2D MMP, Electrodynamics, Efficiencies of periodic structures

Ch. Hafner, Computational Optics Group, IFH, ETH, 8092 Zurich, Switzerland

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The original MaX-1 Tutorial contains several examples of 2D
gratings. Some of them compute the efficiencies for the different transmitted
and reflected orders. Within the MMP models, 2D Rayleigh expansions are
used. For all propagating orders, each Rayleigh order is a plane wave
propagating in a direction that is computed using the Floquet theorem (using
data of the incident plane wave and of the periodicity). Therefore, the
squares of the amplitudes of the each Rayleigh order are proportional to the
power flux density of the corresponding plane wave. Since the amplitude of the
incident plane wave is zero, the squares of the Rayleigh amplitudes indicate
also the strength of the outgoing plane waves with respect to the incident one,
i.e., one directly obtains the reflection and transmission coefficients of the
propagating terms. In other words, the square of any propagating Rayleigh term A^{2}
is equal to S/S_{inc}, where S is the time average of the Poynting
vector of the corresponding plane wave and S_{inc} is the time
average of the Poynting vector of the incident plane wave. These coefficients
were computed and plotted in some examples of the Tutorial. Dr. J. Ruoff told me
that this is not the standard definition of the efficiencies used in grating
theory. For a 2D grating in the xz plane one usually uses eff=S_{y}/S_{y
inc}, i.e., the y components that are perpendicular to the grating plane
rather then the lengths of the Poynting vectors. In order to obtain these
values, one can scale both the incident wave and the Rayleigh terms using the
directions of the corresponding plane waves. For doing this, an additional
parameter has been introduced in the plane wave and Rayleigh expansion
definition (see Expansion dialog). By default, the scaling is turned on for all
Rayleigh expansions, but it is turned off for the plane waves, because these are
more frequently used for non-grating applications, where no scaling is desired.
In order to obtain the standard efficiencies, you therefore should turn on the
plane wave scaling for the incident wave (set corresponding parameter 1 instead
of 0).

In order to write the square of any Rayleigh order to a function file, you could use the directive "WRIte FUNction xxx PARameter n m SQUare", where xxx is the file name, n the expansion number, and m the parameter number. For the propagating Rayleigh terms, you can write "WRIte FUNction EFFiciency n m" instead. When you omit the number m, MaX-1 will add all terms of the Rayleigh expansion together. When this is the transmitted field, you directly obtain the total of the transmitted power with respect to the incident power (both in y direction). When you omit both n and m, you obtain the total scattered power of all Rayleigh terms (non-propagating, i.e., evanescent terms are not counted), which should be 1 when the incident plane wave is scaled, provided that the grating is loss-free. This is a simple check of the power consumption law.

In order to illustrate the procedure, the SING100 project of the original tutorial was modified and can be downloaded here. Don't forget to download the MaX-1 upgrade before you run this project!