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High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators

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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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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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