The numerical model

• The numerical model
• • A note on sign conventions
• Modal decompositions
• • Multi-mode guided
  • Single-mode guided
  • Radially symmetric free-space
  • Three-dimensional free-space
• Nonlinear responses
• • Kerr effect
  • Photoionisation & plasma
  • Raman response

Luna can solve several different variations of the unidirectional pulse propagation equation (UPPE). All of these variations have this basic form in common:

\[\partial_z E(\omega, \mathbf{k}_\perp, z){\mathrm{d}z} = \mathcal{L}(\omega, \mathbf{k}_\perp, z)E(\omega, \mathbf{k}_\perp, z) + \frac{i\omega}{N_{\mathrm{nl}}} P_{\mathrm{nl}}(\omega, \mathbf{k}_\perp, z)\,,\]

where $E(\omega, \mathbf{k}_\perp, z)$ is the electric field in "reciprocal space", i.e. frequency and transverse spatial frequency, $\omega$ is angular frequency, $\mathbf{k}_\perp$ is some generalised transverse spatial frequency, $z$ is the propagation direction, $\mathcal{L}(\omega, \mathbf{k}_\perp, z)$ is a linear operator describing dispersion, loss and diffraction, $P_{\mathrm{nl}}(\omega, \mathbf{k}_\perp, z)$ is the nonlinear polarisation induced by the field $E(\omega, \mathbf{k}_\perp, z)$, and $N_{\mathrm{nl}}$ is a normalisation factor. Since calculating the nonlinear polarisation directly in the frequency domain is not feasible, this is done in the real-space-time domain instead, and $P_{\mathrm{nl}}(\omega, \mathbf{k}_\perp, z)$ is obtained by transforming back:

\[P_{\mathrm{nl}}(\omega, \mathbf{k}_\perp, z) = \int_{-\infty}^{\infty} \mathcal{T}_\perp\Big[\mathcal{P}_\mathrm{nl}(E(t, \mathbf{r}_\perp, z)\Big](t, \mathbf{k}_\perp, z)\mathrm{e}^{i\omega t}\mathrm{d}t\,,\]

where $\mathcal{T}_\perp$ is a transform from (transverse) real space to reciprocal space (i.e. spatial frequency), $\mathbf{r}_\perp$ is the transverse spatial coordinate, $t$ is time, and $\mathcal{P}$ is an operator which calculates the nonlinear response of the medium given an electric field. Naturally, the real-space field $E(t, \mathbf{r}_\perp, z)$ first has to be obtained from $E(\omega, \mathbf{k}_\perp, z)$:

\[E(t, \mathbf{r}_\perp, z) = \mathcal{T}_\perp^{-1}\Big[E(\omega, \mathbf{k}_\perp, z)\Big]\,,\]

where $\mathcal{T}_\perp^{-1}$ is simply the inverse of $\mathcal{T}_\perp$ so transforms from transverse reciprocal space to real space. The chief difference between variations of the UPPE implemented in Luna is the definition of $\mathbf{k}_\perp$ and $\mathcal{T}_\perp$, that is, the choice of Modal decompositions of the field.

A note on sign conventions

In optics, a plane wave is usually written as

\[E(t, \mathbf{r}) = \mathrm{e}^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\]

and hence a general field, the superposition of many plane waves, is

\[E(t, \mathbf{r}) = \int_{-\infty}^\infty \tilde{E}(\omega, \mathbf{k})\mathrm{e}^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\,\mathrm{d}\omega\mathrm{d}^3\mathbf{k}\,,\]

which means that for the time-domain Fourier transform, the sign convention is opposite to that used in mathematics, with the forward and inverse transforms given by

\[\tilde{E}(\omega) = \mathcal{F}_t\left[E(t)\right] = \int_{-\infty}^\infty \!\! E(t)\mathrm{e}^{i\omega t}\,\mathrm{d} t \\ \\ E(t) = \mathcal{F}^{-1}_\omega\left[E(\omega)\right] = \frac{1}{2\pi}\int_{-\infty}^\infty\!\! E(\omega)\mathrm{e}^{-i\omega t}\,\mathrm{d} \omega \,.\]

In this convention with one sign in the exponent for space and the opposite for time, positive group-velocity dispersion (GVD) is indeed a positive parabola ($1/2\,\beta_2(\omega-\omega_0)^2$ with positive $\beta_2$), waves with positive wave vectors move to larger $\mathbf{r}$ for larger times $t$ and so forth. However, fast Fourier transforms (FFTs) use the mathematics convention. For complex (envelope) fields, this could be circumvented by simply using ifft instead of fft and vice versa, but this is not possible for real-valued fields using real FFTs (rFFT). The sign conventions in Luna are:

1. All physical expressions and quantities (propagation constants, dispersion, nonlinear phases etc.) are given in the optics convention, i.e. as they would be found in a textbook.
2. The fields in the actual simulation are given in the mathematics convention as required for FFTs. This leads to the appearance in additional minus signs in the linear operator, see e.g. make_const_linop. Similarly, to add e.g. some dispersion to a field used in or returned by a Luna simulation, the sign of that dispersion has to be flipped.