An optical tweezer is a single-beam optical trap that can localize objects in three dimensions using forces exerted by a strongly focused beam of light [1]. Recently, Stilgoe et al. reported calculations of the axial trapping efficiency of optical tweezers for dielectric spheres, which emphasized the important role of Mie resonances on the stability and strength of optical traps, particularly for high-index particles [2]. These calculations are based on the Lorenz-Mie theory of light scattering by spheres [3], and so should provide exact results for all sphere sizes and compositions. This approach also is numerically efficient, and can be generalized to compute the forces and torques due to arbitrary light fields [4] on objects of arbitrary shape [5, 6]. The calculations reported in [2] were performed with the “optical tweezers toolbox” software suite [7] described in [5], which is formally correct, but relies on numerically inaccurate Hankel functions provided by Matlab^® (versions R2007a through R2008b, The MathWorks, Natick, MA). Consequently, some of the results reported in [2] are incorrect, particularly for very small spheres.

The incident light field, E ₀, the scattered wave, E_s, and the field within the sphere, E_i, are all described in Lorenz-Mie theory as expansions in vector spherical harmonics, [3]

where k = 2πn_m/λ is the wavenumber of light in a medium of refractive index n_m, and where m_p = n_p/n_m is the particle’s relative refractive index.

The incident field’s coefficients, a_nm and b_nm, were computed in [2] through the over-determined point-matching method [8] but can be derived equally well with the Debye-Wolf path integral formalism [4], which we adopt here. The scattering coefficients, p_nm and q_nm, depend on the on the size and composition of the sphere, and on the structure of the light field illuminating it [2, 4, 5]. The interior field’s coefficients, c_nm and d_nm, similarly are computed to satisfy continuity conditions at the sphere’s surface [6]:

where x = ka, and where Ψn(x) and ξ_n(x) are Ricatti-Bessel functions.

Both E ₀(r) and E_s(r) contribute to the Maxwell stress tensor on the sphere’s surface, whose integral yields the force. References [2] and [5] follow Crichton and Marston [9] by performing the integral analytically. The result for the axial trapping efficiency in an optical trap of total power P = ∑^∞ _n=1 ∑ⁿ _m=-n |a_nm|² + |b_nm|² is then [9]

 

Fig. 1. State diagram for dielectric spheres in an optical tweezer of numerical aperture 1.3. Colors in the main diagram correspond to the maximum axial trapping efficiency at each set of conditions, computed on a 144×60 grid. White regions denote conditions for which particles are not stably trapped. Images (a) and (b) show the magnitude of normalized surface fields for the particular conditions indicated on the diagram. (a) corresponds to an unstable point. The sphere at (b) is stably trapped.

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Equation (10) of [5] misstates this result, inconsistently taking the real part of the second term. The distributed software, however, is formally correct [7].

The central results of [2] rely on numerical convergence of the sums in Eq. (6). Unfortunately, the scattering coefficients, p_nm and q_nm, are notoriously difficult to compute accurately [3, 6] because the Ricatti-Bessel functions in which they are expressed have numerically unstable recurrence relations. The expansions’ accuracy can be assessed by checking continuity of the field at the surface using Eqs. (3) through (5). In our implementation, the relative error in the computed fields is smaller than 10^-5, with correspondingly small errors in the force.

Figure 1 shows the maximum axial trapping efficiency, Q_max as a function of the sphere’s radius, a and relative refractive index, m, in an optical tweezer of numerical aperture 1.2. These conditions correspond to those of Fig. 2(b) of [2]. The principal qualitative difference is that Fig. 1 predicts stable trapping for spheres smaller than a = 0.2λ, while [2] does not. The present results are consistent with an analytic computation of the force on a Rayleigh particle in a Gaussian trap [10], and thus would appear to be correct. They also agree quantitatively with results obtained by numerically integrating the Maxwell stress tensor over the surface of the sphere [4], which confirms the correct implementation of Eq. (6). Although these results suggest that even vanishing small spheres can be trapped, the maximum trapping force is limited in practice by saturation of the sphere’s polarization, a nonlinear effect not considered in the linear light-scattering formalism discussed here and in [2].

Explicitly computing the surface fields, such as the examples presented in Fig. 1(a) and (b), reveals smoothly varying field distributions with only subtle differences differentiating stable and unstable points. The results in Fig. 1 therefore highlight both the need for numerical accuracy in predicting optical traps’ performance, and also the importance of small differences in optical and material properties for trapping high-index particles.

Major qualitative and quantitative discrepancies we observe with the results of [2] can be attributed to the computational toolbox’s use of the standard Matlab routine besselh(), which implements the Hankel function H_n ⁽²⁾ (x) that is used to compute the spherical Hankel function ******$h_n^{(2)}(x)=\sqrt{\pi /(2x)}{H}_{n+\frac{1}{2}}^{(2)}(x)={\xi }_n(x)/x$. Taking x = 0.118 as a test case, accurate calculation of Q_max requires the sum in Eq. (6) to be computed to at least n = 5 [3, 6], with h ⁽²⁾ ₅(0.118) = 2.1996469 × 10^-9 -i3.5032872 × 10⁸. The corresponding result obtained with Matlab is 2.7184 × 10^-8 - i3.5033 × 10⁸, whose real component is in error by a factor of ten. Such numerical errors diminish with increasing x and are negligible for x > 1. Other quantitative differences between Fig. 1 and Fig. 2(b) of [2] are likely due to the resolution of the sampling grid and slight differences in the description of the incoming wave.

This work was supported in part by the NSF through Grant Number DMR-0606415 and in part by a grant from the Keck Foundation. B.S. acknowledges support from a Kessler Family Foundation Fellowship.