A tightly focused Gaussian beam of waist w₀ has radius w(z) = w₀√(1+(z/z_R)²), where the Rayleigh range z_R = πw₀²/λ marks the distance over which the beam stays roughly focused. Two axial optical forces act on a dielectric bead sitting on this axis:
Gradient force F_grad(z) ∝ −(z/z_R) / (1+(z/z_R)²)² (restoring — pulls the bead toward the focus)
Scattering force F_scat(z) ∝ 1 / (1+(z/z_R)²) (always forward — radiation pressure pushes downstream)
Net force F_net(z) = F_grad(z) + F_scat(z)
Near the focus the forward scattering push always wins (F_grad = 0 at z = 0), so the bead is never trapped exactly at the waist — the stable equilibrium z_eq sits slightly downstream, where the growing restoring gradient force first overtakes the falling scattering force. Near z_eq the net force is linear, F_net(z) ≈ −k(z − z_eq), a Hookean spring with trap stiffness k = −dF_net/dz — the same quantity measured experimentally by tracking a bead's thermal (Brownian) fluctuations and applying the equipartition theorem, ½k⟨z²⟩ = ½k_BT.
- Numerical aperture (NA) — a higher NA gives a smaller waist and shorter Rayleigh range, which sharpens the axial intensity gradient. This is exactly why real single-beam ("Type II") optical tweezers need a high-NA oil-immersion objective (NA ≳ 1.2 in water): below a critical NA the gradient force can never catch up with the scattering force at any z, the curve above never crosses zero, and the bead is simply blown downstream — a real, well-documented limitation of single-beam trapping.
- Laser power — both forces scale linearly with intensity, so power sets the overall force scale and hence trap stiffness (stiffer trap, same equilibrium point) without changing whether the trap is stable.
- Bead radius — the scattering force grows faster with particle size than the gradient force does, so larger beads are harder to trap axially at fixed NA and power — a real constraint on the usable size range of optical tweezers (typically ~0.2–2 µm beads).
- Brownian motion — toggles thermal kicks from the surrounding fluid; watch the bead jitter around z_eq inside the force well instead of sitting perfectly still, exactly as a real trapped bead does.
The equilibrium and stiffness above are found numerically each frame by scanning F_net(z) for its first zero-crossing — no closed-form shortcut, the same way you'd read it off a measured force curve.