Transition radiation and di raction radiation. Similarities and di erences

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1 Nuclear Instruments and Methods in Physics Research B 145 (1998) 169±179 Transition radiation and di raction radiation. Similarities and di erences A.P. Potylitsyn 1 Institute for Nuclear Physics, Tomsk Polytechnic University, pr. Lenina 30, Tomsk, Russian Federation Received 13 December 1997; received in revised form 08 April 1998 Abstract The characteristics of di raction radiation (DR), i.e. radiation of the charged particle moving near conducting target have been considered for an ultrarelativistic case. The simple expressions for DR elds for the semi-in nite ideal conducting target have been derived. The close connection between transition radiation (TR) and DR has been shown. The e ect of nite transversal sizes of target on TR characteristics has been evaluated. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: Nr 1. Introduction As is generally known, transition radiation (TR) is generated when a charged particle crosses the interface between two media even in the case of its rectilinear motion. TR is caused by a change in the optical properties of the medium along the charge path. On the other hand, having a rectilinearly moving constant-velocity charge in vacuum in close proximity to a target (optical inhomogeneity), there also emits an electromagnetic radiation (the so-called di raction radiation, DR), if the distance to the target h (the impact parameter) satis es the condition h < ck=2p: 1 Here c is the Lorentz factor of the particle and k the DR wavelength. In both cases the radiation produced is due to the presence of optical inhomogeneities in the space (1), where the eld of the traveling charge induces, on the surface of the target, currents changing in time, and it is these currents that eventually give rise to the radiation. 1 Tel.: ; fax: ; pap@phtd.tpu.edu.zu X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S X ( 9 8 ) X

2 170 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169±179 The classical theory of TR reported, for instance, in [1±3] is being used to an advantage in various elds of experimental physics such as constructing elementary particle detectors (X-ray range), diagnostics of charged particle beams (visible light) or generation of coherent radiation in microwave range. It is noteworthy that the TR theory [1±3] has been formulated for an in nite boundary between two media or, in other words, for the case when the following condition holds valid: ck a; 2 where a is the transversal dimension of the target crossed by the particle. The rst theoretical works on DR, i.e. the radiation produced when a charged particle moves near a conducting screen, were published over 30 years ago [4±6]. Later, in a review [7] and in a monograph by M.L.Ter-Mikaelian [1] the DR characteristics were studied in greater detail, however no experimental work on DR of relativistic particles has appeared till recent time. This fact seems somewhat odd, since in its mechanism DR is comparatively close to TR, hence the similarity in properties and characteristics of both radiation types. The present paper considers the connection between TR and DR for an ultrarelativistic case and shows that the major DR characteristics (directivity, width of radiation cone, intensity) are close to those of TR. 2. DR for semi-in nite screen An exact theory of DR for a tilted ideally conducting semi-in nite plane was designed in [6]. Fig. 1(a) depicts the geometry used in the work cited. The angle between the trajectory of the particle and the screen plane is termed as h 0 and the minimum distance to the edge as a. According to [6] the spectral-angular density for the DR intensity can be written in the following form: q d 2 W dxdx ˆ a exp x x c 1 b 2 c 2 cos 2 w 2p 2 sin w n cos 2 u 2 cos2 w 1 b sin w cos h 0 c 2 b 2 cos 2 w sin 2 u o 2 1 b sin w cos h 0 ( " c 2 b 2 cos 2 w sin w cos u cos h # ) 2 0 c 2 b 2 cos 2 w sin 2 h 1 0 b : 3 Here a is the ne structure constant, x c ˆ cb=2a (throughout the paper h ˆ m ˆ c ˆ 1 ; b the particle velocity, the angles w and u are shown in Fig. 1(a). Characteristic energy x c is determined by the particle energy and the distance to the screen and for reasonable values of c and a it corresponds to the range where the ideal screen approximation is valid. It can be easily shown that the density reaches its maximum value (3) at minimum values of the brackets in the denominator. Hence it follows cos w ˆ 0; cos u ˆ cos h b : 4 Conditions (4) determine the directions where the DR intensity would be maximum. For relativistic particles c 1 from (4) we deduce two solutions w ˆ p 2 ; 5a b 2 u ˆ h 0 : 5b

3 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169± Condition (5a) determines the ``di raction'' plane which contains a normal to the surface of the screen and two ``DR rays'' conditioned by (5b). The rst of the ``rays'' coincides with the direction of the initial particle u ˆ h 0 and the second corresponds to the mirror re ection from the screen u ˆ h 0 (see Fig. 1(a)). Let us consider the angular distribution of DR across the ``re ected'' ray. To do so let us use a more convenient system of coordinates (the z-axis along the ray and the x-axis along the edge of screen): h x ˆ w p 2 1; h y ˆ u h 0 1: 6 Upon inserting (6) into (3) followed by trivial simpli cation procedures we have (neglecting the terms higher than the second-order in nitesimal): d 2 W dxdx ˆ a 4p exp x q 1 c 2 x 2 h 2 c 2 2h 2 x x c c 2 h 2 x c 2 h 2 x h2 y : 7 It follows from the above that for h x ˆ h y ˆ 0 in the angular spread there is only one maximum: d 2 W m dxdx ˆ a 4p 2 c2 exp x ; x c 8 Fig. 1. Geometry of DR near the semi-in nite target; (a) projection of electron momentum onto the target plane is perpendicular to the edge; (b) projection is parallel to the edge.

4 172 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169±179 whose magnitude di ers from the maximum q value of the TR intensity by an exponential term (note that TR reaches its maximum value at h ˆ h 2 x h2 y ˆ c 1. For an ideal screen expression (7) describes the spectral-angular density of both rays. From (7) one can derive the DR energy for each cone W ˆ Z 1 1 Z 1 Z 1 dh x dh y 1 0 dx d2 W dxdx ˆ 3 8 ax c: 9 It is worth mentioning that the result obtained is not a ected by the tilt angle of the target h 0. The overall DR energy is obtained through doubling expression (9) W P ˆ 3 4 ax c: 10 This result was obtained by authors of [6] but after cumbersome calculations. For real targets the formulas (9) and (10) will be valid only when x c < x p, where x p is the plasmon energy in the material of the target. In [7] (see also [1]) another geometry of DR was discussed, where the projection of the charged particle momentum onto the plane of the target was parallel to the edge of the target (see Fig. 1(b)). For this geometry exact solutions were obtained for the spectral-angular distribution of DR which are similar to (3): d 2 W dx ˆ ab sin h 0 1 b cos h 0 cos w exp 4ph 2p 2 sin w bk sin h 0 q 1 b cos h 0 cos w 2 b 2 sin 2 h 0 sin 2 w cos2 u cos w b cos h sin 2 u 1 b cos h 2 0 cos w 2 b 2 sin 2 h 0 sin 2 wš 1 b cos h 0 cos w 2 b 2 sin 2 h 0 sin 2 wš 1 b 2 sin 2 h 0 sin 2 w cos 2 u 1 b cos h 0 cos w 2 b 2 sin 2 h 0 sin 2 wš ; 11 the angles w and u in (11) are plotted in Fig. 1(b). It is evident that for the geometry considered expression (11) also has two maxima determining the plane of ``di raction'': w ˆ h; u ˆ p 2 : 12 w ˆ h; u ˆ p 2 : Now, passing over to the angular variables h x and h y that are determined with respect to the ``re ected'' ray as before: h x ˆ u p sin h 0 ; h y ˆ w h 0 : 13 2 One can derive instead of (11): d 2 W dxdx ˆ a 4p exp x q 1 c 2 x 2 h 2 c 2 2h 2 x x c c 2 h 2 x c 2 h 2 y h2 x : 14 After comparison (14) with (7) one may see that the spectral-angular density of DR is the same for both geometries. Result of calculations is presented in Fig. 2. By way of comparing with TR for the case of oblique incidence onto an ideal screen let us resort to the results cited in [1] (see formulas and 25.28). Upon a substitution similar to (13) one can write the following expression:

5 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169± Fig. 2. Angular distribution of DR near the semi-in nite target for k ˆ 0:4ph=c in the plane perpendicular to the edge (solid line) and in the parallel plane (dotted line). Fig. 3. Angular distribution of TR for in nite target for oblique incidence c ˆ 100, h 0 ˆ 45.

6 174 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169±179 d 2 W dxdx ˆ a h 2 x h2 y 1 p 2 c 2 h 2 x h2 y 2 1 h y ctg h 0 ˆ d2 W? 1 2 dxdx 1 h y ctg h 0 : 15 2 In (15) the rst cofactor coincides with the TR intensity for normal intersection of the screen, while the second multiplier describes the angular distribution asymmetry in the plane normal to the target. Fig. 3 shows the TR angular distribution calculated following (15) for h 0 ˆ 45 and c ˆ 100. From (15) one may estimate the asymmetry. r ˆ I R I L 1 I 2 R I L ; where I R;L is the TR intensity in the right and left maxima, i.e. for h y ˆ c 1. In our case for c ˆ 100 we have r 4c 1 ctg h 0 ˆ 4%: Note that distribution (15) agrees well with the numerical results [9] for the optical transition radiation in an aluminium target (the shape of angular distribution is identical to and the absolute value of the yield surpasses the result of [9] by 10%). This striking contrast between the angular distributions of DR and TR results from the asymmetric nature of the problem. For the case of symmetry (passing through the annular center hole or the slit center) DR will have a ``lobe-shaped'' structure in an absolute analogy with TR [1,4,5] Transversal edge e ect in transition radiation Using the results of [6] it is easy to derive the TR characteristics for a particle crossing an ideally conducting semiplane near its edge. For the sake of simplicity let us consider a case of perpendicular crossing h 0 ˆ 90. The TR eld strength for this case can be obtained using the well known principle from the classical optics (see Fig. 4) ~E e h ˆ ~E 1 ~E DR h : 18 Fig. 4. Connection between elds of TR for nite size target ~E e h, for in nite target ~E 1 and eld of DR ~E DR h.

7 In (18) ~E e h ; ~E 1 is the eld strength of TR for intersecting a semi-in nite and in nite target, ~E DR h the eld strength of DR and h the impact parameter. Following [6] we may obtain an expression for transversal components of DR eld for the given geometry: r i 1 cos u p 1 2p cos w c E DR;x ˆ B qx p 2 b 2 cos 2 w q ; k sin w 1 cos u i c 2 b 2 cos 2 w b sin w cos u q r 2p sin u b sin w i c 2 b 2 cos 2 w E DR;y ˆ B qx p q ; 19 k sin w 1 cos u i c 2 b 2 cos 2 w b sin w cos u B qx ˆ ie r r q bk b sin w i c 4p 2 2p 2 b 2 cos 2 w exp j : j ˆ 2p q k h c 2 b 2 cos 2 w; When using standard variables (6) A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169± h x ˆ w p 2 ; h y ˆ u p 2 ; 20 expression (19) is substantially simpli ed: B qx ˆ ie exp j q ; 4p 2 2p k E DR;x ˆ ie h x exp j q q ; 21 4p 2 c 2 h 2 x c 2 h 2 x ih y E DR;y ˆ ie exp j q ; 4p 2 i c 2 h 2 x h y j ˆ 2p q k h c 2 h 2 x: The eld of TR for the case when a charged particle crosses an ideally conducting in nite plane is written in a well-known form [1]: E 1;x ˆ ie h x 2p 2 c 2 h 2 x h2 y ; 22 E 1;y ˆ ie h y 2p 2 c 2 h 2 x h2 y ; Then from (21) and (22) we may readily obtain: 8 q 9 E e;x ˆ ie h < x c 2 h 2 x ih y = 4p 2 c 2 h 2 x 2 exp j h2 : y c 2 h 2 ; ; x E e;y ˆ ie q 1 4p 2 c 2 h 2 x 2h y exp j c 2 h 2 h2 x h y : 23 y

8 176 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169±179 Thus, the spectral-angular density of TR including the edge e ects of the target is written as follows: ( ) d 2 W e h dxdx ˆ 4p2 je e;x j 2 je e;y j 2 ˆ a 1 4p 2 c 2 h e j h 2 e 2j c 2 h 2 c 2 2h 2 x c 2 h 2 x ; 24 where h 2 ˆ h 2 x h2 y. For high values of the impact parameter h ck the spectral-angular density is formally obtained when h! 1: d 2 W e h! 1 dxdx ˆ a p 2 h 2 c 2 h 2 2 ; which agrees with the results for an in nite ideal plane as expected. On the contrary, when h ˆ 0 j ˆ 0 we have the expression d 2 W e h ˆ 0 dxdx ˆ a c 2 2h 2 x 4p 2 c 2 h 2 c 2 h 2 x Near the edges of the target the radiation is formed either due to TR (when the particle intersects the target) or due to DR (when the particles moves near without intersecting the target). The result obtained is consistent with the continuity requirement. Spectral-angular distribution of DR (7) for null impact parameter coincides with expression (26) as expected. Fig. 5 shows the angular distributions of TR for di erent impact parameters h. One may see the transformation of a deep to the maximum with decreasing of an impact parameter. For h ˆ 0 we have the Fig. 5. Angular distribution of TR for nite size target; h ˆ ck=0:4p ± dotted line. h ˆ 0:5 ck=4p ± dashed±dotted line. h ˆ 0 ± solid line.

9 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169± Fig. 6. Spectral-angular density both for DR (j < 0 and TR j > 0). h x ˆ 0, h y ˆ 0 ± solid line. h x ˆ 0, h y ˆ 1:5c 1 ± dashed±dotted line. same distribution for both TR and DR. It is shown in Fig. 6 the yield of TR for j < 0 and DR for j > 0. The value j ˆ 0 h ˆ 0 coincides with edge of target. As an illustration let us consider the in uence of the transversal edge e ect on the TR characteristics for a charge passing normally through round target with the radius R. In [4,5] the authors derived exact expressions for the DR eld with the charged particle passing through the center of an annular opening. For the ultrarelativistic case we have (see [1]): E DR;x ˆ ie h x 2p 2 c 2 h 2 x h2 x J 0 E DR;y ˆ ie h y 2p 2 c 2 h 2 x h2 y J 0 2pR k h 2pR k h ; 27 ; Here J 0 x is the modi ed Bessel function of the zeroth order. Again applying the same approach as before we obtain: ~E TR R ˆ ~E TR;1 ~E DR R : 28 In (28) E TR is TR eld for an in nite boundary surface (see formula (22). From (27) and (28) one easily derives: d 2 2 W e dxdx ˆ d2 W 1 dxdx 1 J 2pR 0 k h : 29 Fig. 7 depicts the angular distributions of TR in a round target with a radius of 50 mm for the particles with c ˆ 2000 and various wavelengths.

10 178 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169±179 Fig. 7. Spectral-angular density of TR for round target with R ˆ 50 mm; c ˆ 2000; k ˆ 0:05 mm ± solid line. k ˆ 0:1 mm ± points and dots. k ˆ 0:25 mm ± points. Dashed±dotted line ± the spectral-angular density of TR for in nite target. As it should be expected from the symmetry considerations, we have a zero density (29) at a zero angle. For the range of angles h c 1, given the relation 2pRh=k 1, from (29) we obtain d 2 4 W e dxdx d2 W 1 prh d2 W 1 dxdx k dxdx : 30 In addition, in this case the positions of the maxima in the angular distribution of TR are shifted into the region of angles h > c 1. In [11] it was suggested to use in coherent TR spectrum of short bunches in order to determine the longitudinal size of the latter. As a TR spectrum of a single particle use was made of the TR spectrum for a target of an in nite transversal dimensions. It is likely, that the above e ect may introduce a systematic experimental error which would grow with increasing c. 4. Conclusion Di raction radiation of non-relativistic electrons is widely used in various radio-engineering devices [12], however, no experiments have been conducted to study the relativistic DR till recently. Ref. [13] seems to be the rst work where DR was experimentally studied in the mm - range for the relativistic electron beam passing through an annular opening. The experimental set up, however, included a mirror on the electron beam path, which brought about the interference between DR and TR generated in the mirror material. It is believed that DR with the relativistic electrons moving close to the inclined screen (see Fig. 1) may be a promising convenient object of experimental studies, because in this case the screen itself acts as a mirror, while the beam moves in vacuum.

11 A.P. Potylitsyn / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 169± It should be emphasized that optical DR is a good candidate for non-destructive beam diagnostics, including the proton beams [14,15]. When choosing x c in the visible range, e.g. x c ˆ 1:6 ev k ˆ 0:8 mcm and the electron energy E ˆ 1 GeV (c ˆ 2000), the impact parameter can be taken of about 0.3 mm. The DR photon yield in this case would be an order of magnitude lower than that of optical TR. Furthermore, DR generated by a beam of particles passing through a slit in a titled screen may provide an additional information on the beam parameters due to interference of the DR elds from both edges of the slit. Acknowledgements The author is grateful to Prof. M. Ikezawa, Prof. X. Artru and Dr.Y. Shibata for stimulating discussions. The author also appreciates the assistance of T.D. Litvinova, L.V. Puzyrevich and O.V. Chefonov in preparing the text of the paper. References [1] M.L. Ter-Milaelian, High-Energy Electromagnetic Processes in Condensed Media, Wiley/Interscience, New York, [2] G.M. Garibian, C. Yuan, Rentgenovskoe perekhodnoe izluchenie, Erevan, 1983 (in Russian). [3] V.L. Ginzburg, V.N. Tsytovich, Transition Radiation and Transition Scattering, Adam Hilger, Bristol, [4] Yu.N. Dnestrovskii, D.P. Kostomarov, Sov. Phys. Dokl. 4 (1959) 132. [5] Yu.N. Dnestrovskii, D.P. Kostomarov, Sov. Phys. Dokl. 4 (1959) 158. [6] A.P. Kazantsev, G.I. Surdutovich, Sov. Phys. Dokl. 7 (1963) 990. [7] D.M. Sedrakian, Izv. AN ArmSSR, 17 (N4) (1964) 103 (in Russian). [8] B.M. Bolotovskii, G.M. Voskresenskii, Sov. Phys. Usp. 9 (1966) 73. [9] L. Wartski, S. Roland, J. Lassale et al., J. Appl. Phys. 46 (1975) [10] Y. Shibata, K. Ishi, T. Takahashi et al., Phys. Rev. E 49 (1994) 785. [11] Y. Shibata, T. Takahashi, T. Kanai et al., Phys. Rev. E. 50 (1994) [12] V.P. Shestopalov, Di raktsionnaya electronika, Kharkov, 1976 (in Russian). [13] Y. Shibata, S. Hasebe, K. Ishi et al., Phys. Rev. E. 52 (1995) [14] D.W. Rule, R.B. Fiorito, W.D. Kimura, Non Interceptive Beam Diagnostics Based on Di raction Radiation, to be published in the Proceedings of the Seventh Beam Instrumentation Workshop, Argonne IL, [15] M. Castellano, Nucl. Instr. and Meth. A 394 (1997) 275.

A New Non Intercepting Beam Size Diagnostics Using Diffraction Radiation from a Slit

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