Abstract

The technique of transformation optics (TO) is an elegant method for the design of electromagnetic media with tailored optical properties. In this paper, we focus on the formal structure of TO theory. By using a complete covariant formalism, we present a general transformation law that holds for arbitrary materials including bianisotropic, magneto-optical, nonlinear and moving media. Due to the principle of general covariance, the formalism is applicable to arbitrary space-time coordinate transformations and automatically accounts for magneto-electric coupling terms. The formalism is demonstrated for the calculation of the second harmonic wave generation in a twisted TO concentrator.

© 2012 OSA

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    [CrossRef]

2011

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[CrossRef]

U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011).
[CrossRef] [PubMed]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[CrossRef]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[CrossRef]

2010

R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
[CrossRef]

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[CrossRef]

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

2009

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
[CrossRef]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[CrossRef]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

2008

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33, 1584–1586 (2008).
[CrossRef] [PubMed]

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008).
[CrossRef]

2007

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

2006

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 312, 977–980 (2006).
[CrossRef]

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

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

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[CrossRef]

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[CrossRef]

1995

1993

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
[CrossRef] [PubMed]

1989

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

1988

1976

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48, 1–32 (1976).
[CrossRef]

1968

D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

1967

N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
[CrossRef]

1966

W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966).
[CrossRef]

1965

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Agrawal, G.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

Alfano, R. R.

Alitalo, P.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[CrossRef]

Baldeck, P. L.

Bergamin, L.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[CrossRef]

Bloembergen, N.

N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
[CrossRef]

Bosenberg, W. R.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Briane, M.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[CrossRef]

Butkus, R.

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[CrossRef]

Byer, R. L.

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

Chan, C. T.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

Chen, H.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

Cheng, D.

D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

Cheng, Q.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

Chin, J. Y.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Cui, T. J.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Cummer, S. A.

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[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,” Science 312, 977–980 (2006).
[CrossRef]

Dubietis, A.

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[CrossRef]

Eckardt, R. C.

Evans, M.

M. Evans and S. Kielich, Modern Nonlinear Optics (Wiley, 1997).

Faust, W. L.

W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966).
[CrossRef]

Fejer, M. M.

Frauendiener, J.

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[CrossRef]

Genov, D. A.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[CrossRef]

Gordon, C. L.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
[CrossRef] [PubMed]

Gu, C.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Guenneau, S.

Han, T.

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[CrossRef]

Henry, C. H.

W. L. Faust and C. H. Henry, “Mixing of visible and near-resonance infrared light in gap,” Phys. Rev. Lett. 17, 1265–1268 (1966).
[CrossRef]

Jiang, W. X.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[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,” Science 312, 977–980 (2006).
[CrossRef]

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kielich, S.

M. Evans and S. Kielich, Modern Nonlinear Optics (Wiley, 1997).

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

Kmetec, J. D.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
[CrossRef] [PubMed]

Kong, J.-A.

D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[CrossRef]

Kundtz, N.

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

Kundtz, N. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Landau, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).

Landy, N. I.

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011).
[CrossRef] [PubMed]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

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

Li, Z.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Lifshitz, E.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).

Liu, R.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
[CrossRef]

Lu, L.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Ma, H.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Ma, Y. G.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Macklin, J. J.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, “High-order harmonic generation using intense femtosecond pulses,” Phys. Rev. Lett. 70, 766–769 (1993).
[CrossRef] [PubMed]

Manassah, J. T.

Milton, G. W.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (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,” Science 312, 977–980 (2006).
[CrossRef]

Myers, L. E.

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Nicolet, A.

Novitsky, A.

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
[CrossRef]

Olsson, N.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

Ong, C. K.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Pendry, J.

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[CrossRef]

Pendry, J. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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 312, 977–980 (2006).
[CrossRef]

Pierce, J. W.

Piskarskas, A.

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[CrossRef]

Pitaevskii, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1984).

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Dover Publications, 1997).

Qiu, C.

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[CrossRef]

Qiu, C.-W.

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
[CrossRef]

Qu, S.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Rousseaux, G.

G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008).
[CrossRef]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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 312, 977–980 (2006).
[CrossRef]

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A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

Shen, Y. R.

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

Y. R. Shen, “Recent advances in nonlinear optics,” Rev. Mod. Phys. 48, 1–32 (1976).
[CrossRef]

Smith, D.

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
[CrossRef]

Smith, D. R.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

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, 063903 (2008).
[CrossRef] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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 312, 977–980 (2006).
[CrossRef]

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 312, 977–980 (2006).
[CrossRef]

Tang, X.

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[CrossRef]

Thompson, R. T.

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[CrossRef]

R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010).
[CrossRef]

Tretyakov, S. A.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
[CrossRef]

Tyc, T.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Urzhumov, Y. A.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

Wang, B.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Wang, J.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Wang, X.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Werner, D. H.

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Willis, J. R.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[CrossRef]

Wu, X.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Xia, S.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Xu, Z.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Zhang, S.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[CrossRef]

Zhang, X.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[CrossRef]

Zhou, H.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

Zolla, F.

Zouhdi, S.

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
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Am. J. Phys.

N. Bloembergen, “The stimulated raman effect,” Am. J. Phys. 35, 989–1023 (1967).
[CrossRef]

Appl. Phys. Lett.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
[CrossRef]

Europhys. Lett.

G. Rousseaux, “On the electrodynamics of minkowski at low velocities,” Europhys. Lett. 84, 20002 (2008).
[CrossRef]

IEEE J. Quantum Electron.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Dubietis, R. Butkus, and A. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006).
[CrossRef]

J. Opt.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[CrossRef]

T. Han, C. Qiu, and X. Tang, “An arbitrarily shaped cloak with nonsingular and homogeneous parameters designed using a twofold transformation,” J. Opt. 12, 095103 (2010).
[CrossRef]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Mater.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Nat. Photonics

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

Nat. Phys.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687–692 (2009).
[CrossRef]

Nature

U. Leonhardt, “To invisibility and beyond,” Nature 471, 292–293 (2011).
[CrossRef] [PubMed]

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

New J. Phys.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006).
[CrossRef]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[CrossRef]

A. Novitsky, C.-W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11, 113001 (2009).
[CrossRef]

Opt. Lett.

Photon. Nanostruct. Fundam. Appl.

X. Wang, S. Qu, S. Xia, B. Wang, Z. Xu, H. Ma, J. Wang, C. Gu, X. Wu, L. Lu, and H. Zhou, “Numerical method of designing three-dimensional open cloaks with arbitrary boundary shapes,” Photon. Nanostruct. Fundam. Appl. 8, 205–208 (2010).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of maxwells equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Phys. Rev. A

R. T. Thompson, “Transformation optics in nonvacuum initial dielectric media,” Phys. Rev. A 82, 053801 (2010).
[CrossRef]

Phys. Rev. B

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the kerr effect,” Phys. Rev. B 84, 205103 (2011).
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W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008).
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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, 063903 (2008).
[CrossRef] [PubMed]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105, 193902 (2010).
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Proc. IEEE

N. Kundtz, D. Smith, and J. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99, 1622–1633 (2011).
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Science

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 312, 977–980 (2006).
[CrossRef]

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

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
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Figures (2)

Fig. 1
Fig. 1

Visualization of the twisted cylindrical concentrator. The mesh grid indicates lines with constant values of x and y plotted in the coordinate system of x′ and y′. The radius of the three displayed circles are R1 = 3, R2 = 4.5 and R3 = 6.

Fig. 2
Fig. 2

Second harmonic wave generation in a nonlinear cylindrical concentrator. (a) Electric field of the fundamental wave in the uniform space (unprimed system) and (b) in the inhomogeneous, physical space (primed system). (c) Electric field of the second harmonic wave in the unprimed system and (d) in the primed system.

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

div B = 0 , rot E + 1 c B t = 0 ,
div D = 4 π ρ , rot H 1 c D t = 4 π c j
F μ ν = ( 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 )
𝒟 μ ν = ( 0 D x D y D z D x 0 H z H y D y H z 0 H x D z H y H x 0 ) ,
μ F ν σ + σ F μ ν + ν F σ μ = 0 ,
μ 𝒟 μ ν = 4 π c j ν
𝒟 μ ν = F μ ν + 4 π 𝒫 μ ν
𝒫 μ ν = ( 0 P x P y P z P x 0 M z M y P y M z 0 M x P z M y M x 0 ) .
D = E + 4 π P , H = B 4 π M .
𝒫 μ ν = χ μ ν σ κ F σ κ + χ μ ν σ κ α β F σ κ F α β + χ μ ν σ κ α β γ δ F σ κ F α β F γ δ + = n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n
χ μ ν α 1 β 1 α 2 β 2 = χ ν μ α 1 β 1 α 2 β 2 = χ μ ν α 2 β 2 α 1 β 1 = χ μ ν β 1 α 1 α 2 β 2 .
𝒟 μ ν = F μ ν + 4 π 𝒫 μ ν = F μ ν + 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n
𝒟 μ ν = 4 π χ μ ν σ κ F σ κ .
( D H ) = ( ε ξ ζ μ 1 ) ( E B ) .
ε i j = 8 π χ 0 i 0 j , ξ i j = 4 π g m n j χ 0 i m n , ζ ij = 4 π g m n i χ m n 0 j , ( μ 1 ) i j = 2 π g m n i g k l j χ m n k l
𝒫 ( 2 ) μ ν = χ μ ν σ κ α β F σ κ F α β .
P ( 2 ) i = χ 0 i σ κ α β F σ κ F α β = 4 χ 0 i 0 k 0 m F 0 k F 0 m + 4 χ 0 i 0 k m n F 0 k F m n + χ 0 i k l m n F k l F m n = a i j k E j E k Pockels , effect , multi-wave mixing + b i j k E j B k Faraday effect + c i j k B j B k
P ( 3 ) i = χ 0 i σ κ α β μ ν F σ κ F α β F μ ν = a i j k l E j E k E l Kerr effect + b i j k l E j E k B l + c i j k l E j B k B l Cotton-Mouton effect + d i j k l B j B k B l ,
x α x α ( x α )
A α α A β β A γ γ = A α β γ α β γ .
F μ ν = A μ ν μ ν F μ ν , 𝒟 μ ν = | A | 1 A μ ν μ ν 𝒟 μ ν .
μ F ν σ + σ F μ ν + ν F σ μ = 0 , μ 𝒟 μ ν = 4 π c j ν
χ μ ν α 1 β 1 α n β n = | A | 1 A μ ν α 1 β 1 α n β n μ ν α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n .
A α 1 β 1 α n β n α 1 β 1 α n β n A α 1 β 1 α n β n α 1 β 1 α n β n = A α 1 β 1 α n β n α 1 β 1 α n β n = 1
𝒟 μ ν = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n | A | 1 A μ ν μ ν 𝒟 μ ν 𝒟 μ ν = 4 π n = 1 | A | 1 A μ ν μ ν A α 1 β 1 α n β n α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n χ μ ν α 1 β 1 α n β n A a 1 β 1 α n β n α 1 β 1 α n β n F α 1 β 1 F α n β n F α 1 β 1 F α n β n 𝒟 μ ν = 4 π n = 1 χ μ ν α 1 β 1 α n β n F α 1 β 1 F α n β n .
A α α = ( γ γ u / c 0 0 γ u / c γ 0 0 0 0 1 0 0 0 0 1 ) , | A | = 1
χ μ ν σ κ = A μ ν σ κ μ ν σ κ χ μ ν σ κ .
χ μ ν σ κ α β = A μ ν σ κ α β μ ν σ κ α β χ μ ν σ κ α β
t = t , x = x ( x , y , z ) , y = y ( x , y , z ) , z = z ( x , y , z )
A 0 0 = t t = 1 and A i 0 = c t x i = 0.
ε i j = 8 π χ 0 i 0 j = 8 π | A | 1 A μ ν α β 0 i 0 j χ μ ν α β = 8 π | A | 1 A i j i j χ 0 i 0 j = | A | 1 A i j i j ε i j .
μ i j = | A | 1 A i j i j μ i j , ξ i j = | A | 1 A i j i j ξ i j , ζ i j = | A | 1 A i j i j ζ i j .
χ μ ν σ κ α β = | A | 1 A μ ν σ κ α β μ ν σ κ α β χ μ ν σ κ α β
a i j k = 4 χ 0 i 0 j 0 k = 4 | A | 1 A μ ν σ κ α β 0 i 0 j 0 k χ μ ν σ κ α β = 4 | A | 1 A i j k i j k χ 0 i 0 j 0 k = | A | 1 A i j k i j k a i j k
r = { R 1 R 2 r 0 r R 2 R 3 R 1 R 3 R 2 r R 2 R 1 R 3 R 2 R 3 R 2 r R 3 r otherwise ϕ = { ϕ + π 2 cos 2 ( π r 2 R 3 ) 0 r R 3 ϕ otherwise .
A g = ( r r r ϕ ϕ r ϕ ϕ ) = { ( R 1 R 2 0 π 2 4 R 3 sin ( π r R 3 ) 1 ) 0 r R 2 ( R 3 R 1 R 3 R 2 0 π 2 4 R 3 sin ( π r R 3 ) 1 ) R 2 r R 3 diag ( 1 , 1 ) otherwise .
| A g | = { R 1 R 2 0 r R 2 R 3 R 1 R 3 R 2 R 2 r R 3 1 otherwise .
( x , y ) f ( r , ϕ ) g ( r , ϕ ) h ( x , y ) .
A f g h = A f A g A h .
| A f g h | = | A f | | A g | | A h | = r r | A g |
a 3 3 3 ( r ) = { a 0 0 r R 1 0 otherwise
E ω ( x , y , t ) = 𝒠 ω e i ( k ω x x + k ω y y ω t )
E 2 ω ( x , y , t ) = 𝒠 2 ω ( x , y ) e i ( k 2 ω x x + k 2 ω y y 2 ω t )
( k 2 ω x x + k 2 ω y y ) 𝒠 2 ω ( x , y ) = κ 𝒠 ω 2 a 3 3 3 ( x , y )
E z = F 0 3 = A 0 3 μ ν F μ ν = F 03 = E z
E ω ( x , t ) = 𝒠 ω e i ( k ω x x ω t )
x 𝒠 2 ω ( x , y ) = κ c 2 ω 𝒠 ω 2 a 333 ( x , y )
𝒠 2 ω ( x , y ) = κ c 2 ω 𝒠 ω 2 x d s a 333 ( s , y ) .
a 3 3 3 = | A | 1 A i j k 3 3 3 a i j k = | A | 1 a 333
a 333 = r r | A g | a 3 3 3 = { ( R 1 R 2 ) 2 a 0 0 r R 2 0 otherwise .
E 2 ω ( x , y , t ) = 𝒠 2 ω ( x , y ) e 2 i ω ( x / c t ) .
𝒟 μ ν = F μ ν + 4 π χ μ ν σ κ F σ κ = ( 1 2 ( η μ σ η ν κ η ν σ η μ κ ) + 4 π χ μ ν σ κ ) F σ κ .
F i j = 1 2 ( F i j F j i ) = 1 2 ( δ i j m n δ j i m n ) F m n = 1 2 g i j k g k m n F m n = g i j k B k
D i = 𝒟 0 i = 4 π χ 0 i σ κ F σ κ = 8 π χ 0 i 0 j F 0 j 4 π χ 0 i m n F m n = 8 π χ 0 i 0 j E j + 4 π g m n j χ 0 i m n B j
H i = 1 2 g m n i 𝒟 m n = 2 π g m n i χ m n σ κ F σ κ = 4 π g m n i χ m n 0 j F 0 j 2 π g m n i χ m n k l F k l = 4 π g m n i χ m n 0 j E j + 2 π g m n i g k l j χ m n k l B j .
D i = ε i j E j + ξ i j B j H i = ζ i j E j + ( μ 1 ) i j B j ,
r = x 2 + y 2 ϕ = arctan ( y / x )
A f = ( r x r y ϕ x ϕ y ) = ( x r y r y r 2 x r 2 ) , | A g | = 1 r .
x = r cos ϕ y = r sin ϕ
A h = ( x r x ϕ y r y ϕ ) = ( cos ϕ r sin ϕ sin ϕ r cos ϕ ) , | A h | = r

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