7 Waveguides and Resonators Kenneth R. Demarest The University of Kansas 7.1. INTRODUCTION Any structure that transports electromagnetic wave s can be considered as a waveguide. Most often, however, this term refers to either metal or dielectric structures that transport electromagnetic energy without the presence of a complete circuit path. Waveguides that consist of conductors and dielectrics (including air or vacuum) are called metal waveguides. Waveguides that consist of only dielectric materials are called dielectric waveguides. Metal waveguides use the reflective properties of conductors to contain and direct electromagnetic waves. In most cases, they consist of a long metal cylinder filled with a homogeneous dielectric. More complicated waveguides can also contain multiple dielectrics and conductors. The conducting cylinders usually have rectangular or circular cross sections, but other shapes can also be used for specialized applications. Metal waveguides provide relatively low loss transport over a wide range of frequencies— from RF through millimeter wave frequencies. Dielectric waveguides guide electromagnetic waves by using the reflections that occur at interfaces between dissimilar dielectric materials. They can be constructed for use at microwave frequencies, but are most commonly used at optical frequencies, where they can offer extremely low loss propagation. The most common dielectric waveguides are optical fibers, which are discussed elsewhere in this handbook (Chapter 14: Optical Communications). Resonators are either metal or dielectric enclosures that exhibit sharp resonances at frequencies that can be controlled by choosing the size and material construction of the resonator. They are electromagnetic analogs of lumped resonant circuits and are typically used at microwave frequencies and above. Resonators can be constructed using a large variety of shaped enclosur es, but simple shapes are usually chosen so that their resonant frequencies can be easily predicted and controlled. Typical shapes are rectangular and circular cylinders. 7.2. MODE CLASSIFICATIONS properties are constant along the waveguide (i.e., z) axis. Every type of waveguide has an Lawrence, Kansas 227 © 2006 by Taylor & Francis Group, LLC Figure 7.1 shows a uniform waveguide, whose cross-sectional dimensions and material infinite number of distinct electromagnetic field configurations that can exist inside it. Each of these configurations is called a mode. The characteristics of these modes depend upon the cross-sectional dimensions of the conducting cylinder, the type of dielectric material inside the waveguide, and the frequency of operation. When waveguide properties are uniform along the z axis, the phasors representing the forward-propagating (i.e., þz) time-harmonic modes vary with the longitudinal coordinate z as E,H/ e z , where the e j!t phasor convention is assumed. The parameter  is called the propagation constant of the mode and is, in general, complex valued:  ¼  þj ð7:1Þ where j ¼ ffiffiffiffiffiffiffi 1 p ,  is the modal attenuation constant, which controls the rate of decay of the wave amplitude,  is the phase constant, which controls the rate at which the phase of the wave changes, which in turn controls a number of other modal characteristics, including wavelength and velocity. Waveguide modes are typically classed according to the nature of the electric and magnetic field components that are directed along the waveguide axis, E z and H z , which are called the longitudinal components. From Maxwell’s equations, it follows that the transverse compo nents (i.e., directed perpendicular to the direction of propagation) are related to the longitudinal componen ts by the relations [1] E x ¼ 1 h 2  @E z @x þ j! @H z @y  ð7:2Þ E y ¼ 1 h 2  @E z @y  j! @H z @x  ð7:3Þ H x ¼ 1 h 2 j!" @E z @y þ  @H z @x  ð7:4Þ H y ¼ 1 h 2 j!" @E z @x þ  @H z @y  ð7:5Þ where, h 2 ¼ k 2 þ  2 ð7:6Þ k ¼ 2f ffiffiffiffiffiffi " p is the wave number of the dielectric, f ¼!/2p is the operating frequency in Hz, and  and " are the permeability and permittivity of the dielectric, respectively. Similar Figure 7.1 A uniform waveguide with arbitrary cross section. 228 Demarest © 2006 by Taylor & Francis Group, LLC expressions for the transverse fields can be derive d in other coordinate systems, but regardless of the coordinate system, the transverse fields are completely determined by the spatial derivatives of longitudinal field components across the cross section of the waveguide. Several types of modes are possible in waveguides. TE modes: Transverse-electric modes, sometimes called H modes. These modes have E z ¼0 at all points within the waveguide, which means that the electric field vector is always perpendicular (i.e., transverse) to the waveguide axis. These modes are always possible in metal waveguides with homogeneous dielectrics. TM modes: Transverse-ma gnetic modes, sometimes called E modes. These modes have H z ¼0 at all points within the waveguide, which means that the magnetic field vector is perpendicular to the waveguide axis. Like TE modes, they are always possible in metal waveguides with homogeneous dielectrics. EH modes: These are hybrid modes in which neither E z nor H z is zero, but the characteristics of the transverse fields are controlled more by E z than H z . These modes usually occur in dielectric waveguides and metal waveguides with inhomogeneous dielectrics. HE modes: These are hybrid modes in which neither E z nor H z is zero, but the characteristics of the transverse fields are controlled more by H z than E z .Like EH modes, these modes usually occur in dielectric waveguides and in metal waveguides with inhomogeneous dielectrics. TEM modes: Transverse-electromagnetic modes, often called transmission-line modes. These modes can exist only when more than one conductor with a complete dc circuit path is present in the waveguide, such as the inner and outer conductors of a coaxial cable. These modes are not considered to be waveguide modes. Both transmission lines and waveguides are capable of guiding electromagnetic signal energy over long distances, but waveguide modes behave quite differently with changes in frequency than do transmission-line modes. The most important difference is that waveguide modes can typically transport energy only at frequencies above distinct cutoff frequencies, whereas transmission line modes can transport energy at frequencies all the way down to dc. For this reason, the term transmission line is reserved for structures capable of supporting TEM modes, whereas the term waveguide is typically reserved for structures that can only support waveguide modes. 7.3. MODAL FIELDS AND CUTOFF FREQUENCIES For all uniform waveguides, E z and H z satisfy the scalar wave equation at all points within the waveguide [1]: r 2 E z þ k 2 E z ¼ 0 ð7:7Þ r 2 H z þ k 2 H z ¼ 0 ð7:8Þ where r 2 is the Laplacian operator and k is the wave number of the dielectric. However, for þz propagating fields, @ðÞ=@z ¼ðÞ, so we can write r 2 t E z þ h 2 E z ¼ 0 ð7:9Þ Waveguides and Resonators 229 © 2006 by Taylor & Francis Group, LLC and r 2 t H z þ h 2 H z ¼ 0 ð7:10Þ where h 2 is given by Eq. (7.5) and r 2 t is the transverse Laplacian operator. In Cartesian coordinates, r 2 t ¼ @ 2 =@x 2 þ @ 2 =@y 2 . When more than one dielectric is present , E z and H z must satisfy Eqs. (7.9) and (7.10) in each region for the appropriate value of k in each region. Modal solutions are obtained by first finding general solutions to Eqs. (7.9) and (7.10) and then applying boundary conditions that are appropriate for the particular waveguide. In the case of metal waveguides, E z ¼0and@H z =@p ¼ 0 at the metal walls, where p is the direction perpendicula r to the waveguide wall. At dielectric–dielectric interfaces, the E- and H-field components tangent to the interfaces must be continuous. Solutions exist for only certain values of h, called modal eigenvalues. For metal waveguides with homogeneous dielectrics, each mode has a single modal eigenvalue, whose value is independent of frequency. Waveguides with multiple dielectrics, on the other hand, have different modal eigenvalues in each dielectric region and are functions of frequency, but the propagation constant  is the same in each region. Regardless of the type of waveguide, the propagation constant  for each mode is determined by its modal eigenvalue, the frequency of operation, and the dielectric properties. From Eqs. (7.1) and (7.6), it follows that  ¼  þj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2  k 2 p ð7:11Þ where h is the modal eigenvalue associated with the dielectric wave number k. When a waveguide has no material or radiation (i.e., leakage) loss, the modal eigenvalues are always real-valued. For this case,  is either real or imaginary. When k 2 > h 2 ,  ¼0 and >0, so the modal fields are propagating fields with no attenuation. On the other hand, when k 2 < h 2 , >0, and  ¼0, which means that the modal fields are nonpropagating and decay exponentially with distance. Fields of this type are called evanescent fie lds. The frequency at which k 2 ¼h 2 is called the modal cuttoff frequency f c . A mode operated at frequencies above its cutoff frequency is a propagating mode. Conversely, a mode operated below its cutoff frequency is an evanescent mode. The dominant mode of a waveguide is the one with the lowest cutoff frequency. Although higher order modes are often useful for a variety of specialized uses of wave- guides, signal distortion is usually minimized when a waveguide is operated in the frequency range where only the dominant mode is propagating. This range of frequencies is called the dominant range of the waveguide. 7.4. PROPERTIES OF METAL WAVEGUIDES Metal waveguides are the most commonl y used waveguides at RF and microwave frequen- cies. Like coaxial transmission lines, they confine fields within a conducting shell, which reduces cross talk with other circuits. In addition, metal waveguides usually exhibit lower losses than coaxial transmission lines of the same size. Although they can be constructed using more than one dielectric, most metal waveguides are simply metal pipes filled with a homogeneous dielectric—usually air. In the remainder of this chapter, the term metal waveguides will denote self-enclosed metal waveguides with homogeneous dielectrics. 230 Demarest © 2006 by Taylor & Francis Group, LLC Metal waveguides have the simplest electrical characteristics of all waveguide types, since their modal eigenvalues are functions only of the cross-sectional shape of the metal cylinder and are independent of frequency. For this case, the amplitude and phase constants of any allowed mode can be written in the form:  ¼ h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  f f c  2 s for f < f c 0 for f > f c 8 > < > : ð7:12Þ and  ¼ 0 for f < f c h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f f c  2 1 s for f > f c 8 > > > < > > > : ð7:13Þ where f c ¼ h 2 ffiffiffiffiffiffi " p ð7:14Þ Each mode has a unique modal eigenvalue h, so each mode has a specific cutoff frequency. The mode with the smallest modal eigenvalue is the dominant mode. If two or more modes have the same eigenvalue, they are degenerate modes. 7.4.1. Guide Wavelength The distance over which the phase of a propagating mode in a waveguide advances by 2 is called the guide wavelength l g . For metal waveguides,  is given by Eq. (7.13), so l g for any mode can be expressed as l g ¼ 2  ¼ l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ðf c =f Þ 2 q ð7:15Þ where l ¼ðf ffiffiffiffiffiffi " p Þ 1 is the wavelength of a plane wave of the same frequency in the waveguide dielectric. For f f c , l g l. Also, l g !1as f ! f c , which is one reason why it is usually undesirable to operate a waveguide mode near modal cutoff frequencies. 7.4.2. Wave Impedance Although waveguide modes are not plane waves, the ratio of their transverse electric and magnetic field magnitudes are constant throughout the cross sections of the metal waveguides, just as for plane waves. This ratio is called the modal wave impedance and has the following values for TE and TM modes [1]: Z TE ¼ E T H T ¼ j!  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ðf c =f Þ 2 q ð7:16Þ Waveguides and Resonators 231 © 2006 by Taylor & Francis Group, LLC and Z TM ¼ E T H T ¼  j!" ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f c =fðÞ 2 q ð7:17Þ where E T and H T are the magnitudes of the transverse electric and magnetic fields, respectively, and  ¼ ffiffiffiffiffiffiffiffi =" p is the intrinsic impedance of the dielectric. In the limit as f !1, both Z TE and Z TM approach . On the other hand, as f ! f c , Z TE !1and Z TM ! 0, which means that the transverse electric fields are dominant in TE modes near cutoff and the transverse magnetic fields are dominant in TM modes near cutoff. 7.4.3. Wave Veloci ties The phase and group velocities of waveguide modes are both related to the rates of change of the modal propagation constant  with respect to frequency. The phase velocity u p is the velocity of the phase fronts of the mode along the waveguide axis and is given by [1] u p ¼ !  ð7:18Þ Conversely, the group velocity is the velocity at which the amplitude envelopes of narrow- band, modulated signals propagate and is given by [1] u g ¼ @! @ ¼ @ @!  1 ð7:19Þ Unlike transmission-line modes, where  is a linear function frequency,  is not a linear function of frequency for waveguide modes; so u p and u g are not the same for waveguide modes. For metal waveguides, it is found from Eqs. (7.13), (7.18), and (7.19) that u p ¼ u TEM ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f c =fðÞ 2 q ð7:20Þ and u g ¼ u TEM ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f c =fðÞ 2 q ð7:21Þ where u TEM ¼ 1= ffiffiffiffiffiffi " p is the velocity of a plane wave in the dielectric. Both u p and u g approach u TEM as f !1, which is an indication that waveguide modes appear more and more like TEM modes at high frequencies. But near cutoff, their behaviors are very different: u g approaches zero, whereas u p approaches infinity. This behavior of u p may at first seem at odds with Einstein’s theory of special relativity, which states that energy and matter cannot travel faster than the vacuum speed of light c. But this result is not a violation of Einstein’s theory since neither information nor energy is conveyed by the phase of a steady-state waveform. Rather, the energy and information are transported at the group velocity, which is always less than or equal to c. 232 Demarest © 2006 by Taylor & Francis Group, LLC 7.4.4. Dispersion Unlike the modes on transmission lines, which exhibit differential propagation delays (i.e., dispersion) only when the materials are lossy or frequency dependent, waveguide modes are always dispersive, even when the dielectric is lossless and walls are perfectly conducting. The pulse spread per meter Át experienced by a modulated pulse is equal to the difference between the arrival times of the lowest and highest frequency portions of the pulse. Since the envelope delay per meter for each narrow-band components of a pulse is equal to the inverse of the group velocity at that frequency, we find that the pulse spreading Át for the entire pulse is given by Át ¼ 1 u g     max  1 u g     min ð7:22Þ where 1=u g   max and 1=u g   min are the maximum and minimum inverse group velocities encountered within the pulse bandwidth, respectively. Using Eq. (7.21), the pulse spreading in metal waveguides can be written as Át ¼ 1 u TEM 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f c =f min ðÞ 2 q  1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f c =f max ðÞ 2 q 0 B @ 1 C A ð7:23Þ where f min and f max are the minimum and maximum frequencies within the pulse 3-dB bandwidth. From this expression, it is apparent that pulse broadening is most pronounced when a waveguide mode is operated close to its cutoff frequency f c . The pulse spreading specified by Eq. (7.23) is the result of waveguide dispersion, which is produced solely by the confinement of a wave by a guiding structure and has nothing to do with any frequency-dependen t parameters of the waveguide materials. Other dispersive effects in waveguides are material dispersion and modal dispersion. Material dispersion is the result of frequency-dependent characteristics of the materials used in the waveguide, usually the dielectric. Typically, material dispersion causes higher frequencies to propagate more slowly than lower frequencies. This is often termed normal dispersion. Waveguide disper sion, on the other hand, causes the opposite effect and is often termed anomalous dispersion. Modal dispersion is the spreading that occurs when the signal energy is carried by more than one waveguide mode. Since each mode has a distinct group velocity, the effects of modal dispersion can be very severe. However, unlike waveguide dispersion, modal dispersion can be eliminated simply by insuring that a waveguide is operated only in its dominant frequency range. 7.4.5. Ef fects of Losses There are two mechanisms that cause losses in metal waveguides: dielectric losses and metal losses. In both cases, these losses cause the amplitudes of the propagating modes to decay as e az , where  is the attenuation constant, measured in units of Nepers per meter. Typically, the attenuation constant is considered as the sum of two components:  ¼ d þ c, where  d and  c are the attenuation constants due to dielectric and metal losses alone, respectively. In most cases, dielectric losses are negligible compared to metal losses, in which case   c . Waveguides and Resonators 233 © 2006 by Taylor & Francis Group, LLC Often, it is useful to specify the attenuation constant of a mode in terms of its decibel loss per meter length, rather than in Nepers per meter. The conversion formula between the two unit conventions is  ðdB=mÞ¼8:686  ðNp=mÞð7:24Þ Both unit systems are useful, but it should be noted that  must be specified in Np/m when it is used in formulas that contain the terms of the form e z . The attenuation constant  d can be found directly from Eq. (7.11) simply by generalizing the dielectric wave number k to include the effect of the dielectric conductivity . For a lossy dielectric, the wave number is given by k 2 ¼ ! 2 " 1 þ =j!"ðÞ, where  is the conductivity of the dielectric, so the attenuation constant  d due to dielectric losses alone is given by  d ¼ Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2  ! 2 " 1þ  j!"  s ! ð7:25Þ where Re signifies ‘‘the real part of ’’ and h is the modal eigenvalue. The effect of metal loss is that the tangential electric fields at the conductor boundary are no longer zero. This means that the modal fields exist both in the dielectric and the metal walls. Exact solutions for this case are much more complicated than the lossless case. Fortunately, a perturbational approach can be used when wall conductivities are high, as is usually the case. For this case, the modal field distributions over the cross section of the waveguide are dist urbed only slightly; so a perturbational approach can be used to estimate the metal losses except at frequencies very close to the modal cutoff frequency [2]. This perturbational approach starts by noting that the power transmitted by a waveguide mode decays as P ¼ P 0 e 2 c z ð7:26Þ where P 0 is the power at z ¼0. Differentiating this expression with respect to z, solving for  c , and noting that dP/dz is the negative of the power loss per meter P L ,it is found that a c ¼ 1 2 P L P ð7:27Þ Expressions for  c in terms of the modal fields can be found by first recognizing that the transmitted power P is integral of the average Poynting vector over the cross section S of the waveguide [1]: P ¼ 1 2 Re ð S E T H  E ds  ð7:28Þ where ‘‘*’’ indicates the complex conjugate, and ‘‘E’’ and ‘‘ T’’ indicate the dot and cross products, respectively. 234 Demarest © 2006 by Taylor & Francis Group, LLC Similarly, the power loss per meter can be estimated by noting that the wall currents are controlled by the tangential H field at the conducting walls. When conductivities are high, the wall currents can be treated as if they flow uniformly within a skin depth of the surface. The resulting expression can be expressed as [1] P L ¼ 1 2 R s þ C H jj 2 dl ð7:29Þ where R s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi f = p is the surface resistance of the walls ( and  are the permeability and conductivity of the metal walls, respectively) and the integration takes place along the perimeter of the waveguide cross section. As long as the metal losses are small and the operation frequency is not too close to cuttoff, the modal fields for the perfectly conducting case can be used in the above integral expressions for P and P L . Closed form expressions for  c for rectangular and circular waveguide modes are presented later in this chapter. 7.5. RECTANGULAR WAVEGUIDES A rectangular waveguide is shown in Fig. 7.2, consisting of a rectangular metal cylinder of wi dth a and height b, filled with a homogenous dielectric with permeability and permittivity  and ", respectively. By convention, it is assumed that a b. If the walls are perfectly conducting, the field components for the TE mn modes are given by E x ¼ H 0 j! h 2 mn n b cos m a x  sin n b y  exp j!tr mn zðÞ ð7:30aÞ E y ¼H 0 j! h 2 mn m a sin m a x  cos n b y  exp j!tr mn zðÞ ð7:30bÞ E z ¼ 0 ð7:30cÞ H x ¼ H 0  mn h 2 mn m a sin m a x  cos n b y  exp j!tr mn zðÞ ð7:30dÞ H y ¼ H 0  mn h 2 mn n b cos m a x  sin n b y  exp j!tr mn zðÞ ð7:30eÞ H z ¼ H 0 cos m a x  cos n b y  exp j!tr mn z ðÞ ð7:30fÞ Figure 7.2 A rectangular waveguide. Waveguides and Resonators 235 © 2006 by Taylor & Francis Group, LLC The modal eigenvalues, propagation constants, and cutoff frequencies are h mn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m a  2 þ n b  2 r ð7:31Þ  mn ¼  mn þ j mn ¼ jð2f Þ ffiffiffiffiffiffi " p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  f c mn f  2 s ð7:32Þ f c mn ¼ 1 2 ffiffiffiffiffiffi " p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m a  2 þ n b  2 r ð7:33Þ For the TE mn modes, m and n can be any positive integer values, including zero, so long as both are not zero. The field components for the TM mn modes are E x ¼E 0  mn h 2 mn  m a  cos m a x  sin n b y  exp j!tr mn z ðÞ ð7:34aÞ E y ¼E 0  mn h 2 mn n b sin m a x  cos n b y  exp j!tr mn zðÞ ð7:34bÞ E z ¼ E 0 sin m a x  sin n b y  exp j!tr mn zðÞ ð7:34cÞ H x ¼ E 0 j!" h 2 mn n b sin m a x  cos n b y  exp j!tr mn zðÞ ð7:34dÞ H y ¼E 0 j!" h 2 mn m a cos m a x  sin n b y  exp j!tr mn zðÞ ð7:34eÞ H z ¼ 0 ð7:34fÞ where the values of h mn ,  mn , and f c mn are the same as for the TE mn modes [Eqs. (7.31)–(7.33)]. For the TM mn modes, m and n can be any positive integer value except zero. The dominant mode in a rectangular waveguide is the TE 10 mode, which has a cutoff frequency of f c 10 ¼ 1 2a ffiffiffiffiffiffi " p ð7:35Þ The modal field patterns for this mode are shown in Fig. 7.3. the cutoff frequencies of the lowest order rectangular waveguide modes (referen ced to the Figure 7.3 Field configuration for the TE 10 (dominant) mode of a rectangular waveguide. (Adapted from Ref. 2 with permission.) 236 Demarest © 2006 by Taylor & Francis Group, LLC showsTable 7.1 [...]... 7: 46Þ 7: 47 The dominant mode in a circular waveguide is the TE11 mode, which has a cutoff frequency given by 0:293 fc11 ¼ pffiffiffiffiffiffi a "" 7: 48Þ The configurations of the electric and magnetic fields of this mode are shown in Fig 7. 7 Table 7. 2 shows the cutoff frequencies of the lowest order modes for circular waveguides, referenced to the cutoff frequency of the dominant mode The modal field patterns of. .. & o 7: 65Þ Requiring continuity of the transverse electric and magnetic fields at the cylinder endcaps z ¼ Æ d/2 yields the following resonance condition [11]:    d ¼ 2 tan þl%  À1 7: 66Þ where l is an integer Using Eqs (7. 63) and (7. 65), Eq (7. 66) can be solved numerically for ko to obtain the resonant frequencies The lowest order mode (for l ¼ 0) exhibits a less-than-unity number of half-wavelength... can be arranged in a variety of ways Figure 7. 16 shows three possibilities In the case of Fig 7. 16a, a coaxial line is positioned such that the E field of the desired resonator mode is tangential to the center conductor probe In the case of Fig 7. 16b, the loop formed from the coaxial line is positioned such that the H field of the desired mode is perpendicular to the plane of the loop For waveguide to... þ À 2 a 2 k n2 b2 þ m 2 a2 TE modes 7: 36Þ and mn ¼ 2Rs À Á1=2 b 1 À h2 =k2 mn  n2 b3 þ m 2 a3 n2 b2 a þ m 2 a3  TM modes 7: 37 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Rs ¼ %f "=' is the surface resistance of the metal,  is the intrinsic impedance of the dielectric ( 377  for air), "0m ¼ 1 for m ¼ 0 and 2 for m > 0, and the modal eigenvalues hmn are given by Eq (7. 31) Figure 7. 5 shows the attenuation constant for... frequency ranges, whereas each waveguide mode can exist over a broad range of frequencies 7. 9.1 Cylindrical Cavity Resonators A cylindrical cavity resonator is shown in Fig 7. 13, consisting of a hollow metal cylinder of radius a and length d, with metal end caps The resonator fields can be considered to be combinations of upward- and downward-propagating waveguide modes If the dielectric inside the resonator... jnm z e j!t hnm &  à Hz ¼ H0 Jn ðhnm &Þ cos n0 Aþ eÀjnm z þ AÀ e jnm z e j!t E0 ¼ H0 7: 51aÞ 7: 51bÞ 7: 51cÞ 7: 51dÞ 7: 51eÞ Here, the modal eigenvalues are hnm ¼ p0nm =a, where the values of p0nm are given by Eq (7. 40) To insure that E& and E0 vanish at z ¼ Æd/2, it is required that AÀ ¼ Aþ (even Figure 7. 13 A cylindrical cavity resonator © 2006 by Taylor & Francis Group, LLC 246 Demarest modes)... wavelength Similarly, for TMnml modes [9], 8 pnm > > 2%ð1 þ 2a=dÞ > <  Q ¼  à lo > p2 þ ðl%a=dÞ2 1=2 > nm > : 2%ð1 þ 2a=dÞ l¼0 TMnml modes 7: 61Þ l>0 Figure 7. 15 shows the Q values of some of the lowest order modes as a function of the of the cylinder radius-to-length ratio Here it is seen that the TE012 has the highest Q, which makes it useful for applications where a sharp resonance is needed This... of several lower order modes are shown in Fig 7. 8 Figure 7. 7 Field configuration for the TE11 (dominant) mode in a circular waveguide (Adapted from Ref 2 with permission.) © 2006 by Taylor & Francis Group, LLC Waveguides and Resonators 241 Table 7. 2 Cutoff Frequencies of the Lowest Order Circular Waveguide Modes fc/fc11 Modes 1.0 1.3 07 1.66 2.083 2.283 2 .79 1 2.89 3.0 TE11 TM01 TE21 TE01, TM11 TE31 TM21... impedance of the waveguide and lowers its phase velocity This reduced phase velocity results in a lowering of the cutoff frequency of the dominant mode by a factor of 5 or higher, depending upon the dimensions of the ridges Thus, the dominant range of a ridge waveguide is much greater than that of a standard rectangular waveguide However, this increased frequency bandwidth is obtained at the expense of increased... Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Wiley: New York, 1 972 , 358– 374 Kretzschmar, J Wave propagation in hollow conducting elliptical waveguides IEEE Transactions on Microwave Theory and Techniques 1 970 , 18: 5 47 554 Montgomery, J On the complete eigenvalue solution of ridged waveguide IEEE Transactions on Microwave Theory and Techniques 1 971 , 19, 4 57 555 Collin, . configurations of the electric and magnetic fields of this mode are shown in Fig. 7. 7. referenced to the cutoff frequency of the dominant mode. The modal field patterns of Figure 7. 7 Field configuration. polarization of the incident field is critical. Figure 7. 10 Coaxial-to-rectangular waveguide transition that couples the coaxial line to the TE 10 waveguide mode. Figure 7. 11 Coaxial-to-rectangular. minimized, and Tabl e 7. 2 Cutoff Frequencies of the Lowest Order Circular Wave- guide Modes. f c /f c11 Modes 1.0 TE 11 1.3 07 TM 01 1.66 TE 21 2.083 TE 01 ,TM 11 2.283 TE 31 2 .79 1 TM 21 2.89 TE 41 3.0