^{1,2,a)}, Alexander Mehlich

^{2}, Benjamin Pelz

^{2}, Matthias Rief

^{2}and Roland R. Netz

^{1,2}

### Abstract

The thermal fluctuations of micron-sized beads in dual trap optical tweezer experiments contain complete dynamic information about the viscoelastic properties of the embedding medium and—if present—macromolecular constructs connecting the two beads. To quantitatively interpret the spectral properties of the measured signals, a detailed understanding of the instrumental characteristics is required. To this end, we present a theoretical description of the signal processing in a typical dual trap optical tweezer experiment accounting for polarization crosstalk and instrumental noise and discuss the effect of finite statistics. To infer the unknown parameters from experimental data, a maximum likelihood method based on the statistical properties of the stochastic signals is derived. In a first step, the method can be used for calibration purposes: We propose a scheme involving three consecutive measurements (both traps empty, first one occupied and second empty, and vice versa), by which all instrumental and physical parameters of the setup are determined. We test our approach for a simple model system, namely a pair of unconnected, but hydrodynamically interacting spheres. The comparison to theoretical predictions based on instantaneous as well as retarded hydrodynamics emphasizes the importance of hydrodynamic retardation effects due to vorticity diffusion in the fluid. For more complex experimental scenarios, where macromolecular constructs are tethered between the two beads, the same maximum likelihood method in conjunction with dynamic deconvolution theory will in a second step allow one to determine the viscoelastic properties of the tethered element connecting the two beads.

The authors thank M. Hinczewski, J.-C. Meiners and E. Schäffer for stimulating discussions and helpful comments. Financial support from the DFG (SFB 863), from the Elitenetzwerk Bayern in the framework of CompInt (Y.v.H.), and from the Nanosystems Initiative Munich (A.M., B.P., M.R.) is acknowledged.

I. INTRODUCTION

II. DUAL TRAP OPTICAL TWEEZERS

III. FORCE RESPONSE AND THERMAL MOTION

A. General formulation

B. Hydrodynamically interacting beads

IV. LOW REYNOLDS NUMBER HYDRODYNAMICS

A. Flow-field around an oscillating sphere

B. Frequency dependence of self- and cross-mobilities

V. SIGNAL PROCESSING IN A DUAL TRAP OPTICAL TWEEZER EXPERIMENT

A. Crosstalk, (parasitic) filtering, and noise

B. Finite time resolution and measurement time

VI. RELATING EXPERIMENT AND THEORY

A. Statistical properties of auto- and cross-periodograms

B. Maximum likelihood fits of PSDs

C. Deviations of fits and data

D. Power spectral analysis in a nutshell

VII. CALIBRATION OF DUAL TRAP OPTICAL TWEEZERS

VIII. DISCUSSION

A. Retarded vs. instantaneous hydrodynamics

B. Instrumental effects and finite statistics

IX. CONCLUSIONS

### Key Topics

- Optical tweezers
- 25.0
- Calibration
- 22.0
- Polarization
- 19.0
- Hydrodynamics
- 16.0
- Spectrum analysis
- 14.0

##### F03H3/00

## Figures

(a) Schematic of the experimental setup described in Sec. II. (b) Zoom-in into the sample chamber: The notation for the coordinates of the beads (blue) relative to the trap centers (gray) is used throughout the manuscript.

(a) Schematic of the experimental setup described in Sec. II. (b) Zoom-in into the sample chamber: The notation for the coordinates of the beads (blue) relative to the trap centers (gray) is used throughout the manuscript.

Cross sections of the flow field around an oscillating no-slip sphere in an incompressible fluid for various driving frequencies: (a) ω = 10^{−4}/τ_{ a }, (b) ω = 10^{−2}/τ_{ a }, (c) ω = 1/τ_{ a }. Streamlines indicate the direction of the velocity field and the color coding its magnitude, where *F* _{0} is the driving force amplitude and μ_{0} = 1/(6πη*a*) the quasi-static sphere mobility. Spatial coordinates are given in units of the sphere radius *a*, frequencies in units of , the kinematic viscosity of the fluid being denoted by ν. Amplitudes of the driving force acting on and the velocity of the sphere are shown as red and blue arrows within the spheres. The figures above show the configuration of the flow-field, when the sphere velocity is maximal; animations showing the time evolution of these flow fields are available online.^{27}

Cross sections of the flow field around an oscillating no-slip sphere in an incompressible fluid for various driving frequencies: (a) ω = 10^{−4}/τ_{ a }, (b) ω = 10^{−2}/τ_{ a }, (c) ω = 1/τ_{ a }. Streamlines indicate the direction of the velocity field and the color coding its magnitude, where *F* _{0} is the driving force amplitude and μ_{0} = 1/(6πη*a*) the quasi-static sphere mobility. Spatial coordinates are given in units of the sphere radius *a*, frequencies in units of , the kinematic viscosity of the fluid being denoted by ν. Amplitudes of the driving force acting on and the velocity of the sphere are shown as red and blue arrows within the spheres. The figures above show the configuration of the flow-field, when the sphere velocity is maximal; animations showing the time evolution of these flow fields are available online.^{27}

Dependence of real and imaginary parts of the self-mobility of a sphere on the driving frequency ω. In the low-frequency limit, the quasi-stationary self-mobility μ_{0} ≡ 1/(6πη*a*) of a sphere of radius *a* in a fluid of shear viscosity η is recovered. On the lower axis, the angular frequency ω is rescaled by τ_{ a } ≡ *a* ^{2}/ν, the upper frequency scale corresponds to the experimental case where τ_{ a } ≈ 0.26 μs (beads of radius *a* = 500 nm in water). The exact result by Stokes (dashed lines, Eq. (10)) and the approximate expressions (solid lines, Eq. (12)) nicely match over the entire range of frequencies resolved in our experiment (*f* ≲ 100 kHz).

Dependence of real and imaginary parts of the self-mobility of a sphere on the driving frequency ω. In the low-frequency limit, the quasi-stationary self-mobility μ_{0} ≡ 1/(6πη*a*) of a sphere of radius *a* in a fluid of shear viscosity η is recovered. On the lower axis, the angular frequency ω is rescaled by τ_{ a } ≡ *a* ^{2}/ν, the upper frequency scale corresponds to the experimental case where τ_{ a } ≈ 0.26 μs (beads of radius *a* = 500 nm in water). The exact result by Stokes (dashed lines, Eq. (10)) and the approximate expressions (solid lines, Eq. (12)) nicely match over the entire range of frequencies resolved in our experiment (*f* ≲ 100 kHz).

Frequency dependence of cross-mobilities parallel (γ = ∥) and perpendicular (γ = ⊥) to the inter-bead axis for different ratios of inter-bead separation *R* to bead radius *a*: (a) *R*/*a* = 20, (b) *R*/*a* = 3. Cross-mobilities for spheres of finite radius (solid lines, Eq. (13)) are compared to the Oseen result for point-like particles (dashed lines, Eq. (15)) and to quasi-static HI (horizontal arrows, Eq. (14)). Cross-mobilities are given in units of the quasi-static Stokes self-mobility μ_{0} ≡ 1/(6πη*a*) for a sphere of radius *a* in a fluid of viscosity η; the angular frequency is rescaled by the characteristic time scale τ_{ a } for vortex diffusion over the length scale of the bead's radius *a*, the upper frequency scale corresponds to experimental conditions (τ_{ a } ≈ 0.26 μs).

Frequency dependence of cross-mobilities parallel (γ = ∥) and perpendicular (γ = ⊥) to the inter-bead axis for different ratios of inter-bead separation *R* to bead radius *a*: (a) *R*/*a* = 20, (b) *R*/*a* = 3. Cross-mobilities for spheres of finite radius (solid lines, Eq. (13)) are compared to the Oseen result for point-like particles (dashed lines, Eq. (15)) and to quasi-static HI (horizontal arrows, Eq. (14)). Cross-mobilities are given in units of the quasi-static Stokes self-mobility μ_{0} ≡ 1/(6πη*a*) for a sphere of radius *a* in a fluid of viscosity η; the angular frequency is rescaled by the characteristic time scale τ_{ a } for vortex diffusion over the length scale of the bead's radius *a*, the upper frequency scale corresponds to experimental conditions (τ_{ a } ≈ 0.26 μs).

Visualization of the signals in a typical dual trap optical tweezer experiment as described in Secs. V and VI A. (a) Signal processing as described by Eqs. (17) and (18); averaged periodograms obtained via DFT from the discretely sampled stochastic trajectories yield experimental estimates for the aliased PSDs . (b) Relation between auto- and cross-PSDs of the beads' thermal motion and aliased electrical PSDs according to Eqs. (22) and (23). Unknown parameters in the PSDs can be determined via a global maximum likelihood fit to as explained in Sec. VI B.

Visualization of the signals in a typical dual trap optical tweezer experiment as described in Secs. V and VI A. (a) Signal processing as described by Eqs. (17) and (18); averaged periodograms obtained via DFT from the discretely sampled stochastic trajectories yield experimental estimates for the aliased PSDs . (b) Relation between auto- and cross-PSDs of the beads' thermal motion and aliased electrical PSDs according to Eqs. (22) and (23). Unknown parameters in the PSDs can be determined via a global maximum likelihood fit to as explained in Sec. VI B.

Results of the three calibration measurements described in Sec. VII: (a) no trapped beads, (b) first trap occupied and second one empty, (c) vice versa. Averaged experimental periodograms are denoted by symbols, theoretical aliased PSDs , based on a global maximum likelihood fit to the data as described in the text, are shown as lines, where broken lines correspond to negative values of the PSDs; fit values are given in Table I. Results for the spatial coordinates parallel to the inter-trap axis (γ = ∥) are shown in the upper row, those for the perpendicular direction (γ = ⊥) in the lower one.

Results of the three calibration measurements described in Sec. VII: (a) no trapped beads, (b) first trap occupied and second one empty, (c) vice versa. Averaged experimental periodograms are denoted by symbols, theoretical aliased PSDs , based on a global maximum likelihood fit to the data as described in the text, are shown as lines, where broken lines correspond to negative values of the PSDs; fit values are given in Table I. Results for the spatial coordinates parallel to the inter-trap axis (γ = ∥) are shown in the upper row, those for the perpendicular direction (γ = ⊥) in the lower one.

Auto- and cross-PSDs of the fluctuations parallel to the inter-trap axis for various trap separations *R*: Symbols denote averaged experimental periodograms , lines the theoretical predictions for the aliased PSDs according to Eqs. (22) and (23): black lines include the full frequency dependence of HI (Eq. (13)), light blue lines correspond to instantaneous HI (Eq. (14)). Positive values of the averaged experimental periodograms are denoted by circles, negative ones by squares; similarly, solid and broken lines, respectively, denote positive and negative PSD values.

Auto- and cross-PSDs of the fluctuations parallel to the inter-trap axis for various trap separations *R*: Symbols denote averaged experimental periodograms , lines the theoretical predictions for the aliased PSDs according to Eqs. (22) and (23): black lines include the full frequency dependence of HI (Eq. (13)), light blue lines correspond to instantaneous HI (Eq. (14)). Positive values of the averaged experimental periodograms are denoted by circles, negative ones by squares; similarly, solid and broken lines, respectively, denote positive and negative PSD values.

Same as Fig. 7 but for the spatial component perpendicular (⊥) to the inter-trap axis.

Same as Fig. 7 but for the spatial component perpendicular (⊥) to the inter-trap axis.

Influence of polarization crosstalk: Theoretical PSDs accounting for polarization crosstalk are shown as black lines (same as in Fig. 8), theoretical PSDs neglecting polarization crosstalk are denoted as purple lines; since auto-PSDs are only marginally affected by polarization crosstalk, black and purple lines for *S* _{11} and *S* _{22} overlap. Symbols denote experimental data (same as in Fig. 8).

Influence of polarization crosstalk: Theoretical PSDs accounting for polarization crosstalk are shown as black lines (same as in Fig. 8), theoretical PSDs neglecting polarization crosstalk are denoted as purple lines; since auto-PSDs are only marginally affected by polarization crosstalk, black and purple lines for *S* _{11} and *S* _{22} overlap. Symbols denote experimental data (same as in Fig. 8).

Attenuation resulting from various filtering sources as discussed in Appendix B: (a) parasitic filtering of the position sensing devices (Eq. (B1)) using the parameters from Table I for the ⊥ direction, (b) different electric low-pass filters (Eqs. (B3) and (B4)), (c) uniform averaging of signals over a time window of duration (Eq. (B5)), (d) time delay between the two recorded signals (Eq. (B7)). Positive values of are drawn as solid lines, negative ones are dashed.

Attenuation resulting from various filtering sources as discussed in Appendix B: (a) parasitic filtering of the position sensing devices (Eq. (B1)) using the parameters from Table I for the ⊥ direction, (b) different electric low-pass filters (Eqs. (B3) and (B4)), (c) uniform averaging of signals over a time window of duration (Eq. (B5)), (d) time delay between the two recorded signals (Eq. (B7)). Positive values of are drawn as solid lines, negative ones are dashed.

## Tables

Calibration results: Best fit parameters obtained from global maximum likelihood fits to the averaged periodograms in Fig. 6 as described in Sec. VII.

Calibration results: Best fit parameters obtained from global maximum likelihood fits to the averaged periodograms in Fig. 6 as described in Sec. VII.

Overview of the notation used throughout the manuscript; sub- and super-scripts γ are omitted in the text, if the discussion equally applies to all spatial directions.

Overview of the notation used throughout the manuscript; sub- and super-scripts γ are omitted in the text, if the discussion equally applies to all spatial directions.

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