Rhee, M J

Schneider, R F

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Particle Accelerators, 1986, Vol. 20, pp. 133-1410031-2460/86/2002-0133$15.00/0© 1986 Gordon and Breach Science Publishers S.A.Printed in the United States of AmericaTHE ROOT-MEAN-SQUARE EMITIANCE OF ANAXISYMMETRIC BEAM WITH A MAXWELLIANVELOCITY DISTRIBUTIONM. J. RHEE and R. F. SCHNEIDERtElectrical Engineering Department and Laboratory for Plasma and Fusion EnergyStudies, University of Maryland, College Park, MD 20742(Received June 17, 1985; in final form January 3, 1986)A method of analyzing the rms emittance for an axisymmetric and nonrotating beam with aMaxwellian velocity distribution is described. The resulting emittance is expressed in terms ofequations for the beam parameters, which may be easily constructed from the experimentalmeasurements. It is also shown that for beam parameters of simple form, the emittance can also becalculated analytically.1. INTRODUCTIONThe root-mean-square (rms) emittance, proposed by Lapostolle l and Sacherer,2has been widely used as an alternative means of defining beam quality.3,4 If abeam has no sharp envelope, as is often the case in practical situations, the rmsemittance provides a useful means of describing the rms envelope of the beam,whereas use of the emittance (the area in two-dimensional trace space) isambiguous, since it cannot be clearly defined. In practice, the rms emittance alsoaccounts for the effective increase due to the filamentation phenomenon in thepresence of nonlinear systems, in contrast to the emittance, which is invariant. 5The rapid development of new pulsed power technology in the past twodecades has led to the production of high-current electron beams and ion beamsfor a variety of applications. 6 Some applications, such as heavy-ion inertialconfinement fusion? and free-electron lasers,8 require the use of high-qualitybeams. The evaluation of rms emittance, however, often requires an elaborateeffort of data taking and analysis. It is thus imperative to find a simpler method ofanalyzing the rms emittance of such beams, which are mostly of a single-pulsenature.In this paper, we describe a method of finding the rms emittance fromexperimental data obtained by using a simple emittance meter for an axisym-metric and nonrotating beam with a Maxwellian transverse velocity distribution.In the following sections, the definition of rms emittance is given, and a fewpractical assumptions are imposed on the distribution function, so as to utilizeresults of simple emittance meters. Subsequently, a method of analyzing dataobtained by a simple emittance meter is described in detail. The resultant rmst Present address: Naval Surface Weapons Center, White Oak, Silver Spring, MD 20903.133134 M. J. RHEE AND R. F. SCHNEIDERemittance is expressed in terms of integrals of empirical equations deduced fromthe experimental data. Also demonstrated are simple examples in which the rmsemittance can be calculated analytically.2. DEFINITION AND ASSUMPTIONThe rms emittance is defined as1,2(1)where x' and y' denote the gradients of the particle trajectories, given byx' = dx/dz and y' = dy/dz. The coordinates (x, y, x', y') in the "trace space,,9 areused throughout. The brackets «"») denote values averaged over the four-dimensional trace space, defined as(ep) = ~ f epP4(X, y, x', y') dx dy dx' dy', (2)where N is the total number of particles, given by N = f P4 dx dy dx' dy', and P4is the density distribution in four-dimensional trace space, also known as themicroscopic brightness. Since the quantity of interest, <j>, in Eq. (1) is a secondmoment of x - x' coordinates only, Eq. (2) may be reduced to a more convenientform:(ep) =~fepP2(x,x')dxdx', (3)where P2(X, x') = f P4 dy dy', the projected density on two-dimensional tracespace. It is necessary for rms emittance evaluation that either P4 or P2 bedetermined by experimental measurement. As will be described in the nextsection, both can be directly measured by certain types of emittance meters.However, such emittance meters may not be practical; in particular, for beams ofthe single-pulse type. In order to utilize a simple emittance meter which cancomplete the measurement in a single pulse, it appears inevitable to make someassumptions regarding the form of the distribution P4' such that P4 can beuniquely determined from the experimental measurements.We assume a Maxwellian distribution of P4 in x' - y' space. This may be apractical assumption,3 since the Maxwellian distribution is that of a thermalequilibrium state. Actual beams, however, are not exactly in a thermalequilibrium state; nevertheless, they have a similar distribution, which inherentlyoriginates from their sources, such as thermionic cathodes and plasma ionsources. With this assumption, the distribution is then written asP4(X, y, x', y') = g(x, y) exp [_(X'2 + y'2)/2a2], (4)where the components of the random angle (velocity) are given by X' = x' - i'and Y' = y' - y', with i' and y' being mean values of angles (velocities) overx' - y' space due to diverging or converging of the beam; i.e., i'(x, y) =n-1 f X'P4 dx' dy' and y'(x, y) = n-1 f y'P4 dx' dy', where n(x, y) = f P4 dx' dy'.RMS EMITIANCE OF AXISYMMETRIC BEAMS 135In Eq. (4) a is the rms value of angle (velocity) over x' - y' space, a(x, y) =[n- 1 f (X,2 + y,Z)P4 dx' dy']1/2, and can be related to an equivalent thermaltemperature as a2= kT/(mof32c2). We also assume that the beam is axisymmetricand nonrotational, Le., g(x, y), a(x, y) = (i,2 + y'2)1I2, and a(x, y) are functionsof r = (x2+ y2)112 only, and the mean angle a has only the radial component,Le., i'/y'=x/y, or i'=ax/r and y'=ay/r. With these assumptions, thefour-dimensional trace-space distribution function can be rewritten asP4(X, y, x', y') = g(r) exp (- y2/2a2),where y2 = (x' - ax/r)2 + (y' - ay/r)2.3. EMITTANCE METERS(5)(6)Although a number of different principles are employed in various emittancemeters, the devices commonly used generally consist of two aperture plates and adetector (see Fig. 1). The upstream and downstream plates may have one ormore holes or slits. The detector, placed downstream of both plates, measures thefraction of beam passed through both apertures. The emittance meters may beclassified into four categories, named after the combination of the aperture types,as two-slit, two-hole, hole-slit, and slit-hole meters. Suppose a hole is placed onthe upstream or downstream plates. Then each hole defines a pair of coordinates(x, y) or (;,11) on the upstream or downstream plate, respectively. The upstreamaperture defines the spatial coordinates of a beamlet formed through it and allowsan angular dispersion due to the transverse. velocity distribution of the beam as itmoves towards the downstream plane. The downstream aperture then analyzesthe angular (velocity) distribution by further sampling the beam. One can relatethese coordinates to that of the four-dimensional trace space byx' = (; -x)/L,y' = (11 - y)/L,SYMMETRYAXISy(x,Y) -- ------""'-""""-"-L -,SI r.rr-Jr.I~Stoo~oFIGURE 1 Trace-space coordinates in relation to a typical emittance meter.136 M. J. RHEE AND R. F. SCHNEIDERwhere L is the distance between the upstream and downstream plates. With theserelations it can be readily shown4,lo that the intensity distribution measured bythree methods (two-hole, two-slit, and hole-slit) can be expressed in terms ofP4(X, y, x', y') or its integral. The two-hole method directly measuresP4(X,y,X',y'), the two-slit method measures p2(x,x')=fp4dydy' [orP2(y, y')], and the hole-slit method can measure P3(X, y, x') = f P4 dy'. Here, wehave made use of the fact that a slit provides an effective integral of intensitydistribution in the direction of the slit (f dy or f dy'). In this analysis, we assumean ideal hole or slit whose size is infinitesimally small. In practice, however, onecan choose any finite size, so long as the intensity distribution measured isproportional to the slit width or hole area, Le., /4 == P4 ilx ily ~x' ily', where /4 isthe measured intensity distribution, and ilx, Lly, Llx', and Lly' refer to the size ofaperture. Since the emittance depends only on the form of distribution but not onthe magnitude of the distribution, one can use any relative intensity distribution /for the P in this analysis without knowing the magnitude.It is emphasized here that the slit-hole method can be utilized conveniently forthe si~gle-pulsebeam,11 although the measured intensity distribution appears notto be related in a straightforward way to the distribution P4' Suppose the detectormeasures particles passing through a slit (parallel to y, say) and a pinhole at;, 1] = O. The measured intensity distribution is f P4(X, y, x', y') dYe Noting that yand y' are related to each other by Eq. (6) as y = - Ly', since 1] = 0, and thusdy = - L dy' , the measured distribution may be rewritten asL f P4(X, Ly', x', y') dy'. If P4 is a slow function of y and is nearly constantwithin a range y = ±La, where the a is the rms value of y', then the measureddistribution L f P4 dy' is considered to be approximately the same as thatobtained by the hole-slit method, excepting the factor L. This condition can easilybe met when the beam has a small diverging angle (a« 1) or by choosing thedistance between the aperture plates very small, such that (ap4/ ay)La « 1.4. EMITTANCE ANALYSISNow we describe a method of analyzing experimental data obtained by theslit-hole type emittance meter. This slit-hole method provides relatively accurateinformation on emittance. A typical configuration of the system consists of anarray of slits and a detector plane as depicted in Fig. 2. The slits at Xi sample thebeam and produce sheet beamlets. These beamlets on the detector plane revealangular distribution. This distribution is scanned along the line of 1] = 0 by a smallpinhole and a detector (e.g., a Faraday cup, or film sllbsequently scanned by amicrodensitometer) , allowing construction of the density distributions f3(x;)corresponding to the slit positions Xi' where x; = (~i -xi)/L.Each f3(x;) is assumed to be Maxwellian and thus characterized by the peakheight f3i' rms width ai' and mean angle (velocity) Xi' From these discrete set~ ofexperimental data, one can construct empirical equations for f3(x), a(x), andi'(x) as functions of x, which smoothly connect (or curve fit) data points f3i' ai'and Xi' respectively, as shown in Fig. 3.RMS EMITIANCE OF AXISYMMETRIC BEAMS 137L (=1)--------......... x------... ,FIGURE 2 Schematic representation of the slit-hole method. Shown is an intensity distributionP(xD after slit Xi' whose mean angle, peak height, and rms width are i;, Pi' and ai' respectively. (Forsimplicity the distance L between the two plates has been chosen to be unity.)f3(x), f3(r)a),II'~~",At'.....x·x,rIo-(x),o-(r)b)."K~.......-.......x,rx·Ii'(x ),a(r)c)---.........-~_L--~~.....a...---L-_...I...---~x , r/AI'.".~/,,/IFIGURE 3 Construction of empirical equations of beam parameters: (a) The peak value of anindividual distribution as a function of X or r, (b) the rms width as a function of X or r, and (c) themean (diverging) angle as a function of x or r.138 M. J. RHEE AND R. F. SCHNEIDERNoting (i) that the peak value of each distribution f3i is the measure of particlesof velocity equal to i' passed through the ith slit located at Xi and arriving at1] =0, and (ii) that the slit in the Y direction effectively provides the integralf dy', we may relate the empirical functions to each other by the four-dimensional distribution functionf3(x) =Jp4(x,y=O,x'=i',y')dy'. (7)By virtue of the Gaussian form, the integral f dy' in Eq. (7) is replaced byV2ir a(x), yielding a simple relation,f3(x) = V2ir a(x)g(x). (8)Thus, the four-dimensional distribution in terms of measured empirical para-meters may be written asP4(X, y, x', y') =vt!;(r) exp {-[(x' - i')Z + (y' - y,)Z]/2aZ}, (9)where Eq. (8) has been used with the argument r = (x2+ y2)112 in place of xbecause ofaxisymmetry. One can easily recognize the spatial density distributionin this case asn(x, y) = JP4 dx' dy' = V2ir f3(r)a(r). (10)Finally, the two-dimensional projected distribution on the x-x' trace space isfound to bePZ(x, x') = Jf3(r) exp [-(x' - ax/rfl2a2] dy, (11)(12)where f3(r), a(r), and £1'(r) are empirical parameters which are functions ofr = (x2+ y2)1I2. The isodensity contours of P2(X, x') in x-x' space, known as theemittance plot, provide very useful information about the beam.For evaluation of rms emittance, three average values of second moments,(x2), (X'2), and (xx'), are needed. Upon substituting the two-dimensionaldistribution function given by Eq. (11) in Eq. (3), the averaging integral in x'space is replaced by V2ir a because of the Gaussian distribution, and theexpression for (x2 ) is obtained asV2nfoo(x2 ) =- x2f3(r)a(r) dx dYeN -00Similarly (X'2) and (xx') are given byV2nfoo { (a'X)2}(x'Z) =N -00 [a(rW+ -; f3(r)a(r)dxdy,( ') =V2irfoo x2£1'(r)f3(r)a(r) d dxx N x y,-00 r(13)(14)RMS EMITTANCE OF AXISYMMETRIC BEAMSand the total number of particles N isN= V2ii: LooJJ(r)a(r) dx dy.It is of interest to note that Eqs. (11)-(15) are of the Abel type,12 i.e.,foo foo R(r)I(x) =2 0 R(r)dy=2 x V(r2 _x2)rdr.Thus, these equations can be rewritten in a more useful form:( ') _ 2foo f3(r) exp [-(x' - ax/r)2/2a2] dP2X,X - ~~ r r,x yr- - x-( 2) = 4V2ii:ioo2foo f3ar dr dx N x ~~ x,o x y r- - x-( '2) =4V2ii:foo [foo f3a3r dr 2foo a2f3a dr ] dx N ~~+x ~~ x,° xyr-x- xryr-x( ') _4V2ii:foo 2foo af3a dr dxx - N x ~~ x,o x yr- - x-foofoo f3ar drN=4V2ii: w=xz dx.° x r-x139(15)(16)(17)(18)(19)(20)The beam parameters a(r), f3(r), and a(r), since they are determined byexperimental measurements, could be any functions, including empirical curves.But a simple numerical method may be employed for integration of either Eqs.(11)-(15) or Eqs. (16)-(20) to evaluate the emittance.It is important to note that if a(r) is proportional to r, i.e., for linear focusing,then Eq. (17) times the second term of Eq. (18) is equal to the square of Eq.(19), regardless of the forms of functions f3(r) and a(r), thus canceling thebeam-diverging effect in Eq. (1). Thus, the rms emittance is independent of linearfocusing. When the beam is passing through a linear lens system, the spatialcoordinates of beam particles remain unchanged, but the angle (velocity) ischanged in proportion to the radial position of each particle. Thus, the rmsemittance may be regarded as an invariant of motion through a linear lenssystem.It should be mentioned that the analysis in this section is equally applicable tothe experimental data obtained by the hole-slit type of emittance meter.5. ANALYTICAL EXAMPLESFor a certain type of integrand in Eqs. (16)-(20), the integration can beperformed analytically. We take three simple examples which are of practicalimportance. Fjrst, suppose the following beam parameter equations: a(r) = ria,f3(r) = f3oH(b2- r2), and a(r) = ao, where H is a Heaviside unit step function.140 M. J. RHEE AND R. F. SCHNEIDERThis set of functions may approximately describe a beam which is emitted by auniform hot cathode, measured immediately downstream of an anode with a holeof radius b. One can easily obtain pz(x, x') from Eq. (16) aspz(x, x') = 2f3oYbz - xZexp [-(x' - x/a)z/2a~]. (21)It is also straightforward to find the integrals in Eqs. (17)-(20), obtaining as meanvalues N = v2 n 3/zf3aobz, (xZ) = bZ14, (x'Z) = aZ+ (bI2a)Z, and (xx') = bZl4a.The rms emittance is then given byCnns = 2bao· (22)As mentioned earlier, this emittance is independent of £1'(r) = ria. The sameresult can be expressed in terms of the thermal temperature of the beam3 asCrms = 2b(kTlmof3Zc2)1/2, which is evidently true, even if the beam is focused (aslong as the focusing is of linear form).The second example is for a beam with a Gaussian spatial distribution. Thebeam parameters are given by £1'(r) = ria, f3(r) = f30 exp (-r2/2b Z), and a(r) = ao.This case may approximately represent the same beam as the previous example,but measured farther downstream. We obtainpz(x, x') = y'2;; f3oao exp [-xZI2b2- (x' - xla)2/2a~]. (23)The mean values given by Eqs. (17)-(20) are easily found as N = sy'2;; nf3oaob,(x2) = bZ, (x'Z) = a~ + (bla)Z, and (xx') = bZla. The rms emittance from Eq. (1)is crms = 4bao. (24)Notice that this result is also independent of ria.Our third example is a beam with a linear focusing, a Gaussian spatial profile,and.a Gaussian spatial dependence of rms angle (transverse velocity). This beamis described by the parameter equations £1'(r) = ria, f3(r) = f30 exp (-rZI2b 2), anda=aoexp(-r2/2c2). If we make the substitutions Ilp2=l/b2+1/cz and IIqZ = I/b z + 3/cz, then Eqs. (17)-(20) may be easily solved to give (x2) =p2,(X'2) = (aoqlp)2 + (pla)Z, and (xx') = p2la. Hence,(25)It can easily be seen that in the limit c~ 00 this is identical to the previousexample.In all examples, (X'2) increases, since the transverse velocity increases due tothe linear focusing £1'(r) = ria. However, this velocity is not a random one; it islinearly correlated with x. As defined by Eq. (1), this effect is reduced by thelinear correlation term (xx'), resulting in the rms emittance being independent ofthe linear focusing, as mentioned earlier.6. CONCLUSIONFor an axisymmetric and nonrotating beam with a Maxwellian velocity distribu-tion, we have described a method of determining the rms emittance fromRMS EMITIANCE OF AXISYMMETRIC BEAMS 141experimental data which can be easily obtained by using simple emittance meters.The resultant emittance and two-dimensional projected distribution in x-x' tracespace are expressed in terms of empirical equations of beam parameters. We havealso demonstrated that for simple types of beam parameter equations, the rmsemittance can be calculated analytically.ACKNOWLEDGMENTSThe authors are grateful to W. Namkung for useful discussions. This work wassupported by the Air Force Office of Scientific Research and the U.S.Department of Energy.REFERENCES1. P. M. Lapostolle, IEEE Trans. Nucl. Sci. NS·18, 1101 (1971).2. F. J. Sacherer, IEEE Trans. Nucl. Sci. NS·18, 1105 (1971).3. J. D. Lawson, The Physics of Charged Particle Beams (Clarendon Press, Oxford, 1977), 172-215.4. C. Lejeune and J. Aubert, in Applied Charged Particle Optics, A. Septier, Ed. (Academic Press,New York, 1980), pp. 159-259.5. J. D. Lawson, P. M. Lapostolle, and R. C. Gluckstern, Particle Accelerators 5, 61 (1973).6. J. Nation, Particle Accelerators 10, 1 (1979).7. W. Chupp, A. Faltens, E. Hartwig, E. Hoyer, D. Keefe, C. Kim, M. Lampel, E. Lofgren, R.Nemetz, S. S. Rosenblum, J. Shiloh, M. Tiefenback, D. Vanecek, and W. Herrmannsfeldt, IEEETrans. Nucl. Sci. NS·28, 3389 (1981).8. P. A. Sprangle, R. A. Smith, and V. L. Granatstein, in Infrared and Submillimeter Waves, K.Button, Ed. (Academic Press, New York, 1980), Vol. 1, p. 279 and references.9. J. Fronteau, in Focusing of Charged Particles, A. Septier, Ed. (Academic Press, New York,1967), Vol. 2, p. 421.10. A. Van Steenbergen, Nucl. Instrum. Methods 51, 245 (1967).11. M. J. Rhee, G. T. Zorn, R. C. Placious, and J. H. Sparrow, IEEE Trans. Nucl. Sci. NS·18, 468(1971).12. W. Lochte-Holtgreven, in Plasma Diagnostics, W. Lochte-Holtgreven, Ed. (American Elsevier,New York, 1968), p. 184.

The root-mean-square emittance of an axisymmetric beam with a maxwellian velocity distribution

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