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 OSA

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  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]
  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. 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]
  6. M. Takahashi and H. Goto, Progress in Nonlinear Optics Research (Nova Science Publishers, 2008).
  7. M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003).
    [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]
  10. R. Boyd, Nonlinear optics, (3rd ed.) (Academic, 2008).
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    [CrossRef]
  12. P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  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 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]
  18. K. B. Wolf, Geometric optics on phase space, (Springer, 2004).
  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]
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    [CrossRef]
  24. S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1-1, pp. 177–178.
  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]
  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]
  29. A. Gutman, “Modified Lüneburg lens,” J. Appl. Phys. 25, 855–859 (1954).
    [CrossRef]

2010 (3)

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]

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

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]

2009 (1)

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]

2008 (2)

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

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]

2007 (1)

2006 (1)

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]

2005 (1)

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]

2004 (1)

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]

2003 (4)

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]

M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003).
[CrossRef]

2002 (1)

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]

2001 (1)

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]

1999 (1)

1983 (1)

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

1979 (1)

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

1978 (1)

1968 (1)

1965 (1)

P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

1954 (1)

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

1944 (1)

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

Agrawal, G.

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

Alfimov, G. L.

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]

Anderson, D.

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

Barbastathis, G.

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]

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg 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]

Bartal, G.

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]

Bastiaans, M.

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

Bermel, P.

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]

Birks, T. A.

Boyd, R.

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

Bravo-Abad, J.

Breinbjerg, O.

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]

Chang, C.

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1-1, pp. 177–178.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Christodoulides, D. N.

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]

Cummer, S. A.

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]

Dunn, E.

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]

Dylov, D. V.

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

Efremidis, N. K.

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]

Fan, S.

Fleischer, J. W.

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

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]

Gao, H.

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]

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]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Genov, D. A.

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]

Goto, H.

M. Takahashi and H. Goto, Progress in Nonlinear Optics Research (Nova Science Publishers, 2008).

Gutman, A.

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

Ibanescu, M.

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]

Jiao, Y.

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]

Jin, J.-M.

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]

Joannopoulos, J.

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]

M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

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]

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]

Justice, B.

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]

Kelley, P. L.

P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Kevrekidis, P. G.

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]

Kivshar, Y.

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

Konotop, V. V.

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]

Lederer, F.

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

Liang, C. S.

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]

Lüneburg, R. K.

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

Luo, C.

Miller, D. A. B.

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]

Mock, J.

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]

Mortensen, N. A.

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]

Mosallaei, H.

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]

Oskooi, A. F.

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]

Pendry, J.

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]

Rahmat-Samii, Y.

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]

Roundy, D.

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]

Rozendal, T.

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]

Russel, P. S. J.

Salerno, M.

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]

Schurig, D.

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]

Segev, M.

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]

Sigmund, O.

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]

Silberberg, Y.

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

Smith, D.

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]

Soljacic, M.

Starr, A.

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]

Streater, D. A.

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]

Takahashi, M.

M. Takahashi and H. Goto, Progress in Nonlinear Optics Research (Nova Science Publishers, 2008).

Takahashi, S.

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg 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]

Tian, L.

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]

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]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Ulin-Avila, E.

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]

Valentine, J.

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]

Walther, A.

Wolf, E.

Wolf, K. B.

K. B. Wolf, Geometric optics on phase space, (Springer, 2004).

Yang, S. Y.

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies,” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1-1, pp. 177–178.

Zentgraf, T.

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]

Zhang, B.

Zhang, S.

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).
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Figures (6)

Fig. 1
Fig. 1

Simulation setup, a Lüneburg lens with a Gaussian source illumination. The black circle encloses the Lüneburg lens region, and the red curved lines denote the general profile of the Gaussian beam propagation. Blue dashed lines denote the waist of input/output Gaussian beams.

Fig. 2
Fig. 2

Subwavelength aperiodic Lüneburg lens structure.

Fig. 3
Fig. 3

Block diagram of the iterative method based on Hamiltonian ray tracing and the Wigner distribution function.

Fig. 4
Fig. 4

Subwavelength Lüneburg lens with a Gaussian source of beam waist 36a0. Linear case: intensity profile generated from nonlinear iterative method (a) and beam propagation from FDTD method (b). Both show the focal shift beyond the edge. Nonlinear case: intensity profile generated from nonlinear iterative method (c) and beam propagation from FDTD method (d). Both show the shift is compensated by Kerr effect. (e) Effective refractive index difference between the nonlinear and linear subwavelength aperiodic Lüneburg lens. (f) Comparison of the cross sections of the focal planes between the linear and nonlinear Lüneburg lens.

Fig. 5
Fig. 5

Relationship between focal points, sources and optical intensities. (a) Simulation setup and beam propagation envelope. (b) Focal point location z′ resulting from different sources and intensities. The input beam waist is always located at z = 2R. In both figures, the following color scheme is adopted. Red dashed line: point source case; red dash-dot line: ideal plane wave case; blue dashed line: Gaussian source with waist 9a0. Horizontal blue line corresponds to the right edge of the lens.

Fig. 6
Fig. 6

Hamiltonian ray-tracing results near the focal point for the original nonlinear Lüneburg lens (a) and the modified nonlinear Lüneburg lens (b). (c) The radii distributions of the dielectric rods across the diameter of both subwavelength nanostructures. (d) Comparison of the cross sections of the focal planes between the original and modified nonlinear Lüneburg lens. The beam waist is chosen as 125a0.

Equations (1)

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d x d σ = ω ( x , k ) k , d k d σ = ω ( x , k ) x ,

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