Automated SelfConvolution Applied to Synchrotron Radiation Spectra

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Automated Self-Convolution Appliedto Synchrotron
Radiation Spectra
Shane Koscielniak, TRIUMF, Vancouver, B.C., Canada
Abstract Convolution integrals are widespread in
physics and mathematics in particular the
self-convolution of single-trial probability
density functions give the distributions of the
cumulative sums of multiple trials. We give a
method of general applicability, based on the
Tchebychev-Hermite expansion, for automatically
computing very high order analytic approximations
to convolution integrals and apply it to the
particular case of the cumulative photon emission
spectrum from synchrotron radiation. The
turn-by-turn longitudinal tracking of electrons
in storage rings proceeds by iterating finite
difference equations (FDEs). Radiation effects
are implemented by using a classical, continuum
description for the damping effect and a
stochastic description for the quantum
excitation. We have found the single fat photon
energy loss and lumped phase advance that have
equivalent effect to the stochastic emission of
many photons. Finally we give a set of FDEs that
includes radiation damping and quantum excitation
treated in a far more precise manner than
previous works, but with almost no additional
computational cost.
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Motivation Whats New
To simulate the longitudinal dynamics of an
electron beam in a storage ring. The beam will
emit synchrotron radiation.
? We use finite difference equations (FDEs) to
model the damping effect. ? We use random
variables to model the quantum excitation. Typica
lly, hundreds of photons are emitted per particle
per turn. Because FDEs are iterated once each
turn it is desirable to substitute the many
photons by a single macro-photon of equivalent
cumulative effect. The energy spectrum of the
macro-photons is given by the multiple
self-convolution of the single-emission spectrum.
Even for the LEP, this distribution is not
gaussian and does not become so until ? 2.7?106
trials. We give a method based on the
Tchebychev-Hermite expansion, for automatically
computing this cumulative photon emission
spectrum. We give a set of FDEs that
includes radiation damping and quantum excitation
treated in a far more precise manner than
previous works, but with almost no additional
computational cost.
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Convolution (Faltung)
Except for special simple cases, convolution is
always accomplished by numerical methods and is
therefore, in practice, approximate.
Self-Convolution
OR symbolically
Multiple Self-Convolution
and so on
Examples
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Cumulative Probability Density Distributions
Random variable x drawn from probability
density distribution Fx Cumulative sum of two
trials is X2 x1 x2 where the subscript is the
trial index.
The probability density distribution of the
cumulative sum is given
by the sum of the probabilities of all the
combinations of x1 and x2 that sum to X2.
The distribution of the cumulative sum of n
trials is given by the n-fold self-convolution of
the probability density for a single trial.
Fourier Transform (FT)
The Fourier Transform of a probability
distribution F(t) is called the characteristic
function f(s).
Forward transform
Inverse transform
The transform of a convolution is equal to the
product of the individual transforms
And thus ?
This product theorem is the basis for multiple
self-convolution
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Synchrotron Radiation
In an electron storage ring (such as ESRF)
typically a few hundred photons are emitted each
turn by each electron.
The single-event emission-energy distribution is
a very bizarre function.
K is the modified Bessel function of the 2nd kind
of fractional order 5/3.
x ?E/uc
Fraction of photons
Suppose we wish to find a single macro-photon
of equivalent effect to the many photons emitted
by a single electron.
How do we find, say, the two-hundred -fold
self-convolution of such a function?
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Road Map for analytic convolution
? Adopt appropriate coordinates ? Construct f(s)
as a power series from the normalized moments of
F(t) ? Write f(s) in the form eB(s) where B is a
power series ? Form f(s)n enB(s) ? Introduce
reduced variable z
We want
How do we get there?
? Form the inversion integral and manipulate via
a coordinate transformation ? Expand exponential
and perform the derivatives to make a power
series expansion about a gaussian kernel, i.e.
construct ? Automatize
User supplies moments of the single-trial density
function, desired n-value accuracy.
i.e. turn it into a program
Program produces a high order approximation to
the n-fold self-convolution suitable for
numerical calculations.
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Coordinates Defnitions
ti individual random variables in world
coordinates drawn from distribution Ft(t) with
r.m.s. value ?. Tn cumulative sum of n trials,
with distribution FTn Step ? When we employ the
dimensionless variable xt/?, the moments become
scale invariant. Xn normalized sum. Zn
reduced variable length scale of distribution
FZn becomes independent of n - to be used later.
i trial index
Fourier Transform expressed as power series
Step ? The characteristic function is composed of
all the moments of F(t), and so we write
are moments
Origin of t chosen so that distribution mean
is zero.
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Semi-invariants
Step ? Let us find a function B such that
f(s)expB(s)
The task is given a power series f
a0a1salsl of order l, find the power series
Bb0b1sblsl such that f eB up to order l.
Each semi-invariant bk is obtained by taking the
kth derivative of f and B, setting s0 and
comparing the two expressions.
The procedure is followed with k ascending and
back-substitution of all previous results.
By suitable choice of the origin, a10. When
normalized moments are used, a2-1/2. Thus a01,
b0ln(a0)0, b1a10, b2-1/2, b3a3,
b4a4 -1/8, b5a5a3/2, b6a6(a4-a32)/2 -1/24,
etc..
Step ?
powers become additions.
A parenthesis The n-fold self-convolution is
given by the inverse transform
Evidently as n increases, so the length scale of
increases as ?n or faster. Thus it is
inevitable to introduce the reduced variable
ZnXn/?n such that has length scale
independent of n.
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Inverse transform and use of reduced variable
Cumulative sums
world
normalized
reduced
Step ?
When variables change, the distribution
transforms as follows
?
The n-fold self-convolution is given by the
inverse transform
Step ?
The key step
?
make the replacement s ? s/?n
Comparison of ? and ? implies
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Tchebychev-Hermite polynomial expansion about
gaussian kernel
Continued from previous slide
Step ?
Now expand the exponential in powers of the
polynomial
Now there is the general theorem
Hence it follows
The derivatives of order k generate a polynomial
of order k. Though we have independently
discovered this expansion for ourself, it was
derived by Tchebychev Hermite in the 1880s.
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Algorithm, program example
Using the program Mathematica, we have created an
expansion up to order z39/?n13 with the first 15
normalized moments of Fx(x) as free variables.
Once n and the moments are substituted, the
enormous symbolic expansion collapses to a simple
Taylor series in powers of z x/?n. Unspecified
moments are replaced with gaussian padding.
Step ? Fortran subroutines (8500 lines) were
auto-generated by Mma (and edited) to compute the
coefficients for the Taylor series and to
evaluate the expansion. Both the probability
density and (its cumulative integral) the
distribution function may be obtained.
Example distribution of the energy spectrum
arising from the cumulative sum of many (n 270)
single photon emission events. Despite the CLT,
this spectrum is clearly not gaussian! (n 270
appropriate to the Canadian Light
Source) abscissa in reduced coordinates. Total
trials 105.
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Definitions for longitudinal motion
synchrotron radiation
E? mean radiation loss per turn Es
synchronous energy uc critical photon energy Js
longitudinal partion h harmonic number ?s
slip factor V peak accelerating voltage per
turn
? the energy relative to a reference particle
that follows an exponential saw-tooth between the
RF cavities. ? the RF phase relative to the
reference particle at the synchronous phase ?s
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Longitudinal FDEs for synchrotron radiation
30 years old FDEs of Bassetti Renieri - still
widely used
y is a dimensionless random variable drawn from a
gaussian distribution
They omit effect of radiation on phase advance
during the turn in which it is emitted. They
assume a constant radiation rate between RF
cavities.
The difference equations from the kth
to (k1)th turn.
x is a dimensionless random variable drawn from
the cumulative sum of n emissions.
The new FDEs include the effect of both continuum
and stochastic radiation on phase advance during
the turn in the photons are is emitted. They
include an exponentially damped radiation rate
between RF cavities, as is evident in machines
with large radiation loss such as the LEP.
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Simulations of the CLS longitudinal dynamics
First 5 ms after injection
Second 5 ms after injection
Scales have been zoomed
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Selected References
Tchebychev-Hermite expansion Gnedenko
Kolomorogrov Limit Distributions for Sums of
Independent Variables, Addison-Wesley Pub. Co.
FDEs for longitudinal motion A. Renieri Problems
in single-particle dynamics specific to
electrons, CERN-77-13
TRIUMF design notes Automated multiple
self-convolution TRI-DN-00-01 Exact Finite
Difference Equations TRI-DN-00-12 Modeling
Quantum Excitation TRI-DN-00-14
Conference Paper Longitudinal Motion With
Synchrotron Radiation Modelled by Fat
Photons'', Proc. of the 7th EPAC, Vienna
Austria, June 2000
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