## Perfect cylindrical lenses

Optics Express, Vol. 11, Issue 7, pp. 755-760 (2003)

http://dx.doi.org/10.1364/OE.11.000755

Acrobat PDF (85 KB)

### Abstract

A slab of negatively refracting material is known to focus light and if *n*=-1 the focussing will be perfect, producing an image which is an exact replica of the object. Magnifying the image requires a new design concept in which the surface of the negatively refracting lens is curved. Here we show how a hollow cylinder of material can be designed to magnify an image but otherwise with the same perfection as the original lens. Curvature requires that ε and µ are now a function of position.

© 2003 Optical Society of America

## 1. Introduction

1. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI **10**, 509 (1968). [CrossRef]

*n*=-1and behave as a lens. The negative refractive index was subsequently confirmed by Shelby, Smith and Schultz in 2001 [2

2. R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science **292**, 79 (2001). [CrossRef]

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

4. D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett **76**, 2480 (1996). [CrossRef] [PubMed]

5. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett **76**4773 (1996) [CrossRef] [PubMed]

5. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett **76**4773 (1996) [CrossRef] [PubMed]

9. R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. **49**, 1262–1265 (2001). [CrossRef]

10. M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” *Science* **291**848–851 (2001). [CrossRef]

7. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques **47**, 2075 (1999). [CrossRef]

*z*=

*x*+

*iy*resulted in a cylindrical lens which reproduced the contents of the smaller cylinder in magnified but undistorted form outside the larger cylinder. This transformation preserves the solutions of Laplace’s equation and leaves the values of ε, µ, the electrical permittivity/magnetic permeability, unchanged in their respective domains. However Laplace’s equation is a valid description of the fields only in the electrostatic or magnetostatic limit where the electric and magnetic fields separate. Lenses defined by conformal transformations are only valid provided that all dimensions are much less than the wavelength of light which is a somewhat limiting condition. Fortunately there exists a more general theory of arbitrary coordinate transformations [13

13. A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics **43**773–793 (1996). [CrossRef]

13. A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics **43**773–793 (1996). [CrossRef]

*l*coordinate is oriented along the radial direction. Now if we make the choice of

*l*

_{0}=1 and,

*l*ϕ

*Z*frame,

_{z}, µ

_{z}now depend on the radius as

*r*

^{-2}. The original paper assumed that the electric field was confined to the

*xy*plane and therefore ε

_{z}was irrelevant. It also assumed the electrostatic limit so that now magnetic fields were present and hence µ

_{z}was also irrelevant. hence in that limit we retrieve our original result.

## 2. A perfect crescent lens

## 3. Conclusions

*compensated for*by a reciprocal change in ε and µ. Applying these ideas to the perfect lens generates a whole new class of lens with the same remarkable capability to focus at the sub wavelength level. In the examples we present here curvature in the

*xy*plane is compensated for by changes in ε

_{z},µ

_{z}in the direction normal to the plane of curvature. This ability to transform a concept from one geometry to another can be expected to have applications beyond the present case, for example in the field of photonic crystals.

## Acknowledgements

## References and links

1. | V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI |

2. | R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science |

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

4. | D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett |

5. | J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett |

6. | J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter] |

7. | J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques |

8. | D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,” |

9. | R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. |

10. | M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” |

11. | Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications |

12. | J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter] |

13. | A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics |

14. | R.H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Focus Issue: Negative refraction and metamaterials

**History**

Original Manuscript: January 27, 2003

Revised Manuscript: March 4, 2003

Published: April 7, 2003

**Citation**

J. Pendry, "Perfect cylindrical lenses," Opt. Express **11**, 755-760 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-755

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### References

- V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Soviet Physics USPEKHI 10, 509 (1968). [CrossRef]
- R.A. Shelby, D.R. Smith, S. Schultz, "Experimental verification of negative index of refraction," Science 292, 79 (2001). [CrossRef]
- J.B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85 3966 (2000). [CrossRef] [PubMed]
- D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, "3D Wire Mesh Photonic Crystals," Phys. Rev. Lett. 76, 2480 (1996). [CrossRef] [PubMed]
- J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76 4773 (1996) [CrossRef] [PubMed]
- J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, "Low Frequency Plasmons in Thin Wire Structures," J. Phys. [Condensed Matter] 10, 4785 (1998).
- J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, "Magnetism from Conductors and Enhanced Non-Linear Phenomena," IEEE Trans. Microwave Theory Techniques 47, 2075 (1999). [CrossRef]
- D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. "High impedance electromagnetic surfaces with a forbidden frequency band," 47, 2059-2074 (1999). [CrossRef]
- R.F.J. Broas, D.F. Sievenpiper, E. Yablonovitch "A high-impedance ground plane applied to a cell phone handset geometry," IEEE Trans. Micr. Theory and Tech. 49, 1262-1265 (2001). [CrossRef]
- M.C.K. Wiltshire, J.B. Pendry, I.R.Young, D.J. Larkman, D.J. Gilderdale and J.V. Hajnal., "Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging," Science 291 848-51 (2001) [CrossRef]
- Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos and J.B. Pendry, "All-Angle Negative Refraction without Negative Effective Index," Phys. Rev. Rapid Commun. B65, 201104(R) (2002).
- J.B. Pendry and S.A. Ramakrishna "Near Field Lenses in Two Dimensions," J. Phys. [Condensed Matter] 14 1-17 (2002).
- A.J. Ward and J. B. Pendry, "Refraction and Geometry in Maxwell's Equations," J. Modern Opt. 43 773-93 (1996). [CrossRef]
- R.H. Ritchie, "Plasma Losses by Fast Electrons in Thin Films," Phys. Rev. 106, 874 (1957). [CrossRef]

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