Noise in the Measurement of Light with Photomultipliers
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AE-323 UDC 621.383.29.521.391,S22 535.08 AE-323 Noise in the Measurement of Light with Photomultipliers F. Robben AKTIEBOLAGET ATOMENERGI STOCKHOLM, SWEDEN 1968 AE-323 NOISE IN THE MEASUREMENT OF LIGHT WITH PHOTOMULTIPLIERS F. Robben ABSTRACT In order to be able to compare measurements derived from the anode current of a photomultiplier with measurement derived from photoelectron pulse counting, a systematic investigation of the prop erties of some photomultiplier tubes has been made. This has led to a correlation of the properties of a photomultiplier based on the quan tum efficiency T|, the gain G, a photoelectron loss factor S and an effective dark rate D. In terms of these quantities the signal to noise ratio of an experimental measurement can be calculated, given the light flux and measurement technique. The fluctuations in a photomultiplier output are divided into two parts; Poisson fluctuations, and those due to excess noise. It is ex perimentally shown, from measurements on a 931A photomultiplier, that the excess noise exceeds the Poisson fluctuations only at very low frequences, or long DC measurement times (> 10 s), for both pulse counting and anode current measurements . The Poisson fluctua tions are found to be approximately the same for both pulse counting and anode current measurements, at both high light levels where the dark current, or dark pulses, are negligible, as well as at low light levels where the dark current is dominant. The excess noise is found to be somewhat greater in the case of anode current measurements. Thus both pulse counting and anode current measurement techniques have nearly identical noise properties, as far as the photomultiplier is concerned, and selection of either experimental technique depends primarly on the properties of the electronic equipment. By use of a synchronous detection technique, the variance of the pulse count was measured experimentally to an accuracy of ± 4 %, and was shown to be in agreement with that predicted by Poisson statistics. Printed and distributed in May, 1968. LIST OF CONTENTS Page Preface 3 1. Summary of Statistical Formulas 5 1. 1 Definition of the Average, Variance, and Standard Deviation 5 1.2 Gaussian Statistics and the Normal Error 7 1.3 Poisson Statistics 10 1. 4 Fourier Analysis of Fluctuating Quantities and the Power Spectral Density Function 11 2. A Photomultiplier Model: the Photoelectron Loss Factor S and the Effective Dark Rate D 15 2. 1 The Quantum Efficiency 71 and the Gain G 15 2.2 The Photoelectron Loss Factor S 18 2. 3 The Effective Dark Rate D 23 3. Methods for Determination of T|, G, S, and D: Fluctua tions in Measurements of Light Signals 27 3. 1 The Determination of 71, G, S, and D from the Photomultiplier Pulse Spectrum 27 3. 2 Noise in DC Measurements 28 3. 3 Noise in AC Measurements with Synchronous Detection 31 3.4 Formulas for S and D from Fluctuation Measure ments 33 4. Discriminator Design and a 931A Photomultiplier Pulse Charge Spectrum 38 4. 1 Discriminator Design 38 4. 2 931A Photomultiplier Pulse Charge Spectrum 40 4. 3 Values of S and D for the Model Photomultiplier 46 5. Fluctuation Measurements and the Inferred Values of S and D 51 5. 1 Variance of the Pulse Count for Short Gate Times 51 5. 2 Fluctuation Measurements: Inferred Values of S 53 5. 3 Fluctuation Measurements : Inferred Values of D 61 6. Fluctuation Measurements with Phase Sensitive Detection: Verification of Poisson Statistics 65 7. Conclusions and Recommendations 71 References 73 - 3 - PREFACE When these experiments were begun it was intended only to make a rather simple comparison between the signal to noise ratio of a pulse counting measurement and that of an anode current measurement. The equipment was built for measurement of the afterglow decay of a pulse discharge, and pulse counting was chosen for two reasons; 1) with dig ital readout, it gave a simpler electronic setup, and 2) it was believed that a somewhat better signal to noise ratio could be obtained. Very little additional equipment was required for the experiments described here, but the analysis and presentation proved to be a matter of some complexity. A report was finally attempted which would be understand able to an experimentalist who needs to use a photomultiplier, and is interested in achieving the best possible signal to noise ratio, but who does not have any special background in the theory of random signals. It is hoped that the result fulfills that goal. The shortcomings of the various data which photomultiplier tube manufacturers give for their products has long been known. For exam ple, although lighting engineers are naturally interested in lumens, to physicists who work with photon intensities this unit is most frustrating. It appears simpler to deal with quantum efficiencies, gain, and dark counting rates, since the subsequent conversion of the data to sensitiv ities in amperes/lumen, for instance, is straight forward. Thus in this report I have tried to present a set of simple and intuitively meaningful parameters to characterize a photomultiplier. Not all aspects are covered by the proposed four parameters Tt, G, S, and D; as, for in stance, there is also the pulse rise time and the photosensitive area. It is hoped that a change will be made in the future to a better presenta tion of photomultiplier tube characteristics. The equipment for these experiments was designed and constructed by Bjorn Fjallstam, who also made most of the earlier measurements. His careful work is greatly appreciated. Dr. Kurt Nygaard deserves special thanks for our many discussions of the nature of random noise, as well as for his suggestions regarding the construction of experimental equipment. The support of Dr. Josef Braun is gratefully acknowledged, and thanks are extended to Dr. Olav Aspelund for the loan of his voltage - - 4 - to -frequency converter. Finally, I would like to thank Dr. Jacob Weitman for his detailed criticisms of an earlier, shorter version of this report. These criticisms led to a more careful development of the statistical theory, the discovery of several new relationships, and an improved presentation of the data. - 5 - 1. SUMMARY OF STATISTICAL FORMULAS Measurements which result in a set of numbers, produced at regular invervals of time, are first treated. The relations which are more general and apply to a normal error distribution (Gaussian statis tics) are presented first. The specialization to Poisson statistics, where the measurement consists of counting a series of random events, is then made. A brief summary of the Fourier analysis of fluctuating quantities is then given. For a complete presentation ref. 1 is recommended. The power spectral density function for a signal consisting of delta functions occuring at random times is derived, thus providing a spectral descrip tion of counting a series of random events. 1.1 Definition of the Average, Variance, and Standard Deviation Assume that a measurement of a quantity which is constant in time, but which has fluctuations of some sort, results in a number. Let this measurement process be repeated, and let the result of each measurement be X^, i = 1, 2, ... N, where the value of i denotes the sequence of measurements. We will define the average value of by N <xi>==ill xi (i-D i= 1 as the result of a practical measurement. If the measurement of <X.>, according to Eq. (1-1), is repeated, the values will not in general be the same, and thus <X.> is also subject to fluctuations. In principle, however, if we let N-* oo, <X^> will approach a limiting value which we will denote by <X> without any subscript, N £ I <X> = lim Xi • (1-2) N-*oo i=l - 6 - If, as in this work, one is interested in obtaining an experimental value for <X>, and of course can only measure X^, some information is needed about the statistical deviations of X^ from <X> in order to estimate the accuracy of the assumption that X. (or <X^> ) is equal to <X>. This can be found from the mean square deviation, or variance, of X, given by N Var(X) = lim ^ £ (X. - <X>)2 . (1-3) N-»03 . i We might note that <X2> = <X>2 + Var(X) , (1-4) 2 where the mean square of X^, <X >, is defined by N <X2> = lim iV (X2) . (1-5) 6i In actual determinations of the variance from a finite number of measurements we will define N Var(X.) (Xi - <Xi>)2 » (1-6) i= 1 where, as before, the subscript will indicate that the variance is the result of a measurement and subject to statistical fluctuations. The root mean square deviation e(X), or standard deviation, will be defined by o(X) = [Var(X)]l/2 , (1-7) and the same convention regarding subscripts will be used. On occasion we will also refer to the normalized standard deviation, given by cr(X)/<X>, which is unchanged if the values of X^ are multiplied by a constant. 7 1, 2 Gaussian Statistics and the Normal Error If the statistical fluctuations are normally distributed, then the probability distribution of the measurements X^ is given by the Gaussian distribution function , _-(X. - <x>)2 n P(xi) = -----------iJl— exp I ----------^"2---------------- • (1-8 ) 1 (2 tt)172 a -2a J The probability that X^ lies within a certain range of values is given by the integral of Eq. (1-8) over this range. In Table I the probability that X. lies outside the range of values <X> ± n<j is given, where it is seen that the probability of an error exceeding 4<j is very small.