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I'm trying to solve the linear non-paraxial pulse propagation equation
$$\partial_z\hat{E}=ik_z\hat{E}$$ for a field defined as $$E=E(r, t, z)$$ The equation given above uses $$\hat{E}=\hat{E}(k_\perp, \omega, z)$$. Therefore, I have to use a FFT twice, once, for going from $r$ to $k_\perp$, and once for going from $t$ to $\omega$. This also means that I have to limit the used frequencies, after a FFT going from $t$ to $\omega$ will only support a certain amount of frequencies, depending on the resolution of $t$ and the limits of $t$, i.e. $t_{min}$ and $t_{max}$.
To properly resolve longer pulses at short wavelengths, this makes rather high resolutions necessary to capture all involved frequencies, and thereby increasing computation time. Therefore I had the idea to set $t$ to $0$, and $\omega$ to $\omega_0$ and thereby effectively reducing the 3d-system $(x, y, t)$ to a 2d-system $(x, y)$. When transforming $E(k_\perp, t, z)$ to $E(k_\perp, \omega, z)$ this would require a FFT over a single number (after $t$ only contains one value), and therefore $E(k_\perp, t, z)\equiv E(k_\perp, \omega, z)$.
Does that make sense, or am I introducing additional errors here? Or are there other methods to reduce the amount of variables resulting from the 3d-system, to speed up computation?

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  • $\begingroup$ Probably any method reducing a 3D system to 2D would essentially amount to expanding the 3D solution f(x,y,t) in terms of some 2D basis functions g_k(x,y)*T_k(t). With some some insight on what the solution looks like and a good choice of basis functions one may end up with only few k terms. $\endgroup$ – Maxim Umansky Apr 29 '20 at 3:31

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