Contains research objectives and reports on one research project.Lincoln Laboratory, Purchase Order DDL B-00368U. S. Air Force under Air Force Contract AF19(604)-7400U. S. NavyU. S. Arm

Haus, Hermann A.

Penfield, Paul L., Jr.

Rafuse, R. P.

Rummler, W. D.

Contains research objectives and reports on two research projects

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VII. NOISE IN ELECTRON DEVICESProf. H. A. HausProf. P. L. Penfield, Jr.Prof. R. P. RafuseRESEARCH OBJECTIVESThe purpose of the work on the noise performance of multiterminal-pair networks(Quarterly Progress Report No. 68, pages 53-59) has been to ascertain the limiting noiseperformance, as well as to determine appropriate measures of noise performance foramplifiers with more than two terminal pairs. This phase of the work has been com-pleted and the results were submitted to the Department of Electrical Engineering,M.I.T., June 1963, as part of an Sc.D. thesis by W. D. Rummler. This thesis is to bepublished as Technical Report 417 of the Research Laboratory of Electronics, M.I.T.Our future effort will be concentrated in three areas. The first concerns the noiseperformance of devices in which quantum effects are expected to be detectable or, infact, dominant. Because amplitude noise measurements on the c.w. gaseous maseroscillator are relatively convenient to make, the theoretical and experimental work willbe concentrated around this device.(a) A quantum-mechanical analysis of the noise in optical maser oscillators will beundertaken. The purpose of the analysis is to determine the limitations of the semi-classical analysis, briefly summarized in this report.(b) Measurements on the amplitude noise in optical maser oscillators will continue.Our intent is to determine the noise level of "quiet" optical maser oscillators by refine-ment of the existing apparatus.(c) The sensitivities of various experimental methods for the determination ofamplitude noise of optical waves will be compared experimentally.The second area of interest is noise in frequency multipliers, especially those madefrom varactors. We hope to develop simple parameters to indicate the noisiness of asignal and of a multiplier through which this signal passes, and then to compare the noiseperformamce of various types of multiplier chains.The third area of interest is the noise contribution of double-sideband degenerateparametric amplifiers in a variety of applications.H. A. Haus, P. Penfield, Jr., R. P. RafuseA. ANALYSIS OF NOISE IN OPTICAL MASER OSCILLATORWith the aid of a semiclassical analysis, Lamb has shown that a gaseous laser witha Doppler-broadened line obeys equations similar to those of the van der Pol oscillator.Because of these results, it is reasonable to attempt to relate the existing work on noisein van der Pol oscillators to fluctuations in the optical maser. If one considers a maseroscillating in a single mode and if one dis-regards the coupling to other modes of theoptical cavity, one may represent the dynamicC L G_ G in behavior of the amplitude of the oscillationby the equivalent circuit of Fig. VII-1. Thenonlinear negative conductance of the maserFig. VII-1. Equivalent circuit. material has the assumed voltage dependenceQPR No. 72(VII. NOISE IN ELECTRON DEVICES)G_ = G (1-a< V2 >), (1)-mwhere the angular brackets represent a(short-)time average over a few cycles of theoptical frequency, and a is a constant. The conductance G represents the losses andexternal loading of the optical cavity. The inductance L and capacitance C of the cir-cuit resonate at the optical-cavity frequency o ; their magnitude with respect to G isdetermined by the "cold" cavity Q. If one intends to represent microphonics as causedby the vibration of the mirrors, an important contribution to the frequency modulationof the optical maser output, one would have to vary L and C in time, keeping their ratiofixed if the Q remains unaffected by these variations. Here we shall disregard sucheffects in the belief that they may not contribute to the amplitude modulation of the maseroutput. The noise current generator incorporates the fluctuations associated with thermaleffects (negligible in the case of an optical maser) and with the emission of the activematerial. If one disregards changes in i that are due to the nonlinearities of G , one2-4may use for it the well-known value associated with the spontaneous emission in alinear, active material. The mean-square fluctuations within the frequency incrementAv are2i = a4G hvA v. (2)n mHere, a(>, 1) is a factor accounting for the incomplete inversion in the maser medium.1a =+ , (3)exp - 1where T is the (equivalent) negative temperature of the maser medium. To the equiv-m 5alent circuit of Fig. VII-1 one may apply the theory of noise in a van der Pol oscillator.One assumes the time dependence of the voltageV(t) = R(t) cos [Wot-O(t)]. (4)Furthermore, one notes that R(t) and 0(t) are relatively slow functions of time. Finally,one treats the case of low noise, for whichR(t) =R + R(t). (5)Here,R (t) << R0(t) = O0 + 1(t), (6)where 0 1 (t) is a slow function. Under these assumptions, for the steady-state ampli-tude R one findsoQPR No. 72(VII. NOISE IN ELECTRON DEVICES)G -GR = m (7)o aGmand for steady-state phase 0oo = const. (8)The noise terms R1 and 01 satisfy the (approximate) differential equationsa 1 diSR1 + dtn sin (wot-o)) (9)oRe = t cos ( t-6 0) , (10)wherewC0 (11)Q=G -G"mHere, Q' is the "hot Q" of the oscillator. Near threshold it varies between a value muchgreater than the "cold Q" to a value comparable with it. The spectra of R1 and 01 canbe obtained from Eqs. 2, 9, and 10. Defining the spectral density of the driving noisecurrent generator, Wo(V), by.21W (v) = n (12)o 2A v'one has1TWW (v) 2 o (13)1 C 2 o 2+w /Q,2with w = 2rrv, and1-WW (v) =2 0 (14)1 R2C2oA measurement with a photocell or photomultiplier on the light output of a single maser2detects only WR (v). The mean-square fluctuation R 1 may be obtained from the integral1 2over all frequencies of WR (v). For the modulation ratio m one finds12 R ahv Gm - 1 (15)2 1 2 G m - Go 2 oQPR No. 72(VII. NOISE IN ELECTRON DEVICES)The output power may be expressed in terms of an external Q, Qe1 2 WoCP = ,-R Q (16)2 oQOne may assign a bandwidth Awe to 1/Q e byA e = Wo/Qe. (17)Then the modulation ratio becomes2 ahv A Go e mP G - G' (18)mOne can make estimates as to the magnitude of m 2 . Taking the He-Ne visible maser7line at 6328 A, Aw = 10 mc, and an output power of 1 1.w, one haseG2 m - 5m =aG m 1. 9 10 (19)mThe factor aG /(G -G) is greater than unity. We believe that a value of m as obtainedmmhere is within reach of achieved experimental sensitivities.H. A. HausReferences1. W. H. Lamb, Theory of an optical maser (to be published in Phys. Rev.).2. M. W. Muller, Noise in a molecular amplifier, Phys. Rev. 106, 8-13 (1957).3. R. V. Pound, Spontaneous emission and the noise figure of maser amplifiers,Ann. Phys. 1, 24-32 (1957).4. H. A. Haus and J. A. Mullen, Noise in optical maser amplifiers, Presented at thePolytechnic Institute of Brooklyn Symposium on Optical Masers, New York, April 1963.5. J. A. Mullen, Background noise in nonlinear oscillators, Proc. IRE 48, 1467-1473 (1960).B. OPTIMUM NOISE PERFORMANCE OF MULTITERMINAL AMPLIFIERSThis report is an abstract of an Sc.D. thesis, Department of Electrical Engineering,M.I.T., August 19, 1963, which will appear as Technical Report 417, in January 1964.The problem is to find the optimum noise performance that can be achieved from alinear multiterminal-pair signal source used in the most general linear way with a linearmultiterminal-pair amplifier; this optimum performance is achieved at a single-outputterminal-pair in a narrow band of frequencies. The most general linear way in which touse such a source and amplifier is to imbed both networks in an arbitrary linear losslessnetwork and take the output through this network. The optimum noise performance ofQPR No. 72(VII. NOISE IN ELECTRON DEVICES)such a system is defined as the maximum signal-to-noise ratio that can be achieved atlarge exchangeable signal power. It is shown that this is a meaningful criterion for anysystem that is to deliver ultimately an amount of signal power considerably greater thankT Af, the noise power available from a room-temperature resistor in the same band-owidth. The techniques described can be used, however, to determine the maximumsignal-to-noise ratio that can be achieved at any value of exchangeable (or available)signal power.The discussions are simplified by presenting the results in a plot with noise-to-signal ratio and the reciprocal of exchangeable signal power as coordinates - the noise-performance plane. This plot is useful for examining the noise performance of anylinear system. The noise performance of a conventional two-terminal-pair amplifier isexamined on this plane. Such a system defines two points in the noise-performanceplane: one by the noise-to-signal ratio and exchangeable signal power of the source; andanother by the noise-to-signal ratio and exchangeable signal power at the output. Interms of these two points, noise figure, exchangeable-power gain, and noise measureare given a geometrical interpretation in the noise-performance plane.It is shown that the noise performance - signal-to-noise ratio and exchangeablesignal power - that can be achieved by imbedding a multiterminal-pair source and amultiterminal-pair amplifier in an arbitrary lossless network can also be achieved byimbedding the source in an arbitrary lossless network to reduce it to a one-terminal-pair source and using this one-terminal-pair source to drive an arbitrary losslessreduction of the amplifier. Thus the optimization problem can be solved by consideringseparately the noise performance of the source and the noise performance of the ampli-fier. For this reason, the problem is attacked by considering a set of problems ofincreasing difficulty. First, the noise performance that can be achieved with a noisy(positive or negative) resistance used in conjunction with a one-terminal-pair source isexamined. The points in the noise-performance plane that can be achieved at the outputof such a system all lie on a straight line through the source point with a slope equal tothe exchangeable noise power of the resistance. Subsequently, the problem of the noiseperformance of a multiterminal-pair amplifier used with a one-terminal-pair sourceis considered. After the noise performance of a multiterminal-pair source imbeddedin an arbitrary lossless network has been examined, the general noise-performanceproblem can be solved by inspection.With this work used as a basis, definitions are given for the noise figure,exchangeable-power gain, and noise measure of a multiterminal-pair amplifier that isused with a multiterminal-pair source. These quantities are derived and/or interpretedby comparing a system with its noiseless and "gainless" equivalent - its equivalentsource network.The optimization procedure is extended to multifrequency networks coupled in anQPR No. 72(VII. NOISE IN ELECTRON DEVICES)arbitrary manner by a lossless nonlinear device that obeys the Manley-Rowe formulas.In this context, the optimum noise performance of an ideal (lossless) double-sidebandparametric amplifier is examined for both the cases in which the noises in the two side-bands are uncorrelated and partially correlated.W. D. RummlerQPR No. 72

Noise in Electron Devices

XII. NOISE IN ELECTRON DEVICESProf. H. A. Haus Prof. R. P. RafuseProf. P. L. Penfield, Jr. W. D. RummlerRESEARCH OBJECTIVESThe research on noise in electron devices has two objectives:(i) The study of specific devices, such as the parametric amplifier and the maser,to determine the physical sources of noise and the limitations they impose on the noiseperformance of the amplifier.The parametric amplifier with coherent input signals at both the signal and idlerfrequency, and the degenerate parametric amplifier are the impetus for the secondobjective.(ii) The determination of a measure of the optimum noise performance of multiter-minal linear amplifiers. The optimum noise performance of a linear twoport amplifieris known to be expressible in terms of its minimum noise measure, a quantity that ischaracteristic of the amplifier. The optimum noise performance of a multiterminalpair amplifier excited by a multiterminal pair source is at present under study for thepurpose of devoloping an analogous measure for the optimum noise performance ofsuch an amplifier.H. A. Haus, P. Penfield, Jr., R. P. RafuseA. SOLUTIONS TO THE PROBLEM OF THE OPTIMUM NOISE PERFORMANCEOF MULTITERMINAL AMPLIFIERSIn a previous report1 we showed that under certain restrictions the stationary valuesof signal-to-noise ratio which can be obtained for a specified value of exchangeablesignal power at a single-output terminal pair by imbedding an n-terminal pair sourceand an m-terminal pair amplifier in an n + m + 1 terminal pair lossless network aregoverned by two coupled matrix equations - Eqs. I and 2.Ena E t  + 2k(Z + Zt) x = 0 (1)na na 2 a a 2EEtx + i E + n 2k(Z+Zt)l = 0 (2)ss 1 n nBy a simpler derivation, it has been established that Eqs. 1 and 2 are valid regardlessof the nature of the matrices characterizing the source and amplifier networks; how-ever, for simplicity we shall assume here that EsE is positive definite, Z + Zf ispositive definite, and Z + Zt is indefinite.a aComponents of the vectors xl and x 2 may be interpreted as complex voltage ratios.For instance, with all voltage sources in the source and amplifier short-circuited,This work was supported in part by Purchase Order DDL B-00368 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 ContractAF19(604)-7400.QPR No. 68(XII. NOISE IN ELECTRON DEVICES)except the source represented by Es 1 , which is the open-circuit signal voltage at thefirst terminal pair of the source network, the open-circuit voltage at the output termi-nal pair is x 1 1 E s 1 , where x1 1 is the first component of the vector xI. Similarly, thecomponents of x2 relate the open-circuit noise voltage at the output terminal pairs to theopen-circuit voltages of the amplifier. Imbedding the source and amplifier networks ina given n + m + 1 terminal pair lossless network enables us to define a unique pair ofvectors xl and x2 . The reverse is not true; it can be shown that there are an infinitenumber of lossless networks corresponding to a particular pair of vectors xl and x2.(i)For Eq. 1 there are n eigenvalues Xi and eigenvectors x2  We may express theeigenvalue X. in terms of its eigenvectors as1(i)tE  Et x (i)2 na na 21 2x Z +Z (i)a a) x2For convenience later, we shall label the smallest positive eigenvalue as X1 and theremaining positive eigenvalues in ascending order, and we shall label the smallest neg-ative eigenvalues Xm and the remaining negative eigenvalues in descending order.Equation 2, on the other hand, has n solutions for each of the m eigenvalues k. of1Eq. 1; this gives us a total of n X m sets of solutions to Eqs. 1 and 2. Using adouble-subscript notation for these eigenvalues and eigenvectors, we may express theeigenvalue pij in terms of . and the eigenvector x(ij)as(ij) I j) + 2 x (ij)t(z+zt (ij)Sx 1  EE nnx + 2.x (Z+Z )xSj n (4)ij xiJ) E Etx(ij)1 s sl1Here, we number our eigenvalues for a given ki by the order of the value of their recip-rocals; the minimum value of 1/ i (which may be negative) is 1/ il and the maximumis 1/pin.We would now like to relate the quantities of interest, namely the signal-to-noise ratio at the output and the exchangeable signal power at the output,to the eigenvalues derived from Eqs. 1 and 2. For the optimal imbedding net-work corresponding to a set of values X. and 4ij we write the signal-to-noiseratio asx(ij) EE x(iJ)1 S= Es s l".. -(5)1 n n 1 2 na na 2and the exchangeable signal power asQPR No. 68(XII. NOISE IN ELECTRON DEVICES)(6)x~ii) E Et x ~ (i1 ss 2x (Z + ) + xx2x (Z+Zt) )xj + 2x(')t Z ,+Z4 x1 1 2 ( 2We now see why these quantities do not appear directly as eigenvalues. In Eq. 3 we seethat X. is independent of the magnitude of x 2 ; similarly, tij in Eq. 4 is independent ofthe magnitude of x . On the other hand, both of the quantities defined in Eqs. 5 and 6(i) a i j )are dependent on the relative magnitude of x 2 and xl ) We can explicitly demonstratethe consequences of this by considering the particular realization of the optimal networkthat places this fact in evidence.Consider an imbedding of the m-terminal pair amplifier in an arbitrary m + 1 ter-minal pair lossless network. We know that we can always pick this network in such amanner that the mean-square voltage and the real part of the impedance at the outputterminal pair are x na naE and 1/2 x Z + Z  , respectively, where(i)' (i)x 2  is a vector that is proportional to x2 Similarly, we can losslessly reduce thesource network to a one-terminal pair network with mean-square signal voltageIDEALI:nFig. XII-1. Realization of the optimal network.QPR No. 68Pij =REDUCED AMPLIFIER NETWORKX' i ) ' t E  E t X i ) '2  na na  2X i)' t Za + Za ) X2 2 (XII. NOISE IN ELECTRON DEVICES)x(i)t EEEx j), mean-square noise voltage x E x , and impedanceI s s1 1 n n 11/2 x i ) (Z+ Z) x i ) . Combining these two reduced networks as shown in Fig. XII-1,we see that the signal-to-noise ratio and the exchangeable signal power at the output are(i) G)given by Eqs. 5 and 6, respectively, where x2 = nx This network is optimal in thesense of Eqs. I and 2 for any value of n. We see that varying the transformer ratio isequivalent to varying the ratio of Ix )J to Ix~i)l . Physically, then, when we vary nor x(/ xij, we are changing the amount of our use of the amplifier, since we arechanging the exchangeable signal power at the output and, thereby, also changing thesignal-to-noise ratio at the output. This interpretation is actually completely generaland independent of the particular imbedding network that we are using. However, ingeneral, variation of the ratio of Ix()l to Ix(lJ)l can correspond to some complexvariation of the imbedding network because of the multiple feedback loops that mayexist between the source and amplifier networks. Figure XII-1, then, just gives aconvenient way of visualizing the effects of varying the ratio of IxM)I to I xl JWe see then that setting the ratio of I x') to xl equal to zero or, equivalently,setting n=O is the same as throwing away the amplifier. The values of aij and pij forthis limit are just the signal -to -noise ratio and the exchangeable power of the sourcenetwork alone. We designate these two quantities asx(iJ)t E Et x(i j )S=1 ssl(7)i, S x(ij)t E Et x j)1 nnlxij)t Et x(ij)Pij, s 1 s s l ,2x(iJ) (Z+Z ) x(ij)respectively.It follows that for a given pair of eigenvalues ki and Lij we may vary ij and pijby varying the ratio of the magnitudes of the eigenvectors, that is, by varying only thetransformer ratio in Fig. XII-1. Such a variation enables us to plot the stationaryvalues of signal-to-noise ratio as a function of the exchangeable signal power at the out-put corresponding to this particular solution to Eqs. 1 and 2. For this purpose we needthe relation1/crij= ij - ki/Pij, 9)which may be verified by using Eqs. 3-6. From Eq. 9 we see that it is more conven-ient to plot characteristic curves of 1/aij as a function of 1/p... In the I/ij. - 1/pijplane these characteristic curves are straight lines with slopes -X. and intercepts1/ i j with the 1/rij axis. Thus a set of eigenvalues k. and Lij merely determine a13 13 1 iQPR No. 68(XII. NOISE IN ELECTRON DEVICES)characteristic line in the 1/hij - 1/Pij plane; moreover, we see that for the statedproblem there are n X m such lines.It must be pointed out, however, that only one-half of a characteristic line is real-izable. Comparing Eqs. 5 and 7, we see that1 1- , (10)1] 13, sand, using Eqs. 4, 7, 8, and 10 in Eq. 9, we find that either. > 0 and i/pij < 1/Pij,or (11). < 0 and i/Pij > 1/piThe equality signs in Eqs. 10 and 11 will hold only when I x') /I x ij) is zero, that is,the realizable one-half of the characteristic line ends at the point determined by thesource - the point whose coordinates are 1/i, s and 1/pij . Hence, in the 1/0-ij - 1/pijij, s ij, splane we can realize only the one-half of the characteristic line that is above and to theleft of the source point for negative Xi, and above and to the right of the source point forpositive Xi. We can only achieve a signal-to-noise ratio equal to 4ij at infinite exchange-able signal power if 1/pij = 0 satisfies one of the inequalities of Eq. 11.In displaying the solution to the optimization problem in the 1/cr. - 1/p. plane, we13 ijshall show only those solutions for a given Xi which correspond to 1/pil, the minimumvalue of 1/. i , and 1/pin, the maximum value of 1/pi. We would like to find where theend points of these characteristic curves lie in the 1/ij - 1/pij plane. If we considerhow these end points change as k. changes, we find that they generate a closed curve.This is illustrated in Fig. XII-2, in which several of these characteristic curves areshown as they must appear for positive X's and minimum 1/ i.  With referenceto Fig. XII-1, when we vary ki we are changing amplifiers and then reoptimizing thesource for use with this new amplifier and thereby obtaining a new source point.32 3 >  2 >  X1 01/P. iFig. XII-2. Characteristic curves for Xi > O, i/pi lQPR No. 68(XII. NOISE IN ELECTRON DEVICES)Fig. XII-3. Typical set of characteristic curves.The closed curve generated is tangent to each of the characteristic lines. Physically,this curve represents the boundary of the region of values of the noise-to-signal ratioand the reciprocal of the exchangeable signal power that may be obtained at the outputby imbedding only the source network, and not the amplifier, in an arbitrary losslessimbedding network.In Fig. XII-3 we show a typical plot of characteristic curves for a positive definitesource and an amplifier having both positive and negative eigenvalues. We have shownonly those characteristics corresponding to extremal values of 1/i. for four values ofX. - the two smallest positive eigenvalues and the two smallest negative eigenvalues.(All solutions corresponding to intermediate eigenvalues of 1/L i would give rise tocharacteristic curves that terminate at points inside the source region. ) All of thesecharacteristics may be interpreted. The line characterized by Xi and 4ij is the trueoptimum-performance curve. It is the curve of the maximum signal-to-noise ratio asa function of exchangeable signal power for exchangeable signal power greater than pa'The line characterized by XI and ln represents the worst way of increasing exchange-able signal power with the best reduction of the amplifier. The line characterized byX2 and L21 is just a locus of points of stationary signal-to-noise ratio for fixedexchangeable signal powers. This same statement applies to all of the curves shown.The line characterized by Xm and Lml represents the best way of reducing theQPR No. 68(XII. NOISE IN ELECTRON DEVICES)exchangeable signal power below pp. Since Xm is negative here, we are using theamplifier as a positive resistor; for that matter, we are using the least noisy positiveresistor appearing in the cannonic form of the amplifier.W. D. RummlerReferences1. W. D. Rummler, Optimum noise performance of multiterminal amplifiers,Quarterly Progress Report No. 66, Research Laboratory of Electronics, M.I.T.,July 15, 1962, pp. 71-77.QPR No. 68

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Noise in Electron Devices

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