< Engineering Physics I >
<Lasers - Principles and Einstein’s coefficients>
Introduction
Learning Objectives
On completion of this session you will be able to:
1. Explain the principles of “spontaneous emission” and “stimulated emission”
2. Explain the characteristics of Laser.
3. Understand Einstein’s A and B coefficients and derive a relation between them.
Spontaneous and Stimulated emission
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<Lasers - Principles and Einstein’s coefficients>
Spontaneous emission
•
•
•
•
It is the process by which a light source such as an atom, molecule, nanocrystal
or nucleus in an excited state undergoes a transition to the ground state and
emits a photon.
The atoms in the excited state drop to the lower energy state after they have
stayed in the excited state for a short duration of time called their life time.
During this process photons of energy h‫ = ע‬E2 - E1, i.e. energy difference
between the two states of the atom, are emitted.
Spontaneous emission of light or luminescence is a fundamental process that
plays an essential role in many phenomena in nature and forms the basis of many
applications, such as fluorescent tubes, television screens, plasma display
panels, lasers and light emitting diodes.
Stimulated absorption
•
•
•
For the photon-atom interaction to occur, the energy of the interacting photon
h‫ ע‬should match with the energy difference between the two states of the atom
involved in the interaction, i.e. h‫ = ע‬E2 - E1.
When the atom at the lower energy level E1 absorb the incident energy with the
corresponding frequency, it jumps to the upper energy level E2, this is called
stimulated absorption.
This process reduces the lower level population and increases the upper level
population.
Stimulated emission
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•
•
When photon interacts with atom which is in the excited state E2, then deexcitation of this atom to the lower energy state E1 occurs with emission of
photon of energy h‫ע‬. Therefore, stimulated emission is the process by which,
when perturbed by a photon, matter may lose energy resulting in the creation of
another photon.
The perturbing photon is not destroyed in the process, and the second photon is
created with the same phase, frequency, polarization, and direction of travel as
the original.
The process can be thought of as "optical amplification" and it forms the basis of
both the laser and maser.
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<Lasers - Principles and Einstein’s coefficients>
Characteristics of LASER
The following are the properties of laser, which differentiate it from ordinary light.
1. Monochromaticity: The light emitted from a laser is monochromatic, i.e. it is of one
wavelength (colour). But in case of ordinary light, many wavelengths of light are
emitted.
Ordinary light
laser light
2. Directionality: Laser emits light that is highly directional. Laser light is emitted as a
relatively narrow beam in a specific direction. Ordinary light such as coming from the
sun, a light bulb, or a candle, is emitted in many directions away from the source.
The degree of directionality is expressed in terms of divergence. The divergence tells
how rapidly the beam spreads when it is emitted from the laser.
The angle of divergence can be expressed in degrees as
Φ = ( a2 – a1 ) / 2( d2 – d1 ) ,
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where d1 and d2 are the distances from the laser window to the two spots and a1 and a2
are the diameter of the spots.
3. Coherence: The light from a laser is said to be highly coherent, which means that
the waves of laser light are in same amplitude and phase. There are two types of
coherence, namely temporal coherence and spatial coherence.
Temporal coherence: Temporal coherence is a measure of the correlation between the
phases of a light wave at different points along the direction of propagation. Temporal
coherence tells us how monochromatic a source is.
Temporal coherence is measured in terms of coherent time.
Coherent time = Coherent length/ Velocity of light.
Coherent time is the indication of the total time in which the wave passing through any
point is continuous.
Spatial coherence: Spatial coherence is a measure of the correlation between the
phases of a light wave at different points transverse to the direction of propagation.
Spatial coherence tells us how uniform the phase of the wave front is.
More precisely, the spatial coherence is the cross-correlation between two points in a
wave for all times.
If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially
coherent.
4. Intensity: Lasers are bright and intense light sources. This is because of coherence
and directionality.
The intensity is enormous due to the coherent addition of amplitude and negligible
divergence.
Einstein’s A and B coefficients
Einstein proposed a mathematical expression for the existence of stimulated emission
of light, which is known as Einstein’s expression.
In atomic system, all three transition processes - stimulated absorption, spontaneous
emission and stimulated emission occur simultaneously. When photons interact with
atoms, both upward (absorption) and downward (emission) transitions occur. At
equilibrium these transition rates must be equal.
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<Lasers - Principles and Einstein’s coefficients>
Let N1 and N2 be the number of atoms per unit volume with energy E1 and E2
respectively. Let ‘n’ be the number of photons per unit volume at frequency ‫ ע‬so that
h‫ = ע‬E2 – E1. Then energy density of interacting photons ρ(‫ )ע‬is given as
ρ(‫ = )ע‬nh‫ע‬
Upward transition
The rate of stimulated absorption depends on the number of atoms available in the
lower energy state and the energy density of the interacting photons.
Stimulated absorption rate α N1
α ρ(‫)ע‬
= N1 ρ(‫ )ע‬B12
where B12 is the constant of proportionality known as Einstein coefficient of stimulated
absorption.
Downward transition
When the atoms are excited by stimulated absorption, they remain in the excited state
for a short duration of time called the life time of the excited state. After their life
time they fall to the lower energy level spontaneously by emitting photons. The rate of
spontaneous emission depends on the number of atoms in the excited energy state.
Spontaneous emission rate α N2
= N2 A21
where the constant of proportionality A21 is the Einstein coefficient of spontaneous
emission.
Before the atoms in the excited state de-excite to their lower energy states by
spontaneous emission, they may interact with photons resulting in stimulated emission
of photons. So rate of stimulated emission depends on the number of atoms available in
the excited state and the energy density of interacting photons.
Stimulated emission rate α N2
α ρ(‫)ע‬
= N2 ρ(‫ )ע‬B21
where the constant of proportionality B21 is the Einstein coefficient of stimulated
emission.
During stimulated emission, the stimulating photon and the stimulated photon are in
phase with each other.
During stimulated absorption, the photon density decreases but during stimulated
emission, the photon density increases.
When the system is in equilibrium, the upward and downward transition rates must be
equal.
N1 ρ(‫ )ע‬B12 = N2 ρ(‫ )ע‬B21 + N2 A21 ………..
(1)
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<Lasers - Principles and Einstein’s coefficients>
Hence
= N2 A21 / N1 B12 - N2 B21 ………..
ρ(‫)ע‬
(2)
The Einstein relations are given by the equations 3 and 4.
g1B12 = g2B21
……..
(3)
A21/B21 = 8πh‫ע‬3/c3 ……..
(4)
Note: The derivation for Einstein relations can be referred in section 5.2.1 of
“Engineering Physics” by ‘P.K. Palanisamy’.
Check your understanding
1. Choose the right answer from the options given below:
When radiation interacts with atom, the energy of the interacting photon h‫ע‬
should be
a) greater than the energy difference between the two states of the atom.
b) lesser than the energy difference between the two states of the atom.
c) equal to the energy difference between the two states of the atom.
2. State if the following statement is true or false?
An one milliwatt He-Ne laser is 100 times brighter than the sun.
d) True
e) False
3. Fill in the blank with the right answer.
The rate of stimulated emission depends on _________ and ____________.
Check the correct answers on page _9_.
Summary
On completion of this chapter you have learned that:
1.
• When the atom at the lower energy level E1 absorb the incident energy with the
corresponding frequency, it jumps to the upper energy level E2 by a process
called stimulated absorption.
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< Engineering Physics I >
•
•
<Lasers - Principles and Einstein’s coefficients>
If the radiation interacts with atoms which are already in the excited state, then
atoms lose energy and drop to the lower energy state with emission of photons.
This process is called stimulated emission.
state
Spontaneous emission is a process in which the atoms in the excited
drop to the lower energy state after they have stayed in the excited state for a
short duration of time called their life time.
2. The properties of laser such as monochromaticity, directionality, coherence and
high intensity differentiates it from ordinary light.
3.
•
•
All three transition processes - stimulated absorption, spontaneous emission and
stimulated emission occur simultaneously. At equilibrium upward (absorption)
and downward (emission) transition rates must be equal.
Einstein coefficient of stimulated absorption, spontaneous emission and
stimulated emission are related to each other.
Activity
1. Search the web and learn about the history of development of lasers.
2. Illustrate stimulated absorption, spontaneous emission and stimulated emission
through energy level diagrams.
Suggested Reading
1. ‘Engineering Physics’ by P.K. Palanisamy.
2. ‘A textbook of Optics’ by Brij Lal and Subrahmaniam.
Answers to CYU.
1. c
2. d
3. number of atoms in the excited state and energy density of interacting photons.
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