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[en] Histograms of counted events are Poisson distributed, but are typically fitted without justification using nonlinear least squares fitting. The more appropriate maximum likelihood estimator (MLE) for Poisson distributed data is seldom used. We extend the use of the Levenberg-Marquardt algorithm commonly used for nonlinear least squares minimization for use with the MLE for Poisson distributed data. In so doing, we remove any excuse for not using this more appropriate MLE. We demonstrate the use of the algorithm and the superior performance of the MLE using simulations and experiments in the context of fluorescence lifetime imaging. Scientists commonly form histograms of counted events from their data, and extract parameters by fitting to a specified model. Assuming that the probability of occurrence for each bin is small, event counts in the histogram bins will be distributed according to the Poisson distribution. We develop here an efficient algorithm for fitting event counting histograms using the maximum likelihood estimator (MLE) for Poisson distributed data, rather than the non-linear least squares measure. This algorithm is a simple extension of the common Levenberg-Marquardt (L-M) algorithm, is simple to implement, quick and robust. Fitting using a least squares measure is most common, but it is the maximum likelihood estimator only for Gaussian-distributed data. Non-linear least squares methods may be applied to event counting histograms in cases where the number of events is very large, so that the Poisson distribution is well approximated by a Gaussian. However, it is not easy to satisfy this criterion in practice - which requires a large number of events. It has been well-known for years that least squares procedures lead to biased results when applied to Poisson-distributed data; a recent paper providing extensive characterization of these biases in exponential fitting is given. The more appropriate measure based on the maximum likelihood estimator (MLE) for the Poisson distribution is also well known, but has not become generally used. This is primarily because, in contrast to non-linear least squares fitting, there has been no quick, robust, and general fitting method. In the field of fluorescence lifetime spectroscopy and imaging, there have been some efforts to use this estimator through minimization routines such as Nelder-Mead optimization, exhaustive line searches, and Gauss-Newton minimization. Minimization based on specific one- or multi-exponential models has been used to obtain quick results, but this procedure does not allow the incorporation of the instrument response, and is not generally applicable to models found in other fields. Methods for using the MLE for Poisson-distributed data have been published by the wider spectroscopic community, including iterative minimization schemes based on Gauss-Newton minimization. The slow acceptance of these procedures for fitting event counting histograms may also be explained by the use of the ubiquitous, fast Levenberg-Marquardt (L-M) fitting procedure for fitting non-linear models using least squares fitting (simple searches obtain ∼10000 references - this doesn't include those who use it, but don't know they are using it). The benefits of L-M include a seamless transition between Gauss-Newton minimization and downward gradient minimization through the use of a regularization parameter. This transition is desirable because Gauss-Newton methods converge quickly, but only within a limited domain of convergence; on the other hand the downward gradient methods have a much wider domain of convergence, but converge extremely slowly nearer the minimum. L-M has the advantages of both procedures: relative insensitivity to initial parameters and rapid convergence. Scientists, when wanting an answer quickly, will fit data using L-M, get an answer, and move on. Only those that are aware of the bias issues will bother to fit using the more appropriate MLE for Poisson deviates. However, since there is a simple, analytical formula for the appropriate MLE measure for Poisson deviates, it is inexcusable that least squares estimators are used almost exclusively when fitting event counting histograms. There have been ways found to use successive non-linear least squares fitting to obtain similarly unbiased results, but this procedure is justified by simulation, must be re-tested when conditions change significantly, and requires two successive fits. There is a great need for a fitting routine for the MLE estimator for Poisson deviates that has convergence domains and rates comparable to the non-linear least squares L-M fitting. We show in this report that a simple way to achieve that goal is to use the L-M fitting procedure not to minimize the least squares measure, but the MLE for Poisson deviates.
[en] A new algorithm is introduced for computing correlations of photon arrival time data acquired in single-molecule fluorescence spectroscopy and fluorescence correlation spectroscopy (FCS). The correlation is first rewritten as a counting operation on photon pairs. For each photon, the contribution to the correlation function for each subsequent photon is calculated for arbitrary bin spacings of the correlation time lag. By retaining the bin positions in the photon sequence after each photon, the correlation can be performed efficiently. Example correlations for simulations of FCS experiments are shown, with comparable execution speed to the commonly used multiple-tau correlation technique. Also, wide bin spacings are possible that allow for real-time software calculation of correlations even for high count rates (∼350 kHz). The flexibility and broad applicability of the algorithm is demonstrated using results from single molecule photon antibunching experiments
[en] We study protein and nucleic acid structure and dynamics using single-molecule fluorescence resonance energy transfer measurements with alternating-laser excitation. Freely diffusing molecules are sorted into subpopulations based on stoichiometry, detecting donor and acceptor coincidence for periods over 100 (micro)s-1 ms. Faster (< 100 (micro)s) fluctuating distance distributions are studied within these subpopulations using time-resolved single photon counting measurements. We find that short double-stranded DNA (dsDNA) is more flexible than expected from persistence lengths measured on long dsDNA. We find that the electrostatic portion of the persistence length of single-stranded poly-dT varies as the ionic strength (I) to the -1/2 power (I-1/2). Lastly, we find that the unfolded protein Chymotrypsin Inhibitor 2 (CI2) is unstructured at high denaturant. However, in the presence of folded CI2 (at lower denaturant), unfolded CI2 is more compact and displays larger distance fluctuations, possibly due to unsuccessful attempts to cross the folding barrier
[en] Modern single molecule fluorescence microscopy offers new, highly quantitative ways of studying the systems biology of cells while keeping the cells healthy and alive in their natural environment. In this context, a quantum optical technique, photon antibunching, has found a small niche in the continuously growing applications of single molecule techniques to small molecular complexes. Here, we review some of the most recent applications of photon antibunching in biophotonics, and we provide a guide for how to conduct photon antibunching experiments at the single molecule level by applying techniques borrowed from time-correlated single photon counting. We provide a number of new examples for applications of photon antibunching to the study of multichromophoric molecules and small molecular complexes
[en] The surface of the layered III-VI chalcogenide semiconductor GaSeTe was treated with (NH4)2S at 60 C to modify the surface chemistry and determine the effect on transport properties. Room temperature photoluminescence (PL) measurements were used to assess the effect of the (NH4)2S treatment on surface defect states. Evaluation of the subsequent surface chemistry was performed with high-resolution core-level photoemission measurements. Metal overlayers were deposited on the (NH4)2S treated surfaces and the I-V characteristics were measured. The measurements were correlated to understand the effect of (NH4)2S modification of the interfacial electronic structure with the goal of optimizing the metal/GaSeTe interface for radiation detector devices.
[en] The microstructural thermal response of fused silica subjected to laser-induced breakdown was investigated. Rapid thermal annealing of laser modified material at the surface was achieved using a CO2 laser and the relaxation response of photoluminescence, infrared (IR) reflectance, electron and white light microscope images were recorded. Subsequent nanosecond-pulsed laser damage threshold measurements revealed thermally driven kinetics which were dominated by absorbing defect annealing at heat-treatment temperatures (THT) below ∼1200 K and material toughening at higher THT. A decrease in the peak photoluminescence lifetime with THT revealed two types of defects which were correlated with non-bridging IR vibrational modes. Near the glass transition temperature, a weakening of the laser modified material was observed and explained in terms of a residual compressive stress relaxation. A nonlinear absorption model was used to predict optical breakdown threshold and compared with critical fracture predictions based on crack tip annealing. Combined with a qualitative stress relaxation analysis, our model agrees well with the experimental data and yields insight to the rate-limiting contributions driving the onset of laser-induced breakdown in defective silica. (paper)