Generalized nonlinear Schrödinger equation

📜 Generalized NLSE Used in the Code

The evolution of the complex envelope $A(x, y, t; z)$ along the propagation axis $z$ is governed by:

$$\boxed{ \begin{aligned} \frac{\partial A}{\partial z} &= \underbrace{i \frac{1}{2k_0} \left( \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} \right)}_{\textbf{(1) Diffraction}} \underbrace{-i \frac{\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + \frac{\beta_3}{6} \frac{\partial^3 A}{\partial t^3}}_{\textbf{(2) Dispersion (GDD + TOD)}} \underbrace{-i \gamma \left(1 - \frac{i}{\omega_0} \frac{\partial}{\partial t}\right)\left(|A|^2 A\right)}_{\text{(3) Kerr + self-steepening}} \underbrace{-\frac{\alpha}{2} A}_{\text{(4) Linear absorbtion}} \underbrace{+ i \gamma T_R A \frac{\partial}{\partial t}\left(|A|^2\right)}_{\text{(5) Raman}} \end{aligned} }$$

where:

• $A(x,y,t;z)$: complex field envelope
• $k_0 = \omega_0 / c$: central wavenumber
• $c$: speed of light in vacuum
• $\beta_2, \beta_3$: group delay dispersion (GDD) and third-order dispersion (TOD)
• $\gamma$: Kerr nonlinear coefficient
• $\omega_0$: central angular frequency
• $\alpha$: linear absorption coefficient
• $T_R$: Raman response

This makes our system capable of modeling ultrashort laser pulse propagation in nonlinear, lossy media. There are two ways to derive the GNLSE: from the Agrawal book and from the Ursula Keller book, because it depends on the Fourier transform convention. I choosed Ursula Keller's notation, because python's FFT and IFFT library incorporates the Ursula Keller's convention.

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