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erlp-ofel-rpt-0001Version: 0.1Date: August 12, 2003

ERL Prototype Free-Electron Laser

Neil Thompson

Document Change Record:

Version Date Section/Sheet Comment0.1 August 12, 2003

Responsible Author: Neil Thompson

Authorised by: J. Clarke

2 erlp-ofel-rpt-0001

Abstract

This report is split into sections so that the reader need only consult the

part that suits their interests. Section 1 explains how a free-electron laser

(FEL) works. Section 2 gives the methods used for the calculation of various

FEL output parameters, such as gain and efficiency. Section 3 gives a sug-

gested optimum design for the proposed infra-red oscillator free-electron laser

(IRFEL) on the ERL Prototype and looks at how and why the performance of

the FEL changes as parameters are moved away from their optimum values.

Section 4 discusses the effect of the FEL interaction on the downstream elec-

tron beam. Finally, Section 5 gives a comprehensive summary of calculated

output parameters from the ERL Prototype IRFEL, assuming the optimum

suggested parameters can be achieved.

Contents

1 The FEL mechanism—how an FEL works 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Small-signal low-gain regime . . . . . . . . . . . . . . . . . . 41.1.2 High-Gain Regime . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Universal Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Phase Space Representation . . . . . . . . . . . . . . . . . . . . . . . 7

2 Predicting FEL performance 102.1 Semi-Analytic Gain Calculation . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Gain degradation due to beam energy spread . . . . . . . . . 112.1.2 Gain degradation due to emittance . . . . . . . . . . . . . . . 122.1.3 Gain reduction due to slippage . . . . . . . . . . . . . . . . . 13

2.2 Numerical Gain Calculation . . . . . . . . . . . . . . . . . . . . . . . 152.3 Calculating Output Characteristics . . . . . . . . . . . . . . . . . . . 16

2.3.1 Power and Efficiency . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Linewidth and Brightness . . . . . . . . . . . . . . . . . . . . 182.3.3 Gaussian Beam and Resonator Equations . . . . . . . . . . . 18

3 ERL Prototype: FEL Optimisation 203.1 Suggested Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Electron Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Bunch charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Bunch length . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.4 Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.5 Energy Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.6 β function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Optical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Cavity Length and Mirror Radii of Curvature . . . . . . . . . 253.3.2 Mirror Aperture and Reflectivity . . . . . . . . . . . . . . . . 28

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 ERL Prototype: Effect of FEL Interaction on Electron Beam 304.1 Induced Energy Spread . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

erlp-ofel-rpt-0001 3

5 ERL Prototype: FEL Output Characteristics 32


1 The FEL mechanism—how an FEL works

1.1 Introduction

In a free-electron laser a beam of relativistic electrons co-propagates with an opticalfield E through the spatially periodic magnetic field of an undulator. The initialoptical field is the spontaneous emission with a spectrum of peak wavelength on-axisof

λr =λw

2γ2r

(1 +

K2

2

)(1)

for the fundamental mode of a planar undulator. Here λw is the period of theundulator, γr is the energy of the electrons relative to their rest mass, and K isthe undulator deflection parameter. The undulator causes the electrons to undergotransverse oscillations. The transverse electron velocity can then couple to thetransverse electric component of the optical field E allowing energy transfer. With-out an undulator no energy transfer is possible because the gain in energy of theoptical field is given by

4W = −e

∫E · ds = −e

∫v ·Edt (2)

where v is the electron velocity, so when this velocity is entirely longitudinal thenv ·E = 0.

The simplicity of this basic mechanism makes it easy to understand that thestrength of the coupling is inversely proportional to the beam energy—a higherenergy beam has more rigidity so the electrons have a lower transverse velocityleading to weaker coupling.

It turns out that for electrons injected into the undulator with resonant energyγr the relative phase between the transverse oscillations of the electrons and theoptical field remains constant. This phase constancy is maintained by the electronsslipping back one radiation wavelength per undulator period, because although theelectrons are relativistic their velocity is still slightly less than c and their path lengthis increased by the transverse oscillations. Depending on the value of this constantphase, the electrons can either transfer energy to the field and decelerate, thusproviding gain, or take energy from the field and accelerate, giving absorption. Foran input electron beam at the resonant energy, where the electrons are distributedrandomly and evenly in phase, both these processes occur together and initiallythere is no net gain.

1.1.1 Small-signal low-gain regime

For an oscillator FEL where the length of undulator is relatively short and thecurrent is low, it is found that for the gain process to dominate over the absorp-tion process the electrons should be injected at an energy slightly higher than theresonant energy. This corresponds to a frequency/energy detuning given by thedetuning parameter

ν = 2πNωr − ω

ωr' 4πN

γr − γ

γr, [(γr − γ) ¿ γr]. (3)

Depending on their phase, some electrons gain energy and some lose energy, but onaverage there is a net transfer of energy to the optical field. The single-pass gain vs


−3 −2 −1 0 1 2 3

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

ν / π

Gai

n

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

ν / π

f(ν)

Figure 1: The top plot shows the small signal single pass gain curve, as a functionof the detuning. The bottom curve shows the spontaneous emission spectrum.

detuning1 is shown in the top plot of Figure 1. The maximum gain is for a detuningν = 2.6 (ν/π = 0.8).

A significant feature of the gain curve is that it is proportional to the negativederivative of the spontaneous emission spectrum shown in the bottom plot of Figure1. This is the essence of the Madey Theorem. It means that any effect that causesa broadening of the natural linewidth will also cause a reduction in the small signalgain.

In this low-gain regime the electrons can be considered as independent particlesand the optical field constant over one pass through the undulator. A high intensityfield is obtained by containing the emitted radiation within an optical cavity of anappropriate length such that the optical pulse coincides on its next pass throughthe undulator with a fresh electron bunch and is further amplified.

The energy modulation of the electrons leads to a phase modulation such thatthe electrons start to bunch on the scale of one radiation wavelength. This effectincreases with successive electron bunches as the field intensity increases. However,the coupled optical/electron system exhibits feedback. As the optical power in thecavity increases, the amount of energy transferred from the electrons to the fieldalso increases towards its maximum value. Because the power loss from the cavityis proportional to the power within the cavity the absolute power loss also increasesuntil it matches the power extracted from the electron beam. At this point thesystem is saturated.

Saturation is described equivalently in terms of the interaction reducing thesingle pass gain until it equals the cavity losses, but this gives a misleading view ofevents—the gain is the relative increase in power per pass, so as the absolute power

1This gain curve is calculated from the suggested parameters of the ERL Prototype IRFEL

given in Table 2 on p21.


in the cavity builds the gain must fall even if the amount of power being extractedfrom the electron beam is constant.

1.1.2 High-Gain Regime

For an FEL with a longer undulator and/or higher current, the gain can be muchhigher and for a single pass through the device the electrons can no longer beconsidered as independent particles and collective effects become important—theelectrons communicate with each other via the common radiation field and theenergy modulation of the electrons turns into a phase modulation. The electronsself-bunch on the scale of a radiation wavelength at a phase which corresponds togain rather than absorption. Since most of the electrons are nearly at the samephase they emit coherent synchrotron radiation. In this regime maximum gain isobtained at an injection energy much closer to the resonant energy and saturationoccurs when electrons that have given up energy to the radiation field start tore-absorb it.

These ideas are well illustrated in Section 1.3 by considering the motion of theelectrons in longitudinal phase space.

1.2 Universal Scaling

The likely parameters for the ERL Prototype infra-red oscillator FEL show thatthe device will operate in a low-gain high-gain transition regime. For this reasonthe approach of Bonifacio et al [7] is considered appropriate—a relatively simpleset of ‘universally-scaled’ FEL equations (a system of coupled ODEs) are used todescribe the FEL interaction in any gain regime. The equations contain no freeparameters so are independent of the undulator and electron beam specifications.The universal solution is related to the real machine via the scaling of the variables.

The derivation of the equations is lengthy and not repeated here. The mainassumptions are that the dynamics are one-dimensional (along the axis of propaga-tion) and that the electron beam is continuous and infinitely long. The relativisticelectrons ‘see’ the transverse, spatially periodic magnetic field due to the undula-tor of wavenumber kw = 2π/λu and the plane wave radiation field of wavenumberk = 2π/λr = ω/c. By considering these fields in terms of their vector potentialsA they can be combined to give a resultant potential experienced by the electronsAtot(z, t) = Au(z) + A(z, t), where Au(z) is the potential of the static (hence not dependence in the lab frame) undulator potential, and A(z, t) is the potentialdue to the radiation field. This combined potential is known as the ponderomotivepotential of the ponderomotive field, and the ponderomotive field is a beat wavepropagating in the z-direction at the phase velocity

vp =ω

k + kw. (4)

Further detailed analysis leads eventually to the 1D steady state universallyscaled FEL equations:

dθj

dz= pj , (5)

dpj

dz= −(A exp[iθj ] + c.c.) (6)

dA

dz= 〈exp[−iθ]〉 ≡ b. (7)


where j = 1 . . . Ne and Ne is the number of electrons and c.c. means complexconjugate. The variable θj is the phase of the jth electron relative to the combinedponderomotive field and is defined as θj = (k + kw)z − ωtj . The scaled electronenergies are given by

pj =γj − γr

ργr(8)

so that the scaled energy/frequency detuning is represented by the parameter δ

which is included in the equations implicitly via the mean value of the energyvariable:

δ = 〈p 〉o =〈γ〉 − γr

ργr≡ ν

G, G = 4πNρ. (9)

The parameter ρ is the fundamental FEL parameter defined for a planar undulator[18] as

ρ =1γr

(awωpfB

4ckw

)2/3

(10)

with aw = eBw/√

2mckw being the undulator deflection parameter and Bw theundulator peak magnetic field. The plasma frequency ωp is defined as

ωp =(

e2ne

ε0m

)1/2

(11)

with ne the electron number density, and the factor fB is given by fB = J0(ζ) −J1(ζ), where ζ = a2

w/2(1+ a2w). The equations (5-7) are valid for values of ρ . 0.01.

G is the FEL Gain Parameter given by

G = 4πρN (12)

and is useful for defining different gain regimes [17]. High gain is defined for G > 1,and low gain for G < 1, with a transition region around G ' 1.

A is the scaled complex radiation field amplitude related to the real field ampli-tude E by the expression

|A|2 =ε0|E|2

ρneγrmc2(13)

and z is a dimensionless length defined as z = 2kwρz. The undulator is of lengthLu = Nλw so the value of z at the end of the undulator is equal to G.

Finally, the RHS of (7) is the bunching parameter— if b = 0 this indicates theelectrons are distributed uniformly in phase, and if b = 1 this indicates the electronsall have the same phase.

1.3 Phase Space Representation

It is useful to plot the electron distribution in longitudinal phase space to illustratethe mechanism. The notation of Section 1.2 is used. A free-electron laser simulatorprogram has been written in MATLAB to solve the 1D FEL equations (5-7) nu-merically. The program allows an initial distribution of macro-particles2 in (p, θ)phase space to be set up, then plots the evolution of the phase space, the bunchingparameter, the optical field intensity, the single pass gain and the electron energyspread. Figure 2 shows the results for a single pass through the undulator, usingthe ERL Prototype suggested parameters of Table 2 on p21. The scaled detuning

2Each macro-particle represents a small region of phase space filled with many electrons


0 2 4 6

2.36

2.37

2.38

θp

Electron Phase Space after 1 pass

2.35 2.36 2.37 2.38 2.39 2.40

200

400

600

800Energy Histogram after 1 pass

p

no. e

lect

rons

0 0.2 0.4 0.6 0.8 1

1.1

1.2

1.3

x 10−4

z bar

|A|2

0 0.2 0.4 0.6 0.8 1

1

2

3

4

x 10−3

z bar

|b|

0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

x 10−3

z bar

st d

ev o

f |p|

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

z bar

G

Figure 2: ERL Prototype IR-FEL: A single pass though the undulator. The toptwo plots show the phase space distribution and energy histogram at the end of theundulator. The other four plots show the evolution of the field intensity, bunching,energy spread and gain as the electrons pass along the undulator which has a scaledlength of z = 1.09. The input energy is p = 2.37.

for maximum gain works out at 2.37, so the input beam is distributed evenly inphase with a scaled energy of p = 2.37. For the sake of illustration the input energyspread is set to zero. It is seen that after one pass electrons have either gained orlost energy according to their phase with respect to the ponderomotive potential.On average the electrons have lost a small amount of energy (although this is hardto detect in the energy histogram plot). The field amplitude has grown accordinglyand the increase in the bunching parameter shows that a small amount of bunchinghas occurred. The single pass gain is seen to be approximately 40%.

To follow the evolution of an oscillator the output field amplitude is fed backinto the equations for another pass, after reducing its intensity to account for cavitylosses and outcoupling. The electron distribution is refreshed for each pass. Figure3 shows the situation after 40 passes, assuming cavity losses of 1% and outcouplingof 7%. It is seen that the electrons are starting to rotate clockwise in phase space,due to the start of oscillations in the ponderomotive potential, and losing significantenergy on each pass to the optical field. A significant energy spread has been in-duced in the electron beam and the bunching is strong indicating coherent emissionbecause the electrons are in phase.

After 150 passes the field intensity has saturated as shown in Figure 4. Eachelectron bunch passing through the wiggler is losing energy to the field, but thesingle-pass gain has reduced to the total cavity losses (passive losses + outcoupling)so the net gain is zero.


0 2 4 6 8−2

0

2

4Phase Space (undulator exit)

θ

p

−2 0 2 40

100

200

300

400Energy Histogram (undulator exit)

p

no.e

lect

rons

0 10 20 30 400

1

2

3

No of passes

|A|2 (

cavi

ty)

0 10 20 30 400

0.2

0.4

0.6

0.8

No of passes

|b|

0 10 20 30 400

0.5

1

1.5

No of passes

std

dev

p

0 10 20 30 400.36

0.37

0.38

0.39

0.4

No of passes

G

Figure 3: ERL Prototype IR-FEL: The top two plots show the phase space dis-tribution and energy histogram at the undulator exit after 40 passes. The inputenergy is p = 2.37 so clearly the electrons are losing energy per pass. The lower 4plots show the field intensity, bunching, energy spread and gain evolution over 40passes.

0 2 4 6 8−10

−5

0

5

10Phase Space (undulator exit)

θ

p

−10 −5 0 5 100

200

400

600Energy Histogram (undulator exit)

p

no.e

lect

rons

0 50 100 1500

20

40

60

No of passes

|A|2 (

cavi

ty)

0 50 100 1500

0.2

0.4

0.6

0.8

No of passes

|b|

0 50 100 1500

1

2

3

4

No of passes

std

dev

p

0 50 100 1500

0.1

0.2

0.3

0.4

No of passes

G

Figure 4: ERL Prototype IR-FEL: The top two plots show the phase space distribu-tion and energy histogram at the undulator exit after 150 passes. The input energyis p = 2.37 so clearly the electrons are losing significant energy per pass. The lower4 plots show the field intensity, bunching, energy spread and gain evolution over150 passes. The system is at equilibrium and the gain has reduced to the level ofthe total cavity losses.


2 Predicting FEL performance

There are many parameters to consider when optimising a free-electron laser design,but good criteria for defining the quality of the design are the gain and the efficiency,which should be as high as possible.

The single pass gain is defined as the relative amount by which the optical fieldintensity increases with one pass of the electron bunch through the undulator. Foran oscillator FEL to lase the gain must exceed the cavity losses. If the differencebetween the gain and the losses can be maximised then the number of passes to reachsaturation will be minimised. In addition, maximising the gain helps to maximisethe efficiency. The efficiency is defined as the fraction of electron beam power (inputpower) converted into laser power (output power).

In the following sections the methods used to predict gain and efficiency aregiven. Included in these sections are discussions of suitable values for some beamparameters (such as energy spread, emittance and bunch length) because theseparameters directly affect the gain. However these parameters are discussed againin Section 3 in the wider context of the design of the whole device. Formulae arealso given for the calculation of optical characteristics of the output laser beam.

2.1 Semi-Analytic Gain Calculation

The maximum gain is found by first calculating an ideal small signal single passgain parameter g0 applicable to a monoenergetic, zero emittance, infinitely longelectron beam, then applying various corrections to model a real beam [1, 9]. Someof these corrections are numerical fits to experimental data or simulations, so themethod is semi-analytic.

The maximum gain is given by [9]

Gmax = 0.85g0 + 0.19g20 + 4.12× 10−3g3

0 . (14)

whereg0 = g0 × Ce × Cx × Cy × Cc. (15)

The parameters Ce, Cx,y and Cc account for gain degradation due to energy spread,emittance and slippage respectively, and are discussed in the following sections.

The gain parameter g0 (for a planar undulator) is given by

g0 =16π

γλR[m]Lu[m]

I

IA

N2

ΣξF (ξ), ξ =

14

K2

1 + K2/2. (16)

I is the peak current, IA is the Alfven Current (17.03kA) and N is the number ofundulator periods. F (ξ) = (J0(ξ) − J1(ξ))2 is a parameter that accounts for gainreduction in a planar undulator due to longitudinal electron oscillations.

The factor Σ is the effective combined area of the optical and electron beams.It is given by [3]

Σ = 2π

√(w2

4+ σ2

x

)(w2

4+ σ2

y

). (17)

The parameter w denotes the mean value of the optical mode size averaged alongthe undulator:

w =w0

Lu

∫ +Lu/2

−Lu/2

√1 +

(λRz

πw20

)2

dz (18)


and w0 is the mode size at the waist.It should be noted that the distribution functions for the electron beam cross

section and the optical mode are of different forms. For a round beam the electronbeam distribution is a gaussian of form fe ∼ exp(−r2/2σ2) so that defining thebeam radius as the point at which the distribution falls to 1/e of its peak valuegives rb =

√2σ and a beam area of Σe = πr2

b = 2πσ2. The optical mode hasa power distribution of form fL ∼ exp(−2r2/w2) so that the distribution falls to1/e of peak value at rL = w/

√2 and the optical beam cross section is therefore

ΣL = πr2L = πw2/2. It can be seen then that in the limit of small electron beam

cross section the combined area Σ defined in Equation (17) reduces to the opticalmode cross section, and in the limit of a small optical mode it reduces to the electronbeam cross section.

2.1.1 Gain degradation due to beam energy spread

From Figure 1 it is seen that the range of energy detuning corresponding to positivegain has a width of ∆ν ' 2π, equivalent to an energy deviation of ∆γ = 1/2N .Any electrons with energy in this range produce positive gain. Increasing the beamenergy spread causes more electrons to fall outside this positive gain region. Anequivalent view is that small deviations in electron energy result in a spread in thefrequency of the emitted radiation (see (1)), thus broadening the natural linewidthand reducing the gain according to the Madey Theorem.

As a guide [5, 21] gain reduction can be neglected if the maximum energyvariation in the beam satisfies

∆γ

γ<

14N

(19)

Assuming a Gaussian energy spread of the form

E(ν) ∼ exp(−(ν − 2.6)2

2σ2

)(20)

and noting that 99.7% of electrons are contained within±3σ the beam energy spreadshould satisfy 6σγ/γ < 1/4N giving

σγ

γ<

124N

. (21)

Using the suggested parameters of the ERL Prototype, this gives a maximum rela-tive energy spread, to avoid gain degradation, of 0.1%(rms).

To estimate the gain reduction as a function of the energy spread, the overlapbetween a gaussian energy distribution E(ν) and the gain curve G(ν) can be cal-culated. Both functions are normalised to a maximum value of unity and the gainreduction factor given by

Coverlap =

∫ +∞−∞ E(ν)G(ν)dν∫ +∞−∞ E(ν)dν

(22)

For the calculations in this report the gain reduction due to energy spread is givenby the simpler expression [1]

Ce =1

1 + 1.7µ2e

, µe = 4Nσγ

γ. (23)

which reproduces Coverlap very closely as shown in Figure 5, using the parametersof the ERL Prototype FEL. Again it is seen that to avoid gain degradation themaximum energy spread of the ERL Prototype should be 0.1%(rms).


10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

Energy Spread (%)

Gai

n R

educ

tion

via

Ove

rlap

10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

1

Energy Spread (%)

Gai

n R

educ

tion

Fac

tor

Ce

Figure 5: ERL Prototype IR-FEL: Comparison of gain reduction factors Coverlap

and Ce

2.1.2 Gain degradation due to emittance

The emittance is a measure of both the beam size and the beam quality. Bothaffect the gain. The first affects the gain via the combined area Σ of the optical andelectron beam cross-sections. The second affects the gain because off axis electronsexperience a non-ideal undulator field.

Beam Overlap The beam radius varies as the square root of the emittance. Alarger beam is reflected in a larger combined area Σ in the definition of the smallsignal gain parameter g0 given by (16), thus reducing the strength of couplingbetween electron beam and optical field and reducing the gain. Figure 6 showshow g0 varies with normalised emittance, using the suggested parameters of theERL Prototype FEL. This plot indicates that the normalised emittance should beas small as possible.

Beam Quality The emittance is also a measure of the beam quality and a smalleremittance equates to a higher quality beam. As emittance increases (and the beamquality decreases) electrons move more off axis. The undulator magnetic field has asinusoidal z-dependence on the axis of the undulator so off-axis electrons experiencea slightly different field and thus a slightly different K—this causes a wavelengthshift and a broadening of the natural linewidth. Similarly, electrons moving at anangle to the axis give rise to a different wavelength. Again the Madey Theoremtells us that the gain is reduced by these effects. Various upper limits on emittancefor good coupling have been proposed [21, 5] but these make assumptions aboutrelative sizes of beam and optical cross-sections. Instead a gain correction factorCx,y [1] is used here to determine at what value emittance starts to cause gaindegradation due to poor beam quality, using the parameters specific to the ERL


10−8

10−7

10−6

10−5

10−4

10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

g 0

Normalised emittance (pi m rad)

Figure 6: ERL Prototype IR-FEL: Variation of g0 with normalised emittance.

Prototype. The correction is given by

Cx,y =1

1 + µ2x,y

(24)

where

µx = πNεn

λu

K

1 + K2/2, µy =

√2µx. (25)

The expressions are valid for a 100% coupled cylindrical beam so that εn = εn,x =εn,y. Figure 7 shows Cx against normalised emittance for the ERL Prototype pa-rameters. It is stressed that Cx only shows gain reduction due to beam quality anddoes not depend on the relative sizes of electron and optical beams in the cavity.The Figure suggests a normalised emittance satisfying εn < 10−5 m-rad would causenegligible gain degradation due to the effects discussed.

2.1.3 Gain reduction due to slippage

As electrons pass along the undulator they slip back one radiation wavelength perundulator period. The total slippage length per undulator traverse is thus given by

∆ = Nλr. (26)

When the electron bunch is short and/or the wavelength is long the slippage lengthcan be of the order of the electron bunch length. In this case the available interactionlength is reduced and the gain degraded.

The FEL interaction also has the effect of reducing the phase velocity of theoptical pulse below its vacuum value. The electrons slip back over the pulse asthey pass along the undulator. The bunching also increases as they pass along theundulator and the maximum emission occurs in the final periods where the bunchingis strongest. As a result the optical pulse peaks at the rear so that the centroid ofthe pulse moves at a velocity slightly less than c. This is known as lethargy.


10−8

10−7

10−6

10−5

10−4

10−3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cx

Normalised emittance (pi m rad)

Figure 7: ERL Prototype IR-FEL: Variation of Cx with normalised emittance.

These effects can be offset to some extent by cavity length detuning. For normalperfect synchronism between electron bunch and optical pulse the cavity lengthshould satisfy

D =nc

2f(27)

where n is an integer and f is the electron bunch repetition frequency. To allow forslippage and lethargy the cavity length is slightly reduced by δL. It is found howeverthat it is impossible to maintain synchronism during both the growth phase andthe saturated phase with a fixed cavity length because the optical group velocityreturns to its vacuum value as the gain decreases and the power saturates [14]. Asa result there is an optimum detuning given by

δLopt =0.456g0∆4(1 + µc

3 )(28)

at which the maximum gain is reduced by a factor

Cc =(1 +

µc

3

)−1

, µc =∆σz

. (29)

For the parameters of ERL Prototype Cc is shown as a function of bunch length inFigure 8. It is seen that to avoid any gain degradation the bunch duration shouldbe greater than 1.8ps. This gives σz = 0.54mm and Le =

√2πσz = 1.35mm which

compares to a slippage length of 0.08mm. A bunch duration of 0.1ps would matchthe slippage length. At this point the gain would be reduced due to slippage andlethargy by about 0.5.

However, shorter bunches provide higher peak current which provide higher gain.This can more than offset the reduction due to slippage. In addition, an oscillatorFEL operating with bunch lengths comparable to the slippage length can produceinteresting effects including superradiant spikes, limit cycle oscillations and chaoticbehaviour [12], as well as enhanced extraction efficiency [8, 6].

Recommendations for suitable bunch lengths to explore these behaviours aregiven in Section 3.2.3.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−12

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Cc

Bunch length (s)

Figure 8: ERL Prototype IR-FEL: Variation of Cc with bunch length

2.2 Numerical Gain Calculation

The MATLAB simulation program can also be used to calculate the gain, as illus-trated in Figures (2-4). The following expression is used:

G =|A|2 − |A0|2

|A0|2 . (30)

The model is 1D but the combined area Σ of the electron and optical beams can beincluded via g0 by expressing ρ as

ρ =(πg0)1/3

4πN. (31)

Multiplying g0 by some correction factor C is equivalent to multiplying ρ by C1/3

so gain reduction factors can be accounted for by using an effective FEL parameter.For example, the effect on the interaction due to slippage can be estimated by usingρeff = ρ× C

1/3c .

Beam energy spread σγ can be included by specifying an energy spread in theinitial electron distribution. The program assumes a gaussian distribution. Emit-tance can then be included [18] as an effective energy spread σε, expressing the totalenergy spread as

σeff =√

σ2ε + σ2

γ (32)

where

σε =εna2

wk2wβ

4γr(1 + a2w)

(33)


2.3 Calculating Output Characteristics

2.3.1 Power and Efficiency

The equilibrium intracavity peak intensity is the value of the intensity for whichthe net gain of the system is zero. It is given by [10]

Ie,i =2π

1− η

ηG

{1− exp

[− 1.8

1 + G

G(1− η)− η

η

]}Is (34)

where Is is the intracavity saturation intensity given by

Is[MW/cm2] = 6.9× 102( γ

N

)4 1(λu[cm]K)2

1F (ξ)

(1 + 2µ2

e

1 + 0.12g0

). (35)

This expression accounts for the dependence3 of Is on g0 and µe. η is the totalcavity losses and G the maximum small signal gain. The saturation intensity is thevalue of the intensity at which the gain drops to half of its initial value.

Assuming the intensity Ie is the peak on-axis value it can be converted from apower density to a power using

Pe = Ie · Σ. (36)

where Σ is the combined area of the electron and optical beams given by (17).By writing η = L + α, where L represents the passive losses (diffraction, reflec-

tivity) and α the outcoupling fraction, the output power is given by

Pout = α× Pe. (37)

and the efficiency is just the ratio of the output and electron beam powers:

E =Pout

Pbeam. (38)

The theoretical efficiency can be derived by considering the small signal gaincurve in Figure 1 on p.5. An electron injected at the peak of the detuning curvewhere the gain is maximum has an energy corresponding to ν = 2.6. As the electronpasses through the undulator it can continue to lose energy to the optical field untilits energy is reduced to ν = 0 where there is no longer any gain. This gives

∆ν = 4πN∆E

E= 2.6 (39)

so that∆E =

2.64πN

E. (40)

Dividing by time, and recognising that at equilibrium the intracavity power is con-stant, so that the power extracted from the electron beam is approximately theoutput power (i.e. neglecting passive losses), gives

Pout =2.6

4πNPbeam ' 1

5NPbeam (41)

and an efficiency of

E =1

5N. (42)

3g0 used in this expression should not be previously corrected for energy spread.


2 4 6 8 10 12 14

100

200

300

400

500

600

700

Intr

acav

ity P

ower

(M

W)

Outcoupling (%)

2 4 6 8 10 12 148

9

10

11

12

Pow

er O

ut (

MW

)

Outcoupling (%)

Figure 9: Variation of equilibrium intracavity power and output power with out-coulping fraction α.

There is an optimum outcoupling fraction which gives the highest output powerand hence the highest efficiency. This is shown is Figure 9 which uses the suggestedparameters of the ERL Prototype FEL. The equilibrium intracavity power andoutput power are plotted against outcoupling fraction, and the optimum outcouplingis seen to be 7%. The value of the optimum α can be estimated from the followingexpression [10]:

αopt = G0.86

(0.141+G

) 12 − 0.28

0.862 − 0.56(1 + G). (43)

For an oscillator FEL operating with bunch length shorter than the slippagelength the efficiency is maximum when the cavity length detuning is very small, forsmall losses. In this case the emission is superradiant with an efficiency given by [6]

Esuperradiant ∼ ρ

√Lb

ηLc(44)

where Lb =√

2πσz is the electron bunch length, η is the total cavity losses andLc = λ/4πρ is the cooperation length.

The numerical solution of the universally scaled FEL equations (5-7) can also beused to estimate the efficiency of an oscillator FEL. In this formulation the efficiencyof a single pass amplifier FEL is given by Eamp = ρ|A|2 so that

Prad = ρ|A|2Pbeam. (45)

However for an oscillator the equilibrium intensity |A|2 within the cavity is constantby definition, so that the power delivered to the cavity must equal the outcoupledpower and the losses. This means that the radiation power due to a single pass is

Prad,pass = ηρ|A|2Pbeam (46)


of which the fraction just due to outcoupling is given by the cavity efficiency

Ecav =α

L + α=

α

η(47)

so that the intensity output power is

Pout = Ecavηρ|A|2 = αρ|A|2 (48)

giving the oscillator efficiency as

Eosc = αρ|A|2 = ρ|A|2out. (49)

At saturation the optical pulse duration matches the electron bunch duration, sothe mean output power and equilibrium intensity can be calculated by multiplyingthe peak values by the electron beam duty cycle. The energy per pulse is then themean output power divided by the number of pulses per second.

2.3.2 Linewidth and Brightness

Following the discussion in [9] the laser bandwidth is given by(

∆ω

ω

)

L

∼ λr

2πσz(50)

The output brightness BL is the output spectral flux divided by the phase spacevolume of the effective source dimensions [4]:

BL =N

4π2ΣxΣyΣx′Σy′. (51)

Assuming the electron beam emittance is negligible and that the optical beam isdiffraction limited this expression reduces to

BL =4N

λ2r

(52)

and the flux N is found by dividing the output power by the energy of a singlephoton:

N =Pout

hc/λr(53)

2.3.3 Gaussian Beam and Resonator Equations

The theory of Gaussian beams and resonators is covered in the review article byKogelnik and Li [15]. The principal equations are given here.

For the lowest order Gaussian mode (TEM00), assuming a symmetric cavity oflength D and mirror radii of curvature R, the mode size (radius) at a distance z

from the waist is given by

w2(z) = w20

(1 +

(λrz

πw20

)2)

(54)

where the mode size at the waist is

w2(0) = w20 =

λr

2π

√D(2R−D). (55)


The Rayleigh length, the distance from the waist over which the mode area doubles,is

zR =πw2

0

λr. (56)

The stability parameter is given by

g = 1− D

R(57)

and the cavity is stable if 0 < g2 < 1. Symmetric cavities at either end of thisstability range are the concentric cavity, where R = D/2, and the confocal cavitywhere R = D. Near concentric cavities have a large spot size at the mirror and asmall waist, whereas near confocal cavities have a smaller mirror spot size and alarger waist.

The region specified byz À zR (58)

is known as the far field. In practice [20], the strength of the inequality is understoodto be 20 to 50 times the value of zR. In this region the divergence half-angle is givenby

θ =λr

πw0(59)

The angular alignment tolerance for the mirrors is given by [2]

θm ¿ (2λr/πD)1/2(1− g)1/4(1 + g)3/4 (60)

The optical axis is defined as the line passing through the centres of curvature ofthe mirrors. For a near concentric cavity the centres of curvature are very close andthe optical axis tilts by a large amount when either mirror is tilted. Therefore thisangular alignment tolerance is very strict for near concentric cavities and less strictfor near confocal cavities.

Alternatively, and again for symmetric cavities, the mode size can be expressedin terms of the stability parameter g. The mode waist is given by

w0 =

√Dλr

2π

(1 + g

1− g

) 14

(61)

and the mode size at the mirrors is given by

w1 = w2 =

√Dλr

π

(1

1− g2

) 14

. (62)


3 ERL Prototype: FEL Optimisation

3.1 Suggested Parameters

The ERL Prototype IR-FEL will use the J-LAB wiggler, with parameters given inTable 1, so the parameters within our control relate to the electron beam and theoptical cavity. A suggested set of achievable parameters is given in Table 2. Theseparameters are believed to offer a near optimal solution and are discussed in turnto determine how their variation, whilst keeping other parameters constant, affectsFEL performance. The focus is on gain and efficiency. Suitable values for someelectron beam parameters (such as energy spread, emittance and bunch length)have already been discussed in previous sections but only in relation to their effecton the gain. In this section these parameters are discussed in the wider context ofthe whole device and what we would like to achieve with it.

Parameter ValuePeriod 0.027 mGap 12 mmNumber of periods 40Length 1.08 mK 1.0

Table 1: The Main Parameters of the JLab Wiggler.

Using the suggested parameters in Table 2 the maximum gain is calculated as37.5%, but it must be stressed that this must be treated as a best estimate. In a realmachine the gain is likely to be lower (and the cavity losses higher) for a numberof reasons (mirror alignment errors, timing jitter leading to poor synchronisationbetween optical pulse and electron bunch, beam instabilities etc.) so a good safetymargin must be allowed for—the FEL on the ERL Prototype must work. There-fore, when discussing each parameter a threshold value is identified at which thetheoretical maximum gain falls below 20%. Even allowing a factor of two reduc-tion in a real machine this would still allow the FEL to operate. This thresholdvalue therefore defines the absolute minimum (or maximum) value that should beobtained, assuming all other parameters are at their optimum values.

3.2 Electron Beam

3.2.1 Energy

The electron beam energy for the ERL Prototype seems to be set at a nominal valueof 50MeV. At this energy the FEL output wavelength is 2µm as given by (1) on p4.The J-Lab wiggler is a permanent magnet fixed gap insertion device so has a fixedK parameter. The only way to tune the output wavelength is to change the beamenergy. Because one of the attractions of a free-electron laser is its tunability thebeam transport system should allow for a range of beam energies. Figure 10 showshow the gain, efficiency, output power and wavelength alter as the beam energy isreduced from 50MeV to an arbitrary 25MeV.


Electron Beam parametersEnergy 50MeVBunch charge 80pCBunch length 0.6psNormalised emittance 5 mm-mradEnergy spread 0.1%Bunch repetition rate 162.5MHzβ function at undulator 2m

Optical Cavity parametersLength 7.3795mMirror radius of curvature 3.8839mMirror aperture (radius) 0.015mMirror reflectivity 99.5%

Table 2: ERL Prototype IR-FEL: Suggested electron beam and optical cavity pa-rameters

25 30 35 40 45 50

0.4

0.45

0.5

0.55

0.6

Gm

ax

E (MeV)25 30 35 40 45 50

0.36

0.38

0.4

0.42

0.44

0.46

0.48

Effi

cien

cy (

%)

E (MeV)

25 30 35 40 45 50

3

4

5

6

7

8

wav

elen

gth

(mic

rons

)

E (MeV)25 30 35 40 45 50

5

6

7

8

9

10

11

12

x 106

Pea

k O

utpu

t Pow

er (

W)

E (MeV)

Figure 10: ERL Prototype: Gain, efficiency, output power and wavelength as afunction of beam energy. All other parameters are as listed in Table 2.


20 40 60 80 100 120 140 160 180 200

0.2

0.4

0.6

0.8

1

Gm

ax

Bunch charge (pC)

20 40 60 80 100 120 140 160 180 200

0.38

0.4

0.42

0.44

0.46

0.48

0.5

Effi

cien

cy (

%)

Bunch charge (pC)

Figure 11: ERL Prototype: Gain and efficiency as a function of bunch charge. Allother parameters are as listed in Table 2.

3.2.2 Bunch charge

The small signal gain parameter g0 is proportional to the peak current so we requirethe bunch charge to be as high as possible. Figure 11 shows the gain and efficiencyas the bunch charge varies from 20pC to 200pC, assuming a constant bunch length.Clearly the bunch charge must be as high as possible. The suggested value, consis-tent with the predicted gun output, is 80pC. Below 45pC the gain falls below 20%.For this reason a bunch charge of 45pC is suggested as a minimum value for goodFEL operation. As the bunch charge increases the efficiency tends asymptoticallytowards the theoretical maximum of E ' 1/5N = 0.5%.

3.2.3 Bunch length

Figure 12 shows how the gain, efficiency, gain degradation due to slippage and out-put power vary with bunch length, assuming a beam energy of 50MeV. The bunchlength is inversely proportional to the peak current, so although the gain degrada-tion increases with shorter bunches the maximum gain increases. To achieve 20%gain the bunch length must be below 1ps (2.5ps FWHM) at the FEL, demonstrat-ing that the proposed >10ps(FWHM) bunch from the injector must be compressedby at least a factor of 4. There is a peak in efficiency at a bunch length of 0.8pswhere the gain is 28.5% and the peak output power is 9.7MW. However, the peakoutput power increases at shorter bunch lengths due to the increased peak current.Reducing the bunch length to 0.6ps would increase the gain from 28.5% to 37.5%and increase the peak power by 3.1MW to 12.8MW with minimal loss of efficiency.A target bunch length of 0.6ps is thus suggested for normal operation at 50MeV.

As mentioned in Section 2.1.3, interesting behaviours are observed at saturationwhen the bunch length is equal to, or less than, the slippage distance. At 50MeV theoutput wavelength is 2µm giving a slippage length of 0.08mm. For a bunch lengthmatching the slippage length the bunch duration needs to be 0.1ps. At 25MeVthe radiation wavelength is 8µm giving a slippage length of 0.32mm. This can be


0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6G

max

Bunch length (ps)0.5 1 1.5 2 2.5 3

0.3

0.35

0.4

0.45

Effi

cien

cy (

%)

Bunch length (ps)

0.5 1 1.5 2 2.5 3

0.6

0.7

0.8

0.9

Cc

Bunch length (ps)0.5 1 1.5 2 2.5 3

1

2

3

4

x 107

Pea

k O

utpu

t Pow

er (

W)

Bunch length (ps)

Figure 12: ERL Prototype: Effect of varying bunch length at 50MeV. All otherparameters are as listed in Table 2.

matched by a bunch duration of 0.4ps.To summarise:

• For the FEL to achieve a theoretical maximum gain of 20% per pass the bunchlength at the FEL must be shorter than 1ps (2.5ps FWHM) necessitating atleast a factor of four bunch compression of a 10ps(FWHM) bunch from theinjector. Longer bunches from the injector will require more compression.

• For operation near the peak of efficiency with a gain of 37.5% the bunch lengthshould be 0.6ps (1.5ps FWHM).

• To enable exploration of short pulse effects at a reduced beam energy of 25MeVthe bunch length should be compressed to less than 0.4ps (1ps FWHM).

3.2.4 Emittance

Figure 13 shows the variation of gain and efficiency as the normalised emittance isvaried. Also shown are the gain reduction factors Cx,y which quantify the effect ofbeam quality on the gain. It is seen that the suggested emittance of 5mm-mradis well below the threshold value that would cause gain degradation due to beamquality via electrons moving off-axis along the undulator. However, as discussedin Section 2.1.2 the gain also depends on the combined area of the optical andelectron beam cross sections. This dependence is relevant at emittances below thethreshold value for beam quality. It is seen that to increase the gain and efficiencythe emittance should be as low as possible. The suggested value is thought to be


10−7

10−6

10−5

10−4

0.1

0.2

0.3

0.4

0.5

0.6

Gm

ax

Normalised emittance (m−rad)10

−710

−610

−510

−40.15

0.2

0.25

0.3

0.35

0.4

0.45

Effi

cien

cy (

%)

Normalised emittance (m−rad)

10−7

10−6

10−5

10−4

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

Cy

Normalised emittance (m−rad)10

−710

−610

−510

−4

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

Cx

Normalised emittance (m−rad)

Figure 13: ERL Prototype: Effect of normalised emittance on gain, efficiency andgain degradation due to beam quality. All other parameters are as listed in Table2.

not too demanding a target, but any further reduction would be beneficial. Thegain falls below 20% for normalised emittance greater than 15mm-mrad so this isgiven as the threshold value.

3.2.5 Energy Spread

Figure 14 shows the effect of an increasing beam energy spread. It is clear thatthe energy spread should be as low as possible. A suggested value of 0.1% causesvery little gain degradation but a spread of up to 0.3% still allows optimum outputpower at a reduced gain of 27%. The gain falls below 20% at an energy spread of0.45%, so this is suggested as the threshold value.

3.2.6 β function

The beam transport system must deliver a dispersion free beam to the undulatorentrance, with a horizontal beam waist at the centre of the undulator and a verticalbeam waist at the entrance to the undulator. This arrangement minimises the effectof emittance on the FEL interaction. The suggested β function value of 2m is anaverage along the undulator in both planes so that βx = βy = 2. The smaller theβ function at the waist the smaller the necessary distance between the quadrupoleseither side of the undulator. As a general rule this distance between quads is aboutequal to the waist β function. The undulator is 1.08m long so a β function of 2m isthought to be a reasonable minimum. If the β function is increased above 2m the


0 0.2 0.4 0.6 0.8 1

0.1

0.15

0.2

0.25

0.3

0.35

Gm

ax

Energy spread (%)0 0.2 0.4 0.6 0.8 1

0.36

0.38

0.4

0.42

0.44

0.46

0.48

Effi

cien

cy (

%)

Energy spread (%)

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

Ce

Energy spread (%)0 0.2 0.4 0.6 0.8 1

0.95

1

1.05

1.1

1.15

1.2

1.25

x 107

Pea

k O

utpu

t Pow

er (

W)

Energy spread (%)

Figure 14: ERL Prototype: Variation of gain, efficiency, gain degradation and peakoutput power as a function of energy spread

beam cross section increases, increasing the combined area Σ and decreasing thegain. This is shown in Figure 15. The gain falls below 20% at β = 6m defining thisas the threshold value.

3.3 Optical Cavity

3.3.1 Cavity Length and Mirror Radii of Curvature

Assuming the RF frequency is fRF = 1.3GHz, convenient frequencies for the pho-tocathode laser are subharmonics of fRF . The most suitable frequency for thephotocathode laser is then fl=162.5MHz [13]. Optical pulses are generated in theFEL cavity at this frequency and the round trip frequency of each pulse must bean integer n subharmonic of fl for each pulse to coincide with an electron bunch oneach trip through the cavity. This defines the allowed cavity lengths via

D =nc

2fl(63)

There will be n independent optical pulses propagating in the cavity.4 The allowedcavity lengths for n = 1 . . . 20 are given in Table 3.

The proposed layouts allow a minimum possible cavity length of around 7m ifthe mirrors are placed in the chicanes so the allowed cavity length of 7.37951m issuggested.

The mirror radii of curvature must then be determined. It is accepted [22, 16, 19]that the optimal interaction between electron beam and optical beam occurs whenthe Rayleigh length zR ' 0.25Lu. However it is found that the required radii of

4An issue for further study is the effect on the electron beam of Compton backscattering within

the cavity.


2 4 6 8 10

0.15

0.2

0.25

0.3

0.35

Gm

ax

Mean Beta function (m)2 4 6 8 10

0.42

0.43

0.44

0.45

0.46

0.47

0.48

Effi

cien

cy (

%)

Mean Beta function (m)

Figure 15: ERL Prototype: Variation of gain and efficiency as the mean β functionwithin the undulator is altered. All other parameters are as listed in Table 2.

n Cavity Length D (m) n Cavity Length D (m)1 0.922438 11 10.14682 1.84488 12 11.06933 2.76731 13 11.99174 3.68975 14 12.91415 4.61219 15 13.83666 5.53463 16 14.7597 6.45707 17 15.68148 7.37951 18 16.60399 8.30194 19 17.526310 9.22438 20 18.4488

Table 3: ERL Prototype: Allowed cavity lengths for the oscillator FEL.


4 5 6 7

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Gm

ax

R(m)4 5 6 7

0.42

0.44

0.46

0.48

Effi

cien

cy (

%)

R(m)

4 5 6 70

0.2

0.4

0.6

0.8

Sta

bilit

y pa

ram

eter

g2

R(m)4 5 6 7

1.1

1.15

1.2

1.25

1.3x 10

7

Pea

k O

utpu

t Pow

er (

W)

R(m)

Figure 16: Variation of gain, efficiency, cavity stability parameter g2 and outputpower as the radius of mirror curvature is varied over the full range of stable valuesfrom a concentric cavity (R = D/2) to a confocal one (R = D). All other parametersare as listed in Table 2.

curvature to achieve this lead to a near-concentric cavity which is very close tounstable and sensitive to mirror alignment errors. Figure 16 shows how the gain,efficiency, cavity stability parameter g2 and output power vary as the radius ofmirror curvature is varied over the full range of stable values from a concentriccavity (R = D/2) to a confocal one (R = D). It is seen that gain and efficiency arehigher for the cavities nearer concentric. Restricting the cavity stability parameterto g2 < 0.81, which is within the range of other FEL cavities [22], gives an optimumradius of curvature of R=3.8839m.

It is possible that the mirrors will not be able to be sited within the bunchcompression chicanes and that one mirror will be outside the return arc. For thisreason the gain and efficiency for all allowed cavity lengths up to 19m has beencalculated, adjusting the mirror radii of curvature for each cavity length to keep thesame cavity stability parameter. The results are shown in Figure 17.

It is seen that the gain and output powers decrease as the cavity length increases.This is explained by expressing the optical mode waist (assumed to be at the centreof the undulator) as follows:

w0 =

√Dλ

2π

(1 + g

1− g

) 14

(64)

so that for constant g the optical mode area is proportional to the cavity length.The combined area Σ thus increases with increasing cavity length, decreasing thesmall signal gain parameter.

To keep the gain and output power as high as possible the cavity length shouldbe as short as possible with 7.37951m given as the suggested length. Furthermore, ashorter cavity length is advantageous as it reduces the requirements on mechanicalstability and minimises the required size of the mirrors. 5

5One issue which has not been considered here is the power density loading on the mirrors


6 8 10 12 14 16 18 2020

25

30

35

40

Cavity Length (m)

Gai

n (%

%)

6 8 10 12 14 16 18 200.45

0.455

0.46

0.465

0.47

0.475

0.48

0.485

Cavity Length (m)

Effi

cien

cy (

%%

)

Figure 17: Gain and efficiency for allowed cavity lengths. All other parameters areas listed in Table 2.

3.3.2 Mirror Aperture and Reflectivity

If the beam energy is reduced to 25MeV the wavelength increases to 8µm. From(62) on p19 the mode size at the mirrors is proportional to the square root ofthe wavelength. The mirror aperture should therefore be large enough to providenegligible diffraction loss at 8µm. As a general rule the mirror aperture (radius)should be at least twice the mode size (radius), giving an aperture of 13.4mm for amode size at the mirror of 6.7mm at 8µm. Results of calculations using the methodsfrom [23] are shown in Figure 18 and indicate that the minimum mirror apertureshould be a radius of 15mm.

Mirror reflectivity should be as high as possible to reduce passive loss. In generalreflectivity exceeds R = 99.5% at normal incidence for Far-IR to Near-IR [19] so99.5% reflectivity is used in this report.

The method of outcoupling has not been discussed. If partially transmittingmirrors are required then their reflectivity should be chosen to provide the op-timum outcoupling. Hole coupling and insertable scaper mirror outcoupling areother possibilities and are discussed in [23]. A scraper mirror seems a good choicebecause it allows easy extraction from the side of the cavity leaving the straight-onpath at the arc free for undulator radiation from U5. Also, because the optimumoutcoupling fraction depends on the gain (see Equation (43)), the size of the scraper

leading to thermal distortion. It is possible that if calculations show the power density is too

high a longer cavity might be advantageous. Alternatively the mirror radii of curvature could be

adjusted to give a smaller waist and a larger mirror spot size, at the expense of increasing the

cavity stability parameter towards the unstable region.


0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.021

2

3

4

5

6

Pas

sive

loss

(%

)

Mirror Aperture (m)

0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Effi

cien

cy (

%)

Mirror Aperture (m)

Figure 18: Cavity passive loss and efficiency variation with mirror aperture (radius)at a wavelength of 8µm. The beam energy here is 25MeV and other parameters areas listed in Table 2.

mirror could be varied experimentally to maximise the output power depending onthe measured gain.

3.4 Summary

A summary of the suggested parameters is given in Table 4. Also given, whereappropriate, are the threshold parameter values at which the gain drops below 20%.It is stressed that these threshold parameter values assume all other parameters areat their suggested values. If more than one parameter was at its threshold value thegain would be reduced well below 20%. If all parameters were at their thresholdvalues the FEL would not work at all.


Parameter Suggested Value Value for gain > 20%

Electron Beam Parameters

Energy 50MeV -

Bunch charge 80pC >45pC

Bunch length 0.6ps (rms) <1ps (rms)

[<0.4ps (rms) for short pulse effects]

Normalised emittance 5 mm-mrad <15 mm-mrad

Energy spread 0.1% (rms) <0.45% (rms)

Bunch repetition rate 162.5MHz -

β function at undulator 2m <6m

Optical Cavity Parameters

Length 7.3795m -

Mirror radius of curvature 3.8839m -

Mirror aperture (radius) 0.015m -

Mirror reflectivity 99.5% -

Table 4: ERL Prototype IR-FEL: Electron beam and optical cavity parameters—suggested values and threshold values. The threshold parameter values assume allother parameters are at their suggested values. If more than one parameter was atits threshold value the gain would be reduced well below 20%.

4 ERL Prototype: Effect of FEL Interaction on

Electron Beam

4.1 Induced Energy Spread

The primary effect on the beam due to the FEL interaction is an induced energyspread. As can be seen from Figure 4 on p9 the induced beam energy spreadincreases as the power in the cavity builds up, saturating when the cavity power isincreasing at its maximum rate. The final beam energy spread σex is due both to theinitial energy spread σe(0) and the induced energy spread σi(x), and is estimatedby the following expression [11]

σ2ex = σ2

e(0) + σ2i (x) (65)

where

σi(x) ' 0.433N

e−0.4x

(βx

1− e−βx− 1

) 12

, x =I0

Is, β =

π

21.0145. (66)

The final energy spread can also be estimated from the numerical solution of theFEL equations (5-7) on p6.

The return arc must have an energy acceptance sufficient to transport all theelectrons, so the full energy spread is of more interest. Assuming a gaussian distri-bution a range of ±3σ represents 99.7% of the electrons, so a good estimate of thefull FEL exhaust energy spread is given by 6σe. This has been calculated using theformula given here and numerically via the 1D MATLAB code, using the optimumsuggested parameters of Table 4 and a range of input energy spreads. The resultsare shown in Figure 19.

It is seen that for an input energy spread of 0.1%(rms) at 50MeV the full exhaustenergy spread due to the FEL interaction will be approximately 4%, so this should


0 0.1 0.2 0.3 0.4 0.53.5

4

4.5

5

5.5

Exh

aust

E s

prea

d (6

σ)(%

)

Input energy spread (%)

Figure 19: Estimation of the full (6σ) exhaust energy spread from the FEL atsaturation. The solid line is the prediction of Equation (65) and the squares showthe numerical solution of the 1D FEL equations.

be seen as a lower bound on the energy acceptance of the beam transport returnarc.

If the FEL is used to investigate short pulse effects enhanced efficiencies maybe obtained leading to a larger induced energy spread. For a bunch length ofσz = 0.4ps and a beam energy of 25MeV the superradiant efficiency (as given byEquation (44)) is about 1.5%, a factor of 2-3 increase over the long-pulse efficiency.The induced energy spread is proportional to the efficiency so this indicates theenergy acceptance of the return arc will need to be as high as 8-12% for full energyrecovery during short pulse operation.

4.2 Other Effects

These are expected to be minor compared with the induced energy spread. Thebunch length is expected to remain essentially unchanged because the electron dis-tribution is modulated on the scale of a radiation wavelength (2µm) which is smallcompared to the bunch length (σz=0.18mm for 0.6ps duration). The effect of theFEL interaction on the emittance has yet to be studied.


5 ERL Prototype: FEL Output Characteristics

The semi-analytic and numerical methods have been applied to the suggested pa-rameters from Table 4 to calculate a comprehensive range of electron beam, cavityand output values which are given in Table 5. Most values are from the semi-analyticformulae. The values in brackets are the numerical results.

The electron beam temporal structure is assumed to be as follows: bunch repeti-tion frequency 162.5MHz, with 100µs macropulses every 1ms. These are the figuresproposed recently by the Central Laser Facility.

Semi-analytic (numerical) calculationg0 0.48Maximum gain Gmax 37.5(37.5) %σx,y 0.32 mmσz 0.18 mmIpeak 53 AIaverage 1.3 mADuty cycle 2.44 · 10−5

Duty cycle within macropulse 2.44 · 10−4

Electron beam power Pbeam 2.65 GWExhaust energy spread (σex) 0.67 (0.67) %Full exhaust energy spread (6σex) 4.0 (4.0) %Intracavity equilibrium intensity Ie 1.13 · 1014 W/m2

Intracavity peak power Pe 186 MWIntracavity average power Pe 4.54 kWOutput peak power Pout 12.8 MWOutput average power Pout 312 WEnergy per pulse 19 µJExtraction efficiency E 0.48 (0.64) %Optimum outcoupling α 6.8 %Peak cav. equil. brightness Be 1.8 · 1027 photons/s/mm2/mrad2/0.1%bwPeak Output brightness BL 1.2 · 1026 photons/s/mm2/mrad2/0.1%bwAverage Output brightness BL 3.0 · 1021 photons/s/mm2/mrad2/0.1%bwBandwidth (∆ω/ω)L 0.18 %Divergence half angle θ 0.89 mradRayleigh length zR 0.85 mw0 0.75 mmw at undulator entrance 0.80 mmw at mirror 3.37 mmPassive loss L 1.0 %Stability parameter g −0.9ρ parameter 2.29 · 10−3

Gain parameter G 1.15

Table 5: FEL parameters and output values calculated from the suggested param-eter values in Table 4, using the semi-analytic (numerical) methods in this report.


References

[1] Rene Bakker. Free electrons as a versatile source of coherent radiation. PhDThesis, 1993.

[2] Charles A. Brau. Free–Electron Lasers. Academic Press Inc, 1990.

[3] A. Chesworth. The FEL gain equation (Issue 2). DL/EUFEL/99/05, 1995.

[4] J. A. Clarke. Undulators and Wigglers: Their Radiation Properties and Mag-netic Design. To be published.

[5] G. Dattoli and A. Renieri. Laser Handbook Volume 4. North Holland, 1985.

[6] N. Piovella et al. Analytical theory of short-pulse free electron laser oscillators.Phys. Rev. E, 52(5):5470, 1995.

[7] R. Bonifacio et al. Physics of the high-gain fel and superradiance. Rivista DelNuovo Cimento, 13(9), 1989.

[8] R. Hajima et al. Demonstration of a high-power fel oscillator with high extrac-tion efficiency. PAC Proceedings, 2001.

[9] A. Torre A. Renieri F. Ciocci, G. Dattoli. Insertion Devices for SynchrotronRadiation and Free Electron Laser. World Scientific, 2000.

[10] L. Giannessi G. Dattoli and A. Torre. Saturation and cavity-loss optimizationin free-electron lasers. Phys Rev E, 48(2):1401–1403, 1993.

[11] P. L. Ottaviani G. Dattoli, L. Giannessi and A. Torre. Parameterization of theelectron beam induced energy spread in free electron laser dynamics. J. Appl.Phys., 76(1), 1994.

[12] S. J. Hahn and J. K. Lee. Nonlinear short-pulse propagation in a free-electronlaser oscillator. Phys. Rev. E, 48(3), 1993.

[13] G. Hirst. private communication.

[14] S. Khodyachykh. Experimental Study of the FEL with a Tapered Undulatorand Numerical Simulations of Short Pulse Free Electron Lasers. Ph. D. Thesis,Technischen Universitat Darmstadt, 2002.

[15] Kogelnik and Li. Laser Beams and Resonators. SPIE MS68, 1993.

[16] P. Luchini and H. Motz. Undulators and Free-Electron Lasers. OUP, 1990.

[17] B. W. J. McNeil. A simple model of the free electron laser oscillator from lowinto high gain. IEEE Journal of Quantum Electronics, 26(6):1124, 1990.

[18] B. W. J. McNeil. 4GLS Feasibility Report - The Free Electron Laser Interaction.2002.

[19] J. M. Ortega. CAS, 90-03.

[20] F. L. Pedrotti and L. S. Pedrotti. Introduction to Optics, 2nd Edition. PrenticeHall, 1996.


[21] J. B. Murphy & C. Pellegrini. Laser Handbook Volume 6. North-Holland, 1990.

[22] K. Saeki. NIM, A375:10–12, 1996.

[23] Neil Thompson. 4GLS Infra-Red Free-Electron Laser Design Study. fgls-fel-rpt-001, 2003.

erl prototype free-electron laser -...

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