= Solution
A tightly focused laser produces a strong intensity gradient. A small polarizable particle experiences a gradient <force> toward increasing intensity when the real <polarizability> has the appropriate positive sign; <radiation pressure> from scattering and absorption pushes predominantly along the beam. A high-numerical-aperture objective creates lateral confinement and a sufficiently strong opposing axial gradient for a three-dimensional <optical trap>. Equilibrium is generally shifted from the intensity maximum by <radiation pressure>.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-optical-trap.png]
{title=Gradient forces restore a displaced particle while radiation pressure shifts the axial equilibrium}
{source=paper-2-optical-trap.py}
For a Rayleigh sphere, $\alpha=4\pi\epsilon_m a^3(\epsilon_p-\epsilon_m)/(\epsilon_p+2\epsilon_m)$ in an absolute-permittivity convention, and $\mathbf F_{\rm grad}\simeq\tfrac14\operatorname{Re}\alpha\nabla|E|^2$ for peak complex field amplitude. Near a stable point, $F_x\simeq-k_xx$; the trap depth must exceed <thermal fluctuations> sufficiently to retain the particle. Viscous drag opposes motion, Brownian forcing produces fluctuations, and gravity/buoyancy are normally small at this scale.
For \b[absorbing gold], use its complex wavelength-dependent permittivity: scattering, absorption and heating can frustrate trapping. The existence of an ideal dielectric gradient-force trap does not guarantee stable trapping of every metal <nanoparticle> at every wavelength or power.<Observation of a single-beam gradient force optical trap for dielectric particles by Arthur Ashkin and others>[General <optical trapping> principle; gold-specific absorption qualification is separate]{ref}
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