## Abstract

We demonstrate layered superlensing in two-dimensional photonic crystals structured by both square and triangular lattices. In virtue of equifrequency contour analysis and FDTD calculation, both near field and far field imaging are displayed. Layered superlensing consisting of only triangular lattice photonic crystal is also studied and it exhibits more flexibility than the single layer counterpart. That is, the objective distance can be changed freely while keeping the image distance constant and vice versa. Hence, farther field imaging is achieved.

©2006 Optical Society of America

## 1. Introduction

Materials with negative refractive index predicted by Veselago in 1968 [1] attracted lots of attention in recent years. These materials, generally referred to as left-handed materials (LHM) by Veseloago, double negative materials by Ziolkowski [2], and backward-wave media by Lindell et al. [3], was first realized in 2000 by Smith et al. [4] as metamaterials composed of wires [5] and split ring resonators [6]. Later, variations of such structure were designed and experiments were performed [7]. One of the most interesting properties of such LHM is the superlensing effect suggested by Pendry [8]. Meanwhile, an effective negative refractive index in the vicinity of photonic band gap in photonic crystals was shown theoretically by Notomi [9]. Later, negative refraction of electromagnetic waves in such photonic crystals was demonstrated numerically and experimentally by Soukoulis et al. [10–12].

To our knowledge, negative refraction in 2D PCs generally occur under two conditions. One is that, as Notomi suggested [9], the EFC (equifrequency contour) of the PC is rounded and its radius shrinks as the frequency approaches to the band gap, which is common for frequencies above the first band near the Brillouin-zone center (Γ), so the wave vector and the group velocity are antiparallel. The other condition lies in frequencies where the EFC is hyperbolic like and the normal components of wave vector and group velocity are parallel, which usually occurs near a Brillouin-zone corner farthest from the center (Γ) [13]. Yet, there is a striking difference between the two cases, i.e. the refractive index is negative for the first case and positive for the second case. It is anisotropy that restricted far field image formation in the second case [13] and imaging of the first case was first demonstrated by Wang et al. [14]. Recently, image quality, transmission efficiency, surface termination and lattice orientation were well studied [15–20]. However, a clear and explicit distinction between the concepts of negative refraction, negative refractive index, and superlensing was not drawn until the work of Li et al. [21, 22]. Superlensing in the sense of Pendry is due to interface modes which can be of at least two types: plasmons in metal layers, or defect interface layer in photonic crystals. Only when the surface is carefully terminated, exciting the surface modes and an improvement of the image is achieved and hence the superlensing effect, not owing to negative permittivity and negative permeability, or inward directed equifrequency contours.

However, these lenses were all composed of single PC slab and layered superlens haven’t yet been demonstrated. Lately, image transfer with very little deterioration of the resolution by a cascaded stack consisting of two or three two-dimensional PC slabs separated by air is demonstrated [23].

In this paper, we propose a layered superlens composed of both square lattice and triangular lattice PC as well as a layered superlens composed of air sandwiched by triangular lattice PC.

## 2. Numerical method

Plane wave expansion method [24, 25] was employed to calculate photonic band diagrams and equifrequency contours, in which Bloch waves are expanded by 625 plane waves. We examine mainly TM modes (electric field lies perpendicular to the 2D plane). To solve the Maxwell equations, the finite-difference time-domain (FDTD) method [26, 27] was used with uniaxial perfectly matched layer (UPML) boundary conditions [28]. Distances were normalized by units of the lattice constant *a*. Time and frequency are then expressed in units of *a*/*c* and *c*/*a*, respectively, where *c* denotes the velocity of light in vacuum. *f*=*ωa*/2*πc* is the dimensionless frequency. Square meshes with mesh size *δ*=0.02*a* and time step Δ*t*=0.95*δ*/√2*c* were employed. More than 10000 time steps were run to reach the steady state.

## 3. Layered superlens structured by triangular lattice PC

Consider photonic crystal slabs made of a triangular lattice of air holes with radius *r*=0.4*a* (*a* is the lattice constant) embedded in dielectric matrix with permittivity 12.96. Its photonic bands (TM modes) are illustrated in Fig. 1(a) showing negative refraction of the first case at frequency 0.306 (normalized by *ωa*/2*πc*) with effective refractive index *n*=-1 [9]. In addition, both slab terminations are identical and are chosen to ensure high transmission efficiency and interface modes which are critical to the superlensing effect. To this end, the distance between the centers of the outmost holes and the surface is (√3*a*/4-0.2)*a*.

As we know, a single triangular PC slab could act as a superlens and the relation *D _{o}*+

*D*=

_{i}*T*holds [14], where

*D*is the distance from the object to one side of the PC slab (we called it objective distance hereafter),

_{o}*D*is the distance from the other side of the PC slab to the image (we called it image distance hereafter), and

_{i}*T*is the thickness of the slab. If the objective distance

*D*is greater than the thickness of the PC slab

_{o}*T*, the image won’t appear.

Yet, a closer inspection shows that a virtual image is formed and we can use a second PC slab to reconverge it to form a real image as pictured in Fig. 2(d). Actually the function of the objective lens (with thickness *T _{o}*) is to draw the object nearer by a distance of 2*

*T*and a virtual image is formed, hence we call the first lens objective lens. The image lens sees the virtual image and a real one is formed, hence we call the second lens image lens. So, even the objective distance is larger than the thickness of objective lens a real image is still formed which farther free the object from near field imaging. Likewise, even the image distance is greater than the thickness of the image lens a image is still formed as pictured in Fig. 2(a).

_{o}It is true that a thicker slab could form a image far from the slab, but with the increase of the slab thickness, resolution dropped. This is also the reason why image transfer rather than a thicker slab was used to increase the distance between the image and the object as addressed in Ref. [23]. Moreover, layered superlens provides more flexibility than the single layer counterpart since we could easily change, holding the distance from the object to the image constant though, the objective distance or image distance freely while keeping the other constant, which is rather useful in optical imaging system and particularly optical lithography. As illustrated in Fig. 2, since the distance between the object and the image are two times the total thickness of objective lens and image lens, it is free to change the objective distance and image distance respectively without change in image position and loss in resolution which is always about 0.44λ (full width at half maximum) in the transversal direction.

## 4. Layered superlens structured by both square and triangular lattice PC

The triangular lattice photonic crystal consist of air holes with radius 0.4 *r*=*a _{tri}* (

*a*is the lattice constant) embedded in dielectric matrix with permittivity 12.96 while the square lattice photonic crystal consists of a periodic array of infinitely long, cylindrical dielectric rods with radius

_{tri}*r*=0.3

*a*(

_{squa}*a*is the lattice constant) and permittivity 14 embedded in air. Their photonic bands (TM modes) are shown in Fig. 1, respectively. the triangular lattice PC show negative refractive of the first case at frequency 0.306 (normalized by

_{squa}*ωa*/2

_{tri}*πc*) with effective refraction index

*n*=-1 [9], while the square lattice PC exhibit a narrow frequency range around 0.192 (normalized by

*ωa*/2

*squa**πc*) of negative refraction of the second case discussed above [13]. What if we bring them together by scaling

*a*, pegging 0.306*

_{tri}*ωa*/2

_{tri}*πc*to 0.192*

*ωa*/2

_{squa}*πc*. Specifically, the triangular lattice constant

*a*is 0.306/0.192 that of the square lattice constant

_{tri}*a*.

_{squa}As we know, the square lattice PC could act as a superlens in the ΓΜ direction only in the near field since its frequency contour is hyperbola like rather than rounded [13]. However, the square-triangular layered superlens could form far field images since the electromagnetic waves undergo a refraction on the square-triangular lattice PC interface, redirecting the wave for ease of convergence. Applying parallel wave vector conservation, refraction on the squaretriangular lattice PC interface is illustrated in Fig. 3. The wave undergoes three refractions, first on the air-square PC interface, second on the square-triangular PC interface, and third on the triangular PC-air interface, which are negative refraction, positive refraction, and again negative refraction, respectively. At last a image is well formed whose full width at half maximum (FWHM) is about 0.46λ in the transversal direction and about 1.75λ in the longitudinal direction by plotting electric field intensity distribution across the image. Aberration in the longitudinal direction which is also observed in previous literature is probably induced by the deviation of *n* form -1 for different angles of incidence. Anyway, superlensing effect of square-triangular layered PC was observed in the transversal direction and the field propagation map are shown in Fig. 4(a) and Fig. 4(b). Similarly, the triangularsquare lattice PC is also studied and the field propagation maps are displayed in Fig. 4(c) and Fig. 4(d).

## 5. Discussions and conclusions

When the electromagnetic wave transports through the layered samples, the scattering loss by the interface is inevitable. Although the transmission efficiency is rather low, there is no pronounced decrease in the resolution for such two layered superlenses since the surface termination and lattice orientation are carefully chosen to ensure high transmission efficiency and interface modes. As for superlenses consisting of more layers, the image quality requires the solution of an optimization problem involving the frequency of the light, the distance between the slabs, the surface termination and lattice orientation, and the thickness of the photonic crystal slabs. In conclusion, we present the study of layered superlening effect in two dimensional photonic crystals including square-triangular layered, triangular-square layered and triangular-air-triangular sandwiched superlenses. Both the square-triangular layered and triangular-square layered superlenses are able to form either near field or far field image. And the triangular sandwiched superlenses provide more flexibility than the single layer counterpart in changing the objective and image distances which is useful in the potential subwavelength optical lithography.

## Acknowledgments

This work is supported by the Key Project of National Science Foundation of China (NSFC) under Contract No. 60531020, and in part by NSFC60671003 and ZJNSF R105253.

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