Licence Science de la matière École Normale Supérieure de Lyon Université Claude Bernard Lyon I
Study of the polarization of Smith-Purcell radiation
Solène Le Corre L3 Physique Internship 2011-2012
Abstract
During my internship I had the opportunity to work with the E-203 collaboration at SLAC, a linear accelerator near San Francisco. The collaboration is working on the measurement of the longitudinal length of the beam using the Smith-Purcell radiation. In this report, I will expose what is the Smith-Purcell radiation and how it is used to reconstruct the bunch pro le. I will also explain my work on the polarization of this radiation and how it is inserted in the work of the E-203 collaboration. Internship supervised by: Nicolas Delerue
[email protected] / tel: (+33) 01 82 52 22 51
Laboratory:
LAL, Université Paris-Sud
Bât. 209 A BP 34-F-91898, Orsay http://www.lal.in2p3.fr/
1
Thanks I would like to thank the E-203 member: George Doucas, Armin Reichold, Ivan Konoplev, Scott Stevenson and Heather Andrews, with who I worked during three intense weeks at FACET. There, I had the opportunity to see how an experiment was run and how a collaboration worked, which was very rewarding. I thank also Christine Clarke for the amount of work she did and for enduring us, despite the e-log being never updated in time. A great thank too to Nuria Fuster, for her work and her good company. And a special thank to my supervisor, for all the hours spent with him talking of polarization, induction and gratings, hoping for understanding our horizontal polarization; and above all, for succeeding in giving me self-con dence.
Contents 1
Introduction
3
2
The E-203 experiment
4
2.1
Smith-Purcell radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
FACET
4
2.3
Experimental set-up
3
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Online data analysis
5
7
3.1
Calibration of the position of the gratings
. . . . . . . . . . . . . . . . . . .
7
3.2
Calibration of the trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
Problem of saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.4
Problem of position from the beam
9
. . . . . . . . . . . . . . . . . . . . . . .
Study of polarization of the Smith-Purcell radiation ,
10
4.1
Problem of the horizontal polarization
. . . . . . . . . . . . . . . . . . . . .
10
4.2
Ratio of SP radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.3
E ciency of the grating
13
4.4
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1
E ciency from Loewen's paper
. . . . . . . . . . . . . . . . . . . . .
13
4.3.2
E ciency from GFW program . . . . . . . . . . . . . . . . . . . . . .
15
Reconstruction of time-pro le
. . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion
17
19
2
1
Introduction
In a particle accelerator, it is quite easy to measure the size of an electron bunch in the transverse direction, but the longitudinal pro le is much more di cult to measure. One of the solution to reconstruct this pro le is to use the Smith-Purcell radiation. This radiation was discovered in 1953 by Smith and Purcell [1]. When an electron bunch passes near a grating, it induces surface charges in the grating, which will emit light. This light is then spectrally dispersed by the grating, which allows to detect, if we only take into account the rst order of diraction, only one wavelength at a given observation angle ( gure 1). The radiation gives access to the module of the Fourier Transform of the longitudinal pro le of the bunch. Then the phase of each Fourier component has to be recovered to reconstruct the pro le. Several methods are available, among them the Kramers Kronig phase recovery method [2], which is the most popular method in particle accelerators, and the iterative shrinkwrap methods, often used in crystallography.
Figure 1: Smith-Purcell radiation [3]
The reconstruction of the longitudinal pro le of the beam had been initiated about 20 years ago by George Doucas, from Oxford, in the picoseconds range. Since 2010, the E203 collaboration, between the University of Oxford and LAL (Laboratoire de l'Accélérateur Linéaire) has set up an experiment at SLAC, a particle accelerator laboratory near San Francisco which can produce femtoseconds beams. During my internship, I had the opportunity to spend three weeks at SLAC to take part in this experiment. There I helped taking data and worked on the online and o ine analysis of the resulting data. The most important work I did is for the study of the polarization of Smith-Purcell radiation, which had been studied mostly theoretically. In this report, I will expose rst the principle of the Smith-Purcell experiment we made at SLAC. Then I will talk about the preliminary results we had and some corrections we had to make. Finally I will expose in detail my work on the Smith-Purcell radiation polarization.
3
2
The E-203 experiment
2.1 Smith-Purcell radiation When a charged particle passes near a metallic grating, it will induce image charges in the
l of b = vc ,
grating which then emits a radiation, named Smith-Purcell radiation. The wavelength the emitted radiation depends on the velocity of the bunch, expressed in terms of on the period
l
of the grating, the angle of observation
j and the the order of diraction n
according to the equation:
l = nl
1
b − cos j
(1)
The angular and frequency distribution of the radiated intensity is given by the formula:
with
Ne
d² I dWdw
(W, w) =
Ne
dI
dWdw
(W, w) . Ne Sinc + αNe2 |F (w)|²
(2)
1
Sinc the incoherent part of the radiation, α a factor depend F (w) the Fourier transform of the bunch shape, and dWdIdw 1
the number of electrons,
ing on the coherent radiation,
the angular distribution of intensity emitted by a single electron (see appendix 1 for more details) [5, 6]. For su ciently short beams, the coherent term dominates and:
d ²I dWdw
(W, w) '
Ne
dI
dWdw
(W, w) .αNe ²|F (w)|²
(3)
1
From the Smith-Purcell radiation, we can therefore know the module of the Fourier transform of the bunch shape and, with a method such as Kramers Kronig reconstruction, we are able to nd the original shape of the bunch.
2.2 FACET The experiment is installed on the FACET accelerator at SLAC. SLAC, with its 3.2 kilometers length, was the longest linear accelerator in the world. This accelerator was built in 1962 and extended in the 90's with the SLC, whose goal was to study the Z boson, and with two damping rings. It is one of the few accelerators having a compression chicane.
Now, the accelerator is split in two ; one part, LCLS, delivers X-
Rays beams with a free-electron laser.
The second one, FACET, where the Smith-Purcell
experiment takes place, delivers ultra-short high energy electron beams. At FACET, the electron bunch is about 100 fs long, with an energy of 20 GeV. The electrons are injected in the Linac by a thermionic electron gun, cooled in a damping ring and then accelerated with accelerating cavities. For most of our acquisition, we used a beam with a repetition rate of 10 Hz. measurement is based on the average of 100 pulses.
4
Each
Figure 2: The SLAC accelerator.
2.3 Experimental set-up The device we used is made up of four gratings xed to a carousel, a ladder with ve rows where lters or polarizers can be put, and fourteen detectors (eleven detecting the SmithPurcell radiation, and three measuring the X-rays background).
row of filters
x z
40°
140°
detectors
Figure 3: Position of the dierent elements of the Smith-Purcell experiment. [3]
The gratings can be moved backward and forward and rotated in order to change the grating facing the beam. We used three aluminium gratings with period of 250µm, 500µm and 1mm respectively,
and one blank piece of aluminium ( gure 4). Their blaze angle was respectively 30°, 40° and 35°.
5
Rotation
500 µm
250 µm
Blank
1 mm Figure 4: Gratings xed on a carousel.
As the blank will not emit Smith-Purcell radiation, it allows to measure the background in the accelerator. For a given angle, the wavelength emitted by the grating will depend on its period. Therefore we used a dierent row of lters for each grating, each lter being matched to the wavelength emitted by that grating at that angle ( gure 5). We put vertical polarizers in the fth row in order to study the polarization of the SmithPurcell radiation, supposed to be vertically polarized. The fourth row was empty at rst, but on two occasions we put horizontal polarizers and took measurements to complete the study of the polarization.
row 5 (vertical polarizers) row 4 (holes / horizontal polarizers) row 3 (1mm filters) row 2 (250µm filters) row 1 (500µm filters)
40°
50°
70° 80° 90° 100° 110°120° 130° 140° θ
60°
z
Direction of the beam
x
Grating
Figure 5: Rows of lters and polarizers
After the lters, the Winston cones collected the radiation toward the pyroelectric detectors ( gure 6). The signal was then transmitted toward a digitiser which allowed us to record the data for later analysis. The digitiser was part of our data acquisition system (DAQ).
6
y
z
x
φ
beam grating
Figure 6: Winston cones [4]
3
Online data analysis
3.1 Calibration of the position of the gratings One of the rst challenge we had was to know precisely the position of our gratings with respect to the beam. Once the device was in the tunnel, we only had access to a value of a voltage representing the position of the grating in the vacuum chamber. Therefore we needed to know the exact relation between the position of the grating and the voltage we could read on a multimeter. Nuria Fuster, a Spanish student, and I worked on this problem and on the evaluation of the error on the position (for more details, see appendix 2).
3.2 Calibration of the trigger The value of the amplitude we measured was the dierence between the amplitude of the signal coming from the detectors before and after the passage of the beam.
The DAQ
computed this dierence. To know when the beam passes the detector, the DAQ needs to receive a trigger signal at the correct time. The value of the trigger was entered manually in a software. To calibrate the trigger, I modi ed the acquisition code in order to show the value of each detector against time. The beam was supposed to pass at the 16th data point. Therefore we needed to change the value of the trigger so that the fall of the amplitude of the signal delivered by the detectors occurred between the 14th and 18th data point (see gure 7). With this method, it was easier to calibrate the trigger.
7
Figure 7: Calibration of the trigger: amplitude of the signal of detectors against time. Left: bad trigger calibration ; Right: good trigger calibration.
3.3 Problem of saturation
Figure 8: Graphs of the data acquisition - raw data. right: 250
µm
Upper left: 500
µm
grating; Upper
grating; Bottom left: 1 mm grating; Bottom right: blank. Red curve: signal
with lter row matching with 1mm grating; Blue curve: same, with 500 curve: same, with 250
µm
µm
grating; Green
grating; Purple curve: signal with row of vertical polarizers; Cyan
curve: signal with open holes.
8
The DAQ acquires the amplitude of the signal for each detector and each pulse.
Then it
makes the mean of these values for each detector. Based on these values, we are able to plot graphs as gure 8. The signal from the dierent gratings correspond to the sum of the background coming from the accelerator and the Smith-Purcell radiation. Because the blank is not corrugated, it does not produce Smith-Purcell radiation.
The signal from the blank is therefore only
background. Having a closer look at the 500
µm grating with the row of open holes (cyan curve), we can
see important falls of the amplitude. We assumed that it could be a problem of saturation. I looked at the raw data and saw that, for some detectors, there were many pulses given a negative amplitude (see gure 9).
Figure 9: Distribution of the amplitude for the dierent detectors
I made a histogram to see the distribution of the positive values. These values never 15 exceeded 33000, which is approximately equal to 2 . I deduced that it was just a problem of 0 15 coding: when the amplitude is too high, the value of the amplitude a is coded as a = 2 −a. I made therefore a correction in the code to take into account this problem: when the positive value of a detector is su ciently high, or if there is no positive values, all the negative 0 15 0 amplitudes a for this detector are corrected with the formula: a = 2 + |a | with a the new value of the amplitude. Nevertheless, in some graphs, the amplitude of the signal seemed to 4 accumulate at a value of ∼ 4.10 , despite the correction for the negative values. We supposed that this phenomenon was due to the saturation of the detectors themselves. This hypothesis was con rmed by the plot of the value of the amplitude of the detectors against time (see gure 21 in appendix 4). This problem could not be xed. Therefore we had to take data far enough from the beam not to saturate the detectors.
3.4 Problem of position from the beam The Smith-Purcell radiation can be extracted from graphs as gure 8 by subtracting from the signal taken with a grating the signal from the blank with the same lter row. For instance, if we take the 500
µm
grating, we take the blue curve ( lter row matching with the 500
µm
grating) and subtract the blue curve from the blank (which correspond to the background). We need therefore a signal of Smith-Purcell radiation su ciently high not to be mixed with the background. Despite the problem of saturation of the detectors, if we were too far away from the beam, the Smith-Purcell signal was too low and it was di cult to have a good extraction of the Smith-Purcell radiation. Most of our measurements were taken assuming
9
that the beam was approximately at the position of the beam axis and that the voltage given by the hand held multimeter , transformed in distance via my matlab routine, corresponded to the real distance between the beam and the gratings. (The routine giving the position of the beam from the beam axis was not yet operational). Nevertheless the Smith-Purcell signal was very low and the extraction was not giving good results. That is why we nally used an other method: before each serie of data taking, we moved the gratings forward until they reached the beam. Thus we knew the position of the beam and we had just to take our gratings a few millimeters away to take quite good measurements.
4
Study of polarization of the Smith-Purcell radiation
1,2
4.1 Problem of the horizontal polarization According to theoretical papers [5, 6], the Smith-Purcell radiation should be vertically polarized. During the second week at SLAC, we had the opportunity to make an access to the accelerator tunnel to change some lters. For this occasion, we took horizontal polarizers in the row of open holes (4th row) and removed the row of the 250µm lters to keep a row of open holes. We took a set of measurements with this con guration of lters and polarizers before coming back to the previous con guration. The study of polarization represents an other way to reconstruct the signal of the SmithPurcell radiation.
There was two important things to check: rst, that the two curves of
polarization from the blank were almost equal, i.e. that the background was not polarized, and then that the signal of horizontal polarization from the gratings was only background. With the hypothesis that the signal of Smith-Purcell is only vertically polarized, we deduce that:
SP = V Pgrating − BV P with
SP
the signal of the Smith-Purcell radiation,
from the grating, and
BV P
V Pgrating
(4) the signal of vertical polarization
the background in the signal of vertical polarization. We assumed
that the background is given by the signal from the blank. If the background is not polarized, we have therefore
B V P = B HP
(5)
and, if the horizontal polarization from the grating is only background, we have also
HPgrating = B HP
(6)
From the equation (4), (5) and (6), we deduce that
SP = V Pgrating − HPgrating 1 If
(7)
not speci ed, the graphs in that section comes from the analysis of the data taken the 26-06-2012 at 23h55 (scan at maximum compression). 2 Because of a lack of polarizers, the two last detectors, at 130° and 140°, were open holes.
10
With equation (7), we can see all the interest of the polarization: we do not need the blank to extract the signal of Smith-Purcell radiation. Once we had made the acquisitions, I wanted to work on the polarization to verify these results. First I looked at the blank data and calculated the degree of polarization given by the −HP d = VV PP +HP with d the degree of polarization, V P the signal from the blank vertically polarized, and HP the signal from the blank horizontally polarized. For an unpolarized
formula : signal,
d ' 0.
We can see in gure 10 that the background is clearly unpolarized. 4
4
blank
x 10
background for the vertical polarizers background for the horizontal polarizers signal (ua)
3
2
1
0 40
50
60
70
80
90 detectors
100
110
120
130
140
degree of polarization for each detector
degree of polarization
1
0.5
0
−0.5
−1 40
50
60
70
80 angle
90
100
110
120
Figure 10: Up: polarizations from the blank ; Down: degree of polarization for each detectors.
Then I wanted to check if the signal horizontally polarized from the grating was only background. I subtracted the signal from the blank from the signal from the gratings. But the resulting signal is far from zero (see gure 11).
11
500um grating
250um gratin
8000
3000
signal
signal
6000 4000
1000
2000 0 40
2000
60
80 100 120 angle horizontal polarizers − blank
0 40
60
80 100 120 angle horizontal polarizers − blank
1mm grating 12000
signal
10000 8000 6000 4000 2000 40
60
80 100 120 angle horizontal polarizers − blank
Figure 11: Horizontal polarization from the grating minus horizontal polarization from the blank One hypothesis was that the horizontal polarization only came from a problem of transmittivity of the polarizers: if the polarizers were perfect, only the signal polarized perpendicularly to the wires should pass through. In fact, it allows also a part of the signal polarized in the same direction as the wires to pass.
I corrected the signal taking into account the 0 0 transmittivity of the polarizers and obtained two new curves, V P and HP , according to 0 0 the formulas: V P = V P − εV ⊥ HP and HP = HP − εH⊥ V P with εH⊥ and εV ⊥ the transmittivity of polarizers with horizontal and vertical wires respectively. But even taking into account this transmittivity, the horizontal component of the polarization was quite important and could not be neglected (see gure 22, in the appendix). We had therefore to take into account the horizontal polarization for the rest of the study.
4.2 Ratio of SP radiation If we cannot neglect the horizontal polarization, we have to correct the formulas (4) to (7) (see part 4.1). We have now, with the same notation as part 4.1:
SP = SPq + SP= with
SP
SPq = V Pgrating − BV P the vertical comSP= = HPgrating − BHP its horizontal component,
the signal of the Smith-Purcell radiation,
ponent of Smith-Purcell radiation and
V Pgrating ings, and BV P
with
(8)
HPgrating the signal of vertical and horizontal polarization from the gratand BHP the background of the signal of vertical and horizontal polarization,
and
assumed to be equal to the signal from the blank. Furthermore, we know that we have a relation like:
SPq = cos α SP 12
and
SP= = sin α SP with
α
a xed angle.
Because the background is not polarized,
BV P = BHP .
We have
therefore:
V Pgrating − HPgrating = SPq − SP= = (cos α − sin α) SP i.e., with
β
a constant:
SP = β (V Pgrating − HPgrating )
(9)
With formulas 8 and 9, we have two dierent ways to obtain the signal of Smith-Purcell radiation (see gure 12). If we make the ratio of the two curves, we should obtain a constant ratio (see gure 12, bottom right). 4
4
4
500um grating
x 10
6 4 Signal AU
Signal AU
3
2
1
0 40
250um grating
x 10
2 0 −2
60
80
100 Angle
120
140
−4 40
60
80
100 Angle
120
140
2012 06 26 23 55 55 k=method A/ method B
Signal AU
8 Smith Purcell radiation (method A) : (vp − bkg) + (hp − bkg) Smith Purcell radiation (method B) : vp − hp 6 4 1mm grating x 10 5 4 0
−5 40
500um 250um 1mm
2
60
80
100 Angle
120
140
0 40
60
80
100
120
Figure 12:
Signal of Smith-Purcell radiation obtained with equation 8 (method A) and method A equation 9 (method B) ; Bottom right: ratio k = for the three gratings. method B We can see that for the detectors above 90°, the ratio is almost constant ; but for 70° and
80°, the value of the ratio is very high. This anomaly can be seen for all the dierent scans of polarization we made. Because of this anomaly, we tried to understand better the theory of gratings and found that their e ciency took a very important place in our study.
4.3 E ciency of the grating 4.3.1
E ciency from Loewen's paper
The amount of radiation emitted by the grating depends on the blaze angle of the grating, the wavelength, the polarization (see gure 13, left) and, if the grating is used in a more
13
classical way with an incident light, the angle of incidence. A theoretical paper [7] expose the shape of the e ciency curves of the two polarization for dierent blaze angles (see gure 13, right).
Figure 13: Left: polarization P (Transverse Electric) and S (Transverse Magnetic) ; Right: e ciency of a grating with a blaze angle
j=26°45' for the polarization P (dashed line) and S
(solid line) in the Littrow con guration [7]
λ with λ the wavelength of the radiation d and d the period of the grating. We can see that the solid line, which is the e ciency for the λ ' 0.7, which S polarization (in our case, the vertical polarization), shows an anomaly at d λ is very close to the value ' 0.66 for our detectors at 70° (see appendix 3 for the dierent d λ values of ). We therefore thought that this anomaly in the e ciency of the grating could d explain the anomaly we had seen in the ratio at 70° (for more details on anomaly on e ciency The gure 13, right, has a horizontal scale in
pro les, see reference [8]) However, there were two important problems in this method: rst, we did not have the e ciency for the blaze angle of our gratings. The nearest we had was 26°45' instead of 30°
(250µm grating), 35° (1mm) and 40° (500µm). The second problem was that the e ciency curves was made in a Littrow con guration, that is when the angle of incidence is equal to the angle of diraction. In our case, we don't have any incident light. In the rest of the study, we assumed that the Littrow con guration can be applied to our gratings, but currently we are not able to know if it is justi ed or not. If the e ciency is applied, we come back to the signal before the in uence of the grating. We should therefore nd the same amplitude for the two signals of polarization. I applied the e ciency to the curves of polarization without the background and obtained 0 0 two new curves, V P and HP , according to the formulas:
V P0 =
V P − BV P S
HP 0 =
HP − BHP P
and
with
S
and
P
the e ciency of the grating for the S and P polarization respectively (see
gure 14a). I also calculated the error for each point, using the error on the raw data (I used
14
the standard deviation on the 100 pulses for each acquisition) and the estimated error on Loewen e ciency (see in appendix 3 for more details). As it was expected, we can see that the two curves of polarization are very close from each other, except for the 60° detector; but the error on this point is too important to take this point into account. 4
4
500 um grating
x 10
4
500um grating
250um grating
4
4
3
3
0
−2
0
ratio
2 signal
signal
2
250 um grating
x 10
ratio
4
2
1
−4 40
60
4
4
80 angle
100
120
−4 40
60
80 angle
100
120
0 40
1 mm grating
x 10
1
60
80 angle
100
120
0 40
60
80 angle
100
120
1mm grating 4
2
3
0
ratio
signal
2
−2
−2
2 1
−4 40
60
80 100 120 angle horizontal polarizers with Loewen efficiency vertical polarizers with Loewen efficiency
0 40
60 80 100 120 angle SP from the polarizers / SP from the filters with Loewen efficiency
(a) Curves of polarization corrected with Loewen ef ciency for each grating, with the formulas: V P 0 = V P −BV P HP and HP 0 = HP −B . S P
0
VP (b) Ratio k = HP 0 for each grating with the formuV P −BV P 0 HP las: V P = and HP 0 = HP −B . S P
Figure 14
V
I also compared the Smith-Purcell signal obtained with the polarizers, SPpolarizers = F −Bf ilter with Fgrating the P 0 + HP 0 and the one obtained with the lters, SPf ilters = grating grating f ilters
signal from a grating with the set of lters associated,
Bf ilter
the background for this set of
lters, f ilters the e ciency of the lters and grating the mean of S and SP k = SPpolarizers , which is expected to be constant (see gure 14b). f ilters
P .
I plot the ratio
Taking into account the error bars, we can consider that the ratio is almost constant for angles above 70°; below, the error is too much important to have very conclusive results.
4.3.2
E ciency from GFW program
In his paper [5], George Doucas, who is a member of the E-203 collaboration, proposes an 2 other formula for the grating e ciency (R ). However, he only considers the radiation emitted in the plan
ϕ=0
(see gure 6), where the emission is only vertically polarized. Hence he
does not consider in this paper the e ciency for the horizontal polarization. Nevertheless, in our case, we had clearly seen that the signal had a strong component horizontally polarized. Theoretically, the Smith-Purcell radiation is vertically polarized only in the plan
ϕ = 0;
otherwise, the two components of polarization exist. We assumed that we were able to see horizontal polarization because the radiation capted by the detectors did not come only from the plane
ϕ = 0.
George Doucas had made a program named GFW calculating the e ciency of gratings depending on the angles
θ and ϕ.
I ran this program to obtain the e ciency curves we needed
(see appendix 4, gures 23 and 24) and applied it to the polarization and the lters as in part 4.3.1 (see gures 15a and 15b).
15
x 10
4
500um grating 14
500um grating
250um grating
x 10
1
0.5
4
3
3
10
ratio
amplitude signal
amplitude signal
12 1.5
250um grating
4
8 6
ratio
5
2
2
1
2
1
4 0 60
80
100
120
2 80
90
angle 5
x 10
110
0 40
120
60
80 angle
100
120
0 40
60
80 angle
100
120
1mm grating
1mm grating
4 vertical polarizers with gfw efficiency horizontal polarizers with gfw efficiency
GFW efficiency 3
10
ratio
amplitude signal
15
100 angle
2
5
1
0 60
80
100
0 40
120
60
angle
(a) Curves of polarization corrected with GFW e ciency for each grating, with the formulas: V P 0 = HP −BHP V P −BV P 0 GF W k and HP = GF W = .
80 angle
(b) Ratio k = V P0 =
100
120
V P0 HP 0
V P −BV P GF W k
for each grating with the formulas: −BHP and HP 0 = HP GF W = .
Figure 15 With GFW e ciency, we have a value of e ciency for the detectors from 40° to 60°, but their value is very low, which can explain why the amplitude of the signal corrected for these detectors is so high. More over, the two curves of polarization do not seem to t well each V P0 other; that is why I also plotted the ratio for Loewen and GFW e ciency, which, in both HP 0 case, is expected to be equal to 1, and compared them (see gure 16). In fact it is di cult to say which method gives the best ratio. 250um grating 2
1.5
1.5 ratio
ratio
500um grating 2
1
0.5
0 40
1
0.5
60
80 angle
100
0 40
120
60
80 angle
100
120
1mm grating 2
ratio vp/hp with Loewen efficiency ratio vp/hp with GFW efficiency
ratio
1.5
1
0.5
0 40
60
80 angle
Figure 16: Ratio
100
120
VP for Loewen and GFW e ciency. HP
In conclusion, we can say that we have di culties to understand the horizontal component
16
of polarization of the Smith-Purcell radiation.
The two dierent models of e ciency give
dierent results, but none of them is very conclusive. That is why, for the reconstruction of time-pro le with the polarization data, I decided to use only vertical polarization, assuming that, once corrected with Loewen e ciency, it represented half of the signal emitted. Nevertheless, there might be a method to check if the e ciency curves given by GFW are correct: we can use a column of polarizers and put masks with a slit to keep only the radiation emitted at a given angle
f.
Thus we can compare the amplitude of the signal
against phi obtained experimentally with the e ciency given by GFW. This will be done during the next E-203 run.
4.4 Reconstruction of time-pro le For the reconstruction of the time-pro le, I used a program named Cauchy, written by George Doucas. This program makes reconstruction based on raw lters data, and makes itself the corrections due to the e ciency of the lters, the windows, the detectors and the Winston cones. The program calculates rst the signal of Smith-Purcell radiation according to the formula:
SP = with
SP
Fgrating
Fgrating − Bf ilter GF W corrections
(10)
the signal of Smith-Purcell radiation used by the program for the reconstruction,
the signal from the grating with the appropriate set of lters,
of the set of lters,
GF W
Bf ilter
the background
the grating e ciency obtained with GFW program, and
corrections
the other corrections made by the program. For the reconstruction with the lters, I assumed that we had:
V P − BV P SP = 2 Loewen with
VP
the signal vertically polarized coming from the grating,
the signal vertically polarized and
Loewen
(11)
BV P
the background of
the grating e ciency for the vertical polarization
obtained from Loewen's paper. We only give
Fgrating
and
Bf ilters to the program to make the reconstruction. I had to F pol and B pol for the polarizers. Combining the equations 10
know what was the equivalent,
and 11, I deduced that we had:
F Because
B pol
pol
= 2.
V P − BV P Loewen
.GF W corrections + B pol
is given to the program and immediately subtracted from
(12)
F pol ,
I could take
any value; I therefore put it equal to zero. An other problem with the reconstruction with the polarization data was that the detectors at 130° and 140° were open holes, we did not have information of the grating e ciency
for the 40° and 50° detectors and the e ciency for the 60° detector was very low. I decided to use four dierent methods for these points:
The detectors at 40°, 50°, 60°, 130°, 140° at 10% of their value (it produced problems in the reconstruction if the values were at 0) ( gure 17, upper left);
17
The detectors at 40°, 50°, 60° at 10% of their value, the detectors at 130° and 140° at
the same value as the 120° detector ( gure 17, upper right);
The detectors at 40°, 50°, 60° at the same value as the 70° detector, the detectors at 130° and 140° at 0 (( gure 17, bottom left);
The detectors at 40°, 50°, 60° at the same value as the 70° detector, the detectors at
130° and 140° at the same value as the 120° detector ( gure 17, bottom right);
With formula 12, I could make a reconstruction with the polarization data (see gure 17). I used three dierent scans from the 26-06-2012, at 18h51 (no compression), 21h23 (half compression) and 23h55 (maximum compression).
preliminary data ; vertical polarization data 40, 50, 60, 130, 140°detectors at 10%
preliminary data ; vertical polarization data 40, 50, 60° at 10% ; 130, 140° = 120° 0.4
no compression medium compression maximum compression
0.3
beam profile (arbitrary unit)
beam profile (arbitrary unit)
0.4
0.2
0.1
0
0
2
4 6 time (ps)
8
0.3
0.2
0.1
0
10
preliminary data ; vertical polarization data 40, 50, 60° = 70° ; 130, 140° at 0
4 6 time (ps)
8
10
0.4 beam profile (arbitrary unit)
beam profile (arbitrary unit)
2
preliminary data ; vertical polarization data 40, 50, 60° = 70° ; 130, 140° = 120°
0.4
0.3
0.2
0.1
0
0
0
2
4 6 time (ps)
8
0.3
0.2
0.1
0
10
0
2
4 6 time (ps)
8
10
Figure 17: Reconstruction of time-pro le with polarization data.
We can see on gure 17 that the time-pro le is very large (almost 1ps). The shape does not look like a simple Gaussian, but the sum of two Gaussians. Moreover, we can clearly see that the compression has an in uence on the reconstruction: the more the beam is compressed, the higher is the beam intensity, producing a reduction of the longitudinal size of the beam when the compression increases. I made also reconstruction with the lters data without any correction for the grating e ciency (see gure 18).
18
preliminary data reconstruction with the filters (raw data) 0.6 no compression (filters) medium compression (filters) maximum compression (filters) 0.5
beam profile (arbitrary unit)
0.4
0.3
0.2
0.1
0
−0.1 −1
0
1
2
3
4 time (ps)
5
6
7
8
9
Figure 18: Reconstruction of time-pro le with lters data.
This time the second Gaussian does not appear clearly in the reconstruction (except for the lowest compression). The reconstruction is larger than for the previous reconstructions (almost 2ps). The evolution of the amplitude is less important than for the reconstruction with the polarization data, but we see clearly the shape of the beam becoming narrower. The results are not exactly similar between the two methods of reconstruction, but these results are very preliminary. We did not have yet some parameters of the beam the program asked, as the transverse size of the beam or the position from the grating; once we have them, we would be able to obtain results more accurate. Nevertheless we are able to obtain results with polarization data which are physically acceptable and which seem to agree with the preliminary results of the other groups of research working on the reconstruction of time-pro le with other methods.
5
Conclusion
The most important part of my work in this internship was to understand the polarization of the Smith Purcell radiation. There is still a lot of work to do to understand both grating theory and Smith-Purcell radiation theory and link them together; but I have shown some important things, as the background unpolarized and the existence of a component of the radiation horizontally polarized, which may be important for the development of a new device, using a single shot acquisition.
19
References [1] S.J. Smith and E.M. Purcell, Visible light from localized surface charges moving across a grating,
Phys. Rev. 92, pg. 1069, (1953).
[2] O. Grimm and P. Schmüser, Principles of longitudinal beam diagnostics with coherent radiation, TESLA FEL 2006-03. [3] V. Blackmore, Determination of the Time Pro le of Picosecond-Long Electron Bunches through the use of Coherent Smith-Purcell Radiation, Ph.D. Dissertation, University of Oxford, 2008. [4] V. Blackmore, G. Doucas, C. Perry, and B. Ottewell, First measurements of the longitudinal bunch pro le of a 28.5 GeV beam using coherent Smith-Purcell radiation,
review special topics - accelerators and beams
Physical
12, 032803 (2009).
[5] .J.H. Brownell and G. Doucas, Role of the grating pro le in Smith-Purcell radiation at high energies, Physical review special topics - accelerators and beams
8, 091301 (2005).
[6] G. Doucas and M.F. Kimmitt, Determination of longitudinal bunch shape by means of coherent Smith-Purcell radiation, Physical review special topics - accelerators and beams
5, 072802 (2002). [7] E.G. Loewen, M. Nevière and D. Maystre, Grating e ciency theory as it applies to blazed and holographic gratings, Applied Optics (October 1977, Vol. 16, No. 10). [8] Erwin G. Loewen and Evgeny Popov,
Diraction Gratings and Applications,
Francis (1997).
20
Taylor &
Appendix
Appendix 1: Angular distribution of Smith-Purcell radiation The angular and frequency distribution of the radiated intensity is given by the formula:
with
Ne
d ²I dWdw
Ne
electron:
λe = λ √ 2π
d ²I dWdw
dI
1
βγ 1+β ²γ ² log sin ²θ sin ²φ
θ and φ φ).
dWdw 1
the total length of the grating, gure 6 for
dWdw 1 ´∞ 1
(W, w) . Ne Sinc + Ne2 Scoh
coherent part of the radiation with f(t) the distribution of the parti-
cles in the time domain, and
grating and
dI
Sinc = √2πσ 0 e−2x/λe e−[(x−x0 )²/2σ²x ] dx the incoherent part x √ 1 ´ ∞ −x/λe −[(x−x0 )²/2σ²x ] 2 √ 1 ´ ∞ −ky y −[(y−y0 )²/2σ²y ] 2 dx × 2πσ −∞ e dy × e = 2πσ 0 e e x y
the number of electrons,
of the radiation, Scoh ´ 2 ∞ −iωt f (t)dt the −∞ e
with
(W, w) =
l
the angular distribution of intensity emitted by a single
n²b² 2x0 Z R² exp − = 2pq² l² (1 − b cos j)3 le the evanescence wavelength, the period of the grating,
R²
q
the charge of the electron,
Z
a parameter depending on the
angles determining the position of observation (see gure 5 for
θ
and
Appendix 2: Estimation of the position of the grating Relation between position and the high precision multimeter First, we used a survey system which calculates the position of the grating in 3D. This allowed us to know the x position of the grating and the angles made by the plane of the grating in the (x,y) plane and the (x,z) plane. For each position, we noted the value of the resistance given by a rst multimeter (named high precision multimeter).
Relation between the high precision multimeter and the hand held multimeter In the control room, where we took data, we had to use an other multimeter (named hand held multimeter) to read the position of the gratings. Therefore we had to know the relation between the value of the resistance from the high precision multimeter and the hand held one.
Relation between resistance and voltage The DAQ we used also measured the position of the grating in volts. Hence it was interesting to read also the voltage of the hand held multimeter. We therefore estimated the relation between, on one hand, the resistance and the volts given by the hand held multimeter and on the other hand the volts given by the DAQ (named EPICS) and the resistance given by the hand held multimeter.
21
Estimation of the error on the position With Nuria Fuster, a Spanish student who worked also on the experiment, we estimated the error on the grating position.
I will only describe the part of the work I did, that is the
error in the relation between the position and the value of resistance of the high precision multimeter. For the measurements of the position, we put a protective plate on each grating in order to have a plane surface and not a blazed one.
Therefore I corrected the raw values by
subtracting the thickness of the protective plate.
Then, for each grating, I plotted the
position corrected against the resistance measured with the high precision multimeter and took a linear function approximation to obtain the equation of each curve (see gure 19).
Figure 19: Position of the grating with the corrections due to the protective plates
From these four equations, I had dierent ways to modelise the position of the gratings. The most physical was to modelise the position of the gratings with the same slope (all the gratings are supposed to cover the same distance for a same dierence of resistance), but with dierent osets (which can be physically explained by a dierence of thickness between the gratings or a slightly dierent tting on the carousel). But the error on this method was
22
quite important: about 170µm taking the largest error, and about 75µm with the mean error (see below for more details about the error). I then intented to see if there was an hysteresis in the movement of the carousel which could explain the important error I found. Unfortunately, because of a lack of time during the data collection, we took measurements for the both movements, backward and forward, for only one grating. I studied the residuals for this grating, but I don't think we are able to conclude with a presence or not of hysteresis. During the acquisitions of the Smith-Purcell radiation, we needed to know approximately the position of the grating from the beam axis. That is why I made a program giving, from the value of the voltage from epics or the hand held multimeter, the position of the grating from the beam axis. I wrote a simpli ed program: assuming that the dierence between the position of two dierent gratings for a same value of voltage was negligible in our situation, I took the model of same slope and same oset for each grating (see gure 20 for the nal curve).
Figure 20: Position of the grating with respect to the beam line against voltage (multimeter).
Summary of the dierent error on the position
W
Error for the conversion between volts (from epics or hand held multimeter) and k (from hand held multimeter)
23
Largest error Error (k
Multimeter
Epics
Multimeter
0.048
0.012
0.0008
0.0014
0.0852
0.0213
0.0014
0.0025
W)
Error (mm) (model same slope, dierent osets)
Mean error
Epics
Table 1: Error for the conversion between volts (from epics or hand held multimeter) and k
W (from hand held multimeter).
Error for the conversion between the small and high precision multimeter:
W
Error in k
Error in mm
Largest error
Mean error
0.0065
0.0025
0.0115
0.0044
Table 2: Error for the conversion between the small and high precision multimeter.
Error for the conversion between the high precision multimeter and the position:
Largest error
Mean error
Model with same slope, dierent osets (mm)
0.168
0.0746
Model for dierent slopes, dierent osets (mm)
0.0526
0.0228
Table 3: Error for the conversion between the high precision multimeter and the position.
Total error:
Largest error Model with same slope, dierent osets (mm) Model with dierent slopes and dierent osets (mm)
Mean error
Epics
hand held multimeter
Epics
hand held multimeter
0.1876
0.1685
0.0747
0.0748
0.1008
0.0579
0.0233
0.0234
Table 4: Total error
Position from the beam axis My work on the position of the grating was assuming that the beam was moving exactly in the center of the cavity (i.e. on the beam axis). In fact the beam is not exactly at this position. We could not know directly the position of the beam at the position of our experiment, but we were able to know its position at some points along the propagation axis using the BPMs (Beam Position Monitor). Based on these informations, Nuria Fuster worked to deduce the position of the beam from the beam axis at our position.
24
Appendix 3: Loewen e ciency λ for each detector we used and the corresponding e ciency for d the S and P polarization I took from Loewen's paper. The values for 40° and 50° detectors λ are arbitrary values (the graphs did not give e ciency for low values of ). The column in d red correspond to the 70° detector, were we saw an anomaly in the ratio of the two dierent You can nd the value of
methods of calculation of the Smith-Purcell radiation (see part 4.2, gure 12). I also estimated the error I made when I noted down the value of e ciency.
l/d
40°
50°
60°
70°
80°
90°
100°
110°
120°
130°
140°
0.23
0.36
0.50
0.66
0.83
1.00
1.7
1.34
1.5
1.64
1.76
E ciency (P polarization)
0.30
0.30
0.28
0.95
0.79
0.53
0.36
0.24
0.15
0.10
0.06
Error (P polarization)
0
0
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
E ciency (S polarization)
0.45
0.45
0.033
0.65
0.99
0.980
0.93
0.9
0.9
0.95
1.000
Error (S polarization)
0
0
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
Angle of the detectors
Table 5: Values of Loewen e ciency for our detectors
Appendix 4: graphs Problem of saturation We can see that the saturation is also due to the saturation of the detectors.
Figure 21: Saturation of detectors
25
Problem of the horizontal polarization Even if we assume that a part of the vertical polarization pass through the horizontal polarizers, the horizontal component of the polarization is still too important to be neglected. 4
4
3.5
x 10
2
x 10
3 1
2
signal
signal
2.5
1.5 1
0
−1
0.5 0 40
60
80 angle
100
120
−2 40
60
80 angle
100
signal
4 Polarization and polarization with correction due to the efficiency of the polarizers x 10 4 horizontal polarizers horizontal polarizers corrected 3.5 vertical polarizers 3 vertical polarizers corrected
2.5 2 1.5 1 0.5 40
60
80 angle
100
120
Figure 22: Polarization taking into account the transmittivity of the polarizers
26
120
E ciency curves from GFW program
250um grating 0.4
0.3
0.3 R2
R2
500um grating 0.4
0.2 0.1 0
0.1
0
50 100 150 angle (degree) 1mm grating
0
200
0.3 0.2 0.1 0
0
100 angle (degree)
0
50 100 150 angle (degree)
200
P1, phi=0 P1, phi=0.4 P1, phi=0.8 P1, phi=1.2 P1, phi=1.6 P1, phi=2 P1, phi=2.4 P1, phi=2.8 P1, mean
0.4
R2
0.2
200
Figure 23: E ciency curve from GFW program; vertical polarization.
27
250um grating 0.4
0.3
0.3 R2
R2
500um grating 0.4
0.2 0.1 0
0.1
0
50 100 150 angle (degree) 1mm grating
0
200
0.6 0.4 0.2 0
0
100 angle (degree)
0
50 100 150 angle (degree)
200
P2, phi=0 P2, phi=0.4 P2, phi=0.8 P2, phi=1.2 P2, phi=1.6 P2, phi=2 P2, phi=2.4 P2, phi=2.8 P2, mean
0.8
R2
0.2
200
Figure 24: E ciency curve from GFW program; horizontal polarization.
28
