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Crystalline undulator radiation of microbunched beams taking into account the medium polarization
Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Crystalline undulator radiation of microbunched beams taking into account the medium polarization L.A. Gevorgian, K.A. Ispirian ⇑, A.H. Shamamian Alikhanian National Laboratory, Yerevan Physics Institute, Br. Alikhanian 2, Yerevan 0036, Armenia
a r t i c l e
i n f o
Article history: Received 25 November 2012 Received in revised form 31 January 2013 Accepted 4 February 2013 Available online 22 March 2013 Keywords: Crystalline undulator radiation Microbunched beams Medium polarization
a b s t r a c t Analytical and numerical results are obtained on the angular and spectral distributions of the number of photons as well as on the total number of the photons of the coherent X-ray crystalline undulator radiation (CXCUR) produced by microbunched beams passing through crystalline undulators (CU). These results show that one can use CXCUR for study of the microbunching process in XFELs and for production of additional monochromatic intense beams. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction At the end of the long undulators of FELs the electron beams undergoes longitudinal density modulation or microbunching (MB) with period equal to the wavelength of the FEL SASE radiation (see, for instance, [1,2]). Since the coherent radiation and exponential growth of the FEL radiation intensity arise due to MB, the study of the MB process is important for diagnostics of FEL beams and for the increase of FEL efficiency. At present using the coherent transition radiation (CTR) of photons softer than the optical photons the experimental study of MB has been carried out by the method for the study of MB proposed in [3] only on FELs with electron beams of energies less than 1 GeV [4,5]. The 10 GeV electron microbunched beam of SLAC XFEL, LCLS is sent to dump without any application [1], though a method based on coherent X-ray transition radiation (CXTR) has been proposed [6]. Moreover, there is a proposal [7] how to study MB at SLAC LCLS with the help of CXTR, and even some preparatory works have been carried out [8]. As the theory and simulations show MB of the electron beams is spoiled after the long undulators. There are proposals [9,10] in which it is shown that for the further use of these microbunched beams one can deflect them with the help of especial magnetic systems conserving the MB. Such deflected microbunched electron beams not accompanied with intense SASE X-beams can be used for additional production and applications of various types coherent X-ray beams, in particular, CXTR beams [6]. In the works [11,12] it is shown that the not deflected microbunched beams
⇑ Corresponding author. Tel.: +374 10 34 46 98. E-mail address: [email protected] (K.A. Ispirian). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.02.034
accompanied with SASE radiation can be used for production of more intense so called self stimulated undulator radiation. After the prediction [13] and theoretical and experimental study of the parametric X-ray radiation (PXR) of single charged particles, PXR has found application for solving various problems and production of monochromatic beams [14]. In the work [15] it has been studied the properties of the bunch coherence in coherent PXR (CPXR) produced by microbunched beams. As an alternative to CXTR it has been proposed to use CPXR for the study of the process of MB without numerical calculations. Moreover, at present it has been developed the theories [16,17] of the coherent Xray bremsstrahlung (CXBR), resonance transition radiation (CXRTR) and X-ray diffraction radiation (CXDR) produced by microbunched beams. On the other hand, after the first theoretical [18] and experimental [19] works on X-ray undulator radiation (XUR) of not microbunched electron beams, the theory of UR has been developed in many works (see [20]). It has been theoretically [21] and experimentally [22] investigated the undulator radiation produced in gas filled undulators. The theory of the predicted in [23] socalled crystalline undulator radiation (CUR) has been further developed without [24] and with [25] taking into account the polarization of the medium. Compared with the UR produced in magnetic undulators in the case of CUR the particles are channeled between the crystallographic planes, and due to the strong field of the crystalline planes one has the advantage of production of much shorter periods of oscillations or radiation wavelength. However, since the electrons in CU move in condensed matter just as in the case of bremsstrahlung [13] at GeV electron and keV radiated photon energies it is imperative to take into account the influence of the medium.
64
L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
At present besides the ultrasonic method of preparation of CU considered still in [23] there are published a few methods of preparation of CU [26–30]. Silicon CU have been prepared by scratching with periods L = (50–100) lm, oscillation amplitudes ACU = (20– 100) A° and number of periods NCU 10. Supperlattice CU of GexSi1x have been prepared by epitaxial methods with L = (9.9– 50) lm, ACU = (4–90) A° and NCU 4. Experimental results on CUR [31–33] have been obtained with electrons and positrons with energies (0.3–10) GeV. However, they need a more correct comparison with the existing theoretical results, and the possibility of construction of CU for electrons is doubtful because of short dechanneling length of electrons [34]. In the next Section of this work with the purpose to find another method of study of MB it is described the derivation of the formulae of the spectral and angular distribution of CXCUR. In Section 3 some numerical results are presented, and finally in the last section the discussion of the obtained results are given.
In (3) the following notations [25] are used:
a ¼ e2 =hc; 1 ¼ c0 ¼
xP X
L ; kP
2pAU c ; q¼ L /ð1Þ ¼ Q
r¼
X¼
;
2p V ; L
u ¼ ch;
pffiffiffi
g ¼ 2pAU =kP ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 ; kp ¼ 2pc=xp ; xP ¼ 4pne2 =m; c q2 g2
Q ¼1þ
2
ð1 11 Þð12 1Þ
12
¼1þ
r2
:
ð4Þ
ð5Þ
Here a is the fine structure constant, 1 and u are the dimensionless frequency and polar angle of CUR; X is the frequency of the oscillations of particles in CU with amplitude ACU and period L, q is the undulator parameter value, kp and xP are the CU plasma wavelength and frequency, c0 ,g, /ð1Þ are suitable notations and 1 varies in an interval between the following maximal and minimal values
2. Calculation of CXCUR One can write the spectral-angular distribution d N CXCUR =ðdxdhÞ of the number of photons of CXCUR produced by a microbunched electron macrobunch passing through an und2 ulator with the help of known d N CUR =ðdxdhÞ for single particle in the form [3–6,35]
¼
x Xc2
11;2 ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r2 Q : Q
ð6Þ
2
2
According to d-function in (3) the dimensionless frequency and angle, 1 and u, are connected by the relation
u¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q 1
2
d NCXCUR d NCUR ffi N2B Fðx; #Þ : dxdh dxdh
ð1Þ
In (1) Fðx; hÞ is the form factor of the microbunched beam [36]: 2 r2 x2 r2z x z kr Fðx; hÞ ¼ exp ðkrr sin hÞ2 expð Þ þ b1 exp 2 V 2 2V 2 r2 2 x z þ kr ð2Þ þ b1 exp V 2
In (1), (2) N b is the number of electrons in the macrobunch, k ¼ x=c ¼ 2p=k and kr ¼ xr =c ¼ 2p=kr are the wave numbers of the produced CXCUR and SASE radiation, rr and rz are the transversal and longitudinal dispersions of the electron Gaussian distribution in the bunch, V is the velocity of the electrons, b1 is the SASE parameter or the MB modulation depth. According to (1) and (2) the physics of the production of coherent radiation by microbunched beams, in brief, is in the following: Despite the spikes let us assume that the macrobunch consists of M microbunches each containing nb electrons with relativistic factor c ¼ E=mc2 ¼ ð1 b2 Þ1=2 (b ¼ V=c), so that Nb ¼ Mnb . If the parameters of the FEL are chosen in such a manner that the SASE resonance frequency xr is within the spectral distribution of the broad spectrum of the incoherent radiation of single particle radiation, then for the corresponding coherent radiation with wavelength larger than the length of the microbunch k P kr , all the nb electrons of the first microbunch oscillate synchronically and radiate photon number of the coherent radiation proportional to n2b . The following next microbunches also radiate in the same way. However, since the microbunches are periodical, due to interference, the summary number of the radiated photons of the macro2 bunch is proportional to N 2b b1 . In this work for a single particle it is reasonable to use the following formula of the angular-spectral distribution of the first harmonic of CUR in dipole approximation per unit length of a medium filled CU obtained after series expansion and integration over azimuth angles of the formula (32) of the work [21]: 2 i d NCUR pag2 h ¼ 2 u 1 þ ð1 u2 1Þ2 d u2 /ð1Þ : d1du r L
ð3Þ
11 1
12 1 : 1
ð7Þ
As it has been shown in [25] compared with the spectrum of undulators without medium filling the CUR spectrum of single particles is narrowed within the interval from 11 up to 12 . The CUR photons with frequencies 11 and 12 are emitted under zero angles with respect to the particle momentum. The formula (3) corresponds to the formula (14.114) of [37]. In particular, integrating (3) over angles one obtains the formula (17) of the work [25] for the spectral distribution of CUR produced by a particle. It is well known that the dipole approximation is applicable for all frequencies if the undulator parameter q < 1. However, formula (3) is also applicable in certain frequencies 1 even when q > 1, but if 1qu << 1 , when one can make series expansion of the Bessel functions in formula (32) of the work [21]. Using the above expressions in the region close to the 1 ¼ xr=Xc2 one can present the angular-spectral distribution of CXCUR of a macrobunch in the form 2
2
d NCXCUR d NCUR ¼ N2b Fð1; uÞ ; d1du d1du
ð8Þ
in which now neglecting the contribution of the first and the third terms in (2), using the dimensionless frequencies and angles (see (4)) and the notations
A¼
2pc2 rz L
2 ;
B¼
2 2pcrr L
the form factor (2) can be presented as product of longitudinal and transversal form-factors
n h io 2 Fð1; uÞ ¼ F ‘ ð1ÞF ? ð1; uÞ ¼ b1 exp Að1 1r Þ2 þ B12 u2
ð9Þ
By an appropriate choice of the experimental parameters one can have 1r > 11 , from which it follows the condition L < 2k2P =kr . Then substituting (3) and (9) into (8) one obtains the expression for the spectral angular distribution of CXCUR. 2 i d NCXCUR pag2 2 2 h ¼ 2 Nb b1 u 1 þ ð1 1u2 Þ2 d1du r L n h io exp Að1 1r Þ2 þ B12 u2 d u2 /ð1Þ
ð10Þ
L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
Fig. 1. The dependence of dNCXCUR =d1 (a) and dN CXCUR =du (b) upon
After corresponding integration over frequencies and angles in (10) one derives the following formulae for the frequency and angular distributions of CXCUR produced in a unit length CU
2 n h io o dNCXCUR pa Nb b1 g n ¼ 1 þ ½1 1/2 exp Að1 1r Þ2 þ B12 / d1 2L r ð11Þ 2 2 o dNCXCUR pa Nb b1 g 13u n u 1 þ 1 1u u2 ¼ 2 du 2L r r 1u n h io exp Að1u 1r Þ2 þ Bu2 12u
ð12Þ
in which one can present the relation between the angle and frequency in the form
1ðuÞ ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðQ þ u2 Þr 2 : Q þ u2
ð13Þ
! 2 h i dNCXCUR pa Nb b1 g B2 exp Að1 11 Þ2 ; ¼ exp d1 L r A dNCXCUR 2pa ¼ du L
! 2 Nb b1 g B2 A14 u4 exp 1 11 u exp : r A 4
ð14Þ
ð15Þ
Integrating (14) over1, or (15) over u one derives the following expression for the total number of CXCUR photons.
NCXCUR ¼
2
p3=2 a Nb b1 g pffiffiffi L A
r
B2 exp A
1 and u of CXCUR photons.
photon energy equal to E = 13.6 GeV and hxr ¼ 8:3 keV the condition 1r 11 is satisfied for Si CU with AU = 3 nm and L = 20.7 lm. The below given numerical results have been calculated with the help of (14)–(16) for the following beam parameters of LCLS. The particle beam parameters are: NB = 1.56 109, b1 = 1, E = 13.6 GeV, h xr = 8.3 keV, rz = 9 104 cm, rr = 6.12 104 cm. The CU parameters are: particles are channeled between the (1 1 0) crystallographic planes of a silicon crystal with interplanar distance d = 0.192 nm, the period and oscillation amplitude of CU are equal to L = 20.7 lm and AU = 3 nm, respectively. In the above calculations and results given for unit length CU it has not been taken into account the absorption of the CXCUR photons produced homogeneously along their entire path in CU. Having the above calculated number N CXCUR for unit length of CU one can calculate N CXCUR ðLCU Þ, the number of photons coming out from a CU with length LCU using the formula [38].
NCXCUR ðLCU Þ ¼ NCXCUR Labs ½1 expðLCU =Labs Þ
As it is seen from (9)–(12) the CXCUR intensities are large if simultaneously 1 1R and u 0, which is possible if 1R 11 . One can show that such conditions will be fulfilled with necessary accuracy, if by an appropriate choice of the available parameters it is possible to obtain 1R ¼ 11 þ B=A. Then, taking into account that for real parameters 1/ << 1 and 1ðuÞu2 << 1, as well as that the spectral distribution at 11 is not sharp, but have finite slope with width much wider than the distributions (11) and (12) due to finite length of the undulator, Dc=c, etc., one can present the spectral and angular distributions of CXCUR in the forms.
! ð16Þ
3. Numerical results and discussion Before beginning the description of the calculations let us note that the CXCUR intensities, calculated with the help of the formulae (11) and (12) when 1r is significantly larger than 11 , are very low because of the second terms in the exponents. For this reason in the below calculations with appropriate choice of amplitude AU and period L of CU for the given particle energy we try to achieve 11 close to 1r and use the formulae (14)–(16). For particle and SASE
65
ð17Þ
where NCXCUR is given either by (14) or by (15), and Labs is the absorption length of the produced monochromatic photons. As it follows from (17) the number of the produced photons increases with the increase of CU length LCU and then is saturated when LCU P ð3 4ÞLabs giving N CXCUR ðLCU Þ N CXCUR Labs . Therefore, it is not reasonable to take LCU greater than 4Labs . For CU made of Si and hxr = 8.3 keV one has Labs ð8:3 keVÞ = 77.5 lm. Therefore, we take ðLCU ¼ 4 Labs = 310 lm giving CU oscillation number Nosc 15, though during the calculations of the number of the CXCUR photons we correctly have taken Leff ¼ Labs = 77.5 lm. Let us note that the dechanneling length for 13.6 GeV electrons is equal to LeDech 13ðlm=GeVÞEe ðGeVÞ = 177 lm and LeDech 90 lm according to [34] and CATCH simulations [39], respectively. Unfor tunately for electrons the experimental data on LeDech are sparse, but even these data show once more that it is not serious to speak about CU for electrons. Therefore, our further considerations concern positrons. For 13.6 GeV positrons channeled between (110) of Si crystal LeDech 8160 lm, much larger than ðLCU = 310 lm.Therefore, one can neglect dechanneling in the above calculations. Fig. 1 shows the spectral and angular distributions of CXCUR produced in a ðLCU = 310 lm long CU calculated with the help of (14) and (15) for the above parameters. As it is seen maximum of the angular distribution of CXCUR is at the same angles and it has the same width as the maxima and widths of the angular distributions of CXTR, CXRTR and CXDR (see [16,17]). Again the emission angles of CXCUR are much smaller than those of UR and bremsstrahlung, XTR, RTR angular distributions which are around u 1 or h 1=c. The total number of CXCUR photons calculated by numerical integration of the curves of Fig. 1 and with the help of the formula (16) is equal to N CXCUR = 1.2 1010. Let us note that this number is
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L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
higher than the achievable numbers of CXTR, CXDR [16,17] and lower than for CXRTR [16,17]. Therefore, one can conclude that CXCUR can be successfully used for the study of the microbunching parameters of XFELs using arrangement schematically shown in [16]. Acknowledgment The authors thank M.A. Aginian and M.K. Ispiryan for help and discussions. References [1] (a) P. Emma et al., Proc. FEL 2009 (2009) 397; (b) P. Emma et al., Nat. Photonics 4 (2010) 641; (c) P. Emma et al., LCLS, Design Study Report (1998). SLAC-R-521. [2] B.D. Patterson et al., New J. Phys. 12 (2010) 035012. [3] J. Rosenzweig, G. Travish, A. Tremaine, Nucl. Instr. and Meth. A 365 (1995) 255. [4] Y. Liu et al., Phys. Rev. Lett. 80 (1998) 4418. [5] A. Tremaine et al., Phys. Rev. Lett. 81 (1998) 5816. [6] E.D. Gazazian, K.A. Ispirian, R.K. Ispirian, M.I. Ivanian, Pisma Zh. Eksp. Teor. Fiz. 70 (1999) 664; Nucl. Instr. Meth. B173 (2001) 160. [7] A.H. Lumpkin, W.M. Fawley, D.W. Rule, Proc. FEL (2004) 515. [8] A.H. Lumpkin, J.B. Hastings, D.W. Rule, SLAC-PUB-12451, 2005. [9] Y. Li, W. Decking, B. Faatz, J. Pflueger, Phys. Rev. STAB 13 (2010) 080705. [10] G. Geloni, V. Kocharyan, E. Saldin, Arxiv 1106 (2011) 1776. [11] G.N. Kulipanov, A.S. Sokolov, N.A. Vinokurov, Nucl. Instr. Meth. A 375 (1996) 576. [12] E.G. Bessonov et al., Proceedings of the conference CHANNELING 2010, to be publ. in Nuovo Cimento, J. Instrum. 12 (2010). [13] M.L. Ter-Mikaelian, The Influence of the Medium on High Energy Processes at High Energies, Publishing House of Academy of Science of Armenia, Yerevan, 1969, High Energy Electromagnetic Processes in Condensed Media, Wiley Interscience, New York, 1972. [14] V.G. Barishevski, I.D. Feranchuk, A.P. Ulyanov, Parametric X-Ray Radiation in Crystals: Theory Experiments and Application, Springer Tracts in Modern Physics, Heidelberg, 2005.
[15] X. Artru, K.A. Ispirian, Proc. NATO Workshop Electron-Photon Interaction in Dense Media, Nor-Hamberd, Yerevan, Armenia 25–29 June, 2001, p 191: ArXive ph/0208018, 2002. [16] K.A. Ispirian, Proceedings of the International Conference on Electron, Positron, Neutron and X-Ray Scattering under External Influences, Yerevan-Megri, Armenia, October, 2011, p.197. [17] M.A. Aginian, K.A. Ispirian, M.K. Ispiryan, Proceedings of NAS Armenia, Physics 47 (2012) 83. [18] N.A. Korkhmazian, Fizika 5 (1970) 287. 5 (1970) 418. [19] A.I. Alikhanian, S.K. Esin, K.A. Ispirian, et al., Pisma Zh. Eksper. Teor. Fiz. 15 (1972) 142; K.A. Ispirian, A.G. Oganesian, Lectures on VII Intern. School on Experimental and Theoretical Physics, Yerevan, 1971, Preprint Yerevan Physics Institute, EFIME-4(71) 1971. [20] D.F. Alferov, Yu.A. Bashmakov, E.G. Bessonov, in: N.G. Basov (Ed.), Proc. Lebedev. Phys. Inst. Ser. 80, 1976, p. 97. [21] L.A. Gevorgian, N.A. Korkhmazian, Zh. Eksp, Teor. Fiz. 76 (1979) 1226. [22] R.H. Pantel et al., Nucl. Instr. and Meth. A 250 (1986) 312. [23] V.V. Kaplin, S.V. Plotnikov, S.A. Vorobev, Zh. Tekh. Fiz. 50 (1980) 1079. [24] A.A. Korol, A.V. Solovev, W. Greiner, J. Phys. G 24 (1998) 45. [25] R.O. Avakian, L.A. Gevorgian, K.A. Ispirian, R.K. Ispirian, Pisma Zh. Eksp. Teor. Fiz. 68 (1998) 437; R.O. Avakian, L.A. Gevorgian, K.A. Ispirian, R.K. Ispirian, Nucl. Instr. Meth. B 173 (2001) 112. [26] U. Mikkelsen, E. Uggerhoy, Nucl. Instr. Meth. A 483 (2002) 455. [27] R.O. Avakian, K.A. Avetian, K.A. Ispirian, E.M. Melikian, Nucl. Instr. Meth. A 492 (2002) 11. A508 (2003) 496. [28] S. Bellucci et al., Phys. Rev. Lett. 90 (2003) 03801. [29] S. Bellucci et al., Phys. Rev. STAB 7 (2004) 023501. [30] V. Guidi et al., Appl. Phys. Lett. 91 (2007) 061908. [31] B.T. Baranov et al., Pisma Zh. Eksp. Teor. Fiz. 82 (2005) 638. [32] H. Backe et al., Proceedings of International Conference on Charged and Neutral Particles Channeling Phenomema, CHANNELING 2008, p. 281. [33] H. Backe et al., Proceedings of International Conference on Charged and Neutral Particles Channeling Phenomema, CHANNELING 2010, p. 157. [34] V. Biryukov, ArXiv0712.3904, 2007. [35] N.A. Korkhmazian, L.A. Gevorgian, M.L. Petrosian, Zh. Tekh Fiz. 47 (1977) 1583. [36] L.A. Gevorgian, A.H. Shamamian, Intern. J. Mod. (Proc. Suppl.), Phys. A 2B (1993) 1175. [37] J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York, 1999. [38] A.I. Alikhanian, F.R. Arutiunian, K.A. Ispirian, M.L. Ter-Mikaelian, Zh. Eksp. Teor. Fiz. 41 (1961) 2002. [39] V.M. Biryukov, Phys. Rev. E51 (1995) 3522.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Crystalline undulator radiation of microbunched beams taking into account the medium polarization L.A. Gevorgian, K.A. Ispirian ⇑, A.H. Shamamian Alikhanian National Laboratory, Yerevan Physics Institute, Br. Alikhanian 2, Yerevan 0036, Armenia
a r t i c l e
i n f o
Article history: Received 25 November 2012 Received in revised form 31 January 2013 Accepted 4 February 2013 Available online 22 March 2013 Keywords: Crystalline undulator radiation Microbunched beams Medium polarization
a b s t r a c t Analytical and numerical results are obtained on the angular and spectral distributions of the number of photons as well as on the total number of the photons of the coherent X-ray crystalline undulator radiation (CXCUR) produced by microbunched beams passing through crystalline undulators (CU). These results show that one can use CXCUR for study of the microbunching process in XFELs and for production of additional monochromatic intense beams. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction At the end of the long undulators of FELs the electron beams undergoes longitudinal density modulation or microbunching (MB) with period equal to the wavelength of the FEL SASE radiation (see, for instance, [1,2]). Since the coherent radiation and exponential growth of the FEL radiation intensity arise due to MB, the study of the MB process is important for diagnostics of FEL beams and for the increase of FEL efficiency. At present using the coherent transition radiation (CTR) of photons softer than the optical photons the experimental study of MB has been carried out by the method for the study of MB proposed in [3] only on FELs with electron beams of energies less than 1 GeV [4,5]. The 10 GeV electron microbunched beam of SLAC XFEL, LCLS is sent to dump without any application [1], though a method based on coherent X-ray transition radiation (CXTR) has been proposed [6]. Moreover, there is a proposal [7] how to study MB at SLAC LCLS with the help of CXTR, and even some preparatory works have been carried out [8]. As the theory and simulations show MB of the electron beams is spoiled after the long undulators. There are proposals [9,10] in which it is shown that for the further use of these microbunched beams one can deflect them with the help of especial magnetic systems conserving the MB. Such deflected microbunched electron beams not accompanied with intense SASE X-beams can be used for additional production and applications of various types coherent X-ray beams, in particular, CXTR beams [6]. In the works [11,12] it is shown that the not deflected microbunched beams
⇑ Corresponding author. Tel.: +374 10 34 46 98. E-mail address: [email protected] (K.A. Ispirian). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.02.034
accompanied with SASE radiation can be used for production of more intense so called self stimulated undulator radiation. After the prediction [13] and theoretical and experimental study of the parametric X-ray radiation (PXR) of single charged particles, PXR has found application for solving various problems and production of monochromatic beams [14]. In the work [15] it has been studied the properties of the bunch coherence in coherent PXR (CPXR) produced by microbunched beams. As an alternative to CXTR it has been proposed to use CPXR for the study of the process of MB without numerical calculations. Moreover, at present it has been developed the theories [16,17] of the coherent Xray bremsstrahlung (CXBR), resonance transition radiation (CXRTR) and X-ray diffraction radiation (CXDR) produced by microbunched beams. On the other hand, after the first theoretical [18] and experimental [19] works on X-ray undulator radiation (XUR) of not microbunched electron beams, the theory of UR has been developed in many works (see [20]). It has been theoretically [21] and experimentally [22] investigated the undulator radiation produced in gas filled undulators. The theory of the predicted in [23] socalled crystalline undulator radiation (CUR) has been further developed without [24] and with [25] taking into account the polarization of the medium. Compared with the UR produced in magnetic undulators in the case of CUR the particles are channeled between the crystallographic planes, and due to the strong field of the crystalline planes one has the advantage of production of much shorter periods of oscillations or radiation wavelength. However, since the electrons in CU move in condensed matter just as in the case of bremsstrahlung [13] at GeV electron and keV radiated photon energies it is imperative to take into account the influence of the medium.
64
L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
At present besides the ultrasonic method of preparation of CU considered still in [23] there are published a few methods of preparation of CU [26–30]. Silicon CU have been prepared by scratching with periods L = (50–100) lm, oscillation amplitudes ACU = (20– 100) A° and number of periods NCU 10. Supperlattice CU of GexSi1x have been prepared by epitaxial methods with L = (9.9– 50) lm, ACU = (4–90) A° and NCU 4. Experimental results on CUR [31–33] have been obtained with electrons and positrons with energies (0.3–10) GeV. However, they need a more correct comparison with the existing theoretical results, and the possibility of construction of CU for electrons is doubtful because of short dechanneling length of electrons [34]. In the next Section of this work with the purpose to find another method of study of MB it is described the derivation of the formulae of the spectral and angular distribution of CXCUR. In Section 3 some numerical results are presented, and finally in the last section the discussion of the obtained results are given.
In (3) the following notations [25] are used:
a ¼ e2 =hc; 1 ¼ c0 ¼
xP X
L ; kP
2pAU c ; q¼ L /ð1Þ ¼ Q
r¼
X¼
;
2p V ; L
u ¼ ch;
pffiffiffi
g ¼ 2pAU =kP ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 ; kp ¼ 2pc=xp ; xP ¼ 4pne2 =m; c q2 g2
Q ¼1þ
2
ð1 11 Þð12 1Þ
12
¼1þ
r2
:
ð4Þ
ð5Þ
Here a is the fine structure constant, 1 and u are the dimensionless frequency and polar angle of CUR; X is the frequency of the oscillations of particles in CU with amplitude ACU and period L, q is the undulator parameter value, kp and xP are the CU plasma wavelength and frequency, c0 ,g, /ð1Þ are suitable notations and 1 varies in an interval between the following maximal and minimal values
2. Calculation of CXCUR One can write the spectral-angular distribution d N CXCUR =ðdxdhÞ of the number of photons of CXCUR produced by a microbunched electron macrobunch passing through an und2 ulator with the help of known d N CUR =ðdxdhÞ for single particle in the form [3–6,35]
¼
x Xc2
11;2 ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r2 Q : Q
ð6Þ
2
2
According to d-function in (3) the dimensionless frequency and angle, 1 and u, are connected by the relation
u¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q 1
2
d NCXCUR d NCUR ffi N2B Fðx; #Þ : dxdh dxdh
ð1Þ
In (1) Fðx; hÞ is the form factor of the microbunched beam [36]: 2 r2 x2 r2z x z kr Fðx; hÞ ¼ exp ðkrr sin hÞ2 expð Þ þ b1 exp 2 V 2 2V 2 r2 2 x z þ kr ð2Þ þ b1 exp V 2
In (1), (2) N b is the number of electrons in the macrobunch, k ¼ x=c ¼ 2p=k and kr ¼ xr =c ¼ 2p=kr are the wave numbers of the produced CXCUR and SASE radiation, rr and rz are the transversal and longitudinal dispersions of the electron Gaussian distribution in the bunch, V is the velocity of the electrons, b1 is the SASE parameter or the MB modulation depth. According to (1) and (2) the physics of the production of coherent radiation by microbunched beams, in brief, is in the following: Despite the spikes let us assume that the macrobunch consists of M microbunches each containing nb electrons with relativistic factor c ¼ E=mc2 ¼ ð1 b2 Þ1=2 (b ¼ V=c), so that Nb ¼ Mnb . If the parameters of the FEL are chosen in such a manner that the SASE resonance frequency xr is within the spectral distribution of the broad spectrum of the incoherent radiation of single particle radiation, then for the corresponding coherent radiation with wavelength larger than the length of the microbunch k P kr , all the nb electrons of the first microbunch oscillate synchronically and radiate photon number of the coherent radiation proportional to n2b . The following next microbunches also radiate in the same way. However, since the microbunches are periodical, due to interference, the summary number of the radiated photons of the macro2 bunch is proportional to N 2b b1 . In this work for a single particle it is reasonable to use the following formula of the angular-spectral distribution of the first harmonic of CUR in dipole approximation per unit length of a medium filled CU obtained after series expansion and integration over azimuth angles of the formula (32) of the work [21]: 2 i d NCUR pag2 h ¼ 2 u 1 þ ð1 u2 1Þ2 d u2 /ð1Þ : d1du r L
ð3Þ
11 1
12 1 : 1
ð7Þ
As it has been shown in [25] compared with the spectrum of undulators without medium filling the CUR spectrum of single particles is narrowed within the interval from 11 up to 12 . The CUR photons with frequencies 11 and 12 are emitted under zero angles with respect to the particle momentum. The formula (3) corresponds to the formula (14.114) of [37]. In particular, integrating (3) over angles one obtains the formula (17) of the work [25] for the spectral distribution of CUR produced by a particle. It is well known that the dipole approximation is applicable for all frequencies if the undulator parameter q < 1. However, formula (3) is also applicable in certain frequencies 1 even when q > 1, but if 1qu << 1 , when one can make series expansion of the Bessel functions in formula (32) of the work [21]. Using the above expressions in the region close to the 1 ¼ xr=Xc2 one can present the angular-spectral distribution of CXCUR of a macrobunch in the form 2
2
d NCXCUR d NCUR ¼ N2b Fð1; uÞ ; d1du d1du
ð8Þ
in which now neglecting the contribution of the first and the third terms in (2), using the dimensionless frequencies and angles (see (4)) and the notations
A¼
2pc2 rz L
2 ;
B¼
2 2pcrr L
the form factor (2) can be presented as product of longitudinal and transversal form-factors
n h io 2 Fð1; uÞ ¼ F ‘ ð1ÞF ? ð1; uÞ ¼ b1 exp Að1 1r Þ2 þ B12 u2
ð9Þ
By an appropriate choice of the experimental parameters one can have 1r > 11 , from which it follows the condition L < 2k2P =kr . Then substituting (3) and (9) into (8) one obtains the expression for the spectral angular distribution of CXCUR. 2 i d NCXCUR pag2 2 2 h ¼ 2 Nb b1 u 1 þ ð1 1u2 Þ2 d1du r L n h io exp Að1 1r Þ2 þ B12 u2 d u2 /ð1Þ
ð10Þ
L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
Fig. 1. The dependence of dNCXCUR =d1 (a) and dN CXCUR =du (b) upon
After corresponding integration over frequencies and angles in (10) one derives the following formulae for the frequency and angular distributions of CXCUR produced in a unit length CU
2 n h io o dNCXCUR pa Nb b1 g n ¼ 1 þ ½1 1/2 exp Að1 1r Þ2 þ B12 / d1 2L r ð11Þ 2 2 o dNCXCUR pa Nb b1 g 13u n u 1 þ 1 1u u2 ¼ 2 du 2L r r 1u n h io exp Að1u 1r Þ2 þ Bu2 12u
ð12Þ
in which one can present the relation between the angle and frequency in the form
1ðuÞ ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðQ þ u2 Þr 2 : Q þ u2
ð13Þ
! 2 h i dNCXCUR pa Nb b1 g B2 exp Að1 11 Þ2 ; ¼ exp d1 L r A dNCXCUR 2pa ¼ du L
! 2 Nb b1 g B2 A14 u4 exp 1 11 u exp : r A 4
ð14Þ
ð15Þ
Integrating (14) over1, or (15) over u one derives the following expression for the total number of CXCUR photons.
NCXCUR ¼
2
p3=2 a Nb b1 g pffiffiffi L A
r
B2 exp A
1 and u of CXCUR photons.
photon energy equal to E = 13.6 GeV and hxr ¼ 8:3 keV the condition 1r 11 is satisfied for Si CU with AU = 3 nm and L = 20.7 lm. The below given numerical results have been calculated with the help of (14)–(16) for the following beam parameters of LCLS. The particle beam parameters are: NB = 1.56 109, b1 = 1, E = 13.6 GeV, h xr = 8.3 keV, rz = 9 104 cm, rr = 6.12 104 cm. The CU parameters are: particles are channeled between the (1 1 0) crystallographic planes of a silicon crystal with interplanar distance d = 0.192 nm, the period and oscillation amplitude of CU are equal to L = 20.7 lm and AU = 3 nm, respectively. In the above calculations and results given for unit length CU it has not been taken into account the absorption of the CXCUR photons produced homogeneously along their entire path in CU. Having the above calculated number N CXCUR for unit length of CU one can calculate N CXCUR ðLCU Þ, the number of photons coming out from a CU with length LCU using the formula [38].
NCXCUR ðLCU Þ ¼ NCXCUR Labs ½1 expðLCU =Labs Þ
As it is seen from (9)–(12) the CXCUR intensities are large if simultaneously 1 1R and u 0, which is possible if 1R 11 . One can show that such conditions will be fulfilled with necessary accuracy, if by an appropriate choice of the available parameters it is possible to obtain 1R ¼ 11 þ B=A. Then, taking into account that for real parameters 1/ << 1 and 1ðuÞu2 << 1, as well as that the spectral distribution at 11 is not sharp, but have finite slope with width much wider than the distributions (11) and (12) due to finite length of the undulator, Dc=c, etc., one can present the spectral and angular distributions of CXCUR in the forms.
! ð16Þ
3. Numerical results and discussion Before beginning the description of the calculations let us note that the CXCUR intensities, calculated with the help of the formulae (11) and (12) when 1r is significantly larger than 11 , are very low because of the second terms in the exponents. For this reason in the below calculations with appropriate choice of amplitude AU and period L of CU for the given particle energy we try to achieve 11 close to 1r and use the formulae (14)–(16). For particle and SASE
65
ð17Þ
where NCXCUR is given either by (14) or by (15), and Labs is the absorption length of the produced monochromatic photons. As it follows from (17) the number of the produced photons increases with the increase of CU length LCU and then is saturated when LCU P ð3 4ÞLabs giving N CXCUR ðLCU Þ N CXCUR Labs . Therefore, it is not reasonable to take LCU greater than 4Labs . For CU made of Si and hxr = 8.3 keV one has Labs ð8:3 keVÞ = 77.5 lm. Therefore, we take ðLCU ¼ 4 Labs = 310 lm giving CU oscillation number Nosc 15, though during the calculations of the number of the CXCUR photons we correctly have taken Leff ¼ Labs = 77.5 lm. Let us note that the dechanneling length for 13.6 GeV electrons is equal to LeDech 13ðlm=GeVÞEe ðGeVÞ = 177 lm and LeDech 90 lm according to [34] and CATCH simulations [39], respectively. Unfor tunately for electrons the experimental data on LeDech are sparse, but even these data show once more that it is not serious to speak about CU for electrons. Therefore, our further considerations concern positrons. For 13.6 GeV positrons channeled between (110) of Si crystal LeDech 8160 lm, much larger than ðLCU = 310 lm.Therefore, one can neglect dechanneling in the above calculations. Fig. 1 shows the spectral and angular distributions of CXCUR produced in a ðLCU = 310 lm long CU calculated with the help of (14) and (15) for the above parameters. As it is seen maximum of the angular distribution of CXCUR is at the same angles and it has the same width as the maxima and widths of the angular distributions of CXTR, CXRTR and CXDR (see [16,17]). Again the emission angles of CXCUR are much smaller than those of UR and bremsstrahlung, XTR, RTR angular distributions which are around u 1 or h 1=c. The total number of CXCUR photons calculated by numerical integration of the curves of Fig. 1 and with the help of the formula (16) is equal to N CXCUR = 1.2 1010. Let us note that this number is
66
L.A. Gevorgian et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 63–66
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