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Nuclear Instruments and Methods in Physics Research A 375 (1996) 314-316NUCLEAR

INSTRUMENTS

& M E T H O DS

IN PHYSICS

RESEARCHSectIon

Effects of undulator interruptions on the performance of high-gainFEL amplifiers *

K.-J. Kim, M. Xie, C. PellegrinibLawrence Berkeley National Laboratory, Berkeley. CA 94720, USA

hUniversity of California, Los Angeles, Los Angeles, CA 90024. USA

AbstractThe construction as well as the operation of a long undulator required for short wavelength generation with high-gain

free-electron lasers will become easier if the undulator could be interrupted with drift sections every few gain lengths. Toevaluate the effect of such interruption on the FEL performance, we study the following three effects; (i) the diffraction loss,(ii) the free space slippage and (iii) the phase de-coherence. The effect (i) is found to be small, the effect (ii) gives acondition on the length of the drift section and the effect (iii) is small, but not negligible.

1. Introduction

The high-gain amplifiers for short wavelength free-electron lasers (FELs) [I] require a long undulator. Theconstruction of such an undulator as well as the FELoperation would become easier if the undulator could beinterrupted with drift sections every few gain lengths. Wehave investigated the influence of such interruption on theFEL performance. These effects are considered: (i) thediffraction loss, (ii) the free space slippage and, (iii) thephase de-coherence due to velocity spread and to disper-sion errors. The effect (i) is the loss during the process inwhich the optical mode in a section of the undulator leavesthe undulator, propagates through the free space and thenre-enters and re-adjusts itself in the next section. The effect(ii) is the fact that the phase of the optical beam is delayedwith respect to the electrons density modulation foroptical FEL interaction due to the slippage of the electronbeam in the interruption region. The effect (iii) is the factthat the electrons velocity spread, emittance, and disper-

sion due to misalignment of the quadrupoles used foradditional focusing lead to a reduction of the bunchingfactor. We present an approximate analysis of these effects.When applied to the LCLS parameter [2], we find that theeffect (i) is negligible, the effect (ii) gives a condition onthe length of the drift section, and the effects (iii) aresmall, but could be non-negligible if there is a sufficient

* This work was supported by the Director, Office of HighEnergy and Nuclear Physics, U.S. Department of Energy, underContract No. DE-AC03-76SF00098 and DE-FG-3 = 92ER406923.

* Corresponding author. Tel. + 1 5 10 486 7224, fax + I 5 10 4866485, e-mail kwangje@lbl.gov.

number of interruptions and if the quadrupole misalign-ment exceeds a certain tolerance.

2. The diffraction loss

Referring to Fig. 1, we consider an undulator terminatedat 1, followed by a free space of length d, and anotherundulator starting at 2. Consider the dominant guidedmode at 1. The transverse profile of the mode may beapproximated by the fundamental Gaussian mode

I2

- exp ( >ikr91 %I (1)Here

1 1 i

91 R, P,

is the complex beam parameter with R, = the radius of the

curvature of the wavefront and p, the beam envelopeparameter such that (T, =dm is the rms beam size atlocation 1 ((TV W, /2 in the normal laser jargon). Theminimum value of j3 is usually known as the Rayleigh

length Z,.

In the drift section between 1 and 2, the free space

_lLLuLLl l L L L _ L u L_ _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ ~ - - -I 21 I I I I I Irl-i-llllll r

Fig. 1. Interrupted wiggler.

01689002/96/$15.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reservedSSDI 0168-9002(95)01209-5

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K.-J. Kim et al. f Nucl. Instr. and Meth. in Phvs. Res. A 375 (1996) 314-316 315

diffraction is described by the transformation q, + q2 =q, + d. Thus the transverse profile at 2 becomes

1- expq2

(2)

At this point, the optical beam needs to be matched back

into the guided mode with the beam parameter q,.A rigorous calculation of FEL mode involves a self-

consistent treatment of both optical mode and modulationof electron beam, and is rather involved [3]. Here we willestimate the magnitude of the diffraction effect in asimpler way by assuming the interruption due to the driftsection is small. Thus we expand the beam profile at 2 asfollows:

Here the terms in are the higher order Gaussian modes

with beam parameter q,, which are orthogonal to thefundamental mode. The coefficient C is determined fromthe orthogonality of the Gauss-Laguerre modes. i.e.. bymultiplying exp(ikr*/2qT)/qy and integrating over rdr;

&f-41 e-q,47 - 42 YT-4, -d

(4)

Thus, the quantity ICI measuring the diffraction loss isgiven by

IclZ= l1 + (d/2 Im q,)2 .

(5)

The wavefront of the guided mode at the undulator endI may be represented by an equivalent free space modewith waist at a distance Z,, inside of the undulator, shownas the dotted curves in Fig. 2. Thus, the wavefront of theguided mode may be parametrized by Z,, and the Rayleighlength Z, of the equivalent free space mode as follows:

~,=Z,[l+($j], R,=Zz = 0.999

I+ 22,

for d = 25 cm. Indeed, the diffraction loss is very small.

3. The free space slippage

In the free space, the optical phase seen by an electron is

$ = k: - wt + tan-(z/Z,) (7)

The last term. in which z is measured from the waist

location, is due to the fact that the optical field is close to aGaussian mode rather than a plane wave. The phasedifference, or phase slippage over a distance d is

A#==&++&. (8)

where A4P = (I - 1 P)kd is the slippage with respect tothe carrier plane wave, and Aq%,, tan-((Z,, + d)/Z,) -tan- (Z,,/Z,) is due to diffraction, p = speed of electron/c. The first term is the dominant one, and it needs to beclose to an integer times 27~. Thus

LL2n77.2y

Using the FEL resonance equation, we obtain the condition

d=nh,(l +a;,. (10)

With the LCLS parameters given before, we find d =25 cm for n = I. A distance of 25 cm is a reasonable lengthfor the drift section.

The second term of Eq. (8) becomes

A4,, = tan-d

1 + z,,(Z,, + d)lZZ, 1 o.12,which is indeed much smaller than the first one.

It is interesting to note that the length of the drift sectiond is independent of the wavelength if the second term, thephase slippage due to diffraction. is negligible.

4. The loss of the phase coherence

The electrons velocity spread leads to phase de-coher-ence. Let us consider the bunching factor (e@) at the exitof the first undulator 1. After passing through the driftsection, the bunching factor will change as follows:

V. SHORT WAVELENGTH FELs

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316 K.-J. Kim et al. I Nucl. Imtr. and Metlt. in Phxs. Rrs. A -17-F 1996) .?I&.116

(e)~(e+HI),

If we assume that the statistics of 6, and A0 are in-dependent, we obtain

(e,,#+Ui,) = (e)(eH),

Here

(II)

Averaging over the electron distribution, we obtain thereduction in the intensity due to the phase de-coherence.

(12)

Here g,, is the rms angular divergence. in the I-direction,which is assumed to be the same as that in the v-direction,and g6 is the rms relative energy spread.

For LCLS parameters given before and with d = 25 cm,we obtain

In calculating the second number. we have used a, -p -IO-, the value of the energy modulation expected wellinto the exponential gain regime. Thus. the phase de-coherence due to the energy spread is negligible. On theother hand the phase de-coherence due to the angularspread is not entirely negligible with a reduction factor

1~ = 0.87.( 1.07)?

The intensity reduction per interruption is about 13%.In some cases we would like to use quadrupoles

between the undulator modules to provide additionalfocusing for the beam. A quadrupole misalignment errorwould then produce a dispersion which would producedebunching. This effect can be taken into account bymodifying Eq. ( 1 ) as

where D and p are the average dispersion and the bendingradius due to the misalignment errors.

For this effect to be small we need to have D/p < 1 y,To estimate the effect we assume to have a quadrupoledoublet with a gradient G. and a misalignment error St. Wetreat the doublet as a FODO array with bending dipolesand assume that it produces a small phase advance per cell(this condition is a necessary to avoid a gain reduction dueto betron oscillations). To first approximation the conditionD/p < I/y can then be written as SC< IO /G(T/m).which indicates that for G - IO T/m we have a toleranceS[< IO m.

Acknowledgements

We wish to thank Bill Fawley for discussions on thissubject.

References

[I] For a review see. K.-J. Kim, Nucl. Instr. and Meth. A 3%(1995) 31.

[2] H. Winick et al., Nucl. Instr. and Meth. A 347 (1994) 199.[3] M. Xie. to be published.