Infrared Dielectric Resonator Metamaterial
James C. Ginn, Igal Brener, David W. Peters, Joel R. Wendt, Jeffrey O. Stevens, Paul F. Hines, Lorena I. Basilio, Larry K. Warne, Jon F. Ihlefeld, Paul G. Clem, Michael B. Sinclair
IInfrared Dielectric Resonator Metamaterial
James C. Ginn ∗ and Igal Brener Sandia National Laboratory, Albuquerque, NM 87185, USA. andCenter for Integrated Nanotechnologies, Sandia National Laboratory, Albuquerque, NM 87185, USA.
David W. Peters, Joel R. Wendt, Jeffrey O. Stevens, Paul F. Hines, Lorena I.Basilio, Larry K. Warne, Jon F. Ihlefeld, Paul G. Clem, and Michael B. Sinclair
Sandia National Laboratory, Albuquerque, NM 87185, USA. (Dated: May 21, 2018)We demonstrate, for the first time, an all-dielectric metamaterial resonator in the mid-waveinfrared based on high-index tellurium cubic inclusions. Dielectric resonators are desirable comparedto conventional metallo-dielectric metamaterials at optical frequencies as they are largely angularinvariant, free of ohmic loss, and easily integrated into three-dimensional volumes. With theselow-loss, isotropic elements, disruptive optical metamaterial designs, such as wide-angle lenses andcloaks, can be more easily realized.
PACS numbers: 81.05.Xj, 78.67.Pt, 85.50.-n
The unique properties of metamaterials have yieldedmany exciting electromagnetic phenomena including sub-diffraction-limited imaging [1], cloaking [2], and perfectabsorption [3]. In spite of the rapid advances in thisfield, passive metamaterials at optical frequencies haveoften proven impractical due to significant conductor lossfrom the metallic resonators comprising these volumes[4]. Additionally, the inherent geometrical asymmetry ofthese resonators further restricts metamaterial behaviourto a small range of incident angles even when assembledinto three-dimensional structures [5]. Three-dimensionaldielectric resonators, unlike their metallic counterpart,have significantly less material loss, support resonantmodes that are invariant to the excitation angle, and canbe easily integrated into thick volumes. In this letter, wedescribe the development of a dielectric resonator basedmetamaterial in the infrared with spectral regions of neg-ative magnetic and electric effective properties. Addi-tional insight is also provided in addressing material lim-itations imposed on dielectric metamaterials at opticalfrequencies.Lewin [6] theoretically demonstrated that an array ofsub-wavelength dielectric resonators can exhibit spectralregions of Lorentzian-like effective permittivity and per-meability. Following Mie theory [7], the effective electricand magnetic polarizabilities of a densely packed arrayof sub-wavelength spheres can be altered by changing thedimensions, composition, and packing fraction of the in-clusions. Similar behavior can be achieved using cubicdielectric resonators (CDRs) [8] which, unlike spheres,are compatible with existing nano-scale lithography tech-niques and can be integrated into multi-layer compositesthrough repeated steps of thin-film deposition, etching,and planarization. Since there are no analytical expres-sions for predicting the behavior of a cubic scatterer [9]and approximations are problematic due to the high in- ∗ Present Address: Plasmonics Inc, Orlando, FL 32826, USA.;Electronic Mail: [email protected]
FIG. 1. (color online). (a) Excitation configuration of anisolated sphere (top row) and a 1:1 periodic cube array (bot-tom row). (b) normalized electric field distribution for thelowest-order magnetic and (c) normalized electric field distri-bution for the lowest-order electric mode in the sphere andcubic resonators. Dotted white lines indicate field direction. dex dispersion of materials in the infrared, numericalcomputational electromagnetic approaches must be usedfor analysis. In Fig. 1 the analytically determined on-resonance field distribution of an isolated sphere is com-pared to that of a cubic resonator array calculated usingthe commercially available rigorous coupled wave anal-ysis (RCWA)[10] package, GDCALC. Like the sphericalresonator, the lowest-order mode of a CDR is a mag-netic dipole (TE ) and the second-lowest mode is anelectric dipole (TM ). However, in contrast to tradi-tional metallic resonators the electromagnetic responsesof single spherical inclusions are isotropic, and cubic res-onators represent only a minor perturbation from thisspherical symmetry. Also, whereas significant dampingdue to ohmic loss is unavoidable for metallic resonatorsin the infrared [11], the damping of dielectric resonatorscan be quite low provided the resonator material lacksactive carriers or phonon modes in the band of inter- a r X i v : . [ phy s i c s . op ti c s ] A ug FIG. 2. (color online). (a) Scanning electron micrograph of fabricated CDR. (b) Measured reflection and transmission coeffi-cients for CDR. Field patterns from Fig. 1 are shown above each corresponding resonance. est. The scattering cross-sections exhibited by such un-damped systems are known to be equal to the fundamen-tal limits imposed by Mie theory [12].Thus, identification of candidate low-loss resonatormaterials is critical in designing a practical infrared CDRmetamaterial. In addition, the CDR material must pos-sess a large index of refraction to ensure that the dimen-sions of the resonator and array spacing are sufficientlysmall (non-diffracting) compared to the operating wave-length. Only two classes of dielectrics exhibit positiveindices of refraction greater than three in the infrared:highly crystalline polaritonic materials and narrow band-gap ( < µ m[18]. For CDR applications, a polycrystalline morphol-ogy is preferable which yields a crystal-averaged index ofrefraction of 5.3 at 10 µ m, with an extinction coefficientof less than 10 − [18]. Barium fluoride (BaF ) was se-lected as the optimal substrate due to its low refractiveindex (n ∼ µ m.Through simulation, a 1.7 µ m CDR with a 3.4 µ m unit- cell spacing (1:1 duty-cycle) was chosen to center the re-flection peak of the magnetic resonance at 10 µ m. Priorto patterning, a 1.7 µ m thick film of Te was depositedon the surface of a 25 mm diameter BaF optical flatvia electron-beam evaporation. Both x-ray diffractionand variable angle spectral ellipsometric analysis veri-fied that the film was predominately polycrystalline, andellipsometry analysis yielded a fitted complex index ofrefraction of n = 5.02 + 0.04j. The Te film was pat-terned using electron beam lithography and etched usinga reactive ion etching (RIE) process. A scanning elec-tron micrograph of the etched pattern is shown in Fig.2a. The etching process resulted in excellent uniformityover a 1 cm area, with only a slight over-etching of thepattern. The final CDR element was 1.7 µ m tall witha 1.53 x 1.53 µ m base and a 10 degree sidewall slope.The overall process required significantly less steps thanexisting three-dimensional metallo-dielectric lithography[5] and the feasibility of planarizing the patterned re-gion for multi-layer fabrication was verified by success-fully spin-coating a thin-film of polynorbornene on thesurface of the CDR array [19]. Although the CDRs havean isotropic resonance mode, multi-layer fabrication isnecessary for an angular-independent array response.Following fabrication, the patterned wafer was char-acterized using a hemispherical directional reflectometer(HDR) at an angle of incidence of seven degrees. Themeasured collimated transmission and specular reflectionof the array are plotted in Fig. 2b. As expected from thefabricated topology and measured index of the Te film,the reflection peak of the magnetic resonance occurredat 9 µ m and the reflection peak of the electric resonanceoccurred at 7.5 µ m. The resonances are well defined andoccur at wavelengths above the diffraction cut-off limit.In the spectral region between the two resonances, loss (1reflection transmission) drops to less than 8%. We notethat no artificial correction factors were used to accountfor reflection and absorption losses due to the substrate.The optical response of the fabricated cube array was FIG. 3. (color online). (a) Plot of simulated reflectionand transmission for CDR. (b) Plot of calculated impedancephase, permittivity, and permeability for the simulated CDRarray. Real values are in red and imaginary values are in blue. simulated (Fig. 3). The simulated and measured co-efficients show overall good agreement (Fig. 3a), withdifferences primarily arising from the asymmetry andnon-uniformity of the as-fabricated cube. The calculatedsurface impedance (Fig. 3b) indicates two distinct reso-nances with inverted phase delays. Positive phase delayoccurs when the electric field leads the magnetic field inphase (permeability less than zero) and negative phasedelay occurs when the electric field follows the magneticfield (permittivity less than zero). Using a standard re-trieval algorithm [20] for the planar array, the effectivepermittivity and permeability were calculated for thesetwo regions (inset 3b). In both cases, the extracted pa-rameters reach values of less than -1, and the loss tangentof the permeability falls to 0.48 when the real part of per-meability is equal to -1. We note that the extinction coef-ficient of the as deposited film is more than two orders ofmagnitude larger than the literature value for Te. Thus,we anticipate that significantly lower-loss metamaterialswill be achievable as the Te loss is minimized.To further investigate the features of infrared CDRs,a series of RCWA simulations was run for a 1:1 duty-cycle array while only varying the refractive index of thecubes. The solid lines in Fig. 4 denote the normal-ized wavelength at the points of peak reflectivity (due
FIG. 4. (color online). Design metric for 1:1 CDR metama-terials. Solid lines correspond to the lowest-order magnetic(red) and electric (blue) resonances. The top dotted linedefines where peak positive permeability/permittivity occursand the bottom line defines where permeability/permittivityis equal to zero. The indices of several materials at 10 µ m arealso labeled. to the two primary resonances), versus index of refrac-tion (restricted to a range of values that is realistic forknown optical materials). These lines also directly corre-spond to the point where the CDR array switches fromsupporting a propagating mode (positive effective per-meability/permittivity) to a plasma mode (negative ef-fective permeability/permittivity). The effective perme-ability and permittivity were also calculated using theretrieval algorithm [20] assuming a fixed extinction coef-ficient of 0.001 for algorithm stability. From these cal-culated values, the points of peak effective positive per-meability/permittivity and the zero crossing of the ef-fective permeability/permittivity curves were determinedand plotted as dotted lines on Fig. 4.Several critical performance metrics for optical CDRscan be gleaned from Fig. 4 and related analysis. Asexpected, the resonant wavelength of a CDR will scalelinearly with the dimensions and index of refraction ofthe cube. This behavior explicitly limits the practicalityof negative permitivity metamaterials in the mid-waveinfrared based around traditional high-index materialssuch as silicon and germanium. This limitation is alsomore severe in the optical regime when compared to therf and THz portions of the spectrum where materialswith indices surpassing thirty exist [21]. These arraysalso exhibit appreciable spatial dispersion [22] that de-creases asymptotically with increasing normalized wave-length (the array period is decreasing relative to the reso-nant wavelength). Spatial dispersion dominates when theeffective index of the CDR array exceeds its normalizedwavelength and field homogenization breaks down. Inthis regime, a photonic crystal band-gap mode is excitedand retrieved optical properties no longer hold physicalmeaning. Consequently, the peak effective positive per-meability/permittivity curve defines the largest index of FIG. 5. (color online). Specular (red line) and diffuse (blueline) transmission for measured CDR. refraction supported by the array at normal incidenceand the start of the band-gap region. This regime per-sists until the resonance line is crossed and the behav-ior of the CDR array becomes dominated by the plasmamode for which spatial dispersion can largely be ignored.More exotic behaviors, such as doubly negative parame-ters, can be realized by mixing two dielectric resonatorswith different indices of refraction or different dimensionsin a multi-layer composite.One of the advantages of characterizing the fabricated metamaterial with an HDR is that it allowed for exper-imental validation of the onset of spatial dispersion inthe fabricated CDR. Fig. 5 shows a comparison of themeasured specular and diffuse (diffracted) transmission.From the figure, the region where the device is dominatedby the photonic crystal band-gap mode corresponds di-rectly to the appearance of appreciable coupling of in-cident light into diffracted orders. This behavior alsomanifests near in the center of the metamaterial’s reso-nance and is stronger for the electric mode, as expectedfrom theory. Furthermore, this confirms the effective lossof the CDR is much larger than loss associated with ma-terial absorption.In this letter, we have described the design, fabrica-tion, and characterization of a dielectric cubic resonatormetamaterial with electric and magnetic activity in themid-infrared. Through theory and simulation, a general-ized design approach for metamaterial surfaces compris-ing of cubic resonators at optical frequencies was devel-oped. This work represents a first step toward the devel-opment of passive low-loss, multi-layer, isotropic meta-material devices in the infrared.This research was supported by the Laboratory Di-rected Research and Development program at Sandia Na-tional Laboratories. This work was performed, in part,at the Center for Integrated Nanotechnologies, a U.S. De-partment of Energy, Office of Basic Energy Sciences userfacility. 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