LECTURE 1.
2
This topic is mapped with CO2 and CO3 of the course
LASER
LASER stands for ‘Light Amplification by Stimulated
Emission of Radiation’
Rate Equations of Absorption, Spontaneous Emission and
Stimulated Emission
(a) Absorption
The probability of occurrence of absorption process from state 1 to state 2 is
proportional to the energy density u(ν) of the radiation
P12=B12u(ν)
Where, B12 is proportionality constant and is known as Einstein coefficient of absorption of
radiation
Figure 1 Mechanism of absorption process
The rate of stimulated Absorption of photons is given by
Rab = N1P12 = N1B12u(ν)
Where N1 is the population of ground state i.e. number of atoms per unit volume occupying the
ground state.
(b) Spontaneous Emission
An excited atom can stay in the higher energy state only for the time of 10-8 s. After this
time, it returns back to the lower energy state by emitting a photon.
The probability of occurrence of spontaneous emission transition from state 2 to state 1
depends only on the properties of states 2 and 1 and is given by
P21=A21
Where, A21 is proportionality constant and is known as Einstein coefficient of absorption of
radiation. Its reciprocal gives the life time of the excited state.
Figure 2 Mechanism of spontaneous emission process
The rate of spontaneous emission of photons is given by
Rsp = N2P21 = N2A21
Where, N2 is the population of excited state i.e. number of atoms per unit volume occupying
the excited state.
a) Stimulated emission
When photon of suitable frequency is made to fall on an excited atom in the higher
energy state, the atom falls back to the ground state by emitting a photon of energy which is in
phase with the stimulating (incident) photon.
The probability of occurrence of stimulated emission process from state 2 to state 1 is
proportional to the energy density u(ν) of the radiation
P’21=B21u(ν)
Where, B21 is proportionality constant and is known as Einstein coefficient of stimulated
emission of radiation
Figure 3 Mechanism of stimulated absorption process
The rate of stimulated emission of photons is given by
Rst = N2P’21 = N2B21u(ν)
Where, N2 is the population of excited state i.e. number of atoms per unit volume occupying
the excited state.
Relation Between Einstein’s Coefficients
Let N1 and N2 be the number of atoms at any instant at any instant in the state 1 and 2,
respectively. The probability of absorption for number of atoms from state 1 to 2 per unit
time i.e. rate is given by
Rab = N1P12 = N1B12u(ν)…….(1)
The total probability of transition for number of atoms from state 2 to 1, either by spontaneous
or by stimulated emission per unit time i.e rate of emissions is given by
Rsp + Rst = N2A21 + N2B21u(ν)……(2)
In thermal equilibrium at temperature T, the absorption and emission probabilities are equal
Thus equating (1) and (2)
Rab = Rsp + Rst
N1B12u(ν) = N2A21 + N2B21u(ν)……..(3)
After rearranging the terms, we obtain the expression for photon density as
u(ν) = N2A21/( N1B12- N2B21)….(4)
Divide R.H.S of above equation by N2 B12 in numerator and denominator
u(ν) = A21/ B12 ( N1/N2 - B21/B12)….(5)
According to Boltzmann’s law, the distribution of atoms among the energy states E1 and E2 at
the thermal equilibrium at temperature T is given
−𝑬𝟏/𝑲
𝑵𝟏 𝒆 𝑻
= −𝑬𝟐/𝑲
𝑵𝟐 𝒆 𝑻
where, E=E2-E1=hν
Making these substitutions in (5) we obtain
u(ν) = A21/ B12 ( ehv/KT - B21/B12)…….(6)
Comparing equation number (6) with the Planck’s radiation law for black body radiations as
given in equation (7)
u(ν) = 8πhν3/c3 ( ehv/KT - 1)……………(7)
We obtain
A21/ B12= 8πhν3/c3………………..(8)
And B21/B12=1 or B12=B21...........(9)
Equations (8) and (9) are relations between Einstein’s coefficients
Significance
1. It signifies that probability of absorption is same as that of stimulated emission
2. It signifies that B21/A21 is inversely proportional to frequency of the resonant
radiation. Therefore, higher the frequency smaller is the value of B21. That is, it is
comparatively difficult to obtain the stimulated emission of higher frequencies.
Relevant Books for this topic
1.Laud B.B., Laser and Non -Linear Optics, Edition 1st, (1991), Wiley Eastern Ltd.
https://www.amazon.in/Lasers-Non-Linear-Optics-B-B-
Laud/dp/8122430562/ref=sr_1_2?dchild=1&keywords=laser+and+nonlinear+optics&qid=159294
8185&s=books&sr=1-2
2. Ghatak A. (2012) Optics, McGraw Hill Education. ISBN: 978-1259004346.
https://www.amazon.in/Optics-Old-Ajoy-Ghatak/dp/1259004341
Relevant Web links for this topic
1. http://www.physics-and-radio-electronics.com/blog/absorption-of-radiation-spontaneous-
emission-and-stimulated-emission
2. http://www.nitttrchd.ac.in/sitenew1/app_sc/ppts/laser/Lasing%20%20Principles%20&%20Condi
tions.pdf
Relevant Videos for this topic
1. https://www.youtube.com/watch?v=2g25zn-ofck
2. https://www.youtube.com/watch?v=5Kia0HHmkHY
3. https://pt.coursera.org/lecture/leds-semiconductor-lasers/einstein-coefficients-part-ii-tWhnc
