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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 24411–24423
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Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths

Yu-Ju Tsai, Stéphane Larouche, Talmage Tyler, Guy Lipworth, Nan M. Jokerst, and David R. Smith  »View Author Affiliations


Optics Express, Vol. 19, Issue 24, pp. 24411-24423 (2011)
http://dx.doi.org/10.1364/OE.19.024411


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Abstract

We demonstrate the design, fabrication and characterization of an artificially structured, gradient index metamaterial with a linear index variation of Δn ∼ 3.0. The linear gradient profile is repeated periodically to form the equivalent of a blazed grating, with the gradient occurring across a spatial distance of 61μm. The grating, which operates at a wavelength of 10.6μm, is composed of non-resonant, progressively modified “I-beam” metamaterial elements and approximates a linear phase shift gradient using 61 distinguishable phase levels. The grating structure consists of four layers of lithographically patterned metallic I-beam elements separated by dielectric layers of SiO2. The index gradient is confirmed by comparing the measured magnitudes of the −1, 0 and +1 diffracted orders to those obtained from full wave simulations incorporating all material properties of the metals and dielectrics of the structures. The large index gradient has the potential to enable compact infrared diffractive and gradient index optics, as well as more exotic transformation optical media.

© 2011 OSA

1. Introduction

Graded index of refraction (GRIN) materials can control and manipulate light, and are particularly useful in imaging and waveguiding applications. Typical GRIN optics are formed using a variety of self-assembly techniques or diffusion processes, which typically lead to Δn values smaller than 0.05, with highly constrained spatial distributions. Since the greater advantages in GRIN optics come with larger gradients and controlled spatial distributions, there is a compelling opportunity for structured materials to be applied in the development of GRIN optics. As just one example, a GRIN lens has been synthesized by leveraging polymer forced assembly of nanolayered films with incremental differences in the refractive index [1

1. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. 103, 1834–1841 (2007). [CrossRef]

, 2

2. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16, 11540–11547 (2008). [PubMed]

]. Alternatively, subwavelength structures have been designed to achieve gradient index profiles based on effective medium theory [3

3. P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnol. 8, 53 (1997). [CrossRef]

7

7. H. Elfström, M. Kuittinen, T. Vallius, B. H. Kleemann, J. Ruoff, and R. Arnold, “Fabrication of blazed gratings by area-coded effective medium structures,” Opt. Commun. 266, 697–703 (2006). [CrossRef]

]. Though the index range is limited by the polymer constituents or the dielectric host materials, the process in principle could be applied to generate larger index gradients with flexible distributions, and illustrates the potential of employing artificial structuring to achieve improved optics. In one recent 2D waveguide study, a refractive index that varied from 1.4 to 2.8 has been experimentally achieved and used to form a planar Luneburg lens [8

8. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011). [CrossRef] [PubMed]

].

The term metamaterial is now broadly used to refer to artificially structured media that derive their properties primarily from the geometry of mesoscopic inclusions [9

9. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

12

12. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

]. Metamaterials can be used to realize a host of unique optical properties, such as very large refractive index gradients [13

13. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef] [PubMed]

], and negative index of refraction [14

14. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

]. Optical designs using metamaterials can precisely control the constitutive parameters point-by-point throughout a medium, and thus novel gradient index media can be realized, with index gradients typically an order of magnitude or more larger than what is currently available using even the state-of-the-art techniques [15

15. 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]

, 16

16. 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]

]. Moreover, given the flexibility associated with metamaterials, including the control over magnetic response and anisotropy, the design technique of transformation optics has also achieved relevance. Transformation optics can be used to design unprecedented optical devices, such as invisibility cloaks, which have functionality unlike any conventional lens or optic [17

17. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

]. Even if the material specifications for transformation optical media are highly demanding, metamaterial constructs have been demonstrated at microwave frequencies that reasonably approximate the salient features of transformation optical designs, and illustrate the potential for metamaterials to enable new classes of optical media [18

18. 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]

].

Large refractive index gradients have been demonstrated using metamaterials in the microwave and THz regimes [19

19. J. Neu, B. Krolla, O. Paul, B. Reinhard, R. Beigang, and M. Rahm, “Metamaterial-based gradient index lens with strong focusing in the THz frequency range,” Opt. Express 18, 27748–27757 (2010). [CrossRef]

, 20

20. R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Opt. Express 17, 21030–21041 (2009). [CrossRef] [PubMed]

], however, metamaterials are difficult to implement in the infrared (IR) and visible regimes. The reasons for this difficulty are twofold. First, metals—which are a key component of metamaterials in these regimes—behave as lossy dielectrics at frequencies above a few terahertz. Structures that utilize metal in the metamaterial thus tend to exhibit large absorption losses, particularly in the IR and visible regimes, which become severe if light needs to be transmitted over many hundreds of wavelengths. Though numerous demonstrations of intriguing metamaterials at IR and visible wavelengths have been reported, those structures have tended to be a single or just a few planar layers, with large losses noted in most cases [21

21. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–37 (2008). [CrossRef]

23

23. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]

]. In addition, for IR wavelengths, many non-metallic materials that might be used as supporting matrices, including ceramics and polymers, have absorption bands that can impact the function of the device. A second area of difficulty in the practical realization of visible and IR metamaterials is the challenge of fabricating feature sizes must be smaller than the unit cell sizes that are at largest λ/10. To achieve the resolution necessary to resolve the structural details often necessary for metamaterial designs, nanofeature fabrication using optical, electron beam or focused ion beam lithographies must be used. Thus, as the wavelength of operation decreases, so does the feature size.

The material and fabrication challenges associated with scaling metamaterials toward visible and IR wavelengths constrain the format of potentially useful optical metamaterial devices. Clearly, in the absence of loss compensation [24

24. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006). [CrossRef] [PubMed]

], absorption is an issue needing mitigation. Metamaterial elements based on resonant structures, while providing the greatest and most varied electromagnetic response, also produce significant losses and are best avoided for optical metamaterials. Fortunately, a class of non-resonant metamaterials can be employed, which can provide a large electric response with controlled anisotropy. Such metamaterials have been used at low frequencies for gradient index media [25

25. R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett. 94 (2009).

] as well as for transformation optical media that have been optimized to eliminate the need for magnetic response [26

26. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). [CrossRef] [PubMed]

,27

27. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010). [CrossRef]

]. While optimized transformation optical media have also been demonstrated using dielectric only materials at IR and visible wavelengths in a 2D waveguide format [28

28. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009). [CrossRef] [PubMed]

, 29

29. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009). [CrossRef]

], the inclusion of metal elements can greatly extend the potential range of permittivity or refractive index values. For example, in recent work, Choi et al. reported a metamaterial based on a lithographically patterned gold I-beam structure, experimentally demonstrating a refractive index of as large as n = 8 over a band of terahertz frequencies [13

13. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef] [PubMed]

]. Because the structure was operated far from resonance, the effective refractive index was minimally dispersive (i.e., broad band) and exhibited relatively low loss.

The metamaterial described herein utilizes metallic I-beam structures that operate off resonance to minimize absorption losses. The I-beam metamaterial response arises from the strong polarizability of the I-beam element, which can be tuned by adjusting the size and shape of the I-beam element. In particular, the arms of the I-beams provide capacitive coupling between adjacent elements, which increases the polarizability. Therefore, the effective refractive index is larger when the electric field is polarized along the axis of the I-beam. In addition, in multilayer structures, if the I-beam elements in adjacent layers exhibit strong capacitive coupling, the effective refractive index can be increased to large values. It should be noted, however, that when strong coupling occurs between elements in adjacent layers, much of the simple effective medium theoretical framework breaks down; in particular, the concept of a well-defined wave impedance becomes nebulous, and a description of the system in terms of an electric permittivity and magnetic permeability is called into question. Even for strongly coupled structures, an effective refractive index can always be defined, as is the case with the fish-net metamaterial structure [30

30. C. Rockstuhl, C. Menzel, T. Paul, T. Pertsch, and F. Lederer, “Light propagation in a fishnet metamaterial,” Phys. Rev. B 78 (2008). [CrossRef]

], resulting in layered metamaterial structures that are relevant for gradient index optics [31

31. S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36, 451–453 (2011). [CrossRef] [PubMed]

].

In the present study, we design, fabricate, and characterize a normal incidence graded I-beam metamaterial having an index variation as large as Δn ∼ 3.0 at a wavelength of 10.6μm. The metamaterial was characterized by using a grating structure that operated as a diffractive optic, preferentially directing incident energy into the first order (+1) diffracted beam. The grating is formed by introducing a linear gradient of Δn ∼ 3.0 in the refractive index over a distance of 61μm, and then repeating the linear gradient 12 times to form the equivalent of a blazed grating. To characterize this large refractive index gradient, the theoretical diffraction efficiency predicted from full-wave numerical simulations (that include all geometrical and material aspects of the fabricated structures) was compared with the experimental characterization of the metamaterial grating structure. Because the properties of the constituent element and substrate materials are more difficult to determine at IR wavelengths, obtaining good agreement between simulations and measurements confirms that the constituent materials are being modeled accurately, providing a level of confidence that the techniques can be applied towards the design of more sophisticated IR metamaterial devices. We also introduce a method of multilayer fabrication that can be extended to the large scale manufacture of bulk metamaterial samples.

2. Design and simulation

The design of a gradient-index (GRIN) device can be achieved by progressively varying the geometry of successive metamaterial inclusions as a function of position [32

32. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E 71, 036609 (2005). [CrossRef]

]. For the present work, an I-beam element provides the underlying electric polarization response, with the height of the element and the widths of the I-beam cross bars varied to achieve a range of effective index values. Several variants of the I-beam structure are illustrated in Fig. 1. In a simple sense, as the dimension of the metallic inclusions increases, the polarizability and hence the effective index of the metamaterial increases as well. A nearly arbitrary gradient index profile can thus be designed by introducing a set of metamaterial elements with geometrical parameters that correspond to a large range of index values. As can be seen in Fig.1, low index elements are effectively small disks, while large index elements have I-beam shapes. The I-beam geometry can achieve large values of refractive index well below the resonant frequency of the structure, resulting in a relatively low-loss structure that is also tolerant of fabrication constraints (such as slightly rounded I-beam corners).

Fig. 1 Relationship between the effective refractive index and the geometric parameters. This large span of the refractive index is achieved by making use of 3 geometries: small disk, rectangular bar and I-beam structure, shown in the inset above.

As a preliminary step in the design of the metamaterial grating, the relationship between the effective refractive index and the geometry of metamaterial elements was determined. For this metamaterial design, 75 nm thick gold I-beams embedded in SiO2 was utilized. This metal thickness is a tradeoff between low metamaterial loss (thinner metals) and the maximum achievable refractive index (thicker metals). A 3 nm Cr layer were used as an adhesion layer below each Au layer. The metamaterial elements are parameterized by the height, width, and length of the cross bars of the I-beams, as indicated in Fig. 1. Full wave simulations were performed using COMSOL Multiphysics (a finite-element based solver) on a number of exemplary elements that were embedded in a 1μm × 1μm × 500nm cuboid of SiO2 [See Fig. 3(d)].

Fig. 3 (a,b,c) Retrieved effective constitutive parameters of the I-beam structure shown in (d).

To achieve an accurate prediction of the effective index for each element, it is crucial to include the precise electromagnetic properties of all materials within the simulation volume, notably those of the gold film and of the SiO2 background dielectric. The properties of gold are well known; we made use of the values tabulated in [33

33. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22, 1099–1119 (1983). [CrossRef] [PubMed]

]. The dielectric properties of the SiO2 background dielectric layers were more difficult to describe, since deposition conditions can affect the absorption resonances. To characterize the structures, broadband reflection measurements over the wavelength range 8 – 12μm were performed using a FT-IR spectrometer (Bruker Vertex 80v) at a set of four angles (30, 40, 50, and 60 degrees from normal) on a planar SiO2 film that was 1μm thick, deposited using the same vacuum deposition method as that used to fabricate the grating structure. By fitting the measured reflectance values as a function of angle for each frequency, we were able to obtain the complex index spectrum of the SiO2. A 3 nm thick Cr adhesion layer between the Au and SiO2 was included in the simulation, but it has a negligible effect upon the optical properties of the metamaterial structure. Figure 2 provides a summary of the measured material properties of the vacuum deposited SiO2 and those of the gold (from [33

33. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22, 1099–1119 (1983). [CrossRef] [PubMed]

]). The SiO2 exhibits an absorption resonance around 9.5μm, where the imaginary part of the permittivity peaks. While the absorption resonance renders SiO2 a lossy candidate as a broadband infrared dielectric, the losses drop considerably above 10μm. Since the grating sample was designed to operate at the CO2 laser line of 10.6μm, the losses associated with the SiO2 dielectric are acceptable.

Fig. 2 Material properties: (a) relative permittivity of SiO2; (b) relative permittivity of gold.

To obtain the effective refractive index as a function of these geometrical parameters, we performed a standard retrieval using the simulated reflection and transmission coefficients [34

34. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65 (2002). [CrossRef]

, 35

35. X. D. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70 (2004). [CrossRef]

] for several elements, as shown in Fig. 1. A sufficient number of simulations were performed so that an accurate interpolation of the effective index could be performed as a function of varying geometrical parameters. In the full wave simulations to determine the reflection and transmission coefficients, periodic boundary conditions were applied such that the unit cell was infinitely repeated in the lateral directions. The unit cell was illuminated with a plane wave that was polarized along the axis of the I-beam structure. To more accurately model the realistic fabricated structures, we rounded the corners and edges of the elements (radius of curvature = 30nm). Figure 1 shows the simulated relationship between the I-beam geometry (height (H) and width (W)) and the effective refractive index of the I-beam structure at an input wavelength of 10.6μm. To obtain intermediate refractive index values, the I-beam height (H) and width (W) were varied simultaneously. The dimensions used in the simulations were all within the resolution of the electron beam lithography (EBL) nanolithography system used to fabricate the structures, and resulted in extracted refractive indices that span the range from n = 2.0 to n = 5.5. The minimum value of the refractive index was approximately equal to the background dielectric SiO2 index of n ∼ 2 (at a wavelength of 10.6μm). The maximum index value of the I-beam design is limited by the physical gap between adjacent I-beam elements. The gap chosen for this study was 100nm, which corresponded to a n = 5.5 maximum refractive index value.

The grating structure consists of sequences of unique elements, so there is no repeated “unit cell” as in typical periodic structures. Nevertheless, for the purposes of design, numerical simulations were performed using an infinite periodic 2D sheet of identical elements to determine the effective index of the element. Although the structure was not periodic, we assumed that the extracted refractive index represents the metamaterial elements because the neighboring elements are very similar, and metamaterial experiments at other frequencies have validated this approach [26

26. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). [CrossRef] [PubMed]

, 32

32. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E 71, 036609 (2005). [CrossRef]

].

Many applications in diffractive and GRIN optics rely on modulating the phase of a wave front that transits the optical device. An index modulation can translate to a phase modulation, since the wave at each point acquires a phase proportional to (nk0d), where n is the refractive index, d is the path length, and k0 is the wavenumber in the free space, assuming that multiple reflections inside the slab can be neglected. The optimal performance for a diffractive optical element occurs when the phase difference between the highest index cell and the lowest index cell reaches 2π. For the modest index gradients available using conventional GRIN optic approaches, a relatively long optical path length is required to achieve the optimal condition; however, for the very large index gradient associated with the metamaterial described herein, the optimal grating performance can be achieved for only approximately six I-beam layers.

The equivalent of a blazed diffraction grating can be formed by introducing a linear index gradient that spans a distance of one or more wavelengths. A blazed grating with a linear saw-tooth profile either in its topography or in its index profile can preferentially direct an incident light beam into a selected diffracted order with considerable efficiency. For example, a saw-tooth index profile can be designed such that all incoming light is coupled into the first diffracted order [36

36. B. Kress, Applied digital optics : from micro-optics to nanophotonics (Wiley, Chichester, U.K., 2009).

], so the fraction of an incident beam diffracted into different orders can be used to characterize the index variation in a structure. Thus, even for non-optimized meta-material grating samples, the light coupled into the various diffracted orders provides a nearly direct measurement of the linear index gradient of the structure. Given the relatively large index gradients achievable using metamaterials, a useful sample characterization can be made with only a few metamaterial layers.

The blazed grating sample designed herein had a period of Λ = 61μm. For the indicated grating period, the first order diffracted beams were at ±10° and the second order diffraction beams were at ±20.3°, according to the grating equation θm = sin−1(/Λ), where m is the diffracted order. These diffraction angles were suitable for measurements to characterize the metamaterial sample, since the separation between beams was much larger than the diffraction limited spot size and any associated side lobes. A full-wave simulation of a grating having the expected saw-tooth index profile achievable from the I-beam metamaterial was performed, again using Comsol Multiphysics. A field plot showing the incoming wave (excited at the plane (C)) and the diffracted wave is shown in Fig. 4. The metamaterial blazed grating was approximated in the simulation by a slab with the effective permittivity and permeability continuously graded along one period (Fig. 4 (b)). This approximation is a very important feature of metamaterials because only one unit cell element needs to be simulated; therefore, the simulation speed dramatically improves. Perfectly matching layers (PMLs) were placed both at the top and the bottom of the simulation domain to prevent unwanted reflections at the boundaries. Periodic conditions were applied at the left and right boundaries to simulate a grating with an infinite number of periods. Because of the applied periodic conditions, only one grating period needs to be simulated. Figure 4 shows the result of such simulation, conducted using three periods. The simulation result indicates the diffracted beam is shifted preferentially to the +1st diffracted order.

Fig. 4 (a) Finite element simulations of the out-of-plane electric field patterns from a 6-layer grating with three periods (Λ = 61μm). (A): The plane used to calculate the complex transmittance for the grating; (B): The metamaterial grating region; (C): The excited surface current. The simulation shows that the optical beam is shifted to the positive direction. (b)The refractive index profile of the grating.

To compare this simulation to sample measurements, the far-field diffraction pattern (Fraunhofer regime) was calculated by Fourier transforming the complex transmittance that was numerically obtained from the plane a few wavelengths away from the grating (plane (A) in Fig. 4):
E(νx)=E(x)exp(j2πνxx)dx.
(1)
The diffraction angles can be calculated by θ = sin−1(λνx) for normal incidence [37

37. B. E. A. Saleh, Fundamentals of photonics, Wiley series in pure and applied optics (Wiley-Interscience, Hoboken, N.J., 2007).

], where νx is the spatial frequency and λ is the free space wavelength. The far-field intensity profile can be obtained by multiplying Eq. (1) with its complex conjugate. Using this Fourier transform method, a six layer metamaterial blazed grating approaches optimum performance or, the overall efficiency of the +1st diffracted order is at a maximum. However, due to the severe impedance mismatch and significant absorption in the high index regime, the overall diffraction efficiency of the +1st diffracted order was limited to 76%.

It should be noted that the current metamaterial grating is dispersive. Like any phase grating, the design of the metamaterial grating relies on the phase gained by the wave passing through the gratings, which is wavelength dependent. Furthermore, the metamaterial itself has significant dispersion, as shown in Fig. 2. As the wavelength increases, both the grating and material dispersion contribute to the decreases of the phase contrast between the highest and the lowest refractive index elements, resulting in a decrease in the blazing efficiency.

3. Fabrication

The metamaterial grating sample was fabricated using electron beam lithography to pattern Au I-beams and vacuum deposited SiO2 dielectric layers. Fabrication began with the choice of a suitable infrared transparent substrate; in this case, double-side polished germanium (< 100 > oriented, undoped, ρ > 40Ω·cm, ∼ 325μm thick) was used. Electron beam vacuum evaporated SiO2 was selected as the interlayer dielectric, due to ease of deposition and good infrared transmittance at λ = 10.6μm. The multilayer metamaterial design required 500 nm of interlayer dielectric separating each nanopatterned metal I-beam layer. To maintain symmetry, the initial and final layers of SiO2 were 250 nm thick, while each SiO2 layer separating adjacent metamaterial layers was 500 nm thick, as illustrated in Fig. 5. Electron beam lithography (EBL, Elionix ELS-7500) was used to pattern the metamaterial elements, filling an area of 732μm × 732μm (Fig. 6). Electron beam evaporation was then used to deposit a 3 nm Cr film (adhesion layer) and a 75 nm thick gold film, with a lift-off step used to define the desired metal nanofeatures comprising a single layer of the metal in the metamaterial grating.

Fig. 5 Process flow for the fabrication of the multi-layer metamaterial blazed grating. In step (i) the first SiO2 layer was deposited and the first metamaterial layer was patterned. Step (ii) shows the conformal nature of the evaporated SiO2 inter-layer dielectric. In the next step, (iii), a sacrificial planarizing layer of BCB was deposited. Reactive ion etching then removed the BCB and a thin layer of SiO2, as shown in step (iv). A second metamaterial layer was then patterned (step (v)). This process was repeated to build up multiple layers. As a final step, the upper layer of metamaterial was coated with a capping layer of SiO2 (250 nm thick, similar to the initial layer shown in step (i)).
Fig. 6 Scanning electron micrographs of the fabricated metamaterial diffraction grating. Top: entire 732μm × 732μm device area. Lower-left: Three periods of the periodic index profile; lower-right: High magnification of fabricated metamaterial I-beam elements (the elements are spaced with periodicity of 1μm).

4. Sample characterization

To characterize the metamaterial grating, the relative diffraction efficiencies of incident light into the different diffracted orders were measured. The experimental setup consisted of a CO2 Laser (Lasy3S, Access Laser Company), the fabricated metamaterial grating sample, and an infrared camera (FLIR photo 640) that was mounted on a rotational arm. As depicted in Fig. 7(a), the diffraction grating was illuminated with a linearly polarized beam at a wavelength of 10.6μm, which was generated by placing a linear polarizer at the output of the CO2 laser source. The laser power was adjusted such that the pixel counts of the 0th and +1st diffraction spots were well within the linear range of the camera. This adjustment was achieved by modulating the duty cycle of the laser pulse. A neutral density (ND) filter was used to further attenuate the laser power. Additionally, we placed the ND filter at an angle so that we could make use of the reflected light to monitor the laser power fluctuations; the measured fluctuations were less than ±5% throughout the experiments. Since the laser spot size was larger than the metamaterial grating, a 725μm × 725μm square aperture (gold surrounded the aperture window) was placed in front of the grating sample to block the incident light outside of the 732μm × 732μm grating area. The aperture was aligned approximately 1.5 mm from the metamaterial sample, which delivered an estimated 94% of the incident light passing through the aperture to the grating area. Figure 7(b) shows an experimentally obtained image of three diffraction spots, the −1st, 0th, and +1st diffracted orders, which were measured by placing an infrared camera at a distance of approximately 3.5 cm from the grating sample. The image clearly shows that the brightest spot occurs at the +1st diffracted order, consistent with the designed sample operation. In Fig. 7(c), we plotted the intensity profile around these three major peaks by binning the image (b) to a 1D array and normalizing to the +1st diffracted order spot.

Fig. 7 (a) The experimental setup for measuring grating efficiency; (b) Experimentally obtained image of the diffraction spots from the −1st, 0th and +1st diffracted orders; (c) The intensity profile of image (b).

We moved the camera farther away from the grating sample, to a distance of approximately 11 cm, to ensure the measurements were performed in the far-field, Fraunhofer regime, for comparison with the simulations using the Fourier transformation (Eq. (1)). This distance is much larger than the Fraunhofer criteria of a2/4λ ∼ 1cm [38

38. M. Born, Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, Cambridge ; New York, 1999). [PubMed]

], where a is the width of the grating sample. This larger distance resulted in a larger beam waist (55 pixels, or equivalently 1.4 mm) and larger spacing between spots; therefore, only one spot can be imaged in the field-of-view of the camera. Since the distance between spots now extended beyond the camera field-of-view, we rotated the arm so that the camera could measure the spots from each diffracted order. Each diffracted order was imaged separately. The background noise was measured by a single shot in the absence of laser illumination. In post processing, the background noise was subtracted from each measured diffracted signal image on a pixel-by-pixel basis. The relative intensity of each diffracted order was calculated by integrating the intensity of the main lobe and the first side lobes (altogether occupying an area of 200 × 200 pixels). The intensities were normalized to the +1st order, and then compared to the simulation results, as shown in Fig. 8.

Fig. 8 Experiment vs. simulation results for a 4-layer metamaterial grating.

The four layer metamaterial grating structure shifted a significant fraction of the beam into the +1st diffracted order, as shown in Fig. 8. Both the simulation and experimental results were normalized to the +1st diffracted order to obtain the normalized diffraction efficiency. Sources of the errors in the experiment include ±5% laser fluctuations and alignment inaccuracies; sources of the errors in the simulation include ±5% possible variations of the thickness of the SiO2 in a unit cell. It is worth noting that a decrease in the SiO2 thickness not only reduces the optical path length in the SiO2, but also increases the effective refractive index. Therefore, the overall phase shift (nk0d) is not a linear function of the SiO2 thickness, resulting in the error bars in the simulation being not symmetrically distributed.

To extract the index variation of the fabricated metamaterial structure, we compared the experimentally measured 0th order normalized efficiency (∼ 0.8) with a range of relative efficiencies in the numerical simulations (assuming linear index gradients). The normalized efficiency from the 0th diffracted order agrees well with the experimental results for an index variation of Δn = 3.0. This demonstrates index variation in the metamaterial grating is the highest reported for a 3D infrared device, and further, demonstrates that the refractive index can be spatially controlled using metamaterial inclusions.

5. Summary

In conclusion, we have designed, fabricated, and quantitatively characterized a metamaterial with a Δn = 3.0 using a blazed grating to characterize the metamaterial structure. The saw-tooth gradient refractive index profile was implemented with a set of metamaterial elements ranging from small disks to I-beam structures. These 61 distinguishable elements approximate a linear phase shift gradient with 61 steps. Electron beam lithography and vacuum deposition were used to fabricate the four layers of the metamaterial structure. The metamaterial sample characterization resulted in good agreement with the numerical simulations of the structure, as determined through a comparison of the 0th order and +1st order diffraction efficiencies. The significance of this report is two-fold. First, a refractive index variation of Δn = 3.0 is, to our knowledge, the largest for an IR device yet reported in the literature. Over the grating period of 61μm, this corresponds to a refractive index gradient of 492cm−1. Second, the combination of fabrication techniques introduced here can achieve spatial control of the refractive index in a single device, which suggests the possibility for the implementation of large refractive index gradient and transformation optical metamaterial structures. The use of metal inclusions is particularly relevant for gradient structures with controlled anisotropy, as might be required to manage polarization in multifunctional optical devices. Combining the large index gradients with nanolithogrphic fabrication, the results presented suggest a path towards extremely complex media relevant to advanced IR optics.

Acknowledgments

References and links

1.

Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. 103, 1834–1841 (2007). [CrossRef]

2.

G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16, 11540–11547 (2008). [PubMed]

3.

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnol. 8, 53 (1997). [CrossRef]

4.

P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998). [CrossRef]

5.

U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A 22, 724–733 (2005). [CrossRef]

6.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. 98, 243901 (2007). [CrossRef] [PubMed]

7.

H. Elfström, M. Kuittinen, T. Vallius, B. H. Kleemann, J. Ruoff, and R. Arnold, “Fabrication of blazed gratings by area-coded effective medium structures,” Opt. Commun. 266, 697–703 (2006). [CrossRef]

8.

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011). [CrossRef] [PubMed]

9.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

10.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004). [CrossRef] [PubMed]

11.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004). [CrossRef] [PubMed]

12.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

13.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef] [PubMed]

14.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

15.

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]

16.

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]

17.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

18.

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]

19.

J. Neu, B. Krolla, O. Paul, B. Reinhard, R. Beigang, and M. Rahm, “Metamaterial-based gradient index lens with strong focusing in the THz frequency range,” Opt. Express 18, 27748–27757 (2010). [CrossRef]

20.

R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Opt. Express 17, 21030–21041 (2009). [CrossRef] [PubMed]

21.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–37 (2008). [CrossRef]

22.

G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic metamaterial,” Opt. Lett. 32, 551–553 (2007). [CrossRef] [PubMed]

23.

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]

24.

A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006). [CrossRef] [PubMed]

25.

R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett. 94 (2009).

26.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). [CrossRef] [PubMed]

27.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010). [CrossRef]

28.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009). [CrossRef] [PubMed]

29.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009). [CrossRef]

30.

C. Rockstuhl, C. Menzel, T. Paul, T. Pertsch, and F. Lederer, “Light propagation in a fishnet metamaterial,” Phys. Rev. B 78 (2008). [CrossRef]

31.

S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36, 451–453 (2011). [CrossRef] [PubMed]

32.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E 71, 036609 (2005). [CrossRef]

33.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22, 1099–1119 (1983). [CrossRef] [PubMed]

34.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65 (2002). [CrossRef]

35.

X. D. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70 (2004). [CrossRef]

36.

B. Kress, Applied digital optics : from micro-optics to nanophotonics (Wiley, Chichester, U.K., 2009).

37.

B. E. A. Saleh, Fundamentals of photonics, Wiley series in pure and applied optics (Wiley-Interscience, Hoboken, N.J., 2007).

38.

M. Born, Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, Cambridge ; New York, 1999). [PubMed]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(110.2760) Imaging systems : Gradient-index lenses
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: September 27, 2011
Revised Manuscript: November 2, 2011
Manuscript Accepted: November 3, 2011
Published: November 14, 2011

Citation
Yu-Ju Tsai, Stéphane Larouche, Talmage Tyler, Guy Lipworth, Nan M. Jokerst, and David R. Smith, "Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths," Opt. Express 19, 24411-24423 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24411


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References

  1. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci.103, 1834–1841 (2007). [CrossRef]
  2. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express16, 11540–11547 (2008). [PubMed]
  3. P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnol.8, 53 (1997). [CrossRef]
  4. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett.23, 1081–1083 (1998). [CrossRef]
  5. U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, “Implementation of a graded-index medium by use of subwavelength structures with graded fill factor,” J. Opt. Soc. Am. A22, 724–733 (2005). [CrossRef]
  6. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98, 243901 (2007). [CrossRef] [PubMed]
  7. H. Elfström, M. Kuittinen, T. Vallius, B. H. Kleemann, J. Ruoff, and R. Arnold, “Fabrication of blazed gratings by area-coded effective medium structures,” Opt. Commun.266, 697–703 (2006). [CrossRef]
  8. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express19, 5156–5162 (2011). [CrossRef] [PubMed]
  9. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305, 788–792 (2004). [CrossRef] [PubMed]
  10. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science303, 1494–1496 (2004). [CrossRef] [PubMed]
  11. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306, 1351–1353 (2004). [CrossRef] [PubMed]
  12. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett.95, 137404 (2005). [CrossRef] [PubMed]
  13. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011). [CrossRef] [PubMed]
  14. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001). [CrossRef] [PubMed]
  15. 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]
  16. 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]
  17. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
  18. 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,” Science314, 977–980 (2006). [CrossRef] [PubMed]
  19. J. Neu, B. Krolla, O. Paul, B. Reinhard, R. Beigang, and M. Rahm, “Metamaterial-based gradient index lens with strong focusing in the THz frequency range,” Opt. Express18, 27748–27757 (2010). [CrossRef]
  20. R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Opt. Express17, 21030–21041 (2009). [CrossRef] [PubMed]
  21. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater.7, 31–37 (2008). [CrossRef]
  22. G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic metamaterial,” Opt. Lett.32, 551–553 (2007). [CrossRef] [PubMed]
  23. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett.30, 3356–3358 (2005). [CrossRef]
  24. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett.31, 2169–2171 (2006). [CrossRef] [PubMed]
  25. R. P. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett.94 (2009).
  26. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323, 366–369 (2009). [CrossRef] [PubMed]
  27. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9, 129–132 (2010). [CrossRef]
  28. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8, 568–571 (2009). [CrossRef] [PubMed]
  29. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3, 461–463 (2009). [CrossRef]
  30. C. Rockstuhl, C. Menzel, T. Paul, T. Pertsch, and F. Lederer, “Light propagation in a fishnet metamaterial,” Phys. Rev. B78 (2008). [CrossRef]
  31. S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett.36, 451–453 (2011). [CrossRef] [PubMed]
  32. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71, 036609 (2005). [CrossRef]
  33. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt.22, 1099–1119 (1983). [CrossRef] [PubMed]
  34. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65 (2002). [CrossRef]
  35. X. D. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E70 (2004). [CrossRef]
  36. B. Kress, Applied digital optics : from micro-optics to nanophotonics (Wiley, Chichester, U.K., 2009).
  37. B. E. A. Saleh, Fundamentals of photonics, Wiley series in pure and applied optics (Wiley-Interscience, Hoboken, N.J., 2007).
  38. M. Born, Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, Cambridge ; New York, 1999). [PubMed]

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