Electromagnetic field of a charge traveling into an anisotropic medium

PHYSICAL REVIEW E 84, 056608 (2011)

Electromagnetic field of a charge traveling into an anisotropic medium Sergey N. Galyamin* and Andrey V. Tyukhtin† Physics Department, Saint Petersburg State University, Saint Petersburg 198504, Russia (Received 5 August 2011; published 15 November 2011) We analyze the electromagnetic field generated by a point charge intersecting the interface between vacuum and a nonmagnetic anisotropic medium with a plasma-type dispersion of the dielectric permittivity tensor. After penetrating the medium, the charge moves along its main axis. The total field is presented as a sum of a self-field (i.e., a charge field in a corresponding unbounded medium) and a scattered field associated with the boundary influence. We show that the self-field in the considered anisotropic medium is divided into a quasistatic field and a wave field (the so-called “plasma trace” is absent in the case under consideration). Under certain conditions, the Vavilov-Cherenkov radiation generated in the medium is reversed (i.e., the energy flux density vector forms an obtuse angle with the direction of the charge motion). Accordingly, so-called reversed Cherenkov-transition radiation (RCTR) can be generated. We analytically and numerically investigate both the scattered field and the total one, and we show that RCTR exists in the vacuum region if the charge velocity exceeds a certain threshold value associated with total internal reflection. Computations of the Fourier harmonics of the field as well as the total field itself demonstrate that RCTR in vacuum can be a dominant effect. Some properties of RCTR can be useful for diagnostics of particle bunches and determination of medium characteristics. DOI: 10.1103/PhysRevE.84.056608

PACS number(s): 41.20.−q, 41.60.Bq, 41.60.Dk

I. INTRODUCTION

Radiation processes in anisotropic media exhibit important distinct characteristics as compared with the case of isotropic surroundings [1–6]. One essential distinction consists in the noncollinearity of the energy flux density vector and the wave vector. In the case of generation of Vavilov-Cherenkov radiation (VCR) in an anisotropic medium, it is possible that the Poynting vector of the VCR wave forms an obtuse angle with the direction of the charge motion. In this situation, it is expedient to call the VCR reversed. The extreme case of reversed VCR in which the Poynting vector is opposite to the wave vector was theoretically predicted to occur in an isotropic left-handed medium (LHM) [7]. Later, this effect was actively investigated [8–11] and experimentally proven [12,13]. This attractive phenomenon shows promise for a number of applications, including reversed Cherenkov detectors [8,11]. Considerable interest is paid in the scientific literature to the analysis of electromagnetic processes accompanying a charge movement in the presence of a boundary with an LHM [14–17]. As one example, a detailed investigation of the electromagnetic field in the case of a charge trajectory orthogonal to the interface between vacuum and an LHM was recently performed [16]. Because the VCR in an LHM is reversed, it reaches the interface and produces reflected and transmitted waves. These waves were termed reversed Cherenkov-transition radiation (RCTR) [15,16]. It was shown that RCTR can dominate transition radiation at certain parameters. Moreover, the estimation of RCTR decay due to losses in the LHM gives hope that this radiation can be detected in experiments. One of the most significant properties of RCTR in the vacuum region that was observed was the presence of lower

* †

[email protected] [email protected]

1539-3755/2011/84(5)/056608(16)

and upper thresholds in both the frequency and charge velocity domains. These thresholds carry information concerning both the charge velocity and the parameters of the LHM; thus, they can be used both for diagnostics of particles in accelerators and for characterization of LHM. However, the most significant problem with the abovementioned attractive phenomenon resides in the practical realization of an isotropic LHM. Recent progress in the design and fabrication of artificial metamaterials has resulted in a series of materials with rather exciting electromagnetic properties, including left-handed properties [18]. However, metamaterials characterized by isotropic effective parameters are still difficult to realize. Therefore, it is useful to analyze fields generated by a moving charge in the presence of an anisotropic medium, which nevertheless can support reversed VCR and therefore RCTR. It is remarkable that most metamaterials are substantially anisotropic. Moreover, Cherenkov detectors based on anisotropic metamaterials with left-handed properties can promise a wider variety of possibilities than those based on an isotropic LHM [11]. Here we investigate the electromagnetic field generated by a point charge passing from vacuum into a nonmagnetic electrically anisotropic uniaxial medium with a plasma-type frequency dispersion of the dielectric permittivity tensor components. After penetrating the medium, the charge moves along the main crystal axis with a constant velocity. As is known [4–6], for certain relationships among the anisotropic medium parameters, the VCR is reversed and the RCTR effect should therefore occur. This situation is analyzed in detail in the present paper. II. GENERAL RESULTS

In the present paper, we investigate the electromagnetic field of a charge traveling from vacuum into an anisotropic plasmalike medium. Let a point charge q move along the z axis of the Cartesian frame x, y, z with the constant 056608-1

©2011 American Physical Society

SERGEY N. GALYAMIN AND ANDREY V. TYUKHTIN


be written (in the cylindrical frame ρ, ϕ, z) in the following form: ⎧ (1,2) ⎫ ⎧ q(1,2) ⎫ ⎧ b(1,2) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎨ Eρ ⎬ ⎪ ⎨ Eρ ⎬ ⎨ Eρ ⎪ (1,2) Ez = Ezq(1,2) + Ezb(1,2) , (7) ⎪ ⎭ ⎪ ⎩ q(1,2) ⎪ ⎭ ⎪ ⎩H b(1,2) ⎪ ⎭ ⎩ (1,2) ⎪ Hϕ Hϕ ϕ FIG. 1. (Color online) Geometry of the problem.

velocity V = β c, where c is the speed of light in vacuum. The charge’s location at a time t is determined by the equations x = y = 0 and z = V t. The half space z > 0 is filled with a uniaxial medium characterized by the diagonal tensor of permittivity ⎞ ⎛ ε⊥ 0 0 εˆ 2 = ⎝ 0 ε⊥ 0 ⎠ (1) 0 0 ε and by the permeability μ2 = 1 (see Fig. 1). We suppose that the components of tensor εˆ 2 possess a plasma-type frequency dispersion: ε = 1 −

2 ωp

ω2 + 2iωd ω

,

ε⊥ = 1 −

2 ωp⊥

ω2 + 2iωd⊥ ω

,

(2)

where ωp and ωp⊥ are “plasma frequencies” and the parameters ωd ωp and ωd⊥ ωp⊥ are responsible for absorption. The half space z < 0 is described by the permeability μ1 = 1 and the plasmalike dielectric permittivity ε1 = 1 −

2 ωp1

ω2 + 2iωωd1

, ωd1 ωp1 ,

(3)

with ωp1 ωp⊥ ,ωp . Further, in the analytical manipulations we suppose that the frequency under consideration fulfills the inequality |ω| ωp1 . This means that the half space z < 0 is, in fact, a vacuum with ε1 ≈ 1, while the plasmatype dispersion (3) is a convenient way to introduce small losses. Moreover, we suppose that the following relations take place: ω max{ωd1 ,ωd ,ωd⊥ },

where the indices 1 and 2 correspond, respectively, to half spaces z < 0 and z > 0. The upper index “q” corresponds to the “incident” field (self-field of a charge in the corresponding unbounded medium): ⎧ q(1,2) ⎫ ⎧ q(1,2) ⎫ ⎪ ⎪ ⎨ Eρ ⎬ +∞ ⎪ ⎨ Eρω ⎪ ⎬ iωζ q(1,2) q(1,2) dω, (8) = exp Ez Ezω ⎪ V −∞ ⎪ ⎩ q(1,2) ⎪ ⎭ ⎩ q(1,2) ⎪ ⎭ Hϕ Hϕω q is1,2 (ω) (1) H [s1,2 (ω)ρ], 2c βε1,⊥ (ω) 1 c q q(1,2) Ezω s 2 (ω)H0(1) [s1,2 (ω)ρ], (9) (ω) = − 2c ωε1, (ω) 1,2 q q(1,2) is1,2 (ω)H1(1) [s1,2 (ω)ρ], (ω) = Hϕω 2c (1) where ρ = x 2 + y 2 , ζ = z − V t, H0,1 (kρ ρ) are Hankel functions and ω2 ω2 ε s1 = (ε1 β 2 − 1), s2 = (ε⊥ β 2 − 1). 2 (cβ) (cβ)2 ε⊥ q(1,2) Eρω (ω) =

(10) The upper index “b” corresponds to the “free” field or the field scattered from the boundary: ⎧ b(1,2) ⎫ ⎧ b(1,2) ⎫ E E ⎪ ⎪ ⎨ ρ ⎬ +∞ ⎪ ⎨ ρω ⎪ ⎬ b(1,2) b(1,2) Ez Ezω = exp(−iωt) dω, (11) ⎪ −∞ ⎪ ⎩ b(1,2) ⎪ ⎭ ⎩ b(1,2) ⎪ ⎭ Hϕ Hϕω ⎧ 2 (1) ⎫ kρ H1 (ρkρ ) ⎪ ⎧ b(1,2) ⎫ ⎪ ⎪ ⎪ ⎪ E ωε1,⊥ ⎪ ⎪ ⎪

+∞ ⎪ ⎨ ⎬ ⎨ ρω ⎪ ⎬ 3 (1) q ikρ H0 (ρkρ ) b(1,2) Ezω = ∓ ωε1, kz(1,2) ⎪ ⎪ ⎪ ⎪ ⎩ b(1,2) ⎪ ⎭ 2πβ −∞ ⎪ 2 (1) ⎪ k (ρkρ ) ⎪ ⎪ ⎪ Hϕω ⎩ ∓ ρ H1 (1,2) ⎭

(4)

|ω − ωp | ωd , |ω − ωp⊥ | ωd⊥ .

ckz

×B

Consequently, the following inequalities are fulfilled: ε1,⊥, |ε1,⊥, |, ε1 ε⊥, ,

(5)

B

(1)

where prime and double prime mean, respectively, real and imaginary parts: ε1 ≈ 1 − ε⊥,

≈1−

2 ωp1

ω2 2 ωp⊥, ω2

, ε1 ≈ 2

,

ε⊥,

2 ωp1 ωd1

≈2

B

(2)

,

ω3 2 ωp⊥, ωd⊥, ω3

(6) .

kz(1)

The general solution of the problem for arbitrary frequency dispersion is given, for example, in monograph [19]. Expressions for the nonzero electromagnetic field components can 056608-2

=

k (1) = z g3 k (2) = z g3

(1,2)

exp(ikz(1,2) |z|) dkρ ,

βε1 kz(2) − ωc ε⊥ kρ2 − s12

ε⊥ β 2 ω c

− βkz(1)

ω2 ε1 − kρ2 , kz(2) = c2

−

+

ε⊥ ε−1 ε1 β 2

(13)

,

(14)

ω2 2 , ε − k ρ c2

(15)

+ βkz(2)

kρ2 − s22 ε⊥ ε

,

ω c

βkz(1) ε⊥ + ωc ε1

(12)

g3 (kρ ,ω) = ε1 kz(2) + ε⊥ kz(1) .

(16)

ELECTROMAGNETIC FIELD OF A CHARGE TRAVELING . . .

Note that the “physical” branch of the radicals in Eqs. (10) and (15) should be defined for real values of ω for physical 2 reasons. Because both media exhibit absorption, Im s1,2

= 0 (1,2) 2 and Im(kz ) = 0 for real ω, and one must demand that Im s1,2 > 0,

Im kz(1,2) > 0.

(17)

The conditions (17) guarantee the exponential decay of the Fourier harmonics (9) with increasing distance ρ from the charge trajectory and the exponential decay of the Fourier harmonics (12) with increasing distance |z| from the boundary. Further, in the analytical manipulations, losses are assumed to be infinitely small; they are used to identify the relative position of the integration path and the integrands’ singularities only. However, the numerical approach described below can be applied for arbitrary losses, and the presented numerical calculations were performed for finite losses. III. SELF-FIELD OF A CHARGE A. Analytical approach to the investigation of the self-field components

The self-field of a charge (8) can be analyzed with the use of the conventional approach suggested in our earlier papers for several cases of different isotropic media [16,20–22]. The essence of this approach is to consider the integrands of (8) in the complex plane ω and to use methods of complex function theory for the calculation of the integrals. The branching functions s1,2 should be determined in the complex plane ω in such a way that the first inequality in Eq. (17) is fulfilled on the real axis. It is expedient to draw cuts of functions s1,2 in segments where Im s1,2 = 0 and to fix the “physical” sheet of the Riemann surface by the rule Im s1,2 > 0. Next, we may take advantage of the following helpful properties (for more detail see [16,20–22]). If the integration path consists of two parts, with one of them ( + ) lying in the domain Re ω > 0, another ( − ) lying in the domain Re ω < 0, and the total being symmetrical with respect to the imaginary axis, then, for the integrals (8), we obtain

iωζ iωζ dω = 2 dω , Fωq exp Re Fωq exp V V + (18) q

where Fω is any of functions (9). Later we use integration paths possessing the required symmetry. Next, based on the asymptotic expression for s1,2 , s1,2 −−−−→ V −1 1 − β 2 iωsgn (Re ω) , (19) |ω|→∞

one can determine areas in the ω plane where the integrands of (8) vanish at |ω| → ∞: Im ω > −|Re ω|ρ 1 − β 2 ζ −1 for ζ > 0, (20) Im ω < |Re ω|ρ 1 − β 2 |ζ |−1 for ζ < 0. Asymptotes of the steepest-descent path (SDP) for integrands of (8) lying within the regions defined by (20) are


determined by the equation ρ 1 − β 2 Im ω = ζ |Re ω| .

(21)

Further manipulations are based on proper transformation of the initial integration path in Eq. (8). This allows both the attainment of conventional analytical expressions and the development of effective numerical algorithms for calculation of the self-field components (8). To illustrate the possibilities of the mentioned approach, we consider the self-field of a charge in the anisotropic medium described by (1) and (2). When losses are neglected (ωd = ωd⊥ = 0), the function s22 (ω) can be presented in the form 2 2 2 2 1 − β 2 ω − ωp ω − ωc 2 s2 (ω) = − 2 2 , (22) 2 β c ω2 − ωp⊥ where ωc = iωp⊥ β 2 /(1 − β 2 ). As one can see from (22), the frequency ranges of propagating waves (we call them the VCR frequency bands) are intervals of real frequencies between plasma frequencies [Fig. 2(I)]. The complex plane ω contains both cuts along the VCR bands (on the real axis) and cuts along segments [|ωc |,+∞), [−|ωc |,−∞) of the imaginary axis [Fig. 2(II)]. Assuming that the medium exhibits nonzero losses and then making passage to the limit ωd = ωd⊥ = 0, one can show that the integration path γ0 passes cuts along the VCR bands on the upper banks, where sgns2 = −sgnω for ωp⊥ < ωp and sgns2 = sgnω for ωp⊥ > ωp . Thus, in the case of ωp⊥ < ωp , the phase velocity V ph is directed toward the line x = y = 0; that is, Vphρ = s2 ω(s22 + ω2 V −2 ) < 0, while in the case of ωp⊥ > ωp , the phase velocity V ph is directed away from the source, Vphρ = s2 ω(s22 + ω2 V −2 ) > 0. This result is in agreement with that obtained using Mandelshtam’s concept. This concept dictates that the group velocity of a physical wave must be directed away from the source [23–25]. For the case under consideration, this condition means that the group velocity must be directed away from the charge’s motion line: Vgρ > 0 [25]. The dispersion relation in the medium described by (1) has the form (23) = kx2 ε + kz2 ε⊥ − ω2 c2 = 0. The group velocity can be calculated with the use of the formula ∂/∂ k dω . (24) =− V g = ∂/∂ω d k Using (23) and (2), one obtains (for ω > 0) ∂ kz2 ∂ε⊥ kx2 ∂ε 2ω ∂ , =− 2 + 2 + 2 = − ∂ω c ∂ω ε ∂ω ε⊥ ∂ω ∂ −1 2kx 2kz Vg = e x + e z . ∂ω ε ε⊥

(25) (26)

Hyperbolas corresponding to Eq. (23) for min (ωp ,ωp⊥ ) < ω < max (ωp ,ωp⊥ ) are shown in Fig. 3. Thin arrows indicate the group velocity direction in accordance with (26) (it is known [25] that the vector of the group velocity is orthogonal to these curves). The z component of the VCR wave vector is equal to ωV −1 [in accordance with (8)]. The wave vector,

056608-3



FIG. 2. (Color online) Schematic view of the dispersion characteristics (I) and the complex plane ω (II) for the plasmalike medium (1), (2) in the case of ωp < ωp⊥ (a) and ωp⊥ < ωp (b). The lower row (II) shows the right half plane only because the cuts are situated symmetrically with respect to the imaginary axis. Punctured points are branch points, solid lines are cuts (along lines Im s2 = 0), γ0 is the initial integration path, and γ± are contours used for numerical calculations in the domains ζ > 0 and ζ < 0, respectively.

which corresponds to the group velocity directed away from the charge trajectory (Vgρ > 0), is shown by a bold arrow. As one can see from Fig. 3, the phase velocity of this wave is directed toward the source (toward the charge motion line) in the case of ωp⊥ < ωp [Fig. 3(a)], while it is directed away from the source (away from the charge motion line) in the case of ωp⊥ > ωp [Fig. 3(b)]. This result coincides with that discussed in the previous paragraph. Thus, we have proven the equivalence of results obtained with the use of Mandelshtam’s concept and with the use of the “decay concept” (17). Moreover, as one can see, in the case of ωp⊥ > ωp , the z projection of the group velocity is negative; that is, the VCR is reversed.

Based on (20), the initial integration path in Eq. (8) can be complemented by a half circle of infinite radius, in the upper half plane for ζ > 0 and in the lower half plane for ζ < 0. Thereafter, the field components (8) can be presented as a sum of integrals along cuts [see Fig. 2(II)]. For ζ > 0, only the cut situated on the positive imaginary semiaxis contributes to the field. This contribution describes the quasistatic (“quasi-Coulomb”) field of the charge. For ζ < 0, both the cut on the negative imaginary semiaxis and cuts along the VCR frequency bands contribute to the field. The contribution of cuts lying on the real axis describes a wave field, that is, the VCR field. Performing a series of manipulations, one can obtain the following

FIG. 3. (Color online) Isofrequency curves [green (light gray) lines] and direction of the group velocity (thin short arrows) for an infinite its projections medium described by Eqs. (1) and (2) in the case of ωp⊥ < ωp (a) and ωp⊥ > ωp . (b) Bold arrows show the VCR wave vector k, on the x and z directions, and corresponding group velocity V g directed away from the charge motion line (Vgx > 0). 056608-4


expressions:


⎧ q(2) ⎫ ⎧ q(2) ⎫ ⎪ E ⎪ ⎪ E ⎪ ⎪ ⎨ ρC ⎪ ⎨ ρW ⎪ ⎬ ⎪ ⎬ q(2) q(2) = EzC + Ezq(2) , Ez W ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q(2) ⎭ ⎩ q(2) ⎭ ⎩ q(2) ⎭ HϕC HϕW Hϕ

(27)

⎧ q(2) ⎫ ⎧ π|s (i ω)| ⎫ 2 ˜ J1 [ρ |s2 (i ω)|] ˜ ⎪ ⎪ E ⎪ ⎪ ⎪ ⎪ ε⊥ (i ω) ˜ ⎪

∞ ⎨ ρC ⎪ ⎨ ⎬ ⎬ |ζ | q q(2) πβc 2 = × ε (i ω) d ω, ˜ exp −ω˜ EzC |s |s (i ω)| ˜ J (i ω)|] ˜ sgn(ζ ) [ρ 2 0 2 ˜ ˜ ω ⎪ ⎪ ⎪ πβc |ωc | ⎪ V ⎪ ⎪ ⎪ ⎩ q(2) ⎪ ⎩ ⎭ ⎭ HϕC ˜ J1 [ρ |s2 (i ω)|] ˜ πβ |s2 (i ω)|

(28)

⎧ q(2) ⎫ ⎫ ⎧ π|s2 (ω)| J [ρ|s2 (ω)| sin ω |ζV | ⎪ ⎪ ε⊥ (ω) 1 ⎪ EρW ⎪ ⎪ ⎪ ⎪

ωp⊥ ⎪ ⎨ ⎬ ⎨ |ζ | ⎬ 2q q(2) −πβc 2 dω,

(−ζ ) × |s (ω)| J [ρ|s (ω)|) cos ω EzW = 0 2 ωε (ω) 2 V ⎪ ⎪ ⎪ πβc ωp ⎪ ⎪ ⎪ ⎪ |ζ | ⎩ q(2) ⎪ ⎭ ⎭ ⎩ HϕW πβ|s2 (ω)|J1 [ρ|s2 (ω)|] sin ω V

(29)

⎧ q(2) ⎫ ⎪ E ⎪ ⎪ ⎬ ⎨ ρ ⎪

where (ξ ) is the Heaviside step function. Note that dissipation is neglected in Eqs. (27)–(29), that is, ωd = ωd⊥ = 0. The index “C” is assigned to the quasistatic parts, and the index “W” is assigned to the wave parts of the field components. A quasistatic field exists both behind and before the moving charge and quickly diminishes with distance from it. A wave field (VCR field) exists behind the charge only and oscillates with distance ζ . It should be noted that, in the case of an isotropic medium, the representation analogous to (27)–(29) contains (along with quasistatic and wave terms) a so-called “plasma trace” which is associated with the contribution of poles ε(ω) = 0 [5,16,20–22]. In the present case, in spite of the fact that the q(2) q(2) expressions for Eρω and Ezω in Eq. (9) contain the multiplier −1 −1 2 −1 ε⊥ (s2 ε ∼ ε⊥ ), (27) is free from this plasma trace. Taking

and

⎧ q(2) ⎫ ⎪ ⎪ ⎪ Eρω ⎪ ⎨ q(2) ⎬ Ezω ∼ |exp(iρs2 )| = exp(−ρIm s2 ) −−→ 0. β→0 ⎪ ⎪ ⎪ ⎭ ⎩ q(2) ⎪ Hϕω

As all real media exhibit nonzero losses, VCR is absent at small velocity and the full field is determined by the quasistatic part, (28). It should be noted that a representation similar to (27) can be obtained for the self-field of the charge in vacuum [which is described by the permittivity given by (3) in the frame of the present paper]. In this case, the charge’s field contains only the quasistatic term described by formula (28), wherein one should substitute 2 → 1 and ε,⊥ → ε1 .

q(2)

EzW as an example and considering the case ωp⊥ < ωp , let us show that the integrands in Eq. (29) are free of any singularity as ω → ωp⊥ . Indeed, in the vicinity of ωp⊥ (ω = ωp⊥ + δ, δ → +0), the integral (29) has the form

q(2) EzW

∼

+∞ √ 1/ δ

2 2 ωp −ωp⊥ ωp⊥ π cos ρ ξ − 4 2β 2 c2 ξ 3/2

dξ,

√ with ξ = 1/ ω − ωp⊥ , and converges at the upper limit. In a q(2) similar way, one can demonstrate that EρW is free from any singularity as ω → ωp⊥ as well. As follows from the above considerations, the VCR frequency range does not depend on the charge velocity β, and thus VCR exists at arbitrarily small β. This paradox can be resolved by taking losses into account. For ω > 0 in the VCR frequency range ωp⊥ < ω < ωp (we suppose for definiteness that ωp⊥ < ωp ), one obtains Re s2 −−→ −ω(cβ)−1 |ε |/ε⊥ , β→0 Im s2 −−→ ω(2cβ)−1 |ε |/ε⊥ (ε /|ε | + ε⊥ /ε⊥ ), β→0

B. Numerical approach and numerical results

Another approach to calculating the charge’s field in an unbounded medium consists of the following. Based on (18), one can perform the integration over positive ω only. For large values of |ω|, it is reasonable to transform the initial integration path γ0 to pass parallel to the SDP asymptote, (21). For relatively small values of ω, it is reasonable to bypass singularities at a certain distance in the complex plane to ensure smooth behavior of the integrands. In the region before the charge (ζ > 0), it is convenient to use a ray [γ+ in Fig. 2(II)], while in the region behind the charge (ζ < 0), a trapezoidal contour can be utilized [γ− in Fig. 2(II)]. One can fit the parameters of contour γ− to achieve the best convergence of the integrals (8). Moreover, one can apply the described approach to a medium with arbitrarily large losses [recall that the analytical decomposition given by (27) is valid at negligible losses]. Below, some results of the numerical calculation are preq(2) sented. Figure 4 shows the dependence of the Hϕ component on the relative distance ζ at certain offsets ρ from the charge motion line. The bold solid line in Fig. 4 represents the total magnetic field, while the dotted and dashed lines correspond to the quasistatic and wave parts, calculated with the use of formulas (28) and (29), respectively. As one can see, for relatively small distances, the quasistatic field exhibits a huge peak

056608-5



FIG. 4. (Color online) Dependence of the total magnetic field Hϕq(2) (solid green line) and its quasistatic (dotted red line) and wave (dashed blue line) parts on relative distance ζ for q = −1 nC and different β and ρ in the unbounded medium described by Eqs. (1) and (2) with the following parameters (in units of ωp = 2π × 1010 s−1 ): ωp⊥ = 1.5 and ωd = ωd⊥ = 10−3 .

near the charge motion plane ζ = 0 and quickly diminishes away from it [see Fig. 4(a)]. Moreover, this quasistatic peak grows with β. As ρ increases, the quasistatic peak becomes comparable with the wave field [Fig. 4(b)], and for relatively large ρ, it becomes inconsequential on the background of the wave field [Fig. 4(c)]. The oscillating wave field plays the main role at sufficiently large ζ behind the charge; its magnitude tends to slightly increase as β increases, while the frequency of oscillations tends to decrease as ρ increases. IV. SCATTERED FIELD

For investigation of the scattered field described by (11), we use two approaches as well. In the frame of an analytical approach, we construct asymptotic expressions for the Fourier harmonics (12) in the far-field zone using the steepest-descent technique. In the frame of a numerical approach, we develop an algorithm for the calculation of both the Fourier harmonics (12) and the total scattered field described by (11). First, we describe an analytical approach. As one can see, the following helpful property may be used [it is the consequence of the reality of the field components in Eq. (11) as well as Eq. (18)]:

Fωb exp(−iωt) dω = 2 Re Fωb exp(−iωt) dω . (30)

+

Here is the integration path symmetrical with respect to the imaginary axis, + is the part of lying in the domain Re ω > 0, and Fωb is any of the functions (12). Further, we use the real axis ω as the integration path in Eq. (11). Thus, according to (30), it is enough to consider the functions (12)

at positive frequencies only. Moreover, we consider only frequencies from the VCR band (between plasma frequencies) and compare two cases: ωp < ωp⊥ and ωp > ωp⊥ . The integrands in Eq. (12) contain two pairs of branch points ±k1,2 , where k1 = ω2 c−2 ε1 , k2 = ω2 c−2 ε , Im k1,2 > 0, (31) corresponding cuts along lines Im kz(1,2) = 0, and three types of poles being the nulls of denominators. For convenience, we call these types “first,” “second,” and “third” poles. First poles are nulls of the following equations: kρ2 − s12 = 0,

(half space z < 0),

ωc−1 − βkz(1) = 0,

(half space z > 0).

(32) (33)

The solution of (32) and (33) is kρ = ±s1 ; in the case of the half space z > 0, the poles are situated on those banks of the cut Im kz(1) = 0 where arg kz(1) = 0. Second poles satisfy the following equations: ωc−1 + βkz(2) = 0,

(half space z < 0),

(34)

−

(half space z > 0).

(35)

kρ2

s22

= 0,

In both cases, the solution is kρ = ±s2 , but in the case of the half space z < 0, the poles lie on banks of the cut Im kz(2) = 0 with arg kz(2) = π . Third poles ±kρP3 satisfy the following equation:

056608-6

g3 kρP3 = ε1 kz(2) kρP3 + ε⊥ kz(1) kρP3 = 0.

(36)



Let us determine the positional relationship of poles and cuts in the kρ plane. For the sake of simplicity, while constructing an asymptotic representation, we take into account only the main terms of the spatial radiation—spherical and cylindrical waves—which depend on distance, respectively, as ∼R −1 and ∼ρ −1/2 . All other waves exhibiting faster decay will be neglected in the asymptotic representation. Rigorous consideration shows that only the “second” poles satisfy this condition, while the contribution of the “first” and the “third” poles exhibit exponential decay with increasing ρ and |z|, respectively. In the case of a “first” pole, its contribution describes a quasistatic field, while in the case of a “third” pole, it represents a surface wave. In both cases, these terms are neglected in the asymptotic representation. Thus, the only essential poles are kρ = ±s2 , situated on banks of the cut Im kz(2) = 0. In the end, therefore, what we must do is determine the configuration and positional relationship of the cuts Im kz(1) = 0 and Im kz(2) = 0. The cuts Im kz(1,2) = 0 satisfy the following system: 2 2 (37) Im kz(1,2) = 0, Re kz(1,2) > 0. For kz(1) , correct to first order with respect to ε1 /ε1 , we obtain 2kρ kρ = ω2 c−2 ε1 , ω2 c−2 ε1 − (kρ )2 > 0.

(38)

The line satisfying (38) is part of a hyperbola, the asymptotes of which coincide with the kρ and kρ axes. The branch points ±k1 lie, correspondingly, in the I and III quarters of the kρ plane. /ε⊥ ∼ ε /ε , For kz(2) , correct to first order with respect to ε⊥ we obtain ε 2kρ kρ 1 ε⊥ 2 (k − ) + ρ ε ε⊥ ε ε ε 1 ε⊥ ω2 ε⊥ 2 (k − − ) − (39) ρ = 0, ε ε⊥ ε c2 ε⊥ 2 ε ε − ε1 (kρ )2 + ε2 ε⊥ − ε kρ kρ + ε1 (kρ )2 + ωc2 > 0.

⊥

Analyzing system (39), one can show that the cuts of the radical kz(2) are parts of a hyperbola, the asymptotes of which are rotated with respect to the kρ and kρ axes through an angle α, determined by the formula tan 2α = (ε /ε − ε⊥ /ε⊥ ). A general view of cuts and poles for frequencies within the VCR band in two comparable cases is presented in Fig. 5. A. Asymptotic representation of the field in the vacuum region

Here we construct the asymptotic representation for the Fourier harmonics (12) of the scattered electromagnetic field in the vacuum half space z < 0. For simplicity, we work √ in the limit ε1 → 0 and impose k1 = ωc−1 ε1 > 0, where 2 ε1 = 1 − ωp1 /ω2 . However, losses in the half space z > 0 are still taken into account. As usual [16,26] we introduce new spatial variables R and θ (see Fig. 1), such that ρ = R sin θ and |z| = R cos θ (in the vacuum half space 0 < θ < π/2). We also introduce the new variable of integration ψ, such that kρ = k1 sin ψ and consequently kz(1) = k1 cos ψ.

FIG. 5. (Color online) General view of the disposition of branch points ±k1 and ±k2 , corresponding cuts (dashed lines) and poles ±s2 on the kρ complex plane for frequencies within the VCR band in the case of ωp⊥ < ωp (a) and ωp < ωp⊥ . (b) Integration path goes along the real axis.

A general view of the complex plane ψ is presented in (1) corresponds to the initial integration Fig. 6. The contour 0ψ path (real axis). The poles ±s2 transform to the points ±ψ2(1) (sin ψ2(1) = k1−1 s2 ), the branch points ±k1 disappear, and the branch points ±k2 transform to the points ±ψb2 (sin ψb2 = k1−1 k2 ). The expressions in Eq. (12) take the form ⎧ ⎫ ⎧ b(1) ⎫ b(1) ⎪ eρω (ψ,ω)H1(1) (Rk1 sin θ sin ψ) ⎪ E ⎪ ⎪ ⎪ ⎪

ρω ⎨ ⎬ ⎬ ⎨ (1) b(1) b(1) Ezω = ezω (ψ,ω)H0 (Rk1 sin θ sin ψ) (1) ⎪ ⎪ ⎪ 0ψ ⎪ ⎭ ⎩ b(1) ⎪ ⎩hb(1) (ψ,ω)H (1) (Rk sin θ sin ψ)⎪ ⎭ H ϕω

ϕω

1

1

× exp(iRk1 cos θ cos ψ) dψ, (40) ⎧ ⎫ ⎫ ⎧ b(1) 2 c(k sin ψ) k cos ψ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ωε1 ⎬ ⎨ eρω (ψ,ω) ⎪ ⎬ q ⎨ 3 b(1) ic(k1 sin ψ) ezω (ψ,ω) = B (1) (k1 sin ψ,ω). − ωε1 ⎪ ⎪ ⎪ ⎪ ⎭ 2πβc ⎪ ⎩ b(1) ⎪ ⎩ ⎭ hϕω (ψ,ω) −(k1 sin ψ)2 (41) The following manipulations consist of finding the SDP passing through the saddle point and transforming the initial (1) path 0ψ to the SDP [25,26]. Using asymptotic expressions for Hankel functions, one can extract the exponential term from (40): ∼ exp [1 ϕ1 (ψ,θ )], where 1 = k1 R and ϕ1 (ψ,θ ) = i cos(ψ − θ ). The SDP passing through the saddle point ψS1 = θ (dϕ1 /dψ|ψ=ψS1 = 0) is

056608-7



The pole ψ2(1) is captured on the sheet with Im kz(1) < 0, so that Re kz(1) (s2 ) > 0 and Im kz(1) (s2 ) < 0. As a result, we have ⎧ b(1) ⎫ ⎧ b(1)S ⎫ ⎧ b(1)P ⎫ ⎪ ⎬ ⎪ ⎨ Eρω ⎪ ⎬ ⎪ ⎨ Eρω ⎪ ⎬ ⎨ Eρω ⎪ b(1) b(1)S b(1)P Ezω ≈ Ezω + Ezω , (43) ⎪ ⎭ ⎪ ⎩H b(1)S ⎪ ⎭ ⎪ ⎩ b(1)P ⎪ ⎭ ⎩H b(1) ⎪ H ϕω ϕω ϕω where the first term is the integral along the SDP, ⎧ ⎫ ⎧ b(1)S ⎫ (1) b(1) ⎪ ⎪ e (ψ,ω)H (Rk sin θ sin ψ) 1 E ⎪ ⎪ ⎪ ⎪ ρω 1

⎨ ⎨ ρω ⎬ ⎬ (1) b(1)S b(1) Ezω = ezω (ψ,ω)H0 (Rk1 sin θ sin ψ) (1) ⎪ ⎪ ⎪ SDP ⎪ ⎩H b(1)S ⎪ ⎭ ⎩hb(1) (ψ,ω)H (1) (Rk sin θ sin ψ)⎪ ⎭ ϕω ϕω

1

1

× exp (iRk1 cos θ cos ψ)dψ,

(44)

while the second one corresponds to the pole contribution. It should be noted that, as it is known [26], integrals along SDP’s + b2 represents the lateral wave, whose decay with distance is faster than that of spherical and cylindrical waves. Thus, this wave is excluded from the representation (43). Below we find the rigorous condition of the ψ2(1) pole capturing. First, for the significance of the wave corresponding to this pole, the following conditions must be fulfilled: Re s2 Im s2 , or Re s22 Im s22 ,

Re kz(1) (s2 ) Im kz(1) (s2 ), 2 2 Re kz(1) (s2 ) Im kz(1) (s2 ) ,

which is equivalent to (correct to first order with respect to ε⊥ /|ε⊥ | ∼ ε /|ε |) |ε⊥ |β 2 +1

ε1 β − 2

FIG. 6. (Color online) General view of the disposition of branch points ±ψb2 , corresponding cuts (dashed lines), and poles ±ψ2(1) on the ψ plane for frequencies within the VCR band in the case of (1) is the initial integration path, ωp⊥ < ωp (a) and ωp < ωp⊥ (b). 0ψ (1) SDP is the SDP.

Re ϕ1 (ψ,θ ) < 0.

|ε⊥ |

(|ε⊥ |β 2

+ 1)

ε⊥ 2 (|ε⊥ |β +1)+ , ε |ε⊥ |

ε

|ε⊥ |

ε

(|ε |β 2 ε ⊥

(45) ε⊥ + 1) + . |ε⊥ |

The first condition in Eq. (45) is fulfilled for arbitrary ω within ωp < ω < ωp⊥ and arbitrary β. The second condition in → 0), Eq. (45), for fixed velocity β and negligible losses (ε,⊥ can be written as ωp < ω < (β),

(46)

where (β) is determined from 2 4 2 ω (1 − β ) ωp (1 − β 2 )2 p 2 2 2 (β) = ωp⊥ β 2 . (47) + + ωp 2 4

determined by the following conditions: Im ϕ1 (ψ,θ ) = 1,

ε

ε

(42)

(1) In Fig. 4, the SDP is marked SDP . In the most interesting case of ωp < ωp⊥ [see Fig. (6b)], (1) (1) the transformation 0ψ → SDP captures the pole ψ2(1) . After transformation, the integration path consists of two parts. The (1) , while the second one goes along the first one goes along SDP + SDP’s b2 , originating from the branch point ψb2 . Note that the integration path lies on a Riemann sheet with Im kz(2) > 0 between points A1 and A2 , while the rest of it lies on a sheet with Im kz(2) < 0. This path passes through the saddle point on the sheet with Im kz(1) > 0 and Im kz(2) > 0.

The second condition in Eq. (45), for fixed ω, takes the form 2 2 2 ω ω − ωp 2 . β > βRCTR (ω) = (48) 2 ωp⊥ − ω2 ωp

= 0), the intervals If losses are taken into account (ε,⊥ represented by (46) and (48) become narrower. For example, at fixed ω, for the significance of pole ψ2(1) , it is necessary that the following inequality be fulfilled along with (48): βRCTR ε ε1 ε⊥ β − βRCTR + ; (49) 2 ε ε1 − ε |ε⊥ |

056608-8


that is, the velocity β must be sufficiently far from the threshold value βRCTR . As rigorous consideration shows (see Appendix), the ψ2(1) pole capturing condition has the form θ > θ1 , where ε 1 + |ε⊥ |β 2 θ1 = θ10 + δθ1 , sin θ10 = , ε1 |ε⊥ |β 2 (50) δθ1 = 2/k1 R1 , ! "−2 ε 1 8 ε⊥ R1 = + . (51) tan θ10 k1 ε |ε⊥ | 1 + |ε⊥ |β 2 Expressions for the pole contribution take the form ⎧ s k(1) (s ) (1) ⎫ ⎧ b(1)P ⎫ 2 z 2 H1 (s2 R sin θ )⎪ ⎪ ⎪ ⎪ ⎪ g3∗ (ω) ⎨ ⎨ Eρω ⎪ ⎬ ⎬ iq −is22 b(1)P (1) Ezω = H (s2 R sin θ ) g3∗ (ω) 0 ⎪ ⎪ βc ⎪ ⎪ −s ⎩ b(1)P ⎪ ⎪ ⎭ ⎩ ⎭ (1) 2 ωε1 Hϕω H (s R sin θ ) ∗ 2 cg3 (ω) 1 × exp ikz(1) (s2 )R cos θ (θ − θ1 ), ωε1 + ε⊥ kz(1) (s2 ), g3∗ (ω) = − cβ

(52) (53)

⎧ b(1) ⎫ ⎪ ⎨ eρω (θ,ω) exp (−3π i/4) ⎪ ⎬ b(1) ezω (θ,ω) exp (−π i/4) ⎪ ⎪ ⎩ b(1) ⎭ hϕω (θ,ω) exp (−3π i/4)

× exp (ik1 R)R,

tedious calculations and merely write down the final results; penumbra regions are determined by the inequalities |θ − θ10 | θ1 , R R1 , √ θ1 = 2(1 − R/R1 )/k1 R.

(55)

The asymptotic representation given by (43), (54), and (52) is valid outside the penumbra region, where the inequality R R1 or the inequality |θ − θ10 | θ1 is fulfilled. In the case of ωp > ωp⊥ [see Fig. 6(a)], the transformation (1) (1) → SDP captures no poles, and asymptotic representation 0ψ in the far-field zone outside the penumbra regions (55) is expressed by Eq. (43) without the second term in the right-hand side. Thus, in this case, RCTR is absent and the field is determined by the spherical wave of TR.

B. Asymptotic representation of the field in the medium

where (ξ ) is Heaviside’s step function and Im kz(1) (s2 ) < 0. This term in Eq. (43) describes the VCR refracted through the interface from the anisotropic medium to vacuum. The radiation described by (52) can be named reversed Cherenkov-transition radiation (RCTR) (this term has been introduced earlier for the case of the boundary with a LHM [15,16]). There are two reasons for this. First, this radiation occurs due to the reversed directionality of the VCR. As was shown, in the frequency range under consideration, ωp < ω < ωp⊥ , we have ε > 0 and ε⊥ < 0, and the group velocity vector (along with the flux density vector) of the VCR is directed toward the interface. It is this fact that is responsible for the RCTR effect. Second, this radiation exists given that the interface is present. For the integral along the SDP, (44), in the far-field zone 1 1, one obtains the following representation: ⎧ b(1)S ⎫ √ ⎪ ⎬ ⎨ Eρω ⎪ 2h1 b(1)S Ezω ≈ ⎪ ⎪ k1 sin θ ⎩H b(1)S ⎭ ϕω


Here we briefly describe the asymptotic representation of the field components in the half space z > 0. We deal with the most interesting case of ωp < ωp⊥ , where ε⊥ 0 in the VCR band. Similar to the case of the z < 0 half space, we introduce a new variable ψ, such that kρ = k2 sin ψ. The root kz(2) given by (15) is expressed as kz(2) = k2 ε⊥ /ε cos ψ, where Im ε⊥ /ε > 0, thus fulfilling the requirement that Im kz(2) > 0 transforms to Re(k2 cos ψ) > 0. The electromagnetic field is expressed by formulas analogous to (40) and (41): ⎧ ⎫ ⎧ b(2) ⎫ b(2) ⎪ eρω (ψ,ω)H1(1) (Rk2 sin θ sin ψ) ⎪ E ⎪ ⎪ ⎪ ⎪

ρω ⎨ ⎨ ⎬ ⎬ (1) b(2) b(2) Ezω = ezω (ψ,ω)H0 (Rk2 sin θ sin ψ) (2) ⎪ ⎪ ⎪ 0ψ ⎪ ⎩ b(2) ⎪ ⎭ ⎩hb(2) (ψ,ω)H (1) (Rk sin θ sin ψ)⎪ ⎭ Hϕω 2 ϕω 1 × exp(iRk2 ε⊥ /ε cos (π −θ ) cos ψ)dψ, ⎫ ⎧ b(2) ⎪ ⎬ ⎨ eρω (ψ,ω) ⎪ ε q b(2) ezω (ψ,ω) = ⎪ ⎪ 2πβc ε ⊥ ⎭ ⎩ b(2) hϕω (ψ,ω)

⎧ ⎫ c(k2 sin ψ)2 k2 cos ψ ⎪ ⎪ ⎪ ⎪ ωε ⎨ ⎬ ⊥ ⎪ ⎪ ⎩

ic(k2 sin ψ)3 ωε (k2 sin ψ)2

(56)

⎪ ⎪ ⎭

× B (2) (k2 sin ψ,ω).

(57)

Using the asymptotic representation of the Hankel functions, the exponential term of the integrand in Eq. (56) can be expressed in the following form: (54)

∼ exp [2 ϕ2 (ψ)], 2 = |k2 |R, ϕ2 (ψ,θ ) = i exp(i arg k2 )M(θ ) cos(ψ − ψS2 ), M(θ ) = M 2 (θ ), Im M(θ ) > 0,

where h1 = −2/ϕ1 (θ ) (here double prime means the second derivative with respect to ψ), arg h1 = −π/4. Expression (54) represents a spherical wave of transition radiation (TR). Strictly speaking, if conditions (46) or (48) are fulfilled, the saddle point θ may occur near the pole ψ2(1) . In this case, one should use the uniform asymptotic representation for the integral in Eq. (44) in the far-field zone [25]. Rigorous consideration shows that the representation given by (43), (54), and (52) is valid outside a certain region, which can be called the penumbra region. The penumbra region is a transition region between the domain in which RCTR exists and domain in which RCTR is inessential. Here we omit the

(58) (59) (60)

M 2 (θ ) = sin2 θ + ε⊥ ε−1 cos2 θ ≈ cos2 θ [tan2 θ − tan2 θ2∗

cos ψS2 =

056608-9

+ i tan2 θ2∗ (ε /ε + ε⊥ /|ε⊥ |)], tan θ2∗ = |ε⊥ |ε ,

ε⊥ cos(π − θ ) , ε M(θ )

sinψS2 =

sin (π − θ ) . M(θ )

(61) (62) (63)



Because tan2 θ < tan2 θ2∗ in the angular region θ > π − θ2∗ , Re M 2 < 0 and Re M Im M. As one can show, the transformation of the initial integration path to the SDP is not accompanied by pole capturing. Moreover, as seen from (59), the contribution of the saddle point ψS2 is exponentially small, and the field is essentially absent. This fact can be explained with the use of isofrequency curves [Fig. 3(b)]. According to the Mandelshtam condition, the group velocity of the waves of the scattered field should be directed away from the interface, Vgz > 0. As can be seen from Fig. 3(b), only parts of hyperbolas lying in the second and third quarters satisfy this requirement. The inclination of the corresponding group velocity vector to the positive z direction in these regions varies from π/2 (at kz = 0) to θ2∗ (at kz → −∞) and cannot be less then θ2∗ . Thus, there are no waves transferring energy in the region θ > π − θ2∗ . In the angular interval π/2 < θ < π − θ2∗ , the situation is similar to that of the vacuum half space. The transformation of the initial path to the SDP captures the pole ψ2(2) (sin ψ2(2) = k2−1 s2 ), and the asymptotic representation in the far-field zone (2 = |k2 |R 1) takes the form ⎧ b(2) ⎫ ⎧ b(2)S ⎫ ⎧ b(2)P ⎫ ⎪ ⎨ Eρω ⎪ ⎨ Eρω ⎪ ⎬ ⎪ ⎬ ⎪ ⎬ ⎨ Eρω ⎪ b(2) b(2)S b(2)P Ezω E ≈ + Ezω , (64) zω ⎪ ⎩H b(2)S ⎪ ⎩H b(2)P ⎪ ⎭ ⎪ ⎭ ⎪ ⎭ ⎩H b(2) ⎪ ϕω ϕω ϕω

|, θ2 = θ20 + δθ2 , tan θ20 = tan θ2∗ 1 + β 2 |ε⊥ δθ2 ∼ 2/R2 , 3 8cβ |ε⊥ | 1 + tan2 θ2∗ (1 + β 2 |ε⊥ |) R2 = ω(1 + β 2 |ε⊥ |) −2 ε 1 + 2β 2 |ε⊥ | ε⊥ × + . ε |ε⊥ | 1 + β 2 |ε⊥ |

(67)

Penumbra regions (transition regions between the domain in which reflected RCTR contributes and the domain in which it is insignificant) are determined by the inequalities |θ − (π − θ20 )| θ2 , R R2 , θ2 ∼ 2(1 − R/R2 )/R.

(68)

Outside the penumbra regions described by (68), that is, at R R2 or |θ − (π − θ20 )| θ2 , one obtains the following expression for the TR spherical wave:

where the first term corresponds to a spherical wave of TR, while the second one corresponds to a cylindrical wave of reflected RCTR (constant coefficients are omitted): ⎧ b(2)P ⎫ ⎧ ⎫ (1) ⎪ ⎨ Eρω ⎪ ⎬ ⎨H1 (s2 R sin θ )⎬ ω (1) b(2)P Ezω ∼ H0 (s2 R sin θ ) exp i R cos θ ⎪ βc ⎩H b(2)P ⎪ ⎭ ⎩H (1) (s R sin θ )⎭ 2 1 ϕω × [(π − θ2 ) − θ ],

(66)

(65)

⎧ b(2) ⎫ ⎧ b(2)S ⎫ eρω (ψ2S ,ω) exp −3πi ⎪ ⎪ 4 ⎬ ⎨ ⎨ Eρω ⎬ 2 b(2) b(2)S (ψ2S ,ω) exp −πi ezω Ezω = 4 ⎩ b(2)S ⎭ k2 sin θ ⎪ ⎪ ⎭ ⎩ b(2) Hϕω hϕω (ψ2S ,ω) exp −3πi 4 π exp (ik2 RM) , (69) × exp − i 4 R M ≈ sin2 θ − cos2 θ tan2 θ2∗ ε⊥ 2 2 ∗ ε i cos θ tan θ2 ε + |ε⊥ | + . (70) 2 sin2 θ − cos2 θ tan2 θ2∗

FIG. 7. (Color online) Spatial structure of the Fourier harmonics of the full field in the case of a lossless medium. The Fourier harmonic frequency lies inside the interval defined by (46). Solid lines are parallel to the Poynting vector S of the VCR, dashed lines are parallel to the Poynting vector S r of the RCTR in the medium (reflected RCTR), and dotted lines are parallel to the Poynting vector S t of the RCTR in the vacuum (transmitted RCTR). Penumbra regions are hatched, and dash-dotted lines indicate the boundaries of the penumbra regions. 056608-10


FIG. 8. (Color online) An explanation of the RCTR decay in vacuum. C. Structure of the Fourier harmonics of the field and the impact of losses

We now analyze how losses in the medium affect the Fourier harmonics of the radiation field, that is, the Fourier harmonics of the TR spherical waves and the VCR and RCTR cylindrical waves. First, however, we illustrate the situation in which losses the in medium are negligible (ε,⊥ → 0). We suppose that the condition for RCTR presence in the vacuum expressed by (46) or (48) is fulfilled. The field in the vacuum is practically determined by the scattered field, and the asymptotic representation of the field outside the penumbra regions described by (55) is given by formulas (43), (52), and (54). The essential field in the medium exists only


in the region π/2 < θ < π − θ2∗ and is the superposition of the self-field described by (9) and the scattered field. The asymptotic representation of the scattered field is given by formulas (64), (65), and (69). The spatial structure of the Fourier harmonics of the full field in this case is presented in Fig. 7. Naturally, in the lossless case, the Fourier harmonics are free from exponential decay. In this case, we say that the corresponding waves have an infinite area of significance. In Fig. 7, areas where cylindrical waves of VCR and RCTR exist are indicated with lines parallel to the corresponding Poynting vectors. The existence of spherical waves of TR is indicated with the abbreviation “TR.” The self-field and corresponding VCR exist in the whole half space z > 0. The RCTR field exists in the medium at arbitrarily large R in the angular interval π/2 < θ < π − θ20 and interferes there with the VCR field (shadowed area in Fig. 7). RCTR waves exist in the vacuum at arbitrary R inside the angular interval θ10 < θ < π/2. Recall that TR spherical waves are absent in the medium inside the angular interval near the charge motion line (θ > π − θ2∗ ), because there are no waves transferring energy in this region [this can be seen from Fig. 3(b)]. As one can see from Fig. 7, areas of significance of TR, VCR, and RCTR, as well as penumbra regions, do not have any restrictions with respect to distance. When losses in the medium are taken into account, both the TR spherical wave and the RCTR cylindrical wave in the medium exhibit exponential decay. As can be seen from (69) and (70), in the angular interval π/2 < θ < π − θ2∗ , where the TR spherical wave is only essential, its decay associated with the small imaginary parts of M and k2 is described by the

FIG. 9. (Color online) Spatial structure of the full field Fourier harmonics in the case of a dissipative medium. Notation is the same as in Fig. 7. (Inset) Maximum distance of the RCTR significance in vacuum R1 (in units of c/ω) versus charge velocity β for ωp⊥ = 1.5ωp , ωd⊥ = ωd = 0.001ωp , ω = 1.1ωp , and βRCTR = 0.5. 056608-11


following expressions: ⎧ b(2)S ⎫ ⎪ ⎨ Eρω ⎪ ⎬ E b(2)S ∼ exp − 2R , zω ⎪ R2S (θ ) ⎩ b(2)S ⎪ ⎭ H


calculate the distance ρ0 in the orthogonal direction which a VCR wave will traverse in the medium before penetrating the vacuum and reaching the point (R, θ ) (see Fig. 8): (71)

ϕω

R2S (θ ) = ω

ε

4c tan (π − θ ) − ε cos(π − θ ) ε tan2 θ

tan θ2∗ ε + |ε⊥ | ⊥

tan2 θ2∗

.

(72)

We define the region of TR significance by the inequality R R2S (θ ).

(73)

At the “edge” of this region [R = R2S (θ )], the Fourier harmonics of TR are diminished by a factor of exp(2). As can be seen from (65), the exponential decay of the RCTR cylindrical waves in the medium associated with the small imaginary part of s2 is described by the multiplier ⎧ b(2)P ⎫ ⎪ ⎨ Eρω ⎪ ⎬ E b(2)P ∼ exp − 2ρ , (74) zω ⎪ ρ2 ⎩ b(2)P ⎪ ⎭ Hϕω −1 ε 1 β 2 |ε⊥ | 4c ε⊥ ρ2 = . + 2 |ε⊥ | 1 + β 2 |ε⊥ | ω ε 1 + β |ε⊥ | ε

ρ0 = R sin (θ − θ10 ) cos−1 θ10 . Substituting ρ0 into (74) results in Eq. (77). The closer the observation point to the interface (at R = const), the larger is the distance that the VCR wave propagates in the medium. RCTR waves experience the least attenuation at θ = θ1 , that is, at the edge of the area of RCTR existence. The inequalities of (79) can be presented in the following form: ρ > |z| tan θ1 ,

ρ ρ2 + |z| tan θ10 .

(80)

The system defined by (80) means that RCTR waves on the plane (ρ, z) are significant in the region between two rays. The “lower” ray originates from the point ρ = 0, z = 0 and is inclined at the angle θ1 to the negative direction of the z axis. The “upper” ray originates from the point z = 0, ρ = ρ2 and is inclined at the angle θ10 to the negative z direction. The structure of the Fourier harmonics of the full field when

(75) The region of RCTR significance in the medium is, by definition, determined by the inequalities ρ ρ2 , π/2 < θ < π − θ2 .

(76)

In a similar way, the region of VCR significance is determined by the first inequality in Eq. (76). The region of RCTR significance in the vacuum is determined by the exponential decay of the s2 pole contribution (52): ⎧ b(1)P ⎫ ⎪ ⎬ ⎨ Eρω ⎪ E b(1)P ∼ exp − 2R , (77) zω ⎪ R1P (θ ) ⎭ ⎩ b(1)P ⎪ Hϕω √ 2R1 R1P (θ ) = √ , θ > θ1 . (78) k1 sin (θ − θ10 ) Thus, the vacuum RCTR is significant in the region R R1P (θ ), θ > θ1 .

(79)

As one can see from (78), the distance of significance R1P is a monotonically decreasing function of the angle θ . It has a maximum at θ = θ1 , R1P (θ )|max = R1P (θ1 ) = R1 , and a minimum at θ = π/2, R1P (θ )|min = R1P (π/2) = ρ2 . The exponential decay of the RCTR waves in the vacuum can be explained through simple geometrical consideration. The VCR waves, which in turn induce the RCTR waves, traverse a certain path in a dissipative medium. One can

FIG. 10. (Color online) Spatial distribution of the Fourier harmonics of the magnetic field [Re Hϕω (Am−1 s) versus θ at R = 14 cm] for q = −1 nC and different values of β. The medium parameters are (in units of ωp = 2π × 1010 s−1 ): ωp⊥ = 1.5, ωd = ωd⊥ = 10−3 ; ωp1 = 10−2 , and ωd1 = 10−6 . The frequency of the harmonics lies in the interval defined by (46): ω = 1.13 and βRCTR = 0.6. The solid green line is calculated with the use of the numerical algorithm, the dashed black line is calculated with asymptotic formulas, and shadowed areas correspond to penumbra regions.

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losses are taken into account is shown in Fig. 9. It is notable that the region defined by (80) is strongly extended, because the angles θ1 and θ10 differ by the small value δθ1 given by (50). D. Numerical approach and numerical results

Along with the asymptotic representations presented in the previous sections, the numerical approach was also used for the investigation of the scattered field. This is based on the numerical calculation of both the integrals for the Fourier harmonics given by (12) and the total field integrals given by (11). Recall that the self-field was investigated in detail in Sec. III. Our numerical algorithm is based on the direct calculation of the integrals in Eqs. (11) and (12). Before calculating the Fourier harmonics integrals in Eq. (12), we analytically investigate the behavior of the integrands and then restrict the essential integration range and choose the appropriate integration step size. Moreover, as can be seen from Fig. 5(b) in the case of ωp < ωp⊥ , the poles ±s2 are situated near the integration path. This leads to rather abrupt behavior of the integrands in Eq. (12) in a certain range of the integration variable. To overcome this difficulty and provide a correct calculation, the analytical estimation of this range is performed and a finer step size is chosen inside it. When calculating the full scattered field integrals (11), the analogous estimations are performed numerically. Below, the most interesting numerical results are presented. Figure 10 shows the spatial distribution of the Fourier harmonics of the full field, that is, the dependence of Re Hϕω q b (Hϕω = Hϕω + Hϕω ) on θ at fixed frequency within the range


defined by (46) and R = const (along the dashed line in Fig. 1) at different charge velocities β. The solid line in Fig. 10 is obtained with the use of the described numerical algorithm, while the dashed line is calculated via the exact formulas (9) (for the self-field) and the asymptotic formulas (43), (52), and (54) for the half space z < 0 and (64), (65), and (69) for the half space z > 0 (for the scattered field). First, as can be seen from Fig. 10, both curves are in good agreement through the whole angle range except the penumbra regions given by (55) and (68) (these regions are shadowed in Fig. 10). Recall that the aforementioned asymptotic representations are valid outside the penumbra regions only. Moreover, the good agreement between the analytical and numerical results indicates that the terms neglected in the asymptotic representation [recall that only spherical (∼R −1 ) and cylindrical (∼ρ −1/2 ) waves are taken into account in the asymptotic representation] are not essential under the given conditions. Clearly defined oscillations of the field in Fig. 10 are associated with the contribution of poles and correspond to the RCTR waves. As formulas (52) and (48) predict, RCTR exists in the vacuum in the angular interval near the interface at β = βRCTR and above. All these features are clearly demonstrated in Fig. 10. Figure 11 shows the spectral distribution of the Fourier harmonics of the full field, that is, the dependence of Re Hϕω on ω at fixed R, different θ in the half space z < 0, and different charge velocities β (values of β are marked on the vertical axis). The RCTR frequency range is clearly manifested in Fig. 11; the field experiences several oscillations within it. Dash-dotted lines indicate the lower and upper boundaries of this range as functions of β, in accordance with (46). As one can see, the position of the RCTR range is in good agreement

FIG. 11. (Color online) Spectral distribution of the Fourier harmonics of the magnetic field [Re Hϕω (Am−1 s) versus ω/ωp ] for q = −1 nC at R = 14 cm; θ = 45◦ , θ = 60◦ , and θ = 75◦ ; and different values of β indicated on the vertical axis. The medium parameters are the same as in Fig. 10. 056608-13



with theoretical predictions. Moreover, as can be seen from Fig. 11, the magnitude and number of the oscillations increase as the angle θ increases. The width of the RCTR spectrum is maximal at β → 1. Because the boundaries of the RCTR effect are clearly defined in the spectrum, they could, in principal, be measured with good accuracy. Thus, the RCTR spectrum can be used for the determination of the small bunch velocity. It is interesting to compare the results presented in Fig. 11 with the analogous results for the case of an interface between vacuum and an isotropic LHM [16]. The main difference is that, in the present case, only the upper frequency threshold monotonically increases with β, while in the LHM case, both thresholds increase. Moreover, the lower velocity threshold βRCTR is associated with the total internal reflection, while in the LHM case, the lower threshold coincides with the VCR threshold and the upper one is associated with the total internal reflection. Recall that the maximum distance at which the vacuum RCTR is still essential is determined by the parameter R1 , given by (51). The inset of Fig. 9 shows the dependence of R1 on β within the RCTR range defined by (48) for some realistic medium parameters, including dissipation. As one can see, R1 is an almost linearly increasing function of β with a magnitude on the order of 1000 wavelengths in free space. This estimation shows that that RCTR can be observed in real experiments. With the use of the aforementioned numerical algorithm, the time evolution of the field was investigated as well. Figure 12 shows the dependence of the magnetic field Hϕ on time t at a number of fixed points in the vacuum. The solid line in Fig. 12 is the result of the exact numerical calculation of the double integral in Eq. (11), while the dashed line is the result of the integration of the pole contribution (52) over frequency ω as if it were alone. As one can see, both lines are more or less in agreement starting from 0.3 ns. It is notable that the larger the distance |z|, the better the agreement between curves. This

FIG. 12. (Color online) Time evolution of the total magnetic field (solid green line) and the RCTR magnetic field (dashed red line) at β = 0.99 in several spatial points in the vacuum. The medium parameters are the same as in Fig. 10.

means that the pole contribution (52), which corresponds to the vacuum RCTR, gives the most essential contribution to the total field starting from 0.3 ns and above. In other words, the RCTR in the vacuum region can be the dominant radiation in certain spatial regions. V. CONCLUSION

We have analyzed in detail the electromagnetic field generated by a point charge traversing the interface between vacuum and a nonmagnetic anisotropic uniaxial medium, with dielectric permittivity tensor components possessing a plasmalike frequency dispersion. First, in the case of negligible losses, we have analytically decomposed the self-field of the charge into a sum of a quasistatic part and a wave part corresponding to VCR. We have analyzed the spatial distribution of both the total field and its quasistatic and wave parts for different parameters of the problem. Moreover, we have elaborated an effective numerical algorithm for the calculation of the total self-field which can be used for a medium with arbitrary losses. Further, we have analyzed the field scattered by the boundary in two ways. In the first type of analysis, we have analytically investigated the scattered field in the far-field zone with the use of the steepest-descent technique. We have constructed asymptotic expressions taking into account only the main terms of the spatial radiation: spherical and cylindrical waves. The possibility of RCTR has been clearly demonstrated, and its attractive properties have been mentioned. It has been shown that RCTR has both lower and upper thresholds in the frequency domain and only a lower threshold in the charge velocity domain. The spatial structure of the Fourier harmonics of the total field has been analyzed in detail for both a lossless medium and a dissipative one. It is notable that dissipation in the medium leads to attenuation of the RCTR wave, not only in the medium, but in the vacuum as well. The vacuum RCTR wave experiences the least attenuation in the penumbra regions, with the maximum distance of RCTR significance being on the order of 1000 wavelengths in free space. This result shows that RCTR can be observed in a real experiment, at least at gigahertz-order frequencies. In the second type of analysis, we have elaborated an algorithm for computation of both the Fourier harmonics of the scattered field and the total scattered field. With this approach, both the spatial and spectral distributions of the Fourier harmonics have been investigated. The presented numerical results are in a good agreement with the asymptotic formulas in the domain of the validity of the latter. Moreover, it has been shown that the RCTR spectrum is clearly defined on the background of the total spectrum. The threshold frequencies can, in principle, be measured with a good accuracy, and therefore they can be used for determination of the charge velocity or medium parameters. Thus, the RCTR spectrum can be useful for both diagnostics of charged particle beams in accelerators and characterization of complex media, including modern metamaterials. Finally, the time evolution of the total field in the vacuum has been numerically investigated. As has been shown, RCTR is the dominant radiation at certain parameters.

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ELECTROMAGNETIC FIELD OF A CHARGE TRAVELING . . . ACKNOWLEDGMENTS

This research was supported by Saint Petersburg State University and the Dmitry Zimin “Dynasty” Foundation.


The upper index in Eqs. (A1)–(A8) represents the order of smallness with respect to the parameters ε (1 ε ε ε (1 |ε⊥ | ε

APPENDIX: CONDITION FOR s2 POLE CAPTURING

Given that (46) or (48) is fulfilled, the following decompositions take place: s2 ≈ s2(0) + s2(1) + s2(2) , (0) (1) (2) kz(1) (s2 ) ≈ kz(1) (s2 ) + kz(1) (s2 ) + kz(1) (s2 ) ,

(A1) (A2)

where |β 2 ω 1 + |ε⊥ ε , s2(0) = c |ε⊥ |β 2 |β 2 ε 1 iω 1 + |ε⊥ ε⊥ , = ε + 2c ε |ε⊥ | 1 + |ε⊥ |β 2 |ε⊥ |β 2

s2(1)

s2(2)

kz(1) (s2 ) =

kz(1) (s2 )

[1] [2] [3] [4] [5] [6]

kz(1) (s2 )

(0)

=

c

ε1 1 − ε

|β 2 + |ε⊥ , |ε⊥ |β 2

ε⊥ 1 (1 |ε⊥ | 1+|ε⊥ |β 2

|β 2 ) |ε⊥

ε

−iω

ε

ε ε

2c

(2)

+

|ε⊥ |β 2

ε1 ε

−

+

1+|ε⊥ |β 2 |ε⊥ |β 2

+ |ε⊥ |β 2 ) +

ε1 β 2 −

ε (1 |ε⊥ |

(A3) (A4)

ε⊥ |ε⊥ |

ε⊥ |ε⊥ |

+ |ε⊥ |β 2 )

1, (A9) 1,

for s2 and kz(1) (s2 ), respectively. Let us find the angle θ1 , for which the pole ψ2(1) lies exactly (1) on the SDP SDP . In this case, ψ = ψ2(1) and θ = θ1 satisfy the system defined by (42). Using (A1)–(A8), one can represent this system in the following form: U1(1) (θ1 ) > 0,

(A10)

where U1(0) (θ ) = k1 [1 − cos (θ − θ10 )] , (A11) 1 k1 tan θ10 sin (θ − θ10 ) ε ε⊥ (1) , U1 (θ ) = + 2 ε |ε⊥ | 1 + |ε⊥ |β 2 U1(2) (θ )

=−

(2) kz(1) (s2 )

(A12) cos θ −

s2(2)

sin θ,

(A13)

where θ10 is defined by (50). As one can see, the following properties arise:

(A6)

(1)

1 + |ε⊥ |β 2

U1(0) (θ1 ) + U1(2) (θ1 ) = 0,

2 ε |β 2 1 ω 1 + |ε⊥ ε⊥ = ε − 8c ε |ε⊥ | 1 + |ε⊥ |β 2 |ε⊥ |β 2 2 4 ε⊥ , (A5) − |ε⊥ | 1 + |ε⊥ |β 2 ω

+ |ε⊥ |β 2 ) +

U1(0) (θ10 ) = U1(1) (θ10 ) = 0,

(A14)

∂ 2 U1(0) ∂U1(0) (θ10 ) = 0, (θ10 ) = k1 , ∂θ ∂θ 2 "2 ! ∂U1(1) (θ10 ) = 2k1 U1(2) (θ10 ). ∂θ

(A15)

Using the method of successive approximations, one can obtain the following expression for the desired θ1 : ,

(A7)

θ1 = θ10 + δθ1 , where δθ1 is given by

#

2s2(0) s2(2) 1 =− (1) 2 kz (s2 ) (0) & (0) %2 (0) 2 $ (1) + s2 (1) 2 kz (s2 ) s2 + . $ (1) (0) %3 kz (s2 )

δθ1 =

2 (2) U1 (θ10 ) k1

(A16)

(A17)

(A8)

and can be expressed in terms of R1 , defined by (51), via formula (50). As can be seen from Fig. 6, the pole ψ2(1) is captured when it is situated on the left side of the SDP. Thus, the condition for its capture takes the form θ > θ1 .

V. L. Ginzburg, J. Phys. USSR 3, 101 (1940). A. A. Kolomenskii, C. R. Acad. Sci. USSR 86, 1097 (1952). V. E. Pafomov, Sov. Phys. JETP 5, 597 (1956). V. E. Pafomov, Sov. Phys. JETP 5, 307 (1957). B. M. Bolotovskii, Usp. Fiz. Nauk 62, 201 (1957). V. P. Zrelov, Vavilov-Cherenkov Radiation in High-Energy Physics (Israel Program for Scientific Translations, Jerusalem, 1970).

[7] V. G. Veselago, Phys. Usp. 10, 509 (1968). [8] J. Lu, T. M. Grzegorczyk, Y. Zhang, J. P. Jr, B. I. Wu, and J. A. Kong, Opt. Express 11, 723 (2003). [9] Z. Y. Duan, B.-I. Wu, S. Xi, H. S. Chen, and M. Chen, Prog. Electromagn. Res. 90, 75 (2009). [10] M. I. Bakunov, R. V. Mikhaylovskiy, S. B. Bodrov, and B. S. Luk’yanchuk, Opt. Express 18, 1684 (2010). [11] H. Chen and M. Chen, Mater. Today 14, 34 (2011).

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[12] B.-I. Wu, J. Lu, and J. A. Kong, J. Appl. Phys. 102, 114907 (2007). [13] S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B.-I. Wu, J. A. Kong, and M. Chen, Phys. Rev. Lett. 103, 194801 (2009). [14] Y. O. Averkov, Telecommun. Radio Eng. 63, 419 (2005). [15] S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, Phys. Rev. Lett. 103, 194802 (2009). [16] S. N. Galyamin and A. V. Tyukhtin, Phys. Rev. B 81, 235134 (2010). [17] Y. O. Averkov, A. V. Kats, and V. M. Yakovenko, Phys. Rev. B 79, 193402 (2009). [18] C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, J. Phys. Condens. Matter 20, 304217 (2008).

[19] V. L. Ginzburg and V. N. Tsytovich, Transition Radiation and Transition Scattering (Hilger, London, 1990). [20] S. N. Galyamin and A. V. Tyukhtin, Bulletin of Saint Petersburg State University, Ser. 4 1, 21 (2006) [in Russian]. [21] A. V. Tyukhtin and S. N. Galyamin, Tech. Phys. Lett. 33, 632 (2007). [22] A. V. Tyukhtin and S. N. Galyamin, Phys. Rev. E 77, 066606 (2008). [23] L. I. Mandelshtam, Zh. Eksp. Teor. Fiz. 15, 475 (1945). [24] B. M. Bolotovskii and S. N. Stolyarov, in Problems of Theoretical Physics. Collection of I. E. Tamm’s Memory (Nauka, Moscow, 1972), p. 267. [25] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley Interscience, New Jersey, 2003). [26] L. M. Brekhovskih, Waves in Layered Media (Academic Press, New York, 1980).

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