Abstract

Results of detailed experimental investigations of the power and sweeping speed dependent resonance bandwidth and resonance wavelength shift in microsphere resonators are presented. The experimental manifestations of the nonlinear effects for the different sweeping modes are considered and a possibility of separation between the Kerr and thermal nonlinearities is discussed. As it follows from the detailed comparison between theory and experiments, a single mode theoretical model, based on the mean field approximation, gives a satisfactory description of the experimental data only at small coupling powers and fast sweeping. For example, the values of Kerr nonlinearity, obtained through the fitting of the experimental data, are far from the expected, commonly used ones.

© 2008 Optical Society of America

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  1. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
    [CrossRef]
  2. T. J. Johnson, M. Borselli, and O. Painter, "Self-induced optical modulation of the transmission through a high-Q silicon microdisc resonator," Opt. Express 14, 817-831 (2006).
    [CrossRef] [PubMed]
  3. V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, "Microtorus: a high-finess microcavity with whispering-gallery modes," Opt. Lett. 26, 256-258 (2001).
    [CrossRef]
  4. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).
  5. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
    [PubMed]
  6. B. E. Little, J. P. Laine, and H. A. Haus, "Analytical theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
    [CrossRef]
  7. V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
    [CrossRef] [PubMed]
  8. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
    [CrossRef]
  9. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
    [CrossRef] [PubMed]
  10. L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
    [CrossRef]
  11. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
    [CrossRef] [PubMed]
  12. V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
    [CrossRef]
  13. V. S. Ilchenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004-1009 (1992).
  14. J. A. Stratton, Electromagnetic theory (McGrow-Hill, New York, 1941).
  15. A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, V. S. Ilchenko, "Nonstationary nonlinear effects in optical microspheres," J. Opt. Soc. Am. B 22, 459-465 (2005).
    [CrossRef]
  16. T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behaviour and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
    [CrossRef] [PubMed]
  17. Yu. A. Kuznetcov, Elements of Applied Bifurcation Theory, 2nd ed. (Springer-Verlag, New-York, 1998), p. 135.
  18. R. W. Boyd, Nonlinear Optics, Second Edition, (Academic Press, 2003).
  19. http://de.wikipedia.org/wiki/quarzglas (last access 10.11.2006).
  20. F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

2006

2005

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, V. S. Ilchenko, "Nonstationary nonlinear effects in optical microspheres," J. Opt. Soc. Am. B 22, 459-465 (2005).
[CrossRef]

2004

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behaviour and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

2003

L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
[CrossRef]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

2002

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
[CrossRef] [PubMed]

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

2001

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, "Microtorus: a high-finess microcavity with whispering-gallery modes," Opt. Lett. 26, 256-258 (2001).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

1999

1998

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

1992

V. S. Ilchenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004-1009 (1992).

1989

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

1980

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

Armani, D. K.

L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

Arnold, S.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Asano, T.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

Borselli, M.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Braun, D.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Carmon, T.

Follmer, V.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Fomin, A. E.

Gorodetskii, M. L.

V. S. Ilchenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004-1009 (1992).

Gorodetsky, M. L.

Grudinin, I. S.

Hare, J.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Haroche, S.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Haus, H. A.

Ilchenko, V. S.

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, V. S. Ilchenko, "Nonstationary nonlinear effects in optical microspheres," J. Opt. Soc. Am. B 22, 459-465 (2005).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, "Microtorus: a high-finess microcavity with whispering-gallery modes," Opt. Lett. 26, 256-258 (2001).
[CrossRef]

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

V. S. Ilchenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004-1009 (1992).

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Johnson, T. J.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

Koshsima, M.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Laine, J. P.

Lefevre-Seguin, V.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Libchaber, A.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Little, B. E.

Maleki, L.

Noda, S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

Painter, O.

Raimond, J.-M.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Roch, J.-F.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Song, B.-S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

Teraoka, I.

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Thorne, K. S.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Treussart, F.

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behaviour and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

Vorontsov, Y. I.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Yang, L.

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behaviour and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004).
[CrossRef] [PubMed]

L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
[CrossRef]

Yao, X. S.

Appl. Phys. Lett.

L. Yang, D. K. Armani, and K. J. Vahala, "Fiber-coupled Erbium Microlaser on a chip," Appl. Phys. Lett. 83, 825-826 (2003).
[CrossRef]

V. Follmer, D. Braun, A. Libchaber, M. Koshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Eur. Phys. J. D

F. Treussart, V. S. Ilchenko, J.-F. Roch, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, "Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).

J. Lightwave Technol.

J. Opt. Soc. Am. B

Laser Phys.

V. S. Ilchenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004-1009 (1992).

Nature

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 412, 925-928 (2001).

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double heterostructure nanocavity," Nature 425, 944-947 (2003).
[PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultra-low threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-632 (2002).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Lett. A

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Phys. Rev. A

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Phys. Rev. Lett.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity induced optical parametric oscillation in a ultra-high-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Science

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum-nondemolition measurements," Science 209, 547-557 (1980).
[CrossRef] [PubMed]

Other

J. A. Stratton, Electromagnetic theory (McGrow-Hill, New York, 1941).

Yu. A. Kuznetcov, Elements of Applied Bifurcation Theory, 2nd ed. (Springer-Verlag, New-York, 1998), p. 135.

R. W. Boyd, Nonlinear Optics, Second Edition, (Academic Press, 2003).

http://de.wikipedia.org/wiki/quarzglas (last access 10.11.2006).

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Figures (8)

Fig. 1.
Fig. 1.

Experimental setup used for microsphere resonator measurements.

Fig. 2.
Fig. 2.

High-Q resonance transmission dip measured in a microsphere with a diameter of 410 µm and a coupling power of 7.4 µW.

Fig. 3.
Fig. 3.

Different characteristic time scales for the high-Q resonance shown in Fig. 2.

Fig. 4.
Fig. 4.

Transmission resonance shapes for two coupling power at the different sweeping speeds. Right column - 0.34 mW, left column - 0.07 mW. Experimental data are depicted by black points, red solid line - simulation with Kerr and thermal effects, blue dashed line - simulation without Kerr effect (thermal nonlinearity only).

Fig. 5.
Fig. 5.

Dependence of the bandwidth of a high-Q resonance (λresonance=1550.6600 nm) on sweeping speed for two values of coupling power: black/red solid lines (modeling with/without Kerr effect) with squares (experimental results) - 0.34 mW; black/red dashed lines (modeling with/without Kerr effect) with triangles (experimental results) - 0.07 mW.

Fig. 6.
Fig. 6.

Dependence of nonlinear broadening of a high-Q resonance (λresonance=1550.6541 nm) on coupling power (squares & dashed black line - experiments, solid red line - numerical modeling with Kerr effect, dashed blue line - numerical modeling without Kerr effect).

Fig. 7.
Fig. 7.

Visualized transmission dips broadening of the high-Q resonance shown in Fig. 2resonance=1551.22579 nm) for two different input powers. Small oscillations attributed to the Andronov-Hopf bifurcation are clearly seen on the right side of the solid black curve.

Fig. 8.
Fig. 8.

Nonlinear shift of a high-Q resonance (λresonance=1550.6541 nm) depending on coupling power (solid black line with full rombs - sweeping speed 5 nm/s, dashed red line with empty rombs - 1 nm/s). Dashed line parallel to the x-axis indicates the wavelength positioning accuracy of the laser.

Tables (2)

Tables Icon

Table 1. Parameters for best fitting.

Tables Icon

Table 2. Parameters for best fitting in Fig. 6.

Equations (11)

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

T t k th ρ 𝖢 ~ Δ T = α ρ 𝖢 ~ E 2
{ C lm ( t ) t = i [ Δ + γ el , lm C lm ( t ) 2 + γ th , lm Q ( t ) ] C lm ( t ) i p l A lm ( t ) Q ( t ) t = δ th Q ( t ) + B C lm ( t ) 2
τ th = ( δ th ) 1 ρ C ~ b 2 2 k th
Q = ω l τ mode = λ l δ λ l
{ i [ Δ ( t ) + γ el , lm C lm ( t ) 2 + γ th , lm Q ( t ) ] C lm ( t ) = ip l A lm Q ( t ) t = δ th Q ( t ) + B C lm ( t ) 2
i [ Δ ( t ) + ( γ el , lm + γ th , lm B δ th ) C lm ( t ) 2 ] C lm ( t ) = ip l A lm
Q ( t ) = B 0 t C lm ( t ) 2 d t
i [ Δ ( t ) + γ el , lm C lm ( t ) 2 + γ th , lm B 0 t C lm ( t ) 2 d t ] C lm ( t ) = ip l A lm
0 t C lm ( t ) 2 d t C lm 2 ¯ . τ scan
{ I 3 + u 1 I 2 + u 2 I = u 3 Q ( t ) t = δ th Q ( t ) + BI
Δ λ l = λ l ( 1 n ( n T ) + α V ) Δ T

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