Eq. 10, we find that a light path of geometric length L in a medium of
refractive index µ produces the same phase change as a light path of length µL in a vacuum.
Therefore, in the study of optics we always must calculate the optical paths travelled by
light rays.
The path difference between two in-phase waves may be zero or an integral multiple of a
wavelength, λ and the path difference between two opposite-phase waves will be λ/2 or an odd
integral multiple of λ/2.
Optical path difference and the consequent phase difference may arise due to two reasons. One
reason is the difference in the optical paths and the other is due to reflections at optical interfaces.
(a) Phase difference due to optical path difference: Let us consider two sources of light S1 and
S2 , as shown in Fig. 15.
Figure 15: The light waves from two light sources meet at point P.
Let us assume that the sources are identical and produce waves of same wavelength and
that their vibrations are in the same phase at S1 and S2 . Light from these sources travel
14
in air along different paths, S1 P and S2 P; and meet at a point P. The path lengths S1 P
and S2 P are different and contain different number of waves. The geometric path difference
between the waves at P is (S2 P - S1 P) and the optical path difference = µ(S2 P - S1 P). Since
the optical paths contain different number of waves, the optical path difference will be equal
either to a few integral number of waves or an integral number of wavelengths plus a fraction
of one wavelength. This optical path difference leads to a phase difference between the waves
meeting at P. It means that though the waves started with the same phase, they arrive at P
with different phases because they travelled along different path lengths. Using Eq. 10, the
phase difference between the waves at P may be expressed as
δ=
2π
µ(S2 P − S1 P )
λ
(11)
(b) Phase difference due to reflection at boundaries of optical interfaces: Light waves may also
undergo phase change due to reflection at some point in their path. If the waves are reflected
at a rarer-to-denser medium boundary, the reflected waves suffer a phase change of π rad or
180◦ compared to the incident waves (See Fig. 16).
Figure 16: Phase difference due to reflection.
It is seen from Eq. 9 that a phase change of π rad is equal to a path change of λ/2. Therefore,
we must add (or subtract) λ/2 in the calculation of true optical path difference whenever a
reflection occurs at a denser medium.
15
7.3
Techniques of obtaining interference
The phase relation between the waves emitted by two conventional light sources fluctuates rapidly
and therefore they can never be coherent, though they are identical in all respects. However, two
coherent sources are derived from a single source by techniques, which can be divided into two
broad classes.
• Division of wave front: One of the methods consists in using a narrow slit as the source and
subsequently, the wave front is divided. For example, in the Youngs double slit experiment,
a wave front emerging from the slit S is divided into two parts by the double slit S1 S2 .
Fresnels biprism, Lloyds mirror, etc are the other examples where the division of wave front
method is used.
• Division of amplitude: In this method, amplitude of the light beam is divided by partial
reflection into two or more beams. Thin films (wedge, Newtons rings etc.), interferometers
such as Michelsons interferometer etc. utilize this method in producing interference.
7.4
Interference by division of wave front
Fresnel used a biprism to show interference phenomenon. The biprism consists of two prisms of
very small refracting angles joined base to base. In practice, a thin glass plate is taken and one of
its faces is ground and polished till a prism (Fig 17 a) is formed with an obtuse angle
Figure 17: Fresnel biprism and formation of virtual sources.
of about 179◦ and two side angles of the order of 30◦ .
When a light ray is incident on an ordinary prism, the ray is bent through an angle called the
angle of deviation. As a result, the ray emerging out of the prism appears to have emanated from
a source S’ located at a small distance above the real source, as shown in Fig. 17 (b). We say that
16
the prism produced a virtual image of the source. A biprism, in the same way, creates two virtual
sources S1 and S2 , as seen in Fig. 17 (c). These two virtual sources are images of the same source
S produced by refraction and are hence coherent.
7.4.1
Optical Path Difference
Let S1 and S2 be the two virtual images of the source S. Let d be the distance between S1 and
S2 . The fringes are formed on a screen T kept at a distance D from the biprism. The point O
on the screen is equidistant from S1 and S2 . Hence, the waves arrive from S1 and S2 arrive at
O simultaneously and the point O is always bright. The point O corresponds to the position of
central bright fringe. On both sides of O, alternate bright and dark fringes.
Figure 18: Interaction of light waves at point P.
Let P be an arbitrary point on screen (Fig. 18). Let θ be the angle that MP makes with the
horizontal line MO. Let S1 N be a normal on to the line S2 P. The distances PS1 and PN are equal.
The waves emitted at the slits, S1 and S2 are initially in phase with each other. The difference in
the path lengths of these two waves is S2 N. We assume that the experiment is carried out in air.
Therefore, the optical paths are identical with geometrical paths. The nature of the interference
of the two waves at P depends simply on how many waves are contained in the length of the
path difference S2 N. If S2 N contains an integral number of wavelengths, the two waves interfere
constructively, producing a maximum in the intensity of light on the screen at P. If it contains
an odd number of half-wavelengths, the waves interfere destructively and produce a minimum
17
intensity at P.
Let the point P be at a distance x from O (Fig. 18). Then PE = x - d/2 and PF = x + d/2.
d
d
(S2 P )2 − (S1 P )2 = [D2 + (x + )2 ] − [D2 + (x − )2 ]
2
2
(12)
(S2 P )2 − (S1 P )2 = 2xd
(13)
2xd
S2 P + S1 P
(14)
Or,
Or,
S2 P − S1 P =
We can approximate that S2 P ∼
= S1 P ∼
= D. Therefore,
Path difference = S2 P − S1 P =
xd
D
(15)
• Bright Fringes:
Bright fringes occur wherever the waves from S1 and S2 interfere constructively. The first
place this occurs is at O, the axial point. There, the waves from S1 and S2 travel the same
optical path length to O and arrive in phase. The next bright fringe occurs when the wave
from S2 travels one complete wavelength further the wave from S1 . In general constructive
interference occurs if S1 P and S2 P differ by a whole number of wavelengths. The condition
for finding a bright fringe at P is that
S2 P − S1 P = mλ
(16)
xd
= mλ
D
(17)
Using Eq. 15, we have
where m is called the order of the fringe. The bright fringe at O, corresponding to m = 0,
is called the zero-order fringe. The first-order bright fringe from the axis corresponds to m
18
= 1 and the second order bright fringe to m = 2 and so on.
• Dark Fringes:
The first dark fringe occurs when (S2 P - S1 P) is equal to λ/2. The waves are now in opposite
phase at P. The second dark fringe occurs when (S2 P - S1 P) equals 3λ/2. The mth dark
fringe occurs when
S2 P − S1 P = (2m + 1)λ/2
(18)
The condition for finding a dark fringe is
xd
λ
= (2m + 1)
D
2
(19)
The first-order dark fringe from the axis corresponds to m = 1 and the second order dark
fringe to m = 2 and so on.
• Separation between Neighbouring Bright Fringes: The mth order bright fringe occurs
when
xm =
mλD
d
(20)
and the (m+1)th order bright fringe occurs when
xm+1 =
(m + 1)λD
d
(21)
The bright fringe separation, β is given by
β = xm+1 − xm =
λD
d
(22)
The same result will be obtained for dark fringes. Thus, neighbouring bright and dark fringes
19
are separated by the same amount everywhere on the screen. The separation β is called the
fringe width.
7.5
7.5.1
Interference by division of wave amplitude
Thin film interference
An optical medium is called a thin film when its thickness is about the order of 1 wavelength of
light in visible region. Thus, a film of thickness in the range 0.5 µm to 10 µm may be considered
as a thin film. A thin film may be a thin sheet of transparent material such as glass, mica, an air
film enclosed between two transparent plates or a soap bubble. When light is incident on such a
film, a small part of it gets reflected from the top surface and a major part is transmitted into
the film. Again, a small part of the transmitted component is reflected back into the film by the
bottom surface and the rest of it emerges out of the film. A small portion of the light thus gets
reflected partially several times in succession within the film (see Fig. 19).
Figure 19: Transmission and reflection of light from thin film.
In transparent thin films, the two bounding surfaces Transmitted rays strongly transmit light
and only weakly reflect the incident light. Therefore, only the first reflection at the top surface
and the first reflection at the bottom surface will be of appreciable strength. For example, if we
20
consider a glass plate, having a refractive index 1.52, the reflectivity of the top surface is given by
1.52 − 1
r=
1.52 + 1
2
= 0.042
(23)
It means that about 4% of the incident light is reflected by the top surface of the glass plate,
while 96% of it is transmitted into the plate. Out of the light reaching the bottom surface, again
3.8% is reflected and 92% is transmitted out of the plate. Then, again out of the 3.8% of the light
0.15% is reflected at the inner boundary of the top surface and about 3.65% is transmitted out into
the air. After two reflections, the intensity will become insignificantly small. At each reflection,
the intensity and hence the amplitude of light wave is divided into a reflected component and a
refracted component. The reflected and refracted components travel along different paths and can
be brought to overlap to produce interference. Therefore, the interference in thin films is called
interference by division of amplitude.
7.5.2
Interference due to thin film of uniform thickness
Let us consider a transparent film of uniform thickness t bounded by two parallel surfaces as shown
in Fig. 20.
Figure 20: Interference due to thin film of uniform thickness.
Let the refractive index of the material be µ. The film is surrounded by air on both the sides.
Let us consider plane waves from a monochromatic source falling on the thin film at an angle of
incidence i. Part of a ray such as AB is reflected along BC, and part of it is transmitted into the
film along BF. The transmitted ray BF makes an angle r with the normal to the surface at the
21
point B. The ray BF is in turn partly reflected back into the film along FD while a major part
refracts into the surrounding medium along FK. Part of the reflected ray FD is transmitted at the
upper surface and travels along DE. Since the film boundaries are parallel, the reflected rays BC
and DE will be parallel to each other. The waves travelling along the paths BC and BFDE are
derived from a single incident wave AB. Therefore they are coherent and can produce interference
if they are made to overlap by a condensing lens or the eye.
(i) Geometrical Path Difference: Let DH be normal to BC. From points H and D onwards,
the rays HC and DE travel equal path. The ray BH travels in air while the ray BD travels
in the film of refractive index µ along the path BF and FD. The geometric path difference
between the two rays is
BF + FD BH.
(ii) Optical Path Difference: Optical Path Difference is given by
∆a = µL = µ(BF + F D) − 1(BH)
(24)
In the ∆BFD, ∠BFG = ∠GFD = ∠r
BF = F D
FG
t
BF =
=
cos r
cos r
∴
BF + F D =
2t
cos r
Also,
BG = GD
BD = 2BG
∴
BG = F G tan r = t tan r
∴
BD = 2t tan r
22
(25)
In the ∆BHD
∠HBD = (90 − i)
∠BHD = 90◦
∴
∴
∠BDH = i
BH = BD sin i = 2t tan r sin i
(26)
From Snell’s law,
sin i = µ sin r
∴
BH = 2t tan r(µ sin r) =
2µt sin2 r
cos r
(27)
Using Eq. 26 and Eq. 25 into Eq. 24, we get
∆a = µ[
∴
2t
2µt sin2 r
]−[
]
cos r
cos r
2µt
=
[1 − sin2 r]
cos r
2µt
=
cos2 r
cos r
∆a = 2µt cos r
(28)
(iii) Correction on account of phase change at reflection: When a ray is reflected at the
boundary of a rarer to denser medium, a path-change of λ/2 occurs for the ray BC (see
Fig. 20). There is no path difference due to transmission at D. Including the change in path
difference due to reflection in Eq. 28, the true path difference is given by
∴
∆t = 2µt cos r − λ/2
23
(29)
7.5.3
Conditions for Maxima (Brightness) and Minima (Darkness)
Maxima occur when the optical path difference ∆ = µλ. If the difference in the optical path
between the two rays is equal to an integral number of full waves, then the rays meet each other in
phase. The crests of one wave falls on the crests of the others and the waves interfere constructively.
Thus, when
2µt cos r − λ/2 = mλ
(30)
the reflected rays undergo constructive interference to produce brightness or maxima at the point
of their meeting.
2µt cos r = mλ + λ/2
Or 2µt cos r = (2m + 1)λ/2 Condition for Brightness
(31)
Minima occur when the optical path difference is ∆ = (2m + 1) λ/2. If the difference in the
optical path between the two rays is equal to an odd integral number of half-waves, then the rays
meet each other in opposite phase. The crests of one wave falls on the troughs of the others and
the waves interfere destructively. Thus, when
2µt cos r − λ/2 = (2m + 1)λ/2
the reflected rays undergo destructive interference to produce darkness. Therefore,
2µt cos r = (m + 1)λ
The phase relationship of the interfering waves does not change if one full wave is added to or
subtracted from any of the interfering waves. Therefore (m + 1)λ can be as well replaced by mλ
for simplicity in expression. Thus,
2µt cos r = mλ Condition for Darkness
24
(32)
7.5.4
Interference at wedge-shaped film
A wedge is a thin film of varying thickness having a zero thickness at one end and progres- sively
increasing to a particular thickness at the other end. A thin wedge of air film can be formed by
two glass slides resting on each other at one edge and separated by a thin spacer at the opposite
edge.
When the light is incident on the wedge from above, it gets partly reflected from the glass-to-air
boundary at the top of the air film. Part of the light is transmitted through the air film and gets
reflected partly at the air-to-glass boundary, as shown in Fig. 21.
Figure 21: Interference at wedge-shaped film.
The two rays BC and FE, thus reflected from the top and bottom of the air film, are coherent as
they are derived from the same ray AB through division of amplitude. The rays are close enough
if the thickness of the film is of the order of a wavelength of light. For small film thickness the rays
interfere producing darkness or brightness depending on the phase difference. The thickness of
the glass plates is large compared with the wavelength of the incident light. Hence, the observed
interference effects are entirely due to the wedge-shaped air film.
The optical difference between the two rays BC and FE is given by
∆ = 2µt cos r − λ/2
(33)
where λ/2 takes account the gain of halfwave due to the abrupt jump of π radians in the phase of
the wave reflected from the bottom boundary of air-to-glass.
Maxima occur when the optical path difference ∆ = mλ. If the difference in the optical
path between the two rays is equal to an integral number of full waves, then the rays meet each
other in phase. The crests of one wave falls on the crests of the others and the waves interfere
25
constructively. This needs that
2µt cos r = (2m + 1)λ/2
(34)
Minima occur when the optical path difference is ∆ = (2m + 1)λ/2. If the difference in the
optical path between the two rays is equal to an odd integral number of half-waves, then the rays
meet each other in opposite phase. The crests of one wave fall on the troughs of the others and
the waves interfere destructively. It needs that
2µt cos r = mλ
(35)
The Fringe width, β, is given as
β=
λ
2µ tan θ
(36)
∴
β=
λ
2µθ
(37)
For small values of θ, tan θ ≈ θ.
According to Eq. 37, an increase in the angle θ makes the fringes move closer. At an angle θ ≈
1◦ , the interference pattern vanishes. On the other hand, if θ is gradually decreased, the fringe
separation increases and ultimately the fringes disappear since the faces of the film become parallel
at θ = 0◦ .
8
Newton’s rings
Newtons rings are another example of fringes of equal thickness. Newtons rings are formed when
a plano-convex lens L of a large radius of curvature placed on a sheet of plane glass AB. The
combination forms a thin circular air film of variable thickness in all directions around the point of
contact of the lens and the glass plate. The locus of all points corresponding to specific thickness
of air film falls on a circle whose centre is at O. Consequently, interference fringes are observed in
the form of a series of concentric rings with their centre at O (Fig. 22).
26
Figure 22: Newton’s rings.
Newton originally observed these concentric circular fringes and hence they are called Newtons
rings.
The experimental arrangement for observing Newtons rings is shown in Fig. 23.
Figure 23: Experimental arrangement to observe Newton’s rings.
Monochromatic light from an extended source S is rendered parallel by a lens L’. It is incident
on a glass plate inclined at 45◦ to the horizontal, and is reflected normally down onto a planoconvex lens placed on a flat glass plate. Part of the light incident on the system is reflected from
the glass-to-air boundary, say from point D (Fig. 24).
27
Figure 24: Reflection from Air film.
The remainder of the light is transmitted through the air film. It is again reflected from the
air-to-glass boundary, say from point J. The two rays reflected from the top and bottom of the air
film are derived through division of amplitude from the same incident ray CD and are therefore
coherent. The rays 1 and 2 are close to each other and interfere to produce darkness or brightness.
The condition of brightness or darkness depends on the path difference between the two reflected
light rays, which in turn depends on the thickness of the air film at the point of incidence.
8.1
Condition for Bright and Dark Rings
The optical path difference between the rays is given by ∆ = 2µt cos r - λ/2. Since µ = 1 for air
and cos r = 1 for normal incidence of light,
∆ = 2t − λ/2
(38)
Intensity maxima occur when the optical path difference ∆ = mλ. If the difference in the optical
path between the two rays is equal to an integral number of full waves, then the rays meet each
other in phase. The crests of one wave falls on the crests of the others and the waves interfere
constructively. Thus, if 2t - λ/2 = mλ
2t = (2m + 1)λ/2
28
(39)
bright fringe is obtained.
Intensity minima occur when the optical path difference is ∆ = (2m + 1)λ/2. If the difference
in the optical path between the two rays is equal to an odd integral number of half-waves, then the
rays meet each other in opposite phase. The crests of one wave fall on the troughs of the other
and the waves interfere destructively. Hence, if
2t − λ/2 = (2m + 1)λ/2
Or,
2t = mλ
8.2
(40)
Circular Fringes
In Newtons ring arrangement, a thin air film is enclosed between a plano-convex lens and a glass
plate. The thickness of the air film at the point of contact is zero and gradually increases as we
move outward. The locus of points where the air film has the same thickness then fall on a circle
whose centre is the point of contact. Thus, the thickness of air film is constant at points on any
circle having the point of lensglass plate contact as the centre. The fringes are therefore circular.
8.3
Radii of Dark Fringes
Let R be the radius of curvature of the lens (Fig. 25). Let a dark fringe be located at Q. Let the
thickness of the air film at Q be PQ = t. Let the radius of the circular fringe at Q be OQ = rm .
Figure 25: Planoconvex lens and glass plate.
29
By the Pythagorus theorem,
P M2 = P N2 + MN2
2
R2 = rm
+ (R − t)2
∴
∴
2
= 2Rt − t2
rm
(41)
2 ∼
rm
= 2Rt
(42)
As R >> t, 2Rt >> t2 ,
∴
The condition for darkness at Q is that
2t = mλ
2
rm
=∼
= mλR
∴
Or,
√
rm =
mλR
(43)
Thus, the radius of the mth dark ring is proportional to square root of wavelength.
The diameter of mth dark ring is given
√
Dm = 2rm = 2 mλR
9
(44)
Anti-Reflecting Coatings
Optical instruments such as telescopes and cameras use multicomponent glass lenses. When light
is incident on the lens, part of the incident light is reflected away and that much amount of light
is lost and wasted. When more surfaces are there, the number of reflections will be large and
the quality of the image produced by a device will be poor. In case of solar cells, which operate
on sunlight (daylight), the electrical energy produced will be less because of the loss of part of
30
light energy due to reflection, at the cell surface. It is found that coating the surface with a thin
transparent film of suitable refractive index can reduce such loss of energy due to reflections at
surface. Such coatings are called antireflection coatings. Thus,
Antireflection (AR) coatings are thin transparent coatings of optical thickness of one quarter
wavelength given on a surface in order to suppress reflections from the surface.
Alexander Smakula discovered in 1935 that the reflections from a surface can be reduced by
coating the surface with a thin transparent dielectric film.
A thin film can act as an AR coating if it meets the following two conditions:
• Phase condition: The waves reflected from the top and bottom surfaces of the thin film
are in opposite phase such that their overlapping leads to destructive interference.
• Amplitude condition: The waves have equal amplitudes.
The above conditions enable us determine respectively (a) the required thickness of the film and
(b) the refractive index of the material to be used for forming the film.
(i) Phase condition and minimum thickness of the film: Let the thickness of the film
be t and the refractive of the film-material be µf . The phase condition requires that the
waves (ray 1 and ray 2) reflected from the top and bottom surfaces of thin film be 180◦ out
of phase. It requires that the optical path difference between the two rays must equal one
half-wave or an odd number of half-waves.
Figure 26: Reflection from antireflection coatings.
Referring to Fig. 26, the optical path difference between ray 1 and ray 2 is
∆ = 2µf t cos r − λ/2 − λ/2
31
The first λ/2 corresponds to the π change at the top surface of the film (air-to-film boundary)
and the second λ/2 to the π change that occurs at the film-to glass boundary because
µf < µg . If we assume normal incidence of light, cos r = 1 and the above equation reduces
to
∆ = 2µf t − λ = 2µf t
We wrote the above equality remembering that an addition of a full wave or subtraction of
a full wave from a train of waves does not affect the original phase relation. The ray 1 and
ray 2 interfere destructively if the optical path difference satisfies the condition that ∆ =
(2m+1)λ/2.
Thus, it requires that 2µf t = (2m+1)λ/2.
For the film to be transparent, its thickness should be a minimum, which happens when m
= 0.
2µf tmin = λ/2
Or,
tmin =
λ
(µf < µg )
4µf
(45)
It means that the optical thickness of the AR coating should be of one-quarter wavelength.
Such quarter-wavelength coatings suppress the reflections and cause the light to pass into
the transmitted component.
(ii) Amplitude condition: The amplitude condition requires that the amplitudes of reflected
rays, ray 1 and ray 2 are equal. In this condition,
µf =
√
µg
(46)
It implies that the refractive index of thin film should be less than that of the substrate and
possibly nearer to its square root
In case of glass, if we take µg = 1.5, µf =
√
µg = 1.22.
32
10
Ultrasonic interferometer
The human ear is sensitive to sound waves of frequencies ranging from 16 Hz to 20 kHz. Waves
of frequencies beyond the upper audible limit (f > 20kHz) are called ultrasonic waves.
The velocity of ultrasonic waves is determined using an ultrasonic interferometer.
The interferometer consists of an ultrasonic generator having a liquid cell connected to its tank
circuit. The cell, T, is a vertical cylindrical tube filled with the liquid medium under test (Fig. 27).
Figure 27: Experimental set up for ultrasonic interferometer.
A piezoelectric crystal C is mounted at the bottom of the cell. Reflector, R is a metallic
plate, mounted at the top end of the cell and it can be moved parallel to itself with the help of
a micrometer screw. The surface of crystal C and the reflector are made exactly parallel to each
other.
When the piezoelectric crystal, C is excited ultrasonic waves of known frequency are produced
and propagate through the liquid. These waves are reflected at the reflector R and travel back to
C. The position of the reflector can be adjusted such that the forward and backward waves form a
standing wave pattern in the medium. As the reflector moves, the reading of the microammeter in
the circuit fluctuates. The readings of microammeter are recorded and plotted against the position
of R. The plot will pass through maxima and minima as shown in Fig. 28.
33
Figure 28: Ammeter readings as a function of position R.
The distance between two consecutive minima or maxima is λu /2. From this, we get the value
of the wavelength λu of the ultrasonic wave in the medium.
Then, knowing the value of the frequency of the ultrasonic waves used, the velocity of the
ultrasonic wave in the liquid vu is computed from the relation
vu = f λu
11
(47)
DIFFRACTION
Diffraction phenomenon is a common characteristic of all kinds of waves. It is a matter of common
experience that sound waves readily bend around walls and buildings. When waves pass near an
obstacle (barrier), they tend to bend around the edges of the obstacle. The bending of waves
around an obstacle and deviation from a rectilinear path is called diffraction.
11.1
Dependence of the phenomenon on wavelength
Fig. 29 illustrates the passage of waves through an opening. When the opening is large compared
to the wavelength, the waves do not bend round the edges.
34
Figure 29: (a) λ << d (b) λ = d (c) λ >> d.
When the opening is small, the bending effect round the edges is noticeable. When the opening
is very small (of the order of one wavelength), the waves spread over the entire surface behind the
opening. The opening acts as an independent source of waves, which propagate in all directions.
The diffraction effect is observable quite close to the opening when the size of the opening is
very small. When the opening is large, diffraction effect is observed at greater distances from the
opening. In general diffraction of light waves become noticeable only when the size of the obstacle
is comparable to a wavelength of light.
11.2
Distinction between interference and diffraction
The main differences between interference and diffraction are as follows:
35
11.3
Types of diffraction
The diffraction phenomena are broadly classified into two types: Fresnel diffraction and Fraunhoffer diffraction.
1. Fresnel diffraction: In this type of diffraction, the source of light and the screen are
effectively at finite distances from the obstacle (see Fig. 30).
Figure 30: Condition for Fresnel diffraction.
Lenses are not used to make the rays parallel or convergent. The incident wave front is not
planar. As a result, the phase of secondary wavelets is not the same at all points in the
plane of the obstacle. The resultant amplitude at any point of the screen is obtained by the
mutual interference of secondary wavelets from different elements of unblocked portions of
wave front. It is experimentally simple but the analysis proves to be very complex.
2. Fraunhoffer diffraction: In this type of diffraction, the source of light and the screen are
effectively at infinite distances from the obstacle (see Fig. 7.4b).
36
Figure 31: Condition for Fraunhoffer diffraction.
The conditions required for Fraunhoffer diffraction are achieved using two convex lenses, one
to make the light from the source parallel and the other to focus the light after diffraction
on to the screen. The incident wave front as such is plane and the secondary wavelets, which
originate from the unblocked portions of the wave front, are in the same phase at every point
in the plane of the obstacle. The diffraction is produced by the interference between parallel
rays, which are brought into focus with the help of a convex lens. This problem is simple to
handle mathematically because the rays are parallel.
11.4
Condition of Maxima and Minima
Fig. 32 shows a plane wave front (parallel rays) incident on the slit AB. A small part AB is sliced
off from the incident wave front.
37
Figure 32:
Fraunhofer diffraction at a single slit. The rays parallel to axis come to focus at P
giving a bright band.
According to Huygens principle, each point on AB acts as a source of secondary wavelets. It
would then be appropriate to replace the wave front AB with a string of point sources. As all
points on AB are in phase, the point sources will be coherent. Hence, light from one portion of
the slit can interfere with light from another portion and the resultant intensity on the screen will
depend on the direction q. The secondary wavelets travelling parallel to OP come to focus at P.
The waves from points equidistant from O and situated in the upper and lower halves OA and
OB start in phase. They will travel the same distance in reaching P. The optical path difference
is therefore zero and the waves will be in phase at P. They reinforce each other to produce an
intensity maximum at P. It is at the centre of the diffraction pattern and is called zero order
central maxima.
For any other point like Q on the screen (Fig. 33), the light from different parts of the aperture
travels different distances
Figure 33: Conditions at the first minimum of diffraction pattern-Each point on the sliced portion
AB of the wave front acts as a point source. Any two waves that originate at points separated by
d/2 distance are 180◦ out of phase and interfere destructively.
Now consider the secondary waves travelling in a direction making an angle q with OP. These
38
secondary waves are brought to focus by the lens at a point Q, which will have a maximum
or minimum intensity, depending on the path difference between the waves arriving at Q from
different points on the wave front AB. It is convenient to divide the wave front AB into two halves
AO and OB. A line AM is drawn perpendicular to the direction of the diffracted rays. Waves ON
and BM are in phase at the slit. Wave BM travels farther than ON.
The path difference between these wavelets = ON = (d/2) sin θ.
If ON = λ/2, the two waves interfere destructively and produce darkness at Q. This is true for
any two waves that originate at points separated by (d/2), as the path difference between from
such points will be λ/2. For every point in the upper half OA, there is a corresponding point in
the lower half OB. The path difference between the waves from these corresponding points will be
λ/2. Hence, the waves from the upper half AO interfere destructively with waves from the lower
half OB, if
d
λ
sin θ =
2
2
i.e.,
sin θ = λ/d
(48)
Therefore, the intensity at Q is zero and a dark band called the first order minimum is produced at
Q. A similar dark band occurs at Q’ below P at an angular distance θ governed by the diffraction
pattern equation 48. It is also called the first order minimum. We may divide the slit into four
parts, six parts and so on. Arguments similar to the above show that a dark band occurs whenever
sin θ = 2λ/d, 3λ/d, 4λ/d, 5λ/d, .... etc.
They are known as second order minimum, third order minimum, etc. In general minima
appear if the following condition is satisfied:
sin θm = mλ/d Condition for minimum
(49)
where m = 1, 2, 3,...
More generally, the condition for minima may be expressed as
d sin θm = ±mλ
39
(50)
In addition to the central maximum there are secondary maxima, which lie in between the secondary minima on either side of the central maximum. These are located in a direction in which
the path difference ON is an odd multiple of λ/2. Hence for secondary maxima
ON = d sin θ = (2m + 1)λ/2
In general the secondary maximum are given by
sin θm = (
2m + 1 λ
)
d
2
Condition for maxima
(51)
Thus, the diffraction pattern due to a single slit consists of a central bright maximum flanked by
secondary maxima and minima on both the sides.
11.5
Intensity Distribution in Diffraction Pattern Due to a Single Slit
Let a plane wave be incident normally on a long narrow slit of width d. Let us imagine that this
slit width d is divided into N parallel strips of each of width ∆x.
Intensity distribution is given by
I = I0 [
sin α 2
]
α
(52)
where I0 is the intensity of principal maximum at θ = 0.
Thus, the intensity at any point on the screen is proportional to ( sinα α )2 .
α=
π
a sin θ
λ
(53)
2
Thus, the value of α depends on the angle of the diffraction θ. The value of ( sinα2 α ) for different
values of θ gives the intensity at the point under consideration. Fig. 34 represents the intensity
distribution.
40
Figure 34: Intensity distribution.
It is seen that most of the light is confined to the central band and its intensity is far grater
than that of any other maxima. The intensity of the secondary maxima falls off rapidly as one
moves away from the centre. The intensity of the first secondary maximum is about 1/22 and
that of the second is 1/61 of the intensity of the principal maximum. The secondary maxima are
too faint to be visible ordinarily.
11.6
Diffraction grating
Let us now consider the diffraction pattern produced by N-slits, each of width a. The separation
between consecutive slits is d = a+b, where a is the width of the open portion and b is the width of
the opaque portion. Such a device consisting of a large number of parallel slits of equal width and
separated from one another by equal opaque spaces is called a diffraction grating. The distance d
between the centres of the adjacent slits is known as the grating period.
Let us consider the plane transmission grating held normal to the plane of the page (Fig. ??(a))
and represented by the section ABC...H.
41
(a)
(b)
Figure 35: Diffraction grating.
Let the width of the transparent portion AB be equal to a and opaque portion BC be equal to
b. The distance (a + b) = d and is called the grating constant or grating period. Let a parallel
beam of monochromatic light of wavelength λ be incident normally on the grating surface. Then
all the secondary waves travelling in the same direction as that of the incident light will come to
focus at the point P on the screen. The screen is placed at the focal plane of the collecting lens,
L. The point P where all the secondary waves reinforce one another corresponds to the position
of the central bright maximum.
Now let us consider the secondary waves travelling in a direction inclined at an angle θ with the
direction of the incident light (Fig. 35(b)). The waves travel different distances and it is obvious
that there is a path difference between the waves coming out from each slit and bending at an
angle θ. These secondary waves come to focus at the point Q on the screen. The intensity at Q will
depend on the path difference between the secondary waves originating from the corresponding
points A and C of two neighbouring slits. In the Fig. 35(b), AB = a and BC = b. The path
difference between the secondary waves starting from A and C is equal to AC sin θ.
But AC = AB + BC = a + b
Path difference = AC sin θ = (a + b) sin θ
The point Q will be of maximum intensity if this path difference is equal to integral multiples
of λ. It means that all the secondary waves originating from the corresponding points of the
neighbouring slits reinforce one another and the angle θ gives the direction of maximum intensity.
In general
42
(a + b) sin θm = mλ
(54)
where θm is the direction of the mth principal maximum. If (a + b)sin θ = λ, we obtain
maximum intensity at Q. When (a + b)sin θ = 2λ, there will be again a maximum and so on.
Between the central maximum P and the first maximum at Q there will be minimum intensity
and so on.
Similar maxima and minima are obtained on the other side of central maximum. Thus, on
each side of the central maximum at P, principal maxima and minimum intensity are observed
due to diffracted light. The position of mth minimum is given by
(a + b) sin θm = (2m + 1)λ/2
43
(55)
A
Refractive index
A material through which light propagates is called an optical medium. Depending on the passage
of the light through the materials, they are divided into the following categories:
• Transparent materials: The materials which allow complete transmission of light are called
transparent materials. The examples are air, water, glass, etc.
• Translucent materials: The materials which partially allow the transmission of light but
scatter most of it are called translucent materials. The examples are butter paper, oiled
paper, plastic thin sheets, etc.
• Opaque materials: The materials which do not allow transmission of light are called opaque
materials. We cannot see through opaque materials. The examples are concrete, wood,
metals, etc.
Our study is related to the propagation of light through transparent media. When light travels
through any medium, its velocity reduces. The dependence of velocity of light on the medium is
characterized by the quantity called optical density, which is expressed in terms of the absolute
refractive index of the medium. The absolute refractive index µ of an optical medium is defined
as the ratio of the velocity of light in a vacuum to the velocity of the light in the medium.
µ=
Velocity of light in a vacuum
c
=
Velocity of light in the medium
v
(56)
The refractive index is a dimensionless number greater than unity since v is always less than c.
The refractive index is 1 for a vacuum and it is equal to 1.0003 for air. Therefore, the velocity
of light in air is considered to be equal to c for all practical purposes. An optical medium with a
relatively high refractive index is said to have a high optical density and one with a low refractive
index has a low optical density.
We know that
c = νλ
(57)
where c is the velocity of light, ν is the frequency and λ is the vacuum wavelength of light. As
per eq. 57, a decrease in the velocity of light in a medium can be attributed to a drop in either
wavelength or frequency, or both. However, in view of the law of conservation of energy, the
44
frequency must remain constant. It implies that the decrease in velocity of light in a medium
must lead to a decrease in the wavelength of light. Therefore, the velocity of light in an optical
medium may be expressed as
v = νλm
(58)
where λm is the wavelength of light in the medium. Using Eq. 57 and Eq. 56, we get
µ=
λ
λm
45
(59)