MODE MATCHING MODE MATCHING The use of mode-matching techniques to analyse the performance of circular horns, particularly corrugated horns, has been well known for many years [1], [2], [3], [4]. The method is now mature and robust and horns are routinely manufactured by many companies using this technique. Recent advances in the theory have allowed the technique to be applied to more complex structures as varied as septum polarisers and small reflectors with splash-plate feeds. Comparison with measurement has shown the same excellent agreement as has been obtained with corrugated horns.
The essence of the mathematics is that the structure is subdivided into cylindrical slices. The modes which can propagate in each section, including in some cases evanescent modes, are set up and the complex amplitudes of the modes are matched across the boundary of the slice to the next slice, bearing in mind the boundary conditions (Figure 1). Clearly the mathematics will be written out rather differently for circular and rectangular components, since cylindrical coordinates are natural for circularly components and cartesian for rectangular components. The example below is for circular waveguide geometry and assumes that dielectrics are present.
FIGURE 1 Diagram showing the division of the circular waveguideinto cylindrical sections which may be loaded with dielectric (shown in purple).The dashed lines show the boundaries between cylindrical sectionsacross which the waveguide modes must be matched. |
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In order to solve for the complex amplitudes, it is necessary to evaluate eigen-modes of circular cross-sections loaded with dielectrics. The dielectric loading must be axially symmetric. Electric and magnetic field components within each loaded region are expressed in terms of Bessel functions. By matching tangential fields at the interfaces, a set of linear equations can be derived. Both the propagation coefficients and field distributions of eigen modes can then be obtained by solving the simultaneous equations [2]. The modes can either be transverse electric/magnetic or hybrid depending on the presence of dielectrics. Once the modal distributions are known coupling integrals between a pair of modes, one each from either sides, are evaluated at a junction between two cylindrical sections. The resulting complex coupling coefficients are cast into a matrix form and by performing algebraic matrix operations the multi-mode scattering matrix of the junction can be obtained. The overall scattering matrix of the structure can subsequently be obtained by cascading the scattering matrices. The desired complex amplitudes at the output of the structure are then obtained by multiplying the overall scattering matrix with the input excitation coefficients.
The method is easily extended to coaxial geometries and lossy dielectrics may be included. Since each 'slice' may contain a different geometry and/or dielectric, this program is ideally suited to the analysis of coaxial junctions. Many components and horns have been built based on designs from this type of software.
Because the modes have to be computed in different coordinate systems depending on the local structure, it is very difficult to write a general program to deal with absolutely every case and therefore a set of different programs applicable to different geometries has been written.
RECTGEN extends the application of rectangular modes in waveguide to several rectangular waveguide segments. The structure to be analysed is modelled as a number of discrete rectangular or ridged waveguide segments supporting propagating as well as evanescent modes. Multiple waveguide aperture segments can coexist side by side. This flexibility extends the scope of the analysis software to include many waveguide network structures. In order to achieve an optimal run-time, modes within waveguide segments can be selected in a flexible manner and various symmetry conditions can be applied. For more details on the program, RECTGEN, click here
Typical components are
AXIAL can be used to analyse combined circular and multiple coaxial structures with partial dielectric loading(optional) such as corrugated horns with ring loaded or V shape slots, axially corrugated horns, horns with dielectric rings, coaxial orthomode junctions (OMJs), multi-band coaxial feeds, dielectrically loaded horns and polyrod feeds and small splash-plate fed Cassegrainian antennas (see structure examples below). In principle, any circularly symmetric structure which can be decomposed into circular and coaxial regions can be modelled. The dominant TE11 mode excitation of either a circular or coaxial input A typical example is an axisymmetric Cassegrainian reflector with a splashplate feed [6], [7]. The exciting waveguide with its dielectric is modelled as a circular waveguide while the exterior of the waveguide and the reflector are modelled as a coaxial section (Figure 5). The great advantage of this type of analysis is that is allows an optimisation to be carried out which is much more accurate than the usual ray tracing. The result is an ability to design a high efficiency (65%) reflector with sidelobes of -22 dB and low spillover (<-40 dB) when the aperture is relatively small in wavelengths. Examples up to 12 wavelengths in diameter have been designed and manufactured and are now in production.
For more details on the program, AXIAL, click here.
RTCC deals with mixed rectangular and circular waveguide. A dominant mode excitation is assumed at the input waveguide (TE10 for rectangular and TE11 for circular). Structures which can be analysed (see below) include
Figure 6, Figure 7 and Figure 8 show results from this program. An optimiser program, OPTRTCC, can be used to optimise dimensions for a specified RF performance. An example of a polariser from EASAT Antennas Limited is shown in Figure 9.
For more information on the programs, RTCC and OPTRTCC, click here.
These programs are available on several platforms.
 FIGURE 2 Geometry of Septum Polariser from RECTGEN |
 FIGURE 3 Differential Phase Performance of a Dual Port Septum Polariser |
Predicted (RECTGEN) and Measured Coupling Response of a Waveguide Hybrid Coupler. Measured results from [5].
Cross-section of a Cassegrainian Reflector with splashplate feed (AXIAL). Graphics captured from WINDOWS
Return Loss from a disc cavity fed by rectangular waveguides.
Solid Line -measured results from [8]. Red squares predicted by RTCC
 FIGURE 8 Computed radiation patterns for an E-plane sectoral horn (RTCC). Green - E-plane; Blue - H-plane: Red - D-plane; Purple - Crosspolarisation See Figure 7 for geometry. Measured results of [9] are indistinguishable from these. |
Predicted and Measured Performance of a Polariser in Rectangular Waveguide with two opposed walls corrugated
