Abstract

We investigate lasing thresholds in a representative photonic molecule composed of two coupled active cylinders of slightly different radii. Specifically, we use the recently formulated steady-state ab initio laser theory (SALT) to assess the effect of the underlying gain transition on lasing frequencies and thresholds. We find that the order in which modes lase can be modified by choosing suitable combinations of the gain center frequency and linewidth, a result that cannot be obtained using the conventional approach of quasi-bound modes. The impact of the gain transition center on the lasing frequencies, the frequency pulling effect, is also quantified.

© 2014 Optical Society of America

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  13. B. Peng, S. K. Ozdemir, 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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  15. 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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    [CrossRef]
  32. D. Gagnon, J. Dumont, and L. J. Dubé, “Multiobjective optimization in integrated photonics design,” Opt. Lett. 38, 2181–2184 (2013).
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  37. A further technical aspect of the implementation should be noted. Although we have not encountered any instabilities in our calculations, it should be acknowledged that for high accuracy work, Eqs. (12) and (13) are not well suited (exponential decay or growth of Tll′nn′ with the indices l and l′). This difficulty has been recognized before [17] and solved generally in [36].
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    [CrossRef]
  40. G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.
  41. S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).
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    [CrossRef]

2014 (3)

C. Wang and C. P. Search, “Nonlinearly enhanced refractive index sensing in coupled optical microresonators,” Opt. Lett. 39, 26–29 (2014).
[CrossRef]

B. Peng, S. K. Ozdemir, 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]

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

2013 (6)

2012 (3)

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]

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

D. Gagnon, J. Dumont, and L. J. Dubé, “Beam shaping using genetically optimized two-dimensional photonic crystals,” J. Opt. Soc. Am. A 29, 2673–2678 (2012).
[CrossRef]

2011 (4)

P. Ramachandran and G. Varoquaux, “Mayavi: 3D visualization of scientific data,” Comput. Sci. Eng. 13, 40–51 (2011).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[CrossRef]

2010 (2)

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179–191 (2010).
[CrossRef]

2009 (2)

H. G. Schwefel and C. G. Poulton, “An improved method for calculating resonances of multiple dielectric disks arbitrarily positioned in the plane,” Opt. Express 17, 13178–13186 (2009).
[CrossRef]

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[CrossRef]

2008 (2)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef]

2007 (2)

C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express 15, 3864–3875 (2007).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

2006 (3)

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B 23, 1565–1573 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

2005 (3)

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

1998 (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

1992 (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

1881 (1)

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73) 81–101 (1881).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Andreasen, J.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Arnold, S.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef]

G. Griffel and S. Arnold, “Synthesis of variable optical filters using meso-optical ring resonator arrays,” in 10th Annual Meeting IEEE Conference Proceedings of the Lasers and Electro-Optics Society, LEOS ‘97 (IEEE, 1997), p. 165.

Asatryan, A. A.

Baba, T.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Bender, C. M.

B. Peng, S. K. Ozdemir, 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]

Benson, T. M.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Boriskina, S. V.

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B 23, 1565–1573 (2006).
[CrossRef]

S. V. Boriskina, Photonic Molecules and Spectral Engineering, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), Chap. 16, pp. 393–421.

Botten, L. C.

Byelobrov, V. O.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

Byrne, M. A.

Cao, H.

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Cerjan, A.

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]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Chong, Y. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

Ctyroky, J.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Donegan, J. F.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179–191 (2010).
[CrossRef]

Dubé, L. J.

D. Gagnon, J. Dumont, and L. J. Dubé, “Multiobjective optimization in integrated photonics design,” Opt. Lett. 38, 2181–2184 (2013).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Beam shaping using genetically optimized two-dimensional photonic crystals,” J. Opt. Soc. Am. A 29, 2673–2678 (2012).
[CrossRef]

G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.

Dumont, J.

D. Gagnon, J. Dumont, and L. J. Dubé, “Multiobjective optimization in integrated photonics design,” Opt. Lett. 38, 2181–2184 (2013).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Beam shaping using genetically optimized two-dimensional photonic crystals,” J. Opt. Soc. Am. A 29, 2673–2678 (2012).
[CrossRef]

G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.

Elsherbeni, A. Z.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

Esterhazy, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Fan, S.

B. Peng, S. K. Ozdemir, 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]

Gagnon, D.

D. Gagnon, J. Dumont, and L. J. Dubé, “Multiobjective optimization in integrated photonics design,” Opt. Lett. 38, 2181–2184 (2013).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Beam shaping using genetically optimized two-dimensional photonic crystals,” J. Opt. Soc. Am. A 29, 2673–2678 (2012).
[CrossRef]

G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.

Ge, L.

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]

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. dissertation (Yale University, 2010).

Geissbuehler, M.

Gianfreda, M.

B. Peng, S. K. Ozdemir, 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]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Gouesbet, G.

Griffel, G.

G. Griffel and S. Arnold, “Synthesis of variable optical filters using meso-optical ring resonator arrays,” in 10th Annual Meeting IEEE Conference Proceedings of the Lasers and Electro-Optics Society, LEOS ‘97 (IEEE, 1997), p. 165.

Harayama, T.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[CrossRef]

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Heiss, W. D.

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

Holzwarth, R.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Ikeda, K. S.

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Imamoglu, A.

A. Imamoglu, Quantum Computation Using Quantum Dot Spins and Microcavities (Wiley, 2005), Chap. 14, pp. 217–227.

Ishii, S.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Johnson, S. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Kim, S. W.

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[CrossRef]

Kippenberg, T. J.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Kishk, A. A.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

Labonté, L.

Lasser, T.

Lee, S. Y.

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[CrossRef]

Lei, F.

B. Peng, S. K. Ozdemir, 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]

Li, Z.

Liertzer, M.

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]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Liu, D.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Lock, J. A.

Long, G. L.

B. Peng, S. K. Ozdemir, 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]

Makris, K. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Marciniak, M.

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

Melenk, J. M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Monifi, F.

B. Peng, S. K. Ozdemir, 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]

Nakagawa, A.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Natarov, D. M.

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Nojima, S.

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

Nori, F.

B. Peng, S. K. Ozdemir, 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]

Nosich, A. I.

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, 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]

Painchaud-April, G.

G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.

Peng, B.

B. Peng, S. K. Ozdemir, 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]

Peng, C.

Poulton, C. G.

Raeis-Zadeh, S. M.

S. M. Raeis-Zadeh and S. Safavi-Naeini, “Multipole-based modal analysis of gate-defined quantum dots in graphene,” Eur. Phys. J. B 86, 1–7 (2013).
[CrossRef]

Rakovich, Y. P.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179–191 (2010).
[CrossRef]

Ramachandran, P.

P. Ramachandran and G. Varoquaux, “Mayavi: 3D visualization of scientific data,” Comput. Sci. Eng. 13, 40–51 (2011).
[CrossRef]

Rayleigh,

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73) 81–101 (1881).
[CrossRef]

Rotter, S.

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]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Ryu, J. W.

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[CrossRef]

Safavi-Naeini, S.

S. M. Raeis-Zadeh and S. Safavi-Naeini, “Multipole-based modal analysis of gate-defined quantum dots in graphene,” Eur. Phys. J. B 86, 1–7 (2013).
[CrossRef]

Sanderson, C.

C. Sanderson, “Armadillo: an open source C++ linear algebra library for fast prototyping and computationally intensive experiments,” Technical Report (NICTA, 2010).

Sauleau, R.

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Schwefel, H. G.

Search, C. P.

Sebbah, P.

Sewell, P.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Shim, J.-B.

Shinohara, S.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Smotrova, E. I.

E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

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]

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

Sunada, S.

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Tandy, R. J.

Türeci, H. E.

Vanneste, C.

Varoquaux, G.

P. Ramachandran and G. Varoquaux, “Mayavi: 3D visualization of scientific data,” Comput. Sci. Eng. 13, 40–51 (2011).
[CrossRef]

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef]

Wang, C.

Wiersig, J.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Xu, A.

Yang, L.

B. Peng, S. K. Ozdemir, 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]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Nakagawa, S. Ishii, and T. Baba, “Photonic molecule laser composed of GaInAsP microdisks,” Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

Comput. Sci. Eng. (1)

P. Ramachandran and G. Varoquaux, “Mayavi: 3D visualization of scientific data,” Comput. Sci. Eng. 13, 40–51 (2011).
[CrossRef]

Eur. Phys. J. B (1)

S. M. Raeis-Zadeh and S. Safavi-Naeini, “Multipole-based modal analysis of gate-defined quantum dots in graphene,” Eur. Phys. J. B 86, 1–7 (2013).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

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

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

J. Appl. Phys. (1)

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

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

Laser Photon. Rev. (2)

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photon. Rev. 4, 179–191 (2010).
[CrossRef]

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[CrossRef]

Nat. Methods (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef]

Nat. Phys. (1)

B. Peng, S. K. Ozdemir, 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]

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Philos. Mag. (1)

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12(73) 81–101 (1881).
[CrossRef]

Phys. Rev. A (2)

J. W. Ryu, S. Y. Lee, and S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858 (2009).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

Phys. Rev. E (1)

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Plasmonics (1)

D. M. Natarov, R. Sauleau, M. Marciniak, and A. I. Nosich, “Effect of periodicity in the resonant scattering of light by finite sparse configurations of many silver nanowires,” Plasmonics 9, 389–407 (2014).
[CrossRef]

Science (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef]

Other (12)

A. Imamoglu, Quantum Computation Using Quantum Dot Spins and Microcavities (Wiley, 2005), Chap. 14, pp. 217–227.

S. V. Boriskina, Photonic Molecules and Spectral Engineering, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), Chap. 16, pp. 393–421.

G. Griffel and S. Arnold, “Synthesis of variable optical filters using meso-optical ring resonator arrays,” in 10th Annual Meeting IEEE Conference Proceedings of the Lasers and Electro-Optics Society, LEOS ‘97 (IEEE, 1997), p. 165.

L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. dissertation (Yale University, 2010).

Since ω=ck, we will refer generically to both quantities as eigenfrequencies.

A. E. Siegman, Lasers (University Science Books, 1986).

A further technical aspect of the implementation should be noted. Although we have not encountered any instabilities in our calculations, it should be acknowledged that for high accuracy work, Eqs. (12) and (13) are not well suited (exponential decay or growth of Tll′nn′ with the indices l and l′). This difficulty has been recognized before [17] and solved generally in [36].

G. Painchaud-April, J. Dumont, D. Gagnon, and L. J. Dubé, “S and Q matrices reloaded: applications to open, inhomogeneous, and complex cavities,” in 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–4.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “A scalable numerical approach for the Steady-State Ab-Initio Laser Theory,” arXiv:1312.2488 (2013).

C. Sanderson, “Armadillo: an open source C++ linear algebra library for fast prototyping and computationally intensive experiments,” Technical Report (NICTA, 2010).

We use the denomination GLMT in accordance with [29] to mean “theories dealing with the interaction between electromagnetic arbitrary shaped beams and a regular particle, allowing one to solve the problem by using the method of separation of variables.” However, as a matter of historical precision, it could be argued that a theory, dealing specifically with the scattering by many cylinders, should be called “generalized Rayleigh theory” in honor of the first calculation of scattering by one single cylinder by Rayleigh [45].

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

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Figures (8)

Fig. 1.
Fig. 1.

Geometry of the diatomic photonic molecule used in this work. The cylinders radii are u1 and u2, with u2=0.8908u1. The center-to-center distance is R12=2.448u1, and the relative permittivity of the cylinders is ϵc=4. This geometry is also used in [14].

Fig. 2.
Fig. 2.

Profile of four TM-polarized QB states of a diatomic photonic molecule composed of two cylinders of different diameters, as shown in Fig. 1. The z-coordinate is proportional to the intensity (arbitrary units).

Fig. 3.
Fig. 3.

(a) Map of log|det[T]| in the complex k plane for each of the four QB states of the photonic molecule. Eigenvalues correspond to the zeros of the function (dark spots) and are located at: M1, kQBu1=5.3830i0.0122; M2, kQBu1=5.3958i0.01756; M3, kQBu1=5.3993i0.0154; M4, kQBu1=5.4078i0.0133. (b)–(c) Map of log|det[T]| in the complex K plane for two different values of the exterior frequency k (purely real). Each QB state can be associated to a unique CF state, which allows the use of the same labels for QB and CF states.

Fig. 4.
Fig. 4.

Profiles of two TM-polarized CF states of a diatomic photonic molecule, which are counterparts to the QB states M1 and M4 shown in Fig. 2. These profiles were computed for the same values of the exterior frequency used in Figs. 3(b) and 3(c). The z-coordinate is proportional to the intensity (arbitrary units).

Fig. 5.
Fig. 5.

Evolution of complex D0 values of the four CF states shown in Fig. 3 for different values of the gain transition central frequency ka. The threshold for each mode is given by D0μ when Im[D0μ]=0; this is indicated by circles on the curves. Note the reversal of M1 and M4 as the first lasing mode when kau1 is increased.

Fig. 6.
Fig. 6.

Evolution of complex D0 values of the four CF states shown in Fig. 3 for different values of the gain transition width γa. The threshold for each mode is given by D0μ when Im[D0μ]=0; this is indicated by circles on the curves.

Fig. 7.
Fig. 7.

Evolution of lasing thresholds of modes M1 and M4 as a function of the gain center frequency ka and gain width γa. The thresholds of modes M2 and M3 are higher for this range of parameters (data not shown).

Fig. 8.
Fig. 8.

Frequency pulling effect in a simple photonic molecule for γau1=5.4×102. (a) Evolution of the lasing frequency of each TLM as a function of the gain center frequency ka. (b) Magnitude of the line-pulling effect.

Equations (17)

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

[2+ϵ(r)k2]φ(r)=0,
[2+ϵ(r)K2(k)]φ(r)=0,rC,
[2+ϵ(r)k2]φ(r)=0,rC,
{2+[ϵ(r)+γaD0F(r)kka+iγa]k2}φ(r)=0,
γaD0kka+iγa=ϵc(K2k21).
φ(r)=n=1Nl=bnlHl(+)(k0ρn)eilθn,
φ(r)=l=cnlJl(knρn)eilθn,
φ(r)=l=[anlJl(k0ρn)+bnlHl(+)(k0ρn)]eilθn.
Hl(+)(k0ρn)eilθn=l=ei(ll)ϕnnHll(+)(k0Rnn)Jl(k0ρn)eilθn,
φ(r)=l=bnlHl(+)(k0ρn)eilθn+l=nnl=bnlei(ll)ϕnnHll(+)(k0Rnn)Jl(k0ρn)eilθn.
anl=nnl=ei(ll)ϕnnHll(+)(k0Rnn)bnl.
det[T(kn,k0)]=0,
Tllnn(kn,k0)=δnnδll(1δnn)ei(ll)ϕnnHll(+)(k0Rnn)snl(kn,k0).
snl(kn,k0)=Jl(k0un)ΓnlJl(k0un)Hl(+)(k0un)ΓnlHl(+)(k0un),
Γnl=ξn0knJl(knun)k0Jl(knun)
T=[T11T12T21T22]
D02εcRe[K]Im[K](kka)2γa2k2.

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