Comparison between rigorous light-scattering methods

A modal model for diffraction gratings

“…The degeneracy can be split in this case by choosing the lines and spaces to be very slightly different. One way to deal with near collisions of eigenvalues is to choose a very small time step when integrating (29). This can be quite time consuming in the general case however.…”

“…In [9,30] an approach was presented which was based on contour integral theorems for analytic functions. The approach used here is a generalization of the TayebPetit method [11] which was also used successfully in [12,13,[26][27][28][29]. In order to solve the general eigenvalue problem one starts with a known solution and then varies the parameters of the problem slowly and continuously to change the object from the known one to the desired case, all the while tracking the continuous motion of the eigenvalues.…”

“…A procedure that works well for solving this equation is to use (29) computed numerically to integrate a certain number of time steps, and then to periodically interrupt the integration in order to refine the accuracy of k x by using a Newton's method to minimize F as a function of k x for fixed t. It is important to choose the time steps to be small enough so that nearly degenerate eigenvalues can be resolved numerically. Degenerate cases are hard to avoid though, and some means for removing degenerate eigenvalues is very helpful.…”

“…Suppose that two nearby eigenvalues have been confused by the integration procedure (29) together with Newton's method. So two eigenvalues are incorrectly thought to be the same.…”

“…The modal method has been applied in this context in [26][27][28][29], although the author was not aware of the earlier work at that time, and so the method was referred to as the Analytic Waveguide Method with the tradename Metrologia. Recent interest in grating simulation has increased because of the use of diffraction grating scatterometry for measuring linewidths and cross-sections in semiconductor manufacturing.…”