Motivation
The planar dielectric waveguide consists of a dielectric plate that guides the energy. If the domains on both sides of the plate have the same material properties, the guide is symmetric. From the mirror principle, you essentially obtain the field of a symmetric dielectric plate waveguide for a structure consisting of a PEC plate coated with a dielectric layer. You have already used such a structure in a scattering problem with a rib in a planar structure. Therefore, it is good to study it carefully. It is up to you whether you prefer modeling a symmetric dielectric waveguide or a coated plate.
Comparison with the parallel plate waveguide
Just as for the parallel plate waveguide, you have essentially a one-dimensional resonance problem in the y direction to be solved - provided that the boundaries of the waveguide are also parallel to the x axis. Therefore, you can also distinguish between primary and secondary modes. As soon as you have found a primary mode, you can construct a secondary mode as for the parallel plate waveguide. Therefore, secondary modes are not explicitly considered in the following.
The new element of the planar dielectric waveguide is its dielectric boundary that replaces the PEC boundary of the parallel plate waveguide. Consequently, the field may extend to infinity in the y direction. Is there still a discrete spectrum of modes? Essentially, the total reflection of the waves that hit the PEC plate must now replaced by a total reflection at the boundary between two different dielectrics. A total reflection is only possible for waves in the optically more dense dielectric. When the permeability of both dielectrics is the same, the dielectric with the higher permittivity is more dense. Thus, the dielectric that extends to infinity must have the lower permittivity to obtain guided waves.
Propagation constants
The normalized propagation constant of a plane wave in a dielectric with free-space permeability is equal to the square root of the relative permittivity. Let eps1 be the relative permittivity of the first dielectric that extends to infinity and eps2 the relative permittivity of the second dielectric. To obtain guided waves, you must have eps2>eps1. Where should you search for the normalized propagation constant gam of the guided waves? You can answer this question by trying several search intervals and running MMP. When you do this, you are probably never sure if you have found all the modes. When you analyze the total reflection of a plane wave at a planar boundary between two dielectrics, you find that eps2>gam>eps1 must hold. When gam is equal to eps2 or eps1, numerical overflow problems can occur. Avoid these problems by appropriately setting the search interval in the Project dialog.
Expansions
As for the parallel plate waveguide, you use harmonic expansions to model the field of the planar dielectric waveguide. The x dependence of the primary modes is a constant, i.e., 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 y dependence of the harmonic expansions in the second dielectric is sin(ky*y) or cos(ky*y) just as for the parallel plate waveguides. The y dependence in the first dielectric is exp(i*ky*y) or exp(-i*ky*y), i.e., type 3 or 4. The type of the expansions in the first dielectric must be set in such a way that the field decays exponentially when y goes to infinity. Note that ky for guided waves is imaginary in the first dielectric. MMP always computes ky in such a way that the imaginary part is positive. Thus, exp(i*ky*y) is the correct y dependence for a domain that extends to infinity in the +y direction.
Wave types and modes
The distinction of E and H waves and the numbering of the modes is similar to the case of the parallel plate waveguide and requires no further explanation. Can you find a fundamental mode with zero cutoff frequency?
PLAW10?x.PRO
On your CD-ROM you will find the projects PLAW10?, where ? denotes a digit from 0 to 4. These projects contain simple models for exploring planar dielectric waveguides consisting of a PEC plate coated with a dielectric layer. Modify these projects for a more detailed analysis.
Frequency dependence, cutoff frequencies
To obtain the frequency dependence of the propagation constants of the planar waveguide modes, you can proceed as for the parallel plate waveguide. Read and adapt the directive file PARW130.DIR to find the cutoff frequencies of higher order modes and to trace them while increasing the frequency.
