Abstract

We report parity-time (PT) symmetry breaking in electrically injected, coherently coupled, vertical cavity surface emitting laser arrays. We predict beam steering, mode evolution, and mode hopping as consequences of the non-Hermiticity of the array, analyzed by the temporal coupled-mode theory with both an asymmetric gain distribution and local frequency detuning. We present experimental confirmations of the predicted mode evolution, mode hopping, and PT symmetry breaking with quantitative agreement with the theory.

© 2017 Optical Society of America

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    [Crossref]
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2016 (3)

Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
[Crossref]

Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, J. Ren, W. E. Hayenga, M.-A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Design considerations for single-mode microring lasers using parity-time symmetry,” IEEE J. Sel. Top. Quantum Electron. 22, 1–7 (2016).
[Crossref]

2015 (4)

P. Qiao, L. Zhu, W. C. Chew, and C. J. Chang-Hasnain, “Theory and design of two-dimensional high-contrast-grating phased arrays,” Opt. Express 23, 24508–24524 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

S. T. M. Fryslie, M. P. Tan, D. F. Siriani, M. T. Johnson, and K. D. Choquette, “37-GHz modulation via resonance tuning in single-mode coherent vertical-cavity laser arrays,” IEEE Photon. Technol. Lett. 27, 415–418 (2015).
[Crossref]

S. T. M. Fryslie, M. T. Johnson, and K. D. Choquette, “Coherence tuning in optically coupled phased vertical cavity laser arrays,” IEEE J. Quantum Electron. 51, 1–6 (2015).
[Crossref]

2014 (7)

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

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

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

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

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

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

2013 (6)

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 21651–21668 (2013).
[Crossref]

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

M. T. Johnson, D. F. Siriani, M. Peun Tan, and K. D. Choquette, “Beam steering via resonance detuning in coherently coupled vertical cavity laser arrays,” Appl. Phys. Lett. 103, 201115 (2013).
[Crossref]

M. T. Johnson, D. F. Siriani, and K. D. Choquette, “High-speed beam steering with phased vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 19, 1701006 (2013).
[Crossref]

H. Dalir and F. Koyama, “29  GHz directly modulated 980  nm vertical-cavity surface emitting lasers with bow-tie shape transverse coupled cavity,” Appl. Phys. Lett. 103, 091109 (2013).
[Crossref]

2012 (8)

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 4–9 (2012).
[Crossref]

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

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (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, 108–113 (2012).
[Crossref]

M.-A. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref]

H. Benisty and M. Besbes, “Confinement and optical properties of the plasmonic inverse-rib waveguide,” J. Opt. Soc. Am. B 29, 818–826 (2012).
[Crossref]

H. Benisty, C. Yan, A. Degiron, and A. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: analysis and examples,” J. Lightwave Technol. 30, 2675–2683 (2012).
[Crossref]

M. T. Johnson, D. F. Siriani, J. D. Sulkin, and K. D. Choquette, “Phase and coherence extraction from a phased vertical cavity laser array,” Appl. Phys. Lett. 101, 031116 (2012).
[Crossref]

2011 (3)

H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004–18019 (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, 213901 (2011).
[Crossref]

D. D. F. Siriani, P. S. Carney, and K. D. Choquette, “Coherence of leaky-mode vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 47, 672–675 (2011).
[Crossref]

2010 (2)

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

D. Siriani, K. Choquette, and P. Carney, “Stochastic coupled mode theory for partially coherent laser arrays,” J. Opt. Soc. Am. A 27, 501–508 (2010).
[Crossref]

2009 (2)

A. L. Harren, K. Choquette, and P. Carney, “Partial coherence in coupled photonic crystal vertical cavity laser arrays,” Opt. Lett. 34, 905–907 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2008 (2)

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

2007 (3)

L. D. A. Lundeberg and E. Kapon, “Mode switching and beam steering in photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 90, 2005–2008 (2007).

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

A. Lehman, J. J. J. Raftery, P. S. Carney, and K. D. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25–30 (2007).
[Crossref]

2005 (1)

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

2004 (1)

L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
[Crossref]

2003 (1)

A. J. Danner, J. C. Lee, J. J. Raftery, N. Yokouchi, and K. D. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323–1324 (2003).
[Crossref]

2002 (1)

C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A 35, L467–L471 (2002).
[Crossref]

2001 (1)

B. Lücke, G. Hergenhan, U. Brauch, and A. Giesen, “Phase tuning of injection-locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[Crossref]

2000 (1)

1999 (1)

1998 (1)

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

1992 (1)

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160–1162 (1992).
[Crossref]

1991 (1)

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

1990 (1)

D. Botez, L. J. Mawst, G. L. Peterson, and T. J. Roth, “Phase-locked arrays of antiguides: model content and discrimination,” IEEE J. Quantum Electron. 26, 482–495 (1990).
[Crossref]

1988 (1)

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H. Hodaei, A. U. Hassan, J. Ren, W. E. Hayenga, M.-A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Design considerations for single-mode microring lasers using parity-time symmetry,” IEEE J. Sel. Top. Quantum Electron. 22, 1–7 (2016).
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Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
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M. T. Johnson, D. F. Siriani, and K. D. Choquette, “High-speed beam steering with phased vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 19, 1701006 (2013).
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E. Kapon, J. Katz, S. Margalit, and A. Yariv, “Longitudinal-mode control in integrated semiconductor laser phased arrays by phase velocity matching,” Appl. Phys. Lett. 44, 157–159 (1984).
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E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
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E. Kapon, “Control of mutual phase locking of monolithically integrated semiconductor lasers,” Appl. Phys. Lett. 43, 421–423 (1983).
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E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
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E. Kapon, J. Katz, S. Margalit, and A. Yariv, “Longitudinal-mode control in integrated semiconductor laser phased arrays by phase velocity matching,” Appl. Phys. Lett. 44, 157–159 (1984).
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H. Hodaei, A. U. Hassan, J. Ren, W. E. Hayenga, M.-A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Design considerations for single-mode microring lasers using parity-time symmetry,” IEEE J. Sel. Top. Quantum Electron. 22, 1–7 (2016).
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H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
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L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160–1162 (1992).
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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, 213901 (2011).
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A. J. Danner, J. C. Lee, J. J. Raftery, N. Yokouchi, and K. D. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323–1324 (2003).
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A. Lehman, J. J. J. Raftery, P. S. Carney, and K. D. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25–30 (2007).
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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160–1162 (1992).
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Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
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M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
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LiKamWa, P.

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, 213901 (2011).
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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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E. Kapon, Z. Rav-Noy, L. T. Lu, M. Yi, S. Margalit, and A. Yariv, “Phase-locking characteristics of coupled ridge-waveguide InP/InGaAsP diode lasers,” Appl. Phys. Lett. 45, 1159–1161 (1984).
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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, 108–113 (2012).
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L. D. A. Lundeberg and E. Kapon, “Mode switching and beam steering in photonic crystal heterostructures implemented with vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 90, 2005–2008 (2007).

Lupu, A.

Lyu, Q.

Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
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R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
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S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 4–9 (2012).
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M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Margalit, S.

E. Kapon, Z. Rav-Noy, L. T. Lu, M. Yi, S. Margalit, and A. Yariv, “Phase-locking characteristics of coupled ridge-waveguide InP/InGaAsP diode lasers,” Appl. Phys. Lett. 45, 1159–1161 (1984).
[Crossref]

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L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
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Miri, M.-A.

H. Hodaei, A. U. Hassan, J. Ren, W. E. Hayenga, M.-A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Design considerations for single-mode microring lasers using parity-time symmetry,” IEEE J. Sel. Top. Quantum Electron. 22, 1–7 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160–1162 (1992).
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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A. 38, L171–L176 (2005).
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R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160–1162 (1992).
[Crossref]

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Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
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L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[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, 108–113 (2012).
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Onishchukov, G.

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

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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

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

B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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D. Botez, L. J. Mawst, G. L. Peterson, and T. J. Roth, “Phase-locked arrays of antiguides: model content and discrimination,” IEEE J. Quantum Electron. 26, 482–495 (1990).
[Crossref]

Peun Tan, M.

M. T. Johnson, D. F. Siriani, M. Peun Tan, and K. D. Choquette, “Beam steering via resonance detuning in coherently coupled vertical cavity laser arrays,” Appl. Phys. Lett. 103, 201115 (2013).
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Qiao, P.

Raftery, J. J.

A. J. Danner, J. C. Lee, J. J. Raftery, N. Yokouchi, and K. D. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323–1324 (2003).
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Raftery, J. J. J.

A. Lehman, J. J. J. Raftery, P. S. Carney, and K. D. Choquette, “Coherence of photonic crystal vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 43, 25–30 (2007).
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Ragunathan, G.

Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
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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, 213901 (2011).
[Crossref]

Rav-Noy, Z.

E. Kapon, Z. Rav-Noy, L. T. Lu, M. Yi, S. Margalit, and A. Yariv, “Phase-locking characteristics of coupled ridge-waveguide InP/InGaAsP diode lasers,” Appl. Phys. Lett. 45, 1159–1161 (1984).
[Crossref]

Regensburger, A.

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

Ren, H.-Q.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Ren, J.

H. Hodaei, A. U. Hassan, J. Ren, W. E. Hayenga, M.-A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Design considerations for single-mode microring lasers using parity-time symmetry,” IEEE J. Sel. Top. Quantum Electron. 22, 1–7 (2016).
[Crossref]

Roth, T. J.

D. Botez, L. J. Mawst, G. L. Peterson, and T. J. Roth, “Phase-locked arrays of antiguides: model content and discrimination,” IEEE J. Quantum Electron. 26, 482–495 (1990).
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Rotter, S.

B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
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Rout, B.

Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A. 38, L171–L176 (2005).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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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, 108–113 (2012).
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Schöberl, J.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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Segev, M.

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

Siriani, D. D. F.

D. D. F. Siriani, P. S. Carney, and K. D. Choquette, “Coherence of leaky-mode vertical-cavity surface-emitting laser arrays,” IEEE J. Quantum Electron. 47, 672–675 (2011).
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Siriani, D. F.

S. T. M. Fryslie, M. P. Tan, D. F. Siriani, M. T. Johnson, and K. D. Choquette, “37-GHz modulation via resonance tuning in single-mode coherent vertical-cavity laser arrays,” IEEE Photon. Technol. Lett. 27, 415–418 (2015).
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M. T. Johnson, D. F. Siriani, and K. D. Choquette, “High-speed beam steering with phased vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 19, 1701006 (2013).
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M. T. Johnson, D. F. Siriani, M. Peun Tan, and K. D. Choquette, “Beam steering via resonance detuning in coherently coupled vertical cavity laser arrays,” Appl. Phys. Lett. 103, 201115 (2013).
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Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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Song, Q.

Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
[Crossref]

Stone, A. D.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Strasser, G.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Streifer, W.

D. R. Scifres, R. D. Burnham, and W. Streifer, “Phase-locked semiconductor laser array,” Appl. Phys. Lett. 33, 1015–1017 (1978).
[Crossref]

Suchkov, S. V.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of PT-symmetric couplers,” Phys. Rev. A 85, 4–9 (2012).
[Crossref]

Sulkin, J. D.

M. T. Johnson, D. F. Siriani, J. D. Sulkin, and K. D. Choquette, “Phase and coherence extraction from a phased vertical cavity laser array,” Appl. Phys. Lett. 101, 031116 (2012).
[Crossref]

Sun, J.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Tan, M. P.

S. T. M. Fryslie, M. P. Tan, D. F. Siriani, M. T. Johnson, and K. D. Choquette, “37-GHz modulation via resonance tuning in single-mode coherent vertical-cavity laser arrays,” IEEE Photon. Technol. Lett. 27, 415–418 (2015).
[Crossref]

Thompson, B.

B. Thompson, D. F. Siriani, and K. D. Choquette, “Coherent operation of 2×2 vertical cavity laser array by resonance tuning,” in IEEE Photonics Conference (IPC) (IEEE, 2015), Vol. 3, pp. 589–590.

Thompson, B. J.

Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
[Crossref]

Thompson, G.

G. Thompson, “Analysis of optical directional couplers that include gain or loss and their application to semiconductor slab dielectric guides,” J. Lightwave Technol. 4, 1678–1693 (1986).
[Crossref]

Timurdogan, E.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Troshchieva, V. N.

L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
[Crossref]

Türeci, H. E.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Unterrainer, K.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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L. Bao, N. H. Kim, L. J. Mawst, N. N. Elkin, V. N. Troshchieva, D. V. Vysotsky, and A. P. Napartovich, “Near-diffraction-limited coherent emission from large aperture antiguided vertical-cavity surface-emitting laser arrays,” Appl. Phys. Lett. 84, 320–322 (2004).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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Wang, J.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Wang, S. S.

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
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Wang, Y.

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

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref]

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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Xiao, S.

Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
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Xie, Y.-Y.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Xu, C.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Xu, K.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

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, 108–113 (2012).
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Xun, M.

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

Yaacobi, A.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
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Yan, C.

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
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Yariv, A.

E. Kapon, Z. Rav-Noy, L. T. Lu, M. Yi, S. Margalit, and A. Yariv, “Phase-locking characteristics of coupled ridge-waveguide InP/InGaAsP diode lasers,” Appl. Phys. Lett. 45, 1159–1161 (1984).
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E. Kapon, J. Katz, S. Margalit, and A. Yariv, “Longitudinal-mode control in integrated semiconductor laser phased arrays by phase velocity matching,” Appl. Phys. Lett. 44, 157–159 (1984).
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E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9, 125–127 (1984).
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E. Kapon, Z. Rav-Noy, L. T. Lu, M. Yi, S. Margalit, and A. Yariv, “Phase-locking characteristics of coupled ridge-waveguide InP/InGaAsP diode lasers,” Appl. Phys. Lett. 45, 1159–1161 (1984).
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Yilmaz, H.

B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
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Yokouchi, N.

A. J. Danner, J. C. Lee, J. J. Raftery, N. Yokouchi, and K. D. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323–1324 (2003).
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Zhang, N.

Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
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Zhang, X.

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

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M. T. Johnson, D. F. Siriani, M. Peun Tan, and K. D. Choquette, “Beam steering via resonance detuning in coherently coupled vertical cavity laser arrays,” Appl. Phys. Lett. 103, 201115 (2013).
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S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774–1776 (1988).
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M. T. Johnson, D. F. Siriani, J. D. Sulkin, and K. D. Choquette, “Phase and coherence extraction from a phased vertical cavity laser array,” Appl. Phys. Lett. 101, 031116 (2012).
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A. J. Danner, J. C. Lee, J. J. Raftery, N. Yokouchi, and K. D. Choquette, “Coupled-defect photonic crystal vertical cavity surface emitting lasers,” Electron. Lett. 39, 1323–1324 (2003).
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IEEE Photon. J. (1)

M.-M. Mao, C. Xu, Y.-Y. Xie, Q. Kan, M. Xun, K. Xu, J. Wang, H.-Q. Ren, and H.-D. Chen, “Implant-defined 3 × 3 in-phase coherently coupled vertical cavity surface emitting lasers array,” IEEE Photon. J. 5, 1502606 (2013).

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Z. Gao, B. J. Thompson, G. Ragunathan, M. T. Johnson, B. Rout, and K. D. Choquette, “Bottom-emitting coherently coupled vertical cavity laser arrays,” IEEE Photon. Technol. Lett. 28, 513–515 (2016).
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Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, “Experimental demonstration of PT-symmetric stripe lasers,” Laser Photon. Rev. 10, 588–594 (2016).
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Nat. Commun. (1)

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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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, 108–113 (2012).
[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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B. Peng, Ş. K. Ozdemir, 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, 328–332 (2014).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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Other (1)

B. Thompson, D. F. Siriani, and K. D. Choquette, “Coherent operation of 2×2 vertical cavity laser array by resonance tuning,” in IEEE Photonics Conference (IPC) (IEEE, 2015), Vol. 3, pp. 589–590.

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Figures (6)

Fig. 1.
Fig. 1.

Effect of frequency detuning without gain contrast on (a) wavelengths of the coupled modes, (b) gain of the coupled modes, (c) ratio of the field magnitudes in two cavities, and (d) relative phase between the fields in two cavities.

Fig. 2.
Fig. 2.

Effect of gain contrast without frequency detuning on (a) wavelengths of the coupled modes, (b) gain of the coupled modes, (c) ratio of the field magnitudes in two cavities, and (d) relative phase between the fields in two cavities.

Fig. 3.
Fig. 3.

Effect of coexisting gain contrast and frequency detuning on (a) wavelengths of the coupled modes, (b) gain of the coupled modes, (c) ratio of the field magnitudes in two cavities, and (d) relative phase between the field in two cavities. It is assumed that the local changes of the gain and frequency are both linearly dependent on the current difference, with Δ g = 4 Δ ω , and the maximum gain contrast at the edge of the graphs is Δ g max = 4 κ .

Fig. 4.
Fig. 4.

(a) Microscopic photo of the VCSEL diode array under characterization, (b) near-field intensity profiles with extracted relative phase [48], (c) far-field intensity profiles measured at Δ I = 0 , 0.12 , and 0.09    mA , and (d) evolution of the measured and propagated far-field profiles with varying I b and fixed I a . The red circles in (c) denote a 12-deg angular spread in the far field.

Fig. 5.
Fig. 5.

Comparison of measured data (diamond points) with the coupled-mode theory (solid lines) for (a) wavelength, (b) gain, (c) field magnitude ratio, and (d) relative phase. Parameters used in the coupled-mode theory are: κ = 1.5 × 10 11    Hz , maximum gain contrast and frequency detuning at I b I a = 0.2    mA are Δ g max = 1 × 10 12    Hz and Δ ω max = 2 × 10 11    Hz , respectively. The two cavities are assumed to have the same resonant wavelength at I b = I a , and both red-shift when I b is decreased, with ω b shifting more than ω a . The arrows indicate mode hopping. The thermal hysteresis in (a) is the result of increased device heating over the duration of the measurement.

Fig. 6.
Fig. 6.

Measured (a) near field and (b) far field in the PT-symmetry broken regime compared with the simulated (c) near field and (d) far field of the ( 1 , i exp ( θ ) ) eigenmode.

Equations (5)

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

d E a d t = i ω a E a + γ a E a + i κ a b E b d E b d t = i ω b E b + γ b E b + i κ b a E a ,
i d E d t = Ω E ,
( PT Ω ) i , j = Ω 3 i , 3 j * ,
ω = ω a + ω b 2 + i ( γ a + γ b ) 2 ± [ κ a b κ b a + ( ω a ω b 2 ) 2 ( γ a γ b 2 ) 2 + i ( ω a ω b ) ( γ a γ b ) 2 ] 1 / 2 .
A b A a = i κ b a i ( ω ω b ) + γ b .