Abstract

We have calculated the optically-induced force between coupled high-Q whispering gallery modes of microsphere resonators. Attractive and repulsive forces are found, depending whether the bi-sphere mode is symmetric or antisymmetric. The magnitude of the force is linearly proportional to the total power in the spheres and consequently linearly enhanced by Q. Forces on the order of 100 nN are found for Q=108, large enough to cause displacements in the range of 1μm when the sphere is attached to a fiber stem with spring constant 0.004 N/m.

© 2005 Optical Society of America

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Appl. Phys. B (1)

S. Götzinger, O. Benson, and V.andoghdar, �??Towards controlled coupling between a high-Q whispering gallery mode and a single nanoparticle,�?? Appl. Phys. B 73, 825-828 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, �??Slow-light enhancement of radiation pressure in an omnidirectional reflector waveguide,�?? Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

Europhys. Lett. (1)

L. Collot, V. Lefèvre-Seguin, M. Brune, J.M. Raimond and S. Haroche �??Very high Q whispering gallery modes observed on fused silica microspheres,�?? Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

IEEE Phot. Tech. Lett. (1)

J.-P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling, and H. A. Haus, �??Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler,�?? IEEE Phot. Tech. Lett. 12, 1004-1006 (2000).
[CrossRef]

IEEE Trans. on Electron Devices (1)

T. B. Gabrielson, �??Mechanical-thermal noise in micromachined acoustic and vibration sensors,�?? IEEE Trans. on Electron Devices 40, 903-909 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Nature (1)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621-623 (2002).
[CrossRef] [PubMed]

Opt. Commun. (1)

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, �??Coupling and tunability of whispering-gallery modes: a basis for coordinate meter,�?? Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (2)

V. P. Mitrofanov, L. G. Prokhorov, and K. V. Tokmakov, �??Variation of electric charge on prototype of fused silica test mass of gravitational wave antenna,�?? Phys. Lett. A 300, 370-374 (2002).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, �??On the ultimate sensitivity in coordinate measurements,�?? Phys. Lett. A 179, 244-248 (1993).
[CrossRef]

Phys. Lett. A, (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]

Phys. Rev. A (2)

H. M. Lai, P.T. Leung, K. Young, P. W. Barber, and S. C. Hill, �??Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,�?? Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, �??Time-independent perturbation theory for quasinormal modes in leaky optical cavities,�?? Phys. Rev. A 49, 3068-3073 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (2)

M. I. Antonoyiannakis and J. B. Pendry, �??Electromagnetic forces in photonic crystals,�?? Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

H. Miyazaki and Y. Jimba, �??Ab initio tight-binding description of morphology-dependent resonance in a bisphere,�?? Phys. Rev. B 62, 7976-7997 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Valhala, �??Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,�?? Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. Part J. (1)

D. R. Rowland and J. D. Love, �??�??Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,�??�?? Proc. Inst. Electr. Eng. Part J. 140, 177�??188 (1993).

Sensors and Actuators A (1)

J.-P. Laine, H. C. Tapalian, B. E. Little, and H. A. Haus, �??Acceleration sensor based on high-Q optical microsphere antiresonant reflecting waveguide coupler,�?? Sensors and Actuators A 93, 1-7 (2001).
[CrossRef]

Other (1)

J. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1992), Chap. 11.

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Figures (2)

Fig. 1.
Fig. 1.

Force as a function of separation for coupled microspheres for two different angular mode numbers. Negative values indicate attractive forces. Left and bottom axes use dimensionless units. Right and top axes show the force as a function of distance in physical units, assuming a coupled input power of 1mW to each sphere and a Qo of 108. At a wavelength of 1.55 μm, l = m = 111 corresponds to a sphere radius (a) of 19.9 μm and l = m = 184 to a radius of 32.4 μm. Inset shows modal symmetries. For antisymmetric modes (top inset, upper two data curves), the electric field perpendicular to the page points in opposite directions in the two spheres. For symmetric modes (bottom inset, lower two data curves), the opposite is true. Red and blue correspond to electric fields pointing in and out of the page, respectively. Arrows indicate direction of propagation.

Fig. 2.
Fig. 2.

Magnitude of equilibrium displacement of a sphere due to the optical force as a function of the spring constant k of the attached fiber stem. The optical displacement is larger than 1000 times the estimated thermal displacement (solid black line) for spring constants greater than 0.002 N/m.

Equations (21)

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F = 1 ω d ω d ξ U = 1 ω o + Δ ω d ( ω o + Δ ω ) d ξ U 1 ω d Δ ω d ξ U ,
Δ ω ω o = 1 2 d V [ ( ε 2 1 ) E 1 2 + ( ε 1 1 ) E 2 2 ± ( ( ε 1 1 ) E 1 * · E 2 + ( ε 2 1 ) E 2 * · E 1 ) ] d V [ ε 1 E 1 2 + ε 2 E 2 2 ] ,
d V [ ( ε 2 1 ) E 1 2 + ( ε 1 1 ) E 2 2 ] ,
d V [ ( ε 1 1 ) E 1 * · E 2 + ( ε 2 1 ) E 2 * · E 1 ] .
F = ( F λ U ) U λ = ( F λ U ) P Q o ω o λ = ( F λ U ) P Q o 2 π c ,
F = d U d ξ
F = d ( N ħ ω ) d ξ = N ħ d ω d ξ = 1 ω d ω d ξ U
U = 1 4 π 1 2 Re [ d V ( E ξ * · D ξ + B ξ * · H ξ ) ]
× H ξ = i ω D ξ
× E ξ = i ω B ξ
E ξ * · ( × ξ H ξ ) H ξ * · ( × ξ E ξ ) = E ξ * · [ i ξ ω D ξ + i ω ξ D ξ ] + H ξ * · [ i ξ ω B ξ + i ω ξ B ξ ] .
b · ( × a ) = · ( a × b ) + a · ( × b )
ω ξ d V ( E ξ * · D ξ + B ξ * · H ξ ) = i ( ξ ω ) d V ( E ξ * · D ξ + B ξ * · H ξ )
ω ξ U = U ξ ω
F = U ξ = 1 ω ω ξ U ,
× × E i ( ω i c ) 2 ( ε i 1 ) E i = ( ω i c ) 2 E i
( Θ ˆ λ i A ˆ i ) ψ i = λ i ψ i
Θ ˆ ψ λ [ A ˆ 1 + A ˆ 2 ] | ψ = λ ψ ,
λ ( 0 ) { ψ 1 A ˆ 2 ψ 1 ± ψ 1 A ˆ 2 ψ 2 + ψ 1 A ˆ 2 ψ ( 1 ) } + λ ( 1 ) ψ 1 ( 1 + A ˆ 1 ) ψ 1 = 0
λ ( 0 ) { ψ 2 A ˆ 1 ψ 1 ± ψ 2 A ˆ 1 ψ 2 + ψ 2 A ˆ 1 ψ ( 1 ) } ± λ ( 1 ) ψ 2 ( 1 + A ˆ 2 ) ψ 2 = 0
1 2 λ ( 1 ) λ ( 0 ) = Δ ω ω o = 1 2 d V [ ( ε 2 1 ) E 1 2 + ( ε 1 1 ) E 2 2 ± ( ( ε 1 1 ) E 1 * · E 2 + ( ε 2 1 ) E 2 * · E 1 ) ] d V [ ε 1 E 1 2 + ε 2 E 2 2 ] .

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