Solvers

nonlinearmedium currently contains the equation solvers described below, defined in this directory. They are split into two categories.

1. Linear equations, intended for simulation of quantum signals or classical light where the pump may be approximated as undepleted. For these equations, a Green's functions may be computed.
2. Fully nonlinear equations, where the pump is depleted. A Green's function may not be computed.

In these equations the sign of the nonlinear interaction L_NL depends on the poling, if applicable. D̂ represents the differential dispersion operator for a mode (i [β₁ ∂_t + β₂ ∂_t² + β₃ ∂_t³]).

Linear equations

Pump equation

Unless specified otherwise, the pump propagates influenced only by dispersion, and the effective intensity scales according Rayleigh length.

A_p(z, Δω) = A_p(0, Δω) exp(i k(Δω) z) / √1 + ((z-L/2) / zᵣ)²

Chi2PDC

A₀'(z, t) = D̂ A₀ + i L_NL^-1 A_p A₀^† e^{i Δk z}

Degenerate optical parametric amplification.

Chi2PDCII

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A_p A₁^† e^{i Δk z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 A_p A₀^† e^{i Δk z}

Non-degenerate (or type II) optical parametric amplification.

Chi2SFG

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A_p A₁ e^{i Δk z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 A_p^† A₀ e^{-i Δk z}

Sum (or difference) frequency generation.

Chi2AFC

Adiabatic frequency conversion (aka adiabatic sum/difference frequency generation), in a rotating frame with a linearly varying poling frequency built-in to the solver. Same equation as above, but poling is disabled; intended as a faster, approximate version of Chi2SFG applied to AFC.

Chi2SFGII

A₀'(z, t) = D̂ A₀ + i A_p (L_NL0^-1 A₃ e^{i Δk₀ z} + L_NL1^-1 A₂ e^{i Δk₂ z})
A₁'(z, t) = D̂ A₁ + i L_NL0^-1 A_p A₂ e^{i Δk₁ z}
A₂'(z, t) = D̂ A₂ + i A_p^† (L_NL0^-1 A₁ e^{-i Δk₁ z} + L_NL1^-1 A₀ e^{-i Δk₂ z})
A₃'(z, t) = D̂ A₃ + i L_NL0^-1 A_p^† A₀ e^{-i Δk₀ z}

Three simultaneous sum frequency generation processes with one pump among four modes (e.g. of different polarizations).

Chi2SFGPDC

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A_p A₁ e^{i Δk₀ z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 (A_p^† A₀ e^{-i Δk₀ z} + A_p A₀^† e^{i Δk₁ z})

Simultaneous sum frequency generation and parametric amplification.

Chi3

A₀'(z, t) = D̂ A₀ + i L_NL^-1 (2|A_p|² A₀ + A_p² A₀^†)
A_p'(z, t) = D̂ A_p + i L_NL^-1 |A_p|² A_p / (1 + ((z-L/2) / zr)²)

Noise reduction by self phase modulation.

Chi2SFGXPM

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A_p A₁ e^{i Δk z} + 2 i L_NL2^-1 |A_p|² A₀
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 A_p^† A₀ e^{-i Δk z} + 2 i L_NL3^-1 |A_p|² A₁

Sum (or difference) frequency generation with cross phase modulation.

Chi2SFGOPA

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A_p0 A₁ e^{i Δk₀ z} + i L_NL2^-1 A_p1 A₁^† e^{i Δk₁ z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 A_p0^† A₀ e^{-i Δk₀ z} + i L_NL3^-1 A_p1 A₀^† e^{i Δk₁ z}

Simultaneous sum frequency generation and non-degenerate optical parametric amplification with two pumps.

Fully nonlinear equations

In these equations the strength of the interaction L_NL scales according the Rayleigh length (1 / √1 + ((z-L/2) / zr)²).

Chi2DSFG

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A₁ A₂^† e^{i Δk z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 A₀ A₂ e^{-i Δk z}
A₂'(z, t) = D̂ A₂ + i L_NL2^-1 A₁ A₀^† e^{i Δk z}

Sum or difference frequency generation, or optical parametric amplification.

Chi2SHG

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A₁ A₀^† e^{i Δk z}
A₁'(z, t) = D̂ A₁ + ½ i L_NL1^-1 A₀² e^{-i Δk z}

Second harmonic generation.

Chi2ASHG

Adiabatic second harmonic generation, in a rotating frame with a linearly varying poling frequency built-in to the solver. Same equation as above, but poling is disabled; intended as a faster, approximate version of Chi2SHG applied to the adiabatic case.

Chi2SHGOPA

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A₁ A₀^† e^{i Δk₀ z}
A₁'(z, t) = D̂ A₁ + i L_NL1^-1 (½ A₀² e^{-i Δk₀ z} + A₂ A₃ e^{i Δk₁ z})
A₂'(z, t) = D̂ A₂ + i L_NL2^-1 A₁ A₃^† e^{-i Δk₁ z}
A₃'(z, t) = D̂ A₃ + i L_NL3^-1 A₁ A₂^† e^{-i Δk₁ z}

Non-degenerate optical parametric amplification driven by second harmonic generation.

Chi2SHGXPM

A₀'(z, t) = D̂ A₀ + i L_NL0^-1 A₁ A₀^† e^{i Δk z} + i L_NL2^-1 (|A₀|² + 2 |A₁|²) A₀
A₁'(z, t) = D̂ A₁ + ½ i L_NL1^-1 A₀² e^{-i Δk z} + 2 i L_NL2^-1 (2 |A₀|² + |A₁|²) A₁

Second harmonic generation with self and cross phase modulation.

Chi3GNLSE

A₀'(z, t) = D̂ A₀ + i (L _NL0^-1 + i L_NL1^-1 d/dt) [A₀ ∫dt R(t') |A₀(t-t')|²]
R(t < 0) = 0
R(t ≥ 0) = f_r (1 - f_b) (τ₁ / τ₂² + 1 / τ₁) exp(-t / τ₂) sin(t / τ₁) + f_r f_b / (τ₃²) exp(-t / τ₃) (2 τ₃ - t) + (1 - f_r) δ(t)

Generalized nonlinear Schrödinger equation, with Raman response R(t).