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# Analysis of multiple groove guide

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### Analysis of multiple groove guide

1. 1. Analysis of Multiple Groove Guide Hyo J. Eom and Yong H. Cho Department of Electrical Engineering Korea Advanced Institute of Science and Technology 373-1, Kusong Dong, Yusung Gu, Taejon, Korea Phone 82-42-869-3436, Fax 82-42-869-8036 E-mail : hjeom@ee.kaist.ac.kr Abstract Wave propagation along a rectangular multiple groove guide is rigorously studied. The Fourier transform is used to obtain simultaneous equations for the modal coe cients in rapidly-convergent form. The dispersion characteristics of a multiple groove guide and its eld distribution plots are presented. 1 Introduction A rectangular groove guide 1] is a low-loss and high-power guiding structure. A double-groove guide has been extensively studied to assess its utility as a waveguide or power coupler at 100GHz in 2]. It is of practical interest to understand guiding and coupling characteristics of a multiple groove guide which consists of a nite number of parallel rectangular groove guides. The purpose of the present short paper is to present an exact and rigorous solution for a multiple groove guide by utilizing the Fourier transform which was used to analyze a single rectangular groove guide 3]. The Fourier transform approach allows us to represent a solution in rapidly convergent 1
2. 2. series. In next section, we present a rigorous dispersion relation for a multiple groove guide. 2 Field Analysis Consider a multiple rectangular groove guide in Fig. 1 (N : the number of groove guides). Assume the TE-wave propagates along the z-direction such as H (x; y; z) = H (x; y)ei z and the e i!t time-factor is suppressed throughout. In region (I) ( d < y < 0), (II) (0 < y < b), and (III) (b < y < b + d), the Hz components are HzI (x; y) = N 1X X 1 n=0 m=0 n qm cos am (x nT ) cos m(y + d) u(x nT ) u(x nT a)]; 1 ~ ~ HzII (x; y) = 21 Hz ei y + Hz e i y ]e i xd ; 1 N 1 sn cos am (x nT ) cos m (y b d) HzIII (x; y) = m Z + (1) (2) 1 X X n=0 m=0 u(x nT ) u(x nT a)]; q p (3) , = k , k = 0 and u( ) is where am = m , m = k am a n a unit step function. To determine the modal coe cients qm and sn , we enforce m the boundary conditions on the eld continuities. Applying the Fourier transform R ( 1 (:)ei xdx) on the Ex continuity at y = 0, we obtain 1 N 1 n ~z Hz = X X 1 qm m sin( md)Gn ( ); ~ H (4) m n m i where m ia (5) Gn ( ) = i 1 ( 1) e ] ei nT : m am Similarly from the Ex continuity at y = b, N X 1 X 1 n n ~ ~ Hz ei b Hz e i b = (6) i sm m sin( md)Gm( ): n m Multiplying the Hz continuity at y = 0 by cos al (x pT ), (p = 0; ; N 1), and integrating over pT < x < pT + a, we obtain N X X 1 n fqm m sin( md)I + a cos( md) ml np m ] + sn m sin( md)I g = 0; (7) m 2 n m 2 2 2 2 2 2 2 2 1 + =0 =0 2 2 1 + =0 =0 1 1 =0 =0 2
3. 3. where is the Kronecker delta, Z 1 cot( b) Gn ( )Gp( )d ; (8) I = 21 m l 1 Z 1 csc( b) Gn ( )Gp( )d ; (9) I = 21 m l 1 8 < 2 (m = 0); = : (10) m 1 ( m = 1; 2; 3; ) : Applying the residue calculus, we transform (8) and (9) into rapidly-convergent series as I = a m ml npb) 2 m tan( m 1 i X v (( 1)m l + 1)ei v jn pjT ( 1)m ei v j n p T aj ( 1)l ei v j n p T aj] ; bv v ( v am )( v al ) (11) I = a m ml np ) 2 m sin( mb 1 i X( 1)v v (( 1)m l + 1)ei v jn pjT ( 1)m ei v j n p T aj ( 1)l ei v j n p T aj] ; bv v ( v am )( v al ) (12) ml 1 2 1 + ( 2 =0 2 2 ) + ( ) 2 2 + ( 2 =0 q where v = k ( vb ) region (II) and (III), N 1X X 1 2 2 2 2 ) + ( 2 2 . Similarly from the Hz continuity at y = b between a cos( d) m ml np m ]g = 0: 2 n m A dispersion relationship may be obtained by solving (7) and (13) for . n fqm m sin( md)I + sn m m sin( m d)I1 + 2 =0 =0 1 np ;ml np 2;ml 1 1 and 2 2 2 where the elements of ) 1 are = m sin( m d)I1 + = m sin( m d)I2 : = 0; (13) (14) a cos( d) m 2 ml np m ; (15) (16) When N = 1 (single-groove case), (14) reduces to (33) in 3]. When N = 2 (doublegroove case), (14) in a dominant-mode approximation (m=0), reduces to 00 1 00 ; 00 2 00 ; = ( 3 10 1 00 ; 10 2 00 ; ); (17)
4. 4. where sign corresponds to the TE and TE modes, respectively. It is trivial to nd by using a root-searching scheme with an initial guess based on a singlegroove case. Fig. 2 illustrates the dispersion characteristics for a multiple groove guide, con rming that our solution agrees with 2] when N = 2. Our computational experience indicates that a dominant-mode approximation with m = 0 is almost identical with a more accurate solution including 3 higher-modes. Fig. 3 shows the magnitude plots of Hz component for the TE p modes (p = 1; ; 4) where p signi es the number of half-wave variation of Hz component along the x-axis. The eld plots illustrate that Hz remains almost uniform in the x-direction within the groove, thus con rming the validity of a dominant-mode approximation. 12 11 1 3 Conclusion A simple, exact and rigorous solution for the multiple groove guide is presented and its dispersions are numerically evaluated. Our numerical computation for the doublegroove dispersion characteristics agrees with other existing solution. The eld plots for a quadruple groove guide illustrate the eld distributions of various modes existing within the groove guide. References 1] A.A. Oliner and P. Lampariello, "The dominant mode properties of open groove guide: An improved solution," IEEE Trans. Microwave Theory Tech., Vol. MTT33, pp. 755-764, Sept. 1985. 2] D.J. Harris and K.W. Lee, Theoretical and experimental characteristics of double-groove guide for 100GHz operation," IEE Proc., Vol. 128, Pt. H. No. 1, pp. 6-10, Feb. 1981. 3] B.T. Lee, J.W. Lee, H.J. Eom and S.Y. Shin, Fourier-Transform Analysis for Rectangular Groove Guide" IEEE Trans. Microwave Theory Tech., Vol. MTT43, pp. 2162-2165, Sept. 1995. 4
5. 5. y b+d III b PEC III n=0 Region III ..... n=1 n=N-1 Region II z 0 -d a I T T+a I . . . . . (N-1)T PEC (N-1)T+a Region I Figure 1: Geometry of a multiple groove guide. 5 x
6. 6. β, phase constant [ rad/m ] 2028 2026 2026 2024 2024 2022 2022 2020 2020 Triple groove (N=3) TE12 TE13 TE12 TE13 2018 2018 5 5 Double groove (N=2) TE11 Quadruple groove (N=4) TE14 o o Harris experiment in [2] for double groove (N=2) 10 15 20 10 15 20 T, groove separation [ mm ] 25 25 Figure 2: Behavior of phase constant of TE , TE , TE and TE modes versus groove separation for o = 3:08mm; a = 5mm; b = 10mm, d = 2:5mm and N = 2; ; 4. 11 6 12 13 14
7. 7. 1 11 1 0.50.5 0.5 0.5 00 −0.5 -0.5 00 b/2 x 0 -1 −1 b/2 3T -d x 3T 0 0 y -d (a) 1 0 y (b) 1 1 1 0.50.5 0.50.5 00 0 −0.5 -0.5 0 -0.5−0.5 -1 −1 b/2 -1 3T 0 -d −1 x b/2 x 3T 0 0 -d y (c) 0 y (d) Figure 3: Hz eld distributions for (a) TE mode, (b) TE mode, (c) TE mode and (d) TE mode when N = 4. 11 14 7 12 13