Microwave technology finds many applications in the areas of terrestrial/satellite communications, radar, radio astronomy, radiometry, food science, material science, and medicine [1]. Common to all applications is the transport of electromagnetic energy via guiding structures. In addition, most applications involve the propagation of electromagnetic waves in an unbounded medium. Applications that do require the radiation of electromagnetic energy inherently require the use of an antenna [4]. The IEEE Standard Definitions of Terms for Antennas defines the antenna as "a means for radiating or receiving radio waves".

An application of primary importance lies in communication systems. In general, a communication system seeks to transport meaningful (coded) electromagnetic energy from one point to another. This may be accomplished through the use of transmission lines (i.e. wires/cables) or through the use of electromagnetic propagation in an unbounded medium (i.e. wireless). The latter of course requires a transmitting and receiving antenna system. It is worthwhile to note here that the ability of an antenna to focus electromagnetic energy is inversely proportional to the operating wavelength of that energy for a fixed antenna size. Hence higher operating frequencies result in more sharply focused antenna radiation beams. A sharply focused beam provides an efficient transfer of electromagnetic energy allowing for communication over greater distances. Therefore communication systems operating in the microwave spectrum generally offer improved performance. In addition, the atmospheric attenuation is fairly low in the lower microwave frequency range.

A numerical comparison between the two methods of communicating (i.e. wired vs. wireless) reveals that the wireless system is far more efficient (i.e. less loss over the same distance) when long distances are considered. However for short distances, transmission line communication is more efficient [1]. It is also interesting to note that the cost effectiveness of each method follows suit with its call to performance.

In a wireless communication system, electromagnetic energy is carried to the antenna by means of a guiding structure. The guiding structure may be a coaxial cable, stripline, microstrip, or any other form of Transverse Electric and Magnetic (TEM) transmission line (i.e. one that supports a TEM or quasi-TEM mode of propagation). Alternatively the guiding structure may be a waveguide of circular or rectangular cross section. Waveguides of this type are capable of supporting several modes of propagation at once according to the solution of Maxwell’s equations and associated boundary conditions. These modes of propagation are limited to Transverse Electric (TE) and/or Transverse Magnetic (TM) modes only. In fact the solution to the Helmholtz equation for the general case of a closed conductor (i.e. rectangular waveguide) shows that such a structure cannot support TEM wave propagation. In all guiding structures the electromagnetic wave is considered bounded (i.e. confined to propagate within the enclosed area of the guiding structure). Figure 1 below shows a cross sectional view of rectangular waveguide. This choice of coordinate system and design variables will apply specifically to waveguides of rectangular cross section only.

When an antenna is properly designed, the majority of electromagnetic energy present at the antenna’s terminals will be transmitted through the antenna. That is, the energy is radiated into free space in the form of a propagating wave. Hence, as the electromagnetic energy passes through the antenna, it is transformed from a bounded propagating wave to an unbounded propagating wave. In this way the antenna can be considered as a transducer. In the case of a bounded wave, the electric field lines begin and end on conducting surfaces. However, in the case of an unbounded wave the electric fields lines begin and close on themselves (i.e. they form loops) [4].

A small amount of electromagnetic energy is converted to heat within the antenna. This is due primarily to the presence of lossy dielectrics and imperfect electric conductors that make up the antenna. These losses define the radiation efficiency of the antenna. An additional cause of limited power flow is a poorly matched antenna. That is, the impedance seen at the input of the antenna must be conjugate matched to the characteristic impedance of the guiding structure for maxim power flow. For if the antenna is not properly matched to the transmission line, a large amount of power will be reflected in the reverse direction along the transmission line. Consequently this power is not transmitted and can be represented as a loss. If the antenna is not properly designed, the losses due to heat and reflection may deem the antenna unworthy for practical use. Surely an optimum design seeks to minimize these losses. Passive antennas are reciprocal devices. As such, the receiving behavior of an antenna can be explained in an identical manner by simply reversing the order of operations.

As part of the transmission process, antennas are required to focus the radiated energy in specified spatial directions while simultaneously suppressing it in others. Likewise, as part of the receiving process, antennas must provide sensitivity in specified spatial directions while simultaneously providing discretion in others. Hence one characterizing criteria for an antenna is the intensity of radiation emitted (or received) as a function of angular position over all space. This criteria is measured (or predicted) in the form of a radiation pattern.

Integration of an antenna’s radiation pattern over all space yields a figure of merit known as directivity. The gain of an antenna is yet another figure of merit and it is related to directivity. Antenna gain accounts for the dissipation and reflection losses within an antenna and is thus quite important to the system designer. The requirements placed on the radiation characteristics of an antenna depend upon the application itself. To meet the needs of each application the antenna designer must have an in depth understanding of the many types of antenna designs and modeling software packages that exist. In addition he/she must be proficient at manipulating these designs into new or hybrid forms. In fact the solutions to most antenna problems are often innovative, requiring creativity and perseverance.

Antennas may be broken up into several categories, or types, such as wire antennas, aperture antennas, array antennas, reflector antennas, and lens antennas. We shall consider for the moment, aperture antennas. This is a very important class of antennas for several reasons that will soon become apparent. In particular we shall define the electromagnetic horn antenna, it is a subclass of aperture antennas. The geometry of the electromagnetic horn is shown below in Figure 2. More complicated forms take on a pyramidal shape. This is a robust antenna constructed exclusively of metal. Hence the electromagnetic horn does not contain lossy dielectrics. Furthermore, the conductor losses for this antenna are mathematically predictable. These properties allow the gain of the horn antenna to be computed with reasonable accuracy thus making it an excellent calibration tool. Therefore, the electromagnetic horn has been, and remains, the industry standard for measuring the gain of all types of antennas via comparison or "substitution" methods.

These features make the electromagnetic horn a useful antenna in its own right. In addition the horn antenna is often used as a source, or "feed", to distribute electromagnetic energy over the surface of a parabolic reflector or lens. Moreover, horns (and other aperture antennas) are sometimes used as elemental units upon which large arrays are comprised. Hence the horn is an integral part of array, reflector, and lens antennas [4].

The principal behavior of an electromagnetic horn is fairly straightforward and well documented. It is merely a flared extension of the rectangular waveguide that is often used to feed it. The antenna designer carefully chooses the dimensions of the horn and waveguide to elicit a desired response. As described earlier, the rectangular cross section of a waveguide can support several modes of operation simultaneously. Each mode of operation is partly characterized by its lower cutoff frequency. Mode energy associated with frequencies below its own cutoff frequency is rapidly attenuated. Such a mode (i.e. one that is below cutoff) is referred to as a nonpropagating or evanescent mode.

The dominant mode of propagation is that with the lowest cutoff frequency. The dominant mode in rectangular waveguide is the TE₁₀ mode. The cutoff frequency of the TE₁₀ mode is related to the long dimension of the rectangular waveguide. The TE notation is used to signify that the Electric field is always and everywhere Transverse (i.e. orthogonal) to the direction of propagation within the guide. Since the TE₁₀ mode is the dominant mode (i.e. the lowest cutoff frequency), all additional modes have cutoff frequencies greater than that for the TE₁₀ mode. For instance the next higher order mode is the TE₂₀ mode (for standard waveguide sizes of a/b=2) [2]. Higher order mode propagation is generally undesired. Hence the useful operating range of standard waveguide is limited to those frequencies between the TE₁₀ and TE₂₀ mode cutoff frequencies.

In many applications the bandwidth limitation of rectangular waveguide is too restrictive. It is possible to improve the operating bandwidth of rectangular waveguide by capacitively loading the guide along its center. The case of a lumped element model has been analyzed using the transverse-resonance method. The results indicate that the presence of the capacitor is responsible for lowering the TE_m0 cutoff frequencies for odd values of m. However the cutoff frequencies associated with the even values of m remain unaltered. This finding agrees with intuition since the electric field of the TE_m0 mode pattern is zero along the center of the waveguide for even values of m. Hence the TE₁₀ cutoff frequency is lowered while the TE₂₀ cutoff frequency remains unchanged. This lowering of the TE₁₀ cutoff frequency represents a substantial improvement in operating bandwidth over that of standard rectangular waveguide. There is an additional advantage to using capacitively loaded waveguide. The advantage is that a lower TE₁₀ cutoff frequency equates to a smaller cross sectional area of waveguide over that of standard rectangular waveguide for a given design frequency.

One method of realizing capacitively loaded waveguide is through the use of metal ridges. The placement of a metal ridge centered along the upper or lower wall of standard rectangular waveguide is called single ridged waveguide. Insertion of a metal ridge along both the upper and lower walls of the guide constitutes dual ridged waveguide. In both cases the result is a lowered TE₁₀ cutoff frequency as desired. Cohn [7] and Hopfer [8] have solved both the single and dual ridged configurations using the transverse-resonance method. Their work, as cited in Rizzi [1], yields the TE₁₀ and TE₂₀ cutoff wavelengths for several ridge geometries. It is worth while to note that severe amounts of capacitive loading may lower the TE₁₁ cutoff frequency to a value lower than that for the TE₂₀ mode [9]. Figure 3 below shows a cross sectional view of dual ridged waveguide. This choice of coordinate system and design variables will apply specifically to waveguides of ridged cross section only.

In horn antenna applications requiring extended bandwidth, it is necessary to extend the waveguide ridges into the horn itself. The purpose of extending the ridges into the horn is, again, to lower the TE₁₀ cutoff frequency thus improving the operating bandwidth of the horn. In fact, the ridges need only extend a short distance into the horn. That is, until the cross sectional area of the horn is sufficient to support TE₁₀ mode propagation. An antenna of this type is commonly referred to as a dual ridged horn. Figure 4 below shows a cross sectional view of a dual ridged horn. Designing a dual ridged horn is not a trivial task. There is little found in the literature on the design of dual ridged horn antennas despite their widespread use in industry. Most design practices for these antennas are highly proprietary, empirical, and involve several iterations.

Doctoral research is currently being conducted by Charles D. McCarrick, of Seavey Engineering Associates, Inc.. The main goal of this research is to advance the state of dual ridge horn antenna design and analysis. The present aim of this research is to treat the dual ridged horn as a composition of three separate sections. Each section is being analyzed independently of the other two. The first section considered is a dual ridged waveguide of uniform cross section. The analysis of the ridged waveguide must yield the electric field distribution within the waveguide cross section for an arbitrary geometry (this is the input to the antenna). The next section considered is the ridged flare section of the horn itself. The analysis of the flare section must result in a method for determining the electric field distribution at the output plane of the antenna for a given input field distribution (i.e. from the ridged waveguide). The output plane of the antenna is commonly referred to as the antenna aperture. Once the field distribution across the aperture is known, it is then possible to compute the far field radiation pattern of the antenna. This is the final phase, or section, of the analysis.

A review of the literature shows that ridged waveguide has been analyzed by various methods to date. These include the finite element method [10]&[11], the Rayleigh-Ritz variational method [12]&[13], the moment method [15]-[17], the transverse resonance method [7]&[8], and modal analysis [18]. Most analyses seek to find the cutoff frequencies of the various modes and guide impedances for several geometries. In some cases the field distributions have been derived and plotted [9]&[13]. In addition, closed form expressions for the guide impedance and cutoff frequencies have been presented [14]&[19].

The existing methods have been exercised to predict the field distribution within the waveguide cross section. However, some methods seem to generate erroneous results near the ridge discontinuities. One goal of the present doctoral research is to generate the field distribution within a dual ridged waveguide through conformal mapping. In fact an approach has already been outlined and must now undergo rigorous validation. Compiling a sufficient number of validation cases will be an arduous task in itself. Moreover, the methods presented in the literature do not lend themselves to rapid manipulation of the ridged waveguide geometry.

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