Three experimental power spectra, taken with roughly equal trap strengths, using three different laser wavelengths (800, 900, and ). (a) Power spectrum vs frequency . The solid line corresponds to , the behavior of a Lorentzian at large frequencies, and serves to guide the eye. Above approximately , the power spectrum taken with the laser drops off faster than the other two spectra, though it describes the same physical phenomenon. This faster drop-off is similar to the effect of a low-order low-pass filter. (b) Same data plotted as vs frequency to better display the filter effect. The laser wavelength dependence of the filter effect shows that it cannot be explained by the physics of the Brownian motion. The physics of the position detection system is responsible.
Experimental power spectrum obtained with a laser wavelength of and fitted with theoretical power spectrum given in Eq. (5) in the frequency interval . (a) Blocked data points with the fitted shown as a solid line, and a fitted Lorentzian shown as a dashed line. (b) Residual plot, i.e., a plot of experimental values divided by fitted theoretical values, , vs frequency . This plot reveals a significant difference between the experimental spectrum and the theory fitted to it, a difference that is not visible in i (a). The two horizontal dashed lines indicate a standard deviation according to the theory for how the data, shown as dots, should scatter about their expectation value of 1.
Typical cross section of a photodiode operated with a reverse bias. Reproduced from Ref. 27. Not to scale. While the sensitive area measures , is typically some hundred micrometers, and the layer and the depletion zone are typically a few to tens of micrometers thick.
Comparison of experimental data for the characteristic attenuation function with theoretical models for . Systematic deviations of data points from the value of 1 reflect the low-pass filtering caused by the photodiode. The legend gives the correspondence between symbols and laser wavelengths. The theoretical attenuation function does not differ visibly from the constant 1 for . (a) Insufficiency of model given in Eqs. (19), (11), and (13). Points: Blocked experimental spectra for various laser wavelengths, divided by . The parameters in were determined from a fit to the spectra at low frequencies up to or . Lines: of Eq. (19) with . Parameter values were obtained by fitting to data (with the outlier near excluded). This noise peak is caused by some source in the building that we could not get rid of. This results in , , and . The values for and correspond to for a wavelength of . This fit is shown as the solid line through the data. Other lines use the same values for , , and . Upper solid line: , long-dashed line: , short-dashed line: , dotted line: , and dot-dashed line: . (b) Sufficiency of model given in Eq. (19), when used as a phenomenological model without Eqs. (11) and (13). Only two slowest relaxation modes are needed, i.e., . Same experimental data as in (a).
Lin-log plot of the histogram of values visited by one coordinate in the position time series recorded with a laser wavelength of (same data as in Fig. 2), overlayed with a Gaussian of same width and height. The inset shows the same histogram in a lin-lin plot.
Experimental power spectrum obtained with laser wavelength of (same data as in Fig. 2) and fitted with one diffusion mode in the parasitic filter, i.e., in Eq. (20). The frequency range of the fit was . (a) Residual plot of data/fit. The two horizontal lines show a standard deviation known from the theory. The backing of the fit is 82%. Inset: Same residual plot of data/fit after a further block averaging of data. Two horizontal lines show a standard deviation according to theory, here . The data points scatter a little more than ideally expected for normally distributed data. This may be because the data acquisition board is slightly nonlinear, with ripples in its characteristic function of max amplitude, i.e., . We have insufficient precision to resolve such ripples in our power spectrum of the Brownian motion, but we have reached the limit on achievable precision set by these ripples. (b) Experimental power spectrum (data points with error bars) and fitted theory with parasitic filter (solid line). (c) Histogram of experimental power spectral values , measured in units of their expectation values , the latter being the fit shown in panel (b). Dashed line: , the distribution that should follow according to theory (Ref. 11), and is seen here to do, indeed.
Experimental power spectrum obtained with a laser and fitted with , using the phenomenological version of in Eq. (19)without Eqs. (11) and (13), and with just two diffusion modes, . The value of per degree of freedom is 1.023, resulting in a backing of 10%. The maximum frequency fitted to was . (a) Residual plot. All data are shown, but only data of frequencies below (straight vertical line) were fitted to. The dots are . Two horizontal lines show a standard deviation known from the theory for the power spectrum. Note that the data points seem filtered with a roll-off frequency near . Inset: Same residual plot of data/fit after a further block averaging of data. Two horizontal lines show a standard deviation according to theory, here . (b) Power spectrum vs frequency in a log-log plot. Data points with error bars, most of which are too small to be seen. The solid line is the fit. (c) Histogram of experimental power spectral values , measured in units of their expectation values , the latter being the fit shown in panel (b). Dashed line: , the theoretical distribution for .
Fitting parameters that describe the photodiode, obtained from fits with at least 1% support, shown as a function of wavelength. Squares: coordinate. Circles: coordinate. Open symbols: two diffusional modes in the diode filter. Filled symbols: one diffusional mode in the diode filter. (a) The relative amount of the position signal that is detected instantly as a function of laser wavelength. The solid line shows a fit of the theory in Eq. (26). It resulted in and . (b) The relative importance of the slowest relaxation mode in the parasitic filter, shown as its relative amplitude . The solid line shows this ratio for the theory fitted as in (a). [(c) and (d)] Frequencies of the 0th and 1st mode in the diode filter, resulting from fits at various wavelengths. These values are approximately independent of the wavelength, as expected. Note the higher value of for , obtained in fits with only one mode included.
Laser wavelengths and power spectrum frequencies for which the position detection system is not a parasitic filter. Values for for which fits of to the experimental power spectra have at least 1% support. Tested values of were 2, 5, 10, 15, 25, 40, 60, and .
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