## Abstract

The electromagnetic properties of finite checkerboards consisting of alternating rectangular cells of positive refractive index (*ε*=+1, *μ*=+1) and negative refractive index (*ε*=-1, *μ*=-1) have been investigated numerically. We show that the numerical calculations have to be carried out with very fine discretization to accurately model the highly singular behaviour of these checkerboards. Our solutions show that, within the accuracy of the numerical calculations, the focussing properties of these checkerboards are reasonably robust in the presence of moderate levels of dissipation. We also show that even small systems of checkerboards can display focussing effects to some extent.

© 2006 Optical Society of America

## 1. Introduction

Negative refractive index materials (NRM) have become very popular in the past few years (see [1, 2] for recent reviews), due to both their non-intuitive Physics as well as their immense potential for applications[3]. A necessary condition for negative refractive index is that the real parts of both the dielectric permittivity, *ε* (*ω*), and the magnetic permeability, *μ*(*ω*), should be negative at the given frequency, *ω*. It should also be pointed out that negative values for *ε* and *μ* can occur only within a finite band of frequencies, and that the NRM are dispersive and dissipative. One of the effects in NRM that has captured the imagination of the scientific community is a superlens[4] that can give rise to sub-wavelength resolution in an image. In the simplest case, just a slab of NRM acts as a superlens that images a point on one side of it to a point on the other side [5]. The sub-wavelength resolution is possible because the NRM also involves in the imaging process the non-radiative near-field modes of the source, which are associated with the information on sub-wavelength lengthscales, via resonant surface plasmon modes on the NRM surface[4, 6]. The sub-wavelength resolution is, however, limited by the dissipation inherent in NRM[7], but can increase without bound as the absorption in the NRM tends to zero (Perfect Lens).

The slab lens is actually only one member of a whole class of systems of NRM to which a generalised Perfect Lens Theorem applies[8]. A region filled with NRM is, in a sense, complementary to a region of positive index and cancels the effects of propagation in the positive region. Perfect imaging (in the limit of zero loss) has been predicted with cylindrical or spherical shells[8], 2-D corners[8, 9], 3-D corners, and checkerboards of NRM in 2-D and 3-D[10]. Focussing effects in an infinite non-absorptive checkerboard lattice was discussed by us in Ref.[10], where it was shown that a source placed in one cell of the checkerboard reproduces an image in every other cell of the checkerboard. However, there is considerable concern whether the dissipation in the NRM would badly affect this imaging process. It is well known that the sub-wavelength resolution is badly affected by dissipation[7], and the infinities in the density of modes at the sharp corners[8, 9] can make it worse.

In this paper, we discuss focussing effects in finite 2-D checkerboards of dissipative, but homogeneous NRM. We point out that numerical calculations in such singular structures need to be carried out with fine levels of discretisation. Our results show that although absorption affects the imaging, it does not wipe out sub-wavelength focussing effects. Finally, we also discuss focussing with checkerboard structures with small tranverse extents.

## 2. A checkerboard paradox: generalized lens theorem against ray picture

Consider the generalized perfect lens theorem: If *ε*
_{1}(*x*, *y*; *ω*) and *μ*1(*x*, *y*; *ω*) be the material parameters in the region -*d*<*z*<0, and *ε*
_{2}(*x*, *y*; *ω*) and *μ*
_{2}(*x*, *y*; *ω*) be the material parameters in the region 0<*z*<*d*, then for a wave with any parallel wave-vector, *k*_{x}
, and frequency *ω* incident on the system, the complex transmission (*𝓣*) and reflection (*𝓡*) coefficients are

if *ε*
_{2}(*x*, *y*)=-*ε*
_{1}(*x*, *y*), and *μ*
_{2}(*x*, *y*)=-*μ*
_{1}(*x*, *y*). Here we choose *z* to be the imaging axis and note that the system is anti-symmetric about the *z*=0 plane. The functions for *ε* and *μ* could be arbitrary. We will refer to such media with the same transverse variation but with opposite signs as optical complementary media.

Now consider a layer of a 2-D checkerboard along the *x*-*z* directions of thickness *L* along *z* with another layer of its complementary checkerboard also of thickness *L* imbedded in vacuum as shown in Fig. 1. We assume invariance along the *y*-axis. We will now calculate the response for system with the ideal lossless values of *ε*=±1 and *μ*=±1 for the relative permittivity and permeability respectively. In fact, the very results of the generalized lens theorem to this system is not borne out by a ray analysis as seen in Fig. 1, which shows that a ray can either be transmitted or retro-reflected. But the generalised lens theorem which is based on a full wave analysis transcends any such ray analysis. Such contradictions with the ray analysis have been pointed out previously[2, 10] and arise due to the localized fields at the corners which the rays cannot describe.

## 3. Tranfer matrix analysis of transmission and reflection properties of a pair of ideal and complementary checkerboard layers

The numerical solution of this problem is non-trivial. Our initial attempts to solve the problem using FDTD methods did not succeed with divergences arising in the calculations which we later traced to finite differencing effects [11, 12, 13, 14]. We present here numerical solutions obtained using a refined version of the PHOTON codes based on the Transfer Matrix Method[15, 16]. The checkerboard regions are sliced into thin layers along *z* with thicknesses of only 7 to 11 mesh points to avoid numerical instabilities [16]. The transfer matrix across each of these thin layers is calculated by propagating Maxwell’s equations and the matrices for these layers are combined using a multiple scattering formalism[17] to yield the final transfer matrix for the entire system. We use a free space wavelength of λ=4.53*μ*m, periodic boundary conditions along *x* with a period of *d*=1.11*μ*m or 2.22*μ*m, and a typical layer thickness of *L*=0.45*μ*m≃λ/10. The period along *x* is irrelevant to the effects discussed here. Care is taken to ensure the position of the numerical grid with respect to the checkerboard boundaries is such that the averaged values of *ε* and /*mu* at any grid point does not become zero (singular) [11]. The other numerical approximations inherent in the PHOTON codes are well known [16, 17].

In Fig. 2, we present the transmission and reflection coefficients for a plane wave as a function of the parallel wave-vector *k*_{x}
for this ideal checkerboard. In the top panels (a) and (b), we plot the results obtained using 202 points across *d*. Although the transmission is approximately one for propagating waves (*k*_{x}
<*k*
_{0}=*ω*/*c*), the reflectivity and the phase of the transmission are non-zero. The transmission coefficient also becomes very large for subwavelength *k*_{x}
>*k*
_{0} and there appears to be a resonant response whereas the generalised lens theorm predicts no resonance for any finite *k*_{x}
for *ε*=*μ*=±1. This spurious resonance shifts to larger values of *k*_{x}
with decreasing numerical mesh size. In the bottom panels, we plot the response as obtained for better discretization with 262 points across *d*. The spurious resonance now shifts to a point beyond *k*_{x}
=3*k*
_{0}, the transmittivity becomes closer to unity with zero phase shift and the reflectivity is zero as well.

To understand the sensitivity of the solutions to numerical errors and excitation of spurious resonances, consider the dispersion of the finite difference equations on a simple cubic lattice in media with spatially constant material parameters [17]:

where *a* is the numerical grid size. This evidently tends to the dispersion for the continuum Maxwell equations only when *ka*≪1, or particularly, for very small *a* when we consider subwavelength wave-vectors. We can rewrite the above dispersion as

where the second term can be effectively regarded as arising from a deviation, *δμ*, *μ* (or in *ε*) from the ideal case of *ε*=*μ*=±1 that we have in our problem. This deviation,

in the real part of *μ* will give rise to a resonant excitation on a slab of NRM of thickness *t* with a parallel wave-vector (*k*_{x}
)[7],

in the limit of large parallel wave-vector *k*_{x}
≫*k*
_{0}. In the limit *k*_{x}*a*≪1, the parallel wave-vector for this spurious resonance induced by the discretization will be *k*_{x}
~ (a large number)/*t*. In the ideal continuum limit of *a*→0, we have *k*_{x}
→∞ as expected from the continuum equations. Thus, at any level of discretization, one will always excite a spurious resonance of the system for some finite *k*_{x}
in a numerical calculation. The value of the the parallel wave-vector ${k}_{x}^{\left(s\right)}$ for the spurious resonance gives the range of *k*_{x}
significantly smaller than ${k}_{x}^{\left(s\right)}$ for which the calculations would have converged. In our problem of the checkerboard, *t* can be taken to be the larger dimension of the checkerboard cell. Then the expression above in the long wavelength limit gives ${k}_{x}^{\left(s\right)}$~3*k*
_{0} for *t*~*λ*/2 and *k*
^{(s)}
*x*~8*k*
_{0} for *t*~*λ*/4. This approximate estimate is in reasonable agreement with the numerical calculations where retardation effects will also play a role and further the geometry is more complicated than a simple slab.

The other significant deviation, mainly at *k*_{x}
~*k*
_{0}, can be attributed to the numerically approximate impedance matching at the interfaces. Thus, within the accuracy of our numerical scheme, we have verified the generalised lens theorem for this system for *k*_{x}
<2.5*k*
_{0}.

## 4. Analysis of focussing effects through a dissipative checkerboard lens

We then studied the robustness of the focussing effects against dissipation. In Fig. 3, we show the transmission and reflection coefficients for a dissipative checkerboard. With imaginary parts of *ε* and *μ* of 10^{-4}, the results were not very different from that of the lossless system. Even with larger values of 0.1 for both Im(*ε*) and Im(*μ*) (Fig. 3, top), one can see that although the transmission is not total, it is clearly very flat as a function of *k*_{x}
even for sub-wavelength wave-vectors. This implies that the focussing effects in the system will be robust against dissipation, probably as robust as in the single slab lens. This is surprising in view of the divergences in the local density of modes at the corners. Note that the dissipation here is comparable to that in silver which has been suggested[4] and used as a near-field lens[19]. We have also calculated the response of a silver-vacuum checkerboard with *ε*=-1+*i*0.4 and *μ*=+1 for silver, and present its response in Fig. 3(bottom). One can clearly see the resonant excitation of a surface plasmon mode at sub-wavelength wave-vectors for the P-polarized light only. The transmittivity indicates that one could have considerable transfer of subwavelength information.

We finally examined the response of a silver checkerboard lens with finite transverse size. We show in Fig.4, the response of a small, slightly dissipative checkerboard system consisting of thirty cells with *ε*=-1+*i*0.01 and *μ*=+1, for a line source placed at point (0, 0.6) for the P-polarization. We set the numerical problem using the weak form of Maxwell’s system and discretising it with test functions (Whitney 1-forms, or Nédélec edge elements) defined as

on the edge *e* between vertices *i* and *j* of the triangle of the mesh with barycentric coordinates *λ*_{i}
(the FEM mesh generator is a grid-based Delauney triangulation). Such vector fields are known to have a continuous tangential component across interfaces between different media [20] (hence exhibiting two anti-parallel wave-vectors at both sides of interfaces between complementary media). Use of such curl-confirming elements avoids spurious modes [20] and is particularly well suited for high-contrast media and Left-handed materials. The (hypo) ellipticity of Maxwell’s system is ensured by the small dissipation, and existence and uniqueness of the solution is a consequence of outgoing-wave conditions enforced with Perfectly Matched Layers[21] which provides a reflectionless interface between the region of interest (large middle square containing the line source and the silver checkerboard on Fig. 4) and the PML (four elongated rectangles and four small squares) at all incident angles. Here again, we emphasize that use of Nédélec edge elements is particularly well suited for implementation of PML as this can be seen as a geometric transform for which differential forms are well suited (pull-back porperties).

Numerical simulations have been carried out with an adaptive mesh of upto 91780 finite elements to check the convergence of the finite element method (implemented in COMSOL which uses robust ARPACK solvers). One can clearly see that the source is reproduced in the focal plane of the checkerboard lens on upper left panel of Fig. 4 (picture (a)). On bottom left panel of Fig. 4 (picture (c)), we represent the profile of the P-polarized eigenfield computed along a vertical segment running from point (0, 1)*d* to point (0,-1)*d*. One can clearly see an anti-symmetric plasmon mode excited at the upper interface between positive and negative index media (upper boundary of the checkerboard). It is remarkable that the intensity of the P-polarized eigenfield is of the same order in the focal planes of the source and image in diagram (c). On right panel, we consider two pairs of complementary cells (checkerboard of thickness 0.4*d*) and we notice that the focussing effect is still there (the color scale is chosen so that imaging effect is made clearly visible). On picture (d) the 3D plot of the eigenfield exhibits plasmon resonances running along interfaces between complementary media (note the view is rotated by *π* compared with (b)). Some images in the neighborhood of the source are building up in the first two rows of the checkerboard (8 red picks are located in a symmetric fashion within 8 adjacent cells with smaller picks appearing around). An intersting fact is that the imaging effect remains visible when we change the location of the source. Even though the generalised lens theorem predicts only an imaging effect when the source lies in a close neighbourhood of the heterogeneous slab lens, the numerical simulation shows that the imaging is still present when the source lies two cells away from the checkerboard (the nearer the source, the better the imaging)! Note from Fig. 4(b) that the image is slightly closer from the checkerboard slab than the image. Also, it is worthwhile noticing that when we increase the size of the checkerboard, the imaging process is robust against dissipation, as seen on Fig. 4 (b).

## 5. Conclusion

In conclusion we have shown that, within the accuracy of the numerical calculations presented here, focussing effects in checkerboards of NRMare reasonably robust against dissipation. This is surprising in the face of the expectation that dissipation in such singularly degenerate systems can wipe out all such effects. We have also shown some numerical evidence of interesting focussing effects in weakly dissipative checkerboard systems of small extent. We checked with the FEM package that a plane wave merely suffers a *π* phase shift when it moves through the NRM checkerboard when taking negative cells with *ε*=-1+*i*0.01 and *μ*=1 while a point source is neatly imaged on the other side (with same phase shift). But there is even more: anomalous localised resonances studied in great mathematical details for a cylindrical perfect lens in the quasi-static limit in [23] are present in our checkerboards. We observed numerically with the full-wave FEM package that a line source lying in close neighborhood of the checker-board is cloaked by the system so that its radiations in free space are minimized! Such a light confinement has been also observed for line sources in the presence of a slab and a disc made with left-handed material [24, 25].

While some authors consider a slab of NRM as Alice’s mirror [22], we may say NRM checkerboards behave in some way like the famous Alice’s Cheshire cat who has the annoying habit of disappearing and appearing at random times and places. “Well, I’ve often seen a cat with a grin”, thought Alice, “but a cat without a grin is the most curious thing I ever saw in my life”.

## Acknowledgments

SAR and SC acknowledge support from the Department of Science and Technology, India under grant no.SR/S2/CMP-45/2003 and thank Dr. H. Wanare for discussions. SG acknowledges funding from the EC funded project PHOREMOST under grant no. FP6/2003/IST/2-511616. SAR and SC would like to thank the anonymous referees for constructive comments.

## References and links

**1. **S.A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. **68**, 449–521 (2005). [CrossRef]

**2. **J.B. Pendry, “Negative refraction,” Contemp. Phys. **45**, 191–202 (2004). [CrossRef]

**3. **T. Itoh and C. Caloz, *Electromagnetic Metamaterials*, (Wiley-Interscience, New Jersey, 2006)

**4. **J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000) “Negative refraction makes a perfect lens”, Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef]

**5. **V.G. Veselago, Sov. Phys. Solid state8, 2854 (1967); “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Uspekhi 10, 509–514 (1968)

**6. **S.A. Ramakrishna, J.B. Pendry, D. Schurig, D.R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens”, J. Mod. Opt. **49**1747–1762 (2002) [CrossRef]

**7. **D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, and J.B. Pendry, “Limitations on Sub-Diffraction Imaging with a Negative Refractive Index Slab,” Appl. Phys. Lett. **82**, 1506–1508 (2003) [CrossRef]

**8. **J.B. Pendry and S.A. Ramakrishna, “Focussing light using negative refraction,” J. Phys. Cond. Matter **15**, 6345–6364 (2003) [CrossRef]

**9. **S. Guenneau, B. Gralak, and J.B. Pendry, “Perfect corner reflector” Opt. Lett. **30**, 1204–1206 (2005) [CrossRef]

**10. **S. Guenneau, A.C. Vutha, and S.A. Ramakrishna, “Negative refraction in 2D checkerboards related by mirror anti-symmetry and 3D corner lenses,” New J. Phys. **7**, 164 (2005). [CrossRef]

**11. **A.A. Sukhorukov, I.V. Shadrivov, and Y.S. Kivshar, “Waves scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. **19**, 105–117 (2006). [CrossRef]

**12. **X. S. Rao and C.K. Ong,“Subwavelength imaging by a left-handed material superlens,” Phys. Rev. E **68**, 067601 (2003) [CrossRef]

**13. **R. Ziolkowski,“Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express **11**, 662–681 (2003) [CrossRef]

**14. **J. J. Chen, T. M. Grzegorczyk, B. -I. Wu, and J. A. Kong,“Limitation of FDTD in simulation of a perfect lens imaging system,” Opt. Express **13**, 10840–10845 (2005) [CrossRef]

**15. **J.B. Pendry and A. Mackinnon, “Calculation of photon dispersion-relations,” Phys. Rev. Lett. **69**, 2772–2775 (1992). [CrossRef]

**16. **P.M. Bell, J.B. Pendry, L.M. Moreno, and A.J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Commun. **85**, 206–233 (1995). [CrossRef]

**17. **J.B. Pendry,“Photonic band structures,” Jour. Mod. Opt. **41**, No. 2, 209–229 (1994) [CrossRef]

**18. **S. Cummer, “Simulated causal subwavelength focussing by an negative refractive index slab” Appl. Phys. Lett. **82**, 1503–1505 (2003) [CrossRef]

**19. **N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a sliver superlens,” Science **308**, 534–537 (2005). [CrossRef]

**20. **A. Bossavit, “Solving Maxwell equations in a closed cavity, and the question of spurious modes,” IEEE Trans. Mag. **26**, 702–705 (1990). [CrossRef]

**21. **J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phy. **114**, 185–200 (1994). [CrossRef]

**22. **D. Maystre and S. Enoch, “Perfect lenses made with left-handed materials: Alice’s mirror,” J. Opt. Soc. Am. A **21**, 122–131 (2004). [CrossRef]

**23. **G.W. Milton and N.A. Nicorovici “On the cloaking effects associated with anomalous localised resonance,” Proc. Roy. Lond. A **462**, 3027–3059 (2006) [CrossRef]

**24. **T.J. Cui, Q. Cheng, W.B. Lu, Q. Jiang, and J.A. Kong, “Localization fo electromagnetic energy using a left-handed-medium slab,” Phys. Rev. B **71**, 045114 (2005). [CrossRef]

**25. **A.D. Boardman and K. Marinov, “Non-radiating and radiating configurations driven by left-handed materials,” J. Opt. Soc. Am. B **23**, 543 (2006). [CrossRef]