Light scattering by small objects

Although light scattering by small objects has been studied in great detail, analytic results are available only for a few specific cases. Lorenz-Mie theory provides exact expansions in vector spherical harmonics for the scattering of plane waves by spheres. More general systems can be treated with T-matrix theory (23).

From Eq. (17), the object is illuminated by a superposition of plane waves. The component propagating along $\hat{\boldsymbol{s}}$ with polarization $\mu$ has the complex amplitude $A^{\text{I}}_\mu(\hat{\boldsymbol{s}})$ . This component contributes $\vec{A}^{\text{S}}_\mu(\vec{r},\hat{\boldsymbol{s}})$ to the scattered field at point $\vec{r}$ , where the subscript $\mu$ refers to the polarization of the incident light. The total scattered field at $\vec{r}$ is then

$$\vec{A}^{\text{S}}(\vec{r}) = \sum_{\mu = 1}^2 \int_\Omega \vec{A}^{\text{S}}_\mu(\vec{r},\hat{\boldsymbol{s}}) \, d\Omega$$ (19)

and the total vector potential is

$$\vec{A}(\vec{r}) = \vec{A}^{\text{I}}(\vec{r}) + \vec{A}^{\text{S}}(\vec{r}).$$ (20)

This can be substituted into Eqs. (2) through (4) to obtain the optical force and torque on the object due to the particular hologram implemented with the DOE.

In many cases of practical interest, the object may be modeled as a smooth sphere made of a homogeneous isotropic material with complex refractive index $n_p$ . The imaginary part of $n_p$ , also known as the extinction coefficient, characterizes the material's absorptivity. Although absorption plays a critical role in computations of optically-induced torque (24), its influence on the forces exerted by optical traps has received less attention (25,26,27).

Illuminating such a sphere with $A^{\text{I}}_1(\vec{r},\hat{\boldsymbol{z}})$ , which describes a plane wave propagating along $\hat{\boldsymbol{z}}$ and linearly polarized along $\hat{\boldsymbol{x}}$ , gives rise to a scattered field

$$\vec{A}^{\text{S}}(\vec{r},\hat{\boldsymbol{z}}) = \mathbb{LM}_1(\vec{r}) \, \vec{A}^{\text{I}}_1(\vec{r},\hat{\boldsymbol{z}}),$$ (21)

where the Lorenz-Mie scattering tensor for $\hat{\boldsymbol{x}}$ -polarized light is given (28,29) as an expansion,

$$\mathbb{LM}_1(\vec{r}) = \sum_{n=1}^\infty \left( i a_n \, \vec{N}^{(3)}_{e1n}(\vec{r}) - b_n \, \vec{M}^{(3)}_{o1n}(\vec{r}) \right),$$ (22)

in the vector spherical harmonics

$$\vec{M}^{(3)}_{o1n}(\vec{r}) = \frac{\cos\phi}{\sin\theta} \, P^1... ...phi \, \frac{dP^1_n(\cos\theta)}{d\theta} \, j_n(kr) \, \hat{\boldsymbol{\phi}}$$ (23)

and

$$\begin{multline} \vec{N}^{(3)}_{e1n}(\vec{r}) = n(n+1) \, \cos\phi \,P^1_n(\co... ...kr}\frac{d}{dr}\left[r j_n(kr)\right] \, \hat{\boldsymbol{\phi}}. \end{multline}$$

Here, $P^1_n(\cos\theta)$ is the associated Legendre polynomial of the first kind, and $j_n(kr)$ is the spherical Bessel function of the first kind of order $n$ . The expansion coefficients in Eq. (22) are given by (28)

$$a_n = \frac{m^2 j_n(mx) \left[x \, j_n(x)\right]^\prime - j_n(x) ... ..., h^{(1)}_n(x)\right]^\prime - h^{(1)}_n(x) \left[mx \, j_n(mx)\right]^\prime},$$ (24)

where $m = n_p/n_m$ is the particle's relative refractive index relative, $x = ka$ is its size parameter, $h^{(1)}_n(x)$ is the spherical Hankel function of the first type of order $n$ , and where primes denote derivatives with respect to the argument. Similarly,

$$b_n = \frac{j_n(mx) \left[x \, j_n(x)\right]^\prime - j_n(x) \lef... ..., h^{(1)}_n(x)\right]^\prime - h^{(1)}_n(x) \left[mx \, j_n(mx)\right]^\prime}.$$ (25)

Once these scattering coefficients are known, equivalent results for other incident directions, $\hat{\boldsymbol{s}}$ , and polarizations, $\mu$ , can be obtained through coordinate rotations.

Figure 2: (a) Magnitude $Q_{\text {max}}$ and (b) range $r_{\text {max}}$ of the axial trapping force for a sphere of radius $a$ and relative refractive index $n_p/n_m$ in an optical tweezer created from light of wavenumber $k$ . The solid curve denotes marginal trapping conditions with $Q_{\text {max}} = 0$ . The dashed curve in (b) indicates conditions for which the trap's axial well extends 1  $\unit{\mu m}$ .

From the perspective of practical implementation, the sum in Eq. (22) converges after a number of terms, $n_c = x + 4.05 \, x^{1/3} + 2$ , that depends on the particle's size through $x$ (29,23). Computing additional terms does not improve accuracy because of the accumulation of round-off error. Making matters more difficult, the numerical implementations of the Bessel functions in at least some standard mathematical libraries suffer from large errors at large indexes $n$ , and either large or small arguments $x$ . The error can be assessed by computing the discontinuity in the total field just inside and just outside the particle's surface. The implementations in recent versions of such commercial general purpose scientific computing packages as Mathematica, IDL and Matlab lead to relative discontinuities as large as 10 percent at the surface of a 100 nm diameter titania sphere in water. To avoid such errors, we employed the more robust numerical techniques described by (30) and (31) and typically obtain convergences to within $10^{-5}$ over the entire range of sizes considered.