Diffraction and Smith-Purcell radiation on the hemispherical bulges in a metal plate
aa r X i v : . [ phy s i c s . acc - ph ] O c t Prepared for submission to JINST
XII th International Symposium «Radiation from Relativistic Electrons in PeriodicStructures»September 18-22, 2017Hamburg, Germany
Diffraction and Smith-Purcell radiation on thehemispherical bulges in a metal plate
V.V. Syshchenko, E.A. Larikova, Yu.P. Gladkih
Belgorod State University,Pobedy Street, 85, Belgorod 308015, Russian Federation
E-mail: [email protected]
Abstract:
The radiation resulting from the uniform motion of a charged particle near a hemisphericbulge in a metal plane is considered. The description of the radiation process based on the methodof images is developed for the case of non-relativistic particle and perfectly conducting target. Thespectral-angular and spectral densities of the diffraction radiation on the single bulge (as well asthe Smith-Purcell radiation on the periodic string of bulges) are computed. The possibility ofapplication of the developed approach to the case of relativistic incident particle is discussed.
Keywords:
Interaction of radiation with matter; Beam-line instrumentation Corresponding author. ontents
The radiation emitted under charged particle traveling near the boundary of the spatially localizedtarget (without crossing it) is called diffraction radiation (DR), whereas the crossing of the target’sboundary produces the transition radiation (TR).One of the ways to describe these types of radiation is the application of the boundary conditionsto the Maxwell equations’ solutions for the field of the moving particle in two media. It becomesevident that the boundary conditions could be satisfied only after addition the solution of freeMaxwell equations that corresponds to the radiation field, see, e.g. [1].The conditions on the boundary between vacuum and ideal conductor could be satisfied insome cases via introduction of one or more fictitious charges along with the real charged particle;this approach to electrostatic problems is known as the method of images, see, e.g., [2]. Namely themethod of images had been used in the pioneering paper [3] where TR on a metal plane had beenpredicted. The method of images had been used also in [4] for consideration of TR under passageof the particle through the center of the ideally conducting sphere in dipole approximation.DR and TR of a charge incident on a perfectly conducting sphere under arbitrary impactparameter had been studied using the method of images in [5]. Here we consider DR on thehemispherical bulge in a perfectly conducting plane using the same approach.
Remember how the method of images is implied to meet the boundary condition for the electricfield on the conducting surface. Consider the real charge e passing near the grounded conductingsphere of the radius R with the constant velocity v ≪ c , see fugure 1 (a). In this case the zeropotential on the metal surface is acieved via introduction of the single fictitious charge (the “image”)of the magnitude e ( t ) = − e R q b + v t (2.1)– 1 – igure 1 . The real charge e near the grounded sphere and its image (a); the real charge near the hemisphericalbulge in the conducting plane and three images (b) and (c). placed at the point with coordinates x ( t ) = R bb + v t , z ( t ) = R v tb + v t . (2.2)While the incident particle moves uniformly, its image will move accelerated. The radiationproduced by non-uniform motion of the image charge will be described by the well-known formula[6] d E d ω d Ω = π c (cid:12)(cid:12) k × I (cid:12)(cid:12) , (2.3)where k is the wave vector of the radiated wave, | k | = ω / c , and I = ∫ ∞−∞ e ( t ) v ( t ) exp { i ( ω t − kr ( t ))} dt (2.4)(it could be easily seen that it is applicable to the case of time-varying charge e ( t ) as well as to thecase of the constant one). Note that the method of images for the isolated (in contrast to grounded)– 2 –phere requires introducing another fictitious charge of the magnitude − e ( t ) resting at the center ofthe sphere. However, the last equation shows that such rest charge does not produce any radiation.To achieve the zero potential on the conducting plane with hemispherical bulge we need threeimage charges along with the real one: the image of the incident charge by the spherical surface(2.1)–(2.2) and the mirror reflections of both of them by the plane, − e and − e ( t ) , see figure 1 (b).Two of these image charges, e ( t ) and − e ( t ) , will move accelerated under uniform motion of theincident particle and hence produce the radiation.Note, however, that the projectile can fly not only above the top of the hemisphere. Foraccount of this possibility we introduce the two-dimensional impact parameter b = ( b x , b y ) = b ( cos α, sin α ) , see figure 1 (c), so we have x ( ) ( t ) = R b x b + v t , y ( ) ( t ) = R b y b + v t , z ( ) ( t ) = R v tb + v t (2.5)for the image charge e ( t ) and x ( ) ( t ) = − x ( ) ( t ) , y ( ) ( t ) = y ( ) ( t ) , z ( ) ( t ) = z ( ) ( t ) (2.6)for the image charge − e ( t ) .Substituting the proper values into (2.4) and collecting together the contributions from thesetwo image charges, we obtain the following integrals: I x = e b x R v ∫ ∞−∞ exp ( i " ω t − k y b y R b + v t − k z R v tb + v t cos k x b x R b + v t t dt ( b + v t ) / , (2.7) I y = − ie b y R v ∫ ∞−∞ exp ( i " ω t − k y b y R b + v t − k z R v tb + v t sin k x b x R b + v t t dt ( b + v t ) / , (2.8) I z = ie R v ∫ ∞−∞ exp ( i " ω t − k y b y R b + v t − k z R v tb + v t sin k x b x R b + v t ( b − v t ) dt ( b + v t ) / . (2.9)The integrands in (2.7)–(2.9) are smooth functions and the integration can be easily performednumerically, that leads to the spectral-angular density of diffraction radiation in the form d E d ω d Ω = e π c Φ ( θ, ϕ, ω ) , (2.10)where the typical shape of the angular distribution Φ ( θ, ϕ, ω ) is presented in figure 2 (b). We seeapproximate symmetry of the directional diagram around x axis; for higher R ω / v values the smallforward-backward asymmetry increases, see figure 2 (c, d).In non-relativistic ( v ≪ c ) and low-frequency ( ω ≪ cb / R hence λ ≫ π R / b ) case wecan neglect the second and third terms in the exponents in (2.7)–(2.9) as well as put zero thearguments of the trigonometric functions there. In this case I y = I z = d E d ω d Ω = π e ω R c v cos α (cid:16) − sin θ cos ϕ (cid:17) (cid:20) K (cid:18) ω v b (cid:19) (cid:21) , (2.11)– 3 – igure 2 . The angular dependence as direction diagram of DR intensity on the sphere [5] (a) and hemisphere(b) for the passage of the real charge under b x = R + and b y = (sliding incidence, when DR intensity ismaximal for the whole range of wavelengths) and R ω / v = . (this choice is due to the maximum of DRon hemisphere spectrum (see below) falls on ω b / v ≈ . and b = R in the given case). This shape of thedirectional diagram is typical; for higher frequencies the slight forward-backward asymmetry increases, see R ω / v = (c) and R ω / v = (d). ( d E / d ω ) ( π c / e ) R ω / v (a) (b)R ω / v α ( d E / d ω ) ( π c / e ) Figure 3 . (a) DR spectrum computed using approximated analytical formula (2.11) (solid curve for b = R + , b y = and dashed curve for b x = . R , b y = )) and via numerical integration (circles and triangles,respectively). (b) For nonzero b y the radiation intensity decreases approximately as cos α (without substantialchanges in the direction diagram shape) . – 4 –here K is the modified Bessel functions of the third kind. This analytical approximation is rathergood, as can be seen from the numerical spectrum (integrated (2.10) over radiation angles) comparedwith analytical one (figure 3 (a)). For illustrative purposes, we choose the parameters v = . c , R =
20 nm, b = R +
0, for which the DR intensity maximum will lie in the visible spectrum. Theapplicability of our results in this frequency domain will be discussed in the Conclusion.
Now consider the motion of the charge e along the periodic string of N ≫ b → R (when the radiation intensity is high), for the stringperiod large enough, a & R . Then the interference of the radiation produced on the subsequenthemispheres leads to the simple formula for the spectral-angular density of DR: d E d ω d Ω = e π c Φ ( θ, ϕ, ω ) · π N v ω a ∞ Õ m = δ (cid:18) − v c cos θ − m π v ω a (cid:19) , (3.1)where the delta-function means well-known Smith–Purcell condition [7].The spectrum for the string period a small enough consists of separated bands, see figure 4 (a).The bands overlap each other under increase of the string period a gradually forming the spectrumof DR on a single hemisphere (multiplied by the total number of the bulges N ), see figure 4 (b, c,d). The diffraction radiation resulting from the interaction of non-relativistic particle with the hemo-spherical bulge in perfectly conducting plane is considered. The method of images allows theprecise description of the radiation in this case. The integration in the resulting formulae can beeasily performed numerically that permits to compute the spectral-angular density of the diffractionradiation for an arbitrary impact parameter. The approximate analytical formulae for the radiationcharacteristics are obtained for the case of radiation wavelengths exceeding the bulge’s size.The range of validity of our results is determined, first of all, by the validity of the perfectconductor approximation for the metal target that is necessary for the use of the method of images.The perfect conductor approximation means the possibility of the metal’s electrons to trace outinstantly the changes of the external electric field to meet the requirement of zero tangentialcomponent of the electric field on the metal surface. It is valid for the frequencies less than theinverse relaxation time τ − for the electrons in the metal. For instance, τ − = · − sec − forcopper [8], so the results obtained surely could be applied up to THz and far infrared range. On theother hand, Ginzburg and Tsytovich [9] wrote: “ However, a good metal mirror (for instance copperor silver) is in practice fairly close to an ideal mirror for frequencies not higher than the opticalrange .” This means the applicability of the method of images also in the visible domain. However,the surface plasma oscillations also could be important here that needs further investigation.The results obtained are valid only for the non-relativistic particles due to the geometric natureof the method of images: we can fit the image charge to meet the zero boundary condition on the– 5 – ( d E / d ω ) ( π c / e N ) R ω / v visibleregion (a) ( d E / d ω ) ( π c / e N ) R ω / v visibleregion (b) ( d E / d ω ) ( π c / e N ) R ω / v visibleregion (c) ( d E / d ω ) ( π c / e N ) R ω / v visibleregion (d) Figure 4 . DR spectrum on the string of spheres under b = R + R =
20 nm, v = . c , a = sphere for the simple Coulomb field of slow incident particle, not for the relativistically compressedone. The only possibility to extend the method-of-images-based approach to relativistic case is toconsider the high frequency limit, where the characteristic size of the Coulomb field v γ / ω (where γ = ( − v / c ) − / is the incident particle’s Lorentz factor) is much smaller than the sphere radius R . Here we can neglect the metal surface curvature and consider the reflection of the incident fieldin the locally plane mirror (figure 5).However, we have a difficulty also in this case: the kinematic velocity of the image charge(as well as the corresponding Lorenz factor) is not consistent with the degree of the relativisticcompression of the Coulomb field of the incident particle. The resulting discrepancy in Lorentzfactor values of the real and image charges is negligibly small only for the impact parametersextremely close to the sphere radius, b = R + γ − γ image ≪ γ when1 − R / q b + v t ≪ γ − / . (4.1)Due to the constant acceleration of the image charge on the corresponding part of its trajectory (seefigure 5 (c)) one could expect synchrotron-like radiation in this case.– 6 – a) −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.70.750.80.850.90.95 v i m a g e / c v t / R (b) −2 −1.5 −1 −0.5 0 0.5 1 1.5 200.511.52 acce l e r a ti on , a r b it r a r y un it s v t / R (c) −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.40.50.60.70.80.91 v t / R γ i m a g e / γ (d) Figure 5 . Relativistic incident particle ( γ = b = . R ) and its image in the sphere in locally planemirror approximation (a) and the characteristics of the image’s motion (b, c, d). Acknowledgments
This research is partially supported by the grant of Russian Science Foundation (project 15-12-10019).
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