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Diffraction radiation from a charged particle moving through a rectangular hole in a rectangular screen
Nuclear Instruments and Methods in Physics Research B 227 (2005) 198–208 www.elsevier.com/locate/nimb
Diffraction radiation from a charged particle moving through a rectangular hole in a rectangular screen P. Karataev a,*, S. Araki b, R. Hamatsu a, H. Hayano b, T. Muto b, G. Naumenko c, A. Potylitsyn c, N. Terunuma b, J. Urakawa b a
Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo 192-0937, Japan b KEK, Accelerator Test Facility, Oho 1-1, Tsukuba, 305-0801, Ibaraki-ken, Japan c Tomsk Polytechnic University, 634050, pr. Lenina 2a, Tomsk, Russia Received 17 December 2003; received in revised form 2 March 2004
Abstract We developed a new model for calculating diffraction radiation (DR) from an electron moving through a rectangular hole in a rectangular screen. The calculations show that short wavelength DR is very sensitive to the beam size. For example, optical DR (ODR) could be used to measure the beam size as small as 10 lm. Moreover, splitting two polarization components it becomes possible to measure vertical and horizontal beam sizes independently. We have calculated the DR spectra and compared them with TR ones for a finite size target. It is shown that when the DR wavelength is comparable with or longer than the hole size, the photon yield is mostly determined by the outer target dimensions. It means that in case transversal beam dimensions smaller than the observation wavelength the coherent DR could be used for non-invasive bunch length measurements with the same accuracy as the coherent TR techniques. However, the outer target dimensions must be taken into account because the finite target size causes a significant intensity suppression in the long wavelength spectral range as well as distortion of the coherent spectrum. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.75.Ht; 41.85.Qg Keywords: Transition radiation; Diffraction radiation; Electron beam diagnostics
1. Introduction During last several years different techniques for electron beam diagnostics have been intensively developed. For accelerators of the next generation like linear collider or short wavelength
*
Corresponding author. Tel.: +81-298-64-5715; fax: +81298-64-5746. E-mail address: [email protected] (P. Karataev).
free electron lasers precise measurements of such beam parameters as size, emittance, bunch length, which, as a rule, much smaller than ones for usual accelerators, is extremely necessary. Up-to-date transition radiation (TR) monitors have widely been used for beam diagnostics. TR appears when a charged particle crosses a boundary between two media with different dielectric properties. Due to high signal-to-noise ratio TR in an optical wavelength range gives a good possibility to monitor beam profile and emittance for
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.014
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
single shot [1,2]. Coherent TR (the observation wavelength is comparable with or longer than the longitudinal bunch length) in millimeter and submillimeter wavelength range makes it possible to measure bunch length [3,4]. However, to produce TR at the target surface the particles interact with the target material experiencing multiple scattering. It may lead to worsening of the beam emittance. Moreover, due to ionization process the target can be heated up, deformed or even destroyed under an intense focused electron beam. That means that it is extremely necessary to develop a non-invasive method for beam diagnostics. A few non-invasive techniques are under intense study now. For example, synchrotron radiation monitor (interferometer) is very promising for beam diagnostics at circular accelerators and rings [5]. For a linear accelerators a laser wire beam size monitor can be used [6]. However, there is no way to measure the beam size for a single shot with laser wire. Furthermore, it is necessary to collect statistics and all points are measured from different bunches. In this case other beam parameters like beam position and angular jitter, orbit drift, etc. may cause an additional error and complicate the data analysis. We assume that diffraction radiation (DR) effect can be used for non-invasive electron beam diagnostics. DR is generated when a charged particle moves in vacuum close to a conductive medium and interacts with the matter by its electric field only [7]. TR and DR have a relative nature because they both appear as result of dynamic polarization of a medium. DR spectral range is very broad and its characteristic wavelength, k, is defined by Lorentz-factor, c, and the shortest distance between the target edge and particle trajectory, h, as k h=ð2pcÞ. In spite of the fact that the first theoretical considerations on DR effect have appeared over 40 years ago the first observation of coherent DR in mm and sub-mm wavelength range has been performed in 1995 [8]. Recently a great interest in coherent DR phenomenon has appeared because of the possibility of measuring the bunch length in a non-invasive way [9–11]. Nowadays there are a lot of theoretical approaches describing possible ways for transversal
199
beam parameter determination using short wavelength DR [12–14]. However, in contrast to TR DR effect is not very well investigated experimentally. The first observation of incoherent diffraction radiation in optical wavelength range (ODR) from the edge of a semi-plane target we performed last year [15]. In [16] we represent the results of ODR observation from a slit target. The experimental results were in reasonable agreement with the model of ideally conducting (reflecting) infinitely thin target. Now we are working on our equipment to be able to measure a few micron beam size with a proper accuracy. And we are also thinking about the future prospects. We have developed a new model for calculating the DR spectral-angular characteristics from a particle moving through a rectangular hole in a rectangular screen. Our calculations show that short wavelength DR is very sensitive to the transversal beam size. For example, DR in optical wavelength range (ODR) could be used to measure the beam size as small as 10 lm. Moreover, splitting vertical and horizontal polarization components it becomes possible to measure vertical and horizontal beam sizes independently. To be able to evaluate the bunch length from coherent DR spectrum, it is necessary to have a calculated spectrum for a single electron. In [10,11] the authors investigated coherent DR spectra and determined the bunch length but they did not take into account the finite outer target dimensions. One of the great advantages of our model is that it is possible to calculate DR (TR) spectra from a finite size target. In this paper we show the calculated DR spectra and comparison with TR ones. It is shown that in the soft part of the spectrum the photon yield is significantly suppressed. Without including the finite size target effect the information about longitudinal beam size could be wrong.
2. Transition radiation from an electron moving through a rectangular screen Transition radiation (TR) appears when a charged particle crosses a boundary between two media with different dielectric constants. Of course, to derive an exact expression for TR it is
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necessary to solve MaxwellÕs equations. However, in this paper we intend to use a simpler approach for TR and DR based on the scattering of the electron field pseudo-photons from the target surface [7,13]. This approach is based on the wellknown Huygens principle of plane wave diffraction. The difference from the plane wave is that the electron field depends on the distance from the particle trajectory. According to the Huygens principle when the charged particle interacts with the target with its electric field, all points of the target surface are considered as elementary sources. The total field of the radiation for two polarization components from an arbitrarily shaped target can be derived, by integrating over the target surface in the following manner: Ex;y ðkx ; ky Þ ¼ Rx;y ðk; h0 Þ Z Z 1 i 2 Ex;y ðx; yÞeiðkx xþky yÞ dx dy: 4p
The components of the emitted photon wave vector in ultrarelativistic case are determined by the following relations (with accuracy of c2 ): 1 k sin hx cos hy khx ; c1 1 ky ¼ k cos hx sin hy khy ; c1 1 kz ¼ k cos hx cos hy k: c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ¼
ð2Þ
Here c1 ¼ sin2 hx cos2 hy þ cos2 hx is a normalization constant. hx and hy are the angles measured from the direction of specular reflection (see Fig. 1). In an ultrarelativistic case hx ; hy 1. For simplicity we shall assume that the target is ideally reflecting (Rx ¼ Ry ¼ 1). However, as is shown in [16], to achieve a better consistency between experiment and theory the reflection coefficients must be included for some materials. The components of the incident particle field can be expressed by its Fourier transform [7]:
ð1Þ i ðx; yÞ are the components of the incident Here Ex;y charged particle field, k ¼ 2p=k is the wave number, h0 is the target tilt angle, kx and ky are the components of the emitted photon wave vector, x and y indexes of E represent transversal horizontal and vertical polarization components (with accuracy of order of c1 ), which are proportional to the Fresnel reflection coefficients Rx and Ry , respectively [13]. One should notice that there is the longitudinal component, Ez , however, for an ultrarelativistic case it is negligibly small in comparison with the transversal ones and could be omitted. Throughout the paper the system of units h ¼ me ¼ c ¼ 1 is used. In [13] the authors considered the effect of the tilted target. It has been shown that for large target tilt angles and ultrarelativistic case in Eq. (1) kx kx ðkc2 =2Þ cot h0 , which means that the general angular distribution is shifted by the angle of c2 =2 with respect to the direction of specular reflection. Since we are talking about the electron energies higher than 100 MeV (c ¼ 200), and the target tilt angle h0 ¼ 45 deg that term is negligible. Therefore, in the future we shall assume kx kx .
y a/
si n
a2
0
a1
θ0
e -b1
y
b
x
θy b2
θx
x
z
Fig. 1. Geometry of the transition radiation from a particle passing through a rectangular screen. Here z axis is directed along the direction of specular reflection, x ¼ x sin h0 , y ¼ y, dx dy ¼ cos h0 dx dy.
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
i Ex;y ðx; yÞ
ie ¼ 2 2p
Z Z
0
0
0 kx;y eiðkx xþky yÞ dk 0 dk 0 : kx02 þ ky02 þ k 2 c2 x y
ð3Þ
Here e is the electron charge, c is the charged particle Lorentz-factor, kx0 and ky0 are the components of the electron field pseudo-photon wave vector. In principle it is possible to solve the integral in Eq. (1) for any kind of target surface numerically. However, it is always useful to have a more simplified solution for a particular case. For example, substituting Eq. (2) in (1) and integrating it over x and y from 1 to 1 we obtain two delta functions: 2pidðkx0 kx Þ and 2pidðky0 ky Þ. After that the kx0 and ky0 integration can be done trivially. The result is the expression for transition radiation (TR) fields from an infinite boundary: TR Ex;y ¼
ie kx;y : 2 2 2p kx þ ky2 þ k 2 c2
ð4Þ
Now let us derive an expression for TR fields from a particle crossing a rectangular screen. Here we assume that the target is tilted in one direction only, as it is shown in the calculation geometry represented in Fig. 1. The first step to derive the expression, we are looking for, is to integrate the Eq. (1) over x from a1 sin h0 ¼ a=2 ax to a2 sin h0 ¼ a=2 þ ax and over y from b1 ¼ b=2 by to b2 ¼ b=2 þ by : Z Z ie dkx0 dky0 Ex;y ðkx ; ky Þ ¼ 4 8p 0 kx;y ðkx02 þ ky02 þ k 2 c2 Þðkx0 kx Þðky0 ky Þ h 0 i 0 eiðkx kx Þa2 sin h0 eiðkx kx Þa1 sin h0 h 0 i 0 eiðky ky Þb2 eiðky ky Þb1 : ð5Þ
201
integral hasffi three poles: ky0 ¼ ky , ky0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 2 kx þ k c2 . The integration should be performed over positive and negative imaginary halfplanes separately. The integration contours and the detailed solution of the integral can be found in [13]. Using Eq. (2) The final expression for TR from a rectangular target can be represented in the following form: ie hx 2p2 k h2x þ h2y þ c2 8 Z ie < 1 t sin ðak=2ðt hx ÞÞ ikax ðthx Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 3 dt 4p k : 1 ðt hx Þ t2 þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp kðb=2 by Þ t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 = exp kðb=2 þ by Þ t2 þ c2 þ ihy 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 2 ; t þ c þ ihy 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2y þ c2 ihx e 6 exp kða=2 ax Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 4p k h2 þ c2 ih
Ex ðhx ; hy Þ ¼
y
x
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 exp kða=2 þ ax Þ h2y þ c2 þ ihx 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; h2y þ c2 þ ihx
ie hy 2p2 k h2x þ h2y þ c2 8 Z e < 1 sin ðak=2ðt hx ÞÞ ikax ðthx Þ 3 dt e 4p k : 1 ðt hx Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp kðb=2 by Þ t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 = exp kðb=2 þ by Þ t2 þ c2 þ ihy 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t2 þ c2 þ ihy
Ey ðhx ; hy Þ ¼
ie hy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p2 k h2 þ c2 y 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2y þ c2 ihx 6 exp kða=2 ax Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h2y þ c2 ihx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 exp kða=2 þ ax Þ h2y þ c2 þ ihx 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 5: h2y þ c2 þ ihx
Here b and a are the vertical and effective horizontal target dimensions respectively, by and ax are the vertical and effective horizontal offsets with respect to the target center. The real horizontal target size and offset can be found as a= sin h0 and ax = sin h0 , respectively (see, for instance, Fig. 1). Eq. (5) can be integrated over ky0 and partially over kx0 . Let us briefly consider the ky0 integral. It can be done using residue theorem. The equation under
ð6Þ
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
geometries has been considered in [13]. It means that the model presented above is valid. In spite of the infinite integration limits in Eq. (6) the integration can be done numerically in a very narrow interval Dt 1, where the dominant part of the under integral function is confined. The spectral angular distribution can be obtained from the following expression:
First of all we need to examine Eq. (6). That is apparent that the first term in this expression is the TR field from an infinite boundary represented by Eq. (4). Actually, the rest of the equation is the diffraction radiation field from a particle moving through a rectangular hole in a screen of infinite dimensions taken with an opposite sign. As a matter of fact the above expression introduces a well-known BabinetÕs principle from classical optics. Therefore, for convenience we shall represent the Eq. (6) in the form TR DR Ex;y ðhx ; hy Þ ¼ Ex;y ðhx ; hy Þ Ex;y ðhx ; hy Þ:
d2 W dx dX h i 2 2 ¼ 4p2 k 2 jEx j þ jEy j :
Sðk; hx ; hy ; a; ax ; b; by Þ ¼
ð7Þ
lim
a!1
sin ½ak=2ðt hx Þ ¼ pdðt hx Þ; t hx
ð8Þ
Intensity normalized by TR maximum
the integral part is also transformed into the expression for DR field from a particle passing through a slit between two semi-planes, but the projection of the particle trajectory onto the target plane, in this case, is parallel to the slit edges. Both 1.0
(a) 0.8 0.6 0.4 0.2 0.0 0
2
4
γθy
6
ð9Þ
We have developed a computer code to calculate TR and DR spectral angular characteristics from a particle moving through a target of a rectangular shape. Fig. 2(a) illustrates the TR angular distributions calculated at different incident points of the particle with respect to the target center. It is seen that when the particle crosses the target center, which vertical and horizontal sizes are larger than 2ck, the angular distribution coincides with the TR from an infinite boundary. However, when the particle trajectory closes up to the target edge the minimum in angular distribution disappear and it transforms into DR from a particle moving straight through the target edge, which is single mode in contrast to TR. This effect we have observed experimentally in [15]. Fig. 2(b) shows the TR angular distributions for different
It might be interesting to notice that when in the DR expression for diffraction radiation, Ex;y , b ! 1, the integral part of the equation vanishes and the equation is transformed in to the DR field from a particle moving through a slit between two semiplanes and its projection onto the target plane is perpendicular to the target edges. However, if the a ! 1 the last term in the DR field tends to zero. Keeping in mind that
8
Intensity normalized by TR maximum
202
1.0
(b) 0.8 0.6 0.4 0.2 0.0 0
2
4
γθy
6
8
Fig. 2. (a) Transition radiation angular distributions for different offsets with respect to the target center for the target dimensions a ¼ b ¼ 10 mm: by ¼ 0 – solid line, by ¼ b=2 0:15 mm – dashed line, by ¼ b=2 – dashed-dotted line; (b) Transition radiation angular distribution for different target configurations: a ¼ b ¼ 10 mm – solid line; a ! 1, b ¼ 0:4 mm – dashed line; a ¼ 0:3 mm, b ¼ 0:4 mm – dashed-dotted line; a ¼ 0:1 mm, b ¼ 0:4 mm – dotted line. The calculation parameters: hx ¼ 0; c ¼ 2500; k ¼ 0:5 lm. Note: here a and b are the target dimensions.
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target configurations. In [17] some of the present paper authors have developed an approach for calculating the TR characteristics from an electron moving through an infinite strip. The analysis shows that the model presented in this paper fully coincides with the previously developed one (see, for instance, Fig. 2(b), dashed line). It is seen that, when one of the target sizes is finite, the angular distribution becomes broader and the TR intensity decreases. However it is also seen from the picture that when another target dimension is comparable with or smaller than the parameter 2ck, the angular distribution become more wider and the intensity gradually goes down.
1 Gðx; rÞ ¼ pffiffiffiffiffiffi exp r 2p
ð11Þ Here ry and rx are the vertical and horizontal beam sizes, respectively. Fig. 3(a) illustrates DR angular distribution for different vertical beam sizes at zero horizontal one. It is seen that the angular distribution form is deformed when the beam size changes. A similar kind of DR pattern behavior has been noticed in [12–14]. However, in reality an electron beam has two dimensional transversal distribution. In case of a rectangular hole horizontal and vertical beam size effects could be coupled. The question is how strong the coupling is. For calculations we have chosen beam conditions of KEK-ATF, where the horizontal beam size is about 10 times larger than the vertical one [18]. As a criteria for the beam size effect we have chosen the minimum-to-maximum ratio of the DR angular distribution. Fig. 3(b) shows the ratio dependence as a function of the
Minimum-to-maximum ratio
Intensity normalized by TR maximum
(a)
0.4
σy =0
0.2
0.0 0
1
3
2
γθy
4
5
ð10Þ
Gðax ; rx ÞGðby ; ry Þ dby dax :
One of the advantages of the new model is that it is possible to vary the particle trajectory with respect to the slit center in two directions. Actually an electron beam has a finite size and all particles in it have different trajectories. It gives an opportunity to study the beam size effect onto the DR angular distribution. Let us investigate the DR vertical polarization component SyDR ðax ; by Þ ¼ 4p2 k 2 jEyDR j2 only. At first we assume that the vertical and horizontal electron beam size has a Gaussian form
σy =0.04mm
x2 2 : 2r
Now we may perform two dimensional convolution of the DR vertical polarization component with Gaussians corresponding to vertical and horizontal electron beam distributions. Z 3rx Z 3ry 1 DR Sy ðrx ; ry Þ ¼ S DR ðax ; by Þ 2prx ry 3rx 3ry y
3. Diffraction radiation from a particle moving through a rectangular hole and the beam size effect
0.6
203
0.04
(b)
0.03
σx =0
0.02 σx =100µm
0.01
0.00 0
5
10
15
20
25
30
Electon beam size - σy (µm)
Fig. 3. (a) Diffraction radiation angular distribution calculated for two values of the vertical beam size, ry ; (b) the dependence of the angular distribution minimum-to-maximum ratio as a function of the vertical beam size for two values of the horizontal one, rx . The calculation parameters: hx ¼ 0, c ¼ 2500, a ¼ 0:6 mm, b ¼ 0:2 mm. Note: here a and b are the dimensions of the rectangular hole in an infinite screen.
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That is a powerful instrument which can be used for longitudinal bunch shape measurements in a noninvasive way [8–11]. To do that it is necessary to know a single electron DR or TR spectrum. However, some authors do not include the outer target dimensions in their data analysis. As we show in this section, the finite outer target dimensions may significantly distort the TR and DR spectra. DR fields induced by a particle moving through a rectangular hole in a screen of a rectangular shape can be derived using the BabinetÕs principle (see Fig. 4) and Eqs. (6) or (7).
vertical beam size. The solid line represents the dependence at zero horizontal beam size, the dashed line – at 100 lm horizontal beam size. The difference between them is smaller than 5%. The same kind of analysis can be performed for DR horizontal polarization component and estimate the horizontal beam size influence as well as distortion due to the finite vertical one; however, we assume that the effect will be the same. That means that splitting two polarization components it might be possible to measure vertical and horizontal beam sizes independently. In this paper we do not represent any considerations on the sensitivity for measurements of an electron beam size in two dimensions. However, we can definitely say that the sensitivity strongly depends on the target hole sizes and the observation wavelength. Therefore, in principle it is possible to adjust those two parameters and achieve the same sensitivity for both sizes. Moreover, DR is better sensitive to the beam size at shorter wavelengths. As we have shown in [19], the beam size as small as 10 lm could be measured using angular characteristics of DR in the optical wavelength range.
rec Ex;y ðk;aout ; ain ;bout ;bin ; hx ;hy Þ
¼ Ex;y ðk;aout ;bout ;hx ;hy Þ Ex;y ðk;ain ;bin ; hx ;hy Þ DR DR Ex;y ðk;ain ; bin ;hx ;hy Þ Ex;y ðk;aout ; bout ;hx ; hy Þ:
ð12Þ Here ainðoutÞ and binðoutÞ are the target dimensions. The in and out indexes indicate the hole sizes and outer dimensions of the target, respectively. To calculate the spectra it is necessary to take into account the detector angular acceptance. Therefore, first of all we shall analyze the DR angular distribution from a finite target. Fig. 5(a) represents the two dimensional DR angular distribution from a particle moving through a slit between two semi-planes. That is the distribution considered in [12–14] for beam diagnostics. In [20] the authors represented the bunch length measurements using autocorrelation of coherent DR. For calculating the DR spectrum from a single electron they used
4. Transition and diffraction radiation spectra from a particle moving through a finite size target During last few years TR and DR in mm and sub-mm wavelength range was of great interest.
aout
aout ain
bout bin
bin bout
ain
Fig. 4. Geometry representing the BabinetÕs principle.
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
205
Fig. 5. Two dimensional DR angular distribution from a particle moving through: (a) a slit between two semi-planes (bin ¼ 0:1 mm, bout ; aout ; ain ! 1), (b) a slit between two infinite strips (bin ¼ 0:1 mm, bout ¼ 0:4 mm, aout ; ain ! 1), (c) a slit between two finite strips (bin ¼ 0:1 mm, bout ¼ aout ¼ ain ¼ 0:4 mm) calculated for c ¼ 2500 and k ¼ 0:5 lm.
S DR ðk; aout ; ain ;bout ; bin Þ Z Dhx Z Dhy S DR ðk; aout ;ain ; bout ; bin ; hy ; hx Þdhx dhy : ¼4 0
0
ð13Þ
Here Dhx and Dhy are the detector angular acceptances. The integration could be performed numerically. Fig. 6 represents TR (a) and DR (b) spectral distribution as a function of the wave number for different target dimensions and normalized by TR from an infinite boundary. One can see that at large wave numbers TR spectra are saturated to unity, however DR is suppressed and, actually, keep reducing exponentially at shorter wavelengths. That is the effect of the hole existence. In the long wavelength region both TR and DR 1.0
1.0
(a)
Intensity normalized to TR inf
Intensity normalized to TR inf
the model of DR from a particle moving through a slit between two infinite strips (the target size is finite in one direction only). The angular distribution for such geometry is shown in Fig. 5(b). It is obvious that for the target size comparable with or smaller than the parameter 2ck the angular distribution is significantly deformed and the intensity is suppressed. However, if another target size is finite, the angular distribution becomes much wider and the intensity suppression is higher. The same kind of effect has been described in Section 2 for TR from a finite target. That means that the calculation must be performed for proper target configurations. Otherwise the evaluated bunch length could be wrong. The spectra from an electron moving through a rectangular target may be calculated using Eqs. (9) and (12).
60mm
0.8
40mm
0.6 20mm
0.4 0.2 0.0
(b)
60mm
0.8 40mm
0.6 20mm
0.4 0.2 0.0
1
10
100
Wavenumber (cm-1)
1000
1
10
100
1000
Wavenumber (cm-1)
Fig. 6. Transition (a) and diffraction (b) radiation spectra for different target dimensions. Here aout ¼ bout and their values are represented in the picture, for diffraction radiation ain ¼ 0:6 mm and bin ¼ 0:2 mm. Calculation parameters: Dhx ¼ Dhy ¼ 0:1 rad, c ¼ 2500.
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intensities are significantly suppressed. Furthermore, the form of the TR and DR spectra almost coincide in that region. That means that if the observation wavelength is comparable with or longer than the rectangular hole size the photon yield is mostly determined by the outer target dimensions. A similar effect has been considered in [21] for a circular target. Form this point of view it is
Intensity normalized to TR inf
1.0
0.8
0.6
0.4
0.2
0.0 0
100
200
300
400
500
-1
Wavenumber (cm )
1.0
Intensity normalized to TR inf
Intensity normalized to TRinf
Fig. 7. Transition radiation spectra for circular and rectangular targets with the same square. Here solid line – circular target with the radius of 20 mm [21], dashed line – square target with aout ¼ bout ¼ 35:4 mm dimensions, and dash-dotted line is a rectangular target with aout ¼ 100 mm and bout ¼ 12:6 mm. The calculation parameters: c ¼ 2500, Dhx ¼ Dhy ¼ 0:05 rad.
(a) 0.1rad
0.8
0.05rad
0.6
0.02rad
0.4
interesting to compare the TR spectra for circular and rectangular targets of the same square of the surface. The results of the comparison are represented in Fig. 7. Solid line in the picture represents the TR spectrum for a circular target calculated with the model [21]. The dashed line is TR spectrum for the square target. One can see that the spectra are very close. However, if the target has a rectangular non-symmetric shape (dash-dotted line in Fig. 7), the spectrum is very much different from a circular target in spite of the fact that the square of the surface is the same. Therefore, when the measurements of the longitudinal bunch size and profile is performed, in order to achieve a better consistency between experiment and theory, it is necessary to make calculations for a proper target configuration. As is shown in Fig. 5, when the target dimensions are comparable or smaller than 2ck the angular distribution become broader. Of course, one should say that the natural characteristics of any kind of effect are independent of a detector characteristics. However, the finite angular acceptance may significantly distort the measurement results. Fig. 8 represent TR (a) and DR (b) spectra calculated for different angular acceptances. It is seen that the form of the spectra are very different for different angular acceptances. Therefore, for comparison of the theoretical calculations with a real experiment the angular acceptance must be taken into account too.
0.01rad
0.2
1.0
(b) 0.1rad
0.8
0.05rad
0.6 0.02rad
0.4
0.01rad
0.2 0.0
0.0 1
10
100
1000
Wave number (cm-1)
1
10
100
1000
Wave number (cm-1)
Fig. 8. Transition (a) and diffraction (b) spectra calculated for different apertures of the detector. The outer target dimensions are aout ¼ bout ¼ 60 mm, the hole size for DR are ain ¼ 0:6 mm and bin ¼ 0:2 mm. Here Dhx ¼ Dhy and they are represented in the picture. The calculation parameters: c ¼ 2500.
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
5. Conclusion So far there are a few approaches allowing to calculate DR and TR spectra from a finite circular target [21,24]. However, in some experiments on bunch length measurements with coherent TR or DR it could be more convenient to use a rectangular shaped target, as it has been done in [9]. We have developed a new approach for calculating diffraction radiation spectral angular characteristics from a particle moving through a rectangular hole in a rectangular screen. This approach can be used for ultrarelativistic case and large target tilt angles. We have performed the analysis of the DR (TR) angular characteristics as a function of different parameters like position with respect to the target center and finite target dimensions. It has been noted that the angular distribution could be very distorted if the target dimensions are finite. The model is in very good agreement with the previously developed ones. There are two advantages for this model. At first it is possible to estimate the influence of the transversal beam parameters. Here we have dealt with the transversal beam dimensions. It has been noticed that it is possible to measure vertical and horizontal beam dimensions independently by measuring two polarization components. However, it is also possible to estimate the influence of such beam parameters as the angular divergence, angular and position jitter, but it was not the purpose of this paper. In [10,11] the authors performed the coherent DR investigation and evaluation of the longitudinal bunch size. However, performing the data analysis they did not take into account the finite target dimensions. The second advantage of our model is that it is possible to include the finite target size effect. We have calculated the DR spectra and compared them with the TR ones. It has been noted that the finite target causes a significant intensity suppression when the target size is comparable with or smaller than 2ck. Furthermore, if the observation wavelength is larger than the inner hole size, the DR photon yield is mostly determined by outer target dimensions. We have also noted that the finite detector angular accep-
207
tance must be taken into account when the theory is compared with an experiment. We performed some comparison of the TR spectra from a particle passing through a circular and rectangular target. It has been shown that the spectra could be very much different and the calculations always must be performed for a proper target configuration. Finally one should notice that if the detector is placed at the distance shorter than c2 k from the target, in this case so-called pre-wave zone effect appears. It may also lead to distortion of the DR (TR) spectral-angular characteristics. Therefore, a more advanced theory like [22,23] must be used in this case. References [1] R.B. Fiorito, D.W. Rule, in: R.E. Shaffer (Ed.), Beam Instrumentation Workshop, 1993: AIP Conf. Proc., Vol. 319, AIP, NY, 1994, p. 21. [2] M. Ross, S. Anderson, J. Frisch, in: G. Smith, T. Russo (Eds.), Beam Instrumentation Workshop, 2002, AIP Conf. Proc., Vol. 648, AIP, NY, 2003, p. 237. [3] J.B. Rosenzweig, A. Murokh, A. Tremaine, Eighth Advanced Accelerator Concepts Workshop, 1999, in: W. LowsonC. Bellamy, D. Brosius (Eds.), AIP Conference Proceedings, Vol. 472, AIP, NY, 1999, p. 38. [4] A.H. Lumpkin, B.X. Yang, W.J. Berg, J.W. Lewellen, N.S. Sereno, U. Happek, Nucl. Instr. and Meth. A 445 (2000) 356. [5] I. Sakai, Y. Yamamoto, T. Mitsuhashi, D. Amano, H. Iwasaki, Rev. Sci. Instr. 71 (2000) 1264. [6] H. Sakai, Y. Honda, N. Sasao, S. Araki, Y. Higashi, T. Okugi, T. Taniguchi, J. Urakawa, M. Takano, Phys. Rev. ST-AB 4 (2001) 022801. [7] M.L. Ter-Mikaelyan, High Energy Electromagnetic Processes in Condensed Media, Wiley Interscience, New York, 1972. [8] Y. Shibata et al., Phys. Rev. E 52 (1995) 6787. [9] M. Castellano, V.A. Verzilov, L. Catani, A. Cianchi, G. Orlandy, M. Geitz, Phys. Rev. E 63 (2001) 056501. [10] B. Feng, M. Oyamada, F. Hinode, S. Sato, Y. Kondo, Y. Shibata, M. Ikezawa, Nucl. Instr. and Meth. A 475 (2001) 492. [11] C. Settakorn, Ph.D. Thesis, Stanford University, USA, SLAC-Report-576, August 2001. [12] A.P. Potylitsyn, N.A. Potylitsyna, Rus. Phys. J. 43 (2000) 303. [13] R.B. Fiorito, D.W. Rule, Nucl. Instr. and Meth. B 173 (2001) 67. [14] M. Castellano, Nucl. Instr. and Meth. A 394 (1997) 275.
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[15] T. Muto, S. Araki, R. Hamatsu, H. Hayano, T. Hirose, P. Karataev, G. Naumenko, A. Potylitsyn, J. Urakawa, Phys. Rev. Let. 90 (2003) 1048011. [16] P. Karataev, S. Araki, R. Hamatsu, H. Hayano, T. Muto, G. Naumenko, A. Potylitsyn, N. Terunuma, J. Urakawa, Nucl. Instr. and Meth. B, this issue. doi:10.1016/j.nimb. 2004.03.012. Relativistic Electrons in Periodic Structures, 8–11 September 2003, Tomsk, Russia (these proceedings). [17] P. Karataev, G. Naumenko, A. Potylitsyn, Nucl. Instr. and Meth. B 201 (2003) 133. [18] H. Hayano, K. Kubo, M. Ross, N. Toge, J. Urakawa, F. Zimmermann, ATF: Accelerator Test Facility Study Report JFY 1996-1999, KEK Internal 2000-6 August 2000. [19] P. Karataev, R. Hamatsu, T. Muto, S. Araki, H. Hayano, J. Urakawa, G. Naumenko, A. Potylitsyn, T. Hirose, Joint
[20] [21] [22] [23] [24]
28th ICFA Advanced Beam Dynamics and Advanced & Novel Accelerators Workshop on QUANTUM ASPECTS OF BEAM PHYSICS, 7–11 January 2003, Hiroshima University, Higashi-Hiroshima, Japan (to be published with World Scientific). A.H. Lumpkin, N.S. Sereno, D.W. Rule, Nucl. Instr. and Meth. A 475 (2001) 470. M. Castellano, A. Cianchi, G. Orlandi, V.A. Verzilov, Nucl. Instr. and Meth. A 435 (1999) 297. S.N. Dobrovolsky, N.F. ShulÕga, Nucl. Instr. and Meth. B 201 (2003) 123. M. Castellano, V.A. Verzilov, Phys. Rev. ST-AB 1 (1998) 062801. A. Potylitsyn, Nucl. Instr. and Meth. B 145 (1998) 169.
Diffraction radiation from a charged particle moving through a rectangular hole in a rectangular screen P. Karataev a,*, S. Araki b, R. Hamatsu a, H. Hayano b, T. Muto b, G. Naumenko c, A. Potylitsyn c, N. Terunuma b, J. Urakawa b a
Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo 192-0937, Japan b KEK, Accelerator Test Facility, Oho 1-1, Tsukuba, 305-0801, Ibaraki-ken, Japan c Tomsk Polytechnic University, 634050, pr. Lenina 2a, Tomsk, Russia Received 17 December 2003; received in revised form 2 March 2004
Abstract We developed a new model for calculating diffraction radiation (DR) from an electron moving through a rectangular hole in a rectangular screen. The calculations show that short wavelength DR is very sensitive to the beam size. For example, optical DR (ODR) could be used to measure the beam size as small as 10 lm. Moreover, splitting two polarization components it becomes possible to measure vertical and horizontal beam sizes independently. We have calculated the DR spectra and compared them with TR ones for a finite size target. It is shown that when the DR wavelength is comparable with or longer than the hole size, the photon yield is mostly determined by the outer target dimensions. It means that in case transversal beam dimensions smaller than the observation wavelength the coherent DR could be used for non-invasive bunch length measurements with the same accuracy as the coherent TR techniques. However, the outer target dimensions must be taken into account because the finite target size causes a significant intensity suppression in the long wavelength spectral range as well as distortion of the coherent spectrum. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.75.Ht; 41.85.Qg Keywords: Transition radiation; Diffraction radiation; Electron beam diagnostics
1. Introduction During last several years different techniques for electron beam diagnostics have been intensively developed. For accelerators of the next generation like linear collider or short wavelength
*
Corresponding author. Tel.: +81-298-64-5715; fax: +81298-64-5746. E-mail address: [email protected] (P. Karataev).
free electron lasers precise measurements of such beam parameters as size, emittance, bunch length, which, as a rule, much smaller than ones for usual accelerators, is extremely necessary. Up-to-date transition radiation (TR) monitors have widely been used for beam diagnostics. TR appears when a charged particle crosses a boundary between two media with different dielectric properties. Due to high signal-to-noise ratio TR in an optical wavelength range gives a good possibility to monitor beam profile and emittance for
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.014
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
single shot [1,2]. Coherent TR (the observation wavelength is comparable with or longer than the longitudinal bunch length) in millimeter and submillimeter wavelength range makes it possible to measure bunch length [3,4]. However, to produce TR at the target surface the particles interact with the target material experiencing multiple scattering. It may lead to worsening of the beam emittance. Moreover, due to ionization process the target can be heated up, deformed or even destroyed under an intense focused electron beam. That means that it is extremely necessary to develop a non-invasive method for beam diagnostics. A few non-invasive techniques are under intense study now. For example, synchrotron radiation monitor (interferometer) is very promising for beam diagnostics at circular accelerators and rings [5]. For a linear accelerators a laser wire beam size monitor can be used [6]. However, there is no way to measure the beam size for a single shot with laser wire. Furthermore, it is necessary to collect statistics and all points are measured from different bunches. In this case other beam parameters like beam position and angular jitter, orbit drift, etc. may cause an additional error and complicate the data analysis. We assume that diffraction radiation (DR) effect can be used for non-invasive electron beam diagnostics. DR is generated when a charged particle moves in vacuum close to a conductive medium and interacts with the matter by its electric field only [7]. TR and DR have a relative nature because they both appear as result of dynamic polarization of a medium. DR spectral range is very broad and its characteristic wavelength, k, is defined by Lorentz-factor, c, and the shortest distance between the target edge and particle trajectory, h, as k h=ð2pcÞ. In spite of the fact that the first theoretical considerations on DR effect have appeared over 40 years ago the first observation of coherent DR in mm and sub-mm wavelength range has been performed in 1995 [8]. Recently a great interest in coherent DR phenomenon has appeared because of the possibility of measuring the bunch length in a non-invasive way [9–11]. Nowadays there are a lot of theoretical approaches describing possible ways for transversal
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beam parameter determination using short wavelength DR [12–14]. However, in contrast to TR DR effect is not very well investigated experimentally. The first observation of incoherent diffraction radiation in optical wavelength range (ODR) from the edge of a semi-plane target we performed last year [15]. In [16] we represent the results of ODR observation from a slit target. The experimental results were in reasonable agreement with the model of ideally conducting (reflecting) infinitely thin target. Now we are working on our equipment to be able to measure a few micron beam size with a proper accuracy. And we are also thinking about the future prospects. We have developed a new model for calculating the DR spectral-angular characteristics from a particle moving through a rectangular hole in a rectangular screen. Our calculations show that short wavelength DR is very sensitive to the transversal beam size. For example, DR in optical wavelength range (ODR) could be used to measure the beam size as small as 10 lm. Moreover, splitting vertical and horizontal polarization components it becomes possible to measure vertical and horizontal beam sizes independently. To be able to evaluate the bunch length from coherent DR spectrum, it is necessary to have a calculated spectrum for a single electron. In [10,11] the authors investigated coherent DR spectra and determined the bunch length but they did not take into account the finite outer target dimensions. One of the great advantages of our model is that it is possible to calculate DR (TR) spectra from a finite size target. In this paper we show the calculated DR spectra and comparison with TR ones. It is shown that in the soft part of the spectrum the photon yield is significantly suppressed. Without including the finite size target effect the information about longitudinal beam size could be wrong.
2. Transition radiation from an electron moving through a rectangular screen Transition radiation (TR) appears when a charged particle crosses a boundary between two media with different dielectric constants. Of course, to derive an exact expression for TR it is
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necessary to solve MaxwellÕs equations. However, in this paper we intend to use a simpler approach for TR and DR based on the scattering of the electron field pseudo-photons from the target surface [7,13]. This approach is based on the wellknown Huygens principle of plane wave diffraction. The difference from the plane wave is that the electron field depends on the distance from the particle trajectory. According to the Huygens principle when the charged particle interacts with the target with its electric field, all points of the target surface are considered as elementary sources. The total field of the radiation for two polarization components from an arbitrarily shaped target can be derived, by integrating over the target surface in the following manner: Ex;y ðkx ; ky Þ ¼ Rx;y ðk; h0 Þ Z Z 1 i 2 Ex;y ðx; yÞeiðkx xþky yÞ dx dy: 4p
The components of the emitted photon wave vector in ultrarelativistic case are determined by the following relations (with accuracy of c2 ): 1 k sin hx cos hy khx ; c1 1 ky ¼ k cos hx sin hy khy ; c1 1 kz ¼ k cos hx cos hy k: c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ¼
ð2Þ
Here c1 ¼ sin2 hx cos2 hy þ cos2 hx is a normalization constant. hx and hy are the angles measured from the direction of specular reflection (see Fig. 1). In an ultrarelativistic case hx ; hy 1. For simplicity we shall assume that the target is ideally reflecting (Rx ¼ Ry ¼ 1). However, as is shown in [16], to achieve a better consistency between experiment and theory the reflection coefficients must be included for some materials. The components of the incident particle field can be expressed by its Fourier transform [7]:
ð1Þ i ðx; yÞ are the components of the incident Here Ex;y charged particle field, k ¼ 2p=k is the wave number, h0 is the target tilt angle, kx and ky are the components of the emitted photon wave vector, x and y indexes of E represent transversal horizontal and vertical polarization components (with accuracy of order of c1 ), which are proportional to the Fresnel reflection coefficients Rx and Ry , respectively [13]. One should notice that there is the longitudinal component, Ez , however, for an ultrarelativistic case it is negligibly small in comparison with the transversal ones and could be omitted. Throughout the paper the system of units h ¼ me ¼ c ¼ 1 is used. In [13] the authors considered the effect of the tilted target. It has been shown that for large target tilt angles and ultrarelativistic case in Eq. (1) kx kx ðkc2 =2Þ cot h0 , which means that the general angular distribution is shifted by the angle of c2 =2 with respect to the direction of specular reflection. Since we are talking about the electron energies higher than 100 MeV (c ¼ 200), and the target tilt angle h0 ¼ 45 deg that term is negligible. Therefore, in the future we shall assume kx kx .
y a/
si n
a2
0
a1
θ0
e -b1
y
b
x
θy b2
θx
x
z
Fig. 1. Geometry of the transition radiation from a particle passing through a rectangular screen. Here z axis is directed along the direction of specular reflection, x ¼ x sin h0 , y ¼ y, dx dy ¼ cos h0 dx dy.
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
i Ex;y ðx; yÞ
ie ¼ 2 2p
Z Z
0
0
0 kx;y eiðkx xþky yÞ dk 0 dk 0 : kx02 þ ky02 þ k 2 c2 x y
ð3Þ
Here e is the electron charge, c is the charged particle Lorentz-factor, kx0 and ky0 are the components of the electron field pseudo-photon wave vector. In principle it is possible to solve the integral in Eq. (1) for any kind of target surface numerically. However, it is always useful to have a more simplified solution for a particular case. For example, substituting Eq. (2) in (1) and integrating it over x and y from 1 to 1 we obtain two delta functions: 2pidðkx0 kx Þ and 2pidðky0 ky Þ. After that the kx0 and ky0 integration can be done trivially. The result is the expression for transition radiation (TR) fields from an infinite boundary: TR Ex;y ¼
ie kx;y : 2 2 2p kx þ ky2 þ k 2 c2
ð4Þ
Now let us derive an expression for TR fields from a particle crossing a rectangular screen. Here we assume that the target is tilted in one direction only, as it is shown in the calculation geometry represented in Fig. 1. The first step to derive the expression, we are looking for, is to integrate the Eq. (1) over x from a1 sin h0 ¼ a=2 ax to a2 sin h0 ¼ a=2 þ ax and over y from b1 ¼ b=2 by to b2 ¼ b=2 þ by : Z Z ie dkx0 dky0 Ex;y ðkx ; ky Þ ¼ 4 8p 0 kx;y ðkx02 þ ky02 þ k 2 c2 Þðkx0 kx Þðky0 ky Þ h 0 i 0 eiðkx kx Þa2 sin h0 eiðkx kx Þa1 sin h0 h 0 i 0 eiðky ky Þb2 eiðky ky Þb1 : ð5Þ
201
integral hasffi three poles: ky0 ¼ ky , ky0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 2 kx þ k c2 . The integration should be performed over positive and negative imaginary halfplanes separately. The integration contours and the detailed solution of the integral can be found in [13]. Using Eq. (2) The final expression for TR from a rectangular target can be represented in the following form: ie hx 2p2 k h2x þ h2y þ c2 8 Z ie < 1 t sin ðak=2ðt hx ÞÞ ikax ðthx Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 3 dt 4p k : 1 ðt hx Þ t2 þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp kðb=2 by Þ t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 = exp kðb=2 þ by Þ t2 þ c2 þ ihy 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 2 ; t þ c þ ihy 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2y þ c2 ihx e 6 exp kða=2 ax Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 4p k h2 þ c2 ih
Ex ðhx ; hy Þ ¼
y
x
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 exp kða=2 þ ax Þ h2y þ c2 þ ihx 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; h2y þ c2 þ ihx
ie hy 2p2 k h2x þ h2y þ c2 8 Z e < 1 sin ðak=2ðt hx ÞÞ ikax ðthx Þ 3 dt e 4p k : 1 ðt hx Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 exp kðb=2 by Þ t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 t2 þ c2 ihy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 = exp kðb=2 þ by Þ t2 þ c2 þ ihy 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t2 þ c2 þ ihy
Ey ðhx ; hy Þ ¼
ie hy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p2 k h2 þ c2 y 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2y þ c2 ihx 6 exp kða=2 ax Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h2y þ c2 ihx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 exp kða=2 þ ax Þ h2y þ c2 þ ihx 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 5: h2y þ c2 þ ihx
Here b and a are the vertical and effective horizontal target dimensions respectively, by and ax are the vertical and effective horizontal offsets with respect to the target center. The real horizontal target size and offset can be found as a= sin h0 and ax = sin h0 , respectively (see, for instance, Fig. 1). Eq. (5) can be integrated over ky0 and partially over kx0 . Let us briefly consider the ky0 integral. It can be done using residue theorem. The equation under
ð6Þ
P. Karataev et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 198–208
geometries has been considered in [13]. It means that the model presented above is valid. In spite of the infinite integration limits in Eq. (6) the integration can be done numerically in a very narrow interval Dt 1, where the dominant part of the under integral function is confined. The spectral angular distribution can be obtained from the following expression:
First of all we need to examine Eq. (6). That is apparent that the first term in this expression is the TR field from an infinite boundary represented by Eq. (4). Actually, the rest of the equation is the diffraction radiation field from a particle moving through a rectangular hole in a screen of infinite dimensions taken with an opposite sign. As a matter of fact the above expression introduces a well-known BabinetÕs principle from classical optics. Therefore, for convenience we shall represent the Eq. (6) in the form TR DR Ex;y ðhx ; hy Þ ¼ Ex;y ðhx ; hy Þ Ex;y ðhx ; hy Þ:
d2 W dx dX h i 2 2 ¼ 4p2 k 2 jEx j þ jEy j :
Sðk; hx ; hy ; a; ax ; b; by Þ ¼
ð7Þ
lim
a!1
sin ½ak=2ðt hx Þ ¼ pdðt hx Þ; t hx
ð8Þ
Intensity normalized by TR maximum
the integral part is also transformed into the expression for DR field from a particle passing through a slit between two semi-planes, but the projection of the particle trajectory onto the target plane, in this case, is parallel to the slit edges. Both 1.0
(a) 0.8 0.6 0.4 0.2 0.0 0
2
4
γθy
6
ð9Þ
We have developed a computer code to calculate TR and DR spectral angular characteristics from a particle moving through a target of a rectangular shape. Fig. 2(a) illustrates the TR angular distributions calculated at different incident points of the particle with respect to the target center. It is seen that when the particle crosses the target center, which vertical and horizontal sizes are larger than 2ck, the angular distribution coincides with the TR from an infinite boundary. However, when the particle trajectory closes up to the target edge the minimum in angular distribution disappear and it transforms into DR from a particle moving straight through the target edge, which is single mode in contrast to TR. This effect we have observed experimentally in [15]. Fig. 2(b) shows the TR angular distributions for different
It might be interesting to notice that when in the DR expression for diffraction radiation, Ex;y , b ! 1, the integral part of the equation vanishes and the equation is transformed in to the DR field from a particle moving through a slit between two semiplanes and its projection onto the target plane is perpendicular to the target edges. However, if the a ! 1 the last term in the DR field tends to zero. Keeping in mind that
8
Intensity normalized by TR maximum
202
1.0
(b) 0.8 0.6 0.4 0.2 0.0 0
2
4
γθy
6
8
Fig. 2. (a) Transition radiation angular distributions for different offsets with respect to the target center for the target dimensions a ¼ b ¼ 10 mm: by ¼ 0 – solid line, by ¼ b=2 0:15 mm – dashed line, by ¼ b=2 – dashed-dotted line; (b) Transition radiation angular distribution for different target configurations: a ¼ b ¼ 10 mm – solid line; a ! 1, b ¼ 0:4 mm – dashed line; a ¼ 0:3 mm, b ¼ 0:4 mm – dashed-dotted line; a ¼ 0:1 mm, b ¼ 0:4 mm – dotted line. The calculation parameters: hx ¼ 0; c ¼ 2500; k ¼ 0:5 lm. Note: here a and b are the target dimensions.
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target configurations. In [17] some of the present paper authors have developed an approach for calculating the TR characteristics from an electron moving through an infinite strip. The analysis shows that the model presented in this paper fully coincides with the previously developed one (see, for instance, Fig. 2(b), dashed line). It is seen that, when one of the target sizes is finite, the angular distribution becomes broader and the TR intensity decreases. However it is also seen from the picture that when another target dimension is comparable with or smaller than the parameter 2ck, the angular distribution become more wider and the intensity gradually goes down.
1 Gðx; rÞ ¼ pffiffiffiffiffiffi exp r 2p
ð11Þ Here ry and rx are the vertical and horizontal beam sizes, respectively. Fig. 3(a) illustrates DR angular distribution for different vertical beam sizes at zero horizontal one. It is seen that the angular distribution form is deformed when the beam size changes. A similar kind of DR pattern behavior has been noticed in [12–14]. However, in reality an electron beam has two dimensional transversal distribution. In case of a rectangular hole horizontal and vertical beam size effects could be coupled. The question is how strong the coupling is. For calculations we have chosen beam conditions of KEK-ATF, where the horizontal beam size is about 10 times larger than the vertical one [18]. As a criteria for the beam size effect we have chosen the minimum-to-maximum ratio of the DR angular distribution. Fig. 3(b) shows the ratio dependence as a function of the
Minimum-to-maximum ratio
Intensity normalized by TR maximum
(a)
0.4
σy =0
0.2
0.0 0
1
3
2
γθy
4
5
ð10Þ
Gðax ; rx ÞGðby ; ry Þ dby dax :
One of the advantages of the new model is that it is possible to vary the particle trajectory with respect to the slit center in two directions. Actually an electron beam has a finite size and all particles in it have different trajectories. It gives an opportunity to study the beam size effect onto the DR angular distribution. Let us investigate the DR vertical polarization component SyDR ðax ; by Þ ¼ 4p2 k 2 jEyDR j2 only. At first we assume that the vertical and horizontal electron beam size has a Gaussian form
σy =0.04mm
x2 2 : 2r
Now we may perform two dimensional convolution of the DR vertical polarization component with Gaussians corresponding to vertical and horizontal electron beam distributions. Z 3rx Z 3ry 1 DR Sy ðrx ; ry Þ ¼ S DR ðax ; by Þ 2prx ry 3rx 3ry y
3. Diffraction radiation from a particle moving through a rectangular hole and the beam size effect
0.6
203
0.04
(b)
0.03
σx =0
0.02 σx =100µm
0.01
0.00 0
5
10
15
20
25
30
Electon beam size - σy (µm)
Fig. 3. (a) Diffraction radiation angular distribution calculated for two values of the vertical beam size, ry ; (b) the dependence of the angular distribution minimum-to-maximum ratio as a function of the vertical beam size for two values of the horizontal one, rx . The calculation parameters: hx ¼ 0, c ¼ 2500, a ¼ 0:6 mm, b ¼ 0:2 mm. Note: here a and b are the dimensions of the rectangular hole in an infinite screen.
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That is a powerful instrument which can be used for longitudinal bunch shape measurements in a noninvasive way [8–11]. To do that it is necessary to know a single electron DR or TR spectrum. However, some authors do not include the outer target dimensions in their data analysis. As we show in this section, the finite outer target dimensions may significantly distort the TR and DR spectra. DR fields induced by a particle moving through a rectangular hole in a screen of a rectangular shape can be derived using the BabinetÕs principle (see Fig. 4) and Eqs. (6) or (7).
vertical beam size. The solid line represents the dependence at zero horizontal beam size, the dashed line – at 100 lm horizontal beam size. The difference between them is smaller than 5%. The same kind of analysis can be performed for DR horizontal polarization component and estimate the horizontal beam size influence as well as distortion due to the finite vertical one; however, we assume that the effect will be the same. That means that splitting two polarization components it might be possible to measure vertical and horizontal beam sizes independently. In this paper we do not represent any considerations on the sensitivity for measurements of an electron beam size in two dimensions. However, we can definitely say that the sensitivity strongly depends on the target hole sizes and the observation wavelength. Therefore, in principle it is possible to adjust those two parameters and achieve the same sensitivity for both sizes. Moreover, DR is better sensitive to the beam size at shorter wavelengths. As we have shown in [19], the beam size as small as 10 lm could be measured using angular characteristics of DR in the optical wavelength range.
rec Ex;y ðk;aout ; ain ;bout ;bin ; hx ;hy Þ
¼ Ex;y ðk;aout ;bout ;hx ;hy Þ Ex;y ðk;ain ;bin ; hx ;hy Þ DR DR Ex;y ðk;ain ; bin ;hx ;hy Þ Ex;y ðk;aout ; bout ;hx ; hy Þ:
ð12Þ Here ainðoutÞ and binðoutÞ are the target dimensions. The in and out indexes indicate the hole sizes and outer dimensions of the target, respectively. To calculate the spectra it is necessary to take into account the detector angular acceptance. Therefore, first of all we shall analyze the DR angular distribution from a finite target. Fig. 5(a) represents the two dimensional DR angular distribution from a particle moving through a slit between two semi-planes. That is the distribution considered in [12–14] for beam diagnostics. In [20] the authors represented the bunch length measurements using autocorrelation of coherent DR. For calculating the DR spectrum from a single electron they used
4. Transition and diffraction radiation spectra from a particle moving through a finite size target During last few years TR and DR in mm and sub-mm wavelength range was of great interest.
aout
aout ain
bout bin
bin bout
ain
Fig. 4. Geometry representing the BabinetÕs principle.
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Fig. 5. Two dimensional DR angular distribution from a particle moving through: (a) a slit between two semi-planes (bin ¼ 0:1 mm, bout ; aout ; ain ! 1), (b) a slit between two infinite strips (bin ¼ 0:1 mm, bout ¼ 0:4 mm, aout ; ain ! 1), (c) a slit between two finite strips (bin ¼ 0:1 mm, bout ¼ aout ¼ ain ¼ 0:4 mm) calculated for c ¼ 2500 and k ¼ 0:5 lm.
S DR ðk; aout ; ain ;bout ; bin Þ Z Dhx Z Dhy S DR ðk; aout ;ain ; bout ; bin ; hy ; hx Þdhx dhy : ¼4 0
0
ð13Þ
Here Dhx and Dhy are the detector angular acceptances. The integration could be performed numerically. Fig. 6 represents TR (a) and DR (b) spectral distribution as a function of the wave number for different target dimensions and normalized by TR from an infinite boundary. One can see that at large wave numbers TR spectra are saturated to unity, however DR is suppressed and, actually, keep reducing exponentially at shorter wavelengths. That is the effect of the hole existence. In the long wavelength region both TR and DR 1.0
1.0
(a)
Intensity normalized to TR inf
Intensity normalized to TR inf
the model of DR from a particle moving through a slit between two infinite strips (the target size is finite in one direction only). The angular distribution for such geometry is shown in Fig. 5(b). It is obvious that for the target size comparable with or smaller than the parameter 2ck the angular distribution is significantly deformed and the intensity is suppressed. However, if another target size is finite, the angular distribution becomes much wider and the intensity suppression is higher. The same kind of effect has been described in Section 2 for TR from a finite target. That means that the calculation must be performed for proper target configurations. Otherwise the evaluated bunch length could be wrong. The spectra from an electron moving through a rectangular target may be calculated using Eqs. (9) and (12).
60mm
0.8
40mm
0.6 20mm
0.4 0.2 0.0
(b)
60mm
0.8 40mm
0.6 20mm
0.4 0.2 0.0
1
10
100
Wavenumber (cm-1)
1000
1
10
100
1000
Wavenumber (cm-1)
Fig. 6. Transition (a) and diffraction (b) radiation spectra for different target dimensions. Here aout ¼ bout and their values are represented in the picture, for diffraction radiation ain ¼ 0:6 mm and bin ¼ 0:2 mm. Calculation parameters: Dhx ¼ Dhy ¼ 0:1 rad, c ¼ 2500.
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intensities are significantly suppressed. Furthermore, the form of the TR and DR spectra almost coincide in that region. That means that if the observation wavelength is comparable with or longer than the rectangular hole size the photon yield is mostly determined by the outer target dimensions. A similar effect has been considered in [21] for a circular target. Form this point of view it is
Intensity normalized to TR inf
1.0
0.8
0.6
0.4
0.2
0.0 0
100
200
300
400
500
-1
Wavenumber (cm )
1.0
Intensity normalized to TR inf
Intensity normalized to TRinf
Fig. 7. Transition radiation spectra for circular and rectangular targets with the same square. Here solid line – circular target with the radius of 20 mm [21], dashed line – square target with aout ¼ bout ¼ 35:4 mm dimensions, and dash-dotted line is a rectangular target with aout ¼ 100 mm and bout ¼ 12:6 mm. The calculation parameters: c ¼ 2500, Dhx ¼ Dhy ¼ 0:05 rad.
(a) 0.1rad
0.8
0.05rad
0.6
0.02rad
0.4
interesting to compare the TR spectra for circular and rectangular targets of the same square of the surface. The results of the comparison are represented in Fig. 7. Solid line in the picture represents the TR spectrum for a circular target calculated with the model [21]. The dashed line is TR spectrum for the square target. One can see that the spectra are very close. However, if the target has a rectangular non-symmetric shape (dash-dotted line in Fig. 7), the spectrum is very much different from a circular target in spite of the fact that the square of the surface is the same. Therefore, when the measurements of the longitudinal bunch size and profile is performed, in order to achieve a better consistency between experiment and theory, it is necessary to make calculations for a proper target configuration. As is shown in Fig. 5, when the target dimensions are comparable or smaller than 2ck the angular distribution become broader. Of course, one should say that the natural characteristics of any kind of effect are independent of a detector characteristics. However, the finite angular acceptance may significantly distort the measurement results. Fig. 8 represent TR (a) and DR (b) spectra calculated for different angular acceptances. It is seen that the form of the spectra are very different for different angular acceptances. Therefore, for comparison of the theoretical calculations with a real experiment the angular acceptance must be taken into account too.
0.01rad
0.2
1.0
(b) 0.1rad
0.8
0.05rad
0.6 0.02rad
0.4
0.01rad
0.2 0.0
0.0 1
10
100
1000
Wave number (cm-1)
1
10
100
1000
Wave number (cm-1)
Fig. 8. Transition (a) and diffraction (b) spectra calculated for different apertures of the detector. The outer target dimensions are aout ¼ bout ¼ 60 mm, the hole size for DR are ain ¼ 0:6 mm and bin ¼ 0:2 mm. Here Dhx ¼ Dhy and they are represented in the picture. The calculation parameters: c ¼ 2500.
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5. Conclusion So far there are a few approaches allowing to calculate DR and TR spectra from a finite circular target [21,24]. However, in some experiments on bunch length measurements with coherent TR or DR it could be more convenient to use a rectangular shaped target, as it has been done in [9]. We have developed a new approach for calculating diffraction radiation spectral angular characteristics from a particle moving through a rectangular hole in a rectangular screen. This approach can be used for ultrarelativistic case and large target tilt angles. We have performed the analysis of the DR (TR) angular characteristics as a function of different parameters like position with respect to the target center and finite target dimensions. It has been noted that the angular distribution could be very distorted if the target dimensions are finite. The model is in very good agreement with the previously developed ones. There are two advantages for this model. At first it is possible to estimate the influence of the transversal beam parameters. Here we have dealt with the transversal beam dimensions. It has been noticed that it is possible to measure vertical and horizontal beam dimensions independently by measuring two polarization components. However, it is also possible to estimate the influence of such beam parameters as the angular divergence, angular and position jitter, but it was not the purpose of this paper. In [10,11] the authors performed the coherent DR investigation and evaluation of the longitudinal bunch size. However, performing the data analysis they did not take into account the finite target dimensions. The second advantage of our model is that it is possible to include the finite target size effect. We have calculated the DR spectra and compared them with the TR ones. It has been noted that the finite target causes a significant intensity suppression when the target size is comparable with or smaller than 2ck. Furthermore, if the observation wavelength is larger than the inner hole size, the DR photon yield is mostly determined by outer target dimensions. We have also noted that the finite detector angular accep-
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tance must be taken into account when the theory is compared with an experiment. We performed some comparison of the TR spectra from a particle passing through a circular and rectangular target. It has been shown that the spectra could be very much different and the calculations always must be performed for a proper target configuration. Finally one should notice that if the detector is placed at the distance shorter than c2 k from the target, in this case so-called pre-wave zone effect appears. It may also lead to distortion of the DR (TR) spectral-angular characteristics. Therefore, a more advanced theory like [22,23] must be used in this case. References [1] R.B. Fiorito, D.W. Rule, in: R.E. Shaffer (Ed.), Beam Instrumentation Workshop, 1993: AIP Conf. Proc., Vol. 319, AIP, NY, 1994, p. 21. [2] M. Ross, S. Anderson, J. Frisch, in: G. Smith, T. Russo (Eds.), Beam Instrumentation Workshop, 2002, AIP Conf. Proc., Vol. 648, AIP, NY, 2003, p. 237. [3] J.B. Rosenzweig, A. Murokh, A. Tremaine, Eighth Advanced Accelerator Concepts Workshop, 1999, in: W. LowsonC. Bellamy, D. Brosius (Eds.), AIP Conference Proceedings, Vol. 472, AIP, NY, 1999, p. 38. [4] A.H. Lumpkin, B.X. Yang, W.J. Berg, J.W. Lewellen, N.S. Sereno, U. Happek, Nucl. Instr. and Meth. A 445 (2000) 356. [5] I. Sakai, Y. Yamamoto, T. Mitsuhashi, D. Amano, H. Iwasaki, Rev. Sci. Instr. 71 (2000) 1264. [6] H. Sakai, Y. Honda, N. Sasao, S. Araki, Y. Higashi, T. Okugi, T. Taniguchi, J. Urakawa, M. Takano, Phys. Rev. ST-AB 4 (2001) 022801. [7] M.L. Ter-Mikaelyan, High Energy Electromagnetic Processes in Condensed Media, Wiley Interscience, New York, 1972. [8] Y. Shibata et al., Phys. Rev. E 52 (1995) 6787. [9] M. Castellano, V.A. Verzilov, L. Catani, A. Cianchi, G. Orlandy, M. Geitz, Phys. Rev. E 63 (2001) 056501. [10] B. Feng, M. Oyamada, F. Hinode, S. Sato, Y. Kondo, Y. Shibata, M. Ikezawa, Nucl. Instr. and Meth. A 475 (2001) 492. [11] C. Settakorn, Ph.D. Thesis, Stanford University, USA, SLAC-Report-576, August 2001. [12] A.P. Potylitsyn, N.A. Potylitsyna, Rus. Phys. J. 43 (2000) 303. [13] R.B. Fiorito, D.W. Rule, Nucl. Instr. and Meth. B 173 (2001) 67. [14] M. Castellano, Nucl. Instr. and Meth. A 394 (1997) 275.
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