Question

Explain the concept of population inversion on the basis of Einstein’s A & B coefficients.
Discuss its significance for developing a LASER.

Answer 1

Answer on Question #50686, Physics, Optics

Explain the concept of population inversion on the basis of Einstein's A & B coefficients.

Discuss its significance for developing a LASER.

Solution:

Einstein coefficients are mathematical quantities which are a measure of the probability of absorption or emission of light by an atom or molecule. The Einstein A coefficient is related to the rate of spontaneous emission of light and the Einstein B coefficients are related to the absorption and stimulated emission of light.

Spontaneous emission is the process by which an electron "spontaneously" (i.e. without any outside influence) decays from a higher energy level to a lower one. The process is described by the Einstein coefficient $A_{21}$ which gives the probability per unit time that an electron in state 2 with energy $E_2$ will decay spontaneously to state 1 with energy $E_1$ , emitting a photon with an energy $E_2 - E_1 = h\nu$ .

Stimulated emission (also known as induced emission) is the process by which an electron is induced to jump from a higher energy level to a lower one by the presence of electromagnetic radiation at (or near) the frequency of the transition. The process is described by the Einstein coefficient $B_{21}$ , which gives the probability per unit time per unit spectral energy density of the radiation field that an electron in state 2 with energy $E_2$ will decay to state 1 with energy $E_1$ , emitting a photon with an energy $E_2 - E_1 = h\nu$ .

Absorption is the process by which a photon is absorbed by the atom, causing an electron to jump from a lower energy level to a higher one. The process is described by the Einstein coefficient $B_{12}$ .

The three Einstein coefficients are interrelated by:

$$\frac {A _ {2 1}}{B _ {2 1}} = \frac {8 \pi h v ^ {3}}{c ^ {3}}$$ $$\frac {B _ {2 1}}{B _ {1 2}} = \frac {g _ {1}}{g _ {2}}$$

where $g_{i}$ is the degeneracy (also called the multiplicity) of state $i$ .

The rate of spontaneous emission rate to the stimulated emission rate is given by

$$R = \frac {N _ {2} A _ {2 1}}{N _ {2} \rho (v) B _ {2 1}} = \frac {A _ {2 1}}{\rho (v) B _ {2 1}} = e ^ {\frac {h v}{k T}} - 1$$

In practice, the absorption and emission processes occur simultaneously.

Even for sources operating at higher temperature and lower frequencies $h\nu >> kT$ and hence $R >> 1$ . This confirms that under condition of thermal equilibrium, spontaneous emission predominates over stimulated emission.

From the last equation, we understand that to make R smaller, $\rho(v)$ the energy density of interacting radiation has to be made larger. Let us consider the ratio of stimulated emission rate to stimulated absorption rate.

Stimulated emission rate/ Stimulated absorption rate=

$$= \frac {N _ {2} \rho (v) B _ {2 1}}{N _ {1} \rho (v) B _ {1 2}} = \frac {N _ {2}}{N _ {1}} \quad (as\ B _ {2 1} = B _ {1 2} \ ignoring\ degeneracy)$$

Thus at thermal equilibrium stimulated absorption predominates over stimulated emission. Instead if we create a situation that $N_2 > N_1$, stimulated emission will predominate over stimulated absorption. If stimulated emission predominates, the photon density increases and Light Amplification by Stimulated Emission of Radiation (LASER) occurs. Therefore, in order to achieve more stimulated emission, population of the excited state ($N_2$) should be made larger than the population of the lower state ($N_1$) and the condition is called population inversion.

Hence if we wish to amplify a beam of light by stimulated emission then we must

1) create population inversion and

2) increase the energy density of interacting radiation

A population inversion cannot be achieved with just two levels because the probability for absorption and for spontaneous emission is exactly the same, as shown by Einstein and expressed in the Einstein A and B coefficients.

https://www.AssignmentExpert.com