Julie Cass
SLAC SULI Project Summary
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Very nice
What is the frequency spectrum of the transition radiation? I thought it was X-ray not IR.
What is the connection between bunch length and the transition radiation frequency spectrum.
Do you understand what the vector contains?
Is there any correlation between transverse position and time within the bunch?
The LCLS at SLAC National Accelerator Laboratory is a ground-breaking instrument which uses
high-energy electrons to create ultrafast x-ray pulses. For many of the experiments conducted at the
LCLS, however, the mere description of “ultrafast” is not sufficiently precise; it is crucial for the
interpretation and understanding of the data collected here that we know the precise size of these
electrons pulses. It is understood that the pulses are created on the femtosecond scale- a scale far too
small to be detected by existing detectors. Joe Frisch and his collaborators have designed a method
implementing a broadband infrared prism spectrometer in order to obtain a spectrum of optical
transition radiation emitted by the electron beam. Through the implementation of a Fourier transform,
one can use this spectrum to calculate the bunch length of the beam. The goal of my project at the SLAC
SULI program will be to help implement this infrared-spectrometer and specifically to help complete
mathematical calculations that will help us to understand how the beam behaves as it progresses
through the spectrometer, including mathematical analysis of beam propagation through the optical
system, calculation of beam dispersion from the crystal and analysis of the spectrometer resolution.
The current design for the IR spectrometer involves an optical set-up containing off-axis
parabolic mirrors to steer and collimate the beam of light produced by the electron through optical
transition radiation. These optics are particularly difficult to align- a task made more difficult by our lack
of ability to predict how the beam should behave due to our lack of experience working with these
devices. As proper alignment is essential to our project, I will be responsible for using a mathematical
interpretation of the optical system to calculate how the beam ought to as it travels through various
optical elements. In general, we can model the behavior of a beam traveling through an optical system
using ray transfer matrix analysis. Each optical element, such as a parabolic mirror or lens, can be
described by a ray transfer matrix, which contains such variables as the focal length of the instrument.
When multiplied by a vector representing the initial beam, this matrix describes how the beam will be
affected by each optical element. Through the successive multiplication of ray transfer matrices
describing each element in the optical system, way can generate a model for the beam as it would result
if the optical system is properly configured and aligned. Unlike the electron beam in LSLC, our model
uses a Gaussian beam, whose diameter fluctuations we are particularly interested in. To represent this
style of beam we must generated a vector to represent the initial beam in terms of a complex beam
parameter, calculated in terms of the beam wavelength, radius of curvature and diameter. In order to
model the beam in the optical system, I will need to calculate this complex parameter for the initial
beam, use ray transfer matrix analysis to determine this parameter at later points in the system, and
extract the corresponding beam width. This beam width can be checked against the width measured in
our design as a check to ensure that it is properly aligned and configured.
After traveling through the optical system, the beam will be sent through a “prism” of KRS-5
crystal. The index of refraction for this material is wavelength dependent; thus it will output a beam
dispersed angularly by wavelength. The beam is then passed into a line detector so that we may
electronically analyze the spectrum. As part of the calibration of this IR spectrometer, I will need to
calculate this angular dispersion of wavelengths coming out of the prism and determine the appropriate
tilt of the detector to correct for aberrations. This should leave us with a clear and undistorted
spectrum at the detector. Finally, the importance of detecting the spectrum of the beam is to determine
the form factor so that we may conduct an inverse Fourier transform to find the associated bunch
length. My project will include determining the method and mathematical calculations needed in order
to output a bunch length from the spectrum produced by our spectrometer.