Coherence: from Undulators to Free Electron Lasers David Attwood University of California, Berkeley Cheiron School September 25, 2014 SPring-8

Spatial and Temporal Coherence
d · 20 | FWHM == 2 l coh = λ 2 / 2 λλ Tcoh = Ecoh/c ΔE · Δτ | FWHM ≥ 1.82 eV · fsec N uncorrelated emitters Undulators and Free-Electron Laser Undulator | Pinhole N S N S N | S Pinhole sets d · 20 | FWHM SASE FEL S N S N N S S N | S N N N S N S | S | N | After “self-seeding” -/ spectral bandwidth Δω/ω ≈ 1/2000 l coh = λ²/2Δλ  ~ 10³λ Young’s Double Slit (correlated emitters) •Self-interference only •Electric fields chaotic • Intensities add •Radiated power ~ N d N S S N S – No pinhole required as   σσ' = N/4π d · 20 FWHM ~ λ/2 N | N S N | S | S SASE spectral - FEL: bandwidth • Small phase-space l Δωホ• High peak current 400 ω • Long undulator Spatially, S temporally coherent, but jumbled phase Coherence_Collage_Sect2014.ai Attwood_Cheiron_Sept2014_2.ppt 2 N
lay cohesion   2 : to consist of parts that cohere e \ko-'hir-an(t)s, -'her-\ n : the quality or s ner.ent \ko-'hir-ent, -'her-\ adj [MF or L; MF cohe conderent-, conderens, prp. of cohderere] 1 : having the c ering 2 : logically consistent 3 : having waves in Webster ‘s 7th Collegiate Dictionary (1971) co·her·ence (ko·hîr·ens, -hěr'-) also co·her·en·cy (-ən- The quality or state of cohering, a relationship of parts. co·her·ent (kö-hîr'ənt, -hěr'-) adj. 1. Sticking together; co- ng. 2. Marked by an orderly or logical relation of parts 3. Physics. Of or pertaining to waves with a cont tionship among phases. 4. Of or pertaining to a system o American Heritage 2nd College Edition Dictionary (1985)
◾ Coherence time, the time over which a propagating wave (especially a laser or maser beam) may be considered coherent; the time interval within
Wikipedia, the free encyclopedia
Interference and diffraction with partially coherent light 10.1 Introduction So far we have been mainly concerned with monochromatic light produced by a point source. Light from a real physical source is never strictly monochromatic, since even the sharpest spectral line has a finite width. Moreover, a physical source is not a point source, but has a finite extension, consisting of very many elementary radiators (atoms). The disturbance produced by such a source may be expressed, according to Fourier's theorem, as the sum of strictly monochromatic and therefore infinitely long wave trains. The elementary monochromatic theory is essentially concerned with a single component of this Fourier representation. In a Physical sources are of finite size and duration constar field pr the rap The cor which i time in less tha this tin howeve is of the Consider next the light disturbances at t wo points P 1 and P 2 in a wave field produced by an extended quasi-monochromatic source. For simplicity assume that the wave field is in vacuum and that P 1 and P 2 are many wavelengths away from the source. We may expect that, when P1 and P2 are close enough to each other, the fluctuations of the amplitudes at these points, and also the fluctuations of the phases, will not be independent. It is reasonable to suppose that, if Pj and P2 are so close to each other that the difference AS = SP1 - SP2 between the paths from each source point S is small compared to the mean wavelength A, then the fluctuations at P 1 and P 2 will effectively be the same; and that some correlation between the fluctuations will exist even for greater separations of P1 and P2, provided that for all source points the path difference AS does not exceed the coherence length cAt ~ c/Av = i2/Ai. We are thus led to the concept of a region of coherence around any point P in a wave field. In order to describe adequately a wave field produced by a finite polychromatic source it is evidently desirable to introduce some measure for the correlation that exists between the vibrations at different points P 1 and P 2 in the field. We must expect such a measure to be closely related to the sharpness of the interference fringes which would result on combining the vibrations from the two points. We must expect sharp fringes when the correlation is high (e.g. when the light at P 1 and P 2 comes from a very small source of a narrow spectral range), and no fringes at all in the absence of correlation (e.g. when P 1 and P 2 each receive light from a different physical source). We described these situations by the terms 'coherent' and 'incoherent' respectively. In general neither of these situations is realized and we may speak of vibrations which are  partially coherent. Regions of coherence l coh = cΔt = λ²/2Δλ Normalized degree of coherence (E 1 (t + τ)E 2 (t)) Y12(T) = √ (| E 1 |² ⟩ √ (| E 2 |² ⟩ Coherent Incoherent Partially incoherent
Point source oscillator —8 <t < 8 Source of finite size, divergence, and duration Ch08_F01.ai 5
Mutual coherence factor
Normalize degree of spatial coherence (complex coherence factor)
A high degree of coherence (μ → 1) implies an ability to form a high contrast interference (fringe) pattern. A low degree of coherence (μ → 0) implies  an absence of interference, except with great care. In general radiation is partially coherent. Longitudinal (temporal) coherence length
Full spatial (transverse) coherence
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Define a coherence length ℓcoh as the distance of propagation over which radiation of spectral width Δλ becomes 180° out of phase. For a wavelength λ propagating through N cycles
and for a wavelength λ + Δλ, a half cycle less (N –
Equating the two
so that
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•  Associate spatial coherence with a spherical wavefront. •  A spherical wavefront implies a point source. •  How small is a “point source”? From Heisenberg’s Uncertainty Principle (Δx・Δp ≥与), the smallest source size “d” you can resolve, with wavelength λ and half angle θ, is

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Δx · Δp ≥ h/2 (8.4) Standard deviations of Gaussian distributed functions (Tipler, 1978, pp. 174-189) Δx · ĥΔk ≥ ĥ/2 Δx · kΔθ ≥ 1/2 2Δx · Δθ ≥ λ/2π d = 2Δx + Note: Δp = hΔk Δk = kΔθ or Spherical wavefronts occur in the limiting case d · θ = λ/2π quantities /e (spatially coherent) } FWHM quantities (d · 20)FWHM ~ λ/2 10
V ≤ C Note: Angle-dependent doppler shift
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Courtesy of John Madey Ch05_F09VG_revOct05.ai 11
(ΔΑ) (ΔΩ) (Δλ/λ) Ch05L F08VG 1.04.ai 12
Laboratory Frame of Reference
E = Ymc²
N = # periods Frame of Moving e- Frame of Observer Following Monochromator S N e¯ radiates at the Lorentz contracted wavelength:
Doppler shortened wavelength on axis:
typically θcen ~ 10-30 µrad
Bandwidth: ॥ 
׳ﾍ׳ﾍ׳
Accounting for transverse motion due to the periodic magnetic field: Following Albert Hoffman where K = eBoλu /2πmc Ch05 LG186 June2014.ai 13 λ' Il Calculating Power in the Central Radiation Cone: Using the well known “dipole radiation”’ formula by transforming to the frame of reference
x, z, t laboratory frame of reference x', z', t' frame of reference moving with the average velocity of the electron er λύ = >> Determine x, z, t motion: dp dt
★  x ll θcen Y* /N = N πe Y I K² Eoλu (1 + K²I2)² Z Lorentz transformation Dipole radiation: dP dΩ’
Lorentz transformation Z dP dΩ’ cos² φ′) cos² ω′„t′ 4€ρλ² (1 + K /2) 2 Ch05 T4 Feb07.ai
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θcen = Y / N = [Ay] λ Z | 1 [Ay 1 | W , y  J3 Ch05_PwrCenRadCone3.ai 17
Ordinary thermal light source, atoms radiate independently. A pinhole can be used to obtain spatially coherent light, but at a great loss of power. A color filter (or monochromator) can be used to obtain temporally coherent light, also at a great loss of power. Pinhole and spectral filtering can be used to obtain light which is both spatially and temporally coherent but the power will be very small (tiny). All of the laser light is both spatially and temporally coherent*. Arthur Schawlow, “Laser Light”, Sci. Amer. 219, 120 (Sept. 1968)
19 Water-cooled beam defining apertures 8.0 cm period undulator (λ/2π)² (dx0x)(dy0v) spatial filtering
U5 1.9 GeV, 400 mA λυ = 50 mm, N = 89 0.5 ≤ K ≤ 4.0 σχ = 260 μm, σχ = 23 μr σν = 16 μm, σν' = 3.9 μr θcen = 35 µrad η = 10% Ch08 U5 June2014.ai
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8 GeV, 100 mA λu = 32 mm, N = 140 0 ≤ K ≤ 2.46 σχ = 393 μm, σχ = 15.7 μr σν = 4.98 μm, σν′ = 1.24 μr θcen = 6.6 μr η = 10%
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c) • Intensities add • Radiated power ~ N
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Courtesy of Chang Chang, UC Berkeley and LBNL. λ = 13.4 nm, 450 nm diameter pinholes, 1024 x 1024 EUV/CCD at 26 cm ALS, 1.9 GeV, λιι = 8 cm, N = 55 24
Courtesy of Chang Chang, UC Berkeley and LBNL. λ = 13.4 nm, 450 nm diameter pinholes, 1024 x 1024 EUV/CCD at 26 cm ALS, 1.9 GeV, λιι = 8 cm, N = 55 25
Ordinary thermal light source, atoms radiate independently. A pinhole can be used to obtain spatially coherent light, but at a great loss of power. A color filter (or monochromator) can be used to obtain temporally coherent light, also at a great loss of power. Pinhole and spectral filtering can be used to obtain light which is both spatially and temporally coherent but the power will be very small (tiny). All of the laser light is both spatially and temporally coherent*. Arthur Schawlow, “Laser Light”, Sci. Amer. 219, 120 (Sept. 1968)

Laser Cavity
N uncorrelated emitters •Self-interference only •Electric fields chaotic • Intensities add •Radiated power ~ N
Spatial and Temporal Coherence
d · 20 | FWHM = l coh = λ 2 /2 Δλ Tcoh = Ecoh/C ΔE · Δτ | FWHM ≥ 1.82 eV · fsec Undulators and Free-Electron Laser Undulator N S N S | Pinhole d S N | S N Pinhole sets d · 20 | FWHM SASE FEL N S N S N S N | N S N | S | S | SASE spectral FEL: bandwidth • Small phase-space l Δωホ• High peak current 400 ω • Long undulator – No pinhole required as    σσ' = λ/4π N S N S | S | N | After "self-seeding"-/ spectral bandwidth Δω/ω ⇒ 1/2000 l coh = λ 2 /2Δλ  ~ 10 3λ | S S N N S N N Spatially, S temporally coherent, but jumbled phase d · 20 | FWHM ~ λ/2 Coherence_Collage_Sept2014.ai Attwood_Cheiron_Sept2014_2.ppt 27 >
(a) Equilibrium energy distribution 28
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(a) Amplified spontaneous emission in a long active medium (rod) At startup: Gain medium + flash lamp (drives population inversion) 30
(b) • Increased output; many gain lengths • Greatly reduced side losses • θ =무~ 1 mrad;  1000 x diffr. limited Cavity end-mirrors for feedback and enhanced growth
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(d) 20|> Longitudinal mode selector Transverse mode selector Wavelength selective “etalon”, narrows lasing spectrum to Δλ and sets longitudinal coherence length λ² l coh = 2Δλ Mirror Y
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Charles Hard Townes Aleksandr Mikhailovich Prokhorov “For fundamental work in the field of quantum electronics, which has led to the construction of oscillators and amplifiers based on the maser-laser principle"
“Non-linear optics and spectroscopy” Arthur Leonard Schawlow “Spectroscopy in a new light" “For contributions to the development of laser spectroscopy" 35
• Spatial coherence • Temporal coherence • Partial coherence • Full coherence • Spatial filtering • Uncorrelated emitters • Correlated emitters • True phase coherence and mode control • Lasers, amplified spontaneous emission (ASE) and mode control • Undulator radiation • SASE  FEL 10-70 fsec EUV/x-rays • Seeded FEL true phase coherent x-rays • High harmonic generation (HHG) compact fsec/asec EUV • EUV lasers and laser seeded HHG • Applications with uncorrelated emitters • Applications with correlated emitters
IulatorsFELsCoh Sept2

Undulator N S N S d人(  S N N S Δω // N ω Pinhole sets d・20 FWHM (Noise) 2 | > Self-seeding transmission x-ray crystal SASE FEL N s N N N S S N S … .... S N s S N S s | N N SASE spectral- bandwidth Δω l (l 400 ω FEL: • Small phase-space • High peak current • Long undulator N S N S N S N S N N S N S N S After “self-seeding”– spectral bandwidth Δω = 1/2000 ω X
20 ~ 10³λ l coh – No pinhole requiredムσσ΄ ll as 4π (d · 20 FWHM 2) ll S N
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37 Future possibilities: • Is a phase-coherent seed possible at x-ray wavelengths? • Perhaps a seed plus HGHG multiplication, as at FERMI? • Is an x-ray FEL oscillator possible? ^ To S Spatially and temporally coherent, but jumbled phase
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In an undulator with random, uncorrelated electron positions within the bunch, only the radiated self-fields E add constructively. • Coherence is somewhat limited • Power radiated is proportional to N 。 (total # electrons) For FEL lasing the radiated fields are strong enough to form “microbunches” within which the electron positions are well correlated. Radiated fields from these correlated electrons are in phase. The net electric field scales with Nei, the # of electrons in the microbunch, and power scales with N ci 2 times the number of microbunches, ni. • Essentially full spatial coherence • Power radiated is proportional to Nei²ni ~ 106 × 300 ~ 3 × 108
S • Uniformly distributed particles (beam) into undulator. • Emission of radiation (“spontaneous” emission). • Wave grows enough (undulator radiation) to begin affecting. particle dynamics through ma = −eE radiation. • Transverse coupling between Erad and transverse velocity vx (in undulator) leads to energy exchange between fields and particle (zero net at first)   dt = mc² dy = F · v = −e E · v x • Modulated velocities with increments in v x lead to bunching on axis. • Electron density modulation leads to stronger radiation, P rad ∞ q²|a|² ~ (eNe)² · e² = Ne² e4 • Stronger fields (wave) drive stronger transverse velocity. • Stronger v x drives stronger bunching, . . . stronger fields, . .
Magnetic fields in the periodic undulator cause the electrons to oscillate and thus radiate. In the FEL case the undulator is very long, the radiated electric field is now strong enough that it can not be ignored in the momentum equation, and the energy of the electrons is no longer constant. Equations of motion: dp dt
where p = γmv and Ee = mγc². Ch05_F15_Eq16_19.top_Nov2011.ai
N N S S dt S S N N Undulator – uncorrelated electron positions, radiated fields uncorrelated, intensities add, limited coherence, power ~ N. S N N S S N N S N :: : :: S S N S N S N S N Free Electron Laser (FEL) – very long undulator, electrons are “microbunched” by their own radiated fields into strongly correlated waves of electrons, all radiated electric fields now add, spatially coherent, power ~ N 2 dp –e(E + v dt
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N S N S dp dt S N S N Undulator – uncorrelated electron positions, radiated fields uncorrelated, intensities add, limited coherence, power ~ N. N S N S N S N S ... … ::: N 4HHH Coherent seed pulse S S N S N N S N S Free Electron Laser (FEL) – very long undulator, electrons are “microbunched” by their own radiated fields into strongly correlated waves of electrons, all radiated electric fields now add, spatially coherent, power ~ N 2 di −e(E + v × B) Seeded FEL. Initial bunching driven by phase coherent seed laser pulse. Improved pulse structure and spectrum. UndulatorsAndFELs3_June2014.ai
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Electron energies and subsequent axis crossings are affected by the amplitude and relative phase of the co-propagating field
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Courtesy of Sven Reiche, UCLA, now SLS 45
Courtesy of K-J. Kim Gain_Saturation_FEL_graph.ai 46
Equations of motion: dp d (γmv) = −e(E + v × B) dt. eKE dEe _ mc² = v : F = −ev 2mcy dt dt Strength of wave-particle coupling: K² reneλu² / 3 P FEL = 32πγ³ Lasing gain length (power e-folding): LGain = λυ/4π√3 PFEL Saturated lasing power: Psat ~ PFEL · γmc² · Î/e Limited electron energy spread for efficient energy transfer: σγ/γ < PFEL Electron angular limit for constructive interference: σ′ ≤ 1/γ*√N Electron phase-space (“emittance”) match to coherent radiation phase-space: σσ' < λ/4π Relative spectral bandwidth: Ayy ~ PfEl photonics
P. Emma¹*, R. Akre¹, J. Arthur¹, R. Bionta², C. Bostedt¹, J. Bozek¹, A. Brachmann¹, P. Bucksbaum¹, R. Coffee¹, F.-J. Decker¹, Y. Ding¹, D. Dowell¹, S. Edstrom¹, A. Fisher¹, J. Frisch¹, S. Gilevich¹, J. Hastings', G. Hays', Ph. Hering', Z. Huang', R. Iverson', H. Loos', M. Messerschmidt', A. Miahnahri¹, S. Moeller¹, H.-D. Nuhn¹, G. Pile³, D. Ratner¹, J. Rzepiela¹, D. Schultz¹, T. Smith¹, P. Stefan', H. Tompkins', J. Turner', J. Welch', W. White', J. Wu', G. Yocky' and J. Galayda' The recently commissioned Linac Coherent Light Source is an X-ray free-electron laser at the SLAC National Accelerator Laboratory. It produces coherent soft and hard X-rays with peak brightness nearly ten orders of magnitude beyond conventional synchrotron sources and a range of pulse durations from 500 to <10 fs (10 −15 s). With these beam characteristics this light source is capable of imaging the structure and dynamics of matter at atomic size and timescales. The facility is now operating at X-ray wavelengths from 22 to 1.2 A and is presently delivering this high-brilliance beam to a growing array of scientific researchers. We describe the operation and performance of this new 'fourth-generation' light source'. ver the past several decades, intense beams of X-rays from synchrotron light sources have been used quite successfully to reveal the arrangement of atoms in a wide range of materials, including semiconductors, polymers, ceramics and bio- logical molecules. Unlike visible light, X-ray wavelengths are com¬ parable to the interatomic spacings in matter, thus enabling imaging on the atomic scale. However, even so-called 'third-gener- ation' synchrotron sources produce insufficient brightness to resolve the structure of isolated single molecules. Synchrotrons also gener- ate picoseconds-long X-ray pulses, which yield only blurred images of atoms and molecules in motion. In addition, accelerator research 1 has been focused for some time on advancements in applications of accelerators towards coherent emission from a bunch of electrons. These motivations, along with the desire to invent an X-ray laser, have led to the development a 'fourth-generation light source' based on the free-electron laser (FEL). A FEL 2 is a source of coherent radiation that uses the interaction between an electromagnetic wave and a bunch of relativistic elec¬ trons to amplify this wave as the electrons pass through a periodic magnetic array (an undulator). In principle, the FEL wavelength is length with power saturation after 18-20 gain lengths, providing stability of the shot-to-shot FEL power against small variations of the electron beam brightness. The Linac Coherent Light Source (LCLS) 6.7 , first proposed 8 in 1992, is such a light source, with ultrafast pulse durations on the femtosecond timescale 9 . It allows the imaging of single molecules and, working like a high-speed camera, also enables the creation of molecular movies, potentially revealing the details of physical, chemical and biological dynamics on an unprecedented timescale10. The principles of the SASE FEL have been demonstrated at longer wavelengths 11-16 , and the FLASH facility in Hamburg in Germany currently operates such a SASE FEL at wavelengths down to 65 Å (ref. 17). Several hard-X-ray machines have either been proposed or are under construction 18-20 , but the LCLS is the first hard-X-ray FEL in operation and has served users since 1 October 2009. The performance and operation of the LCLS are described here, beginning with a list of typical measured and design parameters (Table 1) for both soft and hard X-ray settings. The X-ray wavelength is easily tunable from 22 to 1.2 A by varying the electron energy in the range 3.5-15 GeV. (LCLS, lasing April 2009, 1st day; saturated lasing 2009; publ. Sept. 2010)
ARTICLES
NATURE PHOTONICS  DOI: 10.1038/NPHOTON.2010.176
Table1 | Design and typical measured parameters for both hard (8.3 keV) and soft (0.8–2.0 keV) X-rays. The 'design' and ‘hard’ values are shown only at 8.3 keV. Stability levels are measured over a few minutes.
*Brightness is photons per phase space volume, or photons s = 1 mm = mrad = 2 per 0.1% spectral bandwidth. 'Slice' refers to femtosecond-scale time slices and 'projected' to the full time-projected (that is, integrated) emittance of the bunch. Figure 4 | FEL gain length measurement at 1.5 Â. Measured FEL power (red points) plotted after continuous insertion of each 3.4-m undulator segment showing saturation at 60 m and with all 33 undulator segments installed. Error bars represent the r.m.s. statistical uncertainty in the measured power when averaging 30 beam pulses. The measured gain length is 3.5 m with a GENESIS simulation overlaid (blue curve) and with consistent electron beam parameters shown. The YAG screen image is shown in the inset with 140-μm r.m.s. round X-ray spot size in this early case (April 2009). λ , is the fundamental FEL radiation wavelength; I ok is the peak current of the electron beam in the undulator; y is the relativistic Lorentz facotr; ε , is the transverse r.m.s. emittance of the electron beam in the undulator; γε,, is the normalized transverse r.m.s. emittance of the electron beam in the undulator; σ ε /E 0 is the r.m.s. relative energy spread of the electron beam in the undulator (that is, the r.m.s. energy spread, σ c , divided by the mean electron energy, E ). NATURE PHOTONICS | VOL 4 | SEPTEMBER 2010 | www.nature.com/naturephotonics
K = 3.5 @ λ = 1.5Å P. Emma et al., First lasing and operation of an ånstrom-wavelength free-electronlaser, Nature Photonics 4, 64 (2010); Proceed. PAC 2009; first lasing April 2009.
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LCLS, 780 eV, 300 fsec, ¼ nC,1mJ/pulse 78% energy in TEM 00 mode
FEL_Ch6_F12_forSlides.ai
Courtesy of I. Vartanyants (DESY) and A. Sakdinawat (SLAC); PRL 107, 144801 (30Sept2011) 51
While “SASE” FEL radiation is to a large extent spatially coherent, the electron bunch length can support many independent coherent micro-bunches, thus temporal coherence is limited and spectral bandwidth relatively broad. This is a natural property of the “Self Amplified Spontaneous Emission” (SASE) FEL process. Coherent seeding with a sufficiently intense laser pulse seeks to overcome this limitation, with future plans largely centered on high harmonics of visible/IR lasers of higher power than presently available. Filtered “self-seeding” is also under consideration. (LCLS: 2.35 σz = 21 μm FWHM vs. ~400 × .15 nm = 0.06 μm) (... Can support 300+ microbunches, depending on bunch/pulse duration) ahPropertiesFEL_6Sept2014.ai
Courtesy of D. Zhu, J. Hastings, W. Feng, (SLAC); APL 101, 034103 (July 2012)
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20 eV 0.4 eV  = 50:1 reduction ectrometer intensity (a Δλ λ
lcoh = 1.5 μm J. Amann, et al., Nature Photonics 6, 693 (Oct 2012) 54
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Science and Technology of Future Light Sources (Argonne, Brookhaven, LBNL and SLAC: Four lab report to DOE/Office of Science, Dec. 2008)
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•  J.M.J. Madey, “Stimulated Emission of Bremsstrahlung in a Periodic Magnetic Field", J. Appl. Physics 42, 1906 (1971). •  A.M. Kondratenko and E.L. Saldin, “Generation of Coherent Radiation by a Relativistic-Electron Beam in an Undulator", Sov. Phys. Dokl. 24, 986 (1979). •  R. Bonifacio, C. Pellegrini and L. Narducci, “Collective Instabilities and High-Gain Regime in a Free Electron Laser", Optics Commun. 50, 373 (1984). •  V. Avazyan et al., “Generation of GW Radiation Pulses from a VUV Free- Electron Laser Operating the Femtosecond Regime", Phys. Rev. Lett. 88, 104802 (2002). •  P. Emma et al., “First Lasing and Operation of an Ångstrom-Wavelength Free-Electron Laser", Nature Photonics 4, 641 (2010). •  P. Schmüser, M. Dohlus, J. Rossbach and C. Behrens, “Free-Electron Lasers in the Ultraviolet and X-Ray Regime" (Springer, 2014). Second edition. •  K.J. Kim, Z. Huang and R. Lindberg, “Synchrotron Radiation and Free Electron Lasers for Bright X-Rays" (to be published).
UC Berkeley www.coe.AST.berkeley.edu/sxr2009 www.coe.AST.berkeley.edu/srms www.youtube.com Soft X-Rays and Extreme Ultraviolet Radiation 58