Abstract

We show that the bidirectionality of optical modes exists in general bianisotropic but reciprocal photonic crystals and related structures. Using this property, we obtain an alternative bidirectionality of modes with sources (gain) in active photonic crystals. In the former, we conclude that degeneracies of modes with complex anti-parallel wave vectors are always identical as long as the structure is reciprocal. With the latter, we setup an associated biorthogonality relation in the Rayleigh–Carson form for modes in active photonic crystals.

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  1. R. E. Collin, Field Theory of Guided Waves (IEEE Press , 1991).
  2. R. Carminati, M. Nieto-Vesperinas, J.-J. Greffet, " Reciprocity of evanescent electromagnetic waves," J. Opt. Soc. Amer. A 15, 706-712 (1998).
  3. R. Carminati, J. J. Sáenz, J.-J. Greffet, M. Nieto-Vesperinas, "Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components," Phys. Rev. A 62, 012712 (2000 ).
  4. R. J. Potton, "Reciprocity in optics," Rep. Progr. Phys. 67, 717 -754 (2004).
  5. S. L. Chuang, "A coupled mode formulation by reciprocity and a variational principle," J. Lightw. Technol. 5, 5-15 (1987).
  6. S. L. Chuang, "A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation," J. Lightw. Technol. 5, 174-183 (1987).
  7. L. Tsang, S. L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightw. Technol. 6, 304- 311 (1988).
  8. D. Michaelis, U. Peschel, C. Wächter, A. Bräuer, "Reciprocity theorem and perturbation theory for photonic crystal waveguides," Phys. Rev. E 68, 065601 (2003).
  9. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. Martijn de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
  10. A. T. Villeneuve, R. F. Harrington, "Reciprocity relationships for gyrotropic media," IRE Trans. Microw. Theory Tech. 6, 308-310 (1958).
  11. A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Microw. Theory Tech. 7, 441-446 (1959).
  12. F. Olyslager, "Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides," IEEE Trans. Microw. Theory Tech. 44, 2064- 2075 (1996).
  13. V. Dmitriev, "Comments on ‘properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides’," IEEE Trans. Microw. Theory Tech. 47, 655-657 (1999).
  14. F. Olyslager, "Author’s reply," IEEE Trans. Microw. Theory Tech. 47, 658-659 (1999).
  15. F. Olyslager, Electromagnetic Waveguides and Transmission Lines (Oxford Univ. Press , 1999).
  16. D. Pissoort, F. Olyslager, "Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem ," IEEE Trans. Microw. Theory Tech. 52, 542-553 (2004 ).
  17. A. D. Yaghjian, "Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides," IEEE Microw. Wireless Compon. Lett. 17, 480-482 (2007).
  18. F. Olyslager, H. Rogier, "Bidirectionality in the waveguide sturm-liouville problem (invited paper)," Proc. EMTS 2007-Int. URSI Commiss. B-Electromagn. Theory Symp. (2007).
  19. H. A. Kramers, "General theory of paramagnetic rotation in crystals ," Proc. Amsterdam Acad. 33, 959 -972 (1930).
  20. E. Wigner, "Über die operation der zeitumkehr in der quantenmechanik," Nachr. Akad. Ges. Wiss. Göttingen 31 , 546-559 (1932).
  21. O. Madelung, Introduction to Solid-State Theory (Springer-Verlag, 1996).
  22. P. J. Chiang, S. W. Chang, "Frequency-domain formulation of photonic crystals using sources and gain ," Opt. Exp. 21, 1972-1985 (2013).
  23. S. W. Chang, "Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from lorentz reciprocity," Opt. Exp. 19, 21 116- 21 134 (2011).
  24. J. R. Carson, "A generalization of reciprocal theorem ," Bell Syst. Techn. J. 3, 393-399 (1924).
  25. J. R. Carson, "Reciprocal theorems in radio communication ," Proc. IRE. 17, 952-956 (1929).
  26. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).
  27. S. H. Friedberg, A. J. Insel, L. E. Spence, Linear Algebra (Prentice–Hall, 1989).

2013 (1)

P. J. Chiang, S. W. Chang, "Frequency-domain formulation of photonic crystals using sources and gain ," Opt. Exp. 21, 1972-1985 (2013).

2011 (1)

S. W. Chang, "Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from lorentz reciprocity," Opt. Exp. 19, 21 116- 21 134 (2011).

2007 (1)

A. D. Yaghjian, "Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides," IEEE Microw. Wireless Compon. Lett. 17, 480-482 (2007).

2004 (2)

R. J. Potton, "Reciprocity in optics," Rep. Progr. Phys. 67, 717 -754 (2004).

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. Martijn de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).

2003 (1)

D. Michaelis, U. Peschel, C. Wächter, A. Bräuer, "Reciprocity theorem and perturbation theory for photonic crystal waveguides," Phys. Rev. E 68, 065601 (2003).

1999 (2)

V. Dmitriev, "Comments on ‘properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides’," IEEE Trans. Microw. Theory Tech. 47, 655-657 (1999).

F. Olyslager, "Author’s reply," IEEE Trans. Microw. Theory Tech. 47, 658-659 (1999).

1998 (1)

R. Carminati, M. Nieto-Vesperinas, J.-J. Greffet, " Reciprocity of evanescent electromagnetic waves," J. Opt. Soc. Amer. A 15, 706-712 (1998).

1996 (1)

F. Olyslager, "Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides," IEEE Trans. Microw. Theory Tech. 44, 2064- 2075 (1996).

1988 (1)

L. Tsang, S. L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightw. Technol. 6, 304- 311 (1988).

1987 (2)

S. L. Chuang, "A coupled mode formulation by reciprocity and a variational principle," J. Lightw. Technol. 5, 5-15 (1987).

S. L. Chuang, "A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation," J. Lightw. Technol. 5, 174-183 (1987).

1959 (1)

A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Microw. Theory Tech. 7, 441-446 (1959).

1958 (1)

A. T. Villeneuve, R. F. Harrington, "Reciprocity relationships for gyrotropic media," IRE Trans. Microw. Theory Tech. 6, 308-310 (1958).

1932 (1)

E. Wigner, "Über die operation der zeitumkehr in der quantenmechanik," Nachr. Akad. Ges. Wiss. Göttingen 31 , 546-559 (1932).

1930 (1)

H. A. Kramers, "General theory of paramagnetic rotation in crystals ," Proc. Amsterdam Acad. 33, 959 -972 (1930).

1929 (1)

J. R. Carson, "Reciprocal theorems in radio communication ," Proc. IRE. 17, 952-956 (1929).

1924 (1)

J. R. Carson, "A generalization of reciprocal theorem ," Bell Syst. Techn. J. 3, 393-399 (1924).

Bell Syst. Techn. J. (1)

J. R. Carson, "A generalization of reciprocal theorem ," Bell Syst. Techn. J. 3, 393-399 (1924).

IEEE Trans. Microw. Theory Tech. (1)

V. Dmitriev, "Comments on ‘properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides’," IEEE Trans. Microw. Theory Tech. 47, 655-657 (1999).

IEEE Microw. Wireless Compon. Lett. (1)

A. D. Yaghjian, "Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides," IEEE Microw. Wireless Compon. Lett. 17, 480-482 (2007).

IEEE Trans. Microw. Theory Tech. (1)

F. Olyslager, "Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides," IEEE Trans. Microw. Theory Tech. 44, 2064- 2075 (1996).

IEEE Trans. Microw. Theory Tech. (2)

F. Olyslager, "Author’s reply," IEEE Trans. Microw. Theory Tech. 47, 658-659 (1999).

D. Pissoort, F. Olyslager, "Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem ," IEEE Trans. Microw. Theory Tech. 52, 542-553 (2004 ).

IRE Trans. Microw. Theory Tech. (1)

A. T. Villeneuve, R. F. Harrington, "Reciprocity relationships for gyrotropic media," IRE Trans. Microw. Theory Tech. 6, 308-310 (1958).

IRE Trans. Microw. Theory Tech. (1)

A. T. Villeneuve, "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media," IRE Trans. Microw. Theory Tech. 7, 441-446 (1959).

J. Lightw. Technol. (1)

L. Tsang, S. L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightw. Technol. 6, 304- 311 (1988).

J. Lightw. Technol. (1)

S. L. Chuang, "A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation," J. Lightw. Technol. 5, 174-183 (1987).

J. Lightw. Technol. (1)

S. L. Chuang, "A coupled mode formulation by reciprocity and a variational principle," J. Lightw. Technol. 5, 5-15 (1987).

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

R. Carminati, M. Nieto-Vesperinas, J.-J. Greffet, " Reciprocity of evanescent electromagnetic waves," J. Opt. Soc. Amer. A 15, 706-712 (1998).

Nachr. Akad. Ges. Wiss. Göttingen (1)

E. Wigner, "Über die operation der zeitumkehr in der quantenmechanik," Nachr. Akad. Ges. Wiss. Göttingen 31 , 546-559 (1932).

Opt. Exp. (1)

S. W. Chang, "Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from lorentz reciprocity," Opt. Exp. 19, 21 116- 21 134 (2011).

Opt. Exp. (1)

P. J. Chiang, S. W. Chang, "Frequency-domain formulation of photonic crystals using sources and gain ," Opt. Exp. 21, 1972-1985 (2013).

Phys. Rev. A (1)

R. Carminati, J. J. Sáenz, J.-J. Greffet, M. Nieto-Vesperinas, "Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components," Phys. Rev. A 62, 012712 (2000 ).

Phys. Rev. E (2)

D. Michaelis, U. Peschel, C. Wächter, A. Bräuer, "Reciprocity theorem and perturbation theory for photonic crystal waveguides," Phys. Rev. E 68, 065601 (2003).

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. Martijn de Sterke, R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).

Proc. Amsterdam Acad. (1)

H. A. Kramers, "General theory of paramagnetic rotation in crystals ," Proc. Amsterdam Acad. 33, 959 -972 (1930).

Proc. IRE. (1)

J. R. Carson, "Reciprocal theorems in radio communication ," Proc. IRE. 17, 952-956 (1929).

Rep. Progr. Phys. (1)

R. J. Potton, "Reciprocity in optics," Rep. Progr. Phys. 67, 717 -754 (2004).

Other (6)

R. E. Collin, Field Theory of Guided Waves (IEEE Press , 1991).

F. Olyslager, H. Rogier, "Bidirectionality in the waveguide sturm-liouville problem (invited paper)," Proc. EMTS 2007-Int. URSI Commiss. B-Electromagn. Theory Symp. (2007).

F. Olyslager, Electromagnetic Waveguides and Transmission Lines (Oxford Univ. Press , 1999).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).

S. H. Friedberg, A. J. Insel, L. E. Spence, Linear Algebra (Prentice–Hall, 1989).

O. Madelung, Introduction to Solid-State Theory (Springer-Verlag, 1996).

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