Abstract

Unidirectional reflectionless phenomenon is reported in periodic ternary layered material (PTLM). The unit of the material is composed of two real dielectric layers and a complex medium (loss or gain) layer. The model is analyzed by coupled mode theory. Because of the asymmetric coupling between the forward and backward propagating modes, the left- and right-side reflectivities of this PTLM are generally unequal. The necessary and sufficient (NS) condition for unidirectional reflectionless phenomenon is presented in a concise formulation. And the underlying physical mechanism of the unidirectional reflectionless phenomenon in this material is revealed by numerical simulations. Both unidirectional reflectionless and symmetric reflection phenomena can be realized by judicious choice of the structural and optical parameters.

© 2016 Optical Society of America

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References

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2012 (3)

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

2011 (2)

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A Math. Theor. 44(48), 485302 (2011).
[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(21), 213901 (2011).
[Crossref] [PubMed]

2010 (3)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

2009 (2)

A. Alù, “Mantle cloak: Invisibility induced by a surface,” Phys. Rev. B 80(24), 245115 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

2006 (1)

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]

2005 (2)

M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[Crossref] [PubMed]

2004 (2)

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Alù, A.

A. Alù, “Mantle cloak: Invisibility induced by a surface,” Phys. Rev. B 80(24), 245115 (2009).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[Crossref] [PubMed]

Azaña, J.

Bélanger, N.

Bender, C. M.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[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(21), 213901 (2011).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Chong, Y. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[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(21), 213901 (2011).
[Crossref] [PubMed]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

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,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Eichelkraut, 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(21), 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Engheta, N.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Ge, L.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Greenberg, M.

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

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]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[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(21), 213901 (2011).
[Crossref] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Kulishov, M.

Laniel, J.

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(21), 213901 (2011).
[Crossref] [PubMed]

Longhi, S.

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A Math. Theor. 44(48), 485302 (2011).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

Miri, M.-A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[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]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Oliveira, J. E. B.

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Orenstein, M.

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,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Plant, D.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Ramezani, 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(21), 213901 (2011).
[Crossref] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Regensburger, A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Scherer, A.

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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Schurig, D.

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]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Smith, D. R.

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]

Stone, A. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Xu, Y.-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,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

J. Phys. A Math. Theor. (1)

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A Math. Theor. 44(48), 485302 (2011).
[Crossref]

Nat. Mater. (1)

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Figures (5)

Fig. 1
Fig. 1 Schematic diagram of the unidirectional reflectionless PTLM and the corresponding refractive index distribution. Deep orange area for the real modulation layers; blue for the imaginary modulation layers and orange for the non-modulation layers. The refractive index is n 0 + n R , n 0 + i n I , and n 0 with thickness is a , b , and ( Λ a b ) respectively. Λ is the period and N is the number of unit cells. The entire structure is laterally infinite.
Fig. 2
Fig. 2 Theoretical results about log 10 R L R R as a function of a / Λ and b / Λ under the condition of n R = n I cot ϕ a . m = 1 - 4 are the cases that the propagation constants of incident wave are around 1st-4th order reciprocal lattice vectors of PTLM respectively. The red lines at a + b = q Λ / m , a = q Λ / m , and b = q Λ / m represent the symmetric reflection cases: log 10 R L R R = 0 ( R L = R R ). The blue lines are also for log 10 R L R R = 0 but do not meet requirement for the degeneration of structures. The two yellow cross markers in subplot of m = 2 denote the parameter coordinates of the two structures simulated in Fig. 3.
Fig. 3
Fig. 3 Theoretical (red curves) and simulated (black curves) reflectivities of both loss and gain unidirectional reflectionless PTLM. (a), (b) for the loss PTLM with a = b = Λ / 8 and n R = n I = 0.009 ; (c), (d) for the gain PTLM with a = b = 3 Λ / 8 and n R = n I = 0.009 . For both structures n 0 = 3.42 , Λ = 300 n m and N = 100 . The peak wavelength here is absolutely decided by Λ and m . Other wavelength can be designed as an example as long as proper Λ and m are chosen.
Fig. 4
Fig. 4 Intensity distribution of λ = 1026.5 n m in the three structures. (a), (b) for the PTLM; (c), (d) for the RBM; (e), (f) for the IBM. And (a), (c), (e), (g) show the left incident cases; (b), (d), (f), (h) show the right incident cases; (g) and (h) are the intensity distributions along z axis in the three structures and they are plotted with black, red, and blue curves respectively. All the plots presented are four-unit-cells length near the two sides of the structures.
Fig. 5
Fig. 5 Quadrant diagrams about the guiding choice of a and b to realize an expectant side reflectionless when the three elementary media are determined. Here m = 2 is taken for instance.

Equations (15)

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n ( z ) = { n 0 + n R j Λ < z a + j Λ n 0 + i n I a + j Λ < z a + b + j Λ n 0 a + b + j Λ < z ( j + 1 ) Λ , ( 0 < a , b < Λ , j = 0 , 1 , 2 , , N 1 ) .
( 2 i β z ε f β 2 ε f ) e i β z + ( 2 i β z ε b β 2 ε b ) e i β z + k 2 n 2 ( z ) E ( z ) = 0.
n 2 ( z ) = 1 2 m = + C m e 2 i β m G z ,
C 0 = 2 n 0 2 + 4 n 0 n R a Λ + 4 i n 0 n I b Λ ,
C m = i π m ( { 2 n 0 n R [ cos ( m 2 π Λ a ) 1 ] + 2 n 0 n I [ sin ( m 2 π Λ ( a + b ) ) sin ( m 2 π Λ a ) ] } + i { 2 n 0 n R sin ( m 2 π Λ a ) + 2 n 0 n I [ cos ( m 2 π Λ ( a + b ) ) cos ( m 2 π Λ a ) ] } ) .
| C m | 2 = 0
( E f ( L 2 ) E b ( L 2 ) ) = ( M 11 M 12 M 21 M 22 ) ( E f ( L 2 ) E b ( L 2 ) ) ,
M 11 = e i β m G L ( cos ( L ( A + Δ β ) 2 + B D ) i ( A + Δ β ) ( A + Δ β ) 2 + B D sin ( L ( A + Δ β ) 2 + B D ) ) ,
M 12 = i B ( A + Δ β ) 2 + B D sin ( L ( A + Δ β ) 2 + B D ) ,
M 21 = i D ( A + Δ β ) 2 + B D sin ( L ( A + Δ β ) 2 + B D ) ,
M 22 = e i β m G L ( cos ( L ( A + Δ β ) 2 + B D ) + i ( A + Δ β ) ( A + Δ β ) 2 + B D sin ( L ( A + Δ β ) 2 + B D ) ) .
{ a + b = 2 q + 1 2 m Λ a n d a , b q m Λ , q = 0 , 1 , m 1 ( 7.1 ) n R = ± n I cot ϕ a , ϕ a = m π Λ a ( 7.2 )
C m = F ( n R ) + G ( n I ) ,
C m = F ( n R ) G ( n I ) .
                       F ( n R ) = i 2 n 0 n R π m { [ cos ( m 2 π Λ a ) 1 ] i sin ( m 2 π Λ a ) } , G ( n I ) = i 2 n 0 n I π m { [ sin ( m 2 π Λ ( a + b ) ) sin ( m 2 π Λ a ) ] + i [ cos ( m 2 π Λ ( a + b ) ) cos ( m 2 π Λ a ) ] } .

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