Optical Force and Torque on Single and Aggregated Spheres: The Trapping Issue

Abstract

Radiation force and radiation torque stem from the conservation theorems governing the interaction between radiation and particles. As a result, even simple plane waves may yield force and torque that are nonnegligible when studying the dynamics of particles. We show how these forces and torques can be put into formulas of practical use starting with the expansion of the electromagnetic field in a series of vector multipole fields, and adapt the formalism for particles that are fairly well modeled as single or aggregated spheres. Then, we extend the formalism to deal with the case the field is a highly focalized laser beam, that, as is well known, may trap particles within the focal region. Finally, we present our calculations performed by the theory we exposed, finding a convincing concordance as long as comparison can be made with available experimental data.

Appendices

Appendix 1

As anticipated in Sect. 6.2, with the help of the asymptotic multipole expansions it is easily proved that the dyadic terms

$$\begin{aligned} {\hat{\mathbf{r}}^{\prime }}\cdot \left[n^2(\mathbf{E}^{\prime }\otimes \mathbf{E}^{\prime \ast })+(\mathbf{ B}^{\prime }\otimes \mathbf{B}^{\prime \ast })\right]=n^2\bigl ({\hat{\mathbf{r}}^{\prime }}\cdot \mathbf{ E}^{\prime }\bigr )\mathbf{E}^{\prime }+\bigl ({\hat{\mathbf{r}}^{\prime }}\cdot \mathbf{B}^{\prime }\bigr )\mathbf{B}^{\prime } \end{aligned}$$

give a vanishing contribution at the radiation force. In fact, a look to (6.8a) and (6.8b) shows that the asymptotic multipole expansions of $\mathbf{E}^{\prime }_\mathrm{ I}$ and $\mathbf{E}^{\prime }_\mathrm{ S}$ contain the transverse harmonics $\mathbf{Z}^{(p)}_{lm}({\hat{\mathbf{r}}^{\prime }})$ which, according to their definition are orthogonal, in the ordinary vector sense, to ${\hat{\mathbf{r}}^{\prime }}$. This decrees the vanishing of the contribution of the dyadic terms for whatever form of the incident amplitudes $W^{(p)}_{\mathrm{ I}lm}$, even when the latter are substituted by the $\fancyscript{W}^{(p)}_{lm}(\mathbf{R}_{O^{\prime }})$.

Now, we show how it happens that the terms $\mathbf{E}^{\prime }_\mathrm{ I}\cdot \mathbf{E}^{\prime \ast }_\mathrm{ I}$ and $\mathbf{B}^{\prime }_\mathrm{ I}\cdot \mathbf{ B}^{\prime \ast }_\mathrm{ I}$ give a vanishing contribution to the radiation force even when the incident field is not a single plane wave but rather a superposition of plane waves with the same magnitude of $k$ but different direction of propagation, i.e., different ${\hat{\mathbf{k}}}$.

Let us thus assume that the incident electric field is a superposition of plane waves of the kind of (6.28). Thus a typical term that would enter (6.1) is

$$\begin{aligned} I_{E}=\mathrm{ Re}\left[r^{\prime 2}n^2\int \limits _{\Omega ^{\prime }}\mathbf{E}^{\prime }_\mathrm{ I}({\hat{\mathbf{k}}})\cdot \mathbf{E}^{\prime \ast }_\mathrm{ I}\left({\hat{\mathbf{k}^{\prime }}}\right){\hat{\mathbf{r}}^{\prime }}\,\mathrm{ d}\Omega ^{\prime }\right], \end{aligned}$$

Where $\mathbf{E}^{\prime }_\mathrm{ I}\left({\hat{\mathbf{k}}}\right)=E^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right){\hat{\mathbf{u}}}_{{\hat{\mathbf{k}}}}\exp (I\mathbf{k}\cdot \mathbf{r}^{\prime })=\mathbf{ E}^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right)\exp (I\mathbf{k}\cdot \mathbf{r}^{\prime })$. An analogous term $I_{B}$ comes from $\mathbf{B}^{\prime}_\mathrm{ I}$. Since $r^{\prime }$ is large, we can use the asymptotic form of a plane wave [7, 8] so that $I_{E}$ becomes

$$\begin{aligned} I_{E}=\mathrm{ Re}\Bigl \{ \frac{4\pi n^2}{k^2}&\int \limits _{\Omega ^{\prime }}\mathbf{E}^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right)\cdot \mathbf{E}^{\prime \ast }_\mathrm{ PW}\left({\hat{\mathbf{k}^{\prime }}}\right)\\&\times \left[\delta \left({\hat{\mathbf{k}}}+{\hat{\mathbf{r}}^{\prime }}\right)\exp (-ikr^{\prime })-\delta \left({\hat{\mathbf{k}}}-{\hat{\mathbf{r}}^{\prime }}\right)\exp (ikr^{\prime })\right] \\&\times \left[\delta \left({\hat{\mathbf{k}^{\prime }}}+{\hat{\mathbf{r}}^{\prime }}\right)\exp (ikr^{\prime })-\delta \left({\hat{\mathbf{k}^{\prime }}}-{\hat{\mathbf{r}}^{\prime }}\right)\exp (-ikr^{\prime })\right]{\hat{\mathbf{r}}^{\prime }}\,\mathrm{ d}\Omega ^{\prime }\Bigr \}\;. \end{aligned}$$

Due to the properties of the $\delta$-function, the result of the integration is

$$\begin{aligned} I_{E}=\mathrm{ Re}\Bigl \{ \frac{4\pi n^2}{k^2}&\mathbf{E}^{\prime }_\mathrm{ PW}({\hat{\mathbf{k}}})\cdot \mathbf{E}^{\prime \ast }_\mathrm{ PW}\left({\hat{\mathbf{k}^{\prime }}}\right)\left[-{\hat{\mathbf{k}}}\delta \left({\hat{\mathbf{k}}}-{\hat{\mathbf{k}^{\prime }}}\right)+{\hat{\mathbf{k}^{\prime }}}\delta \left({\hat{\mathbf{k}}}-{\hat{\mathbf{k}^{\prime }}}\right)\right.\\ +&\left.{\hat{\mathbf{k}^{\prime }}}\delta \left({\hat{\mathbf{k}}}+{\hat{\mathbf{k}^{\prime }}}\right)\exp (2ikr^{\prime })-{\hat{\mathbf{k}^{\prime }}}\delta \left({\hat{\mathbf{k}}}+{\hat{\mathbf{k}^{\prime }}}\right)\exp (-2ikr^{\prime })\right]\Bigr \}\;. \end{aligned}$$

A quite similar expression, except for the absence of $n$, is obtained for $I_{B}$, so that collecting all the terms we get

$$\begin{aligned} I&= I_{E}+I_{B}\\&= \mathrm{ Re}\Bigl \{ \frac{4\pi }{k^2}\left[n^2\mathbf{E}^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right)\cdot \mathbf{E}^{\prime \ast }_\mathrm{ PW}\left({\hat{\mathbf{k}^{\prime }}}\right)+\mathbf{B}^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right)\cdot \mathbf{B}^{\prime \ast }_\mathrm{ PW}\left({\hat{\mathbf{k}^{\prime }}}\right)\right]\\&\qquad \qquad \times {\hat{\mathbf{k}^{\prime }}}\delta \left({\hat{\mathbf{k}}}+{\hat{\mathbf{k}^{\prime }}}\right)\bigl [\exp (2ikr^{\prime })-\exp (-2ikr^{\prime })\bigr ]\Bigr \} \end{aligned}$$

which is easily seen to vanish, even when ${\hat{\mathbf{k}^{\prime }}}=-{\hat{\mathbf{k}}}$, on account that

$$\begin{aligned} \mathbf{B}^{\prime }_\mathrm{ PW}({\hat{\mathbf{k}}})=-in\,{\hat{\mathbf{k}}}\times \mathbf{E}^{\prime }_\mathrm{ PW}\left({\hat{\mathbf{k}}}\right)\;. \end{aligned}$$

Similar conclusions can be reached, starting from (6.31), for the case of an aberrated laser beam.

Finally we note that, on account of the definition of the amplitudes $\fancyscript{W}^{(p)}_{lm}(\mathbf{R}_{O^{\prime }})$ and of the related amplitudes $\fancyscript{A}^{(p)}_{lm}$, in analogy to Sect. 6.2.2, the radiation torque $\varvec{\Gamma }_\mathrm{ Rad}$ does not get contributions from terms of the form $\fancyscript{W}^{(p)}_{lm}(\mathbf{R}_{O^{\prime }})\fancyscript{W}^{(p^{\prime })\ast }_{l^{\prime }m^{\prime }}(\mathbf{R}_{O^{\prime }})$.

In conclusion, neither a plane wave field nor a focal field gives direct contributions to the radiation force or torque, as they are just the fields in the absence of particles.

Appendix 2

In Sect.6.1 we stated that the calculation of $\mathbf{F}_\mathrm{ Rad}$ and $\varvec{\Gamma }_\mathrm{ Rad}$ requires the knowledge of the field scattered by the particles. We solve the scattering problem for the particles concerned by expanding both the incident and the scattered field in a series of vector multipole fields (see (6.6) and (6.7)). Then, according to Waterman [3] we relate the multipole amplitudes, the incident field $W_{\mathrm{ I}lm}^{(p)}$, to those of the scattered field $A^{(p)}_{lm}$, by the equation [10]

$$\begin{aligned} A^{(p)}_{lm}\left({\hat{\mathbf{u}}}_\mathrm{ I},{\hat{\mathbf{k}}}_\mathrm{ I}\right)=\sum _{p^{\prime }l^{\prime }m^{\prime }}\fancyscript{S}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}W^{(p^{\prime })}_{\mathrm{ I}\,l^{\prime }m^{\prime }}\left({\hat{\mathbf{u}}}_\mathrm{ I},{\hat{\mathbf{k}}}_\mathrm{ I}\right), \end{aligned}$$

(6.38)

that defines the elements of the transition matrix of the particle. Let us stress that, in principle, the sums in (6.6) and (6.7) and thus also (6.38) include an infinite number of terms. In practice, for computational reasons, these sums must be truncated to some suitable $l_\mathrm{ M}$ chosen so as to ensure a fair description of the fields. Therefore, all sums over the multipole order $l$ should be understood to extend up to $l=l_\mathrm{ M}$. Since the particles we deal with either are, or can be modeled as aggregates of spheres, we calculate $\fancyscript{S}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}$ for such aggregates by inverting the matrix of the linear system that is obtained by imposing to the fields the boundary conditions across each of the spherical surfaces [10, 49]. The order of the matrix to be inverted is $2Nl_\mathrm{ M}(l_\mathrm{ M}+2)$, where $N$ is the number of the spheres of the aggregate. The convergence of such kinds of calculations, i.e., the choice of the appropriate $l_\mathrm{ M}$ is studied in Refs. [34]. A comprehensive treatment of all the topics mentioned above related to the calculation of the transition matrix can be found in Ref. [10]. It may be useful to note that for a homogeneous sphere

$$\begin{aligned} \fancyscript{S}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}=\fancyscript{S}^{(p)}_{l}\delta _{pp^{\prime }}\delta _{ll^{\prime }}\delta _{mm^{\prime }}, \end{aligned}$$

and $\fancyscript{S}^{(1)}_l=b_l$, $\fancyscript{S}^{(2)}_l=a_l$, $a_l$ and $b_l$ being the well-known Mie coefficients [4, 10]. The transition matrix is related to the scattering amplitude of the particle $\mathbf{F}({\hat{\mathbf{k}}}_\mathrm{ S},{\hat{\mathbf{k}}}_\mathrm{ I};{\hat{\mathbf{u}}}_\mathrm{ I})$ through the equation

$$\begin{aligned} \mathbf{F}\left({\hat{\mathbf{k}}}_\mathrm{ S},{\hat{\mathbf{k}}}_\mathrm{ I};{\hat{\mathbf{u}}}_\mathrm{ I}\right)=\frac{1}{k}\sum _{plm}(-I)^{l+p}\mathbf{Z}^{(p)}_{lm}\left({\hat{\mathbf{k}}}_\mathrm{ S}\right)A^{(p)}_{lm}\left({\hat{\mathbf{u}}}_\mathrm{ I},{\hat{\mathbf{k}}}_\mathrm{ I}\right) \end{aligned}$$

(6.39)

where the amplitudes of the scattered field are given by (6.38). Equation (6.39) thus allows to express the scattering cross section and the extinction cross section in terms of the elements of the transition matrix.

The usefulness of the transition matrix stems from the its transformation properties under rotation of the coordinate frame. In fact, if the frame $\Sigma ^{\prime }$ attached to the particle rotates to an orientation characterized by the Eulerian angles $\alpha$, $\beta$, and $\gamma$, $\Theta$ for short, the elements of the transition matrix become

$$\begin{aligned} \fancyscript{S}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}=\sum _{\mu \mu ^{\prime }}D^{(l)\ast }_{m\mu }(\Theta )\bar{\fancyscript{S}}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}D^{(l^{\prime })}_{m^{\prime }\mu ^{\prime }}(\Theta ) \end{aligned}$$

(6.40)

where $\bar{\fancyscript{S}}^{(pp^{\prime })}_{lml^{\prime }m^{\prime }}$ denotes the elements of the transition matrix calculated in the frame $\Sigma ^{\prime }$ so that they are independent of the orientation of the particle. As a result, the elements of the transition matrix depend on the orientation through the rotation matrices $D^{(l)}$ only. Then, the averages over the orientation of the particles for all the physically significant quantities result in integrals of products of up to four elements of rotation matrices further multiplied by $P(\Theta )$, the latter being the appropriate function that describes the distribution of orientations. There are relevant cases, e.g., the case of random orientations (random average), or the case of random orientation around a fixed axis (axial average), in which the integration above can be performed analytically. These averaging procedures, although straightforward, yield rather complex formulas so that we refer the interested reader to the details that are fully expounded elsewhere [9, 10, 50].