3D MMP, Electrodynamics, Definition of elliptical boundary, complex origin multipoles and Bessel expansions
Ch. Hafner, Computational Optics Group, IFH, ETH, 8092 Zurich, Switzerland
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Generating an elliptical boundary
MaX-1 allows you to displace the matchingpoints along a C-polygon using a formula as illustrated for the sinusoidal grating, where the C-polygon was a simple line. For defining an ellipse in MaX-1, it is better to use its definition in polar coordinates (r,phi):
Where A and B are parameters that define the ellipse. In this notation, one of the two focus points of the ellipse is located at the origin r=0 and the other one is on the x axis (where phi=0). Some calculation shows that one obtains the parameters A and B from a and b where a is the longer half axis of the ellipse (a is on the x axis) and b is the shorter one:
A=b*b/a and B=sqrt(1-b*b/(a*a)).
Furthermore, the center of the ellipse is located at xc=aB, yc=0.
In order to define an ellipse, for example, with a=1 and b=0.5 in MaX-1 we therefore first compute the parameters A=0.25, B=0.8660254, xc=0.8660254 using the formulae above. Then we define a circle with zero radius (since zero radius causes numerical underflow problems, select a radius that is much smaller than the short half axis b) around the first focus point as a boundary, add the formula inv(add(1,mul(0.8660254,com(2,c1,v)))), set its amplitude equal to A=0.25, and select a sufficiently high number of spline points for it in the Boundary dialog.
It is important to note that this definition causes a nonunifor distribution of the matching points when standard parameters for the generation of the matching points are used in the MMP dialog: Near the first focus point considerably more matching points are generated than near the second focus point. This is not desirable. Therefore, it is reasonable to force a uniform matching point distribution with sufficiently many matching points.
Depending on the incident field, there might be some symmetry planes. Thus, it might be sufficient to discretize only half of the ellipse. This might be done using a C-polygon that defines a semi circle with small radius (a C-polygon with 4 corners). When you are too lazy (as I am) you waist some computation time, but the result should be still OK. Note that you may specify the appropriate symmetry number although the boundary is defined on both sides of the symmetry plane. This saves memory and computation time bacause the expansions may still be simmetrized.
Complex origin multipoles and Bessel expansions
You may use a Bessel expansion and a multipole with sufficiently high orders for the approximation of the field inside and outside the ellipse respectively. Both expansions should be best located in the center of the ellipse when the incident field is a plane wave. This is precisely the modelling you would use for a circle. When the aspect ratio a/b becomes bigger, the number of parameters required increases rapidly, even when a is kept constant. Thus, this standard expansion known from Mie scattering is not appropriate for ellipses in general. Much better results may be obtained with MMP expansions, i.e., a set of multipoles arranged along the boundary. From conformal mapping one finds that the optimal locations of multipoles that approximate the field outside are on co-focal ellipses. This corresponds to concentric circles for the circular case. The most extreme cofocal ellipse is the line from the first to the second focus point - a cut line obtained from conformal mapping. Instead of placing several multipoles along this axis, we can pu only one multipole in the center (as for the Mie scattering) and an additional complex origin multipole that also should have the real part of the origin in the center. When the imaginary part of the y component of the origin yi is equal to c>0 the field pattern is like a beam that exits the cut line from x=-c to x=+c in y direction. When we set c<0, the beam exits in -y direction. To model a scattered field that exits either in +y or in -y direction, we therefore need two complex origin multipoles in the center of the ellipse with opposite values yi. For the ellipse, the optimum values are yi=+yc and yi=-yc.
Similarly, we can define an ordinary and two complex origin Bessel expansions for modeling the field inside the ellipse.
It is important to note that complex origin expansions provide an additional degree of freedom that allows one to solve certain problems more efficiently, but this also requires sufficient experience and understanding of the behavior of these expansions.
The behavior of elliptical particles is interesting, namely when resonances occur. Observe what happens when the angle of incidence varies!
When the ellipse is illuminated by a plane wave perpendicular to the long axis (as shown in the figure above), it may behave like a lens and exhibit some focus. The location of the focus depends on the refractive index, i.e., on the permittivity of the ellipse. By adapting the permittivity, you may push the focus toward the surface of the ellipse and obtain "optical nano jets" as for spherical nano lenses. To get a more realistic picture, you should consider 3D axisymmetric models.