Radiation of charge moving through a dielectric spherical target: ray optics and aperture methods
Andrey V. Tyukhtin, Ekaterina S. Belonogaya, Sergey N. Galyamin, Victor V. Vorobev
RRadiation of charge moving through a dielectric spherical target:ray optics and aperture methods
Andrey V. Tyukhtin, ∗ Ekaterina S. Belonogaya, Sergey N. Galyamin, and Victor V. Vorobev
Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia (Dated: January 13, 2020)Radiation of charged particles moving in the presence of dielectric targets is of significant interestfor various applications in the accelerator and beam physics. The size of these targets is typicallymuch larger than the wavelengths under consideration. This fact gives us an obvious small parameterof the problem and allows developing approximate methods for analysis. We develop two methods,which are called the “ray optics method” and the “aperture method”. In the present paper, weapply these methods to analysis of Cherenkov radiation from a charge moving through a vacuumchannel in a solid dielectric sphere. We present the main analytical results and describe the physicaleffects. In particular, it is shown that the radiation field possesses an expressed maximum at acertain distance from the sphere at the Cherenkov angle. Additionally, we perform simulations inCOMSOL Multiphysics and show a good agreement between numerical and analytical results. ∗ [email protected] a r X i v : . [ phy s i c s . acc - ph ] J a n I. INTRODUCTION
Radiation of charged particles moving in the presence of dielectric objects (“targets”) is of vital interest for variousapplications [1–5]. For example, several experiments have shown that prismatic and conical targets can be prospectivefor both bunch diagnostics and generation of high-power radiation [2–5]. Further development of these topics requiresan accurate calculation of Cherenkov radiation (CR) outside dielectric objects, which is typically impossible to dorigorously due to the complicated geometry of these objects. The only exceptions allowing the construction ofrigorous solutions are the simplest geometries like infinite cylinder [6–9] or sphere [9, 10]. However, the obtainedformulas (infinite series) can be applied to the field analysis only when the wavelength λ is comparable with the targetradius. Moreover, such “practical” modifications of the geometry as a vacuum channel for the charge flight cannot beincorporated in this solution.For the discussed applications, the vacuum channel is needed for the flight of the bunch. Moreover, the targetdimensions are typically much larger than the wavelengths of interest. Therefore, both calculations based on thesolutions mentioned above and numerical simulations are very complicated. However, the mentioned relation between λ and target size gives us an obvious small parameter allowing the development of approximate (asymptotic) methodsfor the analysis of radiation. Recently we have offered and successfully verified two such methods called the “rayoptics method” [11, 12] and “aperture method” [13–19]. They can be divided into three steps.The first two steps are the same for both methods. First, we solve the specific “etalon” problem, which does nottake into account the “external” boundaries of the target. For example, if the charge moves in the vacuum channelinside the target, then in the first step, we consider the problem with the channel in the unbounded medium. In otherwords, we consider only the boundary nearest to the charge trajectory and solve the problem for the semi-infinitemedium.In the second step, we select a part of the external surface of the object which is illuminated by CR and transparentfor CR (so that there is no total internal reflection here). This part of the object boundary is called an “aperture”further. Then we use the fact that the object is large in comparison with the wavelengths under consideration. Moreprecisely, we assume that: (i) the size of the aperture Σ is much larger than the wavelength λ ; (ii) the distance fromthe main part of the aperture to the charge trajectory is also much larger than the wavelength λ .The field obtained in the first step is used as the incident field on the aperture. Due to point (ii), we can neglectthe quasi-static (quasi-Coulomb) part of this field and use corresponding asymptotic approximation, which is a quasi-plane wave (more precisely, a cylindrical wave with a small curvature of the wave front). Further, we decompose thiswave into a superposition of vertical and horizontal polarizations (concerning the plane of incidence) and calculatethe field on the external surface of the aperture using Snells law and the Fresnel equations.The third step is different for the two methods. The ray optics method uses the ray optics laws for calculationof the wavefield outside the object [11, 12]. However, this technique has essential limitations. First, the so-called“wave parameter” W ∼ λL/ Σ ( L is a distance from the aperture to the observation point) should be small W (cid:28) L can be much larger than λ but cannot be larger than Σ /λ . In particular,we cannot consider the important Fraunhofer area where W (cid:29)
1. Second, the observation point can not be in theneighborhood of focuses and caustics, where ray optics is not applicable.The aperture method is more general [13–19] than the ray optics. It is valid for observation points with arbitrarywave parameter W , including the Fraunhofer (far-field) area and neighborhoods of focuses and caustics. In the thirdstep of this technique, we calculate the field outside the target using Stratton-Chu formulas (“aperture integrals”).These formulas allow determining the field in the surrounding space if tangential components of the electric andmagnetic fields on the aperture are known.This paper is devoted to the study of CR from a solid dielectric sphere with the radius being much larger than thewavelength and having an axisymmetric vacuum channel where the charge moves. Such a target can be manufacturedwith high accuracy and can be a prospective candidate for the aforementioned applications. We apply both methods forthe spherical target and compare the obtained analytical results with the results of COMSOL Multiphysics simulations. II. THE FIELD ON THE BALL SURFACE
Here we consider a dielectric ball with the radius R having the cylindrical vacuum channel with the radius a (Fig. 1, left). In accordance with point (ii) in the Introduction, it is assumed that kR (cid:29)
1, where k = ω/c ( ω is thefrequency, c is the speed of light in vacuum). The ball material is characterized by permittivity ε , permeability µ , andthe refractive index n = √ εµ (the conductivity is considered to be negligible). The channel axis ( z -axis) coincideswith the ball diameter. The charge q moves with constant velocity (cid:126)V = cβ(cid:126)e z along the z -axis, and this velocityexceeds the “Cherenkov threshold”, i.e. β > /n . a small curvature of the wavefront). Further, we decompose this wave into a superposition of vertical and horizontal polarizations (concerning the plane of incidence) and calculate the field on the external surface of the aperture using Snell’s law and the Fresnel equations. The third step is different for the two methods. The ray optics method uses the ray optics laws for calculation of the wavefield outside the object [11, 12]. However, this technique has essential limitations. First, the so-called “wave parameter” ~ / D L ( L is a distance from the aperture to the observation point) should be small D . Note that this means that the dis-tance L can be much larger than but cannot be larger than / . In particular, we cannot consider the important Fraunhofer area where D . Second, the observation point can not be in the neighborhood of focuses and caustics, where ray optics is not applicable. The aperture method is more general [13–19] than the ray optics. It is valid for observation points with arbitrary wave parameter D , including the Fraunhofer (far-field) area and neigh-borhoods of focuses and caustics. In the third step of this technique, we calculate the field out-side the target using Stratton-Chu formulas (“aperture integrals”). These formulas allow deter-mining the field in the surrounding space if tangential components of the electric and magnetic fields on the aperture are known. This paper is devoted to the study of CR from a solid dielectric sphere with the radius being much larger than the wavelength and having an axisymmetric vacuum channel where the charge moves. Such a target can be manufactured with high accuracy and can be a prospective candidate for the aforementioned applications. We apply both methods for the spherical target and compare the obtained analytical results with the results of COMSOL simulations. The field on the ball surface
Here we consider a dielectric ball with the radius R having the cylindrical vacuum channel with the radius a (Fig. 1, left). In accordance with point (ii) in the Introduction, it is assumed that kR , where / k c = ( is the frequency, c is the speed of light in vacuum). The ball material is characterized by permittivity , permeability , and the refractive index n = (the conductivity is considered to be negligible). The channel axis ( z -axis) coincides with the ball diameter. The charge q moves with constant velocity z V c e = along the z -axis, and this velocity exceeds the “Cherenkov threshold”, i.e. n . Figure 1. C ross-section of the dielectric ball with vacuum channel (left); the incident and refracted waves (right). Incidence angle i and refraction angle t are positive for the left ray, and negative for the right ray; the illuminated part of the sphere (aperture) is highlighted by the bold red line. R a z r z i t p ' p p p t Figure 1: Cross-section of the dielectric ball with vacuum channel (left); the incident and refracted waves (right).Incidence angle θ i and refraction angle θ t are positive for the left ray, and negative for the right ray; the illuminatedpart of the sphere (aperture) is highlighted by the bold red line.For definiteness, we deal with a point charge having the charge density ρ = qδ ( x ) δ ( y ) δ ( z − V t ) where δ ( ξ ) is theDirac delta function. However, the results obtained further can be easily generalized for the case of a thin bunch withfinite length because we consider Fourier transforms of the field components. Further, we use the spherical ( R, θ, ϕ )and cylindrical ( r, ϕ, z ) coordinate systems.First, we find the “incident” field, i.e., solution of the “etalon” problem (field in the infinite medium with thevacuum channel). For the case under consideration, this field is well known [20]. We are interested in the incidentfield on the ball surface at the point R , θ (cid:48) , ϕ (cid:48) . Considering that kR (cid:29) H ( i ) ϕ (cid:48) ( R , θ (cid:48) ) ≈ qc η (cid:114) s πr (cid:48) exp (cid:110) i (cid:16) sr (cid:48) + ωV z (cid:48) − π (cid:17)(cid:111) , (1) η = − iπa (cid:20) κ − n β ε (1 − β ) I ( κa ) H (1)0 ( sa ) + sI ( κa ) H (1)1 ( sa ) (cid:21) − , (2)where r (cid:48) = R sin θ (cid:48) , z (cid:48) = R cos θ (cid:48) , s ( ω ) = kβ − (cid:112) n β − κ ( ω ) = kβ − (cid:112) − β , I , ( x ) are the modified Besselfunctions, H (1)0 , ( x ) are the Hankel functions. Note that Im s ( ω ) ≥ s ( ω )) = sgn ( ω ) (we exclude the exotic caseof the so-called “left-handed” medium). The result (1) is valid for | sr (cid:48) | (cid:29)
1. The electric field (cid:126)E ( i ) can be easilyfound because vectors (cid:126)E ( i ) , (cid:126)H ( i ) and the wave vector of Cherenkov radiation (cid:126)k ( i ) = s(cid:126)e r + (cid:126)e z ω/V form the right-handorthogonal triad in this area, thus (cid:126)E ( i ) = − (cid:112) µ/ε (cid:104) (cid:126)k ( i ) /k ( i ) × (cid:126)H ( i ) (cid:105) . The angle between the wave vector (cid:126)k ( i ) and thecharge velocity (cid:126)V is θ p = arccos (1 / ( nβ )).Applying Snells law and the Fresnel equations (note that waves have only vertical polarization) one can obtain thefollowing expressions for the field components on the outer surface of the ball: H ϕ (cid:48) ( R , θ (cid:48) ) = T v ( θ (cid:48) ) H ( i ) ϕ (cid:48) ( R , θ (cid:48) ) , E θ (cid:48) ( R , θ (cid:48) ) = H ϕ (cid:48) ( R , θ (cid:48) ) cos θ t ( θ (cid:48) ) , (3)where T v ( θ (cid:48) ) = 2 cos θ i ( θ (cid:48) )cos θ i ( θ (cid:48) ) + (cid:112) ε/µ cos θ t ( θ (cid:48) ) , (4) θ t ( θ (cid:48) ) = arcsin ( n sin θ i ( θ (cid:48) )) , θ i ( θ (cid:48) ) = θ (cid:48) − θ p . (5)Here θ i ( θ (cid:48) ) is the angle of incidence of the wave on the surface, θ t ( θ (cid:48) ) is the angle of refraction (see Fig. 1, right, whererays with positive and negative angles θ i and θ t are shown). III. RAY OPTICS METHOD
According to the ray optics approach, in order to determine the field at the given observation point r, ϕ, z , oneshould first determine the ray which starts at a certain point r (cid:48) ( θ (cid:48) ) = R sin θ (cid:48) , ϕ (cid:48) , z (cid:48) ( θ (cid:48) ) = R cos θ (cid:48) at the apertureFigure 2: The ray picture for ε = 2, β = 0 . ϕ we obtain ϕ = ϕ (cid:48) and r = r (cid:48) ( θ (cid:48) ) + l sin( θ (cid:48) − θ t ) , z = z (cid:48) ( θ (cid:48) ) + l cos( θ (cid:48) − θ t ) , (6)where l is the length of the ray. Therefore for each pair r, z corresponding pairs θ (cid:48) , l should be determined from (6).This problem can be solved numerically.A typical example of the rays structure is shown in Fig. 2. One can see that rays intersect each other (solutionof (6) is not unique) and form caustics. Moreover, the considerable concentration of the rays occurs near the raywhich is not refracted ( θ (cid:48) = θ p ) at some distance from the ball. The area of the concentration of the rays correspondsto the area where the field increases. The field along each ray can be written in the following form (for example, weconsider Fourier-transform of ϕ -component of the magnetic field): H ϕ ( R, θ ) = H ϕ ( R , θ (cid:48) ) (cid:112) D (0) /D ( l ) exp( ikl ) (7)(the electric field is equal to the magnetic field and orthogonal to it and the ray). Here H ϕ ( R , θ (cid:48) ) is the correspondingFourier-transform at the point of the ray exit and D ( l ) is the square of the ray tube cross-section [21]. Square rootin (7) describes the change in the field magnitude due to the divergence (or convergence) of the ray tube. Using (6),Cartesian coordinates of the observation point can be obtained as the function of θ (cid:48) and ϕ (cid:48) : x ( θ (cid:48) , ϕ (cid:48) ) = r ( θ (cid:48) ) cos ϕ (cid:48) , y ( θ (cid:48) , ϕ (cid:48) ) = r ( θ (cid:48) ) sin ϕ (cid:48) . D ( l ) can be calculated as follows [21]: D ( l ) = 1 √ g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ ∗ x κ ∗ y κ ∗ z ∂x/∂ϕ (cid:48) ∂y/∂ϕ (cid:48) ∂z/∂ϕ (cid:48) ∂x/∂θ (cid:48) ∂y/∂θ (cid:48) ∂z/∂θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (8)where g = R · sin θ (cid:48) is a determinant of the metric tensor for the sphere, (cid:126)κ ∗ is a unit vector along refracted ray: κ ∗ x = sin( θ (cid:48) − θ t ) cos ϕ (cid:48) , κ ∗ y = sin( θ (cid:48) − θ t ) sin ϕ (cid:48) , κ ∗ z = cos( θ (cid:48) − θ t ). After a series of transformations the followingexpression can be obtained: D ( l ) = cos θ t − lR (cid:20) sin ( θ t − θ i )sin θ i cos θ t + sin ( θ t − θ (cid:48) ) cos θ t sin θ (cid:48) (cid:21) + (cid:18) lR (cid:19) sin ( θ t − θ i ) sin ( θ t − θ (cid:48) )sin θ i cos θ t sin θ (cid:48) . (9)Numerical results based on (7) will be given in Section V. IV. APERTURE METHOD
The aperture integrals (Stratton-Chu formulas) for Fourier transform of the electric field can be written in thefollowing general form [15–19]: (cid:126)E (cid:16) (cid:126)R (cid:17) = (cid:126)E ( h ) (cid:16) (cid:126)R (cid:17) + (cid:126)E ( e ) (cid:16) (cid:126)R (cid:17) ,(cid:126)E ( h ) (cid:16) (cid:126)R (cid:17) = ik π (cid:90) Σ (cid:110)(cid:104) (cid:126)n (cid:48) × (cid:126)H (cid:16) (cid:126)R (cid:48) (cid:17)(cid:105) G (cid:16)(cid:12)(cid:12)(cid:12) (cid:126)R − (cid:126)R (cid:48) (cid:12)(cid:12)(cid:12)(cid:17) + 1 k (cid:16)(cid:104) (cid:126)n (cid:48) × (cid:126)H (cid:16) (cid:126)R (cid:48) (cid:17)(cid:105) · ∇ (cid:48) (cid:17) ∇ (cid:48) G (cid:16)(cid:12)(cid:12)(cid:12) (cid:126)R − (cid:126)R (cid:48) (cid:12)(cid:12)(cid:12)(cid:17)(cid:27) d Σ (cid:48) ,(cid:126)E ( e ) (cid:16) (cid:126)R (cid:17) = 14 π (cid:90) Σ (cid:104)(cid:104) (cid:126)n (cid:48) × (cid:126)E (cid:16) (cid:126)R (cid:48) (cid:17)(cid:105) × ∇ (cid:48) G (cid:16)(cid:12)(cid:12)(cid:12) (cid:126)R − (cid:126)R (cid:48) (cid:12)(cid:12)(cid:12)(cid:17)(cid:105) d Σ (cid:48) , (10)where Σ is the aperture area, (cid:126)E (cid:16) (cid:126)R (cid:48) (cid:17) , (cid:126)H (cid:16) (cid:126)R (cid:48) (cid:17) is the field on the surface of the aperture, the prime sign indicatesthat operator or coordinate is referred to the surface of an object, k = ω/c , (cid:126)n (cid:48) is the unit external normal to theaperture in the point (cid:126)R (cid:48) , G ( R ) = exp ( ikR ) /R is the Green function of Helmholtz equation, and ∇ (cid:48) is the gradient: ∇ (cid:48) = (cid:126)e x ∂/∂x (cid:48) + (cid:126)e y ∂/∂y (cid:48) + (cid:126)e z ∂/∂z (cid:48) . Analogous formulas are known for the magnetic field as well. Note that (cid:12)(cid:12)(cid:12) (cid:126)E (cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12) (cid:126)H (cid:12)(cid:12)(cid:12) in the region several wavelengths far from the aperture.In the case of the spherical object, it is convenient to write aperture integrals using spherical coordinates R, θ, ϕ .Besides the primary condition kR (cid:29)
1, we impose for simplicity an additional condition k ( R − R ) (cid:29)
1, which meansthat the observation point is located at a distance of no less than several wavelengths from the ball surface. Usingthe cylindrical symmetry of the problem, we can choose an observation point on the plane x, z ( ϕ = 0). As a result,one can obtain from (10) the following expressions for Fourier-transforms of the non-zero electric field components: (cid:126)E = (cid:126)E ( h ) + (cid:126)E ( e ) = (cid:126)E ( h + (cid:126)E ( h + (cid:126)E ( e ) , (11) (cid:40) E ( h r E ( h z (cid:41) = ikR π Θ (cid:90) Θ dθ (cid:48) π (cid:90) dϕ (cid:48) (cid:26) − cos θ (cid:48) cos ϕ (cid:48) sin θ (cid:48) (cid:27) sin θ (cid:48) exp( ik ˜ R )˜ R H ϕ (cid:48) ( R , θ (cid:48) ) , (12) (cid:40) E ( h r E ( h z (cid:41) = ikR R π Θ (cid:90) Θ dθ (cid:48) π (cid:90) dϕ (cid:48) (cid:26) R sin θ (cid:48) cos ϕ (cid:48) − R sin θR cos θ (cid:48) − R cos θ (cid:27) × sin θ (cid:48) (cos θ sin θ (cid:48) − sin θ cos θ (cid:48) cos ϕ (cid:48) ) H ϕ (cid:48) ( R , θ (cid:48) ) exp( ik ˜ R )˜ R , (13) (cid:40) E ( e ) r E ( e ) z (cid:41) = ikR π Θ (cid:90) Θ dθ (cid:48) π (cid:90) dϕ (cid:48) (cid:26) ( R cos θ (cid:48) − R cos θ ) cos ϕ (cid:48) − R sin θ (cid:48) + R sin θ cos ϕ (cid:48) (cid:27) sin θ (cid:48) exp( ik ˜ R )˜ R E θ (cid:48) ( R , θ (cid:48) ) , (14)˜ R = (cid:113) R + R − RR (cos θ cos θ (cid:48) + sin θ sin θ (cid:48) cos ϕ (cid:48) ) . (15)Here E θ (cid:48) ( R , θ (cid:48) ), H ϕ (cid:48) ( R , θ (cid:48) ) are the tangential components of the transmitted field on the surface of the ball deter-mined by formulas (3). The limits of integration in (12) (14) are determined by the conditions that the aperture isa part of the object surface illuminated by CR which does not experience total internal reflection, in other words, | θ i | < θ ∗ ≡ arcsin (1 /n ). One can show that Θ = max { θ p − θ ∗ , arcsin( a/R ) } , Θ = min { θ p + θ ∗ , θ p } . V. NUMERICAL RESULTS
Figure 3 shows the dependencies of the field magnitude on the angle θ obtained with the use of the ray optics (greencurves) and aperture (blue curves) methods. The results of simulations performed with RF module of COMSOLMultiphysics (red curves) are demonstrated as well. We emphasize that all three methods give similar results evenfor not very large targets (the sphere radius is approximately 30 / (2 π ) ≈ R at the angle θ which is equal to the CRangle θ p . In this case, the refracted wave propagates normal to the sphere. It can be seen that with an increase in thedistance from the sphere, the tendency toward an increase in the field is observed (complicated by the oscillations).The condensation of rays, which was noted within the ray optics examination (see Fig. 2), causes the observedphenomenon. The field starts to steadily decrease only after certain distance from the sphere surface. (a) β = 0 . R = 30; (b) β = 0 . R = 30;(c) β = 0 . R = 300; (d) β = 0 . R = 300; Figure 3: Absolute value of Fourier-transform of electric field (V / m · s) depending on the angle θ (grad) for thefollowing parameters: q = 1 nC, a = c/ω , ε = 2, R = 2 R ; β and R (in c/ω units) are given under the plots. Thecurves were obtained on the basis of the ray optics method (dashed green curve) and the aperture one (solid bluecurve). Red curves are the COMSOL simulations. VI. CONCLUSION
In this paper, two approaches have been applied for the analysis of radiation from a dielectric ball: the ray opticsmethod and the aperture method. Each of them demonstrated a good coincidence with COMSOL Multiphysics (in thearea of their applicability). However, the aperture method gave more exact results for the field structure. Numerouscalculations performed for various parameters show that, as a rule, the error of the aperture method in the area ofthe largest magnitudes of the field is less than 10% for the objects having the size of the order of 10 wavelengths. Forlarger objects, the error becomes even smaller. The main physical effects have been described. For example, it hasbeen shown that, in the direction of the Cherenkov angle, the radiation field possesses an expressed maximum.
VII. ACKNOWLEDGMENTS
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