Optimal characterization

Gauging a HOT system's performance numerically and by characterizing the projected intensity pattern does not provide a complete picture. The real test is in the projected traps' ability to localize particles. A variety of approaches have been developed for measuring the forces exerted by optical traps. The earliest involved estimating the hydrodynamic drag required to dislodge a trapped particle (23). This has several disadvantages, most notably that it identifies only the marginal escape force in a given direction and not the trap's actual three-dimensional potential. Complementary information can be obtained by measuring a particle's thermally driven motions in the trap's potential well (24,25,26). For instance, the measured probability density $P(\vec{r})$ for displacements $\vec{r}$ is related to the trap's potential $V(\vec{r})$ through the Boltzmann distribution

$$P(\vec{r}) \propto \exp(-\beta V(\vec{r})),$$ (18)

where $\beta^{-1} = k_B T$ is the thermal energy scale at temperature $T$. Similarly, the power spectrum of $\vec{r}(t)$ for a harmonically bound particle is a Lorentzian whose width is the viscous relaxation time of the particle in the well (24,27).

Both of these approaches require amassing enough data to characterize the trapped particle's least probable displacements, and therefore oversample the trajectories. Oversampling is acceptable when data from a single optical trap can be collected rapidly, for example with a quadrant photodiode (24,25,28,26). Tracking multiple particles in holographic optical traps, however, requires the area detection capabilities of digital video microscopy (29), which yields data much more slowly. Analyzing video data with optimal statistics (30) offers the benefits of thermal calibration by avoiding oversampling.

An optical trap is accurately modeled as a harmonic potential energy well (25,28,26,27),

$$V(\vec{r}) = \frac{1}{2} \, \sum_{i=1}^3 \kappa_i r_i^2,$$ (19)

with a different characteristic curvature $\kappa_i$ along each axis. This form also is convenient because it is separable into one-dimensional contributions. The trajectory of a colloidal particle localized in a viscous fluid by a harmonic well is described by the one-dimensional Langevin equation (31)

$$\dot{x}(t) = -\frac{x(t)}{\tau} + \xi(t),$$ (20)

where the autocorrelation time $\tau = \gamma/\kappa$, is set by the particle's viscous drag coefficient $\gamma$ and by the curvature of the well, $\kappa$. The Gaussian random thermal force, $\xi(t)$, has zero mean, $\left< \xi(t) \right> = 0$, and variance

$$\left< \xi(t) \, \xi(s) \right> = \frac{2 k_B T}{\gamma} \, \delta(t-s).$$ (21)

If the particle is at position $x_0$ at time $t = 0$, its trajectory at later times is given by

$$x(t) = x_0 \, \exp\left(-\frac{t}{\tau}\right) + \int_0^t \xi(s) \, \exp\left(-\frac{t-s}{\tau}\right) \, ds.$$ (22)

Sampling such a trajectory at discrete times $t_j = j \, \Delta t$, yields

$$x_{j+1} = \phi \, x_j + a_{j+1},$$ (23)

where $x_j = x(t_j)$,

$$\phi = \exp\left(-\frac{\Delta t}{\tau}\right),$$ (24)

and where $a_{j+1}$ is a Gaussian random variable with zero mean and variance

$${\sigma_a^2}= \frac{k_B T}{\kappa}\, \left[1 - \exp\left(-\frac{2\Delta t}{\tau}\right)\right].$$ (25)

Because $\phi < 1$, Eq. (23) is an example of an autoregressive process (30), which is readily invertible.

In principle, the particle's trajectory $\{x_j\}$ can be analyzed to extract $\phi$ and ${\sigma_a^2}$, and thus the trap's stiffness, $\kappa$, and the particle's viscous drag coefficient $\gamma$. In practice, however, the experimentally measured particle positions $y_j$ differ from the actual positions $x_j$ by random errors $b_j$, which we assume to be taken from a Gaussian distribution with zero mean and variance $\sigma_b^2$. The measurement then is described by the coupled equations

$$x_j = \phi \, x_{j-1} + a_j y_j = x_j + b_j,$$ (26)

where $b_j$ is independent of $a_j$. We still can estimate $\phi$ and ${\sigma_a^2}$ from a set of measurements $\{y_j\}$ by first constructing the joint probability

$$p(\{x_i\}, \{y_i\} \vert \phi, {\sigma_a^2}, {\sigma_b^2}) = \pro... ...p\left(-\frac{b_j^2}{2{\sigma_b^2}} \right)}{\sqrt{2 \pi {\sigma_b^2}}}\right].$$ (27)

The probability density for measuring the trajectory $\{y_j\}$, is then the marginal (30)

$$p(\{y_j\} \vert \phi,{\sigma_a^2},{\sigma_b^2}) = \int p(\{x_j\},\{y_j\} \vert \phi,{\sigma_a^2},{\sigma_b^2}) \, dx_1 \cdots dx_N$$ (28)

$$= \frac{(2\pi{\sigma_a^2}{\sigma_b^2})^{-\frac{N-1}{2}}}{\sqrt{{\... ...mathbf I}-\frac{{\mathbf A}_\phi^{-1}}{{\sigma_b^2}} \right] \, \vec{y}\right),$$ (29)

where $\vec{y} = (y_1, \dots, y_N)$ with transpose $(\vec{y})^T$, ${\mathbf I}$ is the $N \times N$ identity matrix, and

$${\mathbf A}_\phi = \frac{{\mathbf I}}{{\sigma_b^2}} + \frac{{\mathbf M}_\phi}{{\sigma_a^2}},$$ (30)

with the tridiagonal memory tensor

$${\mathbf M}_\phi =\left( \begin{array}{cccccc} \phi^2 & -\phi & 0... ...-\phi & 1 + \phi^2 &-\phi\\ 0 & 0 & \cdots & 0 & -\phi & 1 \end{array} \right).$$ (31)

Calculating the determinant, $\det({\mathbf A}_\phi)$, and inverse, ${\mathbf A}_\phi^{-1}$, of ${\mathbf A}_\phi$ is greatly facilitated if we artificially impose time translation invariance by replacing $M_\phi$ with the $(N+1) \times (N+1)$ matrix that identifies time step $N+1$ with time step 1. Physically, this involves imparting an impulse, $a_{N+1}$, that translates the particle from its last position, $x_N$, to its first, $x_1$. Because diffusion in a potential well is a stationary process, the effect of this change is inversely proportional to the number of measurements, $N$. With this approximation,

$$\det({\mathbf A}_\phi) = \prod_{n=1}^{N} \left\{ \frac{1}{{\sigma_b^2}} + \frac{1}{{\sig... ..., \left[1 + \phi^2 - 2 \phi \, \cos\left(\frac{2\pi n}{N}\right)\right]\right\}$$ (32) and

$$({\mathbf A}_\phi^{-1})_{\alpha \beta} = \frac{1}{N} \, \sum_{n=1}^N \frac{{\sigma_a^2}{\sigma_b^2}\, \e... ...b^2}\, \left[ 1 + \phi^2 - 2 \phi \, \cos\left(\frac{2\pi n}{N}\right)\right]},$$ (33)

so that the conditional probability for the measured trajectory, $\{y_j\}$, is

$$p(\{y_j\} \vert \phi, {\sigma_a^2}, {\sigma_b^2}) = (2\pi)^{-\fra... ...\left[1 + \phi^2 - 2 \phi \, \cos\left(\frac{2\pi m}{N}\right)\right]} \right),$$ (34)

where $\tilde{y}_m$ is the $m$-th component of the discrete Fourier transform of $\{y_n\}$. This can be inverted to obtain the likelihood function for $\phi$, ${\sigma_a^2}$, and ${\sigma_b^2}$:

$$L(\phi,{\sigma_a^2},{\sigma_b^2}\vert \{y_i\} ) = - \frac{N}{2} \... ...eft[1 + \phi^2 - 2 \phi \, \cos\left(\frac{2 \pi n}{N}\right) \right] \right) .$$ (35)

Best estimates $(\hat{\phi}, \hat{{\sigma_a^2}}, \hat{{\sigma_b^2}})$ for the parameters $(\phi,{\sigma_a^2},{\sigma_b^2})$ are solutions of the coupled equations

$$\frac{\partial L}{\partial \phi} = \frac{\partial L}{\partial {\sigma_a^2}} = \frac{\partial L}{\partial {\sigma_b^2}} = 0.$$ (36)