1
Longitudinal Phase-Space:
How to obtain a full Picture
- by Markus Hüning The motion of the particles inside
the bunches of accelerated beams
can be described in their 6-dimensional phase-space. In common use
the phase-space is divided into
three 2-dimensional sub-spaces: 2 x
transverse (offset - divergence) 1 x
longitudinal (energy - time). Most
measurement devices only can
measure projections of the phasespace (transverse profile, energy profile, longitudinal profile). For the understanding
of effects it is often desireable to measure the phase-space distribution rather than
one single projection. That is when TOMOGRAPHY enters the game!
Related Articles:
•
The Idea of Tomography
•
The ART of Reconstruction
•
How to cool down the Result
•
In Phase Space
•
The Bunches of TTF
•
Wakefields
•
Outlook
http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
2
The Idea of Tomography
The Idea of Tomography
Given a distribution in 2 dimensions
(particle density in phase-space, tissue in human body, ...)
Take Projections with different angles (Radon Transform)
0˚
45˚
90˚
Algorithm
to
reconstruct
original
distribution
from the
projections
135˚
180˚
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
3
The ART of Reconstruction
The ART of Reconstruction
There are many algorithms to perform the reconstruction:
• Filtered backprojection
-> see M. Geitz
• Fourier reconstruction
• Algebraic reconstruction technique (ART)-> discussed here
• Maximum Entropy Method (MENT)
-> coming soon
• ...
The first to methods take advantage of the similarity of Fourier-Transform and
Radon-Transform,
The third method performs an inversion of matrices,
the fourth method additionally maximizes the distribution’s entropy.
Next I will explain the ART-method
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
4
The ART of Reconstruction
The ART Algorithm
Let the phase-space-distribution being given as a row-vector:
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Also the projections may be merged into one single row-vector:
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The projections then can be described as matrix-multiplication:
g = A ⋅ F ⇒ Atells how projections are done
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
5
The ART of Reconstruction
Doing just a simple inversion of A doesn’t work for noisy data
F = A
–1
⋅g
Instead one uses Kaczmarz’s method
T
ω
F j = F j – 1 + ----------- ( g j – a j F j – 1 )a j
2
aj
j = 1…N
with aj being the jth row of A,
gj the jth value of g,
Fj the full vector of F in jth (sub-)iteration
One full turn of j=1...N is one iteration of the algorithm.
equations
The way of
Kaczmarz’s method
F1
http://www.desy.de/~mhuening
F0
Markus Hüning, February 12, 2001
6
The ART of Reconstruction
One example of that algorithm
original
reconstructed
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Student Version of MATLAB
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
7
The ART of Reconstruction
But when you don’t take projections for 180˚
original
reconstructed
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Student Version of MATLAB
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
8
The ART of Reconstruction
What happens there?
Compare the projections (original and reconstructed)
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What looks very small here has a big impact on the reconstructed distribution.
-> It is not sufficient to only fit the projections
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
9
How to cool down the Result
How to cool down the Result
From Thermodynamics you may know the Entropy
S = log ∆Γ
∆Γ: # of states
The Entropy is the bigger the bigger the disorder in the system. For S=0 the system
is perfectly in order, meaning in one well defined state.
Here we define something similar (and also call it entropy):
η( f ) = –
∫ ∫ f (x,y) log ( f (x,y) ) dx dy
f : distribution function
This doesn’t look very similar by first impression, but
η ∼ log ( F )
F =
f dx dy
∫∫
This expression has to be maximized to obtain the solution with the least information content (there is a “-” missing somewhere) -> the simplest description
(expected to be the best)
The algorithm doing this maximization is called MENT.
Another point is the representation of the phase-space:
The MENT algorithm
divides the phase-space
into polygons as
seen here.
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
10
How to cool down the Result
The same example as above:
original
reconstructed
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http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
11
In Phase Space
In Phase Space
So far the the tomography, But how to achieve rotation of phase space?
a) Transverse
Phase Space
Quadrupole
OTR
off
Quadrupole
OTR
Quadrupole
OTR
Quadrupole
OTR
Quadrupole
maximum current
http://www.desy.de/~mhuening
OTR
Markus Hüning, February 12, 2001
12
In Phase Space
b)Longitudinal
rf module
Phase Space
0˚
rf module
30˚
rf module
-30˚
rf module
-90˚
rf module
90˚
=>impossible to rotate 180˚!
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
13
The Bunches of TTF
The Bunches of TTF
One result from longitudinal tomography in the TTF
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∆ E [MeV]
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Population [a.u.]
comparison with interferometer
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Charge Distribution (Current) [a.u.]
Population [a.u.]
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http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
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The Bunches of TTF
∆E [MeV]
What could have been expected
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incoming beam
σe = 100keV
σt = 3ps
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http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
15
The Bunches of TTF
∆E [MeV]
Or maybe this
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http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
16
The Bunches of TTF
∆E [MeV]
Assuming a non-gaussian bunch from Injector:
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http://www.desy.de/~mhuening
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lt=8ps
σe=180keV
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Markus Hüning, February 12, 2001
17
Wakefields
Wakefields
∆E [MeV]
The method of longitudinal tomography is especially interesting for studying
longitudinal wakefields because it can directly probe the wake-potential
- especially when I can switch on/off the wake
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http://www.desy.de/~mhuening
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Markus Hüning, February 12, 2001
18
Wakefields
sandblasted
What has been seen so far
estimate 4σ
regular
estimate 4σ
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estimate
4σ
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Particle
0
-1 Energy [MeV]
-2
-3
Energy Deviation [MeV]
4.5
estimate based on gaussian bunch, σz=250 µm
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001
19
Outlook
Outlook
Phase-Space tomography is a powerfull means to study effects on the beam, especially longitudinal tomography offers the possibility for interesting insight into the
beam dynamics:
• Bunch Compressors
• Coherent Synchrotron Radiation
• Wakefields
• Cross-Calibration
However, there are some open questions
• What’s the resolution?
From energy resolution (7*10-5) I expect 1.3 fs*MeV,
but there might be some additional effect from the reconstruction,
phase errors and position jitter
• Where do the spikes in the reconstruction come from (MENT)?
• What about CSR in the spectrometer dipole?
• What, if I combine longitudinal and transversal tomography?
Is it possible to measure some slice emittance?
• Is it possible to use longitudinal tomography in the Injector?
http://www.desy.de/~mhuening
Markus Hüning, February 12, 2001