## Enhancing imaging systems using transformation optics |

Optics Express, Vol. 18, Issue 20, pp. 21238-21251 (2010)

http://dx.doi.org/10.1364/OE.18.021238

Acrobat PDF (1035 KB)

### Abstract

We apply the transformation optical technique to modify or improve conventional refractive and gradient index optical imaging devices. In particular, when it is known that a detector will terminate the paths of rays over some surface, more freedom is available in the transformation approach, since the wave behavior over a large portion of the domain becomes unimportant. For the analyzed configurations, quasi-conformal and conformal coordinate transformations can be used, leading to simplified constitutive parameter distributions that, in some cases, can be realized with isotropic index; index-only media can be low-loss and have broad bandwidth. We apply a coordinate transformation to flatten a Maxwell fish-eye lens, forming a near-perfect relay lens; and also flatten the focal surface associated with a conventional refractive lens, such that the system exhibits an ultra-wide field-of-view with reduced aberration.

© 2010 OSA

## 1. Introduction

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3. A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. **75**(10), 6036–6038 (1994). [CrossRef]

4. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

5. 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**(5801), 977–980 (2006). [CrossRef] [PubMed]

9. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. **5**(9), 687–692 (2009). [CrossRef]

10. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express **17**(19), 16535–16542 (2009). [CrossRef] [PubMed]

15. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. **8**(8), 639–642 (2009). [CrossRef] [PubMed]

*transformed*field distribution is viewed relative to the

*original*coordinate system, it will appear distorted. By judiciously choosing coordinate transformations such that a field pattern is distorted or modified in some desired manner, an optical device can be intuitively designed. By subsequently applying the coordinate transformation to the constitutive tensors in a manner suggested by the form-invariance of Maxwell’s equations, the specification of a medium is obtained that will physically distort the waves exactly as the coordinate transformation predicts. By implementing the specified constitutive parameters using metamaterials or other means, the transformation optical device transitions from virtual to actual.

10. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express **17**(19), 16535–16542 (2009). [CrossRef] [PubMed]

14. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. **9**(2), 129–132 (2010). [CrossRef]

16. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

6. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

14. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. **9**(2), 129–132 (2010). [CrossRef]

17. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**(7), 568–571 (2009). [CrossRef] [PubMed]

19. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976), 337–339 (2010). [CrossRef] [PubMed]

## 2. The flattened Maxwell fish-eye lens

10. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express **17**(19), 16535–16542 (2009). [CrossRef] [PubMed]

14. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. **9**(2), 129–132 (2010). [CrossRef]

15. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. **8**(8), 639–642 (2009). [CrossRef] [PubMed]

**17**(19), 16535–16542 (2009). [CrossRef] [PubMed]

*a*is the radius of the lens [20]. The fish-eye lens focuses rays emanating from a point on the lens surface to a conjugate point on the opposite side of the lens. All pairs of object and image points are aplanatic, thus the fish-eye lens is a perfect imaging instrument; however, the curved object and image surfaces generally make the fish-eye lens more a curiosity rather than a useful device. By applying a transformation that flattens two sides of the lens, it may be possible to increase the utility of the fish-eye, for use, say, as a perfect relay lens.

*s*, such thatwhere

*i*indexes the general coordinate

*a*into a rectangular region of width

*w*and height

*l*[11

11. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. **10**(11), 115023 (2008). [CrossRef]

21. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**(21), 9794–9804 (2006). [CrossRef] [PubMed]

*Λ*is the Jacobian matrix relating differential distances between the two coordinate systems, and has the explicit formWhile we could evaluate the elements of

*Λ*using the transformation of Eqs. (2), the resulting medium would not be a practical realization, and the effort of writing down the transformation explicitly would not be illuminating. We instead make use of the quasi-conformal TO method to arrive at a numerical transformation that leads to more realistic material parameters. We describe the QCTO method in the following section.

## 3. Quasi-conformal transformations

22. U. Leonhardt, “Optical conformal mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

23. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

23. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

25. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express **18**(6), 6089–6096 (2010). [CrossRef] [PubMed]

23. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express **17**(17), 14872–14879 (2009). [CrossRef] [PubMed]

25. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express **18**(6), 6089–6096 (2010). [CrossRef] [PubMed]

## 3. Quasi-conformal transformation for the fish-eye lens

*y*-values in the virtual space conformed to the circular boundary of the fish-eye (physical space). The

*x*-values are not changed by the initial transformation (

6. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

**9**(2), 129–132 (2010). [CrossRef]

17. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**(7), 568–571 (2009). [CrossRef] [PubMed]

18. L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics **3**(8), 461 (2009). [CrossRef]

27. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(3), 036621 (2006). [CrossRef] [PubMed]

*a*—thus, one half of the flattened lens thickness. The free space above and below the lens is gradually terminated with perfectly matched layers of refractive index n = 1. The simulations in Figs. 5b and 5c demonstrate the expected focusing behavior. Similar TE-wave focusing behavior was observed with point magnetic dipole sources oriented either vertically or horizontally in the plane of propagation (not shown).

## 4. Compensation of field curvature using transformation optics

*i*indexes the two lens surfaces. For the lens shown, the radii of the two surfaces are

*a*and

*b*are constants that are found by fitting the determined locus of focal points. The focal point for rays incident along the optical axis is

13. D. Schurig, “An aberration-free lens with zero *f*-number,” N. J. Phys. **10**(11), 115034 (2008). [CrossRef]

## 5. Conclusion

19. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976), 337–339 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | E. G. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Interscience Publishers, New York, 1962). |

3. | A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. |

4. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. |

5. | 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 |

6. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

7. | V. M. Shalaev, “Physics. Transforming light,” Science |

8. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. |

9. | D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. |

10. | D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express |

11. | D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. |

12. | D. H. Kwon and D. H. Werner; “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express |

13. | D. Schurig, “An aberration-free lens with zero |

14. | N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. |

15. | Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. |

16. | J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. |

17. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. |

18. | L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics |

19. | T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science |

20. | E. W. Marchand, Gradient Index Optics (Academic Press, New York, 1978). |

21. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

22. | U. Leonhardt, “Optical conformal mapping,” Science |

23. | N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express |

24. | P. Knupp, and S. Steinberg, Fundamentals of Grid Generation (CRC Press, Boca Raton, FL, 1993). |

25. | Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express |

26. | N. B. Kundtz, Advances in Complex Artificial Electromagnetic Media, PhD Dissertation, Duke University, Durham, N. Carolina (2009). |

27. | S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

28. | E. Hecht, Optics, 4th Ed. (Addison-Wesley, San Francisco, 2002). |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(220.3620) Optical design and fabrication : Lens system design

(220.3630) Optical design and fabrication : Lenses

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 23, 2010

Revised Manuscript: September 14, 2010

Manuscript Accepted: September 15, 2010

Published: September 22, 2010

**Citation**

David R. Smith, Yaroslav Urzhumov, Nathan B. Kundtz, and Nathan I. Landy, "Enhancing imaging systems using transformation optics," Opt. Express **18**, 21238-21251 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21238

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- E. G. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (Interscience Publishers, New York, 1962).
- A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994). [CrossRef]
- A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996). [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(5801), 977–980 (2006). [CrossRef] [PubMed]
- R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
- V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]
- D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
- D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009). [CrossRef] [PubMed]
- D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008). [CrossRef]
- D. H. Kwon and D. H. Werner; “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009). [CrossRef] [PubMed]
- D. Schurig, “An aberration-free lens with zero f-number,” N. J. Phys. 10(11), 115034 (2008). [CrossRef]
- N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef]
- Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
- L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009). [CrossRef]
- T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010). [CrossRef] [PubMed]
- E. W. Marchand, Gradient Index Optics (Academic Press, New York, 1978).
- D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]
- P. Knupp, and S. Steinberg, Fundamentals of Grid Generation (CRC Press, Boca Raton, FL, 1993).
- Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]
- N. B. Kundtz, Advances in Complex Artificial Electromagnetic Media, PhD Dissertation, Duke University, Durham, N. Carolina (2009).
- S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]
- E. Hecht, Optics, 4th Ed. (Addison-Wesley, San Francisco, 2002).

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