Abstract

We apply the transformation optical technique to modify or improve conventional refractive and gradient index optical imaging devices. In particular, when it is known that a detector will terminate the paths of rays over some surface, more freedom is available in the transformation approach, since the wave behavior over a large portion of the domain becomes unimportant. For the analyzed configurations, quasi-conformal and conformal coordinate transformations can be used, leading to simplified constitutive parameter distributions that, in some cases, can be realized with isotropic index; index-only media can be low-loss and have broad bandwidth. We apply a coordinate transformation to flatten a Maxwell fish-eye lens, forming a near-perfect relay lens; and also flatten the focal surface associated with a conventional refractive lens, such that the system exhibits an ultra-wide field-of-view with reduced aberration.

© 2010 OSA

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2010

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010).
[CrossRef] [PubMed]

2009

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

D. H. Kwon and D. H. Werner; “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[CrossRef]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
[CrossRef] [PubMed]

2008

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008).
[CrossRef]

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

D. Schurig, “An aberration-free lens with zero f-number,” N. J. Phys. 10(11), 115034 (2008).
[CrossRef]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2006

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

1996

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

1994

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Chang, Z.

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Genon, A.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Genov, D. A.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[CrossRef]

Hu, G.

Hu, J.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kundtz, N.

Kwon, D. H.

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008).
[CrossRef]

Landy, N. I.

Legros, W.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Leonhardt, U.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Ma, Y. G.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

Meys, B.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Nicolet, A.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Ong, C. K.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

Padilla, W. J.

Pendry, J.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

Pointras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Popa, B. I.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

Remacle, J. F.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

Roberts, D. A.

Schurig, D.

D. Schurig, “An aberration-free lens with zero f-number,” N. J. Phys. 10(11), 115034 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Tyc, T.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Werner, D. H.

Werner, , D. H.

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, S.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[CrossRef]

Zhang, X.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhou, X.

J. Appl. Phys.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetism,” J. Appl. Phys. 75(10), 6036–6038 (1994).
[CrossRef]

J. Mod. Opt.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

N. J. Phys.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

D. Schurig, “An aberration-free lens with zero f-number,” N. J. Phys. 10(11), 115034 (2008).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimaters, flat lenses and right-angle bends,” N. J. Phys. 10(11), 115023 (2008).
[CrossRef]

Nat. Mater.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef]

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Nat. Photonics

L. H. Gabrielli, J. Cardenas, C. B. Pointras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461 (2009).
[CrossRef]

Nat. Phys.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[CrossRef]

Opt. Express

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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[CrossRef] [PubMed]

Science

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
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[CrossRef] [PubMed]

Other

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E. W. Marchand, Gradient Index Optics (Academic Press, New York, 1978).

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Figures (8)

Fig. 1
Fig. 1

(a) The Maxwell fish-eye lens is a perfect optical instrument, connecting pairs of aplanatic points on one side of the sphere to the other. The fish-eye lens has a radial distribution of index, given by Eq. (1). (b) Since both image and object points lie on the sphere with the rays contained within the medium, the boundary conditions on the surface are immaterial. We can, for example, imagine two sides terminated by conducting boundary conditions, as illustrated in the figure. The boundary and the interior region can then be transformed. (c) Utilizing the transformation provided in Eq. (3), the fish-eye lens can be flattened to form a perfect relay lens.

Fig. 2
Fig. 2

The spatial grid and the components of the dielectric permittivity tensor for the transformation that flattens a cylindrical region. The central region is in the shape of a cylinder with radius a = 1 (arbitrary units). The shaded areas above and below the grids indicate regions of perfect electric conductor (PEC). The plotted regions extend from x = 2 a to x = + 2 a along the x-axis, and from y = a / 2 to y = + a / 2 along the y-axis (outside of the cylindrical region).

Fig. 3
Fig. 3

The spatial grid and the components of the dielectric permittivity tensor for the quasi-conformal transformation that flattens a cylindrical region. The central region is in the shape of a cylinder with radius a = 1 (arbitrary units). The shaded areas above and below the grids indicate regions of perfect electric conductor (PEC). The plotted regions extend from x = 2 a to x = + 2 a along the x-axis, and from y = a / 2 to y = + a / 2 along the y-axis (outside of the cylindrical region).

Fig. 4
Fig. 4

(a) Permittivity distribution in the virtual frame, shown with grid lines representing the inverse transformation. This transformation assumes the interior region to be flattened consists only of free space. (b) Permittivity distribution corresponding to the coordinate transformation combined with the fish-eye permittivity distribution of Eq. (1) (with n 0 = 2 ). This distribution corresponds to the flattened fish-eye. The plotted regions extend from x = 2 a to x = + 2 a along the x-axis, and from y = a / 2 to y = + a / 2 along the y-axis, where a corresponds to the original fish-eye lens radius.

Fig. 5
Fig. 5

Numerical simulations of the flattened fish-eye lens of thickness a. (a) Ray tracing calculation showing three pencils originating from the bottom surface of the lens, on and off (shifted + 0.5a and −0.5a from the center) the optical axis. (b) Full-wave TE-polarization simulations of the same system with a point source located at the bottom surface on the optical axis. The free-space wavelength is 0.3a. Color shows the out-of-plane electric field. (c) Same as (b) but the source is shifted off-axis by 0.5a. (d) Same as (b) but with TM-polarization excitation; color shows out-of-plane magnetic field. (e) Same as (d), but with off-axis displacement 0.5a. In all cases, the interior region corresponding to the lens extends from x = 2 a to x = + 2 a along the horizontal axis, and from y = a / 2 to y = + a / 2 along the vertical axis.

Fig. 6
Fig. 6

(a) Ray trace diagram of a refractive lens. The lens causes rays to be focused to an arc, which intersects the optical axis at zf . The bounded, darker shaded yellow region indicates where the transformation medium will be placed (between zc and zp ). (b) Transformation and inverse transformation leading to the flattening of the focal surface. The inverse transformation shown compresses the region between the focal arc and zc to the region between zc and zp . The transformation can be used to design a TO device that will effectively flatten the focal surface.

Fig. 7
Fig. 7

A compensated refractive lens. The medium in the shaded rectangle is assumed to be derived from the transformation shown in Eq. (17).

Fig. 8
Fig. 8

The three-dimensional generalization of the flattened fish-eye lens obtained by axisymmetric revolution of the index distribution around the optical axis. Full-wave simulation with an electric point dipole source is shown. The free-space wavelength is 0.6a, where a is the lens thickness. (a) The source (vertically-oriented electric point dipole) is on the optical axis. (b) The source is shifted off-axis by 0.25a.

Equations (17)

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n ( r ) = n 0 1 + ( r / a ) 2
d d s ( n ( q ) d q i d s ) = d n d q i ,
x ( x , z ) = w a x z ( x , z ) = z l a 2 x 2 ,
ε = Λ ε Λ T | Λ | n 2 ( x , z ) μ = Λ μ Λ T | Λ | ,
Λ = ( x x x y x z y x y y y z z x z y z z )
Λ = ( x x x y 0 y x y y 0 0 0 1 ) ,
ε = 1 x x y x y x x y ( ( x x ) 2 + ( x y ) 2 x x y x + x y y y 0 x x y x + x y y y ( y y ) 2 + ( y x ) 2 0 0 0 1 ) .
x x y x = x y y y .
f x x = y y ;
1 f y x = x y .
x f x x + y f x y = 0 x 1 f y x + y 1 f y y = 0.
( x x ) 2 + ( x y ) 2 = 1 f 2 [ ( y y ) 2 + ( y x ) 2 ] ,
f = ( y y ) 2 + ( y x ) 2 ( x x ) 2 + ( x y ) 2 .
2 x x 2 + 2 y y 2 = 0 2 y x 2 + 2 y y 2 = 0 ,
( z R i V i ) 2 R i 2 + x 2 R i 2 = 1 ,
z = z 0 + a ( x 2 b 2 ) ,
x = x z = z + a ( x 2 b 2 ) ( z z c z p z c ) 2 ,

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