Design and Analysis of Metallic Photonic Band Gap Cavity for a Gyrotron

Brazilian Microwave and Optoelectronics Society-SBMO received 11 May 2012; for review 18 May 2012; accepted 20 Aug 2012 Brazilian Society of Electromagnetism-SBMag © 2012 SBMO/SBMag ISSN 2179-1074 Abstract — Design and analysis of a 35 GHz metal photonic band gap (PBG) resonant cavity operating in the TE041-like mode has been presented. The dispersion characteristics of a 2D metal PBG structure comprising of triangular array of rods has been obtained using FDTD method. Global band gap regions have been obtained to be used for the PBG cavity design. A mode map has been also generated to examine the occurrence of the possible modes in the PBG cavity. Electromagnetic simulation of the designed PBG cavity has been performed to study the operating modes and quality factors. Role of the number of metal rods layers around the defect in PBG structure has also been illustrated for confining the desired mode and deciding the diffractive quality factor. PBG cavity results have been compared with the analogous cylindrical cavity for the designed mode along with the possible nearby modes to examine the mode competition. It has been found that the mode competition has been successfully reduced in designed PBG cavity and nearly a single mode operation is achieved. It is hoped that present study would be useful for the application of the metal PBG structures in the Gyrotron devices to alleviate the mode competition problem while operating in the higher order modes.

mode operation is another serious limitation. Use of the overmoded structures can surmount these issues, but mode competition becomes a serious problem. The overmoded RF cavities, conventionally, open ended tapered cylindrical resonators, can be replaced by the metallic photonic band gap (PBG) structures to control the problem of mode competition. Additionally, the transverse dimension of PBG cavity is larger than the conventional cavity for the same operating mode and frequency [4]. For the moderate power level at sub-millimeter wavelengths, PBG cavities could be a substitute to achieve a mode selective operation. Sirigiri et al. have successfully demonstrated experimental implementation of a PBG interaction cavity in a 140 GHz gyrotron [4]. Other electron beam devices, like, high gradient accelerator [5], multibeam reflex klystron [6], Cherenkov device using slow-wave structure [7], and backward wave oscillator [8] have been also proposed/realized incorporating the PBG structures for the merit of its mode selectivity.
In this paper, design and analysis of a 35 GHz metal PBG cavity operating in the higher order TE 041 -like mode is presented. The objective of this paper is to demonstrate the mode selective operation of the PBG cavities to be used in the gyrotron devices as its RF interaction structure while operating at the higher order modes. Dispersion diagram, i.e., band structure is obtained using Yeecell based FDTD method in the TE mode of propagation for a two dimensional triangular PBG lattice, realized from the metallic cylindrical rods in vacuum background. Global band gap diagram utility for the design of PBG cavity is demonstrated. Eigenmode simulation of the PBG cavity is also presented using a commercial 3D electromagnetic code, 'CST Microwave Studio'. Scattering parameter of the PBG cavity is also obtained to inquire the mode existence. Resonant frequencies obtained analytically are compared with the simulated values for different ratio of rod radius to lattice constant. The diffractive and ohmic quality factors of the PBG cavity are determined. Effect of structure layers around the defect on quality factors is observed. Further, resonant frequency and quality factor of the designed metal PBG cavity are compared with the analogous conventional cylindrical cavity including other possible nearby modes.

II. PBG STRUCTURE MODEL
Typically, a two dimensional metal PBG structure of triangular lattice is considered here to determine its dispersion characteristics. Since gyrotron is a fast wave device operating in the transverse electric mode, the present analysis is restricted to TE mode. The triangular lattice is For designing a PBG cavity, the choice of rod radius and lattice constant must be proper so that the desired mode at the specific frequency must lie in band gap and other nearby modes in pass band. For the present cavity design, typically, 19 innermost rods are removed from the triangular lattice formed by copper rods and 4 layers surrounding the defect are kept (Fig. 2). Two end plates have holes of different radii. Smaller radius hole ensures prevention of the RF signal propagation toward input side due to the higher cut-off frequency and bigger radius hole at the output end is used for conversion of standing mode into the travelling mode to couple out the RF radiation from the cavity [4].

III. FDTD ANALYSIS
A. Dispersion characteristics Photonic band structure is calculated using Yee-cell based finite difference time domain (FDTD) method and shown in Fig. 3(a). A modified unit cell is considered here as the computational domain for calculating the band structure with appropriate periodic Bloch boundary [9]. Band structure is Obviously, several band gap regions can be seen in the figure ( Fig. 3(a)). Global band gaps are examined for the entire range of r/a (from 0.01 to 0.5). These band gap regions are plotted as Fig.   3(b) and this plot is found identical with that obtained using MIT PBGSS code [10]. In Fig. 3 several band gap regions are observed which could be used as the operating frequencies of the PBG interaction cavities.

B.
Mode spectrum For the confined modes, defect radius of PBG interaction structure can be taken as similar to cylindrical waveguide using the following well known relation [11]: where, c is velocity of light, f r is resonant frequency of interaction structure and x m,n is eigenvalue, which is root of TE mode dispersion relation J' n (x). Here, J' n (x) is the derivative of n th order Bessel function of argument x. Since, gyrotron oscillator operates close to cutoff frequency; this approximation acquired from the cylindrical waveguide cut-off frequency equation is valid. Equation . ( The radius of PBG cavity by removing 19 innermost rods can be approximated as: Therefore, using equations (2)  Now, from equation (4)

IV. RESULTS AND DISCUSSION
The designed PBG cavity is simulated for its electromagnetic behavior using a commercial simulation code "CST Microwave Studio".

A.
Field distribution Simulation is carried out for the frequency range of 33-38 GHz using eigenmode solver to examine the existence of possible confined modes. All metal boundaries are considered with the tangential component of electric field zero (E t = 0). A well confined TE 041 -like mode is observed in the cavity at frequency 34.94 GHz (Fig. 5). In the simulation, few nearby modes are also observed in the cavity but their quality factor is much lower than that of TE 041 mode. Additionally, these modes are widely separated from the TE 041 mode. The ohmic quality factor for the TE 041 mode is found as 27150.

B.
Resonant frequency The normalized resonant frequency (f n ) is calculated analytically by varying r/a ratio for the TE 041like mode (equation (4)). Through EM simulation, the resonant frequencies of the PBG cavity are estimated for the different r/a ratio. Both analytical and simulation results are compared and found to be within 3% variation (Fig. 6). Further, it can be seen in Fig. 5 that the EM fields near metal posts are not perfectly confined in the defect and slightly penetrate into the rods region; hence, the cavity radius considered analytically is smaller than the effective radius. Consequently, simulated resonant frequencies are lower than the analytical values. The band gap width at the design point is 0.0286 (normalized). In the conventional cylindrical cavity, presence of all modes is inevitable, while in the PBG cavity, only that modes will be confined in defect which resonant frequency lie in the global band gap width. Consequently, PBG cavity provides a mode selectivity.

C.
Scattering parameter In order to examine the reflection and transmission behavior of this metal PBG cavity, structure is also simulated using transient solver of 'CST Microwave Studio' (Fig. 7). It can be observed from the figure 7 that only at 34.94 GHz, reflection coefficient (S 11 ) is very low and transmission coefficient (S 21 ) is maximum. This reveals the existence of a propagating mode at this frequency in the PBG cavity with no mode competition. Obviously, existence of some other modes is apparent widely apart with much higher reflection, lower transmission and quality factor. The bandwidth of PBG cavity is around 20 MHz. This small bandwidth provides a mode selectivity as well as higher quality factor in comparison to the cylindrical cavity.

D. Quality factor
In order to estimate the different structure losses in terms of the cavity quality factor (Q), simulations were performed for both diffractive as well as ohmic quality factors using the relevant boundary conditions. The associated structure losses are Q ohmic and Q diffractive , which accounts for losses due to finite conductivity of the metal surface and EM field leakage through the PBG cavity structure, respectively. The overall quality factor can be obtained using well know relation, ( ) ( ) e diffractiv ohmic total Quality factor (Q) of metal PBG cavity is determined using transient solver tool of CST Microwave Studio. For Q diffractive estimation, entire cavity volume with input and output beam holes (shown in Fig.   2) is excited by a discrete current port at its centre for the excitation frequency range 30 -40 GHz.
Field probe is positioned inside the cavity to record electric field spectrum at the maximum value which yields the diffractive quality factor using the full width half maximum. The energy decays very slowly which demonstrate the resonant behavior of the interaction structure ( Fig. 8(a)). The field maximum is observed at the desired of 34.94 GHz. The diffractive quality factor value, with 4 layers of rods around the PBG defect forming the cavity, is obtained as 1750 ( Fig. 8(b)). For the estimation of ohmic quality factor Q ohmic of PBG cavity, eigenmode solver of CST Microwave Studio is used with all metal boundaries. For calculating the ohmic quality factor, the copper material has been considered (electrical conductivity = 5.8x10 7 S/m, permeability = 1, density = 8930 kg/m 3 , thermal conductivity = 401 W/K/m). Surface roughness has been assumed to be zero, i.e., smooth surface has been considered. Temperature variation of rods has not been taken care of in simulation while boundary temperature has been kept isothermal. No cooling of the rods has been involved.

E.
Effect of number of PBG structure layers on the cavity loss and quality factor Diffraction of EM fields from the layers of rods decreases with the increase in number of layers around the cavity. Moreover, this reduction of the field outside PBG cavity results in better confinement of EM fields inside the PBG cavity thereby enhancing the diffractive quality factor of the cavity. The variation of the quality factors with number of layers around the defect are investigated keeping the other structure parameters constant. Fig. 9 demonstrates the field pattern in the PBG cavity at 34.94 GHz frequency. Variation of quality factors with the number of layers of the metal rods is shown in Fig. 10. Sufficient number of rod layers is required to reduce diffraction losses due to EM wave leakage from the gaps between the metal rods. On the other hand, ohmic quality factor of the PBG cavity decreases with the increase in the number of metal rod layers around the defect. This happens because of increase in ohmic losses involved due to increase in the imperfect metal boundaries in the form of rods of the PBG structure. Ultimately, total quality factor of the PBG cavity enhances with the increase in number of rod layers around the defect. So, for well confinement of the desired mode within the defect, sufficient number of rod layers are required which also leads to high cavity quality factor.     To validate the analysis and design of the metallic PBG cavity, EM simulation is also performed using commercial simulation code 'CST Microwave Studio'. Simulation result confirms a well confined TE 041 -like mode at 35 GHz resonant frequency in the designed metal PBG cavity. In this cavity, no other competing mode with appreciable Q-factor is found. The analytical resonant frequency agrees with the simulated value ~ 3%. Scattering parameter curve also supports the existence of a single mode inside the PBG cavity structure. By increasing the number of metal rods layers around the defect in the PBG structure, diffractive quality factor increases and consequently the field confinement is better. Simultaneously, the ohmic quality factor decreases due to increased losses involving rods. Diffraction of EM fields through the lattice of metal rods decreases with the increase in the number of metal rod layers. Consequently, for the well confinement of a desired mode with high Q in defect region; at least 3-4 layers of rods are necessary. Comparing the designed PBG cavity with the analogous cylindrical cavity for the frequency and ohmic quality factor of the designed mode and possible nearby modes, it has been found that PBG cavity do not encourage competing modes.
The present analysis and design study of the metal PBG cavity structure would be useful in understanding the electromagnetic behavior of the metal PBG structure and its application as the RF propagating structures for the active as well as passive devices in the microwave to sub-millimeter wave frequency range.