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An optical gradient force arises from spatial variation in illumination. In a usual optical trap for a suitable transparent particle with higher refractive index than its surroundings, it draws the particle towards high light intensity. It supplies a restoring force near a stable trap, but three-dimensional confinement also depends on the optical scattering force. The direction and size depend on the particle and field; attraction to the brightest point is not universal. ^[ref].

A detector measures a signal, not a force. One proposed arrangement collects transmitted and scattered light onto a quadrant photodiode, using a normalized difference of opposite segments as a position-sensitive signal. A camera can supply an independent image and bead-height check. Near a chosen operating point, a voltage channel may obey $V-V_0=x/\beta$, but its linear range and sensitivity $\beta$ must be measured at that bead size, focus and collection geometry. An arbitrary scattered-light intensity is not automatically proportional to displacement.

A useful proposed control sequence is to record detector offsets with illumination blocked, calibrate image scale against known stage travel, and test the detector response with independently known bead-relative motion. Moving a free trapped bead's chamber is not by itself a known bead displacement: fluid drag and trap stiffness determine the shift. Use a separate immobilized bead or an independently imaged trajectory for the position response, then check whether the response transfers to the actual trapped bead. Repeat the calibration after changing trapping power or optics.

Measure the acquisition frequency response and choose a sample rate that resolves $f_c$, with sufficient signal bandwidth and control of aliasing. Compare thermal calibration of an optical trap with optical trap drag calibration, sweep small positive and negative flow speeds, inspect for multiple beads, and repeat at several wall distances. Check temperature and drift before interpreting a power-dependent stiffness: a changed fluid viscosity can change $f_c$ even at unchanged $k$. These are proposed diagnostics of the model, not procedures attributed to the 1970 or 1986 papers.

Optical tweezers hold and move a small particle using the force exerted by a tightly focused beam of light, usually from a laser. They can position transparent beads in a liquid and manipulate suitable biological cells. Attaching a molecule to a trapped bead turns the bead into a handle: a measured bead displacement, together with a calibrated spring constant, reveals the force exerted on the attachment. This makes microscopic manipulation and measurement possible without a mechanical tool touching the bead. Living biological cells were manipulated with infrared beams by Ashkin, Dziedzic and Yamane in 1987; Svoboda and Block used trapped beads to measure forces from individual molecules in 1994. ^{[ref] [ref]}.

Why can light pull a bead towards a focus? Light carries momentum. A transparent bead with a higher refractive index than the surrounding liquid bends transmitted light rays by refraction, changing their momentum; conservation of momentum requires an opposite transfer to the bead. In the geometrical optics picture, unequal illumination of a displaced bead produces a net restoring force towards the brighter region. For a particle much smaller than the wavelength, a physical optics description replaces that ray picture. The detailed force depends on particle size, refractive index, beam shape and wavelength; the qualitative attraction does not supply a universal formula for every particle.

The optical gradient force draws a suitable transparent bead towards high illumination, while the optical scattering force generally pushes along beam propagation. A stable three-dimensional optical trap needs a balance with a restoring force in each direction: transverse confinement alone does not ensure axial confinement. Strong focusing can make the backward axial gradient contribution large enough to balance the forward scattering contribution, with the equilibrium displaced from the focus. Higher refractive index alone is insufficient; absorbing particles, different index contrasts and different beam geometries can behave differently. ^[ref].

Historically, Ashkin's 1970 experiments accelerated and trapped particles using radiation pressure. The 1986 paper by Ashkin, Dziedzic, Bjorkholm and Chu demonstrated the single-beam gradient optical trap for transparent particles. These are distinct steps: the earlier work established optical manipulation, and the later work supplied the tightly focused single-beam arrangement. ^{[ref] [ref]}.

Near a stable equilibrium, a harmonic optical trap behaves like a spring; thermal calibration of an optical trap uses its Brownian motion to determine the effective spring constant. The later passage through a translating optical trap sections instead consider a moving localized potential energy, which need not remain harmonic throughout a passage.

Because the sphere lies below the optical axis, a ray passing through its upper half samples a brighter part of the Gaussian beam than a comparable ray through its lower half. For a particle with higher refractive index than its surroundings, the stronger upper ray is deflected downward overall and transfers an upward reaction to the particle. The weaker lower ray gives a smaller downward reaction. Their transverse sum is therefore upward, toward the brighter axis.

If the displacement reverses, the force reverses, so the central position is transversely stable. Summing all the rays gives an optical gradient force toward the intensity maximum. There is also a forward optical scattering force. A tightly focused beam can have an axial restoring component that balances it, generally at a small displacement downstream of the ideal focus.^[ref]

Three-dimensional stability requires the restoring axial component to exceed the forward force over part of the trajectory. A Gaussian transverse intensity profile alone does not prove that condition; insufficient focusing can yield transverse confinement while the particle still moves downstream.

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 $\mathbf{p}(t)=\mathrm{Re}[\alpha {\mathbf{E}}_0{e}^{-i\omega t}]$, with ${\mathbf{E}}_0$ the peak complex electric field and $\alpha$ its SI polarizability. In a transparent, nonmagnetic, nondispersive host,where $n_p,n_m$ are particle and host refractive indices, $I$ is cycle-averaged light intensity, ${ϵ}_0$ is the vacuum permittivity and $c$ is the speed of light. The quasistatic electric dipole approximation requires $2\pi n_{m}a/{\lambda }_0\ll 1$, 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 givesThus $n_p>n_m$ 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 $k_{B}T$, 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.

At an equilibrium ${\mathbf{r}}_{\ast }$, 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 $\dot{\delta \mathbf{r}}=MJ\delta \mathbf{r}$, 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.