## Abstract

Based on coordinate transformation incorporated with conformal mapping approach, a sub-wavelength imaging device with magnification, called “planar hyperlens” is designed, capable of realizing far-field plane-to-plane imaging beyond the diffraction limit. The possible implementation method is proposed by using effective anisotropic metamaterial formed by alternating metallic and dielectric thin layers. The magnification performance of the designed multi-layer lensing structure is numerically simulated to confirm our theoretical analysis.

©2008 Optical Society of America

## 1. Introduction

It is well known that the resolution of conventional optical imaging systems are constrained by the diffraction limit of light due to the loss of evanescent waves that carry sub-wavelength information in the far field. The imaging beyond diffraction limit have gained much attention as Pendry [1] proposed the “perfect lens”, a hypothetical isotropic lossless material slab with *ε*=*µ*=-1, in which the evanescent components can be amplified and restored to produce the perfect image below the diffraction limit. But the perfect resolution is difficult to realize for the inevitable losses of the realizable materials [2–5]. The concept of superlens, generally formed by thin slab of material with negative permittivity or permeability, or both [6–9], has also been utilized for magnifying evanescent waves and thus restoring a sub-wavelength image. But the difficulties in controlling permeability without introducing significant loss in optical frequency range has led researchers to focus on the superlensing structures that have unit permeability and only applicable to transverse-magnetic (TM) waves [1,10–12]. Furthermore, the superlensing devices always show their performance in the near field [13], the amplified evanescent waves can not be further brought to the far field to focus by conventional optics. Another kind of device recently termed hyperlens has also gained considerable attention for its ability to form a magnified optical image of a subwavelength object in the far field with the help of anisotropic metamaterials that feature opposite permittivity signs in two orthogonal directions. It is noted that the works on hyperlens reported to date take advantage of the cylindrical geometry to magnify sub-diffraction-limited objects along radial direction to the far field [14–19]. The cylindrical hyperlens has circular inner object and the outer imaging surfaces, the curved surfaces bring troubles in locating objects on the object surface as well as in observing and measuring images at the far-field imaging surface.

In this paper, we propose a new far-field magnifying imaging structure, here we called it “planar hyperlens”, which is capable of magnifying sub-diffraction-limited objects on one flat surface and form an image on another plane with finer resolution below the diffraction limit in the far field. This plane-to-plane imaging structure is expected to provide conveniences for the applications in photolithography, planar integrated optical devices and so on.

## 2. Design method

The proposed hyperlens structure consists of two geometrical parts. The orthogonal oblate cylindrical system is used to build the lower part and enables the object surface to be flat, as applied in our previous paper for the perfect lens design [20], then for the upper part we introduce another orthogonal coordinate system using conformal transformation from the complex space to 3D Cartesian space to make the imaging surface planar.

Next, with the aim of guiding waves to propagate along ray-like trajectories between the two planar surfaces in this structure, a hyperbolic dispersion relation should be satisfied. As summarized in [18], the hyperlens theory is generally based on the hyperbolic dispersion of transverse-magnetic wave (the magnetic field is polarized along z-axis) in cylindrical coordinates, which can be expressed as

where *ε*
* _{r}*<0 and

*ε*

*>0. Actually, similar dispersion characteristics can also be generalized to other orthogonal coordinate system. Given three orthogonal coordinate curves represented by (*

_{θ}*u*,

*v*,

*w*), hyperbolic dispersion relation has the following form for TM wave with magnetic field polarized along w axis

where *ε*
* _{u}*>0 and

*ε*<

*. It is easy to find that in the limit of |*

_{v}*ε*

*|→∞ and*

_{v}*ε*

*→0, a flat dispersion in*

_{u}*k*

*-*

_{u}*k*

*space can be obtained, thus supporting an infinite range of propagating modes. Similar to the propagation property in the cylindrical hyperlens, the EM wave is confined to propagate along v curve like a ray because the tangential component of the wave vector in the*

_{v}*u*direction is totally compressed due to the conservation of angular momentum. Since anisotropic metamaterial composed of alternating metal and dielectric layers based on effective medium theory have proved to be feasible to fulfill the required hyperbolic dispersion relation in cylindrical hyperlens design [14,15,17–19], the same approach can be applied to achieve the desired dispersion relation depicted in Eq. (2). We can integrate the two parts and then discrete the whole domain using multi-layered metamaterial concept to create a planar hyperlens working at optical frequency range.

For the future realizable design of this planar hyperlens, the permittivity of the metal *ε*
* _{m}* and the dielectric

*ε*

*should be properly chosen to satisfy the required anisotropic feature of the metamaterial in the visible range. We use silver (Ag) with*

_{d}*ε*

*=-2.4012+0.2488*

_{m}*i*at the optical wavelength of 365nm [22] and polymethyl-metacrylate (PMMA) with

*ε*

*=2.301 to form the alternating thin films, applying the effective medium theory*

_{d}$${\epsilon}_{v}\approx \frac{\left({d}_{1}+{d}_{2}\right)}{\left(\frac{{d}_{1}}{{\epsilon}_{m}}+\frac{{d}_{2}}{{\epsilon}_{d}}\right)}\phantom{\rule{.2em}{0ex}},\phantom{\rule{.2em}{0ex}}\left(u={u}_{1},{u}_{2}\phantom{\rule{.9em}{0ex}}v={v}_{1},{v}_{2}\right).$$

In the case of the thickness of the thin films (*d*
_{1} and *d*
_{2}) is much smaller than the wavelength, the desired anisotropic metamaterial properties can be achieved. Note that *d*
_{1} and *d*
_{2} should be measured in terms of the *v*
_{1} and *v*
_{2} coordinates.

## 3. Simulation and discussion

To illustrate the validity of our methodology, we simulated the designed structure using commercial COMSOL Multiphysics solver. The structure is 1.35 µm high and 3.25µm wide, the whole simulated domain is filled by thin Ag (24 layers) and PMMA (24 layers) films, each is about 30nm thick. The testing sources in one pair spaced 70nm. The simulation result is shown in Fig. 3.

It is noted in Fig. 3 that the TM wave propagate along ray-like trajectories in the planar hyperlens and magnified images are also obtained at the top imaging plane. We further demonstrated the magnetic field distributions along the lines of y=0 (the object plane) and y=1.25µm (the imaging plane) in xy-plane, as displayed in Fig. 4. From Fig. 4, we can see that the distance being 70nm that is beyond the diffraction limit at the object plane increases to be about 310nm at the far field imaging plane, which is much larger than *λ*/2, and thus can be manipulated by conventional optics. In this case, the resolution of *λ*/5 is achieved. Actually finer resolution can be obtained if we properly optimize the number and thickness of metal and dielectric films.

Besides, we note in Fig. 4(b) that the contrast between images and background is rather low. The main reason should be the energy loss inside the hyperlens, another reason lies in the fact that. With 365nm incidence, silver has the real part of the permittivity being -2.4012, which is not exactly matched to the dielectric medium (PMMA) used in our simulation. It is possible to apply some gain materials [25] to reduce the energy loss so as to improve the output of light field intensity. It is also noticed in Fig. 4(b) that sources at different location in the object plane correspond to different magnifications in the imaging plane, which can be theoretically verified using similar method as once applied in our previous work [20]. In xy-plane, suppose the magnification of the lower part for an arbitrary set of (*u*
_{1},*v*
_{1})is *M*
_{1}, and *M*
_{2} corresponds to that of the upper part for another arbitrary (*u*
_{2},*v*
_{2}), using the conclusion derived in our previous work, *M*
_{1} can be written as

For the upper part

where *v*
^{in}_{2} represents the value of *v*
_{1} at the inner curve of the upper part. Because the interface of the lower and upper part being thin enough, so that *v*
^{in}_{2}≈*v*
_{1} at the interface (the black region in Fig. 2 (a)), then we have the magnification of the planar hyperlens

It is easily found from Eq. (6) that the value of *M* is always larger than one, any object on the object plane have a magnified image at the imaging plane, besides, the magnification is dependent on the object position *u*
_{1}, the structural shapes of both the lower and upper part characterized by *v*
_{1} and *u*
_{2} respectively.

To clearly demonstrate the magnification characteristic of this structure, we take the planar hyperlens as with the same structural parameters as used for the simulations and further give the magnification *M* along the x axis in Fig. 5.

It is clearly demonstrated that larger magnifications are obtained as the source moves away from the centre along x axis, as the object approach to the edge of the object plane, the magnification becomes increasingly large. We note that the magnification between 0 and 0.15µm on the object plane is slowly varying such that the objects around this area are expected to be imaged on the imaging plane with better fidelity. Besides, an increment of the magnification can also be achieved just by tuning the position of the interface of the lower and upper part of the hyperlens, i.e., the value of *v*
_{1}, as shown in Fig. 5, smaller lower part (corresponding to larger *v*
_{1}) results in larger magnifications. We can optimize structural parameters of the planar hyperlens to satisfy practical needs for better imaging and desired magnification. With other parameters fixed, the number of layers can be controlled to reduce the loss inside the hyperlens, also we can make each film thinner to reduce energy absorption and improve the imaging performance [23, 24]. For example, the lower part of the designed hyperlens can be tuned to have a longer horizontal axis in such a way that the thickness of each layer has relatively small variation along *u* curves, which enables waves to travel along a shorter trajectory, the contrast is expected to be better. But we should note in this case that the magnification capability of the hyperlens becomes smaller. Also the transmitted light may be enhanced by meeting the resonance condition for light propagating inside the lens with proper size. The detailed investigation of the optimization of the planar hyper lens is still in process and would be published elsewhere.

## 4. Conclusion

In summary, we have proposed a metamaterial-based method in association with coordinate transformation and conformal transformation theory to theoretically design the planar hyperlens that is capable of magnifying an object from one plane to another. The possible realization approach have also been demonstrated using alternating thin metal and dielectric films based on effective medium theory. Numerical simulation further verified our design methodology. We believe that the optimization of the structure characteristics of the lower and upper part would be helpful in obtaining better magnification performance, and future realization of this designed planar hyperlens would be possible with the development of nana-fabrication techniques. This kind of flat-to-flat imaging structure is expected to show its convenience in far-field image measurement and have potential applications in real-time far-field sub wavelength imaging, lithography as well as in planar integrate optical devices.

## Acknowledgment

This work was supported by 973 Program of China (No.2006CB302900) and National Natural Science Foundation of China (No.60778018 and No. 60736037).

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