Abstract

Recent experiments demonstrated that chiral symmetry breaking at an exceptional point (EP) is a viable route to achieve unidirectional laser emission in microring lasers. By a detailed semiconductor laser rate equation model, we show here that unidirectional laser emission at an EP is a robust regime. Slight deviations from the EP condition can break preferential unidirectional lasing near threshold via a Hopf instability. However, above a “second” laser threshold, unidirectional emission is restored.

© 2017 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (2)

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

2015 (2)

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photon. 7, 168–240 (2015).
[Crossref]

2014 (2)

2012 (3)

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

2010 (4)

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

J. Zhu, S. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

2009 (2)

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

2008 (2)

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

2004 (1)

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

2003 (3)

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

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

2002 (2)

2001 (1)

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

1997 (1)

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

1993 (1)

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

1988 (1)

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Abraham, N. B.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Ballantyne, J. M.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

Balle, S.

Beausoleil, R. G.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Beri, S.

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

Binsma, H.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Bowers, J. E.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Burkhardt, S.

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

Cao, H.

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Capasso, F.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Chang, R. K.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Chen, W.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

Chern, G. D.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Craft, D. C.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Danckaert, J.

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

de Vries, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Den Besten, J. H.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Diehl, L.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Donati, S.

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Dorren, H. J.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Douglas Stone, A.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Edamura, T.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Fattal, D. A.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Feng, L.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

S. Longhi and L. Feng, “PT-symmetric microring laser-absorber,” Opt. Lett. 39, 5026–5029 (2014).
[Crossref]

Fiorentino, M.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Foreman, M. R.

Ge, L.

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Gelens, L.

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

Geluk, E.-J.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Giuliani, G.

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Harney, H. L.

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

He, L.

Heiss, W. D.

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

Hentschel, M.

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

Hill, M. T.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Hoffer, L. M.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Hohimer, J. P.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Huang, Z.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Huybrechts, K.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Johnson, N. M.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Jung, Y.

Kan, H.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Khoe, G. D.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Kim, M.

Kippenberg, T. J.

Kneissl, M.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Kramer, J.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

Krimer, D. O.

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

Kumar, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Kwon, K.

Lau, S. T.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

Laybourn, P. J. R.

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Leary, M. H.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

Leijtens, X. J.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Liang, D.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Liang, J. J.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

Liertzer, M.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

Lippi, G. L.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Litchinitser, N. M.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Liu, L.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Longhi, S.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

S. Longhi and L. Feng, “PT-symmetric microring laser-absorber,” Opt. Lett. 39, 5026–5029 (2014).
[Crossref]

Mandel, P.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Mello, T.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Mezosi, G.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Miao, P.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Miglierina, R.

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Morthier, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Oei, Y. S.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Özdemir, S. K.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

J. Zhu, S. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

Peng, B.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

Pflügl, C.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Redding, B.

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Regreny, P.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Roelkens, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Rotter, S.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

Scire, A.

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Shim, J.

Smalbrugge, B.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Smit, M. K.

M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Solomon, G. S.

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Song, Q.

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Sorel, M.

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scire, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[Crossref]

Spencer, D. T.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Spillane, S. M.

Spuesens, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Srinivasan, S.

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
[Crossref]

Sun, J.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Swaim, J. D.

Tassin, P.

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

Tureci, H. E.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Unterhinninghofen, J.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Vahala, K. J.

Van der Sande, G.

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

Van Thourhout, D.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
[Crossref]

Vawter, G. A.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Verschaffelt, G.

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

Vollmer, F.

Walasik, W.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Wang, Q. J.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Wiersig, J.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

Yamanishi, M.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Yan, C.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Yang, L.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

J. Zhu, S. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

Yilmaz, H.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

Yu, K.

Yu, N.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

Zeghlache, H.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

Zhang, Z.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Zhu, J.

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (3)

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[Crossref]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Eur. Phys. J. D (2)

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17, 149–151 (2001).
[Crossref]

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–207 (2010).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs/AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

D. Liang, S. Srinivasan, D. A. Fattal, M. Fiorentino, Z. Huang, D. T. Spencer, J. E. Bowers, and R. G. Beausoleil, “Teardrop reflector-assisted unidirectional hybrid silicon microring lasers,” IEEE Photon. Technol. Lett. 24, 1988–1990 (2012).
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J. Phys. A (1)

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

J. Phys. B (1)

G. Van der Sande, L. Gelens, P. Tassin, A. Scire, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[Crossref]

Nat. Photonics (1)

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics 4, 182–187 (2010).
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Nature (2)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
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M. T. Hill, H. J. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (2)

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[Crossref]

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for fully or nearly degenerate cavity modes,” Phys. Rev. A 92, 013847 (2015).
[Crossref]

Phys. Rev. E (1)

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[Crossref]

Phys. Rev. Lett. (3)

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[Crossref]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

B. Redding, L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, “Local chirality of optical resonances in ultrasmall resonators,” Phys. Rev. Lett. 108, 253902 (2012).
[Crossref]

Proc. Natl. Acad. Sci. (2)

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. 107, 22407–22412 (2010).
[Crossref]

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. 113, 6845–6850 (2016).
[Crossref]

Science (1)

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of a microring sustaining cw and ccw TWMs. Control of linear mode couplings κ1,2 based on the use of (b) two Rayleigh scatterers (dots) [19,20], and (c) combined phase-shifted index and loss gratings [21].
Fig. 2.
Fig. 2. Intensity of cw and ccw ring modes versus normalized pump parameter J at an EP (κ2/κ1=0) for α=3. Solid and dashed curves refer to stable and unstable branches of stationary solutions, respectively. For J<J1, the only stable solution corresponds to unidirectional laser emission in the cw mode (E2=0), whereas for J>J1, bistable unidirectional emission, with either dominant cw or ccw modes, is observed. The value of J1 depends on the linewidth enhancement factor α solely according to Eq. (14).
Fig. 3.
Fig. 3. Same as Fig. 2 but for κ2/κ1=0.5, Θ=0, and α=3. For J<J20.33, all stationary solutions are unstable and the unidirectional TWM emission is destabilized by a Hopf instability. Laser emission in the dominant cw mode is realized in the pump parameter range of J2<J<J1.
Fig. 4.
Fig. 4. Behavior of the directionality D versus coupling ratio κ2/κ1 for a linewidth enhancement factor of α=3. The directionality is calculated close to the second laser threshold, i.e., for J=J2+.
Fig. 5.
Fig. 5. Numerically computed behavior of the normalized pump parameter J1, corresponding to the bistability boundary (upper curves, filled circles), and of J2, corresponding to the “second” laser threshold (Hopf instability boundary, open circles), versus κ2/κ1 for Θ=0 and for a few values of the linewidth enhancement factor. (a) α=1, (b) α=2, and (c) α=3. The region embedded between the upper and lower curves corresponds to stable laser emission in the dominant cw mode. The lower and upper curves touch at κ2/κ1=1 (not shown in the figures).
Fig. 6.
Fig. 6. (a) Numerically computed bifurcation diagram showing the extreme (maxima/minima) of amplitudes |E1,2(t)| for cw and ccw TWMs versus normalized injection current μ. Parameter values are given in the text. For μ1.37, a stationary regime, corresponding to almost unidirectional emission in the cw mode, is observed, whereas for μ1.37, the dynamics are oscillatory. (b) Numerically computed time evolution of mode intensities |E1,2(t)|2 for cw (thin solid line) and ccw (thick solid line, with lower amplitude) for increasing values of the normalized injection current μ. The laser is switched on at time t=0, with the initial condition corresponding to small random amplitudes of E1 and E2 and N=0.

Equations (24)

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dE1dt=κ(1+iα)[N(1s|E1|2c|E2|2)1]E1+iκ1E2,
dE2dt=κ(1+iα)[N(1s|E2|2c|E1|2)1]E2+iκ2exp(iΘ)E1,
dNdt=γ[μNN(1s|E1|2c|E2|2)|E1|2N(1s|E2|2c|E1|2)|E2|2],
E1=μ1cos(θ2+π4)exp(iϕ1),
E2=μ1sin(θ2+π4)exp(iϕ2),
dθdt=Jκ1sinθcosθ+(1sinθ)[κ2sin(ψΘ)κ1sinψ]+2κ1sinψ,
dψdt=αJκ1sinθ+κ2cotg(θ+π/22)cos(ψΘ)κ1tg(θ+π/22)cosψ,
Jκκ1c(μ1)(1sc)
dθdt=Jκ1sinθcosθ+κ1(1+sinθ)sinψ,
dψdt=αJκ1sinθκ1tg(θ+π/22)cosψ.
θ(t)=π/2,ψ(t)=ακ1Jt.
dδψ(t)dt=κ12δθ(t)cos(ακ1t),
dδθ(t)dt=κ1Jδθ,
J13.331+α2.
D=|E1|2|E2|2|E1|2+|E2|2,
D=sinθ.
tgψ0=cosθ0[κ2cotg(θ02+π4)κ1tg(θ02+π4)]2ακ1α(κ1κ2)(1sinθ0),
J=(κ1κ2)(1sinθ0)sinψ02κ1sinψ0sinθ0cosθ0.
ddt(δθδψ)=(M11M12M21M22)(δθδψ),
M11=Jcos(2θ0)(κ2κ1)cosθ0sinψ0,
M12=(κ2κ1)(1sinθ0)cosψ0+2κ1cosψ0,
M21=αJcosθ0κ2cosψ02sin2(θ02+π4)κ1cosψ02cos2(θ02+π4),
M22=κ1sinψ0tg(θ02+π4)κ2sinψ0cotg(θ02+π4).
ωHopf=2κ1κ2.

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