Jul 22, 2013 ... Lecture Outline. Electron beams. Photon beams. References: 1. J. D. Jackson,
Classical Electrodynamics (Wiley, New York, third edition, 1999).
Basic concepts in electron and photon beams Zhirong Huang SLAC and Stanford University July 22, 2013
Lecture Outline Electron beams Photon beams References: 1. 2. 3. 4. 5.
J. D. Jackson, Classical Electrodynamics (Wiley, New York, third edition, 1999). Helmut Wiedemann, Particle Accelerator Physics (Springer-Verlag, 2003). Andrew Sessler and Edmund Wilson, Engine of Discovery (World Scientific, 2007). David Attwood, Soft X-rays and Extreme Ultraviolet Radiation (Cambridge, 1999) Peter Schmüser, Martin Dohlus, Jörg Rossbach, Ultraviolet and Soft X-Ray FreeElectron Lasers (Springer-Verlag, 2008). 6. Kwang-Je Kim, Zhirong Huang, Ryan Lindberg, Synchrotron Radiation and FreeElectron 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. 2
William Barletta, USPAS director
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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 frame with the z’ axis will have the length Dz in the lab frame ∆𝑧 ′
∆𝑧 =
𝛾
Time dilation: Two events occuring in the moving frame 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 ′ 6
Energy, Mass, Momentum Energy
𝐸 = 𝑇 + 𝑚𝑐 2 Kinetic energy Rest mass energy
Electrons rest mass energy 511 keV (938 MeV for protons), 1eV = 1.6 × 10-19 Joule
Momentum
𝒑 = 𝛾𝜷𝑚𝑐 Energy and momentum
𝐸 2 = 𝑝2 𝑐 2 + 𝑚2 𝑐 4 , E = 𝛾𝑚𝑐 2 .
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Maxwell’s Equations 𝑫 = 𝜖0 E B = 𝜇0 𝑯
𝑐 = (𝜖0 𝜇0 )−1/2
Wave equation
Lorentz transformation of fields
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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!
A relativistic electron
In electron’s frame, Coulumb field is In lab frame, space charge field is
Guiding beam: dipole Lorentz force
Centrifugal force 𝛾𝑚𝑐 2 𝛽2
𝐹𝑐𝑓 =
(opposite for e-)
𝜌
Bending radius is obtained by balance the forces 1 𝜌
𝑒𝐵 = 𝛾𝛽𝑚𝑐 2
1 𝜌
[m-1]=0.2998
𝐵[T] 𝛽𝐸[GeV]
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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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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
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Beam description
Beam phase space (x,x’,y,y’,Dt, Dg)
Consider paraxial beams such that
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Linear beam transport Transport matrix
Free space drift
Quadrupole (de-)focusing
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Beam properties Second moments of beam distribution rms size rms divergence
correlation
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Beam emittance Emittance or geometric emittance
Emittance is conserved in a linear transport system Normalized emittance is conserved in a linear system including acceleration
Normalized emittance is hence an important figure of merit for electron sources Preservation of emittances is critical for accelerator designs. 25
Beam optics function Optics functions (Twiss parameters)
Given beta function along beamline
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Free space propagation Single particle
Beam envelope
Analogous with Gaussian laser beam 27
FODO lattice Multiple elements (e. g., 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
is the number of electrons per unit phase space volume
Since the number of electrons is an invariant function of z, distribution function satisfies Liouville theorem
equations of motion
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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 beams Wave equation and paraxial approximation Radiation diffraction and emittance Transverse and temporal coherence
Brightness and diffraction limit Radiation intensity and bunching Bright accelerator based photon sources 32
Photon wavelength and energy
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Single pass FELs (SASE or seeded)
Various accelerator and non-acc. sources
Synchrotron radiation FEL oscillators Undulator radiation 34 (High-average power)
Radiation diffraction Wave propagation in free space
Angular representation
General solution
Paraxial approximation (f2
