Representation of synchrotron radiation in phase space
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Representation of synchrotron radiation in phase space
Ivan Bazarov
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Outline
• Motion in phase space: quantum picture• Wigner distribution and its connection to
synchrotron radiation• Brightness definitions, transverse coherence• Synchrotron radiation in phase space– Adding many electrons– Accounting for polarization– Segmented undulators, etc.
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Motivation
• Compute brightness for partially coherent x-ray sources– Gaussian or non-Gaussian X-rays??– How to include non-Gaussian electron beams
close to diffraction limit, energy spread??– How to account for different light polarization,
segmented undulators with focusing in-between, etc.??
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Brightness: geometrical optics
• Rays moving in drifts and focusing elements
• Brightness = particle density in phase space (2D, 4D, or 6D)
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Phase space in classical mechanics
• Classical: particle state
• Evolves in time according to , • E.g. drift: linear restoring force:
• Liouville’s theorem: phase space density stays const along particle trajectories
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Phase space in quantum physics• Quantum state:
or Position space momentum space
• If either or is known – can compute anything. Can evolve state using time evolution operator:
• - probability to measure a particle with
• - probability to measure a particle with
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Wigner distribution
• – (quasi)probability of measuring quantum particle with and
• Quasi-probability – because local values can be negative(!)
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PHYS3317 movies
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Same classical particle in phase space…
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Going quantum in phase space…
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Wigner distribution properties
•
•
•
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• Time evolution of is classical in absence of forces or with linear forces
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Connection to light• Quantum – • Linearly polarized light (1D) – • Measurable – charge density• Measurable – photon flux density• Quantum: momentum representation
is FT of • Light: far field (angle) representation
is FT of
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Connection to classical picture
• Quantum: , recover classical behavior• Light: , recover geometric optics• or – phase space density
(=brightness) of a quantum particle or light• Wigner of a quantum state / light propagates
classically in absence of forces or for linear forces!
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Diffraction limit
• Heisenberg uncertainty principle: cannot squeeze the phase space to be less than since ,
• We call the diffraction limit
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Coherence
• Several definitions, but here is one
• In optics this is known as value
measure of coherence or mode purity
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Useful accelerator physics notations
• -matrix
• Twiss (equivalent ellipse) and emittance
with and or
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Jargon
• Easy to propagate Twiss ellipse for linear optics described by :
• -function is Rayleigh range in optics
• Mode purity
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Hermite-Gaussian beam
, where are Hermite polynomial order in respective plane
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Hermite-Gaussian beam phase space
position xangle qx
x
angle qx
angle qx
position x
position x
x
x
y
y
y
wigner
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Wigner from 2 sources?
• Wigner is a quadratic function, simple adding does not work
• Q: will there an interference pattern from 2 different but same-make lasers?
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Wigner from 2 sources?
- first source, - second source, - interference term
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Interference term
• Let each source have and , with random phases , then Wc = 0 as
• This is the situation in ERL undulator: electron only interferes with itself
• Simple addition of Wigner from all electrons is all we need
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Example of combining sourcestwo Gaussian beams
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Same picture in the phase spacetwo Gaussian beams
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Wigner for polarized light
• Photon helicity or right handed and left handed circularly polarized photons
• Similar to a 1/2-spin particle – need two component state to describe light
• Wigner taken analogous to stokes parameters
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Generalized Stokes (or 4-Wigner)
Total intensity Linearly polarized light (+) x-polarized (–) y-polarized
Linearly polarized light (+) +45°-polarized (–) -45°-polarized
Circularly polarized light (+) right-hand (–) left-hand
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Example Bx(ph/s/0.1%BW/mm/mrad) , Nund = 250
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Synchrotron radiation
Potential from moving charge
with . Then find , then FT to get
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Brightness definitions• Phase space density in 4D phase space =
brightness . Same as Wigner, double integral gives spectral flux– Units ph/s/0.1%BW/mm2/mrad2
– Nice but bulky, huge memory requirements.• Density in 2D phase space = 2D brightness
are integrated away respectively. Easy to compute, modest memory reqs.– Units ph/s/0.1%BW/mm/mrad
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Brightness definitions
• Can quote peak brightness or – but can be negative
• E.g. one possible definition (Rhee, Luis)
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How is brightness computed now
• Find flux in central cone • Spread it out in Gaussian phase space with
light emittance in each plane • Convolve light emittance with electron
emittance and quote on-axis brightness
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Criticism
(I) What about non-Gaussian electron beam?(II) Is synchrotron radiation phase space from a
single electron Gaussian itself (central cone)?
(I) Is the case of ERL, while (II) is never the case for any undulator
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Some results of simulations with synrad
Checking angular flux on-axis
I = 100mA, zero emittance beam everywhere
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Total flux in central cone
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Scanning around resonance…
on-axis
total
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Scanning around resonance…
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Scanning around 2nd harmonic…
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Synchrotron radiation in phase space, back-propagated to the undulator center
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Key observations
• Synchrotron radiation light:- Emittance ~ 3diffraction limit- Ninja star pattern- Bright core (non-Gaussian)
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Emittance vs fraction
• Ellipse cookie-cutter (adjustable), vary from 0 to infinity
• Compute rms emittance inside• All beam
• Core emittance • Core fraction
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Examples
• Uniform: , • Gaussian: ,
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What exactly is core emittance?
• Together with total flux, it is a measure of max brightness in the beam
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Light emittance vs fraction
core emittance is (same as Guassian!), but is much larger
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Optimal beta function
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Light phase space around 1st harmonic
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Checking on-axis 4D brightness
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Checking on-axis 2D brightness
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On-axis over average 2D brightness
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Light emittance
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Optimal beta function
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2 segments, Nu=100 each, 0.48m gapu=2cm, By = 0.375T, Eph= 9533eV
5GeV, quad 0.3m with 3.5T/m
section 1 section 2quad
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flux @ 50m
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flux @ 50m
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Conclusions
• Wigner distribution is a complete way to characterize (any) partially coherent source
• (Micro)brightness in wave optics is allowed to adopt local negative values
• Brightness and emittance specs have not been identified correctly (for ERL) up to this point
• The machinery in place can do segmented undulators, mismatched electron beam, etc.
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Acknowledgements
• Andrew Gasbarro• David Sagan