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Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector
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Image of FIG. 1.
FIG. 1.

Schematic of experimental system: (a) Coupling of trapping and detection laser beams into the optical trap, (b) schematic of imaging setup.

Image of FIG. 2.
FIG. 2.

Dimensions of photodiode array in micrometers.

Image of FIG. 3.
FIG. 3.

Alignment of signal to center of QPD using secondary laser beam. (a) shows bad alignment with the cross segments of the QPD not visible, (b) shows a case of good alignment with the image of the cross brightest and most symmetric.

Image of FIG. 4.
FIG. 4.

Position calibration for Ludl MAC5000 microscope stage. (a) Initial position of stuck bead. (b) New position of bead. (c) Merged image.

Image of FIG. 5.
FIG. 5.

Position calibration for AOD beam deflection. (a) Image showing the extreme positions of trapping laser when it was deflected by an AOD. (b) Position of spot centers.

Image of FIG. 6.
FIG. 6.

Normalized X position signal vs bead displacement for a trapped bead.

Image of FIG. 7.
FIG. 7.

Position fluctuations of 1.1 μm diameter trapped bead for different averaging times.

Image of FIG. 8.
FIG. 8.

Position detector cross talk between X and Y axes.

Image of FIG. 9.
FIG. 9.

Typical power spectra obtained using the QPD for a trapped (a) 1.1 μm bead, (b) 3 μm, and (c) 16 μm bead. All spectra were fit to hydrodynamically correct power spectra (Eq. (5), black lines) as well as Lorentzians (dotted lines), and yielded corner frequency values of (a) 165 Hz, (b) 56.5 Hz, and (c) 11 Hz. The corresponding values of the fit parameter A which gives the diffusion constant D comes out to be (a) 0.093, (b) 0.036, and (c) 0.055. The fit coefficients for the hydrodynamic fits have been included in each plot. The term y0 indicates an offset term that is added to Eq. (5) – this is like a white noise term that represents the measured amplitude of the uncertainty of positions produced in our detection system. The calculated values of f ν and f m are held constant during the fit.

Image of FIG. 10.
FIG. 10.

Residuals of fit data against frequency. The zero line is indicated in both plots. (a) shows an acceptable structure of the residuals with random distribution around zero and no frequency-dependent structure evident, while (b) shows a case of bad fitting with the residuals skewed in the positive side of zero, and also having high amplitude. Peaks at certain frequencies are also seen.

Image of FIG. 11.
FIG. 11.

Linear dependence of corner frequency with trapping laser power.

Image of FIG. 12.
FIG. 12.

Power spectrum of 1.1 μm beads with 100 kHz sampling rate averaged over 49 independent power spectra calculated from consecutive time series.

Image of FIG. 13.
FIG. 13.

Comparison of experimentally measured displacement sensitivity and thermal resolution limit.


Generic image for table
Table I.

Comparison of diffusion constant calculated from fitting the experimental power spectrum and that estimated theoretically from the Einstein equation.

Generic image for table
Table II.

Comparison of measured displacement sensitivity and thermal resolution limit for different averaging times.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector