Optimization of Compact X-ray Free-electron Lasers

Sven ReicheMay 27th 2011

Free-Electron Lasers

Electron Beam Parameters

Almost everything scales with the FEL parameter (1D Model):

The characteristic length is the power gain length:

Increasing the current and/or reducing the emittances increase the performance and reduce the overall required length of the FEL.

Impact of energy spread and emittance needs to be small (Both spreads out any bunching and thus act against FEL process):

(Ideal Case)

Saturation Power and Brilliance Saturation power:

For a given wavelength it favors a higher beam energy and thus a longer undulator period.

Peak Brilliance:

At saturation:

SwissFEL

(transverse coherence)

Optimizing the Focusing

Decreasing the -function (increase focusing), increases the FEL parameter .

Too strong focusing enhances the emittance effect and increasing the FEL gain length.

From 1D Theory:

3D Optimization for SwissFEL

Transverse Coherence (2D FEL Theory) Diffraction Parameter:

Assuming electron size as radiation source size:

Rayleigh Length

Char. Scaling of FEL (=2ku in 1 D model)

x

px

Photon Emittance Diffraction LimitedPhoton Emittance

Constraint for emittance to be smaller than photon emittance for all electrons to contribute on the emission process

Electron

Transverse Coherence (Saldin et al) Growth rates for FEL eigenmodes (r,-decomposition):

Increased gain length due to strong diffraction

Mode competition and reduced coherence

Optimum growth rate of 1D model

Optimum:

Note: fundmental FEL Eigenmode has intrinsic wavefront curvarture and thus correspond to a larger photon emittance

Growth of transverse coherence = spreading the phase information transversely.

Additional Effects

8

Diffraction Convection Spreading

Phase spread by field

Phase spread by electrons

Beam size variation

Emittance constraints is relaxed when realistic electron motion is included

FEL Theory only treats diffraction effects, but spreading is dominating effect in short wavelength FELs

TEM01 Growth rate

FODO Lattice

Rigid

Definition of Coherence (Goodman) Coherence is a stochastic quantity and

correlates the phase relation between two transverse location:

In relation to diffraction experiments: A pulse can be fully coherent without

100% contrast in interference pattern (E0(r1) ≠E0(r2)).

The quasi-monochromatic approximation (E0(t)=E0(t+dt)) is not necessarily fulfilled.

Single spike pulse is not a stationary process and would require ensemble average.

Mutual Coherence Function Normalized Coherence Factor

E0(r2,t-dt)

E0(r2,t)

E0(r1,t)

Example: Coherence for SwissFEL

10

n = 0.43 mm mradSpot Size: 0.0018 mm2

Coherence Area: 0.021 mm2

n = 0.86 mm mradSpot Size: 0.0023 mm2

Coherence Area: 0.015 mm2

=0.86

=0.69

Intensity Mutual Coherence

Intensity

Mutual Coherence

Towards Compact FELs

Maximize the FEL parameter by maximizing the peak current.

Large FEL parameters allows for relaxed values for energy spread and emittance.

“Conventional” undulator design though with a short undulator period.

Reduce undulator period by using a laser wiggler as radiation device.

Provide a low emittance, low energy spread beam.

Moderate peak current which are consistent with the low emittance and energy spread values.

Approach A Approach B

Plasma InjectorField Emitter Source

Case Study:

Laser Wiggler

From XFEL to Laser Wiggler

Due to the scaling, the FEL parameter is always smaller for laser wiggler than for conventional SASE FELs.

In addition, the extreme constraints on the beam quality limits the current to a much smaller value than for SASE FELs.

u: 5 cm 500 nm: 15 GeV 20 MeV

Resonance Condition FEL Parameter

: 5.10-4 5.10-5

Case Study for Laser Wiggler

Emittance and spread has been artificially chosen to allow the FEL to laser with the given peak current.

Energy spread is way beyond state-of-the-art electron beam sources.

Electron emittance much larger than photon emittance.

Energy 6.5 MeV

Current 500 A

Energy Spread 60 eV (!!!)

Emittance 0.1 mm mrad (!)

Laser Period 500 nm

Laser Field Strength 0.5

FEL Wavelength 2 nm

Coherence Properties

An estimate of the coherence can be obtained by the fluctuation in the radiation power.

As expected from the high beam emittance value, only a poor degree of coherence is achieved.

Less than 10% coherence

Pulse profile at saturation

Field stability requirement for all electrons along the interaction length:

Most stringent requirement for transverse dimension over at least 2 of the bunch, assuming a fundamental Gauss mode (alternative is transverse mode stacking):

Required radiation power:

Comments on the Laser Field

Interaction Length

Waist sizeBunch

For the example given the required power is 0.5 PW with a pulse duration of 15 ps !!!!

IncidentLaser

Case Study:

High Current Electron Beam and Short Period

Undulator

Basic Strategy

Very high peak currents yield a very large FEL parameter, which allows relaxed constraints for the electron beam quality (energy spread, emittance).

The beam energy is still reasonable large (>100 MeV) to allow for short wavelength while fulfilling the coherence condition.

Possible sources are plasma injector with an ultra short drive laser to drive a non-linear trapping mechanism of electrons.Currently an active field of research (LOA, LMU, LBNL etc.)

Electron Beam Parameters

Ambitious beam parameters, though the lower end has been successful demonstrated at LOA.

Energy 0.1 - 2 GeV

Current 10 - 150 kA (!)

Emittance 1 mm mrad

Energy spread <1%

Charge 10 – 1000 pC

C. Rechatin et al, Phys. Rev. Lett., 102, 164801 (2009)

Most critical parameter is the energy spread, which requires a FEL parameter on the same order or larger.

Case Study Overly optimistic case (LMU Munich, courtesy of A. Maier/A.

Meseck). E = 2 GeV E/E = 0.04 % (!!!) n = 0.5 mm mrad (!) I = 100 kA (!) Undulator: u = 5 mm, K = 0.65

= 2.5 Å

FEL parameter is ~10-3, thus requiring good beam parameters to make the FEL work. Also coherence is reduced.

A More Reasonable Case

Energy 1 GeV

Current 10 kA

Emittance 1 mm mrad

Energy spread 5 MeV

Effective FEL Parameter 0.1%

Gain Length 0.23 m

Saturation Length 4.8 m

Saturation Power 9 GW

Bandwidth 0.23 % FWHM

Results for = 1 nm

Larger spread and emittance, less current

Optimization for Very High Energy Spreads

Use dispersion to stretch the bunch. Reduces bunch current and energy spread

Energy spread conditions improves due to the I1/3

dependence of the FEL parameter. Helps for very large energy spreads:

Current 10 kA 1 kA

Energy Spread 20 MeV 2 MeV

Sat. Power 1 GW 0.4 GW

Sat. Length 19 m 14 m

Summary

Coherence and peak brightness favors GeVs electron beams with high peak currents.

Plasma injectors are promising candidates for compact x-ray FELs, though the performance seems to be feasible only down to 1 nm

Laser wigglers yield only poor performance (poor coherence) even if the unrealistic beam and laser parameters could be met.

Laser Wiggler

Plasma Beam