Energy loss reduction of a charge moving through an anisotropic plasma-like medium
Aleksandra A. Grigoreva, Andrey V. Tyukhtin, Sergey N. Galyamin, Tatiana Yu. Alekhina
EEnergy loss reduction of a charge moving through an anisotropic plasma-like medium
Aleksandra A. Grigoreva, ∗ Andrey V. Tyukhtin, and Sergey N. Galyamin
Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
We analyze radiation of a charge moving in a vacuum channel in an anisotropic non-gyrotropicmedium with plasma-like components of the permittivity tensor. The dependencies of the energyloss of the charge per the unit path length on the charge velocity and the plasma frequenciesare considered. The most interesting result is that the energy loss is negligible when one of thepermittivity tensor components is equal to 1, and the charge velocity tends to the speed of light invacuum. This effect can be promising for applying in collimators of ultrarelativistic bunches.
INTRODUCTION
Interaction of a charged particle bunch with differentslow-wave structures (including uniform dielectric media)results in generation of various types of radiation. Thisradiation takes the energy away of the bunch, i. e. causesthe loss of the bunch energy. In many devices, such asradiation sources (classical vacuum microwave and THzdevices, as well as modern FELs), detectors and bunchdiagnostis systems, this effect is considered as positive.However, there is another class of devices where the men-tioned issue is parasitic and should be minimized. This isespecially the case for modern particle accelerators wherethe interaction of the bunch with various beamline struc-tures results in undesired decrease in the bunch qualityincluding the particle energy.Typically, the large portion of parasitic effects in ac-celerators and colliders is connected with its collima-tion system [1, 2]. Therefore, new materials are con-sidered for traditional collimator assembly [2] and alter-native dielectric-based collimation systems are also dis-cussed [3, 4]. Moreover, the successful usage of hollowelectron lenses (low-energy hollow electron beams) forefficient halo removal of intense high-energy beams instorage rings and colliders should be particularly men-tioned [5–7]. It is equally important that similar hol-low electron structures (called in that context the hollowplasma channels) were utilized recently in successful ex-periments on plasma wakefield acceleration [8, 9].When a charged particle bunch passes through a chan-nel in a medium, Cherenkov radiation (CR) is generatedand causes the corresponding radiation loss. The CR the-ory for the case of an infinite medium is well known [10].It should be underlined that an increase in charge veloc-ity, as a rule, increases the radiation loss per unit pathlength of the charge. This effect takes place for bothnondispersive isotropic medium and isotropic mediumhaving typical frequency dispersion; however, it is inter-esting that this increase is slower in the second case [11].The main aim of the present paper is to show that thespecific type of anisotropy and dispersion of the mediumresults in a strong decrease of the CR loss for ultrarela- ∗ [email protected] tivistic bunches. This fact allows to minimize the energyloss at collimation of the ultrarelativistic bunches.We consider the electromagnetic field of a charge mov-ing through a vacuum channel in an anisotropic non-gyrotropic uniaxial medium. It is assumed that com-ponents of a permittivity tensor possess plasma-like fre-quency dispersion. As it will be shown these componentshave different signs in the frequency range which is sig-nificant for radiation, i. e. the medium is so-called “hy-perbolic medium” within this range.Such a medium can be implemented in different ways.One of them is the use of metamaterials [12] which areconsidered very prospective during last decades due toa wide range of interesting possibilities they provide forphysics and techniques: negative refraction, focusing ef-fect etc. [13–16]. For the goals of this study, the so-called“hyperbolic metamaterials” (HMM) are appropriate, forexample, a HMM based on a two-dimensional silicon pil-lar array microstructure [16]. Another suitable exampleis a “wire medium”, an artificial lattice comprized of longmetal conductors with small spacings [17, 18]. Unfor-tunately, the latter structure possesses parasitic spatialdispersion which, however, can be suppressed [19]. In re-cent years, metamaterials has attracted an essential in-terest in the context of their implementation in particlebeam physics, including wakefield generation for particledetector design [20] and modern high-gradient accelera-tors [21, 22].A more traditional way to implement the medium withthe desired properties is the use of an electron plasma oran electron flux placed in a strong constant magneticfield. If the electron gyrofrequency is much larger thanthe plasma frequency and the typical frequencies of gen-erated radiation then the permittivity tensor is close tothe desired diagonal tensor. The successes noted abovein the use of the hollow electron beams [5–9] allow usto hope for the successful use of the method proposedhere to reduce the energy loss of bunches subjected tocollimation. ELECTROMAGNETIC FIELD OF A CHARGE
The system under study is an anisotropic uniaxialmedium possessing a vacuum channel with radius a . Theanisotropy axis coincides with the channel axis ( z -axis, a r X i v : . [ phy s i c s . acc - ph ] M a y a z~vq ε , µ ε , µ ε , µ ε , µ FIG. 1. Geometry of the problem: the transverse (left) andlongitudinal (right) cross-sections. see Fig. 1). The permittivities ( ε ) and permeabilities ( µ )in the channel and in the external area are ε = µ = µ = 1 , ˆ ε = ε ⊥ ε ⊥
00 0 ε (cid:107) . (1)Point charge moves along the channel axis with constantvelocity (cid:126)v = v(cid:126)e z . The charge and current densities are ρ = qδ ( x ) δ ( y ) δ ( z − vt ) , (cid:126)j = vρ(cid:126)e z . (2)Note that electromagnetic field structure in the case ofthe charge intersecting a boundary between a vacuumand an anisotropic medium with characteristics (1) wasanalyzed in [23], however the problem of energy loss wasnot considered and the channel radius was not taken intoaccount.The general form of the electromagnetic field compo-nents can be obtained using the Fourier transform [10].Here we give only the final expressions for the longitudi-nal component of the electric field: E z = (cid:90) + ∞−∞ e zω ( ω ) exp (cid:16) iζ ωv (cid:17) dω, (3)where e zω ( ω ) = iqκ πω − K ( κr ) + F F I ( κr ) for r ≤ a,H (1)0 ( sr ) sκaF for r > a, (4) F = ε (cid:107) κK ( κa ) H (1)1 ( sa ) + sH (1)0 ( sa ) K ( κa ) ,F = ε (cid:107) κI ( κa ) H (1)1 ( sa ) − sH (1)0 ( sa ) I ( κa ) , (5) κ = | ω | v (cid:112) − β , s = ω v ε (cid:107) ε ⊥ (cid:0) ε ⊥ β − (cid:1) ,ζ = z − vt, β = v/c. (6)Here c is the light velocity in a vacuum; iκ ( ω ) and s ( ω )are orthogonal components of the wave vector in thechannel and in the exterior area, respectively. The rule for the square root extraction (cid:16) s = (cid:112) s ( ω ) (cid:17) is the fol-lowing: the branch cuts coincide with lines Im ( s ( ω )) = 0and the physical Riemann surface sheet is fixed by therequirement Im ( s ( ω )) ≥ ω ∈ R .We consider the case when the permittivity tensorcomponents have the plasma-like form: ε (cid:107) = 1 − ω p (cid:107) ω + iν (cid:107) ω , ε ⊥ = 1 − ω p ⊥ ω + iν ⊥ ω , (7)where ω p (cid:107) , ω p ⊥ are plasma frequencies and ν (cid:107) , ν ⊥ arevalues responsible for losses. Further we will considerthe loss-free medium (i. e. ν (cid:107) , ⊥ → ω . We restrict the analysis to thecase ω p ⊥ < ω p (cid:107) because the most interesting situation isrealized when ω p ⊥ → s ( ω ) takesthe following form: s ( ω ) = | s ( ω ) | exp ( i arg ( s ( ω ))) , (8)wherearg s = π/ < | ω | < ω p ⊥ and | ω | > ω p (cid:107) ,π for ω p ⊥ < ω < ω p (cid:107) , − ω p (cid:107) < ω < − ω p ⊥ (9)on the integration contour. Figure 2 shows the locationof singularities of the function e zω ( ω ) (4) on the real axisof the complex plane ω (singularities on the imaginaryaxis don’t matter for further analysis). One can showthat the integration path Γ passes along the upper bankof the cuts, which corresponds to Eq. (8). ω p ⊥ − ω p ⊥ ω p k − ω p k Γ ω ω FIG. 2. The complex plane ω : the solid red line with arrowsis the integration path Γ; ± ω p (cid:107) and ± ω p ⊥ are the branchpoints; crosshatched regions are the branch cuts. Note that s ( ω ) is imaginary on the real frequenciesoutside the cuts. Therefore the function H (1)0 ( sr ) ex-ponentially decreases with r for such frequencies. Theseare “evanescent” waves that do not take any energy awayfrom the charge. At the same time, the function s ( ω ) isreal on the real axis in the ranges ω p ⊥ < | ω | < ω p (cid:107) , whichare the radiation frequencies ranges. Naturally, the in-tegrand (4) becomes here the quasi-plane extraordinary ω √ ε ⊥ /cω √ ε ⊥ /c ω/v~kk x ~V g ω/v k x ~k~V g k z k x FIG. 3. The isofrequency surface of extraordinary waves inthe cross-section k y = 0; (cid:126)V g (the group velocity vector) isorthogonal to this surface. Area k z > ω >
0, area k z < ω < wave for | s | r (cid:29)
1. It is interesting to note that sincesgn ( s ( ω )) = − sgn ( ω ) in the radiation ranges then thephase velocity (cid:126)V p of the wave is directed to the chargetrajectory (i.e. V px = ωs/k < k x + k y ε (cid:107) + k z ε ⊥ − ω c = 0 . (10)In the problem under consideration k x + k y = s , there-fore s = k x for k y = 0. Arrows in Fig. 3 illustrate ob-taining the physically correct solution for s (8), (9) usingthe predetermined k z = ω/v . As one can see, the groupvelocity (cid:126)V g (and therefore the Poynting vector which iscollinear to (cid:126)V g ) is directed away from the charge trajec-tory, V gx > RADIATION LOSS OF ENERGY
As it is well known, the energy loss per the path lengthunit are equal to the force acting on the charge with theopposite sign: dWdz = − q E z | ζ = r =0 . (11)Due to the Coulomb singularity of the field, we have totransform the integral assuming that ζ = 0 and only afterthat we can find the limit r →
0. Note that the singularsummand in (4) which is proportional to κ K ( κr ) /ω isan odd function of ω and it does not make a contributionin the integral (3) if ζ = 0. Therefore we can exclude thissingular term. After this, we can put r → dWdz = (cid:90) ∞−∞ f ( ω ) dω, f ( ω ) = − iq π κ ( ω ) ω F ( ω ) F ( ω ) . (12) Using (8), (9) one can obtain that f ( ω ) = − f ( − ω ) onthe real axis outside of the cuts, and therefore the ranges | ω | < ω p ⊥ and | ω | > ω p (cid:107) do not make any contribution inthe integral (12). At the same time, f ( ω ) = f ( − ω ) (theoverline means complex conjugation) on the cuts. Withthe help of this property one can obtain from (12) and(5) the following integral: dWdz = 2Re (cid:90) ω p (cid:107) ω p ⊥ f ( ω ) dω = 2 q π (cid:90) ω p (cid:107) ω p ⊥ κ ( ω ) ω × Im (cid:18) F ( ω ) F ( ω ) (cid:19) dω, (13)Using some properties of cylindrical functions [25] onecan obtain the simpler equivalent formula: dWdz = 4 q π a (cid:90) ω p (cid:107) ω p ⊥ κ ( ω ) (cid:12)(cid:12) ε (cid:107) ( ω ) (cid:12)(cid:12) ω | F ( ω ) | dω, (14)where F = ε (cid:107) κI ( κa ) H (2)1 ( | s | a ) − | s | H (2)0 ( | s | a ) I ( κa ) . (15)Note that another way to obtain the energy loss byradiation is a calculation of the energy flow through aninfinitely long cylinder with an axis coinciding with thecharge trajectory. Due to the absence of dissipation inthe medium, the radius of this cylinder can be arbitrary(the energy flow does not depend on it). This methodafter a series of cumbersome transformations leads to thesame result (14).Figure 4 shows the energy loss dW/dz depending onthe charge velocity β for different values of ω p ⊥ (cid:14) ω p (cid:107) andthe channel radius a . It is natural that the vacuum chan-nel narrowing causes an increase in energy loss. It isworth noting that there are two types of dependences ofthe energy loss on the charge velocity. One of them isa function with an extremum (for relatively small ω p ⊥ ),and the other is a monotonously increasing function (for ω p ⊥ exceeding certain value). Moreover, an increase inthe “orthogonal” plasma frequency causes the increase inthe velocity range with the low-level radiation.The most important phenomenon is that the energyloss vanishes in the ultrarelativistic limit β → ω p ⊥ = 0, i. e. at ε ⊥ = 1. This effect can be explainedanalytically. The integrand in (14) tends to the followinglimit at β → ω p ⊥ : κ (cid:12)(cid:12) ε (cid:107) (cid:12)(cid:12) ω | F | → (cid:12)(cid:12) ε (cid:107) (cid:12)(cid:12) a ω (cid:34)(cid:18) ε (cid:107) a J ( | s | a ) − | s | J ( | s | a ) (cid:19) + (cid:18) ε (cid:107) a Y ( | s | a ) − | s | Y ( | s | a ) (cid:19) (cid:35) − , (16)where | s | = ω p ⊥ c − (cid:16) ω p (cid:107) − ω (cid:17) / (cid:16) ω − ω p ⊥ (cid:17) − / and Y , ( x ) are Neumann functions. As one can see from . . . . . β d W d z (cid:16) q ω p k c (cid:17) a = 0 . . . . . . . β d W d z (cid:16) q ω p k c (cid:17) a = 10 0 . . . . . . β d W d z (cid:16) q ω p k c (cid:17) a = 1 . . . . . . . . . β d W d z (cid:16) q ω p k c (cid:17) a = 2 ω p ⊥ = 0 ω p ⊥ = 0 . ω p ⊥ = 0 . ω p ⊥ = 0 . FIG. 4. Dependence of the energy loss dW/dz (in units of q ω p (cid:107) /c ) on the dimensionless charge velocity β at different valuesof the channel radius a (in units of c/ω p (cid:107) ). Different lines correspond to different values of “orthogonal” plasma frequency ω p ⊥ (in units of ω p (cid:107) ). (16), the energy loss dW/dz tends to some finite valueat β → ω p ⊥ (cid:54) = 0. Otherwise, in the case ω p ⊥ = 0,we have | s | = 0 and expression (16) tends to zero. Thus,the energy loss dW/dz tends to zero at β → ω p ⊥ = 0. This result is confirmed by Figure 4.Although in reality the radiation energy loss is not ex-actly zero (for various reasons) it however can be dramat-ically reduced using the media (or artificial materials)having the properties assumed in this paper. This effectcan be used for the design of collimators with minimalenergy loss.As was already noted, one of the examples of themedium with the required properties is the hollow elec-tron flux placed in the strong longitudinal magnetic field (cid:126)H = H (cid:126)e z . The dielectric constant tensor (1) with ε (cid:107) in the form (7) and ε ⊥ ≈ ω H (cid:29) ω p (cid:107) where ω H = eH (cid:14) ( mc ) is an electron gyrofrequency and ω p (cid:107) = 4 πe N (cid:14) m is a squared plasma frequency ( e , m and N are the charge, mass and concentration of electrons,correspondingly). CONCLUSION