## On optical forces in spherical whispering gallery mode resonators |

Optics Express, Vol. 19, Issue 22, pp. 22337-22349 (2011)

http://dx.doi.org/10.1364/OE.19.022337

Acrobat PDF (849 KB)

### Abstract

In this paper we discuss the force exerted by the field of an optical cavity on a polarizable dipole. We show that the modification of the cavity modes due to interaction with the dipole significantly alters the properties of the force. In particular, all components of the force are found to be non-conservative, and cannot, therefore, be derived from a potential energy. We also suggest a simple generalization of the standard formulas for the optical force on the dipole, which reproduces the results of calculations based on the Maxwell stress tensor.

© 2011 OSA

## 1. Introduction

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

3. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. **5**, 909–914 (2009). [CrossRef]

6. S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature **460**, 724–727 (2009). [CrossRef]

7. O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. **10**, 125021 (2008). [CrossRef]

9. D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. **103**, 207204 (2009). [CrossRef]

10. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. **107**, 1005–1010 (2010). [CrossRef] [PubMed]

15. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. **7**, 527–530 (2011). [CrossRef]

16. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science **321**, 1172–1176 (2008). [CrossRef] [PubMed]

17. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. **58**, 207–323 (2010). [CrossRef]

5. A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. **4**, 415–419 (2008). [CrossRef]

12. P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A **81**, 023826 (2010). [CrossRef]

19. R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A **81**, 063820 (2010). [CrossRef]

20. M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A **362**, 719–737 (2004). [CrossRef]

## 2. Optical force on a small polarizable particle

### 2.1. Gradient force: thermodynamic derivation

### 2.2. Gradient force: direct derivation

*q*, separated by some small, ultimately infinitesimal distance

*d*. The same approach can be used to describe forces on an induced dipole characterized by polarizability

*α*. The dipole is assumed to be placed in some external field

**E**, so that

**p**=

*α*

**E**. We make no demands on

**E**other than that

**p**be defined self consistently with it. In particular,

**E**may be dependent on the dipole itself. For example, if the external field

**E**is due to charges on a conductor, the presence of the dipole will alter the charge distribution and thus

**E**. We make this explicit by writing

**E**=

**E**(

**r**,

**r**

*), where*

_{p}**r**and

**r**

*are respectively a field point and the position vector of the particle. The total force is derived by considering the Coulomb forces at each charge comprising the dipole:*

_{p}**F**=

*q*[

**E**(

**r**

*+*

_{p}**d**/2,

**r**

*) −*

_{p}**E**(

**r**

*−*

_{p}**d**/2,

**r**

*)] (see Fig. 1). By taking the limit |*

_{p}**d**| → 0, keeping |

**p**| =

*q*|

**d**| constant, the force derived in this case is: where ∇

**refers to a gradient with respect to field coordinates**

_{r}**r**. While this result looks similar to Eq. (5), there is an important difference between them. The force given by Eq. (5) implies that before taking the spatial derivative of the field with respect to field coordinates

**r**, one sets coordinate of the particle

*r*to coincide with

_{p}**r**. On the other hand, Eq. (7) requires that this procedure is reversed. Physically, it reflects the fact that the electric force on a dipole results from the spatial variation of the electric field across it. The two equations, Eq. (5) and Eq. (7), produce identical results only if the field exerting the force does not depend upon particle’s position.

*α*|

**E**(

**r**,

**r**

*)|*

_{p}^{2}/2 cannot be interpreted as a potential energy unless

**E**is independent of

**r**

*. Correspondingly, the force calculated according to Eq. (6) does not have to be conservative.*

_{p}### 2.3. Total force on a dipole

*d*

**p**/

*dt*, giving rise to a magnetic force. The total Lorentz force on the particle for incident fields independent of the dipole’s position 𝔈, 𝔅 = −

*i*/

*ω*∇ × 𝔈 have the standard form

**F**= (

**p**·∇)𝔈+

*d*

**p**/

*dt*× 𝔅. The time averaged expression for this force can be rewritten as [24

24. V. Wong and M. A. Ratner, “Explicit computation of gradient and nongradient contributions to optical forces in the discrete-dipole approximation,” J. Opt. Soc. Am. B **23**, 1801–1814 (2006). [CrossRef]

*α*now is a complex valued (with radiative corrections included) polarizability of the dipole. Based upon analysis of the previous sub-section we conjecture that the expression for the force in the case of the field

*dependent on the particle’s position*{

**E**,

**B**} = {

**E**(

**r**,

**r**

*),*

_{p}**B**(

**r**,

**r**

*)} can be obtained from Eq. (9) by substituting 𝔈, 𝔅,∇ with*

_{p}**E**,

**B**,∇̃ respectively. For a standard dipole particle with radius

*R*, refractive index

_{p}*n*, and polarizability where Equation (9) for the force can be re-written as where 〈

_{p}*u*〉 is the average polarization energy of the dipole

*σ*is its scattering cross section

*σ*= ℐ

*m*[

*α*]

*k*/

*ɛ*

_{0}, and 〈

**g**〉 is the average momentum density of the field

*c*is the speed of light in vacuum.

21. J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A **84**, 023844 (2011). [CrossRef]

## 3. Optical force of a WGM resonator

### 3.1. WGMs of a single spherical resonator

*XYZ*system in Fig. 2), its field is described by a single vector spherical harmonic (VSH) characterized by polar, azimuthal, and radial indexes

*l,m,s*respectively and polarization, TE or TM. For concreteness we focus only on TE polarized modes with radial index

*s*= 1 defined as

*e*

^{−iωt}is assumed and suppressed). Here

*E*

_{0}is a normalization factor, ℒ is the scattering amplitude describing response of the resonator to an incident radiation with frequency

*ω*,

*k*is the magnitude of the wavevector

*k*=

*ω*/

*c*. In the close vicinity of a chosen resonant frequency the scattering amplitude can be approximated as with

**L**is the dimensionless angular momentum operator

**L**= −

*i*

**r**

*×*∇, and

*Y*(

_{l,m}*θ*,

*ϕ*) is the scalar spherical harmonic. We focus here on so called fundamental modes with

*m*=

*l*. WGMs with long lifetime are characterized by

*l*≫ 1 and in expressions that follow we neglect terms of order 1/

*l*compared to those or order unity.

*X*′

*Y*′

*Z*′ system in Fig. 2). Such an expression can be obtained using rotational properties of VSH [25] expressed as where

*α*,

*β*,

*γ*are the Euler angles specifying the rotation from the unprimed to primed coordinate system. The D-functions are defined by where the function

*m*=

*l*can be written as In order to find an expression for the WGM in the primed system one has to apply inverse transformation

*γ*is equivalent to shifting the

*ϕ*′ coordinate, so we only consider transformations whith

*γ*= 0. Applying this transformation to a WGM with orbital number

*l*=

*L*we find its representation in the rotated system in the following form: where Here we introduced dimensionless variable

*Z*′ axis (

*θ*′ = 0), all

**X**

*vanish except for*

_{L,m}**X**

_{L,±}_{1}. Assuming

*L*≫ 1, and dispensing with the prime on

**E**′, the field can be written explicitly as: where

*θ̂*and

*ϕ̂*are the spherical coordinate unit vectors referred to the global, unprimed system. The magnetic field

**H**=

**B**/

*μ*

_{0}= −

*i*/

*ωμ*

_{0}∇ ×

**E**is given by: where the prime denotes differentiation with respect to argument

*kr*. We can express the field at any point in space in the form of Eq. (16) and Eq. (17) by changing the Euler angles appearing in

*a*. It is important to note that angles

_{m}*α*and

*β*correspond to the respective angular coordinates

*ϕ*and

*θ*of the

*Z*′ axis as viewed from the unprimed coordinate system.

### 3.2. Calculation of the force neglecting particle-resonator coupling

**r**

*= (*

_{p}*r*,

_{p}*θ*,

_{p}*ϕ*) as defined in the

_{p}*XYZ*coordinate system, lies on the

*Z*′ axis of the

*X*′

*Y*′

*Z*′ system, we can substitute the results of the previous sub-section, Eq. (16) and (17), into Eq. (10) for the force, while replacing the operator of pseudo-gradient with the regular gradient. In the limit

*L*≫ 1,

*m*function

*β*≡

*θ*. Consequently, the field coefficients

_{p}*a*for

_{m}*m*= ±1 can be presented in terms of the

*m*= 0 coefficient

*a*

_{0}as

*a*=

_{m}*a*

_{0}[cot

*θ*/2]

_{p}^{m}. It is clear, therefore, that products of the form

*θ*=

_{p}*π*/2, we only consider

*θ*̄ =

*θ*−

_{p}*π*/2 ≪ 1, and expand

*θ̄*. However, since for

*L*≫

*m*,

*Lθ̄*≫

*θ̄*we shall only expand the terms, which do not contain

*L*. Keeping linear terms in the expansion of [cot

*θ*/2]

_{p}*, and taking into account that for*

^{m}*L*≫ 1 the imaginary part of the Hankel function

*ρ*<

*L*, we obtain the gradient portion of the optical force

**F**

^{(g)}= ∇〈𝔲〉: where and

*n*(

_{L}*kr*) is the spherical Neumann function.

_{p}*θ̄*and can be neglected compared to the scattering force,

**F**

^{(s)}=

*σc*〈

**g**〉, which takes the form: where

*p*= 2

*k*

^{3}

*α*

_{0}/3 ≪ 1. Thus, in this approximation, the gradient force draws the particle toward the resonator radially, and maintains it in the equatorial plane via its

*θ̂*component. It is important to note that its dependence on the particle’s position follows the behavior of the Neumann function

*n*(

_{L}*kr*), which is almost exponential in the considered range of parameters. At the same time, the azimuthal component of the force is of purely scattering nature and proportional to the moment density, which can be presented in the simple form 〈

**g**〉 = (

*L*/

*kr*)

*u*̂, where

_{em}ϕ*u*=

_{em}*ɛ*

_{0}〈|𝔈|

^{2}〉 is the electromagnetic energy density (the use of 𝔈 symbol for the field emphasizes that it is not affected by the presence of the particle).

### 3.3. Effect of the particle-induced modification of the WGM on the optical force

#### 3.3.1. Modification of the WGM by the particle

26. L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A **80**, 061805 (2009). [CrossRef]

27. J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A **81**, 053827 (2010). [CrossRef]

*R*, where

_{p}*kR*≪ 1, and refractive index

_{p}*n*, and the field is found in the form of a general expansion in terms VSHs with all

_{p}*l*,

*m*and polarizations. The particle is found to modify the WGM in two significant ways. First, it creates an additional resonance at frequency

*δω*and

_{L}*δ*Γ

*depending only on*

_{L}*r*. Second, the steady state of the resonator field associated with the

_{p}*ω*resonance is significantly modified compared to the field distribution of the initial WGM. Initially isotropic field turns into a highly directional distribution oriented predominantly toward the particle. Thus, a displacement of the particle in the

_{p}*ϕ̂*direction causes the resonator’s field to move with the particle (see for details Ref. [26

26. L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A **80**, 061805 (2009). [CrossRef]

27. J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A **81**, 053827 (2010). [CrossRef]

*l*and polarization. However, these contributions are small, and can be neglected. In this approximation, the scattered field of the resonator can again be presented in the form of Eq. (14), but with expansion coefficients, which are no longer given by Eq. (15). They have the following form where

*y*= (

*ω*−

*ω*)/Γ

_{p}*. The frequency shift and additional broadening of the resonance are [26*

_{p}26. L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A **80**, 061805 (2009). [CrossRef]

27. J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A **81**, 053827 (2010). [CrossRef]

*V*

_{L,1}(

*kr*) is the VSH translation coefficient [25], which arises when the field scattered by one sphere is expressed in terms of VSH centered about the other, and is given by

_{p}#### 3.3.2. Calculation of the force with particle-modified field

**r**and particle coordinates

**r**even though we calculate the force at the point

_{p}**r**=

**r**. Taking into account that dependence on

_{p}**r**is only contained in the expansion coefficients

_{p}*a*, this procedure becomes rather trivial: one needs to find the gradient of all respective expressions treating these coefficients as constants, and after that equate

_{m}**r**=

**r**. Calculating the required gradients we obtain for the pseudo-gradient component of the force:

_{p}**F**

^{(pg)}≡ ∇̃〈

*u*〉 The scattering component of the force,

**F**

^{(s)}, takes in this case the form of A contribution from the remaining term in Eq. (10), which is proportional to ℐ

*m*(

**E**

*· ∇̃)*

^{*}**E**remains negligible and will not be considered any further.

*y*

_{0}+

*i*)

^{−1}by

*m*[

*a*′] with

_{m}a_{m}*m*≠

*m*′ vanishes. For the same reason the

*θ*-component of the scattering force also vanishes.

**F**=

**F**

^{(pg)}+

**F**

^{(s)}in this case is found by using the correct set of the field coefficients as defined by Eq. (22). The radial component of the force does not change from Eq. (28), while its polar and azimuthal components now can be presented as Terms proportional to

*δω*come from the psuedo-gradient force, while

_{L}*δ*Γ

*terms come from the scattering force. These expressions demonstrate significant deviation of the force from both completely unmodified and frequency-only modified WGM approximations. First, let us note that the radial dependence of the force is determined by the factor (*

_{L}*y*

^{2}+ 1)

^{−1}in addition to the Neumann function in

*δω*and

_{L}*δ*Γ

*. The role of this factor can be seen as follows. The condition*

_{L}*y*= 0 is satisfied for some

*r*=

_{p}*r*

_{0}at which driving frequency

*ω*coincides with the particle-induced resonance. If one linearizes

*y*(

*r*) about this point as

_{p}*y*= (

*r*−

_{p}*r*

_{0})

*y*′(

*r*

_{0}), where If

*p*is sufficiently small, then in the region where

*y*′(

*r*

_{0}) can be considered constant, the factor

*dδω*/

_{L}*dr*in

_{p}*F*is also constant. Therefore the spatial profile of the force has

_{r}*Lorentzian shape*peaked at

*r*=

_{p}*r*

_{0}with width 1/

*y*′. Let us recall that in the unmodified WGM approximation the magnitude of the force monotonically (essentially exponentially) decreases with

*r*.

_{p}*F*is no longer solely due to the scattering contribution. Two different limits are of interest based upon choice of the external driving frequency

_{ϕ}*ω*. In the limit

*F*∝

_{ϕ}*yδω*+

_{L}*δ*Γ

*. The magnitude of the pseudo-gradient term exceeds that of the scattering term unless*

_{L}*y*≪

*p*. This can only happen for very small values of

*δω*, when the magnitude of the force is also very small. When

_{L}*y*≫

*p*, the pseudo-gradient contribution exceeds the scattering force, and the tangential component can be written as

*y*satisfies

*F*becomes negligible as well. In this case the force can be written

_{ϕ}22. S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel—a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express **17**, 6230–6238 (2009). [CrossRef] [PubMed]

*α*

_{0}: while the pseudo-gradient force is linear in this parameter, the scattering force is quadratic.

*ϕ*̂ direction. When the particle moves to a new point, the field re-adjusts so that there is again a field gradient in the

*ϕ*̂ direction. Implicit in this analysis is the assumption that the particle moves slowly enough to consider the field always remaining in the quasi-steady state. Velocity dependent effects can become significant when the time scale of particle motion is comparable to the relaxation time of the resonator, 1/Γ

*.*

_{p}*L/r*)

_{p}*δω*to

_{L}*dδω*/

_{L}*dr*. From the asymptotic expansions for the spherical Nuemann functions and their derivatives in the region

_{p}*L*≫ 1,

*kr < L*, we have

*n*/

_{L}*n*′

*≈ −*

_{L}*cosh*(

*a*) [29] where prime denotes differentiation with respect to argument and

*a*is defined by

*kr*= (

_{p}*L*+ 1/2)

*sech*(

*a*). Since resonances are in the region

*kr*≈

_{p}*L/n*, where

*n*is the refractive index of the resonator, we have

*n*/

_{L}*n*′

*≈ −*

_{L}*n*and thus (

*dδω*/

_{L}*dr*)/(

_{p}*Lδω*/

_{L}*r*) ≈ −2

_{p}*nkr*/

_{p}*L*≈ 2. Thus, assuming we are near a particle induced resonance so that 0 ≤ |

*y*| ≤

*O*(1), the relative magnitude of the forces will be determined by the factor

*F*and

_{ϕ}*F*. If the system is driven at a frequency close to the ideal Mie frequency, so that

_{θ}*y*

_{0}is of order unity, then the radial and azimuthal forces will be of comparable magnitude, while the polar force will be smaller by a factor

*θ*̄. If

*y*

_{0}≫ 1 on the other hand, then the azimuthal force will be smaller than the radial force by a factor 1/

*y*

_{0}, while the polar force is smaller by a factor

*θ*̄

*y*/

*y*

_{0}.

*δ*Γ

*/*

_{L}*δω*=

_{L}*p*≪ 1 (where

*p*= 2

*k*

^{3}

*α*

_{0}/3). In the limit where both the driving frequency and particle induced resonance frequencies are very close to the ideal Mie resonance, so that

*y*,

*y*

_{0}→ 0, the scattering contribution to

*F*becomes appreciable, while it vanishes in

_{ϕ}*F*. This is to be expected given that

_{θ}*y*,

*y*

_{0}→ 0 is the limit where the particle induced modification of the cavity mode becomes vanishingly small, and accordingly the behavior of the forces approaches that of their unmodified forms of Eq. (19) and Eq. (21). This is likely the regime encountered in the experiments of Ref. [22

22. S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel—a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express **17**, 6230–6238 (2009). [CrossRef] [PubMed]

30. J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E **80**, 026607 (2010). [CrossRef]

**81**, 053827 (2010). [CrossRef]

21. J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A **84**, 023844 (2011). [CrossRef]

*L*limit, the forces obtained agree exactly with those calculated from the pseudo-gradient approach validating the latter.

*L*limit of the stress tensor calculations is necessary to maintain consistency with the assumed point-like nature of the particle in the pseudo-gradient approach. To see this, note that for a given resonator of radius

*R*and refractive index

*n*, the lowest order approximation to the resonant frequency is the geometric optical condition

*nkR*≈

*L*. At the same time, a point dipole is defined by the the limit

*R*→ 0 with electromagnetic size parameter

_{p}*ρ*=

*kR*kept constant. Combining these two conditions we have

_{p}*nRρ*=

*LR*, which implies that taking

_{p}*R*→ 0 requires that

_{p}*L*→ ∞.

## 4. Conclusion

## References and links

1. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

3. | G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. |

4. | A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. |

5. | A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. |

6. | S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature |

7. | O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. |

8. | A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. |

9. | D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. |

10. | D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. |

11. | O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. |

12. | P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A |

13. | O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A |

14. | Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A |

15. | T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. |

16. | T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science |

17. | A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. |

18. | V. Braginsky and A. Manukin, |

19. | R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A |

20. | M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A |

21. | J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A |

22. | S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel—a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express |

23. | L. Landau, E. Lifshitz, and L. Pitaevskiĭ, |

24. | V. Wong and M. A. Ratner, “Explicit computation of gradient and nongradient contributions to optical forces in the discrete-dipole approximation,” J. Opt. Soc. Am. B |

25. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

26. | L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A |

27. | J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A |

28. | J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A |

29. | M. Abramowitz and I. Stegun, |

30. | J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(140.3945) Lasers and laser optics : Microcavities

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optomechanics

**History**

Original Manuscript: July 1, 2011

Revised Manuscript: September 20, 2011

Manuscript Accepted: October 1, 2011

Published: October 24, 2011

**Virtual Issues**

Vol. 6, Iss. 11 *Virtual Journal for Biomedical Optics*

Collective Phenomena (2011) *Optics Express*

**Citation**

J. T. Rubin and L. Deych, "On optical forces in spherical whispering gallery mode resonators," Opt. Express **19**, 22337-22349 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22337

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