Study of Phase Reconstruction Techniques applied to Smith-Purcell Radiation Measurements ∗
1
N. Delerue , J. Barros, M. Vielle Grosjean 2 O. Bezshyyko, V. Khodnevych ∗
[email protected] 1 Laboratoire de l’Accélérateur Linéaire (LAL), Université Paris-Sud XI, F-91898 Orsay, France 2 Taras Shevchenko National University of Kyiv, Ukraine Work supported by the French ANR (contract ANR-12-JS05-0003-01), the PICS (CNRS) "Development of the instrumentation for accelerator experiments, beam monitoring and other applications." and Research Grant #F58/380-2013 (project F58/04) from the State Fund for Fundamental Researches of Ukraine in the frame of the State key laboratory of high energy physics."
Coherent Radiation as a bunch profile monitor • Coherent emission encodes the Fourier transform of the bunch longitudinal profile: 2
2
I(λ) = I1 (λ)(N + |F (λ)| N )
Can be used as a diagnostic to measure the longitudinal profile of an electron bunch. Example: Coherent Smith-Purcell radiation produced when a bunch of charged particles passes above a grating.
Simulations G(x) =
i=1 Ai exp
• |F (λ)|2
Phase information lost
• However in an analytical function ( ε(ω) ) there is a relation between phase and amplitude: Kramers-Kronig relations
• Rewrite as log(ε(ω)) = log(ρ(ω)) + iΘ(ω) with ρ(ω) R +∞ ln(ρ(ω)) 2ω0 Then: Θ(ω0 ) = π P 0 dω ω 2 −ω 2
0
• In some cases this can be done using the Hilbert transform
• Simulate multi-gaussian profiles P5
Phase recovery methods
• Hilbert transform directly implemented in Matlab (very fast)
x −( mX −µi )2 2σi2
• Wrote a Matlab implementation for the Kramers-Kronig relations.
A , µ and σ are random numbers with x ∈ [1; mX], A ∈ [0; 1], i
i
i
Statistics
i
µi ∈ 0.5 + [−7.5; +7.5] × 10−4 /mX and σi ∈ [3; 9] × 10−9 ; mX = 65536 • F = kFFT (G) k
∆F W XM
• Sampling at some limited frequencies (33) for example: Fi = F(ωi )
F W XMorig − F W XMreco = MaxX∈rset F W XMorig
• Several sampling models investigated (linear, log, E-203 like,...).
Linear frequency sampling
Reconstructed profiles
FWHM ratio distribution 250
10
−3
Profile
x 10
10
Number of profiles
−3
Profile
x 10
Original Hilbert Kramers−Kronig
Original Hilbert Kramers−Kronig
8
6
4
4
2
2
0
0
150
Hilbert
100
50
8
6
200
Number of profiles
200
Good reconstructions
0
χ2 distribution for Hilbert, Linear sampling
FWHM ratio distribution
250
0.1
0.2
0.3
0.4 0.5 FWHM ratio
0.6
0.7
0.8
Hilbert
100
0
0.9
180
160
160
140
140
120
120
100
100
150
0
0.1
0.2
0.3
0.4 0.5 FWHM ratio
0.6
0.7
KK
80 60
50
0
χ2 distribution for Kramers−Kronig, Linear sampling
180
0.8
0.9
60
40
40
20
20
0
0
0.5
1
1.5
KK
80
2
2.5
0
3
0
0.5
1
1.5
−6
2
2.5
3 −6
x 10
x 10
Effect of sampling and sigma scaling −6
3
x 10
−5
Dependence Chi2 from number of sampling points 1
x 10
0.9 −2 3.26
3.265
3.27
3.275
3.28
3.285
3.29
3.295
−2 3.26
3.3
3.265
3.27
3.275
3.28
3.285
3.29
3.295
4
2.5
3.3
0.8
4
x 10
x 10
0.7
2
Bad reconstructions
χ2
Chi2
0.6 1.5
0.5 0.4
−3
12
−3
Profile
x 10
14 Original Hilbert Kramers−Kronig
Profile
x 10
1 Original Hilbert Kramers−Kronig
12
10
0.3 0.2
0.5
0.1
Hilbert Kramers−Kronig
10
0
8 8
0
50
100 Number of points
150
200
0 −1 10
0
1
10
10
2
10
sigma
6 6 4 4 2 2
0
Discussion
0
−2 3.26
3.265
3.27
3.275
3.28
3.285
3.29
3.295
−2 3.26
3.3
3.265
3.27
3.275
4
3.28
3.285
3.29
3.295
3.3 4
x 10
x 10
• Both methods give good reconstruction accuracy. • Hilbert directly implemented in Matlab => faster
Lorenzian profiles
• More detailed study in progress to find the limits of validity of the methods.
• Instead of multi-guassian, use Lorenzian profiles.
References
Effect of sampling and sigma scaling 2
2
χ distribution for Hilbert, Linear sampling
χ distribution for Kramers−Kronig, Linear sampling
120
120
100
100
80
80
60
60
40
40
20
20
0
0
1
1.5
2
2.5
3
3.5
4
• G. Doucas et al. Reconstruction of the time profile of 20.35 GeV, subpicosecond long electron bunches by means of coherent Smith-Purcell radiation. Phys. Rev. ST Accel. Beams, 17:052802, May 2014.
4.5
5 −6
x 10
1
1.5
2
2.5
3
3.5
4
4.5
5 −6
x 10
• O. Grimm and P. Schmüser Principles of Longitudinal Beam Diagnostics with Coherent Radiation, TESLA FEL 2006-03
