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…
OPA coupled-wave ODE: dA_s/dz = i*kappa*A_i^*; variation-of-parameters closed form
Optical parametric amplification coupled-wave ODEs. Setup: chi^(2) parametric process pumping a signal A_s at omega_s and idler A_i at…
FWM gain matrix eigenvalue bound via Gershgorin discs: |lambda| <= |delta|+|g|
Four-wave-mixing gain matrix and Gershgorin eigenvalue bound. Setup: degenerate FWM in a Kerr fiber produces signal (omega_p + Omega) and…
Modulation instability linear stability: eigenvalue via Courant-Fischer min-max principle
Modulation-instability linear-stability analysis via Courant-Fischer. Setup: nonlinear Schrodinger equation i*partial-t A +…
OPA exponential gain: |A_s(z)| = |A_s(0)|*cosh(|kappa|*z); asymptotic exp(|kappa|*z)/2
OPA exponential gain theorem. From framework F7 (variation-of-parameters on coupled-wave ODEs), the signal amplitude grows as |A_s(z)| =…
FWM matrix eigenvalues: lambda = +-sqrt(delta^2+|g|^2); Gershgorin bound |lambda|<=|delta|+|g|
FWM matrix eigenvalue vs Gershgorin bound. From framework F8 (FWM gain matrix), the exact eigenvalues of M = [[delta, g], [g, -delta]]…
MI peak gain at K_peak = sqrt(gamma*P); g_max = gamma*P via Courant-Fischer
Modulation-instability peak gain (anomalous-dispersion regime, |beta_2| = 1 normalized). From framework F9 (Courant-Fischer), the MI growth…
Zakharov-Shabat 1972 NLS inverse scattering
Zakharov-Shabat 1972: nonlinear Schrodinger equation i u_t + u_xx + 2 |u|^2 u = 0 is integrable via inverse scattering transform; bright +…
Kerr effect (Kerr 1875)
J Kerr 1875 chi(3) electro-optic; modern Kerr-comb microresonators Pasquazi 2018 + chip-scale frequency-comb sources.
Optical solitons (Mollenauer 1980)
Mollenauer-Stolen 1980 fiber-optic solitons; modern soliton-laser + soliton-microcomb.
Supercontinuum (Ranka 2000)
Ranka 2000 PCF supercontinuum; modern octave-spanning + carrier-envelope-phase + frequency-comb metrology.
SPDC (Burnham-Weinberg 1970)
Burnham-Weinberg 1970 SPDC entangled-photon pair; modern Bell-test + QKD source; basis of photonic-quantum-computing.
QPM (Armstrong 1962)
Armstrong-Bloembergen-Ducuing-Pershan 1962 QPM quasi-phase-matching; PPLN; modern PPKTP + PPLN + on-chip integrated.
SHG (Franken 1961)
P Franken 1961 first SHG quartz; modern modern foundational text + KTP / BBO / PPLN + 2024 metasurface-SHG sub-wavelength flat-optics.
Cross-phase (Garmire 1972)
E Garmire 1972 + Stolen 1989 XPM nonlinear-fiber; modern modern foundational text + DWDM nonlinear-impairments + 2024 ML-DWDM compensation.
Zakharov-Shabat (1972)
V Zakharov-A Shabat 1972 NLSE-IST; modern modern foundational text + Hasegawa-Tappert 1973 fiber-soliton + DWDM nonlinear capacity.
Frequency comb (Hall-Hänsch 2005)
J Hall-T Hänsch 2005 (Nobel 2005) optical-frequency-comb; modern modern foundational + 2024 chip-scale microcombs + soliton-microcomb DWDM.
Microcomb (Del'Haye 2007)
P Del'Haye-Vahala 2007 Kerr-microresonator-comb; modern modern foundational + dissipative-Kerr-soliton + 2024 chip-scale telecom…
FWM (Stolen 1974)
R Stolen 1974 four-wave-mixing fiber; modern modern foundational text + 2024 DWDM impairments + ML-mitigation + parametric-amplification…