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

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Before you study the PECT projects, you should be familiar with the simpler PECS projects. The main purpose of this project is to illustrate how you modify an existing model for obtaining a similar model according to your needs. Assume that you want to model the scattering of a laser beam (frequency 6E14) at a metallic sphere with complex relative epsilon equal to –25+2i and diameter 1e-8m. Obviously, the PECSxxx model is quite similar, but the frequency, the material properties, the object size, and the excitation are different. In the following, we adapt these properties step by step. Let us start with the PECS004 project.

Adapting the material properties

First, replace the PEC material properties by the metallic material properties. Note that all perfect conductors have domain number 0 and require no definition in the Domain dialog. Therefore, you now add a new domain (number 2) in the Domain dialog and enter the desired Epsilon = (-25,2) and Mue = 1 in the corresponding boxes. Don't forget to press the Modify button!

In a second step, you must replace the domain number 0 by the domain number 2 for all boundaries in the Boundary dialog. Furthermore, you should know that you now should impose also the boundary conditions for the magnetic field. Since you work with two slices in the xy and yz planes, some of the components are zero in these planes and must not be matched explicitly. For the first slice, i.e., the first boundary, you need the Hz boundary condition in addition to the Et boundary condition and on the second boundary you should impose the Ht boundary condition in addition to Ez. After you have modified the data of a boundary, don't forget to press the Modify button!

Now, you should add an appropriate expansion for domain number 2. Since this is the interior of the sphere, you best add a Bessel type expansion with exactly the same properties as the Hankel type multipole you already have for the scattered field in domain 1, i.e., free space outside the sphere. In the Expansion dialog, you simply select the multipole, press the Copy button once and select the second expansion. To convert it from 3D multipole to 3D Bessel, simply select the corresponding type. Don't forget to modify the domain number for this expansion! It should be 2 rather than 1. If you wish, you can also modify the color, but this has no influence on the computation.

Now, save the project with a new name or number and let MMP compute the results again to make sure that you made no mistakes. Since you have a very high accuracy, you may reduce the maximum orders of both the multipole and the Bessel expansion for saving computation time.

Adapting the frequency

Adapting the frequency is almost trivial. You simply set the desired frequency in the Project dialog. For example, set 6E14 as the real part of the Complex frequency. What you should keep in mind is that the accuracy of the results is decreased when you increase the frequency without modifying anything else. Here, we multiply the frequency with the factor 1E8 and we reduce the radius of the sphere with a factor 1E8. Therefore, we expect to obtain the same accuracy. If we would have increased the frequency by a bigger factor, it would be necessary to increase the maximum order and the number of parameters of the multipole expansion.

Adapting the size

Adapting the size seems to be trivial as well, but this is a source of mistakes because you must adapt the value of many parameters.

First of all, you must adapt the size of all boundaries, i.e., the corresponding corners and radii. Since you have 4 boundaries with 4 corners each, this would be time-consuming when you would do this in the Boundary dialog. You better use the movie directive BLOw BOUndary 0 0 0 1E-8. You may enter this in the corresponding box of the Movie dialog and press the Run button.

In order to make sure that the boundaries have the desired size, inspect the data in the Boundary dialog. When you let MaX-1 draw the boundaries, you will see almost nothing because the boundaries are now very small. This reminds you to adapt also the size of the graphic window in the Window dialog! When you intend to use the OpenGL Graphics window, you should also adapt the corresponding data in the OpenGL window dialog. When you let MaX-1 draw the objects in the OpenGL Graphics window by selecting "draw 3D objects", you will still see nothing because OpenGL uses the old view data until you select "reset to initial view" in the OpenGL Graphics dialog. If you don't see the objects as desired after having selected "reset to initial view", verify the data in the OpenGL window dialog and modify them until you get the desired view. This needs some experience!

Remember that the derived field is computed in a plane that is much too big now. Press the Grid space button in the Field dialog for resizing this plane. When MaX-1 draws arrows, the Maximum length specified in the Field dialog is used. This value should also be reduced by the factor 1E-8. Otherwise the arrows will become much too long.

Furthermore, a second plane for drawing the field is defined as object number 4. Don't forget to adapt also this object in the Object dialog. When you have turned off the Matching point representation, MaX-1 will use a grid on each object for its representation. The corresponding resolution is also much too big. You therefore should reduce the value in the Resolution box for each object by 1E-8.

Finally, MMP generates matching points according to the data in the MMP dialog. In 2D modeling, you could probably resize the boundaries without any modification of the data in the Matching point definition group when the Min. Overdetermination or the Min.points/wavelength was sufficiently high. In the current project, no 2D multipole is defined and therefore the Min. Overdetermination has no influence. Furthermore, the wavelength is quite big compared with the wavelength. Therefore, you will not obtain enough matching points. In order to avoid this, you should either increase the value of the Min.points/segment box or reduce the Max. distance value by the factor 1E-8.

When you now run the MMP solver, you should obtain correct results and the picture shown in Figure 1 – provided that you have adapted the field scaling values in the Min and Max boxes of the Field dialog.

When you did not obtain correct results, you may compare your model with the METS004 project.

*Figure 1**: Time average of the Poynting vector field. Axi-symmetric computation with 2 symmetry planes, two slice model. Plane wave excitation.*

Adapting the excitation

Up to now, the excitation is a plane wave, defined in the Expansion dialog. When you prefer a laser beam instead, you simply select the expansion (number 3) and select 3D Beam type instead of Plane wave. The location and orientation of the beam are the same as for the plane wave, but the definition of the beam requires some additional data, i.e., the Beam radius and order. Select order 5. This is the maximum value and should be good for almost all cases. When you select a higher value, MaX-1 will reduce it because no higher orders are implemented. You might wish to have a beam radius in the order of the radius of the sphere. When you specify the radius 5e-9, MaX-1 will increase it to 1.57045E-7 because this is the smallest radius of a laser beam that may be obtained at the current frequency. If you proceed with this or a higher value, you will obtain essentially the same field as for the plane wave excitation (project METS005) because the laser beam is much broader than the sphere.

Adapting the frequency once more

When you would like to see the field for a laser beam that has a similar radius as the sphere, you must either increase the frequency or the size of the sphere. Since increasing the frequency is much simpler, try this. Note that must increase the frequency quite much in order to obtain a much thinner beam. As a consequence, the results will become considerably less accurate. For obtaining a reasonable accuracy, you should increase the maximum orders of the multipole and Bessel expansion as well as the corresponding maximum numbers of parameters. When you have done this, keep in mind that the matching point density will not be increased automatically. You therefore should also adapt the corresponding data in the MMP dialog. When MMP has found a good solution, make sure that the MMP matrix is sufficiently overdetermined. Note that the matrix size is displayed in the MMP dialog.

Since you now have more 2D matching points along the 2D boundary, it might happen that more than one 3D matching point is generated on the two slices. When this happens, the number of 3D matching points will be bigger than the number of 2D matching points. When this happens, you should make the two slices, i.e., the first two objects in the Object dialog thinner by adjusting the data in the Minimum angle and Sector angle boxes.

Finally, you should obtain results as illustrated in Figure 2, obtained with project METS006.

When you want to compute an even bigger sphere, illuminated by a thin laser beam, it will be useful to work with auxiliary 2D multipoles for obtaining non-uniform matching point distributions as in the PECS01x projects with the dipole illumination.

*Figure 2**: Time average of the Poynting vector field. Axi-symmetric computation with 2 symmetry planes, two slice model. Laser beam excitation at higher frequency.*