Abstract

We demonstrate that due to strong modal interactions through cross-gain saturation, the onset of a new lasing mode can switch off an existing mode via a negative power slope. In this process of interaction-induced mode switching (IMS) the two involved modes maintain their identities, i.e. they do not change their spatial field patterns or lasing frequencies. For a fixed pump profile, a simple analytic criterion for the occurrence of IMS is given in terms of their self- and cross-interaction coefficients and non-interacting thresholds, which is verified for the example of a two-dimensional microdisk laser. When the spatial pump profile is varied as the pump power is increased, IMS can be induced even when it would not occur with a fixed pump profile, as we show for two coupled laser cavities. Our findings apply to steady-state lasing and are hence different from dynamical mode switching or hopping. IMS may have potential applications in robust and flexible all-optical switching.

© 2016 Optical Society of America

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References

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  1. H. Haken, Light: Laser Dynamics, Vol. II (North-Holland Physics Publishing, 1985).
  2. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).
  3. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
  4. K. J. Vahala, Optical Microcavities (World Scientific, 2004).
  5. 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] [PubMed]
  6. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (2008).
    [Crossref]
  7. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [Crossref]
  8. H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
    [Crossref]
  9. 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]
  10. A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455 (2015).
    [Crossref] [PubMed]
  11. 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 (2008).
    [Crossref] [PubMed]
  12. A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-State ab initio laser theory for N-level lasers,” Opt. Express 20, 474 (2012).
    [Crossref] [PubMed]
  13. 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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
    [Crossref]
  14. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
    [Crossref] [PubMed]
  15. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge Univeristy, 1997)
    [Crossref]
  16. H. Kawaguchi, “Semiconductor lasers and optical amplifiers for switching and signal processing,” in Handbook of Laser Technology and Applications: Laser Design and Laser Systems, Vol. II, Colin E. Webb and Julian D. C. Jones, eds. (CRC, 2004).
    [Crossref]
  17. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429 (1964).
    [Crossref]
  18. G. P. Agrawal and N. K. Dutta, “Optical bistability in coupled-cavity semiconductor lasers,” J. App. Phys. 56, 664 (1984).
    [Crossref]
  19. S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” App. Phys. Lett. 87, 181102 (2005).
    [Crossref]
  20. S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
    [Crossref]
  21. L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (2014).
    [Crossref]
  22. H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446 (1991).
    [Crossref] [PubMed]
  23. S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for lasers with fully or nearly degenerate resonator modes,” Phys. Rev. A 92, 013847 (2015).
    [Crossref]
  24. H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. 173, 261 (1963).
    [Crossref]
  25. L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
    [Crossref] [PubMed]
  26. P. Mandel, “Global rate equation description of a laser,” Eur. Phys. J. D 8, 431 (2000).
    [Crossref]
  27. I. V. Koryukin and P. Mandel, “Two-mode threshold of a solid-state Fabry-Perot laser,” J. Opt. B: Quant. Semiclass. Opt. 4, 27 (2002).
    [Crossref]
  28. R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
    [Crossref]
  29. The same behavior happens to I2 in Fig. 4(d) when different initial values of D2 are chosen.
  30. L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.
  31. S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
    [Crossref] [PubMed]
  32. M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
    [Crossref] [PubMed]
  33. S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
    [Crossref]
  34. S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
    [Crossref]

2015 (3)

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

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455 (2015).
[Crossref] [PubMed]

2014 (5)

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

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (2014).
[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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

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

2012 (2)

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] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-State ab initio laser theory for N-level lasers,” Opt. Express 20, 474 (2012).
[Crossref] [PubMed]

2010 (1)

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]

2009 (1)

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[Crossref]

2008 (2)

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (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 (2008).
[Crossref] [PubMed]

2007 (2)

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

2006 (1)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

2005 (2)

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[Crossref] [PubMed]

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” App. Phys. Lett. 87, 181102 (2005).
[Crossref]

2004 (1)

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

2002 (1)

I. V. Koryukin and P. Mandel, “Two-mode threshold of a solid-state Fabry-Perot laser,” J. Opt. B: Quant. Semiclass. Opt. 4, 27 (2002).
[Crossref]

2000 (1)

P. Mandel, “Global rate equation description of a laser,” Eur. Phys. J. D 8, 431 (2000).
[Crossref]

1991 (1)

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446 (1991).
[Crossref] [PubMed]

1984 (1)

G. P. Agrawal and N. K. Dutta, “Optical bistability in coupled-cavity semiconductor lasers,” J. App. Phys. 56, 664 (1984).
[Crossref]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429 (1964).
[Crossref]

1963 (1)

H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. 173, 261 (1963).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, “Optical bistability in coupled-cavity semiconductor lasers,” J. App. Phys. 56, 664 (1984).
[Crossref]

Baba, T.

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” App. Phys. Lett. 87, 181102 (2005).
[Crossref]

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Brandstetter, M.

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

Burkhardt, S.

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

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

Cao, H.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

Cerjan, A.

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455 (2015).
[Crossref] [PubMed]

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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-State ab initio laser theory for N-level lasers,” Opt. Express 20, 474 (2012).
[Crossref] [PubMed]

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] [PubMed]

Chang, R. K.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

Chigrin, D. N.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[Crossref]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

Chong, Y. D.

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

de Vries, T.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Deutsch, C.

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

Deych, L. I.

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[Crossref] [PubMed]

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, “Optical bistability in coupled-cavity semiconductor lasers,” J. App. Phys. 56, 664 (1984).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Fu, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446 (1991).
[Crossref] [PubMed]

Ge, L.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (2014).
[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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (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]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (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] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-State ab initio laser theory for N-level lasers,” Opt. Express 20, 474 (2012).
[Crossref] [PubMed]

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 (2008).
[Crossref] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (2008).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.

Haken, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446 (1991).
[Crossref] [PubMed]

H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. 173, 261 (1963).
[Crossref]

H. Haken, Light: Laser Dynamics, Vol. II (North-Holland Physics Publishing, 1985).

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Ishii, S.

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” App. Phys. Lett. 87, 181102 (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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Kawaguchi, H.

H. Kawaguchi, “Semiconductor lasers and optical amplifiers for switching and signal processing,” in Handbook of Laser Technology and Applications: Laser Design and Laser Systems, Vol. II, Colin E. Webb and Julian D. C. Jones, eds. (CRC, 2004).
[Crossref]

Khajavikhan, M.

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

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Klang, P.

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

Koryukin, I. V.

I. V. Koryukin and P. Mandel, “Two-mode threshold of a solid-state Fabry-Perot laser,” J. Opt. B: Quant. Semiclass. Opt. 4, 27 (2002).
[Crossref]

Krimer, D. O.

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

Kroha, J.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[Crossref]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429 (1964).
[Crossref]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Lavrinenko, A. V.

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Liertzer, M.

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

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

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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (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] [PubMed]

Liew, S. F.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Malik, O.

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (2014).
[Crossref]

Mandel, P.

I. V. Koryukin and P. Mandel, “Two-mode threshold of a solid-state Fabry-Perot laser,” J. Opt. B: Quant. Semiclass. Opt. 4, 27 (2002).
[Crossref]

P. Mandel, “Global rate equation description of a laser,” Eur. Phys. J. D 8, 431 (2000).
[Crossref]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge Univeristy, 1997)
[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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Nersisyan, A.

L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.

Oei, Y.-S.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Oztop, B.

L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.

Redding, B.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

Rotter, S.

S. Burkhardt, M. Liertzer, D. O. Krimer, and S. Rotter, “Steady-state ab-initio laser theory for lasers with fully or nearly degenerate resonator modes,” Phys. Rev. A 92, 013847 (2015).
[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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

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

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] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (2008).
[Crossref]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Sauermann, H.

H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. 173, 261 (1963).
[Crossref]

Schöberl, J.

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

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Smit, M. K.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Solomon, G. S.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

Stone, A. D.

A. Cerjan, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory for complex gain media,” Opt. Express 23, 6455 (2015).
[Crossref] [PubMed]

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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-State ab initio laser theory for N-level lasers,” Opt. Express 20, 474 (2012).
[Crossref] [PubMed]

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] [PubMed]

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]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (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 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Strasser, G.

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

Tandy, R. J.

Tureci, H. E.

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

Türeci, H. E.

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (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] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (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 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.

Unterrainer, K.

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

Vahala, K. J.

K. J. Vahala, Optical Microcavities (World Scientific, 2004).

Zhukovsky, S. V.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[Crossref]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

App. Phys. Lett. (2)

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” App. Phys. Lett. 87, 181102 (2005).
[Crossref]

S. F. Liew, B. Redding, L. Ge, G. S. Solomon, and H. Cao, “Active control of emission directionality of semiconductor microdisk lasers,” App. Phys. Lett. 104, 231108 (2014).
[Crossref]

Eur. Phys. J. D (1)

P. Mandel, “Global rate equation description of a laser,” Eur. Phys. J. D 8, 431 (2000).
[Crossref]

J. App. Phys. (1)

G. P. Agrawal and N. K. Dutta, “Optical bistability in coupled-cavity semiconductor lasers,” J. App. Phys. 56, 664 (1984).
[Crossref]

J. Opt. B: Quant. Semiclass. Opt. (1)

I. V. Koryukin and P. Mandel, “Two-mode threshold of a solid-state Fabry-Perot laser,” J. Opt. B: Quant. Semiclass. Opt. 4, 27 (2002).
[Crossref]

Nat. Commun. (1)

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

Nat. Photon. (1)

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photon. 8, 871 (2014).
[Crossref]

Nature (1)

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. 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 microring lasers,” Nature 432, 206 (2004).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Rev. (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429 (1964).
[Crossref]

Phys. Rev. A (9)

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, “Scalable numerical approach for the steady-state ab-initiolaser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446 (1991).
[Crossref] [PubMed]

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

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[Crossref]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[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]

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

Phys. Rev. Lett. (3)

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902(2007).
[Crossref] [PubMed]

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] [PubMed]

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[Crossref] [PubMed]

Science (1)

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Storng interactions in multimode random lasers,” Science 320, 643 (2008).
[Crossref]

Z. Phys. (1)

H. Haken and H. Sauermann, “Nonlinear interaction of laser modes,” Z. Phys. 173, 261 (1963).
[Crossref]

Other (8)

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge Univeristy, 1997)
[Crossref]

H. Kawaguchi, “Semiconductor lasers and optical amplifiers for switching and signal processing,” in Handbook of Laser Technology and Applications: Laser Design and Laser Systems, Vol. II, Colin E. Webb and Julian D. C. Jones, eds. (CRC, 2004).
[Crossref]

The same behavior happens to I2 in Fig. 4(d) when different initial values of D2 are chosen.

L. Ge, A. Nersisyan, B. Oztop, and H. E. Türeci, “Pattern formation and strong nonlinear interactions in exciton-polariton condensates,” arXiv:1311.4847.

H. Haken, Light: Laser Dynamics, Vol. II (North-Holland Physics Publishing, 1985).

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

K. J. Vahala, Optical Microcavities (World Scientific, 2004).

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Figures (5)

Fig. 1
Fig. 1

Schematics showing modal intensities as a function of the pump strength for (a) interaction-induced mode switching and (c) typical two-mode lasing. (b,d) The corresponding lasing spectra at pump strengths marked by I and II in (a,b).

Fig. 2
Fig. 2

Interaction-induced mode switching in a microdisk laser. Lower panel: Black solid and red dashed lines show the modal intensities of the first two modes. They are whispering-gallery modes with azimuthal quantum numbers m1 = 8, m2 = 7 and lasing frequencies ω1R/c = 5.37, ω2R/c = 4.81, respectively. Upper panel: False color plots of the real part of the electric field in these two modes, where red and blue indicate positive and negative values. Their radial profiles are shown as the inset in the lower panel. The parameters used are: atomic transition frequency ωaR/c = 4.83, longitudinal relaxation rate γR/c = 1 and refractive index n = 2 + 0.01i.

Fig. 3
Fig. 3

(a,c) Same as the modal intensity plot shown in Fig. 2 but with ωaR/c = 0.50 (a) and ωaR/c = 0.48 (c). Single-mode lasing is observed and mode switching does not occur in either case, even when the pump power is high above threshold. (b,d) The modal gain of these two modes in (a,c), confirming that the m2 = 7 mode in (a) and the m1 = 8 mode in (c) do not lase. Modal gain is clamped at 1 for lasing modes and stays below 1 for non-lasing modes [9].

Fig. 4
Fig. 4

Interaction-induced mode switching facilitated by an evolving pumping in two coupled 1D cavities of index n = 3. Their lengths are L1 = 4.2µm, L2 = 5.0µm and the gap width between them is W = 0.8µm. L ≡ L1 + L2 +W = 10µm. (a) Intensity profiles of two modes with wavelength λ1 = 4.15µm, λ2 = 4.28µm. Shaded areas show the two cavities. (b) Their non-interacting thresholds (solid and dashed lines) versus the pump ratio D2/D1. The average pump strength D0 along the pump trajectory used in (c) is shown as the dash-dotted line, and the arrow indicates the direction of increasing D1. The triangle and diamond on it mark the onset of mode 2 and the termination of mode 1. The gain curve is centered at ωaL/c = 15 (λa = 4.19µm) and its width is γL/c = 1. (c) IMS of mode 1 when the pump strength in the right cavity is increased while the pump strength in the left cavity is fixed at D1 = 0.318. Inset: Mode switching does not occur for uniform pumping (D1 = D2). (d) IMS of mode 2 when the pump strength in the left cavity is increased while the pump strength in the right cavity is fixed at D2 = 0.344. Good agreement between SALT (lines) and FDTD simulations (triangles) are shown. γ/γ = 5 × 10−4 is used in FDTD simulations.

Fig. 5
Fig. 5

Invariant shape of the IMS triangle. (a) Same as Fig. 4(c) but with different values of the fixed pump strength D1. Modal intensity of mode 2 is not shown for clarity. Solid lines show the full SALT results and dash-dotted lines show the SPA-SALT approximations [Eq. (13)]. (b) Non-interacting thresholds in terms of D2 when D1 is varied.

Equations (16)

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

cavity d r u ¯ p * ( r ) u q ( r ) = V δ p q
u p ( r , ϕ ) J m ( n Ω p r c ) e i m ϕ , r < R .
u ¯ p ( r , ϕ ) J m ( n * Ω p * r c ) e i m ϕ , r < R .
M ( I 1 I 2 ) = ( D 0 D 0 ( 1 ) 1 D 0 D 0 ( 2 ) 1 ) , M ( Γ 1 χ 11 Γ 2 χ 12 Γ 1 χ 21 Γ 2 χ 22 ) ,
χ p q = 1 V | cavity d r u ¯ p * ( r ) u p ( r ) | u p ( r ) | 2 | .
χ 12 χ 21 cavity d r | J m 1 ( n Ω 1 r c ) J m 2 ( n Ω 2 r c ) | 2
S ˜ 1 = χ 22 χ 12 D 0 ( 1 ) D 0 ( 2 ) χ 22 χ 12 χ 21 χ 11 S 1
D 0 , int ( 2 ) = 1 χ 21 χ 11 D 0 ( 1 ) D 0 ( 2 ) χ 21 χ 11 D 0 ( 1 ) .
χ 21 χ 11 < D 0 ( 1 ) D 0 ( 2 ) < 1
χ 21 χ 11 < χ 22 χ 12 < D 0 ( 1 ) D 0 ( 2 ) .
D 0 , off ( 1 ) = 1 χ 22 χ 12 D 0 ( 1 ) D 0 ( 2 ) χ 22 χ 12 D 0 ( 1 ) .
χ 22 < χ 21 χ 12 < χ 11 .
M ( I 1 I 2 ) = P ( D 1 D 2 ) ( 1 1 ) , P ( W 11 D 0 ( 1 ) W 12 D 0 ( 1 ) W 21 D 0 ( 2 ) W 22 D 0 ( 2 ) ) ,
W μ v = 1 V | cavity d x u μ ( x ) u ¯ μ * ( x ) η v ( x ) |
S ˜ 1 = χ 22 χ 12 D 0 ( 1 ) W 22 D 0 ( 2 ) W 12 χ 22 χ 12 χ 21 χ 11 S 1 ,
W 22 W 12 > D 0 ( 2 ) χ 22 D 0 ( 1 ) χ 12 .

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