1. C. Additional FEL topics

2. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

3. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

4. Self-amplified spontaneous emission (SASE)
In the theoretical treatment, it was always assume that there is a perfect plane Ex wave of correct λl available. In fact, there is not seed laser in the X-ray regime available. So how to we seed the FEL process? The most common solution is to use the spontaneous undulator radiation as seed light (SASE). The problem is, however, that undulator radiation (as ordinary ISR) origins from the random charge distribution in the electron bunch (Shot noise) and is therefore a random process. Hence the seed light is neither monochromatic, nor transverse coherent nor longitudinal coherent. How much of a problem is that in fact.

5. Transverse coherence of X-rays from SASE
Low-gain regime:
The electron beam excites in the beginning not only the TEM00.
Several modes overlap and destroy the transverse coherence (no fixed transverse phase relation).
High-gain regime:
Since the TEM00 is highest on axis, where the beam jz is largest, it grows faster then the other modes (mode cleaning).
In the high-gain regime and in saturation the TEM00 mode is nearly purely present, which results in very high transverse coherence!

6. Longitudinal coherence of X-rays from SASE
Light emission along the beam is a random process and hence there is no defined phase relation all along the electron beam. An appropriate model is the overlap of short coherent wave packages (also called longitudinal modes) that are shifted by random phases. The average length of these individual wave packages is called coherence length and is given by Due to the phase shift jumps between the individual SASE radiation, the X-ray radiation will not be monochromatic anymore. The spectrum is widened (see next page). Also shape of spectrum and the integrated X-ray power will fluctuate (stochastic process).

7. Spectrum of SASE X-rays
Longitudinal spectrum of the X-rays at the end of the undulator.
A wider spectrum with width σω,av is made up by many spikes with width σω,spike.
Properties of spectrum:
The width of the spikes is related to full length of the photon beam
The width of the spectrum is related to the length of the longitudinal modes.
The number of spikes is equal to the number of longitudinal modes
The fluctuations of the integrated X-ray power vary strongly in the exponential gain regime (60-70%), but get reduced in the saturation regime (20%).

8. SASE vs. seeded FEL simulations
GENESIS simulations with realistic beam distributions.
10kW laser used for seeding.
About equivalent to SASE
This already shows that a very strong seed laser is needed if SASE should be suppressed.
SASE and seeded (steady state) simulation give different results in saturation regime.
Reason is a that particles with detuned energy exist and have a high gain (see later more).

9. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

10. Concept 1: Higher harmonic generation (HHG)
SASE FEL radiation has several disadvantages:
Widened bandwidth.
Strong shot-to-shot fluctuations of the X-ray spectrum.
Low longitudinal coherence.
Strong X-ray power fluctuations in the exponential growth regime (less in saturation).
To improve the situation it would be preferable to seed the FEL processes with external longitudinal coherent light.
On concept to create such seed light is to used higher harmonics of a laser:
At SACLA, titanium-sapphire (Ti:Sa) laser was focused on Xenon gas cell where it generates higher order harmonics. Fifth harmonic (160nm) has been used to seed electron test beam.
The smallest reached seeding wavelength has been 38nm demonstrated at FLASH (2013).

11. Concept 2: High-gain harmonic generation (HGHG)
FEL is seeded with available laser at longer λl
In first low gain FEL (modulator), energy modulation is created (only tiny bunching).
This energy modulation is converted to a charge variation in a chicane (bunch compressor principle).
A second high-gain FEL (radiator) is tuned to a higher order of the charge modulation.
If current modulation at higher harmonic is strong enough to overcome SASE it can be used.
Modulation factor ji is reduced for higher harmonics.

12. Concept 3: Echo-enabled harmonic generation (EGHG)
Idea is the same as for HGHG. Difference is that two modulators (and two lasers) are used to create a charge modulation with stronger higher order components.
First chicane over-compresses energy variation and second chicane acts as in HGHG. Result is a very spike charge modulation.
Therefore harmonics of a higher number (compared to HGHG) are still strong enough to seed FEL process. One problem is the adding up of energy spread in the modulators, which increases gain length.

13. Concept 4: Self-seeding
This method does not rely on an external laser. The first undulator is operated in SASE mode for a few gain length.
Then the created light is separated from the beam (chicane) and filtered my a diamond grating monochromator crystal. The filtered light has much better longitudinal coherence.
In a second undulator, the filtered light is used as a seed for the same electron bunch. An additional positive effect is that the micro-bunching of the beam has been washed out in the chicane.
This method can be used at any wavelength, but is limited by the performance of the diamond monochromator. The final spectrum is therefore broader than at HHG, HGHG or EEGH.

14. Overview of advanced seeding schemes
FEL-Oscillator
Direct seeding (HHG)
High gain harmonic generation (HGHG)
Echo-enabled harmonic generation (EEHG)
Self seeding
Light trapped in an oscillator.
Limitations are mirrors.
< 250 nm.
Laser ionizes novel gases and creates higher harmonics.
< 40 nm
First seeding with laser (modulator).
Then lasing at higher harmonic (radiator).
< 10nm
Complex three stage scheme similar to HGHG.
< 1nm
Interesting for soft XFEL design.
Laser creates SASE light in first stage.
Light is filtered.
Second stage for lasing.
No wave-length limitation.
Interesting for soft and hard XFEL design.

15. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

16. Energy detuning 1/2 Assume seeded operation.
Remember: seeding laser wave length determines wave length of X-rays (λl = λs), since FEL acts as light amplifier.
However, Ku, λu and γ determine the resonance wave length λR of the FEL.
Usually choice: λR = λs.
But slight running detuned (via beam energy γ) increases the output power.
Note also the unusual shape in the saturation regime.

17. Energy detuning 2/2
From FEL theory the left gain curves have been derived.
In the low gain regime (very small micro-bunching), no light amplification for λR = λs.
In high gain regime (strong micro-bunching) the maximum gain is close to λR = λs, but at slightly detuned.
This also explains why SASE performs better than a seeded FEL in terms of power. Some frequency components due to shot noise are detuned.

18. Tapering
If the undulator strength K(Z) is weakened along the FEL, the output power can be strongly increased.
This is usually done by changing the gap size of the undulator.
With a linear taper (starting at m) the power can be increase from 1GW to 30-40GW.
With a quadratic taper 50GW can be reached.

19. Tapering 2/2
Many papers claim tapering compensates for the energy loss of the beam.
But this is only a small effect.
More important is that micro-bunches are kept in the right have of the FEL bucket, where they transfer energy to the light wave.
Tapering moves bucket to the left.

20. Circular polarized light and inverse tapering
Nowadays mainly planar undulators in FELs (linear polarized light)
User request also circular polarised light (helical undulator).
An undulator variant for both types of polarization is a long planar undulator (easier to build) with a helical afterburner undulator.
If linear polarised light is desired, jaws of helical undulator are opened and have no effect (only planar undulator).
If circular polarized light is desired, the helical undulator is closed and, light will have a fast power rise due to the already bunched beam from the planar undulator.
But then one gets a mixture or linearly and circular polarised light.
The linear polarized light can be suppressed by an inverse taper of the planar undulator without destroying the bunching.

21. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

22. Motivation Coulomb explosion: Ultra-short X-ray pulses:
Photon beam ionizes probe and hence destroys it completely.
Time scale is about 20-50fs.
Usual FEL pulses are fs.
Picture is smeared out.
Ultra-short X-ray pulses:
If the X-ray pulses can be made short enough (< 1-10fs) compared to the time scale of the Coulomb explosion, then an image can undisturbed image can be made.
Also even fast processes can be time-resolved.

23. Overview of some concepts
Short bunch mode
By reducing bunch charge Q (20pC instead of 200pC), bunches can be made shorter 1-2fs.
Gun laser already produces shorter bunches.
Lower charge makes bunch compression easier.
But lower average brilliance.
Energy modulation from laser
Use laser together with beam in a modulator undulator to create a energy modulation.
Then inject electron bunch into undulator. Beam beam only lase were the energy is close to resonance.
Due to the energy modulation of beam, a series of very short pulses will be created.
If the laser light only includes one optical period, a single spike can be created.

24. Emittance spoiling via foils
Correlation longitudinal and transverse plane (beam tilted). The case in bunch compressors.
Beam is passed through foil with slit.
Emittance is only preserved at slit, but spoiled when passing the foil.
Beam lases only where emittance is unspoiled.

25. C. Additional FEL topics. C. 1 Seeding schemes. C. 1. 1 SASE. C. 2
C. Additional FEL topics C.1 Seeding schemes C.1.1 SASE C.2.2 Improvement of longitudinal coherence C.2 Schemes for increased output power C.3 Ultra-short X-ray pulses C.4 Creation of unusual X-rays

26. Two-color FEL Example2:
The availability of light of two different wavelength is demanded by the user community. This can be realised in different ways:
Example 1:
First SASE undulator is tuned to λ1.
The beam is moved through a chicane to wash out microbunching.
Beam is injected into SASE undulator 2 with a different K to create light with λ2.
Here the light pulses are slightly separated in time.
Example2:
After first undulator light stays bunched and is injected in second undulator which is tuned in higher harmonic of bunching.
Like this the two light pulses are on each other.

27. Application of two color X-rays
Performed at Elettra by K. C. Prince.
Intention: measure the absorption edge more precisely (spectroscopy).
Principle:
Excite electrons with the two wavelength to two different states.
The emitted photons from one state are an s-wave while the other state emits a p-wave.
Each wave itself is symmetric is asymmetric but the overlap is not!
By changing the relative phase of the two X-ray wavelengths, the spatial distribution of emitted light is changed.
Experimental possibilities:
Very precise measurements of absorption edges are possible.
The hope is to measure Wigner times for the first time (time delay of scattering events).

28. Summary
The photon user community has in the order of members and FEL radiation is highly demanded.
The science performed at FELs is of highest impact and only a few users can be supplied at the moment.
Therefore the field of free electron lasers is a very active research topic and of growing interest.
FELs are a very interesting combination of the fields of beam physics, FEL science and photon science and there are many interesting and challenging problems to be solved.
The University of Oslo is currently participating in a collaborative effort to make FELs cheaper and accessible to a wider user group.
If you are interested in working on this subject (Bachelor, Master, PhD theses) please contact us!

29. Optional exercises
You can solve the exercise with or without the hinds on the next page.
Particle motion in an undulator:
Show that the average longitudinal particle speed can be approximated by
3rd order FEL equation:
2. Show that the solution of the 3rd order equation has the following form, if space charge effect are negligible and the FEL process is on resonance.
3. Show that for a seeded laser operation the coefficients
4. Interpret the effect of the three different components of the solution Eq. (1) on the X-ray power in a qualitative way. How can the small-gain at the start of the process be explained from a sum of exponential functions.

30. Optional exercises (hinds)
The instantaneous speed v can be calculated from the γ factor of the particles. Then simplify the √ expression using Taylor expansion. Finally, use the expression for vx by differentiating the known solution for x(t) .
The made assumptions correspond to kp = ηb = 0. Use the Ansatz:
You will have to consider that the FEL process starts in the low-gain regime.
As a help, you can plot the real part of the terms individually for realistic parameters to analyse their behaviour.