We analyze the dependence of the electromagnetic properties of wire array metamaterial media on the choice of metal, and identify promising material combinations for use in the near and mid infrared. We propose a figure of merit for the metal optical quality and consider it as a function of several parameters, such as material loss, wavelength of operation and wire diameter. Accordingly, we select promising material combinations, based on optical quality and fabrication compatibility, and simulate the loss of the quasi-TEM mode, for different wavelengths between 1 and 10 μm. We conclude that wire arrays are unlikely to deliver on their many promises at 1 μm, but should prove useful beyond 3 μm.
© 2015 Optical Society of America
Metamaterials are metal/dielectric composites that exhibit exotic optical properties due to their subwavelength structure. The type of structure varies according to the desired electromagnetic response. In particular, hyperbolic metamaterials are one of the most interesting classes, possessing extreme anisotropy and hyperbolic (or indefinite) dispersion, where a principal component of the permittivity or permeability tensors has the opposite sign to the other components . Wire arrays based on metal wires in a dielectric background behave as hyperbolic metamaterials for frequencies lower than an effective plasma frequency determined by the array’s geometry [1,2], allowing extraordinary waves with a hyperbolic dispersion relation and extremely large transverse spatial frequencies to propagate [3,4]. Consequently, they have many applications such as in high electromagnetic density of states materials [5–7], cloaking [8–10] and hyperlenses [11–13].
Metamaterial hyperlenses utilize the ability to propagate high spatial frequency waves to transport subwavelength information away from the object/source, overcoming the diffraction limit. Such high spatial frequencies containing subwavelength information are evanescent in conventional isotropic materials [14,15]. Furthermore, magnifying hyperlenses can magnify subwavelength features to dimensions within the diffraction limit. Hyperlenses based on wire array media have been demonstrated in the microwave spectrum [12,16,17], and were extended to the THz spectrum using fiber drawing techniques . Extending the operational wavelength range further to the infrared and visible is challenging due the subwavelength requirement of the structure and the losses of the metals at such frequencies.
The fiber drawing technique is a scalable and inexpensive method for the fabrication of subwavelength structures, particularly applicable to wire arrays and high taper ratio wire arrays as required for hyperlenses and magnifying hyperlenses. However, the scaling down of a metallic wire array metamaterial fiber is limited by rheological properties of the materials . Due to the Plateau-Rayleigh instability , the surface tension between the molten metal and the dielectric during drawing can distort the structure or break the wires into droplets . This instability has to date limited the application of this technique to metamaterials for wavelengths longer than 20 μm. However, considering the size of wires that have been drawn with this technique [21,22], it should be possible to produce subwavelength wire array metamaterial fibers for the infrared and visible using glass as the dielectric medium.
In this paper, we identify promising metal/dielectric combinations for the fabrication of wire array metamaterial fibers for the near and mid-infrared (NIR and MIR, respectively). Firstly, in Section 2, we propose a figure of merit for the optical quality of metals specific to wire arrays. Based on optical quality and fabrication feasibility, we identify interesting material combinations for operation around 1, 3 and 10 μm wavelengths. In Section 3, we extend our loss study to parameters such as wire diameter (d), operational wavelength (λ) and the index of the dielectric host (nd). Finally, in Section 4, we study the loss of high-spatial frequency quasi-TEM modes for hexagonal wire array structures with specific material combinations, varying wire diameter, wire spacing and index of the dielectric, in the NIR and MIR.
2. Bulk metal losses and figure of merit
The optical loss of a wave reflecting from, or propagating along, a bulk metal is determined by the imaginary and real part of the metal’s permittivity (εi and εr), which vary with free space wavelength (λ). The dielectric constant εi is related to the metal’s absorption, while the εr is related to the penetration of the wave in the bulk metal. Figures 1(a) and 1(b) show experimental values of εi and εr, respectively, for several metals in the NIR and MIR (data from Refs [23,24].). For all the metals presented, except Bi, εi and the absolute value of εr increases with λ. In a wire array, or even in a single metal wire plasmonic waveguide, εi is also not sufficient to estimate propagation loss of waves in these structures. The optical loss of each mode is also strongly related to the energy distribution between the metal and the surrounding dielectric, meaning that the real part of permittivity (εr) of both materials and the geometry also influence the loss.
An appropriate figure of merit for this type of structure is the loss of the fundamental TM mode in a single metal wire waveguide. While these losses also depend on the wire diameter used, setting a nominal diameter enables comparison between metals. The mode condition equation for the 0th order TM mode of a single wire can be found analytically as ,Eq. (1) can be found numerically and provides the neff of the TM mode, which can be used to calculate the loss in dB/μm using
Figures 1(c) and 1(d) illustrate the loss of the lowest order TM mode calculated using Eqs. (1)-(3), for wire diameters of 250 nm and 500 nm, respectively, embedded in vacuum, as a function of wavelength. The representative metals were chosen for their good optical properties (Au, Ag, Al, Cu) or their convenience for fabrication due to availability and relatively low melting temperature (Bi, Sn, In). As expected, Figs. 1(c) and 1(d) show that the absolute value of the loss changes with wire diameter but the relative loss of the different metals is not qualitatively affected. Consequently, the loss for a nominal diameter is a good figure of merit for a metal’s optical quality. As such, Figs. 1(c) and 1(d) show that loss decreases for longer wavelengths and Au, Ag, Al and Cu are clearly the best metals as far as loss is concerned for the wavelength range between 1 to 10 μm.
Apart from optical loss, fabrication feasibility is another extremely important criterion in the selection of materials for wire array metamaterial media. When fiber drawing is used, the co-drawing of metal and glass is only possible if the materials are chemically compatible and the rheological properties match. The components must not react and the metal must be liquid at the drawing temperature. In addition, the potential formation of solid oxides during drawing can be extremely detrimental to the fabrication.
Considering Fig. 1(c), Al is the best metal in terms of optical quality over almost all the wavelength range considered. However, experimental tests with common silica-based glasses (borosilicate, soda-lime and SiO2) showed high reactivity resulting in the production of Si, through the reaction [26–28]29], however its strong tendency to oxidize when molten can be a problem, especially during preform fabrication. According to our figure of merit, both Au and Ag have similar and high optical quality, but Ag presents higher reactivity. Therefore, among the metals with lowest losses, Au remains the best option once compatibility with the glass drawing process is considered.
The second group of metals, having poorer optical quality, is considered for practicality. Of these, Sn presents low oxidation when molten and high compatibility with common glasses, especially with soda-lime and borosilicate. This compatibility is commercially employed in the fabrication of float soda-lime and borosilicate by the Pilkington process or microfloat process [30–32]. This feature, combined with low cost and low melting point, motivates a deeper analysis of its use in drawn wire arrays. We thus focus further analysis on Au and Sn, as cases of high optical quality (Au) and practicality (Sn). Figures 1(c) and 1(d) show that losses for wire arrays using other interesting metals (Ag and Cu) are expected to lie between these two cases.
In the next section, we extend our loss study of the 0th order TM mode of the single wire waveguide to other parameters such as wire diameter (d), wavelength, and the refractive index of the dielectric host (nd), before studying losses of the full wire array in Section 4.
3. Single metal wire waveguide
The loss of the TM mode of a single metal wire depends on the wire diameter, the wavelength and the index of the surrounding dielectric. Figures 2(a) and 2(b) present this loss for Au and Sn wires with several diameters (from 10 nm to 2.5 μm), in the wavelength region between 1 to 10 μm, calculated with Eqs. (1)-(3) using the complex permittivity of the metals from Figs. 1(a) and 1(b). Vacuum is selected as the dielectric medium in order to analyze, in the first instance, the influence of the metal in isolation.
Figure 2 shows that the losses are higher for the Sn wires, as expected from the figure of merit and because Sn has larger εi (Fig. 1(a)). In addition, the optical losses decrease with longer wavelength and larger wire diameter. This is somewhat counterintuitive in that εi increases with wavelength (Fig. 1(a)), and one could naively expect losses to diminish with wire diameter, as the amount of metal is reduced. In order to clarify this behavior, we calculate the field distribution in the metal.
The fractional energy in the metal is defined as33,34]
Figure 3(a) illustrates how fractional energy in the wire varies with wire diameter for Au and Sn wires embedded in vacuum, for a fixed wavelength of 3 μm. As seen in Fig. 3(a), smaller wire diameters lead to an extremely confined mode, which increases loss . This behavior is consistent with Figs. 1(c) and 1(d). We note that we use the bulk permittivity of the metals for this calculation, which becomes increasingly inapplicable for very thin wires with diameters below tens of nanometers [35,36]. Thus, these results should be treated as an approximation.
The fractional energy analysis cannot be used to explain the wavelength dependence of loss as it does not take into account the dispersion and group velocity. From perturbation theory , the difference in the imaginary part of the mode effective index for the TM mode of a single wire with and without the metallic loss can be expressed to first order asFigure 3(b) shows the factor Δκ/λ, which is proportional to the loss per unit length, for 250 nm Au and Sn wires in vacuum, as a function of wavelength. This is consistent with the loss calculated directly from the complex mode effective index in Figs. 1(c) and 1(d), showing again that loss decreases for longer wavelengths and that Sn is lossier than Au.
The refractive index of the dielectric surrounding the metal also influences the mode energy distribution. For this reason, for a specific material combination, any loss analysis as a function of wavelength must include the dispersion of the complex dielectric index. However, for a fixed wavelength, it is interesting to understand how the loss behaves if only nd varies, which simulates a change of the dielectric. Figure 4 illustrates the loss of the Au and Sn wire with d = 250 nm, at λ = 3 μm, for nd varying from 1 to 3. This range covers the indices of SiO2 (1.41925 ), soda-lime (1.4849 ), borosilicate (around 1.5), and some chalcogenide glasses (up to 3 ) commonly used for the wavelengths in the range of 1-10 µm. In order to simplify the analysis, the loss of the dielectric was omitted, as with appropriate choice of dielectrics the loss of the metal should dominate.
As seen in Fig. 4, the loss and fractional energy in the metal increase for higher dielectric index nd. Consequently, the selection of the dielectric must not only consider its transparency but also the real part of its refractive index. As the loss of the metal is usually much higher than that of the dielectric, the selection of an optically poorer dielectric could result in a lower modal loss if the real part of the dielectric index is low, through reducing the fractional energy in the metal.
In the next section, we study the loss of the high-spatial frequency quasi-TEM mode for the full hexagonal wire array metamaterial media. We propose some material combinations for operational wavelengths in the NIR and MIR, and consider the effect of varying structure parameters such as wire diameter and wire spacing (Λ).
4. Indefinite wire array metamaterial media
The indefinite wire array metamaterial media constitutes a spatially dispersive hyperbolic medium for frequencies below the effective plasma frequency of the structure , and has three types of modes: TE, TM and quasi-TEM (a second TM mode) . When this hyperbolic metamaterial is considered for super resolution imaging, it is the propagation of high spatial frequency modes that is most important, as these contain the subwavelength information usually restricted to the near field. In the absence of spatial dispersion, the extraordinary mode of hyperbolic media has the dispersion relation [1,3,4,18]Eq. (8) is hyperbolic, and has no high spatial frequency cut-off, enabling sub-diffraction imaging. However, because of spatial dispersion not one but two extraordinary modes exist (the TM and quasi–TEM), and isofrequency surfaces are best calculated numerically. Figure 5(a) shows isofrequency curves for the modes in a hexagonal wire array shown in Fig. 5(b), based on Sn and soda-lime, with d = 100 nm, wire spacing Λ = 600 nm and λ = 3 μm, calculated using a commercial finite element solver (COMSOL). As can be seen, the quasi-TEM mode (black dots) allows propagation of high transverse spatial frequencies with (k⊥/k0)>nd, which can enable imaging beyond the diffraction limit. Figure 5(c) shows an example of a calculated quasi-TEM mode with high spatial frequency, where the white arrows represent the E field distribution in the xy plane and the color scale represents the normalized time averaged energy flow in the wire direction.
From homogenization theory , the TE mode (red dots) in this structure does not have a hyperbolic dispersion of the form of Eq. (8) because its polarization is transverse to the anisotropy axis, corresponding to an ordinary wave. As a result, the wire array behaves like a dielectric and the dispersion relation is given by Fig. 5(a). The TE mode is evanescent for (k⊥/k0)>nd, which corresponds to k⊥/k0>1.4849 in Fig. 5(a). The dispersion relation of the TM modes (blue dots), can be approximated from spatially dispersive homogenization models as 19,42]. According to Eq. (10), the TM mode is evanescent for all k⊥ when kp2>εd*(ω/c)2, a condition that is fulfilled in the structure of Fig. 5.
The wire array is more complex than the single wire waveguide as the propagation loss also depends on the distance between the wires Λ and the mode field distribution, in addition to the other parameters already discussed. In the following subsections, we discuss some possible material combinations for wire arrays in the MIR and NIR and analyze their loss for a range of structural parameters. Any effect relying on the hyperbolic nature of the material (e.g. density of states enhancement, sub-diffraction imaging), will require low loss for high transverse spatial frequencies (k⊥/k0)>nd.
We used COMSOL to calculate the isofrequency curves, loss and modal fields of the wire array with a hexagonal lattice as in Fig. 5(b), using Bloch-Floquet boundary conditions. From the complex effective index (neff) of the simulated quasi-TEM mode, Eq. (3) is used to calculate the propagation loss. For each specific structure and material combination, two cases of the quasi-TEM mode were considered, k⊥ = 0 being the lowest transverse spatial frequency, corresponding to k oriented along the wires, and k⊥ = (π/Λ)*(2/√3) = k⊥max which corresponds to the edge of the Brillouin zone for the wire array, and thus the highest transverse spatial frequency that can propagate. Consequently, k⊥max corresponds to the highest resolution possible with the structure for a given Λ. The range of the geometric parameters (d and Λ) for each material combination varies with the operational wavelength and the refractive index of the dielectric host. We limit our study to structures capable of sub-diffraction propagation and thus keep the distance between the wires below the diffraction limit (Λ<λ/2nd). The wire diameters are selected in order to achieve d/Λ between 0.1 and 0.9. The mesh size in the metal wire was set as d/100, while the mesh in the dielectric was set smaller than Λ/50.
4.1 – Near infrared (1μm), System: Au/SiO2
We consider Au wires in SiO2 as the case study for the NIR. Au was shown to be favorable above, and SiO2 is chosen for its high transmission at 1 μm and known compatibility with Au with both co-drawing techniques  and the pressure-assisted melt-filling method [44–46].
Figure 6 presents the loss in dB/μm of the quasi-TEM mode with lowest loss as a function of L, for λ = 1 μm and different wire diameters. In Figs. 6(a) and 6(b), the modes have k⊥ = 0 and k⊥max, respectively. The indices used were 1.45042  for SiO2 and 0.25-6.66i  for Au, considering its bulk complex permittivity from Figs. 1(a) and 1(b). It is important to emphasize that this becomes increasingly inapplicable for very thin wires with diameters below tens of nanometers. According to the literature, the loss of thin gold films is expected to increase or decrease as a function of thickness, depending on the type of film [35,36]. Thus, further work will be required and these simulations at 1 μm wavelength must be treated as an approximation. We limited our study to structures with Λ<345 nm, which corresponds to the diffraction limit (λ/2nd).
As shown in Fig. 6(a), the loss of the quasi-TEM mode decreases with increasing Λ for k⊥ = 0. This is expected since the metal fraction in the unit cell decreases for a fixed wire diameter. When the distance between the wires is constant, the loss increases for larger wire diameters, showing the opposite behavior compared to the single wire case (Figs. 2(a) and 2(b)). This indicates that, for k⊥ = 0, the influence of the metal fraction on the loss is more significant than the influence of the mode confinement when d is changed. On the other hand, the modes with k⊥ = k⊥max follow the behavior of the single wire waveguide, presenting higher loss for smaller wire diameter. This indicates that, for k⊥max, the variation on the mode confinement is more significant than the change in metal fraction when d increases. Indeed, for k⊥ = 0 the quasi-TEM modal fields are predominantly between wires, while for large k⊥ fields localize more strongly at the wire interface.
Figure 7 (a) shows the profile of the normalized electric field across the small diagonal of the unit cell (red line of the inset) of the quasi-TEM modes with k⊥ = 0 and k⊥max, for the cases with d = 10 nm (Figs. 7(b) and 7(c)) and d = 25 nm (Figs. 7(d) and 7(e)), for Λ = 40 nm, Au/SiO2 system, at λ = 1 μm.
According to Figs. 7(a)-7(e), the concentration of electric field in the metal wire for the modes with k⊥max is higher than their respective modes with k⊥ = 0, which explains the difference in the losses between Fig. 6(a) and 6(b). In addition, Fig. 7(a) also shows that, for k⊥ = 0, there is slightly more electric field in the metal for the case with d = 25 nm than the one with d = 10 nm. This explains the unusual behavior presented in Fig. 6(a), where the loss increase for larger wire diameters. From the E and H field distributions and using Eqs. (5)-(6), we calculated the energy density inside the metal for the all the four cases presented in Figs. 7(b)-7(d). The fractional energies in the metal found were 0.004 (Fig. 7(b)), 0.242 (Fig. 7(c)), 0.0325 (Fig. 7(d)) and 0.229 (Fig. 7(e)), which are also in agreement with the losses presented in Fig. 6.
Importantly our results show that while the loss of the quasi-TEM mode with k oriented parallel to the wires (k⊥ = 0) can be low enough to propagate over hundreds of wavelengths, this is not the case for modes with k⊥>ndk0, which are the modes most relevant to any sub-diffraction physics such as hyperlenses and density of states enhancement.
For an imaging application, considering 20 dB as the maximum acceptable propagation loss, Fig. 6(b) shows that the maximum propagation length for the Au/SiO2 system at λ = 1 μm is around 1, 3 and 7 μm for wire diameters of 10, 25 and 50 nm, respectively. These maximum propagation lengths of approximately 1, 3, and 7 wavelengths make wire array based metamaterial hyperlenses unlikely to be of use at 1 μm wavelength.
4.2 – Mid infrared (3 μm), Systems: Au/SiO2 and Sn/soda-lime
For the MIR we consider the Au/SiO2 system again, and also consider Sn for the metal. Its low melting point (232 °C) makes possible the use of drawing methods using common soft-glasses such as soda-lime (drawing temperature around 700 °C), and borosilicate (800 °C). Both have similar optical quality at 3 μm, with a transmission around 50% for 1 mm thickness [47,48]. We select soda-lime for our simulations because of its lower drawing temperature and well known compatibility with Sn [30–32].
Figures 8(a) and 8(b) present the loss for the Au/SiO2 system at λ = 3 μm, for the quasi-TEM modes with k⊥ = 0 and for k⊥max. Similarly, Figs. 8(c) and 8(d) show the equivalent results for the Sn/soda-lime system. The indices used were 1.41925  for SiO2, 1.63-18.6i  for Au, 1.4849  for soda-lime and 4.41-17.78i  for Sn. The structure is limited to L<1 μm, which corresponds to the diffraction limit for λ = 3 μm, given the indices of the glasses.
As shown in Figs. 8(a) and 8(b), the loss of the quasi-TEM modes at 3 μm for both k⊥ = 0 and k⊥max, present the same behavior as the modes at 1 μm when wire diameter and separation are varied. Considering again 20 dB as the maximum acceptable loss, Fig. 8(b) shows that the maximum propagation lengths of the high k⊥ mode for the Au/SiO2 system at λ = 3 μm are around 33 μm (d = 100 nm), 100 μm (d = 250 nm) and 110 μm (d = 500 nm), when the distance between the wires approaches the diffraction limit. These maximum propagation lengths of approximately 10, 30, and 36 wavelengths make the use of wire arrays much more compelling at 3 μm than at 1 μm wavelength. In this regime, depending on the combination of d and L, even distances between the wires around half of the diffraction limit have reasonable propagation lengths. For instance, for L = 500 nm and d = 250 nm, the 20 dB propagation length is around 50 μm.
Regarding the Sn/soda-lime system, Fig. 8(d) shows that the maximum propagation length at λ = 3 μm is around 13 μm (d = 100 nm), 33 μm (d = 250 nm), and 40 μm (d = 500 nm). As expected from the single wire waveguide analysis, Au is optically better than Sn. However, the propagation lengths of approximately 4, 11 and 13 wavelengths described above make this an interesting option for large wire diameters and wire separation.
4.3 – Mid infrared (10 μm), Systems: Au and Sn embedded in a glass with nd = 2.8
At the operational wavelength of 10 μm, the loss of the dielectric also needs to be taken into account in the selection of the materials. Because of their low transparency, silica-based glasses must be excluded. Chalcogenide glasses such as IRG 22 (Schott, Ge33As12Se55) , GLS (Gallium Lanthanum Sulphide) , As2Se3 and As2S3 are better choices because they have a transmission higher than 50% at λ = 10 μm, for 2 mm samples. Therefore, we consider a material with nd = 2.8 as the dielectric, which is representative of the above examples.
The low minimum drawing temperature of the chalcogenide glasses, between 300 to 700°C, makes the co-drawing of these glasses with high melting point metals such as Au, Al, Cu and Ag impossible. This limitation emerges from the fabrication requirement that the minimum drawing temperature of the dielectric must be higher than the melting point of metal. According to our figure of merit (Fig. 1(c)), apart from these high melting point metals, the best options at λ = 10 μm are Sn and In. Because their optical quality is very similar, Sn was selected due its low oxidation and low cost. Even though it is impossible to co-draw chalcogenide glasses with molten Au, we nevertheless consider Au as an ideal scenario for comparison.
Figures 9(a) and 9(b) present the loss for Sn wires at λ = 10 μm, for the quasi-TEM modes with k⊥ = 0 and for k⊥max, respectively. Figures 9(c) and 9(d) show the equivalent results for Au wires. The indices used were 22-46.41i  for Sn and 12.36-55.04i  for Au. The structure is limited to L<1.785 μm, which corresponds to the diffraction limit for λ = 10 μm in a dielectric with nd = 2.8.
Considering again 20 dB as the maximum acceptable loss of the high k⊥ modes, Fig. 9(b) shows that the maximum propagation lengths for the Sn/glass with nd = 2.8 system at λ = 10 μm are around 38 μm (d = 250 nm), 60 μm (d = 500 nm), and 65 μm (d = 800 nm). These propagation lengths of approximately 3.8, 6 and 6.5 wavelengths are not as encouraging as the propagation lengths at λ = 3 μm.
Regarding the Au wires, Fig. 9(d) shows that the maximum propagation lengths are around 83 μm (8 wavelengths, d = 100 nm), 127 μm (12 wavelengths, d = 250 nm), and 140 μm (14 wavelengths, d = 500 nm). As expected by the single wire waveguide analysis, Au is optically better than Sn for the wire array at λ = 10 μm, presenting a factor of 2 improvement on the loss. Although the metals have lower loss at this wavelength compared to 3 μm, the higher index of the dielectric results in an overall increase in the loss due to higher confinement in the metal.
4.4 – Loss as a function of nd
The refractive index of the dielectric (nd) influences the mode energy distribution in the wire array and, consequently, also affects the mode loss. Therefore, the best dielectric in terms of the metamaterial optical performance is not necessarily the one with the lowest extinction coefficient (i.e. the highest transparency).
Figure 10(a) illustrates the loss of the quasi-TEM modes (k⊥ = 0 and k⊥max) as functions of nd at λ = 3 μm, for a fixed structure with Sn wires (d = 250 nm, L = 500 nm). The refractive index of the dielectric is varied from 1 to 3, covering SiO2 (1.41925 ), soda-lime (1.4849 ), borosilicate (around 1.5), and some chalcogenide glasses (up to 3 ).
Figure 10 shows the loss increasing for larger nd for both modes. This behavior is in agreement with the single wire waveguide (Figs. 4(a) and 4(b)) and indicates that larger nd increases the mode confinement in the metal. As a result, if the transparency of the glass candidates is equivalent, the best option is the dielectric with the lower refractive index nd.
Since lower refractive indices lower losses, it is tempting to artificially reduce the effective refractive index of the glass e.g. by adding holes between wires. Simulations to that effect (adding a hole between wires in the unit cell) shows this only reduces loss by ~1%, because electric fields are weakest at the half-way point between wires, where it would be practical to add holes in the structure. Further simulations showed that reducing the amount of metal by replacing wires by metallic coated holes do not reduce loss at all, but rather lead to an increase in losses when the metal thickness is below the skin depth – which can be understood in that the effective resistance increases when reducing the metal cross section.
We have investigated promising metal/glass combinations considering optical quality and drawing feasibility for the fabrication of wire array metamaterial media for the MIR and NIR. The figure of merit based on the 0th-order TM mode loss of a single metal wire waveguide provides a qualitative comparison between the optical quality of the metals. The single wire analysis shows that loss decreases for longer wavelengths, larger wire diameter and smaller refractive index of the surrounding dielectric due to the variations in the mode confinement. Simulations of the full wire arrays show that the loss for the quasi-TEM mode varies greatly with transverse spatial frequency. While it is the high spatial frequency modes that give hyperbolic media their most interesting properties, the propagation loss of the highest spatial frequency (at the Brillouin zone edge) can be orders of magnitude higher than for propagation parallel to the wire. At λ = 1 μm the loss of high spatial frequency modes are high for all material combinations considered. With 20 dB-propagation lengths of a few microns at most among all configurations simulated, wire media are unlikely to yield many applications in the NIR.
The situation is considerably better at longer wavelengths. At λ = 3 μm, Au is optically the best option yielding 20 dB-propagation lengths for high spatial frequency modes up to 110μm, depending on the wire diameter and the distance between the wires. Reasonable losses are still achieved for distances between the wires smaller than half of the diffraction limit, so that sub-diffraction limited wire-array based hyperlenses are in principle feasible at this wavelength. Sn is optically poorer but still usable for large wire diameters and distance between the wires, with maximum 20 dB-propagation lengths up to 40 μm in this limit.
For an optical wavelength of 10 μm, losses are relatively higher, even for Au wire arrays, because of the increased refractive index of suitable host dielectrics. At this wavelength, Sn wire arrays with relatively large wire diameters and distance between the wires marginally below the diffraction limit could also prove useful, with maximum propagation lengths up to 65 μm.
In all cases, the loss of the high spatial frequency modes diminishes rapidly with increased wire diameter and wire-to-wire spacing. Magnifying hyperlenses based on tapered wire arrays will thus suffer much lower losses than those calculated here – as long as the taper angle is steep from the outset. Such configurations require further study.
Our study shows the practical spectral limits of wire array metamaterial media, considering the properties of feasible material combinations, and assuming the need to propagate distances of multiple wavelengths. We conclude there is little possibility for operation wavelengths of 1 μm or shorter for applications relying on transmission of high spatial frequencies, but have identified material combinations that will permit operation in the near infrared to wavelengths as short as 3 μm.
This research was supported by the Australian Research Council (ARC) under the Discovery Project scheme number DP120103942 and DP140104116, and it was performed in part at the Optofab node of the Australian National Fabrication Facility (ANFF) using Commonwealth and NSW State Government funding. The author Juliano Hayashi would like to thank the Science without Borders Program by CAPES (Brazil) for the financial support under grant 9468/13-7.
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