## Performance of a three dimensional transformation-optical-flattened Lüneburg lens |

Optics Express, Vol. 20, Issue 12, pp. 13262-13273 (2012)

http://dx.doi.org/10.1364/OE.20.013262

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### Abstract

We demonstrate both the beam-forming and imaging capabilities of an X-band (8–12 GHz) operational Lüneburg lens, one side of which has been flattened via a coordinate transformation optimized using quasi-conformal transformation optics (QCTO) procedures. Our experimental investigation includes benchmark performance comparisons between the QCTO Lüneburg lens and a commensurate conventional Lüneburg lens. The QCTO Lüneburg lens is made from a metamaterial comprised of inexpensive plastic and fiberglass, and manufactured using fast and versatile numerically controlled water-jet machining. Looking forward towards the future and advanced TO designs, we discuss inevitable design trade-offs between affordable scalable manufacturing and rigorous adherence to the full TO solution, as well as possible paths to mitigate performance degradation in realizable designs.

© 2012 OSA

## 1. Introduction

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

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

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

*ε*and magnetic permeability

*μ*); the optical parameters resulting from TO designs are typically inhomogeneous and anisotropic [6

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

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

8. D. R. Smith, W. J. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

10. W. J. Padilla, D. N. Basov, and D. R. Smith, “Negative refractive index metamaterials,” Mater. Today **9**, 28–35 (2006). [CrossRef]

11. 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]

13. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. **10**, 1991–1997 (2010). [CrossRef] [PubMed]

16. W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. **32**, 1271–1284 (1992). [CrossRef] [PubMed]

*ε*(

*r*) = (2 −

*r*

^{2}). The primary drawback to the Lüneburg lens is that its image surface is spherical [17

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

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

18. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. **1**, 124 (2010). [CrossRef] [PubMed]

18. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. **1**, 124 (2010). [CrossRef] [PubMed]

## 2. Design

*π*steradian hemisphere is flattened - creating a fixed field of view (FOV). The greater the FOV is, the more extreme the resultant optical parameters are [3

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

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

*π*steradian, we follow the procedure of Landy et al. [19

19. N. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasi-conformal coordinate transformations,” Phys. Rev. Lett. **105**, 193902 (2010). [CrossRef]

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

*μ*= 1), isotropic (

_{θ,ϕ,z}*ε*=

_{θ}*ε*), and nonmetallic (all

_{ρ,z}*ε*> 1). Each of these simplifications has negative repercussions in lens performance. The resultant isotropic dielectric profile after these simplifications is shown in Fig. 1(a).

*ε*

_{eff}from the bulk value - in agreement with effective medium theory [20

20. J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London **205**, 237–288 (1906). [CrossRef]

*ε*

_{eff}for polarizations perpendicular and parallel to the cylinders, and later the effects of this choice and its anisotropy will become apparent. To be thorough we perform finite element simulations to obtain exact solutions for the effective parameters of the unit cells. Scattering-parameter results from the finite element simulations are inverted using the procedure of [21

21. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

*ε*(

*ρ,z*) prescription of Fig. 1(a) is broken up into planar sheets, each of which contains holes in FR4 and HDPE mapped from Fig. 1(b). Physically, these holes are made via CNC waterjet - which results in a total-lens cost competitive with existing conventional Lüneburg lenses (often made of nested spherical shells of different density polystyrene) even for this first-generation device. The sheets are stacked to make the complete lens as shown in Fig. 1(c).

## 3. Experiment

22. Rozendal Associates, http://www.rozendalassociates.com/

18. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. **1**, 124 (2010). [CrossRef] [PubMed]

*ε*< 1, which we have omitted to simplify the material requirements and to enable broadband operation (such regions must be resonant). It is know that when these regions are omitted, the effective FOV becomes constricted [23

23. J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. F. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors **11**, 7982–7991 (2011). [CrossRef] [PubMed]

*ε*> 1, not only can rays

*not*be directed beyond ∼30°, attempts to do so result in additional aberrations. The measured directivity for these extreme angle beams is also reduced - although directivity (calculated from peak angular-intensity) becomes a less meaningful measurement for such a deformed beam.

**Re**[

*Ẽ*(

*x,y,z*)] (also at 12 GHz).

*E*(

*x,y*)| on the back plane (using a XY scanner and tapered-waveguide nearfield-probe detector).

25. A. Mojammad-Djafari, N. Qaddoumi, and R. Zoughi, “A blind deconvolution approach for resolution enhancement of near-field microwave images,” Proc. SPIE **3816**, 274–281 (1999). [CrossRef]

*ε*

_{eff}= 1 at the boundary (thus eliminating refraction and reflection), our chosen metamaterial only reaches a minimum of

*ε*

_{eff}= 1.4. This produces aberrations in focus as the refractive effect is not accounted for by TO design machinery. The edge-refraction due to non-unity minimum-dielectric can be eliminated by using a refracting-Lüneburg lens design [26

26. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. **29**, 1358–1368 (1958). [CrossRef]

## 4. Conclusion

*new*applications. Of the material issues mentioned in this work which are detrimental to performance (polarization, anisotropy, unit-cell size, attainable dielectric extrema, etc), there are potential routes to minimize or eliminate most. For instance, a comprehensive design could either avoid geometric anisotropy (using spherical voids, for instance) or alternatively factor the known permittivity anisotropy into the initial design. Similarly, refraction at the edges could be accounted for and the TO prescription modified to maintain imaging performance. More advanced - although not necessarily more expensive - fabrication methods can extend the range of attainable dielectric values.

27. T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, S. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science **325**, 1518–1521 (2009). [CrossRef] [PubMed]

28. M. D. Goldflam, T. Driscoll, B. Chapler, O. Khatib, N. M. Jokerst, S. Palit, D. R. Smith, H. T. Kim, M. Di Ventra, and D. N. Basov, “Reconfigurable gradient index using VO_{2} memory metamaterials,” Appl. Phys. Lett. **99**, 044103 (2011). [CrossRef]

29. N. Kundtz, D. Gaultney, and D. R. Smith, “Scattering cross-section of a transformation optics-based metamaterial cloak,” New J. Phys. **12**, 043039 (2010). [CrossRef]

## References and links

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

2. | N. Kundtz and D. R. Smith, “Experimental and theoretical advances in the design of complex artificial electromagnetic media,” Ph.D. thesis (Duke University, 2009). |

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

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

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

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

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

8. | D. R. Smith, W. J. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

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

10. | W. J. Padilla, D. N. Basov, and D. R. Smith, “Negative refractive index metamaterials,” Mater. Today |

11. | 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 |

12. | V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Broadband transformation optics devices,” Materials |

13. | Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. |

14. | J. C. Maxwell, “Solutions of problems,” Cambridge Dublin Math. J. |

15. | R. Luneburg, |

16. | W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. |

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

18. | H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. |

19. | N. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasi-conformal coordinate transformations,” Phys. Rev. Lett. |

20. | J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London |

21. | D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

22. | Rozendal Associates, http://www.rozendalassociates.com/ |

23. | J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. F. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors |

24. | M. Born and E. Wolf, |

25. | A. Mojammad-Djafari, N. Qaddoumi, and R. Zoughi, “A blind deconvolution approach for resolution enhancement of near-field microwave images,” Proc. SPIE |

26. | S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. |

27. | T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, S. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science |

28. | M. D. Goldflam, T. Driscoll, B. Chapler, O. Khatib, N. M. Jokerst, S. Palit, D. R. Smith, H. T. Kim, M. Di Ventra, and D. N. Basov, “Reconfigurable gradient index using VO |

29. | N. Kundtz, D. Gaultney, and D. R. Smith, “Scattering cross-section of a transformation optics-based metamaterial cloak,” New J. Phys. |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(220.0220) Optical design and fabrication : Optical design and fabrication

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: March 23, 2012

Revised Manuscript: May 15, 2012

Manuscript Accepted: May 16, 2012

Published: May 29, 2012

**Citation**

Tom Driscoll, Guy Lipworth, Jack Hunt, Nathan Landy, Nathan Kundtz, Dimitri N. Basov, and David R. Smith, "Performance of a three dimensional transformation-optical-flattened Lüneburg lens," Opt. Express **20**, 13262-13273 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13262

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- N. Kundtz and D. R. Smith, “Experimental and theoretical advances in the design of complex artificial electromagnetic media,” Ph.D. thesis (Duke University, 2009).
- N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9, 129–132 (2010). [CrossRef]
- R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323, 366–369 (2009). [CrossRef] [PubMed]
- D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express17, 16535–16542 (2009). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14, 9794–9804 (2006). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15, 14772–14782 (2007). [CrossRef] [PubMed]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000). [CrossRef] [PubMed]
- T. Driscoll, D. N. Basov, A. F. Starr, P. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett.88, 081101 (2006). [CrossRef]
- W. J. Padilla, D. N. Basov, and D. R. Smith, “Negative refractive index metamaterials,” Mater. Today9, 28–35 (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,” Science314, 977–980 (2006). [CrossRef] [PubMed]
- V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Broadband transformation optics devices,” Materials3, 4793–4810 (2010). [CrossRef]
- Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10, 1991–1997 (2010). [CrossRef] [PubMed]
- J. C. Maxwell, “Solutions of problems,” Cambridge Dublin Math. J.8, 188–195 (1854).
- R. Luneburg, Mathematical Theory of Optics (Brown University, 1944).
- W. S. Jagger, “The optics of the spherical fish lens,” Vision Res.32, 1271–1284 (1992). [CrossRef] [PubMed]
- D. Schurig, “An aberration-free lens with zero F-number,” New J. Phys.10, 115034 (2008). [CrossRef]
- H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1, 124 (2010). [CrossRef] [PubMed]
- N. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasi-conformal coordinate transformations,” Phys. Rev. Lett.105, 193902 (2010). [CrossRef]
- J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London205, 237–288 (1906). [CrossRef]
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002). [CrossRef]
- Rozendal Associates, http://www.rozendalassociates.com/
- J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. F. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors11, 7982–7991 (2011). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1977).
- A. Mojammad-Djafari, N. Qaddoumi, and R. Zoughi, “A blind deconvolution approach for resolution enhancement of near-field microwave images,” Proc. SPIE3816, 274–281 (1999). [CrossRef]
- S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys.29, 1358–1368 (1958). [CrossRef]
- T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, S. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009). [CrossRef] [PubMed]
- M. D. Goldflam, T. Driscoll, B. Chapler, O. Khatib, N. M. Jokerst, S. Palit, D. R. Smith, H. T. Kim, M. Di Ventra, and D. N. Basov, “Reconfigurable gradient index using VO2 memory metamaterials,” Appl. Phys. Lett.99, 044103 (2011). [CrossRef]
- N. Kundtz, D. Gaultney, and D. R. Smith, “Scattering cross-section of a transformation optics-based metamaterial cloak,” New J. Phys.12, 043039 (2010). [CrossRef]

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