What is the reason for each of the optical elements in Fig. 1?
State the reason for the slit (presumably background reduction)
Why is the beam made different widths at different places?
SIMULATIONS AND ANALYSIS OF AN
INFRARED PRISM SPECTROMETER FOR ULTRA-SHORT
BUNCH LENGTH DIAGNOSTICS AT LCLS
Julie Cass
Office of Science, Science Undergraduate Laboratory Internship Program
University of Notre Dame
SLAC National Accelerator Laboratory
Menlo Park, California
July 18, 2011
Prepared in partial fulfillment of the requirement of the Office of Science, Department of
Energy’s Science Undergraduate Laboratory Internship under the direction of Josef
Frisch in the ICD Accelerator Physics & Engineering Department at SLAC National
Accelerator Laboratory.
Participant:
__________________________________
Signature
Research Advisor:
__________________________________
Signature
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INTRODUCTION
The Linear Coherent Light Source (LCLS) at SLAC National Accelerator Laboratory
is a groundbreaking instrument, implementing high-energy electrons to generate ultrafast Xray pulses with bunch charges as low as 20 pC [1]. This free-electron laser (FEL) is capable
of producing pulses with ultrashort bunch lengths, enabling scientists to image molecules and
atoms in motion and pushing the limits of technology in pursuit of a new level of
understanding of fundamental chemical, biological and physical properties of materials [2].
The interpretation and understanding of the data collected in these experiments relies on a
complete understanding of the parameters used in driving the FEL. While the minimum
achievable bunch length at LCLS has been constrained to a value below 10 fs, the precise
value of this parameter remains unknown. Diagnosis of this parameter requires a mechanism
capable of probing pulses shorter than 10 fs, a scale too small to be resolved by existing time
domain detectors. Collaborators at the LCLS have developed an alternative design
implementing a broadband infrared prism spectrometer to probe for this parameter in the
frequency domain.
This new approach is centered on the concept that information about the bunch length
of the electron beam can be extracted from the spectrum of optical transition radiation
generated by the FEL beam. This transition radiation can be characterized as a relativistic
effect, resulting in the emission of electromagnetic radiation when a charge particle
approaches a foil. The implementation of an inverse Fourier transform of the transition
radiation spectrum allows for the inversion of spectral information for analysis in the time
3
domain. This provides a resolute means for calculation of the FEL bunch length in the femtosecond scale from the information contained in the transition radiation spectrum.
The wavelength range of interest for the transition radiation from the LCLS beam is
expected to fall within the near- to mid-infrared range. This radiation is expanded and
collimated using precision optics [3] and steered to a crystal acting as the spectrometer prism.
The index of refraction of the crystal is wavelength dependent, creating a dispersion of the
wavelengths comprising the radiation beam. The dispersed light will then fall onto the
detecting surface of a PYREOS pyroelectric detector array, designed to register the intensity
of light reaching each of its pixels [4]. Relating the voltage registered by each pixel to the
wavelength of light corresponding to each pixel based on the dispersion relation of the crystal
allows for the interpretation of the spectrum of light reaching the detector [5]. This spectrum
will then be used in the inverse Fourier analysis to determine the FEL bunch length.
It is crucial for the success of this approach that the spectrometer is designed with
precisely aligned optics for propagation of the optical transition radiation from the foil to the
detector. For this purpose, a method for analyzing the behavior of the beam in the optical
system of the spectrometer has been determined. We require an understanding of both how the
beam would behave for an ideal system, containing flawless optical elements and perfect
alignment, and for a system with the introduction of alignment errors due to physical
constraints and aberrations from flaws in the optical materials. Computer programs simulating
this behavior will allow for both a guide for optical alignment and analysis the resolution
possible given the physical constraints of the spectrometer.
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METHODS AND MATERIALS
i. Spectrometer Design
The current design for the infrared spectrometer involves the combination of an
electronic system to run the pyroelectric line detector and an optical set-up employing lenses
and mirrors to expand, collimate and steer the beam towards this detector. A diagram of the
optical system is provided in Figure 1. Many of the mirrors chosen for this optical design are
90-degree gold-coated off-axis parabolic mirrors, chosen for their ability to expand and focus
the infrared beam while steering it through the system. While the functionality of these
mirrors is ideal for our design, they are particularly difficult to align, due to both the challenge
in locating the off-axis central point and our requirement to calculate the tolerances for how
the beam ought to behave. A HeNe laser emitting light in the visible range at 632nm was
initially used for positioning of the system elements, due to the greater ease of working with
visible light. Alignment will be further complicated when a laser of the proper infrared
wavelength is implemented in testing of the optical system for use in the FEL.
ii. MATLAB Simulations for Ideal Modeling
As proper alignment is essential to our project, a mathematical interpretation of the
optical system was programmed in MATLAB [6] to graphically simulate how the beam will
be altered as it is passed through various optical elements. In general, we can model the
behavior of a beam traveling through an optical system using ray transfer matrix analysis [7].
The programming software MATLAB was selected for this simulation due to its capabilities
in matrix-based calculation. In the ray transfer matrix approach to optical analysis, each
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optical element (such as a parabolic mirror or lens) or length free space propagated by the
beam can be described by a matrix. This 2x2 matrix, called a ray transfer matrix, contains
such variables as the focal length of the instrument or the length of free space spanned by the
beam between elements. When a vector representing the initial beam is multiplied by a ray
transfer matrix, the result is a new vector representing the beam as it would be altered by the
optical element of interest. Through the successive multiplication of ray transfer matrices
describing each element in the optical system, we generate a model for the beam as it would
behave throughout the optical system and as it would emerge as it approaches the
spectrometer detector. This calculation is modeled below in Equation 1:
rf  A B  A B  A B ri 
  
 
 ...
 
 f  C DN C DN 1 C D1 i 
(1),
where each 2x2 matrix represents a ray transfer matrix and the two vectors represent the initial
 rays in terms of their height, r, and angle of propagation from the optical axis,
and final light
.
While basic ray transfer matrix analysis is designed for modeling ideal behavior of

light rays of negligible waist size, our system required ray transfer analysis of Gaussian beam
propagation. While the beam at LCLS is not Gaussian in distribution, the HeNe laser used to
model the spectrometer and test alignment produces a Gaussian beam. Thus for modeling of
this system, we required the consideration of the Gaussian effects in beam transmission. To
represent this style of beam rather than a light ray, we generated a vector to represent the
initial beam in terms of a complex beam parameter q, calculated by the following equation:
1 1
i
 
q R w 2
(2),

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where q is the complex beam parameter, R is the beam radius of curvature,  is the
wavelength of the beam and w is the beam waist size [8]. With this Gaussian notation, the

equation modeling the transformation of the initial beam to a final beam through
ray transfer
analysis (Eq. 1) is modified to a new form:
q f  A B A B  A B qi 
  k
 
 ...
 
1  C DN C DN1 C D11 
(3),
where the beam is now represented by a vector containing the complex beam parameter and k
 chosen to maintain the normalization of the bottom component of the beam
is a constant
vector. Thus after the multiplication of the initial vector by each additional ray transfer matrix
in the system, we obtain a new vector containing information about the effected beam waist.
Carrying out this multiplication and solving for the inverse q value leaves us with:
D
1 C  qi

qf A  B
q
(4),
i
giving us a way to find the new waist size from the definition of the beam parameter q and the
 the ray transfer matrices and initial beam parameter. By storing the
elements contained in
inverse q value after each cumulative multiplication, the MATLAB code calculates the beam
waist following each element and at each point in free space at an interval of 10um and
outputs a graph of these waist sizes. Due to the great variation in orders of magnitudes of
these widths, it is also helpful to generate a plot of the logarithm of the widths.
The MATLAB graphical simulation of our optical system was designed with ray
transfer matrices representing ideal versions of each optical element implemented in the
spectrometer, with off-axis parabolic mirrors and any lenses in the system are treated as ideal
thin lenses. The physical constraints in alignment and aberrations in optical surfaces are not
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accounted for; the program merely offers an outline for ideal behavior of the system with
infinite precision as a guideline for verification of the proper positions of all optical elements
for focusing and collimation under ideal conditions. This provides both a guideline against
which to test the alignment of the physical optical system and a means of comparing it to ideal
behavior.
ii. ZEMAX Simulations for Advanced Modeling Tolerance Calculations
For a more accurate and elaborate representation of the physical system, specialized
software called ZEMAX was used. ZEMAX is specifically designed for simulation of such
optical interests as laser beam propagation, stray light and freeform optical design [9]. A
model of ray tracing through a 90-degree ff-axis parabolic mirror designed for our optical setup is provided in Figure 2. Modeling our system in this software offers several enhancements
to the computational modeling of our optical system. ZEMAX is designed for optical analysis,
and provides a visual representation of all optical elements and offers the ability to trace rays
of light through them. Thus, when modeling the propagation of the laser light through the
prism, we can simulation the dispersion of different wavelengths of light through the prism
due to the wavelength dependency of the KRS-5 crystal index of refraction. This allows us to
simulate how the wavelengths will emerge from the crystal for a more precise understanding
of where to place the spectrometer detector. As the detector is designed to register changes of
light intensity at different detector pixels, it is essential to understand where on the detector
each wavelength of light is hitting. By knowing which pixels correspond to which
wavelengths of light, and therefore detecting the relative intensities of different wavelength,
we may properly determine the spectrum of light emitted by the system. Proper calibration of
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the spectrum detection is essential to the functionality of the spectrometer and thus intensive
and precise modeling of this behavior is of upmost importance.
While functionality of the spectrometer depends on this dispersion of wavelengths
through the crystal, this dispersion also introduces chromatic aberrations in the focal plane of
the spectrum, due to the dependence of light deflection angle on wavelength. Previous
simulations by collaborators in the project indicate that due to these chromatic aberrations, the
wavelength-dependent focal plane is not perpendicular to the optical axis; a tilt of the detector
is required to minimize these effects, allowing for the detector to fall as accurately as possible
on the focal plane of all wavelengths of the dispersed light [10]. ZEMAX simulations of our
updated optical system will allow for further investigation of these chromatic effects and the
determination of the proper detector tilt to minimize aberrations and more accurately
determine the transition radiation spectrum.
Finally, the ZEMAX simulation software was used to determine the resolution
obtainable in our optical system by considering physical constraints of the system. By
comparing the idealized MATLAB simulation beam waist results with the measured beam
waist in the physical system, we can determine any points at which the model and system
disagree. It is important to investigate the nature of these discrepancies and determine whether
they are the results of the physical constraints in aligning the optical system or if there is an
unaccounted for effect in either our models or our actual optical layout. Through the
adjustment of several variables in the ZEMAX simulation code, we may determine if
offsetting optical elements by small amounts (within the range of estimated human error in
alignment) results in beam waists that match the beam waist measurements that disagreed
with the MATLAB model. If these effects are obtainable, the simulation indicates that the
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discrepancies are in fact the results of human limitations and we may determine whether they
are of negligible effect in our determination of the radiation spectrum. If we are unable to
simulate the beam behavior found in our physical system, this indicates that we may have a
flaw in the system that has not yet been corrected for, either in the simulation of the physical
system.
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TABLES AND FIGURES
Figure 1. Design of Current Optical Layout
The spectrometer design begins with the shedding of transition radiation
from the electron beam through the use of a foil. This radiation is then
steered and expanded with off-axis parabolic mirrors (black). The
spectrum is then dispersed through a KRS-5 prism and focused onto the
surface of the line array detector. Note: the detector is pictured at a tilt in
accordance with the corrections for chromatic aberrations described in
Methods and Materials section ii.
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Figure 2. ZEMAX Simulation of a 90-degree Off-Axis Parabolic Mirror
ZEMAX Design of a 90-degree off-axis parabolic mirror. This mirror is modeled through the
design of a full parabolic mirror and implementation of an off-centered aperture to extract only
the section of the mirror reflecting light at a ninety degree angle.
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REFERENCES
[1] Y. Ding et al., “Measurements and Simulations of Ultralow Emittance and Ultrashort
Electron Beams in the Linas Coherent Light Source”, PRL 102, 254801, 2009
[2] Linear Coherent Light Source http://lcls.slac.stanford.edu/
[3] (Kiel Williams’ SULI paper)
[4] Pyreos Ltd, http://www.pyreos.com/
[5] (Gilles Dongmo-Momo’s SULI paper)
[6] MATLAB 7.0.4, The MathWorks Inc.
[7] Bahaa E. A. Saleh and Malvin Carl Teich (1991). Fundamentals of Photonics. New York:
John Wiley & Sons. Section 1.4, pp. 26-36
[8] Gaussian Beams http://www.rp-photonics.com/
[9] Radiant ZEMAX LLC, http://www.zemax.com
[10] C. Behrens et al., “Design of a Single-Shot Prism Spectrometer in the Near- and MidInfrared Wavelength Range for Ultra-Short Bunch Length Diagnostics”, DIPAC'11,
Hamburg, Germany, 2011
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