Lattice models of nontrivial “optical spaces” based onmetamaterial waveguidesAlexei I. Smolyaninov* and Igor I. Smolyaninov

Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA*Corresponding author: a.smoly@gmail.com

Received April 7, 2011; revised May 26, 2011; accepted May 30, 2011;posted May 31, 2011 (Doc. ID 145442); published June 21, 2011

Metamaterials are being used to model various exotic “optical spaces” for such applications as novel lenses andcloaking. While most efforts are directed toward the engineering of continuously changing dielectric permittivityand magnetic permeability tensors, an alternative approach may be based on lattices of metamaterial waveguides.Here we demonstrate the power of the latter technique by presenting metamaterial lattice models of variousfour-dimensional spaces. © 2011 Optical Society of AmericaOCIS codes: 160.3918, 000.6800.

Electromagnetic metamaterials can be used to emulatehighly unusual “optical spaces,” enabling such applica-tions as novel transformation optics (TO)-based lensesand cloaking [1–3]. This development is enabled by thenewfound freedom of continuous control of the localdielectric permittivity εik and magnetic permeability μiktensors in electromagnetic metamaterials. Even thoughless prominent, another direction in metamaterial re-search may be characterized as “lattice-based meta-material models.” These models are based on thenetworks of metamaterial waveguides, which controlpropagation of the electromagnetic signals within a giventhree-dimensional (3D) volume. Examples of this ap-proach may be found in a couple of recent experimentalrealizations of electromagnetic cloaking [4,5]. The goal ofthis Letter is to demonstrate that the latter approach to“optical space” design with metamaterials is also verypowerful, and it may supplement the more common ap-proach of continuous engineering of εik and μik tensors.For example, lattice-based metamaterial designs allowstraightforward realization of various non-Euclidean “op-tical spaces,” which were suggested as a pathway tobroadband cloaking [6].We demonstrate the ultimate power of the lattice-

based approach by presenting metamaterial lattice mod-els of various hypercubic four-dimensional (4D) spaces,which are impossible to emulate using conventional TO.While our claim may seem unusual, we must point outthat “hypercubic network architectures” for multiproces-sor computing are being widely described in computerengineering literature. It is well known that hypercubiclattice networks of any spatial dimension D can be pro-jected onto 3D space. Moreover, many of these hypercu-bic networks had been realized in the experiment [7]. Anobvious advantage of implementing a lattice-based 4Dmetamaterial “optical hyperspace” is that any 3D non-Euclidean space may easily fit into such 4D space [8].Therefore, a natural platform for the proposed non-Euclidean broadband TO designs will be provided.The simplest model of a 4D lattice projected onto

regular 3D space is presented in Fig. 1, which shows aperspective projection of an elementary unit of a 4D hy-percubic spatial lattice, and a perspective projection of a2 × 2 × 2 × 2 region of the hypercubic lattice onto a regionof 3D space. The elementary unit is filled with metal, ex-

cept for the black lines, which represent pieces of thinsingle-mode coaxial waveguides, which are engineeredto have the same impedance

Z ¼ 12π

ffiffiffiμε

rlnDd

ð1Þ

and optical length L ¼ aðεμÞ1=2 (where a is the waveguidelength, d is the diameter of the inner conductor, and D isthe inner diameter of the outer electrode). Connection ofthe cables in the vertices is realized using beam splitters.Eight cubic faces of this lattice act as boundaries of theeight neighboring hypercubic cells (similar to the way inwhich each square face in a 3D cubic lattice acts as aboundary of the neighboring cube). Thus, by keepingL and Z constant for each edge in the lattice, we can cre-ate a good lattice model of the 4D space: electromagneticsignals launched into the lattice would travel from vertexto vertex as if they propagate in a 4D hypercubic lattice.Because of metal filling, the cables communicate only atthe vertices.

We have confirmed this approach by electromagneticsimulations performed using the Ansoft HFSS softwarepackage. Ansoft HFSS is an iterative matrix solver of

Fig. 1. (Color online) (a) Perspective projection of an elemen-tary unit of a 4D hypercubic spatial lattice, which shows theðx; y; z; wÞ coordinates of each vertex. The elementary unit isfilled with metal, except for the black lines, which representpieces of thin single-mode waveguides engineered to havethe same impedance and the same optical length. (b) Perspec-tive projection of a 2 × 2 × 2 × 2 region of the hypercubic lattice:three 2 × 2 × 2 elements of the cubic lattice are shifted along theprojected “fourth orthogonal direction.”

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0146-9592/11/132420-03$15.00/0 © 2011 Optical Society of America

Maxwell’s equations that applies the finite elementmethod to a mesh generated for a physical model. Ournumerical data are presented in Figs. 2 and 3. In Fig. 2we used the HFSS simulation engine to model a hyper-cubic lattice with the software suite’s built-in Hertzian di-pole source at the geometric center of the lattice. BothFig. 2(a) and 2(b) used the same technique applied toa slightly different lattice structure, as shown in the fig-ures. Maps of electric field distribution inside 2 × 2 × 2 ×2 and 3 × 3 × 3 × 3 hypercubic lattices of waveguides arepresented in Figs. 2(a) and 2(b), respectively. In bothcases a radiation source was placed at the origin pointð0; 0; 0; 0Þ of the hypercubic lattice. The simulations wereperformed at 0:5THz. For the sake of clarity, in the caseof the 3 × 3 × 3 × 3 lattice, the field magnitude is shownonly for four 3D sections of the hypercube in Fig. 2(b).The effective “4D distance” R inside these lattices can beintroduced as R2 ¼ x2 þ y2 þ z2 þw2, wherew is the dis-tance along the projected “fourth spatial dimension”. Measurements of field intensity I as a function of R

are supposed to determine the effective dimensionalityof the created lattice space. In the 3D space, I ∼ 1=R2,while in the 4D space, I ∼ 1=R3. Data plotted in Fig. 3came directly from the results shown in Fig. 2(b) by tak-ing the magnitude of the E field at the lattice points. Asshown in Fig. 3, field intensity measured at the lattice ver-tices plotted as a function of R−3 is consistent with the 4Dcharacter of electromagnetic field propagation inside thelattice (at small Rs, minor deviation from R−3 behavior isnatural due to near-field effects). These simulations de-monstrate the practical feasibility of creating metamater-ial models of the 4D space. In fact, a similar technique isalso applicable to lattice models of other higher dimen-sional D > 4 spaces since higher dimensional lattices canbe projected onto 3D space in a similar manner [7]. How-ever, we must emphasize that the lattice models used insuch applications as cloaking must be large enough toemulate “continuous” space [4,5].

Even though making very large 4D lattice modelsmay not be easy, a lattice model of the classic five-dimensional (5D) Kaluza–Klein space–time [9]

ds2 ¼ gαβdxαdxβ þ 2gα5dxαdϕþ g55dϕ2 ð2Þ

(where the Greek indices α ¼ 0, 1, 2, 3 indicate coordi-nates of the 4D space–time, the fifth spatial dimensionϕ is compacted, and its circumference g551=2 is small)seems to be feasible using ε near zero waveguidesdescribed by Edwards et al. [10]. These waveguideswould need to connect vertices of the emulated 4D hy-percubic lattice with lattice coordinates ðx; y; z; 0Þ andðx; y; z; g551=2Þ. As a result, spatial geometry of the 4Dmetamaterial lattice space has one spatial dimension,which is compacted. This concept has been demon-strated in our numerical simulations presented in Fig. 4,which shows propagation of a Kaluza–Klein mode insidea lattice of waveguides, which emulates the Kaluza–Klein space–time described by Eq. (2). The results inFig. 4 were generated on the same lattice as Fig. 2(a),but the software suite’s periodic boundary conditionswere used at the vertices to simulate the compacted “fifthdimension.” In good agreement with the theory [9], an

Fig. 2. (Color online) (a) HFSS simulations of electric fieldmagnitude inside the 2 × 2 × 2 × 2 hypercubic lattice of wave-guides. Radiation source is placed at the origin ð0; 0; 0; 0Þ.(b) HFSS simulations of electric field magnitude inside the 3 ×3 × 3 × 3 hypercubic lattice of waveguides. For the sake ofclarity, the field magnitude is shown only for four 3D sectionsof the hypercube.

Fig. 3. (Color online) Field intensity measured at the latticevertices plotted as a function of R−3 is consistent with the4D character of field propagation at large Rs.

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inset in Fig. 4 clearly demonstrates a standing wave char-acteristic of the Kaluza–Klein electromagnetic modealong the projected fifth lattice dimension (an individualwaveguide in the inset is oriented along the projected“fifth spatial dimension”).In conclusion, we have demonstrated that the meta-

material lattice-based approach to “optical space” design

may supplement the more common approach of contin-uous engineering of εik and μik tensors. We demonstratethis by presenting metamaterial lattice models of varioushypercubic 4D spaces, which are impossible to emulateusing conventional TO. These lattice models provide anatural platform for various non-Euclidean broadbandTO designs, as suggested in [6]. In addition, studyingfundamental linear and nonlinear optics in the emulated“4D space” may be quite an interesting fundamentalexercise [11].

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Fig. 4. (Color online) HFSS simulations of a Kaluza–Kleinmode propagation inside a lattice of waveguides, whichemulates the 5D space–time described by Eq. (2). The insetin the top right corner shows field magnitude inside an indivi-dual waveguide, which is oriented along the projected “fifthspatial dimension.” Standing wave character of the Kaluza–Klein mode along the emulated fifth lattice dimension is clearlydemonstrated.

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