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

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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).
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    [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]
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    [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).
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    [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).
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  22. E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6-17 (1978).
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  23. M. Bastiaans, "Transport equations for the Wigner distribution function," Opt. Acta 26, 1265-1272 (1979).
    [CrossRef]
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  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

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

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

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

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]

2006

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

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

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

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

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

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

P. S. J. Russell, and T. A. Birks, "Hamiltonian optics of nonuniform photonic crystals," J. Lightwave Technol. 17, 1982-1988 (1999).
[CrossRef]

1983

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

1979

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

1978

E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6-17 (1978).
[CrossRef]

1968

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
[CrossRef]

1965

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

1964

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

1954

A. Gutman, "Modified Lüneburg lens," J. Appl. Phys. 25, 855-859 (1954).
[CrossRef]

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]

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.

P. S. J. Russell, and T. A. Birks, "Hamiltonian optics of nonuniform photonic crystals," J. Lightwave Technol. 17, 1982-1988 (1999).
[CrossRef]

Bravo-Abad, J.

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]

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]

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.

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]

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]

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

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]

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]

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.

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]

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]

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]

Luo, C.

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

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]

Russell, P. S. J.

P. S. J. Russell, and T. A. Birks, "Hamiltonian optics of nonuniform photonic crystals," J. Lightwave Technol. 17, 1982-1988 (1999).
[CrossRef]

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.

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]

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

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]

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]

Townes, C. H.

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

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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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