Haus, Hermann A.

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VI. NOISE IN ELECTRON DEVICESProf. H. A. Haus Prof. R. P. Rafuse H. J. E. H. PauwelsProf. P. L. Penfield, Jr. V. K. PrabhuA. HIGHER ORDER MOMENTS OF PHOTOELECTRON COUNTSWe report experimentally observed higher order moments of the photoelectroncounts in a photomultiplier illuminated by a maser operating below but near threshold.The results are compared with theory. The photoelectron counts occurring in a set ofpredetermined intervals of length T were recorded on tape and then fed to a computer.The second, third, and fourth order moments of the photoelectron counts as functionsof the observation interval, T, were evaluated. The higher order moments contain infor-mation about the higher order correlation functions of the power P(t) incident upon thephotomultiplier.The probability w(K) of receiving exactly K photoelectrons within a time intervalextending from t = 0 to t = T is'- K -a T P (t) dtw(K) = a S P(t) dt e K! (1)where a is the proportionality factor71a , (2)and 1 is the quantum efficiency of the photosurface. We may now evaluate the factorialmoments of the photoelectron count. We haveK(K-1) ... (K-+) = a P(t) dt = aV dtf... Y dt I P(t l)  p(t ). (3)This expression is obtained by changing the I t h power of the time integral into ank-fold time-ordered integral. Equation 3 can be applied to various assumed time depend-ences of the power emitted by a laser. Comparison of the experimental results withthe theoretical expressions thus obtained provides checks on the correctness of theassumed model. We assume a correlation function for the amplitude of the electricfield of the incident laser light when the laser is operated below threshold in asingle modea(t) a(t+T) = a eo/2 (4)Such an expression is valid when the laser line is Lorentzian. If we use expression(4)in (3) applied to the second-order factorial moment, we findQPR No. 78(VI. NOISE IN ELECTRON DEVICES)2K(K-1) - K -I 1 -oT_P 1_oS coT l 1 - e . (5)K o oWhen operating a laser below threshold, one finds in general that more than one singlemode contributes to the output power. Under the assumptions that all but one of the out-put modes of the laser are broad-band and that the single narrow-band mode contrib-utes a fraction w to the output power, we obtain, instead of Eq. 5, the followingexpression for the normalized second-order factorial moment:2 rK(K-1) -K 1 (- T]2 aP 1 o= w -21 1-eK o oFor a Gaussian distribution of the amplitude of the laser signal below threshold, allhigher order correlation functions of the power can be related to the correlation functionof the amplitude. When this is done, we obtain for the normalized third-order factorialmoment-3K(K-1)(K-2) - KaP-2 WoK o0S2 ( 1 1 o3 ! w 1  +  eo o+ 2w3 e - 0To1 T 2 ( o T+ l-eo (coT0and for the normalized fourth-order factorial moment4K(K-1)(K-2)(K-3) - K3KaP= 4!o-wTw21 1 e22T 2w To o3 T 2 o Z2 1 4 [ 1 4+ 2w + e +  + w +wT 2 2 0 2 T )14 + T) + + 13(Co T) ( T) o (o T)0 O-2 T_o+ e 3(O T )I0O~ (8)6 4In Fig. VI-1 Eqs. 6-8 have been plotted for the values aP = 4. 3 x 106, C = 0.77 X 10 ,and w = 0.44. Experimental points are shown for comparison. The values of aP weredetermined from the observed photoelectron count (since aP is the rate of photoelectronemission in the photomultiplier). The value of wo was determined from a spectral meas-urement that was run simultaneously with the photoelectron counting experiment. Thevalue of w was obtained by matching Eq. 6 to the experimentally observed second-orderQPR No. 78(VI. NOISE IN ELECTRON DEVICES)ORDER MOMENTO SECONDX THIRDA FOURTH1/ w, T = 1.30 x 10- 4T ( seconds )Fig. VI-1. Comparison of theoretical and experimental results.factorial moment. By using these values, the theoretical curves for the third and fourthmoments were evaluated. The check of these theoretical curves with the experimentallyobserved points is gratifying.The experiments were performed at Lincoln Laboratory, M. I. T., jointly withMr. C. Freed.H. A. HausReferences1. L. Mandel, "Fluctuations of Photon Beams and their Correlations," Proc. Phys. Soc.(London) 72, 1037-1048 (1958).QPR No. 78

Noise in Electron Devices

VI. NOISE IN ELECTRON DEVICESProf. H. A. Haus Prof. R. P. Rafuse L. R. CardilloProf. P. L. Penfield, Jr. H. J. E. H. PauwelsA. DETECTION OF LIGHT INTENSITY FLUCTUATIONS THROUGH PHOTO-ELECTRON COUNTSMeasurements on the noise of optical maser oscillators have been in progress forsome time. Such measurements can be performed by using spectral methods, thatis, by detecting the spectrum of the photomultiplier current,2 or they can be made byobserving the statistics of the photoelectron count of a photomultiplier illuminated byoptical maser light. Measurements of the last kind are reported here.It was shown previously3 that the correlation functions of the intensity of a light beammay be detected by measurements of the statistics of the photoelectrons in a photomul-tiplier illuminated by the light. If one observes the number of photoelectrons K in aset of time intervals of fixed length T, one may determine from the observed secondfactorial moment K(K-I) the autocorrelation function of the light intensity. We have 3K(K-1) = 2a 2  (T-Tr) R(T) d-. (1)S0Here, R(T) is the autocorrelation function (short-time average) of the light power, anda is a factor related to the photoefficiency, a = r/hv. An optical maser oscillating ina single mode produces at the photomultiplier surface a field amplitude with a timedependence,A(t) = [A +a(t)] cos [wt+4(t)] (2)The term A is the steady-state amplitude of the field, a(t) is the small-noise excitation,a quantity with frequency components low compared with the optical frequency. Definingthe normalized correlation function of the noise amplitude, we havea(t) a(t+T)pa(7) = (3)a(0)2And the modulation coefficient is22 a(0)m = (4)A2oWe then obtain for the autocorrelation function of the powerR(t) = P[1+4m2pa(T)]. (5)QPR No. 74(VI. NOISE IN ELECTRON DEVICES)Combining these results, we obtain for the mean-square deviation of the photoelectroncount2 -2 2 2 TK -K m1+8 TKK K T0(T-T)pa(T) dT.If we assume that the autocorrelation function pa(T) is of the formPa(T) = ewhich corresponds to a Lorentzian line shape for the noise, we have(7)(8)2 26K 1+8 ma 1 - ,S1 + 8 m 1 1-ewhere a is the photoelectron rate, a = R/T. The bandwidth of the noise expected fora maser oscillating in a single mode will be, in general, narrower than the bandwidthof the optical cavity. The narrowing is due to the negative-resistance effect of the masermaterial. 4 Thus the noise bandwidth is, in general, a function of the excitation levelof the optical maser. Figure VI-1 shows a set of observed values for the mean-squarefluctuation normalized to the photoelectron count. The measurements were taken ona DC excited internal mirror optical maser oscillating at 6328 angstroms.A thousand samples were taken for each counting interval, T. The deviation of theideal value from the experimentally determined value, because of the noninfinite num-ber of samples taken, was approximately 5 per cent. Even though the theoretical4SECONDSFig. VI-1. Experimentally determined mean-square deviation.QPR No. 74(VI. NOISE IN ELECTRON DEVICES)formula predicts that the plot should start out at unity, it was found experimentally thatthe mean-square fluctuation for short observation times was less than this ideal value.It can be shown that the deviation is due to occurrence of multiple pulses in the reso-lution time of the counter.If the light were completely unmodulated, and the photoelectron count were there-fore Poisson, the observed mean-square count with an average count of IT per resolutiontime would have been6K2 -nK =e . (9)KThe resolving time of the counter was approximately 20 nanosec, the count was12. 7 million per second; thus TI = 0. 25. Therefore, the observed mean-square fluc-tuations at short observation times can be accounted for theoretically.The power output of the optical maser was stabilized by an external feedback circuitthat kept the fluctuations in the output power to less than one tenth of one per cent. Thetime constant of the feedback circuit was 1/2 sec, so that rapid fluctuations were notaffected by it. In Fig. VI-1 we have fitted a curve to the experimental data under theassumption that the spectrum of the noise amplitude a(t) has bandwidth wo = 125 rad/sec.The corresponding assumed modulation coefficient is m 2 = 0. 9 X 10 - 5 . During thecounting experiment, the spectrum of the photomultiplier current was monitored. Themaser operated quietly and no gas-discharge fluctuations were visible on the spectrumS analyzer.The experiments were performed at Lincoln Laboratory, M. I. T., in cooperationwith Mr. C. Freed. Mr. L. R. Cardillo, a candidate for the Bachelor's degree, par-ticipated in the measurements.H. A. HausReferences1. Quarterly Progress Report No. 69, Research Laboratory of Electronics, M. I. T.,April 15, 1963, p. 31; No. 70, July 15, 1963, p. 77; No. 71, October 15, 1963, p. 87;No. 72, January 15, 1964, p. 53.2. Quarterly Progress Report No. 71, loc. cit.3. Quarterly Progress Report No. 70, loc. cit.4. Quarterly Progress Report No. 72, loc. cit.QPR No. 74

IX. NOISE IN ELECTRON DEVICESProf. H. A. Haus Prof. R. P. RafuseProf. P. L. Penfield, Jr. Dr. G. FioccoA. HIGHER-ORDER CORRELATION FUNCTIONS OF LIGHT INTENSITYA measurement of the photoelectron count of a photosurface illuminated by laserlight was outlined in Quarterly Progress Report No. 69 (pages 31-33). It was pointedout there that the mean-square deviation of the photoelectron count within a time Tis related to the correlation function of the light intensity R(T). 1 ' 2 We shall now showthat higher-order moments of the photoelectron count are related to the higher-ordercorrelation functions of the light intensity. In the derivation we assume that the coher-ence area of the light beam is larger than the photocathode area.The probability PT(K) of obtaining exactly K counts in the time interval 0 < t < Tfrom a photocathode illuminated by light of intensity (power) P(t) isKnT -nTPT(K) e (1)whereTn T = a P(t) dt. (2)0Here, a is a proportionality factor incorporating the photoefficiency of the cathode. Thefalling factorial moment of kt h order of the photoelectron count, K(K-1) . . . (K-k+l), is0C KSnT -n T  kK(K-1) ... (K-k+1) = K! e K(K-1) . .. (K-k+1) = n . (3)K=1Thus far, we have taken an average with respect to the probability distribution of thePoisson process. If the light intensity itself varies in a statistical manner, an averagehas to be taken over the statistics of the incident light. This average is denoted byangular brackets.(K(K-1)... (K-k+1) = (n kT = ak P(t) d . (4)The right-hand side of Eq. 4 may be writtennk = k! ak t0 P(t) Ptk)dt ... dt k .  (5)Sk =0 1 ( kThe integrand can be put into the form of a (k-1) th-order correlation functionQPR No. 70(IX. NOISE IN ELECTRON DEVICES)by introducing the new variablestl = t t = t + TI ... tk t + Tk- 1nk =k! ak S T = T-Tk-1k-1S 0 (t) .. P(t+ k- 1 )  dT1 ... dtdTk- 11If the light is stationary, the expectation value of the integrand does not depend upon t,and the integration over t can be carried out immediately. Furthermore, noting thatthe integrand is the correlation function of (k-1)th order, Rk- 1 (T . . .Tk-), of the lightpower, one obtainsnk> = k! ak5Tk-1 = 02  (T-Tk-l) Rk- (T' ' . k-1) dT . . . dTk- 1  (7)1=0Differentiating Eq. 7 with respect to T, one obtainsd <k k k TdT T = k! akTk-1 =0S Rk-1 (T k-) dT dTk-1=0The falling factorial moment of k t h order of the photoelectron count within an observa-tion time T is thus simply related to the (k-l)th-order correlation function of the lightintensity. For the factorial moment of second order, k = 2, one obtainsdT2 nT> = ZaR 1(T).dT T1H. A. HausReferences1. E. M. Purcell, The question of correlation between photons in coherent lightrays, Nature 178, 1449 (1956).2. L. Mandel, Fluctuations of photon beams and their correlations, Proc. Roy.Soc. (London) 72, 1037-1048 (1958).QPR No. 70

Contains a report on a research project.Lincoln Laboratory (Purchase Order DDL B-00337)United States ArmyUnited States NavyUnited States Air Force (Contract AF19(604)-7400

IX. NOISE IN ELECTRON DEVICESProf. H. A. Haus Prof. R. P. RafuseProf. P. L. Penfield, Jr. W. D. RummlerA. NOISE FIGURE FOR NEGATIVE SOURCE RESISTANCEWhen parametric amplifiers and Esaki diode amplifiers are used as one stage in acascade of twoport amplifiers (Fig. IX-1), it often happens that the stage following suchan amplifier "looks" into a source impedance (the output impedance of the precedingstage) with a negative real part. For brevity, we shall call a twoport that exhibits anoutput impedance with a negative real part "negative-resistance twoport." The noisefigure for a twoport connected to a source impedance with a negative real part has notbeen defined by the Institute of Radio Engineers. The question arises how one shouldgo about either measuring or computing the noise figure of the entire system, or forany one of the stages.is s Fel'Gel Iout e2'Ge2Fig. IX-1. Cascade of two amplifiers.The measurement of the over-all noise figure of the system does not usually encoun-ter any difficulty because the source (usually an antenna) impedance connected to theinput of a receiving system has a positive real part.When computing the over-all noise figure of a system one may resort to two equiv-alent methods:(a) One may compute the noise figure without separating the system into successiveamplifying stages.(b) One may extend the IRE noise-figure definition to the case of negative-sourceresistance, 1 compute the noise figure for each stage, and evaluate the over-all noisefigure from an appropriate cascading formula.Both approaches are equivalent if one is interested in the over-all noise figure (ornoise temperature) of the system. When one is interested in measuring or computingThis work was supported in part by Purchase Order DDL B-00337 with LincolnLaboratory, a center for research operated by Massachusetts Institute of Technologywith the joint support of the U.S. Army, Navy, and Air Force under Air Force ContractAF 19(604)-7400.(IX. NOISE IN ELECTRON DEVICES)noise parameters for the individual negative-resistance twoport in a cascade it is con-venient to use formulas from which the noise parameters of this twoport may be eval-uated in a systematic way. The noise at a particular frequency of any twoport, whethera negative-resistance twoport or not, is completely described in terms of four param-eters. All of these may be obtained by noise-figure measurements on the twoport forvarious source impedances. In practice, one measures the noise figure of the entiresystem, consisting of the negative-resistance twoport used as the first stage, followedby higher stages, and one must be able to extract from the over-all noise-figure meas-urement the noise figure of the first stage. This can be done conveniently by using thecascading formula, provided that one uses the noise-figure definition that is consistentwith this formula.1 The cascading formula for the two stages in cascade isFez -1Fe Fel + G (1)elHere, the Fe's are the noise figures defined on the basis of exchangeable power, I andGel is the exchangeable gain of the first stage. In order to find F e l from a measure-ment of F e , one must know Gel and (Fe 2 -1) for the same value of source impedance asthat which drives the second stage in the cascade. If this impedance has a negative realpart, one must overcome the practical problem of determining experimentally Fe2 - 1for negative-source impedance, a rather difficult and inconvenient measurement to per-form directly. Furthermore, Gel cannot be obtained from a gain (Ge) measurementof the over-all system unless GeZ (the exchangeable gain of the second stage) is knownfor a negative-source resistance. The purpose of this letter is to show how one candetermine the second term in Eq. 1 from signal measurements on the first stage, andfrom noise measurements on the second stage by using only source admittances witha positive real part. For this purpose, we rewrite the second term in a form containingterms that are more conveniently measured.e2 e 1- 1 F- out-(F 2 -1) G1 out (2)el Ge 1 G1 outwhere G l out is the output conductance of the first stage that acts as the source con-ductance of the second stage.G1 out = Gs 2  (3)It is well known2 that the conventional (IRE definition of the) noise figure of a twoport canbe expressed as a function of the source admittance Y = G + jB for G > 0 as follows:F = F + n (G -G ) + (B -B o )  (4)o G so so (F=F S(IX. NOISE IN ELECTRON DEVICES)The quantities F o , R n , G , and B are the four noise parameters that describe the noiseof the twoport (the second stage). The extended definition of the noise figure, F e , pre-serves the analytic form of F in terms of G s and B for negative values of G . If oneplotsH = (Fe-1) G = (F 0 -1) G + Rn [(Gs-Go ) + (Bs-Bo)2Ias a function of G one finds it to be a parabola with a minimums(F -1)o 2Hmin= Go(Fo-1) - 1)4R + R (Bs-B )min G ( 4 R n s onatF -1G =G 0s o ZRnIt can be shown that Hmin > 0 in all cases. Therefore, a plot of H against G s has theappearance of Fig. IX-2. A measurement of H for positive G s may be extrapolated tonegative values of G s . One simple way of doing this is to plotFH-H vs x G - G + omm s o RnThis plot is a straight line, a fact that can serve as a check on the experimental accu-racy of the measurements with positive values of G s , as well as a simple means ofextrapolating H to negative values of G s . The measurement of G 1 out is straightforward.The experimentally obtained value of G 1 out is used as the source conductance on thetHHminS 0 2R n- GsFig. IX-Z. Plot of H versus G .s(IX. NOISE IN ELECTRON DEVICES)is Ys i o YI outFig. IX-3. Norton's equivalents of source and output of the first stage.previously obtained plot to read off the value of H = (F-1) G s for the second stage.The value of GelG1 out may be obtained from a transducer gain (GT) measurementon the first stage. If we use Norton's equivalents of the source on the one hand, andthe output port of the first stage connected to the source on the other hand (Fig. IX-3),for the transducer gain of the first stage, we havei 2GoGL22(Yo+Y) i 4G 21 .2. . (Y +Y 1 )1 1 1 out Ls s4Gswith G L = Re (YL), where YL is the admittance of the load connected to the output of thefirst stage. The exchangeable gain isG i Gel -7 G out.2 1 out1sTherefore(Y out+YLG G out +LGelG1 out T= GI 4GThus one may obtain GelG1 out from measurements of the transducer gain of the firststage, of the load admittance YL, and of the output admittance of the first stage. Withthese measurements one may determine (Fe2-1)/Gel as it appears in the cascading for-mula. (Fel-l) may thus be evaluated.H. A. HausReferences1. H. A. Haus and R. B. Adler, An extension of the noise figure definition, Letterto the Editor, Proc. IRE 45, 690 (1957).2. A. G. Th. Becking, H. Groendijk and K. S. Knol, The noise factor of 4 terminalnetworks, Philips. Res. Rep., vol. 10, pp. 349-357, 1955, and IRE Subcommittee 7.9on Noise, Representation of noise in linear twoports, Proc. IRE 48, 69-74 (1960).

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Noise in Electron Devices

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