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Translate the liquid past a bead held in a stationary harmonic optical trap. At steady mean position, take $u$ as the local fluid velocity relative to the fixed trap and $x$ positive downstream. Averaging the overdamped force balance givesFor the invented optical trap Langevin calibration example, $u=50.0\mu m/s$ predicts $⟨x⟩=23.562\mathrm{nm}$, $V-V_0=0.23562\mathrm{V}$, and $\zeta u=k⟨x⟩=0.47124\mathrm{pN}$. Reversing $u$ reverses the mean shift but leaves the ideal thermal variance unchanged. A symmetric speed sweep tests the offset and the linear force range; the slope determines $\zeta /k$ if position is calibrated.

Near a wall, $\zeta =6\pi \eta a$ is not the appropriate drag without a boundary correction; record bead height and direction of motion. Stage velocity must correspond to the local fluid velocity after transients have settled. Escape under increasing flow estimates a maximum holding force in that geometry, whereas small-displacement drag calibration estimates the local spring constant. They are different quantities. The atomic force microscopy similarly needs a calibrated spring constant to convert a measured deflection to force, but its cantilever is a physical spring rather than this optical field.

This relates an thermodynamic equilibrium variance to the local stiffness of an appropriate thermodynamic free energy. At fixed bath temperature, suppose the effective thermodynamic free energy $A(x)$ includes the constraints and the measure for a chosen coordinate. Its thermodynamic equilibrium probability density function is proportional to $e^{-A(x)/(k_{B}T)}$. Near a stable minimum $x_0$, writeThe resulting normal distribution gives the Gaussian fluctuation approximationThis is exact for a quadratic thermodynamic free energy with the stated flat coordinate measure; in general it is a small-fluctuation result. A smaller curvature means a softer restoring response and larger fluctuations. Near vanishing curvature, higher-order terms become important, so the quadratic approximation cannot alone predict an actual infinite variance.

For example optical trap Langevin calibration uses the Brownian displacement of a trapped bead: $A=\kappa (x-x_0{)}^2/2$ gives $\kappa =k_{B}T/⟨\Delta x^2⟩$. A camera or position detector records the fluctuations. Finite exposure averages rapid motion and can bias the measured variance; the measurement protocol matters as well as the thermodynamic equilibrium identity.

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.