Part V, Chapter 5

Nonlinear Optics

At high optical intensities, the response of a medium becomes nonlinear, enabling frequency conversion, self-focusing, and optical solitons through second- and third-order susceptibilities.

5.1 Nonlinear Polarization

In the linear regime, the polarization of a medium is proportional to the electric field: P = ε₀χ⁽¹⁾E. At high intensities, higher-order terms become significant:

$$P = \epsilon_0\left[\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots\right]$$

The second-order susceptibility χ⁽²⁾ is nonzero only in non-centrosymmetric media (e.g., KDP, LiNbO₃, BBO). The third-order susceptibility χ⁽³⁾exists in all materials, including glasses and liquids.

Orders of Magnitude

- χ⁽¹⁾ ~ 1 (dimensionless)
- χ⁽²⁾ ~ 10^-12 m/V (LiNbO₃: d₃₃ = 25.2 pm/V)
- χ⁽³⁾ ~ 10^-22 m²/V² (silica: 2.5 × 10^-22 m²/V²)

Despite the tiny values, focused laser beams can reach E ~ 10⁸ V/m, making nonlinear effects substantial.

5.2 Second-Order (χ⁽²⁾) Processes

5.2.1 Second-Harmonic Generation (SHG)

Two photons at frequency ω combine to produce one photon at 2ω. The nonlinear polarization at the second harmonic is:

$$P^{(2)}(2\omega) = \epsilon_0 \chi^{(2)} E(\omega)^2$$

Derivation: SHG Efficiency with Phase Matching

Step 1. The coupled-wave equation for the second-harmonic field A₂(z) in the slowly-varying envelope approximation is:

$$\frac{dA_2}{dz} = i\kappa A_1^2 e^{i\Delta k z}, \quad \kappa = \frac{\omega d_{\text{eff}}}{n_2 c}$$

Step 2. For undepleted pump (A₁ ≈ const), integrating over crystal length L:

$$A_2(L) = i\kappa A_1^2 L \cdot \operatorname{sinc}\!\left(\frac{\Delta k L}{2}\right) e^{i\Delta k L/2}$$

Step 3. The SHG efficiency is:

Result:

$$\eta_{\text{SHG}} = \frac{P_{2\omega}}{P_\omega} \propto d_{\text{eff}}^2 L^2 I_\omega \operatorname{sinc}^2\!\left(\frac{\Delta k L}{2}\right)$$

Maximum efficiency requires phase matching: Δk = k(2ω) - 2k(ω) = 0. This is achieved by birefringent phase matching or quasi-phase matching (periodic poling).

5.2.2 Spontaneous Parametric Down-Conversion (SPDC)

The reverse of SHG: a pump photon at ω_p spontaneously splits into signal (ω_s) and idler (ω_i) photons satisfying energy and momentum conservation:

$$\omega_p = \omega_s + \omega_i, \qquad \mathbf{k}_p = \mathbf{k}_s + \mathbf{k}_i$$

SPDC produces entangled photon pairs and is the primary source for quantum optics experiments, including tests of Bell inequalities and quantum key distribution.

5.2.3 Optical Parametric Amplification

In optical parametric amplification (OPA), a weak signal is amplified at the expense of a strong pump. The parametric gain is:

$$G = 1 + \left(\frac{\kappa L}{\sinh(\kappa L)}\right)^2 \sinh^2(gL), \quad g = \sqrt{\kappa^2 - (\Delta k/2)^2}$$

5.3 Third-Order (χ⁽³⁾) Processes

5.3.1 The Optical Kerr Effect

The third-order nonlinearity causes the refractive index to depend on intensity:

$$n = n_0 + n_2 I, \qquad n_2 = \frac{3\chi^{(3)}}{4n_0^2 \epsilon_0 c}$$

For silica glass, n₂ ≈ 2.6 × 10^-20 m²/W. Despite this tiny value, the Kerr effect is crucial in fibers due to the long interaction lengths and small mode areas.

5.3.2 Self-Phase Modulation (SPM)

A pulse propagating in a Kerr medium acquires an intensity-dependent phase:

$$\phi_{\text{NL}}(t) = -n_2 \frac{\omega_0}{c} I(t) L = -\gamma P(t) L_{\text{eff}}$$

where γ = n₂ω₀/(cA_eff) is the nonlinear parameter and L_eff = (1 - e^-αL)/α. SPM generates new frequencies (spectral broadening) without changing the temporal profile. The nonlinear length is:

$$L_{\text{NL}} = \frac{1}{\gamma P_0}$$

5.3.3 Optical Solitons

When the anomalous GVD (β₂ < 0) exactly balances SPM, the pulse propagates without changing shape: a fundamental soliton. The condition is:

$$N^2 = \frac{\gamma P_0 T_0^2}{|\beta_2|} = 1 \quad \text{(fundamental soliton)}$$

The soliton pulse shape is a hyperbolic secant: E(t) = √P₀ sech(t/T₀). Soliton propagation in fibers was proposed by Hasegawa (1973) and first observed by Mollenauer (1980). Solitons are self-reinforcing: perturbations shed dispersive radiation and the soliton reforms.

5.3.4 Four-Wave Mixing

Three input frequencies can interact via χ⁽³⁾ to generate a fourth: ω₄ = ω₁ + ω₂ - ω₃. In fibers, degenerate FWM (2ω_p = ω_s + ω_i) produces parametric gain and is both a source of crosstalk in WDM systems and a tool for wavelength conversion.

5.4 Python Simulation: SHG, SPM & Solitons

This simulation demonstrates SHG phase-matching curves, self-phase modulation spectral broadening, soliton propagation, and the Kerr lens effect.

Nonlinear Optics: SHG, SPM & Solitons

Python

script.py153 lines

import numpy as np
import matplotlib.pyplot as plt

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: SHG efficiency vs phase mismatch ---
ax1 = axes[0, 0]
dkL = np.linspace(-10*np.pi, 10*np.pi, 1000)
eta = np.sinc(dkL / (2*np.pi))**2  # sinc^2(dkL/2) using numpy sinc convention

ax1.plot(dkL/np.pi, eta, color='#f59e0b', linewidth=2)
ax1.fill_between(dkL/np.pi, eta, alpha=0.15, color='#f59e0b')
ax1.axvline(0, color='#fbbf24', linestyle='--', alpha=0.5)
ax1.set_xlabel('Delta_k * L / pi', color='white')
ax1.set_ylabel('Normalized SHG efficiency', color='white')
ax1.set_title('SHG Phase-Matching Curve', color='#fbbf24', fontsize=13)
ax1.annotate('Perfect phase\nmatching', (0.3, 0.9), color='#fbbf24', fontsize=9)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 2: Self-phase modulation ---
ax2 = axes[0, 1]
t = np.linspace(-5, 5, 1024)
P0 = 1.0  # peak power (normalized)
T0 = 1.0  # pulse width

# Gaussian pulse
pulse = P0 * np.exp(-t**2 / (2*T0**2))

# SPM phase for different nonlinear lengths
for phi_max_val, label in [(0.5*np.pi, 'phi_max=0.5pi'),
                            (2*np.pi, 'phi_max=2pi'),
                            (5*np.pi, 'phi_max=5pi')]:
    phi_nl = phi_max_val * pulse / P0
    E_spm = np.sqrt(pulse) * np.exp(1j * phi_nl)
    spectrum = np.abs(np.fft.fftshift(np.fft.fft(E_spm)))**2
    freq = np.fft.fftshift(np.fft.fftfreq(len(t), t[1]-t[0]))
    spectrum /= np.max(spectrum)
    ax2.plot(freq, spectrum, linewidth=1.5, label=label, alpha=0.8)

ax2.set_xlim(-3, 3)
ax2.set_xlabel('Frequency (normalized)', color='white')
ax2.set_ylabel('Spectral intensity', color='white')
ax2.set_title('Self-Phase Modulation: Spectral Broadening', 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: Soliton propagation (split-step Fourier) ---
ax3 = axes[1, 0]

# NLSE: i dA/dz + (beta2/2) d^2A/dt^2 + gamma|A|^2 A = 0
# Normalized: i du/dxi + (1/2) d^2u/dtau^2 + |u|^2 u = 0
N_t = 1024
tau = np.linspace(-10, 10, N_t)
dtau = tau[1] - tau[0]
dxi = 0.005  # step size in normalized distance
xi_max = 5 * np.pi  # propagation distance (soliton period = pi/2)

# Fundamental soliton: u = sech(tau)
u = 1.0 / np.cosh(tau)
u_init = u.copy()

# Frequency grid
omega = 2*np.pi * np.fft.fftfreq(N_t, dtau)

# Split-step Fourier
n_steps = int(xi_max / dxi)
propagation_distances = []
propagation_profiles = []

for i in range(n_steps):
    # Half nonlinear step
    u = u * np.exp(1j * np.abs(u)**2 * dxi / 2)
    # Full dispersive step
    u_f = np.fft.fft(u)
    u_f = u_f * np.exp(-1j * omega**2 * dxi / 2)  # anomalous dispersion sign
    u = np.fft.ifft(u_f)
    # Half nonlinear step
    u = u * np.exp(1j * np.abs(u)**2 * dxi / 2)

if i % (n_steps // 5) == 0:
        propagation_distances.append(i * dxi)
        propagation_profiles.append(np.abs(u)**2)

colors_sol = plt.cm.autumn(np.linspace(0, 1, len(propagation_profiles)))
for j, (xi_val, profile) in enumerate(zip(propagation_distances, propagation_profiles)):
    ax3.plot(tau, profile, color=colors_sol[j], linewidth=1.5,
             label=f'xi={xi_val:.1f}', alpha=0.8)
ax3.plot(tau, np.abs(u_init)**2, 'w--', linewidth=1, alpha=0.5, label='Initial')
ax3.set_xlim(-5, 5)
ax3.set_xlabel('Normalized time tau', color='white')
ax3.set_ylabel('|u|^2', color='white')
ax3.set_title('Fundamental Soliton Propagation', color='#fbbf24', fontsize=13)
ax3.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=8)
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 4: Quasi-phase matching (periodic poling) ---
ax4 = axes[1, 1]
z = np.linspace(0, 10, 1000)  # normalized crystal length

# Perfect phase matching
A_pm = z
# Phase mismatched (dk*L = 20)
dk = 6
A_mismatch = np.abs(np.sin(dk * z / 2) / (dk / 2))
# Quasi-phase matched (periodic poling with Lambda = 2pi/dk)
Lambda_qpm = 2 * np.pi / dk
# In QPM, effective dk -> 0 with reduced deff by factor 2/pi
A_qpm_envelope = (2/np.pi) * z

# Build QPM oscillation
A_qpm = np.zeros_like(z)
for i in range(1, len(z)):
    domain_sign = 1 if (z[i] % Lambda_qpm) < Lambda_qpm/2 else -1
    A_qpm[i] = A_qpm[i-1] + domain_sign * np.cos(dk * z[i]) * (z[1]-z[0])
A_qpm = np.abs(A_qpm)

ax4.plot(z, A_pm**2/np.max(A_pm**2), color='#f59e0b', linewidth=2, label='Perfect PM')
ax4.plot(z, A_mismatch**2/np.max(A_pm**2), color='#fb923c', linewidth=1.5, linestyle='--', label='Phase mismatched')
ax4.plot(z, A_qpm_envelope**2/np.max(A_pm**2), color='#fbbf24', linewidth=2, linestyle='-.', label='QPM (envelope)')
ax4.set_xlabel('Crystal length (normalized)', color='white')
ax4.set_ylabel('SHG power (normalized)', color='white')
ax4.set_title('Phase Matching Schemes', color='#fbbf24', fontsize=13)
ax4.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#f59e0b')

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

# Numerical examples
n2_silica = 2.6e-20  # m^2/W
A_eff = 80e-12       # 80 um^2 effective area
lam_op = 1550e-9
gamma = n2_silica * 2 * np.pi / (lam_op * A_eff)
print(f"Nonlinear parameter for SMF at 1550 nm:")
print(f"  n2 = {n2_silica:.1e} m^2/W")
print(f"  A_eff = {A_eff*1e12:.0f} um^2")
print(f"  gamma = {gamma:.2f} /W/km = {gamma*1e3:.2f} /W/m")
P_soliton = abs(-21.7e-27) / (gamma * (10e-12)**2)
print(f"\nFundamental soliton power (T0=10 ps):")
print(f"  P0 = {P_soliton*1e3:.1f} mW")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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