Understanding and improving axial detection in optical tweezers based on the interference of forward- and backward- scattered light

TL;DR

This work addresses the axial-detection challenge in optical tweezers by introducing a minimal model that explicitly accounts for backward-scattered light from the bottom interface, which induces standing-wave effects and nonlinearities in the axial signal. The authors derive a nonlinear signal model $s(t) \approx s_0 + \beta z + \gamma(z_0) \sin\left(\omega z + \frac{4\pi}{\lambda} z_0 + \phi_0\right)$ and show that the signal’s autocorrelation and power spectral density remain tractable, with the PSD taking the Lorentzian form $PSD_s(f) \approx [\beta^2 + 2\beta\gamma\omega e^{-\frac{1}{2}\omega^2 k_B T/\kappa} \cos(\frac{4\pi}{\lambda} z_0)] \frac{1}{2\pi^2} \frac{D}{f_c^2 + f^2}$ and $f_c = \kappa D/(2\pi k_B T)$. The theory incorporates Faxén corrections for diffusion near surfaces and provides a practical fitting framework to extract trap stiffness $\kappa$ and diffusion $D$ from PSD data, validated experimentally on beads of $d_p = 0.96 \mu m$ and $0.50 \mu m$ diameter. The study also quantifies how power and geometry influence the backward-forward interference balance, establishing conditions under which near-surface standing-wave effects become significant. Overall, the work enhances OT calibration accuracy in near-surface and generalized conditions, and offers a unified framework for surface-induced interference phenomena in optical detection.

Abstract

Fast and accurate 3D position detection in optical tweezers (OT) is essential for quantitatively monitoring subtle variations in the mechanical properties of microscopic systems ranging from biomolecules to cells and colloids. Because standard OT configurations do not provide direct access to the axial position, axial detection typically relies on temporal fluctuations in forward-scattered optical power to infer the position of the particle. This approach generally assumes a linear-response regime in which the signal arises from the interference between the forward scattered and the nonscattered optical fields; however, under certain conditions, the backward-scattered contribution becomes non-negligible, leading to deviations from the linear response. Here, we present a simple yet comprehensive model for axial detection in standard OT while explicitly accounting for the backward-scattered field. Together with experimental validation, this framework neatly explains the standing-wave response observed when the backward-scattered field interferes with the nonscattered and the forward-scattered components, enabling accurate estimation of trap stiffness and particle diffusion under more general conditions. This work deepens our understanding of the phenomenology observed in real optical-tweezers measurements and extends their capabilities to conditions where standard approaches fail.

Figures (6)

• Figure 1: (a) Experimental configuration and axial ($z$) position detection in an OT. A linearly polarized Gaussian beam is tightly focused from below using a water-immersion objective lens (OL), trapping a particle near the beam waist inside the sample chamber. The outgoing light is collected from above by the lens condenser (LC). The low-spatial-frequency components of the collected field pass through the pupil (Pupil), located at the back focal plane (BFP), and the transmitted light is then relayed by means of a half-wave plate (HWP), a polarizing beam-splitter (PBS) and the lens L1 toward the aperture stop (Aperture), after which it is focused by lens L2 and detected with a photodiode (PD). Red solid lines in the sample and in the detector indicate conjugated planes of detection. The resulting signal is digitized using a data-acquisition card (FPGA). The sample cell (Sample) is mounted on a piezoelectric stage (PS) which controls the particle's axial position relative to the cover glass according to $z_0=z_{\rm stage}-\delta_z$, where $\delta_z$ is a constant offset. (b) Schematic representation of light propagation. The forward-scattered field (blue wavefronts) and the backward-scattered field (red wavefronts, reflected by the bottom cover glass) interfere with the outgoing laser beam at the conjugated plane of detection indicated by the red solid line, which is located at the focal point of the LC, at distance $q$ from the focal point of the OL. Here, $z(t)$ denotes the instantaneous particle displacement relative to its equilibrium position near the beam waist, and $z_0$ is the distance from the cover glass to the equilibrium position. The technical details of the setup are presented in Methods and materials in the Supplementary information.
• Figure 2: Statistical behavior of the photodiode signal as a function of the distance between the trapped particle and the bottom cover glass for particles with diameters $d_p=0.96\,\mu\mathrm{m}$ (a-c) and $0.50\,\mu\mathrm{m}$ (d-f). The particle position $z_{\rm stage}$ approximately corresponds to the distance between the particle equilibrium position and the cover glass $z_0$. (a) and (d) show the probability density function ($\rho_s$) of the signals ($s(t)$) at several evenly spaced positions $z_{\rm stage}$, ranging from $z_{\rm stage}=2.2\,\mu\mathrm{m}$ (bottom) to $z_{\rm stage}=3\,\mu\mathrm{m}$ (top). (b) and (e) display the mean and variance of $s(t)$ as a function of $z_{\rm stage}$. Insets in these figures show the zoomed out plots in the range $z_0=(2-4)\mu\mathrm{m}$. (c) and (f) show the corresponding skewness and kurtosis. Both particles were trapped using the same laser power $P=25 \pm 1 \,m\mathrm{W}$, estimated inside the sample.
• Figure 3: Examples of the power spectral density of the photodiode signal for particles of diameter $0.96\,\mu\mathrm{m}$ (a) and $0.50\,\mu\mathrm{m}$ (b). In each plot, blue dots correspond to $|\hat{s}|^2/T_s$, circles correspond to the expected values of the PSD at low frequencies, in the range $(20-1000) \mathrm{Hz}$, and red line corresponds to the Lorentzian fitting. The particles were located at position $z_{0}\sim 5\,\mu\mathrm{m}$ far from the cover glass, marked with red arrows in Fig. \ref{['fig:Afcsignal']}. Experimental parameters are the same of those in Fig. \ref{['fig:signals']}.
• Figure 4: Amplitude ($A$) and cutoff frequency ($f_c$) of the PSD of the photodiode signal for trapped particles with diameters $d_p=0.96\,\mu\mathrm{m}$ (a,b) and $0.50\,\mu\mathrm{m}$ (c,d). The dots represent experimental estimates obtained from nonlinear least-squares fits of the measured PSD to the Lorentzian model given by Eq. \ref{['eq:acfpsd']}, as shown in Fig. \ref{['fig:PDFs']}. Arrows indicate data points corresponding to the fittings in Fig. \ref{['fig:PDFs']}. The solid dark lines correspond to fits using the theoretical model described by Eq. \ref{['eq:psd_amplitudefc']}. The fitting results are summarized in Table \ref{['tab:tablefit']}.
• Figure 5: Diffusion coefficient ($D$) and trapped stiffness ($\kappa$) for an OT with particles of diameters $d_p=0.96\,\mu\mathrm{m}$ (a,b) and $0.50\,\mu\mathrm{m}$ (c,d). Based on the experimetnal data and the model-fitting parameters summarized in Table \ref{['tab:tablefit']}, several approaches were used to estimate these quantities (see main text for details). The black solid lines correspond to calculations using the lower and upper limits of the particle diameter specified by the manufacturer, incorporating Faxen's correction and assuming $z_0=z_{0,1}$ (Eqs. \ref{['eq:faxen']} and \ref{['eq:stiff0']}). Circles denote experimental estimates according to Eqs. \ref{['eq:modelfit1']} and \ref{['eq:stiff1']}. The red dashed and solid lines in (a) and (c) were obtained using Faxen's correction with $z_0=z_{0,1}$ and the particle diameters obtained from the fitting parameters reported in Table \ref{['tab:tablefit']}, $d_{exp,1}$ and $d_{exp,2}$. Vertical dashed lines indicate the distance from the cover glass equal to the radius of the particle.