AYAN KR. MONDAL
Mob : 8981580693.
Email : ayan.mondal15@ymail.in
QUOTE : In Science there is only Physics, all the rest is Stamp Collecting – LORD KELVIN
PART III (CU)
LASER PHYSICS
MARKS : 5
Intro: LASER is a type of device that can be used to produce high monochromatic coherent beam of light of
very high power density. It is an acronym (BcÉrl¡ - ¢h¢iæ n-ël BcÉrl ¢c-u N¢Wa pw¢rç ¢e-cÑnL në) of
Light Amplification by Stimulated Emission of Radiation.
Principle: LASER works on the phenomenon of ‘stimulated or induced emission’ of radiation process, the
theory which was worked by Einstein in 1917.
Scientific background: (see A.B GUPTA, page 303)
Frequency and Temperature dependence:
Usual light sources provide luminous intensities corresponding to thermal radiation at best ~ 104 K
with lasers intensities corresponding to ~ 1020 − 1030 K are rather obtained easily i.e. a body in this huge
range of temperature emits laser.
Characteristics of LASER: (i) Complete spatial coherence source, where all the waves stay in phase.
(i) It is a plane polarised beam of light.
(ii) It is unidirectional flow of light.
(iii) It has very small broadening.
(iv) Intensity is considerably very high.
(iv) It is monochromatic type.
Absorption and Emission of radiation:
We have three kinds of transition involving electromagnetic radiations are possible between two
energy levels E1 and E2 in an atom. They are,
(i) Induced absorption: Consider an atom is initially in the lower state E1 ,
it can be raised to E2 by absorbing a photon having energy,
E2 – E1 = ℎ𝜈 [𝜈 = frequency].
This process is called induced absorption.
Einstein postulated that the induced absorption transition rate
was directly proportional to the – (i) number of atoms with electrons in
the lower state and (ii) density of incident radiation energy on the atoms.
𝑑𝑁
𝑑𝑁
i.e. 12 ∝ 𝑁1 𝑢(𝜈)
𝑜𝑟, 12 = 𝐵12 𝑁1 𝑢(𝜈)
where,
𝑑𝑁12
𝑑𝑡
𝑑𝑡
𝑑𝑡
= induced absorption transition rate
N1 = number of atoms with electrons in the n=1 state
𝑢(𝜈) = density of the incident electromagnetic radiation (with energy equal to the energy difference
between the two states)
𝐵12 = Einstein coefficient for induced absorption
𝑑𝑁
Definition of 𝐵12 ∶ For N = 1, 𝑢(𝜈) = 1 then 𝑑𝑡12 = 𝐵12 i.e. if the number of atoms in initial state and the
density of the incident electromagnetic radiation be unity, then the transition rate of induced absorption is
known as Einstein coefficient for induced absorption (𝐵12 ).
(ii) Spontaneous emission: An atom stays in the excited state usually for
a short period of 10−8sec. If an atom is initially at the excited state E2 ,
then after 10−8sec it returns to its own initial state E1 . This process is
termed as spontaneous emission or de-excitation of the photon energy.
The energy of the emitted photon is, ℎ𝜈 = 𝐸2 − 𝐸1 .
According to the Einstein postulations the spontaneous emission transition rates (or probability of
spontaneous emission) were proportional to the number of atoms with electrons in the upper state.
𝑑𝑁21
i.e.
= 𝐴21 𝑁2 where 𝐴21 = Einstein co-efficient for spontaneous emission and 𝑁2 = number
𝑑𝑡
of atoms available with electrons in the upper excited energy state.
(iii) Stimulated emission: Say an atom is already is in excited state E2, if a photon of
frequency 𝜈 i.e. of energy ℎ𝜈 (= E2 – E1) irradiates ( ¢LlZhoÑZ Ll¡ , radiates) the
atom, it cannot excite the atom which is already is in excited state. But the
photon produces an equivalent effect by de-excite the atom. So, under the
AYAN KR. MONDAL
Mob : 8981580693.
Email : ayan.mondal15@ymail.in
QUOTE : In Science there is only Physics, all the rest is Stamp Collecting – LORD KELVIN
influence of the electromagnetic field of the incident photon, the previously excited atom makes a transition
to the lower energy state E1, emitting an additional photon of the same frequency 𝜈. Thus we get two
photons, one original and the other emitted one of same frequency. This type of transition is termed as
stimulated or induced emission.
Similarities of the two photons: (i) have same frequency, (ii) will be in phase, (iii) will move in the same
direction.
According to the postulates of Einstein, the transition rate for stimulated emission is proportional to
the – (i) numbers of atom available with electrons in the upper excited state (N2) and (ii) density of
radiation incident on the atoms with energy equal to the difference between two states[u(𝜈)].
𝑑𝑁21
Thus,
= 𝐵21 𝑢(𝜈)𝑁2 where B21 = Einstein coefficient for stimulated emission.
𝑑𝑡
Relation between Einstein A and B coefficients:
Let us consider an assembly of atoms in thermal equilibrium at a temperature T having radiation
frequency 𝜈 and energy density u(𝜈). Say N1 and N2 are the number of atoms per unit volume in energy
states E1 and E2 respectively at any instant of time.
Now the number of atoms in state E1 that absorbs a photon and rise to
the state E2 in unit time is, 𝑁1 𝐵12 𝑢(𝜈)
The number of atoms in state E2 that drops to state E1 by – (i) spontaneous emission is N2A21 or (ii)
stimulated emission of two photons in per unit time is, 𝐵21 𝑢(𝜈)𝑁2
Hence in equilibrium, the absorption and emission rates per unit time must get equal,
𝑁1 𝐵12 𝑢(𝜈) = N2 A21 + 𝐵21 𝑢(𝜈)𝑁2
𝐴21
 𝑢(𝜈) = 𝑁2 𝐴21⁄(𝑁 𝐵 − 𝑁 𝐵 ) =
1 12
2 21
𝐵21
According to 𝑀𝑎𝑥𝑤𝑒𝑙𝑙 − 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑙𝑎𝑤,
Substituting this value in eqn. (i) we get, 𝑢(𝜈) =
𝐴21
1
𝑁2
𝑁1
=𝑒
𝐵21
=
8𝜋ℎ𝜈3
𝑐3
and
−(𝐸2 −𝐸1 )
𝑘𝑇
𝐵12
𝐵21
= 𝑒
−ℎ𝜈⁄
𝑘𝑇
1
𝐵21 𝑒 ℎ𝜈⁄𝑘𝑇 (𝐵12 )−1
𝐵21
8𝜋ℎ𝜈3
Comparing eqn. (ii) with Planck’s radiation formula, 𝑢(𝜈) =
𝐴21
......... (i)
𝑁1 𝐵12
(
)−1
𝑁2 𝐵21
𝑐3
......... (ii)
1
ℎ𝜈
𝑒 ⁄𝑘𝑇
we get,
−1
=1
So we can conclude,
(i)
𝐴21
𝐵21
𝛼 𝜈 3 i.e. the probabilities of spontaneous emission (A21) increase as 𝜈 3 . This implies that laser
action becomes more difficult at higher frequencies of photons because the probabilities of
spontaneous emission rapids with the energy difference between the involved states.
(ii) B12 = B21 i.e. the probabilities of stimulated absorption and stimulated emission are equal.
These two relations are called Einstein’s relations and the A’s and B’s are respectively known as
Einstein’s A and B coefficients. These coefficients cannot be calculated by the classical electromagnetic
theory. We can calculate B by using Dirac’s quantum theory and thence A can be obtained from this.
Q.1. What do you mean by the life-time of higher energy state?
 The reciprocal of A coefficient of Einstein’s relation i.e. 1/A21 is a measure of the lifetime of the upper high
energy state (E2) against the spontaneous decay to the lower state (E2).
Concept of Population Inversion:
Usually in the normal condition of thermal equilibrium, the numbers of atoms in the higher energy
state are considerably smaller compared to the numbers of atoms present in the lower energy level. This is
also true by applying Boltzmann law,
𝑁2
𝑁1
=𝑒
−(𝐸2 −𝐸1 )
𝑘𝑇
as N2 < N1. This says that whether B12 = B21, the
stimulated emission is very little compared with the absorption. Let by some means, the atoms be initially
excited so that we get more atoms in the higher energy state E2 than the lower one i.e. N2 > N1. The situation
in which the number of atoms in the higher energy state exceeds the number of atoms in the lower state is
known as population inversion. The method by which population inversion is achieved is termed as
pumping. There are several methods of pumping like, (i) optical pumping, (ii) excitations by electrons, (iii)
inelastic collisions between the atoms, (iv) chemical excitation.
AYAN KR. MONDAL
Mob : 8981580693.
Email : ayan.mondal15@ymail.in
QUOTE : In Science there is only Physics, all the rest is Stamp Collecting – LORD KELVIN
Main components of a Laser system:
A basic LASER system consists of the following main components,
i) Active or laser Medium: It can be a collection of atoms, molecules or ions of a solid, liquid or gas or
simply semiconductor junction.
ii) External source for Pumping: An external source is used to cause the population inversion between a
pair of energy levels of the atomic system.
iii) Optical resonator: It is usually a pair of mirrors that reflect the light back and forth through the
active medium.
Threshold condition for laser action:
Consider an assembly of atoms with two energy states E1 and E2. Let N1 and N2 be the number of atoms
per unit volume in the states E1 and E2 respectively.
Let a laser beam of energy density 𝑢(𝜈) propagate through an active laser medium along +X direction.
Consider a slab of thickness dx and area of cross-section S as on the figure.
Energy loss per sec in the slab due to stimulated absorption in the volume 𝑆 𝑑𝑥 is, 𝐵12 𝑢(𝜈)𝑁1 (𝑆𝑑𝑥)ℎ𝜈
Energy gain per sec in the slab due to stimulated emission in same volume is, 𝐵21 𝑢(𝜈)𝑁2 (𝑆𝑑𝑥)ℎ𝜈
Here we ignore the radiation due to the spontaneous emission because such radiations are directed at
random.
∴ Rate of increase of energy in the volume Sdx is given by,
[𝐵21 𝑁2 − 𝐵12 𝑁1 ] 𝑢(𝜈) ℎ𝜈. 𝑆 𝑑𝑥 = [𝑁2 − 𝑁1 ]𝐵. 𝑢(𝜈)ℎ𝜈. 𝑆𝑑𝑥 (using, 𝐵12 = 𝐵21 = 𝐵) ............ (i)
Now , the net energy leaving the volume 𝑆 𝑑𝑥 per second is, [ 𝐼𝜈 (𝑥 + 𝑑𝑥) − 𝐼𝜈 (𝑥)] 𝑆
= [𝐼𝜈 (𝑥) +
=
Equation (i) and (ii) must be equal i.e.
𝑑𝐼𝜈
𝑑𝑥
Now the rate of increase of intensity is given by,
where 𝑣 =
For
𝑑𝑥
𝑑𝑡
𝑑𝐼𝜈
𝑑𝑥
𝑑𝐼𝜈
𝑑𝑥
𝑑𝑥 − 𝐼𝜈 (𝑥)] 𝑆
𝑑𝑥 𝑆
.............. (ii)
= [𝑁2 − 𝑁1 ]. 𝐵. 𝑢(𝜈). ℎ𝜈. 𝑆𝑑𝑥
𝑑𝐼𝜈
)
𝑑𝑡 𝑔𝑎𝑖𝑛
(
=
𝑑𝐼𝜈 𝑑𝑥
.
𝑑𝑥 𝑑𝑡
=
𝑑𝐼𝜈
.𝑣
𝑑𝑥
= [𝑁2 − 𝑁1 ]𝐵. 𝑢(𝜈). ℎ𝜈. 𝑣
is the velocity of light in the medium
𝑑𝐼
( 𝑑𝑡𝜈 )
> 0, N2 > N1 which implies that for the growth of laser intensity we must have population
𝑔𝑎𝑖𝑛
inversion.
Now the total loss of intensity is given by,
𝑑𝐼𝜈
)
𝑑𝑡 𝑙𝑜𝑠𝑠
(
=
𝐼𝜈
𝑡𝑝
where tp = life time of the photon in the cavity
𝑑𝐼𝜈
)
𝑑𝑡 𝑔𝑎𝑖𝑛
Thus we get the condition for the sustained laser action as, (
𝑑𝐼𝜈
)
𝑑𝑡 𝑙𝑜𝑠𝑠
≥ (
 (𝑁2 −𝑁1 ). 𝑢(𝜈). ℎ𝜈. 𝐵. 𝑣 ≥
Now
𝐴
𝐵
=
8𝜋ℎ𝜈3
𝑐3
𝐼𝜈
𝑡𝑝
........(iv)
, A=A21 , the spontaneous transition probability from 2 to 1. Also 1/A21 = 1/A = tsp =
spontaneous life time of the upper state and𝐼𝜈 = 𝑢(𝜈). 𝑣. Putting these values we get,
(𝑵𝟐 −𝑵𝟏 ) ≥
𝟖𝝅𝒕𝒔𝒑 𝝂𝟐
𝒄𝟑 𝒕𝒑
This gives the threshold condition for the laser action.
Three level laser system (Pulsed Ruby Laser):
In about 1960 Maiman built the first ruby crystal. Ruby is a type of crystal where 𝐴𝑙2 𝑂3 is doped with
~ 0.04% 𝐶𝑟2 𝑂3 so that some of the aluminium ions in the crystal lattice are replaced by𝐶𝑟 3+ ions.
Construction: These type of laser system consist a cylindrical ruby rod whose both ends are optically flat
and accurately parallel. One end is fully silvered and the other is only partially silvered. The rod is
completely surrounded by a helical xenon flash tube which acts as the optical pumping system.
AYAN KR. MONDAL
Mob : 8981580693.
Email : ayan.mondal15@ymail.in
QUOTE : In Science there is only Physics, all the rest is Stamp Collecting – LORD KELVIN
Working Principle: When xenon lamp is switched on, the flash from it excites a large number of Cr - ions
from ground state E1 to the level E3 by absorption of light of wavelength 550nm from the flash tube. Cr-ions
have an intermediate metastable state E2 , so the excited ions quickly undergo non-radiation/radiationless
transition with a transfer of energy to the lattice thermal motion to the level E2 . Cr-ions remain there
relatively longer for few milliseconds. Now, the population of the E2 level becomes greater than that of the
E1 level, thus ‘population inversion’ is achieved.
Some photons are produced by spontaneous transition of Cr-ions from E2 to E1 having wavelength of
6243Å. This process stimulates the emission of similar other photons. The chain reaction continues and
quickly develops a beam of photons all moving parallel to the rod, which is monochromatic and coherent.
When the beam develops sufficient intensity, it emerges through the partially silvered end of the rod.
Here once all the Cr-ions in the metastabe state have returned tom the ground state, the laser action
stops. It is then necessary to send one more flash of pumping radiation through the rod. Thus we can say a
ruby laser operates only in pulses, for which why we have termed it as ‘pulsed ruby laser’.
Four level laser system (He-Ne laser):
In 1961 Ali Javan and his co-workers developed the first gas laser by Helium and Neon which was the
most popular continuous wave lasers.
Construction: A helium–neon laser or He-Ne laser, is a type of gas laser that consists of a long narrow
discharge tube containing a gas mixture of Helium and Neon gas in a ratio of about 10:1 at a total pressure
of 1 mm of Hg (1 torr). The electrodes are connected to a high voltage dc source. A pair of mirrors – one of
which is a perfect reflector and other is a partial reflector constitutes the resonator system. The end
windows (Brewster’s window) of the tube are made optically flat and set at reflections from the windows
of the tube and makes the output plane polarised.
AYAN KR. MONDAL
Mob : 8981580693.
Email : ayan.mondal15@ymail.in
QUOTE : In Science there is only Physics, all the rest is Stamp Collecting – LORD KELVIN
Working principle: The mixture of He and Ne gas is contained in the discharge tube and it is ionized by
placing a dc through the gas. The electrons in the tube are accelerated and they collide with the helium
atoms. Due to this collision He atoms gets excited to the metastable states and the number of atoms in the
states increases since no radiative transition to lower state is possible. Now energies of the two excited
states 3s and 2s of Ne are slightly lower but close to the energy of the metastable state of the He atoms. So
when the He atoms in the metastable state collide with the heavier Ne atoms in the ground state, energyexchange occurs and thus the Ne atoms are excited to 2s and 3s states while He atoms return to ground
states.
These higher Ne states (2p and 3p) are metastable for Ne and thus the excitation energy transfer
increases the population of the neon in 2s and 3s levels. When the population of these two upper levels
exceeds that of the corresponding lower level neon state, 2p and 3p, to which they are optically connected,
population inversion is occurred. Due to the transitions of Ne atoms from the above excited states to the
lower energy 2p and 3p levels, the medium becomes capable of amplifying light in a narrow band at 1.15 μm
(corresponding to the 2s to 2p transition) and in a narrow band at 632.8 nm (corresponding to the 3s to
2p transition).
Advantages of using gas laser than solid state laser:
(i) Gas lasers are more unidirectional. (ii) More monochromatic than solid state lasers. (iii) Flash lamp in the
solid state laser gets heated but no such incidents happens in gas lasers.