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Transformation Optics in Orthogonal Coordinates

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Transformation Optics in Orthogonal Coordinates Transformation optics in orthogonal coordinates Huanyang Chena, b, * a Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China b Institute of Theoretical Physics, Shanghai Jiao Tong University, Shanghai 200240, China * Corresponding author at: Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. Tel.: +852 2358 7981; fax: +852 2358 1652. E-mail address: [email protected] (Huanyang Chen). Abstract: The author proposes the methodology of transformation optics in orthogonal coordinates to obtain the material parameters of the transformation media from the mapping in orthogonal coordinates. Several examples are given to show the applications of such a methodology by using the full-wave simulations. PACS number(s): 41.20.Jb, 42.25.Fx Keywords: transformation optics; transformation media; cloaking 1 / 23 References: [1] J.B. Pendry, D. Schurig, and D.R. Smith, Science 312, 1780 (2006). [2] U. Leonhardt, Science 312, 1777 (2006). [3] M. Rahm, D. Schurig, D.A. Roberts, S.A. Cummer, D.R. Smith, and J.B. Pendry, Photon. Nanostruct.: Fundam. Applic. 6, 87 (2008). [4] H.Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007). [5] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007). [6] A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007). [7] H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen and J. Wang, Phys. Rev. A 77, 013825 (2008). [8] D.-H. Kwon and D. H. Werner, Appl. Phys. Lett. 92, 013505 (2008). [9] W. Yan, M. Yan, Z. Ruan and M. Qiu, New J. Phys. 10, 043040 (2008). [10] A. Nicolet, F. Zolla, and S. Guenneau, Opt. Lett. 33, 1584 (2008). [11] H. Y. Chen, X. Zhang, X. Luo, H. Ma and C. T. Chan, New J. Phys. 10, 113016 (2008). [12] U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006). [13] U. Leonhardt and T. G. Philbin, to appear in Progress in Optics (2008), http://arxiv.org/abs/0805.4778. [14] D.Schurig, J.B. Pendry, and D.R. Smith, Opt. Express 14, 9794 (2006). [15] G.W. Milton,M. Briane, and J. R. Willis, New J. Phys. 8, 248-267 (2006). [16] S.A. Cummer, B.-I. Popa, D. Schurig, D.R. Smith, and J.B. Pendry, Phys. Rev. E 2 / 23 74, 036621 (2006). [17] H. Y. Chen and C. T. Chan, Phys. Rev. B 78, 054204 (2008). [18] Jensen Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 (2008). [19] H. Y. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma and C. T. Chan, Phys. Rev. B 76 241104(R) (2008). [20] U. Leonhardt and T. Tyc, Science Express Reports, DOI: 10.1126/science.1166332 (2008). I. Introduction The transformation optics [1, 2] have provided a general method to design all kinds of devices with novel properties, such as the invisibility cloak [1], concentrator [3], rotation cloak [4], electromagnetic wormhole [5], and impedance-matched hyperlens [6], etc. The invisibility cloak is the most attracting topics. Cloaks with all kinds of shapes have been designed [3, 7-10]. Specifically, a cylindrical cloak with arbitrary cross section has been given in [10], which has been extended to design the perfect electrical conductor (PEC) reshaper [11] and other devices with arbitrary cross sections. In this paper, I will go further to rewrite the transformation media equations from the Cartesian coordinates to orthogonal coordinates. Several examples will be demonstrated to show the applications of such transformation optics. We note that the technique here is not a new theory but to rewrite the formula in a convenient form and to show an easy way to obtain the material parameters which would be very laborious from original transformation optics in Cartesian coordinate. The reader can also refer 3 / 23 a more general technique proposed by Leonhardt and Philbin to study the “general relativity in electrical engineering” [12, 13], where they started from Maxwell's equations in curved coordinates and used standard results from differential geometry. The technique here focuses to write out the material parameters in a very explicit way for the future study by means of the numerical finite element methods. We should also differentiate the Cartesian coordinates from the general curvilinear orthogonal coordinates because the Cartesian coordinate itself is orthogonal. When we talk about the orthogonal coordinates in this paper, it usually means other orthogonal coordinates but not the Cartesian coordinates. The paper is organized as follows. Section I is the introduction part. Section II is the theory part for transformation optics in orthogonal coordinates. Section III and IV are two applications of the above transformation optics. Section V is a summary of this paper. II. Transformation optics in orthogonal coordinate. Let’s start from the transformation media equations in Cartesian coordinate [14, 15], IIij'' i ' j ' ij i ' εεCartesian= JJ i j Cartesian/det( J i ), (1) IIij'' i ' j ' ij i ' μμCartesian= JJ i j Cartesian/det( J i ), ∂xi' where J i' = is the Jacobian transformation matrix in Cartesian coordinate, i ∂xi I ij I ij εCartesian and μCartesian are the components of the permittivity and permeability I ij'' I ij'' tensors in the original space in Cartesian coordinate, εCartesian and μCartesian are the components of the equivalent permittivity and permeability tensors of the transformation media in Cartesian coordinate to obtain the transformed space [14, 15]. 4 / 23 Now we rewrite the components of the permittivity and permeability tensors in the original space in Cartesian coordinate in the form of the ones in the original space in an orthogonal coordinate, IIij i j kl i εεCartesian= RR00 k l Orthogonal/det( R 0 k ), (2) IIij i j kl i μμCartesian= RR00 k l Orthogonal/det( R 0 k ), ∂xi where R i = is the rotation matrix between the orthogonal coordinate and the 0k hu∂ k uk Cartesian coordinate. The line element in Cartesian coordinate in the original space is 2222 ij ds=++= dx dy dzδij dx dx , while in the orthogonal coordinate in the original space, it is ds2222222=++ h(,, u v w ) du h (,, u v w ) dv h (,, u v w) dw uvw (3) =++=hdu22 hdv 22 hdw 2 2δ hhd udui j . uvw ij uuij Substitute Eq. (2) in to Eq. (1), we have, IIij'' i ' j ' i j kl i ' i εεCartesian= JJRR i j00 k l Orthogonal/det( J i )/det( R 0 k ) ij''I kl i ' = QQk lε Orthogonal/det( Q k ), (4) IIij'' i ' j ' i j kl i ' i μμCartesian= JJRR i j00 k l Orthogonal/det( J i )/det( R 0 k ) ij''I kl i ' = QQk lμ Orthogonal/det( Q k ), ∂∂xii''xx ∂ i with QJRiii''== = . kik0 ∂∂xikhu hu ∂ k uukk We rewrite the components of the equivalent permittivity and permeability tensors of the transformation media from the Cartesian coordinate to the orthogonal coordinate, 5 / 23 IIkl''−− 1 k ' 1' l i ' j ' − 1 k ' εεOrthogonal= RR i'' j Cartesian/det( R i ' ), (5) IIkl''−− 1 k ' 1' l i ' j ' − 1 k ' μμOrthogonal= RR i'' j Cartesian/det( R i ' ), hu∂ k ' with the rotation matrix R−1'k = uk ' . The line element in Cartesian coordinate in i' ∂xi' 2222ij '' the transformed space is ds''''=++= dx dy dzδij'' dx dx , while in the orthogonal coordinate in the transformed space, it is ds'2222222=++ h ( u ', v ', w ') du ' h ( u ', v ', w ') dv ' h ( u ', v ', w ') dw ' uvw''' (6) =++hduhdvh22'' 22 2dw ' 2 =δ h h dudij'' u . uv'''wi''j uuij'' Substitute Eq. (4) in to Eq. (5), IIkl''−− 1 k ' 1' l i ' j ' kl i ' − 1 k ' εεOrthogonal= RRQQ i'' j k l Orthogonal/det( Q k )/det( R i ' ) kl''I kl k ' = TTk lε Orthogonal/det( T k ), (7) IIkl''−− 1 k ' 1' l i ' j ' kl i ' − 1 k ' μμOrthogonal= RRQQ i'' j k l Orthogonal/det( Q k )/det( R i ' ) kl''I kl k ' = TTklμ Orthogonal/det(T k ), kk''i' hukk''∂∂∂x hu with TRQkki'1''==− uu =. Equation (7) is the transformation kik' ∂∂xik' hu hu ∂ k uukk media equation in the orthogonal coordinate. More explicitly, suppose the mapping is written as (,,uvw )⇔ (',',') u v w , ⎧ uuuvw''(,,),= ⎧ uuuvw= (',','), ⎪ ⎪ ⎨ vvuvw''(,,),= or ⎨ vvuvw= (',','), (8) ⎪ ⎪ ⎩wwuvw''(,,),= ⎩wwuvw= (',','), the Jacobian transformation matrix in the orthogonal coordinate is, ⎡⎤hhhuuu'''∂∂∂uuu''' ⎢⎥ hu∂∂∂ hv hw ⎢⎥uvw ⎢⎥hhh∂∂∂vvv''' T = ⎢⎥vvv''', (9) ⎢⎥huuvw∂∂∂ hv hw ⎢⎥hhh∂∂∂www''' ⎢⎥www''' ⎣⎦huhvhwuvw∂∂∂ the transformation media equation is, 6 / 23 IIT εεOrthogonal'/det(),= TTT Orthogonal (10) IIT μμOrthogonal'/det(),= TTT Orthogonal in the form of (uvw ', ', ') . To use the numerical finite element methods (for instance, the COMSOL Multiphysics finite element-based electromagnetics solver), we should obtain the explicit form of the required parameters in the form of (x ',yz ', ') [16]. The rotation matrix is, ⎡⎤1'1'1'∂∂x xx ∂ ⎢⎥ hu∂∂∂'' hv hw ' ⎢⎥uvw''' ⎢⎥1'1'1'∂∂y yy ∂ R = ⎢⎥, (11) ⎢⎥huuvw'''∂∂∂'' hv hw ' ⎢⎥1'1'1'∂∂zz ∂ z ⎢⎥ ⎣⎦huuvw'''∂∂∂'' hv hw ' Finally, the material parameters are, IITT εεCartesian'= RTTRTR Orthogonal / det( ) / det( ), IITT μμCartesian'= RTTRTR Orthogonal / det( ) / det( ). (12) One can find more detailed procedures on using Eq. (12) in the following sections. III. An application of the transformation optics in orthogonal coordinate Now we come to an application of the above transformation optics. Let’s suppose, ⎡⎤ε 00 I u εε= ⎢⎥00, Orthogonal⎢⎥ v ⎣⎦⎢⎥00ε w (13) ⎡⎤μ 00 I u μμ= ⎢⎥00, Orthogonal⎢⎥ v ⎣⎦⎢⎥00μw 7 / 23 the mapping is (uvw , , )⇔ ( u ', v ', w ') with, ⎧ uuu''(),= ⎧ uuu= ('), ⎪ ⎪ ⎨ vvv''(),= or ⎨ vvv= ('), (14) ⎪ ⎪ ⎩www''(),= ⎩www= ('), the Jacobian transformation matrix in the orthogonal coordinate is, ⎡⎤hu ' ∂u ' ⎡ 1 ⎤ ⎢⎥00⎢ 00⎥ hu∂ Q ⎢⎥u ⎢ u ⎥ ⎢⎥h ∂v '1⎢ ⎥ T ==⎢⎥0000,v' ⎢ ⎥ (15) ⎢⎥hvvv∂ ⎢ Q ⎥ ⎢⎥h ∂w' ⎢ 1 ⎥ ⎢⎥00w' ⎢ 00 ⎥ ⎣⎦hww ∂ ⎣ Qw ⎦ from Eq.
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