2.5 Third-order effects

The index also rises with intensity, the optical Kerr effect:

For LN, n₂ = 9.33×10⁻²⁰ m²/W at 1064 nm (Z-scan, 30 ps pulses, e-polarized) P-db, transcribed in refractiveindex.info. The widely tabulated 1.8×10⁻¹⁹ m²/W S is 1.9 times that derived and has no traced primary.

In a high-Q ring, four-wave mixing builds a Kerr frequency comb, and in the right detuning range a dissipative soliton. TFLN rings have produced self-starting solitons (He et al. 2019), combs spanning 4/5 of an octave at 335 GHz spacing with stimulated Raman scattering suppressed reported, full-octave solitons with near-unity device yield reported, and a photorefraction-stabilized soliton running for more than 13 h without feedback reported. One chip has generated a Kerr comb, selected a line with an electro-optic filter and modulated it reported.

A phase-mismatched χ⁽²⁾ process imitates a Kerr effect whose size and sign the designer chooses: cascaded χ⁽²⁾. Drag the mismatch Δk through zero: the cascaded phase grows as 1/Δk, flips sign and decays, while the Kerr phase stays fixed.

Kerr

The nonlinear parameter at 1550 nm with A_eff = 1 µm² template is γ_Kerr = 0.378 /(W m) derived, using n₂ = 9.33×10⁻²⁰ m²/W measured at 1064 nm; the dispersion of n₂ to 1550 nm is not sourced.

Cascaded χ⁽²⁾

For |Δk|L >> 1 the coupled equations of 2.3 L3 give a fundamental-wave phase

Δφ_NL carries the sign of a Kerr phase γPL (positive = index increase); in the field convention of 2.3 L3, arg A_ω = −Δφ_NL. The fundamental couples to a second harmonic detuned above it (Δk > 0) and is pushed down: a negative, self-defocusing nonlinearity. Δk < 0 gives self-focusing (principle: DeSalvo et al. 1992; sign from the level-repulsion picture derived). Along z the undepleted harmonic carries

The cascaded nonlinearity is 11 times the Kerr one while on average only 1.3×10⁻⁵ of the power sits in the harmonic. In nanophotonic PPLN, strong phase mismatch broadens both harmonics with 2 pJ pulses reported.

Lugiato-Lefever equation

Normalized (Lugiato & Lefever 1987; Chembo & Menyuk 2013; Coen et al. 2013):

ζ = 2(ω₀ − ω_p)/κ is the detuning, f the normalized pump, d₂ the normalized second-order dispersion (anomalous: d₂ > 0 here), θ the angle around the ring. Solitons exist on the red-detuned side above the bistability threshold; scanning the laser from blue to red crosses chaotic modulation instability into discrete soliton steps (Herr et al. 2014; review Kippenberg et al. 2018). Press Run: the scan starts on noise at ζ = −2, passes through chaos, and drops onto plateaus whose height counts the solitons; the default ends on one soliton with a sech² spectrum.

• Z-scan n₂ depends on pulse duration (slow thermal and electrostrictive contributions grow with it), wavelength (two-photon absorption raises n₂ toward the band edge) and polarization. No 1550 nm LN n₂ measurement is held.
• Cascaded contamination. An n₂ measured in a non-centrosymmetric crystal can include a cascaded χ⁽²⁾ contribution from residual phase mismatch; whether the tabulated LN values were corrected for it is not stated in the sources held.
• Raman. LN is strongly Raman-active: soliton combs must place no cavity mode under the Raman gain peak or control the dissipation spectrum there reported reported. Raman gain coefficients for TFLN are not sourced here; the soliton simulation has no Raman term.
• LN-specific dynamics. Photorefraction produces a blue shift that counteracts the thermal red shift and extends the soliton existence range, giving spontaneous, deterministic single solitons reported; self-starting LN solitons were first reported by He et al. 2019. The soliton simulation has no photorefractive or thermal term.

n₂ of LN

LLE normalization

Time in units of 2/κ; field, pump and dispersion as in Chembo & Menyuk 2013, with κ = ω₀/Q_L, ψ = √(2g₀/κ) a (|a|² = intracavity photons) and the Kerr frequency shift per photon g₀ from the waveguide γ:

D₂ from ω_μ = ω₀ + D₁μ + D₂μ²/2. Both views of the soliton simulation print this conversion for a template ring template.

Single-soliton approximation

Herr et al. 2014 (Methods); ψ_cw is the lower homogeneous state. The soliton branch requires √(8ζ) ≤ πf, so it ends at ζ = π²f²/8; the mode powers follow from the Fourier series of the sech derived. The dashed soliton-comb spectra follow this envelope; for the default single soliton (ζ = 10, f = 3, d₂ = 0.3) the fitted width is 0.979 of w.

Numerics of the soliton simulation

Strang split-step Fourier on N = 512 points in θ: Kerr rotation exp(i|ψ|²dt/2), then loss, detuning, dispersion and pump integrated exactly per mode, then the second Kerr half step; dt = 10⁻³. The scan advances ζ after every block of 20 steps, so a run is deterministic for a given seed; the first step of each block is the energy-balance check d⟨|ψ|²⟩/dt = −2⟨|ψ|²⟩ + 2f Re⟨ψ⟩ (dispersion and Kerr conserve energy).