Lasers

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.

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.

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:

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

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): $\theta = \lambda/(\pi w_0)$.

The dynamics of laser populations are described by rate equations. For a four-level laser:

$\frac{dN_2}{dt} = R_p - \frac{N_2}{\tau_2} - \frac{N_2}{\tau_{21}} - \sigma c\, n_p (N_2 - N_1)$

$\frac{dN_1}{dt} = \frac{N_2}{\tau_{21}} - \frac{N_1}{\tau_1} + \sigma c\, n_p (N_2 - N_1)$

$\frac{dn_p}{dt} = \sigma c\, n_p (N_2 - N_1) - \frac{n_p}{\tau_p} + \beta \frac{N_2}{\tau_{21}}$

where $R_p$ is the pump rate, $\sigma$ is the stimulated emission cross-section, $n_p$ is the photon density, $\tau_p$ is the photon cavity lifetime, and $\beta$ is the spontaneous emission factor.

The laser threshold is reached when gain equals loss. The threshold population inversion is:

$\Delta N_{\mathrm{th}} = \frac{1}{\sigma L} \left(\alpha_{\mathrm{int}} - \frac{1}{2L}\ln(R_1 R_2)\right)$

where $\alpha_{\mathrm{int}}$ is the internal loss coefficient and $R_1, R_2$ are the mirror reflectivities.

Q-switching produces short, high-energy pulses by modulating the cavity quality factor $Q$. The energy is stored in the gain medium while the cavity is kept low-Q, then released suddenly when Q-switched to high-Q.

Mode locking produces ultrashort pulses by fixing the phase relationship between longitudinal modes. With $M$ locked modes, the pulse duration is $\Delta t \approx 1/(M \Delta\nu)$, which can reach femtoseconds.

• He-Ne laser (gas, 632.8 nm): continuous wave, low power (mW), used in alignment and interferometry.
• Nd:YAG laser (solid-state, 1064 nm): high power, pulsed or CW, used in machining and surgery.
• CO$_2$ laser (gas, 10.6 $\mu$m): very high power, used in cutting and welding.
• Diode laser (semiconductor): compact, efficient, used in telecommunications and barcode readers.
• Ti:sapphire laser (solid-state, tunable 650—1100 nm): mode-locked for femtosecond pulses.

Problem 1. A He-Ne laser cavity is $L = 30$ cm long. Calculate the mode spacing and the number of longitudinal modes under the gain bandwidth $\Delta\nu \approx 1.5$ GHz.

Problem 2. A Nd:YAG laser produces 10 ns pulses at 10 Hz with 100 mJ per pulse. Calculate the peak power and average power.

Problem 3. Show that lasing cannot occur in a two-level system.

Solution. In steady state for a two-level system, $N_1 + N_2 = N$ and detailed balance gives $N_2/N_1 = e^{-\hbar\omega/(k_B T)} < 1$ at any positive temperature. Thus $N_2 \leq N_1$, and population inversion is impossible. $\blacksquare$

The fundamental linewidth of a laser is given by the Schawlow-Townes limit:

$\Delta\nu_{\mathrm{laser}} = \frac{2\pi h\nu (\Delta\nu_c)^2}{P}$

where $\Delta\nu_c$ is the cavity linewidth and $P$ is the output power. Modern lasers can achieve linewidths below 1 Hz, enabling applications in precision metrology and optical clocks.

Semiconductor (diode) lasers use direct bandgap materials like GaAs and InP. The gain is provided by electron-hole recombination across the bandgap. Key parameters:

• Threshold current density: $J_{\mathrm{th}}$ (typically 100-1000 A/cm$^2$).
• Slope efficiency: $\eta_d = \frac{dP}{dI}$ above threshold.
• Modulation bandwidth: up to 40 GHz for direct modulation.

Distributed feedback (DFB) lasers use a built-in Bragg grating to select a single longitudinal mode, essential for wavelength-division multiplexing in fiber communications.

Lasers are classified by power and wavelength:

• Class 1: Safe under all conditions (e.g., DVD players).
• Class 2: Low-power visible (< 1 mW), blink reflex protects.
• Class 3R/3B: Direct intrabeam viewing hazardous (1-500 mW).
• Class 4: High-power (> 500 mW), hazardous to eyes and skin, fire risk.

Problem 4. A He-Ne laser has output power 5 mW at 632.8 nm with beam diameter 0.8 mm. Compute the irradiance (power/area) and determine the laser class.

Problem 5. Calculate the photon flux (photons per second) for the laser in Problem 4.

Problem 6. A Q-switched Nd:YAG laser produces 10 ns pulses with 100 mJ pulse energy at 10 Hz. Calculate the peak power, average power, and photon energy at 1064 nm.