#### 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.

#### 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 (), the eigenfield of electron beam with modulated density and the instantaneous value of the charge density given by the expression represents the -polarized field (, ) and accordingly [4]Here is Dirac’s -function; and are the amplitude and the modulation frequency of the electron beam and is beam’s relative velocity; is the impedance of free space; and are electric and magnetic constants of the vacuum; and are relative permittivity and permeability of the medium in the domain of wave propagation; is a point in the space ; time dependence of the harmonically oscillating fields is determined by the factor . 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 . Then, eigenfield (1) of the electron beam passing in the conventional medium (vacuum) over the periodic boundary (see Figure 1) of the dispersive medium with constitutive parameters defined by relations generates in the domains and the -polarized field with nonzero components represented as follows [8]: Here, and are the period length and height of the corrugation mounts ; and is the length of the electromagnetic waves in free space; and are the characteristic frequencies of dispersive medium; , , and ; and , . Signs of real and imaginary parts of the root are given so that all partial components (spatial harmonics) of the field in the domain are the outgoing plane waves, that is, homogeneous waves (), transferring energy in the direction towards or the inhomogeneous waves (), exponentially decaying when moving in the same direction.

Amplitude coefficients and 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 :Here, is the Kronecker symbol and is one of the three nonzero components of the total field

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] obtained by the application of the Poynting theorem for complex power to the field within the domain . The term in the left part in (6) is the total electromagnetic energy radiated into the half-spaces and . In the approximation of the given current, it is determined by the expression . The values and 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 and/or . The last inequality and the relationship ( is the -component of the Poynting vector for field in the plane averaged over the period ) allow determining unambiguously and quite rigorously the direction of the phase velocity of propagating in the half-space harmonic and the direction in which this harmonic carries the energy. In conventional media, these directions coincide and are given by the vector , . In binegative media , the phase velocity is oriented along the vector , and the energy transfer holds along the vector . In media with only one negative constitutive parameter the harmonics carrying the energy in the direction are forbidden and cannot be excited.

For and fixed, the homogeneous (spectral) problem is obtained from (4a)–(4c). Its nontrivial solutions exist for no more than a countable set of eigenvalues and define the field of eigenwaves of the periodic interface [8, 10]. If several eigenvalues belong to the real axis of the first physical sheet of the surface (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 ), 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] These waves correspond to the propagation constants and velocity of synchronously moving electron beam.

Within the range 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 and .

Figure 2 shows and , propagation constants of eigensurface waves and velocity of electron beam synchronized with these waves versus frequency . All these dependencies are presented within the ranges defined by (7) and (8). For , , and , we have the forward waves (Figure 2(b)). For , , and , the backward waves (Figure 2(a)).

**(a)**

**(b)**

Principal differences in the behavior of these characteristics for forward and backward surface waves are rather obvious. In the frequency range where the backward waves exist, the medium is binegative: and . At the same time in the domain of the forward waves existence, the binegative medium transforms into a medium with and . 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 condition , 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: It can be shown that the inequality always holds if . And as , then . As , . In Figure 2, the VChR domain is darkened.

In the case , the frequency region of the VChR and the backward surface wave region overlap. On the contrary, when , the domain of the forward surface waves is beyond the domain of possible observation of the VChR.

At , only zeroth spatial harmonics with amplitudes and remain in (4a)–(4c). The zeroth spatial harmonic in the dispersive material becomes propagating, that is, transferring the energy infinitely far in the direction provided , or what is the same, provided , 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.

Figure 3 shows the contour plots , describing the radiation intensity of VChR in the half-space . Various fragments correspond to the media with different parameters and . And everywhere, as one would expect, the range of values and in which is bounded by curves restricting the domain of VChR existence.

**(a)**

**(b)**

**(c)**

In the case (Figure 3(b)), a maximum VChR intensity region is shifted towards higher () and it is concentrated in the vicinity of the curve . In the case (Figure 3(a)), a similar area is again located near , but shifted towards . These behavioral features of energy characteristics are observed only in the dispersive medium. For a dispersion-free medium (Figure 3(c)) is attained at any , but keeps unchanged. The value grows with .

The contour plots 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 and (in Figure 4(b)), the lines intersect with straight line (the most rapid variation of VChR intensity) at an almost right angle, while for smaller and (see Figure 4(a)) and the passage along the line does not lead to a noticeable change in the VChR intensity.

**(a)**

**(b)**

#### 4. Physical Results: Periodically Rough Boundary

In the case of a periodically rough boundary (further on we assume ), the radiation field in the domain () consists of spatial harmonics propagating without attenuation. So, it is represented by harmonics with such that (). Obviously, when only harmonics with negative can propagate without attenuation in the domain . In a dispersive medium, the range of and providing the propagation without decay of one or another spatial harmonic can be easily determined using the data on the configuration of their limits (see, e.g., Figure 5(a)).

**(a)**

**(b)**

**(c)**

A periodically rough boundary supports a larger number of surface waves. Their propagation constants in the frequency range of interest () and small are given by the approximate relations and , [8, 10]. Accordingly, we define the sets of approximate values and (see Figures 5(b) and 5(c)). It should be pointed out that all the curves corresponding to velocities and practically merge near the values of the frequency parameter , which for the given dispersion law (2) define accumulation point of the frequency spectra of the electrodynamic structure [13]. At this point, 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 for 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 . Over the whole frequency range considered here, we have and . For parameters and (see Figure 6(a)), a periodic boundary within the considered frequency range and for not very large does not support propagation of surface waves, and behaves almost in the same way as in the case .

**(a)**

**(b)**

For parameters and (see Figure 6(b)), the periodic boundary supports the propagation of backward surface waves (see Figure 2(a)). This fact changes significantly the behavior: at a certain resonance appears due to the excitation of the surface wave with eigenpropagation constant . The phase velocity of this wave is and coincides with the beam velocity .

Resonances in reverse Vavilov-Cherenkov radiation are possible in the region of parameter values and such that and for all (only the fundamental spatial harmonic propagates in the dispersive medium without decay). It is also necessary that in this domain (in the domain ) the synchronism conditions for certain can be implemented.

Naturally, the selected values and should ensure the existence of such a domain. For example, when , the values do not fall into the domain ; but for and in the domain , 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 (Figure 7(c)) and thus at low velocities of the particles beam and (ii) the accumulation point of the frequency spectrum falls into the region 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.

**(a)**

**(b)**

**(c)**

The data presented in Figure 8 allow us to estimate the influence of parameters of the dispersive media and on the intensity of reverse VChR. The lines (Figure 8(b)) are calculated for parameters and from the domain and for , , and . Under such choice of parameters, the region bounded by the curves and is the domain . 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 and ) corresponds to the synchronism of electron beam with the surface wave having phase velocity . Sequence of resonances is clearly displayed in Figure 8(a), where a plot for is presented. The low- resonant fall of intensity of the reverse VChR manifests itself when increases and then as you get closer to the value at which the chosen frequency is an accumulation point. As we have mentioned above and in [13] in the vicinity of accumulation point, the number of high- resonances, causing a sharp drop and a sharp rise in , increases significantly.

**(a)**

**(b)**

Influence of profiling depth 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, , , and , 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 . Two other fragments of Figure 9 allow considering the important details in the behavior of the dependencies (for some fixed values ) and (for some fixed values ) which may be lost in the analysis of the overall picture of the contour plots (Figure 9(b)).

**(a)**

**(b)**

**(c)**

In the case of the dispersive medium and the parameter values , , , and , guaranteeing the existence of direct surface waves, dependencies 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 , , and (), functions 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 for different values of from the frequency interval comprising (Figure 10). Also, it is possible to characterize the behavior and features of calculated for fixed , , and , and for different values (Figure 11). Location of peaks of function is quite sensitive to changes in 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.

#### 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 frequencies and in constitutive parameters of dispersive medium or/and the height of the corrugation mounts 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work has been carried out under financial support of the Ministry of Education and Science of Kazakhstan.