IMPLEMENTATION OF MUR S ABSORBING BOUNDARIES WITH PERIODIC STRUCTURES TO SPEED UP THE DESIGN PROCESS USING FINITE-DIFFERENCE TIME-DOMAIN METHOD

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1 Pogess In Eletomagnetis Reseah, PIER 58, , 006 IMPLEMENTATION OF MUR S ABSORBING BOUNDARIES WITH PERIODIC STRUCTURES TO SPEED UP THE DESIGN PROCESS USING FINITE-DIFFERENCE TIME-DOMAIN METHOD G. Zheng, A. A. Kishk, A. W. Glisson, and A. B. Yakovlev Depatment of Eletial Engineeing Cente fo Applied Eletomagneti Systems Reseah CAESR Univesity of Mississippi Univesity, MS 38677, USA Abstat The finite-diffeene time-domain FDTD method is used to obtain numeial solutions of infinite peiodi stutues without esoting to the omplex fequeny-domain analysis, whih is equied in taditional fequeny-domain tehniques. The field tansfomation method is suessfully used to model peiodi stutues with oblique inident waves/san angles. Maxwell s equations ae tansfomed so that only a single peiod of the infinite peiodi stutue is modeled in FDTD by using peiodi bounday onditions PBCs. When modeling peiodi stutues with the tansfomed fields, the standad Mu seond-ode absobing bounday ondition annot be used dietly to absob the outgoing waves. This pape pesents a new implementation of Mu s seond-ode absobing bounday ondition ABC with the tansfomed fields in the FDTD method. Fo designs that equie multi-paameti studies, Mu s ABCs ae effiient and suffiient bounday onditions. If moe auate esults ae needed, the pefetly mathed laye PML ABC an be used with the paametes obtained fom the Mu solution. 1. INTRODUCTION The finite-diffeene time-domain FDTD method has been widely used fo the analysis of eletomagneti satteing, adiation, and popagation poblems sine it was fist intodued by Yee [1]. The FDTD method is simple and auate without esoting to the somewhat moe omplex taditional fequeny-domain tehniques, suh as Method of Moments MoM. It has the apability to

2 10 Zheng et al. simulate eletomagneti inteations in ompliated geometies that ae extemely diffiult to analyze by othe methods. Due to the limitations of omputational esoues, the FDTD numeial solution equies the use of adiation o absobing bounday onditions ABCs in ode to auately tunate an infinite omputational domain. One of the analytial ABC methods developed fo open-bounday poblems is based on the one-way wave equations OWWEs. Engquist and Majda poposed the use of OWWEs fo tunation of the omputational domain [] and Mu intodued the disetization and appliation of OWWEs to the Yee algoithm [3]. Befoe the Pefetly Mathed Laye PML ABCs wee developed in [4], Mu s seond-ode ABC and its modifiations wee among the most widely used ABCs in the FDTD method. In many eletomagneti appliations, the stutues of inteest have a peiodiity in one o two dimensions, suh as fequeny seletive sufaes FSS [5], eletomagneti bandgap EBG stutues [6], o infinite antenna aays [7]. To apply the FDTD method to the oblique inidene ase fo suh stutues, the field tansfomation method was intodued in [8]. Its futhe extensions an be found in [9, 10]. One of the FDTD disetization methods fo the tansfomed fields is the split-field method [11 13]. The field tansfomation method is applied to tansfom Maxwell s equations fom the E-H domain to the mapped P -Q domain. In addition, this method an be used in FDTD to oveome diffiulties in the implementation of time-advane and time-delay aoss the gid [14]. In this ase, infinite peiodi stutues ae tunated into single-peiod stutues with peiodi bounday onditions PBCs in one o two dimensions. The othe sides of the omputational domain in FDTD must be tunated by ABCs to avoid efletions. Due to stutual peiodiities, the ABCs should have peiodi popeties so that they an popely absob the outgoing waves. In ontast, the standad Mu seond-ode ABC needs pope adjustments in ode to absob the outgoing waves. This pape poposes a new disetization method of Mu s seondode ABC with tansfomed fields in FDTD. A key featue of the method is to sepaate the FDTD updating equations in Mu s seond-ode ABC into two pats. Eah pat is updated by diffeent omponents. In the nomal inidene ase, howeve, the poposed Mu seond-ode ABC in the mapped P -Q domain etains its standad fom as in the E-H domain.

3 Pogess In Eletomagnetis Reseah, PIER 58, FORMULATIONS AND EQUATIONS Conside a peiodi stutue that may ontain lossy, anisotopi mateials with peiodiities in both the y- and z-dietions, and whih is tunated by ABCs in the x-dietion. In the fequeny domain, the field tansfomation method is applied to tansfom the eleti and magneti field omponents fom the E-H domain to the P -Q domain as, P x = Ẽxe jkyy+jkzz, P y = Ẽye jkyy+jkzz, P z = Ẽze jkyy+jkzz, Q x = η 0 Hx e jkyy+jkzz, Q y = η 0 Hy e jkyy+jkzz, Q z = η 0 Hz e jkyy+jkzz. 1a 1b 1 1d 1e 1f The tilde symbol is used to denote the field omponents in the fequeny domain. Afte substituting these tansfomed field omponents into Maxwell s equations and tansfoming them fom the fequeny domain to the time domain, the modified time-dependent Maxwell s equations an be obtained [9, 10] the divation ae given in the Appendix fo onveniene, ε P t + 1 Λ Q = Q R a µ Q t 1 Λ P = P R m b whee P = P xˆx + P y ŷ + P z ẑ Q = Q xˆx + Q y ŷ + Q z ẑ Λ=k y ŷ + k z ẑ k y = sin θ sin φ k z = os θ R = ση 0P R m = σ Q η 0 and θ, φ epesent the inident/san angles in the poblem.

4 104 Zheng et al. Using the modified Maxwell s equations and following the poedue pesented in [], the fee-spae one-way wave equation in the mapped P -Q domain an be obtained as [8, 10], x + y k y y t +k y t + z k z z t +k z t t W =0 3 whee W epesents field quantity that is tangential to the absobing bounday in the P -Q domain. The omputational domain in FDTD is tunated at x = 0 and x = h in the x-dietion with ABCs, and the ABCs with tansfomed fields at these two planes beome W x t 1 1 k y k z and 1 t W x t k y + k z + 1 t W t + W k y y + k W z z W t W y W k y y + k W z z W y + W z =0, at x = 0 4a + W z =0, at x = h. 4b In the following, the absobing bounday at x = 0 is disetized in FDTD as an example. The othe bounday is also developed in a simila way..1. Mu sfist-ode ABCs Beause the fist-ode Mu ABC in FDTD emoves both the y- and z- deivative tems, the peiodi popeties of the stutues in OWWEs will also be emoved. Theefoe, the disetized fom of the fist-ode Mu ABC in FDTD at x = 0 will be the same as the standad one [14], W n+1 0,j,k = W t x n 1,j,k + W n+1 1,j,k t + x W n 0,j,k 5a o W n+1 0,j = W n 1 t x 1,j + W n+1 1,j W n 1 0,j t + x + x t + x W n 0,j + W n 1,j 5b

5 Pogess In Eletomagnetis Reseah, PIER 58, whee i, j, k epesent the gid positions in the x-, y- and z-dietions in the omputational domain, and n epesents the iteation time step... Mu sseond-ode ABCs Following standad poedues in the disetization of Mu s seondode ABC in FDTD eates diffiulties in disetizing some tems in equation 3 that ombine both the y- and z-deivative and the timedeivative, that ae intodued by the field tansfomation method. Theefoe, it is onvenient to ewite equation 3 as W x t 1 1 k y k z W t + W y + W z 1 W k y t y + k W z =0. 6a z Equation 6a is sepaated into two pats, fw = W x t 1 1 k y k z gw = 1 W k y t y + k W z z W t + W y + W z 6b. 6 In this ase, fw an be disetized by using the standad Mu poedue to obtain 1 t x + 1 t 1 k y k z W n+1 0,j,k = 1 t 1 W n+1 1,j,k x + 1 k y k z W n 1 1,j,k W n 1 0,j,k x W n 0,j,k + W n 1 0,j,k t + W n+1 1,j,k W n 1,j,k + W n 1 1,j,k t W n 0,j+1,k W n 0,j,k + W n 0,j 1,k y + W n 1,j+1,k W n 1,j,k + W n 1,j 1,k y

6 106 Zheng et al. W n 0,j,k+1 W n 0,j,k + W n 0,j,k 1 + z + W n 1,j,k+1 W n 1,j,k + W n 7a 1,j,k 1 z In the field tansfomation method, we hoose the dual-time tehnique [9, 11] to update fields in FDTD. Thus, we an update gw as W n+1/ gw = k 1,j+1,k W n+1/ 1,j 1,k W n 1/ 1,j+1,k W n 1/ 1,j 1,k y y t W n+1/ 0,j+1,k + W n+1/ 0,j 1,k W n 1/ 0,j+1,k W n 1/ 0,j 1,k y t W n+1/ + k 1,j,k+1 W n+1/ 1,j,k 1 W n 1/ 1,j,k+1 W n 1/ 1,j,k 1 z z t W n+1/ 0,j,k+1 + W n+1/ 0,j,k 1 W n 1/. 0,j,k+1 W n 1/ 0,j,k 1 z t 7b By ombining the two pats togethe, we obtain the final updating equation fo Mu s seond-ode ABC, 1 t x + 1 t 1 k y k z W n+1 0,j,k = 1 W n+1 1,j,k t x W n 1 1,j,k W n 1 0,j,k x 1 1 k y k z W n 0,j,k + W n 1 0,j,k t + W n+1 1,j,k W n 1,j,k + W n 1 1,j,k t W n 0,j+1,k W n 0,j,k + W n 0,j 1,k y + + W n 1,j+1,k W n 1,j,k + W n 1,j 1,k y W n 0,j,k+1 W n 0,j,k + W n 0,j,k 1 + z + W n 1,j,k+1 W n 1,j,k + W n 1,j,k 1 z

7 Pogess In Eletomagnetis Reseah, PIER 58, W n+1/ 1,j+1,k W n+1/ 1,j 1,k W n 1/ 1,j+1,k W n 1/ 1,j 1,k y t k y W n+1/ 0,j+1,k + W n+1/ 0,j 1,k W n 1/ 0,j+1,k W n 1/ 0,j 1,k y t W n+1/ 1,j,k+1 W n+1/ 1,j,k 1 W n 1/ 1,j,k+1 W n 1/ 1,j,k 1 z t k z W n+1/ 0,j,k+1 + W n+1/ 0,j,k 1 W n 1/. 0,j,k+1 W n 1/ 0,j,k 1 z t 7 3. NUMERICAL VERIFICATIONS In this setion, numeial validations based on the fomulations desibed in Setion ae pesented. In the fist numeial expeiment, we onside a y- and z-peiodi infinite infinitesimal-dipole aay with san angle nomal san angle. The gid sizes in the x-, y-, and z- dietions ae 0.5 mm, and the time step is 0.77 ps. The omputational domain in the x-, y-, and z-dietions is 15 m 5 m 5 m ells. The dimensions in the peiodi dietions ae lage than in the absobing dietion so that the oupling fields fom neighboing unit ells aive to the sampling positions late than fields efleted fom the ABCs. A lage benhmak domain ells tunated by a 10-laye PML is used as the efeene fo the efletion studies of Mu s ABCs. A hoizontal infinitesimal dipole is plaed in the ente of the omputational domain, 15, 5, 5, with a Gaussian wavefom pulse peaking at time-step n = 73 with a peak value of 1 as P y t = exp t t 0 τ 8 whee t 0 is the time at whih the pulse eahes its maximum and τ is a paamete elated to pulse width. To study the efletions fom ABCs, global eo [14] is not used hee due to the limitation of ompute esoues and omplexity of the field tansfomation method. Instead, a loal nomalized eo at the obsevation point 1, 15, 15 and an aveaged nomalized eo at the obsevation plane x = 1 ae defined in 9 and 10 Loal Eot = yt y efeenet max y efeene t 9

8 108 Zheng et al. yt yefeene t Avg Eot =Avg x=1. 10 max y efeene t Sine t x/, the leading edge of the popagating wave geneated by the soue equies appoximately 116 t to popagate fom the soue to the obsevation point 1, 15, 15 and appoximately 103 t to popagate ove the 15 t distane to the ente of the plane gid at x = 1, o 1, 5, 5. The stong oupling fields fom the neaest neighboing unit ells will aive at appoximately 167 t. This allows the outgoing wave to pass though the boundaies and exite the possible efletions fom the boundaies. x Mu 1st ABC Mu nd ABC 10-laye PML Refeene Time Step Figue 1. Time-domain esponse fom Mu s fist-ode, seond-ode ABCs and PML ABCs ove the fist 00 time steps fo nomal san angle. Fig. 1 shows the time-domain esponses fom Mu s fist-ode, seond-ode ABCs and 10-laye PML ABCs fo the ase of nomal san angle. In Fig. 1, the time-domain esult obtained by using the Mu seond-ode ABC is muh lose to the efeene esult than that of the Mu fist-ode ABC, as expeted. Fig. ompaes both the loal eo and aveaged eo among the Mu fist-ode, seond-ode ABCs and 10-laye PML ABCs ove 00 time steps. The Mu seondode ABC yields appoximately 1/3 of loal eo and appoximately 1/ of aveaged eo of the Mu fist-ode ABC. Afte the oupling fields fom the neighboing unit ells beome signifiant, the diffeenes between Mu s fist-ode and seond-ode ABCs ae smalle. The omputational time and memoy equiement fo Mu s fist-ode, seond-ode ABCs and 10-laye PML ABCs ae listed as Table 1.

9 Pogess In Eletomagnetis Reseah, PIER 58, Table 1. Computational esoues equied fo Mu s 1st ode, Mu s nd ode ABCs and 10-laye PML ABCs. Mu s 1 st ode ABCs Mu s nd ode ABCs 10-laye PML ABCs Memoy Mb CPU time s Mu 1st ABC Mu nd ABC 10-laye PML Mu 1st ABC Mu nd ABC 10-laye PML o 10-4 E o 10-4 E Time Step Time Step Figue. Refletion eos fom Mu s fist-ode, seond-ode ABCs and PML ABCs ove the fist 00 time steps fo nomal san angle: a loal eos, b aveaged eos. A seond numeial validation is onduted at 45 san angle with the same omputational domain, exitation fom, and sampling positions as in the pevious ase. The equiements fo stability ondition and the dispesion elation in FDTD at this angle ae the most estitive among all the san angles [13]. The time step is hanged to 0.4 ps fo this ase in ode to satisfy the stability ondition, whih depends on the san angle. The leading edge of the popagating wave geneated by the soue equies appoximately 10 t to popagate fom the soue to the obsevation point 1, 15, 15 and appoximately 187 t to popagate ove the 15 t distane to the ente of the plane gid at x = 1, o 1, 5, 5. The stong oupling fields fom the neaest neighboing unit ells will aive at appoximately 304 t. Fig. 3 shows the time-domain esponse of Mu s fist-ode, seond-ode ABCs and 10-laye PML ABCs fo the ase of the 45- degee san angle. The time-domain esult obtained by using the Mu seond-ode ABC is muh lose to the efeene esult than the esult of the Mu fist-ode ABC, whih is simila to the pevious ase. This

10 110 Zheng et al. 4 x Mu 1st ABC -4 Mu nd ABC 10-laye PML Refeene Time Step Figue 3. Time-domain esponse fom Mu s fist-ode, seond-ode ABCs and PML ABCs ove the fist 400 time steps fo the 45-degee san angle o 10-4 E o 10-4 E Mu 1st ABC Mu nd ABC 10-laye PML Time Step Mu 1st ABC Mu nd ABC 10-laye PML Time Step Figue 4. Refletion eos fom Mu s fist-ode, seond-ode ABCs and PML ABCs ove the fist 400 time steps fo the 45-degee san angle: a loal eos, b aveaged eos. indiates that Mu s seond-ode ABCs geneate less efletion than the fist-ode ABCs fo the oblique san angle. Fig. 4 ompaes both the loal eo and aveaged eo among the Mu fist-ode, seondode ABCs and 10-laye PML ABCs ove 400 time steps. Mu s seond-ode ABC yields appoximately /5 of the loal eo and appoximately /3 of aveaged eo of Mu s fist-ode ABC. The loal and aveaged eos at 45-degee san angle ae lage than those

11 Pogess In Eletomagnetis Reseah, PIER 58, obtained in the ase of the nomal san angle. At 45-degee san angle, the numeial value of wave veloity is not equal to the value of fee-spae wave veloity, whih is used in both Mu s fist-ode and seond-ode ABCs. Fom the updating equations, the effet on Mu s seond-ode ABCs is lage than that of the fist-ode. The efletion eos fom Mu s seond-ode ABC an be edued if the atual numeial value of the wave veloity is used in the updating equations. Anothe eason is that some weake oupling fields have aleady aived at the sampling positions befoe 304 t have affeted the efletion studies. 4. CONCLUSION Mu s fist-ode and seond-ode ABCs with peiodi bounday onditions have been implemented in the FDTD method in ode to analyze peiodi stutues. Based on the numeial expeiments, Mu s fist-ode and seond-ode ABCs equie 50% less memoy, 50algoithm than a 10-laye PML ABCs with the same omputational domain. Mu s seond-ode ABC is moe auate than the fistode ABC and an be suessfully applied fo a multi-paameti quik design with peiodiities. ACKNOWLEDGMENT This wok was patially suppoted by The Amy Reseah Offie unde gant No. DAAD APPENDIX A. MODIFIED MAXWELL S EQUATIONS IN THE P -Q DOMAIN Define the tansfomed fields as P x = Ẽxe jkyy+jkzz, P y = Ẽye jkyy+jkzz, P z = Ẽze jkyy+jkzz, Q x = η 0 Hx e jkyy+jkzz, Q y = η 0 Hy e jkyy+jkzz, Q z = η 0 Hz e jkyy+jkzz, A1a A1b A1 A1d A1e A1f

12 11 Zheng et al. whih ae used in the fequeny domain Maxwells equations to yield jωε x P x + σ x η 0 Px = Q z y Q y z j k y t Q z + j k z Qy Aa jωε y jωε z P y + σ y η 0 Py = Q z x + Q x z j k z Qx Ab P z + σ z η 0 Pz = Q y x Q x y + j k y Qx A jωµ x Q x + σ x η 0 Qx = P z y + P y z + j k y Pz j k z Py jωµ y Q y + σ y η 0 Qy = P z x P x z + j k z Px Ad Ae jωµ z Q z + σ z η 0 Qy = P y x + P x y k y Px Af Tansfomation of equation A fom fequeny domain into time domain yields ε x P x t + σ xη 0 P x = Q z y Q y z k y Q z + k z t ε y P y t + σ yη 0 P y = Q z x + Q x z k z Q x t ε z P z t + σ zη 0 P z = Q y x Q x y + k y Q x t µ x Q x + σ x Q x = P z t η 0 y + P y z + k y P z t k z µ y Q y + σ y Q y = P z t η 0 x P x z + k z P x t µ z Q z + σ z Q y = P y t η 0 x + P x y k y P x t Q y t P y t A3a A3b A3 A3d A3e A3f whee ˆk = sin θ os φˆx + sin θ sin φŷ + os θẑ = k xˆx + k y ŷ + k z ẑ. By ewiting equation A3, a ompat fom of the modified Maxwell s equations in time domain an be obtained as ε P t + 1 Λ Q = Q R A4a µ Q t 1 Λ P = P R m A4b

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