PHYSICAL REVIEW E 89, 013202 (2014)
Radiation of charges moving along the boundary of a wire metamaterial Andrey V. Tyukhtin* and Viktor V. Vorobev† Physical Faculty of St. Petersburg State University, St. Petersburg 198504, Russia (Received 19 November 2013; published 9 January 2014) The electromagnetic fields of charges moving along the boundary of a “wire metamaterial” perpendicularly to the wires are investigated. The metamaterial under consideration represents a volume-periodic structure of thin parallel wires located in a square lattice. This structure is described by an effective permittivity tensor and exhibits both spatial and frequency dispersion. It is shown that the charge generates nondivergent radiation, as in the case of an infinite metamaterial. However, unlike the infinite-metamaterial case, the radiation concentrates near a certain plane (not line) behind the charge, and it is asymmetric with respect to this plane. An algorithm for calculating the wave fields of finite-length bunches is developed, and some typical numerical results are given. They demonstrate that the structure under consideration can be applied to determine the size and form of such bunches. The stopping and deflection forces acting on the charge are also calculated. DOI: 10.1103/PhysRevE.89.013202
PACS number(s): 41.60.Bq, 81.05.Xj, 78.67.Uh
following expression [12–14]:
I. INTRODUCTION
Cherenkov radiation is widely used for the detection of charged particles [1], and it can be applied for diagnostics of particle bunches, as well [2,3]. The prospects for these techniques strongly depend on the material used to generate the radiation. Cherenkov radiation can possess unusual properties when a medium that exhibits dispersion and/or anisotropy is used. Among such media, particular attention has been paid to so-called “metamaterials” which represent composite periodic structures that consist of macroscopic objects placed with relatively small spacing. Under the condition that the wavelengths of the electromagnetic field are greater than this spacing, such a structure can be considered as a solid medium and can be described by effective parameters, such as permittivity and permeability. For example, a so-called “left-handed” medium (LHM), which has both negative permeability and negative permittivity in a certain range of frequencies, can be implemented as a metamaterial. The processes of charged-particle radiation in the presence of a LHM or other metamaterial have been widely investigated previously [4–11]. This paper is focused on the case of a structure that consists of thin parallel wires in vacuum placed in a square lattice (Fig. 1). If the period is small in comparison with the wavelengths, the structure can be described by an effective permittivity tensor of the following form [12–14] (in Cartesian coordinates): ⎛ ⎞ ε (ω ,kx ) 0 0 0 1 0⎠ , εˆ = ⎝ (1) 0 0 1 where the x axis lies along the wires. Generally speaking, wires can possess nonconductive coatings [13,14]; here, however, we consider wires without any coverings. In this case, the longitudinal component of the tensor (1) is given by the
ε (ω ,kx ) = 1 −
ω2p ω2 −c2 kx2 + 2i ω ωd
,
(2)
where kx is the x component of the wave vector. The effective “plasma frequency” squared is related to the geometrical parameters of the structure as follows: ω2p =
d2
2π c2 , [ln (d/r0 ) − C]
(3)
where d is the period of the structure, r0 is the radius of the conductors, c is the speed of light in vacuum, and C is some constant of order 1. Note that structures with different periods along the y and z axes are also described by the tensor (1) with the component (2); a difference exists only in more complex formulas for the plasma frequency [14]. The structure under consideration is frequently called a “wire metamaterial.” As one can see, the effective medium that models this structure possesses both frequency and spatial dispersion of a specific type (there is dependence only on a single component of the wave vector). These features result in unique properties of the radiation of a charged-particle bunch [15–17]. For example, in the case in which the charge moves perpendicularly to the wires inside an infinite metamaterial and in which losses are negligible, radiation is present for any velocity of the charge. In addition, the radiation is nondivergent and concentrates along certain lines behind the charge [15,16]. It has been shown that this phenomenon allows the determination of the sizes of charged-particle bunches. However, wires can have an essential influence on particle bunches. Therefore, it is important to consider the case in which the bunch moves outside the metamaterial. In this paper, we analyze the case in which the charge moves in vacuum along a flat boundary of the metamaterial perpendicularly to the wires. II. TOTAL FIELD OF A POINT CHARGE
*
[email protected] † [email protected] 1539-3755/2014/89(1)/013202(8)
We consider a point charge q moving in vacuum with a constant velocity V = cβez along the flat boundary of a wire metamaterial that fills the area x > 0. The charge moves at a 013202-1
©2014 American Physical Society
ANDREY V. TYUKHTIN AND VIKTOR V. VOROBEV
PHYSICAL REVIEW E 89, 013202 (2014)
In the following discussion, we use a six-component vector Fα , which consists of components of the electric and magnetic fields:
x
Fα = {Exα ; Eyα ; Ezα ; Hxα ; Hyα ; Hzα },
where the index α corresponds to various types of waves. To obtain the solution, we use the decomposition of the total field over plane waves. The incident field (that is, the field of a point charge moving in unbounded vacuum) has the following form: ∞ ∞ ∞ q ω dω dky δ kz − Fi = 2π −∞ cβ −∞ −∞
d FIG. 1. Wire metamaterial.
distance a0 from the material’s border and perpendicularly to the wires (Fig. 2). The corresponding current density has the form j = qcβδ (x + a0 ) δ (y) δ (z − cβt) ez ,
(4)
×
z is a unit vector along the z axis in wave-vector space. where u The standard boundary conditions, which demand the continuity of the tangential components of the electric and magnetic fields, have the following form: Ey,z |x=−0 = Ey,z |x=+0 , (6)
Hy,z |x=−0 = Hy,z |x=+0 .
However, there exist three types of waves in the metamaterial [14] and two waves of different polarizations in the vacuum. Therefore, four boundary conditions (6) are insufficient. Such a situation is typical for problems in which the medium exhibits spatial dispersion [18]. To obtain the unique solution, we must use an additional boundary condition, which is the continuity of the normal component of the electric field [19]: Ex |x=−0 = Ex |x=+0 .
2 fi = sgn(x + a0 )kxo kz ; ky kz ; −kxo − ky2 ;
ω ω − ky ; sgn(x + a0 ) kxo ; 0 , c c ω2 kxo = − ky2 − kz2 . c2 The root (11) is fixed such that √ sgn( · · ·) = sgn(ω) if √ if Im( · · ·) > 0
Fv =
∞
dω −∞
∞
−∞
dky
∞ −∞
(11)
(12)
dkz (ftm Vtm +fte Vte )
× e−ikxo x+iky y+ikz z−i ω t ,
2 ω ω ω 2 − kxo ; ky kxo ; kz kxo ; 0; kz ; − ky , ftm = c2 c c
2 ω ω ω 2 fte = 0; − kz ; ky ; 2 − kxo ; ky kxo ; kz kxo , c c c
On the “microscopic” level, this requirement means that the currents at the wire’s extremes are negligible.
-a0
√ Im · · · = 0, √ Re · · · = 0.
(10)
These rules ensure that excited waves propagate from the charge trajectory, and “evanescent waves” exponentially decrease. The field that is reflected into vacuum consists of two waves of different polarizations:
(7)
x
fi exp{i |x + a0 | kxo + iky y + ikz z − i ω t}dkz , ω kxo (9)
where
where ez is a unit vector along the z axis, and c is the speed of light in vacuum. To proceed, we require the Fourier transform of the current: q ω z, exp (ikx a0 ) u δ kz − (5) jω k = 8π 3 cβ
0
(8)
(13)
(14)
where kxo is represented by the expression given in (11) and fixed by the rules given in (12). The field that is transmitted into the metamaterial consists of three types of waves: an ordinary wave (o), an “extraordinary isotropic” wave (ei), and an “extraordinary anisotropic” wave (ea) [14]. The components of these waves can be written in the following forms:
z V=cβ
Fo =
FIG. 2. (Color online) Movement of a charge along a boundary. 013202-2
∞
dω −∞
∞ −∞
dky
∞ −∞
dkz fo Mo eikxo x+iky y+ikz z−i ω t , (15)
RADIATION OF CHARGES MOVING ALONG THE . . .
PHYSICAL REVIEW E 89, 013202 (2014)
ω ω ω2 2 ; −ky kxo ; kz kxo , fo = 0; − kz ; ky ; 2 − kxo c c c ∞ ∞ ∞ dω dky dkz fei Mei eikxei x+iky y+ikz z−i ω t , Fei = −∞
−∞
−∞
(16)
Fo =
+∞
+∞
dω −∞
−∞
(24) Fei =
+∞ −∞
−∞
fea = {0; ky ; kz ; 0; −kz ky }. The expressions for the components of the wave vector along the x axis can be written as follows: ω2 − ky2 − kz2 , kxo = c2 ω2 − ω2p (18) kxei = − ky2 − kz2 , 2 c ω kxea = , c where the fixing of the roots is provided by the rules given in (12). Thus, the total field is x < 0, Fi + Fv , (19) F = Fo + Fei + Fea , x > 0. The five unknown amplitudes of the waves Vtm , Vte , Mo , Mei , and Mea are determined from the boundary conditions (6) and (7). As a result, one obtains
Vtm
ω2 (kxo − kxei ) ωc + kxo − c2p i Ex0 = ,
2 ω
ω + k − k + k ) (k xo xei xo xei c c
Vte = 0, Mo =
(20)
−βky i , ω2
Ex0 2 kxo c2 − kxo
Mei = ω c
Mea = ω c
− kxei +
ω
2kxo
Ei , + kxo (kxo + kxei ) x0
c ω 2kxo c (kxei − kxo ) i
2 ω
ω
Ex0 , kxo − k − k xo xei c c
+∞
+∞
−∞
−∞
+∞
+∞
dω −∞
−∞
(fea Vea )kz =ω/cβ (26)
where ζ = z − cβt and the x projections of the wave vector are written as kxo =
−
ω2 1 − β 2 − ky2 , c2 β 2
(27) ωp2 ω2 1 − β 2 kxei = − 2 − 2 − ky2 . c β2 c One can see that the radicals in (27) are imaginary. Thus, two waves in the medium (ordinary and extraordinary isotropic) and the wave that is reflected into the vacuum are exponentially decreasing. We are primarily interested in the volume radiation in the medium, which consists of the extraordinary anisotropic wave only. One can see from (26) that the field of this wave is invariant along any line ζ = −βx + const, y = const. III. RADIATION OF A POINT CHARGE
We consider in detail the transmitted extraordinary wave, which has Ey , Ez , Hy , and Hz components (the last two can be expressed as Hy = −Ez and Hz = Ey ). The typical distribution of the electrical field in the Y Z plane is given in Fig. 3. This figure illustrates that the distribution of the field becomes narrower when the velocity of the charge increases. This fact can be explained by the relativistic transverse contraction of the “incident” field (i.e., the field of the charge in unbounded vacuum). Further simplification of expression (26) can be achieved for the case of an ultrarelativistic charge. When β → 1, integration over ω can be performed for both components: +∞ Ey = q ky sin(ky y)e−a0 ky dky 0
ˆ ˆ × (ξ ) e−ky ξ −
(22) + (−ξˆ )
dω
× ei(ω/c)x+iky y+i(ω/cβ)ζ dky ,
(21)
ω i where Ex0 = 2πq ω kz δ(kz − cβ ) exp(ia0 ω2 /c2 −ky2 −kz2 ). Integrating over kz , one obtains the reflected and transmitted waves in the following forms:
Fv =
Fea =
−∞
(17)
(fei Vei )kz =ω/cβ eixkxei +iky y+i(ω/cβ)ζ dky , (25)
ω2 ω ω 2 − kxei ; −ky kxei ; −kz kxei ; 0; kz ; − ky , fei = c2 c c ∞ ∞ ∞ Fea = dω dky dkz fea Mea ei(ω/c)x+iky y+ikz z−i ω t , −∞
+∞
dω
−∞
(fo Vo )kz =(ω/cβ) eixkxo +iky y+i(ω/cβ)ζ dky ,
Ez = q
+∞
013202-3
2ky2 −2ky kyp +(2ky ξˆ +1) ω2p /c2 (ky + kyp )2
e
ky ξˆ
, (28)
ky cos(ky y)e−a0 ky dky
0
ˆ × (ξˆ ) e−ky ξ −
(ftm Vtm )kz =ω/cβ e−ixkxo +iky y+i(ω/cβ)ζ dky , (23)
4ky kyp ˆ e−ξ kyp (ky + kyp )2
+ (−ξˆ )
2 4kyp
(ky + kyp )
e−ξ kyp 2
ˆ
−2ky2 +2ky kyp −(2ky ξˆ +3) ω2p /c2 (ky + kyp )2
e
ky ξˆ
, (29)
ANDREY V. TYUKHTIN AND VIKTOR V. VOROBEV
PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 3. (Color online) Distribution of the radiation field of a point charge in the plane parallel to the boundary. The distance to the boundary is a0 = 0.5c/ ωp ; the electric field is given in units of q ω2p /c2 ; distances are in units of c/ ωp ; the charge’s velocity is β = 0.5 (upper row) and β = 0.95 (lower row).
where kyp = ω2p /c2 + ky2 , ξˆ = ζ + x, and (ξˆ ) is the Heaviside function. Numerical results for the distribution of the field in the plane x = const for different distances from the ultrarelativistic charge to the boundary are given in Fig. 4. The distributions shown in Figs. 3 and 4 represent the radiation field only (which consists solely of the extraordinary anisotropic wave). This part of the field is the dominant component of the total field when x c/ ωp (because of the decrease in the other parts of the field). The radiation field does not change when the observation point is shifted along the line ζ = −βx at any fixed y. Thus, the radiation is nondivergent, as in the case of an infinite metamaterial [15]. The difference is that this radiation concentrates at the semiplane ζ = −βx, y > 0 (not a line) behind the charge. In addition, one can note that the radiation is asymmetric with respect to this semiplane. The component Ey is equal to zero if y = 0, while Ez reaches its maximum. The maximum magnitude of the electric field observed at y = 0, ζ = −βx decreases with the distance a0 , and the “light spot” in the plane x = const essentially widens. This widening along the y direction (orthogonal to the charge velocity) is greater than that along the z direction (this is explained by the relativistic transverse contraction of the incident field). The integral (29) can be analytically calculated in terms of special functions for the case a0 = 0, y = 0 [20]: + − + (−ξˆ )Ezea , Ezea = (ξˆ )Ezea + Ezea
1 1 7 34 114 240 240 −p =q 2 −4 + 2 + 3 + 4 + 5 + 6 e c p2 p p p p p p
6p(p2 +10)K0 (p) + (p4 +27p2 +120)K1 (p) , +8 p5 (30) ω2p
− =q Ezea
ω2p 1 (16p5 +p4 −160p3 +24p2 −960) c2 p 5
+ 2q
ω2p π {p(p4 − 18p2 + 120)[Y0 (−p) + H0 (p)] c2 p 5
+(5p4 − 66p2 + 240)[Y1 (−p) − H1 (p)]},
(31)
where p = ξˆ ωp /c = (z − ct + x) ωp /c, Kν is a modified Hankel function, Yν is a Neumann function, and Hν is a Struve function. Note that in the plane y = 0, the integral given in (28) for Ey is equal to zero. Expressions (30) and (31) can be written as series expansions under the condition p 1, using known expansions for special functions. It turns out that the main terms for both summands in (30) and (31) are the same:
± Ezea
= Ezea
q ωp2 ≈ 2 c2
ωp |ξˆ | 13 + γ˜ + ln , 2c 12
(32)
where γ˜ 0.577 is the Euler constant. One can see that the longitudinal electric-field component has a logarithmic singularity at y = 0, ζ = − βx. The obtained result (32) is similar to the case of a charge moving inside an infinite metamaterial [15]. However, it should be emphasized that the singularity appears only for a charge moving directly at the boundary, that is, for the case a0 = 0. If a0 = 0, the field is continuous everywhere in the region x > 0. It is also interesting to examine the special case in which the relation r0 /d is fixed, but d/a0 → 0, that is, ωp a0 /c → ∞. Such a medium is ideally conducting in only one direction (along the x axis). We can obtain the expressions for the field
013202-4
RADIATION OF CHARGES MOVING ALONG THE . . .
PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 4. (Color online) Radiation field of an ultrarelativistic point charge. The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The distance to the boundary is a0 = 0 (top row), a0 = 0.1c/ ωp (middle row), and a0 = c/ ωp (bottom row).
of radiation as a limiting case of (28) and (29): +∞ ideal Eyea =q ky sin(ky y)e−a0 ky
Thus, we find a simple analytical result for the case of a medium that is ideally conductive in one direction. This approximation can be used under the condition ωp a0 /c 1.
0
× { (ξˆ )e−ky ξ + (−ξˆ )(2ky ξˆ + 1)eky ξ }dky , +∞ =q ky cos(ky y)e−a0 ky ˆ
ideal Ezea
ˆ
0
× { (ξˆ )e−ky ξ − (−ξˆ )(2ky ξˆ + 3)eky ξ }dky , ˆ
ˆ
IV. STOPPING AND DEFLECTING FORCES
(33)
(34)
We considered the field of radiation that is produced by the extraordinary anisotropic wave in a metamaterial. The two other waves inside the metamaterial and the one in the vacuum are surface waves. However, the field in the vacuum is interesting because it affects the charge. A particle’s energy losses for a unit path can be expressed via the stopping force:
The obtained integrals can be analytically evaluated [20]: ideal Eyea =q
dE = Fz = qEvz |x → −a0 , dz0 ζ → 0,y → 0
2y(a0 + |ξˆ |) 2
2 [y 2 + (a0 + |ξˆ |) ]
+ 4q (−ξˆ )
2 y ξˆ [3(a0 + |ξˆ |) − y 2 ] 2 3
(35)
,
[y 2 + (a0 + |ξˆ |) ] 2
ideal Ezea =q
(a0 + |ξˆ |) − y 2 2
2 [y 2 + (a0 + |ξˆ |) ] 2 (a0 + |ξˆ |) − y 2 − 4q (−ξˆ ) 2 2 [y 2 + (a0 + |ξˆ |) ]
+
2 ξˆ (a0 + |ξˆ |)[(a0 + |ξˆ |) − 3y 2 ]
[y 2
2 + (a0 + |ξˆ |) ]
3
(37)
where Ezv is the component of the reflected field (the incident field Ezi does not affect the charge because it is odd over ζ ). The integral (23) for the z component can be taken analytically in the plane y = 0 for the ultrarelativistic case β → 1 [10]: ω2 −2p3 + p2 + 12 2 p Fz = −2q 2 c p4
π + 3 {3p[Y0 (p) − H0 (p)]+(p2 − 6)[Y1 (p) − H1 (p)]} , p
(38) .
(36)
where p = 2a0 ωp /c, Yν is a Neumann function, and Hν is a Struve function. The result of a numerical calculation for Ezv
013202-5
ANDREY V. TYUKHTIN AND VIKTOR V. VOROBEV
PHYSICAL REVIEW E 89, 013202 (2014)
Using the expressions for the components Exv and Eyv , one can determine the deflecting forces in the x and y directions. It turns out that Eyv y=0 = 0, so the charge is not deviated in the y direction (this is natural because of the symmetry of the problem relative to the y axis). At the same time, one can find from (23) that Fx = qExv |y → 0, ζ → 0 = −2Fz >0. Thus, the x → −a0
particle is deflected in the direction of the boundary, and the deflecting force is greater than the stopping force by a factor of 2. FIG. 5. (Color online) Stopping force Fz (in units of q 2 ω2p /c2 ) as a function of the distance from the charge trajectory to the boundary (in units of c/ ωp ).
is presented in Fig. 5. As one can see, the field is limited, while the distance to the boundary is nonzero. When a0 ωp /c 1, the stopping force can be approximated as follows:
a0 ωp 1 q 2 ω2p + γ˜ + Fz ≈ ln . (39) 2 c2 c 4 Therefore, the energy loss has a logarithmic singularity at a0 → 0. Note that in the case a0 0, the losses exhibit asymptotic behavior as 1/a02 . Thus, the stopping force (that is, energy losses) is limited in the case a0 = 0, unlike the result for a charge moving inside an unbounded metamaterial.
V. RADIATION OF A BUNCH OF CHARGED PARTICLES
The results obtained for the point charge, (28) and (29), can be applied for the calculation of the field of a bunch of charged particles as a Green’s function. We consider a thin bunch that has a limited size in the direction of its motion (along the z axis) and is infinitely small in transverse dimensions. The charge distribution is given by the expression ρ (x,y,ζ ) =
1 δ (x) δ (y) (σ − |ζ |) . 2σ
In this case, the field of radiation is expressed as σ
1 Eea x,y,ζ − ζ dζ . Eline (x,y,ζ ) = 2σ −σ
(40)
(41)
Numerical results for ultrarelativistic bunches (β → 1) with relatively large lengths (σ c/ ωp ) are presented in Figs. 6 and 7. They represent the distributions of the radiation
FIG. 6. (Color online) Radiation of an ultrarelativistic charged bunch moving at a distance of a0 = 20c/ ωp from the boundary. The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The width of the bunch is σ = 100c/ ωp (top row), σ = 50c/ ωp (middle row), and σ = 25c/ ωp (bottom row). 013202-6
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PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 7. (Color online) Radiation of an ultrarelativistic charged bunch with σ = 100c/ ωp . The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The distance from the bunch to the boundary is a0 = 30c/ ωp (upper row) and a0 = 100c/ ωp (lower row).
field (that is, the anisotropic extraordinary wave) at some plane x = const > 0. These distributions do not depend on x, as for the point charge. One can see that the distances between the extremes of the field correspond to the length of the bunch. Therefore, the analysis of the field generated by a bunch of charged particles moving near the boundary of a wire metamaterial can help to estimate the size of this bunch. However, the precision of this method depends on the distance from the bunch to the boundary because the “snapshot” of the bunch becomes more fuzzy (mostly in the y direction) as this distance increases (Fig. 7). Note that the length of a short bunch (σ c/ ωp ) cannot be easily evaluated using this method because the snapshots of the wave field are similar to the field of a point charge. VI. CONCLUSIONS
A point charge moving along the boundary of a wire metamaterial perpendicularly to the wires was considered. The radiation field was presented as twofold integrals, which can be reduced to single-fold ones if the charge moves at the speed of light in vacuum. It was shown that the radiation is nondivergent, as in the case of unbounded wire material, but the radiation properties are
[1] V. P. Zrelov, Vavilov-Cherenkov Radiation in High-Energy Physics (Israel Program for Scientific Translations, Jerusalem, 1970). [2] A. P. Potylitsyn, Yu. A. Popov, L. G. Sukhikh, G. A. Naumenko, and M. V. Shevelev, J. Phys.: Conf. Ser. 236, 012025 (2012). [3] A. V. Tyukhtin, Phys. Rev. Spec. Top.–Accel. Beams 15, 102801 (2012).
essentially different. The radiation concentrates near a certain semiplane behind the charge and is characterized by some asymmetry with respect to this plane. The z component of the electric field has a logarithmic singularity if the charge moves directly on the boundary. However, if the distance from the charge trajectory to the boundary is not equal to zero, the radiation field is finite and continuous everywhere in the metamaterial. The typical picture of the field of a bunch demonstrates that the radiation can be used to determine bunch sizes and forms. The influence of the reflected field on the charge was also analyzed. The value of the stopping force was obtained analytically and computed. The stopping force is limited if the distance from the charge to the boundary is finite, and it tends to infinity logarithmically as this distance approaches zero. Moreover, the charge tends to deviate toward the material boundary.
ACKNOWLEDGMENT
This work was supported by a Grant of the Russian Foundation for Basic Research (Grant No. 12-02-31258) and the Dmitry Zimin “Dynasty” Foundation.
[4] J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco, Jr., B.-I. Wu, J. A. Kong, and M. Chen, Opt. Express 11, 723 (2003). [5] Z. Duan, B.-I. Wu, S. Xi, H. S. Chen, and M. Chen, Prog. Electromagn. Res. 90, 75 (2009). [6] S. Antipov, L. Spentzouris, W. Liu, W. Gai, and J. G. Power, J. Appl. Phys. 102, 034906 (2007). [7] S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, Phys. Rev. Lett. 103, 194802 (2009).
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[8] S. N. Galyamin and A. V. Tyukhtin, Phys. Rev. B 81, 235134 (2010). [9] E. S. Belonogaya, S. N. Galyamin, and A. V. Tyukhtin, J. Opt. Soc. Am. B 28, 2871 (2011). [10] S. N. Galyamin and A. V. Tyukhtin, Phys. Rev. E 84, 056608 (2011). [11] S. N. Galyamin, A. A. Peshkov, and A. V. Tyukhtin, Phys. Rev. E 88, 013206 (2013). [12] P. A. Belov, S. Tretyakov, and A. J. Viitanen, J. Electromagn. Waves Appl. 16, 1153 (2002). [13] A. Demetriadou and J. B. Pendry, J. Phys.: Condens. Matter 20, 295222 (2008). [14] A. V. Tyukhtin and E. G. Doilnitsina, J. Phys. D 44, 265401 (2011).
[15] V. V. Vorobev and A. V. Tyukhtin, Phys. Rev. Lett. 108, 184801 (2012). [16] D. E. Fernandes, S. I. Maslovski, and M. G. Silveirinha, Phys. Rev. B 85, 155107 (2012). [17] A. V. Tyukhtin and V. V. Vorobev, J. Opt. Soc. Am. B 30, 1524 (2013). [18] V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, New York, 1966). [19] M. G. Silveirinha, IEEE Trans. Antennas Propag. 54, 1766 (2006). [20] A. P. Prudnikov, Y. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).
013202-8
Radiation of charges moving along the boundary of a wire metamaterial Andrey V. Tyukhtin* and Viktor V. Vorobev† Physical Faculty of St. Petersburg State University, St. Petersburg 198504, Russia (Received 19 November 2013; published 9 January 2014) The electromagnetic fields of charges moving along the boundary of a “wire metamaterial” perpendicularly to the wires are investigated. The metamaterial under consideration represents a volume-periodic structure of thin parallel wires located in a square lattice. This structure is described by an effective permittivity tensor and exhibits both spatial and frequency dispersion. It is shown that the charge generates nondivergent radiation, as in the case of an infinite metamaterial. However, unlike the infinite-metamaterial case, the radiation concentrates near a certain plane (not line) behind the charge, and it is asymmetric with respect to this plane. An algorithm for calculating the wave fields of finite-length bunches is developed, and some typical numerical results are given. They demonstrate that the structure under consideration can be applied to determine the size and form of such bunches. The stopping and deflection forces acting on the charge are also calculated. DOI: 10.1103/PhysRevE.89.013202
PACS number(s): 41.60.Bq, 81.05.Xj, 78.67.Uh
following expression [12–14]:
I. INTRODUCTION
Cherenkov radiation is widely used for the detection of charged particles [1], and it can be applied for diagnostics of particle bunches, as well [2,3]. The prospects for these techniques strongly depend on the material used to generate the radiation. Cherenkov radiation can possess unusual properties when a medium that exhibits dispersion and/or anisotropy is used. Among such media, particular attention has been paid to so-called “metamaterials” which represent composite periodic structures that consist of macroscopic objects placed with relatively small spacing. Under the condition that the wavelengths of the electromagnetic field are greater than this spacing, such a structure can be considered as a solid medium and can be described by effective parameters, such as permittivity and permeability. For example, a so-called “left-handed” medium (LHM), which has both negative permeability and negative permittivity in a certain range of frequencies, can be implemented as a metamaterial. The processes of charged-particle radiation in the presence of a LHM or other metamaterial have been widely investigated previously [4–11]. This paper is focused on the case of a structure that consists of thin parallel wires in vacuum placed in a square lattice (Fig. 1). If the period is small in comparison with the wavelengths, the structure can be described by an effective permittivity tensor of the following form [12–14] (in Cartesian coordinates): ⎛ ⎞ ε (ω ,kx ) 0 0 0 1 0⎠ , εˆ = ⎝ (1) 0 0 1 where the x axis lies along the wires. Generally speaking, wires can possess nonconductive coatings [13,14]; here, however, we consider wires without any coverings. In this case, the longitudinal component of the tensor (1) is given by the
ε (ω ,kx ) = 1 −
ω2p ω2 −c2 kx2 + 2i ω ωd
,
(2)
where kx is the x component of the wave vector. The effective “plasma frequency” squared is related to the geometrical parameters of the structure as follows: ω2p =
d2
2π c2 , [ln (d/r0 ) − C]
(3)
where d is the period of the structure, r0 is the radius of the conductors, c is the speed of light in vacuum, and C is some constant of order 1. Note that structures with different periods along the y and z axes are also described by the tensor (1) with the component (2); a difference exists only in more complex formulas for the plasma frequency [14]. The structure under consideration is frequently called a “wire metamaterial.” As one can see, the effective medium that models this structure possesses both frequency and spatial dispersion of a specific type (there is dependence only on a single component of the wave vector). These features result in unique properties of the radiation of a charged-particle bunch [15–17]. For example, in the case in which the charge moves perpendicularly to the wires inside an infinite metamaterial and in which losses are negligible, radiation is present for any velocity of the charge. In addition, the radiation is nondivergent and concentrates along certain lines behind the charge [15,16]. It has been shown that this phenomenon allows the determination of the sizes of charged-particle bunches. However, wires can have an essential influence on particle bunches. Therefore, it is important to consider the case in which the bunch moves outside the metamaterial. In this paper, we analyze the case in which the charge moves in vacuum along a flat boundary of the metamaterial perpendicularly to the wires. II. TOTAL FIELD OF A POINT CHARGE
*
[email protected] † [email protected] 1539-3755/2014/89(1)/013202(8)
We consider a point charge q moving in vacuum with a constant velocity V = cβez along the flat boundary of a wire metamaterial that fills the area x > 0. The charge moves at a 013202-1
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PHYSICAL REVIEW E 89, 013202 (2014)
In the following discussion, we use a six-component vector Fα , which consists of components of the electric and magnetic fields:
x
Fα = {Exα ; Eyα ; Ezα ; Hxα ; Hyα ; Hzα },
where the index α corresponds to various types of waves. To obtain the solution, we use the decomposition of the total field over plane waves. The incident field (that is, the field of a point charge moving in unbounded vacuum) has the following form: ∞ ∞ ∞ q ω dω dky δ kz − Fi = 2π −∞ cβ −∞ −∞
d FIG. 1. Wire metamaterial.
distance a0 from the material’s border and perpendicularly to the wires (Fig. 2). The corresponding current density has the form j = qcβδ (x + a0 ) δ (y) δ (z − cβt) ez ,
(4)
×
z is a unit vector along the z axis in wave-vector space. where u The standard boundary conditions, which demand the continuity of the tangential components of the electric and magnetic fields, have the following form: Ey,z |x=−0 = Ey,z |x=+0 , (6)
Hy,z |x=−0 = Hy,z |x=+0 .
However, there exist three types of waves in the metamaterial [14] and two waves of different polarizations in the vacuum. Therefore, four boundary conditions (6) are insufficient. Such a situation is typical for problems in which the medium exhibits spatial dispersion [18]. To obtain the unique solution, we must use an additional boundary condition, which is the continuity of the normal component of the electric field [19]: Ex |x=−0 = Ex |x=+0 .
2 fi = sgn(x + a0 )kxo kz ; ky kz ; −kxo − ky2 ;
ω ω − ky ; sgn(x + a0 ) kxo ; 0 , c c ω2 kxo = − ky2 − kz2 . c2 The root (11) is fixed such that √ sgn( · · ·) = sgn(ω) if √ if Im( · · ·) > 0
Fv =
∞
dω −∞
∞
−∞
dky
∞ −∞
(11)
(12)
dkz (ftm Vtm +fte Vte )
× e−ikxo x+iky y+ikz z−i ω t ,
2 ω ω ω 2 − kxo ; ky kxo ; kz kxo ; 0; kz ; − ky , ftm = c2 c c
2 ω ω ω 2 fte = 0; − kz ; ky ; 2 − kxo ; ky kxo ; kz kxo , c c c
On the “microscopic” level, this requirement means that the currents at the wire’s extremes are negligible.
-a0
√ Im · · · = 0, √ Re · · · = 0.
(10)
These rules ensure that excited waves propagate from the charge trajectory, and “evanescent waves” exponentially decrease. The field that is reflected into vacuum consists of two waves of different polarizations:
(7)
x
fi exp{i |x + a0 | kxo + iky y + ikz z − i ω t}dkz , ω kxo (9)
where
where ez is a unit vector along the z axis, and c is the speed of light in vacuum. To proceed, we require the Fourier transform of the current: q ω z, exp (ikx a0 ) u δ kz − (5) jω k = 8π 3 cβ
0
(8)
(13)
(14)
where kxo is represented by the expression given in (11) and fixed by the rules given in (12). The field that is transmitted into the metamaterial consists of three types of waves: an ordinary wave (o), an “extraordinary isotropic” wave (ei), and an “extraordinary anisotropic” wave (ea) [14]. The components of these waves can be written in the following forms:
z V=cβ
Fo =
FIG. 2. (Color online) Movement of a charge along a boundary. 013202-2
∞
dω −∞
∞ −∞
dky
∞ −∞
dkz fo Mo eikxo x+iky y+ikz z−i ω t , (15)
RADIATION OF CHARGES MOVING ALONG THE . . .
PHYSICAL REVIEW E 89, 013202 (2014)
ω ω ω2 2 ; −ky kxo ; kz kxo , fo = 0; − kz ; ky ; 2 − kxo c c c ∞ ∞ ∞ dω dky dkz fei Mei eikxei x+iky y+ikz z−i ω t , Fei = −∞
−∞
−∞
(16)
Fo =
+∞
+∞
dω −∞
−∞
(24) Fei =
+∞ −∞
−∞
fea = {0; ky ; kz ; 0; −kz ky }. The expressions for the components of the wave vector along the x axis can be written as follows: ω2 − ky2 − kz2 , kxo = c2 ω2 − ω2p (18) kxei = − ky2 − kz2 , 2 c ω kxea = , c where the fixing of the roots is provided by the rules given in (12). Thus, the total field is x < 0, Fi + Fv , (19) F = Fo + Fei + Fea , x > 0. The five unknown amplitudes of the waves Vtm , Vte , Mo , Mei , and Mea are determined from the boundary conditions (6) and (7). As a result, one obtains
Vtm
ω2 (kxo − kxei ) ωc + kxo − c2p i Ex0 = ,
2 ω
ω + k − k + k ) (k xo xei xo xei c c
Vte = 0, Mo =
(20)
−βky i , ω2
Ex0 2 kxo c2 − kxo
Mei = ω c
Mea = ω c
− kxei +
ω
2kxo
Ei , + kxo (kxo + kxei ) x0
c ω 2kxo c (kxei − kxo ) i
2 ω
ω
Ex0 , kxo − k − k xo xei c c
+∞
+∞
−∞
−∞
+∞
+∞
dω −∞
−∞
(fea Vea )kz =ω/cβ (26)
where ζ = z − cβt and the x projections of the wave vector are written as kxo =
−
ω2 1 − β 2 − ky2 , c2 β 2
(27) ωp2 ω2 1 − β 2 kxei = − 2 − 2 − ky2 . c β2 c One can see that the radicals in (27) are imaginary. Thus, two waves in the medium (ordinary and extraordinary isotropic) and the wave that is reflected into the vacuum are exponentially decreasing. We are primarily interested in the volume radiation in the medium, which consists of the extraordinary anisotropic wave only. One can see from (26) that the field of this wave is invariant along any line ζ = −βx + const, y = const. III. RADIATION OF A POINT CHARGE
We consider in detail the transmitted extraordinary wave, which has Ey , Ez , Hy , and Hz components (the last two can be expressed as Hy = −Ez and Hz = Ey ). The typical distribution of the electrical field in the Y Z plane is given in Fig. 3. This figure illustrates that the distribution of the field becomes narrower when the velocity of the charge increases. This fact can be explained by the relativistic transverse contraction of the “incident” field (i.e., the field of the charge in unbounded vacuum). Further simplification of expression (26) can be achieved for the case of an ultrarelativistic charge. When β → 1, integration over ω can be performed for both components: +∞ Ey = q ky sin(ky y)e−a0 ky dky 0
ˆ ˆ × (ξ ) e−ky ξ −
(22) + (−ξˆ )
dω
× ei(ω/c)x+iky y+i(ω/cβ)ζ dky ,
(21)
ω i where Ex0 = 2πq ω kz δ(kz − cβ ) exp(ia0 ω2 /c2 −ky2 −kz2 ). Integrating over kz , one obtains the reflected and transmitted waves in the following forms:
Fv =
Fea =
−∞
(17)
(fei Vei )kz =ω/cβ eixkxei +iky y+i(ω/cβ)ζ dky , (25)
ω2 ω ω 2 − kxei ; −ky kxei ; −kz kxei ; 0; kz ; − ky , fei = c2 c c ∞ ∞ ∞ Fea = dω dky dkz fea Mea ei(ω/c)x+iky y+ikz z−i ω t , −∞
+∞
dω
−∞
(fo Vo )kz =(ω/cβ) eixkxo +iky y+i(ω/cβ)ζ dky ,
Ez = q
+∞
013202-3
2ky2 −2ky kyp +(2ky ξˆ +1) ω2p /c2 (ky + kyp )2
e
ky ξˆ
, (28)
ky cos(ky y)e−a0 ky dky
0
ˆ × (ξˆ ) e−ky ξ −
(ftm Vtm )kz =ω/cβ e−ixkxo +iky y+i(ω/cβ)ζ dky , (23)
4ky kyp ˆ e−ξ kyp (ky + kyp )2
+ (−ξˆ )
2 4kyp
(ky + kyp )
e−ξ kyp 2
ˆ
−2ky2 +2ky kyp −(2ky ξˆ +3) ω2p /c2 (ky + kyp )2
e
ky ξˆ
, (29)
ANDREY V. TYUKHTIN AND VIKTOR V. VOROBEV
PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 3. (Color online) Distribution of the radiation field of a point charge in the plane parallel to the boundary. The distance to the boundary is a0 = 0.5c/ ωp ; the electric field is given in units of q ω2p /c2 ; distances are in units of c/ ωp ; the charge’s velocity is β = 0.5 (upper row) and β = 0.95 (lower row).
where kyp = ω2p /c2 + ky2 , ξˆ = ζ + x, and (ξˆ ) is the Heaviside function. Numerical results for the distribution of the field in the plane x = const for different distances from the ultrarelativistic charge to the boundary are given in Fig. 4. The distributions shown in Figs. 3 and 4 represent the radiation field only (which consists solely of the extraordinary anisotropic wave). This part of the field is the dominant component of the total field when x c/ ωp (because of the decrease in the other parts of the field). The radiation field does not change when the observation point is shifted along the line ζ = −βx at any fixed y. Thus, the radiation is nondivergent, as in the case of an infinite metamaterial [15]. The difference is that this radiation concentrates at the semiplane ζ = −βx, y > 0 (not a line) behind the charge. In addition, one can note that the radiation is asymmetric with respect to this semiplane. The component Ey is equal to zero if y = 0, while Ez reaches its maximum. The maximum magnitude of the electric field observed at y = 0, ζ = −βx decreases with the distance a0 , and the “light spot” in the plane x = const essentially widens. This widening along the y direction (orthogonal to the charge velocity) is greater than that along the z direction (this is explained by the relativistic transverse contraction of the incident field). The integral (29) can be analytically calculated in terms of special functions for the case a0 = 0, y = 0 [20]: + − + (−ξˆ )Ezea , Ezea = (ξˆ )Ezea + Ezea
1 1 7 34 114 240 240 −p =q 2 −4 + 2 + 3 + 4 + 5 + 6 e c p2 p p p p p p
6p(p2 +10)K0 (p) + (p4 +27p2 +120)K1 (p) , +8 p5 (30) ω2p
− =q Ezea
ω2p 1 (16p5 +p4 −160p3 +24p2 −960) c2 p 5
+ 2q
ω2p π {p(p4 − 18p2 + 120)[Y0 (−p) + H0 (p)] c2 p 5
+(5p4 − 66p2 + 240)[Y1 (−p) − H1 (p)]},
(31)
where p = ξˆ ωp /c = (z − ct + x) ωp /c, Kν is a modified Hankel function, Yν is a Neumann function, and Hν is a Struve function. Note that in the plane y = 0, the integral given in (28) for Ey is equal to zero. Expressions (30) and (31) can be written as series expansions under the condition p 1, using known expansions for special functions. It turns out that the main terms for both summands in (30) and (31) are the same:
± Ezea
= Ezea
q ωp2 ≈ 2 c2
ωp |ξˆ | 13 + γ˜ + ln , 2c 12
(32)
where γ˜ 0.577 is the Euler constant. One can see that the longitudinal electric-field component has a logarithmic singularity at y = 0, ζ = − βx. The obtained result (32) is similar to the case of a charge moving inside an infinite metamaterial [15]. However, it should be emphasized that the singularity appears only for a charge moving directly at the boundary, that is, for the case a0 = 0. If a0 = 0, the field is continuous everywhere in the region x > 0. It is also interesting to examine the special case in which the relation r0 /d is fixed, but d/a0 → 0, that is, ωp a0 /c → ∞. Such a medium is ideally conducting in only one direction (along the x axis). We can obtain the expressions for the field
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PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 4. (Color online) Radiation field of an ultrarelativistic point charge. The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The distance to the boundary is a0 = 0 (top row), a0 = 0.1c/ ωp (middle row), and a0 = c/ ωp (bottom row).
of radiation as a limiting case of (28) and (29): +∞ ideal Eyea =q ky sin(ky y)e−a0 ky
Thus, we find a simple analytical result for the case of a medium that is ideally conductive in one direction. This approximation can be used under the condition ωp a0 /c 1.
0
× { (ξˆ )e−ky ξ + (−ξˆ )(2ky ξˆ + 1)eky ξ }dky , +∞ =q ky cos(ky y)e−a0 ky ˆ
ideal Ezea
ˆ
0
× { (ξˆ )e−ky ξ − (−ξˆ )(2ky ξˆ + 3)eky ξ }dky , ˆ
ˆ
IV. STOPPING AND DEFLECTING FORCES
(33)
(34)
We considered the field of radiation that is produced by the extraordinary anisotropic wave in a metamaterial. The two other waves inside the metamaterial and the one in the vacuum are surface waves. However, the field in the vacuum is interesting because it affects the charge. A particle’s energy losses for a unit path can be expressed via the stopping force:
The obtained integrals can be analytically evaluated [20]: ideal Eyea =q
dE = Fz = qEvz |x → −a0 , dz0 ζ → 0,y → 0
2y(a0 + |ξˆ |) 2
2 [y 2 + (a0 + |ξˆ |) ]
+ 4q (−ξˆ )
2 y ξˆ [3(a0 + |ξˆ |) − y 2 ] 2 3
(35)
,
[y 2 + (a0 + |ξˆ |) ] 2
ideal Ezea =q
(a0 + |ξˆ |) − y 2 2
2 [y 2 + (a0 + |ξˆ |) ] 2 (a0 + |ξˆ |) − y 2 − 4q (−ξˆ ) 2 2 [y 2 + (a0 + |ξˆ |) ]
+
2 ξˆ (a0 + |ξˆ |)[(a0 + |ξˆ |) − 3y 2 ]
[y 2
2 + (a0 + |ξˆ |) ]
3
(37)
where Ezv is the component of the reflected field (the incident field Ezi does not affect the charge because it is odd over ζ ). The integral (23) for the z component can be taken analytically in the plane y = 0 for the ultrarelativistic case β → 1 [10]: ω2 −2p3 + p2 + 12 2 p Fz = −2q 2 c p4
π + 3 {3p[Y0 (p) − H0 (p)]+(p2 − 6)[Y1 (p) − H1 (p)]} , p
(38) .
(36)
where p = 2a0 ωp /c, Yν is a Neumann function, and Hν is a Struve function. The result of a numerical calculation for Ezv
013202-5
ANDREY V. TYUKHTIN AND VIKTOR V. VOROBEV
PHYSICAL REVIEW E 89, 013202 (2014)
Using the expressions for the components Exv and Eyv , one can determine the deflecting forces in the x and y directions. It turns out that Eyv y=0 = 0, so the charge is not deviated in the y direction (this is natural because of the symmetry of the problem relative to the y axis). At the same time, one can find from (23) that Fx = qExv |y → 0, ζ → 0 = −2Fz >0. Thus, the x → −a0
particle is deflected in the direction of the boundary, and the deflecting force is greater than the stopping force by a factor of 2. FIG. 5. (Color online) Stopping force Fz (in units of q 2 ω2p /c2 ) as a function of the distance from the charge trajectory to the boundary (in units of c/ ωp ).
is presented in Fig. 5. As one can see, the field is limited, while the distance to the boundary is nonzero. When a0 ωp /c 1, the stopping force can be approximated as follows:
a0 ωp 1 q 2 ω2p + γ˜ + Fz ≈ ln . (39) 2 c2 c 4 Therefore, the energy loss has a logarithmic singularity at a0 → 0. Note that in the case a0 0, the losses exhibit asymptotic behavior as 1/a02 . Thus, the stopping force (that is, energy losses) is limited in the case a0 = 0, unlike the result for a charge moving inside an unbounded metamaterial.
V. RADIATION OF A BUNCH OF CHARGED PARTICLES
The results obtained for the point charge, (28) and (29), can be applied for the calculation of the field of a bunch of charged particles as a Green’s function. We consider a thin bunch that has a limited size in the direction of its motion (along the z axis) and is infinitely small in transverse dimensions. The charge distribution is given by the expression ρ (x,y,ζ ) =
1 δ (x) δ (y) (σ − |ζ |) . 2σ
In this case, the field of radiation is expressed as σ
1 Eea x,y,ζ − ζ dζ . Eline (x,y,ζ ) = 2σ −σ
(40)
(41)
Numerical results for ultrarelativistic bunches (β → 1) with relatively large lengths (σ c/ ωp ) are presented in Figs. 6 and 7. They represent the distributions of the radiation
FIG. 6. (Color online) Radiation of an ultrarelativistic charged bunch moving at a distance of a0 = 20c/ ωp from the boundary. The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The width of the bunch is σ = 100c/ ωp (top row), σ = 50c/ ωp (middle row), and σ = 25c/ ωp (bottom row). 013202-6
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PHYSICAL REVIEW E 89, 013202 (2014)
FIG. 7. (Color online) Radiation of an ultrarelativistic charged bunch with σ = 100c/ ωp . The electric field is given in units of q ω2p /c2 , and distances are in units of c/ ωp . The distance from the bunch to the boundary is a0 = 30c/ ωp (upper row) and a0 = 100c/ ωp (lower row).
field (that is, the anisotropic extraordinary wave) at some plane x = const > 0. These distributions do not depend on x, as for the point charge. One can see that the distances between the extremes of the field correspond to the length of the bunch. Therefore, the analysis of the field generated by a bunch of charged particles moving near the boundary of a wire metamaterial can help to estimate the size of this bunch. However, the precision of this method depends on the distance from the bunch to the boundary because the “snapshot” of the bunch becomes more fuzzy (mostly in the y direction) as this distance increases (Fig. 7). Note that the length of a short bunch (σ c/ ωp ) cannot be easily evaluated using this method because the snapshots of the wave field are similar to the field of a point charge. VI. CONCLUSIONS
A point charge moving along the boundary of a wire metamaterial perpendicularly to the wires was considered. The radiation field was presented as twofold integrals, which can be reduced to single-fold ones if the charge moves at the speed of light in vacuum. It was shown that the radiation is nondivergent, as in the case of unbounded wire material, but the radiation properties are
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essentially different. The radiation concentrates near a certain semiplane behind the charge and is characterized by some asymmetry with respect to this plane. The z component of the electric field has a logarithmic singularity if the charge moves directly on the boundary. However, if the distance from the charge trajectory to the boundary is not equal to zero, the radiation field is finite and continuous everywhere in the metamaterial. The typical picture of the field of a bunch demonstrates that the radiation can be used to determine bunch sizes and forms. The influence of the reflected field on the charge was also analyzed. The value of the stopping force was obtained analytically and computed. The stopping force is limited if the distance from the charge to the boundary is finite, and it tends to infinity logarithmically as this distance approaches zero. Moreover, the charge tends to deviate toward the material boundary.
ACKNOWLEDGMENT
This work was supported by a Grant of the Russian Foundation for Basic Research (Grant No. 12-02-31258) and the Dmitry Zimin “Dynasty” Foundation.
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