1High gain metal-only reflectarray antenna composedof multiple rectangular groovesYong Heui Cho , Woo Jin Byuny, Myung Sun ...
2field isHi(x;y;z) = 12eiki r^i ; (1)where the incident angles, i and i, are defined as i =and i = + in terms of the - and -...
3potential A(r). Inserting (16) into (17) yieldsRHmn(x;y;z)= a2m + b2n2 2Z 10sin( z)(k222)( 2 2mn)nZ 11( 1)nfH(y;b; ) fH(y...
4where ^n ^z = ^t illustrated in Fig. 1. The expression forREEmn(x;y;z) is obtained from (29) and (31), and written byREEm...
5(2(p)1(p)mn cos( (p)mnd(p))a(p)b(p)m n ml nk prsin( (p)mnd(p))hi (p)mna(p)b(p)m n ml nk pr+ IHmn;lk(x0;y0) +IEHmn;lk(x0;y...
6T(p) = T, S(p+PX) S(p) = S for all p in (39) and (40),where PX is the number of grooves in the x-axis. A depth forthe pth...
7−90 −60 −30 0 30 60 90010203040Observation angle, θ [Degree]Co−andcross−polegainpatterns[dB]Crosspole, φ = 0°Crosspole, φ...
8to a fast convergent integral without singularities asQm(x; a;y;y0;k2)= sgn(y y0)(ei mjy y0j cosam(x + a)ux(a)i2Z 11ddu e...
9sina(r)l (x x0 + a(r))sinb(r)k (y y0 + b(r)) dydx :(64)Since the integrands from (60) to (64) are composed ofsimple eleme...
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High gain metal only reflectarray antenna composed of multiple rectangular grooves

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High gain metal only reflectarray antenna composed of multiple rectangular grooves

  1. 1. 1High gain metal-only reflectarray antenna composedof multiple rectangular groovesYong Heui Cho , Woo Jin Byuny, Myung Sun SongyDepartment of Electrical and Computer Engineering, University of Massachusetts Amherst, on sabbatical leavefrom School of Information and Communication Engineering, Mokwon University, KoreayMillimeter-Wave Research Team, Electronics and Telecommunications Research Institute, KoreaAbstract—Using an overlapping T-block method based onmode-matching technique and virtual current cancellation, thescattering formulations for a metal-only reflectarray antennacomposed of multiple rectangular grooves are rigorously pre-sented in fast convergent integrals. By matching the normalboundary conditions at boundaries, we get the simultaneousequations of the TE and TM modal coefficients for plane-waveincidence and Hertzian dipole excitation. A metallic-rectangular-grooves based reflectarray fed by a pyramidal horn antennawas fabricated and measured with near-field scanning, thusresulting in 42.3 [dB] antenna gain at f = 75 [GHz]. Our circularreflectarray has 30 [cm] diameter and 5,961 rectangular grooveson its metal surface. The simultaneous equations for a Hertziandipole feed are solved to approximately obtain the radiationcharacteristics of a fabricated reflectarray. The measured resultsare compared with those of the overlapping T-block method andthe FDTD simulation and they show favorable agreements interms of radiation patterns and antenna gain.I. INTRODUCTIONIn the millimeter-wave frequency bands, there have beena lot of interesting applications in the fields of broadbandradio links for backhaul networking of cellular base stations,Gbps-class Wireless Personal Area Network (WPAN), andmillimeter-wave imaging to detect concealed weapons andnon-metal objects [1]-[4]. Especially, 70 and 80 GHz commu-nication systems within 1-mile distance will play an importantrole in the next-generation wireless networks, because the cellsite connectivity will require more than 1 Gbps data rates.For these millimeter-wave bands, a well-designed antennafor narrow beamwidth and high antenna gain is essential tocompensation for severe free-space path loss and to preventionagainst signal interference. Although a high gain antennacan be manufactured by the concept of a parabolic reflectorantenna, a reflectarray antenna has been extensively studiedin [5]-[8], due to the fact that the reflectarray technologyhas several advantages such as low profile, low cost, easymanufacturing, low feeding loss, and simple controllability ofmainbeam and polarization.Th reflectarray antenna in [5] consists of rectangular waveg-uide arrays and a feed waveguide to reflect electromagneticwaves efficiently. The phases of reflected waves are controlledbased on the variation of a surface impedance. In view ofa transmission line theory, the variation of waveguide depthresults in that of surface impedance at the end of each waveg-uide. This indicates that reradiated waves can be designedin the predefined way and then the high-gain antenna with11,µεInfinite flangePECa2b2xy22,µεdiθIncidenceziφφθ, xyFig. 1. Geometry of a metallic rectangular groove in a perfectly conductingplanevery low loss can be easily implemented. Modern reflectarrays[6]-[8] have the same operational principle of the originalreflectarray [5]. In addition, the metal-only high gain antennasin [8], [9] were proposed for the millimeter-wave bands, wherethe loss characteristics are very important to maintain goodcommunication links.In this work, we propose a novel analytic approach suitablefor a metal-only metallic-rectangular-grooves based reflectar-ray. A two-dimensional (2D) metal-only reflectarray antennahas already been analyzed and fabricated in [8]. In orderto analyze the problem of a three-dimensional (3D) metal-only metallic-rectangular-grooves based reflectarray, we willintroduce an overlapping T-block method [8], [10] based onmode-matching technique, virtual current cancellation, andsuperposition principle, thereby obtaining analytic scatteringequations in rapidly convergent and numerically efficient inte-grals. In view of mode-matching and Green’s function, we canrepresent the discrete modal expansions for closed regions andthe continuous wavenumber spectra for open regions. Usingvirtual current cancellation by means of virtual PEC coversrelated to closed regions, the vector potential formulations foropen regions are easily evaluated. To apply the superpositionprinciple, we divide multiple rectangular grooves into severalsimple T-blocks and a source block [8], [10].II. FIELD REPRESENTATIONS FOR SINGLE GROOVELet’s consider the TE plane-wave incidence (Ez = 0) withincident angles, i and i, shown in Fig. 1. The time conven-tion e i!t is suppressed throughout. The incident magnetic
  2. 2. 2field isHi(x;y;z) = 12eiki r^i ; (1)where the incident angles, i and i, are defined as i =and i = + in terms of the - and -axes,ki = k2 (sin i cos i^x + sin i sin i^y cos i^z) (2)^i = cos i cos i^x cos i sin i^y sin i^z ; (3)r = x^x + y^y + z^z, k2 = !p 2 2, and 2 =p2= 2. Theincident electric field for the TE plane-wave is also formulatedasEi (x;y;z) = ( sin i^x + cos i^y)eiki r : (4)In order to match boundary conditions efficiently, we definethe reflected electromagnetic waves from a perfectly conduct-ing plane at z = 0 asHr(x;y;z) = 12eikr r^r (5)Er(x;y;z) = (sin i^x cos i^y)eikr r ; (6)wherekr = k2 (sin i cos i^x + sin i sin i^y + cos i^z) (7)^r = cos i cos i^x + cos i sin i^y sin i^z : (8)Similar to the TE plane-wave, we obtain the TM incident andreflected plane-waves (Hz = 0) as, respectively,Ei(x;y;z) = eiki r^i (9)Er(x;y;z) = eikr r^r : (10)Considering the TE (ui) and TM (vi) polarizations, the inci-dent and reflected electric fields are represented asEi(x;y;z) =h(ui sin i + vi cos i cos i)^x+ (ui cos i vi cos i sin i)^yvi sin i^zieiki r (11)Er(x;y;z) =h(ui sin i + vi cos i cos i)^x(ui cos i vi cos i sin i)^yvi sin i^zieikr r ; (12)where ui and vi are polarization constants for the TE and TMmodes, respectively.Since all electromagnetic fields can be formulated withcorresponding electric and magnetic vector potentials, weintroduce the electric vector potentials for regions (I) (z < 0)and (II) (z 0) illustrated in Fig. 1 asFIz (x;y;z) = 11Xm=01Xn=0qmn cosam(x + a)cosbn(y + b)sin mn(z + d)uxy(a;b) (13)FIIz (x;y;z) = 21Xm=01Xn=0smnhHmn(x;y;z)+ RHmn(x;y;z)i; (14)b2PECsurfacea222 ,µεdVirtualPECcover11,µεRadiationboundaryxn ˆˆ =xyzyn ˆˆ −=Fig. 2. Artificial geometry for virtual current cancellationwhere m+n 6= 0, qmn and smn are the unknown TE modal co-efficients for regions (I) and (II), respectively, am = m =(2a),bn = n =(2b), mn =pk21 a2m b2n, k1 = !p 1 1,uxy(a;b) = u(x+a) u(x a)] u(y+b) u(y b)], and u( )is a unit-step function. Enforcing the Hz field continuity atz = 0 yieldssmn = qmn sin( mnd) ; (15)where we presume that RHmn(x;y;0) = 0,Hmn(x;y;z) = ei mnz cosam(x + a)cosbn(y + b)uxy(a;b) ; (16)and mn =pk22 a2m b2n. Note that Hmn(x;y;z) in (16)is formulated with virtual PEC covers placed at ( a xa;y = b;z > 0) and (x = a; b y b;z > 0)shown in Fig. 2. Even though the virtual PEC covers in Fig.2 are absent from the original geometry illustrated in Fig. 1,the PEC covers are artificially inserted to accommodate thefield formulations which will be described in (17). In orderto make the fields Hmn(x;y;z) + RHmn(x;y;z) continuousfor z > 0, we define a scattered component RHmn(x;y;z)which is implicitly related to Hmn(x;y;z). Adding the vir-tual PEC covers inevitably generates the unwanted surfacecurrent densities on ( a x a;y = b;z > 0) and(x = a; b y b;z > 0). By means of the Green’sfunction, RHmn(x;y;z) is utilized to remove a current densityJ(r) (= ( @2@z2 + k22)Hmn(r)^z ^n) produced by the Hz-fielddiscontinuities. Then,@2@z2 + k22 RHmn(x;y;z)= i! r A(r)z component=Z @2@z02 + k22 Hmn(r0) @@nhGxxA (r;r0)idr0 ; (17)where ^n is an outward normal unit vector denoted in Fig.2 and GxxA (r;r0) indicates the x-directional Green functionexcited by the x-directed source in terms of a magnetic vector
  3. 3. 3potential A(r). Inserting (16) into (17) yieldsRHmn(x;y;z)= a2m + b2n2 2Z 10sin( z)(k222)( 2 2mn)nZ 11( 1)nfH(y;b; ) fH(y; b; )Gm(a; )ei x d+Z 11( 1)mfH(x;a; ) fH(x; a; )Gn(b; )ei y dod ; (18)where k22 = 2 + 2 + 2,fH(y;y0; ) = sgn(y y0)ei jy y0j (19)Gm(a; ) = i e i a ( 1)mei a]2 a2m; (20)and sgn( ) = 2u( ) 1. It should be noted that our previousassumption such as RHmn(x;y;0) = 0 is right because ofsin( z) = 0z=0. To avoid pole singularities at = nmand = k2 on the real axis, we deform the integral pathas = k2v(v i) for v 0 [10]. By using this substitution,a radiation integral (18) can be transformed to that withoutsingularities asRHmn(x;y;z)= k2(a2m + b2n) Z 10(2v i) sin( z)(k222)( 2 2mn)h( 1)nQm(x; a;y;b;k222)Qm(x; a;y; b;k222)+ ( 1)mQn(y; b;x;a;k222)Qn(y; b;x; a;k222)idv ; (21)where a precise and efficient evaluation of Qm( ) is presentedin Appendix A. For numerical computation, a path-deformingvariable v in (21) can be empirically truncated toRVmax0 ( ) dvasVmax = max 10vt;1 ; (22)where max(x;y) is the maximum value of x and y,vt = 120@sjxj2 2+ 1+sjyj2 2+ 11A ; (23)and 2 = 2 =k2. Applying the Green’s second integralidentity, we reduce (16) and (18) as@2@z2 + k22 Hmn(x;y;z) + RHmn(x;y;z)=Z aaZ bb@2@z02 + k22 Hmn(r0)@@z0GxxA (r;r0) dy0dx0z0=0: (24)The integral (24) is numerically efficient for jxj a or jyjb where the Green function GxxA (r;r0) in (24) does not haveany singularity in the region of jx0j a and jy0j b . Using(24), we obtain the asymptotic Fz-potential in region (II) asFIIz (r; ; )eik2ri2 rcosk2 sin2 21Xm=01Xn=0qmn(a2m + b2n)sin( mnd)Gm(a; k2 sin cos )Gn(b; k2 sin sin ) ; (25)where r =px2 + y2 + z2, = cos 1(z=r), and =tan 1(y=x) shown in Fig. 1.In the next step, the magnetic vector potentials for Fig. 1are formulated asAIz(x;y;z) = 11Xm=01Xn=0pmn sinam(x + a)sinbn(y + b)cos mn(z + d)uxy(a;b) (26)AIIz (x;y;z) = 21Xm=01Xn=0hrmnEmn(x;y;z)+ REmn(x;y;z)i; (27)where m n 6= 0, pmn and rmn are the unknown TM modalcoefficients for regions (I) and (II), respectively. Applying the@Ez=@z e= field continuity at z = 0, we getrmn = 21pmn mn sin( mnd) ; (28)where e (= 1EIx;y ^n) is an equivalent electric chargedensity which may be produced by the field discontinuities atboundaries, we assume that @=@zREmn(x;y;z)jz=0 = 0, andEmn(x;y;z) = ei mnzi mnsinam(x + a)sinbn(y + b)uxy(a;b) : (29)Similar to the Green’s function relation already described in(17), we get the formula for REmn(x;y;z) asREmn(x;y;z)=ZJ(r0)GzzA (r;r0) dr0z component= rmnREEmn(x;y;z) ismn! 2REHmn(x;y;z) ; (30)where J(r) = Hx;y(r) ^n. To remove the Hx-field discon-tinuities at ( a x a, y = b, z 0) and the Hy-fielddiscontinuities at (x = a, b y b, z 0), REEmn(r) andREHmn(r) are given by, respectively,REEmn(x;y;z) =Z @@n0hEm(r0)iGzzA (r;r0) dr0 (31)@2@z2 + k22 REHmn(x;y;z)=Z @2@z02 + k22 Hmn(r0) @2@z@tGxxA (r;r0) dr0+Z @2@z0@t0Hmn(r0) @2@z2 + k22 GzzA (r;r0) dr0 ; (32)
  4. 4. 4where ^n ^z = ^t illustrated in Fig. 1. The expression forREEmn(x;y;z) is obtained from (29) and (31), and written byREEmn(x;y;z)= k2Z 10(2v i)cos( z)2 2mnnbn ( 1)nPm(x; a;y;b;k222)Pm(x; a;y; b;k222)+ am ( 1)mPn(y; b;x;a;k222)Pn(y; b;x; a;k222)odv ; (33)where = k2v(v i) and Pm( ) is formulated in AppendixA. Similarly, integrating (32) with (16), we getREHmn(x;y;z)= k2Z 10(2v i)cos( z)2 2mnn( 1)nTmn(x; a;y;b;k2; )Tmn(x; a;y; b;k2; )( 1)mTnm(y; b;x;a;k2; )+ Tnm(y; b;x; a;k2; )odv ; (34)where = k2v(v i) and Tmn( ) is shown in Appendix A.Note that @=@zREE;EHmn (x;y;z)jz=0 = 0 are satisfied due to@=@zcos( z) = 0z=0.Using the Green’s second integral identity, we simplify (29)and (31) asEmn(x;y;z) + REEmn(x;y;z)=Z aaZ bb@@z0hEmn(r0)iGzzA (r;r0) dy0dx0z0=0: (35)Similarly, REHmn(x;y;z) is reduced to@2@x@y@2@z2 + k22 REHmn(x;y;z)= k22Z aaZ bbHmn(r0)b2n@2@x2 a2m@2@y2 GzzA (r;r0) dy0dx0z0=0: (36)In the far-field, the Az-potential in region (II) is asymptoticallyrepresented asAIIz (r; ; )eik2r2 r(2 211Xm=01Xn=0pmn mn sin( mnd)Fm(a; k2 sin cos )Fn(b; k2 sin sin )+ 1i!sin21Xm=01Xn=0qmn sin( mnd)(b2n cot a2m tan )Gm(a; k2 sin cos )Gn(b; k2 sin sin )); (37)),( )1()1(ST),( )2()2(ST ),( )3()3(ST ...)1()2(TT − )2()3(TT −),( )1()1( ++ XX PPST ),( )2()2( ++ XX PPST ),( )3()3( ++ XX PPST),( )12()12( ++ XX PPST ),( )22()22( ++ XX PPST ),( )32()32( ++ XX PPST...............)1()1(SS XP−+xyzFig. 3. Geometry of multiple rectangular grooves in a perfectly conductingplane (PX: the number of grooves in the x-axis and (T(p);S(p)): atranslation point of the pth groove in the x-y plane)whereFm(a; ) = am ( 1)mei a e i a]2 a2m: (38)III. FIELD MATCHING FOR MULTIPLE GROOVESDue to large number of metallic rectangular grooves illus-trated in Fig. 3, scattering formulations and analyses are alittle complicated. These difficulties can be easily overcomewith field superposition principle [10], [11]. In the fieldsuperposition, electromagnetic fields in open region (z 0)related to those of each metallic rectangular groove (jxj a,jyj b, and z < 0) for Fig. 1 are additively superimposed toproduce the total electromagnetic fields for multiple groovesshown in Fig. 3. Based on the superposition principle, the totalelectric and magnetic vector potentials are represented asFtotz (x;y;z) =PTXp=1T(p)H (x T(p);y S(p);z) (39)Atotz (x;y;z) =PTXp=1T(p)E (x T(p);y S(p);z) ; (40)where PT is the total number of grooves, T(p) and S(p) ofthe pth groove are translation positions for the x- and y-axes,respectively, andT(p)H (x;y;z) = FI(p)z (x;y;z) + FII(p)z (x;y;z) (41)T(p)E (x;y;z) = AI(p)z (x;y;z) + AII(p)z (x;y;z) : (42)By matching the @Hz=@z m= and Ez fields continuitiesat z = 0, we can obtain simultaneous scattering equations forarbitrarily polarized plane-wave incidence in (11). Multiplyingthe @Hz=@z m= continuity at z = 0 with cosa(r)l (xx0 + a(r))cosb(r)k (y y0 + b(r)) (l = 0;1; , k = 0;1; ,l+k 6= 0, r = 1;2; ;PT) for the rth groove and integratingover x0 a(r) x x0 +a(r) and y0 b(r) y y0 +b(r)yieldsPTXp=11Xm=01Xn=02(p)1p(p)mn(p)mn sin( (p)mnd(p))IEEmn;lk(x0;y0)+h(a(r)l )2 + (b(r)k )2ii! 2PTXp=11Xm=01Xn=0q(p)mn
  5. 5. 5(2(p)1(p)mn cos( (p)mnd(p))a(p)b(p)m n ml nk prsin( (p)mnd(p))hi (p)mna(p)b(p)m n ml nk pr+ IHmn;lk(x0;y0) +IEHmn;lk(x0;y0)(a(r)l )2 + (b(r)k )2i)= 2i! 2Gl(a(r);k2 sin i cos i)Gk(b(r);k2 sin i sin i)ei 0nuih(a(r)l )2 + (b(r)k )2icot i+ visin ih(b(r)k )2 cot i (a(r)l )2 tan iio; (43)where m (= 2HIIx;y ^n) is an equivalent magnetic chargedensity generated by the field discontinuities between regions(I) and (II) placed at z = 0, ( )(p) is a parameter for thepth groove, a(r)l = l =(2a(r)), b(r)k = k =(2b(r)), 0 =k2 sin i(x0 cos i + y0 sin i), m = m0 + 1, ml is theKronecker delta, and IHmn;lk( ), IEEmn;lk( ), IEHmn;lk( ) are definedin Appendix B. Note that (x0 = T(r);y0 = S(r)) is a centerpoint of the rth groove for field matching and m is aninevitable term which must be included in normal magneticfield matching.Similarly, multiplying the Ez continuity with sina(r)l (xx0+a(r))sinb(r)k (y y0+b(r)) (l = 1;2; , k = 1;2; , r =1;2; ;PT) for the rth groove and integrating with respectto x and y givesPTXp=11Xm=01Xn=0p(p)mn(h(a(p)m )2 + (b(p)n )2icos( (p)mnd(p))a(p)b(p)ml nk pr+ 2(p)1(p)mn sin( (p)mnd(p))h(a(p)m )2 + (b(p)n )2i (p)mna(p)b(p)ml nk pr + JEEmn;lk(x0;y0)i)+ i! 2PTXp=11Xm=01Xn=0q(p)mn sin( (p)mnd(p))JEHmn;lk(x0;y0)= 2i! 2vi sin iFl(a(r);k2 sin i cos i)Fk(b(r);k2 sin i sin i)ei 0 ; (44)where JEEmn;lk( ), JEHmn;lk( ) are defined in Appendix B.IV. NUMERICAL COMPUTATIONS AND MEASUREMENTPlane-wave scattering from a rectangular metallic groovein a perfectly conducting infinite plane is extensively studiedwith numerical [12], [13] and analytic [14] techniques. Inorder to validate our formulations, (43) and (44), we comparedour numerical results for a wide groove (2a = 2:5 0 and2b = 0:25 0 in Fig. 1) with other simulations [12], [14].Fig. 4 illustrates the normalized backscattered co-polarizationradar cross section (RCS) for an incident angle ( i) of a plane-wave illustrated in Fig. 1. All simulated results in Fig. 4 arestrongly consistent for i 70 . In addition, Fig. 4 indicatesthe convergence behavior of our simultaneous equations, (43)0 15 30 45 60 75 90−40−30−20−100Incident angle, θi[Degree]Backscatteredco−poleRCS,σ/λ02[dB]M = 4M = 8M = 12M = 16M = 20[12] FEM[14] Fourier transformFig. 4. Behaviors of the backscattered co-polarization RCS ( = 20) versusa plane-wave incident angle ( i) with PT = 1, N = 2, 2a = 2:5 0,2b = d = 0:25 0, i = 0, ui = 0, vi = 1, 1 = 2 = 0, 1 = 2 = 0Fig. 5. Fabricated 3D metal-only reflectarray composed of rectangulargrooves with a whole diameter (D0) = 30 [cm] and the total number ofgrooves (PT ) = 5,961and (44), where M and N denote the truncation numbers ofmodal coefficients with respect to m and n in (13) and (26) fornumerical computation, respectively. As the number of modesfor m increases, the backscattered RCS converges very fastfor any i. A lower-mode solution (M = 4 and N = 2) isvery good approximation for i 35 .Fig. 5 shows a fabricated three dimensional (3D) metal-onlyreflectarray with prime focus composed of multiple rectangulargrooves. A thick circular metal plate with 30 [cm] diameter(D0) and 1 [cm] thickness contains 5,961 rectangular metallicgrooves. A pyramidal horn antenna used as a feed has 7 [mm]5 [mm] aperture size and 12 [mm] waveguide transition,and an input waveguide for a feed is WR-12 (3.1 [mm] 1.5[mm]). For simple design, we assume that each rectangulargroove illustrated in Figs. 3 and 5 has the same aperture sizeof a(p) = ag, b(p) = bg and the same separations of T(p+1)
  6. 6. 6T(p) = T, S(p+PX) S(p) = S for all p in (39) and (40),where PX is the number of grooves in the x-axis. A depth forthe pth groove d(p) is individually calculated with a formulabased on the phase matching condition [8] asd(p) = g2 2"d0 + f0q(T(p))2 + (S(p))2 + (f0 d0)2#; (45)where f = 78:5 [GHz], g = 2 = , =pk22 =(2a(p))]2,f0 and d0 are a focus and the maximum depth of a paraboloid,respectively, which is effectively formed by a metal-only flatreflectarray in Fig. 5. Because of the periodicity of reflectedphase, the depth in (45) can be limited to half the guidedwavelength [8] ( 2.38 [mm] for 78.5 [GHz]). Consideringthe phase center of a pyramidal horn antenna denoted as f9 mm], the feed in Fig. 5 is placed at (xi = 0;yi = 0;zi+f )where is the focus of a paraboloid, f0 = zi + d0 [8].Fig. 6 presents the H- and E-plane radiation patterns of ametal-only reflectarray in Fig. 5 for 78.5 [GHz] and RA (Rel-ative Aperture) = f0=D0 = 0.75. In our computations, we usedthe simultaneous equations, (43) and (44), with Hertzian dipoleexcitation polarized in the y-axis. This means that the righthand sides of (43) and (44) should be modified for a Hertziandipole. Our formulations based on the overlapping T-blockmethod are compared with planar near-field measurement[8], [16] and numerical simulation [15]. We obtained planarnear-field measurement results with a WR-10 (2.54 [mm]1.27 [mm]) OERW (Open-Ended Rectangular Waveguide)probe and the parameters such as distance between probe andreflectarray = 30.8 [cm], sampling step = 1.71 [mm], andscan range = 52.497 [cm]. In Fig. 6, the radiation behaviorsof our method and FDTD simulation agree very well for allobservation angle. The GEMS parallel FDTD simulation [15]for Figs. 5 and 6 requires the parameters such as the numberof cells = 2 109, total memory = 66 [GB], the number ofparallel processors = 54, and simulation time = 18.7 hours.The FDTD simulation is performed for the geometry shown inFig. 5 without three metallic struts to support a pyramidal hornfeed. In contrast to the FDTD simulation, our calculation timewith CPU 2 [GHz] and RAM 2 [GB] is 4.4 minutes. Fig. 6 alsoshows the discrepancy between simulations and measurementresults in the side-lobe region. This noticeable difference iscaused by our simple feed modeling such as Hertzian dipoleexcitation and finite measurement scan area (52.497 [cm]52.497 [cm]) [16].Fig. 7 shows characteristics of co- and cross-polarizationexcited gain patterns for 78.5 [GHz]. The co- and cross-polarized excitations were simulated with Hertzian dipolespolarized in the y- and x-axes, respectively, when all pa-rameters of a metal-only reflectarray were fixed. In case ofcross-polarized excitation, radiation patterns have the nullpoint at = 0 and their side-lobe levels are approximately15 [dB] higher than those of co-polarized excitation. Fig. 8indicates a succinct comparison of gain behaviors in terms ofour method, FDTD simulation, and near-field measurement.−90 −60 −30 0 30 60 90−10010203040Observation angle, θ [Degree]H−plane(φ=0°)gainpatterns[dB]Overlapping T−blocksGEMS (parallel FDTD)Measurement(a) H-plane ( = 0 )−90 −60 −30 0 30 60 90−10010203040Observation angle, θ [Degree]E−plane(φ=90°)gainpatterns[dB]Overlapping T−blocksGEMS (parallel FDTD)Measurement(b) E-plane ( = 90 )Fig. 6. Characteristics of the H- and E-plane antenna gain patterns versusan observation angle ( ) with f = 78.5 [GHz], PT = 5961, M = 2, N = 1,2ag = 3:2 mm], 2bg = 2:7 mm], T 2ag = S 2bg = 0:5 mm],xi = yi = 0, zi = 193:845 mm], 1 = 2 = 0, 1 = 2 = 0,RA = 0:75, d0 = 25 mm], d(p)obtained from (45)Although the discrepancy among simulated and measuredresults is maximally 3 [dB], overall tendency of gain behaviorsis not significantly different among results. It should be notedthat antenna gain of our method is higher than others, dueto the fact that our computation is based on Hertzian dipoleexcitation and thus cannot include the feed characteristics. Themeasured aperture efficiencies for 75, 77, 78.5 [GHz] are 30.2,27.2, 23.3 [%], respectively.V. CONCLUSIONSRigorous and analytic solutions for scattering from mul-tiple rectangular grooves in a perfectly conducting plane areobtained with the overlapping T-block method based on super-position principle and Green’s function relation. The simulta-neous scattering equations for Hertzian dipole excitation canbe utilized to predict radiation characteristics of a metal-only
  7. 7. 7−90 −60 −30 0 30 60 90010203040Observation angle, θ [Degree]Co−andcross−polegainpatterns[dB]Crosspole, φ = 0°Crosspole, φ = 90°Copole, φ = 0°Copole, φ = 90°Fig. 7. Behaviors of the co- and cross-polarization excited gain patternsversus an observation angle ( ) (The parameters are selected from those inFig. 6)73 75 77 79 81363840424446Frequency [GHz]Co−poleantennagain[dB]Overlapping T−blocksGEMS (parallel FDTD)MeasurementFig. 8. Co-polarization excited antenna gain variations versus a frequency(The parameters are chosen from those in Fig. 6)reflectarray fed by a pyramidal horn antenna. Our simulationswere compared with commercial FDTD computation and near-field measurement and all results show favorable agreements.The mode-matching and Green’s function approach for ametal-only reflectarray with rectangular grooves can be ex-tended to that with non-rectangular grooves by using suitablemodal expansions. In further work, we will investigate the gen-eral feed modeling in near-field region and the correspondingphase matching condition for a metal-only reflectarray.APPENDIX A: INTEGRALS FOR Qm( ), Pm( ), AND Tmn( )The definitions of Qm( ), Pm( ), and Tmn( ) are written byQm(x; a;y;y0;k2)Original path[ ]ηRe[ ]ηIm2kBranch cutDeformed pathk−kFig. 9. Deformed integral path to remove singularities= 12Z 11fH(y;y0; )Gm(a; )ei x d (46)Pm(x; a;y;y0;k2)= 12Z 11ei jy y0ji Fm(a; )ei x d (47)Tmn(x; a;y;y0;k; )=2(a2m + b2n)k2 2 Sm(x; a;y;y0;k2 2)+ 2mnamPm(x; a;y;y0;k2 2) ; (48)where k2 = 2 + 2, am = m =(2a), bn = n =(2b), mn =pk22 a2m b2n, andSm(x; a;y;y0;k2)= 12Z 11ei jy y0jGm(a; )ei x d : (49)To evaluate Qm( ) efficiently, we utilize the residue cal-culus. Thus, (46) can be transformed in terms of pole andbranch-cut contributions asQm(x; a;y;y0;k2)= sgn(y y0)ei mjy y0j cosam(x + a)ux(a)+ 1 Z 10sin (y y0)]2 a2mhsgn(x + a)ei jx+aj( 1)msgn(x a)ei jx ajid ; (50)where ux(a) = u(x + a) u(x a) and k2 = a2m + 2m.However, the integral (50) has rapidly oscillating behaviorswhen jy y0j 1. These oscillatory characteristics can beremoved with proper path deforming. As such, we propose anovel integral path shown in Fig. 9 which always Im ] 0for any u as=8<:k2u(u+ i) (u < 0)k2u2 (0 u 1)k2 u(u+ i) i] (u > 1): (51)Based on the path parameter (51) and Fig. 9, we modify (50)
  8. 8. 8to a fast convergent integral without singularities asQm(x; a;y;y0;k2)= sgn(y y0)(ei mjy y0j cosam(x + a)ux(a)i2Z 11ddu ei jy y0j2 a2mhsgn(x + a)ei jx+aj( 1)msgn(x a)ei jx ajidu): (52)Since the integrand in (52) has complex exponential functionsand the complex numbers, m, , and in (52) have positiveimaginary parts, the integral (52) converges exponentially. Thismeans that the double integral (18) with Qm( ) also convergesvery rapidly. For Gaussian quadrature technique, the integral(52) can be empirically truncated toRUmax+1Umax ( ) du asUmax = max"sut ut + 4 2jyj+ jy0j + v2;1#; (53)where v is defined in (21) as k =pk222 =k2p1 + v2(1 v2) + 2v3i andut = jyj+ jy0j2(jxj+ jaj) : (54)When 0 u 1 and jy y0j 1, the integral (52) still hasunwanted numerical oscillations with respect to u. To avoidnumerical oscillations of integrand in (52), we analyticallyreduce (52) to a finite integral asQm(x; a;y;y0;k2)= ik(y y0)2Z aaH(1)1 kRxy(x0;y0)]Rxy(x0;y0)cosam(x0 + a) dx0 ; (55)where H(1)m ( ) is the mth order Hankel function of the firstkind and Rxy(x0;y0) =p(x x0)2 + (y y0)2. Note that(55) is very efficient for numerical computation when jxx0j 1 or jy y0j 1. Similar to the evaluation of Q( ), weobtain the following integrals asPm(x; a;y;y0;k2)= ei mjy y0ji msinam(x + a)ux(a)+ iam2Z 11dduei jy y0j( 2 a2m)hei jx+aj ( 1)mei jx ajidu(56)Sm(x; a;y;y0;k2)= iamei mjy y0jmsinam(x + a)ux(a)i2Z 11ddu ei jy y0j2 a2mhei jx+aj ( 1)mei jx ajidu :(57)For large argument approximation (jx x0j 1 or jy y0j1), (56) and (57) are also formulated asPm(x; a;y;y0;k2)= i2Z aaH(1)0 kRxy(x0;y0)]sinam(x0 + a) dx0 (58)Sm(x; a;y;y0;k2)= ik2Z aaH(1)1 kRxy(x0;y0)] (x x0)Rxy(x0;y0)cosam(x0 + a) dx0 : (59)APPENDIX B: MATCHING INTEGRALSThe matching integrals for simultaneous modal equations,(43) and (44), are defined asIHmn;lk(x0;y0)=Z x0+a(r)x0 a(r)Z y0+b(r)y0 b(r)@@zRHmn(x;y;z)z=0cosa(r)l (x x0 + a(r))cosb(r)k (y y0 + b(r)) dydx(60)IEEmn;lk(x0;y0)= a(r)lZ x0+a(r)x0 a(r)REEmn(x;y;0)sina(r)l (x x0 + a(r))cosb(r)k (y y0 + b(r)) dxy=y0+b(r)y=y0 b(r)b(r)kZ y0+b(r)y0 b(r)REEmn(x;y;0)cosa(r)l (x x0 + a(r))sinb(r)k (y y0 + b(r)) dyx=x0+a(r)x=x0 a(r)(61)IEHmn;lk(x0;y0)= a(r)lZ x0+a(r)x0 a(r)REHmn(x;y;0)sina(r)l (x x0 + a(r))cosb(r)k (y y0 + b(r)) dxy=y0+b(r)y=y0 b(r)b(r)kZ y0+b(r)y0 b(r)REHmn(x;y;0)cosa(r)l (x x0 + a(r))sinb(r)k (y y0 + b(r)) dyx=x0+a(r)x=x0 a(r)(62)JEEmn;lk(x0;y0)=Z x0+a(r)x0 a(r)Z y0+b(r)y0 b(r)@2@z2 + k22 REEmn(x;y;z)z=0sina(r)l (x x0 + a(r))sinb(r)k (y y0 + b(r)) dydx(63)JEHmn;lk(x0;y0)=Z x0+a(r)x0 a(r)Z y0+b(r)y0 b(r)@2@z2 + k22 REHmn(x;y;z)z=0
  9. 9. 9sina(r)l (x x0 + a(r))sinb(r)k (y y0 + b(r)) dydx :(64)Since the integrands from (60) to (64) are composed ofsimple elementary functions, we can easily evaluate the aboveintegrals in closed form. When 0 =px20 + y20 1, (60)through (64) are approximately formulated asIHmn;lk(x0;y0)(a(p)m )2 + (b(p)n )2 (1 ik2 0)2 k2230Gm(a(p); k2 cos 0)Gl(a(r);k2 cos 0)Gn(b(p); k2 sin 0)Gk(b(r);k2 sin 0)eik2 0 (65)IEEmn;lk(x0;y0)eik2 02 0Fm(a(p); k2 cos 0)Fn(b(p); k2 sin 0)(a(r)l Fl(a(r);k2 cos 0)he ik2 sin 0b(r)( 1)keik2 sin 0b(r)ib(r)k Fk(b(r);k2 sin 0)he ik2 cos 0a(r)( 1)leik2 cos 0a(r)i)(66)IEHmn;lk(x0;y0)eik2 02 0(b(p)n )2 cot 0 (a(p)m )2 tan 0Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)(a(r)l Fl(a(r);k2 cos 0)h( 1)keik2 sin 0b(r)e ik2 sin 0b(r)ib(r)k Fk(b(r);k2 sin 0)h( 1)leik2 cos 0a(r)e ik2 cos 0a(r)i)(67)JEEmn;lk(x0;y0)k22eik2 02 0Fm(a(p); k2 cos 0)Fl(a(r);k2 cos 0)Fn(b(p); k2 sin 0)Fk(b(r);k2 sin 0) (68)JEHmn;lk(x0;y0)k22eik2 02 0(b(p)n )2 cot 0 (a(p)m )2 tan 0Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)Fl(a(r);k2 cos 0)Fk(b(r);k2 sin 0) ; (69)where 0 = tan 1(y0=x0) and ( )(p);(r) are the parametersfor the pth and rth grooves. It should be noted that theformulations in (65) through (69) are very useful to obtainmodal matrixes of the simultaneous scattering equations, (43)and (44) for 0 1. This is because the simplified integralsin (65) through (69) are in closed form without double infiniteintegrals, whereas the original integrals, (60) through (64),still have infinite integrals. By using simplified integrals, (65)through (69), we can compute the modal matrixes for a verylarge metal-only reflectarray very efficiently.ACKNOWLEDGEMENTThis work was supported by the IT R&D program ofMKE/KCC/KCA (2008-F-013-04, Development of SpectrumEngineering and Millimeterwave Utilizing Technology).REFERENCES[1] W.-J. Byun, B.-S. Kim, K. S. Kim, K.-C. Eun, M.-S. Song, R. Kulke, O.Kersten, G. M¨ollenbeck, and M. Rittweger, “A 40GHz vertical transitionhaving a dual mode cavity for a low temperature co-fired ceramictransceiver module,” ETRI Journal, vol. 32, no. 2, pp. 195-203, April2010.[2] W. Byun, B.-S. Kim, K.-S. Kim, M.-S. Kang, and M.-S. Song, “60GHz2x4 low temperature co-fired ceramic cavity backed array antenna,” IEEEAntennas and Propagation Society International Symposium, 2009.[3] D. Lockie and D. Peck, “High-data-rate millimeter-wave radios,” IEEEMicrowave Magazine, vol. 10, no. 5, pp. 75-83, Aug. 2009.[4] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensionalmillimeter-wave imaging for concealed weapon detection,” IEEE Trans.Microwave Theory Tech., vol. 49, no. 9, pp. 1581-1592, Sept. 2001.[5] D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarrayantenna,” IEEE Trans. Antennas Propagat., vol. 11, no. 6, pp. 645-651,Nov. 1963.[6] J. Huang and J. A. Encinar, Reflectarray Antennas, Wiley-IEEE Press,2007.[7] A. G. Roederer, “Reflectarray antennas,” European Conference on Anten-nas and Propagation, pp. 18-22, March 2009.[8] Y. H. Cho, W. J. Byun, and M. S. Song, “Metallic-rectangular-groovesbased 2D reflectarray antenna excited by an open-ended parallel-platewaveguide,” IEEE Trans. Antennas Propagat., vol. 58, no. 5, pp. 1788-1792, May 2010.[9] A. Lemons, R. Lewis, W. Milroy, R. Robertson, S. Coppedge, and T.Kastle, “W-band CTS planar array,” IEEE Microwave Symposium Digest,vol. 2, pp. 651-654, 1999.[10] Y. H. Cho, “TM plane-wave scattering from finite rectangular groovesin a conducting plane using overlapping T-block method,” IEEE Trans.Antennas Propagat., vol. 54, no. 2, pp. 746-749, Feb. 2006.[11] Y. H. Cho, “Transverse magnetic plane-wave scattering equations forinfinite and semi-infinite rectangular grooves in a conducting plane,” IETMicrow. Antennas Propag., vol. 2, no. 7, pp. 704-710, Oct. 2008.[12] K. Barkeshli and J. L. Volakis, “Electromagnetic scattering from anaperture formed by a rectangular cavity recessed in a ground plane,” J.Electromag. Waves Applicat., vol. 5, no. 7, pp. 715-734, 1991.[13] J. M. Jin and J. L. Volakis, “A finite element-boundary integral formu-lation for scattering by three-dimensional cavity-backed apertures,” IEEETrans. Antennas Propagat., vol. 39, no. 1, pp. 97-104, Jan. 1991.[14] H. H. Park and H. J. Eom, “Electromagnetic scattering from multiplerectangular apertures in a thick conducting screen,” IEEE Trans. AntennasPropagat., vol. 47, no. 6, pp. 1056-1060, June 1999.[15] GEMS Quick Start Guide, http://www.2comu.com.[16] A. D. Yaghjian, “An overview of near-field antenna measurements,”IEEE Trans. Antennas Propagat., vol. 34, no. 1, pp. 30-45, Jan. 1986.

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