Proceedings of EPAC 2002, Paris, France

TRANSITION AND DIFFRACTION RADIATION BY RELATIVISTIC IN A PRE-WAVE ZONE

S.N. Dobrovolsky, N.F. Shul'ga, NSC “KIPT”, Kharkov, Ukraine Abstract depended on the transversal size of target in the The long- transition radiation from the millimeter and submillimeter waverange. ultrarelativistic bunched electrons on the thin metallic The sufficiently different situation can happen when we transverse-limited target has been considered. We have have small detector, which is placed not far from the investigated the properties of transition radiation and target. Because the transition radiation is formed on the whole electromagnetic energy flux in a pre-wave zone. target's surface, the effect of interference between The general formula for spectral-angular density of radiation from various elementary radiators is possible. electromagnetic energy flux through the small-size Therefore, the conditions of near-field and far-field detector in the "forward" and "backward" direction is radiation will be determined by diameter of extended obtained and analysed. We have showed that the angular source of radiation. The longitudinal length of near-field distribution of electromagnetic energy flux in the "pre- radiation region has the same value as "coherent length'' wave zone" is strongly distorted in comparison to the case in the "forward” direction. But this interference effect will of transverse-unlimited detector. occur even for "backward" transition radiation alone. Longitudinal size of near-field region for "backward" 1 INTRODUCTION direction will be much larger, than "backward coherent length", which has size nearly λ . The investigation of The long-wavelength transition radiation by relativistic such effect in the case of infinite target was performed electrons has various applications in the modern physics recently for "backward" radiation in small-angle of accelerated particles. Coherent transition radiation approximation [3]. In present paper we will consider emitted by short bunches of charged particles that passed transversal spatial distribution of near-field "backward" through a thin plate, is a powerful tool for non-destructive transition radiation and "forward" electromagnetic energy beam diagnostics. On the basis of transition and flux through small detector, which is placed in the near- emitted by fast particles in the field region for various distances from the target and multilayer plates, the tabletop soft X-ray sources are arbitrary angles of observation. developed. Transition radiation has been used for detection of high-energy particles and diagnostics in the LHC-based experiments. Recently, the coherent transition 2 ELECTROMAGNETIC FIELDS AND and Cherenkov radiation has been used for RADIATION INTENSITY electromagnetic cascades detection in the number of We will consider distribution of electromagnetic fields experiments dealing with high-energy cosmic neutrino after passage through a target. We will detection problem. investigate the typical case of normal electron passage For long-wavelength region the transition radiation through the thin ideally conducted plate. We will consider from relativistic particles is formed in the space region, the case when plate thickness is less than radiated which has macroscopic sizes. It can lead to the situation wavelength λ but much more than electron's field when both transversal and longitudinal sizes exceed the "penetration length" into a metal. We assume that the size of experimental setup. It is well-investigated situation electron is moving along the OZ axis and target is when extended detector is placed in the limits of positioned in the plane z = 0 . Let's write in space z > 0 "coherent length" (longitudinal size of formation region), < + − where the interference between radiation fields and and z 0 the electric fields E (r,t) and E (r,t) in the electron’s own field is occurred. At typical angles form ϑ ≈ γ −1 of transition radiation the "coherent length" is E+ (r,t) = E(e) (r,t) + E′(+) (r,t) , E− (r,t) = E′(−) (r,t) , (1) determined by the relation l ≈ 2γ 2λ [1], where γ is a coh λ ± Lorenz-factor of electron and is a radiated wavelength. where E′( ) (r,t) is a "forward" and "backward" radiation For example, it is obvious that interference effects will (e) happen within tens of meters for far-infrared transition fields and E (r,t) is an electron’s own field, that radiation by 100-200MeV electrons. Recently, we called uniformly moved with velocity v in the vacuum. The attention to the fact, that the transversal size of formation Fourier component of the E (e) (r, t) field with respect to region γλ can reach macroscopic values too [2]. time is determined for relativistic electrons, with accuracy Therefore, the diameter of formation region may be γ −1 comparable with size of target and may strongly affect the of , by the expression spectral-angular density of transition radiation. Transition radiation intensity in this case should be strongly

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ω  ωρ   ω  substitution of electron’s own field and radiation field ()e ()≈ ρ e     Eω r 2K1  exp iz , (2) from (2) and (4) into (6) the following expressions for ρ v 2γ  vγ   v  spatial distribution of "forward" and "backward" electromagnetic energy flux are obtained where e - is an electron's charge, v = v , ρ - transversal = + ⋅ ± coordinate ( r ρ z e z ), K1 – McDonald's function of dS 2 = e 2 ϑ× γ −1 ()ϑγ −1 ⋅Θ − first kind. y cos K1 y sin (z) dωdΩ β 4cπ 2 Target is an ideally reflecting screen, and therefore we 2 , (7) can write in the plane z = 0 next conditions for "forward" 1 ϑ 2 J1 (xysin )  −1 2  (±) − β x dx expiqβ (β 1− x −1) ′ ∫ 2 −2 and "backward" radiation fields Eω (r) +γ   0 x ′ (±) (e) Eω (r) = −Θ(a − ρ)Eω (r) ,0z = , (3) where y = ω ⋅ r ⋅ c −1 , q = y cosϑ , u = ω ⋅ a ⋅ (c ⋅γ )−1 , Θ Θ −1 where (x) - is a Heaviside step function: (x)=1 , if β = v ⋅ c . ≥ Θ x 0 and (x)=0, if x<0. The "forward" and "backward" The obtained expressions describe the energy flux in radiation fields are propagating in positive and negative "forward" and "backward" directions through the small direction of OZ axis as a wave packet of free detector at the point. r . It is obvious that "forward" electromagnetic waves. To determine radiation fields in energy flux is differing from "backward" energy flux by the whole "forward" and "backward" space we have to the presence of electron’s own field term. expand the bounded wave packets (3) to the full Fourier components and perform reverse Fourier transformation into space. By taking into account the (2) and (3) 3 "BACKWARD" AND "FORWARD" equations we will obtain the expression for the "forward" ELECTROMAGNETIC ENERGY FLUX and "backward" radiation fields in the point r Let’s analyze the formula (7). We will consider the behavior of intensity of electromagnetic energy flux for  2  ω 2 J (χρ) ⋅exp± iz ()− χ  various distances from the target. ∞ 1  c  ′(±) 2e ρ 2 First, we will analyze and simplify the value Φ under Eω (r) = − ∫ χ dχ .(4) v ρ  2 various distances from the target. 0 χ 2 + ω   vγ   −1 2  1 J (xy sinϑ) ⋅ expiqβ (β 1− x −1) Here J is a Bessel function of first kind. This expression 1   1 Φ = ∫ x 2 dx . (8) 2 + γ −2 is correct for all distances greater than radiated 0 x wavelength from the target. Now we can consider the electromagnetic energy flux, For distances y >> γ (i.e. not closely to the target) we which crosses over the entire observation time through the small plate that is perpendicular to the vector r and can perform the expansion in series for small x in the placed at the distance r from the target. We should expression (8) and following calculate the value of Poynting vector of electromagnetic γ flux − − J (uη) Φ = γ 1 exp()− iξβ 1 ⋅ ∫ u 2 du 1 exp()− iu 2ξ , (9) 2 + c 0 u 1 S = ∫ (E × H)n d 2σ dt , (5) 4π ϑ ϑ ξ = y cos η = y sin 2 where , . It is necessary to note where n = r / r and d σ is an elementary area with 2γ 2 γ normal vector n . Here the integration is over the surface that ξ is a ratio between longitudinal distance and of detecting plate and the time of observation. Now we 2 can write "backward" transition radiation intensity and longitudinal "coherent length" l = 2γ c ω and η is a summary energy flux in the "forward" direction on a unit ratio between transversal distance and effective diameter of frequency and on the unit of solid angle by the of "formation zone" λγ . Thus, the behavior of spatial following equation distribution depends on “near-field” zone sizes. 2 ± η 2 ξ << ≤ dS cr ± 2 Under conditions 1 and 1 we can obtain the = Ee (ρ, z) ⋅ Θ(z) + E′( ) (ρ, z) ,0ω > , (6). 4ξ ω Ω π 2 ω ω d d 4 final expression for the value Φ by some calculations

Ω = ϑ ϑ ϕ ϑ ϕ − Here d sin d d ( and - is a polar and exp()− iξβ 1  η 2   η 2  ϑ Φ = 2i expi sini  . (10) azimuthal angles) and the angle is counted out γη  8ξ   8ξ  negative and positive directions of OZ-axis for "backward" and "forward" radiation accordingly. After

1868 Proceedings of EPAC 2002, Paris, France

Taking into account the formula (10), let's express the formula (7) for "backward" direction. Thus, under lg(W(Θ)) conditions ϑ <<1, rω cosϑ(2cγ 2 ) −1 < 1, we can write the distribution of "backward" transition radiation intensity

− 2 ϑ  2 ϑ  dS 4e cos 2 y sin ≈ sin   . (11) dωdΩ cπ 2 β 4 sin 2 ϑ  4cosϑ 

This formula gives the spatial distribution of radiation intensity in the pre-wave zone, not closely to the target. We can conclude from (11) that angular maximum of transition radiation intensity is located at angle ϑ ≈ 4y −1 . It is substantially different from well-known case of peak under γ −1 . This result is differing from the Fig.1 Spectral-angular density of electromagnetic 2 -2 case of interference between electron’s own field and energy flux W(Θ)=cπ e (dS/dωdΩ) in the "forward" radiation fields too. Here we are observing the direction for electron with γ = 30 . Solid curve interference between radiation from various elementary corresponds to the case when y = 50 , dotted curve - places of target's surface, i.e. the "transition radiation self- y = 300 and dashed curve - y = 3000 . interference". This effect occurs even for "backward" 2 direction. The near-field condition 2γ λ > z is 4 CONCLUSIONS determined by the size of effective diameter of formation The long-wavelength transition radiation from the λγ zone and characteristic angle of radiation from each ultrarelativistic electrons on the thin metallic transverse- ϑ ≈ γ −1 limited disk has been considered. The spectral-angular elementary radiator eff . For far-field distances distribution of "forward" electromagnetic energy flux 2 2γ λ << z the whole surface can be considered as a through the small detector is analyzed. It is shown that pointed source of radiation and vice versa, but at near- interference between radiation fields from various field zone we must consider it as an extended source. elementary radiators and interference between electron’s Spectral-angular electromagnetic energy own field and radiation fields are occurred. We have distribution in the "forward" direction for the distances showed that the angular distribution of electromagnetic 2γ 2λ > z > γλ is described by the following formula energy flux in the "pre-wave zone" is strongly distorted as + compared to the case of transverse-unlimited detector. dS e2 ≈ y 2 cosϑ × A number of experiments concerning long-wavelength dωdΩ cπ 2 β 4 transition radiation of relativistic electron bunches were 2 . (12) performed in the last years [4-5]. For the typical − − 2  ysinϑ  × γ 1K ()y sinϑγ 1 − β sin  experimental conditions (electrons with γ ≈ 10 ÷100 , 1 ϑ  ϑ  ysin 4cos long-wavelength transition radiation with λ ≈ 1÷10 cm. ) both the longitudinal and transversal sizes of "pre-wave The expression (12) describes the "forward" spectral- zone" are the macroscopic values and can be compared angular distribution of energy flux at the whole with distance to the detecting setup. Thus, the considered observation time through the dot-like detector that is interference effects can play very important role in such placed near-field region. Given density is combined by experiments. Therefore, it is necessary to take into the energy flux from electron’s own field and by the account this effect under experimental data analysis. transition radiation energy flux. Fig.1 illustrates the spatial distribution of “forward” energy flux for different 5 REFERENCES distances from the target. For small-angle region (i.e., when detector is placed closely to the electron trajectory) [1] V.L. Ginzburg, V.N. Tsytovich, “Transition radiation the term that corresponds to electron’s own field and transition scattering” Adam Hilger, Bristol, 1990. contribution will dominate. For large-angle region (i.e., [2] N.F. Shul'ga, S.N. Dobrovolsky, “Theory of when detector is located farther than λγ from electron relativistic electron transition radiation in a thin metal target”, JETP, 90, pp.579-583, 2000. trajectory) the radiation field contribution for the [3] V.A. Verzilov, “Transition radiation in the prewave summary energy flux will determine the distribution zone”, Phys. Lett. A 273, pp.135-140, 2000. behavior. At intermediate angles we can observe [4] X.Artru et al. “Resolution power of optical transition sufficient distortion, which is caused by both the radiation: theoretical considerations” NIM B145, pp.161- interference of radiation from elementary radiators on the 168, 1998. radiated surface (target) and the interference between own [5] Y.Shibata, et al. “Coherent transition radiation in the and transition radiation fields within the "coherent far-infrared region”, Phys. Rev. E49, pp.785-793, 1994. length".

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