Lasers

9.1 Stimulated Emission

Einstein’s coefficients: $A_{21}$ (spontaneous emission), $B_{21}$ (stimulated emission), $B_{12}$ (absorption).

At thermal equilibrium:

$A_{21} + B_{21}\rho(\omega) = B_{12}\rho(\omega) \cdot \frac{g_1}{g_2} e^{\hbar\omega/(k_B T)}$

The relations $B_{21} = B_{12}$ (for non-degenerate levels) and $A_{21}/B_{21} = \hbar\omega^3 n^3/(\pi^2 c^3)$ Follow from detailed balance with the Planck distribution.

9.2 Population Inversion

Laser operation requires population inversion: $N_2 \gt N_1$ where $N_2$ is the population of the Upper laser level and $N_1$ is the lower.

This cannot be achieved in a two-level system at thermal equilibrium. A three-level or four-level laser scheme is needed.

9.3 Laser Cavity Modes

A Fabry-Perot cavity of length $L$ supports longitudinal modes at frequencies:

$\nu_m = m\frac{c}{2nL}, \quad m = 1, 2, 3, \ldots$

The mode spacing (free spectral range):

$\Delta\nu = \frac{c}{2nL}$

For a cavity with mirrors of reflectivity $R$The finesse is:

$\mathcal{F} = \frac{\pi\sqrt{R}}{1 - R}$

9.4 Gaussian Beams

The fundamental TEM$_{00}$ mode of a laser cavity is a Gaussian beam:

$E(r, z) = E_0 \frac{w_0}{w(z)} \exp\left(-\frac{r^2}{w(z)^2}\right) \exp\left(-ikz - ik\frac{r^2}{2R(z)} + i\zeta(z)\right)$

Where:

• Beam waist: $w_0$ (minimum spot size).
• Rayleigh range: $z_R = \pi w_0^2 / \lambda$.
• Beam radius: $w(z) = w_0\sqrt{1 + (z/z_R)^2}$.
• Radius of curvature: $R(z) = z[1 + (z_R/z)^2]$.
• Gouy phase: $\zeta(z) = \arctan(z/z_R)$.

The beam divergence (half-angle, far field): $\theta = \lambda/(\pi w_0)$.