## Abstract

We demonstrate unrestricted superlensing in a *triangular* two-dimensional photonic crystal. We investigate simple two-point light sources imaged by a slab lenses made of this photonic crystal, and show that the refraction of light follows simple rules of geometric optics with the Snell’s-law refraction at each interface, and an effective *isotropic* refractive index *n*=-1 for light propagating inside the crystal. We contrast this behavior with that of a *square* two-dimensional photonic crystal in the first photonic band, where the effective dielectric response is *anisotropic*. This leads to a restricted superlensing, which does not follow the geometric optics.

©2004 Optical Society of America

Materials with isotropic refractive index *n*=-1 have exotic optical properties. Such materials restore not only *phases* of the transmitted propagating waves, but also *amplitudes* of the evanescent waves, that are responsible for the subwavelength details of source geometry, and were proposed to make a superlens capable of subwavelength imaging [1–2]. Since such materials do not exist naturally, proposals have been put forth in recent years to artificially synthesize them. In one scheme [3–6], an effective medium made of metallic rods, and the so-called split-ring resonators, has been shown to act as such a metamedium in the microwave frequency range. Other similar schemes have also been proposed [7–8].

In another approach, it has been shown that a two-dimensional photonic crystal (2D-PC) can act as a metamedium [9–11], however the superlensisng has not been demonstrated. Recently, it was shown that the propagation of light in a square 2D-PC in the first photonic band leads to an all angle negative refraction (AANR), which leads to superlensing, but where an isotropic negative refractive index cannot be defined [12]. The negative refraction and the slab focusing in the second photonic band have recently been demonstrated experimentally in square 2D-PC, again in the microwave frequency range [13]. However, in the theoretical and experimental imaging studies [12–16], the distance between the point source and the edge of the PC slab was less than the lattice constant of the PC. Recently, it was shown that imaging in a square 2D-PC, in the AANR mode, does not follow rules of geometric optics, and that a channeling, rather than isotropic wave propagation occurs inside the PC [17]. This shows that the imaging capabilities are restricted in this PC. In this paper we clarify this issue, and identify a 2D-PC capable of efficient, unrestricted superlensing.

The PCs studied in this letter consist of a periodic (in the x-y plane) array of infinitely long, cylindrical (along z-direction) air-holes of radius *r*, embedded in a thick slab of dielectric matrix. Each PC is assumed to have a finite width in the x-direction, but is unrestricted in the y direction. We obtain photonic band structures by numerically solving the Maxwell equations. We employ here the standard plane wave expansion [18]. To obtain maps of propagating waves, we use the finite-difference time-domain (FDTD) simulations with perfectly matched layer boundary conditions [19–21].

We first study the square PC, which became a structure of choice in the experimental studies of the negative refraction. It was shown, that AANR can occur in this PC [12]. We choose parameters of Ref. [12], i.e., hole radius *r*=0.35*a*, where the lattice constant is a, and the dielectric matrix with dielectric constant *ε*=12. The calculated band structure (normalized frequency *ω=a/λ* vs. wave vector k) for the transverse electric (TE) mode (the magnetic field parallel to air holes) is shown in Fig. 1(a). The AANR occurs at the top of the first band (around the M-point), indicated by the horizontal dashed line for *ω*=*ω*
_{1}. The inset in Fig. 1(a) shows the constant frequency contour obtained by crossing the first band with the equal energy surface for *ω*=*ω*
_{1}. The fact that it is not circular, suggest a strong anisotropy of the wave propagation in this PC. As a result, the PC cannot be represented as an effective medium with an isotropic refractive index, and instead a refractive index tensor must be used. In fact, a channeling of the wave propagation in the ΓM and equivalent directions is expected [17], due to larger group velocities of propagating waves in these directions.

In order to study this effect in detail, we calculate the wave propagation maps. We turn-on a point source of cylindrical waves with frequency *ω*=*ω*
_{1} at time *t*
_{0}, outside and to the left of the slab of the PC, and calculate the field patterns at some later time *t*. Figure 2 shows the calculated magnetic field pattern at the steady state (*t*→∞) for two different thicknesses of the slab, and demonstrates that the positions of the image and the source *do not* obey the geometric optics, since in order for this to happen, the distance of the image from the slab surface would have to be larger for a thicker slab. This is illustrated in Fig. 2 by sketches of geometric optics rays. In fact, a closer inspection of the images shows that instead of the expected focus inside the PC, there is a channel of propagation perpendicular to the slab surface (ΓM direction), in agreement with the argument based on the anisotropy of the equal frequency contour, as discussed above. Increasing the distance of the source from the slab surface, causes the image to disappear inside the slab. By plotting the light intensity across the image, we find that the spatial resolution in this case, as measured by full width at half maximum, is about *t _{s}*=0.3

*λ*. This all suggests, that the superlensing in this crystal is restricted, and the image formation possible only in the highly non-geometric near-lens mode.

Let’s now consider the *triangular* 2D-PC. Now we choose parameters used in Ref. [9], i.e., *r*=0.4*a* and *ε*=12.96. The photonic band structure for the transverse magnetic (TM) modes (the electric field parallel to cylindrical holes) is show in Fig. 1(b). The horizontal dashed line with frequency *ω*
_{2} indicates the best frequency for the superlensing. The corresponding constant frequency contour is now circular. The propagation of the waves is therefore isotropic, and one can define an isotropic effective refractive index. Fig. 3 shows the propagation maps for this case, for various positions of the point source, and thickness of the slab. It can be easily shown, for example by drawing traces of the rays between the sources and the corresponding images, that now the PC acts as a metamedium whit *n*=-1, and therefore the superlensing is *unrestricted*. Again, by plotting light intensity across the image, we find that the spatial resolution in this case is about *t _{H}*=0.4

*λ*.

In order to demonstrate the superlens imaging, we show in Fig. 4 wave propagation maps for a two-point source, differently located in front of the slab of the triangular 2D-PC. One obtains clear images of the two points in either horizontal, or vertical configurations. By reducing the distance between the two point sources well below the wavelength, we can test the possibility of the subwavelength resolution. Fig. 5 shows the light intensity across the image of two vertical point sources for different distances between the sources.

In conclusion, we show that the imaging is restricted only to the near-lens in the *square* crystal in the AANR mode of operation. We trace this to an *anisotropic* geometry of the constant frequency contour in the relevant section of the photonic band structure. In contrast, we show that in a *triangular* 2D-PC, a highly circular constant frequency contour can be chosen in a relevant band, and as a result an *isotropic* propagation of the refracted waves occurs. Thus this PC can act as an isotropic effective metamedium with an effective refractive index *n*=-1, and therefore capable of unrestricted superlensing. The possibility of improving the spatial resolution of the superlensing will be discussed elsewhere.

## Acknowledgments

This work is supported by US Army Research Development and Engineering Command, Natick Soldier Center, under the grant DAAD16-03-C-0052.

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