Abstract

Thermal nonlinearity can produce oscillatory instability in optical microspheres. We experimentally demonstrate this instability and analyze the conditions needed to observe this regime. The observed behavior is in good agreement with the results of numerical simulation. In pure fused silica with low optical absorption the thermal oscillations are suppressed owing to an interaction of thermal and Kerr nonlinearities. We also describe experimentally observed slow and irreversible thermo-optical processes in microspheres.

© 2005 Optical Society of America

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References

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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. V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, "Microtorus: a high-finesse microcavity with whispering-gallery modes," Opt. Lett. 26, 256-258 (2001).
    [CrossRef]
  3. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
    [CrossRef] [PubMed]
  4. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  5. M. L. Gorodetsky and V. S. Ilchenko, "Thermal nonlinear effects in optical whispering-gallery microresonators," Laser Phys. 2, 1004-1009 (1992).
  6. A. E. Salomonovich, "Automodulation at ferroresonance," Zh. Tekh. Fiz. 22, 245-258 (1952) (in Russian).
  7. G. V. Belokopytov, "Electrothermal instability of oscillations in temperature sensitive resonant systems," Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 3, 11-15 (1997).
  8. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, "On the ultimate Q of optical microsphere resonators," Opt. Lett. 21, 453-455 (1996).
    [CrossRef] [PubMed]
  9. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, "Rayleigh scattering in high-Q microspheres," J. Opt. Soc. Am. B 17, 1051-1057 (2000).
    [CrossRef]
  10. M. E. Lines, "Scattering losses in optic fiber materials. I. A new parametrization. II. Numerical estimates," J. Appl. Phys. 55, 4052-4063 (1984).
    [CrossRef]
  11. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers , 2nd ed. (McGraw-Hill, New York, 1968), p. 17.
  12. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory , 2nd ed. (Springer-Verlag, New York, 1998), p. 135.

2003

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

2001

2000

1997

G. V. Belokopytov, "Electrothermal instability of oscillations in temperature sensitive resonant systems," Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 3, 11-15 (1997).

1996

1992

M. L. Gorodetsky and V. S. Ilchenko, "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]

1984

M. E. Lines, "Scattering losses in optic fiber materials. I. A new parametrization. II. Numerical estimates," J. Appl. Phys. 55, 4052-4063 (1984).
[CrossRef]

1952

A. E. Salomonovich, "Automodulation at ferroresonance," Zh. Tekh. Fiz. 22, 245-258 (1952) (in Russian).

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Belokopytov, G. V.

G. V. Belokopytov, "Electrothermal instability of oscillations in temperature sensitive resonant systems," Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 3, 11-15 (1997).

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]

Gorodetsky, M. L.

Gorodetsky , M. L.

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

Gorodetsky, M. L.

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]

Ilchenko, V. S.

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Lines, M. E.

M. E. Lines, "Scattering losses in optic fiber materials. I. A new parametrization. II. Numerical estimates," J. Appl. Phys. 55, 4052-4063 (1984).
[CrossRef]

Maleki, L.

Pryamikov, A. D.

Salomonovich, A. E.

A. E. Salomonovich, "Automodulation at ferroresonance," Zh. Tekh. Fiz. 22, 245-258 (1952) (in Russian).

Savchenkov, A. A.

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Yao, X. S.

J. Appl. Phys.

M. E. Lines, "Scattering losses in optic fiber materials. I. A new parametrization. II. Numerical estimates," J. Appl. Phys. 55, 4052-4063 (1984).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

M. L. Gorodetsky and V. S. Ilchenko, "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 421, 925-928 (2003).
[CrossRef] [PubMed]

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]

Vestn. Mosk. Univ., Ser. 3: Fiz., Astron.

G. V. Belokopytov, "Electrothermal instability of oscillations in temperature sensitive resonant systems," Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 3, 11-15 (1997).

Zh. Tekh. Fiz.

A. E. Salomonovich, "Automodulation at ferroresonance," Zh. Tekh. Fiz. 22, 245-258 (1952) (in Russian).

Other

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers , 2nd ed. (McGraw-Hill, New York, 1968), p. 17.

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

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Figures (9)

Fig. 1
Fig. 1

Nonlinear resonance and oscillatory instability in optical microsphere observed in the experiment. Horizontal axis corresponds to detuning from the resonant frequency; vertical, output intensity.

Fig. 2
Fig. 2

Areas of stability for nonlinear resonance with relaxational nonlinearity. Dark-gray area marks the area of optical bistability; two stable states of output power are possible at given detuning. Dotted curve marks unstable state. Light-gray area shows the area of possible relaxational oscillations that can be observed on the dashed part of the nonlinear resonance curve.

Fig. 3
Fig. 3

Numerical simulation of WGM nonlinear optical resonance in microspheres with thermal nonlinearity: hysteretic response curve. In both optical and numerical experiments, the oscillatory behavior was observed on both branches of response curves.

Fig. 4
Fig. 4

Numerical simulation of WGM nonlinear optical resonance in microspheres with thermal nonlinearity—phase diagram.

Fig. 5
Fig. 5

Numerical simulation of chaotic regime in microsphere with thermal nonlinearity and two close optical modes.

Fig. 6
Fig. 6

Scheme of the experimental setup.

Fig. 7
Fig. 7

Thermal nonlinearity and oscillations in the microsphere 82.5 µm in diameter. The picture represents thermal oscillations on the fraction of the slope of the nonlinear resonant curve shown in Fig. 8.

Fig. 8
Fig. 8

Thermal nonlinearity and oscillations in the microsphere 82.5 µm in diameter. Sweep rate is 2.9 GHz/s.

Fig. 9
Fig. 9

Nonstationary thermal effects and bifurcations in the microsphere with the diameter 90 µm, Q108. The oscillating area is to the top of the curve. Several accompanying peaks in the picture to the left are moving “down the resonant curve” and disappear later, as seen on the picture to the right.

Equations (34)

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ΔEs-(r)c2 2Est2+2δ Est
=4πc2 2Pp(ω)t2+2Pn(ω)t2,
χn=2πβθ+χ(3)(ω)Es2,
E(r, t)=a(t)E0(r)exp(iωt),
|E0|2dr=1,
a˙+aδ+iΔω+2πω0χ(3)n2Veff|a|2+ω0βΘ
=iπω0Wn2VeffQ1/2.
Θ=(T-T0)|E0(r)|2dr,
Veff-1=|E0(r)|4dr.
Tt-DΔT=nαac4πCρ|E|2,
1I dIdtω0Q.
T(t, r)-T0=nαac4πCρ 1(2π)4  I(Ω)G(q)Dq2+iΩ×exp(iΩt+iqr)dΩdq,
G(q)=|E0|2 exp(-iqr)dr.
Θ(t)=(T-T0)|E0(r)|2dr=nαac4πCρ 1(2π)4  I(Ω)|G(q)|2Dq2+iΩ exp(iΩt)dqdΩ=12π Θ(Ω)exp(iΩt)dΩ.
iΩΘ(Ω)+δθΘ(Ω)=nαac4πCρVeffI(Ω),
δθ(Ω)=DVeff  |G(q)|2q2+iΩ/D dq(2π)3-1-iΩ.
δθ2D/b2,
Θt+δθΘ=nαac4πCρVeff|a(t)|2.
u˙+δu-[Δω+μ(u2+v2)+ω0βΘ]v=0,
v˙+δv+[Δω+μ(u2+v2)+ω0βΘ]u=F,
Θ˙+δθΘ=νδθω0β(u2+v2),
a(t)=u(t)+iv(t)
μ=2πω0χ(3)n2Veff,ν=ω0βnαac4πCρVeffδθ,F=πω0Wn2VeffQ1/2.
{δ2+[Δω+(μ+ν)I0]2}I0-F2=0.
χθ=n3αaβc8π2Cρδθ>χ(3).
αa1.1×10-3dBkmexp56µmλ,
ξ=u-u0,η=v-v0,ζ=Θ-Θ0
ξ˙=-(δ-2µu0v0)ξ+(Δn+2µv02)η+ω0βv0ζ,
η˙=-(Δn+2µu02)ξ-(δ+2µu0v0)η-ω0βu0ζ,
ζ˙=2 νδθω0βu0ξ+2 νδθω0βv0η-δθζ,
λ3+(2δ+δθ)λ2+[(μ+ν)(3µ+ν)I02+2Δω(2µ+ν)I0+Δω2+δ2+2δδθ]λ+δθ[3(μ+ν)2I02+4Δω(μ+ν)I0+Δω2+δ2]=0.
Ia3=-[2Δω±(Δω2-3δ2)1/2]×[3(μ+ν)]-1,
IT2={-Δω(2µ+ν-νδθ/2δ)±[Δω2(μ-νδθ/2δ)2-(δ+δθ)2(μ+ν)(3µ+ν-νδθ/δ)]1/2}×[(μ+ν)(3µ+ν-νδθ/δ)]-1.
χ(3)<βn48π2Cρ αaα4×10-14 αaαesu,

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