Lecture 1 Slides

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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


Electron beams Primer on special relativity and E&M Accelerating electrons Transporting electrons

Beam emittance and optics Beam distribution function


Lorentz Transformation


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 =


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


𝒑 = 𝛾𝜷𝑚𝑐 Energy and momentum

𝐸 2 = 𝑝2 𝑐 2 + 𝑚2 𝑐 4 , E = 𝛾𝑚𝑐 2 .


Maxwell’s Equations 𝑫 = 𝜖0 E B = 𝜇0 𝑯

𝑐 = (𝜖0 𝜇0 )−1/2

Wave equation

Lorentz transformation of fields


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-)


Bending radius is obtained by balance the forces 1 𝜌

𝑒𝐵 = 𝛾𝛽𝑚𝑐 2

1 𝜌


𝐵[T] 𝛽𝐸[GeV]


Cyclotron If beam moves circularly, re-traverses the same accelerating section again and again, we can accelerate the beam repetitively




From Cyclotron to Synchrotron Cyclotron does not work for relativistic beams.


Synchrotron GE synchrotron observed first synchrotron radiation (1946) and opened a new era of accelerator-based light sources.


Electron linac



SLAC linac


Beam description

Beam phase space (x,x’,y,y’,Dt, Dg)

Consider paraxial beams such that


Linear beam transport Transport matrix

Free space drift

Quadrupole (de-)focusing


Beam properties Second moments of beam distribution rms size rms divergence



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


Free space propagation Single particle

Beam envelope

Analogous with Gaussian laser beam 27

FODO lattice Multiple elements (e. g., FODO lattice)


FODO lattice II Maximum beta

Minimum beta

When f >> l


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


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


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


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