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A harmonic optical trap approximates one coordinate near a stable equilibrium by a quadratic potential energy. Measure displacement $x$ from that equilibrium, set its potential energy to zero there, and take a positive effective spring constant $k$, also called the trap stiffness:The restoring force points left when $x>0$ and right when $x<0$. This is a local model: a Taylor series of a smooth potential energy about a nondegenerate local minimum gives $k=U^{\prime \prime }(0)>0$ as its leading curvature. A real optical trap has a finite capture region; the endlessly rising parabola does not describe its escape boundary. A three-dimensional optical trap can have different stiffnesses along different axes, and nonconservative optical forces can limit the potential energy description.

A magneto-optical trap combines laser cooling with a position-dependent magnetic field and suitable beam electromagnetic polarization to produce a restoring force. Its spatial confinement is additional to the velocity damping of uniform optical molasses.

Neutral buoyancy means a fully immersed body's upward buoyancy equals its downward force from gravity. For a body of fixed volume in a constant-mass density fluid under a uniform gravitational field, this occurs when its mean mass density equals the surrounding mass density. A $0.10\mathrm{m}$ cube of mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$ in the stipulated water has mass $1.000\mathrm{kg}$, so both forces are $9.81\mathrm{N}$.

Ideally it can remain at rest at any depth for which it stays fully immersed and clear of boundaries. Raising or lowering it changes the absolute pressure, but neither its displaced volume nor the difference between the vertical pressure forces. There is therefore no vertical restoring force toward a preferred depth in this model. Neutral buoyancy is a vertical force balance, not a claim about resistance to tilting or about a compressible body whose volume changes with pressure.

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].

Optical molasses surrounds slow chemical atoms with opposing red-detuned laser beams, usually three pairs along perpendicular axes. In the elementary Doppler cooling picture, each pair damps one component of velocity. The result resembles viscous drag, even though the atoms are moving through a vacuum rather than a liquid.

Uniform balanced optical molasses supplies velocity damping, without a restoring force towards a chosen position. An atom at rest has zero mean force wherever it sits within the uniform illuminated region. Without noise or external forces, $m\dot{v}=-\alpha v$ givesThe atom approaches a position determined by its initial condition, not the beam intersection. Recoil-driven diffusion and gravity eventually remove atoms from finite beams. A long residence time in optical molasses is therefore compatible with the absence of position trapping. An optical trap provides spatial confinement by a different force mechanism; a magneto-optical trap adds a position-dependent magnetic field and suitable beam electromagnetic polarization to create a restoring force.

For a bead in a stationary viscous liquid, linear drag often dominates inertia on the observed timescale. Neglecting both inertia and the random thermal force, the deterministic force balance givesHere $\zeta >0$ is the drag coefficient and $\tau$ the optical trap relaxation time: the time to reduce the deterministic displacement by a factor $e$. The bead approaches equilibrium without oscillation in this overdamped model. The approximation requires times long compared with the inertial time $m/\zeta$ and, for separated timescales, $m/\zeta \ll \zeta /k$, where $m$ is the bead's mass. For a no-slip spherical bead of radius $a$ in an unbounded liquid of dynamic viscosity $\eta$, Stokes drag law gives $\zeta =6\pi \eta a$. Nearby walls change that drag coefficient.

For example, $a=0.5\mu \mathrm{m}$, $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ and $k=0.01pN/nm=10^{-5}N/m$ give $\zeta \approx 9.42\times 10^{-9}\mathrm{N}s/m$ and $\tau \approx 0.942\mathrm{ms}$. A displacement of $100\mathrm{nm}$ gives a restoring force of magnitude $1\mathrm{pN}$, and its deterministic displacement drops to $100/e\approx 36.8\mathrm{nm}$ after one optical trap relaxation time. These are predictions for the stated model, not measured trajectories.

The figure uses $q=x/\ell$, $\kappa =k/k_0$ and $s=t/(\zeta /k_0)$, with arbitrary positive reference length $\ell$ and stiffness $k_0$. Thus $U/(k_0{\ell }^2)=\kappa q^2/2$, $F/(k_0\ell )=-\kappa q$ and, from $q(0)=1$, $q(s)=e^{-\kappa s}$. At fixed drag, greater stiffness gives both a steeper potential energy well and faster relaxation. Thermal Brownian motion is omitted from these deterministic curves.

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.

Average the equations in part b and make exactly the prescribed scale-separation approximation: the retained perturbations vary on lengths much greater than $\ell$, and perturbations at the tangle scale are omitted. Products of two retained perturbations can then be neglected. Writing $\mathbf{v}=\delta \mathbf{u}$, $m=\delta M$ and $\pi =\delta p$, the linearized equations areNo unmodelled small-scale response has been inserted into this closure.

Introduce the fluid displacement $\mathbf{\xi }$ by $\mathbf{v}={\partial }_t\mathbf{\xi }$ and choose $\nabla \cdot \mathbf{\xi }=0$. Integrating the magnetic second-moment tensor equation giveswhere $s$ is a time-independent integration tensor set by the initial stress. For a pure oscillatory normal mode, $s=0$. Its restoring force is therefore $v_A^2{\nabla }^2\mathbf{\xi }$: the isotropic magnetic tangle has the same transverse response as an incompressible elastic medium with shear modulus ${\rho }_0{v}_A^2$.

For a plane wave proportional to $e^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, with $\omega \ne 0$, incompressibility gives $\mathbf{k}\cdot \hat{\mathbf{v}}=0$. The tensor equation and magnetohydrodynamic momentum equation becomeDotting the second equation with $\mathbf{k}$ yields $\hat{\pi }=0$ for $k\ne 0$, since $k_i{k}_j{\hat{m}}_{ij}=0$. Substitution then givesThere are two independent transverse polarizations. The phase speed is $v_A$ in every direction, and the two signs of $\omega$ give group velocities $\pm v_A\hat{\mathbf{k}}$. These magnetoelastic waves in an isotropic magnetic tangle exist because a shear deformation distorts the second moment of the tangled magnetic field, producing a restoring magnetic tension even though its mean is zero.

For comparison, a uniform magnetic field with Alfvén velocity ${\mathbf{B}}_0$ gives the incompressible Alfvén wave relation ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_0{)}^2$. Its frequency depends on the direction of $\mathbf{k}$, and energy propagates along ${\mathbf{B}}_0$. Here the averaged restoring response is isotropic: there is no preferred mean-field direction. Both systems resist transverse deformation through magnetic tension, rather than through compressive sound.

The propagating branch is not the entire perturbation space. For example, $\mathbf{v}=0$, $m_{ij}=f(\mathbf{r}){\delta }_{ij}$ and $\pi =f(\mathbf{r})$ solve the linearized equations without propagation, for any small, slowly varying $f$. Such a static stress perturbation in magnetoelastic wave equations balances a pressure gradient. Thus the requested wave conclusion applies to the oscillatory branch; it must not exclude stationary initial stress perturbations.

By reflection symmetry the vertical gravitational field of the self-gravitating incompressible slab vanishes at its mid-plane. Inside the slab, the gravitational Poisson equation therefore integrates to $d\psi /dz=4\pi G\rho z$. Treat the star as a test particle, so its own mass does not alter that field. Its vertical stellar orbit in a uniform slab obeysHence $z(t)=a\cos (\omega t)$ with $\omega =\sqrt{4\pi G\rho }$. The first mid-plane crossing occurs after one quarter of this simple harmonic motion period:Its independence of $a$ follows from the linear interior restoring force. The calculation neglects drag and changes to the slab, and it uses the infinite planar approximation rather than the field of a finite Galactic disk.

The right spring changes length by $u_{n+1}-u_n$ and pulls atom $n$ to the right with force $C(u_{n+1}-u_n)$. The left spring contributes $-C(u_n-u_{n-1})$. Adding these restoring forces and applying Newton's second law givesUniform displacement makes both spring extensions vanish, as translational invariance requires. Search for a normal mode $u_n=u_0{e}^{i(kna-\omega t)}$. Substitution yieldsOne representative first Brillouin zone is $-\pi /a\le k\le \pi /a$. This is one positive-frequency acoustic branch: near zero wavevector, $\omega \simeq a\sqrt{C/m}|k|$, so the sound speed is $v_s=a\sqrt{C/m}$. Opposite signs of $k$ label opposite propagation directions, not a second polarization branch.

The model assumes small displacements, harmonic nearest-neighbor forces, one scalar displacement per atom and uniform equilibrium spacing. It excludes damping, anharmonic expansion and transverse polarizations. In a three-dimensional monatomic crystal there would normally be three acoustic polarizations; that counting cannot be imported into this explicitly one-dimensional scalar model.

Differentiate the sum of kinetic energy and potential energy using the dot product product rule:The equations of motion make the bracket zero. Hence the mechanical energy is constant in time. The central restoring force can exchange kinetic and potential energy but supplies no change in their sum.

Irrotational motion in this simply connected fluid region has a velocity potential, $u=\nabla \varphi$. Incompressibility gives $\nabla \cdot u=0$, hence $\Delta \varphi =0$.

The moving surface is a material surface: differentiating $y-\eta (x,t)=0$ along a particle gives ${\eta }_t+{\varphi }_x{\eta }_x={\varphi }_y$ at $y=\eta$. Linearizing about a flat stationary surface drops the product term and evaluates at zero, giving ${\eta }_t={\varphi }_y$ there. The Unsteady Bernoulli equation, after dropping quadratic velocity terms and absorbing the time-only constant into $\varphi$, is ${\varphi }_t+p/\rho +gy=0$. The linear capillary pressure at the surface is $p=-T{\eta }_{xx}$ relative to atmosphere, in the sign convention of the source. ThereforeFor $\eta =A\cos (kx-\omega t)$ choose the harmonic potential decaying at infinite depth,Its vertical derivative at zero equals $A\omega \sin (kx-\omega t)={\eta }_t$. Substitution into the dynamic condition gives $-{\omega }^2/k+g+T{k}^2/\rho =0$. ThusThe two positive terms are the gravitational and capillary contributions to the restoring force.

Differentiate the quadratic Lagrangian with respect to coordinates and velocities. The equations areThe velocity-cross terms are Coriolis accelerations; moving $-{\Omega }^2{x}_i$ to the force side identifies the outward centrifugal acceleration. The remaining terms are elastic restoring forces. The sign pair is fixed by the rotating-basis convention in part a.

TODO: biomolecular force assays are user-deferred. The following notes cover the generic physical apparatus, calibrated dielectric handles and synthetic attachments only.

In optical tweezers, a tightly focused laser produces a restoring gradient force on a suitable transparent bead. Scattering also transfers momentum along propagation, so a stable three-dimensional trap needs axial as well as lateral restoration. An objective with high numerical aperture, correctly filled entrance pupil, controlled beam polarization and low-aberration sample chamber are practical design elements. Collect transmitted/scattered light with a condenser and measure bead displacement with calibrated imaging or back-focal-plane photodetection. A steerable beam or translation stage sets relative positions; the force signal comes from bead displacement, not commanded stage displacement alone.

Near a stable operating point, use a harmonic optical trap with $U(x)=k{x}^2/2$. For a slowly changing external force, $F_{\mathrm{ext}}=k⟨x⟩$ in that coordinate; the optical restoring force is $-kx$. Calibrate each axis and the range over which its response is linear. Detector sensitivity must also be known: write $x=\beta (s-s_0)$, with measured detector voltage $s$, offset $s_0$ and position sensitivity $\beta$ in $m/V$. A spring constant in arbitrary detector units is not a force calibration.

Three complementary physical calibrations are useful. At thermal equilibrium the equipartition theorem givesThis needs an independently calibrated position scale and a variance corrected for drift, detector noise and exposure averaging. In an overdamped Newtonian medium, the optical trap Langevin calibration obeysUsing a one-sided position power spectral density per hertz,The corner frequency and spectral amplitude test both relaxation and fluctuation size. If drag is known, $k=2\pi \zeta f_c$. The isolated-sphere value $\zeta =6\pi \eta a$ must be corrected near boundaries; otherwise measured height dependence can be mistaken for a changed trap stiffness. An independent steady optical trap drag calibration moves fluid past the trapped sphere: $k⟨x⟩=\zeta u$. Use small positive and negative speeds to verify slope, offset and force range.

One can also determine position sensitivity without assuming the bulk drag. In a low-frequency sinusoidal-stage optical trap calibration, let the known local fluid velocity amplitude be $u_0$ at drive angular frequency $\Omega$. The coherent bead displacement amplitude isIf the corresponding detector amplitude is $A_s$, thenAfter removing the coherent drive line from the thermal record, combine this sensitivity with equipartition theorem to recover $k$, and then infer $\zeta$ from $f_c$. A published calibration combines thermal spectra and stage-driven motion without taking bead drag as an input.^[ref]

Controls determine whether these formulas measure the apparatus rather than its noise. Record blocked-light offsets and independent fixed-bead noise; calibrate length with known motion; keep sampling and electronics bandwidth adequate around $f_c$; and include exposure filtering and aliasing in a spectral fit. motion blur reduces measured variance and therefore makes naive equipartition theorem overestimate stiffness; additive position noise has the opposite bias. Changing exposure time provides a direct diagnostic.^[ref] Check temperature and laser heating, wall separation, bead size and refractive index, alignment and power stability. Repeat after changing power or optical geometry and compare thermal, spectral and drag estimates. For a synthetic force handle, also measure the attachment's compliance and geometry so that bead motion is not automatically identified with its extension.

A linear spring stores elastic energy $k{x}^2/2$ when displaced by $x$ from its relaxed position. The external force maintaining that displacement is $kx$; its restoring force on the moving attachment has the opposite sign. Its spring constant $k$ has units force per length.

The spring constant is the positive proportionality coefficient in the ideal spring restoring force $F=-kx$. Its SI unit is newtons per metre. With a constant attached mass $m$, it sets the angular frequency $\omega =\sqrt{k/m}$ of simple harmonic motion. Doubling $k$ doubles the restoring force at a fixed displacement, but increases that angular frequency only by a factor $\sqrt{2}$.

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.