---
title: "Optical Tweezers: Brownian Calibration Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/optical-tweezers-calibration/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/optical-tweezers-calibration/"
last_modified: "2026-10-23"
---

B-X2 · A force scale from a moving bead

# 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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Recorded measurements

## From motion to a force scale

Record a sample to view its measurement and analysis here.

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 is enlarged separately at the reported scale, without clipping or a wall boundary.

[Jun et al. (2014) · calibration comparison](https://pmc.ncbi.nlm.nih.gov/articles/PMC4167536/)
[Sarshar et al. (2014) · displacement and spectral methods](https://pmc.ncbi.nlm.nih.gov/articles/PMC4221290/)
[Pérez-García et al. · finite detector exposure](https://arxiv.org/abs/2305.07245)

## Read the experimental evidence

The simulation uses the assumptions stated above. Read how the historical apparatus measured the effect, what its records supported, and where the interpretation was limited.

[Optical tweezers and force calibration](https://sandboxphysics.com/stories/probe-evidence/optical-tweezers-force-calibration/) [All six reading paths](https://sandboxphysics.com/stories/probe-evidence/)

Physics tutorial

## Use thermal motion as a force calibration

**Background** A dielectric bead trapped in water still receives random kicks from the fluid. Near the trap center, its restoring force acts like a spring.

**Why it matters** The amplitude and speed of the bead’s motion provide two ways to estimate stiffness. They depend on different calibration assumptions.

Start with the essentials

- **Focus question**: How does a bead’s thermal motion reveal the stiffness of an optical trap?
- **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 misconception** If the spectrum is smooth, the stiffness must be accurate.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

A trapped bead becomes a force gauge only over a calibrated displacement range and bandwidth.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)