Materializing a binary hyperlens design

We present a design of cylindrical hyperlens made of a layered binary material. The design approach uses an improved effective medium theory to take account of radius-dependent effects due to curvature of material interfaces resulting in nonperiodically distributed thicknesses of the lens layers. The performance of this lens is compared versus the designs with periodically thick layers, which we showed in earlier papers. Detailed quantitative results analyzed for the lenses with the same number and starting order of layers prove better functioning of the lens designed with this approach. © 2009 American Institute of Physics. DOI: 10.1063/1.3081403


Materializing a binary hyperlens design
Alexander V. Kildishev, a͒  We present a design of cylindrical hyperlens made of a layered binary material.The design approach uses an improved effective medium theory to take account of radius-dependent effects due to curvature of material interfaces resulting in nonperiodically distributed thicknesses of the lens layers.The performance of this lens is compared versus the designs with periodically thick layers, which we showed in earlier papers.Detailed quantitative results analyzed for the lenses with the same number and starting order of layers prove better functioning of the lens designed with this approach.© 2009 American Institute of Physics.͓DOI: 10.1063/1.3081403͔ The recent advancement theory of hyperlens and enhanced nanofabrication techniques have provided us with a fresh perspective on magnifying subwavelength imaging devices made of anisotropic materials with hyperbolic dispersion. 1,2In prefabrication design and experiments the required anisotropy was achieved through the fabrication of a cylindrical lamellar structure made of two distinct elementary materials ͑i.e., a binary metal-dielectric composite͒.A periodic distribution of metal-dielectric layers of a fixed thickness has been used.The focus of our study is ͑i͒ how to quantify, at least in quasistatic limit, the genuinely curvilinear interfaces of the cylindrical hyperlens in a radiusdependent effective medium theory ͑EMT͒ and ͑ii͒ if this EMT could improve the performance of the cylindrical hyperlens.The initial analysis indicated that a nonperiodic distribution of thicknesses of the lens layers is required for the improved design.Detailed simulations prove superior functioning of the lens designed with the improved approach versus the designs with the uniformly thick layers if the same number and starting order of layers is taken.The numerical analysis is built on the solution of the wave equation in piecewise homogeneous cylindrical coordinates, which has been widely detailed in literature. 3Our study is using the scalar wave equation; the electric field is assumed to be twodimensional and the magnetic field is the purely scalar quantity.For TM polarization the magnetic field ͑parallel to H ជ = z ˆh͒ can be expanded in azimuthal Fourier modes, h͑ , ͒ = ͚h m ͑͒exp m and the wave equation for the mth mode reduces to

͑1͒
Here the physical Cartesian coordinates ͑x , y , z͒ are defined through the cylindrical coordinates ͑ , , z͒ as x = cos͑͒, y = sin͑͒, and z = z; the prime corresponds to the radial derivative ‫ץ‬ / ‫,ץ‬ and are the only nonzero diagonal components of the anisotropic permittivity tensor, and k is the wavenumber of free-space.While the ideal and simplified hyperlens paradigms 2,4 are based on smooth scaling transformations 5 of Eq. ͑1͒, the proposed design of a manufacturable discrete-layered cylindrical hyperlens does not yet imply quantum size effects and other complications at the metal-dielectric interfaces, and thus each layer is taken to have isotropic dielectric function ͑ = ͒.The transformation of field data within each cylindrical layer to another place is accomplished by representing the field as a summation of spatial harmonics.The harmonic coefficients are determined by matching the fields at the interfaces with the expansion solution of the wave equation.Once these coefficients are found, the field is computed everywhere by using the sums of spatial harmonics.Making use of the cylindrical mode superposition, the source field and scattered fields are decomposed into backward ͑inbound͒ and forward ͑out-bound͒ modes.The transfer matrices are deduced directly from Maxwell equations and are cast in a spatial spectral form and then in matrix relations.Test problems are simulated using our ad hoc online tool, 6 where the hyperlens may consist of any number of layers bounded by infinite concentric cylindrical surfaces.The material properties of the media layers are utilizing the material optical database 7 where the materials are allowed to be dispersive, embracing polaritonic ͑plasmonic or phononic͒ media.Consider a classical hyperlens ͓a circular cylindrical multilayered structure comprising l max cylindrical interfaces of lamellar media ͑Fig.1͔͒.The axes of concentric cylindrical surfaces are assumed to coincide with the z-axis of a common cylindrical coordinate system ͑ , , z͒.Each interface boundary with a radius l separates two homogeneous media with the constant permeability = 1 and a pair of dielectric functions, l and l+1 .In TM formulation for a fixed free-space wavelength , the scalar magnetic field for a given lth interface is described by four components: the transmitted ͑h t 2l ͒ and reflected ͑h r 2l ͒ field at a͒ Electronic mail: kildishev@purdue.edu.
FIG. 1. ͑Color online͒ ͑a͒ Geometry of the concentric circular domains.͑b͒ Example of the test source inside the zoomed internal region of the lens.
the inner side of the boundary, h t 2l and h r 2l field at the inner side of the boundary for p l = n l ¯l and q l = n l+1 ¯l͒.Here, n l = ͱ l is the refractive in- dex of the medium adjacent to the inner side of the lth interface and ¯l =2 l / is a wavelength-normalized radius.The sums in Eqs.͑1͒ and ͑2͒ and in equations below are generally defined as summations over m = ͑−ϱ , ϱ͒.
For testing, the incident field can be, for example, generated by an expansion of the step-function, forcing the incident field to be unity at an arc segment with a constant radius s and angular dimensions = ͓ s,1 , s,2 ͔ ͓Fig.1͑b͔͒.If we apply the complete version of the same addition theorem in h i = H 0 ͑1͒ ͑k i ͒m s , then the incident field is given either by h t 0 = ͚t m 0 e m ͑for sources distributed inside the lens, Ͼ s ͒ or by h r 2l max +1 = ͚r m l max +1 e m ͑for external sources, s Ͼ ͒.Then, for s Ͻ 0 , t m 0 = m , as well as r m l max +1 = m for s Ͼ l max , where for m Ͼ 0, m = ͑e −m s,2 − e −m s,1 ͓͒m͑ s,2 − s,1 ͔͒ −1 , while otherwise, 0 = 1.In practice of designing engineered optical spaces, a smooth ideal distribution is approximated by a few elemental materials.Thus normally, subwavelength thin layers of dispersive and nondispersive elemental materials should be arranged to provide desired effective properties.The effective material properties of a planar binary-phase cylindrical structure can be obtained from the following quasistatic approximations: Since ln͑1+x͒Ϸx +O͑x 2 ͒, then for very thin layers the above relapses into familiar expressions for the effective permittivity of a planar lamellar structure, which could be also used as an effective medium approximation.It is essential to understand which of the two formulations, Eqs.͑3͒ and ͑4͒ or Eqs.͑5͒ and ͑6͒, provides a better performance of a binary circular hyperlens.Field-matching boundary conditions at lth interface are given by h ͒ / ‫ץ‬q l .The above system of equations can be rewritten using a matrix notation v l,m = w l,m v l+1,m , with the components of vector v m l being ͑v m l ͒ 0 = t m l and ͑v m l ͒ 1 = r m l , while the elements of matrix w l,m are given by w l,m +1 and w m = ͟ l=1 l max w l,m .If the ratios r ¯m l = r m l / t m l and t ¯m l = t m l max +1 / t m l are initialized with t ¯m l max +1 = 1 and t ¯m l max +1 = 1, then in a multilayer structure the normalized spatial spectral reflectance and transmittance are obtained through the following recurrences: r ¯m l = ͑w l,m 2,1 + w l,m 2,2 r ¯m l+1 ͒ / ¯m l and t ¯m l = t ¯m l+1 / ¯m l , where ¯m l = w l,m 1,1 + w l,m 1,2 r ¯m l+1 .The performance of the hyperlens is tested for two designs, proposed for two elemental materials SiC and ZnSe to be operational at a wavelength of = 11.061m, with SiC = −4.5+0.271, and ZnSe = 5.5.Note that at this wavelength both dielectric functions satisfy the condition Re͑ SiC + ZnSe ͒ = 1, providing good balance of elemental materials in the structure.The first design ͑lens A͒ is arranged with uniformly spaced interfaces, which in accord with Eqs.͑5͒ and ͑6͒ would give constant values of Re͑ ,l ͒ = 1 and Re͑ ,l ͒ ϳ 10.Though as shown in Fig. 2, lens A gives staggered values of ,l and ,l if calculated for the same uniform layers with Eqs.͑3͒ and ͑4͒.In contrast to lens A, lens B is also obtained by fixing Re͑ ,l ͒ = 1 for each two sequential layers ͑ZnSe-SiC or SiC-ZnSe͒ but without fixing Re͑ ,l ͒.This design rule defines the following three term recurrence for interface radii: Thus for a given number of layers l max in lens B, the entire design is achieved through adjusting the sequence of radii in order to fit the outer radius l max .While compared, both lens designs share a common set of dimensions, 0 = , and l max =3, and the same number of elementary layers.It has been observed that a lens with nonuniform thicknesses of layers, lens B always gives a better resolution in comparison with lens A with the same number of layers, and the best performance for lens B is achieved with the odd number of layers with the inner layer made of metal ͑SiC-ZnSe sequence͒.For example, Fig. 3 compares imaging with lens A versus lens B; the normalized distance in Fig. 3 is defined as d e = l max / .For both lenses, the test image is obtained with the best-performing order of elementary layers ͑SiC-ZnSe for l max = 31 and ZnSe-SiC for l max = 32͒.Figures 3͑b͒ and 3͑d͒ show that lens B is already working quite well even for such small numbers of layers, FIG. 2. ͑Color online͒ ͑a͒ Input magnetic field intensity at 0 used for all tests.͑b͒ Comparison of the nonuniform distribution of l max obtained from Eq. ͑9͒ vs the uniform distribution; l max = 34 has been taken for this example.
while lens A is barely resolving the image ͓Fig.3͑c͔͒ or does not resolve it at all ͓Fig.3͑d͔͒.
In the past few years, an improved EMT has been applied to obtain nonlocal corrections to the quasistatic EMT of anisotropic materials made of lamellar structures with subwavelength-thick layers. 8Thus, a near term work could include the incorporation of nonlocal adjustments for cylin-drical system in a way much similar to the techniques introduced for planar structures.

͑2͒
and the transmitted ͑h t 2l+1 ͒ and reflected ͑h r 2l+1 ͒ fields at the outer side of the interface boundary h t 2l+1 = ͚ t m l+1 f ¯m l e m , h r 2l+1 = ͚ r m l+1 b ¯m l e m , ͑3͒ where factors f ¯m l = f m 2l+1 / f m 2l+2 , b ¯m l = b m 2l+1 / b m 2l+2 can be defined using the forward ͓f m ͑.͒ ͔ and backward ͓b m ͑.͒ ͔ radial functions are the Hankel and Bessel cylindrical functions of the first kind.Thus, the even and odd terms are given by f m 2l = H m ͑1͒ ϫ͑p l ͒, b m 2l = J m ͑p l ͒ and f m 2l+1 = H m ͑1͒ ͑q l ͒, b m 2l+1 ͑͒ = J m ͑q l ͒ ͑their derivatives are given as Uday K. Chettiar, Zubin Jacob, Vladimir M. Shalaev, and Evgenii E. Narimanov Birck Nanotechnology Center, School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA ͑Received 23 December 2008; accepted 17 January 2009; published online 17 February 2009͒