Abstract

We demonstrate that the interplay of a magneto-optical layer sandwiched between two judiciously balanced gain and loss layers which are both birefringent with misaligned in-plane anisotropy, induces unidirectional electromagnetic modes. Embedding one such optically active non-reciprocal unit between a pair of birefringent Bragg reflectors, results in an exceptionally strong asymmetry in light transmission. Remarkably, such asymmetry persists regardless of the incident light polarization. This photonic architecture may be used as the building block for chip-scale non-reciprocal devices such as optical isolators and circulators.

© 2012 OSA

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  2. D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
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  3. T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).
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  5. There are several ways to address the problem with absorption. One approach is to replace a uniform magnetic material with a slow-wave magneto-photonic structure [6]. Under certain conditions, such a structure can enhance asymmetric transmitance effects associated with magnetism, while significantly reducing absorption. The problem with the above approach is that it does not apply to infrared and optical frequencies it can only work at MW frequencies. Another approach is to incorporate gain and loss together with non-linearity [19]. In this case, however, optical isolation occurs only for specific power ranges.
  6. A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008).
    [CrossRef]
  7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
    [CrossRef] [PubMed]
  8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).
  9. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys.6, 192 (2010).
    [CrossRef]
  10. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
    [CrossRef]
  11. 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, 093902 (2009).
    [CrossRef] [PubMed]
  12. Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
    [CrossRef]
  13. H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
    [CrossRef]
  14. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett106, 213901 (2011).
    [CrossRef] [PubMed]
  15. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
    [CrossRef]
  16. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett.103, 123601 (2009).
    [CrossRef] [PubMed]
  17. H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
    [CrossRef]
  18. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012).
    [CrossRef] [PubMed]
  19. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
    [CrossRef]
  20. S. Longhi, “𝒫𝒯 -symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
    [CrossRef]
  21. Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
    [CrossRef] [PubMed]
  22. Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
    [CrossRef] [PubMed]
  23. A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
    [CrossRef]
  24. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001).
    [CrossRef]
  25. P. A. Mello and N. Kumar, Quantum transport in mesoscopic systems: complexity and statistical fluctuations : a maximum-entropy viewpoint, Volume 4 of Mesoscopic Physics and Nanotechnology (Oxford University Press, India, 2004).
  26. M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
    [CrossRef]
  27. M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006).
    [CrossRef]
  28. A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009).
    [CrossRef]

2012 (5)

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
[CrossRef]

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012).
[CrossRef] [PubMed]

2011 (3)

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
[CrossRef] [PubMed]

2010 (5)

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
[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,” Nat. Phys.6, 192 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[CrossRef]

S. Longhi, “𝒫𝒯 -symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

2009 (3)

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett.103, 123601 (2009).
[CrossRef] [PubMed]

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, 093902 (2009).
[CrossRef] [PubMed]

A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009).
[CrossRef]

2008 (3)

A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

2006 (2)

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006).
[CrossRef]

2005 (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
[CrossRef]

2001 (1)

A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001).
[CrossRef]

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, 093902 (2009).
[CrossRef] [PubMed]

Aktsipetrov, O.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Baryshev, A.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Bauters, J.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
[CrossRef]

Bowers, J. E.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
[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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
[CrossRef] [PubMed]

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Christodoulides, D. N.

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[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,” Nat. Phys.6, 192 (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, 093902 (2009).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

Dai, D.

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
[CrossRef]

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, 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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[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,” Nat. Phys.6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

Ellis, F. M.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

Fedyanin, A.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008).
[CrossRef]

A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001).
[CrossRef]

Fujikawa, R.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
[CrossRef] [PubMed]

Ge, Li

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Granovsky, A.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Günther, U.

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

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, 093902 (2009).
[CrossRef] [PubMed]

Inoue, M.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Khanikaev, A.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

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,” Nat. Phys.6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
[CrossRef]

Kottos, T.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[CrossRef]

Kovanis, V.

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012).
[CrossRef] [PubMed]

Kovanis, Vassilios

T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).

Kumar, N.

P. A. Mello and N. Kumar, Quantum transport in mesoscopic systems: complexity and statistical fluctuations : a maximum-entropy viewpoint, Volume 4 of Mesoscopic Physics and Nanotechnology (Oxford University Press, India, 2004).

Levy, M.

M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006).
[CrossRef]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

Li, R.

M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006).
[CrossRef]

Lim, P. B.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Lin, Z.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

Liu, Jia Ming

T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).

Lockwood, D. J.

L. Pavesi and D. J. Lockwood, Silicon photonics (Springer, Germany, 2004).

Longhi, S.

S. Longhi, “𝒫𝒯 -symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett.103, 123601 (2009).
[CrossRef] [PubMed]

Lozovik, Yu.E.

A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009).
[CrossRef]

Makris, K. G.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

Mello, P. A.

P. A. Mello and N. Kumar, Quantum transport in mesoscopic systems: complexity and statistical fluctuations : a maximum-entropy viewpoint, Volume 4 of Mesoscopic Physics and Nanotechnology (Oxford University Press, India, 2004).

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, 093902 (2009).
[CrossRef] [PubMed]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
[CrossRef]

Murzina, T.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

Pavesi, L.

L. Pavesi and D. J. Lockwood, Silicon photonics (Springer, Germany, 2004).

Ramezani, H.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012).
[CrossRef] [PubMed]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
[CrossRef]

Ruter, C. E.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys.6, 192 (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, 093902 (2009).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New York, 1991).
[CrossRef]

Schindler, J.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

Segev, M.

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

Simpson, T. B.

T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).

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, 093902 (2009).
[CrossRef] [PubMed]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
[CrossRef] [PubMed]

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New York, 1991).
[CrossRef]

Uchida, H.

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Usechak, Nicholas

T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).

Vinogradov, A.

A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009).
[CrossRef]

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (2012).
[CrossRef] [PubMed]

A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008).
[CrossRef]

Vitebsky, I.

A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001).
[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, 093902 (2009).
[CrossRef] [PubMed]

Xu, Z.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
[CrossRef]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

Appl. Phys. Lett. (1)

M. Levy and R. Li, “Polarization rotation enhancement and scattering mechanisms in waveguide magnetopho-tonic crystals,” Appl. Phys. Lett.89, 121113 (2006).
[CrossRef]

J. Phys. A: Math. Gen. (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171 (2005).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

M. Inoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina, and A. Granovsky, “Magnetophotonic crystals,” J. Phys. D: Appl. Phys.39, R151 (2006).
[CrossRef]

Light: Science and Applications (1)

D. Dai, J. Bauters, and J. E. Bowers, “Passive technologies for future large-scale photonic integrated circuits on silicon: polarization handling, light non-reciprocity and loss reduction,” Light: Science and Applications1, 1 (2012).
[CrossRef]

Nat. 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,” Nat. Phys.6, 192 (2010).
[CrossRef]

Phys. Rev. A (7)

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101(R) (2011).
[CrossRef]

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of 𝒫𝒯 -symmetric scattering,” Phys. Rev. A85, 050101(R) (2012).
[CrossRef]

H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with 𝒫𝒯 symmetry,” Phys. Rev. A85, 062122 (2012).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear 𝒫𝒯 -symmetric optical structures,” Phys. Rev. A82, 043803 (2010).
[CrossRef]

S. Longhi, “𝒫𝒯 -symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with 𝒫𝒯 symmetry,” Phys. Rev. A85, 013818 (2012).
[CrossRef]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in 𝒫𝒯-symmetric optical couplers,” Phys. Rev. A82, 043818 (2010).
[CrossRef]

Phys. Rev. B (2)

A. Vinogradov and Yu.E. Lozovik, “Inverse Borrmann effect in photonic crystals,” Phys. Rev. B80, 235106 (2009).
[CrossRef]

A. Figotin and I. Vitebskiy, “Absorption suppression in photonic crystals,” Phys. Rev. B77, 104421 (2008).
[CrossRef]

Phys. Rev. E (1)

A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001).
[CrossRef]

Phys. Rev. Lett (2)

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “𝒫𝒯 -symmetric Talbot effects,” Phys. Rev. Lett109, 033902 (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. Lett106, 213901 (2011).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett.103, 123601 (2009).
[CrossRef] [PubMed]

Y. D. Chong, Li Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “𝒫𝒯-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett.106, 093902 (2011).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

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, 093902 (2009).
[CrossRef] [PubMed]

Other (6)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” ibid. 100, 030402 (2008).

T. B. Simpson, Jia Ming Liu, Nicholas Usechak, and Vassilios Kovanis, Tunable photonic microwave oscillator self–locked by polarizationrotated optical feedback, Frequency Control Symposium (FCS), IEEE International (2012).

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New York, 1991).
[CrossRef]

There are several ways to address the problem with absorption. One approach is to replace a uniform magnetic material with a slow-wave magneto-photonic structure [6]. Under certain conditions, such a structure can enhance asymmetric transmitance effects associated with magnetism, while significantly reducing absorption. The problem with the above approach is that it does not apply to infrared and optical frequencies it can only work at MW frequencies. Another approach is to incorporate gain and loss together with non-linearity [19]. In this case, however, optical isolation occurs only for specific power ranges.

P. A. Mello and N. Kumar, Quantum transport in mesoscopic systems: complexity and statistical fluctuations : a maximum-entropy viewpoint, Volume 4 of Mesoscopic Physics and Nanotechnology (Oxford University Press, India, 2004).

L. Pavesi and D. J. Lockwood, Silicon photonics (Springer, Germany, 2004).

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Figures (3)

Fig. 1
Fig. 1

(a) Scattering setup of a micro-cavity with ��̃�� - symmetric gain/loss distribution and asymmetric transport. The left/right (green/red) slab is a lossy/gain non magnetic layer with in-plane anisotropy. The middle (arsenic) slab is a passive ferromagnetic material with magnetization M0 (indicated with the arrows inside the layer). (b) A generalized ��̃�� - symmetric micro-cavity embedded in an anisotropic Bragg reflector.

Fig. 2
Fig. 2

(a) The left/right reflectances vs. ω. (b) The difference |〈Tl〉 − 〈Tr〉| between the left and right transmittances vs. ω. (c) the quality factor QT vs. ω. All layers have the same thickness d, the phase misalignment is Δ ϕ active dielectric = π, the anisotropy is δ = 1, while the gain/loss parameter is γ = 0.0005. Furthermore the real part of permittivity for the birefringent layers is ε = 9, for the magneto-optical layer is ε = 1.525 while we assume that ε0 = 1. The Faraday gyrotropic parameters are α = 0.925, β = 0, and μ = 1.

Fig. 3
Fig. 3

(Upper subfigure) Dispersion relation ω(k) of the infinite periodic anisotropic Bragg reflector with permittivity contrasts ε = 3 and ε = 12 and δ = 1, Δϕ = 0. (Lower subfigure) Quality factor QT (red dashed line) of a ��̃�� -symmetric non-reciprocal magneto-optical cavity when it is embedded in the anisotropic Bragg reflector associated with the set-up of the upper sub-figure. The micro-cavity has the same constitute parameters as in Fig. 2. The transmittance spectrum of the periodic Bragg is also shown in order to identify the pseudo-gap frequency windows (green shadowed areas) where the QT -factor take large values.

Equations (15)

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

ε ^ ( z + ) = [ ε x x ε x y 0 ε x y * ε y y 0 0 0 ε z z ] , ε ^ ( z ) = [ ε x x * ε x y 0 ε x y * ε y y * 0 0 0 ε z z ] ,
μ ^ ( L z L ) = [ μ x x μ x y 0 μ x y * μ x x 0 0 0 μ z z ] ; μ ^ ( z [ L , L ] ) = 1 ^
ε ^ ε ^ , μ ^ μ ^ , α α , β β
𝒫 ˜ 𝒯 ε ^ ( z ) ( 𝒫 ˜ 𝒯 ) 1 = ε ^ ( z ) 𝒫 ˜ 𝒯 μ ^ ( z ) ( 𝒫 ˜ 𝒯 ) 1 = μ ^ ( z )
× E ( r ) = i ω c μ ^ H ( r ) , × H ( r ) = i ω c ε ^ E ( r )
E ( r ) = e i ( k x x + k y y ) E ( z ) , H ( r ) = e i ( k x x + k y y ) H ( z ) .
E l , r ( α , β , ϕ , z ) = A l , r e i k z + B l , r e i k z .
A l , r = [ A x l , r ( α , β , ϕ , ϕ + ) A y l , r ( α , β , ϕ , ϕ + ) ] T B l , r = [ B x l , r ( α , β , ϕ , ϕ + ) B y l , r ( α , β , ϕ , ϕ + ) ] T
[ A r B r ] = M [ A l B l ] , M = [ M 11 M 12 M 21 M 22 ] .
E l , r ( α , β , ϕ , ϕ + ) 𝒫 ˜ 𝒯 ( E l , r ( α , β , ϕ + , ϕ ) ) *
[ A l ( α , β , ϕ + , ϕ ) B l ( α , β , ϕ + , ϕ ) ] * = M [ A r ( α , β , ϕ + , ϕ ) B r ( α , β , ϕ + , ϕ ) ] * .
M ( α , β , ϕ , ϕ + ) M * ( α , β , ϕ + , ϕ ) = 1 ^ .
{ r l = M 22 1 M 21 , r r = M 12 M 22 1 t l = M 11 M 12 M 22 1 M 21 , t r = M 22 1
𝒫 S * ( α , β , ϕ + , ϕ ) 𝒫 S ( α , β , ϕ , ϕ + ) = 1 ^
Q T = | < T l T r > p | < T l + T r > p .

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