Noise model

Background

Noise-seeded processes are central to many phenomena in nonlinear fibre optics: spontaneous Raman scattering seeds Raman combs; noise-seeded four-wave mixing (FWM) drives modulational instability (MI); and the interplay of both produces the shot-to-shot fluctuations characteristic of supercontinuum generation. An accurate noise model is therefore essential for any simulation where noise-initiated nonlinear dynamics play a role.

Luna.jl supports two noise models, selected via the shotnoise keyword argument in prop_capillary and prop_gnlse:

• :input –- the traditional shot-noise approach.
• :modified or true (default) –- the modified shot-noise approach of Chen & Wise [1].

Traditional shot noise (:input)

The traditional approach adds one-photon-per-mode quantum noise directly to the input field at $z = 0$:

\[\tilde{E}(z=0, \omega) = \tilde{E}_{\mathrm{pulse}}(\omega) + \tilde{E}_{\mathrm{noise}}(\omega)\]

where $|\tilde{E}_{\mathrm{noise}}(\omega)| = \sqrt{\hbar\omega\,\Delta\nu}$ with uniformly random spectral phase and $\Delta\nu$ is the frequency bin spacing. The propagation equation then contains only stimulated (deterministic) terms, and all noise-seeded processes arise from the stimulated evolution of the injected noise.

While simple, this approach has several well-known limitations:

1. Artificial FWM phase-matching: The noise field, superimposed on the input pulse, acquires a coherent phase increment from dispersion during propagation. This means the noise can satisfy phase-matching conditions for FWM, which is unphysical –- real vacuum fluctuations are incoherent and cannot maintain phase-matching over extended distances.

2. Elevated noise floor: The noise creates a persistent increase of the spectral intensity at all frequencies that does not decay.

3. No Stokes/anti-Stokes asymmetry: Spontaneous Raman generation exhibits a fundamental spectral asymmetry (Stokes is favoured by an extra quantum of vacuum fluctuation). The traditional model does not capture this.

Modified shot noise (:modified)

The modified shot-noise model [1] resolves all of the above issues by including the noise field in the nonlinear operator rather than in the input field:

\[\partial_z A = \hat{\mathcal{D}}\,A + \hat{\mathcal{N}}(A + A_{\mathrm{noise}})\]

The key structural features are:

• Dispersion acts only on the physical field: $\hat{\mathcal{D}}\,A$, not $\hat{\mathcal{D}}(A + A_{\mathrm{noise}})$. This prevents the noise from acquiring coherent phase increments that would cause artificial FWM phase-matching.

• The noise field is constant: $A_{\mathrm{noise}}(\omega)$ is generated once before propagation with one-photon-per-mode spectral density and random phase, and held fixed throughout. It does not evolve, grow, or accumulate.

• All nonlinear processes are correctly seeded: Both Kerr (FWM/MI) and Raman (spontaneous Stokes/anti-Stokes) processes are seeded through the same mechanism –- the noise enters the nonlinear operator alongside the field.

• No elevated noise floor: Because the noise is not added to the propagating field, the spectrum at $z = 0$ contains only the input pulse. Noise-seeded spectral components grow continuously from zero as the pulse propagates.

This approach is a general-purpose noise model applicable to all $\chi^{(3)}$ propagation problems, including MI-based supercontinuum generation, Raman comb formation, soliton dynamics with self-frequency shift, and combined Kerr–Raman broadening.

Usage

The modified shot-noise model is used by default (shotnoise=true, equivalent to shotnoise=:modified). To use the traditional model instead, pass shotnoise=:input:

# Default: modified shot-noise model (shotnoise=true)
output = prop_capillary(radius, flength, gas, pressure;
    λ0=800e-9, λlims=(200e-9, 4000e-9), trange=400e-15,
    τfwhm=30e-15, energy=1e-6,
)

For ensemble simulations (shot-to-shot statistics), use different random seeds for each shot:

using Random
for shot in 1:100
    rng = MersenneTwister(shot)
    output = prop_capillary(radius, flength, gas, pressure;
        λ0=800e-9, λlims=(200e-9, 4000e-9), trange=400e-15,
        τfwhm=30e-15, energy=1e-6,
        shotnoise=:modified, rng=rng
    )
    # ... process output ...
end

Setting shotnoise=false disables noise entirely. This prevents generation of the noise field when using the modified model, and prevents adding shot noise to the input spectrum when using the traditional model.

Ionization and plasma

In the modified shot-noise model, the combined field $A + A_{\mathrm{noise}}$ is passed to all nonlinear response functions, including Kerr, Raman, and plasma/ionization. This is physically reasonable because the noise amplitude is of order $\sqrt{\hbar\omega\,\Delta\nu} \approx 5 \times 10^{-4}\;\sqrt{\mathrm{W}}$ per frequency mode — roughly $10^{-14}$ of a typical pulse peak power. The tunnelling ionization rate depends on the electric field through a highly nonlinear (exponential) threshold, so the negligible noise field has no measurable effect on the plasma response. Only the Kerr and Raman processes, which are linear in the perturbation field, are meaningfully affected by the noise injection.

Power diagnostics at low signal levels

Because the noise is not added to the propagating field, spontaneously generated spectral components start from exactly zero power at $z = 0$. At very small $z$, the power $|A_{s'}|^2 \propto z^2$ (quadratic growth), which is the expected linearised result. To recover the correct physical power including the vacuum fluctuation contribution, one can temporarily add the noise before computing the power:

\[P_{s'}(z) = |A_{s'}(z) + A_{\mathrm{noise}}|^2 - P_{\mathrm{noise}}\]

This correction is only necessary when the generated power is comparable to or below the noise floor. At higher powers (the stimulated/saturated regime), $|A_{s'}|^2$ directly gives the correct result.

Comparison

References

1. Y.-H. Chen and F. W. Wise, "A simple accurate way to model noise-seeded ultrafast nonlinear processes", arXiv:2410.20567 (2024).

2. Y.-H. Chen and F. W. Wise, "Unified theory for Raman scattering in gas-filled hollow-core fibers", APL Photonics 9, 030902 (2024).