Optical Bloch oscillations in periodic structures with metamaterials
Artur R. Davoyan, Ilya V. Shadrivov, Andrey A. Sukhorukov, Yuri S. Kivshar
OOptical Bloch oscillations in periodic structures with metamaterials
Artur R. Davoyan, Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar
Nonlinear Physics Center, Research School of Physical Sciences and Engineering,Australian National University, Canberra, ACT 0200, Australia
We predict that optical Bloch oscillations can be observed in layered structures with left-handedmetamaterials and zero average refractive index where the layer thickness varies linearly across thestructure. We demonstrate a new type of the Bloch oscillations associated with coupled surface wavesexcited at the interfaces between the layers with left-handed material and conventional dielectric.
Electron oscillations in the presence of a constant elec-tric field were predicted by Bloch in 1928 [1]. Such Blochoscillations become possible due to beating of the local-ized eigenmodes of the structure corresponding to theequidistant eigenstates of the spectrum known as theWannier-Stark ladder [2]. Experimental verification ofthe theory was impossible at that time, since dephasingtime of electrons in crystals is shorter than the period ofthe electron Bloch oscillation. Later, electron Bloch oscil-lations were observed in semiconductor superlattices [3]for which the period was reduced due to a small mini-band width in the artificial structure.Dephasing processes for electromagnetic waves are neg-ligible making the observation of the optical Bloch oscil-lations in photonic systems much easier. The first ex-perimental observation of optical Bloch oscillations wasreported in Ref. [4] for linearly chirped Bragg gratings.Later, several studies reported the observation of opticalBloch oscillations in various structures [5, 6, 7, 8, 9].Recent experimental realization of left-handed ma-terials [10] has opened up many unique opportunitiesto explore novel effects in the structures with nega-tive refractive index. In this Letter we study, for thefirst time to our knowledge, optical Bloch oscillationsin one-dimensional layered structures containing alter-nating layers of left-handed and conventional dielectricslabs. We choose the material parameters in such a waythat the average refractive index ¯ n across pair of theneighboring layers vanishes, thus fulfilling the conditionfor the existence of a novel type of the specific zero-¯ n bandgap [11, 12]. We change the layer thickness lin-early in the structure and observe an optical analogueof the Wannier-Stark ladder in the eigenmode spectrum,and the corresponding Bloch oscillations in the resonanttransmission bands. We reveal that in such structuresthe Bloch oscillations can be observed in three differentregimes. Compared to the photonic Bloch oscillationsin conventional dielectric structures, the metamaterialstructures can support a novel type of the Bloch oscil-lations associated with coupling of surface waves at theinterfaces between left-handed and dielectric layers.We study a one-dimensional layered structure shownschematically in Fig. 1, where the slabs with the width b i are made of metamaterial being separated by a dielectricslab with the width a i . Variation of the refractive index FIG. 1: Schematic of linearly chirped one-dimensional pho-tonic crystal with alternating layers of left-handed metama-terial and dielectric. in the i-th pair of layers can be described as follows: n ( z ) = (cid:26) n r = √ ε r µ r z ∈ ( z i , z i + a i ) n l = −√ ε l µ l z ∈ ( z i + a i , z i + Λ i ) (1)where n l and n r are the refractive indices of metamaterialand dielectric, respectively.We consider TE-polarized waves with the electric fieldhaving one component E = ( E x , , y, z ). In this case, the field distribu-tion can be described by the Helmholtz equation:∆ E x ( y, z )+ n ( z ) E x ( y, z ) − µ dµ ( z ) dz ∂E x ( y, z ) ∂z = 0 , (2)where ∆ is the two-dimensional Laplacian, and the co-ordinates are normalized to c/ω . Firstly we considerperiodic structure. Electric field in an infinite one-dimensional periodic structure can be represented as asuperposition of Bloch eigenmodes [13], with the electricfield envelopes U ( z + Λ) = U ( z ) , where Λ is the struc-ture period. The dispersion relation for the Bloch wavesis found by the transfer matrix method [13],2 cos( K B Λ) = 2 cos( k zr a ) cos( k zl b ) − (3) (cid:18) k zl µ r k zr µ l + k zr µ l k zl µ r (cid:19) sin( k zr a ) sin( k zl b ) , a r X i v : . [ phy s i c s . op ti c s ] N ov FIG. 2: (Color online) Bandgap diagram for the TE-polarizedwaves. Black and white areas correspond to gaps and bands,respectively. Two spectra of the excited Bloch oscillations areshown on the left. The inset shows a magnified part of thespectrum. where K B is the Bloch wavenumber, k zl,zr = ∓ (cid:113) n l,r − k y and k y is the normalized propagation con-stant along the y axis. According to this relation aninfinite stack containing metamaterials exhibits a non-resonant gap for ¯ n ≡ Λ − (cid:82) Λ0 n ( z ) dz = 0, where ¯ n is anaverage refractive index of the structure, i.e. n r a = | n l b | .This condition is easy to fulfil for negative refractive in-dex materials.The bandgap diagram of the layered structure is shownin Fig. 2 for the parameter plane (Λ , k y ). Here we assumethat dielectric is vacuum, ε r = µ r = 1, and that it is twotimes thicker than the second layer, a/b = 2. We choosethe parameters of the left-handed media as follows: ε r = − µ r = − .
8. This set of parameters allows surfacewaves to exist at the interfaces between metamaterialand vacuum [12]. As follows from Fig. 2, for the zero¯ n structure the bandgap spectrum differs substantiallyfrom the case of conventional periodic structures madeof conventional dielectrics [13]. Stack with the averagezero refractive index possesses a complete gap with thetransmission resonances [11, 12] when the optical path ofthe wave in either layer of the period coincides with a halfof the wavelength in the corresponding medium. Thus forthe normal incidence ( k y = 0) transmission is observedonly when n r a = n l b = πm , where m is integer. For theslabs of equal thickness the regions of the transmissionresonances in (Λ , k y ) plane degenerate into infinitely thinlines.We study the propagation of electromagnetic waves insuch a layered structure with zero average refractive in-dex in each pair of layers, when the thickness of layersis chirped linearly, i.e. Λ q = Λ + qδ Λ, where integer q numbers the layers. We are looking for localized solu-tions in the structure, and for numerical simulations we FIG. 3: (Color online) Field distribution in the case of surface-wave-assisted Bloch oscillations. The Wannier-Stark ladderappears for the propagation constants centered around k y =2 .
47, normalized period is L y = 820. consider a finite stack of layers with perfect metal bound-ary conditions, E ( z = 0) = E ( z = L ) = 0, where L is thetotal length of the structure. We assume the Gaussianfield distribution in the plane y = 0 across the layers, andin order to find the electromagnetic field distribution inthe whole stack we look for its eigenmodes by solving theHelmholtz equation (2). Then we decompose the initialfield distribution using the basis of eigenmodes and findthe solution in the whole structure.To find the eigenmodes of Eq. (2) we employ the fol-lowing discretization scheme [14]:2 µ − m − + µ − m +1 (cid:20) U m +1 − U m µ m +1 − U m − U m − µ m − (cid:21) h + (4)+ ε m − + ε m +1 µ − m − + µ − m +1 U m = k y U m , where z m = mh are the mesh points with the discretiza-tion step h and E x ( y, z ) = U ( z ) e − jk y y . Such a discretiza-tion scheme provides an algorithm convergence [14], andit avoids excitation of spurious modes in the structure.In metamaterials the energy flow (cid:82) [ E × H ] dz can benegative, i.e. the energy can propagate in the oppositedirection to the propagation constant k y [15]. Conse-quently, we determine the direction of the energy flowof each eigenmode and choose the sign of the propaga-tion constant such that the energy flows in the positive y -direction. Decomposition of the initial condition in theplane y = 0 in the eigenmode basis is made using theleast squares method.To find propagation constants (values of k y ) which leadto the Bloch oscillations, we analyze the spectrum ofeigenvalues of this layered structure. The Bloch oscil-lations are expected to appear where the spectrum ofeigenmodes is equidistant. Practically for all gradientsof a linear ramp we observe several sets of equidistantstates. The equidistant eigenvalues of k y correspond to FIG. 4: (Color online) Field distribution for the case of guidedwaves. The Wannier-Stark ladder appears for the propagationconstants centered around k y = 1 .
34, period is L y ∼ = 100. a spatial optical equivalent of the Wannier-Stark ladderwhich is associated with the Bloch oscillations.Spectrum of k y can be divided into three different re-gions. First, when k y < n r < | n l | , electromagnetic wavespropagate in both left- and right-handed materials. Inthe second region, n r < k y < | n l | , waves propagatein metamaterial only being evanescent in the vacuumlayers. In this regime, our structure can be consideredas an array of coupled left-handed waveguides. When k y > | n l > | n r , only surface waves may propagate alongthe interfaces separating different materials.We find that the Bloch oscillations can be observedin all three regimes of the wave propagation when thecorresponding set of equidistant propagation constants isexcited. We consider a stack containing 36 pairs of meta-material and dielectric slabs and the normalized period Λvarying from 3.7 to 6. First, we excite the eigenstates cor-responding to the regime of surface waves with the cen-ter of the spectrum at k y = 2 .
47. Figure 3 presents theintensity distribution for the electric field which showsclearly spatially periodic oscillations of the beam posi-tion in the structure. The corresponding spectrum ofeigenstates is shown on the left side of Fig. 2. We notethat the beam reconstructs its shape after each period ofoscillations. The field is highly confined to the interfacesbetween metamaterial and vacuum, demonstrating thatsuch Bloch oscillations exist due to interaction of surfacewaves in the structure. The distance between Wannier-Stark eigenstates ∆ k y defines the period of oscillations, L y = 2 π/ ∆ k y . For this case, we find L y = 820, and thisagrees well with Fig. 3.Bloch oscillations of the beam with the spectrum cor- responding to the coupled waveguide regime, n r < k y < | n l | , are shown in Fig. 4. The equidistant spectrumof eigenstates corresponding to the Wannier-Stark lad-der is also shown in Fig. 2 (top, left). We notice that FIG. 5: (Color online) Field distribution for the case whenwaves propagate in both type of materials. The Wannier-Stark ladder appears with average propagation constant k y = 0 .
8, period of oscillations is L y = 210. oscillations are strongly anharmonic, but they are stillperiodic with the period defined well by the relation L y = 2 π/ ∆ k y , which is less than the period of Blochoscillations associated with surface waves.The regime of Bloch oscillations corresponding to thewaves propagating in both media can be found in adifferent structure with wider transmission resonance.We analyse that the structure consisting of 36 periodsand where the normalized period varies linearly from2.5 to 7.5 (corresponding to the period change gradi-ent δ Λ = 0 . ε = − . µ = − .
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