= Solution
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 $U(x)=kx^2/2$. For a slowly changing external <force>, $F_{\rm ext}=k\langle x\rangle$ 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 $\mathrm{m/V}$. A <spring constant> in arbitrary detector units is not a <force> calibration.
Three complementary physical calibrations are useful. At <thermal equilibrium> the <equipartition theorem> gives
$$
\boxed{k=\frac{k_BT}{\langle(x-\langle x\rangle)^2\rangle}.}
$$
This 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> obeys
$$
\zeta\dot x=-kx+\xi(t),\qquad
\langle\xi(t)\xi(t')\rangle=2k_BT\zeta\delta(t-t').
$$
Using a one-sided position <power spectral density> per hertz,
$$
\boxed{S_x(f)=\frac{k_BT}{\pi^2\zeta(f^2+f_c^2)},\qquad
f_c=\frac{k}{2\pi\zeta},\qquad\int_0^\infty S_x(f)df=\frac{k_BT}{k}.}
$$
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: $k\langle x\rangle=\zeta u$. 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 is
$$
A_x=\frac{\zeta u_0}{\sqrt{k^2+\Omega^2\zeta^2}}
=\frac{u_0}{\sqrt{(2\pi f_c)^2+\Omega^2}}.
$$
If the corresponding detector amplitude is $A_s$, then
$$
\boxed{\beta=\frac{u_0}{A_s\sqrt{(2\pi f_c)^2+\Omega^2}}.}
$$
After 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.<Calibration of optical tweezers with positional detection in the back-focal-plane by Simon F. Tolić-Nørrelykke and others>[2006 author preprint, abstract: combined thermal/stage calibration and drag-independent sensitivity]{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.<The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur by Wesley P. Wong and others>[§II, equations (12)–(18): variance reduction, temporal filter and aliased spectrum]{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.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2012/paper-1-trap-calibration.png]
{title=Optical trap calibration}
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