|Amardeep Singh||Master Project|
|University of the Bundeswehr Munich, Munich, Germany||March 2008|
In Photonic Force Microscopy (PFM), it is possible to trap and track thermally fluctuating probes to measure local interactions. At the core of this work is an analog estimation procedure of the covariance matrix of the three-dimensional position signal of the bead which is used as a probe in the Photonic Force Microscope. We use it to visualize the isoprobability surface of the movement of the bead under the assumption that its position follows a three-dimensional Gaussian distribution. This assumption is justified by the three-dimensional Langevin equation that we use to model the movement of the bead. This isosurface forms an ellipsoid. The length of its axes is related to the eigenvalues, while its orientation in space is related to the eigenvectors of the covariance matrix. Taking into account covariances between the coordinates of the three-dimensional position of the bead has been done for the first time, to the best of our knowledge. The estimation error of the analog estimation procedure is investigated. We model the detection of the photonic force microscope in a way that enables us to show that the analog estimation of the covariance matrix can be related to the stiffness matrix of the three-dimensional Langevin equation. It is also shown that by differentiating the position signal before the analog estimation procedure, one can even obtain the mass matrix. By knowing the stiffness and mass matrix, we can then estimate the last remaining parameter, the friction matrix. Furthermore, we develop an inversion of the detector response that is able to account for its non-linearities in the on- and off-axis regime around the focus of the laser by using cubic B-splines.
We find the generating function of the linear minimum-mean-square-estimator of the onedimensional Langevin equation to be an exponential spline.