Part III, Chapter 3

Laser Physics

Lasers produce coherent, monochromatic, directional light through stimulated emission. Understanding their operation requires Einstein's radiation theory, population inversion, and optical cavity design.

3.1 Einstein Coefficients

In 1917, Einstein introduced three radiative processes between two energy levels E₁ and E₂ (with E₂ > E₁):

Three Radiative Processes

Absorption (B₁₂): A photon of energy hν = E₂ - E₁ is absorbed, promoting an atom from level 1 to 2. Rate: B₁₂ N₁ ρ(ν).
Spontaneous emission (A₂₁): An atom in level 2 spontaneously decays to level 1, emitting a photon in a random direction. Rate: A₂₁ N₂.
Stimulated emission (B₂₁): An incident photon causes an atom in level 2 to emit an identical photon (same frequency, phase, polarization, and direction). Rate: B₂₁ N₂ ρ(ν).

3.1.1 Rate Equation

The population dynamics of the upper level are governed by:

$$\frac{dN_2}{dt} = B_{12} N_1 \rho - (A_{21} + B_{21}\rho) N_2$$

Derivation: Relations Between Einstein Coefficients

Step 1. In thermal equilibrium, dN₂/dt = 0, so:

$$\rho(\nu) = \frac{A_{21}/B_{21}}{(B_{12}/B_{21})(N_1/N_2) - 1}$$

Step 2. Using the Boltzmann distribution N₁/N₂ = (g₁/g₂) exp(hν/k_BT) and comparing with Planck's radiation law:

$$\rho(\nu) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/k_BT} - 1}$$

Step 3. Matching coefficients:

Result:

$$g_1 B_{12} = g_2 B_{21}, \qquad A_{21} = \frac{8\pi h\nu^3}{c^3} B_{21}$$

The A/B ratio scales as ν³, explaining why spontaneous emission dominates at optical frequencies but is negligible at microwave frequencies (masers).

3.2 Population Inversion

For stimulated emission to exceed absorption, we need N₂ > (g₂/g₁)N₁, a condition called population inversion. This cannot be achieved in a two-level system in steady state (optical pumping saturates at N₂ = N₁ for g₁ = g₂).

3.2.1 Three-Level System

In a three-level laser (e.g., ruby laser), atoms are pumped from level 1 to level 3, then rapidly decay non-radiatively to level 2, creating inversion between levels 2 and 1. The threshold pump rate must overcome the spontaneous emission rate:

$$W_p > \frac{A_{21}}{1 + \tau_2/\tau_{32}}$$

3.2.2 Four-Level System

The four-level system (e.g., Nd:YAG) is more efficient because the lower laser level is not the ground state, so it is naturally empty. Inversion is achieved with much lower pump power. The small-signal gain coefficient is:

$$g_0 = \sigma_{21} \Delta N = \sigma_{21} (N_2 - N_1)$$

where σ₂₁ is the stimulated emission cross-section. For Nd:YAG at 1064 nm, σ₂₁ ≈ 2.8 × 10^-19 cm².

3.3 Optical Cavity Modes

A laser cavity (resonator) consists of two mirrors separated by distance L. Standing waves form when the round-trip phase is a multiple of 2π:

$$\nu_q = q \frac{c}{2nL}, \qquad q = 1, 2, 3, \ldots$$

The longitudinal mode spacing is Δν = c/(2nL). For L = 30 cm, Δν = 500 MHz.

3.3.1 Threshold Condition

Lasing occurs when the round-trip gain equals the round-trip loss. For a cavity with mirror reflectivities R₁, R₂ and internal loss α_i:

$$g_{\text{th}} = \alpha_i + \frac{1}{2L}\ln\!\left(\frac{1}{R_1 R_2}\right)$$

3.3.2 Gaussian Beams

The transverse mode of a stable cavity is a Gaussian beam. The fundamental TEM₀₀mode has a field profile:

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

where w(z) = w₀√(1 + (z/z_R)²) is the beam radius, z_R = πw₀²/λ is the Rayleigh range, and ζ(z) is the Gouy phase.

3.4 Laser Systems

He-Ne Laser (632.8 nm)

The helium-neon laser is the most common gas laser. Helium atoms are excited by electrical discharge and transfer energy to neon via collisions (resonant energy transfer). The 632.8 nm line operates on the 3s₂ → 2p₄ transition of neon. Typical output power: 0.5-50 mW. Excellent spatial and temporal coherence.

Nd:YAG Laser (1064 nm)

Neodymium-doped yttrium aluminum garnet is the workhorse solid-state laser. It is a four-level system pumped by flashlamps or diode lasers (808 nm). The ⁴F_3/2 →⁴I_11/2 transition at 1064 nm has a long upper-state lifetime (τ = 230 μs), making it ideal for Q-switching and mode-locking. Frequency-doubled to 532 nm (green) via SHG.

Semiconductor Diode Laser

A p-n junction in a direct-bandgap semiconductor (e.g., GaAs, InGaAsP) achieves population inversion through current injection. The gain medium and cavity (cleaved facets) are integrated on a chip. Wavelength is determined by bandgap: GaAs ∼ 850 nm, InGaAsP ∼ 1.3-1.55 μm (telecom windows). Distributed feedback (DFB) lasers provide single-mode operation for fiber communications.

3.5 Pulsed Laser Operation

3.5.1 Q-Switching

Q-switching produces high-energy nanosecond pulses by modulating the cavity loss (Q-factor). Energy is stored in the population inversion while the cavity Q is low (high loss). When Q is suddenly switched high, the stored energy is released in a giant pulse. Peak powers of megawatts to gigawatts are achievable. The pulse energy is:

$$E_{\text{pulse}} \approx h\nu \cdot \Delta N_i \cdot V_{\text{mode}} / 2$$

3.5.2 Mode-Locking

Mode-locking locks the phases of many longitudinal modes to produce ultrashort pulses. If N modes of equal amplitude and spacing Δν are phase-locked, the pulse duration is:

$$\Delta t \approx \frac{1}{N \Delta\nu} = \frac{2L}{Nc}$$

The repetition rate is f_rep = c/(2L). Ti:sapphire lasers achieve pulses as short as 5 fs by Kerr-lens mode-locking across a bandwidth of ~100 THz.

3.6 Python Simulation: Laser Rate Equations & Mode-Locking

This simulation solves the laser rate equations for a four-level system, shows the threshold behavior, and demonstrates how superposing phase-locked modes produces ultrashort pulses.

Laser Physics: Rate Equations & Mode-Locking

Python

script.py140 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# --- Plot 1: Laser rate equations (relaxation oscillations) ---
ax1 = axes[0, 0]

# Four-level laser rate equations
# dN/dt = Wp - N/tau_f - sigma*c*N*phi
# dphi/dt = (sigma*c*N - 1/tau_c)*phi + beta*N/tau_f
sigma = 2.8e-19 * 1e-4  # stimulated emission cross section (m^2), Nd:YAG
c_light = 3e8
tau_f = 230e-6           # fluorescence lifetime (s)
L_cav = 0.1             # cavity length (m)
R1, R2 = 0.99, 0.95
alpha_i = 0.01           # internal loss (1/m)
tau_c = 2*L_cav / (c_light * (alpha_i*2*L_cav - np.log(R1*R2)))  # photon lifetime
beta = 1e-6              # spontaneous emission fraction into lasing mode
V_mode = 1e-8            # mode volume (m^3)

Wp = 3.0 / (tau_f)      # pump rate (3x threshold)

def rate_eqs(t, y):
    N, phi = y
    dN = Wp - N/tau_f - sigma*c_light*N*phi
    dphi = (sigma*c_light*N - 1/tau_c)*phi + beta*N/tau_f
    return [dN, dphi]

t_span = (0, 200e-6)
t_eval = np.linspace(*t_span, 5000)
sol = solve_ivp(rate_eqs, t_span, [0, 1e10], t_eval=t_eval, method='RK45', rtol=1e-10, atol=1e-6)

ax1.plot(sol.t*1e6, sol.y[1]/np.max(sol.y[1]), color='#f59e0b', linewidth=1.5)
ax1.set_xlabel('Time (us)', color='white')
ax1.set_ylabel('Photon density (normalized)', color='white')
ax1.set_title('Laser Turn-On: Relaxation Oscillations', color='#fbbf24', fontsize=13)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 2: Gain vs pump power (threshold) ---
ax2 = axes[0, 1]
pump_ratio = np.linspace(0, 5, 200)  # W_p / W_threshold
g_th = alpha_i + 1/(2*L_cav) * np.log(1/(R1*R2))

# Below threshold: no output, gain = pump
# Above threshold: gain clamps, output increases linearly
output_power = np.where(pump_ratio < 1, 0, (pump_ratio - 1))
gain = np.where(pump_ratio < 1, pump_ratio * g_th, g_th)

ax2.plot(pump_ratio, output_power, color='#f59e0b', linewidth=2, label='Output power')
ax2.plot(pump_ratio, gain/g_th, color='#fb923c', linewidth=2, linestyle='--', label='Gain / g_th')
ax2.axvline(1, color='#fbbf24', linestyle=':', alpha=0.5)
ax2.annotate('Threshold', (1.05, 3.5), color='#fbbf24', fontsize=10)
ax2.set_xlabel('Pump / Threshold', color='white')
ax2.set_ylabel('Normalized', color='white')
ax2.set_title('Laser Threshold Behavior', color='#fbbf24', fontsize=13)
ax2.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 3: Mode-locking ---
ax3 = axes[1, 0]
t_ml = np.linspace(0, 10e-9, 2000)  # 10 ns window
L_ml = 0.15  # 15 cm cavity
delta_nu = c_light / (2 * L_ml)  # mode spacing ~ 1 GHz
nu_0 = 3e14  # center frequency

for N_modes in [5, 15, 50]:
    E_total = np.zeros_like(t_ml)
    for n in range(-(N_modes//2), N_modes//2 + 1):
        E_total += np.cos(2*np.pi*(nu_0 + n*delta_nu)*t_ml)
    I_total = E_total**2
    I_total /= np.max(I_total)
    ax3.plot(t_ml*1e9, I_total, linewidth=1, label=f'N={N_modes}', alpha=0.8)

ax3.set_xlabel('Time (ns)', color='white')
ax3.set_ylabel('Intensity (normalized)', color='white')
ax3.set_title('Mode-Locked Pulse Train', color='#fbbf24', fontsize=13)
ax3.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax3.set_xlim(0, 5)
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 4: Q-switching dynamics ---
ax4 = axes[1, 1]

# Simplified Q-switch model
def q_switch(t, y, t_switch=50e-6):
    N, phi = y
    # Q switches at t_switch
    if t < t_switch:
        tau_c_qs = 1e-9  # very lossy cavity (low Q)
    else:
        tau_c_qs = tau_c  # normal cavity (high Q)

dN = Wp * 2 - N/tau_f - sigma*c_light*N*phi
    dphi = (sigma*c_light*N - 1/tau_c_qs)*phi + beta*N/tau_f
    return [dN, dphi]

t_span_qs = (0, 100e-6)
t_eval_qs = np.linspace(*t_span_qs, 10000)
sol_qs = solve_ivp(lambda t, y: q_switch(t, y), t_span_qs, [0, 1e8],
                    t_eval=t_eval_qs, method='RK45', rtol=1e-10, atol=1e-4, max_step=1e-7)

ax4_twin = ax4.twinx()
ax4.plot(sol_qs.t*1e6, sol_qs.y[0]/np.max(sol_qs.y[0]), color='#fb923c', linewidth=1.5, label='Inversion')
ax4_twin.plot(sol_qs.t*1e6, sol_qs.y[1]/np.max(sol_qs.y[1]), color='#f59e0b', linewidth=1.5, label='Photon density')
ax4.axvline(50, color='#fbbf24', linestyle='--', alpha=0.5)
ax4.annotate('Q-switch', (50.5, 0.9), color='#fbbf24', fontsize=10)
ax4.set_xlabel('Time (us)', color='white')
ax4.set_ylabel('Inversion (norm.)', color='#fb923c')
ax4_twin.set_ylabel('Photon density (norm.)', color='#f59e0b')
ax4.set_title('Q-Switched Pulse', color='#fbbf24', fontsize=13)
ax4.tick_params(colors='white'); ax4_twin.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#f59e0b')
for spine in ax4_twin.spines.values():
    spine.set_color('#f59e0b')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.show()

print(f"Cavity parameters:")
print(f"  Photon lifetime tau_c = {tau_c*1e9:.2f} ns")
print(f"  Mode spacing = {delta_nu/1e9:.2f} GHz")
print(f"  Threshold gain g_th = {g_th:.4f} /m")
print(f"\nMode-locking (50 modes, L=15cm):")
pulse_dur = 1/(50 * delta_nu)
print(f"  Pulse duration ~ {pulse_dur*1e12:.1f} ps")
print(f"  Repetition rate = {delta_nu/1e9:.2f} GHz")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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