Abstract

Cavity array metamaterials (CAMs), composed of optical microcavities in a lattice coupled via tight-binding interactions, represent a novel architecture for engineering metamaterials. Since the size of the CAMs’ constituent elements are commensurate with the operating wavelength of the device, it cannot directly utilise classical transformation optics in the same way as traditional metamaterials. By directly transforming the internal geometry of the system, and locally tuning the permittivity between cavities, we provide an alternative framework suitable for tight-binding implementations of metamaterials. We develop a CAM-based cloak as the case study.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. C.-H. Su, “Novel quantum technology based on atom-cavity physics”, Ph.D. thesis, The University of Melbourne, Victoria (2010).
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    [CrossRef] [PubMed]
  9. P. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields”, Science312, 1780–1782 (2006).
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  10. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves”, Phys. Rev. Lett.100, 123002 (2008).
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    [CrossRef] [PubMed]

2011

2010

2009

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

A. Faraon and J. Vučković, “Local temperature control of photonic crystal devices via micron-scale electrical heaters,” Appl. Phys. Lett.95, 043102 (2009).
[CrossRef]

S. Tomljenovic-Hanic, A. D. Greentree, C. M. de Sterke, and S. Prawer, “Flexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabslexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabs,” Opt. Express18, 6465–6475 (2009).
[CrossRef]

D. Englund, B. Ellis, E. Edwards, T. Sarmiento, J. S. Harris, D. A. B. Miller, and J. Vuckovic, “Electrically controlled modulation in a photonic crystal nanocavity,” Opt. Express17, 15409–15419 (2009).
[CrossRef] [PubMed]

G. Le Gac, A. Rahmani, C. Seassal, E. Picard, E. Hadji, and S. Callard, “Tuning of an active photonic crystal cavity by an hybrid silica/silicon near-field probe,” Opt. Express17, 21672–21679 (2009).
[CrossRef] [PubMed]

2008

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

M.-K. Seo, H.-G. Park, J.-K. Yang, J.-Y. Kim, S.-H. Kim, and Y.-H. Lee, “Controlled sub-nanometer tuning of photonic crystal resonator by carbonaceous nano-dots,” Opt. Express16, 9829–9837 (2008).
[CrossRef] [PubMed]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves”, Phys. Rev. Lett.100, 123002 (2008).
[CrossRef] [PubMed]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities”, Nat. Phot.2, 741–747, (2008).
[CrossRef]

2006

U. Leonhardt, “Optical conformal mapping”, Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

P. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields”, Science312, 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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

2005

D. Dalacu, S. Frédérick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87, 151107 (2005).
[CrossRef]

2003

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B68, 045115 (2003).
[CrossRef]

2001

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[CrossRef] [PubMed]

1999

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24, 771–713 (1999).
[CrossRef]

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum” Phys. Rev. Lett.83, 967 (1999).
[CrossRef]

1987

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference”, Phys. Rev. Lett.59, 2044–2046, (1987).
[CrossRef] [PubMed]

Aers, G. C.

D. Dalacu, S. Frédérick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87, 151107 (2005).
[CrossRef]

Bulla, D.

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Busch, K.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum” Phys. Rev. Lett.83, 967 (1999).
[CrossRef]

Callard, S.

Cummer, S. A.

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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Dalacu, D.

D. Dalacu, S. Frédérick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87, 151107 (2005).
[CrossRef]

de Sterke, C. M.

S. Tomljenovic-Hanic, A. D. Greentree, C. M. de Sterke, and S. Prawer, “Flexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabslexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabs,” Opt. Express18, 6465–6475 (2009).
[CrossRef]

Edwards, E.

Eggleton, B. J.

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Ellis, B.

Englund, D.

D. Englund, B. Ellis, E. Edwards, T. Sarmiento, J. S. Harris, D. A. B. Miller, and J. Vuckovic, “Electrically controlled modulation in a photonic crystal nanocavity,” Opt. Express17, 15409–15419 (2009).
[CrossRef] [PubMed]

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Faraon, A.

A. Faraon and J. Vučković, “Local temperature control of photonic crystal devices via micron-scale electrical heaters,” Appl. Phys. Lett.95, 043102 (2009).
[CrossRef]

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Frédérick, S.

D. Dalacu, S. Frédérick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87, 151107 (2005).
[CrossRef]

Genov, D. A.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves”, Phys. Rev. Lett.100, 123002 (2008).
[CrossRef] [PubMed]

Gibson, B. C.

Greentree, A. D.

S. Tomljenovic-Hanic, A. D. Greentree, B. C. Gibson, T. J. Karle, and S. Prawer, “Nanodiamond induced high-Q resonances in defect-free photonic crystal slabs,” Opt. Express19, 22219–22226 (2011).
[CrossRef] [PubMed]

J. Q. Quach, C.-H. Su, A. M. Martin, A. D. Greentree, and L. C. L. Hollenberg, “Reconfigurable quantum metamaterials”, Opt. Express19, 11018–11033 (2011).
[CrossRef] [PubMed]

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

S. Tomljenovic-Hanic, A. D. Greentree, C. M. de Sterke, and S. Prawer, “Flexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabslexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabs,” Opt. Express18, 6465–6475 (2009).
[CrossRef]

Hadji, E.

Harris, J. S.

Hatami, F.

Hollenberg, L. C. L.

J. Q. Quach, C.-H. Su, A. M. Martin, A. D. Greentree, and L. C. L. Hollenberg, “Reconfigurable quantum metamaterials”, Opt. Express19, 11018–11033 (2011).
[CrossRef] [PubMed]

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference”, Phys. Rev. Lett.59, 2044–2046, (1987).
[CrossRef] [PubMed]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B68, 045115 (2003).
[CrossRef]

John, S.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum” Phys. Rev. Lett.83, 967 (1999).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B68, 045115 (2003).
[CrossRef]

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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Karle, T. J.

Kim, J.-Y.

Kim, S.-H.

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities”, Nat. Phot.2, 741–747, (2008).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

Le Gac, G.

Lee, R. K.

Lee, Y.-H.

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping”, Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

Liang, Z.

Lu, J.

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B68, 045115 (2003).
[CrossRef]

Luther-Davies, B.

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Makin, M. I.

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference”, Phys. Rev. Lett.59, 2044–2046, (1987).
[CrossRef] [PubMed]

Martin, A. M.

Miller, D. A. B.

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities”, Nat. Phot.2, 741–747, (2008).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference”, Phys. Rev. Lett.59, 2044–2046, (1987).
[CrossRef] [PubMed]

Park, H.-G.

Pendry, J. B.

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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B68, 045115 (2003).
[CrossRef]

Pendry, P. B.

P. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields”, Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Petroff, P.

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Picard, E.

Poole, P. J.

D. Dalacu, S. Frédérick, P. J. Poole, G. C. Aers, and R. L. Williams, “Postfabrication fine-tuning of photonic crystal microcavities in InAs/InP quantum dot membranes,” Appl. Phys. Lett.87, 151107 (2005).
[CrossRef]

Prawer, S.

S. Tomljenovic-Hanic, A. D. Greentree, B. C. Gibson, T. J. Karle, and S. Prawer, “Nanodiamond induced high-Q resonances in defect-free photonic crystal slabs,” Opt. Express19, 22219–22226 (2011).
[CrossRef] [PubMed]

S. Tomljenovic-Hanic, A. D. Greentree, C. M. de Sterke, and S. Prawer, “Flexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabslexible design of ultrahigh-Q microcavities in diamond-based photonic crystal slabs,” Opt. Express18, 6465–6475 (2009).
[CrossRef]

Quach, J.

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

Quach, J. Q.

Rahmani, A.

Rivoire, K.

Sarmiento, T.

Scherer, A.

Schultz, S.

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[CrossRef] [PubMed]

Schurig, D.

P. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields”, Science312, 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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Seassal, C.

Seo, M.-K.

Shambat, G.

Shelby, R.

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[CrossRef] [PubMed]

Shinya, A.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

Smith, D. R.

Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett.105, 163901–163905 (2010).
[CrossRef]

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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

P. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields”, Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[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,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

Stoltz, N.

A. Faraon, D. Englund, D. Bulla, B. Luther-Davies, B. J. Eggleton, N. Stoltz, P. Petroff, and J. Vuckovic, “Local tuning of photonic crystal cavities using chalcogenide glasses,” Appl. Phys. Lett.92, 043123 (2008).
[CrossRef]

Su, C.-H.

J. Q. Quach, C.-H. Su, A. M. Martin, A. D. Greentree, and L. C. L. Hollenberg, “Reconfigurable quantum metamaterials”, Opt. Express19, 11018–11033 (2011).
[CrossRef] [PubMed]

J. Quach, M. I. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A,80, 063838 (2009).
[CrossRef]

C.-H. Su, “Novel quantum technology based on atom-cavity physics”, Ph.D. thesis, The University of Melbourne, Victoria (2010).

Sun, C.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves”, Phys. Rev. Lett.100, 123002 (2008).
[CrossRef] [PubMed]

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities”, Nat. Phot.2, 741–747, (2008).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

Tomljenovic-Hanic, S.

S. Tomljenovic-Hanic, A. D. Greentree, B. C. Gibson, T. J. Karle, and S. Prawer, “Nanodiamond induced high-Q resonances in defect-free photonic crystal slabs,” Opt. Express19, 22219–22226 (2011).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a 2D array of coupled optical cavities that guides electromagnetic waves to form a cloaked region. (b) The inter-cavity coupling κ can be achieved through evanescent field overlap. (c) An infiltrated silicon-based PhC. The refractive index of the liquid crystal infiltrated into air pores (blue) can be tuned with an applied electric field, providing a mechanism to dynamically control local intercavity couplings.

Fig. 2
Fig. 2

Dispersion surface E of a square coupled-cavity lattice system in the first Brillouin zone. The frequency of the contours are labeled with values of (Eω)/κ. The white contour indicates the operating frequency used to simulate the effective point source in Fig. 3(a). The black contour indicates the operating frequency used to simulate the beam source in Fig. 3(c).

Fig. 3
Fig. 3

(a) A quasi-point source forms a spherical wave. The spherical wave bends around the circular cloak region, rendering it effectively invisible. (b) The position of the cavity sites in the circular cloak are transformed according to Eq. (4) with a = 5 and b = 10, making an annulus feature. (c) Cloaking of a normally incident Gaussian pulse, superimposed in time to form a continuous beam. The circular cloak region is formed with a = 50, b = 100. The x and y axis indicate site co-ordinates.

Fig. 4
Fig. 4

(a) Implementation of the cloaking device shown in Fig. 3, using a lattice distribution of inter-site permittivity εb. εb values are denoted by the colored dots between the lattice sites. (b) Variation of the inter-site permittivity as a function of the distance from the cloaking center, in a quadrant (other quadrants are symmetrically equivalent). Near the centre, the intersite permittivity is lower than the baseline permittivity as the spacing between sites are stretched; whereas near the outer edge of the cloaking annulus the permittivity is higher as the sites are squashed closer together. The blue dots indicate the permittivity distribution along co-ordinate x = 25.5.

Equations (6)

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= ω i a i a i κ i , j a i a j ,
E = ω 2 κ [ cos ( k x d ) + cos ( k y d ) ] ,
v g k E = 2 κ d [ sin ( k x d ) x ^ + sin ( k y d ) y ^ ] .
r = b a b r + a , ϕ = ϕ ,
| E Ω ( x ) | = { A cos ( ε a ω c x ) , | x | < w / 2 A cos ( ε a ω w 2 c ) e ε b ω c ( w / 2 | x | ) , | x | > w / 2
κ = ω 2 ( ε a ε b ) w / 2 w / 2 | E Ω ( x ) | | E Ω ( x d ) | d x .

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