Modal decompositions
All Fourier transforms in this section are written in the optics sign convention. See A note on sign conventions for details on what is used in the code.
Multi-mode guided
For propagation in waveguides taking into account multiple modes and the coupling between them, Luna uses the model laid out in Kolesik and Moloney, Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations and Tani et al., Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagomé photonic-crystal fiber. This is implemented in NonlinearRHS.TransModal. The electric field $\mathbf{E}(t, \mathbf{r_\perp}, z)$ is expressed as the inverse Fourier transform in time and the superposition of waveguide modes in space. This means that the transverse wave vector $\mathbf{k}_\perp$ turns into a modal index $j$ (this transform is implemented in Modes.ToSpace and Modes.to_space!):
\[\mathbf{E}(t, \mathbf{r_\perp}, z) = \frac{1}{2\pi} \int_{-\infty}^\infty \mathrm{d} \omega \sum_j \hat{\mathbf{e}}_j(\mathbf{r_\perp}, z) \tilde{E}_j(\omega, z) \mathrm{e}^{-i \omega t}\,,\]
where $\hat{\mathbf{e}}_j(\mathbf{r_\perp}, z)$ is the orthonormal transverse field distribution of the $j^{\mathrm{th}}$ mode and $\tilde{E}_j(\omega, z)$ is the frequency-domain field in mode $j$. The mode fields $\hat{\mathbf{e}}_j(\mathbf{r_\perp}, z)$ are taken to be independent of frequency but can depend on the propagation coordinate $z$ (e.g. in tapered waveguides). They can be vector quantities if polarisations other than purely lineary $x$- or $y$-polarisations need to be taken into account. The modes are normalised such that $\vert \tilde{E}_j(\omega, z) \vert^2$ gives the spectral energy density in mode $j$, and equivalently $\vert E_j(t, z)\vert^2$ gives the instantaneous power. The inverse transform is simply the overlap integral of the total field with each mode combined with the Fourier transform:
\[\tilde{E}_j(\omega, z) = \int_S \mathrm{d}^2\mathbf{r_\perp} \int_{-\infty}^\infty \mathrm{d} t\,\, \hat{\mathbf{e}}_j^*(\mathbf{r_\perp}, z) \cdot \mathbf{E}(t, \mathbf{r_\perp}, z) \mathrm{e}^{i \omega t}\,,\]
where $S$ is the cross-sectional area of the waveguide. This transform is implemented in NonlinearRHS.TransModal for use within simulations and in Modes.overlap for decomposition of existing sampled fields.
The linear operator for a mode $\mathcal{L}_j(\omega, z)$ is given by (see LinearOps.make_const_linop)
\[\mathcal{L}_j(\omega, z) = i\left(\beta_j(\omega, z) - \frac{\omega}{v}\right) - \frac{1}{2}\alpha_j(\omega, z)\,,\]
where $\beta_j(\omega, z)$ describes the phase evolution of the mode, $v$ is a chosen frame velocity (this is the same for all modes) and $\alpha(\omega, z)$ describes the attenuation of the waveguide (i.e. $1/\alpha$ is the $1/\mathrm{e}$ loss-length). This can also be expressed in terms of the effective index of the mode:
\[\mathcal{L}_j(\omega, z) = i \left(\frac{\omega}{c} n_\mathrm{eff}(\omega, z) - \frac{\omega}{v}\right)\,,\]
where $c$ is the speed of light in vacuum and $n_\mathrm{eff}$ is complex, $n_\mathrm{eff} = n + i k$, with $n$ describing the effective refractive index and $k$ describing the attenuation. With the modal power normalisation for $\hat{\mathbf{e}}_j(\mathbf{r_\perp}, z)$, the normalisation factor $N_{\mathrm{nl}}$ comes out as simply $N_{\mathrm{nl}}=4$. The propagation equation, coupling the modes through the nonlinear polarisation, is therefore
\[\partial_z \tilde{E}_j(\omega, z) = i \left(\frac{\omega}{c} n_\mathrm{eff}(\omega, z) - \frac{\omega}{v}\right)\tilde{E}_j(\omega, z) + i\frac{\omega}{4} \tilde{\mathbf{P}}_\mathrm{nl}\,,\]
where $\tilde{\mathbf{P}}_\mathrm{nl}$ is given by
\[\tilde{\mathbf{P}}_\mathrm{nl} = \int_S \mathrm{d}^2\mathbf{r_\perp} \int_{-\infty}^\infty \mathrm{d} t\,\, \hat{\mathbf{e}}_j^*(\mathbf{r_\perp}, z) \cdot \mathbf{P}_\mathrm{nl}\left[\mathbf{E}(t, \mathbf{r_\perp}, z)\right] \mathrm{e}^{i \omega t}\]
and $\mathbf{E}(t, \mathbf{r_\perp}, z)$ is obtained from the set of $\tilde{E}_j(\omega, z)$ as above.
The transverse coordinate $\mathbf{r_\perp}$ for circular waveguides (e.g. hollow capillaries, optical fibres, and anti-resonant fibres) is in polar coordinates, $\mathbf{r_\perp} = (r, \theta)$. For other waveguides (e.g. rectangular), it is Cartesian, $\mathbf{r_\perp} = (x, y)$.
While $\mathbf{r_\perp}$ can be given in either coordinate system, the components of the modal fields $\hat{\mathbf{e}}_j(\mathbf{r_\perp}, z)$ are always given in Cartesian coordinates, i.e. the basis vectors for the polarisation of the field are always $\mathbf{x}$ and $\mathbf{y}$.
Implementation
The modules and functions that define and implement this decomposition for different modes are
- Modes.jl
- Capillary.jl
- RectModes.jl
- Antiresonant.jl
NonlinearRHS.TransModalNonlinearRHS.norm_modalLinearOps.make_const_linopLinearOps.make_linop