Nuclear Instruments and Methods in Physics Research A 407 (1998) 130—135

A simple theory of gain and saturation in FEL oscillators

Vinit Kumar*, Srinivas Krishnagopal

Advanced Beam Physics Section, Center for Advanced Technology, Indore 452013, India

Abstract

We develop a simple theory of gain and saturation in FEL oscillators, using a Maxwell—Lorentz approach. Ouranalysis includes inhomogeneous broadening effects. We derive a simple expression for the FEL gain as a function of: (a)optical intensity, (b) energy detuning, and (c) energy-spread. The analytic results are discussed and are shown to bein good agreement with one-dimensional, time-independent, numerical simulations. ( 1998 Elsevier Science B.V. Allrights reserved.

1. Introduction

As in all lasers, the gain in free-electron lasers(FELs) is an important quantity, since it is thedependence of the FEL gain on intensity that deter-mines the saturated power, and hence the out-coupled power, in an FEL oscillator. The study ofthe dependence of the FEL gain on the intensitytherefore becomes important. The dependence ofthe FEL gain on intensity is known to be a functionof electron beam parameters such as the energydetuning, longitudinal energy spread, transverseemittance, etc. There have been earlier analyticstudies of FEL dynamics and gain in the high gainregime, using a Maxwell—Vlasov approach that isof relevance to FEL amplifiers [1,2]. In FEL oscil-lators, where the gain per pass is typically low butthe optical intensity builds up to a large value over

*Corresponding author. Tel.: #91 731 488018; fax: #91731481525; e-mail: vinit@cat.ernet.in.

many passes, it is the small-gain, large-signal regimethat is important.

In an earlier work [3] we had studied FELdynamics and gain in the one-dimensional, small-gain, large-signal regime. The emphasis there wason the simultaneous dependence of the FEL gainon the intensity as well as the detuning parameter.The electron beam was however assumed to bemonoenergetic and therefore the effect of the finiteenergy-spread present in the beam was not studied.In this paper, we extend our earlier analysis toinclude the effect of finite energy-spread on theFEL dynamics and gain. We stick to the simple,single-particle, approach of Colson [4], i.e., usethe Maxwell—Lorentz equations. Our approachprovides a transparent and physically intuitive un-derstanding of the FEL gain in the small-gain,large-signal regime.

In the next section we derive an analytic expres-sion for the gain as a function of energy detuning,optical intensity as well as energy-spread. In Sec-tion 3 we compare our predictions about these

0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reservedPII S 0 1 6 8 - 9 0 0 2 ( 9 7 ) 0 1 3 7 2 - 7

dependencies of FEL gain with results from multi-particle one-dimensional time-independent numer-ical simulations. We also use our analysis to predictthe saturated power in an FEL oscillator and itsdependence on the detuning parameter as well asthe energy spread. We conclude with some com-ments and discussion.

2. Analytical calculations

We start with the dimensionless Colson equa-tions [4] and solve them perturbatively under thesmall-gain approximation, where we assume thatthe gain in a single pass is small. The dimensionlesslaser field e

Lcan then be treated as a constant

during any given pass, and the Maxwell equationconsequently decouples from the Lorentz equa-tions. Note that the magnitude of e

Lcan still be

large, so that this analysis is valid in the small-gain,large-signal regime. We look for a solution to theLorentz equations in the form of a power series inthe electric field e

L. The perturbative series for the

differential gain dG/dq is then,

dG

dq"

dG0

dq#e

L

dG1

dq#e2

L

dG2

dq#e3

L

dG3

dq#e4

L

dG4

dq,

(1)

where we have kept terms upto the fourth order ineL.Substituting the power series for the dimension-

less energy k (which is the electron energy measuredrelative to its resonant value and normalised to theenergy bandwidth of the FEL) and phase t (whichis electron’s phase relative to the electromagneticfield) in the Lorentz equations, Taylor expanding ineL

and gathering equal powers of eL

on both sides,gives a hierarchy of equations for the varioustn

and kn

which are the coefficients in the powerseries. These are solved using the initial conditions:k0"a, t

0"/

0, k

n;0"0, t

n;0"0. Here a is the

detuning parameter, and /0

is the electron’s initialrandom phase. We assume an ideal monoenergeticunbunched beam. Once these t

nand k

nare known,

they are substituted in the Maxwell equation toderive the expression for the differential gain, and

then Taylor expanded in eL

to get the expressionsfor the dG

n/dq; these will be zero for odd n, since the

gain depends only on the intensity, i.e., on evenpowers of e

L. The total FEL gain (defined here as

the ratio of the increase in intensity to the initialintensity) can be evaluated by integrating the differ-ential gain dG/dq over the entire length of thewiggler (i.e., from q"0 to 1; q being the dimension-less time). This gives us the following equation forthe FEL gain:

G(a, e2L)"2j

%[g

0(a)#e2

Lg2(a)#e4

Lg4(a)], (2)

where j%is the dimensionless electron beam current

density, g0

is the small-signal gain function whichcan be found in any text-book [5], and the func-tions g

2(a), and g

4(a) are calculated and given in

Ref. [3].Next, we introduce the effect of energy-spread on

the expression for the FEL gain. For this, we mustconvolute the gain function G(a, e2

L) with the elec-

tron beam energy distribution to get the grand gainfunction GI (a

0, e2

L, pa):

GI (a0, e2

L, pa)"

1

J2ppaP

=

~=

G(a, e2L)

]expC!(a!a

0)2

2p2a Dda, (3)

where the electrons have been assumed to havea Gaussian energy distribution centred arounda0

and with a spread pa. Here, the relative energyspread pc/c is related to the parameter pa by therelation pa"4pN

Wpc/c,NW

being the number ofwiggler periods.

It is not possible to evaluate the above integralanalytically with the exact form of the functionsg0, g

2and g

4that this analysis yields [3]. We there-

fore try to find some simplified approximate formfor these functions which is numerically very closeto the form in Ref. [3], but analytically simpleenough to facilitate the analytic evaluation of theabove integral. In order to do this, we start with thefollowing approximant which is same as the oneDattoli et al. have used for g

0in Ref. [6], and

extend it to g2

and g4

in our analysis:

hi(a)"a

iexp(!b

ia2)sin(c

ia), i"0, 2, 4, (4)

V. Kumar, S. Krishnagopal/Nucl. Instr. and Meth. in Phys. Res. A 407 (1998) 130—135 131

II. THEORY/MODELING

where the coefficients ai, b

iand c

iare obtained by

Taylor expanding the functions hiand g

iupto fifth

order in a and equating then the two series. Whenwe do so, we get a

0"0.0833, b

0"0.025, c

0"

0.5, a2"!0.000953, b

2"0.0488, c

2"0.859, a

4"0.0000117, b

4"0.07315 and c

4"1.22. With

these numbers, we find that the approximants hiare

numerically in excellent agreement with the orig-inal functions g

ifor relevant values of the detuning

parameter (a"0—6): the typical deviation is lessthan 1%. So, the g

iin Eq. (2) for the gain G(a, e2

L)

can be replaced by the approximants hi. With this

replacement the grand gain function can now beanalytically evaluated. Doing that we get,

GI (a0, e2

L, pa)"2j

%[hI

0(a

0, pa)#e2

LhI2(a

0, pa)

#e4LhI4(a

0,pa)], (5)

where the functions hIi(a

0, pa) are given by

hIi(a, pa)"a

iC1

1#2bipaD

1@2sinC

cia

1#2bip2aD

]expC!2b

ia2#c2

ip2a

2#4bip2a D. (6)

We note that hIi(a, pa)Ph

i(a) in the limit paP0 as

expected. In the limit paPR,hIi(a, pa)P0. (Note

that, for convenience, we replace a0

by a).The above expression for the FEL gain is ex-

pected to be valid only upto some limited value ofintensity e2

Lbeyond which the contribution of the

higher-order terms in intensity that we have drop-ped (e6

Lonwards) becomes important. Extending

the expansion even further is neither very practicalnor tractable. Instead, in order to get a generalisedgrand gain function that is valid for higher inten-sities and into the saturation regime, it is moreproductive to parameterize the gain. We adopt thefollowing three-parameter parameterisation of thegain proposed by us in Ref. [3]:

GI (a, e2L, pa)"2 j

%g0

1!exp[!(g2e2L#g

4e4L)]

g2e2L#g

4e4L

, (7)

where the functions g0(a, pa), g

2(a, pa) and g

4(a, pa),

determined analytically by Taylor-expandingEq. (7) to second order in e2

Land equating with

Eq. (5), are given by

g0(a, pa)"hI

0(a, pa), (8a)

g2(a, pa)"!2

hI2(a, pa)

hI0(a, pa)

, (8b)

g4(a, pa)"

4

3ChI2(a, pa)

hI0(a, pa)D

2!2 C

hI4(a, pa)

hI0(a, pa)D. (8c)

Hence, we finally end up with an expression fora generalised grand gain function which has theintensity, detuning, and energy-spread depend-encies in it, and which is valid at large intensitiesand into the saturation regime.

3. Comparision with numerical simulations

After having obtained the expression for the gen-eralised grand gain function, we now compare thedependencies of the FEL gain as predicted by ouranalysis with the results from numerical simula-tions. We use a one-dimensional, time-independent,multi-particle FEL simulation that solves the non-linear Kroll—Morton—Rosenbluth equations [7]and does not make the small-gain approximation.The FEL parameters chosen are j

R"1lm, j

W"

4 cm, cR"200,a

W"1.0,r

"%!."2mm, N

W"50,

and I"100A. For these parameters j%"0.68, and

intracavity power P"11.97e2LMW.

Fig. 1a and b show, for two different values of thedetuning parameter (a"2.6, and a"4.2), how theintensity dependence of the FEL gain is affected bythe energy spread. We find that the comparisionbetween the analysis and the numerical simulationis generally very good. There are a couple of pointswhich are worth noting. First, the gain decreasessignificantly as the energy-spread increases. Fora"2.6, it falls to &10% of its zero energy spreadvalue as the energy spread approaches &1% (i.e.,1/2N

W). We also find that the fall of the gain with

intensity becomes flatter as the energy spread in-creases. In other words, the gain saturation inten-sity (i.e., the intensity at which the gain halves)increases as the energy spread increases (eventhough the gain itself decreases). For a"4.2, withzero energy-spread we find that the gain first in-creases slightly with intensity before finally falling.

132 V. Kumar, S. Krishnagopal/Nucl. Instr. and Meth. in Phys. Res. A 407 (1998) 130—135

Fig. 1. Plots of the total gain as a function of the (dimension-less) optical intensity e2

Lfor different energy spreads (pc/c), (a) for

a"2.6, and (b) for a"4.2. Solid lines are the analytic curvesand the dotted lines are the simulation curves. The analyticcurves are obtained from Eq. (7)

As the energy spread increases, this non-monotonicbehaviour of the gain gets damped. ComparingFig. 1b with Fig. 1a, we find that for a"4.2, evenwith zero energy spread, the gain is less sensitive to

intensity as compared to a"2.6. When the en-ergy-spread is increased, the slope of the gain curvedoes not change as significantly as it does for thea"2.6 case. In other words, the gain saturationintensity is not as sensitive to energy-spread as it isfor the a"2.6 case.

Fig. 2a and b show, for two different values of theintensity, how the detuning curves (i.e., the gain asa function of the detuning parameter a) change asthe energy-spread is increased. Here too we findgood agreement between the analysis and the simu-lation. Comparision of Fig. 2a and b reveals that asthe intensity builds up, the peak of the detuningcurve shifts towards higher values of a (for pa"0,from a"3 in Fig. 2a to a"3.8 in Fig. 2b). How-ever, this shift is less pronounced at higher energy-spreads. We also find that at very high values of thedetuning parameter (say a"6), the gain increaseswith energy-spread. This is understandable sincethe gain at higher detuning is very small, and if theenergy spread is increased, there will always besome electrons with a low enough to give somedecent contribution to the gain.

3.1. Saturation in an oscillator

Our analysis also makes it possible to calculatethe saturated intracavity intensity e2

L,4!5as a func-

tion of the detuning parameter a as well as energy-spread, by equating the gain at saturation to theloss l, i.e.,

GI (a, e2L,4!5

, pa)"l. (9)

Fig. 3 shows e2L,4!5

as function of the detuning para-meter a, for two different energy-spreads, calculatedusing the above equation, where a round trip loss of8% has been assumed. The FEL parameters are thesame as earlier except that electron beam currentI"200 A. The corresponding numerical simula-tion curve is also shown in Fig. 3. We find thatthere is good agreement between the analysis andthe simulation. The positions of the two thresholdsat which the lasing starts and stops agree extremelywell between analysis and simulation. The differ-ence between the maximum saturated intensity aspredicted by our analysis and the numerical simu-lation is &10%. The saturated intensity falls sig-nificantly with the energy-spread.

V. Kumar, S. Krishnagopal/Nucl. Instr. and Meth. in Phys. Res. A 407 (1998) 130—135 133

II. THEORY/MODELING

Fig. 2. Plots of the total gain as a function of the detuningparameter a for different energy spreads (pc/c), (a) for e2

L"15,

and (b) for e2L"45. Solid lines are the analytic curves and the

dotted lines are the simulation curves. The analytical curves areobtained from Eq. (7).

4. Discussions and conclusions

Our analysis shows that it is possible to get ananalytic expression for the FEL gain as a function

Fig. 3. Plot of the (dimensionless) saturated intensity e2L,4!5

in anFEL oscillator as a function of the detuning parameter a for twodifferent values of the relative energy-spread, pc/c"0.2% and0.4%, respectively. Solid lines are the analytic curves and thedotted lines are the simulation curves. The analytic curves cor-respond to the solution of Eq. (9).

of detuning parameter even away from the small-signal regime, and even including the effect ofa finite energy-spread, using the simple single-par-ticle Colson equations. It should be emphasisedthat our analysis is valid in the small-gain, large-signal regime which is the regime of operation ofFEL oscillators where the gain can be quite low butthe intensity can, over many passes, build up toa large value.

Compared to our earlier work [3], here we get anexpression for the gain which is much simpler. Thissimplicity has resulted because we have writtendown a simplified approximate form for the func-tions g

2, and g

4, which are extremely complicated.

Hence, we get a really handy formula for the FELgain as a function of: (a) the detuning parameter, (b)the optical intensity, and (c) the energy-spread.Note that we work in dimensionless variables. Thedependence on the other FEL parameters such aswiggler length ¸

W, wiggler parameter a

W, wiggler

period jW, beam energy E, etc. can be extracted from

our formula by simply writing the dimensionlessvariables e2

Land j

%in terms of the FEL variables.

134 V. Kumar, S. Krishnagopal/Nucl. Instr. and Meth. in Phys. Res. A 407 (1998) 130—135

In our earlier work [3], we had found that withincreasing optical intensity the peak of the detuningcurve shifts towards higher values of the detuningparameter. Since a free-running oscillator picksa detuning at which the gain is maximum, theoperating value of the detuning parameter at satu-ration is going to be higher than its value at small-signal levels. In our analysis here, in Fig. 2b, we findthat at higher values of the detuning parameter (4.2in this case), the slope of the fall of gain withintensity is not that sensitive to the energy-spread.This is in contrast with the situation at a"2.6where the slope is sensitive to the energy-spread.However, in both cases, the gain keeps falling sig-nificantly with energy-spread. Hence, for the oper-ating value of a, the gain is going to be sensitive toenergy-spread, but the gain saturation intensity isgoing to be relatively insensitive to the energy-spread. This distinction, between the functional de-pendences of the gain and the saturation intensity,is an important one for FEL oscillators. It explainswhy, in oscillators, you can start out with lowergain, but end up with higher saturated power.

In conclusion, we have derived, using a pertur-bative Maxwell—Lorentz approach, a simple theoryof the FEL gain and its dependence on the detuningparameter, optical intensity, and energy-spread, in

the small-gain, large-signal regime. The usual small-signal-gain curve is generalised into a family ofdetuning curves for different intensities and energy-spreads. The peak of the detuning curve is found toshift towards higher values of the detuning para-meter as the intensity builds up; this shift is lesspronounced for higher energy-spreads. The gainfalls significantly with energy-spread whereas thegain saturation intensity is relatively less sensitiveto the energy-spread. Our results are in good agree-ment with one-dimensional, time-independent FELsimulations, and can be used for the design andstudy of FEL oscillators.

References

[1] L.-H. Yu, S. Krinsky, R.L. Gluckstern, Phys. Rev. Lett. 64(1990) 3011.

[2] Y.-H. Chin, K.-J. Kim, M. Xie, Phys. Rev. A 46 (1992)6662.

[3] Vinit Kumar, Srinivas Krishnagopal, Phys. Rev. E 55 (1997)1887.

[4] W.B. Colson, Phys. Lett. A 64 (1977) 190.[5] C.A. Brau, Free-Electron Lasers, Academic Press, San

Diego, 1990.[6] G. Dattoli et al., J. Appl. Phys. 77 (1995) 6162.[7] N.M. Kroll, P.L. Morton, M.N. Rosenbluth, IEEE, J.

Quant. Elect. QE 17 (1981) 1436.

V. Kumar, S. Krishnagopal/Nucl. Instr. and Meth. in Phys. Res. A 407 (1998) 130—135 135

II. THEORY/MODELING