Coherence and Lasers
Paper: Optics
Lesson: Coherence and Lasers
Author: Dr. D. V. Chopra
College/Department: Associate Professor (Retired),
Department of Physics and Electronics, Rajdhani College,
University of Delhi
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Coherence and Lasers
Objectives: After studying this chapter you should:
1. Be able to understand the meaning of ‘coherence of a wave’ and its criteria, viz,
temporal and spatial coherence of a wave
2. Be able to explain experiment determination of coherence length and coherence
time
3. Be able to understand the meaning of lateral coherence width
4. Be able to describe experiment on coherence using a laser beam
5. Be able is describe different types of lasers, their properties and applications
6. Be able to know about holography along with its theory as interference between
two plane waves and its applications
7. Be able to solve problems involving coherence and lasers
1 Emission of Photon by Matter:
In this chapter the discussion applies to thermal sources of light, which are non-laser
sources. According to atomic theory of mater, an atom consists of a positively charged
nucleus with electrons moving orbits round it. According to Bohr’s theory of hydrogen
atom, a photon is emitted when an electron jumps
from higher orbit to a lower orbit (which is closer to
the nucleus) or if the electron moves from a point
quite outside the atom to one of its orbits.
The
emission of photon then ceases until another available
place has been made and another electron moves in.
Experimental evidence shows that the duration of an
109 s
unbroken chain of light waves from a source which is
8
9
monochromatic only has a life of about 10 or 10 s.
Then other trains of waves are produced and there is
Fig. 1 A Pulse
no constant phase difference between successive
wave-trains. The us we get a pulse of limited duration,
(as shown in Fig. 1) instead of a continuous wave train
extending from  to +. In a conventional light
source, light is emitted from a very large number of
independent atoms, each emitting a pulse of duration
109 s. Even if the atoms were emitting under similar conditions, light waves from different
atoms would differ in their initial phases. If this light were to illuminate two slits, light
coming out from two holes would have a constant phase relationship for a time duration of
about 109 s. Hence, interference pattern will keep on changing every 10 9 second
duration. Human eye can notice intensity changes which last at least for
of a second
which is equal to persistence of vision, and hence a uniform intensity on the screen would
be observed. However, if we have a camera which can photograph in a time less than
of a second, then the photographic film will record an interference pattern producing
maxima and minima. Consequently it is concluded that, since there are multiples of wavetrains arriving at the screen with no regular phase difference, the screen is illuminated
without any visible redistribution of the light intensity, i.e., no interference effects are
observed. In actual practice it is found essential for the two sources (or two slits in Young’s
double slit experiment) to be identical in all respects on account of the relatively long time
required for an interference effect to be recorded, either on the retina or photographically.
Thus it means that the wave trains necessary to produce interference effects must have a
common origin; such wavetrains are said to be coherent.
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Coherence and Lasers
2 (A)Interference between independent sources of light:
Equation of a wavetrain is
This equation represents a continuous train of waves stretching form
and proceeding in the positive x-direction with speed v.
This is shown in Fig. 2. If we plot time t instead of distance x, then x-axis in Fig. 2 is
replaced
by
time
t-axis.
In
that
case,
time
t
extends
from
t =  to t = +. Fig 2 shows any sine or cosine wave between  to +. Eq,(2-1) may
also be written
as, for two
waves having
y
same
frequency ,
or
t
Fig. 2 A continuous wave-train extending from  to +
Suppose that at a given point on the screen one has superposition of these two waves.
Then, the resultant intensity at that point is governed by
where A is resultant amplitude and A2, its intensity. The intensity is independent of time t.
This gives the time-average of the resultant intensity. It phase difference
remains
constant, amplitude A will also be a constant. That is, the two waves are not in the same
phase, but the phase difference
between them is constant.
The essential and the most sufficient condition for interference is that there must be a
constant phase difference between the two waves emitted by the two sources (real or
virtual). Such sources, having a constant phase difference and same frequency, are called
coherent sources.
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Sum of the separate intensities on superposition due to both waves (given by Eq 2-2
and 2-3) at a point on the screen,
This value is different from the time-average of the resultant intensity given by Eq
(2-3). When intensity on the screen varies according to Eq 2-3, the two waves on
superposition are said to interfere, and they are called coherent. When the intensity on the
screen varies as Eq 2-4, then the two waves do not interfere, and they are called
incoherent. Thus, in other words, when the two intensities given by Eqs 2-3 and 2-4 are
same, the two waves do not interfere. The two interfering wave trains are always derived
from the same source of light so that they may have a constant phase difference. A steady
(or constant or fixed or regular) phase difference gives intensity according Eq. (2-3).
An experimental observation shows that it is impossible to obtain interference fringes
from two separate sources, such as two identical sodium vapour lamps placed side by side.
Of course, two identical lamps ensure waves of the same amplitude and frequency, thus
fulfilling some conditions of interference for producing maxima and minima; even then, we
do not observe interference maxima and minima. This failure is caused by the fact that the
light from any one source is not an infinite train of waves given by Eq. 2-1, as explained
using the structure of atom in section 1. Consequently, the screen is illuminated without
any visible interference maxima and minima, as for no constant phase difference,
will vary between zero and 2, and thus
will vary rapidly in the range +1 and
1. Thus the mean value of
will be zero and Eq.(2-3) will give the mean value
of intensity as
. This value of intensity is same as that of Eq. 2-4, which indicates
that there is no interference. Since every source exhibits random changes of phase, waves
from different sources (or different points of an extended source) are inchoherent.
Thus, it is found that coherent waves interfere and incoherent waves do not. As in
the case of Fresnel’s biprism, the two virtual (coherent) sources S 1 and S2 always have a
point-to-point correspondence of phase, since they are both derived from the same source.
If the phase of the light from a point in S 1 suddenly shifts, that of the light from the
corresponding point in S2 will shift simultaneously. The result is that the phase difference
between any pair of points in the two sources always remains constant.
2(B):
The above discussion applies to thermal sources of light and does not apply to
laser sources and microwaves. In the case of laser sources, or microwaves, the special
arrangements (like Fresnel’s biprism) are not necessary for producing coherent sources of
light. Laser sources themselves are highly coherent, highly directional and monochromatic
beam of light.
Whereas, microwaves which are radiowaves of a few centimeters
wavelength, are produced by an oscillator which is capable of emitting a continuous wave,
the phase of which remains constant over a time long compared with the duration of an
observation.
Thus, two independent microwave sources of the same frequency are
therefore coherent and may be used to demonstrate interference.
2(C) If waves from two coherent sources (such as produced by Fresnel’s biprism) traverse
widely different optical path lengths before being superimposed, they must have been
emitted by the source at widely different times. This follows from the Bohr’s theory of H –
atom. Thus, there will not be any fixed phase relation between these two coherent sources
– they will be incoherent if path difference between two coherent sources is large. In such a
case, there will not be interference maxima and minima on the screen.
3.
Partially Coherent Sources:
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Consider Young’s double slit experiment, as shown in Fig. 3. Point O on the screen is
a point of zero path difference because S1O = S2O. As we move away from O on the screen
and reach point P, the path difference between the two waves is increased from zero. The
same set of atoms is assumed to be illuminating both the slits S 1 and S2 simultaneously,
giving rise to pulses of length L as shown in Fig. 3 and thus producing interference. As we
mover further away from P, the path difference may be increased to such an extent that the
abrupt changes in one wave occur about half – way between those of the other. Then for
half the time the superposed waves are derived from the same wave train (i.e. from the
same set of atoms in S producing light) and for half the time they belong to different wave
– trains. This means that in the former case they are coherent whereas in the latter case
they are incoherent. When averaging over a relatively large period of time one could say
that the waves are half – coherent and half – incoherent, i.e., they are partially coherent.
In such a case, the wave – trains from S1 and S2 partially overlap and the overlapping
portions interfere. The effect of interference is less marked than the wavetrains overlap
alongtheir entire length L as in the case of coherent sources.
Let  be the path differences and
two interfering waves from S1 and S2.
, the corresponding phase difference between the
P
S1
Intensity at a point on the screen
As  increases
decreases.
Thus, as the path difference is increased in steps,
the waves may be said to be coherent over a smaller
fraction of the total time. A stage will reach when
the time average of the intensity gradually changes
from
to
i.e. from
coherent to incoherent waves.
O
S
S2
Fig.3 Young’s double slit
experiment
A truly simple harmonic wave extends to
infinity, and hence one can never have a source of truly monochromatic light. In our
laboratory, a single line in aline specxtrum may correspond to very long wave – trains and it
is seen that this line has a spread called width of spectral line which is equivalent to a
mixture of monochromatic waves, most of which have wavelengths close to a mean value.
If  is the mean wavelengths and  its spread, then this line is equivalent to (). Note
that  is not wavelength difference; it is wavelength spread of single line, also called line –
width.
4
Coherence of a Wave:
Light consists of radiation from individual atoms. Each atomic emission event
produces a train of wave oscillation that has finite length and time duration. In ordinary
sources, these emissions occur randomly from atom to atom. [An exception is the class of
devices that exhibit laser action]. Consequently, the resulting wave trains have random
phase. So far, we have seen that two light waves would interfere only if the wave trains
overlap in space, provided these two waves are coherent. Now, in the present section, we
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Coherence and Lasers
consider the coherence of a single wave. No actual light source, however, emits a perfectly
coherent wave.
A perfectly coherent wave is a wave which appears to be a pure sine wave for an
infinitely large period of time or in an infinitely extended space.
Fig 4 is a schematic representation of a wave train with finite length L. In this
figure, this wave train consists of only a few oscillations, but an actual wave train would
typically have a few million oscillations within its length L. Instead of distance x, one can
plot time t, then this wave train has finite time duration .
Y
Within the length L, the wave train
is essentially sinusoidal. In order for an
oscillation to be a pure monochromatic
B
O A
wave, the frequency must be the same
x
wherever and whenever it is measured;
L
such a wave must have an infinite length.
or
Any wave with finite length, such as the
or
t
wave train in Fig 4, consists of the
()
superposition of waves with slightly
Fig.4 Schematic representation of a wave
different frequencies distributed in such a
train – also called a pulse – with finite
way
that
there
is
a
constructive
interference in the region of length L and
length L or with finite time duration 
destructive interference everywhere else.
[Compare Fig. 4 with that for formation of beats due to superposition of waves slightly
differing in their frequency]. Such waves are said to be quasi – monochromatic (which
means almost monochromatic). The wave trains from ordinary light sources have lengths
that are typically a fraction of a meter to several meters. The loss of coherence along the
path of a wave from a source which is nearly, but not quite, monochromatic can be
understood by supposing that the wave is made up of a large number of wave trains of
finite length, and that a large number of such wave trains passs a point in the time taken to
make an observation of intensity. There is a typical coherence length in the light beam,
which is the length of an elementary wave train. There is also a typical coherence time,
which is the time for the wave train to pass any point.
For any source the average length of a wave train is called the coherence length L,
and the time taken by light to travel this distance (i.e. the interval of time during which the
mean wave train is emitted) is called coherence time . The time during which the field (or
wave train) remains sinusoidal is called coherence time . These are shown in Fig. 4.
5
Two criteria of coherence of a wave. Temporal and spatial coherence of a
wave.
There are two different criteria to measure the coherence of a wave. In a pulse or
elementary wave train such as shown in Fig. 4, if there is a definite relationship between the
phase of the wave at a given time and at a certain time later, then it is referred to as
temporal coherence; and if there is a definite relationship between the phase of the wave at
a given point and at a certain distance away, then it is referred to as spatial coherence.
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6
Temporal Coherence:
An ideal sinusoidal wave as a function of time is shown in Fig. 5(a). It extends from
t =  to t = +. This is a perfectly coherent light
having constant amplitude A while its phase would
y
vary linearly with time.
Actual light wave emitted by an atom is
shown in Fig. 5(b).
This is a pulse of short
duration about 1010 s for sodium atom. This is
because a single photon is emitted when one
electron jumps from higher to lower orbit. There
are billions and billions of atoms emitting such
pulses which do not have any fixed phase
relationship. Thus the wave is sinusoidal only for
short duration (about 109s) after which phase
changes with the emission of new pulse (as shown
by dotted lines in Fig. 5). Coherence time  of the
light beam is defined as the average time – interval
for which definite phase relationship exists. The
coherence length L of the light beam is the
distance for which definite phase relationship exists
(which means the wave remains sinusoidal). If c is
speed of light, the spatial dimension is
A
t
(a)
y

t
(b)
Fig.5
(a)Ideal sinusoidal wave
(b) A pulse
where T is time period of the sinusoidal wave. Duration of an unbroken chain of waves is
about 109s which is .
Coherence length,
For interference to observe, the overlapping of waves must occur. Thus, it is impossible to
observe interference for large path differences (i.e. for path difference greater than 30 cm)
For interference to take place, the path difference  should be much less than coherence
length L. So, the time t should be less than coherence time . Interference of light beams
is impossible if  > L or t > .
The coherence time  is also referred to as temporal coherence of the beam.
7.
Experimental determination of Coherence Length(L)
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Coherence length L and hence coherence time  can be measured by means of the
Michelson interferometer which is shown in Fig. 6 Mirror M 1 is movable while mirror M2 is
fixed. When M1 and M2 are at right angles to each other, mirror image
of M2 is seen in
mirror M1 and
is parallel to M1. Then, the interferometer is adjusted for circular fringes.
If d is the distance M1
and
then the path difference between the interfering beams is
We know that two light waves produce a stationary interference fringes only if there
is
a
definite
amplitude
and
phase
relationship between them.
M1
Movable
Mirror
If path difference 2d is less than
coherence length L, then there will be a
definite phase relationship between the two
interfering beams marked (A) and (B) in
Fig. 6. On the other hand if 2d > L, there
will not be any definite phase relationship
between the two interfering beams.
In
other words, if 2d < L, sustained circular
fringes would be observed, whereas if 2d<
L, no sustained interference pattern would
be obtained.
In the given Michelson
interferometer experiment,
d
(A)
l1
Fixed Mirror
S
G1
(B)
G2
l2
M2
Screen or eye
Fig.6 The Michelson Interferometer
In other words, interference fringes will appear only if the difference in optical paths is less
than the coherence length. Experiment is first set at G1M2=G2M2 which means zero path
difference. As mirror M1 is continuously moved, path difference increases. We observe dark
and bright circular fringes with contrast fading away as the path difference is continuously
increased. Finally, the circular fringes vanish. At this point of disappearance of fringes, the
coherence length is equal to the path difference . Hence,
This point of disappearance of fringes cannot be determined with exactness. Thus,
coherence length L is not a very well defined quantity as there is no definite cut off point at
which circular fringes suddenly become indistinctness. For sodium yellow light,
cm.
For cadmium red light,
cm, which is ten times larger than sodium yellow light. For
laser light, coherence length L can exceed a few kilometers.
8
Determination of Coherence time 
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Coherence time  is time interval of coherence such that when t << , phase
correlation exists; but when
t >> , phase correlation does not exist. Thus the
indistinctness of circular fringes can be interpreted to mean that time difference
is
comparable to . Time difference,
For example, sodium yellow light has
cm. Therefore coherence time,
Corresponding number of oscillations over a time  for which the field remains coherent is
given by
This is the number of oscillation in the wave train such as shown in Fig.4.
As explained in section 3 and 4, a single spectral line due to a quasi –
monochromatic source has a wavelength spread () centered at mean wavelength ().
Due to this spread we observe broadening of spectral line corresponding to wavelength
spread , the line has a frequency spread, denoted by .
It follows from Fig 4, frequency of light emitted ,
when n is the number of waves in AB
Differentiating,
Here
is the uncertainty in determining the frequency called the frequency spread, and
is the uncertainty in counting the number of oscillations. At most we can count the number
of waves to an accuracy of
waves at each end A and B of the pulse (see Fig. 4).
Thus,
 is the time for which the electron had emitted radiation and it is the same as coherence
time. Dominant frequency
is defined as
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The radiation has a band of width
, called frequency broadening centered about the
dominant frequency . This radiation is quasi – monochromatic and the frequency spread
obeys an order of magnitude relation:
In real sources the bandwidth
is no longer of the order of
but has a much greater value.
The time  for which the electron had emitted radiation is much smaller than actual
coherence time. Hence, coherence time may be defined as the time interval needed for the
extreme frequencies of a band to get out of phase by 2.
According to Eq. 8-2, the frequency – spread of a spectral line is of the order of the
inverse of the coherence time. It means that a perfectly sharp monochromatic line ( =0)
would correspond to an infinite interval of time ( = ). A strictly monochromatic light
corresponds to a perfect sinusoidal wave over infinitely long distance (L=) or for infinitely
long interval of time ( = ). In actual practice, we get pulse of light from a thermal source
of light for finite duration. Hence, the strictly monochromatic light is not realisable in actual
practice. Laser light is, however, nearest to the monochromatic light. The laser light is
essentially monochromatic and spatially coherent.
9
Purity of a Spectral Line
A perfect spectral line has only length but no width. Such a sharp monochromatic
line corresponds to a perfect sinusoidal wave which has infinite coherence length and
coherence time. For such a line (or for a perfectly monochromatic wave) as follows from
Eq. 8-1, as ,
. The frequency spread
or wavelength spread
is a measure of spectral width of the line. The ideal sinusoidal wave has no frequency
spread and remains coherent over indefinitely long intervals time. A decrease in the
coherence time corresponds to an increase in the breadth of a spectral line. This fact
follows from Eq. 8-1.
As wavelength spread
decreases, the purity of a line increases. It can be shown
on the basis of quantum theory that the spectral lines have a finite purity Q; defined as
If
which is an ideal case. The purity of sodium light,
105 whereas for
6
Cadmium red light,
10 . This Cadmium line was used for standardization of meter
because its purity Q is very large i.e. its
is small. The concept of coherence length L is
directly related to the purity Q of a spectral line.
For a very sharp spectral line
(monochromatic), we have
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A qualitative picture can be discussed on the basis of circular fringes obtained in Michelson
interferometer. As the path difference between interfering beams is increased by moving
the movable mirror M1, we are able to observe interference circular fringes. If the given
light is a mixture of two wavelength  and +, each wavelength produces its own set of
circular fringes. The condition for interference pattern to disappear is that path difference 
is equal to or exceeds the coherence length L. Thus, the concept of temporal coherence is
directly related to the width (or purity) of the spectral line. Now consider the case of two
closely spaced wavelengths 1 and 2 where 1  2 and 2 > 1. Thus, it follows that when 
= L, the bright circular fringes due to 1 coincides with dark fringes due to 2 so that there
will be uniform illumination i.e. there will be disappearance of fringes.
(Order n is greater for smaller wavelength 1)
If instead of two discrete wavelength
, the beam consists of all wavelengths lying
between
then the pattern would disappear if
Putting
This is the value of wavelength spread
width)
which is a measure of a line width (or spectral
Using Eq. (9-1) and (9-2), we have
Also
For two discrete wavelength
and
, we have, as proved above,
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Assuming
and
This is the equation of wavelength spread for two discrete wavelengths. If, instead of two
discrete wavelengths, we have all wavelengths lying between
, then we have line
width (or spectral width)
Eq. 9-3 and Eq. 9-4 express coherence length L in terms of purity Q and line width
spectral line. We now try to derive the equation in terms of frequency spread
of the
We know
Differentiating this, (ignoring minus sign) we have
(using Eq. 9-4) The quantity
is called frequency spread of the line.
Since
we can also get
This equation is similar to Eq 8-1. Thus, if frequency spread of a line is known, coherence
time can be determined.
The equation
shows that the frequency spread of a spectral line is of the order of he
inverse of the coherence time. It means that a perfectly sharp monochromatic line (
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would correspond to an infinite interval of time (
).
The quantity
represents the
monochromaticity (or the spectral purity) of he source; it represents the frequency stability.
For an ordinary light source it is quite small. For a laser beam having coherence time 
50ns the monochromaticity is
The direct relationship between the finite
coherence
analysis.
time and the spectral width of the source can also be seen using Fourier
From above it follows that the temporal coherence  of the beam is directly related to
the spectral width . This relation is given by
10
Spatial Coherence of a wave
Spatial coherence of a wave is the phase relationship or coherence between the
disturbances at different points in space. This is of importance in the study of coherence of
radiation fields of extended sources. A laser beam is spatially coherent and monochromatic.
In this beam it is not necessary to use a slit source to illuminate Young’s double slit. Let us
first consider thermal sources of light. There can be two ways to study the spatial
coherence, as given in the following section A and B.
A. Longitudinal spatial coherence of the wave.
Consider a single quasi – monochromatic point source
of light S0, as shown in Fig. 7. Consider two equidistant
points S1 and S2 such that S0S1 = S0S2 , and a third
point S3 lying on S0S1 line produced. Let the fields at
S1 ,S2 and S3 be E1 , E2 and E3 respectively. Points S1
and S3 differ in their distance from point source S0.
S2
S0
S1
S3
Fig.7 Spatial coherence
Longitudinal spatial coherence of the field is
measured by the coherence between the fields E1 and
E3. Such a coherence will depend on the distance S1S3 in comparison to the coherence
length L of the field at S0. If distance S1S3<<L, there will be high coherence between E1
and E3. If S1S3 >> L, there will not be any coherence between E1 and E3 on the wave
propagating form S0. The above explanation in terms of coherence time  is as follows:
(c is velocity of light).
between fields E1 and E3 .
B. Lateral spatial coherence of the wave
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Now consider the coherence between the points S 1 and S2 situated at equal distances
from point source S0. In this case, the coherence between the fields E1 and E2 is a
measure of lateral spatial coherences. As S0 is a point source, the fields E1 and E2
will be mutually coherent. If S 0 is not a point source, or rather it is an extended
source, then there may or may not be coherence between fields E1 and E2.
11Lateral Spatial coherence due to an extended source of light.
Let us perform Young’s double slit
experiment using an ordinary source of light S,
as shown in Fig. 10,8. S0 is another slit of
variable width, placed in front of source S. The
slit S0 is placed in front of two identical slits S1
and S2, close to each other, such that S0 is
equidistant from S1 and S2. If the slit S0 is
very narrow, then it is possible that the wave
trains from S1 and S2 behave as coherent
beams with respect to each other, as result of
which interference bright and dark fringes are
obtained on the screen. The intensity of dark
fringes is almost zero and the fringes are quite
distinct.
S1
S
S0
S2
If the width of the slit S0 is gradually
increased, keeping the widths of S1 and S2
unchanged, it will be observed on the screen
Screen
that the intensity of maxima goes on
Fig. 8 Lateral spatial coherence
decreasing while the intensity of minima goes
----- Young’s double slit
on increasing. Note that now the intensity of
Experiment.
dark fringe is no longer strictly zero. It means
that the fringes are less distinct and do not
remain very sharp. If the width of the slit S0 is further increased , a stage will reach that
the intensity of maxima which was decreasing, becomes equal to the intensity of minima
which was increasing with the width of the slit. It means that there would be uniform
illumination on the screen. This leads to conclude that the coherent beams form S 1 and S2
pass continuously from the condition of complete coherence to one of complete
incoherence. Between these two limits, the waves from S 1 and S2 are partially coherent.
This can be explained as follows.
In fact, even one single atom does not emit a continuous wave train. A wave train is
emitted by it only when an electron form of its outer orbits falls on to an inner orbit or when
an electron from quite outside the atom enters into one of its orbits and then suddenly
stops until a similar event occurs again. Thus, an atom emits radiations not continuously
but in sudden bursts (or pulses), each lasting about 109s. During this interval of time S1
and S2 can maintain a constant phase difference between them. After this, there are fresh
bursts of radiation in S which slit S0 gets illumination to illuminate S1 and S2. There is now
an entirely unrelated phase of the fresh bursts with the previous ones. This new phase
difference gives rise to new interference maxima and minima formed by S 1 and S2. This
pattern falls at an altogether different place on the screen. The pattern of maxima and
minima of intensity, i.e., the pattern of interference fringes, thus keeps on shifting about
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109 times in one second. This is
much too rapid for the eye to
follow, with the result that it
sees a uniform illumination all
S1

over and not a trace of
S
d/2
S
interference fringes anywhere.

l
d
O

l

What actually happens in this
P
S0
S0
that if S0 is quite broad, then slit
M 
d/2
a2
S1 is illuminated mostly by one
An extended
a2
S2
source
set of atoms while the other slit
S2 is illuminated by another
a
(a)
independent set of atoms. In
(b)
other words, the two slits S1 and
S2 are completely incoherent
Fig.9 Relation between coherence
because their phases are not
screen
of wave and size of the source
correlated.
If slit S0 is quite
narrow, then, at given instant,
the two slits S1 and S2 are illuminated by the radiation emitted by one particular set of
atoms.
12
Lateral Coherence Width (l )
In this section we shall derive a mathematical relationship between the coherence of
wave and size of the source. So is a point source which has variable width. Its width is
gradually increased to
so that there is uniform illumination on the screen. An extended
source is equivalent to a very large number of independent point sources. We can consider
this problem by having two independent incoherent point sources situated at S 0 and , l
distant apart (S0 = l). Their distance l is such that maxima due to S0 coincides with the
minima due to . This coincidence on the screen shown in Fig 9 in which solid (
) lines
show maxima and dotted (
) lines show minima i.e. the interference pattern is washed
out. Let this distance S0 S = l be the minimum distance for which the interference pattern
on the screen will be washed Mathematically, it is expressed as
which means maximum due to S0 coincides with minimum due to , i.e., the interference
pattern due to S0 and
will be out of step. Note that S1P = S2P. [It may be pointed out
that in the case of two independent incoherent sources S 0 and , we have to add up
intensities (I1+I2) instead of I1+I2+2
where I1 and I2 are intensities of S 0 and ;
the phase difference between S0 and S1. In Fig. 9, various distances shown are as
follows:
where O is mid-point of
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Coherence and Lasers
Draw
In
,
We are to calculate .
From S0MS and MOS2,
This gives,
But
a = a1 + a2
This gives required value of  as
Substituting Eq. 12-3 in Eq. 12-2, we have
Assuming
,
For the interference pattern to be washed out, Eq. 12-1 gives,
This gives
For given set-up of Young’s double slit experiment and wavelength , if the spatial extension
of the extended source exceeds
, then the interference pattern will be washed out
meaning thereby that the interference pattern on the screen will not be observed. We can
re-write Eq. 12-4 as
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When
;
is the angle that S0 and S subtends at mid – point O of S1S2 = d.
is called
lateral coherence width. In order to observe distinct interference fringes on the screen, the
distance d between the slits would have to be much less than lateral coherence width
such that d <<
where
a fraction of a mm or a few mm. Thus,
is the maximum value of the spatial
extension of the extended source for which the interference fringes would be visible in a
Young’s double slit experiment.
For a circular extended source, the lateral coherence width is modified to
where
13
Visibility of fringes and degree of coherence
According to Michelson formula for visibility of fringes, the visibility of fringes, the
visibility (or contrast) of interference fringes is given by
where Imax and Imin are the maximum and minimum intensity respectively in an interference
pattern. Thus visibility of the fringes is defined as the ration of the difference between
maximum intensity and minimum intensity to the sum of these intensities. Thus, this
visibility V is a measure of the degree of coherence of the light waves that produce the
interference pattern. Maximum value of V is unity and minimum value is zero.
For complete destructive interference
Imin = 0; this gives V = 1.
This corresponds to overlapping in their entire (finite) lengths of the two interfering
light beams of equal intensities. These interfering waves have maximum value of V which is
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Coherence and Lasers
unity. They are said to be in perfect coherence with each other and have the highest
degree of contrast.
If the two interfering waves do not superimpose at all,
Imax = Imin . This give
V=0
This means that visibility is minimum. Such waves have lowest degree of contrast.
These will be no maxima and minima and the two waves are said to have no coherence,
i.e., no interference fringes observed.
For 1 > V > 0, the waves superimpose in part, interference is possible, with less
degree of contrast of the fringes.
Relationship between visibility (V) and degree of coherence (C).
It can be proved that the visibility (or degree of contrast) of the interference fringes
due to two light beams of equal intensity is equal to the degree of coherence (C) between
the waves. The proof is given below:
Consider two waves of light beam P and Q, each having equal intensity I0,
illuminating two points on a screen. Let us assume that each beam in P and Q consists of
two parts A1 and A2, parts A1 being completely coherent and parts B being completely
incoherent. Thus, parts A1 in P and Q produce intensities
, given by
Then parts A2 in P and Q would produce intensities
So that
The interference occurs due to parts A 1 only. These parts form fringes whose maximum has
intensity 4
and the minimum intensity is zero.
This is so because,
On the interference pattern is superimposed a uniform intensity due to two A2 parts, of
magnitude
. The pattern on the screen is illuminated with maximum intensity, given by
This gives, using Eq (13-2) and (13-3)
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The minimum intensity is
Substitute Eq. (13-4) and
in Eq. 13-1
This means that visibility V of the fringes produced by interference of two waves of equal
intensities is equal to the degree of coherence (C) between the waves. That is why V is also
called the degree of contrast.
14
Experiment on coherence with a laser beam
One of the differences between laser and ordinary light source is that the phase of
the wave from a laser source is constant over very large interval of time. It remains
constant in the laser beam during a period of the order of 0.01s or even 0.1s or even longer
whereas it is constant in a light beam during a period of 10 9s. With a laser beam, the
interference fringes may be observed by means of very simple devices.
A. Temporal Coherence with a laser beam
For sodium yellow light,
Coherence length, L
Spectral purity, Q
2.5 cm,
Wavelength spread, 
0.06Å
106
In Michelson interferometer, interference circular fringes could be obtained
even for optical path difference as large as 9 m or even more than 9 m. This
indicates that path difference between the two interfering beam is still less than
coherence length.
In 1963, Magyar and Mandel have succeded in recording
interference fringes with spectral purity Q
1014 for  = 11.53  105 cm of laser
beam. Since wavelength spread
This corresponds to the coherence time,
B. Spatial Coherence with a laser beam
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Coherence and Lasers
A laser beam is highly coherent and monochromatic. A parallel laser beam is allowed
to fall directly on two slits S1 and S2 without using a slit S used to illuminate S1 and
S2 as is used in usual Young’s double slit experiment. This is because the laser beam
is spatially coherent and monochromatic. The distance D may be kept several
meters apart.
Interference
equidistant
maxima
and
minima are observed on the screen. If we
close any one of these two slits, interference
fringes on the screen disappear, thus
proving that these fringes are due to
interference. The pattern that now appears
on the screen is due to single slit diffraction
maxima and minima which are not
equidistant. If we bring laser source from
infinity to the two slits S1 and S2 , the
interference fringe pattern remains fixed.
This confirms that the given baser light is
spatially coherent, i.e., there is always a
fixed phase relationship between the beam
that enters each slit.
15
S1
y
Laser beam
S2
D
Fig.8 Spatial coherence with a laser
beam
Laser
Introduction: The word LASER is an acronym for “L(ight) A(mplification) by
S(timulated) E(mission) of R(adiation)”
Laser is a device that amplifies focused light waves and concentrates them in a
narrow, very intense beam. The narrow beam can either, pulsed or continuous. Laser is a
highly intense, monochromatic, coherent and unidirectional beam of light. It is a source of
radiation in the visible, ultraviolet or infrared regions of the electromagnetic spectrum. It
depends on the phenomenon of “stimulated emission”, first proposed by Einstein in 1916.
The property of coherence distinguishes laser radiation from ordinary optical beams. Laser
produces a beam of coherent electromagnetic radiation having a particular, well defined
frequency in that region of the spectrum broadly described as optical. It amplifies light by
means of stimulated emission of radiation. It was first predicted by Schawlow and Townes
in 1958 and first put into operation by Maiman in 1960.
Einstein considered the equilibrium between matter and electromagnetic radiation in
a black-body temperature in which mutual exchange of energy takes place due to
absorption and spontaneous emission of radiation by the atoms. He found that this
equilibrium could not be explained completely on the basis of absorption and spontaneous
emission. He predicted that there must be third process called “stimulated emission” along
with absorption and spontaneous emission.
The principle of laser is based on the
phenomenon of stimulated emission. Various processes involved are explained below
Absorption of radiation
An atom can have a definite fixed energy corresponding to the orbitals in which its
electrons move around the nucleus. It is so because an atom has a number of of
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Coherence and Lasers
quantised energy states and the electron can go to permitted orbital. The lowest
energy state of electrons is called ground energy. When all the electrons in an atom
possess lowest energy, the atom is said to be in ground state. If an atom in its
ground state is exposed to an electromagnetic field, it may absorb a photon of
energy
. In doing so, its electron may go to next permissible higher energy state
by absorbing energy of photon. Such an atom is said to be in its excited state. If E 1
and E2 are energy levels in initial and final states and
the frequency of absorbed
radiation, energy of radiation is
where
is Planck’s constant,
and
is energy of a photon
(or
quantum)
of
electromagnetic
radiation.
There is no permissible state in
between transition 1 and 2.
This
process
of
interaction
with
the
electromagnetic radiation field
is called stimulated or induced
absorption of radiation.
2
1
electron
E2
2
E2>E1
E1
1
(Initial state of atom)
Ground state
electron
E2
E1
(final state of atom)
Excited state
When an assembly of a large
Fig. 9 Absorption of photon
number of atoms is exposed to
an electromagnetic radiation of
photon frequency , they will be raised to higher energy state E2 . The absorbed photon is
called stimulated photon. If the frequency of photon is different from , there is no
interaction at all. The excited atoms (E2) can emit photon of frequency
The probable rate of excitation or absorption from transition 12 is proportional to
1. the number N1 of atoms available for excitation in the lower energy state
2. the energy density
of the incident radiation of frequency on the atom
Probable rate is expressed as
where B12 constant of proportionality which is determined by the nature of the two
combining states 1 and 2. This constant B12 is called Einstein coefficient for absorption of
radiation.
Metastable State
Excitation is the addition of energy to an atom transferring it from its ground state to
a higher energy level. The excitation energy is the difference energy between the ground
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Coherence and Lasers
state and the excited state. Generally an atom in its excited state E2 returns to the ground
state E1 within a very short time of about 10 8s by the emission of photon of frequency
This time is called mean life atom. The mean life of atom is characteristic of the energy
state. It is defined as the average time for which an atom remains in its permissible excited
state. Some excited atoms have comparatively longer mean life. Energy states having
mean life of more than 103s are called metastable states.
Spontaneous Emission
Normally an atom in its excited state remains for only 10 8s. Without any external
stimulus, it, then of its own accord, makes a transition to lower energy state emitting a
photon energy . This is called spontaneous emission of photon
2
E2
2
An assembly of atoms in their
excited state emit photons
which have a random phase.
1
E1
1
Their phases are not correlated
final state
and hence they are incoherent.
initial state
The probability of spontaneous
emission 21 is determined
Fig. 10 Spontaneous Emission of Photon
only by the properties of states
2 and 1.
Einstein gave the
probable rate of spontaneous emission as
E2
E1
where N2 is number of atom in energy state E2. A21 is called Einstein’s coefficient of
spontaneous emission of radiation. The probability rate P 21 (spontaneous) is independent of
energy density of the incident radiation whereas P 12 (see Eq.15-1) depends on energy
density
. Hence, for equilibrium, there must be spontaneous emission transition
depending upon
. Thus the total probable rate denoted by P 21 should be equal to sum of
Eqs. 15-1 and 15-2. This sum is called stimulated emission probability.
Stimulated (or induced) Emission of radiation
An atom in its excited state E2 is capable of emitting a photon of frequency
electron jumps to ground state E1 where
when
Suppose, a photon marked A in Fig. 11 of frequency
is made incident upon such an
excited atom. According to Einstein, now two photons move, one incident photon and the
other (marked B) which is emitted, also of the same frequency
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2
electron
E2 A
2
E
2
Coherence and Lasers
B
A
1
E1
(Initial)
A
1
electron
E1 B
A
(Final)
Fig. 11
This is called stimulated (or
induced)
emission
of
radiation.
The incident
photon is called stimulating photon. When an atom in its excited state interacts with an
incident photon of proper frequency and is thereby induced to move to the ground state E1
by emitting photon of the same frequency
the process is known as stimulated emission of radiation (or negative absorption of
radiation). In such a process (i) The incident stimulating photon (A) and the emitted
induced photon (B) travel in the same direction. (ii) The two photons are in the same phase.
(iii) The two photons have the same state of polarization.
Due to the above
characterization, the two photons are capable of interacting with other excited atoms of
matter and these two photons now act as stimulated (incident) photons. This process builds
up and a sort of chain emission starts in which all the photons are identical, are of the same
frequency, in same phase and travel in the same direction. Since, now photons are very
large in number, the radiation beam is now highly amplified i.e., the beam is highly intense.
Thus, the emitted beam of electromagnetic radiation is highly intense, monochromatic,
coherent and unidirectional. Total probability of emission transition 21 is the sum of
spontaneous and stimulated emission probabilities i.e.
where A21 and B21 are Einstein coefficients
In stimulated emission, the photons multiply in number as 2 n where n = 0, 1, 2, 3,……
where n is the number of stimulated emissions.
1
 2
 4
 8
-------- - - - - etc.
1 (photon  2 (photon)  4 (photon)  8 (photon)  - - - - - - - - - - etc.
The radiation of suitable frequency , so that E =
, can interact with the atomic
system in three ways as shown in Fig. 11(a) where E 2  E1 =
. The balance of radiation
energy density is given by the net effect of the three processes:
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This is shown in Fig. 11(a). Amplification is concerned with first two terms; it requires
N2 > N1. The ratio of the population when the medium is in equilibrium at temperature T is
given by
where 1 and 2 are the statistical weights of the two levels, and E is their energy
separation. For simplicity, we put 1 = 2
According to this equation N2 > N1 cannot occur under equilibrium condition at any
physically real temperature. The ratio
at room temperature as given by the above
equation is of the order of e100 for optical frequencies, so only a small proportion of systems
is in the upper state. If by some means N 2 can be made greater than N1, the levels are said
to be inverted. This is possible if T becomes negative. The injection of energy, so as to,
invert the levels is known as pumping; it corresponds to the power supply in an electronic
amplifier.
Relation between Spontaneous
Stimulated Emission Probabilities
and
Energy
Let N1 and N2 be the number of atoms
in states E1 and E2 respectively at any
instant. The number of atoms in state 1 that
absorb a photon and rise to state 2 per unit
time is
Absorption rate =
E2
spontaneous
stimulated
Absorption
E1
Fig.11(a)
where
is energy density of the radiation of frequency
incident on the atoms. Here
we are considering an assembly of atoms in thermal equilibrium at temperature T with
radiation frequency and energy density
.
The number of atoms in state 2 that emit a photon either by spontaneous or
stimulated emission, and drop to state 1 per unit time is
Emission rate =
At equilibrium, absorption rate = emission rate
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Einstein proved thermodynamically that the probability of stimulated emission is equal to
the probability of stimulated absorption, i.e.,
Total number of atoms in thermal equilibrium at temperature T is
According to Maxwell – Boltzmann distribution law,
where k is Boltzmann’s constant
Since E2 > E1, N2 < N1 …………………….……………(15-5)
Since
which is energy of photon emitted or absorbed,
From Eq. 15-3 and 15-5
This gives energy density of photon of frequency
states 1 and 2, at temperature T.
in equilibrium with atoms in energy
Planck radiation formula is
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Comparison of Eq. 15-7 with Eq. 15-8, we get
This gives the ratio of the spontaneous emission and induced emission coefficients
Since
It shows that the probability of spontaneous emission increases rapidly with the
energy difference between the states
It means that the probability of
spontaneous emission dominates over induced emission more and more as the energy
difference between the two states increases. According to Eq.(15-4), the population of
atoms in higher energy levels is less than that in the lower energy levels.
Population Inversion
An assembly of atoms can be excited by making a beam of electromagnetic radiation
of matching frequency incident upon them. The atoms get excited due to stimulated
absorption. These excited atoms can undergo either spontaneous emission or stimulated
emission. In order to achieve higher probability of stimulated emission, the following two
conditions must be satisfied.
(i). The excited atoms should be in their metastable state i.e., the excited state
should have a longer mean life, greater than 108s.
(ii). N2 > N1 i.e., the number of atoms in excited state E2 must be greater than that
in E1. This is opposite to Eq 15-4.
If N2 > N1, then (P21)stimulated > P12 , as this follows from Eq. 15-3.
The condition having N2 > N1 for E2 > E1 is called population inversion. It is the
condition to have greater number of atom in higher energy state than in the lower energy
state. The population inversion is achieved by a procedure called optical pumping of atoms.
Optical Pumping of Atoms
When an assembly of atoms are irradiated with a matching frequency
of
electromagnetic radiation, atom in lower energy state E1 absorbs photon and goes to higher
energy state E2 where
Such an excited atom returns to lower state E1 by emitting a photon due to spontaneous
emission in a very short time (~108s). This process fails to produce required population
inversion which is achieved in optical pumping. This involves a scheme of three energy
levels, namely, E1, E2, and E3 having population N1, N2 and N3 respectively where E3 > E2 >
E1. Here E1 is the ground state, E3 is a short-lived state and E2 is metastable state
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Spontaneous
Metastate State
Stimulated
emission
Absorption
Stimulated
E2
emission
Shortlived State
E3
N3
N2
Amplified Laser
Radiation of
frequency
Ground State
E1
N1
Fig.12 Optical Pumption of Atoms
Transition E3  E2 is allowed but transition E2  E1 is not allowed. When incident frequency
irradiates assembly of atoms, these atoms get excited to E3, so that
The atoms get excited to E3 by the process of stimulated absorption. Some of these excited
atoms in E3 level jump to intermediate level E2 by spontaneous emission or by a nonradiative process thereby converting their excess energy into vibrational kinetic energy of
the atoms forming the substance. The level E 2 is metastable state in which atoms remain
in it for a comparatively longer time (~103s) as compared to 108s in E3 level. For this time
N2 > N1. This is how population inversion is achieved.
Atom in E2 state can decay to E1 state either by spontaneous emission or by
stimulated emission. In both cases, photon has frequency,
This photon may produce stimulated emission form another atom. It produces two
coherent photons travelling in the same direction. These two photons interact with other
atoms producing two more photons, as result of which an amplified beam is produced.
Finally the atoms are induced by radiation of energy
to drop from energy level E2 to E1
emitting laser frequency
which is less than
. This process is called Laser (i.e. light
amplification by stimulated emission of radiation). This underlines the principle of Laser.
Requirement of Laser Action
(i). The number of atoms in higher energy state must be greater than that in lower
energy state
(ii). The energy density of stimulated emission must be large. This condition gives
(P21)stimulated > (P21)spontaneous
In other words, the stimulated emission exceeds the spontaneous emission.
Principle of Laser: The above two requirements underline the principle of laser. The
above requirement mentioned in (i) is referred to as population inversion. The process of
creating population inversion is called optical pumping.
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Three Basic Components of any laser devices are the active medium, the pumping source
and the optical resonator.
Do you know?
Charles Hard Townes, the co-inventor of the laser and a Noble laureate in physics, has died
at age 99. He had been in poor health before he died on January 27, 2015 in USA. He was a
professor emeritus in the University of California, Berkeley. Townes did most of the work
that would make him one of three scientists (along with two other from Russia) to share the
1964 Nobel Prize in physics for research leading to the creation of the laser while he was a
faculty member of Columbia University.
His research applied the microwave technique used in wartime radar research to the study
of spectroscopy, the dispersion of an object’s light into its component colours.
Townes earned praise and scorn for a series of speeches investigating the similarities
between science and religion.
16
Difference between non – laser and laser light sources:
Nonlaser light sources emit radiation in all directions as a result of the spontaneous
emission of photons by thermally excited solids (filament lamps) or electronically excited
atoms, ions, or molecules (fluorescent lamps, etc.).
The emission accompanies the
spontaneous return of the excited species to the ground state and occurs randomly, i.e. the
radiation is incoherent.
In a laser, the atoms, ions, or molecules are first ‘pumped’ to an excited state and
then stimulated to emit photons by collision of a photon of the same energy. This is called
stimulated emission. In order to use it, it is first necessary to create a condition in the
amplifying medium, called population inversion, in which the majority of the relevant
entitities (e.g. atoms) are excited. Random emission from one entity can then trigger
coherent emission from the others that it passes. In this way amplification is obtained.
17 Different types of Lasers
Some lasers are solid, others are liquid or gas devices. The process of achieving
population inversion is called “pumping” of atoms. There are various types of pumping
process, but the most natural is the “optical pumping” which is utilized in Ruby laser. The
other types of pumping are flash lights chemical reactions, discharge in gases, and
recombination emission in semiconducting materials .
(i). Solid lasers: Most common example is Ruby laser. This is the first laser
developed in 1960. The other example is Y3Al5O12 doped with Nd+3 ions in place of
Y(yllerium). It operates at  = 1.064m which is wavelength of infrared light.
(ii). Gas Lasers : One example is helium-neon gas laser which contains 10 parts of
Ne to one part of He. Its operating wavelengths are 6328 Å (red), 1.15m and 3.3m
(infrared). Another example is CO2.
(iii). One example of semiconductor laser consists of a cube of specially treated
gallium arsenide, (GaAs) which is capable of emitting infrared radiation when a current is
passed through it. This is p – n junction diode laser which is small, robust and cheap.
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(iv). Other examples include chemical lasers and dye lasers.
18 The Ruby Laser
This is the first laser developed by Mariman in 1960. It is solid-state laser consisting
of a ruby rod. Ruby is a crystal of aluminum oxide (Al 2O3) doped with 0.05% of chromium
oxide (Cr2O3) so that some of the Al +++ ions are replaced by Cr+++ ions. These impurity
chromium ions give pink colour to the ruby and give visa to the laser action.
Gas Discharge Tube
(Flash Tube)
Glass Tube
Fig. 13 shows schematic of
Partially Reflected
Coolant
end face
the Ruby Laser. It consists
of a ruby rod AB, about 10
cm in length and 0.8 cm in
Laser beam
B
thick. Both end faces A and
A
B are optically plane parallel
Ruby Rod
and silvered so that end A is
Fully Reflecting
heavily silvered and end B,
thinly silvered. The curved
end face
surface of AB is also made
reflecting surface. Such a
Capacitor
ruby rod AB behaves as a
resonant cavity. The rod AB
is surrounded by a glass
power supply
tube through which coolant
(liquid nitrogen or water)
Fig.13 The Ruby Laser
circulates to keep the rod
cool. A xenon flash lamp
(which is a gas discharge
tube) is wound round the glass tube so that the ruby rod AB lies along its axis. The flash
tube is connected to a suitable power supply. It is made to flash for a few milli seconds.
Only a small part of energy is used in exciting (pumping) the Cr+++ ions; the rest part of
energy heats up the apparatus; that is why coolant is used to take away this heat.
The ruby laser makes use of three energy level scheme of population inversion, as
already explained in schematic of Fig. 12. In the present case, Fig. 14 shows a three level
laser of chromium ion. It consists of a ground state level E1 and an upper short-lived
energy excited level E3, along with an intermediate excited level E2 which is metastable
having a-life time of about 3 millisecond. The level E3 has much shorter life-time about
108s. Mostly most of the Cr+++ ions are in E1.
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E3
Short lived state
2
Matastable state
E2
5500 Å
6943Å
1
Optical Pumping
3
4
6943Å
6943Å
Transition
E1
Ground state
Fig. 14 The energy level diagram of chromium ion
First, the flash tube is switched on. Flash of light lasts only for about a millisecond.
This flash falls upon the ruby rod AB. The Cr +++ ions in AB absorbs photon of =5500 Å
from flash of light and are raised (pumped) from E1 to the excited state E3. This is shown
by transition 1 which is optical pumping transition, in Fig. 14. The pumping ions give part of
their energy to the ruby crystal and decay to the metastable state E2. This decay is shown
by transition 2 from E3 to E2 , which is a radiationless transition in time 10 9s. Since E2 is a
metastable state, it has a much longer life-time. Hence the number of ions in this state E 2
goes on increasing while the number of ions in E1 goes on decreasing. Further it may be
added that the probability of transition from E3 to E2 is much higher than that from E3 to E1
whereas probability of transition from E3 to E1 is much smaller than that from E2 to E1. That
is why E2 becomes more populated than E1. Hence population inversion between E1 and E2
is achieved. This provides proper condition of stimulated emission of photon due to which
light amplification can take place.
An excited ion from E2 goes to E1 shown by transition 3. It emits a photon of  =
6943Å. This photon moves through rod AB, parallel to its axis. It is reflected back and
forth between the ends A and B owing to silvered ends. It, then, stimulates an excited
Cr+++ ion in E2 and causes it to emit a fresh photon in phase with the stimulating photon.
Thus we get two photons, each of wavelength 6943Å, as shown in Fig. 14, by transition 4.
This stimulated transition 4 is the laser transition. Now these two photons move back and
forth between the two ends A and B along the axis of crystal. They cause further stimulated
transition from E2 to E1 resulting in four photons and so on. Thus photons being multiplied
in this way produce sufficiently intense coherent monochromatic (6943Å) and unidirectional
beam. This laser beam emerges through end B because this end is partially silvered. The
photons emitted spontaneously which do not move axially escape through the sides of the
crystal.
Laser action is caused by the flash from the xenon flash lamp. The laser beam
ceases till the next flash repeats the process. Thus the ruby laser gives pulses of laser
beam; it is a pulsed laser. The duration of flash is a few millisecond and that of laser pulse
is less than a millisecond. The instantaneous power output of the emitted laser pulse is a
few megawatts.
Drawback of the Ruby Laser
The laser transition from E2 to E1 makes Cr+++ ions large in number in its ground state. In
order to achieve population inversion more than one – half of the atoms in E1 must be
pumped up to the existed state. Besides this, Cr+++ ions which happen to be in E1 level
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absorbs 6943Å photons from beam as it grows. This is a drawback in the three – level laser
such as ruby.
19 Properties of Laser
Lasers have special properties which are not present in ordinary light. Laser is highly
intense, (spatially) coherent, monochromatic and unidirectional. Coherence, the essential
property of lasers is of two kinds : spatial and temporal. The light waves are perpendicular
to the direction in which they are moving; this is called spatial coherence. They are equally
spaced, so that the time between one wave crest and the next is always the same; this is
called temporal coherence. Because of the spatial coherence, the laser beam has an
extremely small divergence and is therefore highly directional. For example, a ruby laser
beam 2.5 cm in diameter at the source will be about 750 cm across on a surface 15 km
away.
Another important feature of lasers is the enormous power that can be generated. Under
certain operating conditions monochromatic bursts of magawatts can be produced.
A laser beam, because it posses space coherence, can be focused to a spot whose
diameter is of the order of one wavelength of the laser light itself. Enormous power
densities are thus attainable. Extraordinary high temperature, orders of magnitude greater
than that at the sun, can be generated at the small area which absorbs this concentrated
radiation, something that can produce energy by fusion of nuclei possible one day.
Perhaps the most promising potential of lasers come from time coherence. It is this
property that permits the exploitation of radio and microwaves for communications. In fact
one single laser beam has in principle more information – carrying capacity than all the
radio and microwave frequencies in civilian and military use combined.
An interference pattern can be obtained by using two independent sources of lasers.
This is because the laser beam is completely spatially coherent. The laser beam has highly
temporal coherence because it is almost perfectly monochromatic.
Because laser beam is extremely intense, it can vaporize even the hardest metal.
20 Applications of Lasers
Lasers have found many uses since their invention in 1960. Some of these uses are
described below:
(1). Technical and Industrial Application:
Laser beam is used to melt and join two metal rods. It is called laser welding. With
laser, it is possible to weld a joint even after the joint has been sealed inside a glass
envelope. Similar to this application, laser beam can be used for cutting or burning metals.
Laser can even pierce through producing a hole in one of the hardest materials called
diamond. It is used for cutting fabric for clothing on one hand and steel sheets on the
other.
(2). Surgery
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In most of the cases in surgery, laser is like an optical very sharp knife which would
be more accurate and painless drilling. It is used in painless drilling and welding of teeth.
It is used in retinal surgery where it is used for welding the detached retina back into
position and treatment of malignant tumors specially removing eye tumors. It is also used
in the treatment of kidney stone, cancer, tumor and in cutting and sealing the small blood
vessels in brain operation. It is used to perform bloodless surgery and for conducting
difficult operation of the scalp and abdominal skin . Using laser-beam , the surgical
operation is completed in a much shorter time.
(3). Science and Research
Because of its high power, laser can be used for precise ranging and detection over
long distances. Time coherence (monochromaticity) lies at the very basis of wireless
communications system, and information-bearing capacity is proportional to the wave
frequency. In theory, a single laser beam could carry some 10 million television broadcasts
at once! It seems likely that when men first go to Mars, they will send live television back
to earth on laser beam.
The distance between earth and moon has been measured by laser rays to an
accuracy of 15 cm. Laser can be used to determine the temperature of plasma and the
density of electron. Laser-torch is used to see objects at long distances. Radio astronomers
have found lasers highly valuable for amplifying very feeble radio signals from space.
Laser is being used in exploring the molecular structure, Raman spectroscopy, nature
of chemical reaction and precision measurement of length. It has been used to perform
Michelson – Morley experiment which is the building stone of the Einstein’s theory of
relativity.
In space, laser has been used to control rockets and satellites and in directional radio –
communication like fibre-optic telephony.
(4). Holography: Laser is used in holography and non – linear optics. Holography is used
in many areas of physics, chemistry, biology and engineering.
(5). Laser rays are used in printing, optical communication, and the reading of digital
information. They have been used in detecting earthquakes.
(6).
Lasers have found many uses including in detecting nuclear explosions, in
vapourising solid fuel of rockets, in the study of the surface of distant planets and satellites.
In the field of entertainment, laser light shows are common and lasers light up the sky with
intense beam of different colours.
(7). Laser-Fusion:
Laser can generate very high temperature (~108K) at which gas
atoms are fully ionized. Such fully ionized gas is called plasma. Laser can also generate
very high pressure. Thus, laser is very useful in producing controlled fusion reaction which
needs very high temperature and pressure.
21 Holography
Introduction: Images can be formed without the use of lenses by a process known
as holography. This word ‘hologram’ consists of two Greek words holos (means whole) and
gamma (means a letter) – the hologram means the ‘whole message’. This word was
invented by the Hungarian born British Nobel laureate Dr. Dennis Gabor who also invented
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the technique of holography. Gabor developed holography in 1947 to improve the electron
microscope, which views and photographs objects with a probing beam of electrons.
Holography woks with any
holography with light, recall how a
pattern of light scattered from your
light is uniform and without details.
of the objects that scatter them.
waves – electron, sound, or light.
To appreciate
camera works. The camera film records the intensity
face. But if there is no lens, that pattern of scattered
Here a lens focuses light patterns to match the shape
Gabor’s problem was with accuracy of focus of his lenses. Blurred information
seemed to be lost completely, but if the direction as well as intensity of the light could be
recorded, resharpening might be possible. The complete light pattern represented by the
combined information about direction and intensity is called a wavefront.
In conventional photography, a photograph represents a two dimensional recording
of a three dimension object and the photographic film records the amplitude or intensity of
the reflected light from the object. In the case of holography, both the amplitude and
phase of the wave are recorded on the photographic film. This is made possible by using a
coherent light. Further, it records the three-dimensional character of a three dimensional
object on the photographic film without using any lens or camera. Although the principle of
holography was first put forward by Dennis Gabor in 1948, he was awarded nobel prize for
holography in 1971. This invention remained a subject of academic interest because there
was lack of an adequate source of coherent light. The invention of highly coherent light
such as laser in 1960, E.N. Leith and J. Upatrieks using a laser succeeded in performing
experiment on holography. In the case of ordinary photography, ordinary light which is
incoherent light falls on the photographic plate, the eye is able to see only a two-dimension
image of the object. Here, in holography, we shall discuss a radically different concept in
photographic optics – giving a full three – dimensional image including hidden back side of
the object. This is a three – dimensional lensless method of photography which records the
amplitude and phase of the light wave using interferometric techniques.
What is holography ?
Holography is a method of recording and displaying a three-dimensional image of an
object, usually using coherent radiation from a laser and photographic plates without using
lenses or camera.
Holography is a process by which the image of an object can be recorded by the
wave-front construction. It does not record the image of the object on the photographic
film, but records the phases and amplitudes
of the light waves themselves.
The
photographic record of light waves thus produced is called hologram. Hologram does not
bear any resemblance to the original object. It has all the information about the object that
is contained in an ordinary photograph plus some additional information that is not
contained in the ordinary photograph because it cannot be recorded in ordinary
photography. This some additional information when reproduced, gives rise to a three –
dimensional image. This image is reconstructed by placing the hologram in a laser beam;
both a real and a virtual image are formed in depth. The formation of image from a
hologram in complete resemblance to the original object is called the reconstruction
process. It is not essential that the illumination of original object and the illumination of the
hologram used for the reconstruction should have the same wavelengths.
Different
wavelength will alter the magnification of the image. But, it is necessary that the shape of
the wave for the two illuminations as mentioned should be same. When the viewer’s eye is
moved from side to side, the ‘rear’ parts of the three-dimenional scene of the image are
seen to move, relative to the more distant parts. This effect called ‘parallax’ is present in
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holography [Parallax is defined as an apparent displacement of a distant object (with
respect to a more distant background) when viewed by the observer from two different
positions. Such a parallax is not present in ordinary photograph. The theory of holography
is mathematically complicated, but the essentials can be explained and understood from
physical arguments.
Holography is a two-step process of optical imagery (i.e. of image formation), namely,
Step I: Formation of hologram: An object illuminated by coherent light is made to
produce interference fringes in a photosensitive medium, such as photographic emulsion.
Step II: The Reconstruction : In this step, the hologram is illuminated by laser light of
the same wavelength. The reillumination of the developed interference pattern in the
hologram produces a three-dimensional image of the original object.
Principle of Holography:
Light waves diffracted (or scattered or reflected) by an object are characterized by
their amplitude and phase. For recording the diffraction pattern, it is not sufficient to a
place a photographic plate in the path of the diffracted wave. It is so because the
photographic film is sensitive to intensity variation i.e. to the square of the amplitude;
hence it records only the amplitude variation and no phase variation at all. This procedure
cannot be adopted to record phase variation. Gabor solved this problem by superposing to
this diffracted wave a second highly coherent wave (laser light). The interference produced
between the two waves increases the intensity at points where two waves are in phase and
decreases the intensity at points where the said waves reach out of phase. As the light is
from a coherent source, each wave will also be coherent and hence will produce interference
or a diffraction pattern on the photographic plate. The same principle applies to transparent
or semi-transparent objects in which the interference bright and dark fringes are produced
by refraction. The photographic plate of diffraction pattern which when developed is called
a hologram. To reproduce the image of the object, the hologram is illuminated by coherent
light, ideally the original incident coherent light. The hologram produces two sets of
diffracted waves; one set forms a virtual image coinciding with the original object position
and the other forms a real image on the other side of the plate. Both are threedimensional. More recent techniques can produce holograms visible in white light.
Step I : Formation of Hologram
When light falls on an object, light waves from each point on the object are diffracted
(or reflected or scattered). Any object may be considered to be equivalent to an infinite
number of point objects. The point O in Fig.15 shows a point object illuminated by a
parallel beam coherent light ‘S’ whose plane wave fronts have beam shown. This incident
wave is called primary wave or reference plane wave which can be a coherent light wave or
a laser beam or a coherent electron wave. AB is a photographic film.
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A
Plane wave fronts
parallel beam
of coherent light
S
A
Spherical wave fronts
O
O
S
B
Unobstructed plane wave fronts
B
Fig 15(a) Principle of Holography
Fig 15(b)
[AB is photographic film]
Light waves diffracted by point O are called secondary waves which would consist of
spherical wave fronts. These are shown by dotted lines in Fig. 15(a) and they are
concentric around point O. Light waves diffracted are characterized by their amplitude and
phase
For recording the wave pattern, it is not sufficient to place a photographic plate AB in the
path of diffracted waves because the photographic film is sensitive to the intensity i.e. to
the square of the amplitude and cannot be used to record the phase information. Gabor
solved this problem by superposing to this diffracted wave (called secondary wave) another
known wave (or unobstructed incident wave or primary wave). Instead of plane wave
fronts from source, light from a point source S can be made incident on the small object O
located at a short distance away, as shown in Fig.15(b). A small proportion of the
light is diffracted by O in all directions, and on the plane AB the secondary waves from O
are superposed on the strong coherent background provided by the primary wave from S
[see Fig. 15(a) and (b)].
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Diffracted wave
A
A
R
P
Primary wave
O
B
B
Fig17
C
Halogram
showing
interference
bright
and dark fringes
Fig16 Superposition of primary and
secondary waves at P on
photographic film AB
In Fig. 16, P is such a point of superposition of primary wave RP and secondary wave OP.
The point P on the film is one point on a circular arc centered on C along which the intensity
is maximum for objecot point O. The intensity distribution that is produced by object point
O is dterminded both by the brightness of the reflected wave from O and by the phase
relationship between the object wave OP and primary wave RP at the film.
The primary wave is highly coherent and monochromatic. It has a coherence length
of several meters. Consequently, if the scene that is viewed has dimensions of a few
meters or less, the primary wave and the secondary waves will overlap and interfere at film
AB, producing interference fringes on the film AB. If O is a single point but a complex
object (a tree, a building etc) it can be regarded as a collection of a number of points and
the resulting wave pattern diffracted from the surface of the object can be regarded as the
sum of many such sets of spherical waves, each set concentric about its set of origin. The
total pattern recorded on the film is the linear superposition of the contributions from object
points such as O. The pattern is characteristic of the object and will be different for
different objects. If a1 is amplude due to primary wave and a2, that due to secondary wave,
the intensity at any point on the film is determined by
At the plane of the photographic plate AB, a system of bright and dark concentric
circles will be formed due to constructive and destructive interference between the
secondary waves and the direct primary waves. That is to say, on AB the secondary waves
from O are superposed on the strong coherent background provided by the primary wave
from S. Upon development the photographic plate AB is found to contain bright and dark
partially absorbing fringes. The diffraction pattern produced on AB due to superposition
(interference) of waves is called the hologram or the holograph. Since the primary wave is
uniform and much stronger than the secondary wave, the distribution in intensity is
obtained corresponding to the variation in the phase of the secondary wave [i.e. amplitude
a1 is much greater than amplitude a2, variation in intensity I correspond to phase difference
between a1 and a2 in the formula
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This hologram is also called a Gabor zone plate which is similar to a Fresnel zone plate
except that the bright and dark fringes shade continuously into each other. The ring pattern
on the hologram is very much similar to the circular fringes pattern obtained in Michelson
interferometer. A hologram may contain rings along with strips.
It is worth mentioning that for producing a hologram the object must be stationary. Also,
another important point worth mentioning is that the points on the hologram AB act as
diffraction grating.
Step II – The Reconstruction Process (Viewing a hologram)
The formation of an observable image from the hologram is called the reconstruction
process. It is retransformation (reconstruction) of the hologram into the image of the
object. No lens is needed in either step, and the second step can be carried out any length
of time-interval after the first step.
Plane wave fronts of reconstruction waves
C
A
first
order
P
eye
D
E
r
Zero order
F
Q
S
O
Virtual image with lens
O
M
First order
Real image
without lens
eye
B
(Hologram photo plate)
Fig. 18 The Reconstruction Process
Fig. 18 shows the reconstruction or viewing process in which AB is a hologram. See Fig.
17. AB is the transparent positive print of the developed film which contains bright and
dark fringes. A plane wave of monochromatic laser light S with the same wavelength as the
light used to expose the film is incident normally on the hologram AB, but in the absence of
scatterer O. AB acts as diffraction grating. Let P and Q be the two adjacent fringes on AB.
The black dots at P and Q mark points of constructive interference which develop as black
fringes on a hologram. Path difference between AB = .
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P
P
r

L
O

L
N
Q
r


O
Q
R
Fig. 19
From Fig. 19,
Path difference between OQ and OP is
All points from the hologram will produce diffracted light, as obtained in a diffracting
grating. The transmitted light through AB consists of three components, namely,
(i). a reduced – intensity undeviated continuation of the incident plane wave (This is
shown by PE and QF). This corresponds to zero-order diffraction pattern.(see Fig 18)
(ii). A diverging diffracted wave. (This is shown by PC and QD) This corresponds to first
order diffracted rays. (See Fig. 18) This appears to originate in a virtual image O that
corresponds in location and brightness to the object point (for example, point O in Fig 15).
The lens of the eye focuses these waves on the retina, where a real image is formed.
(iii). A diffracted wave that converges to form a real image
of the object (and thereafter
diverges); this image is reversed. By reversal of image we mean that if object is placed like
OR, its image will appear as OR (see Fig. 19).
The real image at
can be photographed or it can be formed on a screen and
photographic plate located there can be developed into a real picture. This image is threedimensional (3-D) and will change as the viewer moves his head. As the viewer moves his
eyes to different positions, he receives light form each and different sections of the
interference fringes pattern on the hologram. This is how he sees the object in different
perspective. If there is another object hidden behind the given object, he can see even this
hidden object just by moving his eyes to different position.
The virtual image at O has all characteristics of the object and can be seen on
looking through the hologram AB. But, this cannot be photographed.
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A
First order
Reconstruction coherent light wave
Zero order
Plane wave fronts
Virtual image
B
Real image
First order
Fig.20 Reconstruction of image from hologram AB
Fig. 20 shows an example of reconstruction of 3-D image. Incident light waves used
to illuminate hologram is called reconstruction waves. If the image is to be of the same size
as the object, the primary and reconstruction waves must both be plane waves with the
same wavelength. Illumination by reconstruction light of wavelength different from the
original will cause both a change in size and displacement of the image.
If the hologram is broken into many small pieces, each piece will act as a hologram
of the complete object scene. However, the perspective will be limited accordingly, and
there may be a loss in resolution.
The Off – Axis Hologram
In the method described above, Gabor found several technical difficulties in making
and then viewing the hologram. One difficulty was, of course, non-availability of coherent
(laser light), second difficulty was in the form of a real image O  caused by light diffracted in
the opposite direction. This image O was generally observed in front of the virtual image O,
and therefore it was in the way when viewing the virtual image (see Fig. 20). The method
described by Fig. 20 is called Gabor’s on – axis hologram which faced the two difficulties as
described above.
With the invention of the laser, the outlook for holographs changed completely.
Leith and Upatnicks in 1962 developed the idea of the off – axis hologram. This is an
extension of the Gabor’s on – axis hologram, using an off – axis section of the holographic
plate. Such a variation was possible by the large coherence length of the laser beam.
These were two advantages in having such a variation:
(i). The real image was observed separate from the virtual image line of sight.
(ii). It made possible for separate handling of the primary (reference) beam and scattered
beam. Thus, the object could now be illuminated from any side or several sides
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is
Plane Mirror
Incident beam
(or Reference wave)
Laser beam

Scattered beam
(object wave)
Object
A
Photographic plate
B
Fig 21 Production of hologram AB by off-axis technique
A
First order
eye
Zero order
Laser beam
First order
B
Hologram
Real Image
Virtual image
Fig 22 Illumination of hologram AB
The off-axis method of producing a hologram is shown in Fig. 21. The incident laser
beam is divided into two beams. One beam is obtained from reflection by a plane mirror
and second beam is the scattered beam from the object.
These two beams are
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superimposed to interfere on a photographic plate AB. The angle  between the scattered
light and the primary beam determines the density of the interference fringes, or spatial
frequency. When this angle  increases, the interference patterns are close.
Fig 22 shows viewing the hologram AB on which laser light is made incident. It
produces two diffracted waves, the first order on each side, as explained already (see Fig.
20). The remainder of the direct incident beam forms the unchanged order.
Hologram as a diffracting grating
The points on the hologram AB act as a diffracting grating. The diffracted waves
through AB contain the phase and amplitude of the waves originally diffracted from the
object when the hologram was made.
The object wave fronts have thus been
reconstructed. One of the diffracted beams forms a real image and another diffracted beam
forms a virtual image. This virtual image can be seen on looking through the hologram AB.
The hologram thus acts like a window through which the image can be observed. By
moving the head while looking through it, one can see more of the object originally hidden
from view. Thus, a three – dimensional view is recorded on a two-dimensional photographic
film. This is so because all parts of the object originally photographed have sent diffracted
(or scattered) waves to the photographic film.
22
Requirements for Holography
The requirements (or condition) for holography are as follow:
(i). Monochramaticity of light: Path difference depends on wavelength of light. The
interference pattern produced on the hologram depends on the wavelength of incident light
used. This condition is satisfied by laser light which is highly monochromatic. If the
primary beam consists of many wavelengths, each wavelength would give rise to its own
interference pattern on the hologram. This would result in overlapping of too many patterns
which would lead to average out the fringes to a smooth distribution.
(ii). Spatial coherence of light is the second requirement. This condition is again satisfied by
laser light which is highly spatially coherent. If the source of light is not spatially coherent
(i.e. if it is broad), each element of the source gives its own interference pattern. The
overlapping of these interference patterns from all the elements of the source produces
uniform illumination i.e. the fringe pattern is absent. For stable interference fringes, the
maximum path difference between the primary wave and the scattered wave from the
object should not exceed the coherence length.
(iii). The object must be stationary during making of hologram. A movement of a fraction of
a wavelength will produce a blurred holographic image. However, a three-dimensional
motion – picture holography may be feasible by using pulsed lasers.
23 Properties (or features) of a Hologram
The following are the main properties of a hologram
(1). A hologram has three-dimensional imaging properties.
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(2). It is a highly exact reproduction of the image
(3). Each part of a hologram, if broken into pieces, can produce the entire image with its
still three-dimensional character although with less sharpness. It is due to the fact that
each part of hologram receives light from all parts of the object and therefore contains, in
an encoded form, the entire image.
(4). The hologram itself would normally be regarded as negative but the image it produces
is a positive (as obtained in the reconstruction process). Such an image is distinguishable
from the image produced by the original.
(5). Multicolour holograms have been prepared by exposing a thick photographic emulsion
using three lasers that emit red, green, and blue light (the colours used for colourtelevision), and then viewing the hologram with white light. A photographic emulsion has
thickness due to which successive layers in emulsion can scatter a particular colour strongly
if their spacing is proper for that colour and angle of illumination.
(6). The holographic reconstruction of a scene has all the visual properties of the original
scene. Any object hidden behind another object can also be seen by moving eyes to
different position while viewing the hologram.
(7). The image can be enlarged simply by viewing the hologram in divergent light of
wavelength longer than with which it was made.
The image is of the same size as that of the object when primary and reconstruction waves
are plane waves of the same wavelength.
(8). A photographic emulsion having a thickness is equivalent to a large number of films
placed one over the other. Waves travelling into the emulsion get reflected from successive
layers in the emulsion and are in a position to interfere with there reflected waves. Their
superposition forms a three-dimensional standing – wave
pattern.
Such volume
holograms, when viewed in white light, give reconstruction in full colour.
Full – colour holography can then be achieved by exposing the film using three lasers
that emit red, green and blue light (the colours used for colour television), and then viewing
the hologram with white light. This method is a variation of the colour-photography process
invented in 1891 by the French Physicist Gabriel Lippmann (1845 – 1921), for which he
received the 1908 Nobel Prize in Physics.
(9). Several images can be recorded in a single hologram (as many as 100 or so). This is
achieved either by placing several objects in differently oriented locations and using a single
reference beam for simultaneous recording, or by exposing the plate in succession each
time turning the hologram plates by an angle.
24 Difference between Conventional Photography and Holography
Conventional Photography
1. A photograph records 2-D image of a
3-D
object,
using
a
lenssystem(camera)
Holography
1. Holography is a 3-D lensless method
of photography. It records 3-D image
of a 3-D object.
2. It is a photography by incoherent
light.
2. It is a photography by highly
coherent and monochromatic light.
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3. It records amplitude (or intensity) of
the wave. Intensity recorded on the
photographic film is
3. It records amplitude and phase of the
wave.
Intensity
recorded
on
photographic emulsion (film having
thickness) is (
)
4. Works with any type of waves – light;
x-ray, electron wave, sound wave.
5. The object to be hydrographic must
be at rest.
4. Works with light wave only.
5. The object to be photographed may
be at rest or in motion.
25 Theory of Holography as interference between two plane waves:
(1)
Recording of hologram:
x
x
In holography, first step
is recording of hologram. Let
AB be the photographic plate
(which
is
to
record
the
interference pattern) lying on
xy-plane (i.e. in the plane z=0),
as shown in Fig. 23. The plane
wave reflected from the object
at the plane z=0 is given by
A
Object plane wave
1
P
Reference plane wave
y
z
z
2
B Photographic plate
(compare with Fig.21)
Fig. 23Recording of the Hologram.
where A1 is amplitude and
is the phase of the wave (Here
). The disturbance
is called the object wave, and is the angle subtended by the propagation direction of the
object wave with the z-axis. The primary wave (also called reference wave) is a plane wave
travelling in the x-z planes, given by
where A2 is the amplitude,
is the phase of the primary wave. There is superposition
of the object wave and reference wave at P, causing interference pattern to be recorded on
the photographic plate AB. The resultant amplitude distribution at the plate is
The resultant intensity distribution at the plate is
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This is the intensity recorded on hologram in which the phase has not been lost. Here the
phase is
alongwith amplitudes A1 and A2. Conventional camera records
intensity
which
is
.
This is independent of phase of the wave
since ordinary photographic plate measures
intentsity.
I(x)
The third term of the Eq (24-3) for I(x)
represents a fringe system recorded on the
photographic plate. The amplitude A2 of the
x
primary wave is much greater than A1 and it
is constant. The interference fringes have
Fig 24 - Intensity distribution
an amplitude proportional to A1 and a phase
. They thus contain all the
information in the original wavefront along with the phase difference
which
is
, between
and
. Thus the blackening of the photographic plate
depends on three terms, i.e.,
distribution on the hologram.
,
and
. Fig.24 shows intensity
(2) Reconstruction:
The hologram as recorded above is illuminated with parallel coherent light, (which is
reconstruction wave) as shown in Fig. 25. The hologram is viewed in transmission. This
gives a wave front of constant phase but with amplitude AR varying as T0I(x), where T0 is a
constant transmission factor
x
A
x
conjugate planewave
(First order)
+
Reconstruction wave

zero-order wave
z
y
B
Primary plane wave
(First order)
Hologram
Fig. 25 Reconstruction by a plane wave
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The first two terms
give a beam in the same direction as the illuminating
wavefront with only a slight diffraction due to A. Since A2 >> A1, the first two terms reduce
to
. This corresponds to zero order wave which is attenuated incident wave. Third term
is rewritten as
This is the original wave front
multiplied by
and its complex conjugate
multiplied by
, (remembering A(x) is real). These two term have
phase shifts linear with x but of opposite sign, and cause two beams to emerge at angles 
to the axis of symmetry. The first is the reconstructed wave front, making an off-axis
virtual image, and the second is the complex conjugate of the reconstructed wavefront and
forms a real image of the object, also off axis, and pseudoscopic. These represent the
different orders of diffraction. Since the hologram is simply a sinusoidal grating, only two
first orders are seen as shown in Fig. 25. This indicates that there is a possibility of
reconstructing the original wavefront from the recording of the intensity pattern.
26 Application of Holography :
Holography has a broad range of applications in science, technology and medicine.
It is superior to ordinary photography for secret works. It is so because hologram contains
interference fringes and does not have image of the object as such. Some of the
applications are mentioned below:
(i). Three-dimensional display:
Three – dimensional displays have increased the use of holography in advertising
business. Multiplex holography is an example.
(ii). Holographic Interferometry:
This is a technique used for measuring small displacements. It is used to observe and
analyse the strain in a fractured bolt, the shock wave from a bullet or hidden flaws in
aircraft tyre. One of the applications of this technique is in the determination of Young’s
modulus of the material by the bending of beam method:
(iii). Hologgraphic Optical Elements:
Conventional optical elements such as lenses and gratings can be replaced by their
holographic counterparts because holographic optical elements can change optical wave
fronts in the same way as do lenses. These are usually used in optical data processing
systems.
(iv). Holographic Optical Memories:
Holograms have brought a major revolution in the size, capacity, speed and
usefulness of computers. A hologram is used as an information storage tool. It provides a
high capacity system for image storage and reexamination.
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Holographic memories are used in modern computers. Hologram may be used to
study transient microscopic events, because it contains the information about the depth of
the object.
(v). Acoustical Holography:
Acoustical holography uses an ultra – high – frequency sound wave (ultra sound) to
create the hologram initially, and a light beam (such as laser beam) then serves to form a
recognizable reconstructed image. The advantages of acoustical techniques over due to the
fact that sound waves can propagate considerable distances in dense liquids and solids
where light cannot. Thus acoustical holograms are quite useful in recording such diverse
things as underwater submarines and internal body organs.
(vi). Holographic beam combiner:
Holographic optical elements (HOEs) can already be found in many aircraft at the
heart of the head – on display, which is a device allowing the pilot to view a computer
display at infinity, superimposed over his normal field of view. The HOE used in this
application is referred to as a holographic beam combiner (HBC). This HBC can be made to
have very high reflecting power over a narrow spectral range and very low reflectance
outside this range. The colour of the computer display is matched with the reflectance of its
beam combiner, this allowing the head-up display a virtually unobstructed view of the
outside world, with the computer information clearly superimposed upon it. Politicians can
use head-up displays while delivering speeches to conferences; this way they can read
their script while facing the audience, without having to mumble over a piece of paper !
27
Solved Examples:
Example 1: A light wavetrain consists of 20 waves having wavelength  = 600nm.
What are its coherence length and coherence time ?
Solution 1:
The length of wavetrain
Coherence length,
L
Coherence time,
Example 2:
The coherence length for the red cadmium line of wavelength 6.438 x
105cm is 30 cm. Calculate the coherence time and the number of oscillation
corresponding to the coherence length.
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Solution :
,
Coherence length, L = 30 cm
Coherence time,
Number of oscillations corresponding to the coherence length
This is also equal to number of waves in coherence length L.
Example 3:
For a light source at mean wavelength 6000 Å, the coherence time is
21010 s. Deduce the order of magnitude of
(i). coherence length
(ii). The spectral width of the line
(iii). The purity factor
Solution :
(i). Coherence length, L = c, whose coherence time  = 21010 s and c = 3108
ms1
L = (21010) (3108) = 6102 m = 6 cm
(ii). Spectral width is given by
where



(iii)
Purity factor is


Example 4: The coherence length of D1 line is 2.5 cm.
5896Å. Calculate
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The wavelength of D1 line is
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Coherence and Lasers
(i). The line width of the line.
(ii). The purity factor.
(iii). The coherence time
Solution:
(i). The line width of the line (also called the spectral width of the line) is given by
where =5896 Å = 5896
And coherence length,
L = 2.5 cm = 2.5
(ii) Purity factor,
(iii) Coherence time,
Example 5:
The figure shows that phase (t) of a source that interferes with another
source which has a constant phase.
P(t)
2
1
2
3
4 t(s)
Fig. 27 Example 5.
(i). Will the eye detect interference ?
(ii). Will phototube detect interference ? (Assume response time of phototube =1ns)
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(iii). Are the two given sources coherent ?
Solution:
(i). Eye is able to detect events if they occur after a few milliseconds, i.e., the eye has
detection time of a few microseconds. The figure shows that (t) changes in about 1s.
The interference fringes will shift position many times during the time necessary for the eye
to perceive them and they will appear as a uniformly illuminated due to shifting of maxima
and minima. Hence, the eye will not detect interference.
(ii). Since phototube has detection time of 1ns = 109s. So it will register the maxima and
minima before they change their positions. Hence the phototube will detect interference.
(iii). For the eye the sources are incoherent for a phototube having detection time much less
than 1s, the two sources are coherent. It may be pointed out that any attempt to produce
coherent sources means production of interference maxima and minima.
Example 6. A monochromatic light of wavelength 5000 Å has the linear dispersion 1mm
per 20Å. This light is passed through an exit slit of 0.2 mm. What is the coherence time
and coherence length of the light ?
Solution :
Linear dispersion is
Reciprocal of liner dispersion is
When the width of slit is = 0.2 mm
Relation between the frequency bandwidth and the wavelength bandwidth is
Coherence time,
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Coherence length,
Example 7:
A He – Ne laser beam of =11.5107m is used to obtain interference fringes
in Michelson interferometer. The fringes remained visible when the path
difference was increased upto 8 m. Deduce the lower limits for
(i). The coherence length
(ii). Coherence time
(iii). Spectral half width
(iv). The purity factor of the source
Solution:
(i). coherence length, L = path difference = 8 m
(ii). Coherence time,
(iii). Spectral width,
(iv). Purity factor,
Example 8:
One of the most ideal line of krypton (orange) has a wavelength 6058Å and
coherence – length 20 cm. Calculate the line-width and coherence – time.
Solution :

Coherence length, L = 20 cm
Line width
Frequency spread
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Coherence time
Example 9: Two light beams having intensities in the ration 1:9 produce interference
fringes of visibility 0.3. What information do we get about the degree of
coherence ?
Solution:
Visibility (V) of the fringes is defined by
Let I1 and I2 be the intensities of the two given beams and let their ratio be 
Given
Observed value of V is Vobs = 0.3
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when V is visibility obtained for two coherent beams and V = C
coherence.
Hence degree of coherence of the beam is only
where C is degree of
or 50%.
Example 10: A ruby laser is aimed at a target 104 km away, in free space. Initially the
size of laser beam is 1.4 cm in diameter. Wavelength of laser beam is 6943 Å. Calculate
how large in diameter will the beam be when it hits the target ?
Solution:
The Fraunhaper diffraction of a circular aperture is given by the following equation
where x is the radius of the first minimum of the circular aperture in Fraunhofer diffraction,
D is the distance between the aperture and the screen and d is diameter of aperture.
Diameter of the spot on the screen to the first minimum is
Example 11: Two pin holes arrangement (similar to Young’s double slit arrangement) are
illuminated with sunlight falling normally on them. There is a filter in front of pin holes so
that light corresponding to  6000Å is incident on the pin holes. On the surface of earth
the apparent angular diameter of the sun is 0.50. What should be the separation between
the two pin holes so that fringes of good contrast are observed on the screen ? [or
calculate lateral coherence width of the sun].
Solution:
It is given that on the surface of earth the sun subtends an angle of 0.50.
The sun appears as a disc rather than a set of slits, so we need an equation that will
measure lateral coherence of a disc. The lateral coherence length (or width) is
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This is the value of lateral coherence width of the sun. Inother words, to get a two – slit
interference pattern from the sun, the two slits should be less than 8.4
mm apart.
Example 12: A narrow band of light with mean wavelength of 520 nm is chopped by a
shutter at a frequency of 40 MHz. Determine the bandwidth (in Å) of the resulting light.
Solution:
If we chop a 520 nm light beam at a frequency of 40 MHz, then coherence time is of the
order of
Bandwidth is
where coherence length, L = c = (3108)(2.5108)
Example 13: A Young’s double slit experiment is arranged such that the source slit S is a
pin-hole of diameter 1mm. What should be the distance d between S 1 and S2 (see Fig.28)
to get distinct interference fringes if SS1 = SS2 = 1m and wavelength of monochromatic
source S is 500nm.
Solution:
Let l be the diameter of pin-hole S. Let d be the distance
between S1 and S2. For a circular extended source, the
lateral coherence width is
S1
d
l
S
S2
a
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Fig 28 Example 13
Coherence and Lasers
Given,
The distance S1S2=d should be smaller than
interference fringes.
in order to obtain distinct
Example 14: In an experiment to demonstrate Young’s fringes light from a source slit falls
on two narrow slits 1mm apart and 100mm from a slit source. The incident wavelength is
5000Å. How wide can the source slit be made without seriously reducing the fringe
visibility?
Solution:
Let l be the width of the slit for which the interference fringes are under observation. When
the width becomes more than ‘l’, no fringes are observable
where given
Example 15: A Young’s double slit experiment is arranged such that the distance between
the centres of the two slits is 2.00mm and the source slit is placed 80cm away from the
double slit. If now the source slit is gradually opened up for what width will the first fringe
disappearance occur ?
Solution:
Given
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where is wavelength of light used. Thus, when the source slit is gradually opened up, the
first fringe disappearance occurs for the width
Example 16: A laser beam of power 100 mW has a wavelength of 7.2107 m and
aperture 5103m. The laser beam is sent to moon, the distance of which from earth is
4108m. Calculate (i) the angular spread (ii) aerial spread when the beam reaches the
moon (iii) and the intensity of the image on moon
Solution:
Given =7.2107m, Distance of moon from earth, D=4108m, and Angular spread=
Diameter of aperture, d=5103m
(i). For circular aperture, the angular spread is given by
(ii)
Aerial spread
(iii) Intensity of laser beam on the surface of the moon is
Summary
According to Bohr’s theory of hydrogen atom, a photon is emitted when an electron jumps
from higher orbit to a lower orbit. The emission of photon then ceases until another
electron jumps in a similar way. Experimental evidence shows that the duration of an
unbroken chain of light waves from a source which is monochromatic only has a life of about
108 or 109 second. Then other trains of waves are produced and there is no constant
phase difference between successive wave – trains.
This violates the condition of
interference and hence no interference maxima and minima are observed. In actual
practice it is found essential for the two sources (or two slits in Young’s double slit
experiment) to be identical in all respects on account of the relatively long time required for
an interference effect to be recorded, either on retina or photographically. This means that
the wavetrains necessary to produce interference effects must have a common origin; such
wave trains are said to be coherent.
Coherent Sources: The essential and the most sufficient condition for interference is that
there must be a constant phase difference between the two waves emitted by the two light
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sources (real or virtual). Such sources, having a constant phase difference and same
frequency, are called coherent sources.
A perfectly coherent wave is a wave which appears to be a pure sine wave for an
infinitely large period of time or in an infinitely extended space. For any source the average
length of a wave train is called the coherence length L, and the time taken by light to travel
this distance is called coherence time.
If there is a definite relationship between the phase of the wave at a given point and
at a certain distance away, then it is referred to as spatial coherence. On the other hand,
if there is a definite relationship between the phase of a wave at a given time and at a
certain time later, then it is referred to as temporal coherence.
Purity of a spectral line: A perfect spectral line has only length but no width.
Such a sharp monochromatic line corresponds to a perfect sinusoidal wave has infinite
coherence length and coherence time. As wavelength spread  decreases, the purity of a
line decreases. On the basis of Quantum Theory, the spectral lines have a finite purity Q,
defined by
If 0, Q, which is an ideal case. The concept of coherence length L is directly
related to the purity of a spectral line since
Also, the concept of coherence time  is directly related to the purity of spectral line
since
It means a sharp monochromatic line (
would correspond to an in infinite
interval of time (
. The quantity
represents the monochromaticity (or the spectral
line) of the source. Lateral coherence width(l) would have to be much greater than the
distance (d) between the two coherent sources in Young’s double slit experiment if we wish
to observe distinct interference fringes on the screen i.e., the condition d<< l is to be
satisfied in order to observe interference pattern.
Visibility of interference fringes is given by
where Imax and Imin are the maximum and minimum intensity respectively in an
interference pattern. The visibility V is a measure of the degree of coherence of the light
waves that produce interference pattern. Maximum value of V is unity and minimum value
is zero. For 1>V>0, the waves superimpose in part, interference is possible, with less
degree of contrast of the fringes.
Laser: Laser is a device that amplifies focused light waves and concentrates them in
a narrow, very intense beam which can be either pulsed or continuous. Laser is a highly
intense, monochromatic, coherent and unidirectional beam of light. The principle of laser is
based on the phenomenon of stimulated emission.
Requirement of Laser Action:
(i) The number of atoms in higher energy state must be greater than that in lower
energy state.
(ii) The energy density of stimulated emission must be large so that the stimulated
emission exceeds the spontaneous emission.
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The above two requirements underline the principle of laser. The requirement
mentioned in item (i) is referred to as population inversion. The process of creating
population inversion is called optical pumping.
Different Types of Lasers are such as
(i) Solid Laser (example: Ruby Laser)
(ii) Gas Laser (example: a mixture of helium-neon gas)
(iii) Semi-conductor Laser (example: Gallium arsenide i.e. Ga As)
(iv) Chemical Laser
(v) Dye-Laser
Lasers have found many application in technical and industrial; surgery; science and
research; holography; etc.
Further, holography is used in many areas of physics,
chemistry, biology and engineering.
Holography:
Holography is a method of recording and displaying; a three dimensional images of an
object, usually using coherent radiation from a laser and photographic plates without using
lenses or camera. Holography is a process by which the image of an object can be recorded
by the wave-front construction. It records the phases and amplitudes of the light waves
themselves. The photographic record of light waves thus produced is called hologram. It
has all the information about the object plus some additional information that is not
contained in the ordinary photograph. This additional information when reproduced, gives
rise to a three-dimensional image. This image is reconstructed by placing the hologram in a
laser beam.
Holography is a two – step process of image formation, viz.,
Step I: Formation of hologram by coherent light
Step II: The Reconstruction by re-illuminating the hologram by laser light of the same
wavelength. This step produces a three – dimensional image of the original object.
If the hologram is broken into many small pieces, each piece will act as a hologram of the
complete object scene. However, the perspective will be limited accordingly, and there may
be a loss in resolution. Hologram acts as a diffraction grating which, when laser light is
made incident, produces two diffracted waves, the first order on each side. The remainder
of the direct incident light forms the unchanged order. The diffracted waves through
hologram contain phase and amplitude of the waves originally diffracted from the object,
when the hologram was made. One of the diffracted beams forms a real image and another
diffracted beam forms a virtual image. By moving the head while looking through it, one
can see more of the object originally hidden from the view. Thus, a three-dimensional view
is recorded on a two-dimensional photographic film.
This is so because all parts of the
object originally photographed have sent diffracted waves to the photographic film
Requirement for Holography: Conditions for holography are as follows:
(i) Monochromaticity of light:
(ii) Spatial coherence of light
(iii) Object to remain stationary during making of hologram.
A laser beam satisfies the above conditions.
Applications of Holography: It has a broad range of applications in science, technology
and medicine.
Exercise
1. How is light emitted by a conventional light source? Explain with example. Does
such a source emit a simple harmonic wave.
[Hint: This can be explained on the basis of Bohr’s theory of hydrogen atom. A
conventional light source (e.g. sodium lamp) does not emit a simple harmonic wave;
it emits a pulse of short duration]
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2. Explain the concept of coherence.
Discuss temporal coherence and spatial
coherence. Illustrate them with the help of suitable experiments
3. Is strictly monochromatic light realizable practically? Comment.
4. Give a brief answer to : (a) Can two independent sources of light produce an
interference pattern? (b) Can two independent laser beams produce an interference
pattern ?
5. Distinguish between:
(i)
‘Coherence between two waves and coherence of a given wave.
(ii)
Temporal coherence and spatial coherence
6. Explain the meaning of ‘Incoherent sources’ and ‘Partially coherent sources’.
Illustrate with an example.
7. What do you mean by width of a spectral line and the frequency stability ?
8. Write the relation between coherence time and coherence length. Describe how
Michelson interferometer may be used in determining the coherence time (or the
coherence length).
9. Explain how the purity of a spectral line is related with the concept of temporal
coherence. Prove that the frequency spread of a spectral line is of the order of the
inverse of the coherence time.
10. What do you understand by degree of contrast of the fringes produced by
interference of two waves ? Show that it is equal to the degree of coherence between
the waves.
11. The orange Krypton line ( = 6058 Å) has a coherence length of about 20 cm.
Calculate the line width and the frequency stability.
(Ans. ~ 0.01 Å ; ~ 1.5 x 106)
[Hint:
Line
8
width

and
frequency
stability
is
where
where
-1
c = 310 ms ]
12. Show that the frequency spread of a spectral line is inverse of temporal coherence.
[Hind : Temporal coherence   and
where  is coherence time]
13. The Young’s double slit experiment with white light produces only a few colured
fringes on the screen of observation. Explain this phenomenon qualitatively on the
basis of coherence length.
[Hint: Visible spectrum extends from 4000Å to 7500 Å. Each wavelength has its own
coherence length and interference fringes are formed due to superposition of these
waves]
14. Define coherence time and coherence length. What do you understand by line-width
and the frequency spread ?
15. Discuss spatial coherence due to a point source and due to an extended source.
Give an example of a source which can have coherence length a few kilometers.
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[Example is laser beam which can have coherence length of the order of a few
kilometers]
16. What is the order of lateral coherent width of sunlight ?
[Ans.: About 1 m]
17. Explain the concept of coherence. What do you mean by temporal coherence and
spatial coherence. Explain them with suitable experiments using light and laser
beams.
18. What do you mean by degree of coherence ? How does the visibility of fringes
depend on the degree of coherence ?
19. Explain how the purity of a spectral line is related to the concept of temporal
coherence. Show that the frequency spread of a spectral line is of the order of the
inverse of the coherence time.
20. What is a laser ? Distinguish between laser and non-laser light sources.
21. A Young’s double slit experiment is arranged such that the distance between the
cenres of the two slits is d and the source slit, emitting light of wavelength  is
placed at a distance x from the double slit. If now the source slit is gradually
opened up, for what width will the fringes first disappear ?
22. Show that if one were to perform the Young’s double slit interference experiment,
then the distance between the two slits would have to be much less than the lateral
coherence width in order to obtain distinct interference fringes.
23. With the help of Young’s double slit experiment find a relation between the linear
dimension and spatial coherence of a source for a given separation between the slits.
24. Explain why the interface effects which can be observed when the light waves from
the same source are superposed becomes less distinct when the optical path
difference traversed by the waves is increased.
25. What is stimulated absorption ? Explain the difference between spontaneous and
stimulated emission.
26. How can we achieve higher probability of stimulated emission as compared to that of
spontaneous emission ?
27. What do you understand by the phenomenon of population inversion? Explain the
procedure to achieve population inversion ?
28. (a). Distinguish between laser and non-laser light sources.
(b). Name some important types of lasers and discuss Ruby laser in detail.
29. (a). Explain briefly the working principle of a laser.
(b). what are the properties of laser ?
(c). Discuss the important application of laser.
30. What are the basic differences between a laser emission and emission of an ordinary
discharge lamp ?
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31. A laser beam has a power of 100 mW. It has an aperture of 5 103m and emits a
light of wavelength 6943Å. The beam is focused with a lens of focal length 0.1m.
Calculate the area and intensity of image.
(Ans.: 2.891010 m2 ; 3.46108 Wm2 .)
[Hint: Angular spread
rad
Area = Areal spread =
Intensity =
]
32. What are the important features of stimulating emission ? Discuss the essential
requirements for producing laser action.
33. (a). What is holography ? What is the difference between ordinary photography and
holography
(b). Why is holography superior to ordinary photography for secret works ?
(c). Give a stepwise method for obtaining holographic photographs.
34. What is the fundamental principle of a hologram ? How is hologram produced and
how is the image reconstructed from it.
35. Explain the theory of holography as interference between two plane waves.
36. (a). What are the essential requirements for holography ?
(b). Explain the theory of holography on which recording of a hologram and
reconstruction of the image are based.
(c). Mention various properties of a hologram.
37. State the principle of holography. Why is a laser beam needed for it ? Can one plate
of photographic emulsion record more than one holograms ?
38. Explain the essential requirements for producing laser action.
requirements are usually obtained.
Outline how these
39. Describe how spatial coherence leads to high directionality of a laser beam.
40. For a source radiating at =5400Å the coherence time  is 21010 s. Deduce the
order of magnitude values of
(i) the coherence length
(ii) the spectral half-width of the radiation
(iii) the purity of the spectral line.
[Ans.: (i) 6 cm ; (ii) 0.05Å;
(iii) 105]
41. Light from a 2.5 mW laser source of aperture 1.8 cm diameter and =5400Å is
focused by a lens of focal length 20 cm. Deduce the area and intensity of the image.
[Ans.: 1.410-6 cm2; 1.7 kWcm2 ]
42. A laser beam of wavelength 7400Å has coherence time 4105s.
order of magnitude of its coherence length and spectral half-width.
[Ans: 12 km; 0.46 106 Å]
[Hint:
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Coherence and Lasers
,]
43. The length of a wave train is 10 long where  is 650nm.
wavelength and in frequency
(Ans.: 65 nm ; 4.621013 Hz
Calculate spread in
[Hint:
]
44. For a commercial available laser beam, coherence time is of the order of 50 ns.
Compare its monochromaticity with the conventional sodium vapour lamp.
[Hint: Monochromaticity is
where
. For Na-lamp,
, whereas for laser
. Assuming 
of the same order for laser and Na-lamp.
45. Explain why a laser beam of 100 mW can be focused to drill holes through a steel
plate whereas a torch beam of even 100W would not do that.
[Hint: Laser beam is highly directional; it continues to travel in the same direction
with very little scattering. Whole of its energy can be focused over a small area
(106 m2 108 m2) . Its intensity =
Wm2 . Torch beam spreads in all
direction over surface area of a sphere of radius r. Its intensity at a distance 10 m is
which is quite small in comparison to
Wm2 for laser beam. That
is why laser can be used as an exceeding effective drill to burn through a target]
46. In a hypothetical Young’s experiment, where one of the two pinholes is new covered
by a neutral density filter that cuts the irradiance by a factor of 10, and the other
hole is covered by a transparent sheet of glass, so there is no relative phase shift
introduced. Computer the visibility of completely coherent illumination.
(Ans.: 0.57) [Hint: See example
The ratio of intensities is
Visibility
V=
47. What are coherent sources ? Is it necessary that coherent sources must result from a
single source of light? If yes, why ? Distinguish between spatial coherence and
temporal coherence. Can two sources of light derived from the same source be
incoherent ?If yes, under what circumstances?
48. What do you understand by the terms temporal coherence and spatial coherence ?
Explain the role played by each in governing the visibility of fringes ?
49. Discuss the basic principle of holography. How can we obtain a three-dimensional
image of an object using it on reconstruction?
50. (a). Define the term temporal coherence, coherent length and coherent time.
(b), discuss spatial coherence. Derive condition for interference for an extended
incoherent source
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Coherence and Lasers
(c) A pin hole of 1mm diameter is used as a source for double slit interference
experiment using sodium light of wavelength =5890Å. If the distance from the pin
hole to the slits is 0.5 m, what is the maximum slit spacing such that the
interference fringes are just observable ?
(Ans.: c) : about 0.04 mm
[Hint: (c):
.
Slit spacing should be smaller than 0.04mm. Hence maximum slit spacing is about
0.04 mm.]
51. Fringes of equal inclination are observed in a Michelson interferometer illuminated by
monochromatic light. What will be observed if one then changes to white light ?
Explain your answer in terms of coherence length.
[Hint: if one changes to white light, the field of view will simply be white with no
fringes visible. The constituent wavelengths produce fringes of different diameters
which get intermingled to such an extent that they are not discernible. In terms of
coherence length, one can say that the effective coherence length is virtually zero
and very much smaller than a path difference 2t(t = thickness of air films) so that no
interference effects are observable];,
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