Application of Transverse Gradient Wigglers in High Efficiency Storage Ring Fel's

APPLICATION OF TRANSVERSE GRADIENT WIGGLERS IN HIGH EFFICIENCY STORAGE RING FEL’s J. Madey

To cite this version:

J. Madey. APPLICATION OF TRANSVERSE GRADIENT WIGGLERS IN HIGH EFFICIENCY STORAGE RING FEL’s. Journal de Physique Colloques, 1983, 44 (C1), pp.C1-169-C1-178. 10.1051/jphyscol:1983116. jpa-00222545

HAL Id: jpa-00222545 https://hal.archives-ouvertes.fr/jpa-00222545 Submitted on 1 Jan 1983

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APPLICATION OF TRANSVERSE GRADIENT WIGGLERS IN HIGH EFFICIENCY STORAGE RING FEL'S

J.M.J. Madey

High Energy Physics Laboratory, Stanford lhziversity, Stanford, Cazifornia 94305, U.S. A.

RksumC - La puissance de sortie, lrefficacitC et le gain d'un laser B Clectrons libres sur anneau de stockage peuvent dtre nettement amCliorCs avec l'utilisation d'un onduleur prCsentant un gradient transversal, "gain-expanded wiggler". Les paramktres critiques pour la rCalisation d'un tel projet sont la dispersion en Cnergie, le courant crdte et l16mittance. Une longue r6gion d'interaction (10 metres) doit aussi Ctre pr6vue.

Abstract - A significant improvement in the power output, efficiency, and gain of storage ring FEL's can be realized through the use of a gain-expanded wiggler. The design of such systems must emphasize energy acceptance, peak current, and emittance. Long (10 m) interaction lengths must also be provided.

I. - INTXODUCTION

Storage rings are an attractive means to generate the electron beam for free electron lasers for a variety of reasons. These include: 1. favorable electron energy and current 2. favorable electron current density 3. favorable electron bunch length 4. low ionizing radiation. Of these factors, the current density is probably the most important. The advantages of the SRFEL configuration can most readily be seen by comparing the current density requirements of the FEL with the current density available using the various accelerator technologies. Assuming the use of the transverse wiggler with an acceptance just equal to the emittance of the electron beam, Smith and ~ade~lhave shown that under optimum conditions the gain per pass is given by:

where X = optical wavelength (cm) magnet period (cm) la -

EX,Ey horizontal, vertical emittance (cm-radians) ipeak = instantaneous peak current (amperes).

Inverting this relation, we can solve for the current density rewired to reach a given gain at a given wavelength with given wiggler parameters K' and X . 9

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983116 CI-170 JOURNAL DE PHYSIQUE

The current density requirements in the wavelength range 100A) < A < 10~are plotted in Figure 1 which shows the current density required to reach 100% gain per pass for the case K2 = 1 for the wigglers with A = 1 cm and 10 cm, respectively. As is apparent in the figure, the current densiey required to operate the FEL increases as A-1/2, and increases sharply as the magnet period is reduced. The value of = 1 was chosen as representative of the magnetic field strength at which the gain is maximized. Figure 1 also plots the current density available from storage rings and other types of accelerators. The current density available from accelerators using thermionic, field emission, and plasma cathodes is defined by the cathode characteristics, space chan e and wake field effects, and aberrations in the electron optics. Lawson and Penner 5 have observed that the characteristics of most existing accelerators using such cathodes can be fit by the r,elation:

< i > = y2 zx E x lo3 amperes (21 Y where < i > is the time averaged current and E and z the area of the phase X Y' space ellipses in the transverse coordinates (x,xt) and (y,yl). (Since the Lawson- Penner relation defines only the average current < i > , the peak current density in RF accelerators will typically exceed this value due to the bunching of the beam by the RF accelerating field. Assuming the microscopic duty cycle of the electron beam is likely to lie within the limits 0.01 to 1.0, the peak current density attainable from accelerators using thermionic field emission and plasma cathodes will lie in the range: i 5 2 lo3 < xe2 < 10 amps/cm . (3) y2z E x Y These limits are indicated by the lower set of dotted lines on Figure 1. The arrows on the vertical scale of Figure 1 indicate the peak current densities secured in the SLAC linac injector in its normal mode, and using a 240 MHz subharmonic buncher. In its normal configuration, the SLAC linac operates at a microsc ic duty cycle of 2p 0.01, producing a peak current density equal to lo4 amperes/cm . Note that while the microscopic duty cycle of RF accelerators can be reduced below 0.01 through the use of sub-harmonic bunching, such measures may not result in a proportional increase in the peak current density. At high peak currents, the electron beam excites a wake-field in the accelerating structure which acts back on the electron beam to raise both the energy spread and the emittance In the case of the SLAC linac, the peak current density remained below lo5 amps/cm2 even when the duty cycle was reduced to 0.001 through the use of the subharmonic buncher. As opposed to linear accelerators, in which the electron's phase-space density is unaffected by the accelerator process, the current density in storage rings can be raised through synchrotron damping, with the final emittance determined by the balance of damping and quantum fluctuations. The upper set of dotted lines indicates the range of current densities expected from the new synchrotron radiation sources at Wisconsin and Brookhaven, and in the SLAC damping ring. (A factor of lok1 is assumed around the design objectives for these machines. Since damping and quantum fluctuations are functions of the specific design of a storage ring, the current density will also depend on the design. Thus for storage rings there can be no equivalent to the universal Lawson-Penner relation for linear accelerators. The machines cited in Figure 1 were each optimized for high current density, and are believed to be representative of what can be accomplished in the energy range around 1 GeV. Note that the design value of the peak current density in these rings exceed the current densities available from thermionic, field emission and plasma cathodes by a factor of lo3 - 10'. It is seen from Figure 1 that the current density requirements for FEL operation at long wavelength (A > 1 p) can be met using either linear accelerator or storage ring technology, but that the requirements at short wavelengths (A < 1 y) probably can not be satisfied with linear accelerator technology. As a point of reference, the circled point in Figure 1 indicates the current density required for 100%gain per pass using the existing Stanford 3.2 pm superconducting wiggler at K2 = 0.5. As can be seen in the figure, this system was designed to operate near the limits of the available current density using linear accelerator technology. Barring further improvements in the current density available from linear accelerators, the Stanford 3.2 pm FEL probably represents (within factors of 2) the practical limit for short- wavelength operation using linear accelerator technology. By comparison the current density available in the new synchrotron radiation sources should be adequate for operation throughout the visible and UV wavelength well below 1000 8, perhaps as short as 100 a. Motivated by these reasons, a number of storage-ring FEL concepts have been proposed. Most notably, these include: 1) devices operated in the small signal regime with linear gain in which the electrons' optical phase is assumed to vary stochastically on successive 2 passes through the wiggler; 2) devices in which the optical phase is preserved from pass to pass, thereby generating a family of closed stable orbits in which the electrons circulate in the optical potential wells created in the wiggler. In such an Isochronous FEL, the energy radiated by the electrons scales as the electric field rather than the intensity.3 3) Devices operated in the large signal regime in which the optical synchrotron frequency is high, and the electrons are tightly bound in the optical potential wells. In such systems, the radiated energy is determined by the variation in resonance energy along the interaction length, as determined by the variation in wiggler period and magnetic field. Such systems i clude the tapered wiggler and phase-displacement acceleration concepts. 8 Of these concepts, the one which has received the most attention to date has been that in which the small signal limit and non-isochronous electron optics have been assumed. In general, this choice appars to offer the best available small signal gain and minimizes the spurious ultraviolet radiation generated at the harmonics of the operating wavelength. The power output in this configuration is expected to scale in proportion to the power emitted by electrons as incoherent synchrotron radiation. For simple constant period undulators, Renieri has shown that:3

where (aE/Eo) is the fractional energy spread induced in the circulating electron beam by the laser interaction. Similar relations are now believed to apply to all storage ring FEL's operating in the linear regime. To maximize the power output of such devices, it is evidently necessary to optimize the energy spread (aE/Eo) which can be accepted by the storage ring and undulator. But for simple constant period wigglers the only means to raise the energy acceptance is to reduce the number of periods, drastically reducing the gain. As an example, consider a SRFEL operating at X = 5000 8 with a wiggler period of A = 5 cm, and K* = 2.0. The small signal gain in such a wiggler is optimized whenqthe wiggler transverse acceptance just equals the emittance of the circulating electron beam. Assuming an emittance E = 1.10-~cm-radians, the optimum number of periods is:

Since the gain curve of a conventional wiggler has a width equal, approximately to 1/2 N, the largest energy spread which could be accomodated in this system would be C1-172 JOURNAL DE PHYSIQUE

(a /E ) % 1.5 x mile such a system could be expected to operate at high E 0 gain, it would, of course, have rather low power output. To raise the laser power output to 10% of the synchrotron radiation in this example, it would be necessary to reduce the number of magnet periods from 432 to 5. Assuming the optical mode parameters were altere to match the reduction in length, this change would reduce the gain by (347/5) 9 % 5000. It is evidently im- possible to simultaneously optimize gain and output in a storage ring FEL using a simple constant period wiggler. Tnis conflict led, in part, to the development of the gain expanded or "transverse gradienttf wiggler (TGW) . In a gain-expanded wiggler, the amplitude of the transverse magnetic field is made a function of transverse positions in the bend plane of the wiggler. Properly adjusted, the nominal longitudinal velocity of electrons moving through such a wiggler is independent of their energy, and phase coherence with the optical field can be maintained during the interaction even for large initial momentwn spreads. While the power output of such a system is still determined by a relation of the form of equation (4), there is no longer any direct connection between the energy acceptance of the wiggler, and the wiggler length and number of periods, and the small signal gain. This property makes it possible to design storage ring FELfs with both reasonable power output and reasonable gain over the broad range in wavelengths in which storage ring FELs are capable of operating. The advantages of the TGW configuration can be illustrated by comparing the gain equations for the TGW and conventional constant period wiggler. Assuming filamentary electron bems,the maximum attainable gain for a conventional wiggler is :

while the maximum available gain in a TGW is: (5)

where is the B-function in the wiggler. In both equations 6 and 7 it is assumed that the optical mode has been chosen to minimize the volume of the mode in the interaction region. Assuming the number of periods in the constant period wiggler is chosen to provide some given energy acceptance AE/Eo

the gain of the transverse gradient wiggler will exceed the gain of the conventional wiggler when the number of periods in the gain-expanded wiggler exceeds: For a system requiring an energy acceptance AE/E = 5% and assuming a transverse gradient wiggler with 0.1 T betatron phase advance per period, the gain expanded wiggler will have superior gain if it has 9 or more periods. A 50 period gain- expanded wiggler with this betatron phase advance would have over 5 times the gain of the conventional wiggler.

11.- SELECTION OF PARAMETERS FOR GAIN-EXPANDED SRFEL's:

Storage Ring: While the basic requirements of the gain-expanded SRFEL for large energy acceptance and high damping rates are common to all SRFEL's operating in the linear regime, the constraints on emittance and the values of Band rl at the ends of the wiggler are specific to the TGW. We may anticipate that TGW's constructed for use in the SRFEL configuration will be designed to operate at a betatron phase advance (within the wiggler) in excess of 2~ radians.in both the vertical and horizontal planes. Miminization of the electron beam cross section in the wiggler will require a 8-function at the entrance and exit to the wiggler which matches the period for betatron oscillation in the wiggler. The vertical and horizontal emittance of the circulating electron beams must evidently also fall within the acceptance of the wiggler. The particular values of emittance required depend on the wiggler parameters, as described below. In constrast to the constant period wigglers,which require a zero dispersion function in the wiggler to minimize the excitation of the transverse coordinates, adient wigglers require a specific non-zero 11 in the bend plane of the wiggler:

where 8 is the value of the bend-plane &function. Considering that the required values of rl are typically of the order Of 1 cm, and the bend plane acceptance of TGW's is typically limited, Wiedeman has suggested that the magnetic field of a transverse grad'ent wiggler in a storage ring FEL he made parallel to the orbit plane of the ring.' The dispersion function in this plane is normally zero, and the required small dispersion 11 can be obtained by deflecting the electron beam out of the plane with a pair of weak dipoles. Wiggler: Assuming the operating wavelength and the storage ring energy are fixed, the wiggler design will be determined by the magnet period,the nnnber of priocls , the betatron phase advance per period, and the optical phase slip per period. Previously published analyses have established that the gain is maximized when the wiggler's net optical phase slip is set equal to the net betatron phase advance, and when the number of periods is set at the largest practical value. Equation (7) for the gain also suggests that the best gain can be obtained at small values of betatron phase per period, X /B. To secure a more comprehgnsive understanding of the effects of wiggler length, period, and betatron phase advance on operation, it is necessary to recast the gain equation, and the equation defining the acceptance, magnitude of transverse gradient, and synchrotron radiation flux in terms of these parameters. Solvinggfor K* in terms of the laser wavelengths, wiggler period, and operating energy, r X L 1 1 Gain a 10 i][<.- J12]i (11) yo CI-174 JOURNAL DE PHYSIQUE

A A Acceptance: Wiggler Bend Plane: E (cm-radians) & (=) (8)

Normal Plane: E (cm-radians) % '. X 2 $1

Synchrotron Radiation Flux:

dP P sync sync dQ (watts/steradian) &? K/~~

the gain equation assumes an optical mode of the minimum possible area and the relations for the acceptance assume use of the beta-functions individually appropriate to the bend and normal plane of the wiggler. Note that, for a high energy ring, it can be assumed that 2 lh K 1, for which case 1 1 >> - - (A)h . 2Y It is apparent from equations (11-14) that the choice of the betatron phase advance and wiggler length will involve some tradeoff among the various design ojectives. In the case of the betatron phase advance per period (A /B), the gain is maximized, and the transverse gradient dB/dx in the wiggler is minhized, for small A /B. However, this choice also reduces the bend-plane acceptance, and the emittanse attainable in the ring will set a lower bound to (A /@). Similarly, in the case of the number of periods (L/h ), the growth of the iacoherent synchrotron radiation dP /dQ with the wiggler length may set a practical limit to the length belows?% length at which the gain is optimized. Only in the case of the wiggler period A is it apparent that all factors favor a single choice, and that laser operation will generally be optimized by using the longest possible period.

Kroll and Rosenbluth 9 have, under certain simplifying assumptions, estimated that the power output of a gain-expanded storage ring FEL will be of the order of 1.5 (aE/Eo). It is further predicted that the energy distribution will be gaussian centered about Eo, and the emittance distribution will be exponential, peaked at A = 0. To check these predictions, we have run Monte-Carlo simulations of the operation of a gain-expanded FEL, including the full un-averaged equation of motion in the wiggler, and the betatron and synchrotron motion in the storage ring. The simulations assume oscillator mode operation in which the optical pulse length contracts to a fraction of the electron bunch length. The energy distribution observed in these simulations is gaussian, and the ratio of laser power output to laser induced energy spread is close to 1.5 (figure 2). The simulations indicate that the Kroll-Rosenbluth relations for the laser power and induced energy spread provide a reasonable estimate of the attainable power output of these systems. For cw operation, in which long beam lifetime is required, the allowable laser- induced energy spread can be estimated from the usual rule of thumb that the energy acceptance of the storage ring should be at least six times the standard deviation o This conditions yields the result: E .

This is similar to the 7.3:2 relation of laser power and synchrotron radiation estimated by Barbini s.,in their LEDA SRFEL design. For pulsed systems, a larger spread is allowable, and the ratio of laser power to synchrotron power could be correspondingly larger. For example, if the laser pulse length does not exceed lo2 damping times, the laser-induced energy spread could be allowed to reach one third the storage ring acceptance, resulting in a laser power output:

1 AE 'laser = 7 (3) sync (pulsed), higher by a factor of 2 than for cw operation. It is interesting that pulsed operation could lead to a higher average power output, as well as a higher peak power. Since the laser induced energy spread can be damped within a few damping times once the laser had been turned off, it is possible that the duty cycle of a pulsed system could be made to approach unity. To illustrate the possibilities for a cw system, it is useful to consider the example outlined in Table 1. We assume a 1 GeV storage ring with 1 ampere average current, and 200 amperes instantaneous peak current. Assuming a + 5% energy acceptance, and a net incoherent synchrotron power of 25 kilowatts, the cw laser power output would be 300 watts. Assuming a betatron phase advance of 0.1 period and a maximum transverse gradient of 5 kilogauss/cm, jt would be possible to use a wiggler period as short as 50 cm, corresponding to K = 7.0. Assuming an overall wiggler length of 20 meters, the gain per pass would be 4.8%.

Table 1

Nominal Energy 1.0 GeV Energy Acceptance + 5% Average Current 1.0 amperes Peak Current 200 amperes Net Synchrotron Power 25 kilowatts

Wiggler Per'iod 50 cm cx2f3' 7.0 3 Gradient 5 x 10 gaussJcm Betatron phase advance 0.1 v/period Optical phase slip 0.1 a/period Wiggler length 20 meters JOURNAL DE PHYSIQUE

Table 1 (cont sd)

Laser wavelength 5000 Small signal gain 4.8%/pass Equilibrium energy spread (uE/Eo) .83% RF-to-laser efficiency 1.25% Laser power output (cw) 300 watts

As described in the accompanying paper by ~iedemann," the s~ecificationsfor the storage ring in this example are consistent with presently available machine tecimology. Higher levels of power output could be second, in Frinciple, throuph the use of a high-field wiggler as proposed by Barbini, =.,I1 to raise the synchrotron damping rate. Higher levels of efficiency would require an increase in the energy acceptance. This work supported in part by AFOSR under contract F49620-GO-C-0068.

1V.- BIBLIOGRAPHY

111. SMITH, T. I.and WEY, J, M. J.,Appl. Phys. B27 (1932) p. 195 [2]. RENIEXI, A., IEEE Trans. Nucl. Sci. 26 (1979). 3827. [3]. DEACON, D.A.G., Phys. Rev. Lett. ~(19~0)p. 449. [4]. KROLL, N. M. , MORTON, P. L., and ROSENBLUTE, N. M., IEEE Trans. Quant . Elect. QE-17 (1981) p. 1436. [S] m, N. ?4., MORTON, P.L., ROSENBLUTH, M. N., ECKSTEIN, J. N., and UDEY, J. W. J., IEEE Trans. Quant. Elect. QE-17 (1981) p. 1496. [6]. Ibid. [7]. ~EPIANN,H., IiEPL Report No. TN-80-8 (Stanford University, June 19E0). 181. MOLL, N. M., MORTON, P. L., ROSENBLUTB, k:. N., ECKSTEIN, J. N., and MDEY, J. t4. J., op. cit. [9]. KROLL, N. M., this volume. 1101. WIEDEIWiN, H., this volume. [Ill. BARBINI, R., DATTOLI, G., LETARDI, T., IMRINO, A., RENIERI, A., and VIGNOLO, G. IEEE Trans. Nucl. Sci. 2 (1S79) p. 3836. CURRENT DENSITY REQUIRED FOR UNITY GAIN / PASS 2 22 2 2 (UNITS = ipeak/Y ExEy 1, U @ = K = aw =I

RANGE OF CURRENT DENSITIES ATTAINABLE

- RING

RANGE OF CURRENT

CONFIGURATION ACCELERATOR TECHNOLOGY

I I I I m IOpm I .Opm O.Ipm 100% OPERATING WAVELENGTH

Figure 1: Current density required to provide 100% gain per pass for free electron lasers operating between 10 and 100 a. The figure assumes a conventional constant period wiggler and a minimum volume optical mode. JOURNAL DE PHYSIQUE

DELTA

Figure 2: Electron energy distribution for a gain-expanded storage ring FEL. The points indicate the distribution function computed in the numerical simulation. The solid line indicates the envelope of a gaussian distribution with the same standard deviation as the simulation.