Kerr Effect · Self-Phase Modulation · Optical Solitons
Nonlinear Schrödinger equation · Split-step Fourier method · Soliton formation
Nonlinearity γ (Kerr)
1.50
Dispersion β₂
-1.0
Pulse width T₀
1.5
Peak amplitude A₀
1.0
Propagation speed
3
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iA_z + (β₂/2)A_tt + γ|A|²A = 0
NLS equation (split-step Fourier)
Soliton: A(z,t) = A₀ sech(t/T₀)
with γA₀² = β₂/T₀² (anomalous)
SPM: φ_NL(t) = γ|A(t)|²z
spectral broadening ∝ φ_max
Split-step: alternate dispersion (frequency domain) and nonlinearity (time domain) in half-steps. Soliton requires anomalous dispersion (β₂<0). SPM creates new frequencies; dispersion spreads/compresses. Balance = soliton.