## Optical lens compression via transformation optics

Optics Express, Vol. 17, Issue 19, pp. 16535-16542 (2009)

http://dx.doi.org/10.1364/OE.17.016535

Acrobat PDF (2962 KB)

### Abstract

Transformation optics is widely associated with the design of unconventional electromagnetic devices, such as electromagnetic cloaks or concentrators. However, a wide range of conventional optical devices with potentially advantageous properties can be designed by the transformation optical approach. For example, a coordinate transformation can be introduced that compresses a region of space, resulting in an overall decrease in the thickness of an optical instrument such as a lens. The optical properties of a transformed lens, such as Fresnel reflection or aberration profile, are equivalent to those of the original lens, while the transformed lens and the bounding transformation optical material are thinner than the original lens. This approach to flattening the profile of a lens represents an advantage over the use of a higher dielectric material because it does not introduce greater Fresnel reflections or require a redesign of the basic optic. Though transformation optical media are generally anisotropic, with both electric and magnetic response, it is possible to arrive at a dielectric-only transformation optical distribution for a lens interacting with transverse-magnetic (TM) polarized light. The dielectric-only distribution can be implemented using broad-band, low-loss metamaterials. Lens designs for both a full transformation and a dielectric-only implementation are discussed and confirmed via finite-element simulations.

© 2009 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. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**(4), 224–227 (2007). [CrossRef]

4. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**(6), 063903 (2008). [CrossRef] [PubMed]

7. D. H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express **16**(23), 18731–18738 (2008). [CrossRef]

8. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express **15**(22), 14772–14782 (2007). [CrossRef] [PubMed]

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

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

19. D.-H. Kwon and D. H. Werner, “Restoration of antenna parameters in scattering environments using electromagnetic cloaking,” Appl. Phys. Lett. **92**(11), 113507 (2008). [CrossRef]

20. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71**(33 Pt 2B), 036617 (2005). [CrossRef] [PubMed]

24. 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]

21. R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. **87**(9), 091114 (2005). [CrossRef]

23. 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]

25. C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Tanielian, and D. C. Vier, “Performance of a negative index of refraction lens,” Appl. Phys. Lett. **84**(17), 3232 (2004). [CrossRef]

26. D. Schurig and D. R. Smith, “Negative index lens aberrations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(6), 065601 (2004). [CrossRef]

27. A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, “Multichannel sampling schemes for optical imaging systems,” Appl. Opt. **47**(10), B76–B85 (2008). [CrossRef] [PubMed]

## 2. Lens compression

_{1}and l

_{2}are the bounds of the transformation in the unprimed space. This expression can be integrated to determine the coordinate map, givingThe integration constant is determined by a boundary condition that can be used to define certain optical positions of the lens, such as setting the position of

*ε*and

*μ*asWe assume the index profile n(

*x,y*) represents a conventional refractive optic, for which the index of refraction occurs via an electric response, with

*n*=

*ε*

^{1/2}. Equation (6) shows that the dielectric response of the lens must be transformed in addition to the free space surrounding it. Additionally, the position of lens is translated and must be defined through the integration constant.

*a*= ½. The integration constant is chosen so that

*x*= 0 is mapped to

*x’*= 0 (i.e.

*c*= 0). This position corresponds to the planar side of the lens. The spherical surface of the lens extends to –

*d*= −0.05 m. It is clear from Eq. (5) that this position is mapped to -

*d*/2 = 0.025 m. The resulting transformation, which is plotted in Fig. 1c for these parameters, is

## 3. Dielectric only implementation

*μ*,

_{z}*ε*and

_{y}*ε*that leave

_{x}*n*and

_{x}*n*invariant will approximately reproduce the same focusing or imaging properties. Thus, if we define both Eq. (11) and Eq. (12) remain invariant, but we arrive at an index only version of the compressed lens.

_{y}*a*is between 0 and 1). Figure 3a shows a COMSOL simulation of the resulting dielectric-only compressed lens. This transformation is no longer reflectionless, as evidenced by the power flow lines shown in the figure.

## 4. Conclusions

## References and links

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

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

3. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

4. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. |

5. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express |

6. | D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. |

7. | D. H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express |

8. | D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express |

9. | A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. |

10. | M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B |

11. | L. Lin, W. Wang, C. L. Du, and X. G. Luo, “A cone-shaped concentrator with varying performances of concentrating,” Opt. Express |

12. | M. Yan, W. Yan, and M. Qiu, “Cylindrical superlens by a coordinate transformation,” Phys. Rev. B |

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

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

15. | D. Schurig, “An aberration-free lens with zero F-number,” N. J. Phys. |

16. | F. M. Kong, B. I. I. Wu, J. A. Kong, J. T. Huangfu, S. Xi, and H. S. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. |

17. | Yu. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the Emission of Electromagnetic Source,” PIERS |

18. | N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express |

19. | D.-H. Kwon and D. H. Werner, “Restoration of antenna parameters in scattering environments using electromagnetic cloaking,” Appl. Phys. Lett. |

20. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

21. | R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. |

22. | T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett. |

23. | 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 |

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

25. | C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Tanielian, and D. C. Vier, “Performance of a negative index of refraction lens,” Appl. Phys. Lett. |

26. | D. Schurig and D. R. Smith, “Negative index lens aberrations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, “Multichannel sampling schemes for optical imaging systems,” Appl. Opt. |

**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 7, 2009

Revised Manuscript: August 15, 2009

Manuscript Accepted: August 17, 2009

Published: September 1, 2009

**Citation**

D. A. Roberts, N. Kundtz, and D. R. Smith, "Optical lens compression via transformation optics," Opt. Express **17**, 16535-16542 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16535

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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]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
- M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]
- D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93(25), 251111 (2008). [CrossRef]
- D. H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16(23), 18731–18738 (2008). [CrossRef]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15(22), 14772–14782 (2007). [CrossRef] [PubMed]
- A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008). [CrossRef]
- L. Lin, W. Wang, C. L. Du, and X. G. Luo, “A cone-shaped concentrator with varying performances of concentrating,” Opt. Express 16(10), 6809–6814 (2008). [CrossRef] [PubMed]
- M. Yan, W. Yan, and M. Qiu, “Cylindrical superlens by a coordinate transformation,” Phys. Rev. B 78(12), 125113 (2008). [CrossRef]
- V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
- D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008). [CrossRef]
- D. Schurig, “An aberration-free lens with zero F-number,” N. J. Phys. 10(11), 115034 (2008). [CrossRef]
- F. M. Kong, B. I. I. Wu, J. A. Kong, J. T. Huangfu, S. Xi, and H. S. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. 91(25), 253509 (2007). [CrossRef]
- Yu. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the Emission of Electromagnetic Source,” PIERS 4(7), 795–800 (2008). [CrossRef]
- N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008). [CrossRef] [PubMed]
- D.-H. Kwon and D. H. Werner, “Restoration of antenna parameters in scattering environments using electromagnetic cloaking,” Appl. Phys. Lett. 92(11), 113507 (2008). [CrossRef]
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005). [CrossRef] [PubMed]
- R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 091114 (2005). [CrossRef]
- T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett. 88(8), 081101 (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(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]
- C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Tanielian, and D. C. Vier, “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84(17), 3232 (2004). [CrossRef]
- D. Schurig and D. R. Smith, “Negative index lens aberrations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(6), 065601 (2004). [CrossRef]
- A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, “Multichannel sampling schemes for optical imaging systems,” Appl. Opt. 47(10), B76–B85 (2008). [CrossRef] [PubMed]

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