Abstract

Light propagation obeys Fermat’s principle, and an important inference of Fermat’s principle is the optical Lagrange equation, from which the light trace can be determined with a given refractive index. Here, we consider the inverse problem of how to derive the refractive index distribution of a planar geometric optical system once the trace is predetermined. Based on the optical Lagrange equation, we propose a dynamic equation model which associates the refractive index with the light trace. With the consideration of a certain trace, we illustrate the process of solving the partial differential equation of refractive index through first integral method. By setting the distribution function of a gradient-refractive-index (GRIN) medium, one can control the light traveling along a desirable curve, adjust the incoming and outgoing rays, and also use the trace to paint geometrics. This method develops the Lagrangian optics in the application of ray dynamic system design, such as lens, beam splitter, metasurface and optical waveguide. It provides a theoretical guidance to manipulate the ray in a GRIN medium.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Multiple Sets of Fringes in the Michelson Interferometer

I. Walerstein and R. A. Woodson
J. Opt. Soc. Am. 26(6) 267-271 (1936)

Graded-index planar waveguide solar concentrator

Sébastien Bouchard and Simon Thibault
Opt. Lett. 39(5) 1197-1200 (2014)

Huygens–Feynman–Fresnel principle as the basis of applied optics

Andrey V. Gitin
Appl. Opt. 52(31) 7419-7434 (2013)

References

  • View by:
  • |
  • |
  • |

  1. R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
    [Crossref]
  2. X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
    [Crossref] [PubMed]
  3. A. I. Hernandez-Serrano, M. Weidenbach, S. F. Busch, M. Koch, and E. Castro-camus, “Fabrication of gradient-refractive-index lenses for terahertz applications by three-dimensional printing,” J. Opt. Soc. Am. B 33(5), 928 (2016).
    [Crossref]
  4. S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
    [Crossref]
  5. J. D. Ellis, D. R. Brooks, K. T. Wozniak, G. A. Gandara-Montano, E. G. Fox, K. J. Tinkham, S. C. Butler, L. A. Zheleznyak, M. R. Buckley, P. D. Funkenbusch, and W. H. Knox, “Manufacturing of Gradient Index Lenses for Ophthalmic Applications.” Design and Fabrication Congress 2017 (IODC, Freeform, OFT) OW1B–3 (2017).
    [Crossref]
  6. H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
    [Crossref]
  7. E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
    [Crossref]
  8. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
    [Crossref] [PubMed]
  9. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
    [Crossref] [PubMed]
  10. D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
    [Crossref] [PubMed]
  11. R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid, Watasenia scintillans,” Vision Res. 44(18), 2129–2134 (2004).
    [Crossref] [PubMed]
  12. D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
    [Crossref] [PubMed]
  13. W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999).
    [Crossref] [PubMed]
  14. S. Ji, M. Ponting, R. S. Lepkowicz, A. Rosenberg, R. Flynn, G. Beadie, and E. Baer, “A bio-inspired polymeric gradient refractive index (GRIN) human eye lens,” Opt. Express 20(24), 26746–26754 (2012).
    [Crossref] [PubMed]
  15. O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson, and V. I. Smirnov, “High-efficiency bragg gratings in photothermorefractive glass,” Appl. Opt. 38(4), 619–627 (1999).
    [Crossref] [PubMed]
  16. O. M. Efimov, L. B. Glebov, V. I. Smirnov, and L. Glebova, “Process for production of high efficiency volume diffractive elements in photo-thermo-refractive glass,” US 09/648,293, (2003).
  17. L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” US 08/017,034 (1994).
  18. T. H. Tomkinson, J. L. Bentley, M. K. Crawford, C. J. Harkrider, D. T. Moore, and J. L. Rouke, “Rigid endoscopic relay systems: a comparative study,” Appl. Opt. 35(34), 6674–6683 (1996).
    [Crossref] [PubMed]
  19. S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
    [Crossref]
  20. S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
    [Crossref]
  21. H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, Mass. 1950)
  22. A. Romano, Geometric Optics: Theory and Design of Astronomical Optical Systems Using Mathematica (Birkhäuser, 2010).
  23. A. L. Rivera, S. M. Chumakov, and K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 12(6), 1380–1389 (1995).
    [Crossref]
  24. J. C. Kimball and H. Story, “Fermat’s principle, Huygens’ principle, Hamilton’s optics and sailing strategy,” Eur. J. Phys. 19(1), 15–24 (1998).
    [Crossref]
  25. D. S. Lemons, “Gaussian thin lens and mirror formulas from Fermat’s principle,” Am. J. Phys. 62(4), 376–378 (1994).
    [Crossref]
  26. V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian optics. (Springer Science & Business Media, 2013).
  27. J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24(25), 29295–29301 (2016).
    [Crossref] [PubMed]
  28. A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
    [Crossref]

2016 (5)

S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
[Crossref]

E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
[Crossref]

S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
[Crossref]

A. I. Hernandez-Serrano, M. Weidenbach, S. F. Busch, M. Koch, and E. Castro-camus, “Fabrication of gradient-refractive-index lenses for terahertz applications by three-dimensional printing,” J. Opt. Soc. Am. B 33(5), 928 (2016).
[Crossref]

J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24(25), 29295–29301 (2016).
[Crossref] [PubMed]

2014 (1)

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

2013 (1)

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

2012 (2)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

S. Ji, M. Ponting, R. S. Lepkowicz, A. Rosenberg, R. Flynn, G. Beadie, and E. Baer, “A bio-inspired polymeric gradient refractive index (GRIN) human eye lens,” Opt. Express 20(24), 26746–26754 (2012).
[Crossref] [PubMed]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

2009 (2)

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

2007 (1)

A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
[Crossref]

2005 (2)

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

2004 (1)

R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid, Watasenia scintillans,” Vision Res. 44(18), 2129–2134 (2004).
[Crossref] [PubMed]

1999 (2)

W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999).
[Crossref] [PubMed]

O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson, and V. I. Smirnov, “High-efficiency bragg gratings in photothermorefractive glass,” Appl. Opt. 38(4), 619–627 (1999).
[Crossref] [PubMed]

1998 (1)

J. C. Kimball and H. Story, “Fermat’s principle, Huygens’ principle, Hamilton’s optics and sailing strategy,” Eur. J. Phys. 19(1), 15–24 (1998).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

D. S. Lemons, “Gaussian thin lens and mirror formulas from Fermat’s principle,” Am. J. Phys. 62(4), 376–378 (1994).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Baer, E.

Beadie, G.

Bentley, J. L.

Brocker, D. E.

S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
[Crossref]

Brongersma, M. L.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

Busch, S. F.

Cai, B. G.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Campbell, S. D.

S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
[Crossref]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Castro-camus, E.

Chumakov, S. M.

Coates, M. M.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Crawford, M. K.

Cui, T. J.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Degertekin, F. L.

S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
[Crossref]

Efimov, O. M.

Erfani, E.

E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
[Crossref]

Erturk, A.

S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
[Crossref]

Fan, P.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

Flynn, R.

Flynn, R. A.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Garm, A.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gislén, A.

R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid, Watasenia scintillans,” Vision Res. 44(18), 2129–2134 (2004).
[Crossref] [PubMed]

Gislén, L.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Glebov, L. B.

Glebova, L. N.

Greegor, R. B.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Harkrider, C. J.

Hasman, E.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

He, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Hernandez-Serrano, A. I.

Hongo, K.

A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
[Crossref]

Huang, T. J.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

Hussain, A.

A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
[Crossref]

Jagger, W. S.

W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999).
[Crossref] [PubMed]

Ji, S.

Jiang, W. X.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Juluri, B. K.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Kimball, J. C.

J. C. Kimball and H. Story, “Fermat’s principle, Huygens’ principle, Hamilton’s optics and sailing strategy,” Eur. J. Phys. 19(1), 15–24 (1998).
[Crossref]

Koch, M.

Kröger, R. H. H.

R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid, Watasenia scintillans,” Vision Res. 44(18), 2129–2134 (2004).
[Crossref] [PubMed]

Lapsley, M. I.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Lemons, D. S.

D. S. Lemons, “Gaussian thin lens and mirror formulas from Fermat’s principle,” Am. J. Phys. 62(4), 376–378 (1994).
[Crossref]

Lepkowicz, R. S.

Li, X.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Lin, D.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

Lin, S. C.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Lin, S. S.

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

Ma, H. F.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Mait, J. N.

Mao, X.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Milojkovic, P.

Moore, D. T.

Nagar, J.

S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
[Crossref]

Naqvi, Q. A.

A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
[Crossref]

Nielsen, J.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Nilsson, D. E.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Niroo-Jazi, M.

E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
[Crossref]

Parazzoli, C.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Ponting, M.

Richardson, K. C.

Rivera, A. L.

Rosenberg, A.

Rouke, J. L.

Sands, P. J.

W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999).
[Crossref] [PubMed]

Shi, J.

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Skogh, C.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Smirnov, V. I.

Smith, D. R.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Story, H.

J. C. Kimball and H. Story, “Fermat’s principle, Huygens’ principle, Hamilton’s optics and sailing strategy,” Eur. J. Phys. 19(1), 15–24 (1998).
[Crossref]

Sun, J.-H.

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

Sun, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Tanielian, M. H.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Tatu, S.

E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
[Crossref]

Tetienne, J. P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Thompson, M. A.

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

Tol, S.

S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
[Crossref]

Tomkinson, T. H.

Weidenbach, M.

Wolf, K. B.

Wu, T.-T.

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

Xiao, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Xu, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Yang, Y.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Zhang, T. X.

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

Zhou, L.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Am. J. Phys. (1)

D. S. Lemons, “Gaussian thin lens and mirror formulas from Fermat’s principle,” Am. J. Phys. 62(4), 376–378 (1994).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

R. B. Greegor, C. Parazzoli, J. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Appl. Phys. Lett. 87(9), 91114 (2005).
[Crossref]

S. Tol, F. L. Degertekin, and A. Erturk, “Gradient-index phononic crystal lens-based enhancement of elastic wave energy harvesting,” Appl. Phys. Lett. 109(6), 063902 (2016).
[Crossref]

Eur. J. Phys. (1)

J. C. Kimball and H. Story, “Fermat’s principle, Huygens’ principle, Hamilton’s optics and sailing strategy,” Eur. J. Phys. 19(1), 15–24 (1998).
[Crossref]

IEEE Trans. Antenn. Propag. (2)

H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, “Three-dimensional gradient-index materials and their applications in microwave lens antennas,” IEEE Trans. Antenn. Propag. 61(5), 2561–2569 (2013).
[Crossref]

E. Erfani, M. Niroo-Jazi, and S. Tatu, “A high-gain broadband gradient refractive index metasurface lens antenna,” IEEE Trans. Antenn. Propag. 64(5), 1968–1973 (2016).
[Crossref]

J. Opt. (1)

S. D. Campbell, J. Nagar, and D. E. Brocker, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. 18(4), 044019 (2016).
[Crossref]

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

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

Lab Chip (1)

X. Mao, S. C. Lin, M. I. Lapsley, J. Shi, B. K. Juluri, and T. J. Huang, “Tunable Liquid Gradient Refractive Index (L-GRIN) lens with two degrees of freedom,” Lab Chip 9(14), 2050–2058 (2009).
[Crossref] [PubMed]

Nat. Mater. (1)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Nature (1)

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005).
[Crossref] [PubMed]

Opt. Express (2)

Phys. Rev. B Condens. Matter Mater. Phys. (1)

S. S. Lin, T. J. Huang, J.-H. Sun, and T.-T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B Condens. Matter Mater. Phys. 79(9), 094302 (2009).
[Crossref]

Prog. Electromagnetics Res. (1)

A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiation characteristics of the Wood lens using Maslov’s method,” Prog. Electromagnetics Res. 73, 107–129 (2007).
[Crossref]

Science (2)

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Vision Res. (2)

R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid, Watasenia scintillans,” Vision Res. 44(18), 2129–2134 (2004).
[Crossref] [PubMed]

W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999).
[Crossref] [PubMed]

Other (6)

O. M. Efimov, L. B. Glebov, V. I. Smirnov, and L. Glebova, “Process for production of high efficiency volume diffractive elements in photo-thermo-refractive glass,” US 09/648,293, (2003).

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” US 08/017,034 (1994).

J. D. Ellis, D. R. Brooks, K. T. Wozniak, G. A. Gandara-Montano, E. G. Fox, K. J. Tinkham, S. C. Butler, L. A. Zheleznyak, M. R. Buckley, P. D. Funkenbusch, and W. H. Knox, “Manufacturing of Gradient Index Lenses for Ophthalmic Applications.” Design and Fabrication Congress 2017 (IODC, Freeform, OFT) OW1B–3 (2017).
[Crossref]

V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian optics. (Springer Science & Business Media, 2013).

H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, Mass. 1950)

A. Romano, Geometric Optics: Theory and Design of Astronomical Optical Systems Using Mathematica (Birkhäuser, 2010).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Light propagating in a GRIN medium with refractive index characterized by Eq. (6). (a) Contour plot for the refractive index function, designed as n(x,y)=ϕ(x+2y)=x+2y+4. If the launch condition satisfies y ( x 0 )=k=2, the ray will travel straightly. Otherwise, it will get close to the asymptotic line with slope k=2. (b) Schematics of transforming the divergent light into directional light. As the result shown, if the GRIN is set as a function of x+ky, ray with launch condition satisfies y ( x 0 )k will travel close to an asymptotic line y=kx+b. This is also the direction of refractive index gradient. Therefore, if the GRIN medium is thick enough, the inclination of emergent rays is almost the same and thus approximately parallel.
Fig. 2
Fig. 2 (a) Refractive index distribution described by Eq. (11). (b) Light propagating in a GRIN medium with refractive index characterized by Eq. (11). If the ray is launched along the tangential direction, the trace is a circle.
Fig. 3
Fig. 3 (a) Refractive index distribution described by Eq. (14). (b) Light propagating in a GRIN medium with refractive index characterized by Eq. (14). If the launch condition satisfies y | x= x 0 = x 0 y 0 , the trace is a branch of the equilateral hyperbola.
Fig. 4
Fig. 4 Schematics of the beam splitter, designed from the optical system shown in Fig. 3. If a wide beam is incident along the direction of hyperbola’s asymptotic line, it will be split into two beams going to opposite directions.
Fig. 5
Fig. 5 (a) Refractive index distribution described by Eq. (18). (b) Light propagating in a GRIN medium with refractive index characterized by Eq. (18). If the launch condition satisfies y| x= x 0 =a x 0 2 and y | x= x 0 =2a x 0 , the trace is a parabola.
Fig. 6
Fig. 6 (a) Refractive index distribution described by Eq. (21). (b) Light propagating in a GRIN medium with refractive index characterized by Eq. (21). If the launch condition satisfies y| x= x 0 = y | x= x 0 = e x 0 , the trace is an exponential curve.

Equations (21)

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

δ l nds =0
δ x 1 x 2 n(x,y) 1+ y 2 dx =δ x 1 x 2 L(x,y, y )dx= 0
d dx L y = L y
y n x n y + y 1+ y 2 n=0
k n x n y =0
n(x,y)=ϕ(x+ky)
x n x +y n y +n=0
r n r +n=0
n= r 0 (θ) r
n(x,y)= r 0 x 2 + y 2
n(x,y)={ 1, x 2 + y 2 > r 0 2 r 0 x 2 + y 2 , r 0 2 25 x 2 + y 2 r 0 2 5, x 2 + y 2 < r 0 2 25
x n x y n y x 2 y 2 x 2 + y 2 n=0
n(x,y)= x 2 + y 2 ϕ(xy)
n(x,y)={ 1, x 2 + y 2 < r 0 2 x 2 + y 2 r 0 , r 0 2 x 2 + y 2 25 r 0 2 5, x 2 + y 2 <25 r 0 2
2ax n x n y + 2a 1+4 a 2 x 2 n=0
2y n x x n y + 2ax 1+4ay n=0
n= 1+4ay ϕ( x 2 +2 y 2 )
n(x,y)= 1+4ay
y n x n y + y 1+ y 2 n=0
n(x,y)= 1+ y 2 ϕ(2x+ y 2 )
n(x,y)= 1+ y 2