error on the "z" and "w0" modulator
The "z" (and "w0") type modulator does not generate the correct sideband power compared against the analytical derivation, which yields the following result after the carrier HGnm passes through the "z" type modulator
\mathcal{U}^{z}\approx \mathrm{E}_{0} U_{\mathrm{n},\mathrm{m}} e^{\mathrm{i} \omega t}
- \mathrm{i} \frac{\lambda m_{z}}{8 \pi w_{0}^{2}} \mathrm{E}_{0} \bigg(A_{n} U_{\mathrm{n}+2,\mathrm{m}} e^{- 2\mathrm{i}\Psi_{1}} + B_{n} U_{\mathrm{n}-2,\mathrm{m}} e^{2\mathrm{i}\Psi_{1}} + A_{m} U_{\mathrm{n},\mathrm{m}+2} e^{- 2\mathrm{i}\Psi_{1}} + B_{m} U_{\mathrm{n},\mathrm{m}-2} e^{2\mathrm{i}\Psi_{1}}\bigg) \Big(e^{\mathrm{i}(\omega+\Omega) t} + e^{\mathrm{i}(\omega-\Omega) t}\Big)
with A_{n} = \sqrt{(n+1)(n+2)}
, B_{n} = \sqrt{n(n-1)}
.