Advanced laser-driven X-ray sources Stefan Karsch - - PowerPoint PPT Presentation

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Advanced laser-driven X-ray sources

Stefan Karsch

Ludwig-Maximilians-Universität München/ MPI für Quantenoptik

Garching, Germany

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… and what do we use them for ?

m mm μm nm pm eV keV MeV meV Energy Wavelength Radio Microwave IR UV XUV X-rays γ-rays

Dental radiography Airport security X-ray CT Cargo scanners X-ray diffraction

science medicine industry

NDT

• EM-radiation in the above-keV range
• X-rays were discovered in 1895
• First medical radiography 1896
• Since then widely used in science,

industry and medicine

• Most applications rely on absorption

properties What are x-rays ?

slide courtesy A. Döpp, LOA

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(metal jet) bremsstrahlung and line (K-α)-sources

target filter plasma reflected pulse a t t

• s

e c

• n

d p u l s e i n c i d e n t p u l s e

high harmonic sources

Laser-driven X- ray sources

„wiggly“ electron - sources relativistic electron beam +

• undulator = undulator source, FEL
• plasma fields = Betatron source
• laser pulse = Thomson/Compton source

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Brilliance = photons mm2 ⋅mrad2 ⋅s⋅0.1% bandwidth

• transv. emittance

(=phase space area)

• long. emittance
• 1. many photons
• 2. small bandwidth
• 3. low divergence
• 4. small source
• 5. short duration

That‘s where LWFA sources excel

their brilliance: Quality scale for x-rays ...

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„Wiggly“ electron X-ray sources: Ingredients: relativistic electron beam +

undulator undulator radiation, FEL 100‘s eV - keV λu≈1cm plasma fields Betatron radiation keV – 10‘s keV λb≈500µm e- laser fields Thomson scattering 10‘s keV - MeV λl≈1µm

λx−ray = λu,b,l 2(4)γ 2 1+ (K,a0)2 2 +γ 2θ 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

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• Relativistic electrons emit orders of magnitude more radiation than non-relativistic electrons

need an accelerator ! MT/m mm-cm TV/m fs 500 T/m 10-1000 m 10-100 MV/m ns-ps

Conventional : Laser-plasma : Accelerating field Duration Focusing field Total size Plasma cavity

Laser-Plasma Accelerators

slide courtesy A. Döpp, LOA

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brilliance

[ph/ (sec mm2 mrad2 0.1% BW)]

costs (size)

[M€ (meter)]

1 10 100 1000 1022 1015 107 1011 undulator deflecting magnet rotating anode 100 kW, Bremsstrahlung average brilliance of laser driven sources peak brilliance of laser driven sources slide courtesy F. Pfeiffer, TUM

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Larmor radiation (see e.g. Jackson) of an accelerated charged particle: Radiation power: Angular distribution: (Hertzian dipole)

P

R =

e2 6πε0m0

2c3

d! p dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

Find relativistic invariant form of Larmor formula: Transform time: and four-momentum: “general radiation formula”

dP

R

dΩ = e2 16π 2ε0m0

2c3

d! p dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

sin2ϕ dt → dτ = 1 γ dt dP

µ

dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

→ d! p dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

− 1 c2 dE dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

P

R =

e2c 6πε0 m0c2

( )

2

d! p dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

− 1 c2 dE dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

→lab. frame

e2cγ 2 6πε0 m0c2

( )

2

d! p dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

− 1 c2 dE dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒

! p=γ ! βmec, E=γ mec2

P

R =

e2 6πε0cγ 6 " # β

( )

2

− ! β × ! # β

( )

2

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

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Longitudinal acceleration: rel. energy-momentum relation:

E2 = m0c2

( )

2 + !

p2c2 →

d dτ

E dE dτ = c2 ! p d! p dτ →

E=γ m0c2," p=γ m0 " v

dE dτ = ! v d! p dτ

P

R =

e2c 6πε0 m0c2

( )

2

d! p dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

− β 2 d! p dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = e2c 6πε0 m0c2

( )

2

d! p γ dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

= e2c 6πε0 m0c2

( )

2

d! p dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

Transverse acceleration: Deflection in magnetic field

dE dt = 0 ⇒ P

R =

e2c 6πε0 m0c2

( )

2

d! p dτ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

=

dτ =1/γ dt

e2c γ 2 6πε0 m0c2

( )

2

d! p dt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

⇓ For relativistic particles and the same acceleration, transverse deflection produces γ2-times stronger radiation

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Larmor radiation for longitudinal (left) and transverse (right) deflection for particle velocities of v = (0,0.3, 0.9, 0.99) c, corresponding to γ = (1, 1.4, 10, 100). Note different scale for yellow and red distributions

dP

!

dΩ = e2 16π 2ε0c " β 2 sin2θ 1− β cosθ

( )

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dP

⊥

dΩ = e2 16π 2ε0c ! β 2 1− β cosθ

( )

3 1−

sin2θ cos2φ γ 2 1− β cosθ

( )

2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

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Longitudinal acceleration: Injection radiation

laser direction density (schematic) driving laser + plasma wave injection radiation

Quasi-monochromatic electron spectrum indicates localized injection:

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Lorentz Force

=

• +

×

• Purely electric:

Betatron radiation Electromagnetic: Thomson/ Compton scattering Purely magnetic: undulator/ wiggler How can we wiggle an electron beam: Insertion devices

Slide courtesy A. Döpp, LOA

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The radiated energy per solid angle is given by the time integral of the radiated power per solid angle: And likewise in the frequency domain: Therefore, the integrand describes the angular and spectral distribution.

dW dΩ = dP dΩ dt =

−∞ ∞

∫

cε0 R ! E t

( )

2 dt −∞ ∞

∫

dW dΩ = 2cε0 R ! " E ω

( )

2

dω

∞

∫

dW 2 dωdΩ = 2cε0R2 ! " E ω

( )

2

Finding the fields from the Lienard-Wiechert Potentials (Jackson):

E = −∇φ − ∂ ∂t ! A B = ∇ × ! A

φ = − q 4πε0 1 1− ! n ⋅ ! β

( )R

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

ret

, ! A = − q 4πε0c ! β 1− ! n ⋅ ! β

( )R

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

ret

⇒ E = e 4πε0 ! n − ! β γ 2 1− ! n ⋅ ! β

( )

3

R2 + ! n × ! n − ! β

( )×

! " β

( )

c 1− ! n ⋅ ! β

( )

3

R ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ret , ! B = 1 c ! n × ! E ⎡ ⎣ ⎤ ⎦ret

Coulomb field radiation field

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FT to frequency domain, neglecting the Coulomb field ∼1/R2γ2:

! " E ω

( ) =

ieω 32π 3cε0R ! n × ! n − ! β

( )×

! # β

( )

c 1− ! n ⋅ ! β

( )

3

R e

iω ′ t +R ′ t

( )/c

( ) d ′

t

−∞ ∞

∫

, ! " B = 1 c ! n × ! " E ω

( )

( )

This field plugged into the integrand of the angular distribution of radiated energy gives the so- called radiation integral (Jackson or Corde, Rev. Mod. Phys. 85 1 2013): For any given trajectory β(t) and observation direction n, this integral describes the radiated energy per solid angle and energy interval. ⇒ Done!

d 2W dωdΩ = e2 16π 3ε0c ! n × ! n − ! β

( )×

! " β

( )

1− ! n ⋅ ! β

( )

2

e

iω t−! n⋅! r /c

( ) dt

−∞ ∞

∫

2

= e2ω 2 16π 3ε0c ! n × ! n × ! β

( )e

iω t−! n⋅! r /c

( ) dt

−∞ ∞

∫

2

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• Jackson, J.D. Classical Electrodynamics, 3rd Edition, Chapter 14: Radiation by Moving Charges

d2I ddΩ = e2 42c

• +∞

−∞

• n × [(

n − ) × ˙

• ]

(1 − · n)2 · ei(t−

n· r(t)/c)dt

• 2

Fourier Transform Calculate Poynting vector (more or less) Change of variables tret to t

Electric field Radiated energy

• E(

x, t) = e

• n −
• 2(1 −

· n)3R2

• ret

+ e c

• n × ((

n − ) × ˙

• )

(1 − · n)3R

• ret

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Conclusion: for the above conditions, the emitted spectrum is just the Fourier transform of the transverse acceleration and therefore the wiggling force. For the simple yet important case of highly relativistic electrons, negligible deflection (β|| = const = c) and on-axis observation, assuming without any loss of generality e.g. a sinusoidal transverse acceleration, we have: and becomes which is essentially just the transverse acceleration again times a constant.

! n = 1 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ , ! β = β" ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ , ! # β= asinγ 2ωut ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ! n × ! n − ! β

( )×

! " β ⎡ ⎣ ⎤ ⎦ 1− ! n ⋅ ! β

( )

2

1 1− β! asin γ 2ωut

( )

What happens if the above assumptions are violated? We have to solve the radiation integral for special cases (or numerically, if we are lazy)

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Electrons on a circular orbit: Bending magnet radiation nonrelativistic relativistic (γ = 1) (γ =3)

• F

C

• vs

y

• F

C

• vs

y

1

y'

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This short flash leads to a typical (“critical“) frequency of

ω c = 2 Δt = 3cγ 3 2R

For a circular motion with radius ρ in the x,z-plane, and the fields in z-direction are: and the radiation integral becomes:

r t

( ) = ρ 1− cos ωt ( )

( ),0,sin ωt

( )

( )

! Ex ω

( ) =

6πe 4π 2cε0 γξ R 1+γ 2θ 2

( )

K2/3 ξ

( ), !

Ey ω

( ) =

6πe 4π 2cε0 iθγ 2ξ R 1+γ 2θ 2

( )

K1/3 ξ

( ), ξ = ω

2ω c 1+γ 2θ 2

( )

3/2

d 2W dωdΩ = 3e2γ 2 16π 3cε0 ω ω c ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

1+γ 2θ 2

( ) 1+γ 2θ 2 ( )K2/3

2

ξ

( )+γ 2θ 2K1/3

2 ξ

( )

⎡ ⎣ ⎤ ⎦

Circular orbit: Radiation lobe crosses the observer‘s FOV. Emission duration is given by time difference of radiation propagating straight from A to B and electron on circular orbit between A and B

• pol. || orbit plane
• pol. ⊥ orbit plane

Δt = te − trad = 2R cβ 1 γ − 2Rsin 1/γ

( )

c ≈ 4R 3cγ 3

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Circular orbit: Radiation polarized in (red) and perpendicular to (blue) orbit plane Circular orbit: An integration of the direction-dependent emission over the full solid angle gives the universal spectral shape

• f synchrotron radiation (Schwinger 1949):

d 2W dω = 3e2γ 4πcε0 ω ω c K5/3 x

( )

ω /ωc ∞

∫

dx

10

−3

10

−2

10

−1

10 10

1

10

−3

10

−2

10

−1

10

Max at ω/ωc ≈ 0.25 S ≈ 1.33 ξ1/3 S ≈ 0.78 ξ1/2e−ξ radiated energy: 50 % 50 % S(ξ)∝ξ ∫ ∞

ξ K5/3(x)dx

ξ=ω/ωc S(ω/ωc)

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log E log N ωU x 2NU

periodic deflection in an undulator: “identical“ emission at each turning point As long as deflection angle is smaller than 1/γ: Spectrum is coherently enhanced at wiggling frequency!

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Periodic deflection in a stationary field (undulator) with a wavelength λu in the limit of zero deflection (weak field): In the laboratory frame, the electron is wiggled at the frequency: which in its rest frame transforms to . Back in the lab frame, the

• bserver sees ω’ Doppler-upshifted:

ωu = 2πv λu = kuβc ′ ω = γ ωu ωemission,undulator = ′ ω c + v c − v = ′ ω 1+ β 1− β = ′ ω 1+ β

( )

2

1− β 2 ≈ 2γ ′ ω = 2γ 2ωu Periodic deflection in a moving field (Thomson scattering): Instead of a simple relativistic length contraction of the wiggling field, the (counterpropagating) laser field with λL has to be Doppler- shifted into the electron rest frame, leading to: ωemission,Thomson = ′ ω c + v c − v = ′ ω 1+ β 1− β = ′ ω 1+ β

( )

2

1− β 2 = 1+ β

( )

2

1− β 2 ω L ≈ 4γ 2ω L The emission frequency scales as the wiggling frequency times γ2

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Strong wiggling fields: Period lengthening due to excursion of electrons from a straight line: Consider a photon observed in the lab frame with ℏωx-ray with a forward (p||) and transverse (p⊥) momentum. Its four-momentum in the proper frame of the electron is then: and the energy becomes angle-dependent: :

′ P

µ =

′ E c ′ px ′ py ′ pz ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ = γ −βγ −βγ γ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ E c psin θ

( )

pcos θ

( )

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ′ E c = γ E c − βγ pcos θ

( )

= γ !ω x−ray c 1− β cos θ

( )

( )

Therefore: With , , we get:

ω x−ray = ′ ω γ 1− β cos θ

( )

( )

= ωu 1− β cos θ

( )

λx−ray = λu 1− β cos θ

( )

( )

cos θ

( ) ≈1−θ 2 / 2

β ≈1− 1+ K 2 2

( ) / 2γ 2

K=eBλu 2πmec

λu 1− β cos θ

( )

( ) = λu 1− 1+ K 2 2

2γ 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1− θ 2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =λu 1− 1− θ 2 2 − 1+ K 2 2 2γ 2 +... ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⇒ λx−ray ≈ λu 2γ 2 1+ K 2 2 +γ 2θ 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Resonance condition for undulator radiation

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λx−ray = λu,L 2(4)γ 2 1+ (K,a0)2 2 +γ 2θ 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Wiggling wavelength Electron gamma factor field strength Observation angle range

X-ray spectrum is influenced by:

• Electron energy and bandwidth
• Wiggling field strength and number of
• scillations
• Observation direction and solid angle
• Wiggling period

The importance of K: In the case of an undulator, is the dimensionless undulator strength parameter. In the case of a laser field, the dimensionless laser amplitude plays the same role. The only difference is the temporal variability of a0(t). Factors entering the resonance condition

K = eBλu 2πmec =

v||=c

eB "ωu "me ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ a0 t

( ) = eE0 t ( )

ω Lmec = eB0 t

( )

ω Lme ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

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Emission into fundamental or harmonics?

• K,a0 < 1, Undulator regime: Electron

deflection angle <1/γ, all radiation is confined to 1/γcone and emitted continuously during wiggling path.

• K,a0 > 1, Wiggler regime: Electron

deflection angle >1/γ, radiation cone sweeps across observer’s field of view twice per period and emits series of short flashes ⇒ harmonic emission Wiggler case: Constructive interference if nλx−ray = λu β − λu cosθ Linewidth: (as in FT of finite length (∼Nu) sine wave) Emission angle: ⇒ In total N2 brightness enhancement over synchrotron radiation. Δλ λ = 1 Nun Δθ = 1 γ 1+ K 2 / 2 Nun ∼ 1 γ Nun

25

Solution of the radiation integral for a sinusoidal trajectory: d 2W dωdΩ θ=0 = e2γ 2Nu

2

4πε0c R NΔω ω1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ nK 1+ K 2 / 2 J n+1

2 α z

( )− J n−1

2 α z

( )

( )

2

α z = nK 2 4 1+ K 2 / 2

( )

, R Nu Δω ω1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = sin2 Nuπ Δω ω1

( )

Nuπ Δω ω1

( )

2

−0.2 0.2 −1 −0.5 0.5 1

zK/γku xK/γku

K=0.02 K=0.6 K=1 K=1.5 K=4 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequency ω/ω1 Spectral Intensity [a.u.]

K=0.02x0.02−2 K=0.6 x0.6−2 K=1.0 x1.0−1 K=1.5 x1.5−1 K=4.0 x4.0−1 ∝ ξ2 K2

2/3(ξ)

Frequency ω/ω1 Spectral Intensity [a.u.]

−0.4 −0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1 K=1.0 Nu=8 K=0.02 Nu=8 K=4.0 Nu=8 K=1.0 Nu=2 K=1.0 Nu=32

For increasing K, the oscillation becomes anharmonic, leading to the emission of harmonics whose intensity follows the universal spectrum

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Which harmonic is dominant?

Normalized Vector Potential/ K-parameter

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Radiation from a bunch: The resulting radiated field from a bunch of electrons is just the vector sum (i.e. regarding the phase) of contributions from the individual electrons. The emitted spectrum is then given by: If βj=β=1, and rj=r+Rj, i.e. the electrons have a fixed distance from the center coordinate r, we can write:

d 2W dωdΩ = e2ω 2 16π 3ε0c ! n × ! n × ! β j

( )e

iω t−! n⋅! rj/c

( )dt

j=1 Ne

∑

−∞ ∞

∫

2

d 2W dωdΩ = e2ω 2 16π 3ε0c e

iω Rj/βc j=1 Ne

∑

2 :=c ω

( )

! " # $ # × % n × % n × % β j

( )e

iω t−% n⋅% r /c

( ) dt

−∞ ∞

∫

2

c ω

( ) =

β→1

Ne + Ne Ne −1

( ) f ω ( );

The Form factor f(ω) is the Fourier-Transform of the longitudinal bunch profile. For a Gaussian bunch with Which is zero for few-fs bunch profiles (µm) and Angström x-ray wavelengths.

S z

( ) =

2πσ e

( )

−1

e

−z2/2σ e

2 → f ω

( ) = e

−4π 2σ e

2/λx 2

Thus, the radiation is the incoherent sum

d 2W dωdΩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Ne

= Ne d 2W dωdΩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Ne=1

28

Brilliance (again): For a photon flux the brilliance is defined as Due to the narrow 1/γemission cone, σ x σ’ ist strongly linked to the electron beam emittance via the total emittance:

B= − dN x dt 4π 2σ x ′ σ xσ y ′ σ y dω ω = number of photons per time s-1 ⎡ ⎣ ⎤ ⎦ source size mm2 ⎡ ⎣ ⎤ ⎦ × divergence mrad2 ⎡ ⎣ ⎤ ⎦ × bandwidth 0.1% ⎡ ⎣ ⎤ ⎦

εtot,x,y = σ b,x,y

2

+σ x,y

2

′ σ b,x,y

2

+ ′ σ x,y

2

Photon number Undulator/wiggler: In uthe undulator case, the average photon energy is 2γ2 ℏ, and the photon number per electron amounts to: For Wigglers, the average photon energy is and the total photon number becomes: But keep in mind that brilliance does care for spectral width, while total photon number doesn’t.. N x−ray

U

= π 3 α f K 2Nu ∼10−2 K 2Nu per e− 8 15 3 !ω c N x−ray

W

= π5 3 6 α KNu ∼ 10−1...1

( )Nu per e−

29

Experiment: Laser Accelerator & Undulator:

Fuchs, M. et al. Laser-driven soft-X-ray undulator source. Nature Physics 5, 826–829 (2009).

Observation angle (mrad) CCD counts (arb. units) CCD counts (arb. units)

a b

Wavelength (nm) 10 10 30 30 20 20 Wavelength (nm) ¬1.0 2.0 1.0 10 10 3 3 20 20 10 40 35 30 25 20 15 5 600 400 200

fund. 2nd harm.

200 MeV / γ=400 electron beam 5 mm wavelength undulator, 60 periods X-ray spectral width determined by electron bunch energy width / quadrupole chromaticity, not native undulator bandwidth