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Principles of high-contrast energy modulation and microbunching of electron beams
Nuclear Instruments and Methods in Physics Research A 429 (1999) 462}470
Principles of high-contrast energy modulation and microbunching of electron beams R. Tatchyn* Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, CA 94309, USA
Abstract Recent advancements in ultra-short-pulse terawatt IR/visible/UV laser technology have made it possible to consider particle-beam energy modulation schemes in which the practical "eld amplitudes of the laser and undulator "elds in an ultrarelativistic electron bunch's average rest frame can attain comparable magnitudes. This parameter regime, well outside the radiation "eld strengths attainable by conventional free electron lasers, makes possible the exploitation of `relativistic interferometrya, viz., the phenomenon of high-contrast interference, to isolate regions in the longitudinal space of an electron bunch on the order of one laser wavelength. In this paper we review selected requirements on the laser and undulator "elds to generate highly e$cient microbunching followed by coherent radiation at a substantially reduced wavelength. Conditions on the electron beam quality, as well as selected possibilities for new modes of operation, are summarized. 1999 Elsevier Science B.V. All rights reserved. Keywords: Microbunching; High-contrast modulation; Free electron lasers; Relativistic interferometry
1. Introduction In practically every free electron laser (FEL) heretofore designed or operated, the radiation "eld amplitude in the rest frame of the electron bunch is overwhelmingly dominated by the "eld amplitude of the insertion device. For example, in a gain-saturated 1.5 As SASE LCLS driven by a 15 GeV beam with a 1 mm mrad normalized emittance [1], an output power of 100 GW and an undulator "eld of 1.3 T in the laboratory frame are transformed, respectively, into bunch-frame ampli-
* Tel.: #1-650-926-2731; fax: #1-650-926-4100. E-mail address: [email protected] (R. Tatchyn)
tudes of &0.0400 T versus &13000 T. In prior work, the relatively recent advent of ultrashortpulse IR/visible/UV lasers in the terawatt range [2] was adopted as a premise for considering the preparation of electron bunch frames in which the practical "eld amplitudes and wavelengths of the laser and insertion device could be made equal, leading to the possibility of inducing high-contrast (HC) interference e!ects, in particular on the spontaneous radiation and particle energy [3] distributions. A basic motivation in these initial studies was } and remains } the generation of ultrashort (subfemtosecond) radiation pulses. In subsequent papers [4,5], the energy modulation of single particles in this regime was analyzed and con"rmed the initiation of conditions for rapid bunching, which, if attained, could be used to generate coherently
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 1 4 2 - 4
R. Tatchyn / Nuclear Instruments and Methods in Physics Research A 429 (1999) 462}470
enhanced radiation at substantially reduced wavelengths. More recently [6], these studies were extended to preliminary simulations of collective dynamics within a cold bunch. In the present paper the basic principles of HC systems are reviewed with the aim of: (1) clarifying the basic physics involved, and (2) deriving simple expressions for the initial evaluation or design of practical con"gurations. Selected possibilities for further developing the HC modulation technique are discussed and some recent preliminary simulations of warm-beam e!ects on the bunching process are presented.
2. Physical principles: single-electron dynamics A layout of the basic constituents of a HC modulation (HCM) and bunching system is schematized in Fig. 1. A high-intensity radiation (e.g., laser) pulse (or synchronized group of superimposed pulses) of wavelength j and normalized vector potential K intercepts a high-quality, high-density electron beam at angle(s) h along the axis of a static or dynamic "eld synthesizer (FS) [7]. The FWHM bunch length is &(2pp and the FS parameters are j and K . We adopt, as a practical working de"nition of HCM, the relation O(K ) O(K ). The polarizations of the "eld sources can be arbitrary, but each one will be assumed linear for the present discourse. An essential attribute of the FS is its ability to generate arbitrary near-axis "eld distributions, and in particular accurately-scaled replications of the superimposed laser-"eld components. Inside the FS, the laser imparts an energy modulation pro"le to a region of the bunch. This modulation induces a (ballistic) bunching drift inside the beam which can be further modi"ed with suitably designed dispersion regions, which can also comprise sections of a FS. The conditions for maximum-contrast interference of the radiation and FS "elds in the rest frame of the electron bunch are
straightforwardly derived to be [8] j "j (1!bH cos h) and
(1)
B B /(1!bH cos h) (2) where bH"(1!cH\, cH"c/(1#K/2, and c is an (incoming) electron's Lorentz factor. For h;1, Eqs. (1) and (2) reduce to j (j /2) ((cH)\#h) and B 2B /((cH)\#h). Upon compression of the modulated region down to a (microbunched) length D , the beam, or a suit ably "ltered portion thereof, is sent through a radiator of period j 54cHD , from which 0
a coherently enhanced radiation pulse of wavelength j ;j is emitted. The maximum number 0 of radiator periods usable for this purpose will in principle be limited by the time it takes for the bunching to relax, di!use, or scatter back to the condition D '(j /2cH). It should be evident
0 from these descriptions that the basic elements and selected processes of the HC system have their counterparts in the optical klystron (OK) [9}11] and free electron laser (FEL) [12]. Indeed, the basic single-electron dynamics underlying the energy modulation can be described by the same equations. However, as will be outlined below, signi"cant di!erences in the detailed physics can arise from the highly disparate parameter regime of HC modulation, as well as from the basically di!erent phenomena involved in the collective (bunching) dynamics. Our single-particle analysis will apply explicitly to linearly polarized "elds, following the approach of Colson [13]. The equations of motion for the transverse (y) and longitudinal (z) trajectory components can be written as d(mccb ) W "qE #cqb B W X V dt
(3)
and d(mccb ) X "!cqb B W V dt
Although in the present article we focus primarily on ultrashort laser pulses (down to the order of a few wavelengths long) and static magnetic-"eld FSs, the general HC modulation and bunching system need not be restricted to these conditions.
463
(4)
along with the energy-transfer equation q c" b E . cm W W
(5)
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Fig. 1. High contrast modulation (HCM) and dispersion-assisted bunching (A) followed by coherence-enhanced radiation (B). For ballistic bunching, K "0. The FS can, in general, generate arbitrary static or dynamic "eld distributions with components along all " three axes.
Taking E "E sin (2p(ct!z)/j # ) and B " W V B sin (2pz/j # )!(E /c)sin (2p(ct!z)/j # ), with h"0 rad, and the bracketed exponents denoting the number of integral sinusoidal cycles, the following expressions for the transverse c-normalized velocity and longitudinal acceleration are derived:
K 2p b "! cos
(ct!z)#
W c j 2p K ! cos
z#
c j and 2pcK K bQ " X j c
(6)
(ct!z) z ! # !
j j 1 4p # sin
(ct!z)#2
2 j 4pz !sin
#2
(7) j where the indexed 's determine the phases of the laser and undulator "elds with respect to the electron in question. Although this does not allow for sin 2p
These "eld distributions are, strictly speaking, unphysical. The approximation is, notwithstanding, reasonable in that: (1) it allows the systematic study of HC modulation by a progressive increase in the number of `singlea cycles, and (2) it can in fact provide insight into the evolution of an actual HC bunching distribution, which, due to the highly non-linear dynamics, can be made to develop a single dominant bunched peak. In the general case more realistic "eld distributions can of course be employed using numerical simulations.
a fully general source decomposition, it su$ces to explore the basic physics of HCM. We note the explicit dependence of both motion components on z. In general, the functional dependence of z on t is complicated, and can ordinarily be expressed as a Fourier series (or, in our limit, a Fourier integral) in t. The signi"cance of higher-order spectral terms is dependent on the amplitude of the coe$cient of Eq. (7), which is seen to be directly proportional to K K and inversely proportional to j c. Although the investigation of Eqs. (6) and (7) for smallthrough-large values of both these factors is of legitimate concern for HCM, we restrict ourselves in this paper to the ranges j c<1 m and 0.005(K &K (0.25, the lower bound of the latter constraint stemming from present-day technological limitations on implementing long-period insertion devices with overly weak "elds. In this regime, z bHct, the z-velocity in the bunch frame, v , is non-relativistic, and the visualization of the X dynamics contained in Eqs. (6) and (7) is correspondingly simpli"ed. Given the above approximations and assumptions, a number of basic results follow. For an initial transverse velocity of zero, the "elds impart no net transverse kick to the electron, irrespective of their phases, so long as their 1st "eld integrals are zero (Eq. (6)). The initial transverse position, however, proportional to the di!erence of the 2nd "eld integrals, need not remain constant [4]. For the condition on j versus j expressed in Eq. (1), the upper expression in square brackets in Eq. (7) approaches zero, and the time-dependent parts of the arguments in the lower bracketed expressions approach equality. Thus, the magnitude and sign of the longitudinal acceleration imparted to a bunch
R. Tatchyn / Nuclear Instruments and Methods in Physics Research A 429 (1999) 462}470
electron are determined primarily by its phase relative to the laser and FS "elds. In this regard we note that the direction of electron motion in z is always directed toward the nearest minimum of bQ , indicatX ing the onset of dynamic bunching. We can now derive a number of useful expressions for assessing HC modulation systems in this parameter regime. Using the Einstein velocityaddition formula [8] and the above equations, the maximum normalized energy and velocity kicks (i.e., for " ! ""p/2) imparted to a bunch elec tron in the lab frame are given, respectively, by: (1) *c 2pcK K , and (2) *b 2pK K /c. The corre X sponding absolute energy and velocity kicks in the bunch frame are: (1) *E 2pm c(K K ) J, and (2) *v 2pcK K m/s. We can now compare seX lected features of HCM versus OK (or FEL) systems running at similar wavelengths. In the latter, K &O(10\}10\) at saturation, and 4}5 orders of magnitude smaller than that at the FEL entrance. Thus, even for a high-K FEL the net longi tudinal kinetic energy extracted from a bunch electron will be considerably smaller than in a typical HCM system with, e.g., K &K &0.1. Further more, the net energy exchange in an OK or FEL accrues over many periods N } typically of the order of 10, whereas the net gain (or loss) in HCM is attained over a substantially smaller number of periods (e.g., for the case being considered, N "1). Thus, the average relative energy transfer per period in a typical FEL is of the order of 1/(N ), or &10\, justifying its characterization as `adiabatica, whereas the HCM loss (or gain) per period, typically O(0.1!0.01), could be more aptly termed as `high-actiona, or `rapida. Based on these observations, a relevant criterion for distinguishing a HCM system versus a FEL or OK modulating at a similar wavelength can be expressed as (K K ) <(K K ) . &!+ $#* -) 3. Physical principles: collective dynamics As opposed to a FEL or OK, in which the radiation scattered o! the electrons and the interparticle Coulomb forces both act, in general, as essential factors in the energy modulation and bunching processes, the scattered radiation in an
465
HCM system plays an initially minor, if not negligible, role. Upon modulation by the (exogenous) radiation "eld, the bunching in the electrons' rest frame is mediated, in the non-relativistic limit by an exchange between the electrons' acquired kinetic energy and the electrostatic potential energy associated with the gradually evolving particle density. Just as in an FEL, however, the dynamics is still governed by the self-consistent interaction among all the particles moving in the combined undulator, Coulomb, and radiation "elds. The combined collective modulation and bunching in the general HCM system can be described by an unrestricted Vlasov equation [14]. If the overall process can be approximated as a sequence of two independent sub-processes, viz., modulation followed by ballistic bunching, the (dissipation-free) Vlasov equation governing the dynamics of the bunching phase can be written as
*f *f e *f * 1 #* ) " ) f (r,*,t) dr d* *t *r 4pe ** *r "r!r "
(8)
where the term on the right describes the interparticle Coulomb interactions, and f, the particle distribution, is assumed to carry the "elds' velocity modulation imprint, *(r(0)), as an initial condition. We point out, in passing, that for arbitrary HCM "elds and distributions f, Eq. (8) cannot in general be linearized and must be solved either numerically or approached with alternative analytical techniques. Prior to performing numerical simulations based on Eq. (8), it is instructive to "rst assess the bunching process for an idealized case. In our treatment, we reduce the dimensionality of the problem from 3-D to 1-D by "rst partitioning the bunch longitudinally into in"nitesimal slices of thickness d . In X view of the "nite bunch radius, r , we assume the In this limit we can disregard interactions stemming from the particles' magnetic "elds. This approximation (equivalent to taking K &K ;1), al though admittedly inexact, is justi"able for the systems analyzed in this paper. Since *v ;c, and the modulation employs single-cycle "elds, the assumption is that the particles have moved over only a small fraction of j ("j /cH) by the time the two "elds have passed through.
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force between any two such slices to vary inversely with (1#(*z /r )), where *z is the inter-slice dis tance in the bunch frame. We next estimate the potential energy of a collapsed sheet containing all the charge from a linearly modulated region of extent j , by integrating work over pairs of charge sheets. We thus "nd that the peak velocity required (assuming linear modulation) to compress the given region into an in"nitesimal sheet (at which point the total kinetic energy will vanish) is approximable by *v
X!0
qr o j j tan\ m/s m ce 2cr
(9)
where q and m are the electronic charge and mass, o is the lab-frame charge density, and e is the permittivity of free space, all in MKS units. We note that for any set (o ,j ,r ,c) there are evidently three relevant modulation regimes, viz.: I. *v (*v (undermodulation); II. *v '*v X X!0 X X!0 (overmodulation); III. *v *v (critical moduX X!0 lation); and that *v varies inversely with X!0 a power of c higher than unity. Using the 1-D model approximation to Eq. (8), several numerical simulations of a typical HCM systems's bunching dynamics using di!erent energy modulation distributions of Gaussian beams [6] were performed. Recent extensions of the model code to include energy spread within the bunch were also calculated. Here we present results for a linear modulation pro"le, *v " X (2cH"*v "/j )z , with z "0 the position midway X+ 6 between the approaching "elds. In the top line of Fig. 2 we plot the evolution of bunching for regimes I. (a), II. (b), and III. (c) in a cold 10 nC beam using velocity modulation amplitudes, "*v ", of 0.1c, X+ 6 0.15c, and 0.12c, respectively. As expected, undermodulation leads to the longest bunching time and results in underbunching, while overmodulation induces the quickest bunching with only a transitory attainment of the bunching peak. In the second line of Fig. 2 we present histograms of the evolving macroparticle distribution in regime III. (corresponding to Fig. 2(c)) for bunch frame times of 0 ps (d), 3.5 ps (e), and 5.25 ps (f ). The last plot (f ) shows only a coarse approximation to the actual theoretical maximum of 1000 since: (1) no search for the
precise time of the maximum peak was made, and (2) the electric "eld #uctuations associated with the model distribution's (relatively large) density #uctuations would in any case tend to inhibit the attainment of a perfectly compressed peak. In Fig. 2(g) the e!ect of an 0.1% 1-p energy spread on the bunching process of (d}f ) is displayed. In Figs. 2(h) and (i) the evolution of the electric "eld distributions in the bunch frame for the regime III case with 0.0% versus 0.1% energy spreads are shown. The graphs reveal the basic features of the bunching process, supporting the results derived analytically in the above-cited work. As our numerical simulation code has not yet been extended to incorporate the modulation interaction or the action of additional dispersion "elds into the bunching dynamics, analytical estimates for the minimum attainable bunch length D in a dispersion section of length
¸ in terms of the electron beam's emittance para" meters (assuming a net linear modulation) have been worked out [5]. The basic results can be summarized by the ratio D 2c
+ (2pp#(p(p #p ))cH (10) C 4 & " j *c , where *c is the total modulation increment, p is , C the standard deviation of the normalized random energy spread, p and p are the standard devi4 & ations of the beam's transverse emittance angles, and cH"c/(1#(K /2), where K is the nor" " " malized vector potential of the dispersion region. Eq. (10) can be used (with K "0) as an approxim" ate check on the peak bunching reduction displayed in Fig. 2(g).
4. Radiation performance Estimates of the radiative output characteristics of the microbunched beam in a radiator "eld ful"lling the condition j 54cHD are readily derived. 0
Here we restrict ourselves to assessments of the total emitted energy and output power in the lab frame. The maximum possible spontaneous emission power can be expressed in MKS units as P N [cq1v 2 /(6pe c)], where N is the num1. W R ber of bunch electrons and the quantity in brackets
R. Tatchyn / Nuclear Instruments and Methods in Physics Research A 429 (1999) 462}470
467
Fig. 2. Bunching and electric "eld distributions in the rest frame of a cylindrical electron beam with longitudinally Gaussian and transversely uniform density pro"les. Beam parameters: c"300; (2pp "0.5 mm; r "100 lm; N "10; number of macro particles 10 (contained in a region of length 10j /cH at the center of the bunch); electrons /macroparticle 8.9;10. To enhance the resolution only the central 2000 macroparticles containing the modulated region are displayed in a}c; in d}i, the central 3700. For graphs a}f, and h, p "0.0; for g and i, p "0.001. C C
is an approximation to the relativistic Larmor formula [15]. The maximum coherent emission power is given by a similar expression, but with the factor N replaced by (N j /(2pp ). The ratio g. of &!+ the total number of coherent photons, N , emit!-& ted by the microbunch to the spontaneous number N emitted by the entire bunch is consequently 1. g. ,(N /N ) N (j /(2pp ). For broad &!+ !-& 1. ranges of readily implemented linac beam parameters, it is readily veri"ed that g. <1. In practi&!+ cal terms, the number of spontaneous photons emitted by the N bunch electrons passing through an N -period radiator can be expressed as 0 N 7.61;10\N KN , and an upper limit on 1. 0 0
the emitted coherent power by P [W] 1.585;10\cg. N (B [T]) . (11) !-& &!+ 0 A set of parameters for a representative HCM buncher/radiator case is listed in Table 1 (cf. Fig. 1 and Eq. (10)). We observe, in Eq. (11), the scaling of P with (B ). This indicates that in cases where !-& 0 K is initially small the coherent in-band power 0 from a single microbunch could be substantially increased } perhaps well toward the multi-gW level } by substantially increasing the radiator "eld. Fields up to the order of 5 T (versus the &1 T assumed in the table) could be readily generated using a pulsed FS.
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Table 1 System and radiation output parameters of a HCM ultra-short pulse buncher/radiator. Longitudinal beam pro"le Gaussian; transverse pro"le #at Ultra-short pulse generator parameters (c"300; r "100 lm; o "10.6 C/m) Bunch charge (2pp j j p p 4 p & K " ¸ " *c/c D
K 0 Compression ratio (* /j )
480 As coherent photons/pulse Peak coherent output power
1 nC 3000 lm 4800 As 8 cm 0.1% 0.1 mr 0.1 mr 1 0.355 m 0.05 240 As &1 6% 7.0;10 0.18 GW
5. Selected aspects of high contrast modulation and bunching Perhaps the most critical aspect of prior and current studies of HCM has been the emphasis on ultra-short pulse generation. This has facilitated the tractability of our exploratory analyses of both modulation and bunching using single-cycle "elds. In actuality, of course, an ultra-short pulse may span more than one cycle and will always have non-instantaneously decaying edge derivatives. If the FS replicates this distribution then it is clear that the modulated region will also extend over more than twice one period. We can note, however, that the non-linearity of the bunching and radiation process will tend to give a strong preference to the global maxima of the velocity interference pattern. Consequently, by using, e.g., Fourier transform techniques to, say, sharply peak the center of a laser pulse containing a small number of cycles, it should be possible to generate a micro-bunched pulse train in which the radiation from a single microbunch is overridingly dominant. In the general case, many other interesting possibilities arise. For example, for su$ciently small K and K the modulation pattern will approximate to the convo-
lution of the laser and FS pulse trains. Thus, two "elds with #at-top envelopes will generate a pattern with a pyramid-shaped envelope. This hints at the wide range of possibilities } and challenges } for imposing arbitrary bunch structures on arbitrarily-distributed beams, e.g., in the event that both K and K are appreciable, it should be evident that for N ,N <1 the dynamics of the system will no $1 longer be separable into independent modulation and bunching phases, the general form of the Vlasov equation will need to be employed in computer simulations, and the modulation/bunching patterns will no longer be approximable, in general, by a straightforward convolution of the system "elds. A second critical point concerns our assumption of h"0 rad for the analyses and calculations described in this paper. By varying this parameter (in conjunction with the k-vector(s) and polarizations of the radiation and FS "elds), substantial reductions in the scaling of the required laser "eld strength and period in relation to those of the FS could be realized, and a broad range of novel variations in the energy modulation and bunching distributions would become accessible. An interesting possibility for extending the present HCM scheme would be to post-accelerate the beam following an initial modulation phase. Eq. (9) reveals that the potential barrier to bunching is lowered inversely with a power of c greater than 2. Thus, tailoring the initial modulation and postaccelerating "eld pro"les could lead to the preparation of optimally microbunched beams at substantially higher energies. Another interesting extension could involve the restriction of FS and laser "elds to the combinations of one or more higher multipole distributions for tailoring the energy modulation in directions transverse to the beam axis. An extension of this technique would be to prepare non-zero "eld integrals of selected multipole components (including the dipole). This would allow the design of transverse velocity modulation pro"les, which could be used for phase-space "ltering of the modulated/bunched region(s) independently of the overall bunch. A potentially important application of HCM would be to attempt the longitudinal compression of a whole electron bunch at a relatively low value
R. Tatchyn / Nuclear Instruments and Methods in Physics Research A 429 (1999) 462}470
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Table 2 Fundamental di!erences between `single-cyclea HCM/bunching and multiple-period FEL (or OK) systems FEL/OK
HCM
Modulation and bunching adiabatic Relative energy change/period &O(10\) Electrons interact over `co-operation lengtha &N j /10 Self-consistent interactions through radiation# Coulomb "elds essential to modulation/bunching
Modulation and bunching rapid Relative energy change/period &O(10\) Electrons interact over O(1) period Electron-scattered radiation plays an initially minor role. Self-consistent interactions through Coulomb "elds dominate Dynamics strongly non-linear Modulation/bunching patterns less restricted
Dynamics can be linearized Modulation/bunching process periodic (quasi-periodic (chirped) in a tapered FEL) Modulated bunch typically
of gamma, followed by acceleration. In this case j '(8pp , implying the use of high-intensity mm-wavelength "elds. Due to the small (and `softa) radiation losses in such a system, the technique, if successfully developed, could perhaps help to circumvent potential limits set by coherent synchrotron radiation (CSR) e!ects in high-energy chromatic compressors [16,17]. Finally, throughout this paper we have emphasized the relationship of HCM to OK and FEL phenomena. We conclude this section with a summarization of some of the basic distinctions between these two physical regimes in Table 2.
6. Summary We have summarized recent studies of HCM systems and pointed out a number of potential applications for particle-beam and radiation source physics. It is perhaps not super#uous to note that HCM, as analyzed in this article, could be considered practicable for beam energies up to the 2}3 GeV range, provided FS structures of up to 2}4 m in length and with "elds controllable down to the order of a few gauss could be implemented. These requirements could of course be substantially eased with the advent of reliable UV lasers
A basic limit on the modulating FS length will be set by the Rayleigh range and waist size of the laser pulse (assuming no additional optics in the interaction region).
Modulated bunch region typically &O(j )
approaching the petawatt range, or, alternatively, by operating at h'1/cH with lower-power sources set in axisymmetric con"gurations. In the shorter term, it appears that the still-limited availability of femtosecond, terawatt lasers, is in part compensated by the fact that such devices are likely to be employed anyway at FEL facilities driven by photocathode RF guns, and there is a reason to expect that these may start becoming more prevalent. Moreover, collateral improvements in laser performance and parameter values [18], coupled with reductions in cost, can be expected to continue. Apart from this, the other basic constituents of an HCM and bunching/radiating system, viz., a high-quality, high current electron beam and a FS insertion device(s) are either in the process of being developed at a number of laboratories [7,19], or can be approximated to with existing resources. For example, even with the substantially poorer compression expected for a storage ring beam, the impressive output predicted by Eq. (11) } coupled with, say, a high-K radiator for harmonic genera0 tion } strongly suggests that HCM could be productively applied to recirculating beams as well. Given the apparent richness of HCM physics, we
For example, for a 2 GeV beam with 3;10 particles/bunch and a bunch length of 3 cm, coherent output powers could approach the TW range. We note, however, that the higher relative energy spread of the circulating beam, which would inhibit the attainment of large wavelength reductions (as predicted by Eq. (10)), would make the attainment of even higher in-band powers substantially more di$cult.
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believe that undertaking such e!orts is likely to further the understanding, control, and metrology of particle beams, and contribute to the development of FEL, OK, and other 4th generation radiation sources.
Acknowledgements This work was supported in part by the Department of Energy O$ces of Basic Energy Sciences and High Energy and Nuclear Physics, and Department of Energy Contract DE-AC0376SF00515. Selected calculations utilized resources of the National Energy Research Scienti"c Computing Center, which is supported by the O$ce of Energy Research of the U.S. Department of Energy.
References [1] R. Tatchyn et al., Nucl. Instr. and Meth. A 375 (1996) 274. [2] C.P.J. Barty, T. Guo, C. LeBlanc, F. Raksi, C. RosePetruck, J.A. Squier, B. Walker, K.R. Wilson, V.V. Yakovlev, K. Yamakawa, Sub-20-fs multiterawatt lasers, Proceedings of Ultrafast Phenomena 1997, OSA Technical Digest Series vol. 8, 1996, p. 162. [3] R. Tatchyn, Quantum limited temporal pulse generation, in: M. Cornacchia, H. Winick (Eds.), Proceedings of the Workshop on Fourth Generation Light Sources, SSRL Report 92/02, p. 482. [4] R. Tatchyn, Nucl. Instr. and Meth. A 358 (1995) 56. [5] R. Tatchyn, SPIE Proc. 2988 (1997) 112.
[6] R. Tatchyn, High-contrast energy modulation of electron beams for improved microbunching and coherence of free-electron radiation sources, presented at the SPIE 1998 Annual Meeting, San Diego Ca, July 19}24, 1998; SPIE Proc. (1998) 3451, in press. [7] R. Tatchyn, Fourth generation insertion devices: new conceptual directions, applications, and technologies, in: M. Cornacchia, H. Winick (Eds.), Proceedings of the Workshop on Fourth Generation Light Sources, SSRL Report 92/02, p. 417. [8] J. Aharoni, The Special Theory of Relativity, 2nd ed., Clarendon Press, Oxford, 1965. [9] P.L. Csonka, Part. Accel. 11 (1980) 45. [10] N.A. Vinokurov, A.N. Skrinsky, Institute of Nuclear Physics, Novosibirsk, USSR, Preprint 77/67. [11] R. Bonifacio, R. Corsini, P. Pierini, Phys. Rev. A 45 (6) (1992) 4091. [12] J.B. Murphy, C. Pellegrini, Introduction to the physics of the free electron laser, in: M. Month, S. Turner (Eds.), Lecture Notes in Physics vol. 296, Springer, Berlin, 1988, pp. 163}219. [13] W.B. Colson, Free electron laser theory, Ph. D. Dissertation, Department of Physics, Stanford University, 1977. [14] T.-Y. Wu, Kinetic Equations of Gases and Plasmas, Addison-Wesley, Palo Alto, 1966. [15] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975 (Chapter 14). [16] B.E. Carlsten, J.C. Goldstein, Nucl. Instr. and Meth. A 393 (1977) 490. [17] The LCLS Design Study Group, Linac Coherent Light source (LCLS) Design Study Report, Section 7, SLAC-R521, 1998. [18] C. Le Blanc, E. Baubeau, F. Salin, J.A. Squier, C.P.J. Barty, C. Spielmann, IEEE J. Selected Topics Quantum Electronics 4 (2) (1998) 407. [19] J.F. Schmerge, D.A. Reis, M. Hernandez, D.D. Meyerhofer, R.H. Miller, A.D.T. Palmer, J.N. Weaver, H. Winick, D. Yeremian, Nucl. Instr. and Meth. A 407 (1988).
Principles of high-contrast energy modulation and microbunching of electron beams R. Tatchyn* Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, CA 94309, USA
Abstract Recent advancements in ultra-short-pulse terawatt IR/visible/UV laser technology have made it possible to consider particle-beam energy modulation schemes in which the practical "eld amplitudes of the laser and undulator "elds in an ultrarelativistic electron bunch's average rest frame can attain comparable magnitudes. This parameter regime, well outside the radiation "eld strengths attainable by conventional free electron lasers, makes possible the exploitation of `relativistic interferometrya, viz., the phenomenon of high-contrast interference, to isolate regions in the longitudinal space of an electron bunch on the order of one laser wavelength. In this paper we review selected requirements on the laser and undulator "elds to generate highly e$cient microbunching followed by coherent radiation at a substantially reduced wavelength. Conditions on the electron beam quality, as well as selected possibilities for new modes of operation, are summarized. 1999 Elsevier Science B.V. All rights reserved. Keywords: Microbunching; High-contrast modulation; Free electron lasers; Relativistic interferometry
1. Introduction In practically every free electron laser (FEL) heretofore designed or operated, the radiation "eld amplitude in the rest frame of the electron bunch is overwhelmingly dominated by the "eld amplitude of the insertion device. For example, in a gain-saturated 1.5 As SASE LCLS driven by a 15 GeV beam with a 1 mm mrad normalized emittance [1], an output power of 100 GW and an undulator "eld of 1.3 T in the laboratory frame are transformed, respectively, into bunch-frame ampli-
* Tel.: #1-650-926-2731; fax: #1-650-926-4100. E-mail address: [email protected] (R. Tatchyn)
tudes of &0.0400 T versus &13000 T. In prior work, the relatively recent advent of ultrashortpulse IR/visible/UV lasers in the terawatt range [2] was adopted as a premise for considering the preparation of electron bunch frames in which the practical "eld amplitudes and wavelengths of the laser and insertion device could be made equal, leading to the possibility of inducing high-contrast (HC) interference e!ects, in particular on the spontaneous radiation and particle energy [3] distributions. A basic motivation in these initial studies was } and remains } the generation of ultrashort (subfemtosecond) radiation pulses. In subsequent papers [4,5], the energy modulation of single particles in this regime was analyzed and con"rmed the initiation of conditions for rapid bunching, which, if attained, could be used to generate coherently
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enhanced radiation at substantially reduced wavelengths. More recently [6], these studies were extended to preliminary simulations of collective dynamics within a cold bunch. In the present paper the basic principles of HC systems are reviewed with the aim of: (1) clarifying the basic physics involved, and (2) deriving simple expressions for the initial evaluation or design of practical con"gurations. Selected possibilities for further developing the HC modulation technique are discussed and some recent preliminary simulations of warm-beam e!ects on the bunching process are presented.
2. Physical principles: single-electron dynamics A layout of the basic constituents of a HC modulation (HCM) and bunching system is schematized in Fig. 1. A high-intensity radiation (e.g., laser) pulse (or synchronized group of superimposed pulses) of wavelength j and normalized vector potential K intercepts a high-quality, high-density electron beam at angle(s) h along the axis of a static or dynamic "eld synthesizer (FS) [7]. The FWHM bunch length is &(2pp and the FS parameters are j and K . We adopt, as a practical working de"nition of HCM, the relation O(K ) O(K ). The polarizations of the "eld sources can be arbitrary, but each one will be assumed linear for the present discourse. An essential attribute of the FS is its ability to generate arbitrary near-axis "eld distributions, and in particular accurately-scaled replications of the superimposed laser-"eld components. Inside the FS, the laser imparts an energy modulation pro"le to a region of the bunch. This modulation induces a (ballistic) bunching drift inside the beam which can be further modi"ed with suitably designed dispersion regions, which can also comprise sections of a FS. The conditions for maximum-contrast interference of the radiation and FS "elds in the rest frame of the electron bunch are
straightforwardly derived to be [8] j "j (1!bH cos h) and
(1)
B B /(1!bH cos h) (2) where bH"(1!cH\, cH"c/(1#K/2, and c is an (incoming) electron's Lorentz factor. For h;1, Eqs. (1) and (2) reduce to j (j /2) ((cH)\#h) and B 2B /((cH)\#h). Upon compression of the modulated region down to a (microbunched) length D , the beam, or a suit ably "ltered portion thereof, is sent through a radiator of period j 54cHD , from which 0
a coherently enhanced radiation pulse of wavelength j ;j is emitted. The maximum number 0 of radiator periods usable for this purpose will in principle be limited by the time it takes for the bunching to relax, di!use, or scatter back to the condition D '(j /2cH). It should be evident
0 from these descriptions that the basic elements and selected processes of the HC system have their counterparts in the optical klystron (OK) [9}11] and free electron laser (FEL) [12]. Indeed, the basic single-electron dynamics underlying the energy modulation can be described by the same equations. However, as will be outlined below, signi"cant di!erences in the detailed physics can arise from the highly disparate parameter regime of HC modulation, as well as from the basically di!erent phenomena involved in the collective (bunching) dynamics. Our single-particle analysis will apply explicitly to linearly polarized "elds, following the approach of Colson [13]. The equations of motion for the transverse (y) and longitudinal (z) trajectory components can be written as d(mccb ) W "qE #cqb B W X V dt
(3)
and d(mccb ) X "!cqb B W V dt
Although in the present article we focus primarily on ultrashort laser pulses (down to the order of a few wavelengths long) and static magnetic-"eld FSs, the general HC modulation and bunching system need not be restricted to these conditions.
463
(4)
along with the energy-transfer equation q c" b E . cm W W
(5)
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Fig. 1. High contrast modulation (HCM) and dispersion-assisted bunching (A) followed by coherence-enhanced radiation (B). For ballistic bunching, K "0. The FS can, in general, generate arbitrary static or dynamic "eld distributions with components along all " three axes.
Taking E "E sin (2p(ct!z)/j # ) and B " W V B sin (2pz/j # )!(E /c)sin (2p(ct!z)/j # ), with h"0 rad, and the bracketed exponents denoting the number of integral sinusoidal cycles, the following expressions for the transverse c-normalized velocity and longitudinal acceleration are derived:
K 2p b "! cos
(ct!z)#
W c j 2p K ! cos
z#
c j and 2pcK K bQ " X j c
(6)
(ct!z) z ! # !
j j 1 4p # sin
(ct!z)#2
2 j 4pz !sin
#2
(7) j where the indexed 's determine the phases of the laser and undulator "elds with respect to the electron in question. Although this does not allow for sin 2p
These "eld distributions are, strictly speaking, unphysical. The approximation is, notwithstanding, reasonable in that: (1) it allows the systematic study of HC modulation by a progressive increase in the number of `singlea cycles, and (2) it can in fact provide insight into the evolution of an actual HC bunching distribution, which, due to the highly non-linear dynamics, can be made to develop a single dominant bunched peak. In the general case more realistic "eld distributions can of course be employed using numerical simulations.
a fully general source decomposition, it su$ces to explore the basic physics of HCM. We note the explicit dependence of both motion components on z. In general, the functional dependence of z on t is complicated, and can ordinarily be expressed as a Fourier series (or, in our limit, a Fourier integral) in t. The signi"cance of higher-order spectral terms is dependent on the amplitude of the coe$cient of Eq. (7), which is seen to be directly proportional to K K and inversely proportional to j c. Although the investigation of Eqs. (6) and (7) for smallthrough-large values of both these factors is of legitimate concern for HCM, we restrict ourselves in this paper to the ranges j c<1 m and 0.005(K &K (0.25, the lower bound of the latter constraint stemming from present-day technological limitations on implementing long-period insertion devices with overly weak "elds. In this regime, z bHct, the z-velocity in the bunch frame, v , is non-relativistic, and the visualization of the X dynamics contained in Eqs. (6) and (7) is correspondingly simpli"ed. Given the above approximations and assumptions, a number of basic results follow. For an initial transverse velocity of zero, the "elds impart no net transverse kick to the electron, irrespective of their phases, so long as their 1st "eld integrals are zero (Eq. (6)). The initial transverse position, however, proportional to the di!erence of the 2nd "eld integrals, need not remain constant [4]. For the condition on j versus j expressed in Eq. (1), the upper expression in square brackets in Eq. (7) approaches zero, and the time-dependent parts of the arguments in the lower bracketed expressions approach equality. Thus, the magnitude and sign of the longitudinal acceleration imparted to a bunch
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electron are determined primarily by its phase relative to the laser and FS "elds. In this regard we note that the direction of electron motion in z is always directed toward the nearest minimum of bQ , indicatX ing the onset of dynamic bunching. We can now derive a number of useful expressions for assessing HC modulation systems in this parameter regime. Using the Einstein velocityaddition formula [8] and the above equations, the maximum normalized energy and velocity kicks (i.e., for " ! ""p/2) imparted to a bunch elec tron in the lab frame are given, respectively, by: (1) *c 2pcK K , and (2) *b 2pK K /c. The corre X sponding absolute energy and velocity kicks in the bunch frame are: (1) *E 2pm c(K K ) J, and (2) *v 2pcK K m/s. We can now compare seX lected features of HCM versus OK (or FEL) systems running at similar wavelengths. In the latter, K &O(10\}10\) at saturation, and 4}5 orders of magnitude smaller than that at the FEL entrance. Thus, even for a high-K FEL the net longi tudinal kinetic energy extracted from a bunch electron will be considerably smaller than in a typical HCM system with, e.g., K &K &0.1. Further more, the net energy exchange in an OK or FEL accrues over many periods N } typically of the order of 10, whereas the net gain (or loss) in HCM is attained over a substantially smaller number of periods (e.g., for the case being considered, N "1). Thus, the average relative energy transfer per period in a typical FEL is of the order of 1/(N ), or &10\, justifying its characterization as `adiabatica, whereas the HCM loss (or gain) per period, typically O(0.1!0.01), could be more aptly termed as `high-actiona, or `rapida. Based on these observations, a relevant criterion for distinguishing a HCM system versus a FEL or OK modulating at a similar wavelength can be expressed as (K K ) <(K K ) . &!+ $#* -) 3. Physical principles: collective dynamics As opposed to a FEL or OK, in which the radiation scattered o! the electrons and the interparticle Coulomb forces both act, in general, as essential factors in the energy modulation and bunching processes, the scattered radiation in an
465
HCM system plays an initially minor, if not negligible, role. Upon modulation by the (exogenous) radiation "eld, the bunching in the electrons' rest frame is mediated, in the non-relativistic limit by an exchange between the electrons' acquired kinetic energy and the electrostatic potential energy associated with the gradually evolving particle density. Just as in an FEL, however, the dynamics is still governed by the self-consistent interaction among all the particles moving in the combined undulator, Coulomb, and radiation "elds. The combined collective modulation and bunching in the general HCM system can be described by an unrestricted Vlasov equation [14]. If the overall process can be approximated as a sequence of two independent sub-processes, viz., modulation followed by ballistic bunching, the (dissipation-free) Vlasov equation governing the dynamics of the bunching phase can be written as
*f *f e *f * 1 #* ) " ) f (r,*,t) dr d* *t *r 4pe ** *r "r!r "
(8)
where the term on the right describes the interparticle Coulomb interactions, and f, the particle distribution, is assumed to carry the "elds' velocity modulation imprint, *(r(0)), as an initial condition. We point out, in passing, that for arbitrary HCM "elds and distributions f, Eq. (8) cannot in general be linearized and must be solved either numerically or approached with alternative analytical techniques. Prior to performing numerical simulations based on Eq. (8), it is instructive to "rst assess the bunching process for an idealized case. In our treatment, we reduce the dimensionality of the problem from 3-D to 1-D by "rst partitioning the bunch longitudinally into in"nitesimal slices of thickness d . In X view of the "nite bunch radius, r , we assume the In this limit we can disregard interactions stemming from the particles' magnetic "elds. This approximation (equivalent to taking K &K ;1), al though admittedly inexact, is justi"able for the systems analyzed in this paper. Since *v ;c, and the modulation employs single-cycle "elds, the assumption is that the particles have moved over only a small fraction of j ("j /cH) by the time the two "elds have passed through.
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force between any two such slices to vary inversely with (1#(*z /r )), where *z is the inter-slice dis tance in the bunch frame. We next estimate the potential energy of a collapsed sheet containing all the charge from a linearly modulated region of extent j , by integrating work over pairs of charge sheets. We thus "nd that the peak velocity required (assuming linear modulation) to compress the given region into an in"nitesimal sheet (at which point the total kinetic energy will vanish) is approximable by *v
X!0
qr o j j tan\ m/s m ce 2cr
(9)
where q and m are the electronic charge and mass, o is the lab-frame charge density, and e is the permittivity of free space, all in MKS units. We note that for any set (o ,j ,r ,c) there are evidently three relevant modulation regimes, viz.: I. *v (*v (undermodulation); II. *v '*v X X!0 X X!0 (overmodulation); III. *v *v (critical moduX X!0 lation); and that *v varies inversely with X!0 a power of c higher than unity. Using the 1-D model approximation to Eq. (8), several numerical simulations of a typical HCM systems's bunching dynamics using di!erent energy modulation distributions of Gaussian beams [6] were performed. Recent extensions of the model code to include energy spread within the bunch were also calculated. Here we present results for a linear modulation pro"le, *v " X (2cH"*v "/j )z , with z "0 the position midway X+ 6 between the approaching "elds. In the top line of Fig. 2 we plot the evolution of bunching for regimes I. (a), II. (b), and III. (c) in a cold 10 nC beam using velocity modulation amplitudes, "*v ", of 0.1c, X+ 6 0.15c, and 0.12c, respectively. As expected, undermodulation leads to the longest bunching time and results in underbunching, while overmodulation induces the quickest bunching with only a transitory attainment of the bunching peak. In the second line of Fig. 2 we present histograms of the evolving macroparticle distribution in regime III. (corresponding to Fig. 2(c)) for bunch frame times of 0 ps (d), 3.5 ps (e), and 5.25 ps (f ). The last plot (f ) shows only a coarse approximation to the actual theoretical maximum of 1000 since: (1) no search for the
precise time of the maximum peak was made, and (2) the electric "eld #uctuations associated with the model distribution's (relatively large) density #uctuations would in any case tend to inhibit the attainment of a perfectly compressed peak. In Fig. 2(g) the e!ect of an 0.1% 1-p energy spread on the bunching process of (d}f ) is displayed. In Figs. 2(h) and (i) the evolution of the electric "eld distributions in the bunch frame for the regime III case with 0.0% versus 0.1% energy spreads are shown. The graphs reveal the basic features of the bunching process, supporting the results derived analytically in the above-cited work. As our numerical simulation code has not yet been extended to incorporate the modulation interaction or the action of additional dispersion "elds into the bunching dynamics, analytical estimates for the minimum attainable bunch length D in a dispersion section of length
¸ in terms of the electron beam's emittance para" meters (assuming a net linear modulation) have been worked out [5]. The basic results can be summarized by the ratio D 2c
+ (2pp#(p(p #p ))cH (10) C 4 & " j *c , where *c is the total modulation increment, p is , C the standard deviation of the normalized random energy spread, p and p are the standard devi4 & ations of the beam's transverse emittance angles, and cH"c/(1#(K /2), where K is the nor" " " malized vector potential of the dispersion region. Eq. (10) can be used (with K "0) as an approxim" ate check on the peak bunching reduction displayed in Fig. 2(g).
4. Radiation performance Estimates of the radiative output characteristics of the microbunched beam in a radiator "eld ful"lling the condition j 54cHD are readily derived. 0
Here we restrict ourselves to assessments of the total emitted energy and output power in the lab frame. The maximum possible spontaneous emission power can be expressed in MKS units as P N [cq1v 2 /(6pe c)], where N is the num1. W R ber of bunch electrons and the quantity in brackets
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467
Fig. 2. Bunching and electric "eld distributions in the rest frame of a cylindrical electron beam with longitudinally Gaussian and transversely uniform density pro"les. Beam parameters: c"300; (2pp "0.5 mm; r "100 lm; N "10; number of macro particles 10 (contained in a region of length 10j /cH at the center of the bunch); electrons /macroparticle 8.9;10. To enhance the resolution only the central 2000 macroparticles containing the modulated region are displayed in a}c; in d}i, the central 3700. For graphs a}f, and h, p "0.0; for g and i, p "0.001. C C
is an approximation to the relativistic Larmor formula [15]. The maximum coherent emission power is given by a similar expression, but with the factor N replaced by (N j /(2pp ). The ratio g. of &!+ the total number of coherent photons, N , emit!-& ted by the microbunch to the spontaneous number N emitted by the entire bunch is consequently 1. g. ,(N /N ) N (j /(2pp ). For broad &!+ !-& 1. ranges of readily implemented linac beam parameters, it is readily veri"ed that g. <1. In practi&!+ cal terms, the number of spontaneous photons emitted by the N bunch electrons passing through an N -period radiator can be expressed as 0 N 7.61;10\N KN , and an upper limit on 1. 0 0
the emitted coherent power by P [W] 1.585;10\cg. N (B [T]) . (11) !-& &!+ 0 A set of parameters for a representative HCM buncher/radiator case is listed in Table 1 (cf. Fig. 1 and Eq. (10)). We observe, in Eq. (11), the scaling of P with (B ). This indicates that in cases where !-& 0 K is initially small the coherent in-band power 0 from a single microbunch could be substantially increased } perhaps well toward the multi-gW level } by substantially increasing the radiator "eld. Fields up to the order of 5 T (versus the &1 T assumed in the table) could be readily generated using a pulsed FS.
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Table 1 System and radiation output parameters of a HCM ultra-short pulse buncher/radiator. Longitudinal beam pro"le Gaussian; transverse pro"le #at Ultra-short pulse generator parameters (c"300; r "100 lm; o "10.6 C/m) Bunch charge (2pp j j p p 4 p & K " ¸ " *c/c D
K 0 Compression ratio (* /j )
480 As coherent photons/pulse Peak coherent output power
1 nC 3000 lm 4800 As 8 cm 0.1% 0.1 mr 0.1 mr 1 0.355 m 0.05 240 As &1 6% 7.0;10 0.18 GW
5. Selected aspects of high contrast modulation and bunching Perhaps the most critical aspect of prior and current studies of HCM has been the emphasis on ultra-short pulse generation. This has facilitated the tractability of our exploratory analyses of both modulation and bunching using single-cycle "elds. In actuality, of course, an ultra-short pulse may span more than one cycle and will always have non-instantaneously decaying edge derivatives. If the FS replicates this distribution then it is clear that the modulated region will also extend over more than twice one period. We can note, however, that the non-linearity of the bunching and radiation process will tend to give a strong preference to the global maxima of the velocity interference pattern. Consequently, by using, e.g., Fourier transform techniques to, say, sharply peak the center of a laser pulse containing a small number of cycles, it should be possible to generate a micro-bunched pulse train in which the radiation from a single microbunch is overridingly dominant. In the general case, many other interesting possibilities arise. For example, for su$ciently small K and K the modulation pattern will approximate to the convo-
lution of the laser and FS pulse trains. Thus, two "elds with #at-top envelopes will generate a pattern with a pyramid-shaped envelope. This hints at the wide range of possibilities } and challenges } for imposing arbitrary bunch structures on arbitrarily-distributed beams, e.g., in the event that both K and K are appreciable, it should be evident that for N ,N <1 the dynamics of the system will no $1 longer be separable into independent modulation and bunching phases, the general form of the Vlasov equation will need to be employed in computer simulations, and the modulation/bunching patterns will no longer be approximable, in general, by a straightforward convolution of the system "elds. A second critical point concerns our assumption of h"0 rad for the analyses and calculations described in this paper. By varying this parameter (in conjunction with the k-vector(s) and polarizations of the radiation and FS "elds), substantial reductions in the scaling of the required laser "eld strength and period in relation to those of the FS could be realized, and a broad range of novel variations in the energy modulation and bunching distributions would become accessible. An interesting possibility for extending the present HCM scheme would be to post-accelerate the beam following an initial modulation phase. Eq. (9) reveals that the potential barrier to bunching is lowered inversely with a power of c greater than 2. Thus, tailoring the initial modulation and postaccelerating "eld pro"les could lead to the preparation of optimally microbunched beams at substantially higher energies. Another interesting extension could involve the restriction of FS and laser "elds to the combinations of one or more higher multipole distributions for tailoring the energy modulation in directions transverse to the beam axis. An extension of this technique would be to prepare non-zero "eld integrals of selected multipole components (including the dipole). This would allow the design of transverse velocity modulation pro"les, which could be used for phase-space "ltering of the modulated/bunched region(s) independently of the overall bunch. A potentially important application of HCM would be to attempt the longitudinal compression of a whole electron bunch at a relatively low value
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Table 2 Fundamental di!erences between `single-cyclea HCM/bunching and multiple-period FEL (or OK) systems FEL/OK
HCM
Modulation and bunching adiabatic Relative energy change/period &O(10\) Electrons interact over `co-operation lengtha &N j /10 Self-consistent interactions through radiation# Coulomb "elds essential to modulation/bunching
Modulation and bunching rapid Relative energy change/period &O(10\) Electrons interact over O(1) period Electron-scattered radiation plays an initially minor role. Self-consistent interactions through Coulomb "elds dominate Dynamics strongly non-linear Modulation/bunching patterns less restricted
Dynamics can be linearized Modulation/bunching process periodic (quasi-periodic (chirped) in a tapered FEL) Modulated bunch typically
of gamma, followed by acceleration. In this case j '(8pp , implying the use of high-intensity mm-wavelength "elds. Due to the small (and `softa) radiation losses in such a system, the technique, if successfully developed, could perhaps help to circumvent potential limits set by coherent synchrotron radiation (CSR) e!ects in high-energy chromatic compressors [16,17]. Finally, throughout this paper we have emphasized the relationship of HCM to OK and FEL phenomena. We conclude this section with a summarization of some of the basic distinctions between these two physical regimes in Table 2.
6. Summary We have summarized recent studies of HCM systems and pointed out a number of potential applications for particle-beam and radiation source physics. It is perhaps not super#uous to note that HCM, as analyzed in this article, could be considered practicable for beam energies up to the 2}3 GeV range, provided FS structures of up to 2}4 m in length and with "elds controllable down to the order of a few gauss could be implemented. These requirements could of course be substantially eased with the advent of reliable UV lasers
A basic limit on the modulating FS length will be set by the Rayleigh range and waist size of the laser pulse (assuming no additional optics in the interaction region).
Modulated bunch region typically &O(j )
approaching the petawatt range, or, alternatively, by operating at h'1/cH with lower-power sources set in axisymmetric con"gurations. In the shorter term, it appears that the still-limited availability of femtosecond, terawatt lasers, is in part compensated by the fact that such devices are likely to be employed anyway at FEL facilities driven by photocathode RF guns, and there is a reason to expect that these may start becoming more prevalent. Moreover, collateral improvements in laser performance and parameter values [18], coupled with reductions in cost, can be expected to continue. Apart from this, the other basic constituents of an HCM and bunching/radiating system, viz., a high-quality, high current electron beam and a FS insertion device(s) are either in the process of being developed at a number of laboratories [7,19], or can be approximated to with existing resources. For example, even with the substantially poorer compression expected for a storage ring beam, the impressive output predicted by Eq. (11) } coupled with, say, a high-K radiator for harmonic genera0 tion } strongly suggests that HCM could be productively applied to recirculating beams as well. Given the apparent richness of HCM physics, we
For example, for a 2 GeV beam with 3;10 particles/bunch and a bunch length of 3 cm, coherent output powers could approach the TW range. We note, however, that the higher relative energy spread of the circulating beam, which would inhibit the attainment of large wavelength reductions (as predicted by Eq. (10)), would make the attainment of even higher in-band powers substantially more di$cult.
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believe that undertaking such e!orts is likely to further the understanding, control, and metrology of particle beams, and contribute to the development of FEL, OK, and other 4th generation radiation sources.
Acknowledgements This work was supported in part by the Department of Energy O$ces of Basic Energy Sciences and High Energy and Nuclear Physics, and Department of Energy Contract DE-AC0376SF00515. Selected calculations utilized resources of the National Energy Research Scienti"c Computing Center, which is supported by the O$ce of Energy Research of the U.S. Department of Energy.
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[6] R. Tatchyn, High-contrast energy modulation of electron beams for improved microbunching and coherence of free-electron radiation sources, presented at the SPIE 1998 Annual Meeting, San Diego Ca, July 19}24, 1998; SPIE Proc. (1998) 3451, in press. [7] R. Tatchyn, Fourth generation insertion devices: new conceptual directions, applications, and technologies, in: M. Cornacchia, H. Winick (Eds.), Proceedings of the Workshop on Fourth Generation Light Sources, SSRL Report 92/02, p. 417. [8] J. Aharoni, The Special Theory of Relativity, 2nd ed., Clarendon Press, Oxford, 1965. [9] P.L. Csonka, Part. Accel. 11 (1980) 45. [10] N.A. Vinokurov, A.N. Skrinsky, Institute of Nuclear Physics, Novosibirsk, USSR, Preprint 77/67. [11] R. Bonifacio, R. Corsini, P. Pierini, Phys. Rev. A 45 (6) (1992) 4091. [12] J.B. Murphy, C. Pellegrini, Introduction to the physics of the free electron laser, in: M. Month, S. Turner (Eds.), Lecture Notes in Physics vol. 296, Springer, Berlin, 1988, pp. 163}219. [13] W.B. Colson, Free electron laser theory, Ph. D. Dissertation, Department of Physics, Stanford University, 1977. [14] T.-Y. Wu, Kinetic Equations of Gases and Plasmas, Addison-Wesley, Palo Alto, 1966. [15] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975 (Chapter 14). [16] B.E. Carlsten, J.C. Goldstein, Nucl. Instr. and Meth. A 393 (1977) 490. [17] The LCLS Design Study Group, Linac Coherent Light source (LCLS) Design Study Report, Section 7, SLAC-R521, 1998. [18] C. Le Blanc, E. Baubeau, F. Salin, J.A. Squier, C.P.J. Barty, C. Spielmann, IEEE J. Selected Topics Quantum Electronics 4 (2) (1998) 407. [19] J.F. Schmerge, D.A. Reis, M. Hernandez, D.D. Meyerhofer, R.H. Miller, A.D.T. Palmer, J.N. Weaver, H. Winick, D. Yeremian, Nucl. Instr. and Meth. A 407 (1988).