Motivation
The rectangular PEC waveguide has a lower symmetry than the circular one, but its geometry is well adapted to Cartesian coordinates. Therefore, this is one of the few waveguides that allows you to find a simple analytic solution. Since harmonic expansions describe the field of the rectangular PEC waveguide exactly, you can use these expansions to obtain very accurate MMP solutions. For each mode, a single harmonic expansion is required. Thus, the situation is similar to the computation of the circular PEC waveguide. MMP has to search only for the correct propagation constant without solving a matrix equation.
Instead of expanding the field with a harmonic expansion, you can also apply Bessel or multipole expansions. Since these expansions do not fit the geometry of the problem, a series with several unknowns is required and the resulting MMP matrix has more than one column. Thus, a matrix equation is involved and the MMP solution proceeds as for arbitrary waveguides. Although this is much more time consuming than searching using the harmonic solution, it is a good opportunity to study true MMP approximations of eigenvalue problems and to gain experience. Comparisons with the accurate harmonic solutions allow you to easily validate the results.
Modes and harmonic expansions
Just as for the circular PEC waveguide, one has Hmn and Emn modes. The field of these modes is characterized by harmonic expansions of the form harm(m*kx0*x)*harm(n*ky0*y), where harm is either sine or cosine. The indices m and n indicate how rapidly the field oscillates in the x and y direction respectively. To obtain an 'analytic' MMP solution, select the appropriate harmonic expansion in the Expansion dialog. Note that you can select m*kx0 and let MMP compute n*ky0 and the propagation constant. For a given value of m*kx0, MMP will find several modes with a differing index n.
To obtain accurate results, set very small values in the Accuracy box and in the Flatness box of the Eigenvalue search group in the Project dialog.
Symmetries
All modes are either symmetric or anti-symmetric with respect to the XZ and YZ planes. When you specify the symmetry numbers, the number of minims of the search function is reduced, which is desirable especially at high frequencies. For a given m*kx0, you will find all modes either with an even or an odd index n, depending on the symmetry with respect to the XZ plane.
Bessel expansions
When you approximate the field with non-harmonic expansions, a single expansion with a single parameter is not sufficient. Instead, you need a series. First of all, you can use a single Bessel expansion in the center of the rectangle. This is the same expansion you have used for the circular PEC waveguide, but now, more than one parameter is required to obtain accurate results.
Symmetries
When you do not take the symmetries into account, one Bessel expansion is used for obtaining all Emn or all Hmn modes. Thus, the number of minims of the eigenvalue search function E2/Amp2 can be so high that it is difficult to detect all minims. Therefore, it is wise to take the symmetries with respect to the XZ and YZ planes into account. This reduces the number of minims and the size of the MMP matrix required for obtaining accurate results.
When you take symmetries into account, the origin of the Bessel expansion should be on the intersection of the symmetry planes, i.e., at the origin of the global coordinate system. In this case, some of the orders of the Bessel expansion have the wrong symmetry and are omitted automatically by the MMP tool. For example, when the symmetry number with respect to the XZ and YZ planes are 2 and 1 respectively, the even Bessel orders are omitted. When you set the Minimum order in the Expansion dialog equal to the default value zero, MMP will start with order 1, but it will leave the value in the Minimum order box unchanged. It is important to know this when you split the Bessel function into two parts.
Amplitude definition
For the model using the harmonic expansion, no amplitude definition was required. As soon as you use Bessel expansions with several parameters, the definition of the amplitude plays an important role. See what happens when you omit the amplitude definition. Then see what happens for different definitions of the amplitude in the Integral dialog. For each amplitude definition you should save the rough search data in a function file and plot the function E2/Amp2(gamma). For example, define the amplitude as the integral of the time-averaged Poynting field over a rectangle that covers the first quadrant of the waveguide. How sensitive are the results when you change the discretization for the evaluation of the integral? Note that the important 'details' of the function E2/Amp2(gamma) are the number of and location of the minims. Can you observe shifts in the minims when you change the amplitude definition? Does the number of minims change? Do you agree that it is important to define the amplitude? How important is the accuracy of the integral to be evaluated? Note that the answer to this question depends on the complexity of the field and of the mode to be searched for. For fundamental modes and low order modes, very coarse integration is sufficient.
Accuracy of the eigenvalues
How many parameters are required, for example, to obtain the propagation constant with 1% accuracy? Note, that you can specify the desired accuracy of the eigenvalue search in the Project dialog. When the number of parameters is small, the desired accuracy might not be reached. Moreover, the stopping criteria is not completely safe. Check the accuracy by comparing with accurate results obtained with harmonic expansions.
Search function, accuracy of the field
When you view the eigenvalue search function E2/Amp2, you can observe very sharp minims. The fine search algorithm requires only a few steps to accurately find the correct location of such minims. When you set a relatively high value for the Accuracy in the Project dialog (for example, 0.01), the fine search stop might stop at a position, where the function E2/Amp2 has quite a large value. Although the propagation constant is sufficiently accurate for your needs, the field may be computed inaccurately. When the MMP errors are large, this can either be caused by an inaccurate MMP expansion or by using too inaccurate an eigenvalue search!
Last parameter and last expansion
It has been mentioned in the introduction to guided waves that the field of the last expansion can play an important role. The field of this expansion should be somehow characteristic for the mode to be searched. At least, the corresponding last parameter should not be much smaller than the other parameters. In the computation of the circular dielectric waveguides, this was no problem because all of the two or four expansions were characteristic. Now, the higher order Bessel functions contribute almost nothing to the field of a low order mode and the corresponding parameters have very small amplitudes. Since MMP starts with the lowest order terms of the Bessel expansion, the last expansion is the highest order Bessel expansion, i.e., an expansion that is not characteristic for low order modes. Although this is not recommended, you can obtain useful results. For getting more experience, try what happens when you put another expansion at the last position.
How can you move the lowest order Bessel function - which is representative for low order modes - to the last position? To do that, you have to split the Bessel expansion into two Bessel expansions. Assume that you have the symmetries mentioned above and that you have defined a single Bessel expansion with 20 parameters in the Expansion dialog. Copy this expansion to obtain two identical expansions. Now, reduce the number of parameters of the first expansion to 19 and set a single parameter for the second expansion. This will be the last expansion. The lowest order that fits the symmetry is one. Thus, set the minimum order to 1 for the second expansion. The second lowest order that fits the symmetry is three. Thus, set the Minimum order 3 for the first expansion. Observe what happens with the eigenvalue search function E2/Amp2.
When you want to move an expansion other than the lowest order Bessel term to the last position, you have to split the Bessel expansion into three parts. Check the influence of the last expansion on the shape of the eigenvalue search function E2/Amp2 by putting different Bessel orders in the last position. Note that the last expansion is not critical here, but you might encounter more difficult eigenvalue problems, where it is helpful to carefully select the last expansion.
Multipole expansions
You may replace the Bessel expansion by appropriate sets of multipole expansions. To check the multipole expansions, proceed as for the Bessel expansions. Now, you have many more possibilities of obtaining useful expansions. Try to find multipole expansions that outperform the Bessel expansion.
A simple set of multipole expansions consists of multipoles on the x and y axes. You can set these multipoles quite far away from the rectangle. When you do that, you can increase the step of the multipole orders in the Expansion dialog to avoid ill-conditioned matrices. Observe what happens!
Ill-conditioned matrices may cause noise in the function to be minimized, and this causes problems for the rough search as well as for the fine search. To obtain very ill-conditioned matrices, you can set two identical multipoles very close together. Move multipoles around and look how the eigenvalue search function E2/Amp2 behaves.
RECW100.PRO
The project RECW100 on the CD-ROM contains all the files needed study the rectangular PEC waveguide with harmonic expansions (RECW100.EXP), with Bessel expansions (RECW101.EXP), and with multipole expansions (RECW102.EXP, RECW103.EXP). Modify these files to find more modes, and to explore the advantages and disadvantages of various expansions and combinations of expansions. Note that this project contains directives for movies that show the frequency dependence of the eigenvalue search function. The corresponding data is automatically written to function files. When you run eigenvalue computations before you let MaX-1 run the directives of this project, the numbers in the function files may not coincide with the numbers assumed in the directive files. To avoid problems with incorrect file numbers, exit MaX-1, start it again and press the generate movie button before you perform any eigenvalue computation. The directive files are chained and you will obtain all the movies for this project.
