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Coherent radiation recoil effect for the optical diffraction radiation beam size monitor at SLAC FFTB
Nuclear Instruments and Methods in Physics Research B 227 (2005) 170–174 www.elsevier.com/locate/nimb
Coherent radiation recoil effect for the optical diffraction radiation beam size monitor at SLAC FFTB A. Potylitsyn a,*, G. Naumenko a, A. Aryshev a, Y. Fukui b, D. Cline b, F. Zhou b, M. Ross c, P. Bolton c, J. Urakawa d, T. Muto d, M. Tobiyama d, R. Hamatsu e, P. Karataev e a
d
Nuclear Physics Institute Tomsk Polytechnic University, Lenin Avenue 30, Tomsk 634034, Russia b University of California at Los Angeles, California, 90095-1547, USA c Stanford Linear Accelerator Center, Stanford University, California, 94505, USA KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 300-0801, Japan e Tokyo Metropolitan University, 1-1 Minamiohzawa, Hachioji, Tokyo 192-0397, Japan Received 17 December 2003; received in revised form 8 June 2004
Abstract A short electron bunch with length r passing through a slit in the diffraction radiation (DR) target generates radiation with a broad spectrum. Optical part of the spectrum (incoherent radiation) may be used for beam size measurements, but in the wavelength range k P r radiation becomes coherent. The coherent DR spectrum per each electron in a bunch is equal to single electron spectrum times by the number of electrons in a bunch Ne and bunch form factor. For SLAC FFTB conditions (Ne 1010, r = 0.7 mm, outer target size R 10 mm, slit width h 0.1 mm) we approximated coherent DR (CDR) spectrum by coherent transition radiation (TR) one because in the wavelength region k r h TR and DR spectra coincide with high accuracy. Changing the DR target by a TR target with projection on the plane perpendicular to electron beam as a circle with radius R 6 20 mm we calculated CDR spectra using simple model. Knowing the CDR spectrum we estimated the energy CDR emitting by each electron in the perpendicular direction (due to target inclination angle 45). It means an electron receives the radiation recoil in this direction. In other words, electron has a transverse kick about 1 lrad that may be considered as permissible. Published by Elsevier B.V. PACS: 41.60.m Keywords: Diffraction radiation; Coherent radiation; Electron beams; Diagnostics
*
Corresponding author. Tel.: +7 3822 41 89 06. E-mail address: [email protected] (A. Potylitsyn).
0168-583X/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.nimb.2004.06.016
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
171
Optical transition radiation is recently widely used for diagnostics of the beams [1]. However, the direct interaction of beam particles with the target material may cause its transformation and distortion of optical characteristics, and for intense focused beams even destruction of the target [2]. When the beam of ultrarelativistic electrons passes in a vacuum near the conductive target the socalled diffraction radiation (DR) is generated, which features allow using it for non-invasive beam diagnostics [3,4]. In [5] we proposed to use the optical DR to create beam size monitor at SLAC Final Focus Test Beam, where the intensity of electron beam is (1–3) · 1010 particles/ bunch with FWHM bunch length equal to 0.7 mm. In this case there is a need to estimate the possible distortions of the beam characteristics due to wakefield deflection. This effect is connected with two kinds of short bunch interaction with conductive target:
angle w = 45 DR is emitted at right angle to the electron momentum (backward diffraction radiation, BDR). Real photons of BDR, having a certain momentum in the specular reflection direction k ? ¼ W cDR (WDR the total BDR energy losses, c-speed of light), transfer momentum to electrons jD~ q? j ¼ k ? (recoil effect). Just this reason leads to beam deflection. The estimation of an electron deflection angle based on the described approach [8] agrees with result obtained from the geometric wakefield model [6]. In order to estimate the radiation losses in the BDR cone we will use the following assumptions:
• geometric wakefield [6]; • resistive wakefield [7].
Authors of [9] pointed out that DR energy losses for a particle moving near a tilted semi-infinite perfect screen are defined by its projection on the plane perpendicular to the particle trajectory. For an ultrarelativistic case DR energy losses were calculated in [10] where no dependence on a tilted angle W was obtained if condition W c1 is fulfilled. One may assume that for a slit DR target there is the same situation. The simplest model of DR from ultrarelativistic particle for the perpendicular target with the finite square (DR for a disk with the hole) is described in [11]. Spectral-angular distribution of DR in such target is described with the following formula:
DR target is made of the highly conductive material that is why the geometric wakefield determines the major impact. The consideration given in [8] shows that the geometric wakefield effect may be interpreted as a recoil effect due to coherent DR. This is shown in Fig. 1. During the passing of the electron through the slit in the inclined target the intrinsic Coulomb field of the particle is reflected from the surface of the target and is transformed into the real photons of DR, which emitted in the cone with an apex angle c1 along the direction of the mirror reflection. For the target inclination
Fig. 1. Generation of forward (FDR) and backward diffraction radiation (BDR) from inclined and perpendicular target.
• perfect conductive target; • BDR energy from the inclined target with the surface St equal to BDR energy from the perpendicular target with the surface S ? t ¼ S t sin W (see Fig. 1).
dW BDR a h h2 xRout xRout xRout h ¼ 2 K1 J0 2 dx dX cc cc p c2 þ h2 c 2 xRin xRin xRin h K1 J0 : ð1Þ cc cc c
Here a – fine structure constant; h – PlanckÕs constant; h – BDR photon outgoing angle; x – photon frequency; c – Lorentz factor; Rout – outer radius of the disk; Rin – radius of the hole; K1(x), J0(x) – Bessel functions. Formula (1) is written for incoherent radiation of electron. During the generation of BDR by
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short electron bunch the intensity of radiation (1) in the region of wavelength comparable with the length of the bunch r increases in proportion to bunch population Ne (radiation becomes coherent). This part of the spectrum gives the main contribution to the radiation losses. For the expected sizes of the target for SLAC FFTB (Rout 10 mm, Rin 0.1 mm) and Lorentz factor c 6 · 104 in the range of wave lengths k 6 1 mm we can neglect the influence of the target hole on the BDR characteristics (J 0 2pRkin h 1 with accuracy better than 1%). Then,having in mind that for considered case xRin;out xRin;out K 1 cc 1 with the same accuracy, we cc can write down: dW DR ðk > 1 mmÞ dW TR ¼ dx dX dx dX 2 a h h2 xRout h 1 J0 2 : p c2 þ h2 2 c
ð2Þ
In other words for the above mentioned parameters the characteristics of DR from the disk with the bore coincide with the characteristics of transition radiation (TR) from the target with the radius Rout [10,12]. Fig. 2 shows the calculation results according to formula (2) of the TR spectra into the cone h 6 hm for Lorentz factor c = 2000 and target radius R = 20 mm. All the curves are normalized to the TR spectrum from the infinite target. It should be mentioned that the obtained results coincide
Fig. 2. TR spectra from a round target into solid angle ph2m , normalized to one from infinite target.
Fig. 3. Angular distribution of DR from target for fixed wavelengths.
within several percent with the calculations of Castellano et al. [12], which are based on the rather complicated model (see Fig. 3 from the quoted article). Therefore all the subsequent computations are given with the use of the simple expression (2). Fig. 3 shows the angular distributions of TR for the finite target for fixed wavelengths k. It is not difficult to show that the angle h0, corresponding to the maximum of angular distribution is linearly dependent from the ratio k/R: h0 ¼
k z0 ; R 2p
ð3Þ
where z0 is the solution of the transcedental equation: 1 J 0 ðzÞ þ zJ 1 ðzÞ ¼ 0:
ð4Þ
In (4) J0(z), J1(z) – Bessel functions. Fig. 4 shows the spectrum of DR for SLAC FFTB case (c = 60,000; Rout = R = 5 mm; hm = 0.2 rad) (lower curve) and spectral distribution on the wave number x/c = 2p/k (upper curve). The intensity of coherent DR per each electron is determined by the number of electrons in a bunch and by a formfactor: xr 2 dW dW CBDR DR ¼ N e F ; ð5Þ c dx dX dx dX where Ne is a bunch population. For Gaussian distribution of electrons in a bunch the formfactor is described by the Gaussian law too:
2 2 xs 2 rx : F ¼ exp c 2c2
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
173
Table 1 Transverse kick for different targets R, mm
WCBDR, keV
Dh?, lrad
5 10 20
1.6 7.5 17.1
0.056 0.26 0.6
and consequently transverse momentum: Dp? ¼
W CBDR : c
The transverse kick can be easily determined using obtained result: Dh? ¼
Fig. 4. DR spectrum from round target into aperture hm = 0.2 rad (upper curve) and same spectrum depending on wave number (lower curve).
Fig. 5 shows the spectra of coherent BDR for different radii of the target for c = 60,000; r = 0.7 mm; hm = 0.2 rad. Using the known spectrum it is easy to calculate radiation losses per electron in a bunch: Z 1 dW CBDR W CBDR ¼ N e dx dx 0
Dp? : P
Table 1 presents the evaluation results for the transverse kick for 3 radii of a target and the bunch population Ne = 1010 e/bunch. Thus for the DR target with the square of 300 mm2 and angle of w = 45 even for the bunch population Ne = 3 · 1010 the expected kick due to the coherent diffraction radiation recoil effect (or, in other words, geometrical wakefield) does not exceed 2 lrad, which may be considered as permissible. The use of the target of such size to measure the size of SLAC-FFTB bunch using an optical DR (k 0.5 l, ck 30 mm) requires the development of the relevant theoretical models of ODR, because existing ones are true for the distance between the target and the detector L c2k. Our numerical model allowing to calculate ODR characteristics in the pre-wave zone [13,14] where L c2k is described in [15]. We plan to study both theoretically and experimentally the effect of pre-wave zone for DR for c = 60,000 at SLAC to estimate the feasibility of the ODR technique for the determination of a micron size beam in the extremely relativistic case.
References
Fig. 5. Coherent DR spectra for SLAC FFTB beam.
[1] M. Ferianis, in: EPAC 98 Proceeding, Vol. 1, 1998, p. 159. [2] P. Piot et al., BIW 96, in: AIP Conference Proceedings, Vol. 390, 1997, p. 298. [3] M. Castellano, V.A. Verzilov, L. Catani, A. Cianchi, G. Orlandi, M. Geitz, Phys. Rev. E 63 (2001) 056501.
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[4] J. Urakawa et al., Nucl. Instr. and Meth. A 472 (2001) 309. [5] Y. Fukui et al., Design of an Optical Diffraction Radiation Beam Size Monitor at SLAC FFTB, PAC03, 2003. [6] K. Bane, P. Morton, in: Proceedings of the 1986 Linear Accelerator Conference, SLAC, 1986, p. 490. [7] P. Tenenbaum et al., in: PAC99 Proceedings, 1999, p. 3453. [8] A.P. Potylitsyn, Phys. Rev. E 60 (1999) 2272. [9] A.P. Kazantsev, G.I. Surdutovich, Sov. Phys. Dokl. 7 (1963) 990.
[10] A.P. Potylitsyn, Nucl. Instr. and Meth. B 145 (1998) 169. [11] A.P. Potylitsyn, Nucl. Instr. and Meth. A 455 (2000) 213. [12] M. Castellano, A. Cianchi, E. Orlandi, V.A. Verzilov, Nucl. Instr. and Meth. A 435 (1999) 297. [13] V.A. Verzilov, Phys. Lett. A 273 (2000) 135. [14] M. Castellano, V. Verzilov, L. Catani, A. Cianchi, G. Orlandi, M. Geitz, Phys. Rev. E 66 (2003) 015501. [15] Y. Fukui et al., in: Proceedings of the 2003 Particle Accelerator Conference, Portland, Oregon, USA, 12–16 May 2003, p. 2709.
Coherent radiation recoil effect for the optical diffraction radiation beam size monitor at SLAC FFTB A. Potylitsyn a,*, G. Naumenko a, A. Aryshev a, Y. Fukui b, D. Cline b, F. Zhou b, M. Ross c, P. Bolton c, J. Urakawa d, T. Muto d, M. Tobiyama d, R. Hamatsu e, P. Karataev e a
d
Nuclear Physics Institute Tomsk Polytechnic University, Lenin Avenue 30, Tomsk 634034, Russia b University of California at Los Angeles, California, 90095-1547, USA c Stanford Linear Accelerator Center, Stanford University, California, 94505, USA KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 300-0801, Japan e Tokyo Metropolitan University, 1-1 Minamiohzawa, Hachioji, Tokyo 192-0397, Japan Received 17 December 2003; received in revised form 8 June 2004
Abstract A short electron bunch with length r passing through a slit in the diffraction radiation (DR) target generates radiation with a broad spectrum. Optical part of the spectrum (incoherent radiation) may be used for beam size measurements, but in the wavelength range k P r radiation becomes coherent. The coherent DR spectrum per each electron in a bunch is equal to single electron spectrum times by the number of electrons in a bunch Ne and bunch form factor. For SLAC FFTB conditions (Ne 1010, r = 0.7 mm, outer target size R 10 mm, slit width h 0.1 mm) we approximated coherent DR (CDR) spectrum by coherent transition radiation (TR) one because in the wavelength region k r h TR and DR spectra coincide with high accuracy. Changing the DR target by a TR target with projection on the plane perpendicular to electron beam as a circle with radius R 6 20 mm we calculated CDR spectra using simple model. Knowing the CDR spectrum we estimated the energy CDR emitting by each electron in the perpendicular direction (due to target inclination angle 45). It means an electron receives the radiation recoil in this direction. In other words, electron has a transverse kick about 1 lrad that may be considered as permissible. Published by Elsevier B.V. PACS: 41.60.m Keywords: Diffraction radiation; Coherent radiation; Electron beams; Diagnostics
*
Corresponding author. Tel.: +7 3822 41 89 06. E-mail address: [email protected] (A. Potylitsyn).
0168-583X/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.nimb.2004.06.016
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
171
Optical transition radiation is recently widely used for diagnostics of the beams [1]. However, the direct interaction of beam particles with the target material may cause its transformation and distortion of optical characteristics, and for intense focused beams even destruction of the target [2]. When the beam of ultrarelativistic electrons passes in a vacuum near the conductive target the socalled diffraction radiation (DR) is generated, which features allow using it for non-invasive beam diagnostics [3,4]. In [5] we proposed to use the optical DR to create beam size monitor at SLAC Final Focus Test Beam, where the intensity of electron beam is (1–3) · 1010 particles/ bunch with FWHM bunch length equal to 0.7 mm. In this case there is a need to estimate the possible distortions of the beam characteristics due to wakefield deflection. This effect is connected with two kinds of short bunch interaction with conductive target:
angle w = 45 DR is emitted at right angle to the electron momentum (backward diffraction radiation, BDR). Real photons of BDR, having a certain momentum in the specular reflection direction k ? ¼ W cDR (WDR the total BDR energy losses, c-speed of light), transfer momentum to electrons jD~ q? j ¼ k ? (recoil effect). Just this reason leads to beam deflection. The estimation of an electron deflection angle based on the described approach [8] agrees with result obtained from the geometric wakefield model [6]. In order to estimate the radiation losses in the BDR cone we will use the following assumptions:
• geometric wakefield [6]; • resistive wakefield [7].
Authors of [9] pointed out that DR energy losses for a particle moving near a tilted semi-infinite perfect screen are defined by its projection on the plane perpendicular to the particle trajectory. For an ultrarelativistic case DR energy losses were calculated in [10] where no dependence on a tilted angle W was obtained if condition W c1 is fulfilled. One may assume that for a slit DR target there is the same situation. The simplest model of DR from ultrarelativistic particle for the perpendicular target with the finite square (DR for a disk with the hole) is described in [11]. Spectral-angular distribution of DR in such target is described with the following formula:
DR target is made of the highly conductive material that is why the geometric wakefield determines the major impact. The consideration given in [8] shows that the geometric wakefield effect may be interpreted as a recoil effect due to coherent DR. This is shown in Fig. 1. During the passing of the electron through the slit in the inclined target the intrinsic Coulomb field of the particle is reflected from the surface of the target and is transformed into the real photons of DR, which emitted in the cone with an apex angle c1 along the direction of the mirror reflection. For the target inclination
Fig. 1. Generation of forward (FDR) and backward diffraction radiation (BDR) from inclined and perpendicular target.
• perfect conductive target; • BDR energy from the inclined target with the surface St equal to BDR energy from the perpendicular target with the surface S ? t ¼ S t sin W (see Fig. 1).
dW BDR a h h2 xRout xRout xRout h ¼ 2 K1 J0 2 dx dX cc cc p c2 þ h2 c 2 xRin xRin xRin h K1 J0 : ð1Þ cc cc c
Here a – fine structure constant; h – PlanckÕs constant; h – BDR photon outgoing angle; x – photon frequency; c – Lorentz factor; Rout – outer radius of the disk; Rin – radius of the hole; K1(x), J0(x) – Bessel functions. Formula (1) is written for incoherent radiation of electron. During the generation of BDR by
172
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
short electron bunch the intensity of radiation (1) in the region of wavelength comparable with the length of the bunch r increases in proportion to bunch population Ne (radiation becomes coherent). This part of the spectrum gives the main contribution to the radiation losses. For the expected sizes of the target for SLAC FFTB (Rout 10 mm, Rin 0.1 mm) and Lorentz factor c 6 · 104 in the range of wave lengths k 6 1 mm we can neglect the influence of the target hole on the BDR characteristics (J 0 2pRkin h 1 with accuracy better than 1%). Then,having in mind that for considered case xRin;out xRin;out K 1 cc 1 with the same accuracy, we cc can write down: dW DR ðk > 1 mmÞ dW TR ¼ dx dX dx dX 2 a h h2 xRout h 1 J0 2 : p c2 þ h2 2 c
ð2Þ
In other words for the above mentioned parameters the characteristics of DR from the disk with the bore coincide with the characteristics of transition radiation (TR) from the target with the radius Rout [10,12]. Fig. 2 shows the calculation results according to formula (2) of the TR spectra into the cone h 6 hm for Lorentz factor c = 2000 and target radius R = 20 mm. All the curves are normalized to the TR spectrum from the infinite target. It should be mentioned that the obtained results coincide
Fig. 2. TR spectra from a round target into solid angle ph2m , normalized to one from infinite target.
Fig. 3. Angular distribution of DR from target for fixed wavelengths.
within several percent with the calculations of Castellano et al. [12], which are based on the rather complicated model (see Fig. 3 from the quoted article). Therefore all the subsequent computations are given with the use of the simple expression (2). Fig. 3 shows the angular distributions of TR for the finite target for fixed wavelengths k. It is not difficult to show that the angle h0, corresponding to the maximum of angular distribution is linearly dependent from the ratio k/R: h0 ¼
k z0 ; R 2p
ð3Þ
where z0 is the solution of the transcedental equation: 1 J 0 ðzÞ þ zJ 1 ðzÞ ¼ 0:
ð4Þ
In (4) J0(z), J1(z) – Bessel functions. Fig. 4 shows the spectrum of DR for SLAC FFTB case (c = 60,000; Rout = R = 5 mm; hm = 0.2 rad) (lower curve) and spectral distribution on the wave number x/c = 2p/k (upper curve). The intensity of coherent DR per each electron is determined by the number of electrons in a bunch and by a formfactor: xr 2 dW dW CBDR DR ¼ N e F ; ð5Þ c dx dX dx dX where Ne is a bunch population. For Gaussian distribution of electrons in a bunch the formfactor is described by the Gaussian law too:
2 2 xs 2 rx : F ¼ exp c 2c2
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
173
Table 1 Transverse kick for different targets R, mm
WCBDR, keV
Dh?, lrad
5 10 20
1.6 7.5 17.1
0.056 0.26 0.6
and consequently transverse momentum: Dp? ¼
W CBDR : c
The transverse kick can be easily determined using obtained result: Dh? ¼
Fig. 4. DR spectrum from round target into aperture hm = 0.2 rad (upper curve) and same spectrum depending on wave number (lower curve).
Fig. 5 shows the spectra of coherent BDR for different radii of the target for c = 60,000; r = 0.7 mm; hm = 0.2 rad. Using the known spectrum it is easy to calculate radiation losses per electron in a bunch: Z 1 dW CBDR W CBDR ¼ N e dx dx 0
Dp? : P
Table 1 presents the evaluation results for the transverse kick for 3 radii of a target and the bunch population Ne = 1010 e/bunch. Thus for the DR target with the square of 300 mm2 and angle of w = 45 even for the bunch population Ne = 3 · 1010 the expected kick due to the coherent diffraction radiation recoil effect (or, in other words, geometrical wakefield) does not exceed 2 lrad, which may be considered as permissible. The use of the target of such size to measure the size of SLAC-FFTB bunch using an optical DR (k 0.5 l, ck 30 mm) requires the development of the relevant theoretical models of ODR, because existing ones are true for the distance between the target and the detector L c2k. Our numerical model allowing to calculate ODR characteristics in the pre-wave zone [13,14] where L c2k is described in [15]. We plan to study both theoretically and experimentally the effect of pre-wave zone for DR for c = 60,000 at SLAC to estimate the feasibility of the ODR technique for the determination of a micron size beam in the extremely relativistic case.
References
Fig. 5. Coherent DR spectra for SLAC FFTB beam.
[1] M. Ferianis, in: EPAC 98 Proceeding, Vol. 1, 1998, p. 159. [2] P. Piot et al., BIW 96, in: AIP Conference Proceedings, Vol. 390, 1997, p. 298. [3] M. Castellano, V.A. Verzilov, L. Catani, A. Cianchi, G. Orlandi, M. Geitz, Phys. Rev. E 63 (2001) 056501.
174
A. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 227 (2005) 170–174
[4] J. Urakawa et al., Nucl. Instr. and Meth. A 472 (2001) 309. [5] Y. Fukui et al., Design of an Optical Diffraction Radiation Beam Size Monitor at SLAC FFTB, PAC03, 2003. [6] K. Bane, P. Morton, in: Proceedings of the 1986 Linear Accelerator Conference, SLAC, 1986, p. 490. [7] P. Tenenbaum et al., in: PAC99 Proceedings, 1999, p. 3453. [8] A.P. Potylitsyn, Phys. Rev. E 60 (1999) 2272. [9] A.P. Kazantsev, G.I. Surdutovich, Sov. Phys. Dokl. 7 (1963) 990.
[10] A.P. Potylitsyn, Nucl. Instr. and Meth. B 145 (1998) 169. [11] A.P. Potylitsyn, Nucl. Instr. and Meth. A 455 (2000) 213. [12] M. Castellano, A. Cianchi, E. Orlandi, V.A. Verzilov, Nucl. Instr. and Meth. A 435 (1999) 297. [13] V.A. Verzilov, Phys. Lett. A 273 (2000) 135. [14] M. Castellano, V. Verzilov, L. Catani, A. Cianchi, G. Orlandi, M. Geitz, Phys. Rev. E 66 (2003) 015501. [15] Y. Fukui et al., in: Proceedings of the 2003 Particle Accelerator Conference, Portland, Oregon, USA, 12–16 May 2003, p. 2709.