We use the theory of inhomogeneous waves to study the transmission of light in μ-near-zero metamaterials. We find the effect of all-angle collimation of incident light, which means that the vector of energy flow in a wave transmitted to a μ-near-zero metamaterial is perpendicular to the interface for any incident angles if an incident wave is s-polarized. This effect is similar to the all-angle collimation of incident light recently found through a different theoretical framework in ε-near-zero metamaterials for a p-polarized incident wave [S. Feng, Phys. Rev. Lett. 108, 193904 (2012)]. To provide a specific example, we consider the transmission of light in a negative-index metamaterial in the spectral region with a permeability resonance, and show that all-angle collimation indeed takes place at the wavelength for which the real part of permeability is vanishingly small.
© 2013 OSA
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 , 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  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 . 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 : 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  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. .
2. Inhomogeneous waves in a lossy metamaterial
Since all-angle collimation of incident light is a consequence of losses in a medium  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 asEq. (2) the phase vector k′ is decomposed as a sum of two vectors, namely, p and q′q̂ (see Fig. 1). Vectors p = [q̂ × [k′ × q̂]] and q′q̂ are, respectively, parallel and perpendicular to the interface. The attenuation vector k″ is always normal to the interface. Thus vectors k′ and k″ are not parallel (the only exception is the case of normal incidence, p = 0) and the transmitted wave is inhomogeneous as mentioned earlier. The normal components of vectors k′ and k″ have magnitudes q′ = (k′ · q̂) and q″ = (k″ · q̂), respectively, which are given by  Equation (3) says that the normal component q′ of the phase vector k′ is positive (i. e., k′ is directed away from the interface) if ξ″ > 0, and negative (i. e., k′ is directed towards the interface) if ξ″ < 0. The latter case corresponds to negative refraction .
The refractive index m′ and the attenuation coefficient m″ are formally defined by and . In practice they can be found from the following equations:Eq. (4), both m′ and m″ depend on the incident angle θ0. The transmission angle θ is the angle between k′ and q̂, and can be found from the Snell’s law, m′ sin θ = n0 sin θ0.
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 = es + ep, where es = s−2(e · s)s and ep = s−2[s × [e × s]] = s−2(e · q̂)[s × k], or6, 7], i. e., Ap = 0 if the incident wave is s-polarized, and As = 0 if it is p-polarized. To find the decomposition of the complex magnetic vector amplitude h, we substitute Eq. (5) into the identity h = (μ0μω)−1[k × e], and obtain
Now, with the help of Eq. (1), we write a complex time-averaged Poynting vector as , 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, , whereEq. (7c), the cross-polarized component Ssp exists only in lossy media where the wave vector k is complex.
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 , whereEquation (8) says that the s- and p-polarized components of the Poynting vector are proportional to the sum of vectors k′ and k″, while the cross-polarized component Psp is proportional to vector s and thus normal to the plane of incidence. The component Psp is responsible for the transversal shift of the transmitted light beam. A similar shift named Imbert-Fedorov shift takes place for the reflected light beam for the case of total internal reflection [10, 11]. Despite the fact that these two shifts look similar, there are several different viewpoints on the component Psp: The authors of  argue that ”there is no mechanism for energy transport in the direction perpendicular to the plane of incidence” and set Psp equal to zero, based on the fact that ”the Poynting vector is defined only up to an arbitrary, additive, solenoidal vector”. Fedorov in his book  believes that this component is real and responsible for the light pressure in the direction perpendicular to the plane of incidence. The authors of  came to the conclusion that the appearance of Psp is caused by excitation of surface electric polariton mode or surface magnetic mode by the resonant or non-resonant manner. In this work, however, we only consider s-(Ap = 0) or p-polarized (As = 0) incident wave for which Psp = 0.
By inspecting Eq. (8), we find that both Ps and Pp 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 .
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 Bs and Bp (h = Bss + Bp[s × k]) instead of As and Ap. 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 ψs and ψp for s- and p-polarized components of the Poynting vector. Angles ψs and ψp are defined as the angles between vectors Ps and Pp, respectively, and the unit normal q̂. Consider, for example, the angle ψs. We can find this angle from the equation, tan ψs = |[Ps × q̂]|/(Ps · q̂). Using Eq. (8) and taking into account that [k′ × q̂] = s and k″ || q̂, we find that tan ψs = μ′|s|/(μ′q′ + μ″q″). Similarly we obtain tan ψp = ε′|s|/(ε′q′ + ε″q″). This form of equations for ψs and ψp was previously obtained in . Using the equations of , , and , we finally obtainEq. (9) says that the transmission angles ψs and ψp are not equal, which means that the direction of the energy flow in a lossy material is different for s- and p-polarized incident wave . However, for any natural material this difference is negligibly small, since usually μ″/μ′ ≪ 1 and ε″/ε′ ≪ 1, which, according to Eq. (9), means that ψs ≃ ψp ≃ θ. Nevertheless, the difference between ψs and ψp can be significant in metamaterials where the above inequalities may not hold.
To be more quantitative we calculate the values of ψs and ψp for the case in which the first medium is vacuum (n0 = 1) and the second medium is the negative-index metamaterial reported in . We retrieve the relevant parameters for the permittivity ε and permeability μ of this metamaterial, performing the parameter fitting for the Drude model . 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.
Using the retrieved functions for ε and μ, we calculate ψs and ψp 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 ψs and ψp on the wavelength λ and the incident angle θ0. In spite of negative refraction, we consider ψs and ψp 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 ψs 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 ψs and ψp. 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 . 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 , 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.
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 , 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.
This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education and Science of Japan. Part of the work by V. Yu. Fedorov was also supported by the Japan Society for the Promotion of Science (JSPS).
References and links
3. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef] [PubMed]
5. S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108,193904 (2012). [CrossRef]
6. F. I. Fedorov, Optics of Anisotropic Media (URSS, 2004), (in Russian).
7. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, 1983).
8. C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011). [CrossRef] [PubMed]
9. V. Yu. Fedorov and T. Nakajima, “Inhomogeneous waves in lossy metamaterials and negative refraction,” http://www.arxiv.org/abs/1305.6393.
10. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972). [CrossRef]
11. F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013). [CrossRef]
12. P. Halevi and A. Mendoza-Hernández, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. 71, 1238–1242 (1981). [CrossRef]
13. N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011). [CrossRef]