Cherenkov radiation and emission of surface polaritons from charges moving paraxially outside a dielectric cylindrical waveguide
A.A. Saharian, L.Sh. Grigoryan, A.Kh. Grigorian, H.F. Khachatryan, A.S. Kotanjyan
aa r X i v : . [ phy s i c s . acc - ph ] S e p Cherenkov radiation and emission of surface polaritonsfrom charges moving paraxially outsidea dielectric cylindrical waveguide
A. A. Saharian , ∗ , L. Sh. Grigoryan , A. Kh. Grigorian ,H. F. Khachatryan , A. S. Kotanjyan , Institute of Applied Problems in Physics NAS RA,25 Hr. Nersessian Str., 0014 Yerevan, Armenia Department of Physics, Yerevan State University,1 Alex Manoogian Street, 0025 Yerevan, Armenia
September 4, 2020
Abstract
We investigate the radiation from a charged particle moving outside a dielectric cylinderparallel to its axis. It is assumed that the cylinder is immersed into a homogeneous medium.The expressions are given for the vector potential and for the electric and magnetic fields. Thespectral distributions are studied for three types of the radiations: (i) Cherenkov radiation(CR) in the exterior medium, (ii) radiation on the guided modes of the dielectric cylinder, and(iii) emission of surface polaritons. Unlike the first two types of radiations, there is no velocitythreshold for the generation of surface polaritons. The corresponding radiation is present inthe spectral range where the dielectric permittivities of the cylinder and surrounding mediumhave opposite signs. The spectral range of the emitted surface polaritons becomes narrowerwith decreasing energy of the particle. The general results are illustrated for a special caseof the Drude model for dispersion of the dielectric permittivity of the cylinder. We showthat the presence of the cylinder may lead to the appearance of strong narrow peaks in thespectral distribution of the CR in the exterior medium. The conditions are specified forthe appearance of those peaks and the corresponding heights and widths are analyticallyestimated. The collective effects of particles in bunches are discussed.
The polarization of a medium by moving charged particles gives rise to a number of radiationprocesses. Examples are the Cherenkov radiation (CR), transition radiation and the diffractionradiation. Among those radiation processes, the remarkable properties of the CR (for reviewssee [1, 2]) have resulted in a wide variety of applications, including counting and identifyingof high-energy particles, cosmic-ray physics, high-power radiation sources in various spectralranges, particle accelerating systems, medical imaging and therapy and so on. These applica-tions motivate the importance of the further investigations for various mechanisms to control ∗ E-mail: [email protected]
Consider a point charge q moving parallel to the axis of a cylinder with dielectric permittivity ε and with the radius r c . The distance of the charge trajectory from the axis will be denotedby r > r c and it will be assumed that the cylinder is immersed into a homogeneous mediumwith dielectric permittivity ε (see figure 1, the magnetic permeabilities for both the cylinder2nd surrounding medium will be taken to be unit). In accordance with the problem symmetrywe will use cylindrical coordinates ( r, φ, z ) with the axis z along the axis of the cylinder. Inthe generalized Lorentz gauge, the vector potential of the electromagnetic field created by thecharge is expressed in terms of the electromagnetic field Green tensor G il ( r , t, r ′ , t ′ ) as A i ( t, r ) = − π c Z dt ′ d r ′ X l =1 G il ( t, r , t ′ , r ′ ) j l ( t ′ , r ′ ) , (2.1)where j l ( t, r ) is the current density for the source. In the problem under consideration the onlynonzero component of the latter is given by j ( t, r ) = qr vδ ( r − r ) δ ( φ − φ ) δ ( z − vt ) , (2.2)with v being the charge velocity. ε Figure 1: The problem geometry and the notations.It is convenient to write the relation (2.1) in terms of the partial Fourier components A l,n ( k z , r ) of the vector potential defined in accordance with A l ( t, r ) = ∞ X n = −∞ e in ( φ − φ ) Z ∞−∞ dk z e ik z ( z − vt ) A l,n ( k z , r ) . (2.3)By using the Fourier expansion G il ( t, r , t ′ , r ′ ) = ∞ X n = −∞ Z ∞−∞ dω Z ∞−∞ dk z G il,n ( ω, k z , r, r ′ ) e in ( φ − φ ′ )+ ik z ( z − z ′ ) − iω ( t − t ′ ) , (2.4)from (2.1) we get A l,n ( k z , r ) = − qvπc G l ,n ( vk z , k z , r, r ) . (2.5)In [14] a recurrence scheme was developed for evaluation of the Green tensor in a mediumwith an arbitrary number of cylindrically symmetric homogeneous layers. In the problem at handthe Green tensor is obtained by using the corresponding tensor in a homogeneous medium. Inparticular, for the region r > r the Fourier components of the Green tensor appearing in (2.5)are given by the expressions [14] G l ,n ( ω, k z , r, r ) = i − l k z r c J n ( λ r c ) H n ( λ r ) α n V Hn X p = ± p l − J n + p ( λ r c ) H n + p ( λ r ) V Hn + p ,G ,n ( ω, k z , r, r ) = π i (cid:20) J n ( λ r ) − H n ( λ r ) V Jn V Hn (cid:21) H n ( λ r ) , (2.6)3here l = 1 ,
2, and λ j = ω ε j /c − k z with j = 0 ,
1. In (2.6), J n ( x ) is the Bessel function, H n ( x ) = H (1) n ( x ) is the Hankel function of the first kind, and we have introduced the notation V Fn = J n ( λ r c ) ∂ r c F n ( λ r c ) − [ ∂ r c J n ( λ r c )] F n ( λ r c ) , (2.7)for F = J, H . The function α n in the expression for the component G l ,n ( ω, k z , r, r ′ ) is given bythe formula α n = ε ε − ε + 12 X l = ± (cid:20) − λ λ J n + l ( λ r c ) H n ( λ r c ) J n ( λ r c ) H n + l ( λ r c ) (cid:21) − . (2.8)The eigenmodes of the dielectric cylinder are determined from the equation α n = 0. They arepoles of the integrand in (2.4) for the corresponding components of the Green tensor.The Fourier components of the vector potential are given by (2.5) with the Green tensorcomponents from (2.6) where now ω = vk z and λ j is given by the expression λ j = k z (cid:0) β j − (cid:1) , (2.9)with β j = ( v/c ) ε j . In the discussion below we will assume that the exterior medium is trans-parent and the permittivity ε is real. The both cases β > β < λ is purely imaginary and its sign is determined in accordance with λ = i | k z | p − β . Note that in the arguments of the Hankel functions λ appears only and withthis choice of the sign they are reduced to the Macdonald functions K ν ( | λ | x ), with x = r, r , r c .For real ε j and β j > λ j = k z q β j − E l ( t, r ) and H l ( t, r ), similar to (2.3), withthe Fourier coefficients E l,n ( k z , r ), H l,n ( k z , r ), for the magnetic field one finds H l,n ( k z , r ) = qvk z i l − c X p = ± p l − f ( p ) n H n + p ( λ r ) , l = 1 , ,H ,n ( k z , r ) = iqvk z c q β − X p = ± pf ( p ) n H n ( λ r ) , (2.10)where for p = ± f ( p ) n = − q β − J n ( λ r ) + H n ( λ r ) V Hn "q β − V Jn + 2 ipk z π J n ( λ r c ) r c α n J n + p ( λ r c ) V Hn + p . (2.11)By taking into account that for the function from (2.7) one has V F − n = V Fn , F = J, H , it can beseen that f ( p ) − n = ( − n f ( − p ) n . (2.12)The Fourier coefficients for the electric field are obtained from the Maxwell equations and aregiven by E l,n ( k z , r ) = qk z i l ε X p = ± p l h(cid:0) β + 1 (cid:1) f ( p ) n − (cid:0) β − (cid:1) f ( − p ) n i H n + p ( λ r ) ,E ,n ( k z , r ) = qk z ε q β − X p = ± f ( p ) n H n ( λ r ) , (2.13)where l = 1 ,
2. From (2.12) we get the following relations for the Fourier components of thefields E l, − n ( k z , r ) = ( − l +1 E l,n ( k z , r ) , H l, − n ( k z , r ) = ( − l H l,n ( k z , r ) , (2.14)4or l = 1 , ,
3. Note that we have also the relations E l, − n ( − k z , r ) = E ∗ l,n ( k z , r ) and H l, − n ( − k z , r ) = H ∗ l,n ( k z , r ), where the star stands for the complex conjugate.The electromagnetic fields for a charge moving in a homogeneous medium with dielectricpermittivity ε are obtained from the expressions given above taking ε = ε . In this limit V Jn = 0 and V Hn = 2 i/πr c , whereas the function α n tends to infinity. Hence, the correspondingFourier components are given by (2.10) and (2.13) with the replacement f ( p ) n → − q β − J n ( λ r ) . (2.15)Now we see that the fields in the exterior region are decomposed into the parts corresponding tothe fields in homogeneous medium with permittivity ε and the part induced by the presence ofthe cylinder. The latter is given by (2.10) and (2.13) excluding the first term in the right-handside of (2.11). The Fourier components have poles at the zeros of the function α n . As it hasbeen mentioned before, those zeros determine the eigenmodes of the cylinder. The expressionsfor the fields in the region r c < r < r are obtained from the corresponding formulas in theregion r > r , given above, by the replacements J → H , H → J in the parts corresponding tothe fields in homogeneous medium with permittivity ε . The cylinder induced contributions aredescribed by the same expressions for all values r > r . The fields inside the cylinder can befound by using the corresponding expressions of the Green tensor components from [14]. Having the electric and magnetic fields in the form of the Fourier expansion we can investigatethe radiation intensity emitted by the charged particle. In the problem at hand we have threetypes of radiations. The first one corresponds to the CR in the exterior medium influenced bythe presence of the cylinder. The second one is the radiation emitted on the guided modes of thecylinder and propagates inside the waveguide. The corresponding fields exponentially decay inthe exterior medium. Under certain conditions on the characteristics of the media one can havealso the radiation in the form of surface polaritons (surface modes). We start our discussionfrom the radiation in the exterior medium at large distances from the cylinder, r ≫ r c . From theexpressions (2.10) and (2.13) it follows that this kind of radiation is present under the condition λ >
0. By taking into account the expression (2.9) the latter condition is translated to β > λ < K n ( | λ | r ), K n + p ( | λ | r ), and the Fouriercomponents exponentially decay at large distances from the cylinder, r ≫ v/ω .We denote by I the energy flux per unit time through the cylindrical surface of radius r . Itis given by the expression I = c π Z π dφ Z ∞−∞ dz r n · [ E × H ] , (3.1)where n is the unit normal to the integration surface. By using the Fourier expansions of thefields we get I = πcr ∞ X n = −∞ Z ∞−∞ dk z n · [ E n ( k z , r ) × H ∗ n ( k z , r )] . (3.2)Under the condition λ >
0, substituting the expressions for the Fourier components, using therelation (2.12) and the asymptotic expressions of the Hankel functions for large arguments, at5arge distances from the cylinder we find I = Z dω dIdω , (3.3)with the spectral density dIdω = q ω vε ∞ X ′ n =0 (cid:20)(cid:12)(cid:12)(cid:12) f (1) n + f ( − n (cid:12)(cid:12)(cid:12) + β (cid:12)(cid:12)(cid:12) f (1) n − f ( − n (cid:12)(cid:12)(cid:12) (cid:21) , (3.4)where in the expressions (2.11) for f ( ± n the quantities λ j are given by (2.9) with k z = ω/v .In (3.3), the integration over ω goes over the part of the region ω ∈ [0 , ∞ ) where the condition β > n = 0 should be taken with an additional coefficient 1/2. An alternative representation for thespectral distribution of the radiation intensity dI/dω , based on the evaluation of the energylosses, will be given below (see (4.9)).From the relation ω = k z v it follows that the radiation described by (3.4) propagates alongthe Cherenkov cone having the opening angle θ = θ Ch with respect to the cylinder axis, wherecos θ Ch = 1 /β . In the limit ε → ε the functions f ( p ) n are given by the right-hand side in (2.15)and from (3.4) the Tamm-Frank formula is obtained for the CR in a homogeneous transparentmedium. In the limit r c → n = 0 , ωr c /v ) . The contributions of the terms with n ≥ ωr c /v ) n . Notethat the quantity ω − dI/dω , that determines the number of the radiated quanta (see below),depends on the frequency and on the cylinder radius in the form of the product ωr c . Hence,the limiting behavior for small r c determines also the behaviour of the radiation intensity forsmall frequencies. Namely, for ωr c /v ≪ ω for the terms with n = 1 , ω n for n ≥ d Ndzdω = 1 ~ ωv dIdω . (3.5)The corresponding quantity for the CR in a transparent homogeneous medium with permittivity ε is given by d N dzdω = q ~ c (cid:18) − β (cid:19) . (3.6)In figure 2 we display the ratio R N = d N/dzdωd N /dzdω (3.7)as a function of ωr c /c for several values of the ratio r /r c (the numbers near the curves). Thegraphs are plotted for the electron energy E e = 2 MeV and for ε = 3 . . . − ). The left and right panels correspondto ε = 1 and ε = 2 . r /r c . In the case corresponding to the left panel of figure 2 the CRinside the cylinder is absent and the oscillations are a consequence of the interference between6 / r c = c ω / c R N r / r c = c ω / c R N Figure 2: The ratio R N as a function of ωr c /c for the electron energy E e = 2 MeV and for ε = 3 .
8. The left and right panels correspond to ε = 1 and ε = 2 . r /r c .the direct CR and radiation reflected from the cylinder. For small wavelengths, compared tothe waveguide diameter, the oscillations enter the quasiperiodic regime. The beginning of thatregime with respect to the radiation wavelength increases with increasing values of the ratio r /r c . For small frequencies the presence of a cylindrical hole in a homogeneous medium leadsto the decrease of the radiation intensity. That is related to the fact that a part of the mediumis excluded from the radiation process. For the example considered on the right panel of figure 2the Cherenkov condition for the cylinder material is obeyed and the interference pattern is morecomplicated. It is formed by the interference of the direct radiation, the radiation reflected fromthe cylinder and the CR formed inside the cylinder. We have ε < ε and, as in the previouscase, here the radiation intensity for large wavelengths is smaller than that for a homogeneousmedium.For graphs in figure 2 we have taken ε > ε . The behavior of the radiation intensity isessentially different for ε < ε . This is seen from figures 3 and 4 where we have plotted R N versus ωr c /c for E e = 2 MeV, ε = 3 . ε = 2 .
2. For the left and right panels of figure 3 wehave taken r /r c = 1 . r /r c = 1 .
1, respectively. Figure 4 is plotted for r /r c = 1 .
05. Aswe see, for the charge trajectory sufficiently close to the cylinder strong narrow peaks appear inthe spectral density of the radiation intensity. The amplification of the radiation intensity forrelatively small values of ωr c /c is related to that now ε > ε and the CR inside the cylinder ismore intense than in an equivalent cylinder with permittivity ε .The appearance of the strong narrow peaks in the spectral distribution of the CR in theexterior medium is an interesting effect induced by the cylinder. Their presence can be under-stood analytically by using the formula (3.4) for the radiation intensity (see also the discussionsin [15] and [16] for the peaks in the angular distribution of the radiation intensity from chargesrotating around/inside a dielectric cylinder along circular and helical trajectories, respectively).First of all it can be seen that the peaks come from the terms in the series on the right-handside of (3.4) with large values of n . For large n one has the following asymptotic expression forthe Neumann function (the leading term in the Debye’s asymptotic expansion, see [17]) Y n ( ny ) ∼ e nζ ( y ) √ πn (1 − y ) / , (3.8)7 c ω / c R N c ω / c R N Figure 3: The same as in figure 2 for ε = 3 . ε = 2 .
2. For the left and right panels r /r c = 1 . r /r c = 1 . c ω / c R N Figure 4: The same as in figure 3 for r /r c = 1 . < y < ζ ( y ) = ln 1 + p − y y − p − y . (3.9)The function (3.9) is positive and monotonically decreasing in the region 0 < y < ζ (1) = 0. For y > n the function Y n ( ny ) exhibits an oscillating behavior (see theanalog behavior for the function J n ( ny ) in (3.10) below). For the Bessel function one has theasymptotics [17] J n ( ny ) ∼ e − nζ ( y ) √ πn (1 − y ) / , < y < ,J n ( ny ) ∼ r πn cos { n [ p y − − arccos(1 /y )] − π/ } ( y − / , y > . (3.10)The key point for our discussion is that the ratio | J n ( ny ) | /Y n ( ny ) is exponentially small forlarge n and for fixed 0 < y <
1. For 0 < y < J n ( ny ) /Y n ( ny ) ∝ e − n [ ζ ( y )+ ζ ( y )] andfor y > | J n ( ny ) | /Y n ( ny ) ∝ e − nζ ( y ) .With these asymptotic estimates, let us return to the expression (2.8) for the function α n .As we have mentioned above, the roots of the equation α n = 0 determine the eigenmodes of thedielectric cylinder. Under the condition λ > α n would take its minimal value. In accordance with (3.4) that could correspond to large intensitiesfor the CR. By taking into account that, in accordance with the asymptotics given above, forlarge n and λ r c < n the ratio | J n ( λ r c ) | /Y n ( λ r c ) is exponentially small, we can expand α n interms of this ratio. In the next-to-leading order we get α n ≈ ε ε − ε + 12 X l = ± g l,n + iπλ r c X l = ± lJ n + l ( λ r c ) g l,n J n ( λ r c ) Y n + l ( λ r c ) , (3.11)where g l,n = (cid:20) − λ λ J n + l ( λ r c ) Y n ( λ r c ) J n ( λ r c ) Y n + l ( λ r c ) (cid:21) − . (3.12)Note that, compared to the first two terms in the right-hand side of (3.11), the last term is ofthe order e − nζ ( λ r c /n ) . From here it follows that near the roots of the equation X l = ± g l,n + 2 ε ε − ε = 0 , (3.13)the function α n is exponentially small, α n ∝ e − nζ ( λ r c /n ) . Of course, this does not yet meanthat the radiation intensity at those points will be large because exponential factors may alsocome from the other functions in the last term of the right-hand side of (2.11).We recall that under the condition λ > α n = 0 has no solutions and thereare no eigenmodes of the cylinder in that region. The mathematical reason is that the functionis complex and the real and imaginary parts do not become zero simultaneously. Unlike thefunction α n , the function g l,n is real and the equation (3.13) may have solutions. In orderto specify the conditions under which the roots exist, first we consider the case λ > λ r c < n , by using theasymptotics (3.8) and (3.10) for the functions Y n ( λ r c ) and J n ( λ r c ), to the leading order, theequation (3.13) is reduced to p n − λ r c / p n − λ r c = − ε /ε . This shows that for large9alues of n and for λ r c < n the equation (3.13) has solutions under the condition λ r c > n .In particular, one should have ε > ε . By making use of the uniform asymptotic expansionfor the modified Bessel function I n ( | λ | r c ), we can see that from (3.13) the same leading orderequation is obtained for λ <
0. From that equation, as a necessary condition for the existenceof the roots in the range λ < < λ one gets ε < − ε . In the leading order, the roots withrespect to the angular frequency are given by ω ≈ cnr c (cid:18) ε ε ε + ε − c v (cid:19) − / . (3.14)Note that the inequality ε < − ε appears also as a necessary condition for the radiation ofsurface polaritons (see below). For the latter modes one has λ < f ( p ) n , it can be seenthat for the appearance of the peaks an additional condition λ r < n is required. Under thiscondition, for the Hankel function in (2.11) one has H n ( λ r ) ≈ iY n ( λ r ) and f ( p ) n ∝ e nζ ( λ r /n ) .As a consequence, the heights of the peaks in the spectral distribution of the radiation intensityare estimated as e nζ ( λ r /n ) . We have numerically checked that the locations of the peaks withrespect to ωr c /c in the graphs above are determined by the roots of the equation (3.13) with highaccuracy. For example, the peaks in figure 4 at ωr c /c = 10 . , . , . , . , . , . n = 15 , , , , ,
20, respectively. On the base of theasymptotic consideration given above the widths of the peaks can be estimated as well. In orderto do that we expand the function α n near the roots of the equation (3.13). By using (3.11)it can be seen that the width of the peaks is determined by the last term in the right-handside and is of the order ∆ ω/ω ∝ e − nζ ( λ r c /n ) . Note that in the estimates given above we haveassumed that the dielectric permittivity ε is real. For complex permittivity ε = ε ′ + iε ′′ , withreal and imaginary parts ε ′ and ε ′′ , the consideration presented is valid under the condition e − nζ ( λ r c /n ) ≫ | ε ′′ /ε ′ | . For e − nζ ( λ r c /n ) < | ε ′′ /ε ′ | the heights and the widths of the peaks aredetermined by the imaginary part of the permittivity.Summarizing the discussion above, we conclude that though there are no eigenmodes of thewaveguide in the range under consideration ( λ > n they approximately obey the equation α n = 0 with exponential accuracy.In this sense, those roots can be termed as ”quasimodes” of the dielectric waveguide (for thediscussion of quasi-bound waves on curved interfaces see, e.g., [18]). Unlike to the guided andsurface modes (see below) which remain coupled to the waveguide during their propagation andexponentially decay in the exterior medium, the radiation on the ”quasimodes” appears in formof the CR giving rise to high narrow peaks in the spectral distribution of the radiation intensityunder the conditions λ r < n < λ r c for λ > λ r < n , ε < − ε for λ <
0. In the latter case, for a given n , the angular frequencies of the peaks are given by(3.14). In the corresponding spectral range one has a quasidiscrete part of the CR. The spectralpeaks appear for large values of n and, hence, this effect is absent in the axially symmetricproblems (coaxial motion of charges and beams) where only the mode n = 0 contributes to theradiation intensity. We expect that similar features of the radiation intensity may appear forother geometries of the interface (see, for example, Ref. [19] for the radiation on a dielectricball). 10 Energy losses
In addition to the radiation propagating at large distances from the cylinder one can haveradiation emitted by the charge on the eigenmodes of the cylindrical waveguide. The total energylosses per unit of path length can be evaluated in terms of the work done by the electromagneticfield on the charge: dWdz = qE | r → r ,z → vt,φ → φ . (4.1)Substituting the analog of the Fourier expansion (2.3) for the z -component of the electric fieldone gets dWdz = 4 q ∞ X ′ n =0 Re (cid:20)Z ∞ dk z E n ( k z , r ) (cid:21) , (4.2)where the relation (2.14) is used for E n . By using the expression for the corresponding Fouriercomponent (2.13), we obtain dWdz = q ∞ X ′ n =0 Re Z ∞ dk z k z ε q β − X p = ± f ( p ) n H n ( λ r ) . (4.3)Note that, unlike the expression (3.4), the functions f ( p ) n enter in the expression of the energylosses linearly.By taking into account the formulas (2.13) and (2.11), the expression ( 4.2) is decomposedinto two contributions: dWdz = dW (0) dz + dW (c) dz , (4.4)where dW (0) dz = − q lim r → r ∞ X ′ n =0 Re (cid:20)Z ∞ dk z k z ε (cid:0) β − (cid:1) J n ( λ r ) H n ( λ r ) (cid:21) , (4.5)corresponds to the losses in a homogeneous medium with permittivity ε and dW (c) dz = 2 q ∞ X ′ n =0 Re (cid:26)Z ∞ dk z k z ε (cid:0) β − (cid:1) H n ( λ r ) V Hn × V Jn + ik z J n ( λ r c ) π p β − r c α n X p = ± p J n + p ( λ r c ) V Hn + p (4.6)is induced by the cylinder. This formula gives the expression for the losses in the general caseof the dielectric permittivity for the cylinder.First we consider the case when the Cherenkov condition for the exterior medium is satisfied, β >
1. The part (4.5) is further simplified by using the formula ∞ X ′ n =0 J n ( λ r ) H n ( λ r ) = 12 H ( λ ( r − r )) . (4.7)The real part of the latter is J ( λ ( r − r )) / r → r in (4.5) we get dW (0) dz = − q c Z β > dω ω (cid:0) − /β (cid:1) , (4.8)11hich gives the standard expression for the Cherenkov radiation in homogeneous medium. Underthe condition β > λ > α n = 0 has nosolutions with respect to k z and the integrand in (4.6) is regular on the positive semiaxis of k z .For real values of ε = ε ( ω ) the energy losses are in the form of the radiation (here and belowwe will not consider the ionization losses that correspond to the zeros of the function ε ). Forthe spectral density of the energy radiated per unit time we find dIdω = − v d Wdzdω = q vc ω (cid:18) − β (cid:19) ( − ∞ X ′ n =0 Re (cid:20) H n ( λ r ) V Hn × V Jn + ik z J n ( λ r c ) π p β − r c α n X p = ± p J n + p ( λ r c ) V Hn + p . (4.9)For β > λ > λ <
0. Note that in (4.9) the contributioncorresponding to the radiation in homogeneous medium (the part with the first term in figurebraces) is explicitly separated.
Now we consider the case when the Cherenkov condition for the exterior medium is not obeyed, β <
1. In this case one has λ = i | λ | and the expression in the square brackets of (4.5) ispurely imaginary. As a consequence, we get dW (0) /dz = 0 and the radiation in a transparenthomogeneous medium is absent. Introducing the modified Bessel functions I n ( x ) and K n ( x ),the expression for the energy losses is presented as dWdz = − q π ∞ X ′ n =0 Im (cid:26)Z ∞ dk z k z ε (cid:0) − β (cid:1) K n ( | λ | r ) V J,Kn × V J,In − k z J n ( λ r c ) r c p − β α n X p = ± J n + p ( λ r c ) V J,Kn + p . (5.1)with ( F = I, K ) V J,Fn = J n ( λ r c ) ∂ r c F n ( | λ | r c ) − F n ( | λ | r c ) ∂ r c J n ( λ r c ) , (5.2)and α n = ε ε − ε + 12 X l = ± (cid:20) l | λ | λ J n + l ( λ r c ) K n ( | λ | r c ) J n ( λ r c ) K n + l ( | λ | r c ) (cid:21) − . (5.3)For real values of ε the integrand in (5.1) is real on the real axis of k z and the nonzero contri-butions to the integral may come from the possible poles on the real axis only. We can see thatthe integral is regular at the zeros of the functions V J,Kn ± and V J,Kn . Hence, the only nonzerocontributions come from the zeros of the function α n . These zeros with respect to k z we willdenote by k z = k n,s >
0, where s = 1 , , . . . enumerates the roots for a given n , k n,s +1 > k n,s .These roots determine the eigenmodes of the dielectric cylinder (the equation α n = 0 is easilytransformed to the form given, for example, in [20]). For n = 0 the equation for those modes issimplified to ε p − β p β − J ( λ r c ) J ( λ r c ) + ε K ( | λ | r c ) K ( | λ | r c ) = 0 . (5.4)12ote that the product k z r c = k n,s r c does not depend on r c and is a function of two parameters, ε /ε and β , k n,s r c = f ( ε /ε , β ) . (5.5)In order to evaluate the integral in (5.1) one needs to specify the integration contour nearthe poles k z = k n,s . In this section we will consider the spectral range where λ >
0. Thecorresponding eigenmodes k n,s are the guided modes of the dielectric cylinder. For those modesthe radial dependence of the Fourier components for the fields inside the cylinder is expressed interms of the Bessel function J n ( λ r ). In order to specify the contour, we note that in physicallyrealistic problems the permittivity ε has an imaginary part, ε = ε ′ + iε ′′ . We consider α n from (5.3) as a function of k z and ε , α n = α n ( k z , ε ). Note that in the presence of dispersionone has ε = ε ( ω ) = ε ( k z v ) and the second argument is a function of k z as well. Assumingthat | ε ′′ /ε ′ | ≪
1, the dominant contribution to the integral in (5.1) comes from the regionnear k z = k n,s , where k n,s is the s th root of the equation α n ( k z , ε ′ ) = 0. First we write α n ( k z , ε ) ≈ α n ( k z , ε ′ ) + iε ′′ ∂ ε ′ α n ( k z , ε ′ ) and then expand near k z = k n,s : α n ( k z , ε ) ≈ ∂ k z α n ( k z , ε ′ ) | k z = k n,s (cid:0) k z − k m,s + iε ′′ b n,s (cid:1) , (5.6)where b n,s = ∂ ε α n ( k z , ε ) ∂ k z α n ( k z , ε ) (cid:12)(cid:12)(cid:12)(cid:12) k z = k n,s ,ε = ε ′ . (5.7)Note that, though ε may depend on k z , the derivative ∂ ε α n ( k z , ε ) is taken for the fixed valueof k z . As for the denominator, ∂ k z α n ( k z , ε ) = ( d/dk z ) α n ( k z , ε ), in the presence of dispersion ε = ε ( k z v ) the derivative is taken with respect to both the arguments. From (5.6) we see thatthe pole of the integrand in (5.1) is located at k z = k m,s − iε ′′ b n,s . We have numerically checkedthat the numerator in (5.7) is negative for λ > b n,s is determined by the signof the denominator. The latter will be denoted as σ n,s = sgn( ∂ k z α n ( k z , ε ′ ) | k = k n,s ) = − sgn( b n,s ).By taking into account that ε ′′ ( ω ) > ω >
0, from here we conclude that for λ > k z = k n,s should be avoided from above for σ n,s < σ n,s > k z . The integrals over these semicircles are expressed interms of the corresponding residues. Returning to the case of real ε , ε = ε ′ , for the energyradiated per unit time we get I = ∞ X ′ n =0 X s I n,s = − v dWdz , (5.8)where the radiation intensity on the angular frequency ω n,s = vk n,s is given by I n,s = − δ n q vε q − β k z K n ( | λ | r ) V J,Kn J n ( λ r c ) r c | α ′ n ( k z ) | X p = ± J n + p ( λ r c ) V J,Kn + p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k z = k n,s . (5.9)Here, α ′ n ( k z ) = ∂ k z α n ( k z , ε ), δ = 1 / δ n = 1 for n = 1 , , . . . . This expression determinesthe radiation intensity on the guided modes of the dielectric waveguide. If the Cherenkovcondition for the surrounding medium is not satisfied the CR emitted inside the cylinder istotally reflected from the separating boundary.The dependence of the radiation intensity on the distance of the charge from the waveguideaxis enters through the function K n ( | λ | r ). For large values of r the intensity is exponentiallysmall. For large values of | λ | r c ≫
1, the intensity is suppressed by the factor e − | λ | ( r − r c ) .Hence, the guided modes of the waveguide are mainly radiated on the frequencies ω n,s . v p − β ( r − r c ) . (5.10)13able 1: The first eigenvalues for k z r c for different values of the azimuthal number n . n k n, r c n ≥ k n +1 , > k n, . In table 1 we present k n, r c for E e = 2 MeV, ε = 3 . ε = 1and for several values of n . As seen, for n ≥ k n, r c is of the order of n .Assuming that | λ j | r c ≫ n , the asymptotic expression for the roots k n,s is found by using theasymptotic formulas for the cylinder functions for large arguments: k n, l +1 r c ≈ p β − " nπ π − arctan ε ε s β − − β ! + πl , (5.11)where l ≫ k n, l < k n, l +1 is close to (5.11). For a given n , the frequency ω n,s of theguided mode increases with increasing s and the upper limit of the summation over s in (5.8) isdetermined from the Cherenkov condition vε ( ω n,s ) /c > k n,s per unit lengthof the charge trajectory: N n,s = I n,s ~ ω n,s v . (5.12)Figure 5 presents the number of the radiated quanta as a function of ω n,s r c /c for given n andfor different values of s . For the parameters we have taken E e = 2 MeV, ε = 3 . ε = 1, r /r c = 1 .
05. The left and right panels correspond to n = 1 and n = 2, respectively. As seen,for fixed n and started from s = 2 the roots k n,s come in pairs which are close to each other.The radiation intensity on the first root in the pair is much smaller than on the second one. Forexample N , /N , ≈ . N , /N , ≈ . n = 0is essentially smaller compared to the cases presented in figure 5. For the same values of theparameters one has ω , r c /c ≈ .
63 and r c N , ≈ . q / ( ~ c ). The corresponding results for n = 5 (circles), n = 10 (diamonds) and n = 20 (squares) are presented in figure 6. ω n,s r c / c ( ℏ c / q ) r c N n , s ω n,s r c / c ( ℏ c / q ) r c N n , s Figure 5: The number of quanta radiated on guided modes of the cylinder versus ω n,s r c /c for n = 1 (left panel) and n = 2 (right panel). The data are presented for E e = 2 MeV, ε = 3 . ε = 1, r /r c = 1 . ◆◆ ◆ ◆ ◆ ◆◆ ◆◆ ◆◆ ◆◆ ◆◆ ◆◆■ ■ ■ ■ ■ ■ ■ ■ ■ ω n,s r c / c ( ℏ c / q ) r c N n , s Figure 6: The same as in figure 5 for n = 5 , , In this section we consider the radiation on the modes of the dielectric cylinder with λ j < j = 0 ,
1, that correspond to surface polaritons. For the Fourier components of the fields with agiven n , the radial dependence is described by the function K n ( | λ | r ) in the region r > r c and bythe function I n ( | λ | r ) inside the cylinder, r < r c , and these modes correspond to surface waves.Depending on the electromagnetic properties of the contacting media, various types of surfacewaves can be excited on the separating boundary. Among them, motivated by wide applicationsin light-emitting devices, surface imaging, data storage, surface-enhanced Raman spectroscopy,biomedicine, plasmonic solar cells, etc., the surface plasmon polaritons have attracted a greatdeal of attention [21]. They are evanescent electromagnetic waves propagating along a metal-dielectric interface as a result of collective oscillations of electrons coupled to electromagneticfield. Among the most important properties of surface plasmon polaritons is the possibility forconcentration of the fields beyond the diffraction limit that enhances the local field strengthsby several orders of magnitude. Other types of active media instead of metals can also sup-port surface polariton modes. Examples are organic and inorganic dielectrics, ionic crystals,doped semiconductors and metamaterials [22]. An important advantage of these materials isthe possibility to control the parameters in the dispersion relations for dielectric permittivityand magnetic permeability. In particular, they can be used for the extension of plasmonics tothe infrared and terahertz frequency ranges.In the problem under consideration, the formula for the energy losses in the form of surfacepolaritons is obtained from (5.1) introducing instead of the functions J n ( λ r c ) and J n ± ( λ r c )the modified Bessel functions I n ( | λ | r c ) and I n ± ( | λ | r c ). Similar to the case of guided modes,we can see that for real k z the integrand is real and, hence, the only nonzero contribution to theintegral comes from the poles of the integrand. As before, the latter correspond to the zeros of15he function α n . In the case under consideration this function is written as α n = ε ε − ε + 12 X l = ± (cid:20) | λ || λ | I n + l ( | λ | r c ) K n ( | λ | r c ) I n ( | λ | r c ) K n + l ( | λ | r c ) (cid:21) − . (6.1)The equation α n = 0 determines the dispersion relation for the surface modes (see, for ex-ample, [24]). By taking into account that the term with the ratios of the modified Besselfunctions is always positive, we conclude that the equation may have solutions if and only if0 < / (1 − ε /ε ) <
1, or, equivalently, under the condition ε /ε <
0. Hence, in order to haveeigenmodes of the cylinder with λ < k n,s the eigenvalues for k z , being the roots of the equation α n = 0. Unlike the case of guided modes, because of monotonicity of the modified Besselfunctions, the equation α n = 0 for surface polaritons has a finite number of solutions. For agiven n we can have one or two roots. This feature is illustrated in figure 7 where the roots withrespect to k z r c are plotted versus ε for ε = 1 and for several values of the ratio v/c (numbersnear the curves). The dashed and full curves correspond to n = 0 and n = 1, respectively. Bytaking into account that the product k n,s r c depends on the parameters through the combinations ε /ε and β , we see that figure 7 describes the distribution of the roots for ε = 1 as well. Inthe limit k z r c → ∞ the curves tend to the limiting value ε = ε ( ∞ )0 which depends on the ratio v/c and does not depend on n . Below it will be shown that ε ( ∞ )0 = − ε − β . (6.2)As seen from the graphs, for n = 0 one has a single root in the region ε < ε ( ∞ )0 and there areno surface modes in the range ε > ε ( ∞ )0 . For n ≥ ε (m)0 ≤ ε < − ε (see the asymptotic analysis below), where the minimal value ε (m)0 depends on n and v/c . For ε close to the minimal value one has two roots, whereas in the remaining rangea single root exists. In the limit v/c → ε (m)0 → − ε and for v/c ≪ ε with the length of the order β .The distribution of the roots presented in figure 7 can be understood qualitatively consideringthe asymptotic behavior of the function α n from (6.1). For k z r c ≫ n + 1, assuming also that | λ j | r c ≫ n + 1, we get α n ≈ ε ε − ε + s − β − β ! − (cid:18) | λ | r c (cid:19) . (6.3)From here it follows that for the graphs in figure 7 one has ε → ε ( ∞ )0 ≡ − ε / (1 − β ) in thelimit k z r c → ∞ . Note that this asymptotic does not depend on n . In the opposite limit of small k z r c ≪
1, we get α ≈ ε ε − ε + 14 (cid:0) − β (cid:1) ( k z r c ) ln( k z r c ) ,α ≈ ε + ε ε − ε − (cid:0) − β (cid:1) ( k z r c ) ln( k z r c ) , (6.4)and α n ≈ ε + ε ε − ε + k z r c n − ε − ( n + 1) ε ] v /c n ( n − , (6.5)16 / c = - - - - z r c ε Figure 7: The localization of the eigenmodes of the cylinder with respect to k z r c for n = 0(dashed curves) and n = 1 (full curves). For the surrounding medium we have taken ε = 1 andthe numbers near the curves are the values of v/c .for n >
1. From these asymptotic expressions it follows that for the roots of the equation α = 0we have ε → −∞ in the limit k z r c →
0. This feature is seen in figure 7 (dashed curves). For n ≥ α n = 0 one has ε → − ε in the limit k z r c →
0. Again, this is confirmed by figure 7 (full curves).In considerations of surface polaritons the allowance for the dispersion of the dielectric per-mittivity of the cylinder, ε = ε ( ω ), is required. Among the most popular models used insurface plasmonics (see, for example, [21, 22]) is the Drude type dispersion ε ( ω ) = ε ∞ − ω p ω + iγω , (6.6)where ε ∞ is the background dielectric constant, ω p is the plasma frequency and γ is the char-acteristic collision frequency or the damping coefficient. The plasma frequency can be tunedchanging the carrier concentrations in the material. For example, in the terahertz range dopedsemiconductors are used. Alternatively, one can control the electromagnetic properties by usingartificially constructed materials.In the discussion below we will ignore the imaginary part in (6.6) assuming that the absorp-tion is small. In the corresponding model the surface polaritons are radiated in the spectralrange ω < ω p / √ ε ∞ . Let us consider the properties of those modes in the asymptotic regions ofthe dimensionless parameter ω p r c /v . For ω p r c /v ≪ | λ j | r c ≪ α we have theasymptotic expression (6.4). As it has been already mentioned, from that asymptotic it followsthat − ε ≫ ω/ω p ≪ n = 0 modes (for composite materials with with high negativepermittivity see, for example, [23] and references therein). For the dispersion (6.6) with γ = 0,from the asymptotic expression of α for the frequencies of n = 0 surface polaritons in the range ω p r c /v ≪ ωω p ≈ ( ω p r c /v ) − p − β exp (cid:20) − ε ( ω p r c /v ) − − β (cid:21) . (6.7)For the modes with n ≥ ω p r c /v ≪ | ε + ε | ≪
1. By taking into account (6.6), for the surface polariton modes with n ≥ ωω p → √ ε ∞ + ε , ω p r c /v → . (6.8)In the opposite limit ω p r c /v ≫ n + 1 one has | λ j | r c ≫ n + 1 and we can use the asymptoticexpression (6.3). From the equation α n = 0, in combination with ε /ε <
0, it follows that ε /ε ≈ − / (1 − β ). For the dispersion (6.6) this gives ω ω p ≈ ε ∞ + ε / (cid:0) − β (cid:1) , (6.9)in the asymptotic region ω p r c /v ≫ n + 1.In the left panel of figure 8, for dispersion law (6.6) with ε ∞ = 1 and γ = 0, we present thefrequencies for the eigenmodes of the cylinder as functions of the plasma frequency. The full anddashed curves correspond to n = 1 and n = 0 respectively. The numbers near the curves are thevalues of ratio v/c . The right panel of figure 8 presents the frequencies of the eigenmodes fordifferent values of n (numbers near the curves). From the data plotted in figure 8 we see thatfor n = 0 the frequencies of surface polaritons are in the range ω < ω p (cid:18) ε ∞ + ε − β (cid:19) − / . (6.10)For the frequencies of the modes with n ≥
1, in addition to the upper limit in (6.10) onehas a lower limit: ω ≥ ω ( m ) . The limiting frequency increases with increasing n and tends to ω p / p ε ∞ + ε / (1 − β ) for large values of n . / c = ω p r c / c ω / ω p n =
12 3 5 10 ω p r c / c ω / ω p Figure 8: Eigenfrequencies of the cylinder corresponding to surface polaritons versus ω p r c /c forthe dispersion law (6.6) with γ = 0 and for ε = 1. On the left panel the full and dashed curvescorrespond to the modes with n = 1 and n = 0, respectively, and the numbers near the curvesare the values of the ratio v/c . The graphs on the right panel are plotted for v/c = 0 . n .Having clarified the distribution of the eigenmodes we turn to the radiation intensity forsurface polaritons. Similar to the case of guided modes, in order to specify the integration contournear the poles of the integrand in (5.1), we introduce an imaginary part of the permittivity ε and use the expansion (5.6). The poles are located at k z = k n,s − iε ′′ b n,s , where b n,s is definedby (5.7). We have checked numerically that for λ < ∂ ε ′ α n ( k n,s , ε ′ ) >
0. From hereit follows that the poles k z = k n,s should be avoided from above for σ n,s > σ n,s < k z . The energy radiated per unit time ispresented as (5.8), where the radiation intensity for surface polaritons of the angular frequency ω n,s = vk n,s is expressed as I n,s = 2 δ n q vε q − β k z K n ( | λ | r ) V Kn I n ( | λ | r c ) r c | α ′ n ( k z ) | X p = ± I n + p ( | λ | r c ) V Kn + p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k z = k n,s , (6.11)where | λ j | = k z q − β j , V Fn = I n ( | λ | r c ) ∂ r c F n ( | λ | r c ) − F n ( | λ | r c ) ∂ r c I n ( | λ | r c ) , (6.12)for F = I, K . Note that one has V Kn <
0. Similar to the case of the guided modes, the radiationintensity is suppressed by the factor e − | λ | ( r − r c ) for the modes with | λ | r c ≫
1. Unlike theguided modes, there is no velocity threshold for the generation of surface polaritons.Let us consider asymptotic estimates of the radiation intensity for the dispersion relation(6.6) with γ = 0. In accordance with the analysis given above, in the limit v → ω → ω p / √ ε + ε ∞ . By taking into account that | λ j | r c ≈ ωr c /v , we see that the argumentsof the modified Bessel functions in (6.11) are large. By using the corresponding asymptoticexpressions we conclude that in the limit v → − ω p ( r − r c ) / ( v √ ε + ε ∞ )]. Now we turn to the behavior of the radiation intensityin the limiting regions of the combination ω p r c /v . In the region ω p r c /v ≪ n = 0 we get I ,s ∝ ( ω/ω p ) / ( ω p r c /v ) , where the ratio ω/ω p is given by (6.7). For the surfacemodes with n ≥ I n,s ≈ q vr c ( r c /r ) n ( ω p r c /v ) n ( ε ∞ + ε ) . (6.13)In the opposite limit, ω p r c /v ≫ n + 1, the radiation intensity is estimated as I n,s ≈ q ω p r ( ω/ω p ) − β / (cid:20) − r − r c ) ωv q − β (cid:21) , (6.14)with the radiation frequency from (6.9).In figure 9 we have displayed the number of the radiated quanta for surface polaritons as afunction of the frequency for the modes with n = 0 and for ε = 1, r /r c = 1 .
05. The numbersnear the curves are the values of the ratio v/c . Note that different frequencies correspond todifferent values of the permittivity ε . The value for ε corresponding to given frequency canbe found from the data depicted in figure 7. We see that the number of the radiated quanta islarge enough compared to the case of the radiation of guided modes.Here, a comment is in order. In the numerical evaluations corresponding to figure 9, for agiven value of ε , with fixed ε and v , we solve the equation α n = 0 with respect to k z r c . Atthis step, for a given ε , the specific form of the dispersion is not required. The latter is neededin the numerical evaluation of the radiation intensity. Indeed, the radiation intensity containsthe derivative α ′ n ( k z ). By taking into account the relation ω = k z v , in the expression for α ′ n ( k z )the derivative ∂ ω ε ( ω ) will enter coming from the terms in (6.1) with λ and from the first termin the right-hand side. Hence, for the evaluation of the radiation intensity on a given frequency ω , in addition to ε ( ω ), the value of the derivative ∂ ω ε ( ω ) is required. Plotting the graphsin figure 9 we have assumed that the dispersion is weak and the part of the derivative α ′ n ( k z )containing ∂ ω ε ( ω ) has been ignored. In the spectral range with ε < n = 1. In the absence of dispersion there exists a special value of ωr c /c (or equivalentlyof ε ) for which the derivative α ′ n ( k z ) becomes zero. This means that the corresponding point isa higher order pole of the integrand in (5.1). One of possible ways to regularize this singularityis to include the imaginary part of the permittivity ε ( ω ). Note that this kind of problem doesnot appear in the problem of radiation from a charge circulating around a cylinder, discussed in[25]. The reason is that in the latter problem, for a given n , the radiation frequency ν = n/T ,with T being the charge rotation period, and k z are independent variables. As a consequenceof this, for evaluation of α ′ n ( k z ) the derivative ∂ ω ε ( ω ) is not required and a given value of ε determine both the eigenvalues of k z and the radiation intensity. ω r c / c ( ℏ c / q ) r c N n , s Figure 9: The spectral distribution of the number of radiated surface polaritons on the modeswith n = 0 for a cylinder immersed in the vacuum. The numbers near the curves are the valuesfor v/c .Given the importance of dispersion in discussing the emission of surface polaritons, in figure10, for the dispersion law (6.6) with ε ∞ = 1, γ = 0, and for ε = 1, the number of theradiated quanta for surface polaritons is presented as a function of the frequency (in units of theplasma frequency) ω/ω p = ω n,s /ω p for n ∈ [0 , r /r c = 1 .
05 and ω p r c /c = 1. The plot markers circles and diamonds correspond to v/c = 0 . v/c = 0 . v/c = 0 .
75 and v/c = 0 . n , ω n,s < ω n +1 ,s . The same data for ω p r c /c = 5are presented in figure 11. As it already has been concluded from the asymptotic analysis, forlarge n the radiation frequencies tend to the value ω p / √ ε ∞ + ε (= ω p / √ v/c .In the discussion above we have considered the radiation from a single point charge. Thecorresponding results for the spectral density of the radiation intensity can be generalized fora bunch containing N q particles. Let us consider a simple case of the bunch with transversebeam size smaller than the radiation wavelength. The z -component of the current density ispresented as j (b)3 ( t, r ) = P N q m =1 j m ( t, r ), where the expression for the current density j m ( t, r )for the m th particle in the bunch is obtained from (2.2) by the replacement z → z − z m , with z m being the z -coordinate of the m th particle at the initial moment t = 0. The expressionsfor the Fourier components of the fields are obtained from the corresponding formulas givenabove for a single charge by adding the factor P N q m =1 e − ik z z m with k z = ω/v . In the expressionfor the radiation intensity the factor (cid:12)(cid:12)(cid:12)P N q m =1 e − ik z z m (cid:12)(cid:12)(cid:12) will appear. The double sum in this20 ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ω / ω p ( ℏ c / q ) r c N n , s ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ω / ω p ( ℏ c / q ) r c N n , s Figure 10: The number of the radiated quanta in the form of surface polaritons, as a function ofthe frequency, for different values of n ∈ [0 ,
20] (for the values of the parameters see the text). ◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ω / ω p ( ℏ c / q ) r c N n , s ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ω / ω p ( ℏ c / q ) r c N n , s Figure 11: The same as in figure 10 for ω p r c /c = 5.21odulus squared, P N q m,m ′ =1 , is decomposed into the incoherent contribution with m ′ = m andthe remaining coherent contribution. Introducing the longitudinal distribution function of thebunch f ( z ) in accordance with P N q m =1 R + ∞−∞ dz δ ( z − z n ) e − ik z z = N q R + ∞−∞ dz f ( z ) e − ik z z , we seethat the radiation intensity from a bunch is obtained from the formulas for a single charge byadding an additional geometrical factor N q h N q − | g ( k z ) | i , (6.15)where g ( k z ) = R + ∞−∞ dz f ( z ) e − ik z z . For a Guassian bunch one has f ( z ) = e − z / σ / ( √ πσ )and | g ( k z ) | = exp( − k z σ ). The second term in the squared brackets of (6.15) presents thecontribution of the coherent effects in the radiation intensity. We have investigated the radiation emitted by a charge uniformly moving outside a dielectriccylinder, parallel to its axis. The electric and magnetic fields are found for general cases ofdielectric permittivities of the cylinder and surrounding medium. First we have investigated thespectral density for the CR intensity in the exterior medium by evaluating the energy flux atlarge distances from the charge. The spectral density is given by (3.4) with functions f ( p ) n from(2.11). It has been shown that the influence of the cylinder on the CR is essentially different inthe cases ε < ε and ε > ε . The characteristic feature in the first case is presented in figure2 with relatively small oscillations of the spectral density of the radiation intensity around thevalue corresponding to the radiation in a homogeneous medium. For wavelengths much smallerthan the cylinder diameter the oscillations enter into quasiperiodic regime. These oscillationsresult from the interference of the direct CR, the CR reflected from the cylinder and also theCR formed inside the cylinder if the corresponding Cherenkov condition is obeyed. In the case ε > ε strong narrow peaks may appear in the spectral distribution of the radiation intensity.We have specified the conditions for the presence of those peaks. They come from the termsof the series in (3.4) with large values of n and are closely related to the eigenvalue equationfor the dielectric cylinder. The equation (3.13) that determines the spectral locations of thepeaks is obtained form the eigenvalue equation α n = 0 ignoring the exponentially small terms ofthe order | J n ( λ r c ) | /Y n ( λ r c ). Under the Cherenkov condition with the dielectric permittivityof the surrounding medium the eigenvalue equation has no solutions and the radiation modescorresponding to the strong peaks could be called as ”quasimodes” of the dielectric cylinder.The radiation on this types of the modes may also appear in the spectral range where ε < − ε .We have analytically estimated the heights and widths of the peaks by using the asymptoticexpressions for the cylinder functions for large arguments.If the Cherenkov condition for the exterior medium is not satisfied, depending on the spectralrange, two types of radiations may appear propagating inside the cylindrical waveguide. Theyhave discrete spectrum determined by the dispersion relation α n = 0. The corresponding fieldsexponentially decay as functions of the distance from the cylinder surface and they correspond toguided modes and to surface polaritons. For guided modes λ > J n ( λ r ) and the radiation intensity isgiven by (5.9). The lower threshold for the guided modes frequency increases with increasing n and the radiation frequency range is determined by (5.10).Unlike the guided modes, there is no velocity threshold for the emmision of surface polaritons.They are radiated in the spectral range where the dielectric permittivities of the cylinder andof the surrounding medium have opposite signs. The corresponding radial dependence of the22adiation fields inside the cylinder is described by the Bessel modified function I n ( | λ | r ) andthe radiation intensity on a given frequency is expressed as (6.11). The dispersion for surfacepolaritons is qualitatively different for the modes with n = 0 and n ≥
1. For n = 0 there is aupper threshold for the values of the permittivity ε (given by (6.2)): the eigenvalue equationhas a single root in the region ε < ε ( ∞ )0 and there are no surface modes in the range ε >ε ( ∞ )0 . In the case n ≥
1, a single or two surface modes exist in the finite range ε (m)0 ≤ ε < − ε with the lower threshold ε (m)0 depending on n and v/c . In the nonrelativistic limit ε (m)0 tends to − ε and the surface modes are present in the narrow range for ε with the lengthof the order β . For illustration of general results, as an example of dispersion for dielectricpermittivity of the cylinder we have considered Drude type model. In the limiting regionsof the dimensionless parameter ω p r c /v the frequencies of the surface modes are estimated by(6.7)-(6.9). The radiation intensities for surface polaritons in those regions are approximatedby (6.13) and (6.14). The spectral range of the generated surface polaritons becomes narrowerwith decreasing v/c . Having the fields and radiation intensity for a single charge, one can obtainthe corresponding result for a bunch of particles. In the simple case of the bunch with smalltransverse size, the effect of the bunch appears in the form of the geometrical factor (6.15)determined by the bunch longitudinal form-factor. Acknowledgement
The work has been supported by the grant No. 18T-1C397 of the Committee of Science of theMinistry of Education, Science, Culture and Sport RA.
References [1] J. V. Jelly,
Cerenkov Radiation and Its Applications (Pergamon, London, 1958); V. P.Zrelov,
Vavilov-Cherenkov Radiation in High-Energy Physics (Israel Program for Scien-tific Translations, Jerusalem, 1970); G.N. Afanasief,
Vavilov-Cherenkov and SynchrotronRadiation (Springer, Netherlands, 2004).[2] B. M. Bolotovskii, Usp. Fiz. Nauk , 201 (1957); Usp. Fiz. Nauk , 295 (1961) [Sov.Phys. Usp. , 781 (1961)].[3] R. Marqu´es, F. Mart´ın, and M. Sorolla, Metamaterials with Negative Parameters: Theory,Design, and Microwave Applications (John Wiley & Sons, Hoboken, NJ, 2008).[4] V. G. Veselago, Sov. Phys. Usp. , 509 (1968).[5] H. Chen and M. Chen, Materials Today , 34 (2011); Z. Su, B. Xiong, Y. Xu, Z. Cai, J. Yin,R. Peng, and Y. Liu, Adv. Optical Mater. 2019, 1801666 (DOI: 10.1002/adom.201801666).[6] J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B.-I. Wu, and J. A. Kong, Opt. Exp. ,723 (2003); S. Antipov, L. Spentzouris,W. Liu,W. Gai, and J. G. Power, J. Appl. Phys. ,034906 (2007); S. Antipov, L. Spentzouris, W. Gai, M. Conde, F. Franchini, R. Konecny,W. Liu, J. G. Power, Z. Yusof, and C. Jing, J. Appl. Phys. , 014901 (2008); S. Xi, H.Chen, T. Jiang, L. Ran, J. Huangfu, B.-I. Wu, J. A. Kong, and M. Chen, Phys. Rev. Lett. , 194801 (2009); Jin-Kyu So, Jong-Hyo Won, M. A. Sattorov, Seung-Ho Bak, Kyu-HaJang, Gun-Sik Park, D. S. Kim, and F. J. Garcia-Vidal, Appl. Phys. Lett. , 151107(2010); V. V. Vorobev and A. V. Tyukhtin, Phys. Rev. Lett. , 184801 (2012).237] E. Fernandes, S. I. Maslovski, and M. G. Silveirinha, Phys. Rev. B , 155107 (2012);S. Smirnov, Phys. Rev. B , 205301 (2013); X. Lu, M. A. Shapiro, and R. J. Temkin,Phys. Rev. Spec. Top. Accel. Beams , 081303 (2015); J. S. Hummelt, X. Lu, H. Xu,I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys. Rev. Lett. , 237701 (2016); Z.Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, Nat. Commun. ,14901 (2017); X. Lu, J. C. Stephens, I. Mastovsky, M. A. Shapiro, and R. J. Temkin, Phys.Plasmas , 023102 (2018); X. Lu, M. A. Shapiro, I. Mastovsky, R. J. Temkin, M. Conde, J.G. Power, J. Shao, E. E. Wisniewski, and Ch. Jing, Phys. Rev. Lett. , 014801 (2019); O.J. Franca, L. F. Urrutia, and O. Rodr´ıguez-Tzompantzi, Phys. Rev. D , 116020 (2019).[8] F. J. Garc´ıa de Abajo and A. Howie, Phys. Rev. B , 115418 (2002); F. J. Garc´ıa de Abajo,A. Rivacoba, N. Zabala, and N. Yamamoto, Phys. Rev. B , 155420 (2004); F. J. Garc´ıade Abajo, Rev. Mod. Phys. , 209 (2010); S. N. Galyamin and A. V. Tyukhtin, Phys. Rev.Lett. , 064802 (2014); E. S. Belonogaya, S. N. Galyamin, and A. V. Tyukhtin, J. Opt.Soc. Am. B , 649 (2015); A. Tyukhtin, V. Vorobev, E. Belonogaya, and S. Galyamin,JINST , C02033 (2018); A. P. Potylitsyn and S. Yu. Gogolev, Phys. Part. Nuclei Lett. , 127 (2019); A. V. Tyukhtin, S. N. Galyamin, and V. V. Vorobev, Phys. Rev. A ,023810 (2019).[9] J.E. Walsh, T.C. Marshall, and S.P. Shlesinger, Phys. Fluid. , 709 (1977); N. Zabala,A. Rivacoba, and P.M. Echenique, Surface Science , 465 (1989); K. L. Felch, K. O.Busby, R. W. Layman, D. Kapilow, and J. E. Walsh, Appl. Phys. Lett. , 601 (1998);P. Schoessow, M. E. Conde, W. Gai, R. Konecny, J. Power, and J. Simpson, J. Appl.Phys. , 663 (1998); Z. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, J. Appl. Phys. , 063303 (2008); G. Adamo, K. MacDonald, Y. Fu, C-M. Wang, D. Tsai, F. Garc´ıa deAbajo, and N. Zheludev, Phys. Rev. Lett. , 113901 (2009); A. M. Cook, R. Tikhoplav,S. Tochitsky, G. Travish, O. Williams, and J. Rosenzweig, Phys. Rev. Lett. , 095003(2009); G. Andonian, O. Williams, X. Wei, P. Niknejadi, E. Hemsing et al. Appl. Phys.Lett. , 202901 (2011); L. S. Grigoryan, H. F. Khachatryan, and S. R. Arzumanyan, NuovoCim. C , 317 (2011); I. V. Konoplev, A. J. MacLachlan, C. W. Robertson, A. W. Cross,and A. D. R. Phelps, Phys. Rev. A , 013826 (2011); S. Liu, M. Hu, Y. Zhang, W. Liu, P.Zhang, and J. Zhou, Phys. Rev. E , 066 609 (2011); L. S. Grigoryan, A. R. Mkrtchyan,H. F. Khachatryan, S. R. Arzumanyan, and W. Wagner, J. Phys. Conf. Ser. , 012004(2012).[10] S. N. Galyamin, A. V. Tyukhtin, S. Antipov, and S. S. Baturin, Opt. Exp. , 8902 (2014);T. Yu. Alekhina and A. V. Tyukhtin, Phys. Rev. Spec. Top. Accel. Beams , 091302(2012); A. Smirnov, Nucl. Instrum. Methods Phys. Res., Sect. A , 147 (2015); V. Bleko,P. Karataev, A. Konkov, K. Kruchinin, G. Naumenko, A. Potylitsyn, and T. Vaughan,J. Phys.: Conf. Ser. , 012006 (2016); S. N. Galyamin, A. V. Tyukhtin, and V. V.Vorobev, Nucl. Instrum. Methods Phys. Res., Sect. B , 144 (2017); S. N. Galyamin, A.V. Tyukhtin, V. V. Vorobev, A. A. Grigoreva, and A. S. Aryshev, Phys. Rev. Spec. Top.Accel. Beams , 012801 (2019); S. Jiang, W. Li, Z. He, R. Huang, Q. Jia, L. Wang, Y.Lu, Nucl. Instrum. Methods Phys. Res., Sect. A , 45 (2019); A. R. Mkrtchyan, L. S.Grigoryan, A. A. Saharian, A. H. Mkrtchyan, H. F. Khachatryan, and V.K. Kotanjyan,JINST , C06019 (2020).[11] Yu. O. Averkov and V. M. Yakovenko, Phys. Rev. B , 205110 (2005); M. I. Bakunov, M.V. Tsarev, and M. Hangyo, Opt. Express , 9323 (2009); I. V. Konoplev, L. Fisher, A. W.Cross, A. D. R. Phelps, K. Ronald, and C. W. Robertson, Appl. Phys. Lett. , 261 101(2010); V. S. Zuev, A. M. Leontovich, and V. V. Lidsky, JETP Letters , 115 (2010); V.24. Zuev, A. M. Leontovich, and V.V. Lidsky, Opt. Spectrosc. , 411 (2011); J. Tao, Q.J. Wang, J. Zhang, and Y. Luo, Scientific Reports , 30704 (2016); P. Kumar, R. Kumar,and S. K. Rajouria, J. Appl. Phys. , 223101 (2016).[12] S. Liu, P. Zhang, W. Liu, S. Gong, R. Zhong, Y. Zhang and M. Hu, Phys. Rev. Lett. ,153902 (2012); T. Zhao, R. Zhong, M. Hu, X. Chen, P. Zhang, S. Gong, and S. Liu, Eur.Phys. J. D , 120 (2015); T. Zhao, M. Hu, R. Zhong, S. Gong, C. Zhang, and S. Liu, Appl.Phys. Lett. , 231102 (2017); C. Yu and S. Liu, Appl. Phys. Lett. , 181106 (2019).[13] N. Zabala, E. Ogando, A. Rivacoba, and F. J. Garc´ıa de Abajo, Phys. Rev. B , 205410(2001); Y.-N. Wang and Z. L. Miˇskovi´c, Phys. Rev. A , 042904 (2002).[14] L. Sh. Grigoryan, A. S. Kotanjyan, and A. A. Saharian, Izv. Nats. Akad. Nauk Arm., Fiz. , 239 (1995) (Engl. Transl.: J. Contemp. Phys.).[15] A. S. Kotanjyan, H. F. Khachatryan, A. V. Petrosyan, and A. A. Saharian, Izv. Nats. Akad.Nauk Arm., Fiz. , (2000) (Engl. Transl.: J. Contemp. Phys.); A. S. Kotanjyan and A.A. Saharian, Izv. Nats. Akad. Nauk Arm., Fiz. , 135 (2002) (Engl. Transl.: J. Contemp.Phys.); A. S. Kotanjyan, Nucl. Instrum. Methods Phys. Res., Sect. B , 3 (2003); A. A.Saharian, A. S. Kotanjyan, L. Sh. Grigoryan, H. F. Khachatryan, and V. Kh. Kotanjyan,Int. J. Mod. Phys. B , 2050065 (2020).[16] A. A. Saharian and A. S. Kotanjyan, J. Phys. A: Math. Gen. , 4275 (2005); S. R.Arzumanyan, L. Sh. Grigoryan, H. F. Khachatryan, A. S. Kotanjyan, and A. A. Saharian,Nucl. Instrum. Methods Phys. Res., Sect. B , 3703 (2008); A. A. Saharian and A. S.Kotanjyan, J. Phys. A: Math. Gen. , 135402 (2009); A. S. Kotanjyan and A. A. Saharian,Nucl. Instrum. Methods Phys. Res., Sect. B , 177 (2013).[17] Handbook of Mathematical Functions , edited by M. Abramowitz and I. A. Stegun (Dover,New York, 1972).[18] M. V. Berry, J. Phys. A , 1952 (1975); M. V. Berry, Eur. J. Phys. , 045807 (2018).[19] L. Sh. Grigoryan, H. F. Khachatryan, S. R. Arzumanyan, and M. L. Grigoryan, Nucl.Instrum. Methods Phys. Res., Sect. B , 50 (2006); S. R. Arzumanyan, L. Sh. Grigoryan,H. F. Khachatryan, and M. L. Grigoryan, Nucl. Instrum. Methods Phys. Res., Sect. B ,2715 (2008); L. Sh. Grigoryan, A. A. Saharian, H. F. Khachatryan, M. L. Grigoryan, A. V.Sargsyan, and T. A. Petrosyan, JINST , C04035 (2020).[20] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).[21] V. M. Agranovich and D. L. Mills (Editors),
Surface Polaritons: Electromagnetic Waves atSurfaces and Interfaces (North-Holland Pub. Co., Amsterdam, 1982); S. A. Maier,
Plas-monics: Fundamentals and Applications (Springer, 2007); M. I. Stockman et al. , Roadmapon plasmonics, J. Optics , 043001 (2018).[22] P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, LaserPhotonics Rev. , 795 (2010); N. C. Lindquist, P. Nagpal, K.M. McPeak, D.J. Norris, andSang-Hyun Oh, Rep. Prog. Phys. , 036501 (2012).[23] X. Yao, X. Kou, and J. Qiu, Organic Electronics , 133 (2016); K. Sun, R. Fan, Y. Yin,J. Guo, X. Li, Y. Lei, L. An, C. Cheng, and Z. Guo, J. Phys. Chem. C , 7564 (2017).2524] J. C. Ashley and L. C. Emerson, Surf. Sci. , 615 (1974); H. Khosravi, D. R. Tilley, andR. Loudon, J. Opt. Soc. Am. A , 112 (1991).[25] A. S. Kotanjyan, A. R. Mkrtchyan, A. A. Saharian, and V. Kh. Kotanjyan, Phys. Rev.Spec. Top. Accel. Beams22