wave amplification by interaction with a stream of electrons

5
PHYSICAL REVIEW VOLUME 76. NUMBER 3 AUGUST 1. 1949 Wave Amplification by Interaction with a Stream of Electrons J. A. ROBERTS* Department of Physics, University of Sydney, Sydney, Australia (Received March 22, 1949) A discussion is given of the propagation of plane electromagnetic waves in a uniform, ionized medium, in which the electrons possess a mean drift velocity. The treatment is based on the general theory developed by V. A. Bailey. It is shown that, for frequencies below a certain critical frequency (of the same order as the electron plasma frequency), one of the eight possible waves in general grows in amplitude as it progresses through the medium. The relation of this result to the theories of solar noise and of the traveling wave tube amplifier is discussed. I. INTRODUCTION R ECENTLY Bailey x has published a general theory of the propagation of plane electromagnetic waves in an ionized gas, in which static electric and magnetic fields are present. It was shown that under certain con- ditions the coefficient of attenuation of a group of such waves becomes negative, and in more restricted circum- stances the coefficient of damping may be negative. For such growth to occur it is necessary that the electrons or positive ions have drift velocities and it has been found that static magnetic fields and random elec- tron velocities are both favorable to the occurrence of the phenomenon. It will be shown here that waves with negative attenuation also exist when the static mag- netic field is zero and the electron temperature neg- ligible, provided only that the direction of propagation is oblique** to the drift velocity, and the frequency is below a certain critical frequency. II. THE DISPERSION EQUATION The theory investigates the propagation of uniform plane harmonic waves of small amplitude, of the form expi(co/—lx) y in a medium which is uniform in the static state, consisting of No electrons and No positive ions per cm 3 , moving with drift velocities Uo, U t o, respectively, together with neutral molecules. When the static magnetic field, random velocities and collisions are negligible*** and the motion of the positive ions is neglected, it was shown 2 that co and / are related by the dispersion equation J R 2 (Z+l)[(i^ 2 -l)(Z+l)+c7 r 2 / 2 ] = 0, (1) where R Uil, Z=P-o> 2 , and Ui, UT are the components of U 0 , respectively, * Commonwealth Research Student. 1 V. A. Bailey, Nature 161, 599 (1948); J. Roy. Soc. NSW. 82, 107 (1948); Austr. J. Sci. Research 1, 351 (1948). ** Oblique is here taken to mean neither parallel nor perpen- dicular. *** The phenomenon of growth is not due to collisions, but arises from the interaction of the wave with the electrons. Bailey has shown that the growth occurs with collisions present provided that the collision frequency is less than a certain value. 2 V. A. Bailey, J. Roy. Soc. NSW. 82, 112 (1948), Eq. (23.4) with Wi, WT, P, and r zero; Austr. J. Sci. Research 1, 351 (1948), Eq. (21) with Oi, QT, V, and r zero. parallel and perpendicular to the direction of propaga- tion. (Ui and UT are taken positive.) In this equation, the unit of velocity is taken as c (the velocity of light in vacuum) and the unit of frequency as the electron density frequency p.**** As the theory is non-relativistic, values of U 0 are restricted by the con- dition |Uo| 2 = W + c V « l . According to Eq. (1), for any assigned value of co there are eight values of /, namely, l^Ur 1 **, (Two), (2) l=±ia, (a 2 =l-co 2 ), (3) and the four roots of the quartic P(«, 0 = UiH* -a 2 (l—C/i 2 )}^ -2coa 2 £/i/-fl 4 = =0. (4) All the corresponding waves are plane polarized, 3 the electric vector lying in the plane containing Uo and the direction of propagation. For the solutions (3) the field is entirely transverse to the direction of propagation. Each of these waves is associated with a finite Poynting vector. If co is real, the two solutions (2) correspond to unat- tenuated waves which have a phase velocity equal to Ui; the second pair (3) represent unattenuated waves when co 2 > 1, but for co 2 <l do not represent true waves. III. GRAPHICAL SOLUTION OF THE QUARTIC The quartic (4) is simpler when expressed in terms of co and the phase velocityf cr l =<a/l. Thus (4) becomes where * 4 (cr)«H-*2Wco 2 -l = 0, 0 4 (<x) = (i--£MV~i), 1 0 2 (<r) = ( 1 - tfier) 2 +l-(r 2 (l- U T 2 ). J (5) (6) As Eq. (5) is a quadratic in co 2 , the (co, /) curve may easily be plotted by solving (5) for co, with selected values of o\ This curve is shown in Fig. 1 for the case **** p= (47rJW/m)*= 5.64X 10W„* rad./sec. 3 V. A. Bailey, J. Roy. Soc. NSW. 82, 110 (1948), Eq. (24.1). t Following a method developed by Bailey, to be discussed in a joint paper on the graphical study of the dispersion of electro- magneto-ionic waves. 340

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Page 1: Wave Amplification by Interaction with a Stream of Electrons

P H Y S I C A L R E V I E W V O L U M E 7 6 . N U M B E R 3 A U G U S T 1. 1 9 4 9

Wave Amplification by Interaction with a Stream of Electrons J. A. ROBERTS*

Department of Physics, University of Sydney, Sydney, Australia (Received March 22, 1949)

A discussion is given of the propagation of plane electromagnetic waves in a uniform, ionized medium, in which the electrons possess a mean drift velocity. The treatment is based on the general theory developed by V. A. Bailey. I t is shown that, for frequencies below a certain critical frequency (of the same order as the electron plasma frequency), one of the eight possible waves in general grows in amplitude as it progresses through the medium. The relation of this result to the theories of solar noise and of the traveling wave tube amplifier is discussed.

I. INTRODUCTION

RECENTLY Baileyx has published a general theory of the propagation of plane electromagnetic waves

in an ionized gas, in which static electric and magnetic fields are present. It was shown that under certain con­ditions the coefficient of attenuation of a group of such waves becomes negative, and in more restricted circum­stances the coefficient of damping may be negative.

For such growth to occur it is necessary that the electrons or positive ions have drift velocities and it has been found that static magnetic fields and random elec­tron velocities are both favorable to the occurrence of the phenomenon. It will be shown here that waves with negative attenuation also exist when the static mag­netic field is zero and the electron temperature neg­ligible, provided only that the direction of propagation is oblique** to the drift velocity, and the frequency is below a certain critical frequency.

II. THE DISPERSION EQUATION

The theory investigates the propagation of uniform plane harmonic waves of small amplitude, of the form expi(co/—lx)y in a medium which is uniform in the static state, consisting of No electrons and No positive ions per cm3, moving with drift velocities Uo, Uto, respectively, together with neutral molecules. When the static magnetic field, random velocities and collisions are negligible*** and the motion of the positive ions is neglected, it was shown2 that co and / are related by the dispersion equation

JR2(Z+l)[(i^2-l)(Z+l)+c7 r

2/2] = 0, (1) where

R Uil, Z=P-o>2,

and Ui, UT are the components of U0, respectively, * Commonwealth Research Student. 1 V. A. Bailey, Nature 161, 599 (1948); J. Roy. Soc. NSW. 82,

107 (1948); Austr. J. Sci. Research 1, 351 (1948). ** Oblique is here taken to mean neither parallel nor perpen­

dicular. *** The phenomenon of growth is not due to collisions, but

arises from the interaction of the wave with the electrons. Bailey has shown that the growth occurs with collisions present provided that the collision frequency is less than a certain value.

2V. A. Bailey, J. Roy. Soc. NSW. 82, 112 (1948), Eq. (23.4) with Wi, WT, P, and r zero; Austr. J. Sci. Research 1, 351 (1948), Eq. (21) with Oi, QT, V, and r zero.

parallel and perpendicular to the direction of propaga­tion. (Ui and UT are taken positive.)

In this equation, the unit of velocity is taken as c (the velocity of light in vacuum) and the unit of frequency as the electron density frequency p.**** As the theory is non-relativistic, values of U0 are restricted by the con­dition

|Uo|2 = W + c V « l .

According to Eq. (1), for any assigned value of co there are eight values of /, namely,

l^Ur1**, (Two), (2)

l=±ia, (a2=l-co2), (3)

and the four roots of the quartic

P(«, 0 = UiH* -a2(l—C/i2)}^ -2coa2£/i/-fl4= =0. (4)

All the corresponding waves are plane polarized,3 the electric vector lying in the plane containing Uo and the direction of propagation. For the solutions (3) the field is entirely transverse to the direction of propagation. Each of these waves is associated with a finite Poynting vector.

If co is real, the two solutions (2) correspond to unat-tenuated waves which have a phase velocity equal to Ui; the second pair (3) represent unattenuated waves when co2> 1, but for co2<l do not represent true waves.

III. GRAPHICAL SOLUTION OF THE QUARTIC

The quartic (4) is simpler when expressed in terms of co and the phase velocityf crl=<a/l. Thus (4) becomes

where *4(cr)«H-*2Wco2-l = 0,

04(<x) = (i--£MV~i), 1 02(<r) = ( 1 - tfier)2+l-(r2(l- UT

2). J

(5)

(6)

As Eq. (5) is a quadratic in co2, the (co, /) curve may easily be plotted by solving (5) for co, with selected values of o\ This curve is shown in Fig. 1 for the case

**** p= (47rJW/m)*= 5.64X 10W„* rad./sec. 3V. A. Bailey, J. Roy. Soc. NSW. 82, 110 (1948), Eq. (24.1). t Following a method developed by Bailey, to be discussed in a

joint paper on the graphical study of the dispersion of electro-magneto-ionic waves.

340

Page 2: Wave Amplification by Interaction with a Stream of Electrons

W A V E A M P L I F I C A T I O N 341

UI=UT=0.1. Figure 2 shows the detail near w=l . In this region it is simpler to use the variables

x=l—u and y=l/x.

Equation (4) then becomes

h(y)x2+2My)x+^o(y)=o, where

My)=(i+uiynf-D, h(y)=-(i+Uiy)(f-uiy-My) = uTy-iUiy-A.

•2) ,

(7)

(8)

From these curves we see that for |co|>«c, where av^l , there are four real values for I (four unattenuated waves), but for |a>| <coc there are two real roots (h, h) and two complex roots (h, h)- As the coefficients in (4) are real, h} h must be complex conjugates

h, h=a±:i$.

Thus £3, h correspond to a pair of waves propagated in the same direction with the same phase velocity (w/a), the attenuation being positive for one wave and negative for the other.ff

The real roots of (4) may be read directly from the graph, and in the region |o>j <o?c the complex roots hy h may be found from the two relations

h+h+h+h^ —coefficient of /3,

hhhh=coefficient of 1°.

a=a>Ur1-Uh+h)J (9a)

and

Thus

/3=( a2). V hh /

(9b)

It is convenient to replace (9a) by

a=J(A/i+A/2), (10) where

Al=la— I,

la being the value of / on the asymptotes

co= I7i/=fc(l— U12- UW(\-U£)-*

corresponding to the given value of co. In the neighbor­hood of coc, Ah is negligible compared with A/i.

Values of a and 0 derived by the graphical method are shown in Fig. 3 for the values U\= £/V=0.01.

In the two limiting cases UT=0 and Ui=0 the real part of the complex roots vanishes, the solutions of the quartic then being, respectively,

UT=0, J=tfr l(«=bl), l=*±io; (ID and

U1=0) / = ± m V - ^ r 2 ) - * , ^=°o . (12)

ft Bailey has shown that wave groups centered about the frequency <a also exhibit negative attenuation in the direction of the group velocity.

where

IV. A CUBIC APPROXIMATION

When x is very small (i.e., co^l) we may obtain good approximations to the complex roots (and the smaller of the two real roots) by neglecting the term in x2 in Eq. (7). This then becomes

P-Al2+Bl-C=0, (13)

A = Url{}Ui*-(l-Ui*)x}, B=x(2-3x), C=-2Ur1x2(l-x).

This cubic may conveniently be solved by the use of tables.4 Values for the complex roots (a±i/3) derived by this method are shown in Fig. 3. In this example the approximation is seen to be valid for a when x<2X 10~4

and for & when #<3X10~3.

V. NEWTONIAN APPROXIMATION

Since £/V2<3Cl, we may take as first approximations the roots of (4) with 27r=0, namely (11), and obtain closer approximations by Newtonian iteration. The second approximations are therefore

lf=l-P(dP/dl)-\ (14)

where / on the right hand side is given the values (11). For the complex roots we thus obtain

a>a2UiUT2

P

UT2-a2(l+ U12)} 2+4Ui2a>2a2

UT2-a2(l+Ui2)

•a\l-\2a>2a2l 2 {C/ r

2-a2(l+C/i2)}2+4t/1

or for small u(a2^>Ui2 and UT2)

a = o>UiUT2/a2, p=a(l+UT

2/2a2).

(15)

(16)

FIG. 1. / as a function of <a. UI=*UT=0.1. (Solutions corresponding to Eq. (4).)

tS

5

0

-5

-fO

—7 1

/ /s^r\

/ f C ^ v i

/ 1 / 1 4 For example, E. Jahnke and F. Emde, Tables of Functions

(Dover Publications, New York, 1945), Addenda pp. 24-30.

Page 3: Wave Amplification by Interaction with a Stream of Electrons

342 J . A . R O B E R T S

1 if] ' •

1 " 1 \ !

1 5 /**

1 -10

n II . ^ r = ^ ! .

Then

FIG. 2. Detail of the («,/) curve near w = 1. ?7i = £/V = 0.1; #=1—w.

Points determined by (15) are included in Fig. 3. It is seen that a single iteration gives a good approximation at the lower frequencies. In the neighborhood of the critical frequency, the iterative process must be repeated to obtain sufficient accuracy.

VI. THE CRITICAL FREQUENCY AND THE MAXIMUM VALUE OF a

In the example of Fig. 3 the variation of a with the frequency is typical of the general behavior. The wave­length 2-w/a of the growing wave is infinite when the frequency 1 — x is zero and decreases steadily as the frequency is increased. When CO=OJC the wave-length attains its minimum value 2-jr/am where am is the maximum value of a in the amplifying range. The cor­responding value of the phase velocity decreases steadily as the frequency increases ("normal" disper­sion) and when CO = OJC has its minimum value which is approximately 2ir/am. The dispersion of the medium is very large.

At the critical frequency (Fig. 2) there are two equal real roots, each equal to am. We may obtain an estimate of am, and of the critical frequency, from the condition that the discriminant of the cubic (13) should vanish.

If we assume, subject to confirmation, that when

| * | < < f t s j t v / ( l - - t f i 2 ) , (17)

then the discriminantal condition reduces to the ap­proximation

x=(l/54)\-(8U1i+9UT

2) M(»U1

2+9UT2r+27UTi(l+U1-WT-))i]. (18)

The positive root is the one of interest here. We consider three special cases:

I. C/r2«£/1

2.

The solution is then

* = UT*/32Ui*, (19)

and the assumption (17) is justified. The corresponding value of a is

a » * 0 . 2 5 t f i - W . (20)

*=0.028[/i2 . (21)

Here x=0.06&, and the assumption (17) is not fully justified. Consequently the approximation may be somewhat rough. With this value for x, a assumes the value

0^=0.26111. (22) I I I . UT

2y>U{K x = 0 . 0 9 6 * 7 r W .

This solution violates the assumption (17) and is therefore inaccurate. This case may, of course, be treated graphically and leads to interesting results (see Fig. 4).

VII. EFFECT OF THE DIRECTION AND MAGNITUDE OF THE DRIFT VELOCITY

The effect of changing the angle <j> between the drift velocity and the direction of propagation, while keeping the magnitude of the drift velocity constant, is illus­trated by the example of Fig. 4. This shows a and 0 as functions of <j> when UQ= 1.414X 10~2. Curves are drawn for four values of the frequency, namely x=0.5 , 10~2, 10 -4, and 10 -5, the last two values being those of the critical frequencies when <£ = 90° and 62°, respectively.

When the frequency is not in the immediate neigh­borhood of the critical frequency (e.g., when #=0.5 and 10~~2), as 4> increases the wave-length 2ir/a at first decreases but reaches a minimum value when 04=55° (by Eqs. (15) and (16)). But when #=10~5 the wave­length of the growing wave decreases steadily as <f> increases up to 62°. At this point a> = wc and amplifica­tion ceases. For values of <j> greater than 62° / is always real. When #=10~4 the behavior is similar, but when 0 = 90° the value of 13 is infinite, as is evident from Eq. (12).

The variation of the phase velocity parallels that of the wave-length. Near the critical frequency the velocity decreases with increasing 4> until the critical frequency is reached, but at lower frequencies the phase velocity reaches a minimum value when ^==55°.

The coefficient of attenuation 0 is approximately independent of <t> except near the critical frequency.

With a fixed value of </>, for frequencies well away from the critical frequency, Eq. (16) shows that a is propor­tional to £/0

3, while $ depends very little on Uo. On the other hand, it is evident from Eqs. (20) and (22) that near the critical frequency a is approximately pro­portional to Uo.

VIII. APPLICATIONS OF THE THEORY

This particular example of wave amplification is of importance primarily for its physical simplicity. Pro­vided that the drift velocity is neither parallel nor perpendicular to the direction of propagation and that the frequency is less than the critical frequency wc,

Page 4: Wave Amplification by Interaction with a Stream of Electrons

W A V E A M P L I F I C A T I O N 343

groups of waves will increase in amplitude as they progress through the medium. As the associated Poyn-ting vector is finite, the energy carried by the wave must also increase as the wave proceeds.

Bailey1 has suggested that such growing waves may, in part, explain the electrical noise from discharge tubes, solar, cosmic and ionospheric noise, and terrestrial magnetic fluctuations. The use of such waves for the amplification of high frequency signals will also be con­sidered.

We shall first apply our results for oblique propaga­tion to the solar atmosphere. The drift velocities of electrons in the solar atmosphere not being accurately known, we may for purposes of illustration compute the wave magnitudes for the velocities 4.2 X103 and 4.2 X104

km/sec , and take </> as 45°. The values of electron density compiled by Smerd5 have been used. Table I shows the critical frequency coc, wave-length and rate of growth of field strength, for various heights above the sun's surface.

I t will be seen from this table that very large ampli­fications are possible in the sun's atmosphere, the lower regions of the chromosphere being most effective. The theory thus provides a very powerful mechanism for explaining the generation of abnormal noise at the fre­quencies which are observed. Thermal and other fluc­tuations at frequencies below the critical frequency will be greatly amplified, and if this radiation escapes from the sun,ftt the equivalent blackbody temperature at radiofrequencies could be much greater than that ob­served at optical frequencies.

These waves are plane polarized and their associated Poynting vectors are finite. I t is implicit in the general theory that in the presence of a static magnetic field the waves are in general elliptically polarized. The non-linearity of the fundamental equations offers a means of limiting the amplitude ultimately attained, but this problem will not be discussed here.

We shall now consider the use of these growing waves in high frequency amplifiers. I t is well known that the boundary conditions imposed on the fieldfttt by the surfaces of a conducting wave guide may be satisfied by a superposition of plane waves traveling in directions inclined to the axis of the guide. If electrons were projected longitudinally down the guide, the possible plane waves would include the growing waves discussed above and we would expect the corresponding modes to exhibit the negative attenuation characteristic of the component waves. If the guide were sufficiently long, the device would act as an amplifier, since the growing waves must ultimately predominate.

A rigorous discussion of the propagation of waves in 5 S. F. Smerd, Australian Council for Scientific and Industrial

Research R.P.L. 14 (1948). fft I t should be noted that if the electrons have a drift velocity,

propagation, with amplification, is often possible in what is otherwise an over-dense medium.

t t t t The boundary conditions imposed on the vibratory motions of the electrons also require consideration.

TABLE I. Characteristics of growing waves in the solar atmosphere, <t> — 45°

Height above surface of sun (km)

Maximum frequency («c) for am­plification (Mc/sec.)

Minimum wave- UQ = 4.2 X103

length (km) km/sec. 1/0=4.2X10*

km/sec.

Wave-length at U0=4.2X 103

half critical km/sec. frequency (km) UQ = 4.2 X104

km/sec.

Rate of growth at £/0 = 4.2X 103

half critical km/sec. frequency t/0 = 4.2Xl04

(db/km) km/sec.

103

5600

0.021

0.0021

81

0.08

4.4X 105

4.4X105

10*

176

0.68

0.068

2.6X103

2.6

1.4X104

1.4X104

106

6.8

18

1.8

6.6X104

66

5.4X 102

5.4X102

a cylindrical guide containing electrons moving parallel to the axis has been given by Hahn,6 in connection with the theory of the klystron. Hahn also considered the effect of an axial magnetic focusing field. Ramo7 has discussed two specific cases of Hahn's theory namely the situation considered above in which there is no static magnetic field, and the corresponding case with an infinite axial magnetic field. In both cases separate E- and i7-modes are possible, each mode of the charge free cylinder splitting into four sub-modes in the presence of the electrons. These correspond to the four values of / given by Eqs. (2) and (3) for the i7-modes, and the four roots of Eq. (4) for the E-modes.J

Although Hahn and Ramo derived the relevant equations they apparently did not discover the waves with negative attenuation. I t may easily be verified that in certain frequency ranges Ramo's Eq. (37) for £-waves with zero static magnetic field, and his Eq. (15) for E-waves with an infinite axial magnetic field,

*AI

FIG. 3. Frequency dependence of the angular wave number a and the coefficient of negative attenuation /3 of the growing waves. Ui=UT = 0M.

6 W. C. Hahn, Gen. Elec. Rev. 42, 258 (1939). 7 S. Ramo, Phys. Rev. 56, 276 (1939). | E-modes have a longitudinal component of the electric but

not the magnetic field; /7-modes have a longitudinal component of the magnetic but not the electric field.

Page 5: Wave Amplification by Interaction with a Stream of Electrons

344 J . A. R O B E R T S

FIG. 4. a and /3 as functions of <f>, the obliquity of propagation. UQ—IAIAXIO"2.

yield conjugate complex values for the axial wave number. The axial magnetic field has the effect of reducing the phase velocity in the guide and may in fact reverse the direction of phase propagation.

We now give an example of the solution of Ramo's Eq. (15). If a tube 0.3 cm in diameter is uniformly filled by a 100-ma beam of 30-volt electrons projected along the tube, and is pervaded by an infinite magnetic focusing field, then at a frequency of 3000 Mc/sec. the growing wave of the EQI mode has a wave-length of 4.14 cm in the guide and the field strength of the wave increases by 76 db/cm.

Amplifiers utilizing these growing waves have been constructed in a number of laboratories8 and theories of their operation have been given by a number of authors.9

In these "traveling wave tube" amplifiers, the cylin­drical wave guide is replaced by a conducting helix,

designed to reduce the longitudinal phase velocity in the charge free guide to approximately the same value as the electron drift velocity. Apparently these inves­tigators believe that this is a necessary condition for amplification, but our analysis shows that amplification is possible in much less special circumstances. An amplifier of the same type, but operating on a somewhat different principle, has been discussed more recently by Haeff, Nergaard, Pierce and Hebenstreit,10 and Hollen-berg.10

A more complete study of propagation in cylindrical guides, based on Bailey's theory1 in which the motion of the positive ions is also considered, will be given in another publication.

ACKNOWLEDGMENT

In conclusion I should like to thank Professor V. A. Bailey, who suggested this investigation, for much helpful advice and assistance. This work was carried out during the tenure of a Commonwealth Research Studentship.

8 R. Kompfner, Proc. I.R.E. 35, 124 (1947); J. R. Pierce and L. M. Field, Proc. I.R.E. 35, 108 (1947); J. S. A. Tomner, Report No. 2 (1947). Research Laboratory of Electronics, Chalmers In­stitute of Technology, Sweden.

9 J. R. Pierce, Proc. I.R.E. 35, 111 (1947); L. J. Chu and D. Jackson, Proc. I.R.E. 36, 853 (1948); O. E. H. Rydbeck, Report No. 1 (1947) Research Laboratory of Electronics, Chalmers Institute of Technology, Sweden; C. Shulman and M. S. Heagy, R.C.A. Rev. 8, 585 (1947). A. Blanc-Lapierre et al., Onde* £lec. 27, 194 (1947).

10 A. V. Haeff, Phys. Rev. 74, 1532 (1948); Proc. I.R.E. 37, 4 (1949). L. S. Nergaard, R.C.A. Rev. 9, 585 (1948). J. R. Pierce and W. B. Hebenstreit, Bell Syst. Tech. J. 28, 33 (1949). A. V. Hollenberg, Bell. Syst. Tech. J. 28, 52 (1949).