Abstract

We present a comprehensive study of second-order nonlinear difference frequency generation in triply resonant cavities using a theoretical framework based on coupled-mode theory. We show that optimal “quantum-limited” conversion efficiency can be achieved at any pump power when the powers at the pump and idler frequencies satisfy a critical relationship. We demonstrate the existence of a broad parameter range in which all triply-resonant DFG processes exhibit monostable conversion. We also demonstrate the existence of a geometry-dependent bistable region.

© 2009 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Boyd, Nonlinear Optics, (Academic Press, CA, 1992).
  2. H. M. Gibbs, G. Khitrova, and N. Peyghambarian, Nonlinear Photonics, (Springer-Verlag, 1990).
  3. R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
    [CrossRef]
  4. J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
    [CrossRef]
  5. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
    [CrossRef]
  6. 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]
  7. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-Phase Matched Optical Parametric Oscillators in Bulk Periodically Poled LiNbO3,” J. Opt. Soc. Am. B 12(11), 2102–2116 (1995).
    [CrossRef]
  8. M. Soljaci? and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
    [CrossRef] [PubMed]
  9. M. Bieler, “THz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14(2), 458–469 (2008).
    [CrossRef]
  10. A. Andronico, J. Claudon, J. M. Gérard, V. Berger, and G. Leo, “Integrated terahertz source based on three-wave mixing of whispering-gallery modes,” Opt. Lett. 33(21), 2416–2418 (2008).
    [CrossRef] [PubMed]
  11. R. E. Hamam, M. Ibanescu, E. J. Reed, P. Bermel, S. G. Johnson, E. Ippen, J. D. Joannopoulos, and M. Soljacic, “Purcell effect in nonlinear photonic structures: a coupled mode theory analysis,” Opt. Express 16(17), 12523–12537 (2008).
    [CrossRef] [PubMed]
  12. M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
    [CrossRef]
  13. L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,” J. Opt. Soc. Am. B 4(10), 1465–1476 (1987).
    [CrossRef]
  14. Z. Y. Ou and H. J. Kimble, “Enhanced conversion efficiency for harmonic generation with double resonance,” Opt. Lett. 18(13), 1053–1055 (1993).
    [CrossRef] [PubMed]
  15. R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
    [CrossRef]
  16. V. Berger, “Second harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14(6), 1351–1360 (1997).
    [CrossRef]
  17. A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
    [CrossRef]
  18. B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22(7), 1378–1383 (2005).
    [CrossRef]
  19. M. Liscidini and L. Claudio Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016613 (2006).
    [CrossRef] [PubMed]
  20. Y. Dumeige and P. Feron, “Whispering-gallery mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74(6), 063804 (2006).
    [CrossRef]
  21. A. Rodriguez, M. Soljacic, J. Joannopoulos, and S. G. Johnson, “?(2) and ?(3) harmonic generation at a critical power in homogeneous doubly resonant microcavities,” Opt. Express 15(12), 7303–7318 (2007).
    [CrossRef] [PubMed]
  22. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
    [CrossRef]
  23. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15(24), 16161–16176 (2007).
    [CrossRef] [PubMed]
  24. H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013812 (2009).
    [CrossRef]
  25. Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
    [CrossRef]
  26. Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501 (1999).
    [CrossRef]
  27. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
    [CrossRef]
  28. S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-4 and 3-5 semiconductors,” Phys. Rev. 171(3), 891–898 (1968).
    [CrossRef]
  29. S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear ?((2)) photonic crystals,” Opt. Lett. 25(16), 1204–1206 (2000).
    [CrossRef]
  30. K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
    [CrossRef]
  31. G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
    [CrossRef] [PubMed]
  32. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “N. loembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
    [CrossRef]
  33. F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
    [CrossRef]
  34. S. Schiller, Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators. PhD thesis, Stanford University, Stanford, CA (1993).
  35. E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
    [CrossRef]
  36. A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Self-pulsing and bistability in nonlinear Bragg gratings,” J. Opt. Soc. Am. B 24(9), 2229–2237 (2007).
    [CrossRef]
  37. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
    [CrossRef] [PubMed]
  38. J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
  39. H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  40. Note that there is a typographical error in the corresponding equation for the degenerate case in [21], where ? is written instead of ?0 in the numerator.
  41. S. Singh, “Nonlinear Optical Materials” in M.J. Weber Ed., Handbook of laser science and technology, Vol. III: Optical Materials, Part I, CRC Press 1986.
  42. I. B. Burgess, M. W. McCutcheon, Y. Zhang, A. W. Rodriguez, J. Bravo-Abad, S. G. Johnson, and M. Loncar, “Efficient terahertz generation in triply resonant nonlinear photonic crystal microcavities” manuscript in preparation (2009).
  43. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).
  44. M. W. McCutcheon, D. E. Chang, Y. Zhang, M. D. Lukin, and M. Loncar, “Broad-band spectral control of single photon sources using a nonlinear photonic crystal cavity” arXiv:0903.4706 (2009).
  45. M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, New York, 1989).
  46. A. Hurwitz, “On the Conditions Under Which an Equation Has Only Roots with Negative Real Parts,” Mathematicsche Annalen 46, 273–284 (1895). Also in selected papers on Mathematical Trends in Control Theory, Dover, New York, 70–82 (1964).

2009 (2)

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013812 (2009).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

2008 (4)

M. Bieler, “THz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14(2), 458–469 (2008).
[CrossRef]

A. Andronico, J. Claudon, J. M. Gérard, V. Berger, and G. Leo, “Integrated terahertz source based on three-wave mixing of whispering-gallery modes,” Opt. Lett. 33(21), 2416–2418 (2008).
[CrossRef] [PubMed]

R. E. Hamam, M. Ibanescu, E. J. Reed, P. Bermel, S. G. Johnson, E. Ippen, J. D. Joannopoulos, and M. Soljacic, “Purcell effect in nonlinear photonic structures: a coupled mode theory analysis,” Opt. Express 16(17), 12523–12537 (2008).
[CrossRef] [PubMed]

A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
[CrossRef]

2007 (4)

2006 (4)

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

M. Liscidini and L. Claudio Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016613 (2006).
[CrossRef] [PubMed]

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

2005 (3)

2004 (2)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[CrossRef]

M. Soljaci? and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501 (1999).
[CrossRef]

1997 (1)

1995 (1)

1994 (1)

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

1993 (1)

1992 (1)

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]

1987 (1)

1982 (1)

E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
[CrossRef]

1979 (1)

R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
[CrossRef]

1976 (1)

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[CrossRef]

1968 (1)

S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-4 and 3-5 semiconductors,” Phys. Rev. 171(3), 891–898 (1968).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[CrossRef]

1965 (1)

J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[CrossRef]

1962 (1)

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

Abraham, E.

E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
[CrossRef]

Aleshkin, V. Y.

Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
[CrossRef]

Andronico, A.

Angelis, C. D.

Armstrong, J. A.

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

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[CrossRef]

Avetisyan, Y. H.

Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501 (1999).
[CrossRef]

Baets, R.

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
[CrossRef]

Bellanca, G.

Berger, V.

Bermel, P.

Bieler, M.

M. Bieler, “THz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14(2), 458–469 (2008).
[CrossRef]

Bienstman, P.

Bliss, D.

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Bloembergen, N.

S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-4 and 3-5 semiconductors,” Phys. Rev. 171(3), 891–898 (1968).
[CrossRef]

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

Bosenberg, W. R.

Boyd, G. D.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[CrossRef]

Bravo-Abad, J.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-Phase Matched Optical Parametric Oscillators in Bulk Periodically Poled LiNbO3,” J. Opt. Soc. Am. B 12(11), 2102–2116 (1995).
[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]

R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
[CrossRef]

Carr, J.

E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
[CrossRef]

Claudio Andreani, L.

M. Liscidini and L. Claudio Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016613 (2006).
[CrossRef] [PubMed]

Claudon, J.

Conforti, M.

Dalacu, D.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Deotare, P. B.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Ducuing, J.

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

Dumeige, Y.

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

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[CrossRef]

Eckardt, R. C.

Fan, S.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[CrossRef]

Fejer, M. M.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-Phase Matched Optical Parametric Oscillators in Bulk Periodically Poled LiNbO3,” J. Opt. Soc. Am. B 12(11), 2102–2116 (1995).
[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]

Felber, F. S.

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[CrossRef]

Fermann, M. E.

Feron, P.

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

Fiedler, K.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

Firth, W. J.

E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
[CrossRef]

Frank, I. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Frederick, S.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Gérard, J. M.

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[CrossRef]

Hamam, R. E.

Harris, J. S.

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Hashemi, H.

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013812 (2009).
[CrossRef]

Hurlbut, W. C.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Ibanescu, M.

Imeshev, G.

Ippen, E.

Jha, S. S.

S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-4 and 3-5 semiconductors,” Phys. Rev. 171(3), 891–898 (1968).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

Johnson, S. G.

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]

Khan, M.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Kimble, H. J.

Kira, G.

Kivshar, Y. S.

Kozlov, V. G.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Krasnikova, I. V.

Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
[CrossRef]

Kuramochi, E.

Kurz, P.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

Lee, Y.-S.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Leo, G.

Liscidini, M.

M. Liscidini and L. Claudio Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016613 (2006).
[CrossRef] [PubMed]

Locatelli, A.

Loncar, M.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Lynch, C.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

Maes, B.

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]

Marburger, J. H.

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[CrossRef]

Matsko, A. B.

A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
[CrossRef]

McCutcheon, M. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[CrossRef]

Mitsugi, S.

Mlynek, J.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

Morozov, Y. A.

Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
[CrossRef]

Myers, L. E.

Nefedov, I. S.

Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
[CrossRef]

Notomi, M.

Ou, Z. Y.

Parini, A.

Paschotta, R.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

Pershan, P. S.

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

Pierce, J. W.

Poole, P. J.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Reed, E. J.

Rieger, G. W.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Rodriguez, A.

Rodriguez, A. W.

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013812 (2009).
[CrossRef]

Saltiel, S.

Shinya, A.

Soljacic, M.

Strekalov, D. V.

A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
[CrossRef]

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[CrossRef]

Tanabe, T.

Trillo, S.

Vodopyanov, K. L.

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[CrossRef]

Williams, R. L.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Wu, L.-A.

Xiao, M.

Young, J. F.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Yu, N.

A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
[CrossRef]

Yu, X.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

G. Imeshev, M. E. Fermann, K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, D. Bliss, and C. Lynch, “High-power source of THz radiation based on orientation-patterned GaAs pumped by a fiber laser,” Opt. Express 14(10), 4439–4444 (2006).
[CrossRef] [PubMed]

Appl. Phys. B (1)

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, “Nonlinear mode coupling in doubly resonant frequency doublers,” Appl. Phys. B 58, 117–122 (1994).
[CrossRef]

Appl. Phys. Lett. (3)

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High Quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89(14), 141119 (2006).
[CrossRef]

F. S. Felber and J. H. Marburger, “Theory of nonresonant multistable optical devices,” Appl. Phys. Lett. 28(12), 731–733 (1976).
[CrossRef]

IEEE J. Quantum Electron. (4)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical Parametric Amplification,” IEEE J. Quantum Electron. 15, 432–444 (1979).
[CrossRef]

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant Optical Second Harmonic Generation and Mixing,” IEEE J. Quantum Electron. 2(6), 109–124 (1966).
[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]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Bieler, “THz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14(2), 458–469 (2008).
[CrossRef]

J. Opt. Soc. Am. B (5)

Nat. Mater. (1)

M. Soljaci? and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (3)

Phys. Lett. A (1)

E. Abraham, W. J. Firth, and J. Carr, “Self-oscillation and chaos in nonlinear Fabry-Perot resonators with finite response time,” Phys. Lett. A 91(2), 47–51 (1982).
[CrossRef]

Phys. Rev. (2)

S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-4 and 3-5 semiconductors,” Phys. Rev. 171(3), 891–898 (1968).
[CrossRef]

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

Phys. Rev. A (3)

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant Kerr cavities,” Phys. Rev. A 79(1), 013812 (2009).
[CrossRef]

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

A. B. Matsko, D. V. Strekalov, and N. Yu, “Sensitivity of terahertz photonic receivers,” Phys. Rev. A 77(4), 043812 (2008).
[CrossRef]

Phys. Rev. B (1)

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frederick, P. J. Poole, and R. L. Williams, “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76(24), 245104 (2007).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Liscidini and L. Claudio Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016613 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. 14(24), 973–976 (1965).
[CrossRef]

Proc. SPIE (1)

Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501 (1999).
[CrossRef]

Semiconductors (1)

Y. A. Morozov, I. S. Nefedov, V. Y. Aleshkin, and I. V. Krasnikova, “Terahertz Oscillator Based on Nonlinear Frequency Conversion in a Double Vertical Cavity,” Semiconductors 39(1), 113 (2005).
[CrossRef]

Other (12)

S. Schiller, Principles and Applications of Optical Monolithic Total-Internal-Reflection Resonators. PhD thesis, Stanford University, Stanford, CA (1993).

R. W. Boyd, Nonlinear Optics, (Academic Press, CA, 1992).

H. M. Gibbs, G. Khitrova, and N. Peyghambarian, Nonlinear Photonics, (Springer-Verlag, 1990).

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

Note that there is a typographical error in the corresponding equation for the degenerate case in [21], where ? is written instead of ?0 in the numerator.

S. Singh, “Nonlinear Optical Materials” in M.J. Weber Ed., Handbook of laser science and technology, Vol. III: Optical Materials, Part I, CRC Press 1986.

I. B. Burgess, M. W. McCutcheon, Y. Zhang, A. W. Rodriguez, J. Bravo-Abad, S. G. Johnson, and M. Loncar, “Efficient terahertz generation in triply resonant nonlinear photonic crystal microcavities” manuscript in preparation (2009).

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (Wiley Interscience, 2007).

M. W. McCutcheon, D. E. Chang, Y. Zhang, M. D. Lukin, and M. Loncar, “Broad-band spectral control of single photon sources using a nonlinear photonic crystal cavity” arXiv:0903.4706 (2009).

M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, New York, 1989).

A. Hurwitz, “On the Conditions Under Which an Equation Has Only Roots with Negative Real Parts,” Mathematicsche Annalen 46, 273–284 (1895). Also in selected papers on Mathematical Trends in Control Theory, Dover, New York, 70–82 (1964).

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 (3)

Fig. 1
Fig. 1

- Schematic of the CMT framework of triply-resonant DFG. Three resonant modes, ak (t), are coupled through a second order nonlinear interaction (satisfying ωT = ω 1 – ω 2). All three modes leak energy at a rate, γk , into the outgoing waves, sk -. The monochromatic input waves which drive the pump (ω 1) and signal (ω 2) modes are represented by the amplitudes, sk +. The coupling strength is described by the constant, β.

Fig. 2
Fig. 2

– Maximum (a) and minimum (b) normalized efficiency parameter (Eff ) for stable CW DFG (Eff = 1 corresponds to quantum-limited conversion), plotted as a function of normalized powers of the pump (P 1) and idler (P 2) input waves. The solid line denotes the critical relationship between P 1 and P 2 where Eff = 1 is possible (P 2 = (1 – P 1/4)2). The dotted line (shown in (a), (b) and (c)) denotes the onset of multi-stability (saddle-node bifurcation). To the left of the dotted line (in all three plots), there is only one steady state solution to the coupled wave equations, which is stable for all geometries (any r 2 and rT ). To the right of the dotted line, there are three steady-state solutions and bistable behavior is observed for small values of r 2 and rT . The two stable solutions are shown in (a) and (b) (the unstable solution is not shown). Stability in this region was assessed using r 2 = r T = 0.4 (typical values for a near-degenerate coupled cavity system where all the modes have similar Q-factors). The stability of the solutions in the multi-stable region depends on the parameters, r 2 and r T (see Fig. 3 for example). (c) Stable conversion efficiency reached after a step-excitation (Uk (T = 0) = 0, Pk (T ≥ 0) = constant, Pk (T < 0) = 0) for r 2 = r T = 0.4. To the left of the dashed line, this solution is stable in all geometries (any r 2, r T).

Fig. 3
Fig. 3

Plots showing the effect of geometry {r 2, r T} on the conversion efficiency and stability of steady-state solutions for P 2 = (1 - P 1/4)2, with (a) r 2 = r T = 0.4 and (b) r 2 = r T = 2. The onset of multiple solutions occurs at P 1 ≈2.55. Solid lines indicate stable conversion efficiencies; dashed lines indicate unstable conversion. The black line indicates the solution that is approached after a step excitation (Uk (T = 0) = 0, Pk (T ≥ 0) = constant, Pk (T < 0) = 0) and red lines are used to denote all other solutions. While the conversion efficiency of all solutions is geometry-independent, note that the stability of quantum-limited efficiency changes with geometry (r 2, r T). In (a) the high-efficiency solution is stable everywhere in the plot, but is unstable for P 1 > ~10 in (b) when r 2 and rT are increased from 0.4 to 2. For sufficiently large r 2 and r T, the system approaches mono-stability for all P 1 (all solutions denoted in red become unstable).

Equations (27)

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

sk=sk++2γk,sak
da1dt=γ1a1iω1β1a2aT+2γ1Γ1s1+
da2dt=γ2a2iω2β2a1aT*+2γ2Γ2s2+
daTdt=γTaTiωTβTa1a2*
ββ1=β2*=βT*=14d3xi,j,kε0χijk(2)E1,i*(E2,jET,k+ET,jE2,k)d3xε|E1|2d3xε|E2|2d3xε|ET|2
|sT|2|s1+|2ωTω1Γ1ΓT
|s2+|2=ω2|14ω1|β|2Q1Q2QTΓ1|s1+|2|216|β|2Q1Q2QTΓ2
Tγ1t=ω1t2Q1
Ak2Q1Q2QTQk[β*+δk,T(iββ*)]ak
Sk4Q1Q2QTΓkωkβ*sk+
Effω1Γ1ΓTωT|sT|2|s1+|2=4UTP1
dA1dT=A1+A2AT+S1
dA2dT=r2[A2A1AT*+S2]
dATdT=rT[ATA1A2*]
A1=S1(1+U2)
A2=S2(1U1)
AT=A1A2
Eff=4U2(1+U2)2
U2=1,U1=UT=P14,P2=(1P14)2
J=[1ATA2r2ATr2r2A1rTA2rTA1rT]
B=1+r2+rT
C=r2rT(1U1)+r2(1+14EffP1)+rT(1+U2)
D=r2rT(1U1+U2+34EffP1)
{Hi,1}=[1B1B(BCD)D]
BCD=r2rT[(r2+rT)(1U1)+2(1EffP14)]+(1+EffP14)(1+r2)r2+(1+U2)(1+rT)rT
U12Q2<<1,U22Q1<<1
P1<<8Q2

Metrics