All-angle zero reflection at metamaterial surfaces
aa r X i v : . [ phy s i c s . op ti c s ] O c t All-angle zero reflection at metamaterial surfaces
Xin Li, Zixian Liang, Xiaohan Liu, , Xunya Jiang, and Jian Zi , ∗ Department of Physics and Surface Physics Laboratory,Fudan University, Shanghai 200433, People’s Republic of China Shanghai Institute of Microsystem and Information Technology,CAS, Shanghai 200050, People’s Republic of China Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People’s Republic of China (Dated: November 2, 2018)The authors study theoretically reflection on the surface of a metamaterial with a hyperbolicdispersion. It is found that reflection is strongly dependent on how the surface is terminated withrespect to the asymptote of the hyperbolic dispersion. For a surface terminated normally to theasymptote, zero reflection occurs for all incident angles. It is exemplified by a metamaterial madeof a periodic metal-dielectric layered structure with its surface properly cut through numericalsimulations.
PACS numbers: 42.25.Gy, 78.67.Pt, 78.20.Ci, 41.20.Jb
Metamaterials are artificially designed composites con-sisting of periodic subwavelength structures. The op-tical response of metamaterials originates from theirstructures instead from their compositions, leading tomany unusual optical properties that do not occur innature. For instance, metamaterials with a negative re-fractive index can produce negative refraction andsuperlensing.
In contrast to conventional materials,the energy transport through negative-refractive-indexmetamaterials is in a direction opposite to the phase di-rection, giving rise to reversed Doppler effects and in-verted Cherenkov cone.
Metamaterials can also be usedto construct invisible cloak.
Reflection is a wave phenomenon occurring for wavesimpinging upon a surface. Our common understanding isthat reflection is inevitable although it can be eliminatedat some special incident angles, e.g., Brewster’s angles.In this Letter, we show theoretically that all-angle zeroreflection can occur at the surface of a metamaterial witha hyperbolic dispersion. It is exemplified by a metamate-rial made of a periodic metal-dielectric layered structurethrough numerical simulations.For any reflection on a surface, the spatial and timevariation of all fields must be the same at the surface.As a result, the in-plane wave vector of an incident waveshould be equal to that of the reflected one, independentof the nature of the boundary conditions. For homo-geneous media, this leads to the fact that the incidentangle is equal to the reflected angle. For anisotropic me-dia, however, incident and reflected angles may not bethe same. For either isotropic media or anisotropic mediawith an elliptic dispersion, reflection always exists due tothe fact that the conservation of in-plane wave vectors ofincident and reflected waves can be always satisfied, re-gardless of the cutting directions of surfaces, as shownschematically in Fig. 1. For an anisotropic medium ∗ Electronic mail: [email protected] (a)(c) (b) (d)
FIG. 1: (Color online) Equi-frequency surface analysis of re-flection on surfaces of metamaterials with (a) an elliptic and(b-d) a hyperbolic dispersion. Surfaces (dash-dotted lines)lie in the horizontal plane. For metamaterials with the hy-perbolic dispersion, surfaces are cut along the principal axis(b), perpendicularly to the asymptote (dashed lines) of thehyperbolic dispersion (d), and obliquely to both the princi-pal axis and asymptote (c). Black (grey) thick arrows denotethe incident (reflected) wave vector. Thin arrows indicate thedirection of the group velocity. Dotted lines illustrate theconservation of the in-plane wave vectors. Note that meta-materials occupy the half-space below the horizontal line andwaves are incident from the metamaterial side. with a hyperbolic dispersion, reflection is sensitive to thesurface termination direction, namely, the surface orien-tation with respect to the asymptote of the hyperbolicdispersion. For surfaces terminated along the principalaxes, reflection always exists with equal incident and re-flected angles. For surfaces oriented obliquely to both theasymptotes and principal axes, the reflected wave vectorstill exists with the reflected angle different from the in-cident angle. If the surface is cut perpendicularly to theasymptote, however, we cannot find reflected wave vec-tor for any given incident wave vector. In this situation,zero reflection is expected for all incident angles as to beshown later. It should be noted that the energy flow di-rection is the same as that of the group velocity, definedby v g = ∇ ω ( k ), where k is the wave vector and ω ( k )is the dispersion relation. From its definition, the groupvelocity direction is perpendicular to the equi-frequencysurface and points to the direction along which ω ( k ) isincreasing.A qualitative analysis of all-angle zero reflection wasgiven hereinbefore. In the following, we would like togive a rigorous proof. We consider an anisotropic meta-material whose permittivity tensor and permeability ten-sor are both diagonalizable. To simply the proceedinganalysis, we assume that the metamaterial is nonmag-netic, namely, the diagonal elements of the permeabil-ity tensor are all equal to 1. Without loss of generality,we assume in our analysis a plane wave with the mag-netic field polarized along the y direction with the form H = H b y exp [ i ( k x x + k z z ) − ωt ], where b y is the unitvector along the y direction. If we choose a Cartesiancoordinate system x - z with its axes along the principalaxes of the metamaterial, this plane wave satisfies thefollowing dispersion relation k x ε z + k z ε x = ω c , (1)where ε x and ε z are the diagonal elements of the per-mittivity tensor. For ε x and ε z with opposite signs, thecorresponding dispersion is hyperbolic. If we choose anew Cartesian coordinate system x ′ - z ′ with the same ori-gin, the permittivity tensor is no longer diagonal and istransformed to (cid:20) ε x ′ x ′ ε x ′ z ′ ε z ′ x ′ ε z ′ z ′ (cid:21) = (cid:20) c ε x + s ε z cs ( ε x − ε z ) cs ( ε x − ε z ) s ε x + c ε z (cid:21) . (2)Here, the parameters c and s are given by c = cos θ, s = sin θ, (3)where θ is the angle between the x ′ and x axes. Thedispersion in the new coordinate system becomes accord-ingly ε x ′ x ′ k x ′ + ε z ′ z ′ k z ′ + 2 ε x ′ z ′ k x ′ k z ′ ε x ′ x ′ ε z ′ z ′ − ε x ′ z ′ = ω c . (4)For a metamaterial with a hyperbolic dispersion, re-flection is strongly dependent on the cutting direction ofthe surface. Without loss of generality, we assume thatthe metamaterial surface is terminated normally to the z ′ axis, i.e., in the x ′ y ′ plane. Consequently, k x ′ is thein-plane wave vector of an plane wave and k z ′ is the per-pendicular component. For a given in-plane wave vector k x ′ of an incident plane wave, we can always find a realsolution of k z ′ for the reflected wave from the conser-vation of the in-plane wave vector, if the x ′ axis is notperpendicular to the asymptote of the hyperbolic disper-sion. For the x ′ axis just perpendicular to the asymptote, no real solution of k z ′ for the reflected wave can be found.In this case, k z ′ should be a complex number possessingboth a real and an imaginary part. Zero reflection isexpected as can be confirmed by calculating the normalcomponent of the reflected Poynting vector with respectto the surface, defined in cgs units by S r ⊥ = 4 πc Re ( E r × H ∗ r ) ⊥ = 4 πc Re (cid:0) E rx ′ H ∗ ry ′ (cid:1) , (5)where E r and H r are the reflected electric and magneticfields with their components related each other by E rx ′ = − cω ε z ′ z ′ k z ′ + ε x ′ z ′ k x ′ ε x ′ x ′ ε z ′ z ′ − ε x ′ z ′ H ry ′ . (6)Suppose a complex k z ′ for the reflected wave and substi-tute it into Eq. (4) we can obtain ε z ′ z ′ Re( k z ′ ) + ε x ′ z ′ k x ′ = 0 . (7)This immediately leads to the fact that E rx ′ does notpossess a real part, leading to S r ⊥ = 0. We can thusconclude that the reflected wave does not carry any en-ergy along the surface normal, implying zero reflection.One feasible realization of a metamaterial with a hy-perbolic dispersion is to adopt a periodic metal-dielectriclayered structure. In the long wavelength limit(the period is much smaller than the operating wave-length), this periodic metal-dielectric layered structurecan be viewed as a effective anisotropic metamaterialwith the permittivity tensor given by ε x ε x
00 0 ε z . (8)For p -polarized waves, ε x and ε z are related to the pa-rameters of the constituents by ε x = ε d + ε d d , (9a) ε z = ε ε dε d + ε d , (9b)where ε , is the dielectric constant of the constituents, d , is the thickness, and d = d + d is the period. Ifwe adopt a Drude model to describe the dielectric con-stant of the metal constituent, namely, ε ( ω ) = 1 − ω p /ω , where ω p is the plasma frequency of the metal, it can beeasily shown that ε x and ε z can have opposite signs atcertain frequencies for a proper choice of the thicknessparameters. This leads to a hyperbolic dispersion for theperiodic metal-dielectric layered structure.To illustrate all-angle zero reflection on the surface of aperiodic metal-dielectric layered structure, we carry outfinite-difference time-domain (FDTD) simulations withperfectly matched layer boundary conditions, shown inFig. 2. Without loss of generality, the periodic metal-dielectric layered structure is assumed to made from Ag FIG. 2: (Color online) FDTD simulations of the magneticfield distributions for a p -polarized Gaussian beam launchedfrom a periodic metal-dielectric layered structure upon aninterface between the structure (grey area) and a dielectricmedium (white area) with a dielectric constant of 2.5. and a dielectric with a dielectric constant of 2.231. Thethickness of the Ag layer is 10 nm and that of the di-electric layer is 50 nm. In our simulations, a p -polarizedGaussian beam is used. Its wavelength is 756 nm, whichis much larger than the period (60 nm) of the periodicmetal-dielectric layered structure, justifying our effectivemedium approximation. An experimental value of therefractive index n = 0 .
03 + i .
242 at 756 nm for Ag isused. These parameters result in Re( ε x ) = 2 .
72 andRe( ε z ) = − .
72, leading to a rectangularly hyperbolicdispersion for the periodic metal-dielectric layered struc-ture.The terminated surface of the periodic metal-dielectriclayered structure is perpendicular to one of the asymp- tote of the hyperbolic dispersion, namely, in an angle of45 ◦ with respect to the periodic direction. The incidentGaussian beam forms an angle of 45 ◦ with respect to thesurface normal and is perpendicular to the periodic di-rection. It is obvious from the FDTD simulations that noreflection occurs at the surface. It should be noted thatzero reflection does not depend on the incident angle.Simulations for other incident angles are also conductedand zero reflection is always found, manifesting all-anglezero reflection.In conclusion, the reflection on the surface of a meta-material with a hyperbolic dispersion is studied theoreti-cally. For the metamaterial surface terminated obliquelyto the asymptote of the hyperbolic dispersion, reflectionis inevitable. However, all-angle zero reflection can occurif the surface is cut perpendicularly to the asymptote.We show this by a rigorous proof that for any incidentangle the surface normal component of the Poynting vec-tor of the reflected wave does not possess a real part,implying zero energy flow along the surface normal. Nu-merical simulations of a periodic metal-dielectric layeredstructure that is cut properly affirm unambiguously ourtheoretical prediction of all-angle zero reflection. Meta-materials with all-angle zero reflection at their surfacescould be exploited in many applications to eliminate un-desirable reflection.This work was supported by the 973 Program (grantnos. 2007CB613200 and 2006CB921700). The researchof X.H.L and J.Z is further supported by NSFC andShanghai Science and Technology Commission is also ac-knowledged. R. A. Shelby, D. R. Smith, and S. Schultz, Science , 77(2001). C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah,and M. Tanielian, Phys. Rev. Lett. , 107401 (2003). A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev.Lett. , 137401 (2003). J. B. Pendry, Phys. Rev. Lett. , 3966 (2000). A. Grbic, and G. V. Eleftheriades, Phys. Rev. Lett. ,117403 (2004). N. Fang, H. Lee, C. Sun, and X. Zhang, Science , 534(2005). V. G. Veselago, Sov. Phys. Usp. , 509 (1968). J. B. Pendry and D. R. Smith, Sci. Am. , 60 (2006). J. B. Pendry, D. Schurig, D. R. Smith, Science , 1780(2006). D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B.Pendry, A. F. Starr, D. R. Smith, Science , 977 (2006). J. D. Jackson,
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