Over the past few years, research on negative refraction of electromagnetic waves has attracted much interest for their important potential applications. Negative refraction can be realized in negative index materials (NIMs, also known as left-handed materials), which exhibit simultaneously a negative permeability and a negative permittivity [1, 2, 3]. In the case of photonic crystals (PhCs) [4, 5, 6, 7, 8], the composing materials all have positive refractive index while the periodicity of the structure gives rise to the effective negative refraction of the Bloch waves. Due to the effect of negative refraction, an NIM planar slab might overcome the known problems of conventional lenses to achieve a “perfect” lens, capable of focusing the evanescent spectrum as well as the propagating spectrum [2], i.e., sub-wavelength imaging. Furthermore, the sub-wavelength resolution of an image due to the negative refraction of PhCs has also been demonstrated experimentally [9].

In this paper, we show that negative refraction can also be realized through surface waves on a structured perfect conductor surface. Additionally, a plane-lens-imaging simulation is performed to demonstrate the sub-wavelength imaging of a point dipole source.

For a metal illuminated by light, the electromagnetic excitation at the metal-dielectric interface, referred to as the surface plasmon (SP), may give rise to some unusual optical phenomena, e.g. extraordinary transmission through subwavelength holes in metal films [10, 11, 12, 13]. At the same time, in the microwave regime, a metal can often be treated as a nearly perfect electric conductor (PEC). Thus, electromagnetic waves can not penetrate into the metal and the metal-dielectric interface can not support SP waves. However, a metal surface with an array of drilled holes [14, 15, 16, 17, 18] can still support surface waves, so-called designer-surface-plasmons (DSPs), which have many important properties in common with the ordinary surface plasmons, including the phenomenon of enhanced transmission through subwavelength holes [19, 20].

Consider a square array of square holes in the PEC surface, as shown schematically in Fig. 1. A dielectric material, dielectric constant ε_h, fills the holes and covers the top surface of the structure. The full-vectorial three-dimensional (3D) finite-difference time-domain (FDTD) method [21] was used to obtain the band structure of surface modes. Fig. 2(a) shows the dispersion surface of the first surface wave band (other surface wave bands have higher frequencies than the first band) in a DSP-structure with ε_h=8 and a=0.85d, where d is the lattice constant and a is the width of the square holes. The equal frequency contour of the first band shows a convex curvature for frequencies around the M point. In particular, the equal-frequency contour becomes all-convex about the M point at the frequency near ω=0.171(2πc/d). This gives rise to negative refraction of the incident wave, as illustrated by the k vector diagram in Fig. 2(b). Since the surface wave contour is slightly larger than the air contour, for all incident angles one obtains only one single, negatively-refracted Bloch wave beam, i.e., an all-angle negative refraction (AANR) case [5, 9, 22, 23].

 

Fig. 1. The schematic of a structured perfect conductor surface, with a square array of square holes. A dielectric material, dielectric constant ε_h, fills the holes and covers the top surface of the structure. The lattice constant is d, and the width of the holes is a.

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Fig. 2. (a) The dispersion surface of the first surface wave band in the DSP-structure with ε_h=8 and a=0.85d. (b) The equal-frequency contours and the wave-propagation diagrams at ω=0.171(2πc/d).

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A plane-lens-imaging simulation is then performed to demonstrate the imaging through the surface wave in such a structure using the 3D FDTD method. The entire structure is surrounded by perfectly matched layers (PMLs) [24] to absorb the outgoing waves. Due to the upper cutoff to the transverse wave vector that can be amplified in the periodic structure, the point source was placed close enough to the air/DSP-structure interface before the evanescent waves decay out [5, 9, 23]. We place a E_z point dipole source of the frequency ω=0.171(2πc/d) in air with a distance S _o=0.7d from the slab structure, in which the interface of the slab is normal to Γ-M direction (see Fig. 3). The PEC metal slab has a thickness of 5.2d and a height of 1.5d. The depth of the holes is h=1.0d into the perfect conductor. The point dipole source is located on the same plane of the dielectric-metal interface (z=1.5d) in order to obtain the maximum energy coupling into the surface waves. Figure 3 shows the snapshots of the E _z field for three different z positions: 1.0d, 1.6d, and 2.75d. As clearly shown, a focused image is obtained at the other side of slab and the imaging distance is about S_i=0.6d from the slab edge.

 

Fig. 3. The snapshots of the E_z field at the frequency ω=0.171(2πc/d) for three different z positions: 1.0d, 1.6d, and 2.75d. The slab has a thickness of 5.2d and the depth of the holes is h=1.0d. The plane of the metal surface is at z=1.5d. The black lines in the each layer give the boundary of the slab structure and the outline of the metal. The inset shows the ray-tracing analysis for focused imaging by a slab lens with the negative refraction.

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Fig. 4. The snapshots of the E_z field along the plane across the source and the image for (a) the DSP-structure and (b) the unpatterned metal structure (just covered with the dielectric ε=8 but without the drilled holes array). The black line gives the outline of metal. The two dash lines give the boundary between the slab and air.

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Fig. 5. The average field intensity at the imaging distance S_i=0.6d, both for the DSP-structure (the solid line) and the metal structure without holes (the dashed line).

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In order to clearly show the role of surface waves, the snapshot of the E _z field along the plane across the source and the image is shown in Fig. 4(a). When compared with the snapshot of the E_z field along the same plane but for an unpatterned metal structure covered with the dielectric ε_h=8 (see Fig. 4(b)), one can clearly see that the electromagnetic fields propagate through the DSP-slab as surface waves, and then form the image on the opposite side. However, for the unpatterned metal structure, the electromagnetic wave emitted by the point dipole source only propagates through the dielectric material, and can not form an image on the other side of the structure. It also reveals that the mechanism of our design is distinct with that of the very recent research, negative refraction for surface wave using a uniform metal-dielectric-metal structure [25], and the most important advantage of our design is that a sub-wavelength imaging is achieved by a patterned PEC surface.

To prove the sub-wavelength imaging through the DSP-structure, we plot the average field intensity at the imaging plane in Fig. 5. Since the image is very close to the slab edge and thus the field on the imaging side may be dominated by the surface states at the plane of the metal surface (z=1.5d), the average intensity of a real image (obtained by the fast Fourier transform of the field) is shown here for z=2.75d. Compared with the case of the unpatterned metal structure just covered with the dielectric ε=8, the full width at half maximum (FWHM) of the average intensity at the imaging is much smaller, with a value of only 0.42λ (see Fig. 5). This is below the conventional diffraction limit (0.5λ) [26]. It is also found that the imaging quality is sensitive to the position of the air/DSP-slab interface termination, since the termination has strong influence on the transmission coefficient at the air/DSP-slab interface [27].

In conclusion, we have demostratrated two phenomena resulting from the propagtion of surface waves: 1) the negative refraction of surface waves on a periodically patterned PEC surface and 2) the sub-wavelength imaging of a point dipole source by exploiting the negative refraction of surface waves. We also believe that for a real metal with a hole array in it, one can still achieve negative refraction for the surface plasmon waves since the structure induced surface waves (i.e., designer surface plasmons) co-exist with the real surface plasmons due to the negative permittivity of the metal, which is important for nanoscale manipulation of optical waves. Patterning a nanoscale surface means a simplification as compared to a photon crystal, and patterned surfaces can be cost-effectively manufactured by the nano-imprinter lithography.

The authors thank Dr. Curtis Neff for helpful discussions. This work is supported by the Swedish Foundation for Strategic Research (SSF) on INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR) under Project No. 2003–5501. Z.C.R. acknowledges the partial support from the National Basic Research Program (973) of China under Project No. 2004CB719800.