Principle of LASER (Amplification by Stimulated Emission and Population Inversion)

Consider an atom having two energy levels \(E_m\) and \(E_n\), where \(E_n > E_m\). Let \(N_m\) and \(N_n\) be the number of atoms in the lower and upper energy levels, respectively.

\[ E_n-E_m=h\nu \]

The rate of absorption of radiation by atoms in the lower energy state is

\[ R_{mn}=N_mP_{mn} \] \[ R_{mn}=N_mB_{mn}u(\nu) \]

The total rate of emission from the upper energy state is the sum of spontaneous and stimulated emission:

\[ R_{nm}=N_nP_{nm} \] \[ R_{nm}=N_n\left[A_{nm}+B_{nm}u(\nu)\right] \]

Therefore, the ratio of total emission rate to absorption rate is

\[ \frac{R_{nm}}{R_{mn}} = \frac{N_n}{N_m} \left[ 1+\frac{A_{nm}}{B_{nm}u(\nu)} \right] \tag{1} \]

For non-degenerate energy levels, Einstein's relation gives \(B_{mn}=B_{nm}\).

Therefore,

\[ 1>>\frac{A_{nm}}{B_{nm}u(\nu)} \tag{1} \]

Thus

\[ \frac{R_{nm}}{R_{mn}}\approx\frac{N_n}{N_m}=e^{-h\nu/k_BT} \]

Case-I

if T=0

\[ \frac{R_{nm}}{R_{mn}}\approx e^{-h\nu/k_BT}= \frac{1}{e^{h\nu/0}}= \frac{1}{e^{∞}}= \frac{1}{∞}=0 \]

Thus

\[ R_{nm}=0 \]

No excited atoms for emission

Case-II

if T=(+)tive

\[ \frac{R_{nm}}{R_{mn}}\approx e^{-h\nu/k_BT}= \frac{1}{e^{h\nu/kT}}< 1 \]

Thus

\[ R_{nm}<R_{mn} \]

Fewer atoms are excited to higher energy levels for emission compared with absorption.

Case-III

if T=∞

\[ \frac{R_{nm}}{R_{mn}}\approx e^{-h\nu/k_BT}= \frac{1}{e^{h\nu/∞}}= \frac{1}{e^{0}}= \frac{1}{1} \]

Thus

\[ R_{nm}=R_{mn} \]

Same rate pf absorption and emission

Case-IV

if T=(-)tive

\[ \frac{R_{nm}}{R_{mn}}\approx e^{h\nu/k_BT}> 1 \]

Thus

\[ R_{nm}>R_{mn} \]

more atoms are excited to higher energy levels for emission compared with absorption.

This is condition of the population inversion

\[ \boxed{N_n>N_m} \tag{2} \]

The condition in which the population of the higher energy state exceeds that of the lower energy state is called population inversion.

Pumping

Population inversion cannot normally be obtained in thermal equilibrium.

Pumping is the general process of supplying energy to a medium to lift its atoms or electrons from a lower energy level to a higher energy level, which helps create a population inversion needed for lasers. Optical pumping is a specific type of pumping that uses light energy—such as from a flashlamp, arc lamp, or laser diode—to excite those electrons.

Metastable state

It is an excited state of an atom with a longer life time than the other excited states. Atoms in the metastable state remain excited for a considerable time in the order of 10-6 to 10-3 seconds. Such relatively long-lived states are called as Metastable state. An atom can exist in a metastable energy level for a longer time before radiating than itcan in an ordinary energy level.