Nuclear Instruments and Methods in Physics Research A 483 (2002) 527–530

New mechanisms of interaction for even harmonic generationin free electron lasers

Ming Xie*

Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 71-259, Berkeley, CA 94720, USA

Abstract

A systematic search for new mechanisms of interaction is carried out by including electromagnetic field components

that were previously neglected in FEL theory. Two new mechanisms are discovered that have major effect on even

harmonic generation in FELs with planar wiggler. A classification of mechanisms of interaction is introduced and a new

formulation of FEL equations is developed in order to treat the more complicated new phenomena. r 2002 Elsevier

Science B.V. All rights reserved.

PACS: 41.60.Cr

Keywords: Even harmonics; Longitudinal coupling; Vector wave equation

1. Introduction

Even harmonic generation in an FEL withplanar wiggler was demonstrated earlier at themicrowave in a waveguide [1]. Recently, it was alsoachieved at the infrared in an open resonator [2].In the latter experiment, however, puzzling phe-nomena were observed that cannot be explainedby current FEL theory.It is well-known that even harmonics can be

generated via two mechanisms in FELs withplanar wiggler: first, when there is a misalignmentof electron and laser beams in the wiggling plane[3], and second, when there is a transverse gradientin laser field [4]. Yet, none of the two mechanismscould explain the mysterious mode profile andpolarization observed recently [2].

In this paper, two new mechanisms, capable ofpredicting unusual phenomena, are discovered byincluding electromagnetic field components thatwere previously neglected in FEL theory. The goalhere is to introduce a systematic exposition andclassification of mechanisms of interaction foreven harmonic generation and develop a newformulation of FEL equations in order to treat themore complicated new phenomena. Detailedcalculation and comparison with the experimentalresults will be carried out in a forthcomingpaper.

2. Classification of interaction mechanism

To search for new mechanism of FEL interac-tion, we start from the general equation governingenergy exchange between an electron and a vector

*Tel.: +1-510-486-5616; fax: +1-510-486-6485.

E-mail address: mingxie@lbl.gov (M. Xie).

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 3 7 4 - 1

electric field with all three components

dgdz

¼ �e

mc2dx

dzEx þ

dy

dzEy þ Ez

� �: ð1Þ

Two components, Ey and Ez; have been neglectedin previous theory for FEL interaction in a planarwiggler field polarized in the y-direction. Whiledoing so is a good approximation at the dominantfundamental frequency, it is no longer acceptableat even harmonics where the effect of interactionwith Ey and Ez can become comparable to thatwith Ex: To discover these previously neglectedeffects, Eq. (1) has to be evaluated properly alongthe unperturbed electron orbit.Taking z as independent variable, the unper-

turbed orbit in a planar wiggler field specified bythe scaled vector potential, aw ¼

ffiffiffi2

paw sinðkwzÞ #x;

can be described by a superposition of slow andfast motions

x ¼ %xðzÞ þ *xðzÞ; y ¼ %yðzÞ; t ¼ %tðzÞ þ *tðzÞ ð2Þ

where the slow part is wiggler averaged and fastpart varies significantly over the wiggler periodlw ¼ 2p=kw; and aw is the rms wiggler parameter.Specifically, we have

%x ¼ x0 þ pxz; %y ¼ y0 þ pyz; *x ¼

ffiffiffi2

paw

gkwcosðkwzÞ

or %tðzÞ ¼ ort0 þ krz 1þ1þ a2w2g2

þp2x þ p2y

2

!

or *tðzÞ ¼ �x sinð2kwzÞ � z cosðkwzÞ

x ¼kra

2w

4g2kw; z ¼

ffiffiffi2

pkrawpx

gkw

px ¼ px0; py ¼ py0; g ¼ g0

where or ¼ krc is the fundamental resonancefrequency to be defined later. Without loss ofgenerality, let us consider a monochromatic wavewith frequency o ¼ kc and specify each fieldcomponent by a complex envelope in the form E ¼Fðx; y; zÞeikz�iot: The field, when evaluated alongthe orbit using Eq. (2), can also be separated intoslow and fast parts

Fðx; y; zÞ ¼ %Fþ *xðzÞq %Fq %x

; %F ¼ %Fð %x; %y; zÞ ð3Þ

kz � ot ¼ ny� Dnkwz � nkwz � nor *tðzÞ

where y ¼ ðkr þ kwÞz � or %tðzÞ; n ¼ o=or; Dn ¼n� n; n is an integer harmonic number, and wehave assumed that amplitude of wiggling motion ismuch smaller than laser mode waist w0; i.e.ffiffiffi2

paw=gkw5w0: The fundamental resonance fre-

quency can be determined by the resonancecondition, dy=dz ¼ 0; yielding

or ¼2g2kwc

ð1þ a2w þ g2p2x þ g2p2yÞ:

Substituting Eqs. (2) and (3) into Eq. (1) andperforming wiggler average denoted by /S; weobtain

dgdz

� �¼ �

e

mc2Rfeiny�iDnkwzðf1 þ f2 þ f3 þ f4Þg ð4Þ

f1 ¼ hd *x

dz

� �%Fx

f2 ¼ h *xd *x

dz

� �q %Fx

q %x

f3 ¼ /hS %Fz þ/h *xSq %Fz

q %x

f4 ¼ /hSðpx %Fx þ py %FyÞ þ/h *xS px

q %Fx

q %xþ py

q %Fy

q %x

� �

hðzÞ ¼ e�inkwzþinx sinð2kwzÞþinz cosðkwzÞ:

Various mechanisms of FEL interaction aregrouped into four classes in Eq. (4). Whereas f1and f2 have been studied before, f3 and f4 are newmechanisms discovered here. Note that the termscontaining factor /h *xS in f3 and f4 are generallysmall and will be dropped from our discussionsnext.For well-aligned electron with px ¼ py ¼ 0; f1 is

nonvanishing only for odd harmonics [5] and f2 isnonvanishing only for even harmonics [4]. How-ever, when longitudinal electric field is taken intoaccount, a new mechanism of interaction isintroduced through f3 which is nonvanishing onlyfor even harmonics. Furthermore, for misalignedelectron with pya0 and px ¼ 0; yet another newmechanism is introduced through f4 for even

M. Xie / Nuclear Instruments and Methods in Physics Research A 483 (2002) 527–530528

harmonics. Interestingly enough, the effects of f3and f4 can be comparable to that of f2:For misaligned electron with pxa0; f1 is

nonvanishing for both odd and even harmonics[3]. Likewise, f2; f3; f4 are all mixed with both oddand even harmonics. Thus, in general, evenharmonics can be generated by all four mechan-isms: f1 and f4 are due to misalignment, f2 is due tofield gradient, and f3 is due to longitudinalcoupling.It is important to note that the previously

known mechanisms, f1 and f2; are limited to theinteraction with electric field linearly polarized inthe wiggling plane. The new mechanisms, f3 and f4;predict FEL gain in modes with different polariza-tion states, and as a result, different profiles thanpreviously anticipated.It is more difficult to obtain clean analytical

results for the mixed case. From now on we willconcentrate on the case with px ¼ 0 ðz ¼ 0Þ: In thiscase, Eq. (4) can be written separately for oddharmonics as

dgdz

� �¼ �

e

mc2R einy�iDnkwz h

d *x

dz

� �%Fx

which is just the usual FEL equation [5] with onedominant term originated from f1; and for evenharmonics as

dgdz

� �¼ �

e

mc2R einy�iDnkwz h *x

d *x

dz

� �q %Fx

q %x

�þ/hSpy %Fy þ/hS %Fz

��which, in addition to the usual term originatedfrom f2 [4], has two new terms originated from f3and f4; where

hðzÞ ¼XN

l¼�N

JlðnxÞeið2l�nÞkwz; /hS ¼ Jn=2ðnxÞ

hd *x

dz

� �¼

awiffiffiffi2

pg½Jðnþ1Þ=2ðnxÞ � Jðn�1Þ=2ðnxÞ

h *xd *x

dz

� �¼

a2wi

2g2kw½Jðnþ2Þ=2ðnxÞ � Jðn�2Þ=2ðnxÞ:

For nonmonochromatic field, the energy equationcan be modified accordingly, using the following

definition for Fourier integral:

Eðr; z; tÞ ¼1ffiffiffiffiffiffi2p

p ZN

�N

Fðr; zÞeiðkz�otÞ do: ð5Þ

3. Generating function for vector field

It has been fortunate that only a scalar waveequation has to be dealt with in FEL theory so far,since only one field component is dominant at thefundamental frequency. However, at even harmo-nics, vector Maxwell equations have to be solved,making FEL theory much more difficult. Never-theless, the simplicity is restored under certaincircumstances when all field components can berelated to a generating function which satisfies ascalar wave equation.Starting from electric Hertz vector P [6],

electromagnetic field is given by

E ¼ rðr � PÞ �1

c2q2Pqt2

; B ¼ r�1

c2qPqt

ð6Þ

and Maxwell equations are expressed as:

r2P �1

c2q2Pqt2

¼ �P

e0; J ¼

qPqt

: ð7Þ

Applying Fourier transform defined by Eq. (5) andmaking slowly varying envelope approximation,Eq. (7) becomes

r2>W þ 2ik

qWqz

¼ �iJ

oe0ð8Þ

J ¼ �effiffiffiffiffiffi2p

pc

Xi

viðzÞd½r� riðzÞe�i½kz�otiðzÞ: ð9Þ

Using the following expansion

d½r� riðzÞ ¼ d½r� %riðzÞ � *xðzÞqqx

d½r� %riðzÞ þ?

Eq. (9) can be written as

J ¼ �eeiDnkwz Sðr; zÞ � *xðzÞqSðr; zÞqx

� �ð10Þ

Sðr; zÞ ¼1

krc

Zd2p dZ hnðzÞvðzÞfnðr; p; Z; zÞ

fnðr; p; Z; zÞ ¼1ffiffiffiffiffiffi2p

p Zdy e�inyFðr; p; y; Z; zÞ

M. Xie / Nuclear Instruments and Methods in Physics Research A 483 (2002) 527–530 529

where Z ¼ ð/gS� g0Þ=g0 and the distributionfunction for wiggler-averaged motion of electronbeam in 6D phase space X fr; p; y; Zg is definedby

FðX ; zÞ ¼ krX

i

d½r� %riðzÞd½p� %piðzÞ

� d½y� yiðzÞd½Z� ZiðzÞ:

The function F can be separated into two partsFðX ; zÞ ¼ F ðX ; zÞ þ f ðX ; zÞ with F ðX ; zÞ andf ðX ; zÞ being the unperturbed and perturbeddistribution, respectively.In the special case with px ¼ 0; the driving

current in Eq. (10) can be expressed for evenharmonics as, after taking the wiggler average

/JxS ¼ iaxeiDnkwz q

qx

Zd2p dZ fn

/JyS ¼ �ayeiDnkwz

Zpyd

2p dZfn

/JzS ¼ �azeiDnkwz

Zd2p dZ fn

ax ¼ea2w½Jðn�2Þ=2 � Jðnþ2Þ=2

2g2krkw; ay ¼ az ¼

eJn=2

kr:

The relationship between /JxS and /JzSsuggests

Cx ¼ �iax

az

qCz

qx: ð11Þ

Furthermore, limiting our case to nondivergentbeam with the same py for every electron, therelationship between /JyS and /JzS implies

Cy ¼ pyCz: ð12Þ

Therefore, we may choose Cz as the generatingfunction for this case. Once Cz is determined bysolving the scalar wave equation from Eq. (8),

r2>Cz þ 2ik

qCz

qz¼ �

i/JzSoe0

other components of the Hertz vector, Cx and Cy;

are given by Eqs. (11) and (12), and electromag-netic fields by Eq. (6).

4. Conclusions

It is interesting to note that the new mechanismsof energy exchange between an electron and a fastwave are in essence based on the principle ofalternating gradient acceleration [7]. In fact, thelongitudinal coupling was first revealed in a variedform as a mechanism of inverse undulator accel-eration [8]. Furthermore, it has been shownrecently that the longitudinal field of a laser beamin open space, though much smaller than thetransverse field, is important for energy exchangesuch as inverse transition acceleration [9].

Acknowledgements

Interesting discussions on longitudinal FELinteraction with M. Zolotorev and on secondharmonic generation experiment with G. Neil andS. Benson are acknowledged. This work wassupported by the US DOE under Contract No.DE-AC03-76SF00098.

References

[1] H. Bluem, et al., Phys. Rev. Lett. 67 (1991) 824.

[2] G. Neil, et al., Second Harmonic FEL Oscillation, Nucl.

Instr. and Meth. A 483 (2002), these proceedings.

[3] W. Colson, et al., Phys. Rev. A 31 (1985) 828.

[4] M. Schmitt, et al., Phys. Rev. A 34 (1986) 4843.

[5] W. Colson, IEEE J. Quantum Electron. QE-17 (1981) 1417.

[6] J. Stratton, Electromagnetic Theory, McGraw-Hill, New

York, 1941 (Chapter 1).

[7] M. Xie, Proceedings of EPAC00, 2000, p. 895.

[8] M. Xie, Proceedings of EPAC98, 1998, p. 830.

[9] M. Xie, Proceedings of PAC01, 2001, p. 3876.

M. Xie / Nuclear Instruments and Methods in Physics Research A 483 (2002) 527–530530