Shot noise limit

Bialkowski, S. E., Data Analysis in the Shot Noise Limit 1. Single Parameter Estimation with Poisson and Normal Probability Density Functions, Anal. Chem. 61, 1989, 2479-2483. 

Nir, E., Michalet, X., Hamadani, K. M., Laurence, T. A., Neuhauser, D., Kovchegov, Y. and Weiss, S. (2006). Shot-noise limited single-molecule FRET histograms Comparison between theory and experiments. J. Phys. Chem. B 110, 22103-24. 

Some detectors for the visible and UV spectral regions can detect individual photons. These detectors are shot-noise limited. X-ray and gamma-ray spectroscopy also detects... 

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

F(x), here, is (Es + AEs)/ Er + A )), as we just noted. In the previous case, the weighting function was the Normal distribution. Our current interest is the Poisson distribution, and this is the distribution we need to use for the weighting factor. The interest in our current development is to find out what happens when the noise is Poisson-distributed, rather than Normally distributed, since that is the distribution that applies to data whose noise is shot-noise-limited. Using P to represent the Poisson distribution, equation 49-59 now becomes... 

Juma, M., Korterik, J. R, Otto, C., and Offerhaus, H. L. 2007. Shot noise limited heterodyne detection of CARS signals. Opt. Express 15 15207-13. 

As these examples illustrate, in vivo CARS microspectroscopy with shot-noise limited detection sensitivity allows the noninvasive quantification of densities, chemical composition, and physical state of molecular species inside biological systems ranging from a single lipid monolayer to a complex living cell. 

When a pump and a Stokes laser beam coincide on the sample and their difference frequency matches a particular molecular vibrational frequency, then SRS appears in the form of a gain of the Stokes pulse intensity and a loss of the pump pulse intensity, as first observed by Woodbury and Ng in 1962 [170] and by Jones and Stoicheff in 1964 [171], respectively (see Fig. 6.1). SRS has long been recognized as a highly sensitive spectroscopic tool for chemical analyses in the condensed and gas phases [172, 173, 29, 174]. For example, a shot-noise limited SRS spectrum of a single molecular monolayer was demonstrated by Heritage and Allara in 1980 [175]. In this section, we discuss the fundamental properties and applications of SRS microscopy, as was first successfully demonstrated by Nandakumar et al. [20] and subsequently reported by several research teams [21, 12, 13, 22]. 

Photon shot-noise or preamplifier shot-noise limitations, or source-flicker-noise limitations but utilizing source compensation" techniques... 

The multiplex gain in the shot-noise limited case is then ... 

For a light source of a specific power, there is a fundamental limit to the precision to which the intensity can be determined within a specific period. This is termed the shot-noise limit. If n photons are accumulated, the variance is /n and the signal-to-noise ratio (S/N) is fnln. Any additional noise source (due to the amphfier, interference or detector imperfections) will degrade this value. Theoretically, shot-noise in coherently squeezed light can be slightly lower than this limit, but ultimately only by a factor of 1/ 2. 

A 1 m W source in the visible region (a modest power for a laser) corresponds to a photon flux of 10 /s. This translates to a shot-noise-limited S/N of 3 x 10 if the measurement is made for 1 s. Dispersed light from a relatively high bandwidth (nm) lamp/monochromator combination can have power of the order /xW in the visible and near UV leading to a hmiting S/N of 10 for a 1 s accumulation at each spectral point. 

It is possible to approach shot-noise-limited performance in many optical experiments. When light levels are low, photomultipliers serve as noise-free quantum amplifiers with a gain of 10 . For absorption measurements, detectors with the highest quantum efficiency and uniformity of response, such as end-on semitransparent photocathode styles, are better than the high gain, opaque photocathode, low dark count types that are used for luminescence measurements. If one needs to measure absorption with a precision of AA 10, then 10 photons need to be accumulated at each data point. At these light levels, the dark count usually may not contribute greatly to the S/N. However, in absorption... 

Shot-noise-limited performance can be conveniently recognized by noting how the S/N of a system changes with light level. The S/N should vary quadratically with intensity if it is shot-noise-limited. Once a measnrement is determined to be shot-noise-limited, the quantification of the noise amplitnde provides a simple, direct, and effective way to estimate the actinic flnx experienced by a sample during a measnrement. ... 

We defined S as the signal, in electrons, resulting from the Raman scattering of the analyte of interest, as described in Chapter 3. In the absence of other noise sources, the standard deviation of S is determined by the shot noise limit ... 

In the shot noise limit, we observe the familiar increase in SNR with It is worth noting that the SNR cannot exceed that given by Eq. (4.6), even for a perfect spectrometer. The best possible Raman measurement, with 4 7r collection, 100 per cent spectrometer transmission, 100 per cent quantum efficiency, and zero background is still limited in SNR by Eq. (4.6). As long as the photons arrive randomly (and any other case is hard to envision), the maximum SNR will be given by Eq. (4.5) and (4.6). Figure 4.4 is an example of a Raman spectrum with an SNR determined mainly by sample shot noise. 

The sample shot noise limit occurs when ob, ad, and (Jr are small compared to as in Eq. (4.11). This limiting case was discussed in Section 4.2.1 and leads to Eq. (4.6). The SNR increases with for a constant S(e sec ). As stated earlier, the shot noise limit yields the highest SNR possible for a given value of S... 

The background shot noise limit is often encountered in samples containing fluorescent materials. A relatively small Raman peak is superimposed on a high background, but the noise originates almost totally in the background. In this case, ag = (Bt) and... 

SNRd is always smaller than that for the sample shot noise limit, all else (other than detector noise) being equal. As (p is decreased by improvements in the detector, SNRd approaches the value determined by sample and background shot noise. 

SNR/i is used to denote the analyte shot noise limit, and a. Da, La refer only to analyte variables. Recall that ts is the measurement time for each resolution element and does not necessarily equal the total measurement time As expected, SNR varies with... 

The analyte shot noise limit illustrates the fundamental Importance of multichannel detection for improving SNR. Equations (4.15) and (4.16) are valid for each resolution element in the system in a single or multichannel spectrometer (but not for a multiplex spectrometer). As noted in Section 3.4 and Eq. (3.9), a spectrometer that can monitor Nk resolution elements simultaneously increases the measurement time for each resolution element by a factor of Nr over the time required for a single-channel system. Since Im = NrIs,... 

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