Optical Tweezers: Brownian Calibration

How does a bead’s thermal motion reveal the stiffness of an optical trap? A stiffer trap confines the bead to smaller thermal excursions and makes its displacement relax faster. At a known temperature, the displacement variance gives stiffness; the relaxation rate gives a second estimate when the fluid drag is known.

Record a bead’s position signal in a liquid, calculate its displacement spectrum and compare two stiffness estimates. Slow sampling and long exposure show where the calibration becomes unreliable.

Watch a trapped bead fluctuate

Drag the bead to relocate the trap; compare soft and tight traps.

Prescribed optical stiffness

\kappa/(\mathrm{pN}/\mu\mathrm{m})

12

Detector sample rate

f_s/\mathrm{Hz}

10000

Exposure as a fraction of interval

100t_e f_s/\%

10

Calibration and advanced settings

Static focus position

x_0/\mathrm{nm}

0

True fluid viscosity

\eta/(\mathrm{mPa}\,\mathrm{s})

1

Viscosity used in analysis

\eta_{\rm fit}/(\mathrm{mPa}\,\mathrm{s})

1

Bead radius

a/\mu\mathrm{m}

0.5

Recorded duration

t/\mathrm{s}

1

Sampling seed

s_0

51

Dragging the bead relocates the trap; every location starts an independent equilibrium simulation. The detector measures motion relative to the beam; no steering transient is modeled.

Model assumptions

A stable optical trap is assumed and its transverse force is harmonic. One overdamped Brownian coordinate evolves with far-wall Stokes drag at a fixed known temperature of 300 kelvin. Detector conversion is known. Samples average a finite exposure of an exact-transition thermal trajectory; the spectrum and estimates use those stored samples. The drawing does not calculate optical forces or establish axial stability. Inertia, wall corrections, hydrodynamic memory, heating, escape, viscoelasticity and molecular tethers are omitted. Apparatus and bead magnification are schematic.

Motion sets a force scale

Simulated measurements

Move a control or reset to start a fresh simulation.

Physical assumptions and sources

One harmonic transverse coordinate at a fixed known temperature. Stable axial trapping is assumed. Far-wall spherical Stokes drag, thermal white noise and known detector distance calibration are used. Exposure is numerically averaged along the physical trajectory. The PSD and both estimates use stored detector samples. There is no molecular force trace or optical-force field calculation. Cell and bead sizes are schematic. Motion and static focus translation use one fixed scene scale across presets, with the displayed magnification relative to bead size. Playback is slowed one thousand times. Moving the focus starts a new equilibrium record; no steering transient or optical-force field is inferred. The QPD signal remains displacement relative to the beam. Faint bead outlines show past recorded samples from the last 2.5 seconds of slowed playback; no extra trajectory is generated.

Jun et al. (2014) · calibration comparison
Sarshar et al. (2014) · displacement and spectral methods
Pérez-García et al. · finite detector exposure

One-sentence intuition: Variance needs temperature and a distance scale; relaxation additionally needs a drag model.

Core mathematical model

Thermal displacement variance

\langle(x-\bar x)^2\rangle=\frac{k_{\rm B}T}{\kappa}

This relation assumes equilibrium and a harmonic coordinate. The measured variance can be reduced by exposure averaging.

Drag and corner frequency

\gamma=6\pi\eta a,\qquad f_c=\frac{\kappa}{2\pi\gamma}

The bead is spherical and far from walls in a Newtonian fluid. The frequency is in cycles per second.

One-sided displacement spectrum

S_x(f)=\frac{k_{\rm B}T}{\pi^2\gamma(f^2+f_c^2)}

The continuous equilibrium reference integrates to the displacement variance. The plotted spectrum is a Welch estimate of the stored voltage samples after distance conversion.

Common difficulties

A good-looking spectrum may still give the wrong force

Typical misconceptionIf the spectrum is smooth, the stiffness must be accurate.

Better mental modelThe displacement calibration, fluid drag and detector bandwidth remain necessary. The simple decay estimator is withheld when the exposure or sampling cannot resolve its assumptions.

Run the experiment

01
Record a resolved trajectory
Use Resolve the corner, acquire a sample and analyze it.
What to observe: The variance and short-lag decay estimate stiffness from the same position samples. Finite duration gives random differences.
02
Reduce the bandwidth
Acquire Slow sampling and Long exposure, then inspect the time series and spectrum.
What to observe: Slow sampling hides the relaxation. Exposure averaging narrows the observed position distribution and biases variance stiffness upward.
03
Use a wrong viscosity
Acquire Wrong assumed drag and compare the two estimates.
What to observe: Changing the viscosity assumed in analysis changes the decay-based stiffness. The variance estimate uses the known temperature instead.

Optical Tweezers: Brownian Calibration

Motion sets a force scale

Displacement from the beam axis

Calibrate from these samples

Use thermal motion as a force calibration

Core mathematical model

Thermal displacement variance

Drag and corner frequency

One-sided displacement spectrum

Common difficulties

A good-looking spectrum may still give the wrong force

Run the experiment

Record a resolved trajectory

Reduce the bandwidth

Use a wrong viscosity

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