Physics Background

This page summarizes the physical models implemented in GNLSE.jl.

The Generalized Nonlinear Schrödinger Equation

The GNLSE governing the evolution of the pulse envelope $A(z, t)$ in the retarded time frame is:

\[\frac{\partial A}{\partial z} = \underbrace{\sum_{n \geq 2} \frac{i^{n+1}}{n!} \beta_n \frac{\partial^n A}{\partial t^n} - \frac{\alpha}{2} A}_{\text{linear: dispersion + loss}} + \underbrace{i \gamma \left(1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}\right) \left[ A(z,t) \int_{-\infty}^{\infty} R(t') |A(z,t-t')|^2 \, dt' \right]}_{\text{nonlinear: Kerr + Raman + self-steepening}}\]

where the nonlinear response function is:

\[R(t) = (1 - f_R) \delta(t) + f_R h_R(t)\]

and $f_R \approx 0.18$ is the fractional Raman contribution.

Dispersion

In the frequency domain (interaction picture), the linear operator is:

\[D(\omega) = i B(\omega) - \frac{\alpha}{2}\]

where $B(\omega - \omega_0)$ is the propagation-constant deviation from the reference frame, computed from one of the available dispersion models:

• Taylor expansion — TaylorDispersion(betas): $B(V) = \beta_1 V + \sum_{n \geq 2} \frac{\beta_n}{n!} V^n$

• Tabulated dispersion — TabulatedDispersion(detuning, beta): Linearly interpolated from measured data.

• Sellmeier model — SellmeierDispersion(B, C): $n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}$

Kerr Nonlinearity

The instantaneous Kerr contribution gives self-phase modulation (SPM). In the frequency domain (after $\mathcal{F}^{-1}$ of the time-domain product):

\[N_{\rm Kerr} = i \gamma \frac{\omega}{\omega_0} \mathcal{F}^{-1}\!\left[ |A(t)|^2 A(t) \right]\]

The ratio $\omega/\omega_0$ is the self-steepening shock term (enabled via self_steepening=true).

The nonlinear coefficient $\gamma$ can be:

• Constant — a scalar value γ [1/(W·m)]
• z-dependent — a function γ(z) for tapered or graded fibers
• Frequency-dependent — FrequencyDependentNonlinearity or derived from effective mode area via NonlinearityFromEffectiveArea

Raman Scattering

Three models for the Raman response $h_R(t)$ are available:

Self-Steepening

Enabling self_steepening=true replaces the constant $\gamma$ prefactor with the frequency-dependent $\gamma \omega/\omega_0$, which accounts for the intensity-dependent group velocity and produces the optical shock effect important for sub-100 fs pulses.

Birefringent Media (Coupled GNLSE)

For birefringent fibers or waveguides, two orthogonal polarization envelopes $A_x$ and $A_y$ are coupled:

\[\frac{\partial A_x}{\partial z} = D_x A_x + i \gamma \left(|A_x|^2 + \frac{2}{3}|A_y|^2 \right) A_x + \frac{i \gamma}{3} A_y^2 A_x^* e^{-2i\Delta\beta_0 z}\]

\[\frac{\partial A_y}{\partial z} = D_y A_y + i \gamma \left(|A_y|^2 + \frac{2}{3}|A_x|^2 \right) A_y + \frac{i \gamma}{3} A_x^2 A_y^* e^{+2i\Delta\beta_0 z}\]

where $\Delta\beta_0 = \beta_{0x} - \beta_{0y}$ is the birefringence phase mismatch, and $D_x$, $D_y$ are independent dispersion operators including any group-velocity mismatch (walk-off) via $\beta_1$.

Active Amplifying Fibers (EDFA / YDFA)

Active rare-earth doped fibers experience single-pass amplification with dynamic gain saturation:

\[g(z, \omega) = \frac{g_0(\omega)}{1 + E_{\text{pulse}}(z) / E_{\text{sat}}}\]

where $E_{\text{pulse}}(z) = \int |A(z,t)|^2 dt$ is the local pulse energy and $E_{\text{sat}}$ is the saturation energy. Spontaneous emission quantum noise (ASE) is seeded at each step:

\[S_{\text{ASE}}(\omega) = n_{\text{sp}} \cdot \hbar \omega_0 \cdot \left(e^{g(z) \Delta z} - 1\right), \qquad n_{\text{sp}} = \frac{10^{F_{\text{dB}}/10}}{2}\]

where $F_{\text{dB}}$ is the amplifier Noise Figure in dB.

Gas-Filled Hollow-Core PCF

Gas guidance in hollow-core anti-resonant / photonic crystal fibers (HC-PCF) combines:

1. Marcatili-Schmeltzer Capillary Loss (opt-in via confinement_loss=true in HollowCoreFiber; false by default):

   \[\alpha(\lambda) = \left( \frac{u_{01}}{2\pi} \right)^2 \frac{\lambda^2}{a^3} \frac{\nu^2 + 1}{\sqrt{\nu^2 - 1}}\]

   where $a$ is the core radius and $\nu$ is the cladding index ratio. This is the bare single-wall thick-capillary bound — it excludes the anti-resonant wall-thickness transmission-window term that real negative-curvature HC-PCF use to suppress loss by orders of magnitude, so at typical HC-PCF core radii (tens of μm) it substantially overestimates loss relative to measured fibers. Treat it as a conservative bound, not a quantitative prediction.
2. Pressure-Dependent Gas Sellmeier:

   \[n(\lambda, P) = 1 + P \cdot \frac{C_1}{C_2 - \lambda^{-2}}\]

   with coefficients from Börzsönyi et al. (2013) for noble gases (:Ar, :Ne, :Kr, :Xe) and molecular gases (:H2, :N2).
3. Molecular Gas Raman Response: Rotational ($S(1)$ shift $17.6\text{ THz}$) and vibrational ($Q(1)$ shift $124.6\text{ THz}$) Raman lines for $\text{H}_2$ and $\text{N}_2$.

Semiconductor Photonics & TPA / Free Carriers

In silicon nanowires, Germanium, and GaAs PIC waveguides, two-photon absorption (TPA) and free-carrier dynamics modify pulse propagation:

1. Two-Photon Absorption (TPA):

   \[\frac{\partial A}{\partial z} \Big|_{\text{TPA}} = -\frac{\alpha_2}{2 A_{\text{eff}}} |A|^2 A\]

2. Free-Carrier Dynamics: Carrier density $N_c(t)$ evolves via the TPA rate equation:

   \[\frac{d N_c}{dt} = \frac{\alpha_2}{2 \hbar \omega_0 A_{\text{eff}}^2} |A(t)|^4 - \frac{N_c(t)}{\tau_c}\]

   producing Free-Carrier Absorption (FCA loss $\sigma_{\text{FCA}} N_c$) and Free-Carrier Refraction (FCR index blue-shifting $-k_{\text{FCR}} N_c$).

Interaction Picture

All solvers work in the interaction picture — the simulation variable $U = e^{-D z} A$ eliminates the linear dispersion operator from the equation of motion. This allows the solver to take larger steps in weakly nonlinear regimes and enables high-accuracy adaptive stepping.

References

• G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019)
• J. M. Dudley, G. Genty & S. Coen, Rev. Mod. Phys. 78, 1135 (2006)
• P. St.J. Russell et al., Nat. Photonics 8, 278 (2014)
• L. Yin & G. P. Agrawal, Opt. Lett. 32, 2031-2033 (2007)
• A. Börzsönyi et al., Opt. Express 21, 21086 (2013)
• Q. Lin & G. P. Agrawal, Opt. Lett. 31, 3086 (2006)
• D. Hollenbeck & C. D. Cantrell, J. Opt. Soc. Am. B 19, 2886 (2002)
• W. H. Blow & D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989)