Brief History of Corrugated Filters
Corrugated waveguide and “waffle iron” filters were first introduced by Conh  in 1948. The structure used there is schematically shown in Figure 1.
Figure 1: Corrugated filter by Conh .
The design approach is based on representing a periodic E-plane corrugated delay structure as a uniform waveguide of a particular characteristic impedance  and matching with waveguide interface by multi-step transformer. Therefore the procedure is based on some uncertainty and involve rather complicated procedures requiring empirical adjustments to experimental filters before a satisfactory final design is achieved . Later design methods using modern circuit theory and synthesis techniques have been fundamentally developed by Levy  and generalized for wide class of tapered corrugated waveguide filters . The filter structure has been tapered by Levy in several different ways and associated with Chebychev and Zolotarev functions response.
Figure 2: Tapering of corrugated waveguide from interface waveguide to smaller central aperture.
Figure 3: Tapered Chebychev function stub filter.
Figure 4: Tapered Zolotarev function stub filter.
Although the theory of corrugated filters had been powerfully boosted and theoretically optimal structures developed, generally tapered corrugated filters are not often used in microwave applications as harmonic filters because of some practical inconveniences.
Modern practical requirements for harmonic filters are mostly based on low cost, small size, low loss, wide stop-band and extremely high power handling. Over these requirements optimal ratio between power handling capacity and spectrum rejection could not necessary require tapering the filter structure. Moreover, the Conh design would definitely be preferred for production [Fig. 3]. Apart from general production complexity some quality assurance issues can be mentioned. If the filter is made by regular machining methods, all cavities are expected to be rounded and random tolerances to be applied to all dimensions of the filter structure. Deviation of scattering properties of each discontinuity over tolerance would be greater, if each discontinuity scatter differently. In other word, there should be uniformity and similarity in deviations of scattering properties of corrugations over tolerance in order to reduce total impact of manufacturing errors. Therefore in my engineering practice I prefer a simplified Levy [see Figure 5] structure, i.e. a format between generally tapered [Levy] and periodic [Conh] structures.
Figure 5: Simplified tapered corrugated filter.
Here gap b and spacing s are selected the same over all corrugations for production preferences. One transformer section should be enough for transition from larger interface for most applications.
I have not found useful making accent on Zolotarev, Chebychev or other special function of similarity frequency response, when designing regular corrugated filter. This is because the conventional synthesis methods are to build a “good looking” pass-band and roll-off rather than harmonic frequencies where the functions do not model the reality. On the other hand the standard methods limit the designer to control other important technical features such as power handling, size, or manufacturability, which are not directly associated with formal prototypes but are very important. In this case the optimal practical filters might be quite different from any classical prototype. For example, the filter response showed in Figure 6 would be preferable for customer if it is cheap and matching the spec.
Figure 6: Non-esthetic filter example
However fast development of computer technologies have made possible direct optimization of whole filter structure over specific requirements. Direct optimization method has many advantages in comparison to conventional synthesis methods because it gives much flexibility for designers and result in an optimal filter comprising of the best ratio of parameters. It is also very beneficial for a design engineer, because it makes him or her real artist. In order to use that power technique the right starting point and optimization constrains must be found. Here, in my Design Studio, the technique is applied to common harmonic corrugated filter to demonstrate how simple and efficient it could be.
In order to use direct optimization methods, a fast computer algorithm should be created in order to run over huge variety of combinations of dimensions increments and frequencies. Unfortunately the modern engineering software based on FEM or other universal numerical methods cannot be used to create an effective variational engine for optimization, because it is very slow and not regular over increments. I have found the old-fashioned variational Ritz-Galerkin or perturbation methods to be perfect to model filters, because they are “smooth” and can be realized in simple math manipulations. However, the accuracy of such type of “formulas” can be even better than accuracy of the most powerful universal numerical technique. For example, the correlation between measured data and data simulated by FEM based on HFSS and three pages FORTRAN code based on variational estimations [Figure 7].
Figure 7: Frequency response of a corrugated filter over WR75 frequency operation range. The black line shows frequency data measured by a HP vector network analyzer. The red line shows data obtained from HFSS, a FEM based simulator, using twenty passes and discrete sweep. The green line shows data simulated by “filter4”, a FORTRAN module based on variational estimations.
The computation time difference impresses more, as it took only 0.5 seconds the FORTRAN model to run over 350 frequency points in comparison with HFSS running 100 points overnight on the same UNIX platform.
Unfortunately the word wide web much limits possibilities to run engineering software on the both server and client side. The servers usually put restrictions on using exe modules and accessing them. The client side is also limited by capability of internet browsers, which are indeed not created to run serious software. However it is possible to run simple algorithms based on scripts on the both sides. The corrugated filter design pages presented here are based on VBScript code supported by any Microsoft Internet Explorer of version better than 3.02. The scripts are about 10 times slower than PC based module compiled from FORTRAN code of the same function. Nevertheless, it is a convenient way to design a filter while surfing internet.
 S.B. Conh “A theoretical and experimental study of a waveguide filter structure”, Office Naval Res., Cruft Lab., Harvard Univ., Cambridge, Mass., Rep. 39, Apr. 25, 1948.
 C.G. Matthaei, L. Young, and E. M.
T. Jones, “Microwave Filters, Impedance Matching Networks, and Coupling
 R. Levy “Tapered Corrugated Waveguide Low-Pass Filter”, IEEE Trans. Microwave Theory Tech., MTT-21, August 1973, pp. 526-532.
Ralph Levy, “Aperiodic Tapered Corrugated Waveguide
Filter”, US Patent 3,597,710,