We perform a series of simulations and experiments at microwave frequencies where we dynamically alter the magnetic resonance of one individual split ring resonator by photodoping a piece of low-doped semiconductor positioned within the gap of the resonator. We predict and experimentally achieve a complete suppression of the resonance amplitude using an 815 nm laser source and then briefly consider the problem of tuning the frequency of an SRR by the same method. We also illustrate the metamaterial approach to active electromagnetic devices by implementing a simple yet efficient optical modulator and a three channel dynamical filter.
©2007 Optical Society of America
Recently, an innovative approach has been proposed to increase the available range of electromagnetic properties in materials. The idea consists of constructing artificially structured composites, or metamaterials, that mimic the behavior of homogeneous media [1–4]. Typically, metamaterials are periodic structures obtained by stacking hundreds of subwavelength scattering objects in close proximity. Given that the scale of inhomogeneity of the metamaterial components is much smaller than the wavelengths of operation, the interaction of metamaterials with an applied electromagnetic field can be described to a good approximation in terms of macroscopic quantities—the electric permittivity, ε, and the magnetic permeability, μ— that are averaged over the composite . By carefully designing the metamaterial unit cells, it is thus possible to construct composites that exhibit effective homogeneous properties unlike those found in naturally occurring materials. For example, it has been experimentally verified  that an array of wires and split ring resonators (SRRs) possesses all the non-intuitive properties predicted by Veselago and Pendry for negative index materials [6, 7] including negative refraction, evanescent wave enhancement and reversal of phase and group velocities [8ߝ15]. In addition to facilitating the development of negative index media, the metamaterial approach has been also successfully applied to design a variety of new structures with unique electromagnetic properties. For example metamaterials can be constructed that exhibit permittivity and permeability tensors having both positive and negative values . These indefinite media have remarkable imaging and scattering properties  that make them of potential interest as transitioning elements in guided wave optics [18, 19]. Another natural application for metamaterials is the development of gradient index media  because the value of the permittivity and permeability can be engineered at virtually any point within the structure by adjusting the scattering properties of each unit cell [21, 22]. By implementing complex gradients independently in the permittivity and permeability tensor components, it has been shown that an entirely new class of materials can be realized by the process of transformation optics [23, 24]. A recent example utilized metamaterials to form an “invisibility cloak” that was demonstrated to render an object invisible to a narrow band of microwave frequencies .
While the possibilities offered by passive composites are already considerable and still growing, recent studies have examined how to dynamically tune or modulate the electromagnetic properties of metamaterials [26–29]. For example, calculations and experiments performed at microwave frequencies have established that metamaterials can be controlled by incorporating variable capacitance diodes (varactors) to their inner structure. This approach has been first applied in the context of metamaterial transmission lines . In this study, a bias voltage was applied to the varactors so as to adjust the distributed transmission line capacitance and inductance that both govern the signal propagation. Varactors have been later employed to tune the resonance frequency of SRRs . Typically, the electromagnetic response of SRRs results from a resonant exchange of energy between the inductive currents in the rings and the electrostatic fields in the capacitive gaps. It is thus possible to gain control over the structure by altering the resonance conditions; in Ref. , for example, the authors placed a varactor in the gap in order to dynamically change the capacitance of the resonator. SRRs can also be controlled using visible light, as demonstrated in the THz regime for a planar array of SRRs patterned on a GaAs substrate . In Ref. , the metamaterial electric response was modulated by shorting the SRR gaps through photo-excitation of free carriers in the substrate. An interesting aspect of this work is that only low levels of photodoping are needed, so that the GaAs substrate remains globally nearly transparent. The SRR response, however, is easily degraded because large field enhancements occur within the SRR gaps, rendering the structure extremely sensitive to the photo-induced changes in the substrate conductivity.
Building on these earlier results, we report an alternative way to control SRRs with light. Our structures consist of individual, isolated SRRs operating at microwave frequencies and containing a piece of low doped, n-type silicon positioned within the gap. By monitoring the magnetic resonance of each SRR when the gap is illuminated with near-infrared laser light (λ = 815 nm), we show that the SRR response can be modulated in amplitude or tuned in frequency by controlling the photo-induced losses in silicon. Having restricted the active region to the gap, we have a finer and more efficient control over the SRR response, which might ultimately broaden the range of potential applications as will be explained below with specific examples.
A difficulty related to the free space characterization of a single SRR is that its overall dimension is typically ten times smaller than the wavelength, so that the resonator constitutes a very minor perturbation to the incident wave. To make the SRR more easily detectable, we simulate—and later measure—the isolated SRR inside a single-mode hollow waveguide defined by two horizontal perfect conducting plates and two vertical absorbing walls. The purpose of the waveguide is to confine the fields in a small region around the SRR, significantly increasing the effective scattering cross-section of the resonator. It follows that under resonant conditions, the SRR removes a substantial amount of energy from the guided mode and a characteristic dip appears in the transmission spectrum of the waveguide. We consider the situation depicted in Fig. 1, where the SRR is placed in the center of the waveguide with its plane parallel to the absorbing walls and its gap oriented downwards. Far from the SRR, the mode propagating through the waveguide has a field pattern close to the TE10 mode of a metal walled waveguide, with the electric field oriented vertically and the magnetic field parallel to the horizontal plates. Such a configuration ensures that the guided mode only excites the magnetic resonance of the structure—and not the other resonances associated with the magnetoelectric coupling inherent to the asymmetric SRR design . The waveguide is terminated on both sides by a wave port through which the electromagnetic waves enter and exit the model. To characterize the SRR, the transmission coefficient (the S21 parameter) is computed at each port and then normalized against the transmission of the unloaded waveguide. The numerical results are generated with commercial code based on the finite element method (Ansoft’s HFSS).
Figure 2 shows a first example of controllable SRR. In our simulations, the SRR is made of Cu (thickness 17 μm) deposited on an FR4 laminate substrate (having relative permittivity εFR4=4.4+0.1i). The SRR gap is 0.15 mm wide and contains a thin piece of silicon whose edges are just long enough to make electrical contact with the arms of the SRR. The silicon considered here has the characteristics of the low doped, n-type silicon used in the experiments; that is, the silicon is assumed to have a conductivity σ0 = 0.127 S.m-1, an initial electron carrier density e 0 = 5.68×1012 cm-3 and an initial hole carrier density h 0 = 3.97×107 cm-3. When the silicon is illuminated with photons of energy similar to its bandgap, an excess carrier density is generated that increases the conductivity in the SRR gap . Because the photo-induced carriers can be considered as an electron-hole plasma, it is possible to relate the level of photodoping to the silicon permittivity εSi using Drude’s theory. We write εSi in its usual form , which can be derived by finding the contribution of the free carriers to the polarization P, or equivalently, to the volume current J:
In this equation, 11.8 is the relative permittivity due to the host lattice, ω is the incident angular frequency (taken as constant in this study, ω = 2π.8.5 × 109 rad.s-1), ωpe = (eq)1/2/(m*eε0)1/2 is the plasma frequency for the free electrons of density e and effective mass m*e, ωpe = (hq)1/2/(m*hε0)1/2 is the plasma frequency for the holes of density h and effective mass m*h, and ve and Vh are the electron and hole collision frequencies.
Experimentally, the photodoping is limited by optical absorption , which causes the photon density—and hence, the probability of creating an electron-hole pair—to decrease as light penetrates deeper in the material. However this dependence is not taken into account in Eq. (1), for this expression has been obtained assuming an isotropic distribution of e and h. In order to qualitatively include the effects of optical absorption in our simulations, we model the silicon slice as a two layer structure (see inset of Fig. 2). The top layer has a thickness of 10 μm—a value comparable to the penetration depth of near-infrared light in silicon—and its permittivity varies with e and h according to Eq. (1). The bottom layer is 90 μm thick and has a constant permittivity corresponding to the initial free carrier density (i.e., e and h are replaced by e 0 and h 0 in Eq. (1)). In the following, all our results are expressed as a function of the conductivity σ = ωε0 Im(εsi) of the top layer.
Figure 2 shows the spectrum of the S21 parameter for various conductivities. In the absence of photoexcitation, the transmission exhibits a sharp minimum at around 8.7 Ghz which is the signature of the magnetic resonance. As the level of photodoping increases, the off-resonance transmission remains globally constant while the peak amplitude drops dramatically—in fact, the resonance is essentially destroyed for conductivities larger than ∼ 50 S.m-1. Clearly then, the SRR gap gradually loses its primary function which is to introduce a capacitance in the structure to make it resonant. This behavior is consistent with the fact that, should the conductivity indefinitely increase, the SRR would end up being a closed metal loop which is known to be non-resonant at those frequencies (see gray curve of Fig. 2). However, Fig. 2 suggests that the resonance disappears for conductivities that are orders of magnitude smaller than those of good conductors. Hence, the question arises as to whether or not the SRR gap can be considered as electrically shorted in our simulations.
In order to clarify this point, we have repeated the simulations for a slightly larger SRR gap (0.18 mm rather than 0.15 mm) so that the silicon layer is now electrically isolated from the SRR. As can be seen from the transmission coefficient plotted in Fig. 3, this minor modification has important consequences. For the lowest levels of photodoping (σ < 75 S.m-1), the magnetic resonance essentially follows approximately the same behavior as before—it primarily decreases in amplitude while conserving its shape and position. It should be noted, however, that the structure is less sensitive than before, since higher conductivities are needed to obtain the same modulation amplitude. Then, as the conductivity further increases, the resonance distinctively shifts to a new position at smaller frequencies while gradually regaining in strength. The peak reappearance suggests that the tiny interstices between the photo-doped silicon and the SRR have enough capacitance to preserve the resonant nature of the structure. In other words, for the highest conductivities, the system begins to behave as a conducting SRR with two capacitive gaps. We have verified this supposition by replacing the top layer of our silicon slice by a piece of copper of same dimensions. The transmission spectrum (gray curve in Fig. 3) confirms that, indeed, the structure exhibits a pronounced peak at nearly the same frequency as for the SRR with photo-doped silicon.
Figure 3 shows that, although the SRR gap cannot be short-circuited, there is an intermediate range of photodoping (σ ∼ 75 S.m-1) for which the magnetic resonance is significantly attenuated. To interpret this behavior, we have plotted the real part of the permittivity as well as the dielectric loss tangent against the photoconductivity (0 < σ < 150 S.m -1) using Eq. (1). Figure 4 reveals that Re(εSi) is essentially constant and positive, thus indicating that the silicon remains non-conducting over the whole conductivity range. The dielectric loss tangent, in contrast, increases by four orders of magnitude so it is likely that the structure loses its resonant behavior because the photo-doped silicon layer acts to damp the strongly localized fields in the SRR gap. This conclusion has an important consequence for our understanding of the curves of Fig. 2, for the modulation was achieved with the same levels of photodoping. It means that the origin of the changes cannot be attributed to a short circuit of the SRR gap because εSi remains positive long after the magnetic resonance has disappeared—in fact, εSi does not become negative until σ reaches 300 S.m-1. In this case also, the SRR behavior seems therefore primarily governed by the photo-induced losses in silicon.
In summary, the initial resonance can be degraded for free carrier densities that are well below those of a conducting state, which renders the controllable SRRs highly sensitive to photoexcitation. This sensitivity is however limited by the fact that the photodoping is restricted to a small volume just under the silicon surface. Figure 5 illustrates this point by comparing the response of a controllable SRR for three photodoping schemes. The SRR parameters as well as the silicon dimensions are the same as in Fig. 2. In case (i), an isotropic conductivity σ = 15 S.m-1 is assumed for the entire semiconductor layer. This configuration appears as the most favorable of all three because the SRR magnetic resonance is completely destroyed. For case (ii), the same level of photodoping is considered but the photodoped region is restricted to a 10 μm thick layer that is in contact with the SRR arms. As discussed earlier, this bilayer approximation constitutes a more realistic description of photodoping in silicon. In this case, the transmission minimum is significantly attenuated but not entirely suppressed, which means that higher conductivities are needed to achieve the same level of modulation as in (i). To gain more insight on that matter, we systematically compared the response of SRRs (i) and (ii) as a function of photodoping (results not shown here). We found that the scattering parameters for both structures systematically superimpose when:
In this relation, 〈σ(ii)〉 is the averaged conductivity over the entire silicon bilayer of SRR (ii), whereas σ(i) is the conductivity of the homogeneous silicon considered in (i). This strict proportionality relation indicates that for case (ii), the effect of photodoping is in fact counterbalanced and averaged with the contribution of the passive silicon region. It should be noted, however, that averaging the conductivity over the entire silicon slice clearly overestimates the influence of the passive layer—otherwise the proportionality factor between 〈σ(ii)〉 and σ(i) would have been equal to 1. Thus, given the respective locations of the photodoped and passive regions, it can be inferred that the conductivity averaging is only effective in a volume close to the SRR gap. This is confirmed with structure (iii), for which the silicon slice is identical to case (ii) except that the photodoped layer is now on the far side with respect to the SRR arms. In this situation, the magnetic resonance is almost unperturbed despite the fact that the photodoped layer is only 90 μm away from the SRR gap. This result is consistent with the fact that, for the SRR geometry, the microscopic fields are maximum in the gap and decay rapidly outside the structure. A practical consequence is that the sensitivity of actual controllable SRRs critically depends on how well the silicon surface facing the illumination source is adjusted in the gap.
In order to test our simulations, we fabricated the controllable SRR of Fig. 2, loaded it in a hollow waveguide, and verified that its magnetic resonance could be modulated by light. As always in this paper, the SRR orientation is that of Fig. 1 so that only its magnetic resonance can be excited. The waveguide in itself is not unlike the model studied in the previous section: it is made of two horizontal Al plates separated by 15.5 mm and two vertical planar absorbers; however in this case both ends are tapered and coupled to the two ports of a vector network analyzer by means of coaxial cables. The first port excites the fundamental mode of the waveguide while the second port records the transmission spectrum in the range 7–10 Ghz.
In practice, such a configuration restricts the choice of illumination sources that can be used for photodoping the structure: on one hand, the light source cannot be located inside the waveguide because the presence of another scattering object beside the SRR would needlessly complicate the experimental data; on the other hand, the waveguide walls are not transparent and so the silicon contained in the SRR gap cannot be directly illuminated from outside. We therefore opted to use an external laser diode (wavelength 815 nm, variable power between 0 and 1 W) coupled to a multimode optical fiber which enters the waveguide through a small aperture and ends a few microns away from the silicon.
To hold the fiber in position, we adhered it to the dielectric substrate of the SRR according to the following procedure. First, we fabricated the SRR on a FR4 laminate substrate using a high precision micromilling machine . During the machining process, we created a 0.2 mm gap in the middle of which we drilled a 150 μm diameter through hole in the FR4 substrate.
The function of this hole was to enable the illumination of the gap region from the bottom side of the FR4 substrate, that is, in our case, by inserting the tip of the multimode fiber in the hole (see Fig. 6(a)). Once the fiber was in place, we froze its position with a tiny drop of epoxy and then covered the SRR gap with a 0.2 mm × 0.2 mm silicon square diced from a n-type silicon wafer (σ0 = 0.127 S.m-1, thickness 100 μm). Finally, an electrical contact between the Si slice and the SRR arms was created by coating the junctions of the two materials with conductive silver epoxy (last picture of Fig. 6(a)). The silver epoxy reduces the width of the SRR gap to approximately 0.15 mm, which was the value used in the simulations. It should be noted that the fiber core diameter (105 μm) is smaller than the lateral size of the silicon slice. However we allowed a small separation between the fiber tip and the semiconductor surface so as to ensure that the laser spot would be large enough to provide a fairly constant illumination over the gap region. A photograph of the fabricated SRR can be seen in Fig. 6. Aside from the fiber and the cubic holder made of expanded polystyrene foam—a material that does not scatter the microwaves—the structure constitutes the practical realization of the simulated SRR of Fig. 2. We note however that we did not try to carefully adjust the position of the silicon in the gap so as to maximize the effect of photodoping (cf. discussion around Fig. 5).
Figure 6(b) shows the spectrum of the S21 parameter of the SRR with the laser power as parameter. In this figure, the laser power represents the total photon flux emerging from the fiber as measured before the fiber was mounted into the SRR structure. When the laser is off, the SRR magnetic resonance manifests itself by a transmission dip very similar to the minimum predicted for the simulated structure—in fact, the main difference is a shift of about 0.5 GHz towards smaller frequencies. Additional experiments and numerical work indicate that this discrepancy mainly accounts for the tape we used to maintain the SRR against the polystyrene holder and also for the optical fiber and epoxies that were not explicitly included in the model. If we now focus on the curves of Fig. 6(b) obtained while gradually increasing the laser power, we observe the modulation in amplitude we expected from our simulations. It is particularly worth noting that the magnetic resonance can be suppressed for rather low laser power—in this example, a little more than 1 mW. Since, in this case, the SRR has almost no effect on the signal, the ratio between on and off transmissions is chiefly determined by the attenuation caused by the structure in its undoped, resonant state. In Fig. 6, this ratio does not exceed 1.5 dB because the waveguide lateral dimensions are too large to funnel all the electromagnetic flux through the volume where the field substantially interacts with the subwavelength SRR. In order to increase the modulation amplitude, one has to maximize the coupling between the guided modes and the SRR, for example by increasing the field confinement around the resonator. Alternatively, good performances should be observed for controllable SRRs assembled in a bulk metamaterial that would entirely fill the waveguide cross-section.
Equally important is the fact that all photo-induced changes were fully reversible by turning the laser off. Although we did not perform any temporal measurements, it is likely that the time required to recover the original resonance is close to the recombination time of the free carriers, which is approximately 10 microseconds for silicon. In principle, it should be possible to substantially improve the switching rate by filling the SRR gap with semiconductor materials possessing much faster carrier trapping time .
We then verified that the effect of photodoping is different on samples for which there is no electrical contact between the silicon and the SRR. Figure 7 shows the evolution of the magnetic resonance for the exact same structure, except that the measurements were taken before the electrical contacts with the silver epoxy were created. It can be seen that in this case, the peak first decreases in amplitude but never completely disappears; then, it shifts towards smaller frequencies and tends to gain back in amplitude, as expected from our previous discussion in Section 2. We thus achieved a tuning rather than a modulation of the magnetic resonance. Of course, the tuning is not entirely satisfying as the peak amplitude undergoes strong variations; in addition, high power levels are required to observe a significant shift in frequency, in good agreement with our numerical calculations. Nevertheless, this set of measurements suggests that it should be possible to implement a variety of optically controlled devices by simply changing the shape and/or position of the silicon layer in the SRR gap.
Having now an experimental confirmation that our approach for controlling SRRs is effective, we present a simple example application, in which we demonstrate optical switching between two dielectric waveguides. Figure 8(a) shows the configuration under investigation: it consists of two lengths of dielectric waveguide separated by a controllable SRR fabricated as described above. In our experiments, the dielectric waveguides are two rods of polycarbonate (n = 1.67) with a square cross-section of 1 cm × 1 cm. To further simplify the problem, the entire experimental setup is inserted between the two horizontal metal plates of a 2D scattering chamber. The distance between the two metal plates is 1.1 cm (i.e. it is only 1 mm larger than the height of the waveguides), therefore the system can be considered as nearly translationally invariant along the vertical direction. The very close separation between the horizontal metal boundaries furthermore ensures that only the fundamental mode is supported in the planar waveguide over the frequency range we consider (7–10 GHz). More specifically, the dielectric waveguides only support a TE mode for which the electric field E is parallel to and constant along the vertical axis.
To investigate the switching effect, we recorded the map of the electromagnetic field in the vicinity of the SRR. As described elsewhere , the microwaves are detected with a coaxial antenna inserted in the upper metal plate of the mapping chamber. The lower plate of the chamber is mounted on a two axis translation stage that allows the experimental setup to move laterally with respect to the stationary antenna. The scanning process, the excitation of the TE mode in the input waveguide and the signal collected by the antenna are digitally controlled by a computer running a custom code. Figures 8(b) and 8(c) show the electric field intensity recorded at the resonance frequency (f = 8.8 Ghz) when the laser illuminating the SRR gap is turned on and off, respectively. According to our previous experiments, the effect of the laser is to suppress the magnetic resonance of the SRR and therefore the transmission from one waveguide to another should be maximized in this case. In fact, the experimental data of Fig. 8(b) reveal that the coupling between the two polycarbonate rods is nearly perfect when the laser is turned on, which not only confirms that the SRR represents a minor perturbation indeed, but also indicates that the distance between the two waveguides is short enough to prevent significant transition losses. By shutting the laser off, the magnetic resonance of the SRR is restored and Fig. 8(c) shows that the transmission drops by a factor of 8 dB. Consequently the modulation achieved here has a larger amplitude than in our first experiments (Fig. 6). This can be explained by the fact that the dielectric waveguides terminate directly into the SRR, thus leveraging a larger perturbation for the guided modes.
While we have just seen that individual SRRs have attractive properties as individually controllable resonators, their rich potential can also be used for developing more complicated components—these include metamaterials of course, but also devices that exploit the discrete nature of these structures. We illustrate this point by considering the dynamic multichannel filter shown in Fig. 9(a). Our structure—a cluster of three controllable SRRs with distinct resonance frequencies—has been designed to control the transmission at the junction of the two polycarbonate rods previously used in the switching experiment. As before, each SRR is connected to an optical fiber that illuminates the silicon positioned within its gap, so that the transmission between the two dielectric waveguides can be independently modulated at three different frequencies. To characterize the dynamic filter, we coupled the input and output waveguides to the ports of the network analyzer and performed a series of transmission measurements in our 2D scattering chamber. In our experiments, the impedance of the network analyzer is not perfectly matched to the dielectric waveguides, leading to unstable extrema in the transmission spectra due to outgoing waves being partially reflected back to the scattering chamber. We minimize this artifact by tilting the output waveguide to an angle of 30 degrees with respect to the axis of the input waveguide. In this configuration, however, the coupling between the dielectric waveguides is both relatively weak and frequency dependent even in the absence of the controllable SRRs. Here we do not focus on this coupling aspect but rather on the response of the dynamic filter; therefore, we normalize all of the data against the transmission spectrum taken before the SRRs were inserted in the scattering chamber.
Figure 9(b) reproduces the spectra of the S21 parameters as measured when the dynamic filter is in the dark and when each SRR is successively illuminated. It can be seen that three channels corresponding to each SRR resonance are added or removed at will, thus illustrating the dynamic filtering effect. The modulation amplitude for each channel is a little less than 5 dB, which is slightly smaller than for the optical switch considered in the previous section. This difference can be explained by the fact that the coupling between SRRs and tilted waveguides is less efficient than for the non-tilted case. Interestingly, the transmission minima that are not suppressed by light do not vary in shape and position from one curve to another. This result is particularly important, as it shows that each SRR can really be controlled separately without perturbing the neighboring resonators. Hence, our structures can be readily used as building blocks for more sophisticated composites.
We have designed and fabricated active split ring resonators whose response to microwave radiation can be dynamically altered using near-infrared light. Although we have only considered a restricted number of SRRs at a time, the main results of this study can be applied to bulk meta-material structures. In particular, the fact that we have demonstrated optical control at the unit cell level suggests that an arrangement of active SRR can be built for which the capacitance of each gap is individually addressed. This would enable the design of much more sophisticated devices such as beam steerers and other active gradient index media.
DRS acknowledges insightful conversations regarding the use of photodoping to tune metama-terials with Ron Tonucci (Naval Research Laboratory). The authors are also grateful to Shih-Yuan Wang (Hewlett-Packard Laboratories) for stimulating discussions and to Michael Garcia (Duke University) for helping us in the experimental characterization of the silicon used in this study. The laser diode used in the experimental section was a gift provided by Hewlett-Packard. Support for this work was provided by the Defense Advanced Research Projects Agency (Contract Number HR0011-05-3-0002) and by a Multidisciplinary University Research Initiative (MURI) from the Air Force Office of Scientific Research (Contract Number FA9550-04-1-0434).
References and links
1. R. M. Walser, “Electromagnetic metamaterials,” in Complex Mediums II: Beyond Linear Isotropic Dielectrics, A. Lakhtakia, W. S. Weiglhofer, and I. J. Hodgkinson, eds.,Proc. SPIE4467,1–15 (2001). [CrossRef]
2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Technol 47,2075–2084 (1999). [CrossRef]
3. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett 84,4184–4187 (2000). [CrossRef] [PubMed]
4. R. W. Ziolkowski and N. Engheta, “Metamaterial special issue introduction,” IEEE Trans. Antennas Propag 51,2546–2549 (2003). [CrossRef]
6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of eand μ,” Sov. Phys. Usp 10,509–514 (1968). [CrossRef]
9. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett 90,107401 (2003). [CrossRef] [PubMed]
11. J.B. Pendry, “Introduction,” Opt. Express 11,639–639 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-7-639. [CrossRef] [PubMed]
12. A. Lakhtakia and M. McCall, “Focus on negative refraction,” New J. Phys 7 (2005). [CrossRef]
13. V. G. Veselago, L. Braginsky, V. Shkover, and C. Hafner, “Negative refractive index materials,” J. Comput. Theoretical Nanoscience 3,189–218 (2006).
14. A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B-Cond. Mat 338,333–337 (2003). [CrossRef]
15. W. T. Lu and S. Sridhar, “Flat lens without optical axis: Theory of imaging,” Opt. Express 13,10673–10680 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-7-639. [CrossRef] [PubMed]
17. D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys 7,162 (2005). [CrossRef]
18. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, Appl. Phys. Lett 84,2244–2246 (2004). [CrossRef]
19. A. Degiron, D. R. Smith, J. J. Mock, B. J. Justice, and J. Gollub, “Negative Index and Indefinite Media Waveguide Couplers,” Appl. Phys. A, in press.
20. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E 71,036609 (2005). [CrossRef]
21. R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett 87,091114 (2005). [CrossRef]
22. T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett 88,081101 (2006). [CrossRef]
25. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science ,314,977–980 (2006). [CrossRef] [PubMed]
26. S. Lim, C. Caloz, and T Itoh, IEEE Trans. Microw. Theory Tech52,2678–2690 (2004). [CrossRef]
27. I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14,9344–9349 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9344. [CrossRef] [PubMed]
28. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical Electric and Magnetic Metama-terial Response at Terahertz Frequencies,” Phys. Rev. Lett 96,107401 (2006). [CrossRef] [PubMed]
30. J. García-García, F. Martín, J. D. Baena, R. Marqués, and L. Jelinek, “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys 98,033103 (2005). [CrossRef]
31. P. Bhattacharya, “Semiconductor Optoelectronic Devices,” (Prentice Hall, Upper Saddle River, 1997).
32. C. H. Lee, P. S. Mak, and A. P.De Fonzo, “Optical control of millimeter-wave propagation in dielectric waveguides,” IEEE J. Quantum. Electron 16,277–288 (1980). [CrossRef]
33. B. J. Justice, J. J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14,8694–8705 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-19-8694. [CrossRef] [PubMed]