Study of the Parameters Affecting Cylindrical Cloak Using Dispersive FDTD

Transcription

1 Intenational Jounal of Otis and Photonis (IJOP) Vol. 9, No., Winte-Sing, 5 Study of the Paametes Affeting Cylindial Cloak Using Disesive FDTD Hossein Vejdani Shoja a,*, Ali Aab a, and Minoo Shojae Fa b a Deatment of Alied Physis, Malek-Ashta Univesity of Tehnology, Shahin-Shah, Ian b Payame-Noo Univesity, Shahin-Shah, Ian * Coesonding Autho vejdani.shojae@gmail.om ABSTRACT In this ae, an ideal ylindial metamateial invisibility loak with infinitelength whih its eleti emittivity and magneti emeability maed to the Dude disesion model is simulated. The sinusoidal lane waves with miowave fequenies used as soues. To this end, the disesive finitediffeene time-domain method (FDTD) used with Convolutional Pefetly Mathed Layeed (CPML) absobing boundaies onditions. A omaison efomed between satteing of loaked and non-loaked PEC ylinde. And finally, the influene of inident wave fequeny, thikness of loak and obseve angle elative to the oagation line to efomane of loak, was suveyed. KEYWORDS: FDTD, Finite-Diffeene Time- Domain, CPML, Invisibility Cloak, Metamateial. I. INTRODUCTION In eent yeas, metamateial whih is an atifiial geometi omosite mateial, attats wide attention beause of its novel featues, suh as negative efative index [, ], sub-diffation imaging [3], and eletomagneti loaking [, 5] and so on. Sientists showed muh inteest to loaking devies afte exeimental veifiation of a negative index of efation [6]. Pendy et al. oosed an inhomogeneous and anisotoi eletomagneti invisibility loak using tansfomation otis [7]. Thei theoy veified exeimentally in miowave fequenies based on Slit Ring Resonatos (SRRs) at 6 []. It should be noted that the SRRs whih ae often disesive and lossy mediums, ae the basi building blok fo many metamateials [8, 6]. Thee ae vaious numeial methods fo simulation of eletomagneti inteation of wave and mateials exist. The Finite-diffeent time-domain (FDTD) is one of the elatively simle, auate and oweful numeial algoithms fo solving Maxwell s equations [9]. Beause of disesion natue of oosed in-visibility loak in [, 7], a adially deendent disesive FDTD algoithm that offeed in [], was emloyed in this wok fo simulation of an infinite-length ylinde metamateial loak. It is used fo study of vaious aametes effets to loaking efomane suh as: inident wave fequeny, ylinde adius and loak thikness. Also, the effiient Convolutional Pefetly Mathed Layes (CPML) absobing bounday onditions is emloyed in ode to avoid unwanted efletions fom omutational boundaies [9, ]. Afte designing the oblem and esentation of basi fomula in setion, the esults of simulation will be disussed in setion 3. II. DESIGN AND THEORY Conside an infinite length Pefet Eleti Conduto (PEC) ylinde along -axis, whih oated with an ideal metamateial loak and laed in the fee sae, as shown in Fig.. The full set of eletomagneti aametes of the ideal loaking stutue, in ylindial oodinates, is given by: 3

2 H. Vejdani Shoja et al. Study of the Paametes Affeting Invisibility of Cylindial Cloak R R () () R R (3) R R whee R and R ae the inne and oute adius of the loak, esetively [7]. The values of,,, and ae less than one, thus they should be maed into Dude lossless disesion model fo FDTD simulation []. fo examle in the ase of an be wite: () be used fo solving the Maxwell equations in disesive media and hene emloyed by []. In this wok, whee the eletomagneti aametes of medium ae geate than one, the onventional ADE FDTD udating equations: n n n / B B t. E D D t. H n n n/ (5) (6) will be used as well as the onstitutive elations D E and B H whee and ae exessed by Eq. (-3), is the disete ul oeato, t is the disete FDTD time ste and n is the numbe of the time stes. Also, whee the eletomagneti aametes of medium ae less than one, the following equations will be used fo eleti fields alulations []: whee ω is lasma fequeny of the medium. Fo the ase of othe eletomagneti aametes that always ae geate than one, the onventional FDTD method with onstant mateial aametes will be efomed. E [ D D D D n n n n x x x x 5 n n n n 6 y 5 y 7 x 8 x D D E E ]/ E [ D D D D n n n n y 3 y y 3 y 5 D D E E ]/ n y n n n n 6 x 5 x 7 y 8 y 8 8 n x (7) (8) whee field quantities D x and D y ae loally aveaged values of D x and D y like as []. The i oeffiients in Eq. (7) and (8) ae: Fig. Calulation sheme fo ylindial loak stutue with PEC oe and CPML boundaies onditions In this ae, onsideed the disesive FDTD method fo a two-dimensional (-D) ase fo simliity, theefoe only E x, E y and H ae non-eo and an be set. The auxiliay diffeential equation (ADE) FDTD is a elatively simle method that an os sin t t os sin 3 t t sin os t t os t t sin (9) () () ()

3 Intenational Jounal of Otis and Photonis (IJOP) Vol. 9, No., Winte-Sing, 5 sin os t t 5 sin os t t 6 7 (3) () (5) t t 8 (6) Notie that in the above elations, the is aimuth angle in ylindial oodinate and is diven fom Eq. (). Also, fo the ase of magneti field, the emeability maed into the lossless Dude model as: m A (7) A R / R R (8) whee m is magneti fequeny of mateial. Theefoe the H udating equation an be deived as []: H B B A n n n n B A H A H / n 3 n (9) () Δt () Δt m 3 () Δt m (3) Δt To ensue auay and stability of alulations, the following ules must be done: A) Using Δ x /35 fo moe modeling auay and avoiding numeial eos []. Whee, is the wavelength at the oeating fequeny. B) Using the Couant-Fiedih-Levy onditions fo numeial stability as [9]: t ( x) ( y) max () whee max is the maximum seed of light in medium. C) Using the numeial oeted lasma fequeny fo inteetion the modeling instability as []: t tan t (5) Finally, fo efomane analysis of simulation esults, the Rada Coss Setion (RCS) and Total Coss Setion (TCS) as: RCS Hs lim π (6) H TCS RCS d (7) wee used []. Whee H s is satteed field at fa and H is the inident wave. III. PROBLEM DESIGN AND THEORY The slit Pefetly Mathed Layes (PML) and Peiodi Bounday Condition (PBC) was emloyed in [] fo simulation of loak, but the CPML elatively is sueio to othe methods like Uniaxial PML (UPML), PML and Absotion Boundaies Conditions (ABC) [9, ]. 5

4 H. Vejdani Shoja et al. Study of the Paametes Affeting Invisibility of Cylindial Cloak Fig. shows the H distibution of a TE oagating lane wave in fee sae, with 5m. It illustates exellent influene of CPML on wave absotion in alulation boundaies. simulated. To this end, the disesive FDTD algoithm was imlemented with a lane wave in.5-.5 GH fequeny ange. Fo alulations stability and avoiding numeial anomalies, the mentioned A, B and C laws in Setion wee alied to simulation. Fig. 3 Total satteing of loaked and un-loaked PEC shell with metamateial in vaious fequenies. Fig. H distibution of TE lane-wave oagating along x-dietion, with alying the CPML boundaies onditions A. Indiation of the Ideal Cloak Effiieny At fist, an unloaked PEC ylinde and then a loaked PEC ylinde with an ideal metamateial was simulated. In this imlementation, the geometi aametes wee R = m (PEC shell adius) and R =.5 m and inident lane wave fequenies was between.5 to.5 GH. Also squae mesh with x y /35 was alied. As shown in Fig.3, the loaked PEC an deease the wave satteing about 6 db elative to unloaked PEC. Also, Fig. dislays the H distibution, whih onstuted fom -omonent of inident and satteed H field. It an be seen that the ossed wave font has only a slight numeial distotions whih is elated to the mesh sie value. With smalle amounts of Δx and Δy, the simulation beomes moe auate and deease the numeial eos, but it makes omutation memoy and CPU time moe exensive. B. Cloak Thikness and Fequeny Effets In the evious subsetion, invisibility effiieny of ideal loak was indiated. Hee, effet of loak thikness to invisibility was studied. Thus, an ideal metamateial loak with 5, 7.5, and.5 m thiknesses whih oated a PEC shell with R= m, was Fig. H distibution of wave aound a) PEC Shell and b) oated PEC shell by ideal metamateial loak. Fig.5 shows the total satteing (TCS) wave of simulated loaks. As illustated in the hats, invisibility of ideal metamateial beomes 6

5 Intenational Jounal of Otis and Photonis (IJOP) Vol. 9, No., Winte-Sing, 5 bette with ineasing the loak thikness. Also, it is obvious that suess of loak deends on inident wave fequeny. Atually, thee is a fequeny limitation to invisibility with ideal loak aametes. Also, the ideal metamateial invisibility loak gives the best esonse in.55 GH in suveyed fequeny anges fo all loak thiknesses. Fig. 5 total satteing of loaked PEC with diffeent thiknesses in vaious fequenies C. The Effet of Obseve Angle In the simulated ases, the loaks with 5 and.5 m thiknesses have the wost and the best invisibility, esetively. Fig.6 shows RCS of these two ases at.55 GH, in omaison of unloaked PEC shell satteing. Fig. 6 RCS diagam of bae and loaked with metamateial PEC with a) t=5 m b) t=.5 m, thiknesses in.55 GH It is obvious fom diagams that the ideal loak deease satteing in most angles, but in some ases the loaked PEC shell satteing is equal o geate than unloaked PEC shell satte. This means that the ideal metamateial loaks efomane deends on angle of obseve. Theefoe, thee is an angle limitation to invisibility suess, moeove to fequeny estition. The RCS has asymmety elative to o 8 degee and those little anomalies elated to meshing sie amount. If Δx and Δy beome smalle, the RCS hats will have omlete symmety. IV. CONCLUSION Afte intoduing the basi algoithm that was emloyed in this wok, efomane of the ideal metamateial loak with vaious thiknesses in.5-.5 GH fequeny ange was suveyed. The CPML absobing bounday ondition as an effiient method used in alulations. At fist, a omaison efomed between total satteing (TCS) of unloaked and loaked PEC shell by ideal metamateial and illustated effiieny of loak in deeasing the TCS. Then the ole of loaks thikness at diffeent inident wave fequenies was analyed and was indiated that the loaks with geate thikness have moe effiieny. In addition, esults show that invisibility efomane of ideal loak, deends on inident wave fequeny. The esults of simulation wee showed that the loak gives the best esonse in.55 GH, elative to othe studied fequenies in this wok. Also, with analysis of RCS diagam fo unloaked and loaked PEC shell, illustated that invisibility deends on obseve angle elative to inident wave dietion. REFERENCES [] V.G. Veselago and G. Vikto, The Eletodynamis of Substanes with Simultaneously Negative Values of ε and μ, Sov. Phys. Us. Vol.,. 59-5, 968. [] D.R. Smith, W.J. Padilla, D.C. Vie, S.C. Nemat-Nasse, and S. Shult, Comosite Medium with Simultaneously Negative Pemeability and Pemittivity, Phys. Rev. Lett. Vol. 8,. 8-87,. [3] N. Fang, H. Lee, C. Sun, and X. Zhang, Sub Diffation-Limited Otial Imaging with a 7

6 H. Vejdani Shoja et al. Study of the Paametes Affeting Invisibility of Cylindial Cloak Silve Suelens, Siene, Vol. 38, , 5. [] D. Shuig, J.J. Mok, B. Justie, S. Cumme, J. Pendy, A. Sta, and D. Smith, Metamateial Eletomagneti Cloak at Miowave Fequenies, Siene, Vol. 3, , 6. [5] J. B. Pendy, A. J. Holden, D.J. Robbins, and W.J. Stewat, Magnetism fom ondutos and enhaned nonlinea henomena, IEEE Tans. Miow. Theoy Teh. Vol. 7,. 75-8, 999. [6] R. Shelby, D. Smith, and S. Shult, Exeimental veifiation of a negative index of efation, Siene, Vol. 9, ,. [7] J. Pendy, D. Shuig, and D. Smith, Contolling eletomagneti fields, Siene, Vol. 3, , 6. [8] T.J. Yen, W.J. Padilla, N. Fang, D.C. Vie, D.R. Smith, J.B. Pendy, D.N. Basov, and X. Zhang, Teahet Magneti Resonse fom Atifiial Mateials, Siene, Vol. 33,. 9-96,. [9] Taflove and S.C. Hagness, Comutational Eletodynamis: The Finite-Diffeene Time- Domain Method, 3 nd Ed. Nowood: Ateh House, 5. [] Y. Zhao, C. Agyooulos, and Y. Hao, Fullwave finite-diffeene time-domain simulation of eletomagneti loaking stutues, Ot. Exess, Vol. 6, no. 9, , 8. [] J.A. Roden and S.D. Gedney, Convolutional PML (CPML): An Effiient FDTD Imlementation of the CFS-PML fo Abitay Media, Miow. Ot. Tehn. Lett. Vol. 7, ,. [] N. Okada and J.B. Cole, FDTD Modeling of a Cloak with a Nondiagonal Pemittivity Tenso, ISRN Ot. Vol., -7,. [3] R. Shelby, D. Smith, and Shult, Exeimental veifiation of a negative index of efation, Siene, Vol. 9, ,. Hossein Vejdani Shoja eeived the M.S. degee in Eleto-Otial Engineeing fom Malek-Ashta Univesity of Tehnology, Shahin-Shah, Ian, in 3. His eseah fields ae omutational eletomagneti, metamateials, otial mateials and eletootial design. D. Ali Aab eeived the Ph.D. degee in Physis fom Isfahan Univesity, Ian, in 9. He is faulty membe of Malek-Ashta Univesity of Tehnology. His eseah fields ae nanohysis, ystal gowth, metamateials, otial and IR mateials, and eleto-otial design. Minoo Shojaei-Fa eeived B.S. degee in Pue Chemisty fom Payame-Noo Univesity, Shahin-Shah, Ian, in. She is inteested to wok on Quantum and Nano Chemisty. 8