Motivation
The parallel plate waveguide is the simplest structure consisting of two parallel PEC plates. You have already studied the analytic solution of this structure and you have used such waveguides in scattering and antenna problems.
Fundamental mode
The fundamental mode of the parallel plate waveguide is a planar wave between the plates with the electric field perpendicular to the plates. This mode has a normalized propagation constant that is always equal to one, provided that the material between the plates is free space. Thus, no eigenvalue search is required and you can set the propagation constant directly in the Project dialog. Moreover, you have to define a single expansion in the Expansion dialog. This expansion is either a plane wave or a harmonic expansion. Its amplitude can be set equal to one. Since neither an eigenvalue nor an unknown parameter has to be found, you do not need to run MMP and you can directly compute the original and derived fields in the Field dialog. Since you did not run MMP before, you should make sure that MaX-1 uses the expansions rather than the formula for computing the original field. For doing this, press the Field formula… button in the Field dialog and check the Ignore formula box in the Field formula dialog.
Field representation
Since the field is non-zero only within the plates, you should select a grid space that has no points on or outside the plates to avoid unnecessary computations. When a grid line is on the boundary of a plate, it depends on numerical details as to whether MMP detects the points on these line to be in the space between the plates or not. Therefore, you should select the grid space in such a way that no grid line coincides with one of the boundaries.
PARW100.PRO
On your CD-ROM you will find the project PARW100. This project contains a simple model for exploring the fundamental mode of a planar waveguide consisting of two parallel PEC plates.
Higher order modes
The field of all higher modes of the parallel plate waveguide can be described by a single harmonic expansion. Select the corresponding expansion in the Expansion dialog and the wave type in the Project dialog. The amplitude of the expansion can be set equal to one and no MMP computation of unknown parameters is required. Since the propagation constant of the mode at a given frequency is unknown, let MaX-1 search for it. First of all, you have to define the search data in the Project dialog. The normalized propagation constant of all higher order modes is between zero and one when you have free space between the plates. Since you have to search along the real axis of the complex gamma plane, set the second value of the search grid and the imaginary parts of the search corners in the Project dialog equal to zero. To avoid numerical problems, set the real parts of the corners not exactly equal to zero and one, but to two values very close to that, for example, 1E-6 and 0.99999. Do not forget to check the Eigenvalue box in the Project dialog and the Rough and Fine search boxes in the MMP dialog before you let MMP evaluate the eigenvalue of a higher mode.
Primary and secondary modes
When the plates of the waveguide are parallel to the x axis, the waveguide is not closed in the x direction. When you find any mode for such a waveguide, you can construct an entire spectrum of modes with the following procedure. Rotate the field mode around the y axis with an angle phi and also with the angle -phi. The superposition of these two modes gives you a new, derived mode. For each angle phi between zero and 90 degrees you obtain another derived mode. The y dependence of these modes is the same. Therefore, it is sufficient to consider the primary modes only, which have no x dependence of the field. The harmonic expansions describing these modes have a zero x component of the wave vector. For defining such a harmonic expansion, select the x type -1 in the Expansion dialog and set the value in the x-per./2 | kx/k0 box equal to zero.
The x dependence of the field of the derived modes is either sine or cosine, i.e., the x type of the corresponding harmonic expansions is -1 or -2. The x component of the wave vector of these modes is kx=sin(phi)*gam, where gam is the propagation constant of the primary mode. Therefore, you have to select kx/k0 within the interval -gam/k0…gam/k0. The upper limit of this interval is the normalized propagation constant of the primary mode.
Cutoff and evanescent modes
What happens, when you set a value outside this interval? Note that MMP does not refuse such values. Thus, you can find out what happens. Essentially, you have a resonance problem in the y direction only. The mode of this resonance problem is characterized by the y component of the wave vector. The square of the wave number is equal to the sum of the squares of all components of the wave vector, i.e., kx2+ky2+kz2=k2. The wave number k is a constant for a given frequency and for given material properties. When kx is bigger than gam, kz becomes imaginary. Since kz is the propagation constant of the secondary mode, this means that a secondary mode with kx>gam is evanescent, i.e., below cutoff.
When you have searched for an evanescent mode along the real axis of the gamma plane as you did for a guided mode, the rough search has found no possible eigenvalues. This is displayed in the Info window. When you search along the imaginary axis instead, MMP can find the propagation constant of the evanescent mode and you can let MMP compute its field.
Since also primary higher order modes have a non-zero cutoff frequency, these modes can also be evanescent and you can search along the imaginary axis for finding evanescent higher order primary modes.
Mode numbering
The harmonic expansion describing the field of different primary modes is essentially the same. You only have two types of primary modes depending on the wave type, i.e., E and H modes. Different modes of the same type are characterized by different propagation constants, i.e., the function E2/Amp2 has more than one minimum. It is reasonable to number the primary modes in such a way that the mode with the lowest cutoff frequency obtains the lowest number. This mode will also have the highest value of the propagation constant. When you search from gamma=1 down to gamma=0, the number of the minimum that is found corresponds to the mode number.
Derived modes are completely described by the corresponding primary mode and the x dependence of the field, which has the form sin(kx*k+a) with the two real parameters kx and a. For the definition of kx see above. The parameter a causes a simple shift in x direction.
Eigenvalue search grid
At higher frequencies, where many modes exist, the grid of the eigenvalue search specified in the Project dialog must be sufficiently fine. Otherwise MMP might miss some of the modes during the rough search. Modes with a propagation constant sufficiently close to one of the two search corners are often not detected. Therefore, it is good to observe carefully the function E2/Amp2. When MMP starts an eigenvalue search, it asks you to specify a function data file when the Save search data box is checked in the Project dialog. Specify any file name and read this file as a function file as soon as the eigenvalue search is finished. Use the function representation tool to view the contents of this file, i.e., the function E2/Amp2(gamma).
PARW110.PRO, PARW120.PRO
On your CD-ROM you will find the projects PARW110 and PAR120. These projects contain simple models for exploring primary higher order modes of planar waveguides consisting of two parallel PEC plates. PARW110 computes the first H wave and PAR120 computes the second H wave. Modify these projects to find other primary modes, secondary modes, and evanescent modes.
Frequency dependence of gamma
Typically, you may wish to observe the frequency dependence of the propagation constant of one or several modes. To obtain a function file that contains the corresponding data, you have to write appropriate MaX-1 directives.
By starting your rough search at the highest frequency of your frequency range, you can find all modes for your model, provided that the search space in the gamma plane is sufficiently large and provided that the search grid is sufficiently fine. For each mode, you can start at the highest frequency and trace the eigenvalue while the frequency is decreased. Turn the eigenvalue search PET on in the MMP dialog. This can considerably reduce the number of iterations. Tracing the frequency downward has two drawbacks. 1) At the highest frequency, several modes can have similar propagation constants, which requires a fine grid for the rough search. Otherwise, the search may miss some of the eigenvalues. 2) Numerical and graphical problems can occur when the cutoff frequency is reached.
These problems are avoided when you start at the lowest frequency, where you have only a small number of modes. When you trace a mode while increasing the frequency, you never run into cutoff, but with this procedure, you miss the modes with a cutoff above the lowest frequency. How can you find these modes?
Cutoff frequencies
The best approach is the following: Close to cutoff, the propagation constant of all modes is almost zero. When you set the normalized propagation constant in the Project dialog close to zero and search for the frequency instead of searching the propagation constant, you obtain a good estimate of all cutoff frequencies in your frequency range. To do that, check the Frequency box in the Eigenvalue search data group, set Corner 1 of the search space equal to the minimum frequency and the Corner 2 equal to the maximum frequency. Set a sufficient number of search grid lines in the real direction and 0 or 1 grid line in the imaginary direction, and select the number of the cutoff frequency to be found in the Search Evl. box. Run MMP. Once you have found the cutoff frequency of a mode, you can trace gamma for increasing frequencies.
PARW130.PRO
The project PARW130 on your CD-ROM contains a directive file for searching the cutoff frequencies of the two higher order H modes that were studied in the projects PARW110 and PARW120. In the inner loop of the directives, the propagation constants of these modes are traced for increasing frequencies. The resulting plot shows the frequency dependence of both modes. Note that this plot does not show the fundamental mode and it does not produce a movie.
