Kerr Effect & Self-Phase Modulation
Nonlinear phase shift φ_NL = γPL · |A(t)|² · spectral broadening
Self-Phase Modulation (SPM): In a Kerr medium, n = n₀ + n₂I. A pulse acquires nonlinear phase φ_NL(t) = γ|A(t)|²L. The instantaneous frequency shift is δω(t) = −∂φ_NL/∂t ∝ −∂|A|²/∂t. This creates new frequency components — spectral broadening.
B-integral: B = ∫γP(z)dz is the peak nonlinear phase. At B=π, strong side lobes appear. At B≫π, the spectrum develops a characteristic M-shaped structure with multiple peaks whose count ≈ B/π + 1.
Left panel: Time-domain intensity (blue) and instantaneous frequency chirp (orange). Right panel: Spectral intensity via FFT of the chirped pulse.
GVD: Group velocity dispersion β₂ competes with SPM — their balance underlies optical solitons (NLS equation).