Abstract

We present a three-dimensional (3D) Luneburg lens design scheme that employs non-resonant spherical scatterers as inclusions in a host medium for the manipulation of electromagnetic waves. The underlying principle is that the volume fraction of the inclusion scatterers can be varied spatially so as to control the effective permittivity for the desired permittivity profile. Specifically, to achieve desired volume fraction values, simple cubic packing, hexagonal close packing and random packing methods were used for scatterer distribution. The proposed analysis features the plasmonic inclusions as a rational alternative for dielectric inclusions to produce a desired effective value of the permittivity in optics. We demonstrate the applicability of the proposed scheme by employing it to design and simulate Luneburg lens (both in microwave and optics) for beam steering applications. The design leads to polarisation independent functionality in the plane tangent to the lens and yields high antenna gain. The scheme provides a useful means to realize many disruptive applications ranging from the microwaves to optics.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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    [Crossref] [PubMed]

2017 (1)

2016 (1)

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

2014 (2)

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

2013 (1)

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

2012 (4)

2011 (4)

2010 (1)

C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Transactions on Antennas Propag. 58, 3055–3059 (2010).
[Crossref]

2009 (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

2008 (1)

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

2007 (3)

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

L. Xue and V. Fusco, “24 GHz automotive radar planar Luneburg lens,” IET Microwaves, Antennas & Propag. 1, 624–628 (2007).
[Crossref]

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

2006 (2)

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

S. Torquato, O. Uche, and F. Stillinger, “Random sequential addition of hard spheres in high Euclidean dimensions,” Phys. Rev. E 74, 061308 (2006).
[Crossref]

2005 (2)

A. Donev, S. Torquato, and F. H. Stillinger, “Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details,” J. Comput. Phys. 202, 737–764 (2005).
[Crossref]

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[Crossref]

2003 (1)

S. Rondineau, M. Himdi, and J. Sorieux, “A sliced spherical Luneburg lens,” IEEE Antennas Propag. Lett. 2, 163–166 (2003).
[Crossref]

2002 (1)

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[Crossref] [PubMed]

2001 (1)

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, “Optical properties of thin films of Au@SiO2 particles,” The J. Phys. Chem. B 105, 3441–3452 (2001).
[Crossref]

1978 (1)

R. Ruppin, “Validity range of the Maxwell-Garnett theory,” Phys. Status Solidi (b) 87, 619–624 (1978).
[Crossref]

1977 (1)

D. McKenzie and R. McPhedran, “Exact modelling of cubic lattice permittivity and conductivity,” Nature 265, 128 (1977).
[Crossref]

1966 (1)

B. Widom, “Random sequential addition of hard spheres to a volume,” The J. Chem. Phys. 44, 3888–3894 (1966).
[Crossref]

1904 (1)

J. C. M. Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Sect A 3, 385–420 (1904).
[Crossref]

1892 (1)

L. Rayleigh, “LVI. on the influence of obstacles arranged in rectangular order upon the properties of a medium,” The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 34, 481–502 (1892).
[Crossref]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Basov, D. N.

Berry, S.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Bosiljevac, M.

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

Caminita, F.

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

Casaletti, M.

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

Chang, K.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Chen, K. C.

Coq, L. L.

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

Demetriadou, A.

Diaz, A.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Dockrey, J.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Donev, A.

A. Donev, S. Torquato, and F. H. Stillinger, “Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details,” J. Comput. Phys. 202, 737–764 (2005).
[Crossref]

A. Donev, “Jammed packings of hard particles,” Ph.D. thesis, Princeton University (2006).

Dong, X.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Driscoll, T.

Duan, X.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Dyke, A.

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

Dyke, H.

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

Falco, A. D.

Fuchs, B.

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

Funston, A. M.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Fusco, V.

L. Xue and V. Fusco, “24 GHz automotive radar planar Luneburg lens,” IET Microwaves, Antennas & Propag. 1, 624–628 (2007).
[Crossref]

Gabrielli, L. H.

Gajic, R.

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[Crossref] [PubMed]

García de Abajo, F. J.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Sect A 3, 385–420 (1904).
[Crossref]

Gbele, K.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Gehm, M. E.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Grbic, A.

C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Transactions on Antennas Propag. 58, 3055–3059 (2010).
[Crossref]

Guérin, C. A.

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[Crossref]

Hao, Y.

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

A. Demetriadou and Y. Hao, “Slim Luneburg lens for antenna applications,” Opt. Express 19, 19925–19934 (2011).
[Crossref] [PubMed]

Haq, S.

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

Hibbins, A. P.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Himdi, M.

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

S. Rondineau, M. Himdi, and J. Sorieux, “A sliced spherical Luneburg lens,” IEEE Antennas Propag. Lett. 2, 163–166 (2003).
[Crossref]

Horsley, S.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Hunt, J.

Jylhä, L.

L. Jylhä, “Modeling of electrical properties of composites,” Ph.D. thesis, Helsinki University of Technology (2008).

Kant, G. W.

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

Kehn, M. N. M.

Kehr, S. C.

Khin, C. F.

Khoo, I. C.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Kildal, P.

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

Kubo, S.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Kundtz, N.

Lafond, O.

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

Landy, N.

Leonhardt, U.

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Liang, M.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Lipson, M.

Lipworth, G.

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

Liz-Marzan, L. M.

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, “Optical properties of thin films of Au@SiO2 particles,” The J. Phys. Chem. B 105, 3441–3452 (2001).
[Crossref]

Liz-Marzán, L. M.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Lockyear, M. J.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Luneburg, R. K.

R. K. Luneburg and H. Max, Mathematical theory of optics(Brown University, 1944).

Maaskant, R.

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

Maci, S.

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

Mallet, P.

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[Crossref]

Mallouk, T. E.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Mateo-Segura, C.

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

Max, H.

R. K. Luneburg and H. Max, Mathematical theory of optics(Brown University, 1944).

Mayer, T. S.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

McKenzie, D.

D. McKenzie and R. McPhedran, “Exact modelling of cubic lattice permittivity and conductivity,” Nature 265, 128 (1977).
[Crossref]

McPhedran, R.

D. McKenzie and R. McPhedran, “Exact modelling of cubic lattice permittivity and conductivity,” Nature 265, 128 (1977).
[Crossref]

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

Mittra, R.

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

Mulvaney, P.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, “Optical properties of thin films of Au@SiO2 particles,” The J. Phys. Chem. B 105, 3441–3452 (2001).
[Crossref]

Myroshnychenko, V.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Naeem, M.

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

Ng, W.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[Crossref] [PubMed]

Novo, C.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Park, Q.-H.

Pastoriza-Santos, I.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Pendry, J. B.

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

Pfeiffer, C.

C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Transactions on Antennas Propag. 58, 3055–3059 (2010).
[Crossref]

Rayleigh, L.

L. Rayleigh, “LVI. on the influence of obstacles arranged in rectangular order upon the properties of a medium,” The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 34, 481–502 (1892).
[Crossref]

Rodríguez-Fernández, J.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Rondineau, S.

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

S. Rondineau, M. Himdi, and J. Sorieux, “A sliced spherical Luneburg lens,” IEEE Antennas Propag. Lett. 2, 163–166 (2003).
[Crossref]

Ruppin, R.

R. Ruppin, “Validity range of the Maxwell-Garnett theory,” Phys. Status Solidi (b) 87, 619–624 (1978).
[Crossref]

Sambles, J. R.

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Schurig, D.

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

Sebastian, M. T.

M. T. Sebastian, Dielectric Materials for Wireless Communication. (Elsevier, 2008).

Sentenac, A.

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[Crossref]

Sipus, Z.

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

Smith, D. R.

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

Sorieux, J.

S. Rondineau, M. Himdi, and J. Sorieux, “A sliced spherical Luneburg lens,” IEEE Antennas Propag. Lett. 2, 163–166 (2003).
[Crossref]

Stillinger, F.

S. Torquato, O. Uche, and F. Stillinger, “Random sequential addition of hard spheres in high Euclidean dimensions,” Phys. Rev. E 74, 061308 (2006).
[Crossref]

Stillinger, F. H.

A. Donev, S. Torquato, and F. H. Stillinger, “Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details,” J. Comput. Phys. 202, 737–764 (2005).
[Crossref]

Tang, Y.

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Torquato, S.

S. Torquato, O. Uche, and F. Stillinger, “Random sequential addition of hard spheres in high Euclidean dimensions,” Phys. Rev. E 74, 061308 (2006).
[Crossref]

A. Donev, S. Torquato, and F. H. Stillinger, “Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details,” J. Comput. Phys. 202, 737–764 (2005).
[Crossref]

Uche, O.

S. Torquato, O. Uche, and F. Stillinger, “Random sequential addition of hard spheres in high Euclidean dimensions,” Phys. Rev. E 74, 061308 (2006).
[Crossref]

Ung, T.

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, “Optical properties of thin films of Au@SiO2 particles,” The J. Phys. Chem. B 105, 3441–3452 (2001).
[Crossref]

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Vasic, B.

Widom, B.

B. Widom, “Random sequential addition of hard spheres to a volume,” The J. Chem. Phys. 44, 3888–3894 (1966).
[Crossref]

Xin, H.

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

Xue, L.

L. Xue and V. Fusco, “24 GHz automotive radar planar Luneburg lens,” IET Microwaves, Antennas & Propag. 1, 624–628 (2007).
[Crossref]

Yang, J. W.

Yang, Y.-C.

Yoo, S.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Zhao, Y.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Zhao, Z.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Zheng, M.

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Zolla, F.

Chem. Soc. Rev. (1)

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

IEEE Antennas Propag. Lett. (1)

S. Rondineau, M. Himdi, and J. Sorieux, “A sliced spherical Luneburg lens,” IEEE Antennas Propag. Lett. 2, 163–166 (2003).
[Crossref]

IEEE Transactions on Antennas Propag. (5)

M. Liang, W. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg lens antenna fabricated by polymer jetting rapid prototyping,” IEEE Transactions on Antennas Propag. 62, 1799–1807 (2014).
[Crossref]

C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Transactions on Antennas Propag. 58, 3055–3059 (2010).
[Crossref]

M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, “Non-uniform metasurface Luneburg lens antenna design,” IEEE Transactions on Antennas Propag. 60, 4065–4073 (2012).
[Crossref]

C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg lens via transformation optics for directive antenna applications,” IEEE Transactions on Antennas Propag. 62, 1945–1953 (2014).
[Crossref]

B. Fuchs, L. L. Coq, O. Lafond, S. Rondineau, and M. Himdi, “Design optimization of multishell Luneburg lenses,” IEEE Transactions on Antennas Propag. 55, 283–289 (2007).
[Crossref]

IET Microwaves, Antennas & Propag. (1)

L. Xue and V. Fusco, “24 GHz automotive radar planar Luneburg lens,” IET Microwaves, Antennas & Propag. 1, 624–628 (2007).
[Crossref]

J. Comput. Phys. (1)

A. Donev, S. Torquato, and F. H. Stillinger, “Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details,” J. Comput. Phys. 202, 737–764 (2005).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

Y. Zhao, Y. Zhang, M. Zheng, X. Dong, X. Duan, and Z. Zhao, “Three-dimensional Luneburg lens at optical frequencies,” Laser Photonics Rev. 672, 665–672 (2016).
[Crossref]

Nano Lett. (1)

S. Kubo, A. Diaz, Y. Tang, T. S. Mayer, I. C. Khoo, and T. E. Mallouk, “Tunability of the refractive index of gold nanoparticle dispersions,” Nano Lett. 7, 3418–3423 (2007).
[Crossref] [PubMed]

Nat. Mater. (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and eaton lenses,” Nat. Nanotechnol. 6, 151 (2011).
[Crossref] [PubMed]

Nature (1)

D. McKenzie and R. McPhedran, “Exact modelling of cubic lattice permittivity and conductivity,” Nature 265, 128 (1977).
[Crossref]

Opt. Express (6)

Philos. Trans. R. Soc. London, Sect A (1)

J. C. M. Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Sect A 3, 385–420 (1904).
[Crossref]

Phys. Rev. B (2)

P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B 72, 014205 (2005).
[Crossref]

J. Dockrey, M. J. Lockyear, S. Berry, S. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013).
[Crossref]

Phys. Rev. E (1)

S. Torquato, O. Uche, and F. Stillinger, “Random sequential addition of hard spheres in high Euclidean dimensions,” Phys. Rev. E 74, 061308 (2006).
[Crossref]

Phys. Rev. Lett. (1)

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002).
[Crossref] [PubMed]

Phys. Status Solidi (b) (1)

R. Ruppin, “Validity range of the Maxwell-Garnett theory,” Phys. Status Solidi (b) 87, 619–624 (1978).
[Crossref]

Science (1)

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

The J. Chem. Phys. (1)

B. Widom, “Random sequential addition of hard spheres to a volume,” The J. Chem. Phys. 44, 3888–3894 (1966).
[Crossref]

The J. Phys. Chem. B (1)

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, “Optical properties of thin films of Au@SiO2 particles,” The J. Phys. Chem. B 105, 3441–3452 (2001).
[Crossref]

The London, Edinburgh, and Dublin Philos. Mag. J. Sci. (1)

L. Rayleigh, “LVI. on the influence of obstacles arranged in rectangular order upon the properties of a medium,” The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 34, 481–502 (1892).
[Crossref]

Other (5)

R. K. Luneburg and H. Max, Mathematical theory of optics(Brown University, 1944).

M. Naeem, R. Maaskant, G. W. Kant, P. Kildal, and R. Mittra, “The method of equivalent dipole moments (MEDM) combined with CBFM for the fast and accurate solution of dielectric scattering problems,” in 2011 International Conference on Electromagnetics in Advanced Applications, (2011), pp. 1013–1016.
[Crossref]

L. Jylhä, “Modeling of electrical properties of composites,” Ph.D. thesis, Helsinki University of Technology (2008).

A. Donev, “Jammed packings of hard particles,” Ph.D. thesis, Princeton University (2006).

M. T. Sebastian, Dielectric Materials for Wireless Communication. (Elsevier, 2008).

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Figures (11)

Fig. 1
Fig. 1 Selection rules for the effective permittivity modelling. A desired value of the effective permittivity can be achieved by using either dielectric inclusions (in microwaves) or plasmonic inclusions (in optics). Dielectric inclusions require higher volume fraction as compared to that of plasmonic inclusions. Both the dielectric and plasmonic inclusions are considered lossless.
Fig. 2
Fig. 2 Illustration of the subtraction step of the innermost layer. Scatterers are first generated in a cube and then discarded if located outside the layer. Building example based on (a) simple cubic packing and (b) hexagonal close packing. The insets demonstrate the fully generated scatterers in a cube.
Fig. 3
Fig. 3 Visualization of 10 GHz Luneburg lens cut in half and viewed from top. (a) Conventional Luneburg lens using 6 dielectric layers. Homogenized Luneburg lens with 6 composite layers based on (b) simple cubic packing, (c) hexagonal close packing and (d) random packing. The insets demonstrate the three designs from another angle.
Fig. 4
Fig. 4 Calculated achieved permittivity based on the MG formula. The lens radius is 2λ. The inclusion scatterer radius is varied from 0.062λ to 0.041λ and has a relative permittivity of 37.1. (a) Simple cubic packing, (b) hexagonal close packing and (c) random packing. (d) Calculated achieved permittivity comparison after calibration.
Fig. 5
Fig. 5 Component y of electric field values when excited by a y-polarised 10 GHz Hertzian dipole placed at the focal point (x = 0, y = 0, z = −0.06) in metre. Three calibrated approaches by (a) simple cubic packing, (b) hexagonal close packing and (c) random packing with inclusion scatterer radius equals 0.041λ. d) Electric field values of a conventional ideal Luneburg lens simulated using the CST studio. Two uncalibrated random schemes with inclusion scatterer radius at (e)0.062λ and (f) 0.050λ.
Fig. 6
Fig. 6 Far-field gain plot of 10 GHz microwave Luneburg lens based on the (a) simple cubic packing, (b) hexagonal close packing, (c) random packing and (d) conventional dielectric layers. A y-polarized dipole is varied on the focal curve in 7 steps (from 180° to 360°) in the azimuthal plane. Labels on the outer circle indicate the ϕ variation (in degrees) in the azimuthal plane, gain labels on circles are in dBi, and the gain values below 0 have been suppressed.
Fig. 7
Fig. 7 Calculated achieved permittivity of the three calibrated approaches at 450 THz. The lens radius is 1.4 µm and the inclusion scatterer radius is 46.5 nm. The inclusion relative permittivity is -13.81 and the imaginary part of permittivity is 0.61.
Fig. 8
Fig. 8 Component y of electric field values when excited by a y-polarised 450 THz Hertzian dipole placed at the focal point (x = 0, y = 0, z = −1.34) in micrometre. Three calibrated approaches by (a) simple cubic packing, (b) hexagonal close packing and(c) random packing.
Fig. 9
Fig. 9 Far-field gain plot of optical Luneburg lens based on the (a) simple cubic packing, (b) hexagonal close packing and (c) random packing. A y-polarized dipole is varied on the focal curve in 7 steps (from 180° to 360°) in the azimuthal plane. Labels on the outer circle indicate the ϕ variation (in degrees) in the azimuthal plane, gain labels on circles are in dBi, and the gain values below 0 have been suppressed.
Fig. 10
Fig. 10 Effective permittivity change of a randomly packed lens due to dispersion of the gold nanopartiles from 300 THz to 450 THz. The lens radius is fixed to 2λ at 450 THz. The inclusion scatterer radius is 46.5 nm; the inclusion relative permittivity is calculated from the Drude model. (a) Calculated permittivity values of the random lens across 6 layers and (b) effective permittivity of gold nanoparticles against frequency.
Fig. 11
Fig. 11 Dipole gain (from 300 THz to 450 THz in 7 equal steps) in the presence of a randomly packed optical Luneburg lens. A y-polarized dipole is placed in the azimuthal plane at ϕ = 360°.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ε r = n 2 μ r = 2 ( r R ) 2 ,
ε n = 2 ( r n R ) 2 ,
ε eff = ε h + 3 f ε h ε i ε h ε i + 2 ε h f ( ε i ε h )
f = ε eff ε h ε eff + 2 ε h ε i + 2 ε h ε i ε h .
ε r ( ω ) ε b ω p 2 ω ( ω + i τ 1 ) ,

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