Abstract

This Letter presents a theory that allows graded index lenses to be mapped onto arbitrary rotationally symmetric curved surfaces. Examples of the Luneburg and Maxwell fish-eye lens are given, for numerous surfaces, always resulting in isotropic permittivity requirements. The performance of these lenses is initially illustrated with full-wave simulations utilizing a waveguide structure. A transformation of the refractive index profiles is then performed to design surface-wave lenses, where the dielectric layer is not only isotropic but also homogenous, demonstrating the applicability and ease of fabrication.

© 2014 Optical Society of America

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  1. T. A. Rhys, IEEE Trans. Antennas Propag. 18, 497 (1970).
    [CrossRef]
  2. M. Sarbort and T. Tyc, J. Opt. 14, 075705 (2012).
    [CrossRef]
  3. M. Sarbort and T. Tyc, J. Opt. 15, 125716 (2013).
    [CrossRef]
  4. R. K. Luneburg, Mathematical Theory of Optics (Cambridge University, 1964).
  5. A. S. Gutman, J. Appl. Phys. 25, 855 (1954).
    [CrossRef]
  6. U. Leonhardt, New J. Phys. 11, 093040 (2009).
    [CrossRef]
  7. J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
    [CrossRef]
  8. S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
    [CrossRef]
  9. X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
    [CrossRef]
  10. L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 2006).
  11. O. Quevedo-Teruel and Y. Hao, Opt. Lett. 38, 392 (2013).
    [CrossRef]
  12. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).
  13. M. V. Berry, J. Phys. A 8, 1952 (1975).
    [CrossRef]
  14. J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
    [CrossRef]
  15. R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
    [CrossRef]
  16. S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
    [CrossRef]
  17. D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
    [CrossRef]

2014 (1)

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

2013 (5)

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

M. Sarbort and T. Tyc, J. Opt. 15, 125716 (2013).
[CrossRef]

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

O. Quevedo-Teruel and Y. Hao, Opt. Lett. 38, 392 (2013).
[CrossRef]

2012 (2)

M. Sarbort and T. Tyc, J. Opt. 14, 075705 (2012).
[CrossRef]

D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
[CrossRef]

2011 (1)

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

2010 (1)

J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
[CrossRef]

2009 (1)

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

1975 (1)

M. V. Berry, J. Phys. A 8, 1952 (1975).
[CrossRef]

1970 (1)

T. A. Rhys, IEEE Trans. Antennas Propag. 18, 497 (1970).
[CrossRef]

1954 (1)

A. S. Gutman, J. Appl. Phys. 25, 855 (1954).
[CrossRef]

Benitez, P.

J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Phys. A 8, 1952 (1975).
[CrossRef]

Berry, S. J.

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Bosiljevac, M.

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

Caloz, C.

D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
[CrossRef]

Casaletti, M.

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

Cui, T. J.

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

Dockrey, J. A.

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Garcia-Vidal, F. J.

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

Gonzalez, J. C.

J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
[CrossRef]

Gutman, A. S.

A. S. Gutman, J. Appl. Phys. 25, 855 (1954).
[CrossRef]

Hao, Y.

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

O. Quevedo-Teruel and Y. Hao, Opt. Lett. 38, 392 (2013).
[CrossRef]

Hibbins, A. P.

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Hooper, I. R.

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

Horsley, S. A. R.

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

Itoh, T.

D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
[CrossRef]

Jackson, D. R.

D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
[CrossRef]

Josefsson, L.

L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 2006).

Leonhardt, U.

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Lockyear, M. J.

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Cambridge University, 1964).

Maci, S.

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

Martin-Cano, D.

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

McManus, T. M.

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

Minano, J. C.

J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
[CrossRef]

Minatti, G.

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

Mitchell-Thomas, R. C.

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

Persson, P.

L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 2006).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Quevedo-Teruel, O.

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

O. Quevedo-Teruel and Y. Hao, Opt. Lett. 38, 392 (2013).
[CrossRef]

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

Rhys, T. A.

T. A. Rhys, IEEE Trans. Antennas Propag. 18, 497 (1970).
[CrossRef]

Sambles, R. J.

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Sarbort, M.

M. Sarbort and T. Tyc, J. Opt. 15, 125716 (2013).
[CrossRef]

M. Sarbort and T. Tyc, J. Opt. 14, 075705 (2012).
[CrossRef]

Shen, X.

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

Tyc, T.

M. Sarbort and T. Tyc, J. Opt. 15, 125716 (2013).
[CrossRef]

M. Sarbort and T. Tyc, J. Opt. 14, 075705 (2012).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett. (1)

S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, IEEE Antennas Wirel. Propag. Lett. 10, 1499 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. A. Rhys, IEEE Trans. Antennas Propag. 18, 497 (1970).
[CrossRef]

J. Appl. Phys. (1)

A. S. Gutman, J. Appl. Phys. 25, 855 (1954).
[CrossRef]

J. Opt. (2)

M. Sarbort and T. Tyc, J. Opt. 14, 075705 (2012).
[CrossRef]

M. Sarbort and T. Tyc, J. Opt. 15, 125716 (2013).
[CrossRef]

J. Phys. A (1)

M. V. Berry, J. Phys. A 8, 1952 (1975).
[CrossRef]

New J. Phys. (2)

J. C. Minano, P. Benitez, and J. C. Gonzalez, New J. Phys. 12, 123023 (2010).
[CrossRef]

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, R. J. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013).
[CrossRef]

Phys. Rev. Lett. (1)

R. C. Mitchell-Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, and Y. Hao, Phys. Rev. Lett. 111, 213901 (2013).
[CrossRef]

Proc. IEEE (1)

D. R. Jackson, C. Caloz, and T. Itoh, Proc. IEEE 100, 2194 (2012).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, Proc. Natl. Acad. Sci. USA 110, 40 (2013).
[CrossRef]

Sci. Rep. (1)

S. A. R. Horsley, I. R. Hooper, R. C. Mitchell-Thomas, and O. Quevedo-Teruel, Sci. Rep. 4, 4876 (2014).
[CrossRef]

Other (3)

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 2006).

R. K. Luneburg, Mathematical Theory of Optics (Cambridge University, 1964).

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Figures (6)

Fig. 1.
Fig. 1.

Illustration of the ray paths that are equated for (a) the flat system and (b) an arbitrary rotationally symmetric curved surface. ni is the refractive index on each surface.

Fig. 2.
Fig. 2.

Index profiles across a normalized lens radius, R0, for (a) the Luneburg lens, and (b) the Maxwell fish-eye lens. Four surfaces are shown for each lens: the flat case, the Luneburg equivalent surface, the sphere case (Maxwell fish-eye-equivalent surface), and a cosine shape.

Fig. 3.
Fig. 3.

Luneburg lenses with a radius of 5λ0, excited with a point source on the right edge of the lens, perfectly mapped onto four different surfaces: (a) a standard flat surface, (b) the equivalent homogeneous surface, (c) a hemispherical surface, and (d) a cosine surface.

Fig. 4.
Fig. 4.

Maxwell fish-eye lens with radius of 5λ0, excited with a point source on the right edge of the lens, perfectly mapped onto four different surfaces: (a) a standard flat Maxwell fish-eye lens, (b) the equivalent homogeneous surface (sphere), (c) an equivalent surface for a Luneburg lens, and (d) a cosine surface.

Fig. 5.
Fig. 5.

(a) Achieved refractive index at 10 GHz of the surface for varying thickness of a dielectric slab with a permittivity of εr=15 which is placed over a metallic surface. (b) Cross section of the required dimensions of the Luneburg and Maxwell fish-eye lenses above the cosine-shaped ground plane.

Fig. 6.
Fig. 6.

Surface-wave propagation for the Luneburg (a), (c), (e) and Maxwell fish-eye lenses (b), (d), (f). All simulations have a cosine-shaped ground plane and a homogeneous dielectric overlayer of εr=15 with a varying thickness. Frequencies are (a), (b) 10.5 GHz, (c), (d) 11 GHz, (e), (f) 11.5 GHz.

Equations (3)

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

n1(r)2πr=n2(θ)2πR(θ)sin(θ)
n1(r)dr=n2(θ)R(θ)2+R(θ)2dθ.
n2(θ)n2(θ)=(1+rn1(r)n1(r))R(θ)2+R(θ)2R(θ)sin(θ)R(θ)cos(θ)R(θ)sin(θ).

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