Trapping and releasing light by mechanicalimplementation in metamaterial waveguides

Yongyao Chen,1 Jianqiang Gu,1,2 X. C. Xie,3 and Weili Zhang1,*1School of Electrical and Computer Engineering, Oklahoma State University, Stillwater,Oklahoma 74078, USA2Center for Terahertz Waves, College of Precision Instrument & Optoelectronics Engineering, and Key Lab of

Opto-electronics Information and Technical Science (Ministry of Education), Tianjin University, Tianjin 300072, China3Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA

*Corresponding author: weili.zhang@okstate.edu

Received July 20, 2010; revised November 5, 2010; accepted November 7, 2010;posted December 16, 2010 (Doc. ID 131969); published February 1, 2011

We show that light trapping and releasing can be switched by amechanic tuning effect inmetamaterialwaveguides.The transitionmechanism between the trapping and releasing states relies on the synergetic effect of the local Braggreflection and cavity resonance in the waveguides. As a proof-of-concept demonstration, a heterostructuredmetamaterial waveguide comprised of dielectric claddings and a tapered metamaterial core formed by arrays ofmetal slats is analytically and numerically investigated. The spatial separation of the trapped light with variousfrequencies and the transition between the trapping and releasing states can be predicted by a “rainbow equation.”The proposed light trapping and releasing scheme based on the mechanical implementation of waveguide geome-trical parameters can be exploited to develop opto-mechanical devices for slow light technology. © 2011 OpticalSociety of America

OCIS codes: 260.0260, 240.0310, 230.7370, 160.3918.

1. INTRODUCTIONMetallic subwavelength structures have recently attractedmuch attention for their great potential in developing noveldevices and artificial materials with unique electromagneticresponse. Notable examples include negative index meta-materials [1–4], effective plasmonic crystals [4–6], and highindex metamaterials [7–9]. Slow light, on the other hand, asan emerging new field is not only conceptually fascinatingbut also has important applications, such as in optical commu-nications, signal data manipulation, and quantum informationprocessing [10]. It is thus of great interest and merit to bridgebetween the fields of metamaterials and slow light, which maypave a way for developing new research directions and impor-tant applications. Recent progress on slow light in artificialplasmonic structures [11,12], electromagnetically inducedtransparency in metamaterials [13–16], and “trapped rainbow”

in tapered negative index waveguides [17,18] have demon-strated new scopes and possibilities in both metamaterialand slow light.

Various schemes in actively controlling light trapping andreleasing have been reported in metamaterial structures.Tsakmakidis et al. proposed a three-layer waveguide hetero-structure, in which the waveguide core with isotropic nega-tive-index materials was sandwiched by regular dielectriccladdings. The scheme for light trapping relies on the negativeGoos-Hänchen lateral displacements at the media interfaces,and the trapped light can be released by locally tuning thestructure to a perturbed light path based on the effectiveGoos-Hänchen lateral displacements [18]. Recently, Gan et al.

proposed a graded metallic grating structure which supportedspoof surface plasmon polaritons, and the light trapping andreleasing were implemented through a thermal-optical effect[12]. At terahertz frequencies, the realization of metamaterial

waveguides with isotropic negative index of refraction re-mains challenging. Because of different material properties,fabrication issues, and increased metal loss, isotropic ne-gative index metamaterials continue to elude the terahertzregime. In addition, to pursue higher speed modulation thanthe thermal tuning, particularly to be able to use micro-electro mechanical systems (MEMS)-based elastic tuning, oneexpects a novel metamaterial approach that enableshigh-speed dynamic mechanical tuning at MHz switchingspeed [19]

In this article, we introduce a mechanism for trapping andreleasing light in metamaterial waveguides through elastic ormechanical implementation. The insight of the proposed tran-sition method between the trapping and releasing states isexplored by examining a heterostructured metamaterialwaveguide comprised of dielectric claddings and a taperedmetamaterial core formed by arrays of metal slats, which isdifferent from the metamaterial waveguides with isotropicnegative index of refraction. It is found that the transition be-tween light trapping and releasing relies on the synergetic ef-fect between local Bragg reflection and cavity resonance ofthe waveguides. Practically, the proposed scheme can over-come the above challenges in slow light metamaterials, sincethe metamaterial waveguides are simply comprised of cut-through metallic structure instead of isotropic negative-indexmaterials, and the transition between light trapping and re-leasing can be realized in metamaterial waveguides throughelastic or mechanical tuning. The proposed light trappingand releasing scheme based on elastic or mechanical tuningcan be exploited in developing a new class of opto-mechanicaldevices for active control of light speed in a broad spec-tral range.

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1084-7529/11/020272-06$15.00/0 © 2011 Optical Society of America

2. HETEROSTRUCTURED METAMATERIALWAVEGUIDEA heterostructured metamaterial waveguide as a typical two-dimensional cut-through metallic system, as illustrated inFig. 1, is comprised of two dielectric claddings and a meta-material core formed by arrays of metal slats. Such a cut-through design, which has no closing metal substrate, enablesmechanical or elastic tuning. Here, we focus on the frequencyband from microwave up to the threshold of the terahertzregime where metals can be treated as perfect conductors.A light wave with transverse magnetic (TM) polarization pro-pagating along the waveguide and the wavelength of light isassumed to be much longer than the slit width. In region I, thefield inside the slits can be expressed as a combination offorward and backward transverse electromagnetic (TEM)subwavelength propagating modes. In regions II and III, thefields are a sum of diffractive waves, which should be evanes-cent in the dielectric claddings to maintain the boundedguiding wave along the waveguide. By matching boundaryconditions for both electric and magnetic fields at the inter-faces between different regions, we can deduce the followingequations:

X∞n¼−∞

ε2dp

sin c2�βnd2

�τð2;nÞ

¼ −

ε1k1

264A sin

�k1L2

�− B cos

�k1L2

�

A cos�k1L2

�þ B sin

�k1L2

�375; ð1Þ

X∞n¼−∞

ε3dp

sin c2�βnd2

�τð3;nÞ

¼ −

ε1k1

264A sin

�k1L2

�þ B cos

�k1L2

�

A cos�k1L2

�− B sin

�k1L2

�375; ð2Þ

where p and L are the periodicity and thickness of the metalslats, respectively, and d is the width of slits. ki ¼ ω

cffiffiffiffiεip

is thewave vector in different dielectric regions with dielectric con-stants εi, and i ¼ 1, 2, 3 represents regions I, II, and III, respec-tively. ω is light frequency and c is the speed of light invacuum. βn ¼ β þ n 2π

p ðn ¼ −∞;…0;…þ ∞Þ is the momentumalong the z axis of the nth diffraction order of guided waveswith β the propagation constant. τ2ði;nÞ ¼ k2i − β2n is the trans-verse momentum along the x direction and βn > ki for evanes-cent diffractive waves. A and B are coefficients. Thedispersion relation of the waveguide can be derived by solvingEqs. (1) and (2).

Now we consider a simplified case in which the metal slatsare placed in vacuum or air, thus forming a free-standing

symmetric device. For TM polarization, such an array of metalslats can be equivalent to a dielectric waveguide with a higheffective index of refraction, which is determined by thegeometrical parameters [7,9]. The even and odd guided modessupported by the symmetric waveguide can be obtained byconsidering the conditions A ≠ 0, B ¼ 0, respectively, inEqs. (1) and (2). We intend to achieve the light trapping andreleasing by utilizing the fundamental even mode. The taperedwaveguide structure is formed by gradually changing thethickness of the metal slats along the wave propagation axis.

3. MECHANISM FOR LIGHT TRAPPINGAND RELEASING IN THE TAPEREDMETAMATERIAL WAVEGUIDEThe mechanism for light trapping and releasing in the meta-material waveguide is governed by both the local Braggcondition and cavity resonance, which exist in the taperedperiodic cut-through metallic system. Our goal is to explorethe insight into the physics of wave propagation in the taperedwaveguide. We first assume that the thickness variation alongthe propagation axis is much smaller than the wavelength, andlight can be assumed to propagate along a locally uniformgrating structure, as depicted in the inset of Fig. 2(a). The

Fig. 1. (Color online) Schematic of the heterostructured waveguideconsists of dielectric claddings and a metamaterial core formed byarrays of metal slats (yellow).

Fig. 2. (Color online) (a) Dispersion relation in the first Brillouinzone of the fundamental mode supported by the metal slat waveguidewith geometry parameters, p

d ¼ 7 and Ld ¼ 45. (b) Light trapping gov-

erned by the local Bragg condition and cavity resonance. The parallelarrows (green and black arrows in color online) represent forwardand backward guided-waves along the z axis; the evanescent fieldsalong the x direction are represented by dashed curves (green andblack dashed curves in color online). The cavity resonances (lighttrapping) inside the metal slits are indicated by vertical arrows(red arrows in color online).

Chen et al. Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. A 273

Bragg condition occurs when the wave vector of the guidedmode is at the Brillouin zone boundaries where the dispersioncurve becomes discontinuous and approaches the band edge[20]. It is noted when light propagates along the waveguide,the tapered structure will gradually couple light into theguided mode with wave vectors approaching the Brillouinzone boundaries, and thus may resolve the issues of modecoupling and momentummismatch. Figure 2(a) shows the dis-persion relation of the fundamental mode in the first Brillouinzone at the zone boundary where kz ¼ π

p.Now, we consider guided waves supported by a section of

local uniformmetal slats, the propagation wave vectors satisfythe Bragg condition, ~kz ¼ βBragg ¼ π

p (the higher order Braggvectors are neglected as an approximation), as shown inFig. 2(b), and the waves will be reflected back during the for-ward propagation along the z axis and form the backward pro-pagation wave, which is equal to the forward wave but with anopposite wave vector kz

←¼ −

πp. The superposition of such for-

ward and backward waves forms a standing wave with sta-tionary fields around the waveguide. Moreover, along the xdirection, the fields inside the metal slits are formed by theoscillating subwavelength modes, and the fields outside theslits are formed by stationary evanescent waves, thus main-taining the high confinement of light around the slits andcreating a nonradiative local cavity resonance inside the slitsand trapping light in these cavities, as shown in Fig. 2(b). Itshould be noted that the similar mechanism for trapping lightin three-dimensional systems can be realized by changing thegeometrical parameters of waveguides [21]. The tuning oftrapped light relies on the dynamic variations of local geome-try which will change both the local Bragg condition andcavity resonance of the waveguide and influence the lightpropagation.

The light trapping in the tapered waveguide can bedescribed by an analytical formula,

�β2Bragg −

�ωc

�2�−12

sin c2�βBraggd

2

�¼ p

d

cot�ω2c L

�ωc

; ð3Þ

where the left-side term is dominated by the local Bragg con-dition βBragg ¼ π

p, and the right-side term represents the cavityresonance depending on the local geometry characterized bythe local periodicity p, thickness L, and slits width d [22].Equation (3) accounts for contributions of both local Braggcondition and cavity resonance. It is derived from Eqs. (1)and (2), in which higher-order diffraction waves are neglectedas an approximation on the condition that the wavelength λoof light is much larger than the slat periodicity p, and by con-sidering the Bragg condition βBragg ¼ π

p as well as the local geo-metrical structure of the waveguide. From Eq. (3) we can findthat different frequency components ω of the guided wavesare trapped at corresponding sections of the waveguide withdifferent thicknesses L. In addition, the “rainbow phenomen-on” of the waveguide which is related to the spatial separationof different light frequencies, can also be verified by the abovelight trapping equation. We assume that the waveguide is lin-early tapered with a small tapering angle α, and the variationin waveguide thickness L along the propagation axis z canbe approximately expressed as L ¼ αz. By substituting theabove relation in Eq. (3), one can obtain the following rainbowequation:

�β2Bragg −

�ωc

�2�−12

sin c2�βBraggd

2

�¼ p

d

cot�ωα2c z

�ωc

: ð4Þ

It predicts the spatial separation of different frequency com-ponents along the propagation axis, z. Figure 3(a) illustratesthe rainbow phenomenon along the z axis, where light trap-ping at different critical waveguide thicknesses is observed, asshown in Fig. 3(b). Here, we employ the normalized lengthLo ¼ 1 in order to facilitate the scaling laws scaled to the fre-quency range from microwave to the terahertz regime. In thepractical application the normalized length Lo will be assignedthe real length units depending on the size of working space. Itshould be noted that in order to accurately determine the cri-tical thickness, one needs to consider higher-order diffractionwaves compared to the approximation in Eq. (3); this is wellverified by the finite-difference time-domain (FDTD) simula-tions. In the two-dimensional FDTD model, the units of thenormalized length is L0 ¼ 102 μm, and the simulated regionis 600 μm × 3000 μm and surrounded by perfectly matchedlayers to absorb the scattering light. The sampling in space ison a subwavelength scale, in which the meshed cell in the si-mulation space are chosen as Δx ¼ 1 μm and Δz ¼ 0:5 μm,respectively. The sampling in time is selected to achieve

Fig. 3. (Color online) (a)Rainbowphenomenonof thewaveguide. (b)Trapping light at different critical waveguide thicknesses. The solidcurve is derived from Eq. (3) as an approximation. The dotted curveconsiders the higher-order diffraction waves for comparison withthe approximation. The geometry parameters are chosen as p ¼50 μm, d ¼ 20 μm, the length of the waveguide along the propagationaxis is 3000 μm including 60 metal slats, and the slat thickness L in-creases from 5 to 300 μm by a step of Δ ¼ 5 μm for each period.(c)–(e) The frequencies of trapping light at f ¼ 0:9, 0.7, and 0:6THzcor-respond to the critical thicknesses L ¼ 140, 990, and 225 μm, respec-tively. The field distributions are represented by the magnetic fieldof light.

274 J. Opt. Soc. Am. A / Vol. 28, No. 2 / February 2011 Chen et al.

the numerical stability of the simulation, and one time stepsatisfied the condition: Δt ¼ c−1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðΔxÞ−2 þ ðΔzÞ−2

p; the total

simulation time is 20 000 time steps which are long enough toget a steady state [23]. The material of metal slats is set asperfect electrical conductors; the material filled in the gapsamong the slats and the background are chosen as air withrefractive index n ¼ 1. In the simulation, the geometry param-eters of the waveguide are chosen as p ¼ 50 μm, d ¼ 20 μm,the length of the waveguide along the propagation axis is3000 μm including 60 metal slats, and the slat thickness L in-creases from 5 to 300 μm by a step of Δ ¼ 5 μm for each per-iod. A monochromatic point source with TM polarization islocated near the tip region of the tapered waveguide to excitethe propagation mode along the waveguide. As shown inFigs. 3(c)–3(e), for instance, the frequencies of trapping lightat f ¼ 0:9, 0.7, and 0:6THz are spatially separated correspond-ing to the critical thicknesses at L ¼ 140, 190, and 225 μm,respectively.

4. TRANSITION BETWEEN THE TRAPPINGAND RELEASING STATES BY AMECHANICAL TUNING EFFECTAfter trapping light in the waveguide, we now show how torelease light using the mechanical tuning effect. The metalslats can be designed to possess elastic or mechanical prop-erty and the geometrical parameters such as periodicity p andslit width d can be varied by compressing or stretching themetal slats along the propagation axis. The effective index n ¼p=d can be regarded as a parameter associated with mech-anical deformation of the waveguide structure. The variationsof local geometry will change both the local Bragg conditionand cavity resonance of the waveguide and influence the lighttrapping effect derived from Eq. (3). Figure 4(a) illustrates thevariation of light trapping at different waveguide thicknessesthrough compression of the tapered section. By compressingthe slats along the z axis, the trapped light at correspondingwaveguide thickness blueshifts. Meanwhile, in the dispersiondiagram of a section of uniform waveguide, as shown inFig. 4(b), for the relaxation state without any compression,the dispersion curve of the trapped light approaches cutofffrequency (band edge) and its slope is nearly flat, indicatinga close-to-zero group velocity. However, by compressing thewaveguide, light releasing occurs as evidenced by the fact thatthe slope of the dispersion curve for the same frequency re-gion becomes obviously tilted. Since the elastic element canbe compressed and restored, light trapping and releasing arereversible states. The dynamic transition between the twostates leads to deceleration and acceleration of light propagat-ing along the waveguide.

In order to implement the transition between the trappingand releasing states, the waveguide design, as shown inFig. 4(c), is comprised of a tapered metal slat waveguide func-tioning as a light trapping device followed by uniform slatswith elastic and compressible properties which can extractlight from the tapered section by the compression process.To explicitly demonstrate the influence of local geometry var-iation on the trapping light effect, the elastic and compressibleproperties are also added on the thick region of the taperedwaveguide, which is a small section containing several metalslats, as shown in the overlap region between the left (red)and right (green) box in Fig. 4(c). The compression at the

thick region modifies the local geometries, thus tuning thelocal Bragg condition as well as cavity resonance and influen-cing light trapping in that region.

The transition between the states of light trapping and re-leasing is modeled by FDTD simulations. The length of thefirst tapered section of the waveguide along the propagationaxis is 2250 μm including 45 metal slats, and the slat thicknessL increases from 5 to 225 μm with an interval of Δ ¼ 5 μm ateach period. Following the tapered section, an array of metalslats with a constant thickness L ¼ 225 μm forms a uniformsection and has the same periodicity and slit width with thoseof the tapered slats. When compression applies, the periodi-city and slit width change from p ¼ 50 to 35 μm and d ¼ 20 to5 μm. As shown in Fig. 4(d), for the relaxation state withoutcompression, the light with frequency 0:6THz is trapped at theend of the taped section, corresponding to the critical thick-ness L ¼ 225 μm. In Fig. 4(e), by compressing the uniform andtapered thick region, as shown in the right (green) box ofFig. 4(c), the trapped light is released from the confinementregion and propagates along the uniform section.

Fig. 4. (Color online) (a) Variation of trapping light at differentwaveguide thicknesses through compressing of the tapered region.The waveguide structure is the same as that in Fig. 3. The periodicityand slit width are changed from p ¼ 50 μm, d ¼ 20 μm (red curve) top ¼ 35 μm, d ¼ 5 μm (green dotted curve) due to compression. (b)Variation of the dispersion diagram through the compression on a sec-tion of uniform metal slat array. The metal slat array has a constantthickness L ¼ 225 μm with same periodicity and slit width in (a). Thered and green dashed lines represent the slope of the dispersioncurves. The transition between light trapping and releasing is rever-sible as indicated by the dashed arrows. (c) The tapered section fortrapping light (the red solid box) is followed by a uniform section forextracting light. The elastic property and compression are added tothe uniform section as well as the thick slats of the tapered section(green dashed box). (d) and (e) Transition between light trapping andreleasing demonstrated by FDTD simulations.

Chen et al. Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. A 275

5. DISCUSSION AND CONCLUSIONSThe focus of our paper is to introduce a mechanism for trap-ping and releasing lightthat relies on the synergetic effect be-tween the local Bragg reflection and cavity resonance, whichare the general properties in metamaterial waveguides com-prised of periodic cut-through metallic systems. Our analyticaland numerical results demonstrate how this general schemeworks on light trapping and releasing by the elastic or me-chanical switching effect in metamaterial waveguides. Theproposed axially tapered metal slates waveguide is a typicalmodel that enables in-depth understanding of the physics as-sociated with wave propagation in metamaterial waveguides.From the point of view of practical implementation of the elas-tic mechanical switching, since the size of the waveguide unitcells are on the order of centimeters to millimeters in the radiowave to microwave regimes, the elastic switching can be im-plemented by connecting the metal slats with elastic springsor reeds which can be mounted on the edge side of the slats.The influence of such elastic elements on the wave propaga-tion along the propagation direction can be neglected. Thesemetal slats could operate in vacuum or air environment, thusforming a free-standing symmetric device in which no realisticmaterial is needed to fill the gaps between the slats to realizethe elastic connection. At terahertz frequencies, however, dueto reduced minimum feature size in the unit cells, the imple-mentation of the elastic switching is indeed challenging butcould be achievable by using laser micromachining andMEMS techniques. In particular, the MEMS technique willbe used to realize the local mechanical or elastic deformationin the metamaterial waveguides. As a possible approach, forexample, a wedged metal plate is cut through with arrays ofsubwavelength periodic slots by laser machining, and the flatbottom of the wedged plate is covered with a high conductiv-ity silicon wafer. The space between the wedged plate andsilicon wafer ranges from 1 to 30 μm and can be dynamicallycontrolled using the MEMS technique. It was found that whenthe space between the metal plate and silicon wafer is gradu-ally increased, the trapped terahretz light can be released andpropagates along the wedged metal plate, thus demonstratingthe phenomenon of light trapping and releasing proposed inthe terahertz regime.

Furthermore, the propagation losses are an important issuein the slow-light metamaterial waveguides. Several loss me-chanisms may degrade the performance of metamaterialslow-light devices. The reflection waves in the metamaterialwaveguides comprised of the tapered periodic metallic systemmay be the main factor for the slow-light propagation loss.Such reflections originated from the backward propagationBloch modes may depend on the structure parameters of thetapered waveguides. When the group velocity of waves ap-proaching zero near the band edge, the backward reflectionmodes can be greatly amplified and may dominate the trans-mission loss [24]. Further, the imperfections such as structureroughness and distortion in shape or size of unit cell can act asscatterers that radiate or backscatter the waves, thus increas-ing the losses [25]. The dissipation of the metal at a high-frequency band may also contribute to the propagation lossmechanism. Although the above factors may contribute tothe losses, they should be examined carefully by using a morerigorous numerical method to gain essential knowledge aboutdeviations between realistic metamaterial slow-light wave-

guides and the ideal devices discussed here. The losses ofslow-light metamatrial waveguides will be examined and anal-yzed elsewhere, in which the above loss mechanisms will beelaborated in more detail.

In conclusion, we demonstrated a mechanical optic-effect-based light trapping and releasing in a metamaterial wave-guide. The mechanism of implementing such a transitionshould not be restricted to the low-frequency regions wheremetals can be treated as perfect conductors. At high frequen-cies such as the visible regime, the Bragg condition and thecavity resonance [26,27], which govern light trapping and re-leasing, should always exist despite of the strong plasmoniceffects in metals. In addition to the elastic tuning property,the cut-through metallic systems also allow many individualmetamaterial waveguides to interact together through evanes-cent field coupling and form a metamaterial waveguide array,which may exhibit useful and even interesting properties.Furthermore, it would be interesting and illuminative toexplore the possibility of manipulating slow light by the“acousto-optic effect” analogous to the elasto-optic effect indielectric microstructures [28]. It invokes a picture of acousticwave in a transparent host dielectric, which includes thetapered waveguide formed by the metallic inclusions; theacoustic wave will spatially and temporally tune the localBragg condition and cavity resonance. Besides consideringthe isomorphism between acoustics and electromagnetics[29,30], the proposed wave trapping and releasing mechanismmay also be exploited for acoustic systems and will lead to anew class of acoustic metamaterials.

ACKNOWLEDGMENTSThe authors acknowledge R. Singh, Z. Tian, X. Lu, and S.Zhang for helpful discussions. This work was partially sup-ported by the National Science Foundation (NSF) and theNational Natural Science Foundation of China (NSFC) (grant61028011).

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