Analytical solution for the stopping power of the Cherenkov radiation in a uniaxial nanowire material

We derive closed analytical formulae for the power emitted by moving charged particles in a uniaxial wire medium by means of an eigenfunction expansion. Our analytical expressions demonstrate that in the absence of material dispersion the stopping power of the uniaxial wire medium is proportional to the charges velocity, and that there is no velocity threshold for the Cherenkov emission. It is shown that the eigenfunction expansion formalism can be extended to the case of dispersive lossless media. Furthermore, in presence of material dispersion the optimal charge velocity that maximizes the emitted Cherenkov power may be less than the speed of light in vacuum.


I. INTRODUCTION
The Cherenkov effect [1][2] has been a topic of continuous interest and research owing to its many applications [3] particularly in particle detection in high energy physics [4], in the development of light sources [5][6][7], in spectroscopy of nanostructures [8], amongst others.
The Cherenkov effect occurs when a charged particle (e.g. an electron) propagates inside a dielectric medium with a velocity larger than the electromagnetic wave phase velocity ph / v c n = of the medium (c is the light speed in vacuum and n is the refractive index of the medium). A particle with velocity exceeding such a threshold gives rise to a conical wave front, being the emitted light launched along the forward direction ph arccos / v v θ = measured with respect to the particle velocity v.
Interestingly, another anomalous property of Cherenkov radiation made possible by metamaterials is the possibility of having Cherenkov radiation with no threshold for the charged particles velocity [17][18]24]. This remarkable property may be useful to improve the characteristics of free-electron lasers [17]. Such threshold-free Cherenkov emission occurs, for instance, in nanowire metamaterials formed by periodic arrays of parallel metallic nanorods [18]. Interestingly, a nanowire metamaterial structure enables the generation of nondivergent Cherenkov radiation [19][20], and the enhancement of the amount of emitted radiation [18]. It was numerically demonstrated in [18] that the stopping power (defined as the average energy loss of the particles per unit of path length) of a nanowire metamaterial can be more than two orders of magnitude larger than in natural media. This paper is organized as follows. In Sec. II, we characterize the wave dynamics in the uniaxial wire medium based on the quasi-static homogenization approach reported in [25].
Then, in Sec. III we derive closed analytical solutions for the stopping power due to the Cherenkov radiation in the two scenarios illustrated in Fig. 1, assuming that the wires are perfectly electrical conducting (PEC). In Sec. IV, we generalize the theoretical formalism to the case of lossless dispersive media. Finally, in Sec. IV the conclusions are drawn.
Throughout this work we assume that in the case of a time harmonic regime, the time dependence is of the form i t e ω − .

II. WAVE DYNAMICS IN UNIAXIAL WIRE MEDIA
In this section, we characterize the free oscillation modes of the electromagnetic field in a uniaxial wire medium [ Fig. 1] by reducing the problem to the calculation of the spectrum of a Hermitian operator. To this end, we rely on the quasi-static homogenization framework introduced in Ref. [25]. Within this approach, the electromagnetic response of the wire medium is expressed in terms of additional variables with known physical meaning, namely an additional potential ϕ related to the average electric potential drop from a given wire to the boundary of the associated unit cell and an additional current I that represents the electric current flowing along the wire [25]. The wave dynamics in the uniaxial wire medium [ Fig. 1] is described by an eight-component state vector ( , , , ) T I ϕ =

F E H
, which consists of the macroscopic electric and magnetic fields, the additional potential and the current. Assuming that the wires are PEC, the state vector satisfies a differential system of the form where L is a first-order linear differential operator (fully independent of the medium response), M is an 8×8 material matrix that describes the response of the wire medium and that depends solely on the geometry of the structure and on the electromagnetic properties of the involved materials. The explicit formulas for L and M can be found in Appendix A.
where x y z V L L L = × × is the volume of the region of interest. In the end, we take the limit V → ∞ . The symbol "*" denotes complex conjugation. Since M is positive definite, it is , and thus really defines an inner product in the space of eight-component vectors. Indeed, F F has the physical meaning of the stored energy normalized to the volume of the system [26]: where 0 ε is the electric permittivity of free-space, h ε is the relative permittivity of the host medium, 0 µ is the magnetic permeability of free-space, 2 c A a = , and w C and w L are the effective capacitance and inductance of the wires per unit length of a wire, respectively [25].
Importantly, it may be checked that the operator Since the set of eigenfunctions n F is complete, a generic eight-component vector F can be expanded as follows , n n n n n The effective medium is invariant to translations, and thus the dependence of the eigenfunctions on the spatial coordinates is of the form i e ⋅ k r . It is evident that for each wave vector k the eigenvalue problem (4) reduces to a standard 8×8 matrix eigensystem. Hence, there are eight different eigenwaves (eigenfunctions). Consistent with [27], it is found that the

III. STOPPING POWER OF UNIAXIAL WIRE MEDIA
Next, the theoretical formalism of Sec. II is used to obtain the power emitted due to the Cherenkov radiation by charged particles moving inside a uniaxial wire medium. In presence of an external source the electrodynamics of the problem is described by the system of equations: where ext J is an eight-component vector given by ext ext ( , ,0,0) T = J j 0 , where ext j is the electric current density and 0 is the zero vector. In the frequency domain Eq. (7) becomes ext, where ω F and ext,ω J are Fourier transforms in time. Expanding ω F as in Eq. (6) and taking into account that ˆn Thus, because of the orthogonality conditions (5) the coefficients n d must satisfy: Thus, we finally conclude that: The stopping power is given by 0 / P v , where 0 P is the total instantaneous power extracted from charged particles moving at a constant velocity [2]. Specifically, being E the total electric field that acts on the charged particles. It should be noted that as the charged particles are not accelerated, the self-field does not contribute to the stopping power [2]. Within the eight-component vector notation, 0 P can be expressed as In what follows, we obtain analytical expressions for the stopping power in the two scenarios illustrated in Fig. 1.

A. Array of charges moving inside the wire medium
Here, we consider the situation wherein a linear array of charged particles moves inside an Next, we write the eigenmodes n F in the form vector independent of r , k is the wave vector associated with the eigenmode, and the index m= TE, TM, TEM or LS, determines the electromagnetic mode type. In the continuous limit (V → ∞ ), the discrete summation in Eq. (13) must be replaced by an integration over k such Hence, it follows that: where m ω k are the resonant frequencies associated with the Floquet eigenmodes with wave vector k . Straightforward simplifications give the final result for ω F : We are now ready to determine the stopping power of the nanowire metamaterial.
Substituting the above formula into Eq. (12) and using ( ) where y L represents the width of the array of charged particles along the y-direction, so that y y y N n L = is the total number of moving charges. Using the identity [28]: where P.V. stands for the Cauchy principal value, we may rewrite Eq. (17) as follows: The first term is pure imaginary and hence the corresponding integral must vanish. Therefore, This formula shows that the natural modes that contribute to the Cherenkov radiation satisfy the selection rules The electric field associated with the TEM modes is of the form TEM, ,0   The property that stands out from Fig. 2 is the fact that there is no threshold for the velocity of the moving charged particles. Therefore, unlike usual dielectric materials, the uniaxial wire medium allows extracting power from the charges even when they are moving at relatively low velocities.
It can be also seen from Fig. 2 that, as the separation between the wires a is reduced, the magnitude of the stopping power increases. In fact, this could be expected from the analytical formula (23) because the plasma frequency is inversely proportional to the lattice spacing, p~1 / a β . Such an enhancement of the stopping power occurs because the photonic states density increases with the density of wires [18], leading to a boost of the number of available radiative channels. On the other hand, Fig. 2 shows that the value of the stopping power also increases with the radius of the wires. This happens because the coupling between the charges and the wires becomes stronger for larger radii.
Even though the main radiative channel in the uniaxial wire medium is related to the TEM mode, it is known [18] that the TM mode can provide a secondary radiative channel and thereby also contribute to the Cherenkov emission. However, the TM radiative channel only becomes available for velocities greater than the threshold h / v c ε > [18]. In particular, when the host medium is a vacuum ( h 1 ε = ), as considered in Fig. 2, the TM mode does not contribute to the stopping power, and Eq. (23) is exact. It may be checked that when h 1 ε > and the host material dispersion is ignored the contribution of the TM mode to the stopping power is infinitely large. Thus, if the host is not a vacuum it is essential to include the effects of material dispersion in the calculation of the stopping power. We discuss how the material dispersion can be taken into account in Sec. IV.

B. Single charge moving inside the wire medium
In what follows, we extend the study of Sec. III-A to the case wherein a single charged particle moves inside the nanowire structure along a straight line with constant y and z (namely, 0 y y = and 0 z z = ) [ Fig. 1(b)]. In this scenario, the current density may be written as Calculating the inverse Fourier transform in time, we obtain: Substituting this result into Eq. (12) and using the relation (18), we obtain the total instantaneous power extracted from the moving charge: Thus, the stopping power for a single moving charge is: Therefore, analogous to what happens in the scenario wherein an array of charges moves inside the nanowire material [see Eq. (23)], the stopping power is also here an increasing linear function of the velocity v of the charged particles. In Fig. 3 we depict the dependence of the stopping power on the velocity v of the charged particle for different structural parameters, calculated using Eq. (29). Similar to the case of a linear array of moving charges, it is seen that the stopping power increases as the separation between the wires a is reduced or the radius of the wires is enlarged.
It is interesting to note that the power extracted per charge in the case of the linear array is given by In particular, if the number of charges per cell is large one has / 1 p y n β << and consequently 0 0 single charge linear array y P P N << . Therefore, one sees that the interference between the fields emitted by a linear array of moving charges contributes to enhance the stopping power. The physical justification is that the interference of emitted fields suppresses radiation channels with 0 y k ≠ , and promotes the emission into the xoz plane which is a more efficient process ( 2 TEM, ,0⋅ k E x is maximum in the xoz plane).

IV. GENERALIZATION TO DISPERSIVE MEDIA
The analysis of Sec. III assumes that both the host medium and the nanowires are dispersionless. However, in practice the permittivity of realistic materials depends on frequency. In general, the effects of material dispersion are essential to obtain a finite emitted power. Next, we explain how the theory can be generalized in a straightforward manner to the case of lossless dispersive media.
It is well known that one can model the electromagnetic response of lossless dispersive dielectrics and metals using a Hermitian formulation (see [32]). Using such an approach, it is possible to describe the wave dynamics as in Eq. (7) Performing the integration in z k it is found that: where ( ) ( ) The key observation is that m m k k F F is precisely the stored electromagnetic energy of the system (per unit of volume), independent of the number of internal degrees of freedom of the pertinent dielectrics and metals. To prove this, we note that Eq. (7) implies that: where the inner product is defined as in Eq. (2). Because the operator 1L − M is Hermitian with respect to the considered inner product, it follows that: Therefore, it is possible to write: Noting that the extended excitation vector with 8 M + components must be of the form ext ( , , 0, 0,....) T ext = J j 0 we finally conclude that: The right-hand side of the above equation is precisely the power pumped into the system by the external electric current, and hence for a lossless system the left-hand side must be the time rate of the stored electromagnetic energy. Therefore, the stored electromagnetic energy is It is well known that the stored energy density in a medium described by the dielectric function ( ) ( ) This demonstrates that the normalization condition 1 m m = k k F F is equivalent to: ( ) In order to validate this generalized formulation, we consider now a wire medium formed by silver (Ag) nanowires. It is assumed that Ag follows the lossless Drude dispersion model ) [37]. The wire medium is characterized using the effective medium model described in Refs. [18,[33][34]. Similar to Sec. III-A, in this case the Cherenkov emission is only determined by the quasi-TEM mode. In Fig. 4, we show that the results obtained with our generalized theory and the numerical results of Ref. [18] (discrete symbols) agree perfectly. It is important to highlight that in presence of material dispersion the stopping power may not vary monotonically with the velocity (curve (i)), and that the optimal velocity value may be less than c (for curve (i) the optimal velocity is 0.75 v c = ).
In summary, we have demonstrated that the stopping power of a nanowire material formed by arbitrary dispersive lossless metals and dielectrics can be computed using Eq.

V. CONCLUSION
In this work, we derived closed analytical expressions for the stopping power associated with the Cherenkov emission of charged particles moving inside a uniaxial wire medium formed by PEC nanowires. Relying on an eigenwave expansion formalism, it was shown that in the absence of material dispersion the stopping power is an increasing linear function the charged particles velocity. In addition, it was explained how the theoretical framework can be generalized to dispersive media. The results are completely consistent with the numerical analysis reported in [18], and provide further physical insights of the mechanisms that enable the threshold-free Cherenkov radiation by the uniaxial wire medium.
The matrix M is Hermitian and real valued, i.e. † = M M .