Patent Publication Number: US-8111115-B2

Title: Method of operation and construction of dual-mode filters, dual band filters, and diplexer/multiplexer devices using half cut dielectric resonators

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is related to and claims the benefit of U.S. Provisional Application 61/135,289, filed Jul. 21, 2008 and entitled “Method of operation and construction of dual mode filters, dual band filters, and diplexer/multiplexer devices using full or half cut dielectric resonators,” the entirety of which is hereby incorporated by reference. 
    
    
     FIELD 
     The embodiments described herein relate to microwave filters, and more particularly to dielectric resonator filters and multiplexers realized using full cylindrical or half-cut dielectric resonators. 
     BACKGROUND 
     Microwave bandpass filters are commonly realized using one or more resonators. Broadly speaking, a resonator is any physical element that stores both magnetic and electric energy in a frequency-dependent way. The resonant frequency of a resonator is defined as any frequency at which the stored electric and magnetic energies in the resonator are equal, and at that frequency the resonator is said to be in resonance. 
     Realizations of microwave resonators, however, are not so limited. At microwave frequencies, potentially any three-dimensional structure can be used to realize a resonator, in which internal electric and magnetic field distributions are generally determined by the shape and size of the overall structure. Some classes of microwave resonators include lumped element, microstrip, coaxial, waveguide, and dielectric resonators. Each class has application specific advantages and disadvantages. 
     In general, a dielectric resonator (DR) cavity comprises a dielectric resonator formed in a high-permittivity substrate mounted inside a metallic housing using a mounting support formed in a low-permittivity substrate. Compared to lumped element and microstrip resonators, dielectric resonators (as well as coaxial and waveguide resonators) tend to be bulkier in size and more complex in design, but offer superior Q values. In present microwave technologies, dielectric resonators offer Q values in the range of 3,000 to 40,000 at 1 GHz. For this reason, dielectric resonator filters are often favoured for use in satellite/space communication and wireless base station applications, where low loss and high power can be overriding design considerations. In addition to the Q values, resonator size and spurious performance (the frequency separation between an operating mode of the resonator and adjacent resonant modes) can also be important design considerations 
     Dielectric resonators are also commonly operated as single-mode resonators, and dual-mode resonators, and less commonly as triple-mode and quadruple-mode resonators. A single-mode resonator supports only a single field distribution at the resonator&#39;s center frequency. Correspondingly, a dual-mode resonator supports two different field distributions and a triple-mode resonator supports three different field distributions. The intention for using a higher number of modes is mainly size reduction, as one physical resonator is overloaded with more than one electrical resonator, and each electrical resonator is supported by a mode distribution. Resonance modes, such as dual and triple-modes, which support a plurality of field distributions at the center frequency, are referred to as degenerate modes. In the usual case, the different field distributions in a degenerate mode are orthogonal modes of a similar field distribution and are created due to symmetries in the resonator. Thus, dual modes have been mainly realized with resonators having 90-degree radial symmetry (cylindrical and rectangular waveguide cavities and resonators), while triple modes are supported for example in cubic waveguide cavities and cubic dielectric resonators. 
     Quadruple-mode dielectric resonators have also been realized, but mainly due to complications in fabrication and tuning, comparatively less interest has been generated in this area. In order to realize a quadruple-mode dielectric resonator, independent or near independent control over the coupling and tuning of each of the four modes is required, which generally results in a complex overall coupling scheme involving a large number of tuning and/or coupling screws. Although tuning and coupling schemes necessary for single-mode and dual-mode dielectric resonators add some design complexity as well, the added design complexities are more pronounced in triple-mode dielectric resonators, and even more pronounced in presently known realizations of quadruple-mode dielectric resonators. Dual-mode, triple-mode, and quadruple-mode resonators remain attractive alternatives to single-mode dielectric resonators, however, because of their associated size reduction, especially considering that dielectric resonators already tend to be bulky. For the applications in which dielectric resonator filters are preferred, e.g. satellite/space systems, size and mass reduction are highly desirable. 
     SUMMARY 
     The embodiments described herein provide in one aspect a dielectric resonator assembly for use in one of a dielectric resonator filter and a dielectric resonator multiplexer, the dielectric resonator assembly comprising: a) a dielectric resonator; b) the dielectric resonator formed in a unitary piece of high-permittivity dielectric substrate into a half-cut cylinder of a selected height and a selected diameter, the half-cut cylinder defined by a parallel pair of semi-circular surfaces, a curved surface extending along respective curved edges of the pair of semi-circular surfaces, and a rectangular surface subtending the curved surface, wherein a first dimension of the rectangular surface corresponds to the selected height and a second dimension of the rectangular surface corresponds to the selected diameter; wherein the dielectric resonator resonates in a plurality of resonance modes comprising a ½HEH11 mode and a ½HEE11 mode and, at the selected height and the selected diameter, the ½HEH11 mode and the ½HEE11 are mode are operating modes of the dielectric resonator assembly. 
     The embodiments described herein provide in another aspect a dielectric resonator assembly for use in one of a dielectric resonator filter and a dielectric resonator multiplexer, the dielectric resonator assembly comprising: a) a dielectric resonator; b) the dielectric resonator formed in a unitary piece of high-permittivity dielectric substrate into a cylinder of a selected height and a selected diameter; 
     wherein the dielectric resonator resonates in a plurality of resonance modes comprising an HEH11 dual mode and an HEE11 dual mode and, at the selected height and the selected diameter, the HEH11 dual mode and the HEE11 dual mode are operating modes of the dielectric resonator assembly. 
     The embodiments described herein provide in another aspect a dielectric resonator filter comprising: a) at least one dielectric resonator assembly comprising a dielectric resonator formed in a unitary piece of high-permittivity dielectric substrate into one of: (i) a half-cut cylinder of a selected height and a selected diameter, the half-cut cylinder defined by a parallel pair of semi-circular surfaces, a curved surface extending along respective curved edges of the pair of semi-circular surfaces, and a rectangular surface subtending the curved surface, wherein a first dimension of the rectangular surface corresponds to the selected height and a second dimension of the rectangular surface corresponds to the selected diameter; and (ii) a cylinder of the selected height and the selected diameter; wherein the dielectric resonator resonates in a plurality of resonance modes comprising operating modes of the dielectric resonator assembly and, at the selected height and the selected diameter, the half-cut cylinder resonates in a ½HEH11 mode and a ½HEE11 mode, and the cylinder resonates in an HEH11 dual mode and an HEE11 dual mode. 
     The embodiments described herein provide in another aspect a dielectric resonator multiplexer comprising: a) at least one dielectric resonator assembly comprising a dielectric resonator formed in a unitary piece of high-permittivity dielectric substrate into one of: (i) a half-cut cylinder of a selected height and a selected diameter, the half-cut cylinder defined by a parallel pair of semi-circular surfaces, a curved surface extending along respective curved edges of the pair of semi-circular surfaces, and a rectangular surface subtending the curved surface, wherein a first dimension of the rectangular surface corresponds to the selected height and a second dimension of the rectangular surface corresponds to the selected diameter; and (ii) a cylinder of the selected height and the selected diameter; wherein the dielectric resonator resonates in a plurality of resonance modes comprising operating modes of the dielectric resonator assembly and, at the selected height and the selected diameter, the half-cut cylinder resonates in a ½HEH11 mode and a ½HEE11 mode, and the cylinder resonates in an HEH11 mode and an HEE11 mode. 
     The embodiments described herein provide in another aspect a method of manufacturing a unitary resonator assembly for use in one of a dielectric resonator filter and a dielectric resonator multiplexer, said method comprising: a) providing a dielectric material; b) forming the dielectric material into full cylinder of a selected height and a selected diameter; wherein the dielectric resonator resonates in a plurality of resonance modes comprising an HEH11 mode and an HEE11 mode and, at the selected height and the selected diameter, the HEH11 mode and the HEE11 mode are operating modes of the dielectric resonator assembly. 
     Further aspects and advantages of the embodiments described herein will appear from the following description taken together with the accompanying drawings. 
    
    
     
       DRAWINGS 
       For a better understanding of the embodiments described herein and to show more clearly how they may be carried into effect, reference will now be made, by way of example only, to the accompanying drawings which show at least one exemplary embodiment, and in which: 
         FIG. 1A  is a perspective view of an exemplary full cylindrical dielectric resonator; 
         FIG. 1B  is a perspective view of an exemplary half-cut dielectric resonator; 
         FIG. 2A  is a top view of the E field lines in the full cylindrical dielectric resonator of  FIG. 1A  for the HEH 11  resonant mode; 
         FIG. 2B  is a side view showing the concentration of E field lines in the full cylindrical dielectric resonator of  FIG. 1A  for the HEH 11  resonant mode; 
         FIG. 2C  is a top view of the E field lines in the full cylindrical dielectric resonator of  FIG. 1A  for the HEE 11  resonant mode; 
         FIG. 2D  is a side view showing the concentration of E field lines in the full cylindrical dielectric resonator of  FIG. 1A  for the HEH 11  resonant mode; 
         FIG. 3A  is a side view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEH 11  resonant mode; 
         FIG. 3B  is a top view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEH 11  resonant mode; 
         FIG. 3C  is a front view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEH 11  resonant mode; 
         FIG. 3D  is a perspective view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEH 11  resonant mode; 
         FIG. 3E  is a side view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEE 11  resonant mode; 
         FIG. 3F  is a top view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEE 11  resonant mode; 
         FIG. 3G  is a front view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEE 11  resonant mode; 
         FIG. 3H  is a perspective view of the E field lines in the half-cut dielectric resonator of  FIG. 1B  for the ½HEE 11  resonant mode; 
         FIG. 4A  is a mode chart for the full cylindrical dielectric resonator of  FIG. 1A  as a function of diameter-to-length (D/L) ratio; 
         FIG. 4B  is a mode chart for the half-cut dielectric resonator of  FIG. 1B  as a function of diameter-to-length (D/L) ratio; 
         FIG. 5A  is a perspective view of an exemplary inter-cavity coupling of two half-cut dielectric resonator assemblies; 
         FIG. 5B  is a perspective view of another exemplary inter-cavity coupling of two half-cut dielectric resonator assemblies; 
         FIG. 5C  is a perspective view of another exemplary inter-cavity coupling of two half-cut dielectric resonator assemblies for the ½HEH 11  resonant mode; 
         FIG. 5D  is a perspective view of the exemplary inter-cavity coupling of two half-cut dielectric resonators of  FIG. 5C  for the ½HEE 11  resonant mode; 
         FIG. 6A  is a top view of an exemplary half-cut dielectric resonator assembly with intra-cavity mode coupling; 
         FIG. 6B  is a perspective view of the exemplary half-cut dielectric resonator assembly of  FIG. 6A  with intra-cavity mode coupling; 
         FIG. 6C  is a front view of an exemplary half-cut dielectric resonator assembly with tuning and intra-cavity mode coupling; 
         FIG. 6D  is a top view of the exemplary half-cut dielectric resonator assembly of  FIG. 6C  with tuning and intra-cavity mode coupling; 
         FIG. 6E  is a perspective view of an exemplary half-cut dielectric resonator assembly with positive mode intra-cavity mode coupling; 
         FIG. 6F  is a perspective view of an exemplary half-cut dielectric resonator assembly with negative mode intra-cavity coupling; 
         FIG. 7A  is a top view of an exemplary half-cut dielectric resonator assembly with input-output coupling; 
         FIG. 7B  is a perspective view of the half-cut dielectric resonator assembly of  FIG. 7A  with input-output coupling; 
         FIG. 7C  is a perspective view of another exemplary half-cut dielectric resonator assembly with input-output coupling; 
         FIG. 8A  is a top view of another exemplary half-cut cylindrical dielectric resonator assembly with input-output coupling; 
         FIG. 8B  is a perspective view of the half-cut cylindrical dielectric resonator assembly of  FIG. 8A  with input-output coupling; 
         FIG. 9A  is a schematic illustration of an exemplary coupling scheme for a dielectric resonator filter; 
         FIG. 9B  is a schematic illustration of another exemplary coupling scheme for a dielectric resonator filter; 
         FIG. 9C  is a schematic illustration of another exemplary coupling scheme for a dielectric resonator filter; 
         FIG. 9D  is a schematic illustration of another exemplary coupling scheme for a dielectric resonator filter; 
         FIG. 9E  is a schematic illustration of an exemplary coupling scheme for an 8-pole dielectric resonator filter realized using 4 half-cut dielectric resonators; 
         FIG. 10A  is a perspective view of an exemplary single-cavity, 4-pole dielectric resonator filter synthesized using a full cylindrical dielectric resonator operating in a quad-mode; 
         FIG. 10B  is a top view of the exemplary single-cavity, 4-pole dielectric resonator filter of  FIG. 10A ; 
         FIG. 10C  is a front view of the exemplary single-cavity, 4-pole dielectric resonator filter of  FIG. 10A ; 
         FIG. 10D  is a perspective view of another exemplary single-cavity, 4-pole dielectric resonator filter synthesized using a full cylindrical dielectric resonator operating in a quad-mode; 
         FIG. 11A  is a plot of transmissions-parameter response versus frequency for the single-cavity, 4-pole dielectric resonator filter of  FIG. 10A ; 
         FIG. 11B  is a plot of reflection and transmission versus frequency for the single-cavity, 4-pole dielectric resonator filter of  FIG. 10D ; 
         FIG. 12A  is a perspective view of an exemplary 3-pole, dual band dielectric resonator filter synthesized using half-cut cylindrical dielectric resonators operating in a dual-band; 
         FIG. 12B  is a top view of the 3-pole, dual band dielectric resonator filter of  FIG. 12A ; 
         FIG. 13A  is a perspective and top view of an exemplary 2-pole, dielectric resonator diplexer synthesized using half-cut cylindrical dielectric resonators operating in a dual-band; 
         FIG. 13B  is a top view of an exemplary 3-pole, dielectric resonator diplexer with improved output port isolation; 
         FIG. 13C  is a plot of reflection and transmission versus frequency for the 2-pole dielectric resonator diplexer of  FIG. 13A ; 
         FIG. 13D  is a plot of reflection and transmission versus frequency for the 3-pole dielectric resonator diplexer of  FIG. 13B ; 
         FIG. 14A  is a top view of the electric field lines in the half-cut dielectric resonator of  FIG. 1B  for a first spurious resonant mode; 
         FIG. 14B  is a front view of the electric field lines in the half-cut dielectric resonator of  FIG. 1B  for a first spurious resonant mode; 
         FIG. 14C  is a perspective view of the electric field lines in the half-cut dielectric resonator of  FIG. 1B  for a first spurious resonant mode; 
         FIG. 15A  is a perspective view of an exemplary slotted half-cut dielectric resonator; 
         FIG. 15B  is a perspective view of another exemplary slotted half-cut dielectric resonator; 
         FIG. 15C  is a perspective view of another exemplary slotted half-cut dielectric resonator; 
         FIG. 15D  is a perspective view of another exemplary slotted half-cut dielectric resonator; 
         FIG. 16A  is a top view of the E field lines in the slotted half-cut dielectric resonator of  FIG. 15B  for a first spurious mode; 
         FIG. 16B  is a perspective view of the E field lines in the slotted half-cut dielectric resonator of  FIG. 15B  for a first spurious mode; 
         FIG. 17  is a perspective view of an exemplary 2-pole, dual-band dielectric resonator filter having improved spurious performance; 
         FIG. 18A  is a perspective view of an exemplary 3-pole, dual-band dielectric resonator filter having an inter-band transmission zero; 
         FIG. 18B  is a top view of the 3-pole, dual-band dielectric resonator filter of  FIG. 18A ; 
         FIG. 18C  is a front view of the 3-pole, dual-band dielectric resonator filter of  FIG. 18A ; 
         FIG. 18D  is a plot of reflection and transmission versus frequency for the 3-pole, dual-band dielectric resonator filter of  FIG. 18A ; 
         FIG. 19A  is a perspective view of an exemplary 4-pole, dual-band dielectric resonator filter; 
         FIG. 19B  is a perspective view of an exemplary 4-pole, dual-band dielectric resonator filter having an inter-band transmission zero; 
         FIG. 19C  is a plot of reflection and transmission versus frequency for the 4-pole, dual-band dielectric resonator filters of  FIGS. 19A and 19B ; 
         FIG. 20A  is a perspective view of an exemplary 4-pole, dielectric resonator diplexer with improved output port isolation 
         FIG. 20B  is a top view of the 4-pole, dielectric resonator diplexer of  FIG. 20A ; 
         FIG. 21  is a flow chart of the steps of a method of manufacturing a half-cut cylindrical dielectric resonator; and, 
         FIG. 22  is a perspective view of an exemplary rectangular dielectric resonator. 
     
    
    
     It will be appreciated that for simplicity and clarity of illustration, elements shown in the figures have not necessary been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. 
     DESCRIPTION OF VARIOUS EMBODIMENTS 
     One of the more popular dielectric resonator topologies is the cylindrical resonator, which may be operated in a single TEH resonant mode, as well as in dual degenerate HEH 11  or dual degenerate HEE 11  resonant modes. By sizing its diameter D and length L to have a particular D/L ratio, however, the dual HEH 11  and HEE 11  modes of the cylindrical resonator can be made to resonate at a common resonant frequency, thereby converting the full cylinder dielectric resonator into a relatively simple and compact quadruple-mode resonator. Single cavity, four-pole filters (and more generally N-cavity, 4N-pole filters) can then be realized using the full cylinder operated in a quad-mode, wherein the centre frequency of the filter is given by the common resonant frequency of the quad-mode. 
     The structure of the quad-mode cylinder can be simplified by cutting lengthwise along its central axis to produce a new class of half-cut cylindrical resonators. Similar to the quad-mode cylinder, by appropriate sizing of its diameter and length, the half-cut dielectric resonator can be operated as a dual-mode resonator, the two modes in the half-cut cylinder corresponding respectively to half of a single component of the degenerate HEH 11  and HEE 11  modes (hereinafter referred to as the “½HEH 11  mode” and the “½HEE 11  mode”). This realization of a half-cut cylindrical resonator is totally different from the image-type realization that uses metals in contact with the resonator along cut lines to simulate an ideal electric wall boundary condition. By exploiting a naturally occurring magnetic wall boundary condition in the HEH 11  and HEE 11  modes, no metals are required for the half-cut dielectric resonator and all losses and design constraints incurred by inclusion of the metals can be saved. Considerable size reductions are achieved, and complex tuning and/or coupling arrangements are largely avoided. The half-cut dielectric resonator can be used to realize a general class of N-cavity, 2N-pole dual-mode filters, as well as other non-fully dual-mode filters. 
     Both the full cylindrical and the half-cut cylindrical resonator have further application in dual-band filters. If the diameter and length of the cylinder are sized differently, the dual HEH 11  and HEE 11  modes (or alternatively the ½HEH 11  and ½HEE 11  modes) will resonate at separate resonant frequencies. The two frequency bands of the dual-band filter can then be carried by a corresponding resonant mode, wherein the center frequencies of the two bands will be given by the different resonant frequencies of the HEH 11  and HEE 11  modes (or alternatively by the ½HEH 11  and ½HEE 11  modes). The full cylindrical resonator can be used to realize N-cavity, dual-band filters with 2N poles in each band, while the half-cut resonator can be used to realize N-cavity, dual-band filters with N poles in each band. As bases for dual-band filters, the full and half-cut cylindrical resonators are versatile in providing full or near full control over the centre frequencies and fractional bandwidths of the two frequency bands, as well as their frequency band separation. Prior dual-band filters that carry the dual-band on physically separate resonators within a single cavity are bulky. Carrying the dual-band instead on orthogonal resonant modes of a single physical resonator offers significant size reductions over prior filter realizations, and also greatly simplifies filter design by permitting essentially independent control of each band. 
     Suitable modification of the basic dual-band filter will also realize a dielectric resonator diplexer. Rather than coupling both bands of the dual-band to a common output channel, each band can be isolated and independently coupled to different output channels. Components of mixed frequency signals failing somewhere within the dual-band can then be separated. Improved output channel isolation can also be achieved by coupling the different channel outputs to resonators enclosed in separate resonator cavities. The basic diplexer concept is extendible to higher order multiplexers. 
     Spurious performance of the half-cut cylinder can also be improved by cutting one or more through-way slots between opposite surfaces. The first spurious mode of the half-cut dielectric resonator is the third eigenmode of the structure, and its E field lines circulate orthogonal to the E field lines in both the ½HEH 11  and ½HEE 11  modes. Cutting a through-way slot generally parallel to the E field lines of the ½HEH 11  and ½HEE 11  modes, but orthogonal to the E field lines in the first spurious mode, therefore, creates a selective barrier terminating the E field lines of the latter, while leaving the former largely undisturbed. The spurious free window of the half-cut dielectric resonator is thereby greatly increased. Cutting a second through-way slot orthogonal to the first will likewise terminate the E field lines of the fourth eigenmode of the structure (the second spurious mode), and thereby provide an even wider spurious free window. 
     These and other aspects of embodiments of the present invention are discussed in greater detail below. 
     Reference is first made to  FIGS. 1A and 1B , which are perspective views of an exemplary full and half-cut cylindrical dielectric resonator, respectively, according to aspects of embodiments of the present invention. The full cylindrical dielectric resonator  1  shown in  FIG. 1A  comprises a generally cylindrical shape of diameter D and length L formed in a unitary piece of suitable high-permittivity dielectric substrate. Accordingly, the full cylindrical dielectric resonator  1  is defined by a parallel pair of circular surfaces  2  connected by circumferential surface  4  at circular edges  6 . The dielectric constant ∈ r  of the high-permittivity material falls in the range 20-100, but preferably in the range 30-50. For example, the full cylindrical dielectric resonator  1  may be formed out of ceramic, but other dielectric substrates may be suitable as well. 
     The half-cut dielectric resonator  10  is formed by cutting the full cylindrical dielectric resonator  1  along its cylindrical axis to produce the half-cylindrical form shown in  FIG. 1B . Ideally the cut will align precisely with the cylindrical axis resulting in a perfect half-cut cylinder. As will be described in greater detail below, however, some margin of error with respect to the location of the cut is tolerable. This half-cylindrical form is defined by a parallel pair of semi-circular surfaces  12 , a curved surface  14  extending along and connected to the pair of semi-circular surfaces  12  at respective curved edges  16 , and a rectangular surface  18  subtending the curved surface  14  and connected to the pair of semi-circular surfaces  12  at diametric edges  20 . The rectangular surface  18  therefore has dimensions of D and L and, in the ideal case, defines a plane that intersects with the cylindrical axis of the full-cylinder. The half-cut dielectric resonator  10  is formed in the same high-permittivity substrate as the full cylindrical dielectric resonator  1 . 
     Reference is now made to  FIGS. 2A-2D , which illustrate top and side views of the E fields in the full cylindrical dielectric resonator  1  for the HEH 11  and HEE 11  resonant modes, according to aspects of embodiments of the present invention. Both components of the dual HEH 11  mode of the full cylinder are illustrated in  FIG. 2A . As can be seen, the two mode components are provided by E field distributions of the same polarization, rotated 90-degrees relative to one another. Thus the two mode components of the dual HEH 11  mode are orthogonal. As shown in  FIG. 2B , the horizontally circulating E fields in the dual HEH 11  mode, though present throughout the full cylinder, are concentrated at the axial midpoint. 
     Similarly,  FIG. 2C  illustrates both components of the dual HEE 11  mode of the full cylindrical dielectric resonator  1 . Again the two mode components are provided by E field distributions of the same polarization, rotated 90-degrees relative to one another. The two mode components of the dual HEE 11  mode are thus also orthogonal. As shown in  FIG. 2D , the vertically circulating E fields in the dual HEH 11  mode are concentrated at the periphery of the cylinder including the axial ends of the full cylinder. 
     As eigenmodes of the full cylinder, the dual HEH 11  and HEE 11  modes are substantially non-interactive. Neither the two components of the dual HEH 11  mode nor the two components of the dual HEE 11  mode couple, as they are all orthogonal to one another. The dual HEH 11  and HEE 11  modes also do not couple each other. The full cylindrical dielectric resonator  1  has a plurality of resonant modes of which the dual HEH 11  and HEE 11  modes represent only two pairs. The single TEH and single TME modes, which are also substantially non-interactive, are two other examples of resonant modes of the full cylinder. 
     It is evident in  FIGS. 2A-2D  that the E field lines in the full cylinder circulate horizontally (parallel to the plane of the page) for the HEH 11  mode and vertically (perpendicular to the plane of the page) for the HEE 11  mode. For one component of each mode (the top views in  FIGS. 2A and 2C ), however, the E field lines circulate tangential to the symmetry plane  25 , which is oriented perpendicular to the plane of page. For the other components (the bottom views in  FIGS. 2A and 2C ), the E fields circulate orthogonal to the symmetry plane  25 . Owing to this symmetry, an ideal magnetic wall boundary condition coincident with the plane  25  would not disturb the tangentially circulating field distributions within the full cylindrical dielectric resonator  1 . In other words, a half-cut cylindrical dielectric resonator  10  with an ideal magnetic wall coincident with the rectangular surface  18  would perfectly simulate the resonance modes of the full cylindrical dielectric resonator  1  that are tangential to the plane, only with half the stored electric and magnetic field energies. These resonant modes can be denoted ideal ½HEH 11  and ½HEE 11  modes. 
     Reference is now made to  FIGS. 3A-3H , which illustrate various views of the E fields in the half-cut dielectric resonator  10  for the ½HEH 11  and ½HEE 11  resonant modes, according to aspects of embodiments of the present invention. The E field distributions shown in  FIGS. 3A-3D  (side, top, front, perspective) correspond to the ½HEH 11  mode, while those in  FIGS. 3E-3H  (side, top, front, perspective) correspond to the ½HEE 11  mode. In the case of half-cut dielectric resonator  10 , the rectangular surface  18  does act as a magnetic wall boundary condition. But because the dielectric constant in real dielectric resonators is finite, the magnetic wall boundary condition will not be a perfect one. Some energy will leak across the rectangular surface  18 . Consequently, the ½HEH 11  and ½HEE 11  modes of the half-cut dielectric resonator  10  do not exactly replicate the ideal ½HEH 11  and ½HEE 11  modes, resulting in slightly higher resonant frequencies than in the ideal case. On the whole, however, the ½HEH 11  and ½HEE 11  modes of the non-ideal half-cut cylinder provide good approximations of the ideal modes, so long as the cut aligns generally with the cylindrical axis of the full cylinder. If the cut is misaligned by too great an extent, the resulting shape will no longer have a surface coincident with the symmetry plane  25  that provides the magnetic wall necessary for the ½HEH 11  and ½HEE 11  modes to be expressed. 
     As described above, both the HEH 11  and HEE 11  modes of the full cylindrical dielectric resonator  1  are dual modes on account of radial symmetry in the cylinder, each comprising two identical mode components. It is evident in  FIGS. 3A-3H , however, that cutting the full cylinder along its cylindrical axis removes its radial symmetry. By removing half of the dielectric material of the full cylinder, the components from each of the HEH 11  and HEE 11  modes that are orthogonal to the symmetry plane  25  (or alternatively that are orthogonal to the rectangular surface  18 ) are deformed to meet the new boundary conditions of the half-cut cylinder, and are thereby lost as higher order resonant modes. These lost components become the spurious mode resonances of the half-cut cylinder. The mode components of the HEH 11  and HEE 11  modes that remain after the cut become the ½HEH 11  and ½HEE 11  modes and are single modes. 
     Reference is now made to  FIG. 4A , which is a mode chart for the full cylindrical dielectric resonator of  FIG. 1A  as a function of diameter-to-length (D/L) ratio. The mode chart  30  plots frequency (GHz) against diameter-to-length (D/L) ratio and corresponds to a cylindrical resonator (D=0.7, ∈ r =38) located in a 1×1×1 in 3  cavity. The length L of the cylinder is the free variable. Curve  32  represents the resonant frequencies of the HEH 11  mode at corresponding D/L ratios, while curve  34  represents the same for the HEE 11  mode. Curve  36  represents the resonant frequencies of the TEH mode at corresponding D/L ratios. It is observed in the mode chart  30  that curves  32  and  34  intersect at point  38 , representing a particular D/L ratio of the full cylindrical dielectric resonator  1  for which the respective resonant frequencies of the dual HEH 11  and HEE 11  modes are equal. In other words, the intersection point  38  represents a D/L ratio for which the dual HEH 11  and HEE 11  modes resonate at a common resonant frequency. The exact D/L ratio for which this relationship holds will vary depending on the selected dimensions of the resonator and cavity. But in general, for a full cylindrical dielectric resonator of a given diameter in free space, there will exist only one unique D/L ratio for which the two dual modes will resonate at a common resonant frequency. 
     Qualitatively, the resonant frequency of a mode can be inversely related to the length of the circulating E field for that mode. Shorter circulation paths correlate with higher resonant frequencies. As the E field in the HEH 11  mode circulates horizontally parallel to the circular surfaces  2 , its path length is strongly dependent on the diameter D, but largely independent of the length L. In contrast, the E field in the orthogonal HEE 11  mode circulates vertically, and thus its path length has a strong dependency on both the diameter D and the length L of the cylinder. Sizing of the length L therefore has an appreciable affect only on the resonant frequency of the HEE 11  mode, while sizing of the diameter D, though some effect will be seen in the resonant frequency of HEE 11  mode, has a proportionately greater effect on the resonant frequency of the HEH 11  mode. These relative dependencies on the dimensions of the cylinder are reflected in the different slopes of curves  32  and  34 , and thus also account for intersection point  38 . Analytic models and mode charts, refined with full wave solvers, may be used for precise determination of the D/L ratio, and corresponding common resonant frequency, at intersection point  38 . It will be appreciated however that setting D/L˜2 provides a good starting estimate for the computation, and that the exact D/L ratio will typically be slighter greater than 2. 
     By solving the D/L ratio at which the two dual modes of the full cylinder resonate at a common frequency, the full cylindrical dielectric resonator  1  can be sized for operation as a quadruple-mode resonator. Of course, it should be appreciated that only the D/L ratio is fixed for quad-mode operation and that the absolute values of D and L remain to be selected (so long as their ratio is preserved) in the design process based on a selected operating frequency. The four modes of the cylindrical quad-mode resonator then correspond to the dual HEH 11  and HEE 11  resonant modes. As these modes are eigenmodes of the structure, and thus orthogonal, the field distributions of the four modes theoretically do not interact or couple. Independent or near independent control over the four modes (coupling, tuning, etc.) is therefore possible. But unlike prior realizations of quad-mode filters, one constructed using a full cylinder dielectric resonator  1  sized for operation in a quad-mode will offer considerable size reductions and have comparatively less complex coupling and tuning mechanisms. Fabrication is simplified as well because cylindrical dielectric resonators with custom height and diameter are widely available commercially. Size reductions are seen equally in single-cavity, 4-pole filters, as in higher order, 4n-pole filters. Size reductions can be achieved for dual-mode filters by extending the quad-mode concept of the full cylinder to the half-cut cylinder. 
     Reference is now made to  FIG. 4B , which is a mode chart for the half-cut dielectric resonator of  FIG. 1B  as a function of diameter-to-length (D/L) ratio. The mode chart  40  also plots frequency (GHz) against diameter-to-length (D/L) ratio, and is generated for a half-cut cylinder (D=0.9 in, ∈ r =45) located in a 1×1×1 in 3  cavity. The length L of the cylinder is again the free variable. Curve  42  represents the resonant frequency of the ½HEH 11  mode for corresponding D/L ratios, while curve  44  represents the same for the ½HEE 11  mode. Curve  46  represents the resonant frequency of a ½TME mode of the half-cut dielectric resonator  10  for corresponding D/L ratios. It is similarly observed in the mode chart  40  that curves  42  and  44  intersect at point  48 , representing a particular D/L ratio of the half-cut dielectric resonator  10  for which the ½HEH 11  and ½HEE 11  modes resonate at a common resonant frequency. The exact D/L ratio for which this relationship holds will again vary depending on selected dimensions of the resonator and cavity, though again there will in general exist only one unique D/L ratio for which the two modes will resonate at a common frequency. 
     It can also be observed that curves  42  and  44  trace out lower order modes than curve  46 . In other words, over the whole range of D/L ratios, the ½HEH 11  and ½HEE 11  resonate at a lower frequency than the ½TME mode, which confirms that the former are the first two eigenmodes of the half-cut cylindrical structure. Of course, the relative ordering of the ½HEH 11  and ½HEE 11  modes depends on the selected D/L ratio of the half-cut cylinder. Each of the ½HEH 11  and ½HEE 11  modes can constitute either the first or the second eigenmode. Similar trends are observed in the mode chart  30 , except that the HEH 11  and HEE 11  modes constitute second and third eigenmodes of the structure. The TEH mode that does not appear in the half-cut cylinder (because its E fields circulate in an azimuthal plane) constitutes the first eigenmode of the full cylinder. 
     As with the full cylinder, resonant frequency is qualitatively related to the length of the circulating E field in a particular mode. Like the HEH 11  and HEE 11  modes, the ½HEH 11  and ½HEE 11  modes of the half-cut cylinder have relative dependencies on the diameter D and length L. The horizontally circulating E field in the ½HEH 11  remains strongly dependent on the diameter D and largely independent of the length L, while the E field in the orthogonal ½HEE 11  mode retains its strong dependency on both these dimensions. Sizing the length L therefore again predominantly influences the resonant frequency of the ½HEE 11  mode, while sizing of the diameter D predominantly influences the resonant frequency of the ½HEH 11  mode, and thus account for the intersection point  48 . Analytic models and mode charts, refined with full wave solvers, again may be used to determine intersection point  48  exactly. But because the rectangular surface  18  provides a relatively good magnetic wall boundary, as with the full cylinder, setting D/L˜2 still provides a good starting estimate for the computation and the exact D/L ratio will still typically be greater than 2. 
     When the diameter D and length L are appropriately selected so that the ½HEH 11  and ½HEE 11  modes resonate at a common resonant frequency, the half-cut cylindrical dielectric resonator can be operated as a dual-mode resonator in a dual-mode filter. Since the two modes are eigenmodes of the structure, their E field distributions are orthogonal and can coexist within the structure without appreciable interaction or coupling. The center frequency of the dual-mode filter will be set by the common resonant frequency of the ½HEH 11  and ½HEE 11  modes. A dual-mode filter realized in this way using an appropriately sized half-cut cylindrical resonator is unlike other realizations of dual-mode filters insofar as the two resonant modes are provided by a single physical resonator and have completely different field distributions. Other realizations of dual-mode filters involve two physically separate resonators resonating in the same mode (i.e. two parallel coupled resonators) or else one physical resonator operating in a degenerate mode. A good example of the latter is the dual HEH 11  or dual HEE 11  modes of the full cylindrical dielectric resonator  1 . Considerable size reductions can be achieved by using the half-cut dielectric resonator  10  operating in a dual-mode instead. Simplified coupling schemes are also made possible by the relative orthogonality of the dual-mode. 
     Although the half-cut dielectric resonator  10  can be made to operate as a dual-mode resonator through appropriate sizing of its D/L ratio, it is possible also to select other D/L ratios in order to synthesize other classes of microwave filters. Accordingly, in some embodiments, the D/L ratio of the half-cut dielectric resonator  10  is selected so that the ½HEH 11  resonates at a first resonant frequency (hereinafter “f H ”), while the ½HEE 11  mode resonates at a second resonant frequency (hereinafter “f E ”) different from the first resonant frequency. By this selection of D/L ratio, the half-cut dielectric resonator  10  can operate as a dual-band resonator for use in a dual-band filter. The two bands of the dual band filter will be carried by the corresponding different resonant modes of the half-cut dielectric resonator  10 . One of the dual bands is thus supported by the ½HEH 11  mode and has center frequency f H , while the other of the two bands is supported by the ½HEE 11  mode and has center frequency f E . Accordingly, the centre frequencies of the dual bands will correspond to the separate resonant frequencies of the ½HEH 11  and ½HEE 11  modes. 
     It is evident from  FIG. 4B  that the resonant frequencies of the ½HEH 11  and ½HEE 11  mode switch relative magnitudes at the intersection point  48 . For the range of D/L ratios below intersection point  48 , f H  is greater than f E , while for the range of D/L ratios above intersection point  48 , f H  is less than f E . Qualitatively, starting from intersection point  48 , where f H =f E , reducing the length L (for a given diameter D) tends to produce a sharp increase in f E , but only a slight increase in f H , thereby creating frequency separation. The same effect will be achieved alternatively by reducing the diameter D (for a fixed length L), which tends to decrease both f H  and f E , but at a faster rate with respect to f E . Accordingly, by appropriate selection of the D/L ratio of the half-cut dielectric resonator, either f H  or f E  can be set larger than the other. Either of the two bands in the realized dual band filter can therefore be carried by either the ½HEH 11  or ½HEE 11  resonant modes. 
     A dual band filter may generally be defined, among other parameters, by the center frequencies of its two bands, f H  and f E , and their frequency separation, Δf=|f H −f E |. By appropriate selection of the diameter D and length L of the half-cut dielectric resonator  10 , the filter parameters f H , f E , Δf can be designed according to meet specification. It should again be appreciated that the diameter D and length L are independent variables. Consequently, f H , f E  and Δf will generally depend, not just on the D/L ratio, but also on their absolute values. Full sweeps of both variables may therefore be required when designing a dual-band filter using half-cut dielectric resonators to meet specifications. As above, analytic models and mode charts, refined with full wave solvers, if necessary, may be used to solve values for D and L that will realize the desired filter specifications (e.g. f H , f E , Δf). 
     When designing and synthesizing microwave filters, such as dual-mode, quad-mode or dual-band filters, it is generally desirable to be provided with independent, or near independent, control over each resonant mode. Many filter synthesis techniques require independent control over resonant mode coupling and tuning for proper placement of the filter&#39;s transmission zeros as a separate step once the resonators have been designed for proper placement of the filter&#39;s poles. Filter synthesis is greatly complicated where independent control over the resonant modes is lacking. The full cylindrical or half-cut dielectric resonators discussed herein largely avoid this complication because each operating resonant mode of these structures is also an eigenmode and thus orthogonal. That property of the full and half-cut dielectric resonators is exploited to realize controllable, effective and relatively straightforward coupling mechanisms for microwave filters, including inter-cavity mode coupling, intra-cavity mode coupling, and input-output mode coupling. Each of these coupling mechanisms, it should be appreciated, is necessary for advanced microwave filter synthesis. In the discussion to follow, these and other aspects of dielectric resonator filters and multiplexers realized using full cylindrical or half-cut dielectric resonators are explained in greater detail. 
     Reference is now made to  FIGS. 5A-5D , which illustrate perspective views of exemplary inter-cavity couplings of two half-cut dielectric resonator assemblies, according to aspects of embodiments of the present invention. As seen, for example, in  FIGS. 5A and 5B , resonator cavity  50   a  encloses half-cut dielectric resonator  10   a . Preferably, resonator cavity  50   a  comprises a metallic housing and provides electromagnetic shielding. The half-cut dielectric resonator  10   a  is of a selected D/L ratio, as described above, for operation as either a dual-mode or dual-band resonator, and is planar mounted on mounting support  52   a  formed from a unitary piece of suitable low-permittivity dielectric substrate (e.g. ∈ r ≦10). For example, the mounting support  52   a  is formed of Teflon. Resonator cavity  50   b  is located adjacent to resonator cavity  50   a  and encloses half-cut dielectric resonator  10   b  planar mounted on mounting support  52   b  formed in a unitary piece of the same low-permittivity dielectric substrate. In some embodiments, half-cut dielectric resonators  10   a ,  10   b  have the same selected dimensions. In other embodiments, however, these dimensions may differ. Resonator cavities  50   a ,  50   b  also have the same dimensions in some embodiments, and different dimensions in some embodiments. 
     A suitable aperture or iris defined in the common wall between resonator cavities  50   a ,  50   b  is used to couple either or both resonant modes of half-cut dielectric resonator  10   a  to corresponding resonant modes of the half-cut dielectric resonator  10   b . The general shape of the aperture determines the resonant mode or modes that are coupled, and its size determines the amount of coupling. This result is intuitive by considering that the aperture behaves like a waveguide subject to cutoff, which consequently passes only one field polarization. The polarization of a resonant mode is therefore a relevant factor in selecting the shape and size of the aperture, and polarization-discriminant apertures can be designed for each resonant mode of the half-cut dielectric resonator  10 . 
     The horizontal iris  54  shown in  FIG. 5A  couples the ½HEE 11  mode, while substantially rejecting the ½HEH 11  mode. Opposite to this action, the vertical iris  56  shown in  FIG. 5B  couples the ½HEH 11  mode, while substantially rejecting the ½HEE 11  mode. Alternatively, the cross-shaped iris  58  shown in  FIGS. 5C and 5D , which includes both a horizontal and a vertical iris component, couples and provides largely independent control over both the ½HEH 11  and ½HEE 11  resonant modes. The vertical component of cross-shaped iris  58  couples the ½HEH 11  mode ( FIG. 5C ), while the horizontal component couples the ½HEE 11  mode ( FIG. 5D ). The dimensions of each component of cross-shaped iris  58  can be independently varied to provide essentially independent control over the amount of coupling of each respective resonant mode. For greater clarity, the vertical component of cross-shaped iris  58  can be sized to provide a desired amount of coupling of the ½HEH 11  mode, and the horizontal component of cross-shaped iris  58  can be sized to provide a desired amount of coupling to the ½HEE 11  mode. The respective dimensions of the vertical and horizontal components do not necessary have to be same. One mode can therefore be coupled by a greater amount than the other, if desired. As an alternative to cross-shaped iris  58 , one diagonally slanted iris (not shown) may be used to couple both resonant modes simultaneously. In general, any suitably shaped inter-cavity aperture may be used to couple resonant modes of adjacent half-cut dielectric resonators. 
     The coupling coefficient of two adjacent resonators can be determined according to different approaches. One approach is to solve the frequencies of the first two eigenmodes of the full-coupled structure. The coupling coefficient is then given by 
                     k   ≈         f   2     -     f   1         f   2         ,           (   1   )               
where f 1  and f 2  are the first and second resonant frequencies of the full-coupled structure. This approach can be extended for the case of a dual-band filter by solving the frequencies of the first four eigenmodes of the full-coupled structure. The coupling coefficient of the lower band is given by Eq. 1, and the coupling coefficient of the upper band is similarly given by
 
                       k   ′     ≈         f   4     -     f   3         f   4         ,           (   2   )               
where f 3  and f 4  are the resonant frequencies of the third and fourth eigenmodes of the full-coupled structure.
 
     In an alternative approach, computational complexity can be reduced by exploiting symmetry in the full-coupled structure and employing even-odd mode analysis. A symmetry plane is placed half way between the two resonators through the middle of the cross-shaped iris  58 . The symmetry plane simulates an ideal magnetic wall in even-mode analysis and an ideal electric wall in odd-mode analysis. The coupling coefficient, k, is then given by 
                     k   =         f   e   2     -     f   m   2           f   e   2     +     f   m   2           ,           (   3   )               
where f m  and f e  are the even-mode and odd-mode resonant frequencies of the full-coupled structure, respectively. The same calculation can be performed to determine the coupling coefficient, k′, for the upper band of a dual-band.
 
     Yet another approach to determining coupling coefficients is the S-parameter approach (e.g. described in R. Cameron, C. Kudsia &amp; R. Mansour,  Microwave Filters for Communication Systems . Hoboken, N.J.: John Wiley &amp; Sons, Inc., 2007). The inter-cavity aperture is modeled as a discontinuity between two transmission lines (corresponding to the two resonator cavities). The coupling coefficient, k, can then be determined by transforming the solved S-parameters of the waveguide discontinuity into an equivalent T-network comprising a shunt impedance inverter. The coupling coefficient is then derived from the inverter impedance. 
     Once the coupling coefficient, k, has been determined, for example using one of the above-described approaches, dimensions for the inter-cavity aperture (width, height, thickness) can be swept in order to design a suitable iris  54 ,  56 ,  58  that provides the desired amount of inter-cavity coupling of adjacent resonators. Clearly this procedure can be repeated for a plurality of adjacent resonator cavities inter-connected by apertures. The coupling-matrix approach to filter synthesis (described in  Microwave Filters ) would then involve designing each iris in the synthesized filter to provide the required amount of coupling as specified in M matrix derived under that approach. Advanced filter synthesis is greatly simplified by the largely independent control over inter-cavity coupling provided by the half-cut dielectric resonator  10 . 
     Reference is now made to  FIGS. 6A-6B , which illustrate top and perspective views of an exemplary half-cut dielectric resonator assembly with intra-cavity mode coupling, according to aspects of embodiments of the present invention. Similar to before, half-cut cylindrical dielectric resonator  10  is mounted on mounting support  52  inside resonator cavity  50  so as to not directly contact the inner walls of resonator cavity  50 , which comprises a metallic housing and provides electromagnetic shielding. The mounting support  52  is again formed from a unitary piece suitable low-permittivity dielectric substrate. 
     Screw  60  is fastened to an inner wall of the resonator cavity  50  and projects interiorly into the cavity. In the presence of electromagnetic fields, and depending on its location, screw  60  attracts fields of one resonant mode and causes them to leak over into other resonant modes, thereby providing a mechanism for intra-cavity coupling of resonant modes. It should be appreciated that screw  60  is formed out of metal in some embodiments, but that other materials may be substituted in other embodiments. When fastened directly to the inner walls of the resonator cavity  50 , metals screws can sometimes give rise to unwanted propagation of a coaxial mode within the resonator cavity  50 . To suppress this spurious resonance mode, therefore, a dielectric-metal screw can be used instead of a metal screw so that direct metal-to-metal contact with the inner wall of the resonator cavity  50  is avoided. It should also be appreciated that the shape of screw  60  is variable, and that rods, poles and other general forms of projections of varying lengths and widths may be substituted. 
     Screw  60  offers a convenient and controllable mechanism for coupling the orthogonal ½HEH 11  and ½HEE 11  modes of the half-cut dielectric resonator  10 . As eigenmodes of the structure, the natural field distributions of ½HEH 11  and ½HEE 11  modes do not appreciably interact or couple. However, a screw  60  located appropriately within the resonator cavity  50  will disturb the natural field distributions of ½HEH 11  and ½HEE 11  modes simultaneously, and thereby couple these two orthogonal and otherwise non-interactive modes. Areas within resonator cavity  50  in which the E fields of both the ½HEH 11  and ½HEE 11  mode are concentrated provide suitable locations for the screw  60 . At these locations, corresponding interactive E fields will be created in the screw  60 , the effect of which is to couple the two resonant modes. However, as will be described in more detail below, the amount of coupling is variable depending on the dimensions, as well as the location and orientation, of the screw  60 . 
     Screw  60  can also be located within the resonator  50  so that only the field distributions of one resonant mode of the half-cut dielectric resonator  10  are substantially perturbed. To the field distributions of the other resonant mode, the screw  60  will appear non-existent. Screw  60  can therefore be located so as to perturb the field distributions of the ½HEH 11  mode only, while the ½HEE 11  mode largely unaffected; and likewise, so as to perturb the field distributions of the ½HEE 11  mode only, while leaving the ½HEH 11  mode largely unaffected. Perturbing the field distributions of a resonant mode will cause a small shift in the resonant frequency of that mode, either up or down, which may be useful to tune the resonant frequency of that mode. Often tuning screws are required to tune the resonant frequency of a cavity to its designed centre frequency. Exactly sized resonators are normally hard to achieve and some tolerance in the resonator&#39;s dielectric constant should be expected. Thus a practical resonator will often not realize its designed centre frequency without the aid of tuning screws. It should be appreciated, however, that the centre frequency is still predominantly determined by the dimensions of the resonator and cavity, and that tuning screws only provide a mechanism for making slight corrections in order to re-align the resonator&#39;s centre frequency with its designed value. 
     Reference is now made to  FIGS. 6C and 6D , which illustrate front and top views of an exemplary half-cut dielectric resonator assembly with intra-cavity coupling and tuning, according to aspects of embodiments of the present invention. Resonator cavity  50  encloses half-cut dielectric resonator  10 , which is again planar mounted on mounting support  52 . Fastened to the inner walls of resonator cavity  50  are coupling screw  62  and tuning screws  64 ,  66 . Coupling screw  62  is located diagonally offset and adjacent to the upper straight edge  20  of half-cut dielectric resonator  10 . In this location, coupling screw  62  couples the ½HEH 11  and ½HEE 11  resonant modes. 
     The amount of intra-cavity resonant mode coupling provided by coupling screw  62  is variable depending its dimensions and location. For example, the distance and angle of the coupling screw  62  relative to the upper straight edge  20  affect the amount of coupling provided. Moving the coupling screw  62  diagonally further away from the half-cut resonator  10  will tend to decrease the amount of coupling provided, and vice versa. Moving the coupling screw  62  horizontally toward the centre of semi-circular surface  12  or vertically toward the centre of rectangular surface  18  will also tend to decrease the amount of coupling provide as the field distributions in these locations tend to be concentrated in one or the other resonant mode only. Accordingly, field mode interaction decreases in both directions. Good coupling of the ½HEH 11  and ½HEE 11  resonant modes is achieved by locating the coupling screw  62 , as shown in  FIG. 6C , just diagonally offset from and adjacent to the half-cut resonator  10 , where the field distributions of these two resonant modes are more than just weakly interactive. 
     In addition to its location and orientation within the resonator cavity  50 , the dimensions of coupling screw  62  also affect the amount of intra-cavity resonant mode coupling provided by coupling screw  62 . Coupling can generally be increased by providing longer and thicker couplings screws. 
     Tuning screw  64  is positioned above the centre of semi-circular surface  12  and tuning screw  66  is positioned adjacent the centre of curved surface  14 . As there is no more than weak interaction between the ½HEH 11  and ½HEE 11  modes in these locations, tuning screws  64 ,  66 , unlike coupling screw  62 , do not provide an appreciable amount of intra-cavity mode coupling. Instead tuning screws  64 ,  66  provide largely independent tuning of the ½HEE 11  and ½HEH 11  modes, respectively. The field distribution of the ½HEE 11  mode is concentrated above the centre of semi-circular surface  12  where tuning screw  64  is located. Accordingly, tuning screw  64  is used to tune the resonant frequency of the ½HEE 11  mode. Likewise, tuning screw  66  is located adjacent the centre of curved surface  14 , where the field distribution of the ½HEH 11  mode is concentrated, and serves the same purpose for the ½HEH 11  mode. Independent or near independent resonant mode tuning is possible because the orthogonal field mode distributions of the two resonant modes are relatively non-interactive in the vicinity of each tuning screw  64 ,  66 . 
     Reference is now made to  FIGS. 6E and 6F , which illustrate perspective views of exemplary half-cut dielectric resonator assemblies with intra-cavity coupling, according to aspects of embodiments of the present invention. Coupling screw  62  (shown again  FIG. 6E ) is located as before diagonally offset from the upper straight edge  20  of the half-cut dielectric resonator  10 . Coupling screw  68  however has been shifted laterally across the semi-circular surface  12  to the other side of the half-cut dielectric resonator  10 , where it is positioned diagonally offset from the curved edge  16 . Shifting the location of the coupling screw from one side of the half-cut dielectric resonator  10  to the other reverses the polarity of the coupling. As indicated by the directions of the white and grey arrows, leakage from the ½HEE 11  mode (grey arrow) into the ½HEH 11  mode (white arrow) circulates in one direction for coupling screw  62  and the opposite direction for coupling screw  68 . It should be appreciated that moving the coupling screw  62  down toward the lower straight edge  20  of the half-cut dielectric resonator  10  will also reverse the polarity of the coupling relative to that reference location. Both positive and negative mode coupling of the half-cut dielectric resonator  10  are thus possible, when two of such cavities are coupled via an appropriate iris. Having control over the polarity of the cross-coupling can be important for the proper placement of transmission zeros in the realized filter, as discussed in greater detail below. 
     The same process followed for determining the coupling coefficient with respect to inter-cavity mode coupling can be followed as well for intra-cavity mode coupling. Joint simulation of the half-cut dielectric resonator  10 , resonator cavity  50  and coupling screw  62  using an eigenmode solver can be used to solve the first two resonant frequencies of the coupled structure. Tuning screws  64 ,  66  may be omitted from the simulation as they compensate for non-ideal effects in real resonators. The coupling coefficient, k, is then given again by Eq. 1. If desired, the coupling coefficient, k′, can also be solved according to Eq. 2. It should be appreciated that even-odd mode analysis may not be available here due to lack of symmetry in the resonator cavity  50 . S-parameter analysis may be performed but with added complexity as coupling here is between two resonant modes of a single physical resonator. Once the coupling coefficient, k, has been determined, parameters of the coupling screw  62  (length, diameter, etc.) can be swept using an appropriate solver (and, if necessary, interpolated) in order to design a coupling screw that provides the desired amount of intra-cavity coupling. This procedure can be repeated as required in the coupling matrix approach to filter synthesis. 
     Reference is now made to  FIGS. 7A and 7B , which illustrate top and perspective views of an exemplary half-cut dielectric resonator filter assembly with input-output coupling, according to aspects of embodiments of the present invention. Input and output mode coupling can be provided using a similar arrangement as the coupling screw  62  used to provide intra-cavity mode coupling. An electromagnetic probe  70  is fed through a small opening in one of the walls of resonator cavity  50  to project interiorly into resonator cavity  50  in like fashion to coupling screw  62 . External connector  72  is in electrical contact with electromagnetic probe  70  and is used to make a connection with an external coaxial cable or other transmission medium for microwave and RF signals. The half-cut dielectric resonator  10  is again planar mounted on mounting support  52  inside resonator cavity  50  so that half-cut dielectric resonator  10  is not in direct contact with the inner walls of resonator cavity  50 . 
     Depending on the location and orientation of electromagnetic probe  70 , one of the ½HEH 11  and ½HEE 11  modes can be coupled to the external connector  72  independently of the other mode. Alternatively both the ½HEH 11  and ½HEE 11  modes can be coupled simultaneously to the external connector  72 . The location and orientation of electromagnetic probe  70  within the resonator cavity  50  affects the amount of coupling of each resonant mode. In general, the electromagnetic probe  70  will couple a resonant mode of the half-cut dielectric resonator  10  when the field distribution of that resonant mode is concentrated in the immediate vicinity. Simultaneous coupling of both the ½HEH 11  and ½HEE 11  modes is achieved by locating the electromagnetic probe  70  diagonally away from the upper straight edge  20  of the half-cut dielectric resonator  10 . As with the coupling screw  62 , the field distributions of both resonant modes are concentrated in this area. Moving the electromagnetic probe  70  diagonally closer to or away from the straight edge  20  again will increase or decrease the amount coupling of the ½HEH 11  and ½HEE 11  modes. 
     The orthogonality of the ½HEH 11  and ½HEE 11  resonant modes permits electromagnetic probe  70  to be located so as to selectively couple only one resonant mode independently of the other. As illustrated in  FIG. 7B , for example, electromagnetic probe  70  is parallel to and adjacent to the centre of the rectangular surface  18  where the field distribution of the ½HEH 11  mode is concentrated. In that location, electromagnetic probe  70  couples the ½HEH 11  mode, while isolating the ½HEE 11  mode. A similar result is achieved by locating the electromagnetic probe  70  adjacent the centre of the curved surface  14  on the other side of the half-cut dielectric resonator  10  (where tuning screw  66  is shown in  FIG. 6C ), but subject to polarity reversal. On the other hand, by locating the electromagnetic probe  70  parallel to and above the centre of the semi-circular surface  12  (where tuning screw  64  is shown if  FIG. 6C ), the ½HEE 11  mode will be coupled, while the ½HEH 11  mode will be isolated. Only the field distribution of the ½HEE 11  mode is concentrated in that area of the cavity  50 . Locating the electromagnetic probe  70  in intermediate positions is also possible and will achieve some unbalanced coupling of each resonant mode. 
     Reference is now made to  FIG. 7C , which illustrates a perspective view of another exemplary half-cut dielectric resonator filter assembly with input-output coupling, according to aspects of embodiments of the present invention. Different orientations of the electromagnetic probe  70 , relative to the half-cut dielectric resonator  10 , can also be used to provide increased mode isolation. Electromagnetic probe  70   a  is oriented horizontally, similar to electromagnetic probe  70  in  FIGS. 7A and 7B , for coupling the ½HEH 11  mode to external connector  72   a . However, the electromagnetic probe  70   b  is oriented vertically, as opposed to horizontally, for coupling the ½HEE 11  mode to external connector  72   b . When coupling the ½HEE 11  mode to the external connector  72   b , orienting the electromagnetic probe  70   b  vertically adjacent to the curved surface  14 , as opposed to horizontally above the semi-circular surface  12 , better isolates of the ½HEH 11  mode. For that particular orientation, the field distributions of the ½HEH 11  mode are even less interactive. Output mode isolation is a potentially relevant design consideration in single cavity resonator filters (where input and output channels are located in the same physical cavity) as well as diplexers and higher order multiplexers (where multiple output channels may be located in the same physical cavity). 
     In addition to its location and orientation with resonator cavity  50 , similar to the coupling screw  62 , the dimensions (length, thickness) of electromagnetic probe  70 ,  70   a ,  70   b  affect the amount of input-output coupling of half-cut dielectric resonator  10 . Longer and thicker tend to achieve greater mode coupling. Full wave solvers, may be used to solve dimensions and an orientation for the electromagnetic probe  70 ,  70   a ,  70   b  to achieve a desired amount of input/output coupling according to design specifications. 
     Reference is now made to  FIGS. 8A and 8B , which illustrate top and perspective views of another exemplary half-cut dielectric resonator assembly with input-output coupling, according to aspects of embodiments of the present invention. As an alternative to the electromagnetic probe  70 , shown in  FIGS. 7A and 7B , input and output mode coupling can be provided instead by a waveguide aperture  80  connecting resonator cavity  50  to input waveguide  82 . Previous discussion in the context of polarization discriminant irises for providing inter-cavity coupling applies also to waveguide aperture  80 , and thus will not be repeated in detail. To reiterate, by including a predominantly vertical component (as shown) in the waveguide aperture  80 , the ½HEH 11  mode will be coupled, while substantially isolating the ½HEE 11  mode. Alternatively, by including a predominantly horizontal component, the ½HEE 11  mode will be coupled, while substantially isolating the ½HEH 11  mode. Alternatively, where the waveguide aperture  80  includes both a substantial horizontal component and a substantial vertical component, such as when waveguide aperture  80  is approximately square-shaped, both the ½HEH 11  and ½HEE 11  modes will be coupled to the input waveguide  82 . Other configurations and shapes for the waveguide aperture  80  are possible as well. The amount of input-output coupling is determined by the dimensions (height, width, thickness, etc.) and orientation of the waveguide aperture  80 . Analytic models and mode charts, refined with full wave solvers, may be used to solve its dimensions to meet design specifications. 
     Reference is now made to  FIGS. 9A-9D , which schematically illustrate exemplary coupling schemes for a 4-pole dielectric resonator filter, according to aspects of embodiments of the present invention. The above-described inter-cavity, intra-cavity and input-output mode coupling mechanisms provide the necessary elements for synthesizing advanced coupling schemes for dielectric resonator filters. Coupling schemes for both straight and folded resonator configurations are achievable.  FIGS. 9A-9C  illustrate some exemplary coupling schemes for a 4-pole dielectric resonator filter, in which: S designates the source, L designates the load, and R 1 -R 4  designate four resonators located in cavities C 1  and C 2 . More specifically, cavity C 1  encloses a first half-cut dielectric resonator whose ½HEH 11  and ½HEE 11  modes respectively provide resonators R 1  and R 2 , while cavity C 2  encloses a second half-cut dielectric resonator whose ½HEE 11  and ½HEH 11  modes respectively provide resonators R 3  and R 4 . Accordingly, resonators R 1  and R 4  resonate in the same mode, as do resonators R 2  and R 3 . Cavities C 1 , C 2  are also located in close physical proximity to allow for inter-cavity coupling using an appropriate inter-cavity aperture. 
     The coupling scheme illustrated in  FIG. 9A  corresponds to a folded 4-pole dielectric resonator filter. Input coupling (S-R 1 ) and output coupling (R 4 -L) are realized using appropriately positioned electromagnetic probes  70  that couple the ½HEH 11  mode of resonators R 1  and R 4 , respectively, while isolating the ½HEE 11  modes. For example, electromagnetic probes  70  can be aligned horizontally adjacent to the centre of rectangular surface  18  of the half-cut dielectric resonator  10 . Intra-cavity mode coupling (R 1 -R 2  and R 3 -R 4 ) is realized using appropriately positioned coupling screws  62 , for exampled aligned diagonally adjacent to the upper straight edge  20  of each half-cut dielectric resonator  10 . Inter-cavity mode coupling (R 2 -R 3 ) is achieved using a suitably shaped iris that couples the ½HEE 11  mode of R 2  and R 3 , while rejecting the ½HEH 11  mode. A horizontal iris  54  of selected dimensions for example would be appropriate. According to this exemplary coupling scheme, resonators R 1 -R 4  are coupled as in a folded 4-pole dielectric resonator. 
     As the resonators R 1 -R 4  are arranged in C 1 , C 2  in folded formation, additional mode cross-couplings (dotted lines) can be introduced in order to realize more advanced filters. These additional available cross-couplings may be useful, for example, to control placement of transmission zeros. The exemplary coupling scheme shown in  FIG. 9B  corresponds to the folded 4-pole coupling scheme of  FIG. 9A , but with additional input cross-coupling (S-R 2 ) and output cross-coupling (R 3 -L). By adjusting the location of the electromagnetic probe  70  in cavity C 1 , the source S can couple both the ½HEH 11  and ½HEE 11  modes of the first half-cut dielectric resonator  10  used to realize R 1  and R 2 . Likewise by adjusting the location of the electromagnetic probe  70  in cavity C 2 , the load L can couple both the ½HEE 11  and ½HEH 11  modes of the second half-cut dielectric resonator  10  used to realize R 3  and R 4 . For example, the electromagnetic probes may be moved closer to the respective upper straight edges  20  of the first and second half-cut dielectric resonator. 
     Inter-cavity cross-coupling of adjacent resonators is possible as well. The exemplary scheme shown in  FIG. 9C  corresponds to the coupling scheme of  FIG. 9B , but with additional inter-cavity mode cross-coupling (R 1 -R 4 ). By using a suitable cross-shaped iris  58 , rather than a horizontal iris  54 , in between cavities C 1  and C 2 , each of the ½HEH 11  and ½HEE 11  modes of the first and second half-cut dielectric resonators  10  can be coupled, thereby realizing the exemplary scheme shown in  FIG. 9C . Sizing the vertical and horizontal components of the cross-shaped iris  58  can achieve different amounts of couplings of each resonant mode. It should be appreciated that changing the location of an electromagnetic probe or coupling screw or the shape of an inter-cavity aperture are independently controllable and independently affect the amount of cross-coupling that is achievable in the exemplary coupling schemes. These different coupling mechanisms are essentially non-interactive. 
     Alternatively,  FIG. 9D  illustrates a dual-branch coupling scheme that is also realizable by the inter-cavity, intra-cavity and input-output coupling mechanisms for the half-cut dielectric resonator filter  10 . Such a dual-branch coupling scheme provides for effective, controllable and relatively straightforward synthesis of a dual-band filter, wherein the two bands in the dual band are carried by different resonance modes. As in  FIGS. 9A-9C , resonators R 1  and R 4  resonate in the ½HEH 11  mode, while resonators R 2  and R 3  resonate in the ½HEE 11  mode, or vice versa. Cavities C 1 , C 2  are also located in close physical proximity to allow for inter-cavity coupling using an appropriate inter-cavity aperture. 
     Input coupling (S-R, S-R 2 ) is realized using an electromagnetic probe  70  in cavity C 1  that couples both the ½HEH 11  and ½HEE 11  modes simultaneously. Similarly output coupling (R 3 -L, R 4 -L) is realized using an electromagnetic probe  70  in cavity C 2  that couples both the ½HEH 11  and ½HEE 11  modes simultaneously. For example, the electromagnetic probes  70  may be located diagonally adjacent the upper straight edge  20  of each respective half-cut dielectric resonator  10 . As each band is carried by a resonator pair resonating in different resonant modes, inter-cavity mode coupling (R 1 -R 4 , R 2 -R 3 ) is provided by a suitable aperture that couples both the ½HEH 11  and ½HEE 11  modes simultaneously, e.g. cross-shaped aperture  58  of selected dimensions. No coupling screws  62  are included in this scheme because no intra-cavity cross-coupling of resonant modes (R 1 -R 2  and R 3 -R 4 ) is needed in the dual-branch scheme. Any number of tuning screws  64 ,  66  could also be included if desired. 
     Reference is now made to  FIG. 9E , which schematically illustrates exemplary coupling schemes for an 8-pole, dielectric resonator filter, according to aspects of embodiments of the present invention. It is evident that the possible coupling schemes for dielectric resonator filters realized using half-cut dielectric resonator  10  can be generalized for any straight or folded 2N-pole, dual-mode filter (or alternatively any straight or folder N-pole, dual-band filter). It should be appreciated that the order of a dual-mode filter constructed from half-cut dielectric resonators  10  will be twice the number of resonators in the realized filter as each operates in a dual-mode, just as the order of a dual-band filter constructed from half-cut dielectric resonators  10  will equal the number of resonators in the realized filter as each operates in a dual-band. 
     All possible couplings and cross-couplings that are achievable for an 8-pole dielectric resonator filter realized using half-cut dielectric resonators  10  are shown in  FIG. 9E . Each cavity C 1 -C 4  encloses a single physical resonator that realizes two resonators in different resonant modes. Specifically, resonators R 1  and R 2  are realized by a first half-cut dielectric resonator in cavity C 1 , resonators R 3  and R 4  by a second half-cut dielectric resonator in cavity C 2 , resonators R 5  and R 6  by a third half-cut dielectric resonator in cavity C 3 , and finally resonators R 7  and R 8  by a fourth half-cut dielectric resonator in cavity C 4 . The solid connection lines (S-R 1 , R 1 -R 2 , R 3 -R 4 , R 4 -R  5 , R 5 -R 6 , R 6 -R 7 , R 7 -R 8 , R 8 -L) correspond to the direct couplings in a folded, 8-pole resonator, which also constitute all possible couplings in a straight, 8-pole resonator. The dashed connection lines (R 1 -R 8 , R 2 -R 7 , R 3 -R 6 ) correspond to cross-couplings that are possible for the folded, 8-pole resonator. The dotted connection lines (S-R 2 , R 1 -R 4 , R 5 -R 8 , R 7 -L) correspond to additional cross-couplings that are possible by the half-cut dielectric resonator  10  operating in a dual-mode. This generalized coupling scheme for an 8-pole, dual-mode filter can be extended for higher order dual-mode or dual-band filters. 
     Of course, it should also be appreciated that not every resonator pair can be cross-coupled. For example, resonators R 1 , R 7  although located in adjacent cavities C 1 , C 4  cannot be cross-coupled because resonators R 1 , R 7  are implemented by orthogonal resonant modes. Moreover, resonators R 1 , R 5  although implemented by parallel resonator modes cannot be cross-coupled because resonators R 1 , R 5  are not located in adjacent cavities. In general, orthogonal resonant modes located in the same cavity, as well parallel resonant modes located in adjacent cavities can be cross-coupled. All other resonator pairs cannot. The source and load can also be coupled to each orthogonal resonant mode in the first and last cavity, respectively. 
     As described herein, the full cylindrical and half-cut dielectric resonators, together with their associated coupling mechanisms, can be used to realize different classes of resonator filters. For example, the full cylindrical dielectric resonator can be used to realize quad-mode resonator filters, while the half-cut dielectric resonator can be used to realize dual-mode resonator filters. Each can also be used to realize dual-band resonator filters, as well as diplexers and higher-order multiplexers. Exemplary realizations of each of these classes of microwave filters will now be described. It should be appreciated, however, that the descriptions to follow are exemplary only and that other possible realizations are within the scope of the disclosure. 
     Reference is now made to  FIGS. 10A-10D , which show various views of exemplary single-cavity, 4-pole resonator filters synthesized using a full cylindrical dielectric resonator operating in a quad-mode, according to aspects of embodiments of the present invention. Dielectric resonator filter  100  comprises full cylindrical dielectric resonator  101  planar mounted on a cylindrical mounting support  152  inside cylindrical cavity  150 . The diameter D and length L of cylindrical dielectric resonator  101  are selected so that each component of the dual degenerate HEH 11  and HEE 11  modes resonates at a common resonant frequency, thereby providing quad-mode operation. The cylindrical cavity  150  has dimensions of diameter D c  and length L c . Mounting support  152  has diameter D s  and height L s  so that full cylindrical dielectric resonator  101  is axially centered within the cylindrical cavity  150  when mounted. It should be appreciated that full cylindrical dielectric resonator  101  is also mounted on mounting support  152  and is normally radially centered within cylindrical cavity  150 . 
     Input and output coupling are provided using electromagnetic probes  170   a  and  170   b , respectively, of length H p  and located a distance X p  away from the central axis of the cylindrical cavity  150 . Electromagnetic probe  170   a  is in electrical contact with external connector  172   a  and electromagnetic probe  170   b  is in electrical contact with external connector  172   b , and there is approximately 90 degrees of radial separation between the two electromagnetic probes  170   a ,  170   b . With that configuration, one component from each of the dual HEH 11  and HEE 11  mode pairs aligns with electromagnetic probe  170   a  on the input channel, and is thereby coupled to the external connector  172   a , while the other component from each of the two mode pairs aligns with electromagnetic probe  170   b  on the output channel, and is thereby coupled to the external connector  172   b . The amount of input and output mode coupling provided by electromagnetic probes  170   a ,  170   b  is determined predominantly by the length H p  and distance X p , which can be varied to provide different amounts of couplings, as needed, to meet design specifications for the filter  100 . 
     As shown in  FIG. 10A , electromagnetic probes  170   a ,  170   b  are inserted through small openings in the cylindrical cavity  150  from opposite ends, such that one projects upwardly and the other projects downwardly. In some embodiments, however, both electromagnetic probes  170   a ,  170   b  are located at the same end of the cylindrical cavity  150  to both project downwardly (or upwardly) into the interior of the cavity  150 . The dielectric resonator filter  100 ′ shown in  FIG. 10D  has this configuration of electromagnetic probes  170   a ,  170   b . The relative orientation of the electromagnetic probes  170   a ,  170   b  affects the number and location of transmission zeros of the realized filter. 
     Resonant mode coupling and tuning is achieved by inclusion of several tuning and coupling screws in dielectric resonator filter  100 . More specifically, screws  104  and  105  located opposite electromagnetic probe  170   a  couple the two mode components (one from each of the HEH 11  and HEE 11  mode pairs) that align with electromagnetic probe  170   b , as well as tune the resonant frequencies of these modes to the center frequency of the quad-mode filter. Likewise, screws  106  and  107  located opposite electromagnetic probe  170   b  couple the two other components of the degenerate HEH 11  and HEE 11  mode pairs that align with electromagnetic probe  170   b , as well as tune the resonant frequencies of these modes to center frequency of the quad-mode filter. Screws  108  and  109  located at 45 degrees from each electromagnetic probe  170   a ,  170   b  couple the two orthogonal mode components from each of the HEH 11  and HEE 11  degenerate mode pairs. This arrangement of coupling and tuning screws  104 - 109 , it should be appreciated, provides coupling of the dual HEH 11  and HEE 11  mode pairs for operation in a quad-mode. Other screw arrangements are also possible to realize the different mode couplings in the filter. 
     Screws  104 ,  106 ,  108  extend horizontally and radially outward from the circumferential surface of full cylindrical dielectric resonator  1  and are axially centered within the cylindrical cavity  150 , equidistant from the top and bottom walls of the cylindrical cavity  102 . Screws  105 ,  107 ,  109  extend vertically from either the bottom (shown) or top (not shown) of the cylindrical cavity  150  at a radial distance X s  away from the central axis of the cylindrical cavity  150 . The amount of tuning and resonant mode coupling provided by screws  104 - 109  is determined by their respective dimensions and locations within the cylindrical cavity  150 . Full wave solvers, may be used in the design and synthesis stages for the filter  100  in order to precisely determine the dimensions and locations of the screws  104 - 109  to meet design specifications. 
     Reference is now made to  FIGS. 11A and 11B , which show plots of reflection and transmission versus frequency for the single-cavity, 4-pole dielectric resonator filters of  FIGS. 10A and 10D . Filter parameters of D=17.145 mm, L=7.747 mm, D c =29.15 mm, L c =27.2 mm, X p =10.57 mm, H p =25 mm, D s =9 mm, and L s =9.73 mm were simulated. Plot  130  corresponds to simulated results for filter  100  (shown in  FIGS. 10A-10C ), in which curve  132  represents reflection (S 11 ) and curve  134  represents transmission (S 21 ). Likewise plot  140  corresponds to simulated results for filter  100 ′ (shown in  FIG. 10D ), in which curve  142  represents reflection (S 11 ) and curve  144  represents transmission (S 21 ). 
     It is evident in plot  140  that the passband of the filter  100 ′ only has a steep out of band rejection on the low side, whereas the passband of the filter  100  in plot  130  has a steep out of band rejection on both sides. The improved performance is due to the fact that arranging electromagnetic probes  170   a ,  170   b  at opposite ends of the cylindrical cavity  150 , as in filter  100 , places transmission zeros on both sides of the passband. In contrast, arranging electromagnetic probes from the same end of cylindrical cavity  150 , as in filter  100 ′, only places a single transmission zero on the low side of the passband. The extra transmission zero can be explained the polarity reversal of the output coupling relative to the input coupling, which creates a 180° out of band phase shift that is subtractive, not additive, at the output. 
     The out of band rejection of the quad-mode filters  100 ,  100 ′ is also affected by the input and output channels (i.e. electromagnetic probes  170   a ,  170   b ) being located in the same physical cavity (i.e. cylindrical cavity  150 ). Out of band rejection is normally improved in higher order filters, such as a dual-cavity, 8-pole filters, where the input and output channels are located in physically separate cavities. Another approach to improving out of band rejection is to design a 6-pole filter in which input and output coupling is made to single-mode cavities coupled to a quad-mode cavity, such as the ones illustrated in  FIGS. 10A-10D . For example, the single-mode cavities can be operated in the TEH mode. The improvement in out of band rejection is traded off against filter size. Thus, overall the out of band rejection seen in the plots  130  and  140  is satisfactory given the extreme compactness of the filters  100  and  100 ′. 
     It should also be appreciated that with suitable modification the quad-mode filters  100 ,  100 ′ can be converted into dual-mode, dual-band filters. It is recalled that a dual-band filter can be realized using the half-cut dielectric resonator  10  by carrying each band on a separate resonant mode, one on the ½HEH 11  mode and the other on the ½HEE 11  mode. The same general concept is applicable to the full cylinder resonating in the degenerate HEH 11  and HEE 11  modes. Thus the synthesized filter will additionally be dual-mode. In the filters  100 ,  110 ′, electromagnetic probe  170   a  couples to one component from each of the HEH 11  and HEE 11  modes, while electromagnetic probe  170   b  couples to the other orthogonal component of these dual modes. Moreover, screws  108  and  109  located at 45 degrees from each electromagnetic probe  170   a ,  170   b  couple the two orthogonal mode components from each of the HEH 11  and HEE 11  degenerate mode pairs. This arrangement of electromagnetic probes and screws, without needed to include screws  104 - 107 , therefore provides a dual-branch coupling scheme required in dual-mode filters. Removing screws  104 - 107  (or else reconfiguring them so as to tune, but not couple the two mode components, one from each of the HEH 11  and HEE 11  mode pairs, that align with a respective electromagnetic probe  170   a ,  170   b ) will thus convert quad-mode filters  100 ,  100 ′ into corresponding dual-mode, dual-band filters. Higher order dual-mode and mixed quad-mode and dual-mode filters are possible as well using this arrangement of screws. 
     Reference is now made to  FIGS. 12A and 12B , which show different views of an exemplary 3-pole, dual-band dielectric resonator filter synthesized using half-cut cylindrical dielectric resonators operating in a dual-band, according to aspects of embodiments of the present invention. The dual-band dielectric resonator filter  200  comprises half-cut dielectric resonators  210   a - 210   c  enclosed in cavities  250   a - 250   c , respectively. Electromagnetic probe  270   a  couples resonator  210   a  to external connector  272   a  on the input side, and electromagnetic probe  270   c  couples resonator  210   c  to external connector  272   c  on the output side. Cross-shaped iris  258   a  couples the respective operating modes of resonators  210   a  and  210   b , and cross-shaped iris  258   b  couples the respective operating modes resonators  250   b  and  250   c . Screws  204  may also be included in the filter, one of their functions being to provide resonant mode tuning for the half-cut dielectric resonator  10   b . Although not expressly shown, resonators  210   a - 210   c  are planar mounted on mounting supports formed in unitary pieces on suitable low-permittivity dielectric substrate. 
     Appropriate sizing of the half-cut dielectric resonators  210   a - 210   c  and selection of a coupling scheme (analogous to the dual-branch scheme illustrated in FIG.  9 D) will realize the 3-pole, dual-band dielectric resonator  200 . The diameter D and length L of each resonator  210   a - 210   c  are selected so that the ½HEH 11  and ½HEE 11  modes resonate at different resonant frequencies, f H  and f E , respectively, corresponding to the centre frequencies of the two bands in the dual-band filter, and with a frequency band separation, Δf. The dimensions D and L may then be swept in order to meet design specifications imposed on f H , f E  and Δf. Each band in the dual-band filter  200  is carried by a corresponding different resonant mode of the resonators  210   a - 210   c.    
     In conforming with the coupling scheme presented in  FIG. 9D  for a dual-band filter, input electromagnetic probe  270   a  is oriented to couple both the ½HEH 11  and ½HEE 11  modes of half-cut dielectric resonator  210   a , just as output electromagnetic probe  270   c  is oriented to couple both the ½HEH 11  and ½HEE 11  modes of half-cut dielectric resonator  210   c . Cross-shaped iris  258   a  simultaneously couples both the ½HEH 11  and ½HEE 11  modes of resonators  210   a  and  210   b , wherein specifically the horizontal component couples the ½HEH 11  mode and the vertical component couples the ½HEE 11  mode. Similarly, cross-shaped iris  258   b  simultaneously couples both the ½HEH 11  and ½HEE 11  modes of resonators  210   b  and  210   c , wherein specifically the horizontal component couples the ½HEH 11  mode and the vertical component couples the ½HEE 11  mode. Thus, the two frequency bands are carried independently within the dual-band filter  200 . Generally intra-cavity coupling screws are not included in the dual-band filter, as the two bands are separate. However, screws  204  are included in resonator cavity  250   b , in part, to adjust the resonant frequency of the ½HEH 11  modes of the resonators  210   b . It will also be appreciated that additional screws (not shown) can be included in any or all of cavities  250   a - 250   c  for providing additional resonant mode tuning, if desired, and that the screws  204  can serve other functions in the filter  200 , in addition to resonant mode tuning. 
     The basic topology of the dual-band filter  200  can also, after suitable modification, realize a 6-pole, dual-mode filter. The diameter D and length L of each resonator  210   a - 210   c  can be adjusted so that the ½HEH 11  and ½HEE 11  modes of each resonate at a common resonant frequency. Appropriate sizing and positioning of electromagnetic probes, screws and inter-cavity apertures can then realize a coupling scheme suitable for a 6-pole, dual-band filter (analogous to the scheme illustrated in  FIG. 9A  for a 4-pole filter). More specifically, coupling screws can be included in each of cavities  250   a - c  and oriented such as coupling screw  62  so that the ½HEH 11  and ½HEE 11  modes of each resonator  210   a - 210   c  are coupled. Next, electromagnetic probe  270   a  can be oriented horizontally adjacent to rectangular surface  218   a  of half-cut resonator  210   a  so as to couple only the ½HEH 11  mode, and electromagnetic probe  270   c  can be oriented vertically adjacent to curved surface  214   c  of resonator  210   c  so as to couple only the ½HEH 11  mode. Finally, cross-shaped iris  258   a  can be replaced with a suitable iris, such as horizontal iris  54 , in order to couple the ½HEE 11  modes of resonators  210   a  and  210   b , and cross-shaped iris  258   b  can be replaced with a suitable iris, such as vertical iris  56 , in order to couple the ½HEH 11  modes of resonators  210   b  and  210   c . This particular configuration of electromagnetic probes, coupling screws and inter-cavity apertures realizes a linear 6-pole dual-mode filter. The locations of electromagnetic probes  270   a ,  270   b  can also be varied to provide different combinations of positive and negative mode coupling for achieving different numbers and locations of transmission zeros in the filter  200 . 
     Reference is now made to  FIG. 13A , which shows perspective and top views of an exemplary 2-pole, dielectric resonator diplexer synthesized using half-cut cylindrical dielectric resonators operating in a dual-band, according to aspects of embodiments of the present invention. The 2-pole dielectric resonator diplexer  300  has a simple realization using two half-cut dielectric resonators  310   a ,  310   b  planar mounted on respective mounting supports (not shown) in cavities  350   a ,  350   b . Electromagnetic probe  370   a  provides a common input channel for a mixed frequency component signal, and electromagnetic probes  370   b ,  370   c  provide isolated outputs channels, each channel corresponding to a different frequency band. Thus the diplexer  300  can be used to separate frequency components of the mixed-frequency input signal failing within the two respective frequency bands. It should be appreciated that the diplexer  300  is similar to a dual-band filter except that two isolated output channels are substituted for the common output channel. 
     Appropriate sizing of the half-cut dielectric resonators  310   a ,  310   b  and selection of a coupling scheme (analogous to the dual-branch scheme illustrated in  FIG. 9D , but subject to the above-noted difference on the output side) will realize the 2-pole, dual-band dielectric resonator diplexer  300 . As is the case for a dual-band filter, the diameter D and length L of resonators  310   a ,  310   b  are selected to provide a dual band defined by f E , f H  and Δf. Each output channel of the diplexer then corresponds to a different frequency band centered at one or the other of f E  and f H  (depending on which resonant mode carries which frequency band). Electromagnetic probe  370   a  is oriented to couple both the ½HEH 11  and ½HEE 11  modes of half-cut dielectric resonator  310   a  to the external connector  372   a , and cross-shaped iris  58  couples both the ½HEH 11  and ½HEE 11  modes of resonator  350   a  to the corresponding modes of resonator  350   b . Electromagnetic probe  370   b  is oriented horizontally adjacent to the rectangular surface  318   b  of half-cut dielectric resonator  310   b  to couple the ½HEH 11  mode to the external connector  372   b , while substantially isolating the ½HEE 11  mode. On the other hand, electromagnetic probe  370   c  is oriented vertically adjacent to the proximal end of curved surface  314   b  of half-cut dielectric resonator  310   b  to couple the ½HEE 11  mode to the external connector  372   c , while substantially isolating the ½HEH 11  mode. By carrying one frequency band on the ½HEH 11  mode and another frequency band on the ½HEH 11  mode, this exemplary arrangement of a common input channel and isolated output channels realizes a dielectric resonator diplexer. It should be appreciated that alternative realizations of a dielectric resonator diplexer are possible, and that one or more tuning screws may be included for providing resonant mode tuning. As before, the dimensions of the resonators, coupling screws, electromagnetic probes can be designed to realize design specifications for the diplexer. 
     Reference is now made to  FIG. 13B , which shows a top view of another exemplary dielectric resonator diplexer perspective and top views of an exemplary 3-pole, dielectric resonator diplexer synthesized using half-cut cylindrical dielectric resonators operating in a dual-band, according to aspects of embodiments of the present invention. The diplexer  400  is somewhat similar to the diplexer  300 , but constitutes an improvement over diplexer  300 . Superior output channel isolation is achieved in diplexer  400  by locating each respective output channel in a separate resonator cavity. 
     As in the diplexer  300 , electromagnetic probe  470   a  couples both the ½HEH 11  and ½HEE 11  modes of resonator  410   a  to the external connector  472   a , and cross-shaped iris  358  then couples the ½HEH 11  and ½HEE 11  modes of resonator  410   a  to the corresponding modes of resonator  410   b . However, unlike the diplexer  300 , diplexer  400  further comprises resonators  410   c ,  410   d  respectively enclosed in resonator cavities  450   c ,  450   d . Horizontal iris  454  couples the ½HEE 11  modes of resonators  410   b  and  410   d , while substantially isolating the ½HEH 11  modes, and vertical iris  456  couples the ½HEH 11  modes of resonators  410   b  and  410   c , while substantially isolating the ½HEE 11  mode. Thus, the joint effect of horizontal iris  454  and vertical iris  456  is to guide the ½HEH 11  resonant mode into resonator cavity  450   c  and the ½HEE 11  resonant mode into resonator cavity  450   d . Electromagnetic probe  470   c  then couples the ½HEH 11  mode of resonator  410   c  to the external connector  472   c , and electromagnetic probe  470   d  couples the ½HEE 11  mode of resonator  410   d  to the external connector  472   d . Alternatively, half-cut dielectric resonators  410   c ,  410   d  can be replaced with full cylinders operating in a single TEH mode, or other resonant mode, as discussed in greater detail below. 
     Reference is now made to  FIGS. 13C and 13D , which show plots of reflection and transmission versus frequency for the dielectric resonator diplexers of  FIGS. 13A and 13D . Plot  130  corresponds to simulated results for diplexer  300  (shown in  FIG. 13A ), in which curve  432  represents reflection (S 11 ), curve  434  represents transmission (S 21 ) of the ½HEH 11  mode to port  2 , and  436  represents transmission (S 31 ) of the ½HEE 11  mode to port  3 . Likewise plot  440  corresponds to simulated results for diplexer  400  (shown in  FIG. 13B ), in which curve  442  represents reflection (S 11 ), curve  444  represents transmission (S 21 ) of the ½HEH 11  mode to port  2 , and  446  represents transmission (S 31 ) of the ½HEE 11  mode to port  3 . 
     It is evident in plot  440  that better output isolation is achieved in the diplexer  400  as compared to the diplexer  300 . In the lower passband (corresponding to transmission of the ½HEH 11  mode to port  2 ), about −25 dB transmission to port  3  is seen in plot  430  as compared to only about −75 dB in plot  440 . Similarly in the upper passband (corresponding to transmission of the ½HEH 11  mode to port  3 ), about −15 dB transmission to port  2  is seen in plot  430  as compared to only about −50 dB in plot  440 . The improved output mode isolation is due to the physical separation of the channels in different resonator cavities. Plots  430  and  440 , it should be appreciated, also confirm that the dual-band is carried on separate resonant modes of the half-cut dielectric resonator  10 . 
     It should be appreciated that a plurality of resonator diplexers can be combined to realize higher-order multiplexers. For example, a plurality of diplexers can be realized, according to the above-described embodiments, wherein the dual-band in each of the diplexers are defined for different centre frequencies to realize a multi-band defined by a plurality of centre frequencies. The input electromagnetic probe can then be coupled to each of the plurality of diplexers, in that way realizing a higher order multiplexer. A forked electromagnetic probe, for example, could be used to couple each of the diplexers to a common input. As before, in each of the plurality of diplexers, the input electromagnetic probe can be oriented to couple to both the ½HEH 11  mode and ½HEE 11  mode of a first resonator. In that way, each of the plurality of diplexers can carry a dual-band on the two resonant modes. 
     In the exemplary embodiments described herein thus far, constructed from the full cylindrical or half-cut dielectric resonator, spurious performance has not been discussed in any length. Spurious performance, it should be understood, relates to the frequency range of a dielectric resonator in which only the resonator operating mode(s) are present, and no unwanted higher or lower order resonance modes appear. Due to the relative orthogonality of the lower order resonant modes of the half-cut dielectric resonator, a simple modification to the basic half-cut offers significant improvements in spurious performance. Exemplary embodiments of modified half-cut dielectric resonators are discussed below. 
     Reference is now made to  FIGS. 14A-14C , which illustrate various views of the E field lines in the half-cut cylindrical dielectric resonator of  FIG. 1B  for a first spurious resonant mode. It is observed that the TEH mode of the full cylindrical dielectric resonator  1  (which is a lower order mode than either the HEH 11  and HEE 11  modes) does not correspondingly appear in the basic half-cut dielectric resonator  10  as a lower order resonance mode because the radial symmetry present in the full cylinder that expresses the TEH mode is not preserved after the cut. The ½HEH 11  and ½HEE 11  modes of the basic half-cut dielectric resonator  10 , therefore, represent the first two eignenmodes of the structure. The mode charts  30  and  40  of  FIGS. 4A and 4B  confirm these observations. The first higher order resonance mode of the half-cut dielectric resonator  10 , corresponding to the third eigenmode of the structure, is the component of the HEE 11  mode that was orthogonal to the symmetry plane  25  and lost due to the cut. Distorted by the boundary contours of the half-cut cylinder and forced to circulate in a shorter path after to the cut, this component of the HEE 11  mode in the full cylinder becomes a distinct mode in the half-cut cylinder. With the ½HEH 11  and ½HEE 11  modes providing the first two eigenmodes of the structure (their relative ordering depending on the sizing of D and L), this new mode constitutes the third eigenmode of the structure. 
     As shown in  FIGS. 14A-14C , the E field lines of this third eigenmode circulate vertically and orthogonal to the rectangular surface  18  tracing out a path that is limited by the surface boundaries of the half-cut cylinder. The E field lines of this third eigenmode, it should be appreciated, are orthogonal to the E field lines in both the ½HEH 11  resonant mode (which circulate horizontally) and the ½HEE 11  resonant mode (which circulate vertically but tangential to the rectangular surface  18 ). On account of the relative orthogonality of the first three eigenmodes of the structure, selective cutting of the basic half-cut dielectric resonator  10  can create dielectric barriers that effectively terminate the E fields of the third eigenmode, but that have nearly no impact on the E fields of the first two eigenmodes. By suppressing the third eigenmode of the structure, the next higher order (i.e. the fourth) eigenmode becomes the first spurious mode. In this way the spurious free window of the filter is widened. 
     Reference is now made to  FIGS. 15A-15D , which illustrate perspective views of exemplary slotted half-cut dielectric resonators according to aspects of embodiments of the present invention. Each slotted half-cut dielectric resonator illustrated is similar to the basic half-cut dielectric resonator  10 , but further comprises at least one through-way slot extending between opposite surfaces of the half-cut dielectric resonator  10 . For example, slotted half-cut dielectric resonator  510  shown in  FIG. 15A  comprises vertical through-way slot  515  extending between the parallel pair of semi-circular faces  512 , while slotted half-cut dielectric resonator  610  shown in  FIG. 15B  comprises horizontal through-way slot  635  extending between the curved surface  14  and the rectangular surface  18 . Preferably the through-way slot  515 ,  635  is located at or near the center of the opposite surfaces between which it extends. However, in some embodiments, the through-way slot  515 ,  635  may not be exactly centered and may be positioned away from the centre of the opposite surfaces between which it extends. The shape and cross-sectional area of the through-way slot are also both variable. In the particular case of a rectangular through-way slot, the cross-sectional length and width of the through-way slot are variable. 
     The number of through-way slots included in the slotted half-cut dielectric resonator and their relative orientations are also variable. For example, slotted half-cut dielectric resonator  710  shown  FIG. 15C  comprises vertical through-way slot  715  extending between the pair of semi-circular surfaces  712 , as well as horizontal through-way slot  735  extending between the curved surface  714  and the rectangular surface  718 . The through-way slots  715 ,  735  clearly intersect somewhere inside slotted half-cut dielectric resonator  710 . Although not illustrated, in some embodiments, the slotted half-cut dielectric resonator comprises multiple parallel through-way slots. For example two or more parallel through-way slots may extend between semi-circular surfaces  712  or, alternatively, between the curved surface  714  and rectangular surface  718 . 
     In some embodiments, surface slots may be used instead of through-way slots. For example, slotted half-cut dielectric resonator  810  shown in  FIG. 15D  comprises surface slot  845  cut into curved surface  814 , but not extending all the way through to rectangular surface  818 . Similarly, a surface slot may be cut into rectangular surface  818  (not extending all the way through to curved surface  814 ). In some embodiments, surface slots may be cut into each of curved surface  814  and rectangular surface  818 , or alternatively into each of the parallel pair of semi-circular surfaces  812 . Any combination of surface slots is possible. Thus, in some embodiments, surface slots may be cut into one or both of the pair of semi-circular surfaces  812  in addition, or as an alternative, to surface slots cut into the curved surface  814  and rectangular surface  818 . These surface slots may cross, merely adjoin, or neither. 
     Reference is now made to  FIGS. 16A and 16B , which show top and perspective views of the E field lines in the slotted half-cut dielectric resonator of  FIG. 15B  for a first spurious mode, according to aspects of embodiments of the present invention. The E field lines illustrated in  FIGS. 16A and 16B  clearly differ from those in  FIGS. 14A-14C  because the horizontal through-way slot  635  cut into the half-cut dielectric resonator  610  terminates the E field lines of the third eigenmode. Although not expressly shown, the E field lines of the ½HEH 11  and ½HEE 11  modes are not appreciably affected by the horizontal through-way slot  635  because they are oriented more or less parallel to the cut. The respective resonant frequencies of the ½HEH 11  and ½HEE 11  modes are thus not appreciably affected either. 
     Accordingly, the E field lines illustrated in  FIGS. 16A and 16B  actually represent the fourth eigenmode of the half-cut cylinder and correspond to the component of the HEH 11  mode (as opposed to the HEE 11  mode) that was orthogonal to the symmetry plane  25  and was lost by the cut. Forced by the boundaries of the half-cylinder to circulate in a new path, that lost component of the HEH 11  mode becomes the fourth eigenmode of the structure. With its shorter circulation path, the fourth eigenmode has a higher resonant frequency than the third eigenmode. This fourth eigenmode of the half-cut cylinder becomes the first spurious mode when the third eigenmode of the structure is lost due to the cut. By leaving the first and second resonant modes largely unchanged and by substituting the fourth eigenmode for the third eigenmode as the first spurious mode of the resonator, the overall effect of cutting the horizontal through-way cut  635  is an increase the spurious free window of the resonator. 
     It will further be appreciated that the E field lines illustrated in  FIGS. 16A and 16B  are orthogonal to the vertical through-way slot  515  as well. Accordingly, supplementing the horizontal through-way slot  635  with an additional vertical through-way slot cut into the resonator  610  (thereby producing the resonator  710  having both a vertical through-way slot  715  and a horizontal through-way lot  735 ) will terminate the E field lines in the fourth eigenmode as well. An even wider spurious free window is thereby achieved. Table I below illustrates the increased spurious window due to inclusion of through-way slots for a dual-band filter with a 4 GHz lower band and a 4.4 GHz upper band. 
                     TABLE I                  SPURIOUS IMPROVEMENT COMPARISON                                         f lower     f upper     f spurious     Δf lower     Δf upper         Type   (GHZ)   (GHz)   (GHz)   (MHz)   (MHz)                                             Basic Half-cut   3.96   4.38   4.56   600   180       Vertical Through-way   3.96   4.38   4.77   810   390       Slot       Horizontal Through-way   4.02   4.39   5.20   1180   810       Slot       Dual Slotted   3.98   4.39   5.33   1350   940                    
It can be seen that the dual-slotted resonator  710  ( FIG. 15C ) outperforms the single slotted resonators  510 ,  610  ( FIGS. 15A and 15B ). The dual-slotted resonator  710  provides a spurious free window of approximately 1.3 GHz for the lower band and 900 MHz for the upper band, as compared to 600 MHz and 200 MHz, respectively, for the basic half-cut dielectric resonator  10  with no through-way slots. The single slotted configurations, it will be appreciated, also compare favourably to the original half-cut resonator, but still do not provide as wide a spurious free widow as the dual slotted resonator  710  provides.
 
     It should be appreciated that through-way slots cut into the full cylindrical dielectric resonator  1  would remove radial symmetry in the structure, and thus would potentially render the full cylindrical resonator unsuitable for quad-mode operation. For example, a vertical through-way slot, similar to though-way slot  515 , cut along the cylindrical axis of the full cylinder would fix a symmetry plane  25  in the structure. One component from each of the HEH 11  and HEE 11  modes would align with the symmetry plane, while the corresponding orthogonal mode components would terminate at the cut. Clearly it would be possible to cut through-way slots into the full cylinder, though doing so would render the full cylinder unsuitable for some applications (i.e. quad-mode operation), while leaving it potentially still suitable for other applications (i.e. dual-mode operation in the two remaining aligned modes). 
     It should also be appreciated that the basic and slotted half-cut dielectric resonators can be used interchangeably in the exemplary dielectric filter and multiplexer realizations discussed herein. Accordingly, for a wider spurious free window, the dielectric resonator filter  200  ( FIGS. 12A and 12B ), as well as the dielectric resonator multiplexers  300  ( FIG. 13A) and 400  ( FIG. 13B ) can be synthesized using slotted half-cut resonators, rather than the basic half-cut resonators as illustrated. The same design and synthesis processes could be followed without substantial modification. Aspects of some still further exemplary realizations of dielectric resonator filters and multiplexers will now be discussed. 
     Reference is now made to  FIG. 17 , which shows a perspective view of an exemplary 2-pole, dual-band dielectric resonator filter having improved spurious performance, according to aspects of embodiments of the present invention. The 2-pole dual-band filter  900  is similar to, but different than, the 3-pole dual-band filter  200  illustrated in  FIGS. 12A and 12B . For example, the respective filters have different orders and are synthesized using different resonators. The dual-band filter  900  in particular is synthesized using two slotted half-cut dielectric resonators  910   a ,  910   b  comprising horizontal through-way slots  935   a ,  935   b , making it a 2-pole filter. No tuning screws are illustrated in  FIG. 17  either, though tuning screws can be included if desired. The coupling scheme synthesized in dual-band filter  900  is otherwise analogous to the one synthesized in filter  200 . Electromagnetic probe  970   a  couples both the ½HEH 11  and ½HEE 11  resonant modes of the resonator  910   a  to the external connector  972   a , cross-shaped iris  958  couples both modes of resonator  710   a  to corresponding modes of resonator  910   b , and electromagnetic probe  970   b  couples both the ½HEH 11  and ½HEE 11  modes of resonator  910   b  to the external connector  972   b . No intra-cavity coupling screws are included. The electromagnetic probes  970   a ,  970  are oriented for positive mode coupling. This coupling scheme is the dual branch scheme illustrated in  FIG. 9D . 
     Reference is now made to  FIGS. 18A-18C , which illustrate various views of an exemplary 3-pole, dual-band dielectric resonator filter, according to aspects of embodiments of the present invention. The dual-band filter  1000  is similar to the 2-pole dual-band filter  900  illustrated in  FIG. 17 , but is a 3-pole dual-band filter. The dual-band filter  1000  is also similar to the dual-band filter  200  of  FIGS. 12A and 12B , but comprises slotted half-cut dielectric resonators and differently positioned electromagnetic probes. Accordingly, half-cut dielectric resonators  1010   a - 1010   c  are enclosed in resonator cavities  1050   a - 1050   c  and also include horizontal through-way slots  1035   a - 1035   c , respectively. Cross-shaped irises  1058   a ,  1058   b  provide inter-cavity coupling of both the ½HEH 11  and ½HEE 11  modes of resonators  1010   a - 1010   c , as described previously, for carrying a dual-band. Support structures  1052   a - 1052   c  are used to mount resonators  1010   a - 1010   c  in planar fashion. 
     Electromagnetic probe  1070   a  couples both the ½HEH 11  and ½HEE 11  modes of resonator  1010   a  to external connector  1072   a , while electromagnetic probe  1070   c  couples both the ½HEH 11  and ½HEE 11  modes of resonator  1010   c  to external connector  1072   c . As mentioned, it can be seen that the dual-band filter  1000  differs from the dual-band filter  900  also in the location of the electromagnetic probes  1070   a ,  1070   b  relative to the half-cut dielectric resonators  1010   a ,  1010   c . Electromagnetic probes  1070   a ,  1070   c  are located diagonally adjacent respective curved edges of the half-cut dielectric resonators  1010   a ,  1010   b  as opposed to diagonally adjacent respective straight edges. Placing the electromagnetic probes  1070   a ,  1070   c.    
     When configured as shown in  FIGS. 18A-18C , the 2-pole filter  1000  has a natural transmission zero located in between the two bands of the dual-band due to the odd order of the filter. In each resonator cavity  1050   a - 1050 , the two resonant modes of the filter  1000  have a phase separation of approximately 180° for frequencies between the two bands. Thus, frequency signals between the two bands undergo one phase reversal for each cavity included in the filter. Because there are an odd number of cavities in the filter  1000 , the total number of phase reversals is odd and the total phase shift is an odd multiple of 180° phase shifts. In this particular phase relation, the two frequency bands are subtractive at the output and thereby create a transmission zero. 
     It should be appreciated that the same result would not correspondingly hold for even order filters. In that case, the total number of phase reversals would be even and the total phase shift would be an even multiple of 180° phase shifts, corresponding to the even number of cavities in the filter. No inter-band transmission zero would occur because the two frequency bands will be in-phase and thus additive, not subtractive, at the output. Inter-band transmission zeros are still achievable in even order filters, however, as will be seen, by introducing an additional single phase reversal to provide an odd number of phase reversals overall. 
     Reference is now made to  FIG. 18D , which shows a plot of reflection and transmission versus frequency for the 3-pole, dual-band dielectric resonator filter of  FIGS. 18A-18C . Plot  1030  corresponds to simulated results for the dual-band filter  1000 , in which curve  1032  represents reflection (S 11 ), curve  1034  represents transmission (S 21 ). It is evident that region  1036  of the curve  1034  corresponds to an inter-band transmission zero of the filter  1000 . 
     Reference is now made to  FIGS. 19A and 19B , which shows perspective views of exemplary 4-pole, dual-band dielectric resonator filters, according to aspects of embodiments of the present invention. The dual-band filter  1200  ( FIG. 19A ) is similar to the 2-pole dual-band filter  900  illustrated in  FIG. 17 , but is a 4-pole dual-band filter. Half-cut dielectric resonators  1010   a - 1010   d  are enclosed in resonator cavities  1050   a - 1050   d  and include horizontal through-way slots  1035   a - 1035   d , respectively. Cross-shaped irises  1058   a - 1058   c  provide inter-cavity coupling of both the ½HEH 11  and ½HEE 11  modes of resonators  1010   a - 1010   d , as described previously, for carrying a dual-band. Electromagnetic probe  1070   a  couples both the ½HEH 11  and ½HEE 11  modes of resonator  1010   a  to external connector  1072   a , while electromagnetic probe  1070   d  couples both the ½HEH 11  and ½HEE 11  modes of resonator  1010   d  to external connector  1072   d . Based on their location, electromagnetic probes  1070   a ,  1070   d  provide positive coupling. Mounting supports  1052   a - 1052   d  are used for planar mounting of the resonators  1010   a - 1010   d.    
     With an even number of poles, the dual-band filter  1200  does not have an inter-band transmission zero. There is an overall even number of phase reversals for inter-band frequencies attributable to inter-cavity coupling, and thus the two modes are in-phase at the output. In contrast, the dual-band filter  1200 ′ ( FIG. 19B ) has an inter-band transmission zero even though it is an even order filter. As can be seen, the locations of electromagnetic probes  1270   a ,  1270   d  do not match. Electromagnetic probe  1270   a  provides negative coupling on the input, while electromagnetic probe  1270   d  provides positive coupling on the output. Even though there is an even number of phase reversal due to inter-cavity coupling (i.e. because there are an even number of cavities), the polarity reversal in the output coupling achieves an overall out-of-phase relation on the output. Consequently a transmission zero is achieved. It should be noted that this technique can also be used to remove the naturally occurring inter-band transmission zero in odd order filters by converting the natural out-out-phase relation of the two resonant modes into the non-transmission zero producing in-phase relation naturally seen in even order filters. 
     Reference is now made to  FIG. 19C , which shows plots of reflection and transmission versus frequency for the 4-pole, dual-band dielectric resonator filters of  FIGS. 19A and 19B . Curve  1232  represents reflection (S 11 ) and curve  1234  represents transmission (S 21 ) for the filter  1200  of  FIG. 19A , while curve  1242  represents reflection (S 11 ) and curve  1244  represents transmission (S 21 ) for the filter  1200 ′ of  FIG. 19B . It is evident that region  1246  of the curve  1244  corresponds to an inter-band transmission zero of the filter  1200 ′, which does not correspondingly appear in the curve  1234 . The frequency characteristics of the two filters  1200 ,  1200 ′ are otherwise commensurate. 
     Reference is now made to  FIGS. 20A and 20B , which show perspective and top views of an exemplary 4-pole dielectric resonator diplexer with improved spurious performance and output mode isolation, according to aspects of embodiments of the present invention. The dielectric resonator diplexer  1300  shown in  FIGS. 20A and 20B  is similar to the dielectric resonator diplexer  400  shown in  FIG. 13B , except is of a different order and provides improved output mode isolating by coupling full cylindrical resonators  1201   d ,  1201   e  operating in single TEH modes to external connectors  1272   d ,  1272   e . The half-cut dielectric electric resonators  1235   a - 1235   c  also include horizontal through-way slots  1235   a - 1235   c . The principles of operation are otherwise as described herein. 
     Resonator cavities  1250   a - 1250   c  enclosing resonators  1210   a - 1210   c  are configured to carry a dual-band. Electromagnetic probe couples external connector  1272   a  to both the ½HEH 11  and ½HEE 11  modes of resonator  1210   a . Cross-shaped irises  1258   a ,  1258   b  couple to dual band to resonator  1210   c  intermediately through resonator  1210   b . Vertical iris  1256  defined in one wall of resonator cavity  1250   c  guides the ½HEH 11  mode into resonator cavity  1250   d  for coupling to the external connector  1272   d . Similarly, horizontal iris  1254  defined in another wall of resonator cavity  1250   c  guides the ½HEE 11  mode into resonator cavity  1250   e  for coupling to the external connector  1272   e . Electromagnetic probes  1270   d ,  1270   e  are oriented to couple the TEH resonant modes of the full cylindrical resonators  1201   d ,  1201   e , though it should be appreciated that they may be oriented otherwise to couple other resonant modes, if desired. For example, electromagnetic probes  1201   d ,  1201   e  could be located to couple either the HEH or HEE modes of resonators  1201   d ,  1201   e.    
     It should also be appreciated that full cylindrical resonator  1201   e  is mounted to a side wall, rather than the floor, of resonator cavity  1250   e  using mounting support  1252   e  in order to couple the ½HEE 11  mode of resonator  1210   c  to the TEH mode of resonator  1201   e . In contrast, full cylindrical resonator  1201   d  is mounted to the floor of resonator cavity  1250   d  using mounting support  1252   d  in order to couple the ½HEH 11  mode of resonator  1210   c  to the TEH mode of resonator  1201   d . These relative orientations of resonators  1201   d ,  1201   e  are determined by the relative polarizations of the coupled modes. If a different mode of the resonators  1201   d ,  1201   e  were to be coupled (for example the HEH or HEE modes), different orientations of the resonators  1201   d ,  1201   e  could be used. 
     Reference is now made to  FIG. 21 , which shows a flow chart of a method of manufacturing a full cylindrical or half-cut cylindrical dielectric resonator, according to aspects of embodiments of the present invention. The method  2100  may be used to manufacture any of the full cylindrical dielectric resonator  1 , the basic half-cut dielectric resonator  10  and the various slotted half-cut dielectric resonators  510 ,  610 ,  710 ,  910 . Accordingly, some of the steps of method  2100  are optional. 
     Method  2100  begins at step  2105 , which comprises providing a block of a suitable high-permittivity dielectric material. In some embodiments, the dielectric constant of the material lies in the range 20&lt;∈ r &lt;100, though in other embodiments the dielectric constant may be higher or lower. The block of dielectric material should have a volume at least that of the dielectric resonator to be manufactured. 
     Step  2110  comprises forming the dielectric material into a cylinder of a selected diameter D and a selected length L. The selected values of D and L may depend on the filter application to which the resonator will be put. For example, if the final resonator will have a full cylindrical shape, D and L may be selected so that it will be suitable for operation in a quad-mode. In this case, D and L may be selected so that the dual HEH 11  and HEE 11  of the full cylindrical dielectric resonator all resonate at a common resonant frequency, and the method  2100  ends after step  2110 . 
     Alternatively, the final resonator may have a half-cut cylindrical form and D and L may be selected so that it will be suitable for operation in a dual-mode. In that case, D and L may be selected so that both ½HEH 11  and ½HEE 11  modes of the half-cut dielectric resonator resonate at a common resonant frequency. Alternatively, the final resonator may have a half-cut cylindrical form and D and L may be selected so that the half-cut dielectric resonator will be suitable for operation in a dual-band. In that case, D and L may be selected so that the ½HEH 11  mode resonates at first resonant frequency and the ½HEE 11  mode resonates at a second frequency different from the first resonant frequency. In these two alternatives, the method  2100  proceeds to step  2115 . 
     Step  2115  comprises cutting the full cylindrical dielectric resonator lengthwise along a central axis to produce a half-cut dielectric resonator. The half-cut dielectric resonator will be of the diameter D and length L selected in previous step  2110 , which may make the resonator suitable for operation in either a dual-mode or a dual-band. If no through-way slots are to be cut, method  2100  ends after step  2115 . Alternatively, method  2100  proceeds to step  2120 , which comprises cutting one or more through-way slots in the basic half-cut dielectric resonator filter. 
     Steps  2105 ,  2110  and  2120  may be performed using any suitable technique for cutting dielectric material. In some embodiments, steps  2105 ,  2110  and  2120  are performed using watercutting, which provides a highly accurate and cost-effective solution. As a result, no special molding or firing is required. Different cutting techniques however may be used in other embodiments. It should be appreciated, moreover, that modifications to method  2100  are possible, and that other methods of manufacturing a half-cut dielectric resonator exist and are within the scope of the disclosure. For example, half-cut dielectric resonators, and even slotted half-cut dielectric resonators, can be directly molded from a suitable high-permittivity dielectric substrate. Cutting a full cylinder into a half-cut cylinder, however, has the advantage of being both highly accurate and cost-effective. 
     Reference is now made to  FIG. 22 , which is perspective views of an exemplary rectangular dielectric resonator, respectively, according to aspects of embodiments of the present invention. The rectangular dielectric resonator  2201  shown in  FIG. 22  comprises a generally rectangular shape of length L and cross-sectional area D×D formed in a unitary piece of suitable high-permittivity dielectric substrate. Accordingly, the rectangular dielectric resonator  2201  comprises parallel square surfaces  2202  connected by four rectangular surfaces  2204 . It may also be formed in a high-permittivity dielectric substrate. 
     It is evident that the rectangular dielectric resonator  2201 , like the full cylindrical dielectric resonator  1 , has 90 degree radial symmetry. Thus, like the full cylindrical dielectric resonator  1 , the rectangular dielectric resonator  2201  can be sized for operation in a quad mode, wherein each of the four modes resonates at a common resonant frequency. Further, the rectangular dielectric resonator  2201  can also be sized for operation in a dual band, wherein each of two dual modes resonate at separate frequencies, one dual mode resonating a first resonant frequency and the other dual mode resonant at a second resonant frequency different from the first resonant frequency. One dual degenerate mode in the rectangular dielectric resonator  2201  will circulate parallel to the square surfaces  2202  (similar to the HEH mode in the full cylinder), and another dual degenerate mode will circulate orthogonal to the square surfaces (similar to the HEE mode in the full cylinder). Thus, again the D/L ratio can be sized so that the circulating paths of the E fields in these two dual modes are equal, in which case the modes will resonate at the same frequency. Alternatively, the D/L ratio can be sized for operation in a dual-band. 
     It should be appreciated that the above-described embodiments of coupling schemes (input-output, intra-cavity, inter-cavity), as well as filter/multiplexer realizations, though expressly described with reference to the full and half-cut cylindrical dielectric resonators, equally can be realized using rectangular dielectric resonators. Thus, filters and multiplexers realized using rectangular resonators are within the scope of the invention as well. It should further be appreciated that through-way slots may also similarly be cut into the rectangular dielectric resonators. 
     Numerous specific details are set forth to provide a thorough understanding of the exemplary embodiments described herein. However, it will be appreciated by those of ordinary skill in the art that the exemplary embodiments described herein may be practiced in some instances without certain of these specific details. In other instances, well-known methods, procedures and components have not been described in detail so as not to obscure other aspects of the embodiments described herein. It will also be appreciated that some features and/or functions of the described exemplary embodiments are amenable to modification without departing from the principles of operation of the described exemplary embodiments. As the description provided herein is merely illustrative of the invention, other variants and modifications may still be within the invention as defined in the claims appended hereto. This description is not to be considered in any way as limiting the scope of the exemplary embodiments described herein.