Photon detection statistics fluctuations

For very low light intensities the quantum structure of light becomes evident by statistical fluctuations of the number of detected photons, which lead to corresponding fluctuations of the measured photoelectron rate (Sect. 7.8). This photon noise, which is proportional to -/N at a measured rate of N photoelectrons per second, imposes a principal detection limit for experiments with low-level light detection [1330]. Additionally, the frequency stabilization of lasers on the millihertz scale is limited by photon noise of the detector that activates the electronic feedback loop [1331]. 

If the modulation frequency X2 is chosen sufficiently high Q > 1000 MHz), the technical noise may drop below the quantum-noise limit set by the statistical fluctuations of detected photons. In this case, the detection limit is mainly due to the quantum limit [6.4]. Since lock-in detectors cannot handle such high frequencies, the signal input has to be downconverted in a mixer, where the difference frequency between a local oscillator and the signal is generated. 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

In the IR with terrestrial backgrounds and cooled infrared sensitive devices, the ultimate limit in performance is set by the shot noise arising from fluctuations in the arrival rate of the background photons. For the wavelengths of interest the background photons follow Poisson statistics, and the standard deviation of the number of collected photoelectrons from a detector in a sampling time will be the square root of the number. For an individual detection cell of area A with quantum efficiency 7, the noise spectrum of the photocurrent will be Sf=e J tjA where is the background photon flux at the detection cell... 

If the frequency spectrum 1(f) of the detector signals is measured with a spectrum analyzer at sufficiently high frequencies /, where the technical noise is negligible, one obtains a noise power spectrum Pn(/), which is essentially independent of the phase 0 (Fig. 14.62b), but depends only on the number of photons entering the interferometer. It is proportional to /N. It is surprising that the noise power density Pn(f) of each detector is independent of the phase 0. This can be understood as follows the intensity fluctuations, because of the statistical emission of photons, are uncorrelated in the two partial beams. Although the mean intensities I ) and I2) depend on 0, their fluctuations do not The detected noise power pn oc Va shows the same noise level pn oc for the minimum of 7(0) as for the maximum (Fig. 14.62b). 

The analysis and interpretation of the fluctuation statistics obtained with FFS measurements requires knowledge of the excitation and detection volume and the detected fluorescence brightness profile, i.e. the molecule detection efficiency of the fluorescent particle at position P. The brightness Q(P) of a single fluorophore is determined by its photophysical properties, the excitation intensity 0(P)hc/A (where 0(r) is the excitation photon flux, h Planck s... 

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