1. Introduction

Refraction of light is the fundamental optical phenomenon. Significant progress in fabrication of nanoscale structures led to creation of optical metamaterials which allow us to manipulate the way light refracts. For example, at the interface of negative-index metamaterials, the angle of refraction turns out to be negative [1, 2]. In [3], an array of optically thin resonators with subwavelength separation was used to modulate the phase of incident light along the interface. It was demonstrated that, depending on the designed phase gradient, the refraction angle can be controlled at will for any incident angle. The authors of [4] showed that a metamaterial with a near-to-zero refractive index acts as an antenna with an extremely high directivity — a source embedded in a slab of such metamaterial emits the waves whose refraction at the interfaces with the surrounding media causes concentration of the outgoing energy in a narrow cone. A phenomenon that is reverse to this directive emission was predicted by Feng in [5]. Feng showed that the direction of the incoming energy flow bends towards the interface normal for any incident angle when p-polarized (transverse magnetic) light enters an ε-near-zero metamaterial (the metamaterial with vanishingly small real part of permittivity). He clarified that such all-angle collimation of the incident light (in the original paper, term ”omnidirectional bending” was used) is a result of material losses.

In our work we show that similar all-angle collimation can be realized with s-polarized (transverse electric) incident light at the interface of metamaterials with a vanishingly small real part of permeability, so-called μ-near-zero metamaterials. Note, however, that the theoretical framework we use is different from the one employed by Feng [5]: To obtain our result we apply the theory of inhomogeneous waves which is commonly used to describe refraction of light in lossy media [6, 7]. Such approach allows us to generalize the result by Feng [5] and shed some light on its polarization dependence. We show that all-angle collimation of incident light in ε-near-zero and μ-near-zero metamaterials is the manifestation of the same phenomenon which takes place under different polarization conditions.

A vanishingly small real part of permeability can be found in negative-index metamaterials near permeability resonances which are used to achieve a negative index of refraction. To confirm our idea about all-angle collimation of incident light in μ-near-zero metamaterials, we calculate the transmission angle of the Poynting vector at the interface with the negative-index metamaterial recently reported by García-Meca et al. [8].

2. Inhomogeneous waves in a lossy metamaterial

Since all-angle collimation of incident light is a consequence of losses in a medium [5] and the propagation of light in lossy media differs from that in lossless media, we first summarize the basic features of light waves in lossy media [6, 7, 9]. Unlike the case of light waves in lossless media, the equiamplitude and equiphase planes of light waves in lossy media are not parallel, and such waves are called inhomogeneous waves. The summarized results in this section will be used in the following sections to determine the direction of the Poynting vector of the wave transmitted through a metamaterial.

We consider an interface between two isotropic media (see Fig. 1). The first medium is a lossless dielectric with a real refractive index n₀ and the second medium is a lossy metamaterial with complex permittivity ε = ε′ − iε″ and permeability μ = μ′ − iμ″. An incident plane wave with a real wave vector k₀ comes from the first medium. The incident angle θ₀ is an angle between k₀ and the unit vector normal to the interface q̂, which is pointing to the second medium. The complex electric and magnetic fields, E and H, respectively, of the transmitted wave are written as

 

Fig. 1 Refraction of the wave at the interface of a lossy material. The incident plane wave has a real wave vector k₀, while the wave vector k of the transmitted wave is complex due to material losses: k = k′ − ik″. In general, the phase k′ and attenuation k″ vectors are not parallel, and the transmitted wave is inhomogeneous.

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The refractive index m′ and the attenuation coefficient m″ are formally defined by $\left|{\mathbf{k}}^{\prime }\right|=m^{\prime }\frac{\omega }{c_0}$ and $\left|{\mathbf{k}}^{\prime }\right|=m^{″}\frac{\omega }{c_0}$. In practice they can be found from the following equations:

3. Poynting vector

In this section we derive the equation for the Poynting vector of the wave transmitted into a metamaterial. For this purpose we first consider the decomposition of the complex amplitude vectors e and h of the transmitted wave into the s- and p-polarized modes (TE and TM modes, respectively). Since the plane of incidence is a plane spanned by vectors k₀ and q̂, the vector s = [k₀ × q̂] = [k × q̂] is normal to the plane of incidence (see Fig. 1). Using vector s, we can decompose the complex electric vector amplitude e into the s- and p-polarized components: e = e_s + e_p, where e_s = s⁻²(e · s)s and e_p = s⁻²[s × [e × s]] = s⁻²(e · q̂)[s × k], or

Now, with the help of Eq. (1), we write a complex time-averaged Poynting vector as $\mathbf{S}=\frac{1}{2}\left[\mathbf{E}\times {\mathbf{H}}^{*}\right]=\frac{1}{2}\left[\mathbf{e}\times {\mathbf{h}}^{*}\right]\text{exp}\left[-2({\mathbf{k}}^{″}\cdot \mathbf{r})\right]$, where ”*” means complex conjugate. Substituting Eqs. (5) and (6) into the last expression, we find that vector S can be written as a sum of three components, $\mathbf{S}=\frac{1}{2}\left({\mathbf{S}}_s+{\mathbf{S}}_p+{\mathbf{S}}_{sp}\right)\text{exp}\left[-2({\mathbf{k}}^{″}\cdot \mathbf{r})\right]$, where

A real time-averaged Poynting vector P corresponds to the real part of S. Expanding the vector products and taking the real parts of Eq. (7), we find that $\mathbf{P}=\frac{1}{2}({\mathbf{P}}_s+{\mathbf{P}}_p+{\mathbf{P}}_{sp})\text{exp}\left[-2({\mathbf{k}}^{″}\cdot \mathbf{r})\right]$, where

By inspecting Eq. (8), we find that both P_s and P_p are parallel to k″ if μ′ or ε′ is equal to zero, respectively. Meanwhile, for any incident angle, the attenuation vector k″ is always normal to the interface [see Eq. (2)]. Therefore, we conclude that, at the interface of a material with a vanishingly small real part of permeability (μ′ = 0), the s-polarized incident wave gives rise to the transmitted wave whose energy flow is directed normally to the interface, irrespective of the incident angle. A similar argument holds for the p-polarized incident wave at the interface of a material with a vanishingly small real part of permittivity (ε′ = 0), as shown by Feng [5].

In conclusion we would like to note that it is possible to express the Poynting vector in Eq. (8) in terms of the s- and p-polarized components of the magnetic field H. The easiest way to do so is to apply the usual electromagnetic duality by replacing ε₀ and ε by μ₀ and μ, and using the corresponding amplitudes B_s and B_p (h = B_ss + B_p[s × k]) instead of A_s and A_p. After such procedures we will find that all-angle collimation in ε-near-zero and μ-near-zero metamaterials happens for s- and p-polarized magnetic field H, respectively, which is opposite to the case of E. However, this is not surprising, since vectors E and H are perpendicular in the incoming wave, and, for example, s-polarization in terms of E corresponds to p-polarization in terms of H.

4. Transmission angles

Now we consider the transmission angles ψ_s and ψ_p for s- and p-polarized components of the Poynting vector. Angles ψ_s and ψ_p are defined as the angles between vectors P_s and P_p, respectively, and the unit normal q̂. Consider, for example, the angle ψ_s. We can find this angle from the equation, tan ψ_s = |[P_s × q̂]|/(P_s · q̂). Using Eq. (8) and taking into account that [k′ × q̂] = s and k″ || q̂, we find that tan ψ_s = μ′|s|/(μ′q′ + μ″q″). Similarly we obtain tan ψ_p = ε′|s|/(ε′q′ + ε″q″). This form of equations for ψ_s and ψ_p was previously obtained in [12]. Using the equations of $\left|\mathbf{s}\right|=m^{\prime }\frac{\omega }{c_0}\text{sin}\theta$, $q^{\prime }=m^{\prime }\frac{\omega }{c_0}\text{cos}\theta$, and $q^{″}=m^{″}\frac{\omega }{c_0}$, we finally obtain

To be more quantitative we calculate the values of ψ_s and ψ_p for the case in which the first medium is vacuum (n₀ = 1) and the second medium is the negative-index metamaterial reported in [8]. We retrieve the relevant parameters for the permittivity ε and permeability μ of this metamaterial, performing the parameter fitting for the Drude model [14]. Figure 2(a) shows the dependence of the real and imaginary parts of the retrieved permeability μ on the wavelength λ. We see that this dependence has a resonance feature and the real part μ′ of the permeability is equal to zero at the wavelengths 732 and 768 nm. At these wavelengths we expect all-angle collimation for incident s-polarized wave.

 

Fig. 2 (a) Variation of the real μ′ and imaginary μ″ parts of the permeability μ as a function of wavelength λ. (b) and (c) The transmission angles ψ_s and ψ_p for the s- and p-polarized components of the Poynting vector as functions of wavelength λ and incident angle θ₀. All above functions are calculated for the interface between vacuum and the metamaterial reported in [8]. The spectral region of negative refraction is located between the two dotted vertical lines.

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Using the retrieved functions for ε and μ, we calculate ψ_s and ψ_p by Eq. (9) where values of m′, m″, and θ have been obtained using the equations in section 2. Figures 2(b) and 2(c) show the dependencies of the transmission angles ψ_s and ψ_p on the wavelength λ and the incident angle θ₀. In spite of negative refraction, we consider ψ_s and ψ_p as angles between two vectors and set them positive.

As expected, we see in Fig. 2(b) that, at wavelengths where μ′ = 0, the transmission angle ψ_s is zero for any incident angle. Therefore, the direction of energy flow in the second medium will be normal to the interface for any incident angle.

By comparing Figs. 2(b) and 2(c), we clearly see the difference between ψ_s and ψ_p. This difference is more significant at the wavelength λ = 732 nm where we have all-angle collimation for the s-polarized component of the Poynting vector. We hope that this observation will encourage experimentalists to verify the difference between the transmission angles for s- and p-polarized incident light.

Before proceeding to the conclusions, we would like to briefly discuss the case in which light propagates to the reverse direction, that is, from ε-near-zero or μ-near-zero lossy metamaterial into an ordinary medium. One may naively think that we will have a directive emission similar to the one in [4]. However, a simple generalization of our result to the reverse problem leads to an unphysical solution: In this work we have assumed that a homogeneous (the attenuation vector equals zero) plane wave comes from a lossless medium to the interface of a lossy metamaterial. Therefore, the most natural way to formulate the reverse problem is to assume that a homogeneous damped (the nonzero attenuation vector is parallel to the wave vector) plane wave comes from a lossy metamaterial to the interface with a lossless medium. This assumption implies that somewhere deep in the metamaterial there is an embedded light source, whose radiation, close to the interface, can be approximated by homogeneous damped waves. Upon arrival to the interface, the attenuation vector of these homogeneous damped waves will have a nonzero tangential component (the only exception is the waves propagating along the normal to the interface). Since the tangential components of attenuation vectors are continuous across any interface [9], the wave outgoing from the metamaterial will have nonzero attenuation vectors, that is, the wave transmitted into the lossless surrounding medium will be inhomogeneous. A closer examination shows that the attenuation vector of the outgoing wave will point towards the interface, which means that the amplitude of the outgoing wave will grow unlimitedly, which is unphysical. Therefore, we conclude that the radiation of a light source embedded in a lossy metamaterial can not be represented by homogeneous damped waves; to find a correct solution one should consider the mechanism of light emission from such source in more details, which is out of scope of this work.

5. Conclusions

In conclusion we have theoretically studied the transmission of light waves in μ-near-zero metamaterials. Similar to the case of ε-near-zero metamaterials [5], we have found the effect of all-angle collimation of incident light in μ-near-zero metamaterials for the s-polarized incident wave. Thus we have provided the generalized footing, based on which we can show, with sufficient clarity, that all-angle collimation of incident light in ε-near-zero and μ-near-zero metamaterials is the manifestation of the same phenomenon under different polarization of incident light. We have presented specific results for the negative-index metamaterial with a permeability resonance where the real part of the permeability becomes zero. Additionally, we have shown that the transmission angle of the Poynting vector depends on the polarization of the incident wave, and this difference is very significant in the spectral region where all-angle collimation of incident light takes place.