Photonic-platform non-linear optics — non-linear wave propagation in fibres, planar waveguides, photonic crystals, and micro-ring resonators. Foundational: Kerr 1875 n(I) = n_0 + n_2·I electro-optic / photoelectric non-linearity;…
nonlinear-photonics
Fibre Kerr: n(I) = n_0 + n_2 I → self-phase modulation φ_SPM = γ P L_eff
Kerr effect in silica (fused SiO_2) fibre: the refractive index is intensity-dependent, n(I) = n_0 + n_2·I, with Kerr coefficient n_2 ≈…
Manley-Rowe 1956: photon-flux conservation in 3-wave mixing ω_1 + ω_2 = ω_3
Manley-Rowe relations (1956, Proc IRE 44:904): in a lossless χ^(2) 3-wave mixing process ω_1 + ω_2 = ω_3 (sum-frequency generation if 3 is…
Bright soliton in anomalous-dispersion fibre: A(z,t) = √P_0 sech(t/T_0) exp(iγP_0 z/2)
Hasegawa-Tappert 1973 bright soliton. In a fibre with anomalous group-velocity dispersion β_2 < 0 and Kerr non-linearity γ > 0, the…
SPM max chirp: δω_max = γ P_0 / T_0 at pulse edges (∂/∂t |A|²) for Gaussian pulse
Instantaneous frequency chirp imposed by self-phase modulation on a pulse of envelope |A(0,t)|² propagating through fibre of effective…
Manley-Rowe photon balance: ΔΦ_1 = ΔΦ_2 = -ΔΦ_3 per 3-wave interaction step
Integral form of the Manley-Rowe 1956 relations. For a lossless χ^(2) 3-wave mixing interaction ω_1 + ω_2 = ω_3 between points z = 0 and z…
NLSE bright soliton area theorem: P_0 T_0² = |β_2|/γ exactly for N=1 fundamental
Fundamental bright-soliton area theorem. Given the non-linear Schrödinger equation i·∂A/∂z − (β_2/2)·∂²A/∂t² + γ|A|²A = 0 with β_2 < 0, γ…