Abstract

We develop an analytical solution based on the cavity coupling that can be used to predict the output performance of the 1D defect modes. This solution gives a concise analytical expression of every emission wavelength of the defect modes with arbitrary defect numbers. The splitting and the resonance modes are explained qualitatively by the proposed theoretical model. The output performance obtained by the analytical solution are in good consistency with that obtained by the numerical simulations using the finite-difference time-domain method. These results may provide a useful alternative to customize the 1D coupled defect mode laser.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  40. S. Zhang, L. Cui, X. Zhang, J. H. Tong, and T. Zhai, “Tunable polymer lasing in chirped cavities,” Opt. Express 28(3), 2809 (2020).
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2020 (2)

Y. Fu and T. Zhai, “Distributed feedback organic lasing in photonic crystals,” Front. Optoelectron. 13(1), 18–34 (2020).
[Crossref]

S. Zhang, L. Cui, X. Zhang, J. H. Tong, and T. Zhai, “Tunable polymer lasing in chirped cavities,” Opt. Express 28(3), 2809 (2020).
[Crossref]

2019 (3)

B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photonics Res. 7(3), 363–367 (2019).
[Crossref]

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

P. Zhou, L. Niu, A. Hayat, F. Cao, T. Zhai, and X. Zhang, “Operating characteristics of high-order distributed feedback polymer lasers,” Polymers 11(2), 258 (2019).
[Crossref]

2018 (2)

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
[Crossref]

T. Zhai, X. Wu, S. Li, S. Liang, L. Niu, M. Wang, S. Feng, H. Liu, and X. Zhang, “Polymer lasing in a periodic-random compound cavity,” Polymers 10(11), 1194 (2018).
[Crossref]

2017 (2)

Z. M. Meng, A. Liang, and Z. Y. Li, “Fano resonances in photonic crystal nanobeams side-coupled with nanobeam cavities,” J. Appl. Phys. 121(19), 193102 (2017).
[Crossref]

A. Y. Sharaevskaya, E. N. Beginin, and Y. P. Sharaevskii, “Defect modes control in coupled magnonic crystals,” IEEE Trans. Magn. 53(11), 1–4 (2017).
[Crossref]

2016 (1)

Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
[Crossref]

2015 (2)

M. Li, N. Zhang, K. Wang, J. Li, S. Xiao, and Q. Song, “Inversed Vernier effect based single-mode laser emission in coupled microdisks,” Sci. Rep. 5(1), 13682 (2015).
[Crossref]

Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
[Crossref]

2012 (2)

Y. Zhang, T. Mei, and D. H. Zhang, “Temporal coupled-mode theory of ring–bus–ring Mach–Zehnder interferometer,” Appl. Opt. 51(4), 504–508 (2012).
[Crossref]

T. Zhai and X. Zhang, “Gain- and feedback- channel matching in lasers based on radiative-waveguide gratings,” Appl. Phys. Lett. 101(14), 143507 (2012).
[Crossref]

2011 (1)

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[Crossref]

2009 (4)

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009).
[Crossref]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram and nanometer scale photonic crystal opto-mechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[Crossref]

H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
[Crossref]

2008 (3)

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
[Crossref]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[Crossref]

Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
[Crossref]

2007 (3)

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

X. Yang and C. Wong, “Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers,” Opt. Express 15(8), 4763–4780 (2007).
[Crossref]

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75(6), 063833 (2007).
[Crossref]

2003 (5)

O. Makoto, K. Shinichi, and N. Susumu, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68(23), 235110 (2003).
[Crossref]

D. D. Smith, H. Chang, and K. A. Fuller, “Whispering-gallery mode splitting in coupled microresonators,” J. Opt. Soc. Am. B 20(9), 1967–1974 (2003).
[Crossref]

S. Fan and W. Suh, “Temporal coupled mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. B 20(3), 569–572 (2003).
[Crossref]

Y. Akahane, T. Asano, B. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

2000 (3)

M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84(10), 2140–2143 (2000).
[Crossref]

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000).
[Crossref]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[Crossref]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref]

1995 (1)

1994 (1)

1992 (1)

M. H. Rose, M. Lindberg, W. W. Chow, S. W. Koch, and M. Sargent, “Composite-cavity-mode approach to single-mode semiconductor-laser feedback instabilities,” Phys. Rev. A 46(1), 603–611 (1992).
[Crossref]

1991 (1)

C. W. Wieman and L. Hollbreg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62(1), 1–20 (1991).
[Crossref]

1989 (1)

1986 (1)

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

1984 (1)

D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20(2), 166–176 (1984).
[Crossref]

1983 (1)

R. Wyatt and W. J. Devlin, “10 kHz linewidth 1.5 µm InGaAsP external cavity laser with 55 nm tuning range,” Electron. Lett. 19(3), 110–112 (1983).
[Crossref]

1981 (1)

M. W. Fleming and A. Mooradian, “Spectral characteristics of external-cavity controlled semiconductor lasers,” IEEE J. Quantum Electron. 17(1), 44–59 (1981).
[Crossref]

1965 (1)

P. W. Smith, “Stabilized, single-frequency output from a long laser cavity,” IEEE J. Quantum Electron. 1(8), 343–348 (1965).
[Crossref]

Akahane, Y.

Y. Akahane, T. Asano, B. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref]

Andrews, J.

H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
[Crossref]

Aoki, K.

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
[Crossref]

Arakawa, Y.

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
[Crossref]

Asano, T.

Y. Akahane, T. Asano, B. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref]

Baer, E.

H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
[Crossref]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84(10), 2140–2143 (2000).
[Crossref]

Beginin, E. N.

A. Y. Sharaevskaya, E. N. Beginin, and Y. P. Sharaevskii, “Defect modes control in coupled magnonic crystals,” IEEE Trans. Magn. 53(11), 1–4 (2017).
[Crossref]

Buchhave, P.

Cai, S.

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
[Crossref]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram and nanometer scale photonic crystal opto-mechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[Crossref]

Cao, F.

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

P. Zhou, L. Niu, A. Hayat, F. Cao, T. Zhai, and X. Zhang, “Operating characteristics of high-order distributed feedback polymer lasers,” Polymers 11(2), 258 (2019).
[Crossref]

Cao, Y.

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
[Crossref]

Capasso, F.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

Chan, J.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[Crossref]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram and nanometer scale photonic crystal opto-mechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref]

Chang, H.

Chen, C.

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

Chen, Y. L.

Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
[Crossref]

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75(6), 063833 (2007).
[Crossref]

Chen, Z.

Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
[Crossref]

Cho, A. Y.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

Chow, W. W.

M. H. Rose, M. Lindberg, W. W. Chow, S. W. Koch, and M. Sargent, “Composite-cavity-mode approach to single-mode semiconductor-laser feedback instabilities,” Phys. Rev. A 46(1), 603–611 (1992).
[Crossref]

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[Crossref]

Colombelli, R.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

Cooper, J.

Cui, L.

Dapkus, P. D.

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Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
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Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
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S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
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S. Fan and W. Suh, “Temporal coupled mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. B 20(3), 569–572 (2003).
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Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
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Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
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M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84(10), 2140–2143 (2000).
[Crossref]

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R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

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R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 µm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
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S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
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Troccoli, M.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
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M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram and nanometer scale photonic crystal opto-mechanical cavity,” Nature 459(7246), 550–555 (2009).
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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
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B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photonics Res. 7(3), 363–367 (2019).
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Wang, F.

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
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Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
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Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
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C. W. Wieman and L. Hollbreg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62(1), 1–20 (1991).
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Wong, C. W.

X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009).
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Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
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T. Zhai, X. Wu, S. Li, S. Liang, L. Niu, M. Wang, S. Feng, H. Liu, and X. Zhang, “Polymer lasing in a periodic-random compound cavity,” Polymers 10(11), 1194 (2018).
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H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
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Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
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Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
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X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009).
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Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
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X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009).
[Crossref]

Zhai, H.

Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
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Zhai, T.

Y. Fu and T. Zhai, “Distributed feedback organic lasing in photonic crystals,” Front. Optoelectron. 13(1), 18–34 (2020).
[Crossref]

S. Zhang, L. Cui, X. Zhang, J. H. Tong, and T. Zhai, “Tunable polymer lasing in chirped cavities,” Opt. Express 28(3), 2809 (2020).
[Crossref]

P. Zhou, L. Niu, A. Hayat, F. Cao, T. Zhai, and X. Zhang, “Operating characteristics of high-order distributed feedback polymer lasers,” Polymers 11(2), 258 (2019).
[Crossref]

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

T. Zhai, X. Wu, S. Li, S. Liang, L. Niu, M. Wang, S. Feng, H. Liu, and X. Zhang, “Polymer lasing in a periodic-random compound cavity,” Polymers 10(11), 1194 (2018).
[Crossref]

T. Zhai and X. Zhang, “Gain- and feedback- channel matching in lasers based on radiative-waveguide gratings,” Appl. Phys. Lett. 101(14), 143507 (2012).
[Crossref]

Zhang, D. H.

Zhang, N.

Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
[Crossref]

M. Li, N. Zhang, K. Wang, J. Li, S. Xiao, and Q. Song, “Inversed Vernier effect based single-mode laser emission in coupled microdisks,” Sci. Rep. 5(1), 13682 (2015).
[Crossref]

Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
[Crossref]

Zhang, S.

S. Zhang, L. Cui, X. Zhang, J. H. Tong, and T. Zhai, “Tunable polymer lasing in chirped cavities,” Opt. Express 28(3), 2809 (2020).
[Crossref]

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

Zhang, X.

S. Zhang, L. Cui, X. Zhang, J. H. Tong, and T. Zhai, “Tunable polymer lasing in chirped cavities,” Opt. Express 28(3), 2809 (2020).
[Crossref]

P. Zhou, L. Niu, A. Hayat, F. Cao, T. Zhai, and X. Zhang, “Operating characteristics of high-order distributed feedback polymer lasers,” Polymers 11(2), 258 (2019).
[Crossref]

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
[Crossref]

T. Zhai, X. Wu, S. Li, S. Liang, L. Niu, M. Wang, S. Feng, H. Liu, and X. Zhang, “Polymer lasing in a periodic-random compound cavity,” Polymers 10(11), 1194 (2018).
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T. Zhai and X. Zhang, “Gain- and feedback- channel matching in lasers based on radiative-waveguide gratings,” Appl. Phys. Lett. 101(14), 143507 (2012).
[Crossref]

Zhang, Y.

Zhang, Z.

B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photonics Res. 7(3), 363–367 (2019).
[Crossref]

Zhao, H.

B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photonics Res. 7(3), 363–367 (2019).
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Zheng, K.

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
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Zhou, J.

H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
[Crossref]

Zhou, P.

P. Zhou, L. Niu, A. Hayat, F. Cao, T. Zhai, and X. Zhang, “Operating characteristics of high-order distributed feedback polymer lasers,” Polymers 11(2), 258 (2019).
[Crossref]

Zou, X. B.

Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
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Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75(6), 063833 (2007).
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Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. Zhai and X. Zhang, “Gain- and feedback- channel matching in lasers based on radiative-waveguide gratings,” Appl. Phys. Lett. 101(14), 143507 (2012).
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R. Wyatt and W. J. Devlin, “10 kHz linewidth 1.5 µm InGaAsP external cavity laser with 55 nm tuning range,” Electron. Lett. 19(3), 110–112 (1983).
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Front. Optoelectron. (1)

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J. Mater. Chem. (1)

H. Song, K. Singer, J. Lott, Y. Wu, J. Zhou, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Continuous melt processing of all-polymer distributed feedback lasers,” J. Mater. Chem. 19(40), 7520–7524 (2009).
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J. Opt. Soc. Am. B (4)

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Nature (2)

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New J. Phys. (1)

Y. F. Xiao, J. Gao, X. B. Zou, J. F. McMillan, X. Yang, Y. L. Chen, Z. F. Han, G. C. Guo, and C. W. Wong, “Coupled quantum electrodynamics in photonic crystal cavities towards controlled phase gate operations,” New J. Phys. 10(12), 123013 (2008).
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Opt. Express (3)

Opt. Lett. (2)

Photonics Res. (1)

B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photonics Res. 7(3), 363–367 (2019).
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M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84(10), 2140–2143 (2000).
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X. Yang, M. Yu, D. L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102(17), 173902 (2009).
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Polymers (3)

S. Zhang, J. Tong, C. Chen, F. Cao, C. Liang, Y. Song, T. Zhai, and X. Zhang, “Controlling the performance of polymer lasers via the cavity coupling,” Polymers 11(5), 764 (2019).
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T. Zhai, X. Wu, S. Li, S. Liang, L. Niu, M. Wang, S. Feng, H. Liu, and X. Zhang, “Polymer lasing in a periodic-random compound cavity,” Polymers 10(11), 1194 (2018).
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Rev. Sci. Instrum. (1)

C. W. Wieman and L. Hollbreg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62(1), 1–20 (1991).
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RSC Adv. (1)

Z. Gu, K. Wang, W. Sun, S. Liu, N. Zhang, S. Xiao, and Q. Song, “Triangular lasing modes in hexagonal perovskite microplates with balanced gain and loss,” RSC Adv. 6(69), 64589–64594 (2016).
[Crossref]

Sci. China Inf. Sci. (1)

S. Cai, Z. Han, F. Wang, K. Zheng, Y. Cao, Y. Ma, and X. Feng, “Review on flexible photonics/electronics integrated devices and fabrication strategy,” Sci. China Inf. Sci. 61(6), 060410 (2018).
[Crossref]

Sci. Rep. (2)

M. Li, N. Zhang, K. Wang, J. Li, S. Xiao, and Q. Song, “Inversed Vernier effect based single-mode laser emission in coupled microdisks,” Sci. Rep. 5(1), 13682 (2015).
[Crossref]

Q. Song, N. Zhang, H. Zhai, S. Liu, Z. Gu, K. Wang, S. Sun, Z. Chen, M. Li, and S. Xiao, “The combination of high Q factor and chirality in twin cavities and microcavity chain,” Sci. Rep. 4(1), 6493 (2015).
[Crossref]

Science (2)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Schematic of 1D defect mode lasers. Inserting the MDMO-PPV polymer layers in the period stacks of SiO2/TiO2 structure, where the PPV acts as both the defect cavity and gain medium. The coupling of defect cavities results in mode splitting, which provides a simple, flexible and versatile approach for the design of high performance laser devices. The behavior of the cavity mode can be easily tuned by adjusting the separation and size of the defects.
Fig. 2.
Fig. 2. The defect cavity are coupled to each other with the coupling coeffient $\kappa $. The coupling of nonadjacent defect mode isn't considered.
Fig. 3.
Fig. 3. Modes splitting for N = 1, 2, 3, 4, 5, 6, 7 defects array with $\kappa = 0.445$. N split modes distribute symmetrically about ${{\omega }_0}$, and ${\omega _0}$ occurs in the center of the band gap when N is odd whereas disappear when N is even.
Fig. 4.
Fig. 4. Comparison of coupled defect mode obtained from the theoretical model developed (the blue ball) and the simulation using FDTD (the red cross) with N = 2, 3, 4, 5, 6, 7 defects for ${\kappa } = 0.445$. The results agree well, and the discrepancy are expected to vanish if the first and last coupling coefficients should be modified due to external losses.
Fig. 5.
Fig. 5. Electric field amplitude profiles of three defects mode laser. (a) Schematic of 1D defect mode lasers. (b) (c) (d) The electric field amplitude profiles at three resonance frequencies are highly confined in the defect layers for lasing. (e) Electric field of other frequency in band-gap are forbidden for propagation in the structure.

Tables (1)

Tables Icon

Table 1. The frequencies of the coupled defect mode for N = 2,3,4,5 defects array respectively.

Equations (13)

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

d a 1 d t = i ω 1 a 1 + i κ 21 a 2 + i κ 31 a 3 + + i κ N 1 a N d a 2 d t = i ω 2 a 2 + i κ 12 a 1 + i κ 32 a 3 + + i κ N 2 a N d a N 1 dt = i ω N 1 a N 1 + i κ 1 N 1 a 1 + + i κ N N 1 a N d a N dt = i ω N a N + i κ 1 N a 1 + + i κ N 1 N a N 1 }
a ( t ) = A ( ω ) exp ( i ω t ) d ω
[ ω 1 ω 0 κ 21 κ 31 κ N 1 κ 12 ω 2 ω 0 κ 32 κ N 2 κ 1 N 1 κ 2 N 1 ω N 1 ω 0 κ N N 1 κ 1 N κ 2 N κ N 1 N ω N ω 0 ] [ A 1 A 2 A N 1 A N ] = 0
D N = | ω 1 ω 0 κ 1 0 0 κ 1 ω 2 ω 0 κ 2 0 0 0 ω N 1 ω 0 κ N 1 0 0 κ N 1 ω N ω 0 | = 0
C N = [ 0 κ 1 0 0 κ 1 0 0 0 0 0 0 κ N 1 0 0 κ N 1 0 ]
D N = | ω ω 0 κ 1 0 0 κ 1 ω ω 0 k 2 0 0 0 ω ω 0 κ N 1 0 0 κ N 1 ω ω 0 | = ( ω ω 0 ) | ω ω 0 κ 1 0 0 κ 1 ω ω 0 κ 2 0 0 0 ω ω 0 κ N 2 0 0 κ N 2 ω ω 0 | N 1 κ N 1 2 | ω ω 0 κ 1 0 0 κ 1 ω ω 0 κ 2 0 0 0 ω ω 0 κ N 3 0 0 κ N 3 ω ω 0 | N 2
D N = ( ω ω 0 ) D N 1 κ N 1 2 D N 2
D 2 n = f 2 n ( ( ω ω 0 ) 2 ) = 0
D 2 n + 1 = ( ω ω 0 ) f 2 n + 1 ( ( ω ω 0 ) 2 ) = 0
D 2 = ( ω ω 0 ) 2 κ 1 2 = f 2 ( ( ω ω 0 ) 2 )
D 3 = ( ω ω 0 ) ( ( ω ω 0 ) 2 κ 1 2 κ 2 2 ) = ( ω ω 0 ) f 3 ( ( ω ω 0 ) 2 )
D 4 = ( ω ω 0 ) 2 f 3 ( ( ω ω 0 ) 2 ) κ 3 2 f 2 ( ( ω ω 0 ) 2 ) = f 4 ( ( ω ω 0 ) 2 )
D 5 = ( ω ω 0 ) f 4 ( ( ω ω 0 ) 2 ) κ 4 2 ( ω ω 0 ) f 3 ( ( ω ω 0 ) 2 ) = ( ω ω 0 ) f 5 ( ( ω ω 0 ) 2 )

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