Abstract

In Hermitian photonic devices, S-parameters, i.e., the elements of a scattering matrix based on integrated power flux and Hermitian modal orthogonality, are used to account for the transmission or reflection of light from one port to another. The definition of S-parameters in Hermitian settings becomes inappropriate in the non-Hermitian optical environment. Here we revisit the fundamental problems associated with extracting the S-parameters of light in photonic 𝒫𝒯-symmetric devices, i.e., waveguides or coupled waveguide-cavity systems, wherein the waveguide ports themselves may also be non-Hermitian. We first use the bi-orthogonal inner product that restores the modal orthogonality on the waveguide ports containing balanced gain and losses to quantify the modal overlapping instead of Hermitian inner product. Secondly, a finite element implementation is proposed and realized to extract the S-parameters on non-Hermitian ports. Lastly, we illustrate our approach of calculating the S-parameters on non-Hermitian ports via two waveguide-lattice structures. The numerical results of S-parameters are validated against the constraints imposed by reciprocity and 𝒫𝒯-symmetry.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

K. X. Wang, “Time-reversal symmetry in temporal coupled-mode theory and nonreciprocal device applications,” Opt. Lett. 43, 5623–5626 (2018).
[Crossref]

2017 (2)

2016 (2)

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

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

2015 (3)

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

X.-F. Zhu, “Defect states and exceptional point splitting in the band gaps of one-dimensional parity-time lattices,” Opt. Express 23, 22274–22284 (2015).
[Crossref] [PubMed]

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

2014 (2)

2013 (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

2012 (1)

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, 023802 (2012).
[Crossref]

2011 (1)

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]

2010 (2)

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

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

2003 (1)

2000 (1)

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Abián, M. A.

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Andrés, M. V.

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Boria, V. E.

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Cao, H.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Chan, C. T.

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

Chen, F.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Chen, W.

Z. Xiong, W. Chen, P. Wang, and Y. Chen, “Classification of symmetry properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry,” Opt. Express 25, 29822–29834 (2017).
[Crossref] [PubMed]

W. Chen, Z. Xiong, J. Xu, and Y. Chen, “Generalized coupled mode formalism in reciprocal waveguides with gain/loss, anisotropy or bianisotropy,” arXiv preprint arXiv:1801.06673 (2018).

Chen, Y.

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, 023802 (2012).
[Crossref]

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]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

Coccioli, R.

G. Pelosi, R. Coccioli, and S. Selleri, Quick Finite Elements for Electromagnetic Waves (Artech House, 2009).

Coldren, L. A.

L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, 2012).
[Crossref]

Corzine, S. W.

L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, 2012).
[Crossref]

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Fan, S.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Feng, L.

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

Ferrando, A.

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Ge, L.

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

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, 023802 (2012).
[Crossref]

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]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Gimeno, B.

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Hou, B.

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Joannopoulos, J. D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

F. Shanhui, S. Wonjoo, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).

Kuhl, U.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Liang, S.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Libisch, F.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Liu, T.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Longhi, S.

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

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012).

Mailybaev, A. A.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Makris, K. G.

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

Mashanovitch, M. L.

L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, 2012).
[Crossref]

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Milburn, T. J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Moiseyev, N.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Pelosi, G.

G. Pelosi, R. Coccioli, and S. Selleri, Quick Finite Elements for Electromagnetic Waves (Artech House, 2009).

Peng, Y.-G.

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2009).

Rabl, P.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Ramezani, H.

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

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Rotter, S.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Selleri, S.

G. Pelosi, R. Coccioli, and S. Selleri, Quick Finite Elements for Electromagnetic Waves (Artech House, 2009).

Shanhui, F.

Shi, C.

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

Silveirinha, M. G.

M. G. Silveirinha, “𝒫 · 𝒯 · 𝒟 symmetry-protected scattering anomaly in optics,” Phys. Rev. B 95, 035153 (2017).
[Crossref]

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E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
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A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012).

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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, 023802 (2012).
[Crossref]

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]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
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Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, and R. L. Taylor, The Finite Element Method (McGraw-HillLondon, 1977).

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Vanwolleghem, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Wang, K. X.

Wang, P.

Wang, S.

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

Wonjoo, S.

Wu, B.

Xiao, J.

Xiong, Z.

Z. Xiong, W. Chen, P. Wang, and Y. Chen, “Classification of symmetry properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry,” Opt. Express 25, 29822–29834 (2017).
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W. Chen, Z. Xiong, J. Xu, and Y. Chen, “Generalized coupled mode formalism in reciprocal waveguides with gain/loss, anisotropy or bianisotropy,” arXiv preprint arXiv:1801.06673 (2018).

Xu, J.

Yu, Z.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

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

Zhang, X.-L.

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

Zhao, D.-G.

Zhu, J.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
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X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “𝒫𝒯-Symmetric Acoustics,” Phys. Rev. X 4, 031042 (2014).

Zhu, X.

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

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

Zhu, X.-F.

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, and R. L. Taylor, The Finite Element Method (McGraw-HillLondon, 1977).

O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, and R. L. Taylor, The Finite Element Method (McGraw-HillLondon, 1977).

IEEE Trans. Microw. Theory Tech. (1)

E. Silvestre, M. A. Abián, B. Gimeno, A. Ferrando, M. V. Andrés, and V. E. Boria, “Analysis of inhomogeneously filled waveguides using a bi-orthonormal-basis method,” IEEE Trans. Microw. Theory Tech. 48, 589–596 (2000).
[Crossref]

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

Nat. Photonics (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is – and what is not – an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Nature (1)

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (3)

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

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, 023802 (2012).
[Crossref]

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯− and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

Phys. Rev. B (1)

M. G. Silveirinha, “𝒫 · 𝒯 · 𝒟 symmetry-protected scattering anomaly in optics,” Phys. Rev. B 95, 035153 (2017).
[Crossref]

Phys. Rev. Lett. (3)

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]

T. Liu, X. Zhu, F. Chen, S. Liang, and J. Zhu, “Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal,” Phys. Rev. Lett. 120, 124502 (2018).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Phys. Rev. X (2)

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

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, “Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point,” Phys. Rev. X 8, 021066 (2018).

Other (7)

W. Chen, Z. Xiong, J. Xu, and Y. Chen, “Generalized coupled mode formalism in reciprocal waveguides with gain/loss, anisotropy or bianisotropy,” arXiv preprint arXiv:1801.06673 (2018).

D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2009).

L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, 2012).
[Crossref]

http://www.comsol.com/ .

O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, and R. L. Taylor, The Finite Element Method (McGraw-HillLondon, 1977).

G. Pelosi, R. Coccioli, and S. Selleri, Quick Finite Elements for Electromagnetic Waves (Artech House, 2009).

A. W. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 2012).

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic diagram of light scattering of a photonic device connected with a few waveguide channels indicated by the light blue segments S. The field inside the waveguide channel S is given by ϕ = [et ht]. (b) Parallel plate waveguide. The light blue segments represent the waveguide channels, ϕ1 and ϕ2 are eigenmodes on the left boundary, while ϕ3 and ϕ4 are eigenmodes on the right boundary.
Fig. 2
Fig. 2 (a) The schematic diagram of a parallel waveguide lattice, and the red and blue waveguides correspond to those with gain and losses respectively. (b) Result of parallel waveguide lattice via Hermitian port. (c) Field amplitude of parallel waveguide lattice via non-Hermitian port. (d) Field amplitude of unparallel waveguide lattice via Hermitian port. (e) Result of unparallel waveguide lattice via non-Hermitian port. (f) Real part of the modal index as a function of distance d. (g) Imaginary part of effective mode index as a function of distance d.
Fig. 3
Fig. 3 (a) S31 and S13 obtained from Hermitian port and non-Hermitian port; (b) S11 and S22 obtained from Hermitian port and non-Hermitian port; (c) S13 and S31 obtained from Hermitian port and non-Hermitian port; (d) S11 and S22 obtained from Hermitian port and non-Hermitian port.
Fig. 4
Fig. 4 The relative error of the ����-symmetry constraint.

Tables (1)

Tables Icon

Table 1 Extracting the scattering matrix element S1j for Hermitian and non-Hermitian port

Equations (11)

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E μ t ( x , y , z ) = ( a μ e i β μ z + b μ e i β μ z ) e μ t ( x , y ) ,
P δ i j = e i t × h j t * + e j t * + h i t d x d y ,
S ¯ S ¯ = I ¯ ,
( β ϕ β ψ ) ψ , ϕ = ( β ϕ β ψ ) ψ T B ¯ ϕ d Ω = 0 ,
P δ i j = e ψ i t × h ϕ j t e ϕ j t × h ψ i t d x d y ,
× ( μ ¯ r 1 × E ) k 0 2 ¯ r E = 0 .
Ω d Ω { × F μ ¯ r 1 × E k 0 2 F ¯ r E } S F n ^ × μ ¯ r 1 × E d s = 0 ,
Ω d Ω { × F μ ¯ r 1 × E k 0 2 F ¯ r E } k = 1 N S k F k n ^ × μ ¯ r 1 × E wg ( k ) d s = 0 ,
S k F k n ^ × E × n ^ d s = S k F k n ^ × E wg ( k ) × n ^ d s ,
S ¯ = S ¯ T ,
𝒫 𝒯 S ¯ 𝒫 𝒯 = S ¯ 1 .