What it does
A bead held in an optical trap jiggles because water molecules keep hitting it, and the trap pulls it back like a spring. The stiffer the trap, the smaller and faster the jiggle. Measuring that motion is the standard way to calibrate an optical trap, that is, to find its stiffness k (the force per nanometre of displacement), so that the trap can then be used to measure forces.
This page does the two classic calibrations on a recorded position trace:
- Power spectrum: the spectrum of a trapped bead’s position is a Lorentzian, flat at low frequency and falling as 1/f² above the corner frequency fc. The stiffness is k = 2π γ fc, where γ = 6πηr is the Stokes drag on the bead. The fit also gives the diffusion coefficient D, which should equal kT/γ; if your positions are in volts, comparing the two gives the detector’s sensitivity β in nm per volt.
- Equipartition: the mean thermal energy in the spring is ½kT, so k = kT / ⟨x²⟩. It needs calibrated positions and is spoiled by drift and noise, so it is a useful cross-check.
How to use it
- Paste a position record (one number per line), open a CSV, TSV, text or Excel file, or keep the example. If the first column is time in seconds, the sample rate is set from it; choose the position column if there are several.
- Set the units (nm, µm or detector volts) and the sample rate.
- Give the bead diameter, temperature, viscosity and, near a cover glass, the height of the bead centre above it (Faxén’s correction raises the drag).
- Choose the segment length (longer gives finer frequency steps but fewer averages), how many neighbouring bins to average, the fit range and the model. The aliased model is exact for sampled data and should be your default.
- Read the stiffness, the fit quality, and the two charts; download the spectrum and fit as CSV, or the report.
Worked example
The example is a simulated 1 µm bead in water at 25 °C in a trap of 0.02 pN/nm, sampled at 10 kHz for 3.28 s (32,768 samples). Cut into 32 segments of 1024 and averaged in blocks of 4, the aliased Lorentzian fit from 10 to 4000 Hz gives a corner frequency of 380 ± 7.2 Hz and a stiffness of 0.02003 pN/nm, with χ²/dof = 1.28. The fitted D of 0.4916 µm²/s matches the expected 0.4907 µm²/s. Equipartition gives 0.01999 pN/nm from a variance of 205.9 nm² (rms 14.4 nm), within 0.2 % of the spectrum. Switch to the plain Lorentzian and the corner frequency is overestimated at 467 Hz (0.0246 pN/nm, χ²/dof 3.67) because the fit runs close to half the sample rate, where aliasing bends the spectrum.
Tips
- Sample at least 5 to 10 times faster than the corner frequency, and fit well below any detector or anti-aliasing filter roll-off.
- Start the fit above low-frequency drift and building vibration peaks.
- A χ²/dof far above 1 means the model does not describe the data: check for peaks, filtering or a wrong fit range.
FAQ
Which value should I trust? The power spectrum method needs only the drag (bead size and viscosity) and is insensitive to detector calibration; equipartition needs calibrated positions and is biased by noise and drift. Agreement within a few percent is a good sign.
Do my data leave the computer? No. Everything is worked out in this page; nothing is uploaded.
Where the numbers come from
- Power spectrum calibration, the aliased Lorentzian for sampled data, the analytical 1/P fit and the expected scatter of averaged spectra: Berg-Sørensen, K. and Flyvbjerg, H. (2004), Review of Scientific Instruments 75, 594. Equipartition and Stokes drag: standard statistical physics. Faxén's correction: Faxén (1923), as given in Svoboda, K. and Block, S. M. (1994), Annual Review of Biophysics and Biomolecular Structure 23, 247.
- Checked against scipy.signal.welch and scipy.optimize.curve_fit on the same record, written for the tests. Boltzmann constant: SI exact value.
- The example record is simulated (made up); the water viscosity 0.89 mPa·s is a rounded value.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.