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Introduction to electron and photon beam physics
Zhirong Huang
SLAC and Stanford University
August 03, 2015
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Lecture Plan
Electron beams (1.5 hrs)
Photon or radiation beams (1 hr)
References:
1. J. D. Jackson, Classical Electrodynamics (Wiley, New York, third edition, 1999). 2. Helmut Wiedemann, Particle Accelerator Physics (Springer-Verlag, 2003). 3. Andrew Sessler and Edmund Wilson, Engine of Discovery (World Scientific, 2007). 4. David Attwood, Soft X-rays and Extreme Ultraviolet Radiation (Cambridge, 1999) 5. Peter Schmüser, Martin Dohlus, Jörg Rossbach, Ultraviolet and Soft X-Ray Free-
Electron Lasers (Springer-Verlag, 2008). 6. Kwang-Je Kim, Zhirong Huang, Ryan Lindberg, Synchrotron Radiation and Free-
Electron Lasers for Bright X-ray Sources, USPAS lecture notes 2013. 7. Gennady Stupakov, Classical Mechanics and Electromagnetism in Accelerator
Physics, USPAS Lecture notes 2011. 8. Images from various sources and web sites.
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Electron beams
Primer on special relativity and E&M
Accelerating electrons
Transporting electrons
Beam emittance and optics
Beam distribution function
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Lorentz Transformation
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Length Contraction and Time Dilation Length contraction: an object of length Dz* aligned in the
moving system with the z* axis will have the length Dz in the
lab frame
∆𝑧 =∆𝑧∗
𝛾
Time dilation: Two events occurring in the moving system at
the same point and separated by the time interval Dt* will be
measured by the lab observers as separated by Dt
∆𝑡 = 𝛾∆𝑡∗
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Energy, Mass, Momentum
Kinetic energy Rest mass energy
Energy
Electrons rest mass energy 511 keV (938 MeV for protons),
1eV = 1.6 × 10-19 Joule
Momentum
𝒑 = 𝛾𝜷𝑚𝑐
𝐸 = 𝑇 + 𝑚𝑐2
Energy and momentum
𝐸2 = 𝑝2𝑐2 + 𝑚2𝑐4, E = 𝛾𝑚𝑐2. 6
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Relativistic acceleration Momentum change
Beam dynamics drastically different for parallel and
perpendicular acceleration!
Negligible radiation for parallel acceleration at high energy 7
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Maxwell’s Equations
𝑫 = 𝜖0E B = 𝜇0𝑯
Wave equation
Lorentz transformation of fields
𝑐 = (𝜖0𝜇0)−1/2
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Field of a moving electron In electron’s frame, Coulomb field is
In lab frame, space charge fields are
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Lorentz Force Lorentz force
Momentum and energy change
Energy exchange through E field only
= 0
No work done by magnetic field!
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Guiding beams: dipole Lorentz force
Centrifugal force
𝐹𝑐𝑓 =𝛾𝑚𝑐2𝛽2
𝜌
Bending radius is obtained by balance the forces
1
𝜌 =
𝑒𝐵
𝛾𝛽𝑚𝑐2
1
𝜌 [m-1]=0.2998
𝐵[T]
𝛽𝐸[GeV]
(opposite for e-)
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Cyclotron If beam moves circularly, re-traverses the same accelerating section again and again, we can accelerate the beam repetitively
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http://www.aip.org/history/lawrence/first.htm
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Lawrence was my teacher when I built the first cyclotron. He got a Nobel prize for it. I got a Ph.D. (- S. Livingston, years later)
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From Cyclotron to Synchrotron Cyclotron does not work for relativistic beams.
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Synchrotron GE synchrotron observed first synchrotron radiation (1946) and opened a new era of accelerator-based light sources.
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Electron linac
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SLAC linac
35 Total 12-m CryoModules for LCLS-II
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Livingston Plot for High-Energy Accelerators
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Beam description
Consider paraxial beams such that
Beam phase space (x,x’,y,y’,Dt, Dg)
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Linear optics for beam transport Transport matrix
Free space drift
Quadrupole (de-)focusing
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Beam properties Second moments of beam distribution
rms divergence
rms size
correlation
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Emittance or geometric emittance
Emittance is conserved in a linear transport system
Normalized emittance is conserved in a linear system
including acceleration
Beam emittance
Normalized emittance is hence an important figure of merit
for electron sources
Preservation of emittances is critical for accelerator designs. 26
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Beam optics function Optics functions (Twiss parameters)
Given beta function along beamline
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Free space propagation
Analogous with Gaussian laser beam
Single particle
Beam envelope
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Multiple elements (e. g., FODO lattice)
FODO lattice
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FODO lattice II
Maximum beta
Minimum beta
When f >> l
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Electron distribution in phase space We define the distribution function F so that
Since the number of electrons is an
invariant function of z, distribution
function satisfies Liouville theorem
equations of motion
is the number of electrons per unit phase space volume
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Gaussian beam distribution Represent the ensemble of electrons with a continuous
distribution function (e.g., Gaussian in x and x’)
For free space propagation
Distribution in physical space can be obtained by integrating
F over the angle
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Photon or radiation beams
Introduction to radiation
Radiation diffraction and emittance
Coherence and Brightness
Accelerator based light sources
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Radiation intensity and bunching
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Photon wavelength and energy
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FEL oscillators
(High-average power)
Single pass FELs
(SASE or seeded)
Synchrotron radiation
Undulator radiation Various accelerator
and non-acc. sources 35
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Radiation from Accelerated Electrons
The field outside of the circle of radius ct “does not know” that the charge has been moved.
If the charge was moved twice, then the field lines at time t > t1 would look like this—there will be two spheres, with the radiation layers between them
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Three forms of synchrotron radiation
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Shintake Radiation Demo Program
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Radiation propagation and diffraction Wave propagation in free space
Paraxial approximation (f2 << 1)
General solution
Angular representation
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Gaussian beam and radiation emittance
At arbitrary z
Analogous with electron beam
Single electron radiation can be approximated by Gaussian
beam Gaussian fundamental mode at waist z=0
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Coherence
41 R. Ischebeck
E(x,t) at location z z
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Transverse (Spatial) Coherence Transverse coherence can be measured via the interference pattern in Young's double slit experiment.
Near the center of screen, transverse coherence determines fringe visibility
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Phase space criteria for transverse coherence
• Initial phase space area 4pR >> l
• Coherent flux is reduced by MT • Show this criteria from physical optics argument
• Final phase space area
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Temporal Coherence
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Chaotic light Radiation from many random emitters (Sun, SR, SASE FEL)
Correlation function and coherence time
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Temporal mode and fluctuation
1
L
W
W M
D
Number of regular temporal regions is # of coherent modes
Intensity fluctuation
Total # of modes
Same numbers of mode in frequency domain
Fourier limit, minimum longitudinal phase space
Longitudinal phase space is ML larger than Fourier limit
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Light Bulb vs. Laser
A. Schawlow (Nobel prize on laser spectroscopy), Scientific Americans, 1968
Radiation emitted from light bulb is chaotic.
Pinhole can be used to obtain spatial coherence.
Monochromator can be used to obtain temporal coherence.
Pinhole and Monochromator can be combined for coherence.
Laser light is spatially and temporally coherent.
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Brightness
D
Dx
Units: photons/s/mm2/mrad2/0.1%BW
2)divergence size (source
unit timein range spectralunit in Photons
B
Peak
Average 48
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Incoherent radiation from many electrons Such a beam can be described by the convolution of the
coherent Gaussian beam with the electron distribution in
phase space
Effective source size and divergence
When electron beam emittance sxsx’>>l/(4p)
# of transverse modes
x
x’
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Evolution of X-ray Light Sources GE synchrotron (1946) opened a new era of accelerator-based light sources.
These light sources have evolved rapidly over four generations.
The first three-generations are based on synchrotron radiation.
The forth-generation light source is a game-changer based on FELs.
The dramatic improvement of brightness and coherence over 60 years easily outran Moore’s law.
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Synchrotron Radiation Facilities
State-of-art storage rings have pulse duration ~10 ps, emittance ~1 nm. Diffraction-limited storage rings with emittance ~10 pm are under active R&D.
SSRF (2009)
MAX-IV (2016)
NSLS-II (2015)
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When
intensity from many electrons add incoherently (~Ne)
Radiation intensity What if emitters are not random in time
For an electron bunch with rms bunch length se
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Bunching and coherent radiation
Form factor or bunching factor
If the bunch length is shorter than the radiation wavelength
Radiation intensity from many electrons add coherently
(~Ne2)
Another way to produce bunching from a relatively long
bunch is through so-called microbunching
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