## Design of oblate cylindrical perfect lens using coordinate transformation

Optics Express, Vol. 16, Issue 11, pp. 8094-8105 (2008)

http://dx.doi.org/10.1364/OE.16.008094

Acrobat PDF (391 KB)

### 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

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]

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]

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]

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]

## 2. Circular and oblate cylindrical perfect lens

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]

*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:

*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.

*â*,

_{ξ}*â*and

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

_{z}*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

*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:

*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*=1+(sinh

_{z}^{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

*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.

*b*to

*a*, which is exactly the same as in the spherical perfect lens.

*b*≠0), and then select half of the perfect lens to see how it works with flat object plane (

*b*=0).

*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

*x*,

*y*,

*z*) system.

*T*is the transformation matrix from the oblate cylindrical(

*ξ*,

*η*,

*z*) to Cartesian (

*x*,

*y*,

*z*) system and

*is the material tensor from Eq. (6). Substituting*ε ^

*ε*with

*µ*in Eq. (10) completes the transformation for the permeability.

*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

*ξ*′

_{ex},

*η*′) and (

*ξ*′

_{in},

*η*′). We will discuss more about this in the case of

*b*=0.

*a*,

*b*and

*p*, the corresponding angular magnification as a function of

*η*′ is plotted in Fig. 7.

*η*′ reaches to zero.

*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.

*ξ*′

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

_{in}*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

*η*′ is also depicted in the case of

*a*=8

_{ex}*µm*and

*b*=

_{ex}*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

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.-Usp. |

2. | R. C. McPhedran, N. A. Nicorovici, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B |

3. | G. W. Milton and N. A. Nicorovici “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Lond. A |

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 |

5. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

6. | J. B. Pendry and S. A. Rmakrishna, “Focusing light using negative refraction,” J. Phys.:Condens. Matter |

7. | S. A. Rmakrishna and J. B. Pendry, “Spherical perfect lens: solutions of Maxwell’s equations for spherical geometry,” Phys. Rev. B |

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. |

9. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

10. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science |

11. | A. Salandrino and N. Engheta, “Far-field sub-diffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

12. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

13. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

14. | I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying Superlens in the Visible Frequency Range,” Science |

15. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

16. | Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. |

17. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

18. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. |

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 |

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 |

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. |

**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

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys.-Usp. 10, 509-514 (1968). [CrossRef]
- 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]
- 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]
- 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]
- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- J. B. Pendry and S. A. Rmakrishna, "Focusing light using negative refraction," J. Phys.:Condens. Matter 15, 6345-6364 (2003). [CrossRef]
- 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]
- 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]
- D. O. S. Melville and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005). [CrossRef] [PubMed]
- 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]
- A. Salandrino and N. Engheta, "Far-field sub-diffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006). [CrossRef]
- 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]
- 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]
- I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, "Magnifying Superlens in the Visible Frequency Range," Science 315, 1699-1701 (2007). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007). [CrossRef]
- H. Chen, X. Jiang, and C. T. Chan, "On some constraints that limit the design of an invisibility cloak," arXiv:0707.1126v2.
- 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]
- 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]
- M. Tsang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," arXiv: 0708.0262v1.
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New. J. Phys. 8, 247 (2006). [CrossRef]

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