## Aperiodic subwavelength Lüneburg lens with nonlinear Kerr effect compensation |

Optics Express, Vol. 19, Issue 3, pp. 2257-2265 (2011)

http://dx.doi.org/10.1364/OE.19.002257

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### Abstract

We introduce a Lüneburg lens design where Kerr nonlinearity is used to compensate for the focal point shift caused by diffraction of a Gaussian source. A computationally efficient iterative method introduced in [Opt. Lett. **35**, 4148 (2010)] is used to provide ray diagrams in the nonlinear case and verify the focal shift compensation. We study the joint dependence of focal shift on waist size and intensity of Gaussian source, and show how to compensate spherical aberration caused by the nonlinearity by a small perturbation of the Lüneburg profile. Our results are specific to Lüneburg lens but our approach is applicable to more general cases of nonlinear nonperiodic metamaterials.

© 2011 Optical Society of America

## 1. Introduction

3. H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Lüneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antenn. Propag. **49**, 60–69 (2001). [CrossRef]

4. C. S. Liang, D. A. Streater, J.-M. Jin, E. Dunn, and T. Rozendal, “A quantitative study of Lüneburg-lens reflectors,” IEEE Antennas Propag. Mag. **47**, 30–42 (2005). [CrossRef]

5. N. A. Mortensen, O. Sigmund, and O. Breinbjerg, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. **4**, 09008 (2009). [CrossRef]

8. J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. **25**, 2539–2546 (2007). [CrossRef]

9. D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics **4**, 323–328 (2010). [CrossRef]

14. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

15. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

16. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A **27**, 3135–3145 (1983). [CrossRef]

17. G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E **66**, 046608 (2002). [CrossRef]

1. H. Gao, L. Tian, B. Zhang, and G. Barbastathis, “Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function,” Opt. Lett. **35**, 4148–4150 (2010). [CrossRef] [PubMed]

20. P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. **17**, 1982–1988 (1999). [CrossRef]

21. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. **58**, 1256–1259 (1968). [CrossRef]

23. M. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta **26**, 1265–1272 (1979). [CrossRef]

25. H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, “Nonlinear Kerr effect aperiodic Lüneburg lens,” in *Optical MEMS and Nanophotonics*, (IEEE Photonics Society, 2010), Paper Th1-2, pp. 179–180. [CrossRef]

26. D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977 (2006). [CrossRef] [PubMed]

27. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature **455**, 376–379 (2008). [CrossRef] [PubMed]

## 2. Setup

*n*

_{0}is the ambient index outside the lens,

*R*is the radius of the lens, and

*r*is the distance to the lens center. To investigate the focal shift due to diffraction and focal shift compensation by the Kerr effect, a Lüneburg lens illuminated by a Gaussian beam is studied (Fig. 1). The radius of the lens is

*R*, and distances between the beam waist of input/output and the left/right edge of the lens are

*z*and

*z*′, respectively.

*n*= 3.46) centered at each square lattice with lattice constant

*a*

_{0}=

*λ*/8, where

*λ*= 1550 nm is the free space wavelength. The rod radii profile is

*a*

_{1}= 0.367

*a*

_{0},

*a*

_{2}= −0.101

*a*

_{0}and

*R*= 30

*a*

_{0}. The ambient medium is air (

*n*= 1). Since

*λ*is much larger than the lattice constant, the nanostructure could be treated as an effective medium. In this case, effective refractive indices are locally-modulated by controlling the radii of the rods, mimicking the Lüneburg lens index profile. Effective refractive index distribution of this lens is

*n*

_{eff}_{1}= 1.9239 and

*n*

_{eff}_{2}= −0.0678. Throughout this paper, our discussions are based on this 2D subwavelength aperiodic nanostructure example.

## 3. Method

1. H. Gao, L. Tian, B. Zhang, and G. Barbastathis, “Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function,” Opt. Lett. **35**, 4148–4150 (2010). [CrossRef] [PubMed]

1. H. Gao, L. Tian, B. Zhang, and G. Barbastathis, “Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function,” Opt. Lett. **35**, 4148–4150 (2010). [CrossRef] [PubMed]

23. M. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta **26**, 1265–1272 (1979). [CrossRef]

19. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612 (2004). [CrossRef]

*ω*is frequency, and

**x**and

**k**are position and momentum, respectively, along the path and

*σ*parameterizes the ray trajectories. Here

*ω*(

**x**,

**k**) is a function determined by the dispersion diagram of the unit cell. In the third step, a new estimate of the intensity distribution is generated from the ray trajectories as the projections of the WDF along the momentum direction. Besides, a new estimate of refractive index distribution is also calculated based on Kerr effect for use in the next iteration.

## 4. Simulation results

28. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. **181**, 687–702 (2010). [CrossRef]

*a*

_{0}and peak intensity

*I*

_{0}is illuminating the subwavelength Lüneburg lens from the left. Without the Kerr effect, i.e. in the linear Lüneburg lens case, a focal point shift to the right can be clearly seen as shown in Figs. 4(a) and 4(b). The focal shift is due to diffraction of the input Gaussian source as compared with the ideal plane wave input case. Therefore, with a Gaussian beam input, the property that Lüneburg lens produces a geometrical focal point exactly at the opposite edge of the lens is invalid.

^{−14}cm

^{2}/W [10]. For the iterative method simulation, grid size used is 0.03

*a*

_{0}× 0.03

*a*

_{0}. After six iterations, the beam propagation profile converges. For the FDTD simulation, grid size used is 0.083

*a*

_{0}× 0.083

*a*

_{0}.

29. A. Gutman, “Modified Lüneburg lens,” J. Appl. Phys. **25**, 855–859 (1954). [CrossRef]

*z*= 2

*R*. For plane wave illumination, which is an extreme case of Gaussian source with infinite size of waist, the focal point is at the opposite edge when the intensity

*I*approaches 0. Increasing the intensity gradually moves the focal point inside the lens, due to the resulting higher Kerr nonlinearity. For the point source case, which is another extreme case of Gaussian source where the waist approaches zero, focal point without nonlinearity is at

*z*′ = 0.5

*R*. This matches the prediction from geometrical imaging condition for Lüneburg lens under paraxial approximation, i.e.

*zz*′ =

*R*

^{2}. Similar to the plane wave case, the focal point also moves towards the left when the intensity is higher. However, it will not reach the edge within the intensity range discussed here since diffraction is too large to be compensated. Between the two extremes is Gaussian incidence case. We show an example of a Gaussian beam with waist of 9

*a*

_{0}. The nonlinear effect exactly cancels the diffraction when

*I*= 2.8

*I*

_{0}, i.e. the output beam focuses at

*z*′ = 0. In general, the smaller the beam waist is, the higher the intensity is required for the cancellation between diffraction and nonlinearity.

## 5. Modified Lüneburg lens

*a*

_{0}and peak intensity

*I*

_{0}is incident on the nonlinear Lüneburg lens. The beam waist is chosen so that it is much larger than the diameter of the lens. The spherical aberration is shown by the ray tracing results in Fig. 6(a). The aberration is resulting from non-uniform nonlinearity-induced refractive index gradient. As a result, rays near the center experience more nonlinearity-induced bending than the off-center ones, thus resulting in in spherical aberration.

## 6. Conclusion

## Acknowledgments

## References and links

1. | H. Gao, L. Tian, B. Zhang, and G. Barbastathis, “Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function,” Opt. Lett. |

2. | R. K. Lüneburg, Mathematical Theory of Optics (Brown U.P., Providence, 1944). |

3. | H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Lüneburg and two-shell lens antennas: radiation characteristics and design optimization,” IEEE Trans. Antenn. Propag. |

4. | C. S. Liang, D. A. Streater, J.-M. Jin, E. Dunn, and T. Rozendal, “A quantitative study of Lüneburg-lens reflectors,” IEEE Antennas Propag. Mag. |

5. | N. A. Mortensen, O. Sigmund, and O. Breinbjerg, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. |

6. | M. Takahashi and H. Goto, |

7. | M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. |

8. | J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. |

9. | D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics |

10. | R. Boyd, Nonlinear optics , (3rd ed.) (Academic, 2008). |

11. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. |

12. | P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. |

13. | Y. Kivshar and G. Agrawal, Optical solitons , (Academic, 2003). |

14. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

15. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature |

16. | D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A |

17. | G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E |

18. | K. B. Wolf, |

19. | Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E |

20. | P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. |

21. | A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. |

22. | E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. |

23. | M. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta |

24. | S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in |

25. | H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, “Nonlinear Kerr effect aperiodic Lüneburg lens,” in |

26. | D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

27. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

28. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. |

29. | A. Gutman, “Modified Lüneburg lens,” J. Appl. Phys. |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(190.0190) Nonlinear optics : Nonlinear optics

(190.4360) Nonlinear optics : Nonlinear optics, devices

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 10, 2010

Revised Manuscript: January 17, 2011

Manuscript Accepted: January 17, 2011

Published: January 24, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Hanhong Gao, Satoshi Takahashi, Lei Tian, and George Barbastathis, "Aperiodic subwavelength Lüneburg lens with nonlinear Kerr effect compensation," Opt. Express **19**, 2257-2265 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2257

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### References

- H. Gao, L. Tian, B. Zhang, and G. Barbastathis, "Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function," Opt. Lett. 35, 4148-4150 (2010). [CrossRef] [PubMed]
- R. K. Lüneburg, Mathematical Theory of Optics (Brown U.P., Providence, 1944).
- H. Mosallaei, and Y. Rahmat-Samii, "Nonuniform Lüneburg and two-shell lens antennas: radiation characteristics and design optimization," IEEE Trans. Antenn. Propag. 49, 60-69 (2001). [CrossRef]
- C. S. Liang, D. A. Streater, J.-M. Jin, E. Dunn, and T. Rozendal, "A quantitative study of Lüneburg-lens reflectors," IEEE Antennas Propag. Mag. 47, 30-42 (2005). [CrossRef]
- N. A. Mortensen, O. Sigmund, and O. Breinbjerg, "Prospects for poor-man’s cloaking with low-contrast all dielectric optical elements," J. Eur. Opt. Soc. Rapid Publ. 4, 09008 (2009). [CrossRef]
- M. Takahashi, and H. Goto, Progress in Nonlinear Optics Research (Nova Science Publishers, 2008).
- M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2003). [CrossRef]
- J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, "Modeling nonlinear optical phenomena in nanophotonics," J. Lightwave Technol. 25, 2539-2546 (2007). [CrossRef]
- D. V. Dylov, and J. W. Fleischer, "Nonlinear self-filtering of noisy images via dynamical stochastic resonance," Nat. Photonics 4, 323-328 (2010). [CrossRef]
- R. Boyd, Nonlinear optics, (3rd ed.) (Academic, 2008).
- R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-Trapping of Optical Beams," Phys. Rev. Lett. 13, 479 (1964). [CrossRef]
- P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965). [CrossRef]
- Y. Kivshar, and G. Agrawal, Optical solitons, (Academic, 2003).
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behavior in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983). [CrossRef]
- G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential," Phys. Rev. E 66, 046608 (2002). [CrossRef]
- K. B. Wolf, Geometric optics on phase space, (Springer, 2004).
- Y. Jiao, S. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method," Phys. Rev. E 70, 036612 (2004). [CrossRef]
- P. S. J. Russell, and T. A. Birks, "Hamiltonian optics of nonuniform photonic crystals," J. Lightwave Technol. 17, 1982-1988 (1999). [CrossRef]
- A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968). [CrossRef]
- E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6-17 (1978). [CrossRef]
- M. Bastiaans, "Transport equations for the Wigner distribution function," Opt. Acta 26, 1265-1272 (1979). [CrossRef]
- S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, "Design and fabrication of dielectric nanostructured Lüneburg lens in optical frequencies," in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1-1, pp. 177-178.
- H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, "Nonlinear Kerr effect aperiodic Lüneburg lens," in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1-2, pp. 179-180. [CrossRef]
- D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977 (2006). [CrossRef] [PubMed]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-379 (2008). [CrossRef] [PubMed]
- A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, "MEEP: A flexible free software package for electromagnetic simulations by the FDTD method," Comput. Phys. Commun. 181, 687-702 (2010). [CrossRef]
- A. Gutman, "Modified Lüneburg lens," J. Appl. Phys. 25, 855-859 (1954). [CrossRef]

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