Abstract

In this article, we developed a generalized coupled-mode theory for mixing an isolated state with a continuum having an intrinsic energy gap, which dubbed as “the bound states in the gapped continuum” (BIGC). We investigated the mixture interaction by mimicking the Su-Schrieffer-Heeger model in an optical coupled waveguide array (WA), and presented a unified engineering mechanism for topologically-protected zero modes, Fano resonance, and Tamm surface states, even though those phenomena are diverse in topological insulators, atomic physics and semiconductors, respectively. By tuning the on-site potential and coupling strength of the isolated state, we found the unified operating characteristics for zero modes, Fano resonance, and Tamm states, with demonstrating their localization, transmission spectra, and distinct evolution dynamics explicitly. As an extension for triple-modes coupling, two special sandwich-like configurations are studied: the isolated-continuous-isolated and continuous-isolated-continuous configurations lead to adiabatic eliminations and domain walls, respectively, revealing possible applications and wide connections in many fields of physics and optics.

© 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 (7)

E. Smith, V. Shteeman, and A. A. Hardy, “Analysis of optical characteristics of holey photonic crystal devices with the extended coupled-mode formalism,” Opt. Quantum Electron. 50(11), 424 (2018).
[Crossref]

N. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

D. V. Nesterenko, S. Hayashi, and Z. Sekkat, “Asymmetric surface plasmon resonances revisited as Fano resonances,” Phys. Rev. B 97(23), 235437 (2018).
[Crossref]

Z. Feng, J. Ma, Z. Yu, and X. Sun, “Circular Bragg lasers with radial PT symmetry : design and analysis with a coupled-mode approach,” Photonics Res. 6(5), A38–A42 (2018).
[Crossref]

S. Stützer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, “Photonic topological Anderson insulators,” Nature 560(7719), 461–465 (2018).
[Crossref]

S. Longhi, “Probing one-dimensional topological phases in waveguide lattices with broken chiral symmetry,” Opt. Lett. 43(19), 4639 (2018).
[Crossref]

L. E. I. W. Ang, W. E. I. C. Ai, M. E. B. Ie, X. I. Z. Hang, and J. X. U. Ingjun, “Zak phase and topological plasmonic Tamm states in one-dimensional plasmonic crystals,” Opt. Express 26(22), 28963–28975 (2018).
[Crossref]

2017 (5)

M. A. Gorlach and A. N. Poddubny, “Topological edge states of bound photon pairs,” Phys. Rev. A 95(5), 053866 (2017).
[Crossref]

B. P. Nguyen and K. Kim, “Anderson localization and saturable nonlinearity in one-dimensional disordered lattices,” J. Mod. Opt. 64(19), 1923–1929 (2017).
[Crossref]

Z. Gao, S. T. M. Fryslie, B. J. Thompson, P. S. Careny, and K. D. Choquette, “Parity-Time symmetry analogs in coherently coupled vertical cavity laser arrays,” Optica 4(3), 323–329 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

R. Buschlinger, M. Lorke, and U. Peschel, “Coupled-Mode Theory for Semiconductor Nanowires,” Phys. Rev. Appl. 7(3), 034028 (2017).
[Crossref]

2016 (8)

A. Khandelwal, A. Syed, and J. Nayak, “Limits imposed by nonlinear coupling on rotation sensitivity of a semiconductor ring laser gyroscope,” Appl. Opt. 55(19), 5187–5191 (2016).
[Crossref]

R. Gansch, S. Kalchmair, P. Genevet, T. Zederbauer, H. Detz, A. M. Andrews, W. Schrenk, F. Capasso, M. Lončar, and G. Strasser, “Measurement of bound states in the continuum by a detector embedded in a photonic crystal,” Light: Sci. Appl. 5(9), e16147 (2016).
[Crossref]

Y. F. Bai, P. Xu, L. L. Lu, M. L. Zhong, and S. N. Zhu, “Two-photon Anderson localization in a disordered quadratic waveguide array,” J. Opt. 18(5), 055201 (2016).
[Crossref]

P. A. Kalozoumis, C. V. Morfonios, F. K. Diakonos, and P. Schmelcher, “PT -symmetry breaking in waveguides with competing loss-gain pairs,” Phys. Rev. A 93(6), 063831 (2016).
[Crossref]

D. Leykam and Y. D. Chong, “Edge Solitons in Nonlinear-Photonic Topological Insulators,” Phys. Rev. Lett. 117(14), 143901 (2016).
[Crossref]

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

2015 (3)

M. Mrejen, H. Suchowski, T. Hatakeyama, C. Wu, L. Feng, K. O’Brien, Y. Wang, and X. Zhang, “Adiabatic elimination-based coupling control in densely packed subwavelength waveguides,” Nat. Commun. 6(1), 7565 (2015).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a Topological Transition in the Bulk of a Non-Hermitian System,” Phys. Rev. Lett. 115(4), 040402 (2015).
[Crossref]

R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit, and M. I. Molina, “Observation of Localized States in Lieb Photonic Lattices,” Phys. Rev. Lett. 114(24), 245503 (2015).
[Crossref]

2013 (2)

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A. Tünnermann, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, “Compact surface fano states embedded in the continuum of waveguide arrays,” Phys. Rev. Lett. 111(24), 240403 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111(24), 243905 (2013).
[Crossref]

2012 (1)

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
[Crossref]

2011 (1)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

2010 (4)

M. Z. Hasan and C. L. Kane, “Colloquium : Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105(16), 163905 (2010).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

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

2008 (3)

R. A. A. M. E. Sasin, R. P. Seisyan, M. A. Kalitteevski, S. Brand, A. V. K. J. M. Chamberlain, A. Yu. Egorov, A. P. Vasil’ev, and V. S. Mikhrin, “Tamm plasmon polaritons: Slow and spatially compact light,” Appl. Phys. Lett. 92(25), 251112 (2008).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

2007 (1)

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric Bragg mirror,” Phys. Rev. B 76(16), 165415 (2007).
[Crossref]

2005 (1)

X. Liu, X. Zhou, and C. Lu, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72(1), 013811 (2005).
[Crossref]

2003 (1)

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref]

1999 (1)

P. Tong, B. Li, and B. Hu, “Wave transmission, phonon localization, and heat conduction of a one-dimensional Frenkel-Kontorova chain,” Phys. Rev. B 59(13), 8639–8645 (1999).
[Crossref]

1991 (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1518 (1991).
[Crossref]

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in Polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

1975 (2)

J. W. Bray, H. R. Hart, L. V. Interrante, I. S. Jacobs, J. S. Kasper, G. D. Watkins, S. H. Wee, and J. C. Bonner, “Observation of a Spin-Peierls Transition in a Heisenberg Antiferromagnetic Linear-Chain System,” Phys. Rev. Lett. 35(11), 744–747 (1975).
[Crossref]

F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
[Crossref]

1973 (1)

A. Yariv, “Coupled-Mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[Crossref]

Abram, R. A.

M. Kaliteevski, I. Iorsh, S. Brand, R. A. Abram, J. M. Chamberlain, A. V. Kavokin, and I. A. Shelykh, “Tamm plasmon-polaritons: Possible electromagnetic states at the interface of a metal and a dielectric Bragg mirror,” Phys. Rev. B 76(16), 165415 (2007).
[Crossref]

Adams, M. J.

N. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

Ai, W. E. I. C.

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[Crossref]

An, F. A.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7(1), 13986 (2016).
[Crossref]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref]

Andrews, A. M.

R. Gansch, S. Kalchmair, P. Genevet, T. Zederbauer, H. Detz, A. M. Andrews, W. Schrenk, F. Capasso, M. Lončar, and G. Strasser, “Measurement of bound states in the continuum by a detector embedded in a photonic crystal,” Light: Sci. Appl. 5(9), e16147 (2016).
[Crossref]

Ang, L. E. I. W.

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Bai, Y. F.

Y. F. Bai, P. Xu, L. L. Lu, M. L. Zhong, and S. N. Zhu, “Two-photon Anderson localization in a disordered quadratic waveguide array,” J. Opt. 18(5), 055201 (2016).
[Crossref]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref]

Bonner, J. C.

J. W. Bray, H. R. Hart, L. V. Interrante, I. S. Jacobs, J. S. Kasper, G. D. Watkins, S. H. Wee, and J. C. Bonner, “Observation of a Spin-Peierls Transition in a Heisenberg Antiferromagnetic Linear-Chain System,” Phys. Rev. Lett. 35(11), 744–747 (1975).
[Crossref]

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
[Crossref]

Brand, S.

R. A. A. M. E. Sasin, R. P. Seisyan, M. A. Kalitteevski, S. Brand, A. V. K. J. M. Chamberlain, A. Yu. Egorov, A. P. Vasil’ev, and V. S. Mikhrin, “Tamm plasmon polaritons: Slow and spatially compact light,” Appl. Phys. Lett. 92(25), 251112 (2008).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. The generalized coupled mode theory (CMT) for isolated-isolated states, continuous-continuous states, and mixed isolated-continuous states. (a) The symmetric and antisymmetric modes in waveguide systems, (b) gap generation in semiconductor physics, and (c) Fano resonance in scattering phenomena are unified in the CMT framework.
Fig. 2.
Fig. 2. Eigenmode spectra for the mixing between the isolated state (n = 0) and the gapped continuum (2N = 40). The isolated state lies (a) in the gap (<Δ, in the inset), (b) in the bulk (∼ω, in the inset), and (c) outside the bulk spectra (>ω, in the inset), respectively. The insets in (a) and (c) show the confinement configuration of the eigenstates for the protected zero mode and Tamm surface state in lattice space.
Fig. 3.
Fig. 3. Evolution dynamics in the WA for zero mode excitation (a), the bound state in the continuum (b), and Tamm state excitation (c), demonstrating even waveguide number occupation, decay in the bulk and exponential decay, respectively. Their respective output distributions are shown in (d–f).
Fig. 4.
Fig. 4. Fano resonance is antisymmetric in the transmission spectrum when the isolated state couples with the gapped continuum. (b) The relative energy level of isolated state is δβ=0.85 which lies in the range of bulk band from 0.4 to 1.0. The black curve for the uncoupled case (V0=0). (c) The interference origin of Fano resonance and (d) the evolution dynamics of Fano resonance when the 1st waveguide is excited at the input.
Fig. 5.
Fig. 5. Configuration of “isolated-continuum-isolated” (ICI) sandwich-like structure for two zero modes (a) and two Tamm states (c). The adiabatic elimination is achieved for zero modes (b) (δβ=0.1) and Tamm states (d) (δβ=1.1) when total waveguide number is 2N + 2 = 8, and coupling strengths are V1= V8=0.1.
Fig. 6.
Fig. 6. Configuration of a “continuum-isolated-continuum” (CIC) domain wall structure for two zero modes (a) and two Tamm states (c). The solitary dynamics is achieved for (b) Zero modes (δβ=0) and (d) Tamm states (δβ=1.1) when the total waveguide number is 40 and the coupling strengths are V0 = 0.3 for both cases.

Equations (12)

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H C = n = 1 2 N β 0 | ϕ n | 2 + n = 1 2 N ( κ 0 + ( 1 ) n δ κ ) ϕ n + ϕ n 1 + h . c .
H I = ( β 0 + δ β ) | ϕ 0 | 2
H int = ( V I C ϕ k + ϕ 0 + V I C ϕ 0 + ϕ k ) d k = V 0 ( ϕ 0 ϕ 1 + h . c . )
H = n = 1 2 N β 0 | φ n | 2 + n = 1 2 N ( κ 0 + ( 1 ) n δ κ ) φ n + φ n 1 + ( β 0 + δ β ) | ϕ | 2 + V 0 ϕ φ 1 + h . c .
i φ n z = ( κ 0 ( 1 ) n δ κ ) φ n + 1 + ( κ 0 + ( 1 ) n δ κ ) φ n 1 + V 0 φ 0 δ n 0
i ϕ z = ( β 0 + δ β ) ϕ + V 0 φ 0
ω A n = ( κ 0 ( 1 ) n Δ κ ) A n + 1 + ( κ 0 + ( 1 ) n Δ κ ) A n 1 + V 0 B δ n 0 ω B = E F B + V 0 A 0
ω A n = ( κ 0 ( 1 ) n Δ κ ) A n + 1 + ( κ 0 + ( 1 ) n Δ κ ) A n 1 + V 0 2 ω E F A 0 δ n 0
( A n + 1 A n ) = M ( A n A n 1 )
M n = ( ω κ 0 + δ κ V 0 2 ( κ 0 + δ κ ) ( ω E F ) V 0 2 ( κ 0 + δ κ ) ( ω E F ) κ 0 + ( 1 ) n δ κ κ 0 ( 1 ) n δ κ 2 ω κ 0 + δ κ ) ( n 1 ) M F = ( E F κ 0 + δ κ 1 1 0 ) ( n = 1 )
A n = { e i k n + r e i k n n < 0 t e i k n n > 0
T = 4 sin 2 k | M 11 e i k + M 12 M 21 M 22 e i k | 2

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