Particle Accelerators1975, Vol. 7~ pp. 9-17

© Gordon and Breach Science Publishers Ltd.Printed in the United Kingdom

X- AND y-RAY PRODUCTION BYTEMPLATE MODULATED COHERENT LIGHT

PAUL L. CSONKA

Institute of Theoretical Science and Department of PhysicsUniversity of Oregon

Eugene, Oregon 97403, U.S.A.

(Receired July 10, 1974,. infinalform March 31,1975)

In this proposed device coherent electromagnetic radiation, modulated by an optical element called template, producesa field at the trajectory of charged particles whiCh are accelerated by the field and, emit electromagnetic radiation. Thetemplate may be a holograph. The spectrum of this radiation depends on the original coherent radiation, the templatearid the position of the particle in the field. For existing machines high intensity X- and y-ray production is possibleand the frequency and polarization spectrum can be controlled. Coherent production (between the charged particles)enhances the radiation particularly at lower energies. This suggests the possibility of producing essentially monochro­matic photons. Coherent (between photons) radiation production seems feasible. For a suitably chosen field the dominantpart o(the radiative reaction force goes to zero near the beam axis so that it does not interfere with the earlier suggestedmethod of particle acceleration with template modulated coherent light.

In this paper we briefly described a proposed devicecapable of producing X-rays and y-rays with atunable frequency and polarization spectrum.The device is to be used in conjunction with highenergy accelerators or storage rings.

From here on we will use the phrase "y-ray"to mean both X-rays and y-rays, unless explicitlyspecified otherwise.

In the proposed device, y-rays are to be producedby high energy charged particles traversing asuitably shaped electromagnetic field. The pro­duction mechanism is related to, but differentfrom, the one by which high energy charged parti­cles produce synchrotron radiation. The differenceis the following. The intensity and, to some extent,the polarization spectrum ofthe radiation producedby the device to be described are variable and canbe adjusted. By contrast, for any accelerator orstorage ring and a given particle energy, the syn­chrotron radiation spectra are fixed and often in­convenient. The proposed device is also related tobut distinct from a class of devices which we willrefer to as undulators. In an undulator chargedparticles pass through a static magnetic field whichvaries as a function of position along the particletrajectory. When this magnetic field is insertedinto a particle accelerator or storage ring, the fieldexerts a force on the charged particles and the soaccelerated particles emit electromagnetic radia­tion. The change in the particle trajectory caused

9

by the magnetic field can have the shape of a"break" or "wave.,,1,2 The intensity of the magneticfield can be varied. The device to be describeddiffers from an undulator, in that in it no staticfield is to be used, but instead a radiation fieldshaped by an optical template. This makes itpossible to increase the power radiated in theform of y-rays, to produce Fourier componentswith a much shorter wavelength than static mag­netic fields would permit, and to combine andadjust Fourier components in an essentially arbi­trary manner, the precise meaning of this phrasebeing given below. The electromagnetic field at thetrajectory in turn determines the intensity andpolarization spectra of the radiation produced bythe device here described. By appropriately choos­ing the former, one can select an intensity and, tosome extent, the polarization spectrum for thelatter.

In the template radiator described here, highvelocity (in the laboratory) charged particles areaccelerated by an electromagnetic field. Duringacceleration, these particles radiate y-rays. Thecharged particles may be electrons,protons orother particles. We need not specify their exactnature at this time, and until we do, we will referto them simply as "particles."·

The accelerating electromagnetic field is pro­duced in two steps. In the first step, coherentelectromagnetic radiation is generated by a source

10 P. L. CSONKA

which may be a laser. We need not now specify theexact nature of the source, and until we do, we willrefer to it simply as "light source" and to the radia­tion produced by it as "light." Ip the second stepthe light produced by the light source passesthrough a template and is then focused on theparticle trajectory. The template3 is an opticalele~ent which modulates both the intensity and thephase of the light passing through it, and it may be aholograph. By suitably choosing the template, anyelectromagnetic field can be constructed at theparticle trajectory, provided only that it satisfiesMaxwell's equations and the boundary conditionthat it is produced at the template by a convergingmonochromatic field. In practice it will sometimesbe convenient not to insist on creating the mostdesirable field at the trajectory, but one whichdiffers from it by the appearance of certain un­wanted but relatively harmless Fourier com­ponents. When the template is a simple holograph,then these unwanted Fourier components form theso called "twin image" of the desired "correctimage" and can indeed be quite harmless as will bediscussed below.

Let us denote the wavelength in vacuum of the(by assumption monochromatic) light producedby the light source by Ao, and OJ = 2nc/Ao. Werestrict the following discussion to the specialcase when the trajectory of the particle before enter­ing the field produced by the template, is horizontaland straight, and we choose the z axis to be parallelto it. We assume that the electric field E producedby the template is cylindrically symmetric aroundthe z axis. It follows that the magnetic induction,B, is also cylindrically symmetric around the sameaxis. Generalization to arbitrary geometry isstraightforward. We denote by r the radius vectormeasured from the z axis (see Figure 1). Thehorizontal and vertical Cartesian components ofr are x and y, respectively. We choose the originof our coordinate frame so that at the pointwhere a particle enters the electromagnetic fieldproduced by the template, its z coordinate iszero. To specify E, we could now give its threeCartesian components Ex, Ey , and Ez. Instead,for any point P whose coordinates are (z, y, x)we will specify Ez, Er and Es' Where Er is the Ecomponent parallel to r, the s is defined as

(2)

./: radiation from

~ i_ ~~the light source

,::.:::L·.·:::~:·+·::~..!. radiation modifiedy 1 i r I! 2 ~by the template

I '\ ! I lor..:.:····· cylindrical templateI \; ....? I : ~I \:x'/ I I I

I ~.~ i.l.--- ~ -L .: .----+--particle bunchI ...... -1' :' "~\-----"ir-

or-,---::.,:v~::..~~·::·Q'*"-~--+----+! J ..r:-bi!~: I: I:/.~

"'~r---DI

We also assume that the electromagnetic field isperiodic along z with a period 2L. Since L isarbitrary, this assumption imposes no restrictionon the field, but allows us to write sums instead ofintegrals.

The most general electromagnetic field satisfyingthese four conditions (varies as eiwt

, cylindricalsymmetry, Es = 0, and 2L periodic along z) is

FIGURE 1 The figure shows the x, y, z Cartesian coordinatesof an arbitrary point P, its radius vector r, and the sunit vectordefined at P by z x r = s. The electromagnetic radiation fromthe light source passes through a cylindrical template whichmodifies it so as to produce a suitable electromagnetic field.This field extends from 0 to D along the z axis and acceleratescharged particles thereby inducing them to radiate. The particlesenter the field with a common velocity v, and come in buncheswhose length along z is bz , and whose horizontal and verticaldiameter (not shown in figure) are bx and by.

the time dependence of E and B is given by thefactor eiwt

, and by the assumed cylindrical sym­metry E and B depend on z and r, but not on s.

The most general electromagnetic field satisfyingthese conditions is an arbitrary linear superpositionof two types of fields. For the first type, Es = 0,while for the second type Bs = 0. The second typecan be obtained from the first type by the substitu-tion E~ B, B~ - E. Thus it suffices to know themost general field of the first type. Since fields of thesecond type are of little interest for our presentpurposes, we will impose the condition

(1)ZXf=S,

and E s is the component parallel to S. The Bvector will be specified similarly. By assumption,

x- AND y-RAY PRODUCTION IN COHERENT LIGHT 11

{w "~ "k W/C )}Bs(t,z,r) = Re _e- uot l...JCnelz,nZ-Jl(Knr ,C n K n

(3c)

Es(t, z, r) = Bz(t, z, r) = Br(t, Z, r) = 0, (3d)

where

Kn=[(~Y -k;,nJ/2

, (3e)

J 0 and J 1 are cylindrical Bessel functions, C is thevelocity of light in vacuum, kz,n = nn/L with nsome integer because the field is periodic along z,and the Cn are Fourier coefficients.

The particle moves with a velocity w(t) in thelaboratory. The particle enters the electromagneticfield at time to. At that time its velocity is w(to) = v.By definition v is parallel to z. After to, the particle isaccelerated by the field, and w(t) will change. Incases ofmost interest to us, the particle will oscillatewhile crossing the field. While oscillating, it willradiate, thereby loose energy, and slow down. Wewill assume that this slowing down is negligiblewithin the time the particle crosses the field, so thatthe particle velocity can be written as

w(t) = v + u(t), (4a)

where u(t) is simply a periodic function of t, andthus the average (over one oscillation) velocity vis constant in time.

We denote the inertial frame which moves withthe constant velocity v with respect to the labora­tory by K v • For obvious reasons, we may call thisframe the average (over oscillations) restframe of theparticle. We denote quantities measured in K v bya prime, e.g. in this frame the particle velocity is

w'(t) = u'(t), (4b)

and v' = 0 by definition. The components of theelectromagnetic field can be obtained by a Lorentztransformation from Eq. (3). Defining v == Iv I andy == (1 - V 2/C 2)-1 /2,

(8)

(10)

if Iu I ~ c, (7)E'(t', ·z', r')

~q----

rno

B~(t', z' ,. r')

22[d J2P' = _L - u'(t') .3c3 dt'

This expression will be a good approximationprovided that for every Fourier component of E

hw~ ~ rno c2; for all n, (11)

is satisfied. In the cases of interest to us, Eq. (10)holds.

E'(t', z'(t'), r'(t')J ~ E'(t', z~, r~), (9)

where z~ and r~ are the average (over one oscilla­tion) of the z' and r' coordinates of the. particle, sothat by the definition of K v and Eq. (4b) they areconstants in time.

To calculate the power, P', radiated by theoscillating particle in K v , we use the classicalformula

where q, rn and rno are respectively the charge,mass and rest mass of the particle.

Denote by I1r and I1z the amplitude of particleoscillation in K v along rand z respectively. Assume

= Re{L cne-i(wt-kz,nz)~[~ - ~ kz,nJ.L J l(Knr)},n C C C K n

(5c)

E~ = B~ = B~ = 0 (5d)

where t', z' and r' are the coordinates in K v of thatspace time point whose coordinates in the labora­tory are t, z, and r.

We assume that while the particle is the electro­magnetic field,

so that during one particle oscillation along r, thevalue of J 1 changes only by a small amount whichwe neglect. Similarly, we neglect the change in E'due to particle oscillations parallel to z, and canwrite

u'(t) ~ c. (6)

Then in K v the dominant force on the particle willbe electric, as opposed to magnetic, so that

d E'(t' z' r')- u'(t' z' r') ~ q "dt' " rn

(5a)E~(t', z', r') = Ez(t, z, r),

E~(t', z', r')

12 P. L. CSONKA

(13b)

We now focus our attention on the interestingcase when

(15)

(14)

w~' = w~.

1, 1-2 Kn = -2( Kn •n n'

When the particle moves in the field of these pho­tons, it will radiate photons which have in K v

angular frequency w~', momentum k~' with com­ponents, k~:n and K~'. By the classical approxi­mation permitted by Eq. (11),

tively a cylindrical wave converging on the z axiswith radial momentum - Kn/2n and a wavediverging with radial momentum + K n/2n.

Performing a Lorentz transformation to K v , wefind that the Fourier component proportional to Cn

describes photons which have angular frequencyand momentum components

w~ = y(w - vkz,n) ~ y(w - ckz,n),

2~ k~,n = ;n y0z,n - ;2 w) ~ y;n (kz,n -~}

The probability of photon emission by a particlelocated at ro [and, by Eq. (13b) essentially oscilla-,ting radially along roJ, in the k~' direction, isproportional to (k~' x r')2:

P'(k~') ""-I (k~' x r~)2 (16)

The polarization of the emitted photon is parallelto (k~' x r~) x k~'.

So far we considered the radiation emitted byone particle only. When two particles of the sametype are moving parallel to the z axis, both withvelocity v', then they will both radiate and theradiated field emitted by them may interfere con­structively or destructively at any point at I'.Denote the position of the two particles bya'l and a~, d' = a~ - a~. The Cartesian com­ponents of a'l are a'lx, a'ly, a'lz, and a'lr == (a'lx, a'ly)'Similarly for a~, d', etc. The phase of the field ata'l, due to photons produced by the template withw~ and k~ differs from the phase at a~ by (- k~, n • d~

- K~' d~), i.e. the phase of the two fields emittedby the two particles differs by this much at theirrespective points of emission. Since the distanceII' - a'l\ may differ from II' - a~ I, the phase of theradiation reaching I' from the two particles may befurther shifted. Assuming II' I ~ Id' I, this shift isapproximately - (wellc) (d' - 1')1 II' I so that thephase at l' of the two radiations differs by a total of

!lcp = (-k~,n d~ - K~ d~) + we' (d' ·1')/11' I. (17)C

(12)

(13a)v ~ C,

The field produced by the template is a super­position of several Fourier components. Thecomponent proportional to Cn describes photonswhich in the laboratory frame have angularfrequency w, and whose momentum along the Z

axis is kz , n/2n. The absolute value of their momen­tum in the radial direction is Kn/2n. Along theradial direction the momentum of half of thesephotons is Kn/2n, and for the other half it is - Kn/2n,because the cylindrical Bessel functions for largeargument K n r can be written as a sum of two Hankelfunctions, one with argument - Knr and the otherwith argument +Knr, and these describe respec-

and when y is large enough, so that the J6 termcan be neglected. This last assumption is equivalentto neglecting the radiation due to E~ in comparisonwith the radiation due to E~. In other words, weassume that the emitted radiation by a particlelocated at r0, is mainly due to radial oscillationsalong r o.

Substituting Eq. (7) into Eq. (10) and usingapproximation (9), we obtain

2 q2P' ""-I [E'2(' , ') + E,2(, , '/)J

""-I-

3Z3 z l,zo,ro r t,zo,lo'moc

For anyone particle the z~ and r~ do not changein time, by assumption. Therefore, in the labora­tory frame ro is also constant in time, whileZ = [zo + v(t - to)Jy. Equation (5) shows that forfixed ro and linearly varying z, the fields E~ andE~ oscillate harmonically as a function of time dueto the factor e-i[wt-kz,nz(t)]. Thus the time averageof E~2 is one half of the maximum value (in time) ofE~2, and similarly for E~2. Therefore, we find fromEqs. (5) and (12) that the time average of P', is

1 q4 (W)2<P') = -32:3 L Icn l2 -

moc n C

X {J5(Knr) + [k2 ,n - ~~J[:/l(Knr)J}(13)

x- AND y-RAY PRODUCTION IN COHERENT LIGHT 13

d' nc< 2w' then /).<p < n,

otherwise constructive interference may occurwhen I' is almost parallel to k~'.

The values w~ and k~ in the laboratory frame canbe found by Lorentz transforming w~' and k~', andusing v ~ c: The average angular frequency is one half of the

maximum, because in K v all emitted photons havethe same energy: their energy in the laboratoryframe depends on the direction of their emission,but since f o is perpendicular to v, Eq. (16) showsthat for every photon with momentum k~' n

there is one with momentum - k~: no. The polar~ization of the y rays emitted by a particle at f 0,

is predominantly parallel to fa in the laboratory.By Eq. (20), (18) and (14), the radiation emitted bythose particles which in the K v frame lie within asphere of diameter

d' < n'" 2y(w/c - kz,nJ (23)

will add constructively everywhere.Concerning the light source, we assume that it

emits radiation impulses each of which last 11seconds. During the pulse it emits an instantaneouspower W The light illuminates a section of the zaxis whose length is D, i.e. the particle travels adistance D in the field produced by the template.(See Fig. 1.) Since the energy density of the electro­magnetic field in vacuum is E2/4n, we can expressthe electric field in terms of Wand D. UsingEqs. (3) and (20) we can then express Icno I in termsof Wand D:

ICnol = 2(nW/Dw)1/2. (24a)

is when the light converges in the form of a cylin­drical wave on the cylindrical template. In practiceit will be easier to split the light beam into three ormore branches and direct them onto the templatefrom three or more directions. Of course, theelectromagnetic field on the outside surface of thetemplate depends on the manner in which the lightconverges, but in any case the template is so chosenas to achieve a desired field configuration in thevicinity of the z axis. Inside the template one mayvisualize the electromagnetic energy as beingconcentrated in a cylindrically converging wavetrain, the energy density reaching its maximumwithin a radius of order 2n/Kno measured from the zaxis.

Now Eq. (13b) reduces to

< ' _ 1 q4y2 W - ckz,no 2(W)2 2p> - -3 """32 + k Icnol - J 1(Kno r) (21)c rno w C z,no C

According to Eq. (19) the maximum angularfrequency

(18)

(20)

(19)

d' < nc~ 2w

Clearly (d'· 1')/11'1 S Id'l, and by Eq. (3e)

I(k~ n • d~ + K~ d~) I ~ w' Id' I,, c

so that whenever

i.e. the interference is constructive everywhere. If2w' d/c > n then the interference may still beconstructive when I' is almost parallel to k~'.

Similarly, if several particles of the same typeare moving parallel to the z axis, all with a velocityv, then radiation fields produced by all of them willadd constructively everywhere, provided that all ofthem lie within a sphere of diameter

One simple way to produce this is by imprinting acylindrically symmetric reflecting surface on atransparent one and let the reflexivity as a functionof z vary as cos kz,noz. The hologram produced bythis template will contain, in addition to the desiredFourier component which varies as exp{kz n z},another which varies as exp{ -kz,noz}: the "t~inimage." In most cases this component will produceundesired low energy y rays, which can be ignored.If their presence is harmful, then they have to beeliminated by constructing a more sophisticatedtemplate. Theoretically the easiest case to consider

The probability of emission for a photon with k~

can be obtained from Eq. (16), and depends on thevalues of Cn. In general, most of the y rays will beemitted within an angle of l/y radians measuredfrom the z axis. If the emitting particle is locatedat f 0' the polarization of the y rays will be predomi­nantly parallel to f 0 .

Next we make some numerical estimates toevaluate the capabilities of such a device. Weconsider the special case when only one Fourieramplitude is non-zero

14 P. L. CSONKA

Substituting Eq. (24a) into Eq. (21), we find thevalue in K v of the power radiated by one particlewhile it passes through the electromagnetic field.

<P') _ 1 q4y2 OJ - ckz , no 4nWOJ 2( '")---- ---J K 1

3 c3m2 OJ + ck c2D 1 no ,o z, no

(while light pulse lasts) (24)

Concerning the particles, we assume that theyare produced in bunches, each bunch containing Nparticles. The density of the particles can be writtenas

p(t, z, y, x) = p(t, z)p(x, y), (25)

where p(t, z) in the laboratory is a "step functionmoving with velocity v" along z and is nonzeroover a z interval whose length is bz • The p(x, y) isleft unspecified (see Fig. 1).

Let ~t be the time a particle spends radiating yrays, i.e. the time the particle takes to pass throughthe field produced by the template. The samequantity measured in K v is then ~t' = ~t/y. Thetotal energy, lff~, radiated out by the particle duringone passage through the field, is clearly measuredt6 be ~t'. <P') in K v • The total momentum radiatedout in K v is zero. The total energy radiated out bythe particle is measured to be lffy in the laboratoryframe:

lffy = ylff~ = ~t<P'). (26)

To obtain the total energy lffy,t radiated out by allthe particles in a bunch we multiply lff y by Nandaverage Ji over the cross section of the beam. Wedenote the resulting average of Ji by

Ji(Knor) == (f p(x, y)Ji(Knor) dx dy)

Ifp(x, y) dx dy, (27)

and obtain from Eqs. (24), (26) and (27)

4n OJ - ckz,no q4y2 W J-2( )N Alffy t = - 52 OJ - 1 Knor ut.

, 3 OJ + ckz , no C mO D(28)

So far we assumed that Eq. (7) holds, i.e., that inK v magnetic forces can be neglected in comparisonwith electric forces. We will now evaluate the energyradiated out by the particles as a result of theiracceleration due to magnetic forces: ~~. Sincein cases of interest to us Eq. (6) holds, the term tobe evaluated will be only a small correction to

lffy,t given in Eq. (28). We denote by u~, u~ andu~ the corresponding components of u', assume

(29)

and neglect u~. In evaluating lff~t we may proceedexactly as we did for lff y, t except that we have touse Eq. (5c) instead of Eqs. (5a) and (5b), and thatthe force will be proportional to Iu' 1/c, so that theenergy radiated will be proportional to I u' 1

2/c2. In

the limit of high y, stated after Eq. (13a), the result issimply

lffy~ = (~):fy, to (30)

where (u'/c);v is the average of (U'/C)2 taken over allparticles, and time. (Since u' may change in time.)

DISCUSSION

1) It was suggested in another paper3 thattemplate modulated coherent light can be used toconstruct particle accelerators. The objection wasraised that the high intensity electromagneticfields in the accelerator will unavoidably lead tounwanted interactions between those photons andthe accelerated particle, causing it to radiate energyand slow down. Since this effect would be moreimportant at higher particle energies, it would makeit impossible to accelerate particles to high energiesby the suggested method.

We are now in a position to show that thisobjection is unfounded. Indeed, the field con-,figuration suggested for the accelerator is the onegiven by Eq. (3). Therefore, the dominant term­i.e. the one due to electric forces in Kv-in theenergy radiated out by the charged particles isgiven by a sum (over the various Fourier com­ponents) of terms of the form shown in Eq. (28).Assume that the particles lie within a cylinder ofradius r m from the z axis, and that Knrm ~ 1.Then we may approximate Ji(Knr) ~ (Knr/2)2, andlffy,t ~ 0, as r m ~ O. Furthermore, by Eqs. (28)and (30)

M (U')2 (U')2 - (U')2 2lCy,t = - lffy,t f'.I - J 1(Kn r m) rv - rm .c av C av C av

(31)

Here the subscript "av" means averaging over allparticles and also over the time interval I1t,because u' may be a function of time. Thus, thelff y, t and lC~t can both be made arbitrarily small

x- AND y-RAY PRODUCTION IN COHERENT LIGHT 15

provided only that rm is small enough, i.e., providedthat the particles are confined close to the z axis.Turning to the nondominant first term in thesquare bracket in Eq. (13), one sees that it is pro­portional to J6, and does not go to zero as rm

does. However, this term being independent of y,does not increase with energy. Since the forcesused for acceleration in the suggested particleaccelerator are largest near the z axis, while theforces which would be dangerous at high energiesto to zero near the z axis, it seems that accelerationto high energies by the suggested method is indeedpossible.

2) To estimate- the electric fields which can beachieved by the method suggested here, we assume

that the light source is a laser. For u-: Tz, D and Aowe assume the values given in Table I. For sim­plicity we assume that the template produces onlyone Fourier component. We arbitrarily choosethat component for which

(32)

simply because for this one Ikz , no/Kno I = 1 is aneasy number to use in calculations. Substitutingthese values into Eq. (24a), and (5b), we evaluateIE~(t', z', r') I. Multiplying this bye, the electriccharge of the electron, we obtain the absolute valueof the force exerted by the field on a particle ofcharge e. For example, for the first column ofTable I,

TABLE I

In the first part of Table I is a list of the parameters assumed for the light source which in this case is a laser;the template; and the particle beam. The last four lines give the maximum force on a particle exerted bythe electromagnetic field created by the template, and some parameters of the y radiation produced duringone passage of a particle bunch through the electromagnetic field. The * means that while the dominantFouriercomponent has K no = -kz,no' other components also exist because the field is focused by a largetemplate on a section of only D = 3 cm long.

LaserEnergy output per pulseInstantaneous power, WDuration of pulse, ~Wavelength in vacuum, Ao

TemplateR(1 - a)-lLength of illuminated trajectory, DK no =

5.104 J1012 W50 nsec10- 4 cm

15 cmNo optical resonance103 cm-kz,no

103 J2.1010 W50 nsec10- 4 cm

15 cm50103 cm-kz,no

5.104 J101~ W50 nsec10- 2 cm

15 cmNo optical resonance3cmdominant:* - kz , no

Particles(1 - v2 jc2 )-1 / 2 == yspread in y (related to velocity spread) in bunchParticle is anNumber of particles per bunch, NBunchlength in time, ~Bunchlength along z axis, bz

Horizontal emittance in- bunch, AxVertical emittance in bunch, A y

Horizontal beam diameter at waist of bunch, bx

Vertical beam diameter at waist of bunch, byParticle density in bunch, pTime spent by particle in field f1t = Djv

(Maximum force acting on a particle exerted byfield, IF~ ImaJjy

Total energy radiated while bunch traversesfield once, $y, t

Maximum energy of emitted y-ray photon,(nOJ~o)max

Number of y-ray photons produced whilebunch traverses field once, Ny

3.104

10- 3 y = 3·10electron5.1011

10- 1 nsec3cm1C' 1.5.10- 5 cm rad1C. 10- 7 cm rad1.5.10- 1 cm2.10- 2 cm5.55.1013 cm- 3

33.3 nsec

2.12.10- 1 GeVjcm

2.01 . 107 ergs

3.81 GeV

6.59.109

102

10- 1

electron5.1010

10- 1 nsec3cmn . 1.5 . 10- 7 cm radn· 10- 9 cm rad10- 3 cm10- 4 cm1.67.1017 cm- 3

10- 1 nsec

3.87.10- 1 GeVjcm

2.26.10- 1 ergs

4.23.102 eV

6.67 .108

16 P. L. CSONKA

IF~(t', r', z') I = IE~(t', r', z') . e I

= 30 64o

lO- 1Y!J{fiJx COs[~(ct -~)]~~V (33a)

This force oscillates periodically in time withamplitude which depends on r, and reaches itsmaximum IF~lmax, at r = 4.05.10- 5 cm, whereJ 1(wr/j2c) = 0,582, and

IF~lmax = 2.12 ·10- 1y GeV/cm, (33)

a large force by most standards.The required power output of the light source

can be reduced by the use of optical resonance.This is the same principle which is also useful inconnection with particle acceleration by templatemodified coherent light. To achieve such reso­nance, one coats the inside surface of the templateby a partially reflecting mirror surface whose re­flectivity is Ll, so that a photon impinges on it(1 - Ll)-1 times on the average, before penetratingit. We choose the radius of the cylindrical templateto be R, where R satisfies the resonance condition.Since coherence of the radiation must be preservedinside the template, the Ll and R can not exceedthe limits imposed by the coherence length of thelaser light and the maximum smoothness ofthe mirror surfaces. Furthermore, the field at thetemplate must not exceed about E = 1.5 . 103 cgsunits depending on the template material, becausehigher fields would damage the template. Theparameters Ll and R listed in Table I satisfy theseconditions. Table I shows that in this manner onecan achieve considerable decrease in the requiredW For example, if (1 - Ll)-1 = 50, R = 15 cm,then W can be reduced by a factor of 50, withoutreducing IF~lmax.

3) The values assumed for the parameterscharacterizing the particle beam are listed inTable I.

For y = 3 . 104, the parameters are similar

(but not identical) to the ones already achieved orplanned for electrons with energy up to 4.5 GeVat SPEAR and up to 15 GeV at PEP, except thathere we assume a simple step function densitydistribution (as a function of x, y and z). Thequantities bx, Ax and D are chosen so that thehorizontal beam diameter varies only by a smallfraction while the particle passes through thefield, and so does the vertical beam diameter. We

neglect these varIatIons and consider bx and byto be the corresponding diameters everywhere inthe field produced by the template.

For these parameters p(x, y) = p(x)p(y), and thebeam cross section is a quadrangle whose hori­zontal and vertical sides are respectively 1.06·103K;/ and j2. 102K;/. We evaluate Ji forthis p(x, y) from Eq. (27). We use the approximation

J1(Knor) ~ Cr~or}12 COS(Knor _ 3:}

observe that averaging over cos2 gives a factor 1/2when by ~ K~ 1

, and evaluate Ji approximately:-2 -3J 1(Kno r) ~ 10 . (34)

Substituting Eq. (34) into (28) we find ~y,t, theenergy produced during the time when one bunchof particles passes once through the field producedby the template. The result is listed in Table I.From Eqs. (14), (22), and (32), we find the maximumangular frequency (as measured in the laboratoryframe) of the emitted y rays. For example, whenA = 10- 4 cm, then

(w~o)max = 3.41 y 2 wno = 6.43· 1015y2 sec- 1

, (35)

which means that the maximum energy of theemitted y ray photons is 3.81 GeV in the laboratoryframe, and 63.5 keVin K v • Using ~y,t, we find thatthe number, Ny, of photons emitted while onebunch of particles traverses the field producedby the template is about 6.5.109 . The y-rayphotons are emitted within a typical angle ofy-1 measured from the direction of the velocity ofthe emitting particle. According to Table I, theparticle velocities themselves are parallel withinan angle of 10- 4 and 5.10- 6 radians respectivelyin the horizontal and vertical direction, so thatt)1e y rays will be emitted within a typical angle of10- 4 radians in the horizontal direction, and3 . 10- 5 radians in the vertical direction.

The parameters for y = 102 are extrapolatedfrom the parameters for y = 3 . 103 using thefact that the equilibrium electron beam diametersare proportional to y. (The extrapolated beamdamping time, proportional to y- 3 would then beabout 45 minutes, but with a smaller radius oforbit curvature, it can be reduced). The N isassumed to be reduced by a factor of ten.

When y = 102, then bx , by ~ K~l and the beam

can be arranged so that it passes at a distancero ~ 1.8K~ 1 from the axis, where Ji reaches itsmaximum value of 0.3387. For Ao = 10- 2 cm, the

x- AND y-RAY PRODUCTION IN COHERENT LIGHT 17

so that the number of particles inside a sphere ofdiameter d', is

maximum energy of the emitted photons is2.5 . 102 eV, and the radiation is emitted withina typical angle of 10- 2 radians.

4) According to Eqs. (23) and (32), those particleswhich in K v lie within a sphere of diameter d'will radiate coherently, provided that

d' ;5 n[(2 + j2)YW/C]-l. (36)

When y = 102, and Ao = 10- 2 cm, then

d' = 1.46 . 10- 5 cm, (37)

and from Table I we find

p' = y-1 p = 5.55 .1014 cm- 3 , (38)

Since n is proportional to A3, it can rapidly reach

high values. These n particles will radiate co­herently. The total radiated intensity coming fromthese particles will be about n times that whichappears in Table I. This radiation will not beobservable in the high energy region of the emittedradiation, unless the particle bunches have asubstructure. The coherent radiation will be bestobserved in the high energy part of the emittedspectrum, if the substructure consists of smallbunches of particles, d'/y long in the laboratory,each containing about n particles. (If no sub­structure exists, then in the laboratory frame thecoherent radiation will be emitted predominantlywith a momentum and energy equal to the mo­mentum and energy of the photons produced bythe template).

5) The large enhancement in $y, t which can beachieved by coherent scattering by the electrons,opens up the possibility of producing essentiallymonochromatic y rays. Indeed, Table I shows that

4nn = - (d')3 p' ~ 2.5.

3(39)

for high values of n, most of the photons producedby the template are transformed into y-ray photonsnear the surface of the particle beam. If the bx andby beam diameters are large compared to thewavelength A~ measured in K v of the photonsproduced by the template, then we see that they rays are produced simply by reflection from a fastmoving mirror surface: the surface of the particlebunch. When the front of the particle bunch issufficiently well defined, production of high energymonochromatic y rays is possible in the'directionwhich in the laboratory is almost parallel to v.

6) So far we discussed coherence referring toproduction of y rays by several electrons simul­taneously. Next we will discuss the production ofcoherent y ray radiation, that is radiation in whichthe y ray photons are monochromatic and in phasewith each other.

To produce coherent y radiation, one can use theproduced y-rays (whether or not they are essentiallymonochromatic) to irradiate and invert the electronpopulation of a target, by putting it into a higher,possibly metastable energy level. Choosing they-ray frequency appropriately, this method wouldbe much more energy efficient than the usualmethod of pumping with (low energy) laserphotons. This method of producing coherenty-ray radiation will be discussed elsewhere.

ACKNOWLEDGEMENTS

I wish to thank Paul Channell, Phil L. Morton, John R. Rees,N. Webb, and Hermann Winick for discussion concerning thissubject.

REFERENCES

1. Paul L. Csonka, Phys. Lett., 24B, 625 (1967); CERN ReportNo. 67-15; Phys. Rev., 10D, 251(1974).

2. P. Channell, private communication.3. Paul L. Csonka, Particle Accelerators, 5, 129 (1973);

University of Oregon, Institute of Theoretical Sciencepreprint, No. N.T. 149/72.