Abstract

In the presence of loss and gain, the coupled mode equation on describing the mode hybridization of various waveguides or cavities, or cavities coupled to waveguides becomes intrinsically non-Hermitian. In such non-Hermitian waveguides, the standard coupled mode theory fails. We generalize the coupled mode theory with a properly defined inner product based on reaction conservation. We apply our theory to the non-Hermitian parity-time symmetric waveguides, and obtain excellent agreement with results obtained by finite element fullwave simulations. The theory presented here is typically formulated in space to study coupling between waveguides, which can be transformed into time domain by proper reformulation to study coupling between non-Hermitian resonators. Our theory has the strength of studying non-Hermitian optical systems with inclusion of the full vector fields, thus is useful to study and design non-Hermitian devices that support asymmetric and even nonreciprocal light propagations.

© 2015 Optical Society of America

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References

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    [PubMed]
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2015 (2)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Photon.  9, 388–392 (2015).
[Crossref]

J. Xu, B. Wu, and Y. Chen, “Elimination of polarization degeneracy in circularly symmetric bianisotropic waveguides: a decoupled case,” Opt. Express 23, 11566–11575 (2015).
[Crossref] [PubMed]

2014 (7)

P. Pintus, “Accurate vectorial finite element mode solver for magneto-opti and anisotropic waveguides,” Opt. Express 22, 15737–15756 (2014).
[Crossref] [PubMed]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Y. Shen, X. H. Deng, and L. Chen, “Unidirectional invisibility in a two-layer non-PT-symmetric slab,” Opt. Express 22, 19440–19447 (2014).
[Crossref] [PubMed]

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

2013 (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

2011 (3)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

J. Mu and W. P. Huang, “Complex coupled-mode theory for tapered optical waveguides,” Opt. Lett. 36, 1026–1028 (2011).
[PubMed]

G. Zhu, “Pseudo-hermitian hamltonian formalism of electromagnetic wave propagation in a dielectric medium-application to nonorthogonal coupled-mode theory,” J. Lightw. Technol. 29, 905–911 (2011).
[Crossref]

2010 (1)

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

2008 (1)

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

2007 (1)

2003 (1)

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. B 20, 569–572 (2003).
[Crossref]

2000 (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).
[Crossref]

1999 (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[Crossref]

1994 (1)

1987 (2)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variation principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

1980 (1)

C. H. Chen and C.-D. Lien, “The variational principles for non-self-adjoint electromagnetic problems,” IEEE Trans. Microwave Theory Tech. 28, 878–886 (1980).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

1972 (1)

1971 (1)

D. Marcuse, “The coupling of degenerate modes in two parallel dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1816 (1971).
[Crossref]

1956 (1)

A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
[Crossref]

1955 (1)

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).
[Crossref]

1954 (1)

V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
[Crossref]

Alaeian, H.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Bahrampour, A. R.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Berk, A. D.

A. D. Berk, “Variational principles for electromagnetic resonators and waveguides,” IRE IEEE Trans. Antennas Propag. 4, 104–111 (1956).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Chen, C. H.

C. H. Chen and C.-D. Lien, “The variational principles for non-self-adjoint electromagnetic problems,” IEEE Trans. Microwave Theory Tech. 28, 878–886 (1980).
[Crossref]

Chen, L.

Chen, Y.

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Christodoulides, D. N.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

R. EI-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.,  32, 2632–2634 (2007).
[Crossref]

Chuang, S. L.

S. L. Chuang, “A coupled mode formulation by reciprocity and a variation principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

Deng, X. H.

Dionne, J. A.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[Crossref]

Eichelkraut, T.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

EI-Ganainy, R.

El-Ganainy, R.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Fan, S.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Photon.  9, 388–392 (2015).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. B 20, 569–572 (2003).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[Crossref]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Friedman, B.

B. Friedman, Principles and Techniques of Applied Mathematics (John Wiley & Sons, 1962).

Golshani, M.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[Crossref]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields, 2rd (Wiley-IEEE, 2001).
[Crossref]

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

H. A. Haus, “Electron beam waves in microwave tubes,” Proc. Symp. Electronic Waveguides, Polytechnic Inst. of Brooklyn, NY, 1958.

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Huang, W.

Huang, W. P.

J. Mu and W. P. Huang, “Complex coupled-mode theory for tapered optical waveguides,” Opt. Lett. 36, 1026–1028 (2011).
[PubMed]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Jafari, Kh.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. B 20, 569–572 (2003).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[Crossref]

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Khajavikhan, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

Khan, M. J.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[Crossref]

Khazaei, Nezhad M.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Kip, D.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Langari, A.

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).
[Crossref]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).
[Crossref]

Lien, C.-D.

C. H. Chen and C.-D. Lien, “The variational principles for non-self-adjoint electromagnetic problems,” IEEE Trans. Microwave Theory Tech. 28, 878–886 (1980).
[Crossref]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Louisell, W. H.

W. H. Louisell, Coupled-Mode and Parametric Electronics (Wiley, 1960).

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Mahdavi, S. M.

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M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
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L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
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Nature Photon (1)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Photon.  9, 388–392 (2015).
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Nature Phys. (1)

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
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V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. 94, 1483–1491 (1954).
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H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
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[Crossref]

Phys. Rev. E (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).
[Crossref]

Phys. Rev. Lett. (3)

M. Golshani, S. Weimann, Kh. Jafari, Nezhad M. Khazaei, A. Langari, A. R. Bahrampour, T. Eichelkraut, S. M. Mahdavi, and A. Szameit, “Impact of loss on the wave dynamics in photonic waveguide lattices,” Phys. Rev. Lett.,  113, 123903 (2014).
[Crossref] [PubMed]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
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X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).

Science (1)

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time -symmetric microring lasers,” Science 346, 975–978 (2014).
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Figures (2)

Fig. 1
Fig. 1 Real part and imaginary part of effective mode indices (neff) versus Δε using GCMT (a,b) and CCMT (c,d). Gray solid lines are calculated from fullwave simulation. Inset shows the schematic diagram of two coupled core layers with loss and gain, surrounded by air. Dimensions are h = 0.2λ0, w = 0.3λ0, d = 0.03λ0 (blue solid circles, green open circles) or 0.05λ0 (red diamonds, magenta crosses), ε0 = 10. λ0 is vacuum wavelength.
Fig. 2
Fig. 2 Real part of neff versus Δε. Gray solid lines are calculated from fullwave simulations. Blue open circles (red crosses) represent results derived by GCMT(CCMT). d = 0.03λ0. Other parameters are the same as Fig. 1.

Equations (24)

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

( F b , S a ) = F b , σ S a
( F b , S a ) = d r [ E b H b ] σ [ J a M a ]
L ¯ F = W ¯ S ,
( F a , S b ) = ( S a , F b ) ,
H ¯ F = S ,
( ψ , H ¯ ϕ ) = ( H ¯ ψ , ϕ ) .
H ¯ 2 d ϕ 2 d = 0
H ¯ 2 d a ψ 2 d = 0
( ψ 2 d , H ¯ 2 d ϕ 2 d ) = 0 = ( H ¯ 2 d a ψ 2 d , ϕ 2 d ) ,
Y = ( δ ψ 2 d , H ¯ 2 d ϕ 2 d ) + ( H ¯ 2 d a ψ 2 d , δ ϕ 2 d ) = 0.
( ψ 2 d , H ¯ 2 d ϕ 2 d ) ( H ¯ 2 d # ψ 2 d , ϕ 2 d ) = ( Δ H ¯ ψ 2 d , ϕ 2 d ) .
( ψ 2 d , H ¯ 2 d ϕ 2 d ) ( H ¯ 2 d # ψ 2 d , ϕ 2 d ) = 0.
t × e 0 , i + i β 0 , i z × e 0 , i + = i k 0 μ ¯ r 0 h 0 , i + ,
t × h 0 , i + i β 0 , i z × h 0 , i + = i k 0 ε ¯ r 0 e 0 , i + ,
t × Σ j a j e 0 , j + i β z × Σ j a j e 0 , j = i k 0 μ ¯ r Σ j a j h 0 , j ,
t × Σ j a j h 0 , j + i β z × Σ j a j h 0 , j = i k 0 ε ¯ r Σ j a j e 0 , j ,
{ E q . ( 13 a ) Σ j a j h 0 , j E q . ( 14 b ) e 0 , i + + E q . ( 13 b ) Σ j a j e 0 , j E q . ( 14 a ) h 0 , i + } d x d y
Σ j a j [ k i j + b i j i ( β β 0 , i ) p i j ] = 0
b i j = i ( β 0 , j β 0 , i ) p i j .
Σ j a j ( β β 0 , j ) p i j = Σ j a j k i j .
( ψ 2 d * , H ¯ ϕ 2 d ) ( ( H ¯ # ψ 2 d ) * , ϕ 2 d ) = 0 ,
Σ j a j ( β * β 0 , j * ) p i j = Σ j a j k i j .
[ β 0 , 1 p 11 i k 11 β 0 , 2 p 12 i k 12 β 0 , 1 p 21 i k 21 β 0 , 2 p 22 i k 22 ] [ a 1 a 2 ] = β [ p 11 p 12 p 21 p 22 ] [ a 1 a 2 ]
[ β 0 , 1 * p 11 i k 11 β 0 , 2 * p 12 i k 12 β 0 , 1 * p 21 i k 21 β 0 , 2 * p 22 i k 22 ] [ a 1 a 2 ] = β * [ p 11 p 12 p 21 p 22 ] [ a 1 a 2 ] .

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