Simulation of Coherent Diffraction Radiation Generation by Pico-Second Electron Bunches in an Open Resonator

Transcription

1 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 Simulation o Coherent Diraction Radiation Generation by Pico-Second Electron Bunches in an Open Resonator L G Sukhikh, A P Potylitsyn and D A Verigin Tomsk Polytechnic University, Lenin avenue, 3, Tomsk, 6345, Russia sukhikh@tpu.ru Abstract. In this report we present new approach or calculation o processes o diraction radiation generation, storage and decay in an open resonator based on generalized surace current method. The radiation characteristics calculated using the developed approach were compared with those calculated using Gaussian-Laguerre modes method. The comparison shows reasonable coincidence o the results that allows to use developed method or investigation o more complicated resonators. 1. Introduction Stimulated radiation is responsible or optical and inrared laser operation principle based on atomic properties o various materials. Later the term laser was used to speciy light generated in undulators in electron accelerators known as Free Electron Lasers (FEL) [1, 2]. In this case electrons moving on a periodical path generate synchrotron radiation. The radiation traveling along the beam in the undulator starts interacting with the electrons stimulating the production o the radiation with the same wavelength and phase, which are the main properties o the laser light. In the experiments [3, 4] the authors observed Stimulated Transition Radiation (TR) and Stimulated Synchrotron Radiation respectively, which, in contrast to undulator radiation, had a continuous spectrum. Authors o Re. [3] used a special 4-mirror resonator in order to ocus coherent TR generated by the electron bunch on the both suraces o the target alternatively in the moment o passage o the subsequent electron bunch. In papers [4, 5] the authors proposed a concept o the so-called broadband FEL based on stimulated TR in the closed cylindrical resonator where resonator mirrors were used at the same time as TR targets. Aterwards the radiation is recirculated in the resonator. As a result every subsequent bunch generates radiation in the presence o radiation ield rom all preceding bunches stimulating the radiation production. In Re. [6] we proposed to generate intense THz radiation in an semi-conocal resonator to stimulate coherent diraction radiation (DR) generated both by the lat mirror in orward direction (FDR) and by the concave mirror in backward direction (BDR). Due to DR noninvasive nature [7] there was no direct interaction o the electron beam with the mirrors resulted in ine background conditions and the beam parameters remaining almost the same as the initial ones. It was demonstrated that the radiation intensity stored in such a resonator growed up exponentially as a unction o the number o bunches, which has an obvious analogy with Content rom this work may be used under the terms o the Creative Commons Attribution 3. licence. Any urther distribution o this work must maintain attribution to the author(s) and the title o the work, journal citation and DOI. Published under licence by Ltd 1

2 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 Figure 1. Simulation scheme. undulator FELs, where the intensity grows up exponentially as a unction o the number o undulator periods [6]. The aim o this paper is to show the analytical way or calculation o diraction radiation characteristics in the open semi-conocal resonator taking into account radiation generation in the presence o external ield, its storage and resonator ree runs. 2. Theoretical model o DR generation in open resonator Passing through openings in the irst and second resonator mirrors point-like electron bunches with a charge q generate FDR and BDR, consequently. The irst bunch passes through the opening in the irst mirror generating FDR that travels along z-axis together with the electron. The slippage eect could be neglected due to a broadband DR spectrum. The bunch reaches the second mirror and generates BDR in presence o the FDR rom the irst mirror. Both BDR and relected FDR travels back and reaches the irst mirror at the moment when the second bunch arrives. Thus, both FDR and BDR are generated in the presence o the external electric ield that was generated previously. Let us ormulate the problem as ollows. Train o the point-like electron bunches with 43 MeV energy travels along z-axis passing through an open resonator that consists o lat and ocusing mirrors(seeigure1). Thedistancebetweenbunchesinthetrainisequalto84mm(2.8ns)while the distance between resonator mirrors is equal to 42 mm. The second mirror is paraboloidal onewithradiusocurvatureequaltor = 84mm(ocal distance = 42mm). Inthetransverse planebothmirrorsareassumed tobecircles withadiameter equal to a = 1mm. Theopenings in the mirrors are assumed to be round ones. The opening diameter in the ocusing mirror is equal to a 1 = 5mm, in the lat mirror a = 5mm. The resonator parameters coincide with ones rom the previous experiment [6]. For our calculation we assumed that both mirrors were perect conductors. According to generalized surace current method [8] the electric ield o FDR generated by the irst bunch E (1) could be written as integration o the double cross-product: E (1) (r,ω) = 1 ds 1 [n 1,E e (r,ω)] G(r,r,ω), (1) 2π S 1 where r is the observation point; r is the radiation point; ω is the radiation angular requency (ω = 2πc/λ, c is the speed o light); n 1 = {,,1} is the surace normal to the irst mirror; E e (r,ω) is the electric ield o the bunch; G(r,r,ω) is the gradient o ree space Green unction. Theintegration iscarriedoutover thesuraceothelatmirrors 1, whereds 1 = ρdρdφ. 2

3 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 The electron bunch ield E e (r,ω) could be written as ollows [8]: E e = q ω πc 2 β 2 γ ei ω βc z { ρ ρ K 1 [ ] ωρ i [ ]} V ωρ βcγ γ βc K. (2) βcγ Here q is the bunch charge, β = v/c is the electron reduced velocity, i = 1, ρ = {x = ρcosφ,y = ρsinφ}, V = {,,βc}, K n is the modiied Bessel unction o the second kind o the n-th order. The gradient o the ree space Green unction G(r,r,ω) could be written as ollows: G(r,r,ω) = (r r ) exp[ i ω c r r ] r r 3 ( i ω ) c r r 1 Ater the FDR is generated by the irst bunch it travels in the resonator along z-axis. Both electron bunch and FDR reach the second mirror almost at the same time. In this case BDR is generated in the presence o external ield. Thus, electric ield o BDR E (1) b (r,ω) could be written as ollows: E (1) b (r,ω) = 1 2π { } where n 2 = 2 ρcosφ 2, ρsinφ 2, x 2 +y 2 S 2 ds 2 (3) [ n 2,(E e (r,ω)+e (1) ] (r,ω)) G(r,r,ω), (4) is the normal vector to the parabolic mirror surace [9]. The 4 integration is perormed over the surace o the parabolic mirror S 2, ds 2 = 2 +ρ 2 ρdρdφ. 4 Ater generation o BDR the irst bunch leaves the resonator. The BDR travels back to the irst mirror and reaches it almost in the same time with second bunch resulting in the generation o the FDR in the presence o the external ield. In this case the ormula to ind ield o FDR looks similar to equations (1),(4): E (2) (r,ω) = 1 [ ds 1 n 1,(E e (r,ω)+e (1) b ] 2π (r,ω)) G(r,r,ω), (5) S 1 Thus, the problem consists o the calculation o radiation spatial distributions on the mirror suraces generated by the electron bunch train. In the case o i-th bunch the FDR and BDR could be calculated in the ollowing way: E (i) (r,ω) = 1 [ ds 1 n 1,(E e (r,ω)+e (i 1) ] b (r,ω)) G(r,r,ω), 2π S 1 E (i) b (r,ω) = 1 [ ds 2 n 2,(E e (r,ω)+e (i) ] (r,ω)) G(r,r,ω) 2π S 2 The equations (6) allow to calculate radiation characteristics in the resonator or all electron bunches one by one. Ater the bunch train leaves the resonator the radiation enhancement stops and decay process starts (resonator ree runs). The decay o stored radiation could be calculated using the same ormalism as described beore but assuming that the bunch ield is equal to zero. There exists a signiicant diiculty that makes it almost impossible to carry out direct calculation o radiation ield in the resonator using the proposed approach. Each radiation generation event adds double integration in the inal result. For example, in order to calculate FDR rom the second bunch (equation (5)) one needs to calculate six-dimensional integral. In order to overcome this problem the ollowing assumption was made. Calculating BDR rom the irst electron bunch (equation (4)) one needs to know distribution o the FDR rom the irst bunch on the surace o the ocusing mirror. Such a distribution could be calculated easily using (6) 3

4 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 Figure 2. Comparison o calculated ield distribution and its interpolation. Let plot real part o E x -component, right plot imaginary part o E x -component. Red circles cross-section o calculated E x -component, blue line cross-section rom interpolation 2D surace, y =, λ = 3.5mm. equation (1) that allows numerical calculation o the spatial distribution o all three components o the electrical ield on the mirror surace. Practically these ield components are calculated in a inite number o points. Amount o such points should be enough or correct representation o the ield. Thus, one can easily calculate the FDR ield distribution in the speciied number o points situated on the mirror surace and to interpolate obtained distribution assuming its smooth character. In this case the numerical calculation o BDR will include just double integration over the ocusing mirror surace that could be easily carried out numerically. Our calculation was carried out using Wolram Mathematica 7. sotware [1]. The FDR ield distribution on the surace o ocusing mirror was calculated in the range 5 x,y 5mm with the step equal to 2mm. The calculated ield consists o 6 components, namely: E (1) (r,ω) = {R(E x )+ii(e x ),R(E y )+ii(e y ),R(E z )+ii(e z )} (7) Each ield component was interpolated by 2D surace using spline interpolation procedure in Wolram Mathematica 7.. Figure 2 demonstrates comparison o calculated ield distribution (x-component o FDR ield) and results o interpolation by 2D suraces in the orm o crosssections along x-axis (y = ) both or real and imaginary parts o the ield. The calculation was carried out at λ = 3.5mm. One can see in igure 2 that coincidence between calculated ield distribution and its interpolation is good and obtained distribution o the ield is a smooth unction that allows to use proposed method or calculation o urther relections o the ield in the resonator avoiding multiold integration. The spatial distributions o DR on the mirror suraces has azimuthal symmetry as it was expected rom the problem geometry [7]. Both distributions have minimum in the center and one maximum, i.e. one could observe double-lobe distribution expected rom the characteristics o DR. The position o maximum diers or lat mirror and concave mirror due to the act that concave mirror naturally ocuses the radiation on the surace o the lat mirror. 3. Gaussian-Laguerre modes in semi-conocal resonator In a resonator electromagnetic wave is conined by its mirrors. In this case it could be described by superposition o plane waves or transverse electromagnetic modes which occurs in the resonator. Dierent solutions or modes are used or dierent types o symmetry. Gaussian- Laguerre modes (GLM) are used or calculation o transverse distributions o radiation stored in conocal resonators with axial symmetry. In order to apply these modes to our problem we expanded our semi-conocal resonator and assumed that we have conocal resonator that consists o two ocusing mirrors with curvature radii equal to R = 84mm. The intermirror 4

5 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 Figure 3. Comparison o radial distribution o GLM (blue dashed curve) and DR in the resonator ater 15 electrons (red solid curve). The calculation was carried out at λ = 3.5mm distance is also equal to L = 84mm. Using Gaussian-Laguerre modes one could estimate the radiation spatial distribution on the surace o ocusing mirror and in the center o resonator where the radiation waist is situated. According to [11] the amplitude o GLM m,n in the radiation waist could be described as ollowing: ( ) m ] ρ 2 Ψ mn exp[ (ρ,ϕ) = ρ2 w w 2 L m n ( 2ρ2 w 2 ) e imϕ. (8) Here w = Lλ 4π, Lm n is the generalized Laguerre polynomial. Fresnel transormation o GLM allows to obtain amplitude o radiation at any distance z rom the waist position inside resonator: Ψ mn (ρ,ϕ,z) = w ( ) m ρ 2 w(z) w(z) exp [ ρ2 w 2 (z) + ikρ2 2R(z) +imϕ i(2n+m+1)arctan ( 4z L ) ] L m n ( 2ρ 2 w 2 (z) ( ) 4z 2 ( ) ] L 2 Here w(z) = w 1+, R(z) = z[ 1+, k = 2πλ 1. L 4z Due to the act that the DR distribution has double-lobe structure [7] we took into account only GLM Ψ m. Figure 3 shows comparison o radial distribution o GLM and DR calculated using equations (6) or 15 point-like electron bunches on the suraces o both mirrors. GLM on the surace o concave mirror was calculated in the orm Ψ 1 2 assuming z = 42mm. GLM on the surace o lat mirror was calculated in the orm Ψ 1 2. During the calculation we keep λ = 3.5mm. In igure 3 one can see that radial distribution o DR calculated on the surace o concave mirror coincides rather well with Gaussian-Laguerre mode Ψ 1 2. Coincidence o radial distributions o DR and GLM Ψ 1 2 on the surace o lat mirror is not so good. The GLM distribution is narrower than DR distribution, however peak positions are close to each other. Thus, one could say that use o GLM gives reasonable agreement with DR approach and could be used or resonator investigation. ). (9) 5

6 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/ Quality actor o semi-conocal resonator During experimental investigation o coherent DR generation in semi-conocal resonator its quality actor (Q-actor) was deined to be equal to Q = 72.9 [6]. The Q-actor was deined using the ollowing procedure. Ater all electron bunches leaved the resonator the radiation stored in it decayed ollowing exponential law. The part o radiation that came out the resonator rom the opening in the lat mirror was measured as unction o time and was observed as a peak structure with time distance between peaks equal to distance between electron bunches ( t = 2.8ns). In order to estimate Q-actor exponential it o this signal tail was done by the unction A(t) = A e t/t, where A is ree scaling it parameter and t deines Q-actor in the ollowing orm [6]: Q = 2π t t. (1) In our theoretical investigation we tried to estimate the value o Q-actor or the semiconocal resonator using both DR approach and Gaussian-Laguerre mode approach. During the experiment [6] the lat mirror consisted o glass substrate with relecting coating and had the opening diameter a = 5mm. However, the relecting layer was deposited not on the whole substrate surace. The area inside the circle with diameter 15 mm was not deposited and thus consisted o pure glass. Using both theoretical approaches we could not take into account this part with no relecting coating. In order to estimate resonator Q-actor we assumed two cases. In the irst case the opening diameter was assumed to be equal to 5mm that should result in the upper limit o Q-actor. In the second case the opening diameter was assumed to be equal to 15mm that should result in the lower limit o Q-actor. In order to estimate Q-actor using DR approach we calculated a train consisted o 15 pointlike electron bunches. The DR was calculated using equations (6) at radiation wavelength equal to λ = 3.5mm or both opening diameters. Ater each third bunch starting rom third one 25 relaxation ree runs were calculated using again equations (6) but assuming that electron ield is absent (E e (r,ω) = ). In order to calculate the part o radiation that comes out o the resonator the DR intensity distribution on the lat mirror surace was integrated over the opening area. Thisvalue was assumedto bea signal that was measured over time. Figure 4 show an example o radiation yield vs. time that was calculated or a = 15mm. Cavity ree runs were calculated Figure 4. Example o time dependence o radiation yield. Red dots radiation yield electrons, blue dots resonator ree runs ater 15 bunches, green dots resonator ree runs ater 9 bunches. The calculation was carried out at λ = 3.5mm and a = 15mm 6

7 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 or 9 electron bunches in the train and 15 bunches in the train. In the case o DR calculation approach the resonator Q-actor was estimated exactly like it was done during the experiment [6]. The values o resonator Q-actor calculated at a = 5mm are shown in igure 5 by blue triangles and at a = 15mm by red dots. The errors shown in igure 5 are errors caused by it errors o parameter t. It is obvious that Q-actor should not depend on number o electron bunches in the train. In order to demonstrate this the value o Q- actor in igure 5 was plotted versus number o bunches, i.e. resonator ree runs were calculated ater each third bunch. The resonator Q-actor value Q = 72.9 measured experimentally is shown by black solid line. Cavity Q-actor could be also calculated using GLM as ollowing: Q = 2π 1 c Σ, (11) where c Σ is the total losses o energy in the resonator. In our case we took into account geometrical losses (c G ) and diraction losses (c D ). According to [12]: where indexes, c deine lat and concave target, respectively. According to [13], diraction losses could be estimated as: c Σ = 1 (1 c D )(1 cc D ), (12) c D = 1 (1 c G ) 2 (13) Geometrical losses could be calculated due to the act that we know spatial distributions o Figure 5. Dependence o the resonator Q-actor on number o electrons. Red dots DR approach, a = 15mm; blue triangles DR approach, a = 5mm; red dashed line GLM approach, a = 15mm, blue dashed line GLM approach, a = 5mm; black solid line Q = 72.9, experimental result [6] 7

8 RREPS215 Journal o Physics: Conerence Series 732 (216) 1219 doi:1.188/ /732/1/1219 radiation intensity on the mirror suraces [14]: a /2 c G = ρdρ 2π Ψ 1 2 dϕ+ ρdρ 2π c c G = a1 /2 ρdρ 2π Ψ 1 2 dϕ, Ψ 1 2 dϕ Ψ 1 2 dϕ+ a/2 ρdρ 2π Ψ 1 2 dϕ ρdρ 2π, Ψ 1 2 dϕ a/2 ρdρ 2π (14) where a = 5mm or a = 15mm (see urther), a 1 = 5mm, a = 1mm. Combining equations (11) (13) one could obtain resonator Q-actor value. In igure 5 the calculated values o Q-actor are shown by red dashed line at a = 15mm (Q = 22.5) and by blue dashed line at a = 5mm (Q = 43.9). 5. Conclusion In conclusion we would like to note that the well-known theory o Gaussian beams in open resonators and developed computer code allowing to simulate DR characteristics in a semiconocal resonator give the coinciding results or transversal ield components. This act conirms the validity o the code based on the generalized surace current model [8]. The uniied ormulation or DR generation in resonator mirrors and the radiation storage can be applied or calculation o DR characteristics at any resonator coniguration including the conocal one. The preliminary estimations o Q-actor or such a resonator show a possibility to achieve much higher values o Q-actor than in the semi-conocal resonator considered above. Acknowledgments The work was partially supported by the Russian Ministry o Education and Science within the program Nauka, Grant No /K an by RFBR grant No Reerences [1] Dattoli D, Renieri A and Torre A 1993 Lectures on Free Electron Laser Theory and Related Topics (Singapore: World Scientiic) [2] Gover A, Dyumin E 26 Proceedings o Free Electron Laser conerence FEL 26 (Berlin, Germany) p 1 [3] chi-lihn H, Kung P, Settakorn C, Wiedemann H and Bocek D 1996 Phys.Rev.Lett [4] Shibata Y et al 1997 Phys.Rev.Lett [5] Sasaki S, Shibata Y, Ishi K, Ohsaka T, Kondo Y, Hinode F, Matsuyama T and Oyamada M 22 Nucl. Instr. Meth. A [6] Aryshev A et al 214 Nucl. Instr. Meth. A [7] Potylitsyn A, Ryazanov M, Strikhanov M and Tishchenko A 21 Diraction Radiation rom Relativistic Particles (Berlin: Springer) [8] Karlovets D V and Potylitsyn A P 29 Phys.Lett.A [9] Naumenko G A, Cha V A, Kalinin B N, Popov Yu A, Potylitsyn A P, Saruev G A and Sukhikh L G 28 Nucl. Instr. Meth. B [1] [11] Allen L, Beijersbergen W, Spreeuw R J C and Woerdman J P 1992 Phys. Rev. A [12] Kubarev V V 2 Kvantovaya Elektronika [13] Kubarev V V, Persov B Z, Vinokurov N A, Davidov A V 24 Nucl. Instr. Meth. A [14] Gurov M G and Dmitriev A K 29 Russian Physics Journal