Abstract

The exceptional point (EP) is one of the typical properties of parity–time-symmetric systems, arising from modes coupling with identical resonant frequencies or propagation constants in optics. Here we show that in addition to two different modes coupling, a nonuniform distribution of gain and loss leads to an offset from the original propagation constants, including both real and imaginary parts, resulting in the absence of EP. These behaviors are examined by the general coupled-mode theory from the first principle of the Maxwell equations, which yields results that are more accurate than those from the classical coupled-mode theory. Numerical verification via the finite element method is provided. In the end, we present an approach to achieve lossless propagation in a geometrically symmetric waveguide array.

© 2017 Chinese Laser Press

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References

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Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

A. Cerjan and S. Fan, “Eigenvalue dynamics in the presence of non-uniform gain and loss,” Phys. Rev. A 94, 033857 (2016).
[Crossref]

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled microcavities,” Opt. Express 24, 6916–6930 (2016).
[Crossref]

B. Wu, B. Wu, J. Xu, J.-J. Xiao, and Y. Chen, “Coupled mode theory in non-Hermitian optical cavities,” Opt. Express 24, 16566–16573 (2016).
[Crossref]

2015 (5)

N. X. A. Rivolta and B. Maes, “Symmetry recovery for coupled photonic modes with transversal PT symmetry,” Opt. Lett. 40, 3922–3925 (2015).
[Crossref]

J. Xu and Y. T. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23, 22619–22627 (2015).
[Crossref]

S. N. Ghosh and Y. D. Chong, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” Sci. Rep. 6, 19837 (2015).
[Crossref]

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

2014 (6)

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

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

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

2013 (2)

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
[Crossref]

S. Zheng, G. Ren, Z. Lin, and S. Jian, “Mode-coupling analysis and trench design for large-mode-area low-cross-talk multicore fiber,” Appl. Opt. 52, 4541–4548 (2013).
[Crossref]

2012 (3)

M. Kulishov and B. Kress, “Free space diffraction on active grating with balanced phase and gain/loss modulation,” Opt. Express 20, 29319–29328 (2012).
[Crossref]

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

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[Crossref]

2011 (1)

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

2010 (1)

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, 192–195 (2010).
[Crossref]

2009 (2)

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]

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

2008 (1)

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]

2007 (2)

1998 (1)

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

1994 (1)

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]

Bender, C. M.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

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

Benisty, H.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[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, 167–171 (2012).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 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 structure,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Cerjan, A.

A. Cerjan and S. Fan, “Eigenvalue dynamics in the presence of non-uniform gain and loss,” Phys. Rev. A 94, 033857 (2016).
[Crossref]

Chan, C. T.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Chen, H.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Chen, Y.

Chen, Y. T.

Chong, Y. D.

X. Zhou and Y. D. Chong, “PT symmetry breaking and nonlinear optical isolation in coupled microcavities,” Opt. Express 24, 6916–6930 (2016).
[Crossref]

S. N. Ghosh and Y. D. Chong, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” Sci. Rep. 6, 19837 (2015).
[Crossref]

Christodoulides, D. N.

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

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

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

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structure,” Phys. Rev. Lett. 106, 213901 (2011).
[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, 192–195 (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]

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]

R. El-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]

Degiron, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

Ding, K.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Dmitriev, S. V.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[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]

Eichelkraut, T.

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

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, 192–195 (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]

R. El-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]

Fan, S.

A. Cerjan and S. Fan, “Eigenvalue dynamics in the presence of non-uniform gain and loss,” Phys. Rev. A 94, 033857 (2016).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Feng, L.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Ghosh, S. N.

S. N. Ghosh and Y. D. Chong, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” Sci. Rep. 6, 19837 (2015).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[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]

Hassan, A. U.

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

Heinrich, M.

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

Hodaei, H.

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

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

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Huang, W. P.

Jian, S.

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Kang, M.

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
[Crossref]

Khajavikhan, M.

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

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

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, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[Crossref]

Kottos, T.

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

Kress, B.

Kulishov, M.

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Li, H.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Li, J.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
[Crossref]

Liertzer, M.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Lin, Z.

S. Zheng, G. Ren, Z. Lin, and S. Jian, “Mode-coupling analysis and trench design for large-mode-area low-cross-talk multicore fiber,” Appl. Opt. 52, 4541–4548 (2013).
[Crossref]

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

Litchinitser, N. M.

C. Ma, W. Walasik, and N. M. Litchinitser, “Meta-PT symmetry in asymmetric directional couplers,” arXiv:1512.06875 (2015).

Liu, F.

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
[Crossref]

Liu, X.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Liu, Z.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Long, G. L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Lupu, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

Ma, C.

C. Ma, W. Walasik, and N. M. Litchinitser, “Meta-PT symmetry in asymmetric directional couplers,” arXiv:1512.06875 (2015).

Ma, G.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Maes, B.

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, 192–195 (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]

R. El-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]

Malomed, B. A.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[Crossref]

Miri, M.

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

Miri, M. A.

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

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

Moiseyev, N.

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

Monifi, F.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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]

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]

R. El-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]

Nixon, S.

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

Nori, F.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

Onishchukov, G.

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

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Özdemir, S. K.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Peng, B.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Peschel, U.

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

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structure,” Phys. Rev. Lett. 106, 213901 (2011).
[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, 167–171 (2012).
[Crossref]

Ren, G.

Rivolta, N. X. A.

Rotter, I.

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

Rotter, S.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

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, 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, 093902 (2009).
[Crossref]

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, 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, 093902 (2009).
[Crossref]

Suchkov, S. V.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[Crossref]

Sun, Y.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Tan, W.

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[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]

Walasik, W.

C. Ma, W. Walasik, and N. M. Litchinitser, “Meta-PT symmetry in asymmetric directional couplers,” arXiv:1512.06875 (2015).

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Wu, B.

Xiao, J. J.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Xiao, J.-J.

Xiao, M.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Xu, J.

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Yang, J.

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

Yang, L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Yao, Y.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Yilmaz, H.

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Zhang, Q.

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Zhang, X.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Zhang, Z. Q.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Zheng, S.

Zhou, X.

Appl. Opt. (1)

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

J. Phys. A (1)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Nat. Phys. (2)

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, 192–195 (2010).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Nature (1)

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

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (6)

A. Cerjan and S. Fan, “Eigenvalue dynamics in the presence of non-uniform gain and loss,” Phys. Rev. A 94, 033857 (2016).
[Crossref]

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

M. Kang, F. Liu, and J. Li, “Effective spontaneous PT-symmetry breaking in hybridized metamaterials,” Phys. Rev. A 87, 053824 (2013).
[Crossref]

A. U. Hassan, H. Hodaei, M. A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 033825 (2012).
[Crossref]

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91, 053825 (2015).
[Crossref]

Phys. Rev. Lett. (5)

Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

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

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

Phys. Rev. X (1)

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Rep. Prog. Phys. (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

Sci. Rep. (2)

S. N. Ghosh and Y. D. Chong, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” Sci. Rep. 6, 19837 (2015).
[Crossref]

Z. Liu, Q. Zhang, X. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref]

Science (3)

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

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

B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, and F. Nori, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Other (4)

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

C. Ma, W. Walasik, and N. M. Litchinitser, “Meta-PT symmetry in asymmetric directional couplers,” arXiv:1512.06875 (2015).

http://www.comsol.com/ .

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the proposed structure consisting of four coupled waveguides in a square array and the coupled-mode schemes supported in this structure. (a) Top view of the cross section profile, (b)–(e) four coupled-mode fashions between the modes supported in the diagonal (A) and off-diagonal waveguides (B). The geometrical parameters are r=0.2  μm, d=0.5  μm. The relative permittivity of the diagonal (A) waveguides and off-diagonal (B) waveguides are ϵA=ϵco+jϵA,I and ϵB=ϵco+jϵB,I, respectively. Here, ϵco=12.25 corresponds to silicon, and the background medium (silica) has the dielectric function ϵb=2.25.
Fig. 2.
Fig. 2. Propagation constants β± as a function of ϵI for different mode-coupling cases C1 (black solid curve), C2 (green solid curve), C3 (blue dashed curve), and C4 (red dashed curve). (a), (c) The real parts; (b), (d) the imaginary parts. The corresponding structure has ϵB,I=0 and ϵA,I=ϵI<0.
Fig. 3.
Fig. 3. Propagation constants as a function of ϵI for different modes supported in the diagonal (off-diagonal) waveguides: M1 (black circles, real; black solid curve, imaginary), M2 (red squares, real; red dashed curve, imaginary), M3 (green plusses, real; green solid curve, imaginary), and M4 (blue X’s, real; blue dotted curve, imaginary). (a) FEM results, (b) general CMT results. The left y axis corresponds to the real parts of the propagation constants, and the right y axis is for the imaginary parts. Here, ϵA,I=ϵI.
Fig. 4.
Fig. 4. Phase diagram in the parameter space (ϵI, α): the solid and dashed lines mark the classical CMT phase boundaries (e.g., one of the supermodes has a real-valued propagation constant) based on Eq. (2), and the symbols represent the FEM results.
Fig. 5.
Fig. 5. (a) Evolution of the intensity |aA|2 (red dashed line) and |aB|2 (black solid line) as functions of z for the initial condition that only the lossy waveguide mode aA is excited, (b) the total intensities (|aA|2+|aB|2) for the cases where the loss (red dashed line) and gain (black solid line) waveguides modes are excited. (a) and (b) show the system at the α-point (ϵI,α)=(0.6,0.0.9459); (c) and (d) are similar to (a) and (b), but for the point (ϵI,α)=(0.5,0.9459) in phase I.
Fig. 6.
Fig. 6. Propagation constant β as a function of ϵI: (a), (c), (e) the real parts; (b), (d), (f) the imaginary parts. The black solid lines correspond to the FEM results, the dashed red lines show the classical CMT results, and the symbols represent the general CMT results. The insets in (b) and (d) are zoomed-in views of the respective panels for relatively small ϵI values. (a), (b) α=0.9; (c), (d) α=1.08; (e), (f) α=1.3.

Tables (1)

Tables Icon

Table 1. Coupled-Mode Components and the Coefficients of the Hamiltonian in Eq. (3) for Four Combinations of Modes Coupling, i.e., Cases Shown in Figs. 1(b)1(e)

Equations (19)

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

H=[β1+jγ1κ12κ21β1+Δβ+jγ2],
β±=β1+Δβ+j(γ1+γ2)2±[Δβ+i(γ2γ1)]24+κ12κ21.
jddz(aAaB)=(βA+jγAκAκBβB+jγB)(aAaB),
κA(B)=ωϵ02SA(B)(ϵcoϵb)uA(B)*·uB(A)dxdy,
γA(B)=ωϵ02SA(B)ϵA(B),IuA(B)*·uA(B)dxdy.
t×em+jβmz×em+=jωμhm+,
t×hm+jβmz×hm+=jωϵ0ϵrem+,
E=(nanen)exp[j(ωt+βz)],
H=(nanhn)exp[j(ωt+βz)].
t×(nanen)+jβz×(nanen)=jωμ(nanhn),
t×(nanhn)+jβz×(nanhn)=jωϵ0(ϵr+Δϵ)(nanen).
nan[j(ββm)pmn+bmn+jkmn]=0.
bmn={·(hm+×en)·(hn×em+)}dxdy,
pmn={z·(hm+×en)z·(hn×em+)}dxdy,
kmn=ωϵ0Δϵen·em+dxdy.
nan[(ββn)pmn+kmn]=0.
H·a=H21·H1·a=βa,
am[(ββm)pmm+kmm]=0.
[β1p11k11β2p12k12β1p21k21β2p22k22][a1a2]=β[p11p12p21p22][a1a2].

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