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.

Symmetry decomposition of the excitation

A perfectly conducting torus is a simple 3D object, but it is obviously more complicated because then a sphere of its topology. Note that a sphere has a higher symmetry then a torus, but both objects are axisymmetric. Since we mainly took the axisymmetry into account for the modeling of the sphere, we essentially can proceed as for the sphere. Of course, we will not repeat the time-consuming brute-force model and try directly a model with two thin slices of the boundary. Note that this is only possible when the incident wave has an angular dependence of the form cos(k*phi), where k is an integer number and phi is the angle around the symmetry axis. As in the PECS project, we assume that the symmetry axis is the y axis. Remember that a plane wave propagating along the y axis as well as a dipole excitation on the y axis with an orientation perpendicular to the axis have this kind of symmetry with k=1, whereas a dipole on the axis oriented in y direction has k=0. When the excitation has no such angular dependence, one can use Fourier analysis to decompose the excitation into appropriate component. Each of these components can then be analyzed essentially as indicated below. Finally, the resulting scattered field is obtained by a superposition of the results obtained for all components. If a high number of Fourier components must taken into account, this may be time-consuming, but it usually is still much more efficient then a brute-force procedure that ignores the symmetry of the torus. However, we explicitly only consider a simple plane wave excitation propagating along the y axis in the following.

Ring multipoles

The main problem now is to model the scattered field. Obviously, we cannot do this with a single multipole on the symmetry axis as in the spherical case. It would be natural to use a set of N multipoles, uniformly distributed along the center line of the torus, but this would not take the symmetry into account and it would be both time- and memory consuming. In order to obtain the desired symmetry, we let N go to infinity. At the same time, we assume that the amplitudes of the multipole orders and degrees are not independent. Otherwise, we would obtain an infinite set of unknown parameters for each order and degree. Theoretically, we could consider an arbitrary phi dependence of the amplitude, but it is natural to use a Fourier basis, i.e., cos(k*phi) and sin(k*phi) dependence. Let us start with a dipole oriented in y direction, placed on the (circular) center line of the torus. When we move this dipole along the centerline while we change its amplitude with the angular dependence cos(k*phi) or sin(k*phi) and integrate the resulting field, we obtain a field with cos(k*phi) or sin(k*phi) symmetry. We call this a ring dipole of degree k or a ring multipole of order 1 and degree k. Similarly, we can obtain ring multipoles of higher order.

For defining the ring multipole, we need a local coordinate system as for all 3D expansions. It is natural to define its origin on the axis. Therefore, the field of a ring multipole is not singular in its origin. It is singular on a circle with radius R around the origin. In addition to the coordinate system and the orders and degrees, one therefore must also specify the radius R of a ring multipole. Since the standard axis for axi-symmetric problems is the y axis, the "singular circle" of a ring multipole is located in the XZ plane of the ring multipole an its center is at the origin X=0, Y=0, Z=0 of the local coordinate system of the ring multipole.

For avoiding numerical problems when you compute the field of a standard 3D multipole, you may specify a minimum radius for the field computation. When the distance of a field point from the origin is smaller than the minimum radius, MaX-1 will not compute the multipole field in this point. Similarly, you may specify a minimum radius for a ring multipole. MaX-1 will not compute the field when the distance of the field point from the "singular circle" of the ring multipole is less than the minimum radius.

Unfortunately, no analytic solution for the ring multipole field is known. Therefore, the integration must be done numerically, i.e., one adds the fields of a finite set of N multipoles uniformly distributed along the "singular circle". Thus an additional parameter for the ring multipole is N. When you select a small value, the computation is fast but inaccurate, whereas the computation becomes time-consuming for high N. The optimal number N also depends on the distance of the field point from the "singular circle". The shorter this distance is, the higher N should be. Assume that you use a ring multipole for modeling a torus with a radius of 0.5m for the center line and 0.1m for the cross section. Then, the shortest distance of a field point from the ring multipole is 0.1m. The length of its "singular circle" is the same as the length of the center line of the torus, i.e., approximately 3.14m. When you would place a discrete set of multipoles along the center line, you would probably use approximately 3.14/0.1, i.e., approximately 30 multipoles. This is the minimum number N you should use for a ring multipole. Usually, you will take considerably higher number, for example, 120. When you consider a more complicated object, the distance of the points of the "singular circle" of the ring multipole to the matching points is not constant. In this case, you should set N > Length of "singular line" / shortest distance. The same also holds when you want to visualize the field of a ring multipole in different field points. Since it is not reasonable to use very high N also for matching points or field points far away from the ring multipole, you may also specify a negative number in the N multipoles box of the Expansion dialog, when you define a ring multipole. In this case, MaX-1 will use "up to" –N multipoles, i.e., it will use fewer multipoles for computing the field far away. Therefore, you best work with –120 multipoles in our example.

The ring multipole concept can be generalized for finite sections of a ring. Therefore, you must also specify the Minimum angle and the Sector angle of the ring multipole

Rotational symmetry without symmetry planes: PECT000.PRO

The ring multipole concept can be generalized for finite sections of a ring. Therefore, you must also specify the Minimum angle and the Sector angle of the ring multipole. Since you want to model an entire ring, you specify the Minimum angle 0 and the Sector angle 360 degrees in the Expansion dialog. Make sure that the Degree factor has the default value 1. This parameter will be discussed in the following section. After you have specified the ring multipole and the incident plane wave in the Expansions dialog, you proceed exactly as when you model a sphere. You may work with a two slice model as well as with a one slice model. Try the two slice model first. Finally, you should obtain a field as illustrated in Figure 1.

Note that a very high field is obtained on the left hand side inside the torus. The reason for this is the following: For each field point, MaX-1 evaluates the corresponding domain number from the boundary information of the objects. Since you have only modeled two slices of the entire torus, MaX-1 searches for the nearest boundary point on these slices and finds that all points of the xy plane are outside the slices except those points that are inside the first slice near x=0.5m. Thus, the domain numbers are not correctly detected everywhere. In order to let MaX-1 evaluate the correct domain numbers for all field points, you must specify an entire torus before you let MaX-1 compute the field in the xy plane. As in the PECSxxx projects, you best define a dummy object that has no influence on the MMP computation of the parameters of the ring multipole. Note that you might even obtain a singular field value when the field point happens to be exactly on the ring. This problem is removed when you have specified a finite, non-zero value for the Minimum radius of the ring multipole in the Expansion dialog.

*Figure 1**: Time average of the Poynting vector field. Axi-symmetric computation without symmetry planes. Wrong field evaluation near x=-0.5m due to a wrong evaluation of the domain numbers in this area because of the incomplete discretization of the surface of the ring.*

Symmetry planes: PECT001.PRO

As in the PECSxxx projects, you may take the xy and yz symmetry planes into account by properly setting the symmetry numbers in the Project dialog. This considerably reduces the computation time because you now must specify only a quarter of the ring multipole. As a consequence, the integration of the ring multipole that is approximated by a summation over N multipoles requires only a quarter of the computation time. In order to obtain a quarter of a ring, you now reduce the Sector angle of the ring from 360 degrees to 90 degrees. Now, you should know the following: for the computation of the angular (phi) dependence of the ring multipole degrees, MaX-1 uses a Fourier decomposition with the terms cos(k*phy) and sin(k*phi). For a full ring with Sector angle 360 degrees, this expansion is complete when k is an integer number: k=0,1,2,… For an incomplete ring with a Sector angle <360 degrees on could use a polynomial basis instead of the Fourier basis as in other implementations of ring multipoles. In MaX-1, still a Fourier Basis is used, but now, the terms cos(k*phi*F*S/180) and sin(k*phi*F*S/180) are used, where S denotes the Sector angle (in degrees). F is the Degree factor that may also be specified in the Expansion dialog. Assume that S is small and that F is equal to 1. Then, the cos term for k=0 is more or less equal to 1, i.e., the zero order of a power series. The cos and sin terms for k=1 have the values 1 and 0 at the two end points of the segment and are monotonic in between, i.e., these terms correspond to the first order of a power series. In general, such a modified Fourier series can be applied to more complicated "distributed multipoles" that are composed, for example, by a sequence of straight line elements and sections of rings. When we return to our quarter ring section, we see that we obtain the same Fourier basis as for the entire ring (together with the symmetry operations) only when we set F=2, i.e., when we specify the Degree factor 2. Once we have done this, we can proceed as before and obtain essentially the same results as with the first model, but with shorter computation time.

*Figure 2**: Time average of the Poynting vector field. Axi-symmetric computation with 2 symmetry planes, two slice model. Dummy ring for the field evaluation on the surface of the ring and for the correct evaluation of the domain numbers in all field points.*