Metamaterials and imaging with sub-wavelength resolution

Yuan Zhang, Daohua Zhang School of Electrical and Electronic Engineering

Nanyang Technological University Singapore

Michael A. Fiddy

Center for Optoelectronics and Optical Communications

University of North Carolina at Charlotte Charlotte NC, USA mafiddy@uncc.edu

Abstract—we examine the possible use of a planar negative index superlens to recover subwavelength scale information about a three dimensional penetrable object. In particular, we analyze the role of the amplified evanescent waves and show that it may not be possible to get high resolution in three dimensions from one experiment. There is trade-off between the transverse and longitudinal resolution one can obtain.

superlens; negative refraction; super resolution;

I. INTRODUCTION Negative index metamaterials have attracted

much attention in recent years. An interesting and potentially high impact application of these artificial materials is the pontential development of an imaging system that provides a higher resolution image of an object than that obtained in a diffraction limited system. Achieving superresolution has been the goal for much research over the last 60 years, and many numerical methods have been developed to achieve this, but they are typically ill-conditioned and often rely on incorporating some prior knowledge about the object. Assuming one can fabricate a bulk material having a refractive index close to -1, it has been shown that an image plane is formed in which all spatial frequencies, however large, contribute. We do not discuss here the design and fabrication of negative index materials, which is itself a major challenge, but instead use simulations to consider the performance of an imaging system incorporating such a metamaterial with ideal properties. Following the pioneering work of Veselago [1], the paper by Pendry [2] fuelled enormous interest in validating the “perfect” lens concept. Efforts continue to try to fabricate negative index metamaterials, especially operating at optical frequencies (e.g. [3]). Design and fabrication issues remain in order to realize a practically useful negative index metamaterial with good

homogeneous optical properties and, importantly, minimal losses [4]. Losses reduce resolution dramatically leading to an effective limit of 2�d/|ln(�’’)| where d is the thickness of the negative index slab and �’’ is the imaginary part of the metamaterial’s permittivity. Losses may be diminished through the use of gain but the ability to engineer a metamaterial for a superlens [2], with an index exactly equal to -1, actually still remains highly controversial [5]. Nevertheless, some encouraging results have been reported [6, 7] and achieving any improvement over the diffraction limit is a worthwhile pursuit. Pendry’s planar superlens, however, essentially transfers a field from one plane to another with all spatial frequency content preserved. While the merits of this for lithography can be argued, a perfect image of the field in a single 2D plane behind some penetrable scattering object of non-zero thickness might not provide us with the information we want. For all but very weakly scattering objects, there is no simple relationship between its scattered field and the quantity of interest, namely the (high resolution) spatial distribution of permittivity or refractive index. Moreover, for a thick penetrable object, the exit field that is perfectly imaged has to somehow be numerically processed, or additional experiments conducted, to retrieve the full 3D structure. In this paper, we study the superresolution ability of a planar superlens for 3D imaging.

II. 3D IMAGING ABILITY OF SUPERLENS

A. Field masking by a planar superlens The field generated by a point source can be

divided into propagating waves (with kx < k0) and evanescent waves (kx > k0). Without loss of

Figure 1. Schematic diagram of exponentially decaying-increasing-decayingprocess when evanescent wave go throught superlens.

generality, we consider the plane parallel to the

interface of the planar superlens, converting the 3D problem to a 2D one for the purposes of illustration. To achieve superresolution, the evanescent waves are necessary in order to replicate the sub-wavelength features of the object, thus we mainly study the behavior of the evanescent components of the electromagnetic field.

In figure 1, suppose we have a negative index lens (with index n = -1) located between z = -0.05m and z = 0.05m (i.e. it has a thickness of 0.1m where m denotes some unit of distance). The illuminating wavelength is 0.2m in this paper. For a point scatterer located before the superlens (take point A at z = -0.1m for example), there will be an infinite number of evanescent waves generated. As shown in Ref. [1], for the special case of � = μ = -1, such a negative index lens has the ability to amplify these evanescent waves. Once any albeit exponentially decaying evanescent field reaches the left interface of the superlens, its amplitude is increased exponentially when passing through the superlens, after which it exponentially decays again when it leaves the lens. Since the index of the superlens is well matched with the surrounding media (we assume air and n = 1), the field strength for all propagating and evanescent waves will have the same amplitude in the image plane as in the object plane. By contributing the correct amplitude for all spatial frequencies, a perfect image is formed at A’ (z = 0.1m). The exponentially decaying-increasing-decaying field has a coefficient of exp (±kzz) where kz is real and positive, and the positive sign applies

within the lens while the negative sign corresponds to air. This process is illustrated by the solid (A) and dashed (B) curves in Fig. 1.

As already stated, the image of A at A’ will be perfect if the index of the superlens is exactly minus one and without any loss. But that is not the whole story, and we find that the image of B is not perfect. For a point object located before the lens, we would hope to always get a perfect point image as well, for all object-image distances given by d + d/|n|. In Fig. 1 we plot the field amplitude distribution of the evanescent components generated from point source A that pass through the negative index slab and reach the position of its image at A’ in red and blue solid curves for kx=1.5k0 and kx=3k0 , respectively. The amplitude is normalized to the strength at point A. As expected, the field strength will decrease exponentially when leaving the source point A (decreasing on both sides of A), while at the image space, the situation is quite different. For positions located close to the lens, i.e. behind A’ (we use ‘behind’ to indicate those positions with smaller z values), the field distribution is totally different, the evanescent fields decreasing from a very high level back to an amplitude of unity when they reach A’. This unfortunate but necessary behavior means that only the region at and beyond A’ are perfectly replicated. Consequently, all images of object structures behind A will be disturbed by the field of image A’. We illustrate this in Fig. 1 by introducing another object B behind A. An example evanescent field amplitude of the scattered field from B is plotted with dashed curves. If we suppose that the two point scatterers have equal brightness, then we see that the image at B’ will indeed perfectly replicate B since every evanescent wave will be restored to the correct unit amplitude, but the field of image A’ will completely swamp the image of B’, possibly making it impossible to observe B’ at all. The position difference between A and B means that the tail of the field of A’ will be exp (kzd’) times the recovered field amplitude at B’, where d’ is the distance between A and B. Comparing the two dashed curves and solid curves, we also see that when kx is larger, this degrading effect is much worse, being an exponential dependence.

Figure 2. Distributions of patterns of electrical field strength at differentviewing positions: (a) kc = 2k0 ; (b) kc = 5k0 .

Figure 3. (a) |E| distrabution when a scattered object with subwavelength features is located before a superlens, kc=2k0; (b) field distributions when the veiwing position is along the dashed curve in (a) for different values of kc.

We have studied the field pattern for image A’ at different positions in Fig.2. In practice we can expect that a real negative index metamaterial will have intrinsic properties that can only support a limited range of kx increasing exponentially. Also, it is obviously impossible to simulate an infinite number of kx generated by a point object. Consequently, we apply a cut-off to the total range of -� < kx < +� at some kc, (i.e. -kc < kx < +kc), where kc could, for example, be determined either by the finite size of the unit cell from which the metamaterial is constructed, or as a result of the inevitable material loss [4, 6]. In Fig.2 (a), we plot the field distribution at z = 0.1m, 0.09m, and 0.075m for kc = 2k0. The field at z = 0.1m illustrates the superresolved image, having a sub-wavelength feature ~ �/4. When the viewing plane is closer to the negative index lens, the field strength increases and the side lobes become larger, and our ability to observe the sub-wavelength features disappear. We note that the value of the field strength is always larger at smaller z coordinates, which means all images behind A’ will be lost by the field envelope of A’. While it is difficult to resolve the object features behind A, for those objects behind A but which are also shifted along the x direction, there is still some chance that they can be resolved. This is

illustrated by the purple dashed curve, but there is still an unavoidable disturbance from A’. Fig. 2 (b) show the corresponding case when kc=5k0, we see that the field behind the image A’ is now increasing more rapidly than kc=2k0, and for the image positions behind A’ it is harder (or even impossible) to resolved features, despite an x shift. B. Field replication of scattering surface

From Fig. 2, we see that when the 3D features in an object have various an x shifts, it might be possible to resolve them. In Fig. 3 (a) we consider a dielectric object with a random stepped surface in front of a superlens (the lens is located between z = 0 and z = -0.1m, the area between the two white lines). A plane wave is incident from the bottom of the figure with �=0.2m, and it is scattered by the rough surface which is designed to have sub-

wavelength features. Because of the imaging properties of a perfect lens, the field distribution along the undulating (exit) surface of the object is exactly copied to where the black dashed segments replicating the surface profile are indicated. This holds if there is not any disturbance from the image neighboring segments, which from the discussion of last section is highly likely. In Fig. 3 (b) we plot the field distribution along the observing path (black dashed segments in Fig. 3 (a)) for different values of kc. For comparison, we also plot the field pattern on object surface (black curve) in the same figure. We can see that for small step depths of the segments (which we consider as representing the focal depth of the superlens) or for small kc, the field pattern is well copied. However, when the step depth or kc is increased, the field patterns are increasingly masked and difficult to recover. kc is defines the resolution of the superlens along the x direction, while the step depth indicates the depth of focus in the z direction (i.e. it represents the resolution along the z direction). Thus for a superlens there is a trade-off possible between the transverse (x) and vertical (z) resolution limits.

III. CONCLUSION To summarize, we investigated the imaging

property of a superlens for the specific purpose of

imaging 3D objects with subwavelength resolution. We show that the object features closest to the front interface of the superlens will lead to amplified fields that destroy the image quality of those objects behind it. While it is extremely difficult to image a 3D object with subwavelength resolution perfectly, we have found that there is a trade-off that can be made between the transverse (x) and vertical (z) resolution.

ACKNOWLEDGMENT The project is supported by the National Research Foundation, Singapore (NRF-G-CRP 2007-01).

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