Experimental and numerical studies of terahertz surface waves on a thin metamaterial film
aa r X i v : . [ phy s i c s . op ti c s ] D ec Experimental and numerical studies of terahertz surface waves on a thinmetamaterial film
Benjamin Reinhard, ∗ Oliver Paul, Ren´e Beigang,
1, 2 and Marco Rahm
1, 2 Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany Fraunhofer Institute for Physical Measurement Techniques IPM, 79110 Freiburg, Germany
We present experimental and numerical studies of localized terahertz surface waves on asubwavelength-thick metamaterial film consisting of in-plane split-ring resonators. A simple andintuitive model is derived that describes the propagation of surface waves as guided modes in awaveguide filled with a Lorentz-like medium. The effective medium model allows to deduce the dis-persion relation of the surface waves in excellent agreement with the numerical data obtained from3-D full-wave calculations. Both the accuracy of the analytical model and the numerical calculationsare confirmed by spectroscopic terahertz time domain measurements.
Metamaterials have gained a great deal of interest in thelast decade due to their important role as designer ma-terials with tailorable electric and magnetic properties.In this context, metamaterials provide a comprehensivetool box for the design of optical components with avery specific and in some cases even exotic electromag-netic behavior. From the conceptional point of view,the development of such optics requires the design ofindividual subwavelength-sized unit cells. When assem-bled properly, these subwavelength units stamp theirelectromagnetic characteristics on the effective macro-scopic properties of the resulting metamaterial. Thatway, it is possible to create optics with a very specificfunctionality.While free-space optics requires three-dimensionalbulk metamaterials in most of the cases, it was shownthat the surfaces of (thin) metamaterials (also termedmeta-surfaces) can support the propagation of surfacewaves under well-defined conditions. Compared to thehighly sophisticated methods that must be employedfor the fabrication of bulk metamaterials, the tech-niques for the fabrication of most types of meta-surfacesare considerably less involved. As an intriguing anal-ogy, the observed surface waves on meta-surfaces arevery similar in their properties to surface plasmon po-laritons on interfaces between a metal and a dielectric.In contrast to metals yet, the meta-surfaces can bespecifically designed as to their effective material pa-rameters. In consequence, it is possible to tune the dis-persion characteristics and spatio-temporal behavior ofsurface waves on metasurfaces by design.For example, it has already been successfully demon-strated that the structuring of metal surfaces by smallsubwavelength slits or holes can lead to an improvedconfinement of surface waves. By this method, it waspossible to localize surface plasmon polaritons near themetal surfaces at terahertz (THz) and microwave fre-quencies where metals almost behave like perfect elec-tric conductors [1, 2, 3]. The same concept was appliedto the design of waveguides that consisted of structuredmetal sheets [4, 5, 6]. Recently, in a similar approach, awaveguide structure based on complemetary split-ringresonators was proposed [7]. In an early experimen-tal work, it was shown that magnetic surface plasmons ∗ Electronic address: [email protected] (cid:1)(cid:2)(cid:3) (cid:1) (cid:2) (cid:3)(cid:4) (cid:1)(cid:4)(cid:3) (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:5) (cid:6)
Figure 1: (a) Schematic of one unit cell of the split-ringarray, a = 41 µ m, r = 17 . µ m, w = 7 . µ m, g = 3 . µ m.(b) Microscope image of a sample with grating structure. can exist on metamaterials with negative permeability[8]. Moreover, it was discussed in a theoretical workthat materials with simultaneously negative permittiv-ity and permeability can also sustain surface polaritons[9]. Facing the possibility of designing meta-surfaceswith well-defined spatio-temporal and spectral proper-ties, it becomes obvious that such metamaterials offerthe possibility to create compact, integrated plasmonicdevices. However, to exploit the full potential of meta-surfaces with respect to the design process of tailoredplasmons, it is necessary to understand the physicalmechanisms in the interaction of meta-surfaces and sur-face waves.In this letter, we experimentally demonstrate theexcitation of terahertz surface waves on thin mag-netic metamaterial films and provide a simple, intuitivemodel to describe the effective properties of the meta-surface. While previous experimental investigations inthe THz regime focussed on structured metal sheets[3, 4, 6], the magnetic metamaterials employed in thiswork consisted of a single layer of in-plane split-ringresonators (SRRs). The unit cell size of the structurewas 41 µ m × µ m, which corresponds to approx. λ/ n = 1 .
63. The siliconsubstrate which was used during the fabrication processwas removed in a post-process. The resulting sampleswere free-standing metamaterial membranes with a to-tal thickness of approximately 50 µ m. An exemplarymicroscope image of a sample is shown in Fig. 1(b).Since the dispersion relation of the localized surfacewaves is settled to the right of the light line of thesurrounding material (air), incident THz waves can-not be directly coupled to the surface modes of themetamaterial structure. Therefore, we created an ar-tificial grating coupler by skipping every sixth row ofsplit rings in x direction to convert THz waves to boundsurface states. Due to diffraction at the grating struc-ture, the incident THz waves experience an additionalmomentum of k x = m × π/ (6 a ) ( m : order of diffrac-tion, 6 a : grating period) parallel to the surface of themetamaterial. This allows that both the momentumand the energy are conserved in the conversion pro-cess if the frequency of the incident THz wave is be-low a specific threshold value given by the grating pe-riod 6 a . In vacuum, for example, a parallel momentum k x = 2 π/ (6 a ) with a = 41 µ m corresponds to a fre-quency of ν ≈ .
22 THz. This means that below thisfrequency, the momentum of the first-order diffractedbeam parallel to the surface is above threshold and theTHz wave can couple to surface modes of the meta-surface.To gain a basic understanding of the propagation oflocalized surface modes on magnetic meta-surfaces, wedeveloped a simple and intuitive model. For this pur-pose, we treated the metasurface as a thin slab waveg-uide with effective electric and magnetic material pa-rameters. In the model, the waveguide extended from x, y = −∞ . . . ∞ and was confined in the z -directionbetween − d < z < + d such that the slab waveguidethickness was 2 d . We assumed that the material washomogeneous and isotropic. To account for the reso-nant magnetic response of the split-ring resonators, wedescribed the effective permeability µ eff using a Lorentzmodel µ eff = 1 + F ω ω − ω − iΓ ω + F ω ω − ω − iΓ ω (1)with resonance frequencies ω j , geometry factors F j , andcollision frequencies Γ j [11]. The effective permittiv-ity ǫ eff was kept at a constant value that was equal tothe permittivity of BCB. The material outside the slabwas described by a permittivity ǫ o and permeability µ o ( ǫ o ≈ µ o ≈ E = E exp [i( k x x + k z z − ωt )] (2)propagating in the x – z plane with a wave vector k =( k x , k z ) and a frequency ω . It should be noted that inthis case k x is a complex quantity describing the phasepropagation and absorption of the wave. The char-acteristics of the guided modes in the waveguide aremainly determined by the reflection at the boundariesbetween the slab material and the surrounding mate-rial. A stable mode pattern can only exist if the phaseadvance after two reflections at opposite boundaries ofthe material slab is an integer multiple of 2 π . Since F r equen cy ( T H z ) ck x /2 π (10 s −1 )CST calculationswaveguide model 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5(a) radiating modes localized modes mode 2mode 1 m = 1 m = 2 (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:10) (cid:1) (cid:2) (cid:1)(cid:11)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:10) (cid:3) (cid:2) (cid:4) (cid:1) (cid:12)(cid:13)(cid:14) Figure 2: (a) Dispersion relation of the localized wave. (b)Magnetic field normal to the metamaterial surface ( z com-ponent) in the x – z plane and (c) in the x – y plane. a meta-surface only consists of a single layer of reso-nant elements, only mode distributions with a singlemaximum of the field amplitude in the center of thewaveguide are meaningful solutions. These modes areobtained if the phase advance corresponds to 1 × π perround trip. Also, we only considered transverse elec-tric (TE) polarized modes (electric field vector pointsin y direction) to correctly account for the magnetic re-sponse of the SRRs in z -direction. For TE polarization,the Fresnel coefficient of reflection is given by ρ = µ o k z − µ eff k o z µ o k z + µ eff k o z . (3)The wave number k o z is determined by k o z =[ ǫ o µ o ( ω/c ) − k x ] / and the sign of the square rootis chosen such that | ρ | ≤
1. Finally, the condition forthe dispersion relation of the localized modes can bewritten as arg ( ρ ) + 4 dk z = 2 π . (4)Note that ρ is a function of both ω and k z . The solutionof this equation is a dispersion relation k z ( ω ), which canbe converted to the more intuitive relation k x ( ω ) by useof k x + k z = ǫ eff µ eff ( ω/c ) . The resulting function ω ( k x )is indicated by the blue line in Fig. 2(a).To check the validity of our simple model, we per-formed detailed numerical calculations using CST Mi-crowave Studio . Moreover, to obtain an accurate ref-erence to the experiments described later in this let-ter, we determined the dimensions of the fabricatedmetamaterial structure from a microscope image. Thenumerically calculated dispersion diagram in Fig. 2(a)(red crosses) shows two branches in the frequency rangefrom 0 to 2 THz in very good agreement with the re-sults derived from the waveguide model. It is notablethat the waveguide model enables us to fully describe A m p li t ude t r an s m i tt an c e Frequency (THz)(a) measurementTMTE 0.1 0.5 1 1.5Frequency (THz)(b) numericalcalculationTMTE
Figure 3: (a) Measured and (b) calculated spectra of tera-hertz transmission through the sample. the subwavelength-thick meta-surface as an effectivemedium with effective material parameters.Besides the dispersion relation, we determined themagnetic field distribution of the guided modes fromthe numerical data. The results in Figs. 2(b) and (c)reveal that the magnetic field distributions of the twomode branches in the dispersion relation significantlydiffer near the plane of the split rings. At a given wavevector k x , the magnetic fields inside the rings and inthe space between them oscillate in phase for the lowerfrequency mode, whereas they oscillate out of phase inthe higher frequency mode. In the latter case, this leadsto a stronger confinement of the fields in the immediatevicinity of the split rings since the fields cancel out withincreasing distance from the plane.To support the results by experimental data, we mea-sured the transmission through the meta-surface us- ing a terahertz time-domain spectroscopy setup un-der normal incidence. Figure 3 displays the measuredspectra together with the numerically calculated re-sults. The pronounced transmittance minimum at ap-prox. 1.1 THz for transverse magnetic (TM) polariza-tion (magnetic field vector in y -direction) results fromthe electric field coupling to the fundamental resonanceof the split rings. For TE polarization (electric fieldvector in y -direction), one sharp resonance feature ap-pears at approx. 0.9 THz as well as two weak dips at1.15 THz and 1.2 THz. The frequencies of the transmis-sion minima are directly related to the correspondingeigenfrequencies of the excited surface waves with wavevectors equal to integer multiples of the reciprocal grat-ing vector, k x = m × π/ (6 a ). The stronger minimum at0.9 THz corresponds to the excitation of a lower modesurface wave with m = 1 while the two weaker minimaat 1.15 THz and 1.2 THz correspond to a lower modesurface wave with m = 2 and a higher mode surfacewave with m = 1, respectively (see Fig. 2).In conclusion, we have experimentally demonstratedthe excitation of resonant terahertz surface modes onthin magnetic metamaterials by transmission measure-ments. The observed transmission minima could berelated to the excitation of surface waves by direct com-parison to 3-D full-wave numerical calculations. In ad-dition, we developed a simple and intuitive waveguidemodel to describe subwavelength-thick metamaterialfilms as an effective medium. From this model, we coulddeduce the dispersion relation of the surface waves inexcellent agreement with the numerical and experimen-tal results. The physical insight provided by this workis important with respect to the development of designtools for tailored surface waves on meta-surfaces. [1] J. B. Pendry, L. Mart´ın-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structuredsurfaces,” Science , 847–848 (2004).[2] A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Ex-perimental verification of designer surface plasmons,”Science , 670–672 (2005).[3] C. R. Williams, S. R. Andrews, S. A. Maier, A. I.Fern´andez-Dom´ınguez, L. Mart´ın-Moreno, and F. J.Garc´ıa-Vidal, “Highly confined guiding of terahertzsurface plasmon polaritons on structured metal sur-faces,” Nature Photon. , 175–179 (2008).[4] R. Ulrich and M. Tacke, “Submillimeter waveguidingon periodic metal structure,” Appl. Phys. Lett. ,251–253 (1973).[5] J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism fordesigning metallic metamaterials with a high index ofrefraction,” Phys. Rev. Lett. , 197401 (2005).[6] W. Zhu, A. Agrawal, and A. Nahata, “Planar plas-monic terahertz guided-wave devices,” Opt. Express , 6216–6226 (2008). [7] M. Navarro-C´ıa, M. Beruete, S. Agrafiotis, F. Falcone,M. Sorolla, and S. A. Maier, “Broadband spoof plas-mons and subwavelength electromagnetic energy con-finement on ultrathin metafilms,” Opt. Express ,18184–18195 (2009).[8] J. N. Gollub, D. R. Smith, D. C. Vier, T. Perram, andJ. J. Mock, “Experimental characterization of magneticsurface plasmons on metamaterials with negative per-meability,” Phys. Rev. B , 195402 (2005).[9] R. Ruppin, “Surface polaritons of a left-handedmedium,” Phys. Lett. A , 61–64 (2000).[10] O. Paul, C. Imhof, B. Reinhard, R. Zengerle, andR. Beigang, “Negative index bulk metamaterial at tera-hertz frequencies,” Opt. Express , 6736–6744 (2008).[11] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.Stewart, “Magnetism from conductors and enhancednonlinear phenomena,” IEEE T. Microw. Theory.47