For the calculation of the polarization/non-linear absorption of an electric field (given in the temporal domain, cylindrical coordinate system) as $E(r, t)$ usually the spectral domain is used, i.e. $E=E(k_\perp,\omega)$ with $k_\perp$ the spectral equivalent to $r$ and $\omega$ the frequency corresponding to $t$.
To convert the field from the temporal to the spectral space, I first apply an FFT along the rows, followed by a DHT (discrete hankel transform) along the columns. The backconversion is the DHT along the columns, followed by an FFT along the rows. The DHT is given as a matrix multiplication with a matrix $H$, with both the forward and backward conversion as $H\cdot x$. Therefore, $x=H\cdot H\cdot x$.
Thus, the full conversion can be written as
$$E(k_\perp,\omega)=DHT(F^{-1}(E(r, t)))$$
$$E(r, t)=F(DHT(E(k_\perp,\omega)))$$
Now, to calculate the current generated due to $K$-photon absorption, the equation is (simplified)
$$J=cE^{(2\cdot K-1)}$$
when given $E$ in the temporal domain and $c$ a constant. If $c$ depends on the wavelength, based on my knowledge I have to first convert $E^{(2\cdot K-1)}$ to the spectral domain, multiply it with $c(\omega)$ and convert it back.
This gives the full method as
$$J=F(H\cdot c(\omega)\cdot H\cdot F^{-1}(E^{2K-1}))$$
Unfortunately this calculation is computationally expensive, and thus I wanted to simplify the equation.
I know that the multiplication is commutative, and $H\cdot H=\mathbb{1}$. Therefore, I can reduce the equation above to
$$J=F(c(\omega)F^{-1}(E^{2K-1}))$$
Is that correct?
And if yes, are there further simplifications I can apply to make the computations easier? If not, are there other things I can apply to the initial equation for $J$ to make it computationally cheaper?