PROCEEDINGS OF THE IRE

Parametric Amplification of Space Charge Waves*W. H. LOUISELLt AND C. F. QUATEt, MEMBER, IRE

Summary-The principle of using a variable reactance in a circuitto produce gain can be applied to an electron beam. The beam isstrongly modulated by an rf wave at a frequency equal to twice thesignal frequency. In such a beam the two normal space charge waves

at the signal frequency break into two parts-one grows exponen-

tially with distance and the second is attenuated. Either the "slow"or the "fast" space charge wave can be amplified. When this beam,which has been previously modulated at the double frequency, iscoupled to a slow wave circuit, growing waves are found and againthe "fast" space charge wave can be amplified. We stress the amplifi-cation of the fast wave since the noise theorems which apply toprevious microwave amplifiers and establish a lower limit to the noisefigure do not apply. Thus, in principle, we should be able to obtainlower values for the noise figure.

I. INTRODUCTIONHE devices which amplify at microwave fre-quencies have in general used dc power as theirsource of energy, but it is now recognized that the

required power can equally well come from a continuouswave (cw) frequency which is related to the signal in a

particular way. In one form of amplifier this cw source,

denoted by the term "pump," functions to vary thereactance of a circuit; this variable reactance convertspower from the pumping frequency into the signal fre-quency. The term "parametric amplifier" has been usedto describe these devices. Suhl' has illustrated this prin-ciple with his discussion of the ferrite model. He hasshown that if a ferromagnetic sample is placed in a

cavity resonant at frequencies w, co,, and W2 whereC)-W+W2 oscillation or amplification may occur at fre-quencies wi or W2. This is true provided that power issupplied at co which is the frequency of uniform preces-

sion of the magnetization in a magnetic field H. Bridges2has shown that an electron beam can be used to vary

the reactance of a microwave cavity and therebyachieve "parametric" gain. In this paper we will de-scribe a distributed "parametric amplifier" which uses

an electron beam and derives its power from a cw source

oscillating at a frequency equal to twice the signal fre-quency.

Several early articles3-8 have described this methodof using a variable reactance to amplify signals and the

* Original manuscript received by the IRE, November 26, 1957;revised manuscript received, January 14, 1958.

t Bell Telephone Labs., Inc., Murray Hill, N. J.I H. Suhl, "A proposal for a ferromagnetic amplifier in the micro-

wave range," Phys. Rev., vol. 106, pp. 384-385; April, 1957.2 T. J. Bridges, "An electron beam parametric amplifier," PRoc.

IRE, vol. 46, pp. 494-495; February, 1958.3 W. L. Barrow, D. B. Smith, and F. W. Baumann, "A further

study of oscillatory circuits having periodically varying parameters,"J. Franklin Inst., vol. 221, pt. I, pp. 403-416, pt. II, pp. 509-529;March, 1936.

4 R. V. L. Hartley, "Oscillations in electromechanical systems,"Bell Sys. Tech. J., vol. 15, pp. 441-445; July, 1936.

N. W. McLachlan, "Theory and Applications of Mathieu Func-

current interest has been stimulated by the realizationthat, in principle, one can conceive of a parametric am-

plifier with a very low noise content. In turn this ideahas come from the recent work on masers9"10 which is a

form of amplifier that uses the cw "pump" frequency toinvert the normal distribution of energy levels in theactive material rather than to vary a reactance.An amplifier of the most simple form is shown in

Fig. 1(a). Here we have a single resonant circuit with a

capacitance which is varied at a rate equal to twice thesignal frequency. When the depth of modulation is suchthat"

C=

Q

the power delivered to the circuit from the pump isequal to the losses in the circuit, and the circuit willoscillate at co. Below this value we can use the circuit as

a regenerative amplifier, exhibiting negative resistancetype of gain. A natural descendant of this circuit isshown in Fig. 1(b). In this distributed transmission linewe modulate the shunt capacity with a traveling wave

tions," Oxford University Press, New York, N. Y., pp. 69-70, 274-275; 1947.

6 J. D. Laudon, "The use of ferrite-cored coils as converters,amplifiers, and oscillators," RCA Rev., vol. 10, pp. 387-396; Septem-ber, 1949.

7 E. Goto, "On the application of parametrically excited non-linear resonators," J. Elec. Commun. Engrs. Japan, vol. 38, pp. 770-775; June, 1955.

8 C. F. Edwards, "Frequency conversion by means of a nonlinearadmittance," Bell Sys. Tech. J., vol. 35, pp. 1403-1416; November,1956.

9 J. P. Gordon, H. J. Zeiger, and C. H. Townes, "The maser-newtype of microwave amplifier, frequency-standard, and spectrometer,"Phys. Rev., vol. 99, pp. 1264-1274; August 15, 1955.

10 J. P. Wittke, "Molecular amplification and generation of micro-waves," PROC. IRE, vol. 45, pp. 291-316; March, 1957.

11 From Fig. 1(a), C= Co+ACcos 2coot, the equation for the charge,y, is

+--cos2wo "=OL LC, (1 + C o=0

If

y = v(t) exp [A(LCWo)-2 RL] t'

v(t) is periodic. From the theory of Mathieu equations (McLachlan,op. cit.), since

AC

Co

we find

Re (IA)-4Thus, oscillation occurs when

AC 2R 2

Co cooL Q

1958 707

PROCEEDINGS OF THE IRE

at the "pump" frequency. Tien has shown that growingwaves appear on this circuit provided that the phase4)i, 4)2, ** *, cf are properly chosen. Suhl and Tien'2have discussed the case of two transmission linescoupled with reactive elements. Exponentially growingwaves are obtained when the coupling is modulatedwith the proper traveling wave. Kompfner has con-

sidered the system of an electron beam coupled to a

helix and modulated at twice the signal frequency. Ananalysis of this problem shows that waves, which grow

with distance, occur; in a later section we will evaluatethe various waves in such a system.

First, however, we will treat the problem of a singleelectron stream which is modulated at twice the signalfrequency.

C=Co+ACcos2w t

AC = 2 AT OSCILLATIONcQ-

(a)

Adler'7 has discussed this principle and suggestedmethods of amplifying the "fast" wave with deflectiontype circuits. Also, the work of Wade and Degrasse"8 inwhich they obtain frequency division with feedbackaround two separate helices is quite likely related to the"parametric" gain of the slow wave as described inSection IV.

II. SINGLE ELECTRON STREAM MODULATED AT

TWICE THE SIGNAL FREQUENCY

We will consider the waves on an electron beam whichis modulated with a fast or slow wave at twice the signalfrequency. The modulation will be assumed large as

compared to the signal but still small enough so that itcan be described with the linear theory. The beam willbe assumed to be confined by a very large magneticfield.We start from the three fundamental equations-

namely Poisson's equation

V E p, (1)

Co+ACSIN (2Wt+ON)

(b)Fig. 1-Simple form of parametric amplifier. The single resonant cir-

cuit of (a) will exhibit negative resistance gain below the thresh-old of oscillation while the distributed circuit of (b) will exhibitexponential gain with distance along the circuit.

The two normal space charge waves on such a beameach divide into a wave which grows and a wave whichattenuates with distance. Amplification of the "slow"space charge wave is of no interest since it can more

easily be amplified by coupling the stream to such ele-ments as a slow wave circuit to form a conventionaltwt13 or to a resistive wall amplifier.'4 These amplifiershave a minimum noise figure which is determined bythe noise parameters near the cathode potential mini-mum. At present, these amplifiers have been built withnoise figures less than 4 db.'6 The growing wave as-

sociated with the "fast" space charge wave is importantsince the theorems16 pertaining to the minimum noisefigure of a "slow" wave amplifier do not apply and muchlower noise figures should be possible.

12 P. K. Tien and H. Suhl, "A ferromagnetic traveling wave am-plifier," this issue, p 700.

Is J. R. Pierce, "Traveling Wave Tubes," D. Van Nostrand. Co,Inc., New York, N. Y.; 1949.

14 C. K. Birdsall, G. R. Brewer, and A. V. Haeff, "The resistivewall amplifier," PROC. IRE, vol. 41, pp. 865-875; July, 1953.

15 M. R. Currie and D. C. Forster, "New results in noise reductionin electron beams," presented at Conference on Electron Tube Re-search, Berkeley, Calif., June, 1957.

16 H. A. Haus and F. N. H. Robinson, "The minimum noise figureof microwave beam amplifiers," PROC. IRE, vol. 43, pp. 981-991;August, 1955I

the equation of continuity

ciz at

and the equation of motion

dv cv

di az

(2)

(3)

where v-= e/m is the ratio of charge to mass of the elec-tron and E is the electric field acting on the electronwhich is the gradient of the voltage as defined by

dvE= -- . (4)

s0=the dielectric constant of free space,v- velocity of the electron beam,p =charge per unit length in the stream,i= current in the stream and is given by i = pv.

We define as usual

2= WloXPP

(5)

(6)

Here w, is the plasma frequency for a one-dimensionalbeam of infinite cross section. cq is the plasma frequencyfor the cylindrical beam and is related to wcp by the rela-tion

Wq = RcoP. (7)

17 R. Adler, "A new principle of signal amplification," presentedat Conference on Electron Tube Research, Berkeley, Calif., June,1957.

18 R. W. DeGrasse and G. Wade, "Microwave mixing and fre-quency dividing," PROC. IRE, vol. 45, pp. 1013-1015; July, 1957.

co co+ AC SIN (2wt+o,) + AC SIN (2Wt+02)

rnT%--" rnyl.-. -r - -.40---rn

.z -Z -Z - 40-7- .7- -7 -

--- I I L --

708 A pril

Louisell and Qvate: Parametric Amplification of Space Charge Waves

R is the plasma frequency reduction factor for a beamof finite cross section.'9 Hereafter, we shall use coq inplace of w,. Eq. (3) can be written in the form

o2v a2 (v2)at2 ataz 2J

= 719aOzOt

and (1), (2), (4), and (6) can be combined to give

1d1 - cwq2- (9)dzdt Io

The equation of motion (8) is then

02v 02 V2 uo-+- X2 i (10)dt2 dzat 2J Io

Similarly, if we combine (2) and (5) we can write

ai di dvv2 -= - V -+i- . (11)

az at at

Eqs. (10) and (11) are the familiar equations whichrelate the velocity and current in an electron stream.The normal space charge waves on the stream come di-rectly from (10) and (11) and are of the form

i- -Io + i, exp [-p 1+)z + j6Xot] (12)

-wq UO0Wv = UO + - - exp -jl3e(1+ ) z +twtj, (13)

co Io L co

where

03 = /uo,Io=dc current,uo=dc velocity.These solutions were originally described by Hahn23

and Ramo2' and we will briefly review their character-istics for they are the waves which we propose to am-plify. The upper sign in (12) and (13) indicates a wavethat travels faster than the dc beam velocity. This is thefast wave which will attract most of our attention. Wesee that the current and velocity are 1800 out of phaseas we move down the stream. With our particular defini-tions, a positive value of current indicates a reduction inthe number of electrons below the average value. Thisantiphase condition means that the electron density islargest at the point of highest velocity. Therefore, inthe presence of the fast wave the energy of the beamis increased over and above the dc energy. The secondspace charge wave associated with the lower sign in (12)and (13) has a phase velocity less than the dc velocity,and the rf current and velocity are in phase. The highest

19 G. M. Branch and T. G. Mihran, "Plasma reduction factors inelectron beams," IRE TRANS. ON ELECTRON DEVICES, vol. 2, pp.3-11; April, 1955.

20 W. C. Hahn, "Small signal theory of velocity modulated elec-tron beams," Gen. Elec. Rev., vol. 42, pp. 258-270; June, 1939.

21 S. Ramo, "Space charge and field waves in an electron beam,"Phys. Rev., vol. 56, pp. 276-283; August, 1939.

electron density now occurs at the point of minimumvelocity and the energy of the beam is reduced in thepresence of this wave. These two waves are quiteanalogous to the forward and backward waves on anordinary transmission line. There the current andvoltage are in phase for the forward wave and the powerflow is in the positive direction. The backward wave hasa current and voltage which are out of phase and thepower flow is in the negative direction. Chu22 haspointed out this similarity with the fast space chargewave corresponding to the forward wave and the slowspace charge wave corresponding to the backward wave.

In a klystron, for example, both waves are equallyexcited at the input gap. Thus the beam energy is un-changed and it is commonly known that it requires norf power to bunch a beam. After the beam passesthrough the output cavity the slow wave is much largerthan the fast wave and the total beam energy is de-creased by the energy flow through the output cavity.In a traveling-wave tube, amplification occurs at thepoint where the circuit phase velocity is equal to thevelocity of the slow space charge wave. In this tube theslow wave grows exponentially with a continual de-crease in beam energy. We will now consider the para-metric amplifier and show that either the fast or theslow wave can be amplified independently of each other.

For the parametric amplifier we use a modulation at2w which is impressed on the beam in the form of aspace charge wave

i2w = Im exp [-j2f. (1- z2) + 2jwt]

C0q2 UO[ WQ!2\V2, = - 2 Imexp -j24 I -- z+ j2wt (14)

2w lo \2w

where wq2 is the reduced plasma frequency at the pump-ing frequency. If we take Wq2 as positive we have modu-lation in the form of a fast wave, whereas if we use a

negative value for co,2 we have modulation in the form ofa slow wave. In general, wq2 is not equal to coq, the re-

duced plasma frequency at the signal frequency, and itis necessary to introduce a new parameter, a', whichrelates these two as

, O,q2a' =-

2X<,(15)

The value of a' as calculated'9 is plotted in Fig. 2. Wesee that it can range in value from +1 for a thin beamto a value of 2 for an infinitely thick beam.

Im is the peak ac current at the modulation frequencyand we relate it to the dc current by

Im = mlo (16)

2 L. J. Chu, "A kinetic power theorem," presented at IRE Con-ference on Electron Devices, Durham, N. H.; June, 1951.

7091958

PROCEEDINGS OF THE IRE

a-= .5q20.5

0.4

0.3

0.2

W2 -2 W,0.

00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

b

Fig. 2-A plot of a' vs yb. 7yb is the ratio of the circumference ofthe beam to the rf wavelength on the beam at the signal frequency.a' is equal to one-half of the ratio of the reduced plasma frequencyat the pump frequency to the reduced plasma frequency at thesignal frequency. It is thus a measure of the relative velocity ofthe space charge waves at the pump and signal frequencies.

and

di, ma' 13e- + ja'3qil + j 3il* + j Iovdz 2 uO

m 13 --2 Io-vl* = 0

2 uo(20)

where

COq

oq =

We will first consider the case for a thin beam wherea' is unity. This indicates that the modulation is in theform of a fast wave which travels at the same velocityas the fast space charge wave at the signal frequency.We use (19) and (20) to write

d2'v dv1 m12 3m+ 2j13q-- m2 mV--- Oq2V,* = O.

dz2 dz 4 2(21)

where m expresses the depth of modulation. It is a com-plex number since the modulation can have any phasewith respect to the signal frequency. We assume Im>>iand therefore Im can be considered constant with dis-tance along the beam.We can then write

Eliminating v1 or vi* between (21) and its cc; and as-suming solutions of the form

VI = v1efqz)

we obtain the secular equation for ,u,

A4 + 4 1 - )4A +(I 1) = 0.

(22)

(23)

m / a'xqi= Io+r-Ioexp -2jf( -a 0z+2j-Ct + cc

1 a-' Z ++-il(z) exp jf,I- - z+ jot]+cc (17)

and

alco, m awcv= uo- uo-exp -2jfe 1 z + 2jx + CC

4+ 2 1(z) exp Lizle(1 _ + jwt + cc (18)

where cc indicates the complex conjugate. We must addthe complex conjugates because of the quadratic termswhich appear in the equations and represent couplingbetween the pump and signal frequencies. The form ofthese equations is convenient since it will turn out laterthat in the region of gain, il(z) and v1(z) are real ex-

ponentials which means that the velocity of the signalfrequency waves has shifted to become equal to the

velocity of the pumping wave.

If we substitute (17) and (18) into (10) and (11), we

obtain for the coefficients of the signal frequency terms

In (23) |m| is always less than unity and we find ap-proximate roots of the form

31 mlA i =

4

3l mlY2 =

4

3= - j2,/4 = + j2.

(24)

(25)

We can see from (18) that the first two roots (MI, 12)represent waves traveling at the velocity of the fastwave at the pump frequency, one of which grows ex-ponentially and the other of which attenuates exponen-tially. The third root (/3) is the slow wave and thefourth root (I4) is a new wave which travels faster thanthe fast wave. We will show later that this wave cannotbe excited when the pump frequency modulation is re-moved.

For the case where a'51 we can combine (19) and(20) in the form

d2v1 dvi a lm 12 i]v

dv1 m- + ja'138vi + ja',3 - vi1

dz 2

/3q72 Ui (- I ii-,y (19)

13e, IOm

_Oq2m (2a'2 + 1)V1* = 0.2

(26)

710 A pril>

Louisell and Quate: Parametric Amplification of Space Charge Waves

x,

Fig. 3-The real part of the propagation constant for the two-fre-quency parametric beam. On such a beam waves propagate as

exp-j. (1-a'-) z+13Q(X+_jy)z+jwt.

Note that for Ia'l <1, which means that the space charge wavesat the pump and signal frequencies do not travel at the same ve-locity, there is a minimum value of m which will produce gain.

y 1.6

1.4 . . . .F'cY42cvq

Fig. 4-The imaginary part of the propagation constant for the two-frequency parametric beam. On such a beam the waves propagateas

exp-jMe (I-aa'q x + Oq(x +jy)z +jwt.

For a' positive ya represents the normal slow space charge waveand y4 is a wave which travels faster than the fast wave.

If we again eliminate vi* from (26) and its cc andassume that

VJ = vole (2 7)

we obtain

~~~~ )~2 24+ 2 [af2(1 -)+ 1]2 + af4 41

3 M12f2 2 Im12a'2-2a'2+1- = 0. (28)2 4

The real roots of (28) are plotted in Fig. 3. As thevelocity of the pump and signal waves become unequal,the gain per unit length decreases and there is a thresh-old value of modulation, m, below which no gain occurs.The imaginary parts of 4 are shown in Fig. 4. Thisthreshold value is plotted in Fig. 5. The straight line for/3 indicates a wave with a velocity

uo

0.8 \<_0o7- -I _

m \0.5- - _

0.4 __ _

0.3 \__

-IN0.2 - _ -

0.1 -_ _ ___ u-

0 -I-N

0.5 0.6 0.7 0.8at=- C2 /2wqi

0.9 1.0

Fig. 5-The value of the minimum depth of modulation which willproduce gain vs a'. We see that for a' equal to unity whichrepresents a very thin beam there is no threshold value-whereasfor a' equal to one-half which represents a beam of infinite crosssection the threshold modulation is near unity.

Coq1+

which is the slow wave and for A4 a wave with a velocity

uO

1 - -i (1 + 2a')Un

which is the mirror of the slow wave about the pumpvelocity and travels faster than the fast wave.We see that in (28) only even powers of a' occur. This

means that we obtain the same roots for a' positive(fast wave modulation) or a' negative (slow wave modu-

lation). Of course, the way in which these waves areexcited as determined by an examination of the bound-ary condition will depend upon the sign of a'. In SectionIII this will be illustrated by considering the boundaryconditions for a beam which is modulated with a fastwave at the pump frequency which travels at the samevelocity as the fast signal wave-(a' = + 1).

III. BOUNDARY CONDITIONSAt the input (z =0) we can impress an ac current and

velocity with an arbitrary amplitude and arbitraryphase. The phase of the signal with respect to the pumpfrequiency modulation is a new and critical factor.

1958 71 1

3

PROCEEDINGS OF THE IRE

We can write at z = ,

Vi + V2 + V3 + V4 = Vin

Vi* + V2* + V3 + V4* tVin*,

il + i2 + i13 + i4 = iin,

il* + i2* + i3 + i4* = iin, (29)

where again vi is the gaining wave, V2 is the attenuatedwave, v3 is the ordinary slow wave, and V4 iS the newwave with the very fast velocity.

For the case of fast wave modulation or a' = +1 weuse (19), (20), (24), and (25) and write (29) in terms ofthe four velocity components as follows

Vl + V2 + V3 + V4 = Vin

where m = ml eO. We can see that either the amplifiedwave or the attenuated wave can be set up dependingon the phase of the pump. The critical phase requiredbetween the pump and signal frequencies would, ofcourse, hinder the operation of a practical amplifier.This can be avoided by using a pump frequencywhich is not quite equal to twice the signal frequency.In this case, a third frequency is generated withinthe beam which is equal to the difference betweenthe pump and signal frequencies. In this more gen-eral three-frequency type of operation the amplifiedand attenuated waves are set up equal in amplitude andthe phase of the pump frequency is not important.

Returning to (36), we have for the condition 5 = 3wr/2

V1 = Vin (37)

j1|| (V1- V2) - 8v4 = Vin2

-(VI + V2)

Im 1 Cq Uo

+i 2 (-V2) + v3 - 4v --I-iin2 3 wlo

-jl -| f V2)- 8V4 =- - inC4 c, Io

Eq. (30) can now be solved to give

1 / coq uO.

m *Vl + V2 - vin-- tinJ + 8 vin

2 wc lo 8

m (coq UO ** .V1lV2= 2I% 'tItin nil /

21 mlWq Io

V3= Vin + - Tin2 X Io

and from (17) and (18) we can write

31 ml / C1q\Vout- vi exp 4- wL- L + -

(Vout)2 31minGAIN ( = 8.68 _3qL = 41Nqdb,

sin 4(38)

(30) where Nq is the number of plasma wavelengths along thebeam of length L. Eq. (38) represents the maximumamount of gain; in an actual case the gain will be some-what reduced since the velocity of the fast wave at thepump frequency is not in general equal to the velocity

(31) of the fast wave at the signal frequency.With the slow wave as the input we have the relation

from (13)(32) (q UO .

Vin = - - Sin)co Io

and therefore:

mVl + V2 = - Vin)8

V 1 - V~2 0,

+-3- - iin* -Vfin )

3m =Wq Uo .*V4 = -- Vin* +- I 'tin*J (34)

Now we can consider two different input conditions

at the signal frequency-either the fast wave or the

slow wave. With the fast wave as an input we have the

relation from (13).

(q USo,Vin = - - fin

1o

and

Vin*=Vin and iin*=iin.

From (31)-(35) inclusive, we have

VI = "(1 + jeiO)vinV2 = x(1 - jei14)vin

V3 - Vini

Viny

16(39)

which gives

I'm IVL - 16vin(35)

(40)

If we compare (40) with (37) we see that the ratio ofthe growing wave excited by the fast signal wave tothat excited by the slow signal wave is greater than23 db. Thus, we conclude that we can amplify the fastwave at the signal frequency alone for the slow wave

will excite a growing wave of negligible amplitude. Thisis accomplished by properly modulating the beam

(36) again in the form of a fast wave at the pump frequency.

(33)

712 A pril

Louisell and Quate: Parametric Amplification of Space Charge Waves

IV. PARAMETRIC BEAM COUPLED TO ASLOW WAVE CIRCUIT

We shall use a model in which a lossless transmissionline is capacitively coupled to the electron beam. LetLi and C, be the inductance and capacitance per unitlength of the transmission line, respectively, and let V,and I, be the voltage and current of the line, and i, thecurrent in the electron beam. Then'3

vaVC OIcdz 0t

C,c ov6 Oi. (41)

-=01

- -az at dx)

Let

K /Li characteristic impedance of the line in theVC,Cabsence of the beam and,

Pr = \VLC =line propagation constant in beam absence.

If we write (41) in terms of the circuit voltage andstream current it becomes

02Ve r,2 a2V, d2i-- -Kr, = O.

z'2 w2 0t2 OzOl(42)

This is the equation which describes the coupling ofthe beam to the circuit. The equation of motion (10)must be modified for in addition to the space chargefields the circuit fields act on the beam and we can writeas is done for the voltage, VB,13 which acts on the beam

VB = Vc + V.c (43)where V,s is due to the space charge fields and is givenby (9). With this (10) becomes

0'V 0 2V'\ UO 2V+ _ q2i$= 77 (44)012 az&t \2 /I dzdt

The equation of continuity (11) remains unchangedwith the new coupling. We must now introduce normal-izing parameters which allow us to simplify the problem.We define

C3 = K (45)4VO

which is familiar in the twt theory. Here however wemust be careful to note that K is the circuit impedanceat the signal frequency.The space charge parameter at the signal frequency

which has been denoted by V'4QC we will denote by ao;i.e.,

-= C/4QC = aoC. (46)

We see immediately that the plasma frequency at thepump is given by

- = a'aoC.2co

(47)

Here again, as in the previous discussion, a' can rangefrom +1 to 2 if we modulate the beam at the pumpfrequency in the form of a fast wave and -I to -2 ifwe modulate in the form of a slow wave.We need the parameter which relates the circuit ve-

locity to the beam velocity and we can use b as is con-ventional. Thus

w LDr, = = (1 + Cb).

v UO(48)

Here again if b is negative the circuit velocity is greaterthan uo and if b is positive the circuit velocity is lessthan the beam velocity.We now assume

vc - Vc,(z) exp [-13e(I - a'-) z + jwt]+ cc (49)

and take the currents and velocity in the form of (17)and (18). Substituting for V, and i in the circuit equa-tion (42) and expanding in powers of C we have for thesignal frequency term

dVcl Kd1 + jl3eC(b + a'ao)V,1 =jl3.C-i.dz 2

(50)

Similarly the equation of motion (44) becomes

dv, m*

32. " u

-+ ja'#,2vi + ja'#, -V1 Ij -ildz 2 Os lo

fBe uo= . V Vcl (51)

2 Vo

and the equation of continuity remains (20).Eqs. (50), (51), and (20) together with their complex

conjugates, constitute a set of six equations in v,, V1*,i1, i,", and Vcl, V,,*. After decoupling we can look forsolutions of the form

Vc= Vol exp [CO,[a' - ja'ao]z] (52)

where 5' = x' +jy'.We are forced to use the prime notation since it is not

possible to use the same numbering of the roots as isused in the twt equations. With this background we cannow find the secular equation for 5', from the set of sixequations represented by (50), (51), and (20) as follows

(a' -ja'ao)6 + [2ao2(1 + a'20) + (b + aoa')2] (6'-ja'ao) I

+ 12(b + 5a'ao) + 2ao2[(b + aoa')2 - 4ao2a"2]+ 0[ao4(j + 6a'2) - 4aoa' + 2ao2a'2(b + a'ao)2]

+ ao4a'402} (a'-ja'ao) 2

+ I 1 + (b + a'ao)ao2} {I-8ao2a'2(b + aoa')

+ O[i + ao2(b + aoa')(1 + 6a'2)I}+ 62a'4ao4(b + a'ao)2 = 0 (53)

7131958

PROCEEDINGS OF THE IRE

b

Fig. 6-The propagation constants for the two-frequency parametricbeam which is coupled to a slow wave circuit. In this case wavespropagate as

exp Jf3eZ + 3C(x' + jy')z +jwt].

This is the case for a thin beam with moderate space charge andno modulation. It is presented so as to clearly bring out the featurewherein the three new waves which appear are mirror images ofthe familiar twt waves about the value of y which represents thepump velocity-i.e., in this case the mirroring occurs about theline y = + 1.

Fig. 7-The propagation constants for a two-frequency parametricbeam which is coupled to a slow wave circuit. In Figs. 7 through14 the pumping modulation is assuimed to be in the form of thefast wave (a'>O). For large values of b which indicates that thebeam is decoupled from the circuit we get values of x3' and x6'identical with those of the single beam as in Fig. 3. As b becomessmaller these roots are suppressed except in the vicinity of b = ± 1where islands of gain appear. At b = + 1 the velocity of the slowspace charge wave is equal to the circuit wave and we obtainvalues of xi' and x2' which do not differ from those of Fig. 6 withno modulation. At b= -1 the velocity of the fast space chargewave is equal to the circuit wave and nearby we find anotherisland of gain which is dependent upon the depth of modulation.

b

Fig. 8-The propagation constants for a two-frequency beam whichis coupled to a slow wave circuit. In this case of moderate spacecharge the velocity of the modulating wave at the pump fre-quency is taken equal to the velocity of the space charge waves atthe signal frequency. The beam is fully modulated by the pumpand we note that the maximum value of x for the fast wave gain(b = -1.4) is slightly less than the largest value of x for the slowwave gain b= + 4.

Fig. 9-The propagation constants in this case are similar to Fig. 8.Here the velocity of the wave on the beam at the pumping fre-quency is not equal to the velocity of the space charge waves atthe signal frequency. This would represent a beam with -yb near 1.

where

m 120 1- 1m12

4

The roots of this equation are plotted in Figs. 6through 16 (pp. 714-716). We see in these figures the so-lutions of the single beam alone appear as the circuit isdecoupled from the beam by making b large. The effect

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Louisell and Quate: Parametric Amplifcation of Space Charge Waves

Fig. 10 The propagation constants for this case should be comparedto those of Fig. 7. The increased values of space charge, or QC, re-sults in a larger maximum value of x near the fast wave inter-section.

b

Fig. 11-The propagation constants for a case where the velocity ofthe waves at the pump frequency and signal frequency are notequal. Note the asymmetry of the roots about the point where theisland of gain for the fast wave occurs.

of the double frequency modulation is to split the threenormal waves-i.e., circuit wave, slow space chargewave, and fast space charge wave-into two parts andwe see that the three new waves are obtained as mirrorimages of the three original waves about the velocity ofthe pumping wave. In the single stream case the charac-ter of the roots was independent of the sign of a' whereasin the present case the sign of a' influences the roots.When the fast wave (a'<O) at the pump frequency isused to modulate the beam, the real part of the roots forthe beam alone are suppressed as the circuit velocity

b

Fig. 12-The propagation constants for a thin beam with a largevalue of space charge and a moderate value of modulation. Thecharacter of the roots is similar to those in the previous figures.

b

Fig. 13-Here we have the propagation constants for a thin beamwith large space charge and heavy modulation at the pump fre-quency. The character of the roots is quite different for the gain-ing waves of the beam alone are so strong that coupling to thecircuit does not suppress them. The islands of gain near the pointwhere the circuit wave intersects the fast space charge wave andthe slow space charge wave still appear.

approaches the beam velocity. The islands of complexroots occur at the signal frequency in the vicinity of thecrossing of the fast space charge wave with the circuitwave and again where the slow wave crosses the circuitvelocity. When the pumping modulation is in the formof the slow wave (a'>0) we see that the real part of theroots for the beam alone can either be enhanced by thecircuit coupling as in Fig. 15 or suppressed as in Fig. 16.It depends to a large extent on the value of a' or the

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PROCEEDINGS OF THE IRE

b

Fig. 14-The propagation constants for a case of large space chargeand unequal velocities between the modulating wave and signalwave. The character of the roots is not greatly different fromFig. 12 where the two velocities are equal.

relative velocity of the pump wave as compared to thesignal space charge wave and the depth of modulation.

V. CONCLUSION

It has been found that the normal space charge wavesfound in a drifting electron beam will convert to ex-ponentially growing waves provided that one modulatesthe beam with a pumping frequency equal to twice thesignal frequency. It is important to note that either thefast space charge wave or the slow space charge wavecan be made to grow quite independently of each other.These new waves are of the parametric type and anal-ogous to the growing exponential waves which are foundon a transmission line whose distributed reactance isvaried with a traveling wave at the pump frequency.The character of the growing waves on the electronbeam are such that optimum growth occurs when thevelocity of the pumping wave and the signal wave areequal. As the velocities become unequal a greater depthof modulation is required before the onset of exponentialgain. These factors dictate the use of a thin beam notunlike the beams presently used in present klystronsand twt's.A premodulated beam of this type can also be

coupled to a circuit such as a helix whose phase velocityis reduced below that of free space. In this case we findsix waves rather than three as in the normal twt. Thenature of the roots depends on the form of the pumpingmodulation but in general there is more than one grow-ing wave which occurs in the vicinity of the point wherethe fast space charge wave becomes equal to the circuitwave, and secondly, where the slow space charge wave isequal to the circuit wave.The significance of this type of gain lies in the fact

b

Fig. 15-The propagation constants for a case where the modulationis in the form of a slow wave (a' <0) the gain in the region of thecircuit wave equal to the slow space charge wave is enhanced.In the vicinity of the circuit wave velocity equal to the fast wavethe roots seem to be very little affected by the presence of thecircuit.

b

Fig. 16-A second case for the modulation wave in the form of a slowwave. The two complex roots for the uncoupled beam are notsuppressed at all in this case. The roots are somewhat enhancednear the slow wave interaction point.

that the energy required for building up the signalwaves comes from the pumping oscillator and not thedc beam. This allows for amplification of the fastspace charge waves. Previous microwave amplifiershave amplified only the slow wave and the theoremswhich describe the minimum noise figure of such tubesare based directly on this fact. However, these theoremsare not valid for a "fast" wave amplifier and in principlethis form should prove to be less noisy than the slowwave amplifier.

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