= Three-dimensional optical trap stability
{title2=$F_i\simeq-k_i\delta r_i,\quad k_i>0$}
At an <equilibrium> $\mathbf r_*$, the total <force> vanishes. A small transverse <displacement> must produce a transverse <restoring force>; a small axial <displacement> must also restore the bead. Downstream of a tight focus, the backward <optical gradient force> can oppose the forward <optical scattering force>. Equality at one point is insufficient: the total axial <force> must decrease as the bead moves downstream through that point. The <equilibrium> can therefore lie downstream of the intensity maximum.
For uncoupled overdamped coordinates, $\zeta_i\dot{\delta r_i}=-k_i\delta r_i$ with $\zeta_i>0$ decays only when each $k_i>0$. More generally, linearizing gives $\dot{\delta\mathbf r}=M J\delta\mathbf r$, where $M$ is the <hydrodynamic mobility matrix> and $J_{ij}=\partial F_i/\partial r_j$ at <equilibrium>. All <eigenvalues> of $MJ$ must have negative real parts for linear asymptotic stability. For a conservative <force> with symmetric positive-definite mobility, a positive-definite <Hessian matrix> of the <potential energy> supplies this condition. A general optical <force> need not possess a global <potential energy>.
Increasing <numerical aperture> admits more strongly converging <light rays> and a steeper axial variation of the field. But a larger nominal <numerical aperture> alone does not establish trapping: illumination of the objective pupil, <optical aberration>, particle properties and the actual in-liquid focus also matter. A <harmonic optical trap> describes local confinement after this stability question has been answered; it does not predict capture from arbitrary positions or an infinite holding <force>.
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