Abstract

The Rayleigh scattering has to be largely suppressed in high-Q whispering-gallery modes in microspheres because of restrictions imposed on scattering angles by cavity confinement. Earlier estimates of the fundamental limit for the quality factor in fused-silica microspheres are revisited, and Q1012 is predicted in few-millimeter-size fused-silica spheres, if the surface hydration problem is ovecome. Particular effects of surface scattering losses are analyzed, and the manifestation of scattering in the form of intermode coupling is calculated. The predominant effect of counterpropagating mode coupling (intracavity backscattering) is analyzed in the presence of a mode-matched traveling-wave coupler. As much as 100% resonance reflection regime is shown to be feasible.

© 2000 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, 393–396 (1989).
    [CrossRef]
  2. S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).
  3. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements for fused silica microspheres in the NIR,” Opt. Lett. 23, 247–249 (1998).
    [CrossRef]
  4. H. Mabuchi and H. J. Kimble, “Atom galleries for whispering atoms: binding atoms in stable orbits around a dielectric cavity,” Opt. Lett. 19, 749–751 (1994).
    [CrossRef] [PubMed]
  5. L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
    [CrossRef]
  6. V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
    [CrossRef]
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  9. 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]
  10. M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).
  11. D. S. Weiss, V. Sandoghbar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995).
    [CrossRef] [PubMed]
  12. V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).
  13. M. E. Lines, “Scattering losses in optic fiber materials. I. A new parameterization. II. Numerical estimates,” J. Appl. Phys. 55, 4052–4063 (1984).
    [CrossRef]
  14. M. L. Gorodetsky and V. S. Ilchenko, “Optical resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
    [CrossRef]
  15. Fibre Optic Communication Handbook, F. Tosco, ed. (TAB Books, New York, 1990), pp. 15–17.
  16. S. Sakaguchi and S. Todoroki, “Rayleigh scattering of silica core optical fiber after hear treatment,” Appl. Opt. 37, 7708–7711 (1998).
    [CrossRef]

1999

1998

1996

1995

1994

1993

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

1992

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

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).

1990

V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).

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, 393–396 (1989).
[CrossRef]

1984

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

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, 393–396 (1989).
[CrossRef]

Brune, M.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Collot, L.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Datsyuk, V. V.

V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).

Gorodetsky, M. L.

M. L. Gorodetsky and V. S. Ilchenko, “Optical resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

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]

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).

M. L. Gorodetsky and V. S. Ilchenko, “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, 393–396 (1989).
[CrossRef]

Hare, J.

Haroche, S.

D. S. Weiss, V. Sandoghbar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995).
[CrossRef] [PubMed]

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Hollberg, L.

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Ilchenko, V. S.

M. L. Gorodetsky and V. S. Ilchenko, “Optical resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements for fused silica microspheres in the NIR,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

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]

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).

M. L. Gorodetsky and V. S. Ilchenko, “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, 393–396 (1989).
[CrossRef]

Izmailov, I. A.

V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).

Kimble, H. J.

Kochelap, V. A.

V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).

Lefevre-Seguin, V.

Lefèvre-Seguin, V.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Lines, M. E.

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

Mabuchi, H.

Raimond, J.-M.

D. S. Weiss, V. Sandoghbar, J. Hare, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995).
[CrossRef] [PubMed]

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Sakaguchi, S.

Sandoghbar, V.

Savchenkov, A. A.

Streed, E. W.

Todoroki, S.

Vasiliev, V. V.

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Velichansky, V. L.

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Vernooy, D. W.

Vyatchanin, S. P.

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).

Weiss, D. S.

Yarovitsky, A. V.

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

Appl. Opt.

Europhys. Lett.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. B

Kvantovaya Elektron. (Kiev)

V. V. Datsyuk, I. A. Izmailov, and V. A. Kochelap, “Generation of light on ‘whispering-gallery’ modes in a medium with condensated dispersed phase,” Kvantovaya Elektron. (Kiev) 38, 56–65 (1990) (in Russian).

Laser Phys.

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

Opt. Commun.

V. V. Vasiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, “Narrow-line-width diode laser with a high-Q microsphere resonator,” Opt. Commun. 158, 305–312 (1998).
[CrossRef]

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, 393–396 (1989).
[CrossRef]

Zh. Prikl. Spektrosk.

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280 (1992) (in Russian).

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

Fibre Optic Communication Handbook, F. Tosco, ed. (TAB Books, New York, 1990), pp. 15–17.

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Figures (3)

Fig. 1
Fig. 1

Quality-factor limitation by modified Rayleigh and surface scattering losses in fused-silica spheres; incorporated are literature data13,15 on UV and IR absorption: (1) surface scattering limitation, sphere radius 1 mm; (2) volumetric losses, TM modes; (3) volumetric losses, TE modes; (4) volumetric losses, calculated from plane-wave attenuation in the bulk; (5) experimental data for 600–800 µm spheres.3,9

Fig. 2
Fig. 2

Backscattering in microspheres and the resonance reflection in a traveling-wave evanescent coupler.

Fig. 3
Fig. 3

Calculated normalized resonance reflection in a mode-matched evanescent coupler owing to backscattering in a microsphere, as a function of relative coupling strength: (1) for β=δc+δ0, (2) for β=10δ0, (3) for β=3δ0, (4) for β=δ0, and (5) for β=0.3δ0.

Equations (49)

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Q=(2πn)/(αλ),
IsI=π2 sin2 ϑλ4r2δ(r1)δ(r2)dv1dv2,
Ps=I 8π23λ4δ(r1)δ(r2)dv1dv2.
sin2 γTE=a-da2(1-sin2 ϑ cos2 φ)<1n2,
sin2 γTM=a-da2 sin2 ϑ<1n2,
αis=8π33λ4n8p2κTβT,
Qis=KTE,TM 2πnαisλ.
KTE=2.8,KTM=9.6.
δ(x, y, z)=(0-1)f(x, y)δ(z),
Ps=dxI(y, 0) 16π23λ4(n2-1)πB2σ2dx=Pαdx.
αss=I(y, 0)I(y, z)dz16(n2-1)π3B2σ23λ4,
0ajl2(knr)r2dra2jl2(kna)a2jl2(kna)jl(ρ)ρρ=kna2a(n2-1)2n2.
Qss=KTE1+KTE3λ3a8nπ2B2σ2.
α[0.7 µm4/λ4+1.1×10-3 exp(4.6µm/λ)+4×1012 exp(-56 µm/λ)]dB/km.
f(θ, ϕ)=L,MblYLM(θ, ϕ),
bL2=κTσ˜(L-1)(L+2),
δ=f(θ, ϕ)F(r),
r(θ, ϕ)=a+f(θ, ϕ),
δ=(n2-1)f(θ, ϕ)δ(r-a).
ΔE-0(r)c2+δ(r)c2 2Et2=0.
Ej=exp(-iωjt)ej(r, θ, ϕ),
Δej+0kj2ej=0,
E=exp(-iω0t)Aj(t)ej.
2iω00 dAj(t)dtej+ω02δAj(t)ej
+0j(ωj2-ω02)Aj(t)ej=0.
dAkdt+iΔωkAk=ijAjβjk,
βjk=ω02n2ejδek*dv|ej|2dv.
βjk2=ω024n4δ(r)δ(0)dvVjk,
Vjk=|ej|2dv|ek|2dv|ej|2|ek|2dv.
Veff=|ej|2dv2|ej|4dv.
dA+dt+(δ0+δc+iΔω)A+=iA-β+i Tτ0Bin,
dA-dt+(δ0+δc+iΔω)A-=iA+β,
Bt=1-T2Bin+iTA+,
Br=iTA-,
A+=iT2δcβ(δ0+δc)2+β2-Δω2+i2Δω(δ0+δc)Bin,
A-=-1T2δc(δ0+δc+iΔω)(δ0+δc)2+β2-Δω2+i2Δω(δ0+δc)Bin,
Bt=δ02-δc2+β2-Δω2+i2δ0Δω(δ0+δc)2+β2-Δω2+i2Δω(δ0+δc)Bin,
Br=-i2δcβ(δ0+δc)2+β2-Δω2+i2Δω(δ0+δc)Bin.
Br=i βδ0+δc.
|A+|2=|A-|2=1T2δc2(δ0+δ0)2Bin2,
|Bt|2=δ02(δc+δ0)2Bin,
|Br|2=δc2(δc+δ0)2Bin.
Δωω=2βω0.
Δωωis=n4p2κTβTVeff1/2=3λ4αis8π3n4Veff1/2.
Veff=2.3n-7/6a11/6λ7/6,
Δωωis5×10-7 µm3/2λ11/12α7/12.
Veff,lm=Veff,ll(1+0.5l-m-0.5).
βss2=ω024n4πB2σ2|e|4Veff|e(r)|4dr,
Δωωss1.1σBλ1/4α7/4.

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