1. Introduction

We would like to thank the authors for correcting our error in equation (10) in Ref. [1]. As they point out, the code itself is correct; Crichton and Marston’s expression [2] is used literally, after converting the vector spherical wavefunction (VSWF) [1] amplitudes from our convention to theirs. The reader should beware that there are at least three different normalizations, at least four different sign conventions, and the choice of exp(imϕ) versus cos(mϕ) and sin(mϕ), giving a multitude of possible conventions for the VSWFs; the error noted above resulted from an error in conversion between such conventions.

However, while we do not dispute their comments on the calculation of spherical Hankel functions (the actual values depend on the combination of version and platform), we believe that their conclusions concerning the impact of errors in the calculation of spherical Hankel functions using the standard Matlab (The Mathworks, Natick, MA) function besselh are demonstrably wrong. It should be noted that the convergence criteria cited by the authors are intended for particles of moderate size, while the example case given by the authors is for a size parameter x = ka = 0.118, or a particle of radius a = 0.0188λ. A particle of this size is generally accepted as being sufficiently small so that the Rayleigh approximation—taking only the electric dipole term in the Mie coefficients or T-matrix as non-zero—is reliable [3].

The convergence of the calculations can be easily tested, by changing the n _max at which the T-matrix (or Mie coefficients) is truncated [1]. For the example case of x = 0.118, a relative refractive index of m = 1.59/1.33, and a circularly polarized beam focussed by an objective lens of numerical aperture NA = 1.3, the convergence with increasing n _max is shown in table 1. The error in the calculation of h ⁽¹⁾ ₅(x) is entirely irrelevant, since round-off error ensures that it does not contribute to the final result at all. The results obtained with n _max = 1 are sufficient for almost all practical purposes, especially when the likely uncertainties in particle size and refractive index, the effect of aberrations in the optical system, and so on, are considered. The Rayleigh formula [3] gives Q _max = 2.22 × 10^-5, the same as the tweezers toolbox, and z ₀ = 4.42 × 10^-4 λ, which is about 1 atomic diameter larger. Since the toolbox finds the optical force by calculating the difference between the incoming and outgoing momentum fluxes [1], a relatively large round-off error can result when the force is sufficiently small. This limits the smallest feasible particle radius to about an atomic radius.

Table 1. Convergence of trap strength Q _max and equilibrium position z ₀ with increasing n _max. The differences between the calculated values and the final converged values (Q _max = 2.22×10^-5 and z ₀ = 2.92×10^-4 λ) is shown.

View Table

The genesis of the erroneous conclusion that the results presented in [4] are in error seems to be an interpretation that the white regions in Fig. 2 in Ref. [4] are where the optical forces do not result in trapping. This is incorrect—these regions are where either the optical forces do not result in trapping, or the trap strength is very small (for example, in Fig. 2(c) in Ref. [4], the lowest contour is a trap strength of Q _max = 1.27×10^-3). Although we may have contributed to this by not explicitly stating this in the text, it is indicated in the color scale bars in the figures. We only briefly discussed the limits of trapping of Rayleigh particles due to Brownian motion in the text since this was not relevant to the main topic of the paper, the effect of Mie resonances in optical trapping. In the white region along the left and lower borders of the figures, the toolbox correctly reproduces the forces given by the Rayleigh approximation, and predicts trapping as expected.

It is also worth noting that the trap strength is not necessarily the most useful parameter describing how well Rayleigh particles are trapped. The trapping potential—the energy required for the particle to escape from the trap, which can be compared with the thermal energy—is often far more informative [5].

Finally, errors in the calculation of the special functions can cause serious problems. Errors for very large n and m are not of great concern here, since calculations of optical forces using the toolbox which require such values are, by and large, computationally infeasible, but should be kept in mind when dealing with very large particles. More potentially troublesome is the failure of calculation of spherical Bessel or Hankel functions for strongly absorbing particles of large size. While such particles cannot be trapped in conventional optical tweezers, the forces may well be of interest. The calculated forces on weakly or moderately absorbing particles of moderate size appear to be correct, or at least physically reasonable.