Papers in Physics, vol. 11, art. 110007 (2019) Received: 1 June 2019, Accepted: 23 August 2019 Edited by: A. B. Márquez Licence: Creative Commons Attribution 4.0 DOI: http://dx.doi.org/10.4279/PIP.110007 www.papersinphysics.org ISSN 1852-4249 Theory of terahertz Smith–Purcell radiation from a cylindrical grating Z. Rezaei,1∗ B. Farokhi1 An analysis of an annular electron beam propagating along a cylindrical grating with external magnetic field B0 is presented. The grating comprises a dielectric in its slots. The dispersion relation of the modes is derived. The results demonstrate that the dielectric shifts the frequencies of the system modes to smaller values. The growth rates of the modes which are in phase with the beam are also considered. It is found that the decline in the growth rate is brought about by the dielectric. In addition, increasing the thickness of the dielectric and decreasing the height of the slots cause it to rise. The effect of beam thickness on growth rate is considered too. This is shown to increase and then fall as beam thickness increases. These results show that utilizing cylindrical grating loaded with dielectric has a promising effect on developing new kinds of compact high-efficient THz free-electron lasers based on Smith–Purcell radiation. I. Introduction Plasma and beam devices are employed in ampli- fiers, oscillators, charged particle accelerators, and high power sources of electromagnetic radiation. They are also used to transport electromagnetic energy and charged particles, and for basic plasma physics research [1]. When the electron beam passes near the grat- ing surface (periodic structure), spontaneous radi- ations may be excited. This periodic structure can be a metallic corrugated surface with spatial pe- riodicity d and corrugation height h. This radi- ation was first observed by Smith and Purcell [2] in 1953. The Smith–Purcell radiation (SPR) is a tunable electromagnetic source, which is described by ∗E-mail: z-rezaei@phd.araku.ac.ir 1 Department of Physics, Faculty of Science, Arak Univer- sity, Arak, P.O. Box 38156-8-8349, Iran λ = d n ( 1 β − cos θ ) , (1) where λ is the wavelength of the radiated wave from the grating, n is the order of this radiation, β is the relative velocity of the charge, and θ is the direction of the radiated wave with respect to this charge. In the far-infrared or terahertz (THz) region, sev- eral theories have been proposed to describe the operation of SPR, and also its application, in a free electron laser (FEL). Schachter and Ron proposed a theory based on the interaction of an electron beam with a wave traveling along the grating. They used some approximation to evaluate the reflection ma- trix of the grating and found a cubic equation for growth rate, which is consistent with Cherenkov FEL [3]. Urata et al. considered this phenomenon experimentally, and observed high power coher- ent superradiant SP emission in the far infrared (30−100 µm) region [4]. SP radiation in the ultra- violet and near infrared regions was also detected by Y. Neo et al. [5]. Kim and Song, using the inter- action of the electrons with a traveling wave, solved the initial problem of the sheet-beam and found a 110007-1 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. quadratic equation for the exponential growth rate [6]. Later Andrews and Brau explained Urata’s ex- periment as bunching of the beam electrons due to the interaction of an evanescent wave with this suf- ficiently high-current beam. They also derived the gain of this radiation, which had cube root depen- dence on the beam current [7]. Also in 2004, Freund et al. [8] developed a linear theory of a grating- coupled Smith–Purcell traveling wave in a paral- lel plate wave guide. They found the linearized dispersion relation for the vacuum structures and the wave–particle interaction in an arbitrary mag- netic field. Then, D. Li et al. performed simulation and confirmed the theory of Andrews and cowork- ers about the mechanism of superradiation, which happens at integer multiples of the bunching fre- quency [9]. In addition, the growth rate of SP–FEL was considered by D. N. Klochkov et al. and found to be proportional to the square root of the sheet electron beam current [10]. Loading dielectric is an important physical mech- anism which has been successfully applied to some high power microwave and terahertz systems. A dielectric-loaded grating for 3D Smith–Purcell rect- angular device is proposed by Cao et al. [11, 12]. They used 3D particle-in-cell simulation to find the dispersion relations at the operating point, and the growth rates. W. Liu et al. considered a rectangu- lar grating filled with dielectric. They obtained the minimum current for starting the SP oscillator and deduced that the dielectric will decrease this cur- rent. Also, they explained the effect of changing the beam parameters on the growth rate [13]. There is no edge effect issue in cylindrical grat- ing driven by an annular beam. Therefore, as the cylindrical gratings are more efficient, with fewer losses than rectangular ones, they are more appli- cable in radiation sources and considered in many types of research. H. P. Bluem et al. worked on a cylindrical grating exposed by an annular beam. They observed both superradiance and SP radia- tion [14]. S. Hasegawa et al. considered a cylin- drical corrugation in a waveguide. They reported BWO operation, excited by a cylindrical surface wave in k-band signal region. Also, they increased the voltage of the beam and observed SP radiation in the u-band and E-band of frequency, which was the result of interaction between the higher modes of the waveguide [15]. Here we present a linear theory of an annular Figure 1: The cross section view of the grating, filled with dielectric. In addition, the annular electron beam is drifting along the axial direction with an externally magnetic field B0. electron beam, magnetized, propagating along a cylindrical metallic grating. The slots of this grat- ing are filled with dielectric. The results of this pa- per highlight the basic problem of developing SP– FEL based on cylindrical grating loaded with di- electric. For simplicity, we assume that the system is uniform in the direction parallel to the slots of the grating. The fundamental dynamical equations are presented in section II. The results and discus- sion are given in section III. The conclusions are considered at the end. II. Theory Consider a cylindrical grating which is made of an ideal metal. The inner and outer radii of the grat- ing are R1 and R2, respectively as illustrated in Fig. 1. As shown in this figure, d denotes the pe- riod of the grating, l is the length of the slot open- ings (which will be filled with dielectric) and h is the depth of the slots (R2 −R1). An annular elec- tron beam with inner radius a1 and outer radius a2 = a1 + ∆ in a uniform static axial magnetic field B0 is drifting with velocity v0 along the axis of the grating and very close to it. We assume that there is no transverse disturbed movement in the electron beam. In addition, for simplicity, we assume that the system is uniform in the ϕ direction. i. Dispersion relation The dispersion relation of the modes of this sys- tem is the result of considering Maxwell’s equations with the continuity equation and the relativistic 110007-2 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. momentum equation for electron beams: ∇×E = − ∂B ∂t , (2) ∇×B = µJ + 1 c2 ∂E ∂t , ∂n ∂t + ∇ · (nv) = 0, γ3m0n0 [ ∂ ∂t v + v0 ·∇v ] = −en0(Ez + v×B). As a consequence, we expand all quantities in terms of an unperturbed part plus a small perturbation as follows: n = n0 +δn, v = v0 +δvz, J = J0 +δJ, E = δE and B = B0ẑ + δB, where n and v are the electron density and velocity, respectively. Un- perturbed beam density, n0, is uniform and time- independent, δE and δB are the electric and mag- netic fields, J = env is the density of current, γ = (1 − v2/c2)1/2 is the relativistic factor, and c is the velocity of light in free space. The perturbed density of current is δJ = −e(nδvz + δnv0), (3) which, with the help of the continuity and momen- tum equations is defined as δJ = iωε0 γ3 ω2p (ω −knv0)2 δEz. (4) Here, ω2p = ne2 m0ε0 is the relativistic beam plasma frequency. The fields in the electron beam part will be de- rived from this wave equation ∇2δB + ω2 c2 εΠδB = 0, (5) where εΠ = 1− ω 2 p γ3(ω−knv0)2 is the relative dielectric constant in the electron beam [12]. Also, in region 1 and 3, where there is no electron beam, the electromagnetic fields are the solution of the wave equation of this form (∇2⊥ + β 2 n) { δEz δBz = 0. (6) We suppose that the TM mode propagates in this device. And, by applying Floquet’s theorem, the radiation fields take the general form δf(r, t) = ∞∑ n=0 δfn(r)e i(knz−ωt), (7) where fn, kn = k0 + 2nπ d , and ω represent the Fourier coefficient, wave number in the axial direc- tion and frequency of the nth mode, respectively. Region 1 This region is the vacuum above the electron beam. So the fields in a2 < r can be expressed as below Ez(r,z) = ∞∑ n=−∞ bnK0(βnr)e i(knz−ωt), (8) Bϕ(r,z) = ∞∑ n=−∞ iω c2βn nnK1(βnr)e i(knz−ωt). (9) Region 3 In this region, R2 < r < a1, the fields are Ez(r,z) = (10) ∞∑ n=−∞ [cnl0(βnr) + dnK0(βnr)]e i(knz−ωt), Bϕ(r,z) = (11) ∞∑ n=−∞ −iω c2βn [cnl1(βnr) −dnK1(βnr)]ei(knz−ωt). Region 2 In the electron beam region, a1 < r < a2, the evanescent waves are the solution of Eq. (5), and have the following forms Ez(r,z) = (12) ∞∑ n=−∞ i rωε0εΠ {gn[rκ1nl1(κ1nr) + l0(κ1nr)] + fn[−rκ1nK1(κ1nr) + K0(κ1nr)]}ei(knz−ωt), Bϕ(r,z) = (13) ∞∑ n=−∞ µ{gnl0(κ1nr) + fnK0(κ1nr)}ei(knz−ωt). Where, κ1n = √ k2n −ω2εΠ/c2. 110007-3 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. Regions 4 and 5 The slot openings (region 4) are filled with dielec- tric εr. The solutions of the wave equation in this region are Ez(r,z) = ∞∑ m=0 em [ H0(τmr) (14) − H0(τmR1) G0(τmR1 G0(τmr) ] cos (mπ l z ) , Bϕ(r,z) = ∞∑ m=0 em iωεr c2τm [ H́0(τmr) (15) − H0(τmR1) G0(τmR1 Ǵ0(τmr) ] cos (mπ l z ) . Where, H0(τmr) = { J0(τmr) l0(τ́mr) , G0(τmr) = { N0(τmr) K0(τ́mr) , (16) H́0(τmr) = { −J1(τmr) l1(τ́mr) , Ǵ0(τmr) = { −N1(τmr) −K1(τ́mr) , (17) kz = mπ l , τm = √ εr ω2 c2 −k2z > 0, τ́m = √ k2z −εr ω2 c2 > 0. (18) As we assumed k0d < 2π, it is enough to keep just one mode (m = 0) in the slots, so the standing waves will be the fields in this part of the system. Also, there is no field in the ideal metal of region 5. After applying the continuity conditions for the fields in the border of regions 1–2, 2–3 and 3–4, the dispersion relation will be as below R(ω,kn,ε Π) = 0. (19) In which, R(ω,kn,ε Π) = ∞∑ n=−∞ { −1 k2nβnd [2 − 2 cos(knl)] × [ H0(τ0R2) − H0(τ0R1) G0(τ0R1) G0(τ0R2) ] × αaI1(βnR2) + αbK1(βnR2) αaI0(βnR2) −αbK0(βnR2) (20) − εrl τ0 [ H́0(τ0R2) − H0(τ0R1) G0(τ0R1) Ǵ0(τ0R2) ]} , αa = −α1α6 + α2α5, αb = −α3α6 + α4α5, α1 = K1(βna1) βnεΠa1 [a1κ1nI1(κ1na1) + I0(κ1na1)] + K0(βna1)I0(κ1na1), (21) α2 = K1(βna1) βnεΠa1 [−a1κ1nK1(κ1na1) + K0(κ1na1)] + K0(βna1)K0(κ1na1), (22) α3 = − I1(βna1) βnεΠa1 [a1κ1nI1(κ1na1) + I0(κ1na1)] + I0(βna1)I0(κ1na1), (23) α4 = − I1(βna1) βnεΠa1 [−a1κ1nK1(κ1na1) + K0(κ1na1)] + I0(βna1)K0(κ1na1), (24) α5 = K1(βna2) βnεΠa2 [a2κ1nI1(κ1na2) + I0(κ1na2)] + K0(βna2)I0(κ1na2), (25) α6 = K1(βna2) βnεΠa2 [−a2κ1nK1(κ1na2) + K0(κ1na2)] + K0(βna2)K0(κ1na2). (26) If there is no beam (α1 = α5,α2 = α6), the dis- persion relation will become as R(ω0,k0, 1) = ∞∑ n=−∞ { 1 k2nβnd [2 − 2 cos(knl)] × [ H0(τ0R2) − H0(τ0R1) G0(τ0R1) G0(τ0R2) ] K1(βnR2) K0(βnR2) − εrl τ0 [ H́0(τ0R2) − H0(τ0R1) G0(τ0R1) Ǵ0(τ0R2) ]} = 0. (27) This is similar to the dispersion relation in [14] and [16] in the limit of εr = 1. 110007-4 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. Figure 2: (a) Comparison of dispersion relations for different εrs (solid curves). The beam line with voltage 20 kev is also plotted for reference. The growth rate (δ) for each curve, near the intersection (the circles), is indicated by a dash line. (b) The effect of relative dielectric εr on the growth rate. ii. Growth rate So far, the dispersion relation of the modes in this configuration has been derived. One of these modes can grow if it is in resonance with the electron beam. So, we assume that the frequency of this mode is ω = ωr + δ. Then the Taylor expansion of the dispersion relation about the synchronous point (ωr,kr) will become R(ω,kn,ε Π) = R(ωr,kr, 1) (28) + (ω −ωr) ∂R(ω,kn,ε Π) ∂ω ∣∣∣∣ (ωr,kr,1) + (εΠ − 1) ∂R(ω,kn,ε Π) ∂εΠ ∣∣∣∣ (ωr,kr,1) . By assuming that δ is small, the equation below will be found ( R0x 2 − ω2p γ3 ŔεΠ ) + (2xR0 + x 2Ŕω)δ + 2xŔωδ 2 = 0. (29) In which, x = ωr −krv, ŔεΠ = ∂R(ω,kn,ε Π) ∂εΠ ∣∣∣∣ (ωr,kr,1) , Ŕω = ∂R(ω,kn,ε Π) ∂ω ∣∣∣∣ (ωr,kr,1) , R0 = R(ωr,kr, 1). (30) The growth will occur if δ, the solution of Eq. (29) is imaginary and positive. III. Results and discussion By assuming no beam in the system, the disper- sion relation is calculated by solving Eq. (27) nu- merically. The grating parameters are as follows: R1 = 240 µm, R2 = 400 µm, l = 80 µm, d = 160 µm, a1 = 400 µm, a2 = 480 µm and the beam energy is 20 kev, corresponding to the parameters chosen by Y. Zhou et al. [16]. The effect of εr on the dispersion relation has been shown in Fig. 2(a). It is clear that increasing the εr results in smaller height of the dispersion relation. This means that the modes are propagating with smaller velocities in the system. The intersection points of beam- wave also move down. In this figure, the corre- sponding growth rate for each curve is indicated by dash lines (of the same color). Maximum growth 110007-5 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. rates occur in the vicinity of the synchronous points (ωr = krv0), and have the values: 1.279, 1.129 and 1.065 for εr = 1, 1.5 and 2.7, respectively. The influence of dielectric on the growth rate is clearer in Fig. 2(b), which indicates that dielectrics with higher relative permittivities cause smaller values for the growth rate. The grating parameters are considered in Fig. 3, Fig. 4 and Fig. 5, when εr = 2.7. In Fig. 3 the slot depth has been changed. As depth increases, the dispersion relation becomes flatter, indicating that the effect of grating is in- creasing. The normalized maximum growth rate happens when resonance between the beam and the modes is possible (the circles). So, in these points δ = 1.192, 1.066 and 0.493 when h = 100 µm, 160 µm and 250 µm, respectively. Figure 4 indicates how dielectric thickness has an effect on growth rate. Again, lower frequency modes result from increasing dielectric thickness. However, this time the growth rate will increase by this effect: δ = 0.959, 1.232 and 1.295 for l = 30 µm, 80 µm and 110 µm, respectively. The effect of beam thickness (∆) on growth rate is depicted in Fig. 5. First, increasing ∆ causes the growth rate to rise. Its maximum value is 1.066 at ∆ = 80 µm. This happens because more electrons can participate in the beam wave interaction. Then the growth rate falls. This can be justified by the Figure 3: Comparison of dispersion relation for different grating heights, when εr = 2.7. The growth rate cor- responding to each frequency curve is plotted by dash lines, and maximum values are indicated at the circle points. Figure 4: Comparison of the dispersion relations (solid lines) and the growth rates (dash lines) for different thicknesses of the dielectric εr = 2.7. fact that although the thickness is increasing, the electrons which are far from the grating contribute less to the interaction. IV. Conclusions In this paper, a metallic cylindrical grating filled with a dielectric is proposed. The dispersion re- lation of the modes propagating in this configura- tion with an annular electron beam is derived. It is shown that the dielectric causes modes with smaller frequencies, in comparison with results when it is absent. Then, the growth rate of modes which Figure 5: Dependence of the growth rate on beam thick- ness. εr = 2.7, R1 = 240 µm, R2 = 400 µm, l = 80 µm, d = 160 µm, a1 = 400 µm. 110007-6 Papers in Physics, vol. 11, art. 110007 (2019) / Z. Rezaei et al. are in resonance with the beam is considered. It is found that the growth rate is under the influ- ence of dielectric relative permittivity εr, the depth of the slots of the grating and the thickness of the dielectric (the width of the slots). A lower growth rate is the result of increasing the parame- ters of the dielectric relative permittivity and slot depth, and decreasing the thickness of the dielec- tric. Also, beam thickness can increase and de- crease the growth rate, depending on its amount. As we can see, by changing the grating parame- ters, as well as dielectric permittivity and thick- ness, the growth rate and operating frequencies of the device can be controlled. So, it is possible to make SP–FELs with the desired frequencies and powers. These results can be of considerable inter- est for THz wave source research. [1] B Maraghechi, B Farokhi, J E Willett, The- ory of high-frequency waves in a coaxial plasma wave guide, Phys. Plasmas 6, 3778 (1999). [2] S J Smith, E M Purcell, Visible light from lo- calized surface charges moving across a grat- ing, Phys. Rev. 92, 1069 (1953). [3] L Schachter, A Ron, Smith–Purcell free- electron laser, Phys. Rev. A 40, 876 (1989). [4] J Urata, M Goldstein, M F Kimmitt, A Nau- mov, C Platt, J E Walsh, Superradiant Smith– Purcell emission, Phys. Rev. Lett. 80, 516 (1998). [5] Y Neo, H Shimawaki, T Matsumoto, H Mimura, Smith–Purcell radiation from ultra- violet to infrared using a Si field emitter, J. Vac. Sci. Technol. B 24, 924 (2006). [6] K J Kim, S B Song, Self-amplified spontaneous emission in Smith–Purcell free-electron lasers, Nucl. Instrum. Methods Phys. Res. Sect. A, 475, 158 (2001). [7] H L Andrews, C A Brau, Gain of a Smith– Purcell free-electron laser, Phys. Rev. Spec. Top. Accel. Beams 7, 70701 (2004). [8] H P Freund, T M Abu-Elfadl, Linearized field theory of a Smith–Purcell traveling wave tube, IEEE Trans. Plasma Sci. 32, 1015 (2004). [9] D Li, Z Yang, K Imasaki, G-S Park, Particle- in-cell simulation of coherent and superradi- ant Smith–Purcell radiation, Phys. Rev. Spec. Top. Accel. Beams 9, 040701 (2006). [10] D N Klochkov, A I Artemyev, K B Oganesyan, Y V Rostovtsev, M O Scully, C-K Hu, The dis- persion equation of the induced Smith–Purcell instability, Phys. Scr. T140, 014049 (2010). [11] M Cao, W Liu, Y Wang, K Li, Three- dimensional theory of Smith–Purcell free- electron laser with dielectric loaded grating, J. Appl. Phys. 116, 103104 (2014). [12] M Cao, W Liu, Y Wang, K Li, Dispersion characteristics of three dimensional dielectric- loaded grating for terahertz Smith–Purcell ra- diation, Phys. Plasmas 21, 23116 (2014). [13] W Liu, M Cao, Y Wang, K Li, Start current of dielectric-loaded grating in Smith–Purcell radi- ation, Phys. Plasmas 23, 33104 (2016). [14] H P Bluem, R Jackson, J D Jarvis, A M M Todd, J Gardelle, P Modin, J T Donohue, First lasing from a high-power cylindrical grat- ing Smith–Purcell device, IEEE Trans. Plasma Sci. 43, 9 (2015). [15] S Hasegawa, K Ogura, T lwasaki, K Yambe, Smith–Purcell radiation based on cylindrical surface waves, Fusion Sci. Technol. 63, 259 (2017). [16] Y Zhou, Y Zhang, S Liu, Electron-beam-driven enhanced terahertz coherent Smith–Purcell ra- diation within a cylindrical quasi-optical cav- ity, IEEE Trans. THz Sci. Technol. 6, 262 (2016). 110007-7