Super collimation of the radiation by a point source in a uniaxial wire medium

We investigate the radiation properties of a short horizontal dipole embedded in a uniaxial wire medium. It is shown that the uniaxial wire medium enables a super-collimation of the dipole radiation such that the radiation pattern has a singularity and the radiated fields are non-diffractive in the broadside direction. We derive a closed analytical formula for the power radiated by the dipole. Our theory demonstrates that as a consequence of the ultrahigh density of photonic states of the nanowire array, the power radiated by the dipole is strongly enhanced as compared to that emitted in the dielectric host material.

Numerous theoretical and analytical methods were developed in the last decade to accurately characterize the effective electromagnetic response of wire media [3][4][5][6][7][8]. Such tools make possible, for instance, the study of the wave propagation in wire medium slabs [7,[27][28][29] and solving source-free spectral problems for the natural modes in closed analytical form [30][31][32]. However, the study of the problem of radiation by localized sources embedded in a wire medium background was somehow on the back burner for a long time. Only recently this subject has been investigated in more detail [33][34][35][36]. In particular, the radiation properties of a short vertical dipole embedded in a uniaxial wire medium were investigated in Ref. [35], using both a nonlocal framework [4,6] and a quasi-static approach relying on additional variables that describe the internal degrees of freedom of the medium [8]. The objective of this work is to further study the radiation problem of a short dipole embedded in a uniaxial wire medium. Specifically, we extend the analysis to the scenario wherein the dipole is horizontal with respect to the wires [see Fig. 1] rather than vertical as in Ref. [35].
Importantly, we demonstrate that for a horizontal dipole the radiated fields can be supercollimated by the nanowires leading to a super-directive emission along the axial direction. This paper is organized as follows. In Sec. II, we introduce the radiation problem under study. In Sec. III, we present the solution of the problem in the spectral domain using the nonlocal dielectric function approach. In Sec. IV, we derive a closed analytical formula for the power radiated by the short dipole in the metamaterial. Finally, in Sec. V the conclusions are drawn. Throughout this work we assume a time harmonic regime with time dependence of the form i t e   . Figure 1 illustrates the geometry of the problem under study: a square array of metallic wires embedded in a dielectric host. The spacing between the wires is a and the wire radius is w r . The excitation source is centered at the position ( , , )

II. THE RADIATION PROBLEM
x y z      r and corresponds to a short horizontal dipole described by the electric current density  is the angular frequency of oscillation, e p represents the electric dipole moment, and x is the unit vector along the positive x-axis. Our objective is to characterize the radiation emitted by the short dipole using an effective medium theory.
As a starting point, we remark that the use of effective medium methods requires that the source must be localized in a region with characteristic dimensions larger than the lattice constant a of the metamaterial [see Fig. 1], so that it is possible to assume that only waves can be excited in the metamaterial [35]. This property can be justified by the "uncertainty principle" of the Fourier transform, which establishes that the spreading of a function in the spatial and spectral domains is not independent, and the characteristic widths in the spatial ( x  ) and spectral (  It is known [4,6] that the uniaxial wire medium formed by straight metallic wires is characterized by the effective dielectric function where  denotes the tensor product of two vectors and h  is the relative permittivity of the host material. In case of perfect electrical conducting (PEC) wires the zz susceptibility is given by where p k is the plasma wave number of the wire medium [4,6], and is the wave number in the host region. The formula for zz  in the general case of lossy metallic wires can be easily obtained from Ref. [6]. Note that the effective dielectric function depends explicitly on / z k i z     .
Next, based on this nonlocal framework, we calculate the fields radiated by the horizontal dipole in the unbounded uniaxial wire medium and derive an explicit formula for the radiated power.

III. THE RADIATED FIELDS
The strategy to determine the emitted fields is to solve the radiation problem in the spectral domain (i.e., in the Fourier spatial domain) and then, calculate the inverse Fourier transform to obtain the field distributions in the spatial domain.
It is known [35] that in a spatially dispersive medium, the Maxwell equations may be written in the space domain as follows: The dyadic operator   , i     represents the effective dielectric function of the material and can be written explicitly as a convolution. In the spectral (Fourier) domain, we have the correspondence   , , Hence, after some straightforward manipulations of Eqs.
(2)-(3), one can find that the Fourier transform of the electric field satisfies [35]     x. From Eq. (4), it follows that: Yet, for the analytical developments of the next section it is more useful to work directly with Eq. (5).

A. PEC nanowires
In the following, we focus our analysis on the particular case wherein the nanowires are PEC. In such a case, we get from Eq. (5): Interestingly, it is possible to calculate explicitly the inverse Fourier transform of the electric field in z k . A straightforward analysis shows that the components of the electric field satisfy: waves that propagate in the uniaxial wire medium, and can be written in terms of an electric potential  as follows: The total radiated field has also contributions from the excited TE and TM waves. For long wavelengths, h p k k  , the contribution of the TM mode is negligible in the far-field region because its attenuation constant is very large [29]. On the other hand, the TE waves lead to spherical wavefronts, but it will be shown later that they only transport a small fraction of the total radiated power, and hence in practice they are of secondary importance. Moreover, it can be analytically shown that in the xoz plane the electric field associated with the TE mode decays as 2 1/ r along any observation direction, with the exception of the z-axis.
Notably, it may be verified that the field TEM E is singular along the z-axis such that x E diverges logarithmically as 0   . This unphysical behavior is due to the fact that in the continuous limit (when , x y k k     ) the density of photonic states of the wire medium diverges [20,22], and hence the metamaterial has infinite radiation channels leading to a spatial singularity of the radiated field. This problem can be easily fixed by introducing the spatial cut-off max / k a   such that the integration range in x k and y k is truncated to the first properly considered [20]. The truncation of x k and y k to the first Brillouin zone is also consistent with our assumption that the point source is less localized than the period of the wire medium.

B. Numerical example
To illustrate the non-diffractive nature of the radiation transported by the TEM waves, we represent in Fig. 2 the x-component of TEM E in the 0 y  plane (the E-plane) and in the 0 x  plane (the H-plane). As seen, the field is strongly confined to the z-axis and is guided away from the source without suffering any lateral spreading (no diffraction). This result is a consequence of the channeling properties of the uniaxial wire medium [15][16][17][18], which enable collimating the near field of a source. We also numerically computed the total radiated fields in the E-plane by numerically calculating the inverse Fourier transform of Eqs. (10)  In Fig. 3 we depict the field profiles of x E for different positions along the z-direction, calculated by numerically integrating Eq. (15) with no spatial cut-off (green dot-dashed curves) and with a spatial cut-off (blue solid curves). Figure 3 also shows the individual contribution of the TEM modes (without a spatial cut-off) given by the analytical formula (13) (pink solid curves). As seen, there is an overall good agreement between the three calculation methods, especially for the analytical and the numerical results calculated with no spatial cut-off. Only in the very near field some differences are discernible. This result confirms that the TEM waves determine, indeed, the main emission channel. In particular, the results show a strong confinement of the dipole radiation to the wires axis and the guiding of the emitted fields with no diffraction. As expected, with a spatial cut-off the field singularity along the z-axis disappears. Interestingly, the spatial cut-off also introduces some ripple in the field profiles due to the spatial averaging of the field singularities (in the absence of spatial cut-off) along the z-axis. It is important to highlight that imposing a wave vector cut-off is equivalent to perform a low-pass spatial filtering.
where 0  is the free-space impedance. The spatial derivatives are calculated using finite differences [38]. The result of Fig. 4(b) demonstrates that for h p k k  , the contribution of the TM mode is negligible in the far field. Note that the z E component depends exclusively on the TM modes, and is confined to the near-field region. In contrast, similar to the x E component, the y H field is also characterized by a diffraction-free beam pattern in the far field (see Fig. 4(c)).

IV. THE RADIATED POWER
Next, relying on the eigenwave expansion formalism introduced in Ref. [37], we derive a closed analytical formula for the power radiated by the short horizontal dipole inside the unbounded uniaxial wire medium [see Fig. 1]. To begin with, we present an overview of the eigenfunction expansion formalism. Then, we use this formalism to calculate the radiated power for PEC nanowires.

A. Overview of the eigenfunction expansion formalism
In Ref. [37], we derived a general analytical formulation that enables calculating the power emitted by moving sources in frequency dispersive lossless wire media. This formalism can be applied in a straightforward manner to standard non-moving sources, e.g. to an electric dipole. Specifically, using Eq. (11) of Ref. [37] and ext( ) r r x it is simple to check that the electric field E radiated by the point dipole in time-harmonic regime has the following exact modal expansion: Here x y z V L L L    is the volume of the region of interest, which in the end will be let to approach V   . The symbol "*" denotes complex conjugation. The summation in Eq. (17) 13 is over the electromagnetic (plane wave) modes of the bulk material   , n n E H n=1,2,...., which for dispersive media must be normalized as follows [37]: The frequencies n  are the real-valued eigenfrequencies of the natural modes. Importantly, the summation in Eq. (17) must include both the positive frequency and the negative frequency eigenmodes [37].
Since we are dealing with continuous media, it is clear that the eigenmodes where , n  k are the natural frequencies associated with the plane waves , n k E with wave vector k and index n (n determines the eigenmode type). As discussed in Ref. [37], the eigenmodes not contribute to the radiation field. However, they are mathematically important, since one cannot obtain a complete set of eigenfunctions without them [42].
In the continuous limit (V   ), the summation over k is replaced by an integral and Eq.
(19) becomes: where F is defined as before. Thus, the enhancement of the emitted power in the presence of the PEC nanowires is roughly proportional to the ratio of the density of states TEM P,diel. / D D .
Therefore, the anomalously high density of photonic states of the uniaxial wire medium implies indeed a strong enhancement of the power radiated by the dipole.  In Fig. 5 we show the dependence of the radiated power with the radius of the wires. The radiated power is larger for thicker wires, which can be understood by noting that the coupling efficiency between the point source and the nanowires is better for thick wires.

V. CONCLUSION
In this work, we have studied the radiation of a short horizontal electric dipole embedded in a uniaxial wire medium using an effective medium approach. It was shown that the radiation pattern of a short dipole inside a PEC wire medium corresponds to a non-diffractive beam, and thus the fields are super-collimated along the direction parallel to the nanowires.
Moreover, we derived a closed analytical expression for the power radiated by the dipole relying on an eigenfunction expansion [37]. It was demonstrated that, owing to a singularity in the density of photonic states of the uniaxial wire medium, the power radiated by the dipole is strongly enhanced as compared to the power emitted by the same dipole in the host dielectric. For realistic metallic wires the isofrequency contours will become slightly hyperbolic and hence the radiated beam is expected to be slightly divergent. Finally, we note that when the length of the wire medium is finite along the z-direction the supercollimated beam will create a sharp near-field distribution with subwavelength features at the interface with an air region. Only the spatial harmonics with || / k c   can be coupled to the propagating waves in free-space, and hence the rest of the energy will stay trapped in the wire medium slab.