## Abstract

Using the Finite-Difference-Time-Domain (FDTD) method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute the bound charge and bound current densities, then employ the Lorentz law of force to determine the distribution of force density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers.

© 2005 Optical Society of America

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### Equations (11)

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(1)
$$\mathit{F}={\rho}_{b}\mathit{E}+{\mathit{J}}_{b}\times \mathit{B},$$
(2)
$${\underline{\mathit{J}}}_{b}=\partial \underline{\mathit{P}}\u2044\partial t={\epsilon}_{o}\left(\epsilon -1\right)\partial \underline{\mathit{E}}\u2044\partial t.$$
(3)
$$\nabla \times \underline{\mathit{H}}=\sigma \underline{\mathit{E}}+\partial \underline{\mathit{D}}\u2044\partial t.$$
(4)
$${\epsilon}_{o}\partial \underline{\mathit{E}}\u2044\partial t=\nabla \times \underline{\mathit{H}}-(\sigma \underline{\mathit{E}}+\partial \underline{\mathit{P}}\u2044\partial t)=\nabla \times \underline{\mathit{H}}-\hat{\underline{\mathit{J}}},$$
(5)
$$\hat{\underline{\mathit{J}}}=\nabla \times \underline{\mathit{H}}-{\epsilon}_{o}\partial \underline{\mathit{E}}\u2044\partial t.$$
(6)
$${\epsilon}_{0}{\epsilon}_{\infty}\partial \underset{\mathit{\u0305}}{\mathit{E}}\u2044\partial t=\nabla \times \underset{\mathit{\u0305}}{\mathit{H}}-\sigma \underset{\mathit{\u0305}}{\mathit{E}}-{\underset{\mathit{\u0305}}{\mathit{J}}}_{p},$$
(7)
$$\hat{\underset{\mathit{\u0305}}{\mathit{J}}}=\left(\sigma \underset{\mathit{\u0305}}{\mathit{E}}+{\underset{\mathit{\u0305}}{\mathit{J}}}_{p}\right)\u2044{\epsilon}_{\infty}+\left(1-1\u2044{\epsilon}_{\infty}\right)\nabla \times \underset{\mathit{\u0305}}{\mathit{H}}.$$
(8)
$$<\underset{\mathit{\u0305}}{\mathit{F}}>=(1\u2044T){\int}_{0}^{T}\left(\underset{\mathit{\u0305}}{\mathit{E}}\nabla \xb7{\epsilon}_{o}\underset{\mathit{\u0305}}{\mathit{E}}+\underset{\mathit{\u0305}}{\hat{\mathit{J}}}\times {\mu}_{o}\underset{\mathit{\u0305}}{\mathit{H}}\right)dt.$$
(9)
$${F}^{\left(\mathrm{edge}\right)}=\frac{1}{4}{\epsilon}_{o}\left(\epsilon -1\right){\mid {E}_{o}\mid}^{2}.$$
(10)
$${S}_{z}\left(y,z=0.5\mu m\right)=\frac{1}{2}{E}_{o}{H}_{o}\mathrm{exp}[-2{(y\u2044{y}_{o})}^{2}].$$
(11)
$$\int {S}_{z}\left(y,z=0.5\mu m\right)dx=\sqrt{\pi \u20448}{E}_{o}{H}_{o}{y}_{o}=0.5\times {10}^{-3}W\u2044m.$$