## Abstract

We present here a structure with just a single slit covering the planar anisotropic metamaterial. The metamaterial has hyperbolic dispersion and can be realized using metal-dielectric multilayers. The structure combines the focusing performance of the zone plates and subwavelength resolution of the anisotropic metamaterials so that subwavelength focal spots can be obtained at the focal plane. The relationship between the focal spot size and slit width has been investigated, and a resolution of 36*nm* about 1/10 of 365nm incident wavelength is obtained with a 100*nm* wide single slit.

©2010 Optical Society of America

## 1. Introduction

It is well known that the imaging systems performance of conventional optics is constrained by diffraction so that there is a limitation of resolution about half of the incident wavelength [1]. With the fast development of nanoscale fabrication and characteristic techniques, a breakthrough of diffraction limit is highly desired to achieve subwavelength imaging and light manipulation.

Since J.B.Pendry proposed a perfect lens [2] of a thin slab of negative refraction material [3], many structures such as superlens [4], metallodielectric nanofilms [5] and hyperlens [6] and have been proposed to achieve near-field and far-field subwavelength imaging. Those devices have been demonstrated theoretically and experimentally [7–9]. Other suggested structures include those to achieve far-field subwavelength imaging using transformation optics techniques [10, 11] and those analog to near-field plates [12,13] or using transparent metallodielectric stacks [14] to achieve subwavelength focusing. S.Thongrattanasiri and V.Podolskiy have also proposed a class of structures called hypergratings [15] based on combinations of planar hyperbolic metamaterials and diffraction gratings or zone plates that can achieve far-field subwavelength focusing.

In this paper, we present a kind structure of hypergratings that combines the focusing performance of the first order Fresnel zone [15] and subwavelength resolution of the metal-dielectric multilayers [5, 16]. The structure has just a single slit on the mask and a focal spot size far beyond the diffraction limit can be obtained in the planar metamaterial. By changing the slit width, focal spots of different sizes can also be obtained and the relationship between focal spot size and slit width is investigated. With a proper slit width 100*nm*, a focal spot size of 36*nm* (FWHM) which is about 1/10 of incident wavelength has been observed by numerical simulation. Comparing resolutions of the proposed structures with different materials has also been discussed. In the following, the principle of the structure and characteristics will be described in detail.

## 2. Principle and design of the structure

As shown in Fig. 1
, the proposed structure consists of a planar slab by strongly anisotropic metamaterial and a covering chromium mask which is designed according to the boundaries of the first order Fresnel zone. All the components are treated as semi-infinite in the Y direction. The anisotropic metamaterial can be demonstrated by a layered metal-dielectric system [16] and silver-SiC layers are used in this paper. The relative permittivities for silver and SiC are *ε _{Ag}* = −2.4012 + 0.2488

*i*[17] and

*ε*= 8.2369 [18], respectively.

_{SiC}For an alternatively layered system, if each layer can be described by homogeneous and isotropic permittivity and permeability parameters and the layers are sufficiently thin (<<*λ*
_{0}), then the whole system can be treated as an effective anisotropic medium [19]. The used silver and SiC layer are both isotropic medium and the thicknesses are both 5*nm* or 10*nm* that sufficiently thin, therefore according to the effective medium theory [20], the metamaterial’s permittivity tensor is

*ε*)>0, Re(

_{xy}*ε*)<0 (Re(

_{z}*ε*) represents the real part of

*ε*) whose hyperbolic dispersion relation eliminates high-

*k*modes cutoff for the

_{x}*p*-polarized incident wave. If the width of the slit is designed to be part of or the entire first order Fresnel zone, the waves of different

*k*modes will interfere with each other positively at the focal plane. So there will be a subwavelength focal spot at the focal distance.

_{x}Just like the Fresnel zone plates, difference of the phase shift from boundaries of the *m*th Fresnel zones and the origin to the focal point is given by

*f*is the focal distance. Here,

*x*is the horizontal displacement of the outer boundary of the

_{m}*m*th Fresnel zone from Z axis and

*x*is the origin of the x axis. As the angle between the propagation direction and Z axis in the metamaterial can be given by the ratio of Poynting vector components: $\mathrm{tan}\theta =\mathrm{Re}({\epsilon}_{xy}){k}_{x}/\left(\mathrm{Re}({\epsilon}_{z}){k}_{z}\right)={x}_{m}/f$ [15], and according to the dispersion relation equation of the metamaterial ${k}_{z}{}^{2}/{\epsilon}_{xy}+{k}_{x}^{2}/{\epsilon}_{z}={\omega}^{2}/{c}^{2}={k}_{0}^{2}\left({k}_{0}=2\pi /{\lambda}_{0}\right)$ [6], it can be obtained that ${k}_{z}=\left[\mathrm{Re}({\epsilon}_{xy})f/\sqrt{\mathrm{Re}\left({\epsilon}_{xy}\right){f}^{2}+\mathrm{Re}\left({\epsilon}_{z}\right){x}_{m}{}^{2}}\right]{k}_{0}$ and ${k}_{x}=\left[\mathrm{Re}({\epsilon}_{z}){x}_{m}/\sqrt{\mathrm{Re}\left({\epsilon}_{xy}\right){f}^{2}+\mathrm{Re}\left({\epsilon}_{z}\right){x}_{m}{}^{2}}\right]{k}_{0}$ Thus ${\varphi}_{m}=\sqrt{\mathrm{Re}\left({\epsilon}_{xy}\right){f}^{2}+\mathrm{Re}\left({\epsilon}_{z}\right){x}_{m}{}^{2}}{k}_{0}$ and Eq. (2) turns to be

_{0}*f*to obtain the integer

*m*(

*m*> = 1). But the optimal solution for light focusing occurs as

*m*= 1 (the first diffraction order) and $f=\frac{d}{2}\left(\frac{{W}^{2}}{{a}^{2}}+1\right)$. Here the entire first Fresnel zone can be open (

*m*= 1). For the other case $dW/a<d$, the obtainable maximum order

*m*(

*m*<1) and light focusing appears as $f=dW/a$. Therefore the theoretical relation between the focal distance

*f*and the slit width

*W*is expressed as

*W*≤120

*nm*and at a wider slit the difference between them is still small.

As for the traditional Fresnel zone plate (*ε* = *ε _{xy}* =

*ε*= 1), the focal distance is $f=\frac{m{\lambda}_{0}}{2}\left(\frac{{W}^{2}}{{m}^{2}{\lambda}_{0}{}^{2}}-\frac{1}{2}\right)$ (0<

_{z}*m*≤1) for just keeping all or part of the first Fresnel zone open. It’s similar to the second part of Eq. (4), and for any given

*f*, there is always a

*W*that the entire first Fresnel zone opens. But for our structure with the hyperbolic dispersion relation of the metamaterial, if keeping the entire first Fresnel zone open (

*m*= 1),

*f*must be larger than

*d*obtained from Eq. (3). Thus there is a minimum slit width

*W*=

*a*, and for

*W*<

*a*, only part of the first Fresnel zone keeps open. In the following simulations, the slit width is selected to be 100

*nm*that most of the first Fresnel zone keeps open.

## 3. Numerical simulation and analysis

Figure 3(a)
gives the distributions of normalized magnetic field intensity (|Hy|^{2}) for the single slit structure of the ideal hyperbolic medium, whose relative permittivity tensor is (2.9179 + 0.1244*i*, 2.9179 + 0.1244*i*, −6.7362 + 0.9894*i*) calculated from Eq. (1). A normal plane wave at a wavelength of 365*nm* in *p*-polarization incident from the top side of the mask and the slit width *W* is 100*nm*. As seen from Fig. 3(a), the wave has focused at z = 76*nm*, which is exactly the *f* calculated from Eq. (4). The intensity of cross-section at the focal plane is shown in Fig. 3(b). If full width at half maximum (FWHM) is taken as the focal spot size, a spot of 32nm about 1/11 incident wavelength can be achieved.

As the hyperbolic medium can be realized by multilayer system, a subwavelength focal spot could also be obtained in the multilayered structure as shown in Fig. 3(c), which gives the distributions of normalized magnetic filed intensity (|Hy|^{2}). The thicknesses of the silver and SiC layer are both equal to 5*nm* and there are 60 pairs of silver/SiC layers. Other parameters are the same as mentioned before. Figure 3(d) gives the distributions of intensity of the cross-section at the focal plane. However, a larger focal spot of 36*nm* (FWHM) about 1/10 incident wavelength has been achieved.

## 4. Discussion

*4.1 Ratio of focal spot size to wavelength δ/λ*_{0}*versus W for structures of different materials*

According to the effective medium theory, thinner multilayers have better accuracy of realizing the hyperbolic medium. As seen from the blue and yellow curve in Fig. 4
, the multilayers of 5*nm* thick films (yellow curve) always have better resolutions than those of 10*nm* thick films (blue curve) at corresponding different slit widths. This indicates that the proposed structure can be realized by using multilayers in faith and the resolution is close to prediction of effective medium theory as long as the thickness of layers is small enough.

However, the focal spot of the idea medium (red curve) could be larger than that of the 5nm and 10nm films when the slit width exceeds certain value. This can be understood from the following analysis. In the multilayer metallo dielectric system, the high-*k _{x}* waves propagation is caused by coupling to the surface plasmon polaritons (SPPs) that exist on metal-dielectric interfaces [19, 21,23]. The SPPs excited from edges of the slit are stronger than those from the center, as can be seen in Fig. 5(b)
. As the slit width increases, the edge-excited SPPs contribute more than those from other areas of the slit during forming the focal spot. In the case of idea medium, the diffracted waves from edge and other areas contribute almost equally to form the spot, as illustrated in Fig. 5(a). Actually, this effect can also been observed in Fig. 3(a) and Fig. 3(c). The resolution of the focal spot is mainly determined by transmission coefficient and interference of the high-

*k*waves. When the slit width is small, the effect of the uneven distribution is not obvious. But as the slit is larger than about

_{x}*λ*

_{0}/2 (Fig. 4), the formed focal spot in the multilayered system can be mainly decided by the edge-excited SPPs and may be smaller than that in the ideal medium due to this effect.

#### 4.2 Influence of material absorption

The material absorption is the main limitation to the resolution of the structure and the ability of absorption is mostly represented by the imaginary part of the material permittivity. As seen in Fig. 4, the structure of the ideal medium without absorption (coffee color curve) has a much better resolution than that of the medium (red curve) with normal absorption. Figure 6
gives field profiles at the focal plane for structures of different absorptions. The three hyperbolic mediums’ permittivity tensors are calculated using *ε _{Ag}* = −2.4012 + 0.2488

*i*(blue curve), −2.4012 + 0.02488

*i*(black curve) and −2.4012 (pink curve) from Eq. (1), respectively. The slit width

*W*= 100

*nm*, and other parameters are the same as before. All of them show that the resolution is obviously improved by reducing the material absorption or in other words reducing losses.

#### 4.3 Limit of the maximum number of Fresnel zones

The traditional Fresnel zone plates could theoretically have infinite Fresnel zones if diameter of the plates is large enough, and as the number of opened Fresnel zones increases, the intensity of the focal spot becomes stronger. However, for the structure we presented here, as the metamaterial has a hyperbolic dispersion relation, the shift of phase from the origin (*x* = 0, *z* = 0) to the focal point (*Φ*
_{0}) is maximum. So it can be obtained from Eq. (3) that:

*f/d*. If

*f*<

*d*, the opened zone is just part of the first order Fresnel zone.

## 5. Conclusion

In conclusion, we present a kind of hypergratings structure that combines the focusing performance of the first order Fresnel zone and subwavelength resolution of the metal-dielectric multilayers. Numerical analysis and simulation results have been given to demonstrate the design. Finally, a focal spot size of 36*nm* (FWHM), about 1/10 of incident wavelength could be obtained by the multilayered structure of 5*nm* films, with a slit width 100*nm*. The comparing between ideal mediums and multilayered structure of different thickness films has been given. The effect of the materials absorption has also been discussed. In addition, revising the focusing process may be utilized for magnification or demagnification imaging with subwavelength resolution as well. The proposed structure may be used in nano-scale lithography and microscope with proper photoresist and bottom layer design [22]. It may also be used in high density optical storage by stacking appropriate multilayer structures with photo sensitive layer inside it.

## Acknowledgements

This work was supported by 973 Program of China (No.2006-CB302900) and the Chinese Nature Science Grant(60825405).

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