## Abstract

A circular cylindrical and an oblate cylindrical perfect lens are designed by using coordinate transformation theory. Theoretical analyses are performed to give an insight into the variant angular magnification in the oblate cylindrical perfect lens. We further take advantage of the oblate cylindrical coordinate system to make the object surface flat for future practical imaging and lithography applications. We also for the first time make systematical simulations of various kinds of perfect lens, including numerical confirmation of Mankei Tsang’s statement about the magnification of the planar perfect lens and the imaging and magnifying performance beyond the diffraction limit of our designed perfect lens. All the calculated results agree well with our mathematical derivations, thus verifying the coordinate transformation method in designing perfect lenses.

© 2008 Optical Society of America

## 1. Introduction

Negative refractive index, first proposed by Veselago[1] have enabled researchers to obtain perfect imaging beyond diffraction limit, and perfect lensing structures, generally called “perfecr lens” have drawn much attention. An Australian group discussed cylindrical perfect lens by using the plasmonic resonance properties of a coated cylinder[2–4]. Pendry[5] systematically analyized perfect lens slab in 2000 and consequently developed a series of perfect lensing structures including cylinders, spheres and corner perfect lens based on a generalized lens theorem[6–8]. Another popular structures termed “superlens”[9–14] were consequently proposed to obtain diffraction-free imaging in the case of transverse-magnetic (TM) wave illumination considering the availability of current artificial metamaterials in practical implementation. We note that the recipe for perfect or near-perfect imaging of these devices is the recollection of the evanescent components of the incident wave with the aim of recovering the sub-wavelength information at the image plane. In other words, perfect imaging can be achieved as long as both the propagating wave and the evanescent part from the object can be transmitted to rebuild the image at the imaging surface with perfect fidelity. From this point, the coordinate transformation theory recently proposed by Pendry[15] to control electromagnetic (EM) wave using transformation media[16–21] is more direct and straightforward to design perfect lens and superlens since this technique has the ability to map the EM field from one coordinate system to another by applying metamaterials. Mankei Tsang has suggested planar, magnifying spherical and oblate spheroidal perfect lens by choosing appropriate coordinate systems, intuitively enabling the perfect transmission of EM field by filling required metamaterials between two surfaces[22].

Although perfect lenses based on the coordinate transformation and their possible realization approach have been theoretically proposed in Ref. 22, practical fabrication and functionality measurement of 3D spherical and oblate spheroidal perfect lens could be challenging. In this paper, we focus on oblate cylindrical coordinate system to design oblate cylindrical perfect lenses, the symmetry of the oblate cylindrical perfect lens along z axis in oblate cylindrical coordinate system provide the ease of both analytical derivation and future experimental implementation. The magnifying circular cylindrical perfect lens and the oblate cylindrical perfect lens are designed with Pendry’s theory. For the convenience of future practical application in imaging and lithography, we further yield a magnifying oblate cylindrical perfect lens with a flat object plane. Numerical simulations of coordinate-transformation-based perfect lenses are for the first time systematically performed to confirm the theoretical analyses. We first simulate the non-magnifying planar perfect lens proposed in Ref. 22, and then demonstrate the magnifying and imaging ability of the circular cylindrical and the oblate cylindrical perfect lens. Finally their magnification properties, particularly for the oblate one, are quantitatively discussed.

## 2. Circular and oblate cylindrical perfect lens

Cylindrical-shaped perfect lens and superlens have been investigated in many papers [13–16] for their simple geometry and the experimental feasibility, the main task in these lensing structures is to amplify the decaying evanescent waves carrying sub-wavelength information so that they can contribute to an image on a specified image surface. Actually, with the help of the coordinate transformation, we can avoid considering the propagating waves and the evanescent part separately: the principle behind amplifying evanescent waves and the coordinate transformation approach in cylindrical-shaped perfect lens remains the same but the sub-wavelength imaging with respect to coordinate transformation method converts to another problem that how to make the EM fields on the two circular surfaces identical, this problem can be readily handled as follows:

According to Pendry’s theory, we make the coordinate transformation in terms of cylindrical system, where an arbitrary constant-*r* circular cylindrical surface *r*=*b* can be used to fill the physical space between another two constant-*r*′ surfaces *r*′=*a* and *r*′=*b* in the transformed space presented by (*r*′, *φ*′, *z*′), similar to Mankei Tsang’s spherical perfect lens in spherical system, this space deformation procedure can be easily recorded by coordinate transformation in cylindrical as:

which leads to the following expression for the permittivity and permeability tensor if the original surrounding medium is free space:

The components of the diagonal tensor in Eq. (2) determines the infinite radial, zero angular and z-axial material parameters respectively in the region of *a*<*r*′≤*b*, and the isotropic material with constant permittivity and permeability within the circular cylinder of *r*′=*a* completes the design of the ideal circular cylindrical perfect lens, any EM fields emitted from the object surface *r*′=*a* can be perfectly transmitted to the exterior circular surface *r*′=*b*. We can also expect the same angular magnification on the imaging surface, which can later be confirmed by our 2D numerical simulation.

For the convenience of imaging and future 2D practical realization of the perfect lens, we further design the oblate cylindrical perfect lens with flat object surface in the oblate cylindrical coordinate system. For clarity, we start with a brief introduction to this new orthogonal system as sketched in the following figure:

The oblate cylindrical system has three new coordinate variables ξ, η and z. Constant η and z define the generalized ξ axis, i.e., an elliptical curve. Similarly, constant ξ and z define the generalized η axis, which determines a confocal hyperbola with respect to the elliptical curve. The z axis in (x, y, z) system is preserved in this new system. The unit vectors of these three orthogonal coordinate curves are denoted by *â _{ξ}*,

*â*and

_{η}*â*, respectively. The transformation relations between (x, y, z) and (ξ, η, z) system can be represented as:

_{z}Here, we take advantage of two arbitrary constant-ξ surfaces ξ=a and ξ=b for the perfect lens design, where a two-step space distortion is also applied in (ξ, η, z) system analogous to the procedure occurred in the circular cylindrical perfect lens:

the gray-colored space 0≤*ξ*≤*a* in the original (x, y, z) system (top left in Fig. 3) is squeezed into the region of 0≤*ξ*′<*b* in the transformed space, and then the “emptied” *b*≤*ξ*′<*a* area is filled by the single pink-colored constant *ξ*=*a* elliptical surface, yielding the pink region functioning as the oblate cylindrical perfect lens (upper right in Fig. 3). We denote this space distortion by coordinate transformat on as

the corresponding transformation coefficients of this piecewise expression can be derived as

Outside the region of *ξ*′>*a*, the space keeps unchanged, and if it is set to be free space the required material specification of the perfect lens can be derived as:

where ${t}_{z}=\frac{{\mathrm{sinh}}^{2}\left(\xi \prime +a\right)+{\mathrm{sin}}^{2}\eta \prime}{{\mathrm{sinh}}^{2}\xi \prime +{\mathrm{sin}}^{2}\eta \prime}$ .

We note that the material tensors have concise forms, in particular for the components in 0≤*ξ*′<*b* region, where only the z-direction parameters need to be considered and this permittivity and permeability specification can be greatly simplified as we make the inner object surface flat by reducing *ξ*′=*b* to be zero, that is to say, we have *t _{z}*=1+(sinh

^{2}

*a*/sin

^{2}

*η*′), in this case, the impendence mismatch to the free space always exists at the flat interface, but the imaging performance is still excellent. We can just use half of the perfect lens (the upper part with respect to the y=0 plane) to magnify and transmit the EM fields located on the object plane to the exterior imaging surface without any reflection. We should be aware that both the oblate cylindrical proposed here and the oblate spheroidal perfect lens in Ref. 22 have their azimuthal magnifications that are variant with the EM source location on the object plane, which will be verified later by our theoretical analysis and numerical simulation.

## 3. Simulation and discussion

2D finite element simulations are performed in this part to testify the mathematical derivations and the functionality of various kinds of perfect lens designed on the basis of the coordinate transformation. We start with the planar perfect lens from Ref. 22, and then simulate our current designed structures. All the numerical calculations assume that the perfect lenses have ideal material parameters. Small TE plane wave sources situated at the object surface are employed as objects. The wavelength is much larger than the scale of the perfect lenses considered, which satisfies the requirement of the possible implementation approach with effective metamaterial united by thin layers having alternate signs of permittivity and permeability.

As suggested in Ref. 22, with the help of anisotropic metamaterial designed by coordinate transformation, the fields can perfectly propagate in a slab from one plane to another. The author also argue the discussion of Ref. 23 by asserting an opposite opinion that such a planar perfect lens have no capability of magnifying objects, but rather changes the field depth and focus only. From this point, we simulate the planar perfect lens by applying the material specification provided by Ref. 22, but the coordinate sequence is rearranged here by substituting x, y and z axis with z, x and y respectively in our simulation, which leads to the transformation of x coordinate variable as

So, we have

where δ is a small number with positive value and the ideal perfect lens slab can be obtained as δ goes to zero. Figure 4 shows the calculated result:

Figure 4 clearly shows the performance of the ideal planar slab (left), all the object information, regardless of the propagating wave as well as the decaying evanescent part, are directionally guided along x axis to the right side by the exotic anisotropic metamaterial inside the slab, enabling the perfect imaging below diffraction limit. The object seems to be moved by a distance of *b*, the E field’s feature (blue curve) is totally duplicated to the same pattern marked by the red curve at the right side (right), no matter how long this distance would be. Also, we note in Fig. (4) that the planar perfect lens don’t provide any magnification, further confirming Mankei Tsang’s statement in Ref. 22.

We now focus on the simulation of the magnifying circular cylindrical perfect lens. Actually, our cylindrical perfect lens is an example of a perfect lens with simpler symmetry having similar imaging properties in comparison to spherical perfect lens, thereby facilitating the realization in practice and 2D simulation. The calculated result is shown in Fig. 5.

We can see from Fig. 5 that the ideal circular perfect lens also have the function of transmitting the field from the inner circle to the outer surface without any reflection and the object can be magnified at the imaging surface. It is evident that the angular magnification of this perfect lens is invariant with the location of the object on the inner surface and only depends on the ratio of *b* to *a*, which is exactly the same as in the spherical perfect lens.

Compared with the circular cylindrical perfect lens, the oblate perfect lens in the oblate cylindrical system shows the advantages for its ability to make the object surface flat and for the perfect matched interface with the free space, as analyzed in the previous section of this paper. In this section, we first simulate the general oblate perfect lens (*b*≠0), and then select half of the perfect lens to see how it works with flat object plane (*b*=0).

In the general case, the constants *a* and *b* could be the numbers of any positive value, it is convenient to specify them to be the length of the major axis and the minor axis of the outer ellipse respectively, for which the eccentricity can be defined as

The diagonal material tensors in Eq. (6) are transformed into the standard forms in Cartesian (*x*, *y*, *z*) system.

where *T* is the transformation matrix from the oblate cylindrical(*ξ*, *η*, *z*) to Cartesian (*x*, *y*, *z*) system and *$\widehat{\epsilon}$* is the material tensor from Eq. (6). Substituting *ε* with *µ* in Eq. (10) completes the transformation for the permeability.

With the material tensor obtained from Eq. (10), the E field distribution in the general oblate cylindrical perfect lens in x-y plane is calculated and the result is shown in Fig. 6, where we notice that two TE wave sources symmetrically located at the inner surface with respect to the y axis are guided along the hyperbolic trajectories, forming two magnified images at the outer elliptical surface. It can be expected that the magnification on the constant-*ξ*′ surfaces varies with different *η*′, we now consider the magnification between two ellipse surface *ξ*′=*ξ*′_{ex} and *ξ*′=*ξ*′_{in} in x-y plane with *a _{ex}*=

*p*cosh

*ξ*′

_{ex}and

*a*=

_{in}*p*cosh

*ξ*′

_{in}where the differential lengths of the two curves is written as

The angular magnification can be defined as

Equation (13) clearly shows the dependence of the magnification on the object location determined by a set of (*ξ*′_{ex}, *η*′) and (*ξ*′_{in}, *η*′). We will discuss more about this in the case of *b*=0.

Given the values of parameters *a*, *b* and *p*, the corresponding angular magnification as a function of *η*′ is plotted in Fig. 7.

Evidently, Fig. 7 shows that the object located at (0,0) point have the smallest magnification of 1.6 at the imaging surface. As the object depart away from that point along the object surface, the magnification value increases. The value shows its maximum when *η*′ reaches to zero.

We can yield the inner ellipse if we make the length of the minor axis *b* decrease while keeping *p* fixed in Eq. (9), finally the inner ellipse changes into a line in the limit case of *b*=0 and consequently the object surface becomes a plane. We investigate the upper part of the ellipse by putting several pairs of small line sources against the flat plane with the same interval. The simulation result is demonstrated in Fig. 8.

Compared with the general case, similar propagation feature occurs in this structure. Each source in the seven pairs is amplified to the imaging surface and it is noted from each pair that two close sources can be separated at the outer surface because they follow their own hyperbolic trajectory which is unique for objects at different locations. The angular magnification also differs and increases as the source at the flat object plane moves away from the central y-axis. Quantitatively analysis can be yielded from Eq. (11)–(13), where *ξ*′* _{in}* goes to zero, in this case, we have the magnification:

Equation (14) can be further expressed as:

Now consider the right part of the perfect lens with flat object plane, where *η*′ have the range 0≤*η*′≤*π*/2. According to Eq. (15), we can obtain the magnification along y axis by specifying *η*′=*π*/2

According to Eq. (15), the magnification as a function of *η*′ is also depicted in the case of *a _{ex}*=8

*µm*and

*b*= ${b}_{\mathrm{ex}}=\sqrt{48}\mu m$ in Fig. 9.

_{ex}Equation (16) indicates that the magnification shows its minimum *a _{ex}*/

*p*at the direction parallel to y axis. Due to the monotonicity of cos

*η*′ in the range of 0≤

*η*′≤

*π*/2, the magnification is always larger than 1. In other words, the EM source anywhere on the object plane can be magnified at the imaging surface and the magnification increases as the source depart from the center, which is exactly in accordance with the simulation result in Fig. 8. This magnifying property can be exactly confirmed by Fig. 9, where the angular magnification have the minimum value of

*a*/

_{ex}*p*=2, and as

*η*′ drops to zero, the angular magnification approaches to infinity. We expect the analogous imaging features in spherical perfect lenses, they both have flat object planes but the image is still located at curved surfaces, thereby leading to variant magnification. So, how to make both the object and imaging surface flat and simultaneously make the magnification identical are part of our current research.

## 5. Conclusion

We have designed the circular cylindrical and oblate cylindrical perfect lens based on the coordinate transformation method. The non-magnifying planar perfect lens slab proposed in Ref. 22 was simulated to confirm the author’s opinion. We further simulated our magnifying perfect lens structures and systematically studied their imaging and magnification properties, our theoretical analysis is in great agreement with the simulation results, for which we believe that this perfect lens structure with such a simple symmetry will facilitate practical realization as well as measurements in the future.

## Acknowledgments

This work was supported by 973 Program of China (No.2006CB302900) and 863 Program of China (2006AA04Z310).

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