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  Numerically solving the Paraxial Diffraction Integral for resonators

+ 1 like - 0 dislike

So, here's the deal. I need to solve the paraxial diffraction integral for resonators by using a Gaussian quadrature in order to obtain the modes. Typically, this is done by assuming radial symmetry and transforming it to a 1D integral. However, I can't assume symmetry. 

For 1D (Assuming symmetry):

\[\lambda u_{lp}(r_2)=\int_0^{a_1} K_l(r_1, r_2)u_{lp}(r_1)dr_1\]

Then, we take \(\textbf{r}=[r_1, r_2, ..., r_N]\) and \(\textbf{w}=[w_1, w_2,...,w_N]\) to be the abscissa and the weights for the Gaussian quadrature. If we consider equal domains for the variables \(x_1\) and \(x_2\) and \(\textbf{u}=[u_1, u_2, ..., u_N]\) the column vector of the eigenfunction so that \(u_N = u(x_N)\), we can rewrite the integral as

\[\lambda \textbf{u}=(\textbf{H}*\textbf{W})*\textbf{u}\]

Where is a N x N matrix where \(H_{m,n}=H(r_m, r_n)\) and W is diagonal matrix with the elements of w.

Hence, we only need to find the eigenvalues and eigenvectors for the equation before and it's done. 

However, things aren't that easy for 2D. The integral equation reads as:

\[\lambda u(x_2, y_2)=\int_0^{a_1} K(x_1, x_2, y_1, y_2)u(x_1, y_1)dx_1dy_1\]

I'm not sure how to transform that integral into the matrix equation and what does \(\textbf{H}\) should be in this case, since it needs to be a 4D array? \(H_{xm, xn, ym, yn}=H(x_m, x_n, y_m, y_n)\)?

asked Nov 19, 2021 in Q&A by anonymous [ no revision ]

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