Abstract

We present a theoretical description of on- and off-resonance, 4¯-quasi-phasematched, second-harmonic generation (SHG) in microdisks made of GaAs or other materials possessing 4¯ symmetry, such as GaP or ZnSe. The theory describes the interplay between quasi-phasematching (QPM) and the cavity-resonance conditions. For optimal conversion, all waves should be resonant with the microdisk and should satisfy the 4¯-QPM condition. We explore χ (2) nonlinear mixing if one of the waves is not resonant with the microdisk cavity and calculate the second-harmonic conversion spectrum. We also describe perfectly destructive 4¯-QPM where both the fundamental and second-harmonic are on-resonance with the cavity but SHG is suppressed.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
    [CrossRef]
  2. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
    [CrossRef]
  3. E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO2 laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. 23(19), 1511–1513 (1998).
    [CrossRef] [PubMed]
  4. S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
    [CrossRef]
  5. L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
    [CrossRef]
  6. R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004).
    [CrossRef]
  7. H. Komine, W. H. Long, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998).
    [CrossRef] [PubMed]
  8. C. Simonneau, J. P. Debray, J. C. Harmand, P. Vidakovi, D. J. Lovering, and J. A. Levenson, “Second-harmonic generation in a doubly resonant semiconductor microcavity,” Opt. Lett. 22(23), 1775–1777 (1997).
    [CrossRef] [PubMed]
  9. Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
    [CrossRef]
  10. Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32(7), 826–828 (2007).
    [CrossRef] [PubMed]
  11. P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009).
    [CrossRef] [PubMed]
  12. R. T. Horn and G. Weihs, “Directional Quasi-Phase Matching in Curved Waveguides,” http://arXiv.org/abs/1008.2190v1 .
  13. W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
    [CrossRef]
  14. Z. Yang and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. 32(22), 3296–3298 (2007).
    [CrossRef] [PubMed]
  15. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
    [CrossRef] [PubMed]
  16. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
    [CrossRef] [PubMed]
  17. K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vucković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17(25), 22609–22615 (2009).
    [CrossRef] [PubMed]
  18. A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ((2)) and χ((3)) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15(12), 7303–7318 (2007).
    [CrossRef] [PubMed]
  19. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
    [CrossRef]
  20. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
    [CrossRef]
  21. J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994).
    [CrossRef] [PubMed]
  22. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
    [CrossRef] [PubMed]
  23. Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011).
    [CrossRef]
  24. A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett. 33(18), 2026–2028 (2008).
    [CrossRef] [PubMed]
  25. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005).
    [CrossRef] [PubMed]
  26. C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).
  27. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
    [CrossRef]
  28. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
    [CrossRef]
  29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  30. M. Borselli, Ph.D. thesis (California Institute of Technology, 2006).
  31. T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
    [CrossRef] [PubMed]

2011

Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011).
[CrossRef]

2010

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

2009

2008

2007

2006

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
[CrossRef]

2005

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1530 (2005).
[CrossRef] [PubMed]

2004

R. Haidar, N. Forget, P. Kupecek, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522–1534 (2004).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

2003

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

2002

2001

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

2000

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

1998

1997

1994

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

1988

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
[CrossRef]

1973

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Aitchison, J. S.

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Andronico, A.

Arisholm, G.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Becouarn, L.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Borselli, M.

Brevignon, M.

Bristow, A. D.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
[CrossRef]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Chak, P.

Chang, R. K.

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Ctyroký, J.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Debray, J. P.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Dumeige, Y.

Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011).
[CrossRef]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
[CrossRef]

Ebert, C. B.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Eyres, L. A.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Fang, W.

Favero, I.

Fejer, M. M.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Féron, P.

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
[CrossRef]

Forget, N.

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Gerard, B.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Haidar, R.

Hammer, M.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Harmand, J. C.

Harris, J. S.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Hatami, F.

Hiremath, K. R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Ilchenko, V. S.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

Ito, R.

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

Iyer, R.

Joannopoulos, J. D.

Johnson, S. G.

Johnson, T. J.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Koh, S.

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

Komine, H.

Kondo, T.

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

Kozlovsky, W. J.

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
[CrossRef]

Kuo, P. S.

P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009).
[CrossRef] [PubMed]

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

Kupecek, P.

Lallier, E.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO2 laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. 23(19), 1511–1513 (1998).
[CrossRef] [PubMed]

Lassen, M.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Lehoux, J.

Leo, G.

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Levenson, J. A.

Levi, O.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

Lin, Z.

Long, W. H.

Lovering, D. J.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Maleki, L.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

Marquardt, C.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Masselink, W. T.

Matsko, A. B.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

Nabors, C. D.

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
[CrossRef]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Nockel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Nöckel, J. U.

Painter, O.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Pinguet, T. J.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Prkna, L.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Rivoire, K.

Rodriguez, A.

Rosencher, E.

Savchenkov, A. A.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

Schober, A.

Shiraki, Y.

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

Simonneau, C.

Sipe, J. E.

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Skauli, T.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

Smirl, A. L.

Soljacic, M.

Solomon, G. S.

Stappaerts, E. A.

Stoffer, R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Stone, A. D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994).
[CrossRef] [PubMed]

Strekalov, D. V.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Tourreau, P. J.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Tully, J. W.

van Driel, H. M.

Vidakovi, P.

Vodopyanov, K. L.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

Vuckovic, J.

Yang, Z.

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

Appl. Phys. Lett.

L. A. Eyres, P. J. Tourreau, T. J. Pinguet, C. B. Ebert, J. S. Harris, M. M. Fejer, L. Becouarn, B. Gerard, and E. Lallier, “All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion,” Appl. Phys. Lett. 79(7), 904–906 (2001).
[CrossRef]

Electron. Lett.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[CrossRef]

IEEE J. Quantum Electron.

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24(6), 913–919 (1988).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

J. Appl. Phys.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003).
[CrossRef]

J. Cryst. Growth

S. Koh, T. Kondo, Y. Shiraki, and R. Ito, “GaAs/Ge/GaAs sublattice reversal epitaxy and its application to nonlinear optical devices,” J. Cryst. Growth 227–228(1-4), 183–192 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Z. Yang and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. 32(22), 3296–3298 (2007).
[CrossRef] [PubMed]

A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett. 33(18), 2026–2028 (2008).
[CrossRef] [PubMed]

P. S. Kuo, W. Fang, and G. S. Solomon, “4-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34(22), 3580–3582 (2009).
[CrossRef] [PubMed]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32(7), 826–828 (2007).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19(21), 1693–1695 (1994).
[CrossRef] [PubMed]

C. Simonneau, J. P. Debray, J. C. Harmand, P. Vidakovi, D. J. Lovering, and J. A. Levenson, “Second-harmonic generation in a doubly resonant semiconductor microcavity,” Opt. Lett. 22(23), 1775–1777 (1997).
[CrossRef] [PubMed]

H. Komine, W. H. Long, J. W. Tully, and E. A. Stappaerts, “Quasi-phase-matched second-harmonic generation by use of a total-internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23(9), 661–663 (1998).
[CrossRef] [PubMed]

E. Lallier, M. Brevignon, and J. Lehoux, “Efficient second-harmonic generation of a CO2 laser with a quasi-phase-matched GaAs crystal,” Opt. Lett. 23(19), 1511–1513 (1998).
[CrossRef] [PubMed]

T. Skauli, K. L. Vodopyanov, T. J. Pinguet, A. Schober, O. Levi, L. A. Eyres, M. M. Fejer, J. S. Harris, B. Gerard, L. Becouarn, E. Lallier, and G. Arisholm, “Measurement of the nonlinear coefficient of orientation-patterned GaAs and demonstration of highly efficient second-harmonic generation,” Opt. Lett. 27(8), 628–630 (2002).
[CrossRef] [PubMed]

Opt. Quantum Electron.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1-3), 37–61 (2005).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light eaves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Phys. Rev. A

Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys. Rev. A 83(4), 045802 (2011).
[CrossRef]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
[CrossRef]

Phys. Rev. Lett.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92(4), 043903 (2004).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Science

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280(5369), 1556–1564 (1998).
[CrossRef] [PubMed]

Other

R. T. Horn and G. Weihs, “Directional Quasi-Phase Matching in Curved Waveguides,” http://arXiv.org/abs/1008.2190v1 .

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

M. Borselli, Ph.D. thesis (California Institute of Technology, 2006).

C. R. Pollock, Fundamentals of Optoelectronics (Irwin, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Two views of the zincblende crystal structure with (a) inverted relative to (b). 4 ¯ symmetry means that a 90° rotation combined with a crystal inversion reproduces the original crystal. Equivalently, a 90° rotation about the 4 ¯ axis is the same as a crystal inversion. (c) Waves propagating around a <001>-normal GaAs microdisk effectively experience four 90° rotations and hence, four domain inversions.

Fig. 2
Fig. 2

(a) Microdisk coordinate system. (b) Sketch of coupling between a fiber taper and a disk or ring resonator. ti and κi are the through- and cross-coupling coefficients, respectively; and i represents f or SH.

Fig. 3
Fig. 3

SH conversion efficiency for various pumping wavelengths in a GaAs microdisk (R = 2.609 μm, h = 161nm) that supports doubly resonant SHG at λf = 2λSH = 1998.7 nm (where mf = 13, mSH = 28, Δm = 2). The maximum conversion efficiency is η = 1.2% for 1 mW of incident fundamental power. The top inset shows locations of the fundamental and SH (2λSH ) cavity resonances.

Fig. 4
Fig. 4

SH conversion efficiency for different pumping wavelengths in a GaAs microdisk (R = 2.587 μm, h = 161nm). Near 1990 nm where Δm’ ≈2, the fundamental and SH resonances do not overlap and are separated by λf – 2λSH = −5.4 nm. The maximum SH conversion occurs at a pumping wavelength of 1987.4 nm with η = 1.4 × 10-3% (Pf in = 1 mW). The top inset shows locations of the fundamental and SH (2λSH ) cavity resonances.

Fig. 5
Fig. 5

SH conversion efficiency for various pumping wavelengths in a GaAs microdisk (R = 2.643 μm, h = 161nm). Doubly resonant SHG is achieved at λf = 2λSH = 2186.9 nm where mf = 11, mSH = 20, Δm = −2; and SH conversion is maximized with η = 0.29% (Pf in = 1 mW). The top inset shows locations of the fundamental and SH (2λSH ) cavity resonances.

Fig. 6
Fig. 6

Detailed pump-wavelength dependence of SH conversion efficiency for three different GaAs microdisk sizes (h = 161nm). (a) Double resonance satisfying Δm = 2 is achieved at λf = 2λSH = 1998.7 nm in the 2.609-μm-radius disk, and (b) double resonance satisfying Δm = −2 is achieved at λf = 2λSH = 2186.9 nm in the 2.643-μm-radius disk.

Fig. 7
Fig. 7

(a) Varying disk radius around R 0 = 2.609 μm for fixed disk thickness of 161 nm. When the radius is changed by ± 5 nm, the resonances become detuned by |λf – 2λSH | = 1.3 nm (linewidths of passive cavity resonances are 0.4 nm), which results in a 41-fold reduction in maximum conversion efficiency. (b) Temperature tuning in a R 0 = 2.609 μm, h = 161 nm GaAs microdisk. At T = 10 °C and 50 °C, fundamental and SH resonances are separated by |λf – 2λSH | = 0.5 nm and the peak conversion efficiency is seven times smaller than the peak at T = 30 °C.

Fig. 8
Fig. 8

Overlapping fundamental and SH resonances (λf = 2λSH ) occur in a 2.587-μm-radius, 161-nm-thick GaAs microdisk, but Δm = 4 (mf = 14, mSH = 32), which results in a suppression of SHG due to perfectly destructive 4 ¯ -QPM.

Tables (1)

Tables Icon

Table 1 Comparison between coupled-mode theory (CMT), which utilizes the energy normalization and Eq. (62), and on-resonance generalized waveguide-microresonator theory (GWMT) discussed in Section 2, which utilizes the power normalization and Eq. (63). The incident fundamental power Pf in = 1 mW.

Equations (63)

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

H z f = A f ( θ , t ) Ψ f ( r , z ) e i ( ω f t m f θ ) E z S H = A S H ( θ , t ) Ψ S H ( r , z ) e i ( ω S H t m S H θ ) ,
A S H θ = A f 2 ( K + e i ( Δ m + 2 ) θ + K e i ( Δ m 2 ) θ ) ,
A S H ( 2 π ) A S H ( 0 ) = A f 2 K ˜ ,
K ˜ = 2 π ( K + e i π ( Δ m + 2 ) sinc [ ( Δ m + 2 ) π ] + K e i π ( Δ m 2 ) sinc [ ( Δ m 2 ) π ] ) ,
[ C 1 , i C 2 , i ] = [ t i κ i κ i * t i * ] [ B 1 , i B 2 , i ] .
B 2 , f = α f exp ( i ϕ f ) C 2 , f .
B 2 , S H α S H exp ( i ϕ S H ) C 2 , S H = | C 2 , f | 2 K ˜ .
ϕ i = 2 π m i ' ,
2 π Δ m = ϕ S H 2 ϕ f .
| B 2 , f | 2 = α f 2 ( 1 | t f | 2 ) 1 + α f 2 | t f | 2 2 α f | t f | cos ( ψ f + ϕ f ) | B 1 , f | 2 ,
| C 1 , f | 2 = α f 2 + | t f | 2 2 α f | t f | cos ( ψ f + ϕ f ) 1 + α f 2 | t f | 2 2 α f | t f | cos ( ψ f + ϕ f ) | B 1 , f | 2 ,
t i = | t i | exp ( i ψ i ) .
C 1 , S H = κ S H B 2 , S H C 2 , S H = t S H * B 2 , S H .
| B 2 , S H | 2 = | B 2 , f | 4 | K ˜ | 2 α S H 2 1 + α S H 2 | t S H | 2 2 α S H | t S H | cos ( ψ S H + ϕ S H ) = | B 1 , f | 4 | K ˜ | 2 α S H 2 1 + α S H 2 | t S H | 2 2 α S H | t S H | cos ( ψ S H + ϕ S H ) × ( α f 2 ( 1 | t f | 2 ) 1 + α f 2 | t f | 2 2 α f | t f | cos ( ψ f + ϕ f ) ) 2 .
| C 1 , S H | 2 = | B 2 , S H | 2 ( 1 | t S H | 2 ) .
F i = 2 π 2 Δ ( ψ i + ϕ i ) = π α i | t i | 1 α i | t i | = δ ω F S R Δ ω F W H M .
Q i = ω 0 Δ ω F W H M = F i ω 0 δ ω F S R = π α i | t i | 1 α i | t i | ω 0 δ ω F S R .
1 Q i = 1 Q i 0 + 1 Q i c ,
Q i 0 = π α i 1 α i c λ i δ f i , F S R Q i c = π | t i | 1 | t i | c λ i δ f i , F S R .
P S H o u t = | C 1 , S H | 2 | B 2 , f ( ϕ f ) | 4 × | K ˜ | 2 × | B 2 , S H , p a s s i v e ( ϕ S H ) | 2 .
I S H o u t [ sin ( Δ k L / 2 ) Δ k L / 2 ] 2 [ sin ( N ε / 2 ) sin ( ε / 2 ) ] 2 ,
[ sin ( N ε / 2 ) sin ( ε / 2 ) ] 2 = N 2 sinc 2 ( N ε / 2 ) .
TM = { E z , H r , H θ } TE = { H z , E r , E θ } H r = m r μ 0 ω E z E r = m r ε 0 n 2 ω H z . H θ = 1 i μ 0 ω E z r E θ = i ε 0 n 2 ω H z r
F z exp ( i ω t ) = A ( θ ) ψ ( r ) Z ( z ) exp [ i ( ω t m θ ) ] .
2 Z z 2 + k 0 2 ( n 2 n ¯ 2 ) Z = 0 2 ψ r 2 + 1 r ψ r + k 0 2 n ¯ 2 ψ = l 2 r 2 ψ , 2 A θ 2 2 i m A θ m 2 A = l 2 A
ψ ( r ) = { J m ( k 0 n ¯ q r ) r < R J m ( k 0 n ¯ q R ) exp ( α ˜ ( r R ) ) r > R ,
TM : H θ is continuous TE : E θ is continuous E z r | r < R = E z r | r > R 1 n 2 H z r | r < R = H z r | r > R .
TM: J m ( k 0 n ¯ q R ) [ m R + α ˜ ] = k 0 n ¯ q J m + 1 ( k 0 n ¯ q R ) TE: J m ( k 0 n ¯ q R ) [ m R + α ˜ ( n ¯ n ) 2 ] = k 0 n ¯ q J m + 1 ( k 0 n ¯ q R ) .
Z ˜ q ( z ) = d q Z q ( z ) ψ ˜ p ( r ) = c p ψ p ( r ) .
P c i r c = P θ ^ = 1 2 ( E × H * ) θ ^ d r d z .
TM: P c i r c = 1 2 E z H r * d r d z = | A ( θ ) | 2 m 2 μ 0 ω 1 r | ψ ˜ p ( r ) Z ˜ q ( z ) | 2 d r d z = 1 TE: P c i r c = 1 2 H z * E r d r d z = | A ( θ ) | 2 m 2 ε 0 n 2 ω 1 r | ψ ˜ p ( r ) Z ˜ q ( z ) | 2 d r d z = 1 .
- | Z ˜ q ( z ) | 2 d z = 1 TM: m 2 μ 0 ω 0 1 r | ψ ˜ p ( r ) | 2 d r = 1 TE: m 2 ε 0 n 2 ω 0 1 r | ψ ˜ p ( r ) | 2 d r = 1 .
2 E μ ε 2 E t 2 = 0 ,
2 E μ ε 2 E t 2 = μ 0 2 P N L t 2 .
E z = m ' S H , p ' S H , q ' S H A m ' S H p ' S H q ' S H ( θ ) ψ ˜ m ' S H p ' S H q ' S H ( r ) Z ˜ q ' S H ( z ) e i ( ω S H t m ' S H θ ) + radiation modes .
m ' S H , p ' S H , q ' S H 1 r 2 ( 2 A m ' S H p ' S H q ' S H θ 2 2 i m ' S H A m ' S H p ' S H q ' S H θ ) × ψ ˜ m ' S H p ' S H q ' S H ( r ) Z ˜ q ' S H ( z ) e i ( ω S H t m ' S H θ ) = μ 0 ω S H 2 P z N L .
0 2 π 0 F z , m p q F z , m ' p ' q ' r d r d θ d z δ m m ' δ p p ' δ q q ' ,
0 2 π 0 m ' S H , p ' S H , q ' S H ( 2 i m ' S H A m ' S H θ ) ( ψ ˜ m ' S H p ' S H q ' S H ( r ) ψ ˜ m S H p S H q S H ( r ) r ) × Z ˜ q ' S H ( z ) Z ˜ q S H ( z ) e i ( m S H m ' S H ) θ d r d θ d z e i ω S H t = μ 0 ω S H 2 0 2 π 0 P z N L r ψ ˜ m S H p S H q S H ( r ) Z ˜ q S H ( z ) e i m S H θ d r d θ d z .
m ' S H 0 2 π A m ' S H p ' S H q ' S H θ e i ( m S H m ' S H ) θ d θ 2 π A m S H p ' S H q ' S H θ .
4 π i m S H A m S H p S H q S H θ ( 2 μ 0 ω S H m S H ) e i ω S H t = μ 0 ω S H 2 0 2 π 0 P z N L r ψ ˜ m S H p S H q S H ( r ) Z ˜ q S H ( z ) e i m S H θ d r d θ d z .
P z N L = 2 ε 0 d 14 E x f E y f .
E x = E r cos θ E θ sin θ E y = E r sin θ + E θ cos θ ,
P z N L = 2 ε 0 d 14 [ cos 2 θ E r f E θ f + 1 2 sin 2 θ ( E r f 2 E θ f 2 ) ] .
2 π A m S H p S H q S H θ = d 14 2 ε 0 ω S H ( A m f p f q f n f 2 ) 2 h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z × 0 2 π [ 0 R e i ( Δ m + 2 ) θ r ψ ˜ S H ( m f r ψ ˜ f + ψ ˜ f r ) 2 d r 0 R e i ( Δ m 2 ) θ r ψ ˜ S H ( m f r ψ ˜ f ψ ˜ f r ) 2 d r ] d θ
A m S H p S H q S H θ = A m f p f q f 2 ( K + e i ( Δ m + 2 ) θ + K e i ( Δ m 2 ) θ ) ,
K + p o w = d 14 2 ε 0 ω S H n f 4 h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z 0 R r ψ ˜ S H ( m f r ψ ˜ f + ψ ˜ f r ) 2 d r K p o w = d 14 2 ε 0 ω S H n f 4 h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z 0 R r ψ ˜ S H ( m f r ψ ˜ f ψ ˜ f r ) 2 d r .
h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z d q f 2 d q S H ,
P c i r c = W v g = | A | 2 v g .
v g = r δ ω F S R ,
P c i r c = | A | 2 δ f F S R ,
- | Z ˜ q ( z ) | 2 d z = 1 TM: m 2 μ 0 ω 0 1 r | ψ ˜ p e n ( r ) | 2 d r = δ f F S R . TE: m 2 ε 0 n 2 ω 0 1 r | ψ ˜ p e n ( r ) | 2 d r = δ f F S R
K + e n = d 14 2 ε 0 δ f F S R , S H ω S H n f 4 h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z 0 R r ψ ˜ S H e n ( m f r ψ ˜ f e n + ψ ˜ f e n r ) 2 d r K e n = d 14 2 ε 0 δ f F S R , S H ω S H n f 4 h / 2 h / 2 Z ˜ q S H ( z ) Z ˜ q f 2 ( z ) d z 0 R r ψ ˜ S H e n ( m f r ψ ˜ f e n ψ ˜ f e n r ) 2 d r .
a f t = i ω f a f ( 1 τ f 0 + 1 τ f c ) a f + s f 2 τ f c a S H t = i ω S H a S H ( 1 τ S H 0 + 1 τ S H c ) a S H + s N L .
a i = A i exp ( i ω i t ) .
s N L = a S H τ g .
2 ω i τ g = P g a i n ω i W i ,
P g a i n = 2 α g P c i r c .
τ g = 1 α g v g ,
α g = 1 A S H d A S H r d θ ,
s N L = a f 2 δ ω F S R , S H K ± e n .
P f c i r c = | a f | 2 δ f F S R , f = δ f F S R , f 4 Q f c ω f 1 ( 1 + Q f c / Q c 0 ) 2 P f i n .
P S H o u t = 2 τ S H c | a S H | 2 = 4 Q S H c ω S H ( 1 + Q S H c / Q S H 0 ) 2 ( 4 Q f c ω f ( 1 + Q f c / Q f 0 ) 2 P f i n 2 π δ f F S R , S H | K ± e n | ) 2 .
P S H o u t = ( P f i n ) 2 ( 1 | t S H | 2 ) | 2 π K ± p o w | 2 α S H 2 ( 1 α S H | t S H | ) 2 ( α f 2 ( 1 | t f | 2 ) ( 1 α f | t f | ) 2 ) 2 ,

Metrics