# Resonances in Reverse Vavilov-Cherenkov Radiation Produced by Electron Beam Passage over Periodic Interface

Authors: Gerard Granet; Petr Melezhik; Anatoliy Poyedinchuk; Seil Sautbekov; Yuriy Sirenko; Nataliya Yashina
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

Resonances in reverse Vavilov-Cherenkov radiation produced by the charged particles beam passage over periodic boundary of dispersive left-handed medium are found out and studied. Analysis and modeling are performed on the base of rigorous mathematical approaches. For the first time, several physical peculiarities owing to these effects are considered in the conditions of possible resonant scattering of electromagnetic waves.


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## Body

## 1. Introduction

Vavilov-Cherenkov radiation is appreciated as one of the greatest discoveries of the 20th century. The modern history of Vavilov-Cherenkov radiation (VChR) goes back to Cherenkov’s experiments, 1934–1937 [1]. Tamm and Frank developed this radiation theory in 1937 [2]. In 1958, the work was awarded a Nobel Prize in physics. Since then, VChR has been extensively used in physics and engineering [3]. As materials science and engineering advance, actual left-handed media providing qualitatively new characteristics in the implementation of well-known phenomena come and cause a fresh interest in VChR from both theoretical and applied standpoints.The study presented in the paper considers several still poorly understood peculiarities of the diffraction radiation [4] and, in particular, the direct and reverse VChR [4–7] produced by a beam of charged particles moving above the dispersive medium surface (generally periodic). Depending on frequency, this medium can be right-handed (conventional dielectric), mononegative (either permittivity or permeability is negative), or left-handed (both permittivity and permeability are negative). Emphasis is placed on the radiation characteristics in the conditions of possible resonant scattering of electromagnetic waves when the modulation period of the electron beam is comparable with the interface period and the excitation of eigenwaves supported by this boundary is possible. The study of resonant situations requires adequate research methods and tools that are based mainly on the idea of analytic regularization of boundary value problems in electromagnetic theory of gratings [8, 9].


## 2. Models and Characteristics

In free space (εg=μg≡1), the eigenfield E→ig,k,H→ig,k of electron beam with modulated density and the instantaneous value of the charge density given by the expression ρδz-aexpik/βy-kt represents the H-polarized field (∂/∂x≡0, Exi=Hyi=Hzi=0) and accordingly [4](1)Hxig,k=2πρβ·exp⁡ik2-kβ2z-a+kβy·z-az-a;z≠a,Eyig,k=-η0ik∂Hxig,k∂z,Ezig,k=η0ik∂Hxig,k∂y.Here δ(⋯) is Dirac’s δ-function; ρ and k are the amplitude and the modulation frequency of the electron beam and 0<β<1 is beam’s relative velocity; η0=μ0/ε01/2 is the impedance of free space; ε0 and μ0 are electric and magnetic constants of the vacuum; εg and μg are relative permittivity and permeability of the medium in the domain of wave propagation; g=y,z is a point in the space R2; time dependence of the harmonically oscillating fields is determined by the factor exp⁡-ikt. All physical quantities have the dimension of the SI, only “time” is measured in meters; it is the product of the real time on the speed of light in vacuum.Suppose-2πρβlexp⁡-ka-h1/β2-1=1. Then, eigenfield (1) of the electron beam passing in the conventional medium (vacuum) over the periodic boundary S (see Figure 1) of the dispersive medium with constitutive parameters defined by relations (2)εk=1-kε2k2,μk=1-kμ2k2generates in the domains z≥h and z≤0 the H-polarized field E→sg,k,H→sg,k with nonzero components represented as follows [8]: (3)Hxsg,k=∑n=-∞∞μnyRnkexp⁡iΓnz-h;z≥hTnkexp⁡-iΓnε,μz;z≤0,Eysg,k=-η0ikεg∂Hxsg,k∂z=η0k∑n=-∞∞μny·-RnkΓnexp⁡iΓnz-h;z≥hTnkΓnε,με-1kexp⁡-iΓnε,μz;z≤0,Ezsg,k=η0ikεg∂Hxsg,k∂y=η0k∑n=-∞∞Φnμny·Rnkexp⁡iΓnz-h;z≥hTnkε-1kexp⁡-iΓnε,μz;z≤0.Here, l and 0≤h<a are the period length and height of the corrugation mounts S=g:z=fy,0≤fy≤h; k=2π/λ and λ is the length of the electromagnetic waves in free space; kε and kμ are the characteristic frequencies of dispersive medium; μny=l-1/2expiΦny, Φn=2πn/l+Φ0, and Φ0=k/β; Γn=k2-Φn2 and ReΓn≥0, Im⁡Γn≥0. Signs of real and imaginary parts of the root Γnε,μ=k2εkμk-Φn2 are given so that all partial components UnTg,k=Tnkexp⁡-iΓnε,μzμny (spatial harmonics) of the field E→sg,k,H→sg,k in the domain z≤0 are the outgoing plane waves, that is, homogeneous waves (Im⁡Γnε,μ=0), transferring energy in the direction towards z=-∞ or the inhomogeneous waves (ReΓnε,μ=0), exponentially decaying when moving in the same direction.Figure 1
Geometry of the problem of diffraction radiation.Amplitude coefficientsRnk and Tnk are complicated function of frequency and geometric and constitutive parameters of the media of wave propagation. We find them by solving numerically the following boundary value problem [8] in the strip 0≤y≤l:(4a)∂2∂y2+∂2∂z2+k2εgμgHxg,k=0;0≤z≤h,(4b)Hx∂Hx∂yl,z,k=exp⁡iΦ0lHx∂Hx∂y0,z,k;0≤z≤h,(4c)Hxg,k=∑n=-∞∞μny·δ0nexp⁡-iΓnz-h+Rnkexp⁡iΓnz-h;z≥hTnkexp-iΓnε,μz;z≤0,(4d)Hxg,k,Etgg,tare  continuous  acrossSand  virtual  boundariesz=0,z=h.Here, δmn is the Kronecker symbol and Hxg,k is one of the three nonzero components of the total field (5)E→g,k,H→g,k=E→ig,k,H→ig,k+E→sg,k,H→sg,k;z≥hE→sg,k,H→sg,k;z<h.This problem allows determining (within the approximation of a given current) the electromagnetic field (field of diffraction radiation) generated by density modulated electron beam [4, 8]. The method, usually called as a method of analytic regularization [8, 9], was used for numerical solving of the problem of interest. The application of this method provided already the majority of the physical and applied results of the electromagnetic theory of gratings associated with resonant and anomalous spatial-frequency and spatial-time transformations of electromagnetic fields in periodic structures [8, 10–13].The energy characteristics of diffraction radiation are given by the relation [8] (6)∑n=-∞∞Rn2ReΓn+Tn2ReΓnε,με-1k=2Im⁡R0Im⁡Γ0,obtained by the application of the Poynting theorem for complex power to the field E→g,k,H→g,k within the domain 0≤y≤l×0≤z≤h. The term in the left part in (6) is the total electromagnetic energy W=W↑+W↓ radiated into the half-spaces z≥h and z≤0. In the approximation of the given current, it is determined by the expression 2Im⁡R0Im⁡Γ0. The values WnR=Rn2ReΓn and WnT=Tn2ReΓnε,με-1k characterize the distribution of the energy, lost by the electrons beam, between the channels open for radiation, that is, between the harmonics of the spatial spectrum such that ReΓn≥0 and/or ReΓnε,με-1k>0. The last inequality and the relationship RePyk=ε-1k∑n:ImΓnε,μ=0Tn2Φn (Pyk is the y-component of the Poynting vector P→k for field E→g,k,H→g,k in the plane z=0 averaged over the period l) allow determining unambiguously and quite rigorously the direction of the phase velocity of propagating in the half-space z>0 harmonic UnTg,k and the direction in which this harmonic carries the energy. In conventional media, these directions coincide and are given by the vector Φny→-Γnε,μz→, Γnε,μ>0. In binegative media Γnε,μ<0, the phase velocity is oriented along the vector Φny→-Γnε,μz→, and the energy transfer holds along the vector -Φny→+Γnε,μz→. In media with only one negative constitutive parameter the harmonics UnTg,k carrying the energy in the direction z=-∞ are forbidden and cannot be excited.ForE→ig,k,H→ig,k≡0 and k>0 fixed, the homogeneous (spectral) problem is obtained from (4a)–(4c). Its nontrivial solutions Hxg,Φ-0m exist for no more than a countable set of eigenvalues Φ-0mm∈F and define the field of eigenwaves E→g,Φ-0m,H→g,Φ-0m of the periodic interface [8, 10]. If several eigenvalues belong to the real axis ReΦ0 of the first physical sheet of the surface F (i.e., a Riemann surface onto which the solution of the problem (4a)–(4c) can be analytically continued from the domain of real values of the spectral parameter Φ0), then we are dealing with conventional surface (or regular) waves propagating near the interface without attenuation. Otherwise, leaky waves, piston-type waves, and so forth do exist [8].


## 3. Physical Results: Plane Boundary

A flat boundary separating a conventional environment (vacuum) and a nonabsorbing dispersion medium (the medium sort discussed below) is able to support direct (or forward) surface waves in the frequency range [10] (7)K2=kεkμkε2+kμ2<k<kε2=K1;kε>kμ.These waves correspond to the propagation constants Φ-0direct=kkε-1kε2-k2kε2-kμ2kε2-2k2-1 and velocity β=βdirect=k/Φ-0direct of synchronously moving electron beam.Within the range(8)kε2<k<kεkμkε2+kμ2;kε<kμ,this boundary can support backward waves having phase and group velocities oppositely directed and characterized by anomalous dispersion. For such waves, the velocity of the synchronously moving beam is β=βback=k/Φ-0back and Φ-0back=kkε-1kε2-k2kμ2-kε22k2-kε2-1.Figure2 shows εk and μk, propagation constants of eigensurface waves and velocity of electron beam synchronized with these waves versus frequency k. All these dependencies are presented within the ranges defined by (7) and (8). For kμ=0.5, kε=1.0, and 0.4472<k<0.7071, we have the forward waves (Figure 2(b)). For kμ=1.0, kε=0.5, and 0.3536<k<0.4472, the backward waves (Figure 2(a)).Figure 2
Curves ofεk, μk, Φ-0back, Φ-0direct, βback, and βdirect.
(a)
(b)Principal differences in the behavior of these characteristics for forward and backward surface waves are rather obvious. In the frequency rangeK1<k<K2 where the backward waves exist, the medium is binegative: εk<0 and μk<0. At the same time in the domain of the forward waves existence, the binegative medium transforms into a medium with εk<0 and μk>0. Naturally, these differences also affect the characteristics of the field excited by the electron beam.The VChR into the lower half-space filled with the dispersive medium is possible under the conditionβ2>εkμk-1, which can be satisfied only for the bipositive (conventional medium, direct VChR) or binegative (reverse VChR) media. The dispersion law given by (2) yields the following frequency restriction: (9)k<K0=2kεkμkε2+kμ2+kε2-kμ22+4kε2kμ2/β2.It can be shown that the inequality K0<kεkμ/kε2+kμ2 always holds if 0<β<1. And as β→0, then K0→0. As β→1, K0→kεkμ/kε2+kμ2. In Figure 2, the VChR domain is darkened.In the casekε<kμ, the frequency region of the VChR and the backward surface wave region overlap. On the contrary, when kε>kμ, the domain of the forward surface waves is beyond the domain of possible observation of the VChR.Ath=0, only zeroth spatial harmonics with amplitudes R0k and T0k remain in (4a)–(4c). The zeroth spatial harmonic in the dispersive material becomes propagating, that is, transferring the energy infinitely far in the direction z=-∞ provided Im⁡Γnε,μk=0, or what is the same, provided β2>εkμk-1, which is necessary for the VChR existence. Both the diffraction radiation and the VChR are associated with the transfer of electromagnetic energy in the same direction. Therefore, following [4], we identify the Vavilov-Cherenkov radiation with the diffraction radiation on the fundamental (zeroth) spatial harmonic of the periodic interface.Figure3 shows the contour plots W0Tk,β=const, describing the radiation intensity of VChR in the half-space z<0. Various fragments correspond to the media with different parameters εk and μk. And everywhere, as one would expect, the range of values k and β in which W0Tk,β>0 is bounded by curves P:β=εkμk-1/2 restricting the domain of VChR existence.Figure 3
Contour plots ofW0Tk,β=const, characterizing the radiation intensity in the half-space z<0: (a) kε=0.5 and kμ=1.0; (b) kε=1.0 and kμ=0.5; (c) ε=-2.24, μ=-1.24, and β=εμ-1/2≈0.6.
(a)
(b)
(c)In the casekε>kμ (Figure 3(b)), a maximum VChR intensity region is shifted towards higher β (0.75<β<0.95) and it is concentrated in the vicinity of the curve P. In the case kε<kμ (Figure 3(a)), a similar area is again located near P, but shifted towards 0.4<β<0.75. These behavioral features of energy characteristics are observed only in the dispersive medium. For a dispersion-free medium (Figure 3(c)) maxεμ-1/2<β<1⁡W0Tk,β is attained at any k, but β≈0.83 keeps unchanged. The value maxεμ-1/2<β<1⁡W0Tk,β grows with k.The contour plotsW0Tkε,kμ=const in Figure 4 give a fairly complete picture of the influence of the constitutive parameters of the dispersive medium with a plane boundary on the VChR energy characteristics. Interestingly, in the case of large k and β (in Figure 4(b)), the lines W0Tkε,kμ=const intersect with straight line kε=kμ (the most rapid variation of VChR intensity) at an almost right angle, while for smaller k and β (see Figure 4(a)) and kε>0.8 the passage along the line kε=kμ does not lead to a noticeable change in the VChR intensity.Figure 4
Contour plots forW0Tkε,kμ=const: (a) k=0.3 and β=0.3; (b) k=0.6 and β=0.6.
(a)
(b)


## 4. Physical Results: Periodically Rough Boundary

In the case of a periodically rough boundaryS (further on we assume l=2π), the radiation field in the domain z≥h (z<0) consists of spatial harmonics UnRg,k=Rnkexp⁡iΓnz-hμny(UnTg,k) propagating without attenuation. So, it is represented by harmonics with such n that Im⁡Γn=0 (Im⁡Γnε,μ=0). Obviously, when Φ0=k/β only harmonics with negative n can propagate without attenuation in the domain z≥h. In a dispersive medium, the range of k and β providing the propagation without decay of one or another spatial harmonic UnTg,k can be easily determined using the data on the configuration of their limits Qn:Γnε,μk,β=0 (see, e.g., Figure 5(a)).Figure 5
(a) Configuration of boundariesQn; (b) phase velocities of backward eigenwaves, kε=0.5 and kμ=1.0; (c) phase velocities of direct eigenwaves, kε=1.0 and kμ=0.5.
(a)
(b)
(c)A periodically rough boundary supports a larger number of surface waves. Their propagation constants in the frequency range of interest (k<1) and small h are given by the approximate relations Φ-0direct,±,m=±Φ-0direct+m and Φ-0back,±,m=±Φ-0back+m, m=0,±1,±2,… [8, 10]. Accordingly, we define the sets of approximate values βdirect,±,m=k/Φ-0direct,±,m and βback,±,m=k/Φ-0back,±,m (see Figures 5(b) and 5(c)). It should be pointed out that all the curves corresponding to velocities βdirect,±,m and βback,±,m practically merge near the values of the frequency parameter k=kaccum=kε/2, which for the given dispersion law (2) define accumulation point of the frequency spectra of the electrodynamic structure [13]. At this point, εk=-1 in the vicinity of this frequency, there are an infinite number of resonances in the intensity of diffraction radiation.Consider now the changes in the radiation field arising forh≠0 in that case when only zeroth spatial harmonic propagates in the lower half-space without decay. Figure 6 shows the frequency dependence of the intensity of radiation in the lower half-space for the electron beam velocity β=0.89. Over the whole frequency range considered here, we have εk<0 and μk<0. For parameters kε=1.0 and kμ=0.5 (see Figure 6(a)), a periodic boundary within the considered frequency range and for not very large h does not support propagation of surface waves, and W0Tk behaves almost in the same way as in the case h=0.Figure 6
Change in frequency characteristics of radiation intensity with an increase in the height of corrugation mounts of periodically rough boundary.
(a)
(b)For parameterskε=0.5 and kμ=1.0 (see Figure 6(b)), the periodic boundary supports the propagation of backward surface waves (see Figure 2(a)). This fact changes significantly the W0Tk behavior: at k≈0.4365 a certain resonance appears due to the excitation of the surface wave with eigenpropagation constant Φ-0back,-,1=-Φ-0back+1=0.49. The phase velocity of this wave is βback,-,1=k/Φ-0back,-,1=0.89 and coincides with the beam velocity β.Resonances in reverse Vavilov-Cherenkov radiation are possible in the region of parameter valuesk and β such that β2>εkμk-1 and ReΓnε,μk,β=0 for all n≠0 (only the fundamental spatial harmonic propagates in the dispersive medium without decay). It is also necessary that in this domain (in the domain VChk,β) the synchronism conditions β=βback,±,m=k/Φ-0back,±,m for certain m can be implemented.Naturally, the selected valueskε and kμ should ensure the existence of such a domain. For example, when 0.5<kε<kμ, the values β=βback,±,m=k/Φ-0back,±,m do not fall into the domain VChk,β; but for kε=0.1 and kμ=0.55 in the domain VChk,β, which is marked in Figure 7(b) with oblique hatching, the surface waves whose phase velocity coincides with the velocity of electron beam may already exist. The latter case is particularly interesting because (i) the resonances in the reverse VChR can appear for rather small values β (Figure 7(c)) and thus at low velocities of the particles beam and (ii) the accumulation point of the frequency spectrum kaccum=kε/2≈0.0707 falls into the region VChk,β and within it the existence of a large number of resonances in reverse VChR (Figures 7(a) and 7(c)) is possible. It should be pointed out once again that these resonances appeared due to the implementation of synchronism mechanism that is the coincidence of beam velocity with the phase velocity of one or another surface wave.Figure 7
(a), (c) DependenciesW-0Tk, characterizing the radiation intensity in the half-space z<0 for β=0.89 and β=0.2; (b) domain VChk,β. All for kε=0.1, kμ=0.55, and h=0.4. Here and further on W-0T⋯=W0T⋯/max⋯⁡W0T⋯.
(a)
(b)
(c)The data presented in Figure8 allow us to estimate the influence of parameters of the dispersive media kε and kμ on the intensity of reverse VChR. The lines W0Tkε,kμ=const (Figure 8(b)) are calculated for parameters kε and kμ from the domain 0≤kε≤0.12×0.35≤kμ≤0.7 and for k=0.08, β=0.2, and h=0.4. Under such choice of parameters, the region bounded by the curves Q0 and Q-1 is the domain VChkε,kμ. The first pronounced resonant fall in the intensity of the reverse VChR (the locus of corresponding points forms the straight line in the given range of variables kε and kμ) corresponds to the synchronism of electron beam with the surface wave having phase velocity Φ0back,-,1. Sequence of resonances is clearly displayed in Figure 8(a), where a plot W0Tkε for kμ=0.55 is presented. The low-Q resonant fall of intensity of the reverse VChR manifests itself when kε increases and then as you get closer to the value kε=0.113 at which the chosen frequency k=0.08 is an accumulation point. As we have mentioned above and in [13] in the vicinity of accumulation point, the number of high-Q resonances, causing a sharp drop and a sharp rise in W0Tkε, increases significantly.Figure 8
Influence of parameterskε and kμ onto the intensity of the radiation in the lower half-space: (a) plot W-0Tkε for kμ=0.55; (b) contour plots W-0Tkε,kμ=const.
(a)
(b)Influence of profiling depthh onto the intensity of VChR shows itself differently depending on the values of other parameters and, in fact, depending on whether a periodic boundary between two media at such values of parameters is able to support the propagation of forward or backward surface waves and whether the synchronization of electron beam velocity with phase velocity of these waves is possible.Consider periodic boundary of the nondispersive left-handed medium,μ=-4.84, ε=-0.84, and β=0.89, Figure 9. The resonance dip to zero of intensity of reverse VChR, which is clearly visible in Figure 9(b), is associated with the excitation of a surface wave, and the quality factor of the corresponding resonance is significantly reduced with the increase of h. Two other fragments of Figure 9 allow considering the important details in the behavior of the dependencies W0Tk (for some fixed values h) and W0Th (for some fixed values k) which may be lost in the analysis of the overall picture of the contour plots W0Tk,h=const (Figure 9(b)).Figure 9
(a), (c) PlotsW-0Tk for several fixed values h and plots W-0Th for several fixed values k; (b) contour plots W-0Tk,h=const. All for ε=-0.84, μ=-4.84, and β=0.89.
(a)
(b)
(c)In the case of the dispersive medium and the parameter valueskε, kμ, k, and β, guaranteeing the existence of direct surface waves, dependencies W0Th are almost monotonous. In the region of the backward surface waves, intensity of the reverse VChR varies essentially differently. For example, in the case of parameters already considered above kε=0.1, kμ=0.55, and β=0.89 (kcon≈0.0707), functions W0Th have a pronounced peak due to the coincidence of the electrons beam velocity with phase velocity of one of the surface waves; and this maximum occurs at different values of h for different values of k from the frequency interval comprising kaccum (Figure 10). Also, it is possible to characterize the behavior and features of W0Th calculated for fixed kμ=0.55, k=0.08, and β=0.2, and for different values kε (Figure 11). Location of peaks of function W0Th is quite sensitive to changes in kε value and this fact can be used to solve problems related to the definition of parameters of dispersive materials. The results similar to that shown in Figure 10 may be useful in the selection of depth profiling of left-handed dispersive medium capable of supporting over its surface the propagation of eigenwave of a required type.Figure 10
Curves ofW-0Th for several fixed values of k.Figure 11
Influence of profiling depth of periodic boundary between two media onto the intensity of VChR. PlotsW-0Th for various kε.


## 5. Conclusions

The resonances in reverse VChR produced by the charged particles beam passage over periodic boundary of dispersive left-handed medium are found out and studied.Mathematical modeling and computational experiment are the main methodological components of this work devoted to the study of a series of features in the diffraction radiation generated by a flat, density modulated electron beam, moving over periodically rough boundary between conventional media and media with frequency dependent parameters. Conditions of radiation initiation, radiation intensity, and possibility of the existence of different resonance effects in reverse VChR associated with the excitation of surface waves of the periodic boundary have been studied analytically and numerically in the approximation of a given current. The possibilities and perspectives for efficient control of the characteristics of the reverse VChR, that may be performed by changing, for example, characteristic frequencieskε and kμ in constitutive parameters of dispersive medium or/and the height of the corrugation mounts h of the periodic surface, are found out and examined.The numerical results discussed in the paper concern the limited number of specific situations. But the models and algorithms created for their analysis can be used for a more detailed research into the physics of diffraction radiation, for a correct formulation and efficient solution to a number of inverse problems having considerable practical interest.



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