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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 11 — May. 26, 2008
  • pp: 8094–8105
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Design of oblate cylindrical perfect lens using coordinate transformation

Wei Wang, Lan Lin, Xuefeng Yang, Jianhua Cui, Chunlei Du, and Xiangang Luo  »View Author Affiliations


Optics Express, Vol. 16, Issue 11, pp. 8094-8105 (2008)
http://dx.doi.org/10.1364/OE.16.008094


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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

Although perfect lenses based on the coordinate transformation and their possible realization approach have been theoretically proposed in Ref. 22

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

, 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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

, 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

13. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007). [CrossRef] [PubMed]

] 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:

Fig. 1. Diagram of the circular cylindrical perfect lens. Only the x and y axis are involved in the 2D case of the cylindrical stucture. With the coordinate transformation, the design can be processed in the following two steps: (1) the gray-colored space 0≤rb in (r, φ, z) system (left) is compressed into the 0≤r′<a region in the transformed space, and (2) the “emptied” ar′<b area is then filled by the single pink-colored constant r=b surface, resulting in the pink region (right) functioning as the desired circular perfect lens, any EM source on the inner circle can be magnified and perfect transmitted to the outer surface, as depicted by the red region in the perfect lens(right)

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:

r={(ba)r0r<abarb,φ=φ,z=zrr>b
(1)

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

ε̂=μ̂={[10001000(ba)2]0ra[00000000]a<rb
(2)

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:

Fig. 2. The sketch of the oblate cylindrical coordinate system. Three colored curves comprise the new orthogonal coordinate curves in x-y plane: the ellipse represents η curve (dark red), the confocal hyperbola for ξ curve (blue) and z axis for z curve (green) with unit vectors a⃑η, a⃑ξ and a⃑z respectively.

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 âξ, âη and âz, respectively. The transformation relations between (x, y, z) and (ξ, η, z) system can be represented as:

{x=pcoshξcosηy=psinhξsinη,0ξ<0η2π<z<z=z
(3)

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:

Fig. 3. Diagram of Oblate cylindrical perfect lens. We demonstrate the design in x-y plane for simplicity. First, the general oblate cylindrical perfect lens with b≠0 (top right) is designed by coordinate transformation (top left to top right), and the one with flat object surface is then obtained by making the inner elliptical space degenerate to be a slice (top right to bottom right), corresponding to the limit case of ξ′=b=0. Half of the Oblate cylindrical perfect lens is convenient for practical use (bottom left).

the gray-colored space 0≤ξa 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” bξ′<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

ξ={ξ+a0ξ<babξa,η=η,z=zξξ>a
(4)

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

{Qξ=Qη=sinh2(ξ+a)+sin2ηsinh2ξ+sin2ηQz=1,0ξ<bQξ=0Qη=Qz=1bξa
(5)

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:

ε̂=μ̂={[10001000tz]0ξ<b[00000000]b<ξa
(6)

where tz=sinh2(ξ+a)+sin2ηsinh2ξ+sin2η .

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 tz=1+(sinh2 a/sin2 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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

, 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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

, 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

23. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New. J. Phys. 8, 247 (2006). [CrossRef]

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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

, 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

x={x+bx<0δx+b0xb,y=y,z=zx+δbx>b
(7)

So, we have

εx=μx=1δ,εy=μy=δ,εz=μz=δ
(8)

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:

Fig. 4. The 2D (x-y plane) normalized electric field distribution in the planar perfect lens (right) and the normalized electric field distribution on the object and imaging plane. The width of the slab b=4µm. The TE plane wave propagates along x axis from the left side (the object plane) of the slab to the right (the imaging plane) with perfect fidelity.

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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

.

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.

Figure 5. The 2D (x-y plane) normalized electric field distribution in the circular cylindrical perfect lens. The inner circle with the radius of a=1µm represents the object surface and the outer circle with the radius of b=4µm is the imaging surface. The wavelength used here is much larger than the size of the whole structure.

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.

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

P=a2b2
(9)

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

[εxxεxyεxzεyxεyyεyzεzxεzyεzz]=T·ε̂·T1
(10)

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

Fig. 6. The 2D (x-y plane) normalized electric field distribution in the general oblate cylindrical perfect lens. The inner ellipse with ξ′=ξex is the object surface and the outer ellipse with ξ′=ξin is the imaging surface (a>b>0). For simplicity, we choose aex=8µm and bex= bex=48μm for the exterior ellipse, corresponding p=4µm. The inner ellipse has ain=5µm, bin=3µm and the same p. The wavelength of the TE wave is much larger than aex.

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′=ξex and ξ′=ξin in x-y plane with aex=pcoshξex and ain=pcoshξin where the differential lengths of the two curves is written as

lex=dsex=pcosh2ξexcos2ηdξ
(11)
lin=dsin=pcosh2ξincos2ηdξ
(12)

The angular magnification can be defined as

Mη=lexlin=cosh2ξexcos2ηcosh2ξincos2η=(aexp)2cos2η(ainp)2cos2η
(13)

Equation (13) clearly shows the dependence of the magnification on the object location determined by a set of (ξex, η′) and (ξin, 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.

Fig. 7. The angular magnification as a function of η′. aex=8µm, bex= bex=48μm ain=5µm and bin=3µm are used with the same p=4µm for the exterior and interior ellipse respectively.

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.

Fig. 8. The 2D (x-y plane) normalized electric field distribution in the oblate cylindrical perfect lens with flat object plane. The values of aex=8µm, bex= bex=48μm and p=4µm for the exterior ellipse in previous general case are preserved in this perfect lens structure. Seven pairs of line sources emitting TE plane wave are located on the flat plane with the same interval. The two plane wave sources of each pair are separated by a small distance, which is much smaller than the length of major axis.

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 ξin goes to zero, in this case, we have the magnification:

Mη=lexlin=cosh2ξexcos2ηcosh2ξincos2η=(aexp)2cos2η1cos2η
(14)

Equation (14) can be further expressed as:

Mη=[(aexp)21]+(1cos2η)1cos2η=1+(aexp)211cos2η
(15)

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

M(π2)=aexp
(16)

According to Eq. (15), the magnification as a function of η′ is also depicted in the case of aex=8µm and bex= bex=48μm in Fig. 9.

Fig. 9. The angular magnification as a function of η′ in the case of aex=8µm, bex= bex=48μm and bin=0, in which the object surface becomes to be flat.

Equation (16) indicates that the magnification shows its minimum aex/p at the direction parallel to y axis. Due to the monotonicity of cosη′ 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 aex/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

22. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

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).

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.-Usp. 10, 509–514 (1968). [CrossRef]

2.

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–38482 (1994). [CrossRef]

3.

G. W. Milton and N. A. Nicorovici “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Lond. A 462, 3027–3059 (2006). [CrossRef]

4.

N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonancea,” Opt. Express 15, 6314–6323 (2007). [CrossRef] [PubMed]

5.

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

6.

J. B. Pendry and S. A. Rmakrishna, “Focusing light using negative refraction,” J. Phys.:Condens. Matter 15, 6345–6364 (2003). [CrossRef]

7.

S. A. Rmakrishna and J. B. Pendry, “Spherical perfect lens: solutions of Maxwell’s equations for spherical geometry,” Phys. Rev. B 69, 115115 (2004). [CrossRef]

8.

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative Refraction in 2D checkerboards related by mirror anti-symmetry and 3D corner lenses,” New J. Phys. 7, 164 (2005). [CrossRef]

9.

D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127–2134 (2005). [CrossRef] [PubMed]

10.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science 308, 534–537 (2005). [CrossRef] [PubMed]

11.

A. Salandrino and N. Engheta, “Far-field sub-diffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006). [CrossRef]

12.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–825 (2006). [CrossRef] [PubMed]

13.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007). [CrossRef] [PubMed]

14.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying Superlens in the Visible Frequency Range,” Science 315, 1699–1701 (2007). [CrossRef] [PubMed]

15.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006). [CrossRef] [PubMed]

16.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 (2007). [CrossRef] [PubMed]

17.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]

18.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. 1, 224 (2007). [CrossRef]

19.

H. Chen, X. Jiang, and C. T. Chan, “On Some Constraints That Limit the Design of An Invisibility Cloak,” arXiv:0707.1126v2.

20.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006). [CrossRef]

21.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

22.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” arXiv: 0708.0262v1.

23.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New. J. Phys. 8, 247 (2006). [CrossRef]

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: February 22, 2008
Revised Manuscript: April 27, 2008
Manuscript Accepted: May 12, 2008
Published: May 20, 2008

Citation
Wei Wang, Lan Lin, Xuefeng Yang, Jianhua Cui, Chunlei Du, and Xiangang Luo, "Design of oblate cylindrical perfect lens using coordinate transformation," Opt. Express 16, 8094-8105 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-8094


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys.-Usp. 10, 509-514 (1968). [CrossRef]
  2. R. C. McPhedran, N. A. Nicorovici, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-38482 (1994). [CrossRef]
  3. G. W. Milton and N. A. Nicorovici "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Lond. A 462, 3027-3059 (2006). [CrossRef]
  4. N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonancea," Opt. Express 15, 6314-6323 (2007). [CrossRef] [PubMed]
  5. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  6. J. B. Pendry and S. A. Rmakrishna, "Focusing light using negative refraction," J. Phys.:Condens. Matter 15, 6345-6364 (2003). [CrossRef]
  7. S. A. Rmakrishna and J. B. Pendry, "Spherical perfect lens: solutions of Maxwell??s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004). [CrossRef]
  8. S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, "Negative Refraction in 2D checkerboards related by mirror anti-symmetry and 3D corner lenses," New J. Phys. 7, 164 (2005). [CrossRef]
  9. D. O. S. Melville and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005). [CrossRef] [PubMed]
  10. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
  11. A. Salandrino and N. Engheta, "Far-field sub-diffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006). [CrossRef]
  12. Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical Hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-825 (2006). [CrossRef] [PubMed]
  13. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007). [CrossRef] [PubMed]
  14. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, "Magnifying Superlens in the Visible Frequency Range," Science 315, 1699-1701 (2007). [CrossRef] [PubMed]
  15. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
  16. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal Cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007). [CrossRef] [PubMed]
  17. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
  18. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007). [CrossRef]
  19. H. Chen, X. Jiang, and C. T. Chan, "On some constraints that limit the design of an invisibility cloak," arXiv:0707.1126v2.
  20. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]
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  22. M. Tsang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," arXiv: 0708.0262v1.
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