Basis of the Theory of the Laser: The Einstein Coefficients

Abstract

According to Bohr’s atomic model (1911), which is based on spectroscopic investigations, transitions between discrete energy levels of an atom can lead to emission or absorption of radiation of a frequency that fulfills Bohr’s energy-frequency relation. In an absorption process, a photon is absorbed. In an emission process, a photon is emitted. Einstein found that the emission of a photon is possible by two different processes, spontaneous and stimulated emission, and that the coefficients describing the three processes—absorption, stimulated and spontaneous emission—are related to each other (Einstein relations).

Problems

6.1

Photon density. Estimate the density of photons, present in a lecture room, in a frequency interval of 1 MHz

• (a) At a microwave frequency of 1 GHz.

• (b) At a terahertz frequency of 1 THz.

• (c) At a frequency (500 THz) in the visible.

6.2

Number of thermal photons in a mode of a laser resonator. Calculate the average number of thermal photons in a mode of a laser resonator at room temperature, for different lasers.

• (a) Titanium–sapphire laser (frequency 400 THz).

• (b) CO$_2$ laser (30 THz).

• (c) Far infrared laser (1 THz).

6.3

Einstein coefficients. Determine the Einstein coefficients from spontaneous lifetimes of laser media mentioned in Table 6.1:

• (a) Helium–neon laser; $\tau _\mathrm{sp} =100$ ns.

• (b) CO$_2$ laser; $\tau _\mathrm{sp} =5$ s.

• (c) Nd:YAG laser; $\tau _\mathrm{sp} =230\,\upmu$s.

6.4

Einstein coefficient. Relate the Einstein coefficients $B_{21}^{\omega }$ (for $\rho$ on the $\omega$ scale) and $B_{21}^{\nu }$ (for $\rho$ on the $\nu$ scale).

6.5

Write Planck’s radiation law on the wavelength scale.

6.6

Radiation laws. Derive from Planck’s radiation law other laws:

• (a) Wien’s displacement law on the frequency scale; $\nu _\mathrm{{max}}(T)$

• (b) Wien’s displacement law on the wavelength scale. $\lambda _\mathrm{{max}}(T)$

• (c) Rayleigh–Jeans law ($h \nu \ll k T$).

• (d) Wien’s law ($h \nu \gg k T$).

• (d) Stefan–Boltzmann law. [Hint: $\int x^3 (\mathrm {e}^x -1)^{-1} \mathrm {d}x = \pi ^4/15$.]

6.7

Maximum of the Planckian distribution.

• (a) The spectrum of the cosmic background radiation has a Planck distribution corresponding to a temperature of 2.7 K. Determine the frequency $\nu _\mathrm{max}$ of the maximum of the distribution on the frequency scale and the wavelength $\lambda _\mathrm{max}$ of the distribution on the wavelength scale. [Hint: $\lambda _\mathrm{max} \ne \nu _\mathrm{max}/c$.]

• (b) Determine $\nu _\mathrm{max}$ and $\lambda _\mathrm{max}$ for blackbody radiation emitted by a blackbody at a temperature of 300 K.

6.8

Determine the number of photons contained in a cavity with walls at temperature T. [Hint: $\int _{0}^{\infty }x^2(\mathrm {e}^x-1)^{-1}\mathrm {d}x =2.40$.]

6.9

Derive (6.2) from (6.1).

6.10

Determine the matrix elements for optical transitions of media characterized in Table 6.1.