Report

Smith-Purcell radiation and Reconstruction techniques

Supervisor: Dr.Nicolas Delerue

Student: Vitalii Khodnevych

2014

Aim of my internship was to study some aspects of reconstruction phase methods. And to make small amplification board.

Smith-Purcell radiation Smith-Purcell radiation is emitted when a bunch of electrons passes near a grating. When the parameters are properly chosen some of this radiation will be coherent and the spectrum of the radiation emitted will depends on the square of the Fourier transform of the bunch: 

dI dΩdω



dI = dΩdω 

Ne

 SP

[Ne + Ne2 |F (ω)|]

Therefore a measurement of this spectrum gives some information on the beam longitudinal profile but the phase of the Fourier transform must be recovered.

Simulation For all simulations, I used MATLAB environment. My script I separated into several parts and functions for usability. One of the main features is a function that generates profiles. It generates a profile with 65536 points. Further, this function reads randomly generated numbers in advance. Three of these coefficients are for each Gaussians. Conversion factors from random numbers is chosen from physical considerations. Once a profile has been generated, it takes the Fourier transform, and then the module of last. Then select the interval of spectrum for sampling and take points from it according to sampling method. The sample spectrum have 33 points, like on E203. The first point in all methods is the first point of the spectrum. More details about each: 1) Linear sampling. Sampling points distributed uniformly. 2) E203 + line. Linear sampling at three different pieces. At each piece have it own step. Width of pieces is the same as in E203. 3) The method of the derivative. For this method was first conducted preliminary studies. Considerations is follow: detectors should be placed where the spectrum have greatest changes. Gentle areas recover more easily than sloped. For it I generate 1000 profiles. Then I take from each diff and the sum of abs is the desired distribution. 4) Logarithmic sampling. The points are distributed according to the logarithmic law. 5) Sampling is the same as the E203. 6) The method of valuable information. For this method were also conducted preliminary studies. Considerations include: easily recover Gaussians, which spectrum is also Gaussians. But if we prefer to see a more complex shape that is different from Gaussians, we need more detailed view spectrum. Therefore, the 1000 spectra profiles fitted by Gaussians, sum of each module of differences identified the distribution. Then comes the recovery procedure . Because the sample is being made so as to avoid the low-frequency extrapolation, this step in recovering is missing. Recovery starts with the recovery profile of the spectrum. The first step is interpolation. I used a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). This is followed by a high-frequency extrapolation. I use this HF exptrapolation formula[1]: ρHF (ω) = Aω −4 4 where A = ρH ωH . After that mirrors the spectrum for further action.

There procedure of recovery the spectrum ended and starts the recovery procedure of phase and profile. Here I use two methods: the Kramers - Kronig and Hilbert transform. Since the recovery procedure for KK is slow, I recover only half of phase and the other mirrors. When is only possible to measure the amplitude of the complex signal, it is necessary to recover phase of the available data. When this function is analytic there is a real possibility of recovery phase on amplitude, and hence the signal as a whole. Kramers-Kronig relations helps to restore the imaginary part of an analytic in the upper half-plane function ε(ω) from the real part and vice versa. In case you need to recover the phase on amplitude, investigated function should be written as: log(ε(ω)) = log(ρ(ω)) + iΘ(ω). There ρ(ω) is amplitude and Θ(ω) is phase. But then the analyticity of the function is disrupted and Kramers-Kronig relations should be modified as follows: 2ω0 Z +∞ ln(ρ(ω)) P dω Θ(ω0 ) = π ω02 − ω 2 0 The basis of this relationship are the Cauchy-Riemann conditions (analyticity of function). In this case, the value spectrum can gain value at [0, ∞). As for the Hilbert transform, it is got by convolution and Fourier transform: 1 Z +∞ ln(ρ(ω)) dω. Θ(ω0 ) = − P π −∞ ω0 − ω There is also the requirement of analyticity of the function, but it is problem for recovering phase in the case when ρ = 0. It should be noted that this ratio obtained for other reasons than Kramers-Kronig relations. Unfortunately when dealing with absolute value of Fourier transform, we lose the information about the position and direction of the profile. To recover this, I have another procedure: recovery position and direction. It is necessary to further define the quality of reconstruction. The first step is defined by the center of the weight of both functions (the original profile and restored), followed by the combination. But it is not enough to determine the exact position. For this I use a χ2 test. To calculate χ2 I use the following formula: χ2 =

X

ωi2 (Oi − Ei )2 /N

i

where Oi is the observed frequency in a cell, Ei is the expected √ frequency on the null hypothesis, ωi - weight of the point, N - number of points. I use ωi = 1/ Oi + Ei . So moving one set around a common center of weights I define χ2 at each point near the center. Doing the same procedure for the other direction, I take the smallest χ2 and I consider that at this point and this direction profiles combined in the best way. Also I try others methods do define the position. For example I use correlations between two profiles: ΣOi Ei q R= q . ΣOi2 ΣEi2 But it good correlate with χ2 , so I still use χ2 .

Results of simulation First aim was to find what sampling works better. For this I simulated 1000 profiles. Simulations show, that better of all works linear sampling. This can be explained by the fact that all the mathematical methods work better for uniform distribution. Also Hilbert method works better

(a) Chi2 value for different methods

(b) FWHM value for different methods

(c) Dependence Chi2 from number of (d) Dependence FWHM ratio from numpoints ber of points

and faster (recovering phase with KK takes about 7 minutes and with Hilbret only few ms) than Kramers-Kronig. Methods on the figures are numbered as in the descriplion up. FWHM there is not exactly FWHM but FWHM ratio. I mean, it difference between reconstructed and original FWHM divided by original. Also in my profile generator I change gaussian to lorenzians. In this case it have worse χ2 , but in general it reconstruct good enough. Other part was to find optimal number of detectors for experiment. For this (as shown on figures c-d) I change number of points for linear sampling and calculate χ2 and FWHM ratio. There I used Hilbert method. Also I try to explore what will happends if our system will works with bunches that slightly different than for what system was developed. I change sigma of gaussians in my profile generator, than do everything like in previous simulation and calcutate χ2 . In this case I use linear sampling and two methods. So we see, that also in this case Hilbert method works better than Kramers-Kronig. Also I simulate different noise impact to system. But it require more study. Simulation in progress... Figure 1: Dependense Chi2 from bunch width