= Rayleigh optical gradient force
{c}
{title2=$\mathbf F_{\mathrm{grad}}=\tfrac14\operatorname{Re}(\alpha)\nabla|E_0|^2$}
The <geometrical optics> account of an <optical trap> adds the <momentum> flux changes of refracted and reflected <light rays>. It applies when the bead is large compared with the surrounding medium's <wavelength>. When the bead is much smaller, an induced <electric dipole> gives a different approximation. For a homogeneous <dielectric> <sphere> of <radius> $a$, define $\mathbf p(t)=\operatorname{Re}[\alpha\mathbf E_0e^{-i\omega t}]$, with $\mathbf E_0$ the peak complex <electric field> and $\alpha$ its SI <polarizability>. In a transparent, nonmagnetic, nondispersive host,
$$
\epsilon_m=\epsilon_0 n_m^2,\qquad
\alpha=4\pi\epsilon_m a^3\frac{n_p^2-n_m^2}{n_p^2+2n_m^2},\qquad
I=\frac12n_m\epsilon_0c|\mathbf E_0|^2,
$$
where $n_p,n_m$ are particle and host <refractive indices>, $I$ is cycle-averaged <light intensity>, $\epsilon_0$ is the <vacuum permittivity> and $c$ is the <speed of light>. The quasistatic <electric dipole> approximation requires $2\pi n_ma/\lambda_0\ll1$, with $\lambda_0$ the <vacuum> <wavelength>, and a field nearly uniform across the bead.
For real quasistatic $\alpha$, building an induced <electric dipole> gives instantaneous <potential energy> $-\alpha E^2/2$. Averaging over one optical cycle gives
$$
U_{\mathrm{grad}}=-\frac14\alpha|\mathbf E_0|^2,
\qquad \mathbf F_{\mathrm{grad}}=-\nabla U_{\mathrm{grad}}
=\frac{\alpha}{2n_m\epsilon_0c}\nabla I.
$$
Thus $n_p>n_m$ gives positive <polarizability> and attraction toward larger <light intensity>; $n_p<n_m$ reverses this contribution. The factor $1/4$ uses a peak field: substituting a <root mean square> field without changing that factor would halve the <force> incorrectly. Dimensionally $\alpha|E_0|^2$ is <energy>, so its <gradient> is <force>.
This leading conservative term does not remove the <optical scattering force>. Even a nonabsorbing induced <electric dipole> reradiates <light>: at fixed <wavelength> and index contrast, its <Rayleigh scattering cross-section> scales as $a^6$, while $\alpha$ scales as $a^3$. A smaller bead can therefore improve the ratio of gradient to scattering <forces> while weakening the well relative to $k_BT$, where $k_B$ is the <Boltzmann constant> and $T$ the <temperature>. Optical <force balance> and resistance to thermal escape are separate questions. Intermediate bead sizes require a wave-scattering calculation; neither a ray calculation nor this small-dipole expression is uniformly valid there.
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