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Sam Ocko
Coherence Properties of the LCLS
Note-I am working in word right now, so my equations are really ugly,
but am planning to switch to Latex to make the equations prettier
Introduction:
The LINAC Coherent Light Source (LCLS) has recently come on-line.
The LCLS is a free electron laser (FEL), which can create coherent X-Ray
radiation, which will be able to be used for imaging very small objects, such as
molecules. Since the radiation produced by the LCLS is highly coherent,
objects can be imaged through looking at the interference patterns produced,
as opposed to simply looking at absorption patterns like is done, for example,
with a conventional microscope.
The level of transverse coherence of the LCLS tells us a lot about its
imaging capabilities. Therefore, it is desirable to have an accurate idea of the
level of coherence produced by the LCLS, and many diagnostics are being run
to measure the properties of the beam produced. Traditionally, coherence
properties can be easily measured using a Michelson Interferometer for
temporal coherence, and Young’s double slit experiment for transverse
coherence. However, the LCLS is producing X-Rays, with wavelengths of ~1.5
Angstroms, roughly the size an atom, so coherence properties can’t be easily
measured through these standard techniques.
Consider a flat surface of many sources having varying degrees of coherence
emitting monochromatic light having a wavelength smaller than that of the
surface:
When the sources are in-phase and completely coherent, they will
interfere to form a fairly straight beam pointed orthogonal to the surface, as
when you are far away from the surface in the orthogonal direction, all of the
sources are about the same distance apart and are in phase, and therefore all
positively interfere to create a high intensity. However, if you are far away
from the surface in a non-orthogonal direction, you’re at a different distance
from each source, therefore, most of the sources will destructively interfere.
Therefore, the vast majority of the intensity will be concentrated in this small
beam, which will only be slightly divergent.
When the multiple sources are completely incoherent, the different
sources have a random phase relative to each other. Therefore, the light
produced by each source will not interfere with the light produced by other
sources, as the random relative phase between sources remains a random
relative phase regardless of the phase factor from varying source distances.
Therefore, light will emanate from our set of sources equally in all directions.
Note that this is just a fancy way of describing something very intuitive-Light
produced by bulbs arranged on a surface will not spontaneously interfere to
create a coherent beam.
We see that a completely coherent set of planar sources diverges very
little, and the same set of sources, except when they’re completely incoherent,
produce no beam at all. In the middle of these two regimes, we see that the
level of transverse coherence (A continuous source can be thought of as an
infinite array of infinitesimally small point sources) will determine how quickly
the beam will diverge. A highly coherent source will produce a slightly
divergent beam; a low-coherence source will produce a more divergent beam.
Armed with this notion, we now get a little more mathematical.
The Gaussian-Schell Model
Many of the statistical properties of light can be described by the mutual
coherence function
Rho(r1, r2, tau) = <E(r1, t) E*(r2, t+tau)>
Where r1 and r2 are positions, and E(r, t) is the complex valued electric field at
position r and time t.
We can also consider the coherency between two points limited to a single
frequency basis, which is called the cross-spectral density function.
W(r1, r2, omega) = Integral(Rho(tau)* e^(-i* omega*tau) d tau
The Gaussian-Schell model at the source is
W(r1, r2, omega) = A^2 * e^-(r1^2 +r2^2)/sigmaS * e^-(r1-r2)^2/(sigmaG)
Where z is the beam axis, r1 and r2 are positions on the plane of z=0, sigmaS is
the beam width, and sigmaG is transverse coherence length.
Knowing these variables will dictate the rate of divergence of the beam, so the
appeal of assuming a GSM is that we can predict all coherence properties
through simply measuring the beam width for several values of z (Either
experimentally or through a computer simulation) without ever considering the
details of how the beam was actually produced. The question is, of course,
whether the actual coherence properties are close enough to the GSM to make
this approximation reasonable. Another question is where to assume the beam
source to be, since it doesn't make sense to assume it to be at the end of the last
undulator, since much of the power is generated from deeper within the
Undulators.
Analyzing Simulation Data
Working with simulations of electrons traveling through the Undulators, we
can measure whatever we want, and then make predictions based on that.
One quantity of interest is the degree of transverse coherence, which is
Zeta = double Int(J12^2 dr1 dr2)/ Int(J11 dr1)^2
Where J = Rho(r1, r2, 0)
Note that J (r1, r1) is equal to the intensity at r1; therefore, the term in the
denominator is equal to the total intensity squared.
Zeta is difficult to calculate explicitly, as the size of the space is far too large
(For example, if your electric field is 100 cells wide, then your have 10,000
values of r1. Therefore, you have to calculate 10,000^2 = 100 million quantities
for J12: Note-You can get it down to 50 million using the symmetry of the J
matrix, as J (r1, r2) = J (r2, r1)*, but it's still a very unwieldy calculation.
However, this quantity is susceptible to Monte Carlo techniques. One way of
doing this is to randomly select pairs r1, r2 over an even distribution, and
sample
J12(r1r2)^2/Intensity^2
However, doing a Monte Carlo approach such as this is very wasteful, as most
values of r1 and r2 have very little field strength, and therefore contribute very
little to this sum. However, we can come up with a scheme where we spend
most of our time considering the most important parts of the summation. We
can do this by rewriting the degree of coherence as:
Zeta = Int(J12^2)/Intensity^2 = Int(J12^2/J11J22 * J11J22/Intensity) =
Int(f(r1, r2) * P(r1, r2) dr1 dr2)
Where P(r1, r2) = (J11J22/Intensity2)
f(r1, r2) = J12^2/J11J22
Therefore, if we sample using pairs of r1 and r2 with probability P(r1, r2), we
can get a very accurate estimate of the degree of transverse coherence. This
can be done through Markov-Chain Monte Carlo Techniques, where we sample
according to a weighted random walk.
The pseudocode of the Metropolis Algorithm, which is a certain Markov-Chain
Monte Carlo technique is:
State alpha
alpha' = randomNeighboringState(alpha')
if(rand() < P(alpha')/P(alpha'))
alpha = alpha'
Sample f(alpha)
This is a random walk, where moves to neighboring states are proposed (In my
implementation, the proposed values of r1 and r2 have a Gaussian distribution
around the old values of r1 and r2). A proposed move to a neighboring state of
higher probability is always accepted; a proposed move to a neighboring state
of lower probability is only sometimes accepted. This weighted random walk
will we on state alpha with probability P(alpha), and therefore provides an
unbiased estimator.
This tells us the degree of coherence, but we also want to see whether or not W
conforms to the GSM. Since the beam has a very narrow bandwidth, we can
assume that
W(r1, r2, omega) ~=constant * J(r1, r2),
for all values of omega that fall within our bandwidth.
From doing so and analyzing J, we can tell how well the properties of the
Simulated Beam match the GSM at each point in space. To do that, we
calculate the accuracy of projection of J onto the GSM predicted by the beam
width and degree of coherence. We treat both the measured J values and the
GSM as vectors, and the accuracy of the projection is given by
|J*GSM|/|J||GSM|
NOTE-I’ve covered what I’ve done so far, I’m still working on the project and
am not sure where it’s going to go yet.