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Power spectrum

The power spectrum analysis of the position of a trapped bead is another alternative to calibrate optical tweezers, and, as we will explain later, it is usually considered to be the most reliable option. The main idea behind is that the frequency content of the particle motion is related to the strength of the trap: as the stiffness increases, the high frequency components start to dominate the movement of the object. Under certain conditions, the shape of the power spectrum of a trapped microsphere is known, and the model, which depends on the trap stiffness, can provide a measurement of k by fitting the curve to the experimental data.

Typically, the characterisitic frequencies of the particle motion go up to several kHz, so a detector bandwidth of at least 10-100 kHz is generally used. Although calibration is also possible with high-speed CCDs operating at low frequencies (1-2 kHz) [ref], the most common option is the use of silicon-based photodetectors, which exhibit large bandwidth and provide high spatial resolution when implemented in back-focal-plane interferometry.

With this position detection technique, a typical signal obtained with a position sensing detector is:

The power spectrum of this signal is given by square of its time-Fourier transform:

Px = Sx(f)*·Sx(f)/Tmsr = |Sx(f)|2/Tmsr             (1)

For a Tmsr = 40 s measurement, with a Nyquist frequency (bandwith) of fNyq = 7.5 kHz, the raw power spectrum is:

The power spectrum of the position of a trapped particle has a characteristic Lorentzian shape when the bead is held in a harmonic potential well:

P(f) = D/2π2/(f2 + f2c)                (2)

The corner frequency fc in the Lorentzian equation is the free parameter which allows us to determine k from the fitting to the experimental data through the equation k = 2πγfc, where γ is the friction coefficient. In addition, when the position tracking is carried out through back-focal-plane interferometry, the fitting allows, as well, the calibration of the detection system, that is, the converstion factor b between nm and volts. The power spectrum in this case is given by:

P(f) = |Sx(f)|2/Tmsr = (1/b2)·|x(f)|2/Tmsr = A/(f2 + f2c)            (3)

where A = D/2π2b2 is the second free parameter from which the calibration factor b can be obtained. D is the diffusion coefficient given by Einstein relation D = kBT/γ, kB is the Boltzmann constant and T the temperature.

Generally, one defines the so-called one-sided power spectrum, which corresponds to the positive-frequency version of (2), called two-sided power spectrum.

Both definitions can be used to obtain b and k, but only the first one provides the correct information about the total energy, related with the area under the curve.

This method is usually considered to be the most reliable way to calibrate the system since the analysis in the frequency domain provides information about different sources of noise, which can be either discarded when doing the fitting or can be eliminated from the experiment if possible.

Although equation (2) (or its one-sided version) describes the behavior of the particle in the trap, different experimental effects can slightly change the shape of the Lorentzian, leading to a certain error in the determination of both k and b [ref]. For a proper calibration all these effects must be considered. Next, we show how taking into account these effects, changes the value of the calibration. The experimental data are

The calibration according to eq. (2), without any extra effect is:

Considering aliasing:

Considering the photodetector transparency at the infrared:

Also, the frequency dependence of the friction coefficient:
Thus, the free parameters change according to the results in the following table:

  fc (Hz)
D (V2/s)
 No effects 691.3
 Aliasing 663.3 0.00198
 Aliasing & Detector
 728.4 0.00229
 Aliasing & Detector & Viscosity
 722.6 0.00218

It is worthy to point out that these effects should be taken into account since they can introduce a remarkable change in fc and D.

Finally, the following plot shows a typical result of the power spectrum of a trapped microsphere and the fitting with all the effects together. In this case, a polystyrene 1.16 micrometer bead was held deep in a homemade microchamber filled with a watery buffer. The characteristic roll-off frequency was f
c = 1100 Hz, which corresponds to k = 75 pN/um.