3D MMP, Electrodynamics, Axisymmetric problems, complex origin multipoles and Bessel expansions
Ch. Hafner, Computational Optics Group, IFH, ETH, 8092 Zurich, Switzerland
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Before you start with this projct, you should consider the 2D ellipse project that describes how to obtain an elliptical boundary and how to define the appropriate complex origin expansions in 2D.
Generating an ellipsoid
An ellipsoid is an axisymmetric structure. In the case of an oblate ellipsoid, an ellipse is rotated around its short axis. For axisymmetric problems in MaX-1 it is most convenient to always use the y axis as the axis of rotation. Therefore you first define a 2D ellipse with center on the y axis and long axis in x direction. After this, we define a 3D "torus" object based on this boundary. Since we will select only those expansions with the correct angular dependence, this 3D may be only a very thin slice of the ellipsoid. The location of this slice may be arbitrary but it should not coincide with any symmetry plane of the exciting field. Since we will consider a special case, where the xy and yz planes are symmetry planes, we define the slice with a 45 degree rotation around the y axis. Nothe that this slice should be so thin that only one matching point is generated in the angular direction of the slice.
When the excitation is a plane wave or a Gaussian beam propagating in y direction, the field has even and odd symmetries with respect to the xy and yz planes. To save memory and computation time, you should specify the appropriate symmetry numbers in the Project dialog - even when you work with a complete 2D ellipse for generating the ellipsoid.
The angular dependence of all field components in the xz plane (around the y axis) is either cos(phi) or sin(phi). Thus, you should model the field inside the ellipsoid as well as the scattered field by expansions that also only contain this angular pattern. This may only be done with 3D expansions with origins located on the y axis, namely ring multipoles.
Complex origin expansions
Since the evaluation of ring multipoles is time consuming and because of our good experience with 2D complex origin Bessel and multipole expansions, we now tra a very similar model with a standard multipole expansion and two complex origin multipole expansions located in the center of the ellipsoid. First, it is important to orient these expansions in such a way that their field has the cos(phi) or sin(phi) pattern mentioned above. This may be done only for the standard multipole. The complex origin multipoles ar then oriented in the same direction.
Once the orientation is determined, the imaginary parts of the origins must be properly set. In the 2D case, the imaginary parts were xi=0 and yi=+-yc. As in the 2D case, we want to orient the multipole beams in +-y direction. When the orientation of the multipole is set in such a way that its Z direction (local coordinates) points in y direction (global coordinates), we must set its Imaginary z values equal to +-yc.
The definition of the complex origin Bessel expansions is essentially the same and needs no further explanation.
The behavior of the 3D ellipsoid is similar to the one of the 2D ellipse as one can see from the figure above, but the polarization can also provide substantial effects. Therefore, it is reasonable to not only consider one section. The figure below shows the time-averaged Poynting vector field in the xy and yz planes as well as on the surface of the ellipsoid. For this plot, two additional 3D objects need to be defined:
1) The surface of the ellipsoid is obtained by rotating the elliptical 2D boundary around the y axis. Since this 3D boundary would generate by far too many matching points, it is based on an extra "dummy" 2D boundary with zero weights but the same shape as the original elliptical boundary.
2) To obtain the yz plane, another "dummy" 2D boundary line along the y axis is defined and from this a 3D "cylindrical" object is generated. Note that the domain numbers on both sides of this dummy boundary are negative, i.e., the domain numbers of the field points on this object are computed from the domain numbers of the other objects. The correct computation of these domain numbers is guaranteed by the second "dummy" object, i.e., the ellipsoid mentioned above. When this object would not be present, the thin section of the ellipsoid that is used for the MMP matrix computation would not provide the data for correctly computing the domain numbers everywhere!
Note that you cannot easily modify the direction of the incident plane wave or Gaussian beam. As soon as the propagation direction is different from the y axis, the symmetry is broken. Then either a time-consuming true 3D model or a tedious symmetry decomposition - that leads to a sequence to symmetry adapted problems that nend to be solved separately - are required. However, with the axisymmetric model you can study the influence of the aspect ratio and material parameters of the ellipsoid and you may find nice "optical nano jets".