We describe the properties of birefringent left-handed metamaterials and
introduce the concept of a birefringent perfect lens. We demonstrate that, in a
sharp contrast to the conventional left-handed perfect lens at
$\epsilon=\mu=-1$, where $\epsilon$ is the dielectric constant and $\mu$ is the
magnetic permeability, the birefringent left-handed lens can focus either TE or
TM polarized waves or both of them, allowing a spatial separation of the TE and
TM images. We discuss several applications of the birefringent left-handed
lenses such as the beam splitting and near-field diagnostics at the
sub-wavelength scale.Comment: 4 pages 6 figure

Kivshar, Yuri S.

Noskov, Roman E.

Shadrivov, Ilya V.

Zharov, Alexander A.

Zharova, Nina A.

New Journal of Physics

English

arXiv

We describe the properties of specific non-reflecting birefringent left-handed metamaterials and demonstrate a birefringent perfect lens for vectorial fields. We predict that, in a sharp contrast to the concept of a conventional perfect lens realized at ε  =  μ  =   - 1 (where ε is the dielectric permittivity and μ is the magnetic permeability), the birefringent left-handed slab possesses the property of negative refraction either for TE- or TM-polarized waves or for both of them simultaneously. This allows selective focusing and a spatial separation of the images created at different polarizations. We discuss several applications of the birefringent left-handed lenses such as the beam splitting and near-field diagnostics at the sub-wavelength scale

Zharov, N Alexander

Zharova, Nina

Noskov, Roman E

Shadrivov, Ilya

Kivshar, Yuri

The Australian National University

T h e  o p e n – a c c e s s  j o u r n a l  f o r  p h y s i c sNew Journal of PhysicsBirefringent left-handed metamaterials andperfect lenses for vectorial fieldsAlexander A Zharov1,2, Nina A Zharova1,3, Roman E Noskov2,Ilya V Shadrivov1 and Yuri S Kivshar11 Nonlinear Physics Centre, Research School of Physical Sciences andEngineering, Australian National University, Canberra CT 0200, Australia2 Institute for Physics of Microstructures, Russian Academy of Sciences,Nizhny Novgorod 603950, Russia3 Institute of Applied Physics, Russian Academy of Sciences,Nizhny Novgorod 603600, RussiaE-mail: ivs124@rsphysse.anu.edu.auNew Journal of Physics 7 (2005) 220Received 28 March 2005Published 11 October 2005Online at http://www.njp.org/doi:10.1088/1367-2630/7/1/220Abstract. We describe the properties of specific non-reflecting birefringentleft-handed metamaterials and demonstrate a birefringent perfect lens forvectorial fields. We predict that, in a sharp contrast to the concept of a conventionalperfect lens realized at ε = µ = −1 (where ε is the dielectric permittivity and µis the magnetic permeability), the birefringent left-handed slab possesses theproperty of negative refraction either for TE- or TM-polarized waves or for bothof them simultaneously. This allows selective focusing and a spatial separation ofthe images created at different polarizations. We discuss several applicationsof the birefringent left-handed lenses such as the beam splitting and near-fielddiagnostics at the sub-wavelength scale.Contents1. Introduction 22. Birefringent non-reflecting metamaterial 33. Perfect lenses for vectorial fields 44. Conclusions 8Acknowledgments 9References 9New Journal of Physics 7 (2005) 220 PII: S1367-2630(05)97045-61367-2630/05/010220+9$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft2 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT1. IntroductionCurrent interest in the physics of sub-wavelength imaging is explained by many potentialapplications of this effect, including lithography and data storage. The concept of a perfect lens[1] suggests one of the potential ways for improving the resolution of imaging systems basedon the unique properties of the so-called left-handed metamaterials or materials with negativerefractive index [2, 3]. The possibility of a perfect lens whose resolution is not limited by theclassical diffraction limit has been a subject of intense debates by the scientific community in thepast 5 years, and at first it was met with considerable opposition [4]. However, many difficultiesraised by the critics have been answered and have clarified the concept itself and its limitations [5]through the results of numerical simulations [6] and recent experiments on negative refraction,e.g. with a negative-index material assembled from discrete elements arranged on a planar circuitboard [7].The so-called perfect lens is created by a slab of a left-handed metamaterial with ε =µ = −1, where ε is the dielectric pemittivity and µ is the magnetic permeability. Veselago [2]predicted that such a material would have a negative refractive index of n = −√εµ = −1, anda flat slab of this material would act as a lens refocusing all rays from a point source on one sideof the slab into a point on the other side of the slab (see figure 1). Later, Pendry [1] suggestedthat such a lens can reconstruct the near field of a source, and as a result it can create an idealimage [1].Thus, a slab of the left-handed metamaterial can be used for a sub-wavelength imagingbecause it amplifies all evanescent (near field) modes inside the slab, and therefore it allows topreserve the information about the source structure with a resolution better than the radiationwavelength. However, to satisfy the conditions for such a perfect lens to operate, the distancebetween the source and the slab surface, a, and the distance between the second surface of theslab and the image point, b, should be connected with the slab thickness d by the relation [1](see figure 1),a + b = d. (1)The relation (1) means that it is impossible to create an image at the distances larger than the slabthickness, and this is one of the serious limitations for the applicability of left-handed perfectlenses.In this paper, we demonstrate that birefringent non-reflecting left-handed metamaterialscan be used for birefringent perfect lenses, which can focus vectorial fields. In particular,we show that, in contrast to the condition ε = µ = −1 for the conventional perfect lens, thebirefringent left-handed lens can focus either TE- or TM-polarized waves or both, with a varyingdistance between the TE- and TM-images, and this property allows us to expand dramaticallythe applicability limits of the perfect lenses. We suggest specific structures for the dielectric andthe magnetic tensors that allows to create this type of birefringent media, which is transparentfor an arbitrary angle of incidence. In addition, we show that such a birefringent lens is free fromthe limitations imposed by the condition (1), and we also discuss some other applications of thebirefringent left-handed metamaterials for the beam polarization splitting and sub-wavelengthimaging.New Journal of Physics 7 (2005) 220 (http://www.njp.org/)3 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT-a 0 d d+bz (a)(b)Intensity x Figure 1. Schematic presentation of (a) ray focusing and (b) reconstruction ofevanescent waves by a left-handed lens.2. Birefringent non-reflecting metamaterialAnisotropic metamaterials have been studied previously in several papers [8], where differentregimes of wave propagation are discussed. Here, we consider a specific linear birefringentmedium described by the following tensors of dielectric permittivity ε̂ and magnetic permeabilityµ̂, which in the main axes of the crystal have the formε̂ =A 0 00 B 00 0 A−1 , µ̂ =B 0 00 A 00 0 B−1 , (2)where A and B are generally arbitrary complex functions of the frequency. We substitute theexpressions (2) into Maxwell’s equations and obtain the equations for the transverse spatialharmonics of the monochromatic (∼exp(iωt − ikxx)) electromagnetic waves, for the case of (i)TM polarization, when E = (Ex, 0, Ez) and H = (0, Hy, 0)d2Hydz2+ A2(k20 − k2x)Hy = 0, (3)Ex = − 1ik0AdHydz, Ez = −kxk0AHy (4)and for the case of (ii) TE polarization, when E = (0, Ey, 0) and H = (Hx, 0, Hz)d2Eydz2+ B2(k20 − k2x)Ey = 0, (5)Hx = − 1ik0BdEydz, Hz = kxk0BEy, (6)New Journal of Physics 7 (2005) 220 (http://www.njp.org/)4 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTθTE θTMTE, TMvirtual TE image z 0virtual TM image  d x Figure 2. Ray diagram showing the creation of two separate TE- and TM-polarized virtual images of a source; θTE and θTM are the angles between thedirections of the group velocity and wavevector of the TE and TM waves insidethe slab.where k0 = ω/c is the wavenumber in vacuum, kx is the wave vector component along thex-axes and c is the speed of light. It is easy to verify that the wave impedance of this birefringentmedium matches exactly the impedance of vacuum, for both polarizations with any transversewavenumbers kx, and for arbitrary (including complex) values of A and B. Therefore, the mediumdescribed by the tensors (2) is ideally impedance-matched with vacuum, and it is reflectionless4.Such a birefringent medium was suggested to create a perfectly matched layer in the finite-difference time-domain simulations [10]. In a general case, when the vacuum is substitutedby a medium with some εs and µs, the impedance matching conditions would require somemodification of equation (2), namely ε̂ → εsε̂ and µ̂ → µsµ̂. Below we consider εs = µs = 1without loss of generality.3. Perfect lenses for vectorial fieldsWe consider a slab of the metamaterial with the dielectric and magnetic properties characterizedby the tensors (2). The slab is surrounded by vacuum, and it has the thickness d (0  z  d). Weassume that a point source is located at the distance z = −a from the nearest surface of the slab,as shown in figure 2, the source generates both TE- and TM-polarized waves, and it is describedby the corresponding distribution of the electric field Ey(x, z = −a), for the TE polarization,or the magnetic field Hy(x, z = −a), for the TM polarization, in the plane z = −a. We denotethe spatial spectra of these fields as αe(kx) and αm(kx), respectively. Using equations (3)–(6) fordescribing the electromagnetic field in the slab, and satisfying the boundary conditions for thetangential components of the fields, we obtain general expressions for the spatial harmonics of4 Reflectionless in this kind of medium but with A = B > 0 has been mentioned earlier in [9].New Journal of Physics 7 (2005) 220 (http://www.njp.org/)5 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTFigure 3. Beam scattering by a slab of the birefringent metamaterial (A = −4,B = +2, d = 5). The TM-polarized component experiences negative refraction,while the TE-polarized component refracts normally. Coordinates are normalizedto the free-space wavelength.the fields behind the slab, i.e. for z > d,Hy(z, kx) = αm(kx) exp[−i√k20 − k2x(a + Ad + z′)], (7)for the TM-polarized waves, andEy(z, kx) = αe(kx) exp[−i√k20 − k2x(a + Bd + z′)](8)for the TE-polarized waves, where z′ = z − d. Thus, for real A and B, equations (7) and (8)reproduce the field structure of the source in the region z′ > 0 shifted from the source positionby the distance (A − 1)d (TM waves) or (B − 1)d (TE waves). A typical ray diagram for thiscase is shown in figure 2 for A > 1, B > 1 and A > B, where we show the position of the sourcefor both the polarizations, as well as spatially separated virtual images created by the lens. Ingeneral, for A = B, the virtual images of the TE and TM sources are shifted relative to eachother. For 0 < A and B < 1, the virtual images can be located either between the slab and thesource or inside the metamaterial slab.More interesting cases of the medium (2) correspond to negative values of A and/or B.When A < 0 and B > 0, negative refraction occurs for the TM-polarized waves only, whereasthe TE-polarized waves refract normally, see figure 3. For A > 0 and B < 0, the opposite effectoccurs, i.e. negative refraction is possible for the TE-polarized waves only. This property can beNew Journal of Physics 7 (2005) 220 (http://www.njp.org/)6 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTused for the polarization-sensitive beam separation. Figure 3 shows an example of this separationfor the slab with parameters A = −4 and B = +2. A two-dimensional beam propagates at theangle of incidence 30◦ and is refracted. Initially, the beam is composed of two polarizations withthe same partial intensities. When the beam is refracted at the surface, the TM-polarized waveundergoes negative refraction, and it becomes separated from the normally refracted TE beam.Another specific feature of the birefringent lenses is a possibility to form two separateperfect images for the TE- and TM-polarized waves. This property follows from the results (7)and (8). In particular, for A < 0 and B > 0, the transverse spatial spectrum of the TM-polarizedfield in the plane z′m = |A|d − a coincides with the spectrum of the sourceHy(z′m, kx) = αm(kx), (9)while the TE-polarized component of the beam is positively refracted. In the case A > 0 andB < 0, the image is created by the TE-polarized waves at z′e = |B|d − a,Ey(z′e, kx) = αe(kx), (10)whereas the TM-polarized waves experience positive refraction, and they do not create an image.Thus, in the case of the birefringent lens, additional parameters appear, which mitigate the strictlimitations for the isotropic lens imposed by equation (1). As a result, the source and the imagecan be located further away from the slab. More importantly, when both A and B are negativeand A = B, both TE and TM images appear, and they are separated by the distanceh = |z′e − z′m| = |(|B| − |A|)d|, (11)which, in the absence of dissipative losses, can be arbitrary large. This allows novel possibilitiesfor the sub-wavelength resolution, diagnostics and microscopy.In the case |A| = |B|, the TE and TM images coincide and in a particular case A = −1 andB = −1, we recover the results for the isotropic perfect lens discussed by Pendry [1] andVeselago[2]. In general, the basic physics for operating these birefringent perfect lenses is similar to theisotropic case, and it is defined by two major factors: (i) negative refraction and (ii) amplificationof evanescent waves. Figure 1(b) shows schematically the structure of the evanescent waves inthe slab for the case of Pendry’s lens, which is similar for both isotropic and birefringent left-handed media. Figures 4(a) and (b) show schematically the ray diagram in two special cases,when a single source generates both TE- and TM-polarized waves (see figure 4(a)) creating twoseparate images, and when the TE and TM sources are separated and they create a combinedimage (see figure 4(b)).A possibility of the sub-wavelength resolution of a pair of sub-wavelength sources byusing the birefringent left-handed lens has also been verified numerically, and some examplesare presented in figure 5 for the case of a lossy medium when ε̂l = ε̂ − iδik × 10−8 andµ̂l = µ̂ − iδik × 10−8, where δik = 1 for i = k and it is 0 otherwise. The mixed-polarized sourceconsists of two beams of the width λ/5, separated by the distance 2λ/5, where λ is the free-spacewavelength, metamaterial parameters are A = −2.5 and B = −1.5 and the slab thickness is λ/2.A difference in the resolution for the TE- and TM-polarized waves for A = B is explained bydifferent effective optical thickness of the slab for two polarizations. An increase of the lossesdecreases the resolution abilities of the lens dramatically. Figures 6(a) and (b) show the spatialNew Journal of Physics 7 (2005) 220 (http://www.njp.org/)7 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTz (b)Image TMImage TE Source TE, TMImage TE, TM Source TE (a)d 0 Source TM Figure 4. Ray diagrams of the birefringent left-handed lens imaging. (a) A singlesource imaging by a metamaterial slab characterized by negative A = B. TE andTM images are separated by the distance h defined by equation (11). (b) Separatedsources with two different polarizations can create the images in the same plane,provided the sources are separated by the distance (11).X|Hy|, |Ey| (arb. u.)Figure 5. Transverse structure of the TE and TM fields of the source (——),electric field of the TE image (- - - -) and magnetic field of the TM image (• • •).The coordinate is normalized to the free-space wavelength.New Journal of Physics 7 (2005) 220 (http://www.njp.org/)8 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTFigure 6. Spatial distribution of the absolute values of (a) magnetic and (b)electric fields (logarithmic scale) generated by two sub-wavelength sources ofthe width λ/5 separated by the distance 2λ/5. Parameters are A = −2.5 andB = −1.5, and the slab thickness is λ/2. Solid lines mark the metamaterial slab,dashed lines show the image planes. Coordinates are normalized to the free-spacewavelength.distribution of the magnetic field, Re(Hy(x, z)), for the TM polarization and spatial distributionof electric field, Re(Ey(x, z)), for the TE polarization, respectively.Different examples presented above clearly demonstrate that the birefringent left-handedmetamaterials and birefringent perfect lenses are novel objects with many unusual propertiesand, more importantly, they may demonstrate much broader spectrum of potential applications,in comparison with the isotropic metamaterials and perfect lenses [1, 2].Although the birefringentperfect lenses are not yet realized experimentally, we believe that the ideas and results presentedhere are quite realistic and will initiate strong efforts in creating the composite metamaterialswith substantially birefringent properties, including those that satisfy the specific conditionsfor the tensor components of equation (2). This would require a new thinking in applying thetraditional approaches of the papers [11, 12] where the fabrication of isotropic metamaterialshas been reported. Such an anisotropy can be achieved by using more complicated elementarycells made of wires and split-ring resonators, instead of the traditional symmetric cubic lattice[12], in order to engineer both the electric and magnetic response in three different directions.We also note that in order to realize a birefringent lens, which is able to create an image of athree-dimensional source (compared to the two-dimensional case considered above), one shouldtake the metamaterial with A = B, and it can simplify the design of the composite. Such a lenscreates an image of both polarizations in the same plane.4. ConclusionsWe have introduced a novel type of left-handed media with reflectionless and birefringentproperties for refraction of vectorial fields. In particular, we have shown that the flat-slab focusingNew Journal of Physics 7 (2005) 220 (http://www.njp.org/)9 Institute of Physics DEUTSCHE PHYSIKALISCHE GESELLSCHAFTand negative refraction of electromagnetic waves may occur under different conditions for theTE- and TM-polarized waves or simultaneously with two spatially separated TE and TM images.Our results suggest novel directions in the study of the intriguing properties of left-handedmetamaterials and photonic crystals, and they provide novel ideas for a design of compositematerials with negative refraction.AcknowledgmentsThe authors acknowledge a support of the Australian Research Council and thank AI Smirnovfor useful references. AAZ and REN thank IG Kondrat’ev for discussions. AAZ and NAZacknowledge a warm hospitality of Nonlinear Physics Centre in Canberra. AAZ acknowledgesa support from RFBR (grant N05-02-16357).References[1] Pendry J B 2000 Phys. Rev. Lett. 85 3966[2] Veselago V G 1967 Usp. Fiz. Nauk 92 517Veselago V G 1968 Sov. Phys.—Usp. 10 569 (Engl. Transl.)[3] Smith D R, Pendry J B and Wiltshire M C K 2004 Science 305 788[4] See, e.g., ’t Hooft G W 2001 Phys. Rev. Lett. 87 249701Garcia N and Nieto-Vesperinas M 2002 Phys. Rev. Lett. 88 207403[5] Shen J T and Platzman P M 2002 Appl. Phys. Lett. 80 3286Smith D R, Schurig D, Rosenbluth M, Schultz S, Ramakrishna S A and Pendry J B 2003 Appl. Phys. Lett.82 1506[6] Kolinko P and Smith D R 2003 Opt. Express 11 640Cummer S A 2003 Appl. Phys. Lett. 82 1503[7] Grbic A and Eleftheriades G V 2004 Phys. Rev. Lett. 92 117403[8] Smith D R and Schurig D 2003 Phys. Rev. Lett. 90 077405Kärkkäinen M K 2003 Phys. Rev. E 68 026602Lindell I V, Tretyakov S A, Nikoskinen K I and Ilvonen S 2001 Microw. Opt. Tech. Lett. 31 129Hu L and Lin Z 2003 Phys. Lett. A 316 313Lakhtakia A and Sherwin J A 2003 Int. J. Infrared Millim. Waves 24 19[9] Efimov S P 1978 Izv. Vuzov Radiofizika 21 1318Efimov S P 1978 Radiophys. Quantum Electron. 21 1318 (Engl. Transl.)[10] Sacks Z, Kingsland D, Lee R and Lee J 1995 IEEE Trans. Antennas Propag. 43 1460[11] Shelby R A, Smith D R and Schultz S 2001 Science 292 77[12] Parazzoli C G, Greegor R B, Li K, Koltenbah B E C and Tanielian M 2003 Phys. Rev. Lett. 90 107401New Journal of Physics 7 (2005) 220 (http://www.njp.org/)

Birefringent left-handed metamaterials and perfect lenses for vectorial fields

Birefringent left-handed metamaterials and perfect lenses

Birefringent left-handed metamaterials and perfect lenses

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