Introduction to Free-Electron Lasers Neil Thompson ASTeC
Outline Introduction: What is a Free-Electron Laser? How does an FEL work? Choosing the required parameters Laser Resonators for FELs FEL Output Characteristics FEL vs Conventional Laser Current Trends
What is a Free-Electron Laser?
What is an FEL? A beam of relativistic electrons co-propagating with an optical field through a spatially periodic magnetic field Undulator causes transverse electron oscillations Transverse e-velocity couples to E-component (transverse) of optical field giving energy transfer. Interaction between electron beam and optical field causes microbunching of electron beam on scale of radiation wavelength leading to coherent emission
What is an FEL? Output is radiation that is tunable powerful coherent
Types of FEL AMPLIFIER (HIGH GAIN) FEL Long undulator Spontaneous emission from start of undulator interacts with electron beam. Interaction between light and electrons grows giving microbunching Increasing intensity gives stronger bunching giving stronger emission >>> High optical intensity achieved in single pass OSCILLATOR (LOW GAIN) FEL Short undulator Spontaneous emission trapped in an optical cavity Trapped light interacts with successive electron bunches leading to microbunching and coherent emission >>> High optical intensity achieved over many passes
How does a Free-Electron Laser work?
How does an FEL work? Basic components B y E v x x v S N S N S z N S N S N B field Electron path E field
How does an FEL work? Basic physics: Work = Force x Distance Sufficient to understand basic FEL mechanism Electric field of light wave gives a force on electron and work is done! No undulator = No energy transfer i.e. If electron velocity is entirely longitudinal then v.e = 0 Basic mechanism very simple!!
How does an FEL work? Basic mechanism described explains energy transfer between SINGLE electron and an optical field. But in practice need to create right conditions for: CONTINUOUS energy transfer in the RIGHT DIRECTION with a REAL ELECTRON BEAM
How does an FEL work? Q. How do we ensure continuous energy transfer over length of undulator? A. Inject at RESONANT ENERGY: This is the energy at which the electron slips back 1 radiation wavelength per undulator period: relative phase between electron transverse velocity and optical field REMAINS CONSTANT. Why does it slip back? Because electron longitudinal velocity < c: Not 100% relativistic Path length increased by transverse oscillations
How does an FEL work? Q. Which way does the energy flow? E x v x v x A. Depends on electron phase t1 z Depending on phase, electron either: Loses energy to optical field and decelerates: GAIN t2 Takes energy from optical field and accelerates: ABSORPTION t3
Does an FEL work? Q. What about a real e-beam? A. Electrons distributed evenly in phase: For every electron with phase corresponding to gain there is another with phase corresponding to absorption So at resonant energy Net gain is zero So is this the end of the story??
How does an FEL work No! There is a way to proceed. Energy modulation gives bunching. At resonant energy bunching is around phase for zero net gain. By giving the electrons a bit of an energy kick we can shift them along in phase a bit and get bunching around a phase corresponding to positive net gain.
How does an FEL work? Bunching Phase corresponding to GAIN Phase corresponding to ABSORPTION t 1 t 2 E = RESONANT ENERGY Bunching around phase corresponding to ZERO NET GAIN t 1 t 2 E > RESONANT ENERGY Bunching around phase corresponding to POSITIVE NET GAIN
A Simulation example Inject 20 electrons at resonance energy: zero detuning Detuning gain absorption gain Bunching around phase correponding to zero gain 0 10 electrons lose energy 10 electrons gain energy Phase
A Simulation example Inject 20 electrons above resonance energy: +ve detuning Detuning gain absorption gain Bunching around phase correponding to +ve gain +ve 11 electrons lose energy 0 9 electrons gain energy Phase
How does an FEL work? For GAIN > ABSORPTION inject electrons at at energy slightly higher than resonant energy POSITIVE DETUNING >>>> NET TRANSFER of of energy to to optical field NB: this is only true for LOW GAIN FELs. For a HIGH GAIN FEL the maximum gain occurs at a positive detuning much closer to zero. But that s another story.
Small-Signal Gain Curve Gain Gain Spontaneous emission 2.6 2π Energy detuning δ Energy detuning δ Shows how gain varies as a function of detuning Detuning parameter δ = 4πN( E/E) Maximum gain for δ = 2.6 Madey Theorem Gain curve is proportional to negative derivative of spontaneous emission spectrum Broadening of natural linewidth causes gain degradation
The Oscillator FEL electron bunch output pulse optical pulse Random electron phase: incoherent emission Electrons bunched at radiation wavelength: coherent emission
The Oscillator FEL For oscillator FEL the single pass gain is small. The emitted radiation is contained in a resonator to produce a FEEDBACK system Each pass the radiation is further amplified Some radiation extracted, most radiation reflected Increasing cavity intensity strengthens interaction leading to exponential growth: Energy transfer depends on cavity intensity
Saturation: Oscillator For ZERO cavity loss intensity would increase indefinitely! In practice we have passive loss (diffraction, absorption) and active loss (outcoupling) Power lost proportional to cavity intensity As intensity rises so does power loss. Finite extraction efficiency Eventually power lost = power extracted from electrons No more growth. SATURATION. (Equivalently gain falls until it equals cavity losses)
Saturation: Oscillator Parameters: cavity loss = 4% gain cavity intensity Gain = Losses Intensity Intensity pass 150 Distance along undulator pass 40 Amplifier-like saturation in single pass: electrons have continued to rotate in phase space until they start to reabsorb from the optical field pass number Distance along undulator
Choosing the required parameters
Required Parameters To achieve lasing with our Oscillator FEL the following parameters must be optimised Electron beam parameters Energy, Peak Current, Emittance, Energy spread Undulator parameters K, Number of periods, Period Resonator parameters Length, mirror radius of curvature For lasing must have GAIN > LOSSES
Required parameters To first order Select base parameters to optimise gain To second order Optimise other parameters to minimise gain degradation
Gain Scaling We can get an idea of how the gain should scale with various parameters by looking at our familiar equation Total charge depends on beam current Transverse velocity depends on K and beam rigidity Optical E-field depends on area of optical mode Interaction time depends on undulator length
Small Signal Gain In fact, small signal, single pass maximum gain is given by: Undulator periods Undulator K Peak current Beam Energy Electron beam area Filling factor: averaged over undulator length These parameters varied to tune FEL:
Small Signal Gain To summarise, at a given wavelength we need: A long enough undulator To allow sufficient interaction time A good peak current and a tightly focussed electron beam To provide high charge density A small optical cross section To provide high E field
Electron beam quality Now we ve selected parameters to optimise the gain, we need to minimise the gain degradation Gain is degraded due to energy spread emittance For optimum FEL performance a high quality electron beam is required.
Energy Spread Gain 2π SMALL SIGNAL GAIN CURVE: Small energy range for positive gain If energy spread is too large electrons fall outside detuning for positive gain Energy detuning δ GAIN DEGRADATION
Energy Spread Derivation of Limit on Energy Spread: FEL wavelength given by: ERLP IRFEL: N=42 Gives rms energy spread of < 0.1% for negligible gain degradation Expand this in Taylor series giving linewidth spread due to energy perturbation Spontaneous emission 1/2N Require that spread is less than the natural line halfwidth so that no broadening occurs and gain is not reduced Energy detuning δ
Emittance Emittance controls: 1. BEAM DIVERGENCE: this effects overlap between electron beam and optical mode 2. BEAM SIZE: this affects quality of undulator field seen by electrons We can do a separate analysis for each case to see how small an emittance we need to avoid gain degradation
Emittance 1: overlap We need the electron beam contained within optical beam over whole interaction length optical beam electron beam good Interaction length small emittance large emittance bad
Emittance 1: overlap Electron beam envelope equation at a waist gives electron beam Rayleigh length: Similarly, Gaussian beam equations in laser resonator gives optical Rayleigh length: For electron beam confinement need: So: ERLP IRFEL: Gives normalised emittance of < 10 mm-mrad
Emittance 2 : broadening As emittance increases beam size increases so electrons move more off-axis Undulator field has a sinusoidal z-dependence on axis, so off axis electrons experience a different field (because curl B = 0) and thus a different K. wavelength shift linewidth broadening gain degradation By requiring that linewidth broadening is within the natural linewidth the following restriction can be derived: ERLP IRFEL: Gives normalised emittance of < 500 mm-mrad
Longitudinal effects So far we ve only considered an infinitely long electron beam: we haven t worried about the ends. Known as the STEADY STATE solution In reality we have finite electron bunches Need to extend model accordingly to include effects of PULSE PROPAGATION 2D MODEL (transverse effects) 3D MODEL (transverse + longitudinal effects )
Longitudinal effects Slippage Resonance condition: Electrons slip back by one radiation wavelength per undulator period Slippage per undulator traverse = N x wavelength. This is known as slippage length. For short electron bunch and/or long wavelength we have slippage length bunch length Effective interaction length reduced: GAIN DEGRADED.
Pulse effects: Slippage Slippage Undulator length Slippage length Long electron bunch optical pulse electron bunch Interaction length = undulator length Short electron bunch Reduced Interaction length
Pulse effects: Lethargy The electrons slip back over the optical pulse: bunching increases, and maximum emission occurs at end of undulator where bunching is strongest RESULT: optical pulse peaks at rear and centroid of pulse has velocity < c. Synchonism between pulse and electron bunch on next pass is not perfect Known as laser lethargy ct Pulse centroid vt (v < c)
Pulse effects: Lethargy Lethargy can be offset by reducing cavity length slightly: CAVITY LENGTH DETUNING Centroid of optical pulse is then synchronised with e-bunch on successive passes BUT: as intensity increases back of pulse saturates first then rest of pulse saturates, returning centroid velocity to vacuum value c. So a single detuning can t compensate for lethargy in both growth and saturation phases. Different detunings for gain optimisation and power optimisation
Cavity length detuning one cavity length for maximum gain another cavity length for maximum power
Summary 0D beam: (single electron) 1D beam: 2D beam: transverse effects (Resonance condition) (Injection above resonance for net gain) (Emittance beam overlap and broadening Energy spread) 3D beam: longitudinal effects (Pulse effects slippage and lethargy)
Resonators for Free-Electron Lasers
Optical Resonators Some general properties: Optical resonator consists of two spherical mirrors, radius of curvature R1 and R2, separated by a distance D. R 1 R 2 D
Optical Resonators Eigenmode is not a plane wave, but a wave whose front is curved. At mirrors radius of curvature matches mirror surface Fundamental TEM 00 has a Gaussian transverse intensity distribution Gaussian profile Z=0 planar wavefront Rayleigh length z = z R maximum curvature mirror curvature = wavefront curvature Gaussian profile 2w 0 2w 0 x 2
Optical Resonators More general properties: Length D has certain allowed values: must have optical round trip time = bunch repetition frequency Mode size at waist (R 1 =R 2 =R) : Mode size z from waist: Rayleigh length: So for a particular wavelength and D we can choose R to give required waist: defines both the Rayleigh length and the profile along the whole resonator. BUT only certain combinations of D, R are stable!
Optical Resonators: Stability Wave propagation treated using element by element transfer matrices (analogy with beam optics). For stability (i.e. existence of a stable periodic solution) must have where M is round trip transfer matrix (same as for storage rings) This gives the stability condition
Resonators: Stability Diagram Plot of g 1 vs g 2 can be used to show stable and unstable regions [1] FEL cavities required to focus beam for good coupling, so typically between confocal and concentric FELIX LANL FELI CLIO ERLP? [1] Kogelnik and Li, Laser Beams and Resonators, SPIE MS68, 1993 originally published: Applied Optics, Volume 5, Issue 10, 1550- October 1966
Resonator Choice CONFOCAL PROPERTIES R2 R1 Centres of curvature far apart: insensitive to mirror alignment error Large waist = small filling factor Small mirror spot = high power density + low diffraction loss
Resonator Choice Confocal Angular Alignment Sensitivity R 1 R 2 OPTICAL AXIS OPTICAL AXIS D
Resonator Choice CONCENTRIC PROPERTIES R1 R2 Centres of curvature close = sensitive to mirror alignment error small waist = large filling factor large mirror spot = low power density + high diffraction loss
Concentric Angular Alignment Sensitivity OPTICAL AXIS R 1 R 2 OPTICAL AXIS D
Resonator Choice From gain scaling have that gain proportional to filling factor Filling factor is averaged along undulator length, giving an optimum resonator configuration for maximum gain: optical Undulator length L Undulator length L Undulator length L electron Z R > L/3 Z R ~ L/3 Z R < L/3 OPTIMUM
Resonator Choice For an FEL it is difficult to achieve a small cavity length: Restriction due to electron bunch frequency Space required for dipoles and quads So the small Rayleigh length for optimum coupling must be achieved despite a large D: This is only possible if 2R~D, giving g=1-d/r ~ -1: i.e. on the boundary of the stable region of the stability diagram PERFORMANCE VS STABILITY COMPROMISE
Resonators: outcoupling There are various methods for getting the light out of the cavity: Outcoupling Method Pros Cons Hole in Mirror Partially Transmitting Mirror Brewster Plate Scraper Mirror Broadband. Simple. No mode distortion. Variable outcoupling fraction. No mode distortion. Variable outcoupling fraction. Variable scraper position. Outcoupling fraction changes with wavelength. Mode distortion. Not broadband: only suitable for certain wavelengths. Complicated. Polarisation dependent. Complicated.
Free-Electron Laser Output
ERLP IRFEL: FEL Output: power I = 50A E = 35MeV N = 42 Gives P = 8MW Gain curve can be used to estimate maximum output power: δ = 4πN( E/E) Maximum gain for δ = 2.6 Maximum energy that can be lost by an electron injected at peak of gain curve is δ = 2.6: no more gain after that. Gain Gain 2.6 2π Dividing by time, recognising that at equilibrium power extracted from electron beam = output power Energy detuning δ
FEL Output At saturation pulse length matches electron bunch length Linewidth depends on pulse length (Fourier) Brightness: output is coherent, so assuming diffraction limited beam: ERLP IRFEL: High power enables high brightness! P = 8 MW Wavelength = 4 micron Gives B ~ 10 26
FEL Output 1.E+30 Peak Brightness ph/(s.0.1%bp.mm 2.mrad 2 ) 1.E+26 1.E+22 1.E+18 1.E+14 IR-FEL coherent enhancement XUV-FEL VUV-FEL Undulators Bending Magnet 1.E+10 0.001 0.01 0.1 1 10 100 1000 Photon energy, ev
FEL Output TUNABLE OUTPUT! A typical FEL facility operates at certain fixed beam energies then tunes over wavelength sub-ranges by varying undulator gap (and hence K)
Free-Electron Lasers vs Conventional Lasers
FEL vs Conventional Laser Conventional Laser Light Amplification by Stimulated Emission of Radiation Electrons in bound states - discrete energy levels Waste heat in medium ejected at speed of SOUND Limited tunability Continuous tunability Limited power High power Free-Electron Laser Light Amplification by Synchronised Electron Retardation?? Electrons free - continuum of energy levels Waste heat in e-beam ejected at speed of LIGHT (and recovered in ERL)
Current Trends and the way ahead
Current trends Average power of FELs is increasing JLAB - record continuous average power of 2.13kW (IR) JAERI - 2kW over a 1ms macropulse (IR) High average power goal is several tens of kw JLAB upgrade promises >10kW. Wavelengths decreasing 190nm @ ELETTRA - storage ring oscillator FEL. Mirror technology limiting factor for oscillators 80nm @ TTF1 - amplifier FEL
Applications Medical FELs with high-peak and highaverage power are enabling biophysical and biomedical investigations of infrared tissue ablation. A midinfrared FEL has been upgraded to meet the standards of a medical laser and is serving as a surgical tool in ophthalmology and human neurosurgery.
Applications Power Beaming and Remote Sensing
Applications Micromachining A free-electron laser can be used for to fabricate three-dimensional mechanical structures with dimensions as small as a micron The images above are 12-micron, micro-optical Fresnel Lens components created by laser-micromachining.
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