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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 22337–22349
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On optical forces in spherical whispering gallery mode resonators

J. T. Rubin and L. Deych  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 22337-22349 (2011)
http://dx.doi.org/10.1364/OE.19.022337


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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

Understanding cavity optomechanical phenomena depends on correct representation of the optical force exerted by the cavity modes. In the case of free-propagating optical fields (i.e. laser beams), the force on a subwavelength object (dipole) is naturally separated into gradient and scattering components [20

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

]. The gradient component is analogous to the force on a static dipole, which tends to draw a particle into regions of greater field intensity. It can be presented as the gradient of the electromagnetic energy of the polarized particle and is therefore conservative. The scattering component results from radiation pressure and is expressed in terms of the momentum flux impinging on the particle per unit time. This force is non-conservative because it results from the process of irreversible exchange of momentum and energy between the particle and optical field. Due to its conceptual simplicity and apparent universality, this paradigm has become firmly engrained in the current literature and has been accepted as a framework for calculating optical forces also due to cavity modes. The effects of spatial confinement are taken into account by using cavity modes to represent electromagnetic field while allowing the resonant frequencies of the cavity to depend on mechanical degrees of freedom.

2. Optical force on a small polarizable particle

2.1. Gradient force: thermodynamic derivation

2.2. Gradient force: direct derivation

An alternative derivation, which is also commonly encountered in the textbooks, is based on a model of an electric dipole as a system of equal and opposite charges ±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 = αE. 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,rp), where r and rp 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: F = q[E(rp + d/2, rp) − E(rpd/2,rp)] (see Fig. 1). By taking the limit |d| → 0, keeping |p| = q|d| constant, the force derived in this case is:
F=[(pr)E]r=rp
(7)
where ∇r refers to a gradient with respect to field coordinates 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 rp to coincide with 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.

Fig. 1 Set up for evaluating the force on a dipole modeled as a system of equal and opposite charges. The distance between the charges will be taken to zero.

Equation (7) can also be transformed into a “gradient” form
F=12αr|E(r,rp)|2|r=rp.
(8)
which, however, differs from Eq. (6). Unlike the latter, Eq. (8) involves taking the gradient of the function of two variables, and, therefore, the expression α|E(r,rp)|2/2 cannot be interpreted as a potential energy unless E is independent of rp. Correspondingly, the force calculated according to Eq. (6) does not have to be conservative.

To simplify terminology and notations in the subsequent consideration we will call the operation presented in Eq. (8) a “pseudo-gradient” and will use notation ∇̃ to represent it.

2.3. Total force on a dipole

3. Optical force of a WGM resonator

3.1. WGMs of a single spherical resonator

Fig. 2 Coordinate systems used for evaluation of optical forces together with schematic presentation of the resonator, WGM, and the dipole. The axis always connects the center of the resonator and the point of observation. When using this coordinate system to calculate the forces, this axis passes through the center of the particle.

For calculation of optical forces due to this WGM it is also convenient to find an expression for its field in a coordinate system with polar axis connecting the center of the sphere and the point of observation (XYZ′ system in Fig. 2). Such an expression can be obtained using rotational properties of VSH [25

25. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

] expressed as
Xl,m(θ,ϕ)=m=llDm,m(l)(α,β,γ)Xl,m(θ,ϕ)
(12)
where Dm,m(l)(α,β,γ) is the Wigner D function and α,β,γ are the Euler angles specifying the rotation from the unprimed to primed coordinate system. The D-functions are defined by
Dm,m(l)(α,β,γ)=ei(mα+mγ)dm,m(l)(β),
where the function dm,l(l)(β) for m = l can be written as
dm,l(l)(β)=(2l)!(l+m)!(lm)![cosβ2sinβ2]l[cot(β2)]m
(13)
In order to find an expression for the WGM in the primed system one has to apply inverse transformation Dm,m(l)(γ,β,α)=[Dm,m(l)(α,β,γ)]* to the VSH defined in the unprimed coordinate system. Rotation by the angle 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:
E=E0hL(1)mamXL,m
(14)
where
am=Dm,L(L)(0,β,α)=ieiLαdm,L(L)(β)y0+i
(15)
Here we introduced dimensionless variable y0=(ωωL(0))/ΓL(0) representing the relative detuning of the external frequency from the resonance of the WGM with respect to its width. At any point on the polar Z′ axis (θ′ = 0), all XL,m vanish except for XL,±1. Assuming L ≫ 1, and dispensing with the prime on E′, the field can be written explicitly as:
E=E0hL(1)(kr)L4π(a1ξ^++a1ξ^)
(16)
where ξ±=(iθ^±ϕ^)/2 and θ̂ and ϕ̂ are the spherical coordinate unit vectors referred to the global, unprimed system. The magnetic field H = B/μ0 = −i/ωμ0∇ ×E is given by:
H=E0ɛ0μ0L4π[i2LhL(1)(kr)kra0r^+[krhL(1)(kr)]kr(a1ξ^+a1ξ^)]
(17)
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 am. It is important to note that angles 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

The leading term in σcɛ02ωm[(E*˜)E] is of the second order in θ̄ and can be neglected compared to the scattering force, F(s) = σcg〉, which takes the form:
F(s)=𝔉py02+1LkrpnL(krp)ϕ^
(21)
where p = 2k3α0/3 ≪ 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 nL(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)uemϕ̂, where uem = ɛ0〈|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

The problem of determination of the electromagnetic field of the coupled resonator-dipole system is analytically tractable and was solved in 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

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]

]. The dipole is modeled as a small sphere with radius Rp, where kRp ≪ 1, and refractive index np, and the field is found in the form of a general expansion in terms VSHs with all l,m and polarizations. The particle is found to modify the WGM in two significant ways. First, it creates an additional resonance at frequency ωp=ωL(0)+δωL with width Γp=ΓL(0)+δΓL in addition to the ωL(0) resonance of a single sphere, with δωL and δΓL depending only on rp. Second, the steady state of the resonator field associated with the ωp 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 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

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]

]). In addition, the interaction with the particle excites in the resonator WGMs with different 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
am=ieiLϕpdm,l(l)(θp)×{[y0+i]1m±1ΓL(0)Γp[y+i]1m=±1}
(22)
where y = (ωωp)/Γp. The frequency shift and additional broadening of the resonance are [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

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]

]:
δωL=e[α]k36πɛ0ΓL(0)[VL,1(krp)]2;δΓL=p|δωL|
(23)
where VL,1(krp) is the VSH translation coefficient [25

25. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

], which arises when the field scattered by one sphere is expressed in terms of VSH centered about the other, and is given by
VL,m(krp)=i(1)L+132m2L+1hL(krp)(1)(1)L3L/2nL(krp).

3.3.2. Calculation of the force with particle-modified field

In this part of the paper we assess the effects of the particle-induced shift of the resonance frequency and of the changes in the spatial configuration of the field of the resonator on the optical forces exerted by it. To this end we shall analyze the expressions for the force obtained by evaluating Eq. (10) with the field at the location of the particle given by Eq. (14). The role of the pseudo-gradient operator in this equation is to distinguish between field coordinates r and particle coordinates rp even though we calculate the force at the point r = rp. Taking into account that dependence on rp is only contained in the expansion coefficients am, this procedure becomes rather trivial: one needs to find the gradient of all respective expressions treating these coefficients as constants, and after that equate r = rp. Calculating the required gradients we obtain for the pseudo-gradient component of the force: F(pg) ≡ ∇̃〈u
Fr(pg)=14ɛ0|E0|2Lα0(|a1|2+|a1|2)d|hL(1)(kr)|2dr|r=rp
(24)
Fθ(pg)=14ɛ0|E0|2|hL(1)(krp)|2L2α0rpsinθe[a1(a0*a2*)a1(a0*a2*)]
(25)
Fϕ(pg)=14ɛ0|E0|2|hL(1)(krp)|2L2α0rpm[a1(a0*+a2*)+a1(a0*+a2*)]
(26)
The scattering component of the force, F(s), takes in this case the form of
F(s)=ɛ0|E0|2L2α02k33r|hL(1)(kr)|2{θ^m[(a0*(a1a1)]+ϕ^e[(a0*(a1+a1)]}
(27)
A contribution from the remaining term in Eq. (10), which is proportional to ℐm(E* · ∇̃)E remains negligible and will not be considered any further.

One can see that Eq. (24) and Eq. (27) differ significantly from the respective Eq. (19) and Eq. (21) obtained under the assumption of the unmodified WGM. Further analysis of the obtained expression will be performed in two steps. Since it is often assumed that the main effect of the particle on the cavity mode consists in changing the resonance frequency, we first separate this effect. To achieve this we allow the particle to shift and broaden the resonance according to Eq. (23), but will assume that the field coefficients are given by the unmodified Eq. (15) with replacement of (y0 + i)−1 by (Γp/ΓL(0))(y+i)1. In this case the particle can modify the amplitude of the resonator’s field, but does not change its spatial distribution. Then, one immediately sees that in Eq. (24), Fϕ(pg), which contains terms proportional to ℐm[amam′] with mm′ vanishes. For the same reason the θ-component of the scattering force also vanishes.

This result is a clear demonstration of the fact that the polarization energy of the particle cannot be considered a true potential energy even if one neglects the spatial modification of the cavity mode due its interaction with the particle. However, taking this modification into account results even in more drastic changes in the optical force yielding a non-zero azimuthal component of the pseudo-gradient force and a non-zero polar component of the scattering force. The expression for the total force 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
Fθ=2h¯NLΓprpΓL(0)(y02+1)(y2+1)θ¯[δωL(1+yy0)+δΓL(y0y)]
(30)
Fϕ=2h¯NLΓprpΓL(0)(y02+1)(y2+1)[δωL(y0y)+δΓL(1+y0y)]
(31)
Terms proportional to δωL come from the psuedo-gradient force, while δΓL 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 (y2 + 1)−1 in addition to the Neumann function in δωL and δΓL. The role of this factor can be seen as follows. The condition y = 0 is satisfied for some rp = r0 at which driving frequency ω coincides with the particle-induced resonance. If one linearizes y(rp) about this point as y = (rpr0)y′(r0), where
y(r0)=1ΓpdδωLdrp(1+yp)
If p is sufficiently small, then in the region where y′(r0) can be considered constant, the factor dδωL/drp in Fr is also constant. Therefore the spatial profile of the force has Lorentzian shape peaked at rp = r0 with width 1/y′. Let us recall that in the unmodified WGM approximation the magnitude of the force monotonically (essentially exponentially) decreases with rp.

The azimuthal force Fϕ is no longer solely due to the scattering contribution. Two different limits are of interest based upon choice of the external driving frequency ω. In the limit ωωL(0), FϕyδωL + δΓL. The magnitude of the pseudo-gradient term exceeds that of the scattering term unless yp. This can only happen for very small values of δωL, when the magnitude of the force is also very small. When yp, the pseudo-gradient contribution exceeds the scattering force, and the tangential component can be written as Fϕ=(Fϕ(0)ΓL(0))/(ypΓp), where Fϕ(0) is the scattering force in the unmodified WGM approximation as given by Eq. (21). If y satisfies 1y<ΓL(0)/(pΓp), the tangential force exceeds Fϕ(0). When ωωL(0)ΓL(0) the scattering contribution to Fϕ becomes negligible as well. In this case the force can be written Fϕ=Fϕ(0)ΓL(0)/pΓp, which also exceeds the magnitude that would have been obtained in the unmodified WGM approximation. These results show that the force propelling the particle in the experiments like the one 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]

] is not necessarily of scattering origin and might have a pseudo-gradient contribution. The two can be distinguished by their dependence on α0: while the pseudo-gradient force is linear in this parameter, the scattering force is quadratic.

Even in the range of parameters where the pseudo-gradient contribution to the azimuthal component of the force dominates, it remains non-conservative since it imparts net kinetic energy to the particle moving along a closed orbit around the resonator. This occurs because there is a field gradient which pushes the particle in the ϕ̂ direction. When the particle moves to a new point, the field re-adjusts so that there is again a field gradient in the ϕ̂ 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/Γp.

The relative magnitudes of the force components can be analyzed by comparing (L/rp)δωL to dδωL/drp. From the asymptotic expansions for the spherical Nuemann functions and their derivatives in the region L ≫ 1, kr < L, we have nL/nL ≈ −cosh(a) [29

29. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).

] where prime denotes differentiation with respect to argument and a is defined by krp = (L + 1/2)sech(a). Since resonances are in the region krpL/n, where n is the refractive index of the resonator, we have nL/nL ≈ −n and thus (dδωL/drp)/(LδωL/rp) ≈ −2nkrp/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 (y02+1)1 in Fϕ and Fθ. If the system is driven at a frequency close to the ideal Mie frequency, so that y0 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 θ̄. If y0 ≫ 1 on the other hand, then the azimuthal force will be smaller than the radial force by a factor 1/y0, while the polar force is smaller by a factor θ̄y/y0.

It can also be seen that the scattering contribution to the azimuthal and polar forces is in general smaller than the pseudo-gradient contribution due to the fact that δΓL/δωL = p ≪ 1 (where p = 2k3α0/3). In the limit where both the driving frequency and particle induced resonance frequencies are very close to the ideal Mie resonance, so that y,y0 → 0, the scattering contribution to Fϕ becomes appreciable, while it vanishes in Fθ. This is to be expected given that y,y0 → 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]

].

The results of the calculation of the force within the pseudo-gradient approach can be compared with calculations carried out by integrating the Maxwell stress tensor over a surface of the particle, which is assumed to have a small, but finite size. For a field represented by a VSH expansion, the stress tensor integral over a spherical region can be performed analytically and the force given in terms of the VSH expansion coefficients [30

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]

]. For the present case these coefficients are given in Ref. [27

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]

] while full details of the calculations of the force can be found in Ref. [21

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]

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

It is interesting to note that the large 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 nkRL. At the same time, a point dipole is defined by the the limit Rp → 0 with electromagnetic size parameter ρ = kRp kept constant. Combining these two conditions we have nRρ = LRp, which implies that taking Rp → 0 requires that L → ∞.

4. Conclusion

The results of this work have important implications for the quantum theory of optomechanical interaction, which is commonly based on the assumed potential nature of the gradient force. These results are also of importance for proposed optofluidic sensors which rely on a tangential force to drive the particle in orbit around the resonator.

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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]

28.

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010). [CrossRef]

29.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).

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]

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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