LA-UR 14-23595

Physics of Free-Electron Lasers

One-dimensional FEL Theory

Dinh C. Nguyen & Quinn R. Marksteiner

Los Alamos National Laboratory

US Particle Accelerator School

June 23 – 27, 2014

LA-UR 14-23595

1. Coupled phase-energy equations

2. Wave equation

3. Charge and current density

4. Slowly-varying amplitude

5. Coupled FEL first-order equations

6. Third-order equation for high-gain FEL

7. Analytic solutions to third-order equation

8. Inclusion of 3D effects

2

LA-UR 14-23595 3

• Charge density and fields do not depend on the transverse coordinates (x, y), i.e., the electron beam radius is large.

• The electron bunch is long and end effects can be ignored.

• The radiation amplitude grows slowly with z (ignoring 2nd derivative).

• Diffraction of the FEL light is negligible over a gain length.

• The effects of electron beam emittance and energy spread can be ignored.

• 3D effects (diffraction, emittance, and energy spread) can be added later as corrections to the gain length and saturated power.

LA-UR 14-23595 4

We mentioned earlier that the oscillatory axial motion of the electrons in a planar undulator causes a reduction in the interaction strength. The reduction is expressed in terms of the difference between J0 and J1 Bessel functions of an argument x that depends on K (see textbook for explanation of J0 - J1).

JJ is unity for a helical undulator (no reduction). For a planar undulator, JJ becomes less than unity at large K. For K ≤ 1

The modified undulator parameter, K-hat, is used in calculations that involve the interaction strength, but not for wavelength calculation.

2

2

24 K

K

x

24

12 xx

x JJ

xxx 10 JJJJ

JJKK ˆ

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Ponderomotive phase

Taking the derivative with respect to z

Expressing kr in term of ku using resonance condition

2

2

2

12

Rr uk k

K

0 tzkk rur

z

rur kk

dz

d

21

2

11

2

2

K

ckk

dz

d rur

R

Ru

Ru

Ruu

rur

kkkkdz

d

Kkkk

dz

d

2

21

2

11

2

22

2

2

2

2

For small energy deviations

ukdz

d2

R

R

RR 2

where

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E = Type equation here.

Transverse velocity

Transverse current

zkcK

ux cos

tzkzkecK

EeW rrurx coscos

coscos2

02 ecKE

cme

xx ej

Rate of energy transfer

Transverse electric field of radiation

tzkEtzE rrr cos, 0

cos

2

ˆ

2

0

cm

EKe

dt

d

eR

cos

2

ˆ

22

0

cm

EKe

dz

d

eR

Rewrite in term of relative energy deviation, Rate of change of with respect to z

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Rate of change of the jth electron’s phase with respect to t (left) and z (right) Rate of change of the jth electron’s energy deviation w.r.t. to t (left) and z (right) These two coupled equations can be used to model the phase-space motion of N electrons at a constant radiation field amplitude. This applies to the case where a seed laser is used to induce the energy and density modulations, such as the seeded amplifier or the final few passes of an oscillator. In a high-gain FEL, the radiation intensity (and E0) changes with t and z.

j

eR

j

cm

EKe

dz

d

cos

2

ˆ

22

0

ju

jk

dz

d

2

j

eR

j

cm

EKe

dt

d

cos

2

ˆ

2

0

ju

jck

dt

d

2

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Shift the phase variable by p/2 and call it FEL pendulum equation where W is the synchrotron oscillation frequency. At one-quarter of the synchrotron period, the electrons have sinusoidal energy modulations. At one-half of the period, FEL bunching is maximum. Electrons become over-bunched after one-half of the synchrotron period.

sin

2

ˆ

2

0

Recm

KeE

dt

d

ckdt

du2

0sin2

2

2

W

dt

d

2

02ˆ

Re

u

m

kKeE

W

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The initial distribution (blue line) depicts mono-energetic electrons at time = 0. At ⅟4 of the synchrotron period, the electron distribution develops energy modulations (black line). Electrons trapped inside the separatrix (red) execute close-orbit motions. Electrons outside the separatrix flow over and under the separatrix. These electrons also provide gain (linear regime) until the separatrix grows in height and capture the untrapped electrons, causing them to execute trapped electron phase-space motions (nonlinear regime).

trapped

untrapped

separatrix

untrapped

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The separatrix or bucket (blue) separates the trapped (closed) orbits from the untrapped (open) orbits. Electrons with energy above the resonant energy R move lower and provide FEL gain. Electrons below R absorb FEL radiation power.

W

2cos

cku

S

Equation for the FEL separatrix

Gain function in a low-gain FEL

0

(R/R

Gain

Absorption

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The small-signal gain is calculated from the energy gained by the radiation (also equal to the energy lost by the electrons) in each pass divided by the radiation energy. The small-signal gain as function of the initial energy detuning D is derived using second-order perturbation theory. Start with zeroth-order solutions To the first-order e1 the equations are

...

...

2

2

10

2

2

10

tttt

tttt

ee

ee

00

0

2

.

D

D

uckt

constt

0

2

0

2

1

11

2sinsin

2

DWW

u

u

ckt

ckt

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Solutions to the 1st order equations The phase-averaged energy deviation is zero for the 1st order. To the first-order e2 the equations are For the 2nd order, the phase-averaged energy deviation is

D

D

D

W

DD

W

00

2

1

0

2

1

cos2

sin2sin

cos2cos2

tck

tckt

tckck

t

u

u

u

u

01

2

1

22

2cos

2

DW

u

u

cktt

ckt

D

DD

D

W 12sin

2

22cos

23

34

20

TckTck

TckTck

Tckt u

uu

u

u

...0 2

2 D tt e

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Madey theorem: the line shape of the small-signal gain of a low-gain FEL is the negative derivative of the spontaneous emission curve, a sinc-square function.

W

2

234 sin

4 x

x

xd

dTckxG u

Small-signal gain (stimulated emission) function of a low-gain FEL

D uNpx 2D uNpx 2

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2 2

02 2 2

1( , ) xjE z t

z c t t

We also assume the seed radiation is polarized in the x direction, the same as the electron oscillatory motion in a planar undulator. The x component of the radiation field interacts with the complex transverse current due to electrons’ velocity in x. This interaction causes the radiation amplitude to vary with z, especially in a high-gain FEL where the amplitude grows exponentially with z. We must now find the rate of change of radiation field amplitude with respect to z, unlike the situation in slide 7 where the radiation field amplitude is constant.

One-dimension approximation of the wave equation driven by a transverse complex current of electrons oscillating along the x direction in the undulator

Assume a trial solution to the wave equation in the complex form

tzkizEtzE rrxx exp~

,~

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2

2

022 exp xr

r r r r

jk ik E z E z i k z t

c t

Plugging the trial solution into the wave equation

The complex amplitude varies slowly with z so we can drop the 2nd derivative

Transverse current density is related to the longitudinal current density by

cosx z u

Kj j k z

zktzkit

j

k

Ki

dz

Edurr

z

r

x cosexp

~

2

ˆ~0

tzkit

j

k

i

dz

Edrr

x

r

x

exp

~

2

~0

Rate of change of field amplitude with respect to z

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The longitudinal current density has two components, the static (DC) and time-varying (AC) components.

0 1

0 1

exp

exp

z

z r u r

j j j z i

j j j z i k k z t

Taking the partial derivative with respect to time

1 expzr r u r

ji j z i k k z t

t

Rate of change of the field amplitude with z is proportional to the j1 current, i.e., the first harmonic Fourier component of the AC current.

10 ~

4

ˆ~

jKc

dz

Ed

R

x

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Electron longitudinal distribution

N

n

nS1

Complex Fourier coefficients

1

0 expRe2 k

k ikcc

S

p

pdikSkc

2

0

exp1

Expanding the distribution in complex Fourier coefficients

Fourier coefficient of the 1st harmonic

N

nnic

11 exp

1

p

LA-UR 14-23595 18

Initial DC current density 2

00

c

Ak

ecj

r

First harmonic current density 11

~c

Ak

ecj

r

N

nni

Njj

101 exp

2~

First harmonic current is proportional to the correlation of the phases of N electrons.

LA-UR 14-23595

19

Rate of change of the ponderomotive phase of the nth electron Rate of change of the relative energy deviation of the nth electron Rate of change of radiation field amplitude Rate of change of the j1 current amplitude

nun k

dz

d

2

n

rR

x

R

n ijciEK

cm

e

dz

d

exp

~

2

~ˆRe 1

2

0

2

0

10 ~

4

ˆ~

jKc

dz

Ed

R

x

N

n

niN

jj1

01 exp2~

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Follow the discussion on pp. 52 – 56 of the 2nd edition of the textbook.

Third-order equation

0~

~~

2

~ 2

2

2

23

x

x

FEL

px

FEL

x EiΓ

E

Γ

k

Γ

Ei

Γ

E

Growth rate

Plasma wave-number

FEL rho parameter

3

1

3

2

ˆ

2

1

A

pk

bu

FELI

I

k

K

FEL

u

Γ

p4

ck

u

rp

*2

eR

e

m

en

0

2*

e

Plasma frequency in beam frame

LA-UR 14-23595 21

1Γ

k pIn FEL where we can rewrite the 3rd order equation as

0~

~~

2

~ 2

23

x

x

FEL

x

FEL

x EiΓ

E

Γ

Ei

Γ

E

Special case: mono-energetic electron beam at resonant energy, i.e., 0

0~~ 3 xx EiΓE

Plasma oscillation is important in long-wavelength FELs (Raman regime). In many of today’s high-gain FEL, the plasma wavenumber is much smaller than the growth rate, and thus can be ignored.

We have arrived at a simple third-order differential equation.

LA-UR 14-23595 22

Using the trial solution of the form

one obtains the cubic equation

Three roots of the cubic equation:

1

3

2

i

Im

Re

-1 ½

3

2

-3 2

1

2

Growing mode

2

3

2

i

Decaying mode

3 i Oscillatory mode

3Γi3

αzAzEx exp~

3

LA-UR 14-23595 23

In the first 2-3 gain lengths, the growing, decaying and oscillating modes compete with one another. The sum of three electric fields grows slowly with z.

At z > 3 gain lengths, the exponentially growing mode dominates and the radiation field and intensity as functions of z can be written as

z

ΓEzE in

x2

3exp

3

~

Γziz

Γiz

ΓiEzE in

x exp2

3exp

2

3exp

3

~

1

3cos

3cosh4

3cosh4

9

~ 2

22

zΓ2

zΓ2

zΓ2

EzE in

x

ΓzE

zE in

x 3exp9

~2

2

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0 0.4 0.7 1.1 1.4 1.8 2.2 2.5 2.9 3.2 3.6

1h

Fundamental power

z (m)L

og P

ow

er

At saturation, the electrons have undergone one-half of the synchrotron period. The electrons begin to absorb the FEL radiation, but the electrons’ phase space becomes chaotic so the FEL power is not significantly reduced but oscillates about an average non-zero value. The synchrotron oscillation period is a measure of the FEL bucket height.

Undulator length

Exponential gain

Saturation

Lo

g o

f P

ow

er

FEL power grows exponentially with z Power gain length if no 3D effects FEL power at saturation (no 3D effects)

0

0 exp9 GL

zPzP

p

340

uGL

e

EIP bb

sat

Lethargy

LA-UR 14-23595

The rms radius of a focused electron beam is determined by its un-normalized (geometric) emittance and the average b function of the focusing optics (b is the electron beam’s property analogous to the photon beam’s Rayleigh length. One needs to match the average focusing b to the electron beam’s b).

Optical diffraction is measured by the radiation Rayleigh length, which is given by the square of the electron beam’s rms radius in the undulators divided by the photon beam’s emittance, /4p.

For diffraction 3D effect to be small, the gain length must be shorter than the Rayleigh length

25

RG zL

u

bave

e

b

2

p 24 bRz

LA-UR 14-23595

Normalized emittance is the phase space area of the beam in its rest frame. Un-normalized (geometric) emittance is the phase space area in Lab frame, where b ~ 1 for highly relativistic beams.

26

b

ee n

u

pz px

pz

px

x and z momenta at low energy

x and z momenta at high energy

Acceleration reduces x’ by boosting pz

z

x

p

px

LA-UR 14-23595 27

Converging Waist Diverging

x

x

photons

electrons

x

x

x

x

Photon beam emittance Un-normalized emittance

4

p

p

e

4u

For emittance 3D effect to be small, the electron beam’s geometric

emittance must be smaller than the photon beam emittance.

LA-UR 14-23595 28

where Nc = # of wavelengths in a coherence length

= √3 times the # of periods in one gain length

For 3D effect due to energy spread to be small, the relative rms energy spread must be less than

Electrons must maintain the same axial velocity during the coherence length

cNp

4

1

cc Nl

p4

1cN

cl

LA-UR 14-23595 29

Diffraction Ming-Xie parameters

Emittance

Energy spread

Ming-Xie parameters should be less than 1 to minimize 3D effects.

RD zL 1

R

Dd

z

LX 1

p

e

4u

r

u

ave

DLX

pe

be

41

cNp

4

1

p

u

DLX 14

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642

5311,,aaa

dd XaXaXaXXXF ee

1918171514121198

1613107

aaa

d

aa

d

aa

d

aaXXXaXXaXXaXXa eee

LA-UR 14-23595 31

LA-UR 14-23595 32

• 1D theory involves N first-order equations for N electrons coupled with two first-order equations for radiation field amplitude and 1st harmonic current.

Electron phase

Electron energy

Radiation field amplitude

Harmonic current

• The third-order FEL equation leads to the cubic equation that has three roots corresponding to the growing, decaying and oscillating modes.

• 3D effects (diffraction, emittance, and energy spread) can be included as modifications to the 1D theory via Ming-Xie parameterization.

nun ck

dt

d

2

10 ~

4

ˆ~

jKc

dz

Ed

R

x

N

n

niN

jj1

01 exp2~

n

rR

x

R

n ijciEK

cm

e

dz

d

exp

~

2

~ˆRe 1

2

0

2

0