Abstract

We explore a new class of distributed feedback (DFB) structures that employ the recently-developed concept of parity-time (PT) symmetry in optics. We show that, based on PT-symmetric pure reflective volume gratings, a vertical surface-emitting cavity can be constructed. We provide a detailed analysis of the threshold conditions as well as the wavelength and angular spectral characteristics using the Kogelnik coupled-wave approximation, backed up by an exact solution of the Helmholtz equation. We show that such a PT-symmetric cavity can be configured to support one and only one longitudinal mode, leading to inherently single-mode lasing.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
  3. P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
    [Crossref]
  4. I.-S. Chung and J. Mørk, “Silicon photonics light source realized by III-VI/Si grating-mirror lasers,” J. Appl. Phys. Lett. 97(15), 151113 (2010).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (2)

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

H. F. Jones and M. Kulishov, “Extension of analytic results of a PT-symmetric structure,” J. Opt. 18(5), 055101 (2016).
[Crossref]

2015 (1)

2014 (6)

M. Kulishov, B. Kress, and H. F. Jones, “Novel optical characteristics of a Fabry-Perot resonator with embedded PT-symmetrical grating,” Opt. Express 22(19), 23164–23181 (2014).
[Crossref] [PubMed]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014).
[Crossref] [PubMed]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

H. F. Jones, “Analytic results for a PT-symmetric optical structure,” J. Phys. A 45(13), 135306 (2012).
[Crossref]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

2011 (2)

W. Nakwaski, “VCSEL structures used to suppress higher order transverse modes,” Opto-Electron. Rev. 19(1), 119–129 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

2010 (3)

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

I.-S. Chung and J. Mørk, “Silicon photonics light source realized by III-VI/Si grating-mirror lasers,” J. Appl. Phys. Lett. 97(15), 151113 (2010).
[Crossref]

2009 (1)

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

2005 (3)

2003 (1)

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

2002 (2)

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

V. Finazzi and M. N. Zervas, “Effect of periodic background loss on grating spectra,” Appl. Opt. 41(12), 2240–2250 (2002).
[Crossref] [PubMed]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT-symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

1995 (1)

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

1994 (1)

A. J. Lowery and D. Novak, “Performance comparison of gain-coupled and index-coupled DFB semiconductor lasers,” IEEE J. Quantum Electron. 30(9), 2051–2063 (1994).
[Crossref]

1991 (1)

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

1990 (1)

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Alù, A.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Azaña, J.

Baets, R. G.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

Bélanger, N.

Bender, C. M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT-symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT-symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Borchert, B.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Cardimona, D. A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Chong, Y. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Chung, I.-S.

I.-S. Chung and J. Mørk, “Silicon photonics light source realized by III-VI/Si grating-mirror lasers,” J. Appl. Phys. Lett. 97(15), 151113 (2010).
[Crossref]

David, K.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Fan, J.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Feng, L.

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014).
[Crossref] [PubMed]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Finazzi, V.

Fleury, R.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Gavrielides, A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

Ge, L.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Gustavson, J. S.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Haglund, A.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hosomatsu, H.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

Ieaoka, H.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

Inoue, T.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

Joel, A.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Jones, H. F.

Khajavikhan, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Kovanis, V.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

Kress, B.

Kulishov, M.

Laniel, J.

Larsson, A.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Liertzer, M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Longhi, S.

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

Lowery, A. J.

A. J. Lowery and D. Novak, “Performance comparison of gain-coupled and index-coupled DFB semiconductor lasers,” IEEE J. Quantum Electron. 30(9), 2051–2063 (1994).
[Crossref]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Luo, Y.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[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(6212), 972–975 (2014).
[Crossref] [PubMed]

Miri, M.-A.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Monifi, F.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Mørk, J.

I.-S. Chung and J. Mørk, “Silicon photonics light source realized by III-VI/Si grating-mirror lasers,” J. Appl. Phys. Lett. 97(15), 151113 (2010).
[Crossref]

Morthier, G.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Nakano, Y.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

Nakwaski, W.

W. Nakwaski, “VCSEL structures used to suppress higher order transverse modes,” Opto-Electron. Rev. 19(1), 119–129 (2011).
[Crossref]

Nori, F.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Novak, D.

A. J. Lowery and D. Novak, “Performance comparison of gain-coupled and index-coupled DFB semiconductor lasers,” IEEE J. Quantum Electron. 30(9), 2051–2063 (1994).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Özdemir, S. K.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Pelton, M.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Peng, B.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Plant, D.

Plant, J.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Rotter, S.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Santori, C.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

Sharma, M. P.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

Skold, M.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Slavík, R.

Solomon, G. S.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Sounas, D. L.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Stone, A. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Tada, K.

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

Vankwikelberge, P.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

Vuckovic, J.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Wang, Y.

Westbergh, P.

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Wolf, T.

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[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(6212), 972–975 (2014).
[Crossref] [PubMed]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Yamamoto, Y.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Yang, L.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Yang, S.

Yilmaz, H.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Yin, X.

Zervas, M. N.

Zhang, B.

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Zhang, P.

Zhang, X.

Zhao, Y.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

Zhu, H.

Zhu, L.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

Zhu, X.

Zhu, Y.

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Luo, Y. Nakano, K. Tada, T. Inoue, H. Hosomatsu, and H. Ieaoka, “Purely gain-coupled distributed feedback semiconductor lasers,” Appl. Phys. Lett. 56(17), 1620–1622 (1990).
[Crossref]

IEEE J. Quantum Electron. (3)

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31(1), 60–66 (1995).
[Crossref]

K. David, G. Morthier, P. Vankwikelberge, R. G. Baets, T. Wolf, and B. Borchert, “Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield,” IEEE J. Quantum Electron. 27(6), 1714–1723 (1991).
[Crossref]

A. J. Lowery and D. Novak, “Performance comparison of gain-coupled and index-coupled DFB semiconductor lasers,” IEEE J. Quantum Electron. 30(9), 2051–2063 (1994).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

P. Westbergh, J. S. Gustavson, A. Haglund, M. Skold, A. Joel, and A. Larsson, “High speed low-current density 850 nm VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 694–703 (2009).
[Crossref]

Y. Zhu, Y. Zhao, J. Fan, and L. Zhu, “Modal gain analysis of Parity-Time-Symmetric distributed feedback lasers,” IEEE J. Sel. Top. Quantum Electron. 22(5), 1500207 (2016).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

J. Appl. Phys. Lett. (1)

I.-S. Chung and J. Mørk, “Silicon photonics light source realized by III-VI/Si grating-mirror lasers,” J. Appl. Phys. Lett. 97(15), 151113 (2010).
[Crossref]

J. Opt. (1)

H. F. Jones and M. Kulishov, “Extension of analytic results of a PT-symmetric structure,” J. Opt. 18(5), 055101 (2016).
[Crossref]

J. Phys. A (1)

H. F. Jones, “Analytic results for a PT-symmetric optical structure,” J. Phys. A 45(13), 135306 (2012).
[Crossref]

J. Phys. Math. Gen. (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Nat. Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Nature (1)

K. J. Vahala, “Optical Microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

Opt. Express (6)

Opto-Electron. Rev. (1)

W. Nakwaski, “VCSEL structures used to suppress higher order transverse modes,” Opto-Electron. Rev. 19(1), 119–129 (2011).
[Crossref]

Phys. Rev. A (1)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonian having PT-symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

M. Pelton, C. Santori, J. Vucković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

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(6212), 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Other (2)

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.
[Crossref]

S. Nakajima, T. Inoue, Y. Luo, T. Oki, Y. Nakano, K. Tada, R. Takahashi, and T. Kamia, “Dynamic characteristics of 1.55 μm gain-coupled distributed feedback semiconductor lasers,” Conf. Dig. 13th IEEE Int. Semiconduct. Laser Conf., Takamatsu, Japan, Paper b-4, pp.18–19, Sept. 21–25 (1992).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Schematic view of the PT-symmetric vertical emitting laser based on a planar purely reflective volume grating with index (black color fringes) and gain/loss (red color fringes) modulations in the form decribed by Eq. (1).

Fig. 2
Fig. 2

Reflective wavelength (a, c) and angular (b, d) spectra for ε1 = 1; ε2 = 2.4; ε3 = ∞; L = 72Λ; Λ = 0.21 µm below the threshold (ξ = 0.0078) (a, b) and near the threshold condition: ξ = ξth = 0.0104 (c, d).

Fig. 3
Fig. 3

Reflection spectra for the PT-symmetric grating with ϕ = 0 for even number of periods L = 72Λ (a) and (b) and for odd number of periods L = 73Λ (c) and (d) below the threshold for the first longitudinal mode: ξ< ξ th(1) for (a) and (c) and ξ th(1) <ξ< ξ th(2) for (b) and (d). The blue (dashed) vertical lines show the Bragg wavelength λB = 2Λ/(ε2)1/2 = 632.071 nm when δ = 0. The other parameters: ε1 = 1; ε2 = 2.4; Λ = 204 nm.

Fig. 4
Fig. 4

Possible lasing modes for (a) ϕ = 0, (b) ϕ = π/2. The red bars indicate the possible values of k2L, while the blue curve is the sinc function of Eq. (16).

Fig. 5
Fig. 5

Reflection spectra for the PT-symmetric grating with ϕ = π/2 (a) and (b) and ϕ = -π/2 (c) and (d) for even number of periods L = 72Λ (a) and (c) and for odd number of periods L = 73Λ (b) and (d) below the threshold for the fundamental longitudinal mode. The blue (dashed) vertical lines show the Bragg wavelength λB = 2Λ/(ε2)1/2 = 632.071 nm when δ = 0. The other parameters: ξ = 0.008; ε1 = 1; ε2 = 2.4; Λ = 204 nm.

Fig. 6
Fig. 6

The threshold value as a function of the external dielectric constant; the red and blue curves are based on the approximate Kogelnik method [Eqs. (18) and (19) respectively] and the magenta diamonds and black circles are numerically calculated values based on the exact Bessel function method [27]. The vertical dashed line shows the threshold for zero Fresnel reflection, when ε1 = ε2.

Fig. 7
Fig. 7

Reflection spectra for the PT-symmetric balanced grating (solid, red curves), gain/loss dominated grating (magenta, dashed curves) and index-dominated grating (blue, dotted curves) for different value of the coupling coefficient ξ with ϕ = π/2, L = 72Λ, ε1 = 1; ε2 = 2.4; Λ = 204 nm.

Fig. 8
Fig. 8

Reflective wavelength (a, c) and angular (b, d) spectra for ε1 = 6; ε2 = 2.4; ε3 = ∞; L = 72Λ; Λ = 0.204 µm at ξ = 0.005 (a, b) and at the threshold condition: ξ = ξth = 0.00685 (c, d).

Equations (21)

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ε(x,z)= ε 2 +Δ ε Re cos(2βz+φ)+jΔ ε Im sin(2βz+φ)
E 1 =exp(j k 1 z)+ R L exp(j k 1 z) E 2 = E f (z)exp(j k 2 z)+ E b (z)exp(j k 2 z)
M (1,2) = 1 2 ( ( 1+γ )exp( j k 2 k 1 2 L ) ( 1γ )exp( j k 1 + k 2 2 L ) ( 1γ )exp( j k 2 + k 1 2 L ) ( 1+γ )exp( j k 1 k 2 2 L ) )
M (GR) =( m 11 m 12 m 21 m 22 )
m 11 =( cosh( ρL )+j δ ρ sinh( ρL ) )exp(jδL)
m 12 =j κ 12 ρ sinh( ρL )
m 21 =j κ 21 ρ sinh( ρL )
m 22 =( cosh( ρL )j δ ρ sinh( ρL ) )exp(jδL)
( E f (L/2) E b (L/2) )= M (GR) M (1,2) ( 1 R L )
R L = ( M 11 exp(j k 2 L)+ M 21 )(1+γ)+( M 22 exp(j k 2 L)+ M 12 )(1γ) (1γ)( M 21 +M e 11 xp(j k 2 L))+( M 22 exp(j k 2 L)+ M 12 )(1+γ) e j k 1 L
ε(x,z)= ε 2 ( 1+ ξ 2 exp(j(2βz+ϕ)) )
M (GR) =( 1 m 12 (PT) 0 1 )
m 12 (PT) =j k 2 L sin(δL) 2δL ξexp(jφ)
cos( k 2 L)=0
2γ 1+γ sin( k 2 L)= k 2 L ξ 2 sin(δL) δL
ξ th (r) = (1) m 2γ 1+γ (2r+1) (m+r+1/2)
ε(x,z)= ε 2 (1+j(ξ/2)exp(2jβz))
sin( k 2 L)=0
2 1+γ cos( k 2 L)= k 2 L ξ 2 sin(δL) δL
ξ th (0,mΛ) = (1) m 4 πm(1+γ)
ξ th (0,(m+1/2)Λ) = 2γ π(1+γ) 2 (m+1/2)

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