# Solving a spectral system by reducing it to a single frequency - Feasability of approach?

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?

• 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. – Maxim Umansky Apr 29 '20 at 3:31