Design of fiber metamaterials with negative refractive index in the infrared Scott Townsend,1,∗ Shiwei Zhou2 and Qing Li1 1 School
of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia 2 School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476 Melbourne, VIC 3001, Australia ∗ [email protected]
Abstract: Metamaterials possess intricate, sub-wavelength microstructures, making scalability a salient concern in regard to their practicality. Fiber-drawing offers a route to producing large quantities of material at relatively low cost, though to our knowledge, a fiber-based design capable of negative refractive index behaviour has not yet been proposed. We submit that the electric and magnetic dipole resonance modes of the fiber can be enhanced by including in the fiber aligned metallic inclusions. Addition of a solid metallic core can effect a synchronisation of these modes, allowing a collection of the fibers to possess negative refractive index. © 2015 Optical Society of America OCIS codes: (160.3918) Metamaterials; (260.2065) Effective medium theory; (290.4020) Mie theory.
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13. N. Singh, A. Tuniz, R. Lwin, S. Atakaramians, A. Argyros, S. C. Fleming, and B. T. Kuhlmey, “Fiber-drawn double split ring resonators in the terahertz range,” Opt. Mater. Express 2, 1254–1259 (2012). 14. E. V. Loewenstein, D. R. Smith, and R. L. Morgan, “Optical constants of far infrared materials. 2: Crystalline solids,” Appl. Opt. 12, 398–406 (1973). 15. H. Jiang, X. Han, and R. Li, “Improved algorithm for electromagnetic scattering of plane waves by a radially stratified tilted cylinder and its application,” Opt. Commun. 266, 13–18 (2006). 16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008). 17. E. Kallos, I. Chremmos, and V. Yannopapas, “Resonance properties of optical all-dielectric metamaterials using two-dimensional multipole expansion,” Phys. Rev. B 86, 245108 (2012). 18. P. Mallet, C.-A. Guerin, and A. Sentenac, “Maxwell-garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005). 19. J. Ballato, T. Hawkins, P. Foy, R. Stolen, B. Kokuoz, M. Ellison, C. McMillen, J. Reppert, A. Rao, M. Daw et al., “Silicon optical fiber,” Opt. Express 16, 18675–18683 (2008). 20. G. Taylor, “A method of drawing metallic filaments and a discussion of their properties and uses,” Phys. Rev. 23, 655 (1924). 21. X. Zhang, Z. Ma, Z.-Y. Yuan, and M. Su, “Mass-productions of vertically aligned extremely long metallic micro/nanowires using fiber drawing nanomanufacturing,” Adv. Mater. 20, 1310–1314 (2008). 22. A. Tuniz, B. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett. 96, 191101 (2010). 23. A. Mazhorova, J. F. Gu, A. Dupuis, M. Peccianti, O. Tsuneyuki, R. Morandotti, H. Minamide, M. Tang, Y. Wang, H. Ito et al., “Composite thz materials using aligned metallic and semiconductor microwires, experiments and interpretation,” Opt. Express 18, 24632–24647 (2010). 24. M. A. Ordal, R. J. Bell, R. Alexander Jr, L. Long, M. Querry et al., “Optical properties of fourteen metals in the infrared and far infrared: Al, co, cu, au, fe, pb, mo, ni, pd, pt, ag, ti, v, and w,” Appl. Opt. 24, 4493–4499 (1985). 25. O. Levy and D. Stroud, “Maxwell garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers,” Phys. Rev. B 56, 8035 (1997). 26. S. Townsend, S. Zhou, and Q. Li, “Double-negative metamaterial from conducting spheres with a highpermittivity shell,” Opt. Lett. 39, 4587–4590 (2014). 27. I. Trelea, “The particle swarm optimization algorithm: convergence analysis and parameter selection,” Information Processing Lett. 85, 317–325 (2003). 28. C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” Journal of the atmospheric sciences 43, 468–475 (1986). 29. R. Ruppin, “Evaluation of extended maxwell-garnett theories,” Opt. Commun. 182, 273–279 (2000). 30. V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” Journal of Physics: Condensed Matter 20, 255201 (2008). 31. V. Yannopapas, “Binary alloy of virus capsids and gold nanoparticles as a mie-resonance-based optical metamaterial,” Solid State Communications 204, 51–55 (2015).
Meta-materials are defined as those composites having properties not found in nature. In electromagnetics research, there is great interest in developing composites which display precise values of electric permittivity (ε) and magnetic permeability (µ), for these dictate the movement of electromagnetic radiation in and around the material. Even brief surveys of the literature uncover designs for extraordinary devices made possible by the use of metamaterials, such as the invisibility cloak  and artificial black hole . Particular attention has been directed toward achieving negative values for ε and µ, such that √ the refractive index (n = n0 + in00 = ± ε µ | n00 > 0) also becomes negative, and it is to this end that we devote our efforts. In addition, we will propose designs operating at frequencies lower than that of red light; it is at these frequencies where metals retain a good measure of conductivity, which will prove salient in our designs. At these frequencies, past attempts to induce negative refractive index behaviour can be categorized into two main design routes. The first generates electric current flow in periodicallyarranged metallic patterns (unit cells), which are printed onto dielectric backing material in the style of the famous split-ring resonator [3–5]. The second utilises the Mie resonance in small (usually spherical) particles dispersed in a host material [6–10]. Both share the issue of
#237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18237
scalability, since the unit cell (metallic pattern and Mie particle, respectively) must be smaller than the incident wavelength. In addition, the performance of each composite is generally quite sensitive to small changes in the unit cell geometry. These issues have motivated one particularly novel extension: to design metamaterials which can be manufactured using technology borrowed from the fiber optics industry. A metaldielectric composite preform can be designed, which is then drawn into a long fiber. The fiber is cut and stacked, becoming the unit cell in the metamaterial (see Fig. 1). In this manner, researchers have realized negative permeability (µ 0 < 0) by drawing fibers with cross-sections resembling the split-ring resonator [11–13]. To our knowledge, though, a design capable of simultaneously achieving ε 0 < 0, and thus negative refractive index, is yet to be proposed. Such is the novelty of this article. We propose a design methodology capable of leveraging the advantages of fiber-based manufacturing and which also achieves a negative refractive index. Furthering disparity to the previously-mentioned fiber-based designs, ours is one utilising Mie resonance. We show that a very high refractive index fiber is required to induce negative material properties, and that this can be achieved by introducing aligned metallic wires into a dielectric or semiconductor fiber. We continue by showing that the addition of a metallic core to the fiber can effect a synchronization in electric and magnetic dipole resonance, resulting in a negative refractive index. We conclude by directly simulating a backward wave propagating through the proposed composites.
Fig. 1. The proposed metamaterial. Fibers comprised of aligned metal wires and a solid metal core provide a route to strong electromagnetic resonances which can lead to negative refractive index behaviour.
Theory of fiber-drawn metamaterials
To illustrate the emergence of metamaterial behaviour in a fiber-based composite, we simulated an isolated silicon fiber (material properties from ), exposed to a transverse magnetic (TM) plane wave as shown in Fig. 2(a). As predicted by the Mie theory , at certain (geometrydependent) frequencies the fiber develops strong internal fields, known as magnetic (Fig. 2(b)) and electric (Fig. 2(c)) dipole resonance, respectively. What of a composite material comprising such fibers? It can be shown that the resonant fields developed in the fibers can impart a marked effect on the electromagnetic properties (µ, ε) of the composite as a whole, causing them to adopt a resonant nature as well. The relevant
#237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18238
(a) Silicon fiber (0.65 µm) exposed to a TM plane wave
(b) Magnetic dipole resonance (at 100 THz)
(c) Electric dipole resonance (at 150 THz)
Fig. 2. Metamaterial behaviour from fibers. In the above, the arrows and colour scale rep~ fields, respectively. resent the ~E and H
expressions, known as the extended Maxwell-Garnett theory, are given as [16, 17]: µ −1 2i f = 2 b1 µ + 1 πx ε − εb 4i f = 2 a1 ε + εb πx [mD1 (mx) + 1/x] J1 (x) − J0 (x) b1 = (1) (1) [mD1 (mx) + 1/x] H1 (x) − H0 (x) [D1 (mx)/m + 1/x] J1 (x) − J0 (x) a1 = (1) (1) [D1 (mx)/m + 1/x] H1 (x) − H0 (x)
(1a) (1b) (1c) (1d)
In the above, D1 (z) = J10 (z)/J1 (z), and J(z) and H (1) (z) are Bessel and Hankel fuctions, √ respectively. The parameters x = εb k0 r (where k0 and r are the incident wavenumber and fiber radius, respectively) is the fiber size parameter; m is the fiber refractive index, calculated relative to the background/host medium which has permittivity εb . Such formulae have proven to remain quite accurate even at very high fiber volume fractions, f . It should be understood that µ and ε pertain to the material property values perpendicular to the fiber axis; different values exist parallel to the fibers, though we will not discuss these here. Although not immediately obvious from Eq. (1), the higher the fiber refractive index m, the stronger the electric and magnetic resonances will be (such information is contained in the Mie coefficients a1 , b1 ). This in turn produces greater variations in the properties ε, µ. Of the materials amenable to large-scale fiber-drawing techniques, Silicon [14, 19] has the highest index of which the authors are aware. Though we have found that even a very dense collection of homogeneous silicon fibers does not produce strong-enough resonance to emulate #237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18239
negative material properties: Fig. 3(a) demonstrates one example, where the fiber diameter was selected such that the resonances occured between 100 and 150 THz. Similar results can be produced throughout the IR spectrum. The challenge, then, is to increase the refractive index of the fibers. We propose a method thus: Introduce, into the fiber pre-form, a collection of metallic wires, which should remain in position when the (now composite) fiber is drawn. Such a technique is well-established for range of metals, glasses and semiconductors [20, 21], and has been proposed for use in metamaterials for different applications than are considered here [22,23]. Here we will continue to assume silicon as the fiber material, though the analysis is extensible to other materials in principle. To see why metallic wires should increase the refractive index of the fiber, we can apply Eq. (1) to the fiber material itsself. Asymptotically expanding Eq. (1d) for small inclusion size: a1 = −
iπx2 m2 − 1 + O(x5 ) 4 m2 + 1
Now note that for conductors, |m| 1 and we can use Eq. 2 and 1b to solve explicitly the effective permittivity of a silicon-metallic fiber: εfiber = εSi |m|1
1 + fwire 1 − fwire
The above (and accordingly, the fiber refractive index) becomes very large for high wire filling fractions ( fwire ), and Fig. 3(b) exemplifies negative properties being achieved using this technique, whereby copper wires (material properties from ) were arranged in a circular array such that the volume fraction of the wires ( fwire ) was set to 0.5, a value high enough to demonstrate the effect, though small enough to simulate (and presumably manufacture) efficiently. Note here that we have effectively designed a composite on two length scales simultaneously – the smaller being the silicon-metallic (fiber) material, the larger being a collection of such fibers in a host material (Fig. 1).
(a) Homogeneous silicon fibers (0.65 µm)
(b) Silicon fiber (0.33 µm) with metallic wires (0.03 µm, fwire = 0.5)
Fig. 3. Effective properties of fiber-based composites. In each of the above, the fiber filling fraction is f = 0.7 and the background medium has εb = 1.0.
The limitation on this effect occurs when the small inclusion limit breaks down, which typically occurs when the conductor cross-section exceeds its skin depth . For this reason, the #237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18240
proposed technique will be effective only in regions where the chosen conducting wire material has moderately-high conductivity, as the skin depth will be non-zero. For most metals, this is satisfied in the infrared region, and the copper from the above example could reasonably be substituted for most metals. The higher the conductivity, the smaller the wire diameter must be in order to be effective. 3.
Negative refractive index behaviour
We now turn to the requirement of simultaneously-negative ε and µ. Since the corresponding resonance frequencies can be tuned via the diameter of the fiber, one strategy would be to use two sub-lattices of fibers – one providing ε 0 < 0, the other µ 0 < 0. In this particular case, though, our simulations show this strategy to be futile, the reason being that the maximum volume fraction of a given fiber in a two lattice system is too low to produce negative effective properties. Fortunately, we have previously shown that the addition of a conducting core to a fiber (or sphere) can effect a synchronization in the electric and magnetic dipole resonances . Since now only a single lattice of fibers is required, the volume fraction (and strength of the resulting resonances) can be much higher. It then becomes a task of optimizing the size and volume fractions of the fiber, wires and core in order to achieve a negative refractive index. As an illustrative example, we choose 100 THz as a target frequency and optimized the parameters using a custom particle swarm algorithm . As above, copper was nominated for the core material, though as above, most other metals could equally-well be used. Figure 4 demonstrates the geometry and overlapping resonances of such a composite. Although we chose 100 THz as a target here, this strategy can quite generally be applied to the entire infrared region.
(a) Effective properties of composite
(b) Combined resonance mode (near 100 THz)
Fig. 4. Negative refractive index behaviour. Effective properties were calculated for a composite with εb = 1.0, silicon fiber (0.47 µm, f = 0.7), metallic wires (0.03 µm, fwire = 0.5) and metallic core (0.25 µm)
It should be noted that the sharp increase in n00 , the imaginary component of the refractive index near the resonance frequency, does not imply that the composite material is lossy, as would be the case for a homogeneous material. Rather, as a result of the extended MaxwellGarnett formalism assumed in this article, the peak in n00 encapsulates information regarding both the absorption and scattering from the composite inclusions [28, 29]. This being an article pertaining predominantly to conceptual design, we will not further explore techniques to esti-
#237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18241
Fig. 5. Backward wave propagation in the proposed composite. Inside the composite, the field was measured between the fibers and interpolated elsewhere. The color scale shows Hz0 , and was limited to ±1 for clarity; higher fields exist inside the fibers.
mate the true loss in the material; we refer interested readers to the methodology described in Refs. [30, 31] for such an endeavour. As further verification of the predicted material behaviour, we utilised COMSOL Multiphysics to simulate a plane wave travelling through a (5x5) slab of the proposed composite material as shown in Fig. 5. A backward wave, confirming that a negative refractive index has been achieved, is quite clearly visible in the region of the composite, as evidenced by the reversal of the direction of phase change at the boundaries of the composite. 4.
We have thus successfully demonstrated a method for designing a new class of fiber metamaterials. By leveraging effective medium phenomena at two length scales simultaneously, we are able to enhance and synchronize the Mie resonance modes of the fiber. By fine tuning the design parameters, we have proven capable of achieving a composite with a negative refractive index, a strategy which can quite generally be applied in the IR spectrum using the materials suggested herein. Acknowledgments This work was funded by the Australian Research Council (DP110104698 and FT120100947).
#237975 © 2015 OSA
Received 16 Apr 2015; revised 22 Jun 2015; accepted 23 Jun 2015; published 6 Jul 2015 13 Jul 2015 | Vol. 23, No. 14 | DOI:10.1364/OE.23.018236 | OPTICS EXPRESS 18242