Abstract

A general conformal transformation method (CTM) is proposed to construct the conformal mapping between two irregular geometries. In order to find the material parameters corresponding to the conformal transformation between two irregular geometries, two polygons are utilized to approximate the two irregular geometries, and an intermediate geometry is used to connect the mapping relations between the two polygons. Based on these manipulations, the approximate material parameters for TE and TM waves are finally obtained by calculating the Schwarz-Christoffel (SC) mappings. To demonstrate the validity of the method, a phase modulator and a plane focal surface Luneburg lens are designed and simulated by the finite element method. The results show that the conformal transformation can be expanded to the cases that the transformed objects are with irregular geometries.

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2011 (2)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011).
[CrossRef] [PubMed]

2010 (5)

2009 (1)

2008 (3)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

1996 (1)

T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. 22(2), 168–186 (1996).
[CrossRef]

Cen, Z.

C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. 97(4), 044101 (2010).
[CrossRef]

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Deng, Q.

Di Falco, A.

Dong, X.

Driscoll, T. A.

T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. 22(2), 168–186 (1996).
[CrossRef]

Du, C.

Gao, H.

Jiang, Z. H.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kehr, S. C.

Landy, N. I.

Lederer, F.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[CrossRef]

Leonhardt, U.

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Liu, C.

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

Lu, Y.

Ma, Y. G.

Massoud, A. T.

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Ong, C. K.

Padilla, W. J.

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Psaltis, D.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

Ren, C.

C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. 97(4), 044101 (2010).
[CrossRef]

Rockstuhl, C.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[CrossRef]

Schmiele, M.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[CrossRef]

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Smith, D. R.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Tsang, M.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Turpin, J. P.

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

Varma, V. S.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[CrossRef]

Wang, N.

Werner, D. H.

Werner, P. L.

Xiang, Z.

C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. 97(4), 044101 (2010).
[CrossRef]

Yuan, G.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

ACM Trans. Math. Softw. (1)

T. A. Driscoll, “A MATLAB toolbox for Schwarz-Christoffel mapping,” ACM Trans. Math. Softw. 22(2), 168–186 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with isotropic material via conformal transformation,” Appl. Phys. Lett. 97(4), 044101 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Nanotechnol. (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011), doi:.
[CrossRef] [PubMed]

Opt. Express (4)

Phys. Rev. A (1)

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[CrossRef]

Phys. Rev. B (1)

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Other (2)

T. A. Driscoll and L. N. Trefethen, Schwartz-Christoffel Mapping (Cambridge University Press, 2002).

P. Henrici, Applied and Computational Complex Analysis, Volume 3: Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions (Wiley, 1986).

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Figures (6)

Fig. 1
Fig. 1

The procedure of the CTM. Q 1, Q 2 are two irregular geometries, P 1, P 2 are two polygons, R is the intermediate geometry. Q 1, P 1 are in the virtual space, and Q 2, P 2 are in the physical space. In the CTM, the material parameters corresponding to the mapping from P 1 onto P 2 are used to approximate the ones corresponding to the mapping from Q 1 onto Q 2. ω = f (z) is the conformal mapping from P 1 to P 2. f 1, f 2 can be computed by SC mapping, and g 1, g 2 are the inverse mapping of f 1, f 2, respectively.

Fig. 2
Fig. 2

Design for a phase modulator with the CTM. Q 1, P 1 are in the virtual space, and Q 2, P 2 are in the physical space. Q 1, Q 2 are approximated by the two polygons P 1, P 2, respectively. A rectangle R is used as the intermediate geometry to connect P 1, P 2.

Fig. 3
Fig. 3

The simulation results of the phase modulator. (a) The refractive index distribution in the phase modulator. (b) The z component of the electric field when a point source is on the center of the phase modulator.

Fig. 4
Fig. 4

Design for a PSF Luneburg lens with the CTM. Q 1 is in the virtual space. Q 2, P 2 are in the physical space. P 2 is a polygon and its MN boundary (marked in blue) is composed of 60 line segments used to approximate the MN arc of Q 2.

Fig. 5
Fig. 5

Simulation results of the PFS Luneburg lens. (a) is the refractive index distribution in the PFS Luneburg lens. (b)(c)(d) are the Z component of the electric field when the point source is located at (u, v) = (−0.5, 0), (−0.5, 0.1), (−0.5, 0.2), respectively. (e)(f) are the Z component of the electric field when plane waves incident on the PSF Luneburg lens.

Fig. 6
Fig. 6

The dependence of the width of the distorted region on the approximation in the PFS Luneburg lens. (a)(b)(c) are contours of constant value of the resulting permittivity near the MN arc when the arc is approximated by 15 line segments, 60 line segments, and 90 line segments, respectively. And the widths (marked with red line segments) of the distorted regions of the permittivity distributions are about 0.2cm, 0.02cm, and 0.01cm, respectively.

Equations (5)

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ε = 1 / [ ( x u ) 2 + ( x v ) 2 ] ,       μ = 1   ( T E ) , μ = 1 / [ ( x u ) 2 + ( x v ) 2 ] ,     ε = 1   ( T M ) .
ε = 1 / | f ' | 2 ,       μ = 1   ( TE ) , μ = 1 / | f ' | 2 ,     ε = 1   ( TM ) ,
f ' ( z ) = f 2 ' ( ζ ) g 1 ' ( z ) .
ε = | f 1 ' / f 2 ' | 2 ,       μ = 1   ( T E ) , μ = | f 1 ' / f 2 ' | 2 ,     ε = 1   ( T M ) .
ε L ( r ) = 2 ( r / R ) 2 ,

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