Abstract

In this paper we report on the feasibility of light confinement in orbital geodesics on stationary, planar, and centro-symmetric refractive index mappings. Constrained to fabrication and [meta]material limitations, the refractive index, n, has been bounded to the range: 0.8 ≤ n(r⃗) ≤ 3.5. Mappings are obtained through the inverse problem to the light geodesics equations, considering trappings by generalized orbit conditions defined a priori. Our simulation results show that the above mentioned refractive index distributions trap light in an open orbit manifold, both perennial and temporal, in regards to initial conditions. Moreover, due to their characteristics, these mappings could be advantageous to optical computing and telecommunications, for example, providing an on-demand time delay or optical memories. Furthermore, beyond their practical applications to photonics, these mappings set forth an attractive realm to construct a panoply of celestial mechanics analogies and experiments in the laboratory.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2013 (1)

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

2012 (1)

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

2011 (2)

X. Ni and Y.-C. Lai, “Transient chaos in optical metamaterials,” Chaos 21, 033116 (2011).
[CrossRef] [PubMed]

T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the classroom,” Eur. J. Phys. 32, 747–759 (2011).
[CrossRef]

2010 (4)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light in the Schwarzchild metric,” Phys. Rev. D 81, 084017 (2010).
[CrossRef]

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

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

2009 (5)

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (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, 7807–7817 (2009).
[CrossRef] [PubMed]

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

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

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

2008 (1)

J. Levin and G. Perez-Giz, “A periodic table for black hole orbits,” Phys. Rev. D 77, 103005 (2008).
[CrossRef]

2006 (2)

2005 (1)

B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” J. App. Math. 65, 2049–2066 (2005).

2003 (1)

A. Zakery, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Crystaline Solids 330, 1–12 (2003).
[CrossRef]

2002 (1)

Y. Zarmi, “The Bertrand theorem revisited,” Am. J. Phys 70, 446–449 (2002).
[CrossRef]

2001 (2)

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Piwnicki and U. Leonhardt, “Optics of moving media,” App. Phys. B 72, 51–59 (2001).
[CrossRef]

2000 (1)

U. Leonhardt and Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

1999 (1)

U. Leonhardt and Piwnicki, “Optics of non-uniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

1998 (1)

M. Piana, “On uniqueness for an isotropic inhomogeneous inverse scattering problems,” Inv. Problems 14, 1565–1579 (1998).
[CrossRef]

Alsunaidi, M.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Anjum, D. H.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

Belgiorno, F.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Brenner,

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

Brevik, I.

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Cacciatori, S. L.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Cha, E. J.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Chan, C.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

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, 366–369 (2009).
[CrossRef] [PubMed]

Choi, S. K.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Chung, D. C.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

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, 366–369 (2009).
[CrossRef] [PubMed]

Ergin, T.

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

Faccio, D.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Frauendiener, J.

T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the classroom,” Eur. J. Phys. 32, 747–759 (2011).
[CrossRef]

Gangand, B.

B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” J. App. Math. 65, 2049–2066 (2005).

Genov, D.

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

Gorini, V.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Halnes, G.

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
[CrossRef]

Han, D.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Hedhili, M. N.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

Hioe, F. T.

F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light in the Schwarzchild metric,” Phys. Rev. D 81, 084017 (2010).
[CrossRef]

Jeong, S. Y.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

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, 366–369 (2009).
[CrossRef] [PubMed]

Jordan, D. W.

D. W. Jordan and P. Smith, Nonlinear ordinary differential equations (Oxford University Press, 2007).

Kang, D. W.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Kang, K. Y.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Kang, S. B.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Kuebel, D.

F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light in the Schwarzchild metric,” Phys. Rev. D 81, 084017 (2010).
[CrossRef]

Kwak, M. H.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Kwon, D.-H.

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

Lai, Y.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Lai, Y.-C.

X. Ni and Y.-C. Lai, “Transient chaos in optical metamaterials,” Chaos 21, 033116 (2011).
[CrossRef] [PubMed]

Leonhardt, U.

Piwnicki and U. Leonhardt, “Optics of moving media,” App. Phys. B 72, 51–59 (2001).
[CrossRef]

U. Leonhardt and Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

U. Leonhardt and Piwnicki, “Optics of non-uniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” in Progress in Optics53, 69–153, E. Wolf, ed. (Elsevier, 2008).
[CrossRef]

Levin, J.

J. Levin and G. Perez-Giz, “A periodic table for black hole orbits,” Phys. Rev. D 77, 103005 (2008).
[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, 366–369 (2009).
[CrossRef] [PubMed]

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, 366–369 (2009).
[CrossRef] [PubMed]

Müller, T.

T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the classroom,” Eur. J. Phys. 32, 747–759 (2011).
[CrossRef]

Ng, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Ng, T. K.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Ni, X.

X. Ni and Y.-C. Lai, “Transient chaos in optical metamaterials,” Chaos 21, 033116 (2011).
[CrossRef] [PubMed]

Ooi, B. S.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Ortenzi, G.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Paek, M. C.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Park, B. J.

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Peijun, L.

B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” J. App. Math. 65, 2049–2066 (2005).

Pendry, J. B.

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

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9894 (2006).
[CrossRef] [PubMed]

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

Perez-Giz, G.

J. Levin and G. Perez-Giz, “A periodic table for black hole orbits,” Phys. Rev. D 77, 103005 (2008).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” in Progress in Optics53, 69–153, E. Wolf, ed. (Elsevier, 2008).
[CrossRef]

Piana, M.

M. Piana, “On uniqueness for an isotropic inhomogeneous inverse scattering problems,” Inv. Problems 14, 1565–1579 (1998).
[CrossRef]

Piwnicki,

Piwnicki and U. Leonhardt, “Optics of moving media,” App. Phys. B 72, 51–59 (2001).
[CrossRef]

U. Leonhardt and Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

U. Leonhardt and Piwnicki, “Optics of non-uniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Rizzi, L.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
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S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

S. K.,

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Sala, V. G.

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

San-Román-Alerigi, D. P.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Schurig, D.

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Slimane, A. B.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

Smith, D. R.

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782, 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, 9794–9894 (2006).
[CrossRef] [PubMed]

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D. W. Jordan and P. Smith, Nonlinear ordinary differential equations (Oxford University Press, 2007).

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T. Ergin, N. Stenger, Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
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F. Verhulst, Nonlinear differential equations and dynamical systems (Springer Verlag, 1990).
[CrossRef]

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

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

Xiao, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Yang, X.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

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A. Zakery, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Crystaline Solids 330, 1–12 (2003).
[CrossRef]

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Y. Zarmi, “The Bertrand theorem revisited,” Am. J. Phys 70, 446–449 (2002).
[CrossRef]

Zhang, S.

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

Zhang, X.

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

Zhang, Y.

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Zhang, Z.-Q.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Am. J. Phys (1)

Y. Zarmi, “The Bertrand theorem revisited,” Am. J. Phys 70, 446–449 (2002).
[CrossRef]

App. Phys. B (1)

Piwnicki and U. Leonhardt, “Optics of moving media,” App. Phys. B 72, 51–59 (2001).
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Chaos (1)

X. Ni and Y.-C. Lai, “Transient chaos in optical metamaterials,” Chaos 21, 033116 (2011).
[CrossRef] [PubMed]

ETRI J. (1)

S. B. Kang, M. H. Kwak, B. J. Park, S. K., H. C. Ryu, D. C. Chung, S. Y. Jeong, D. W. Kang, S. K. Choi, M. C. Paek, E. J. Cha, and K. Y. Kang, “Optical and dielectric properties of chalcogenide glasses at terahertz frequencies,” ETRI J. 31, 667–674 (2009).
[CrossRef]

Eur. J. Phys. (1)

T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the classroom,” Eur. J. Phys. 32, 747–759 (2011).
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M. Piana, “On uniqueness for an isotropic inhomogeneous inverse scattering problems,” Inv. Problems 14, 1565–1579 (1998).
[CrossRef]

J. App. Phys. (1)

D. P. San-Román-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. N. Hedhili, M. Alsunaidi, and B. S. Ooi, “Electron irradiation induced reduction of the permittivity of chalcogenide glass (As2S3) thin fim,” J. App. Phys. 113, 044116 (2013).
[CrossRef]

J. App. Math. (1)

B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” J. App. Math. 65, 2049–2066 (2005).

J. Non-Crystaline Solids (1)

A. Zakery, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Crystaline Solids 330, 1–12 (2003).
[CrossRef]

J. Opt. Soc. A (1)

D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of J0-Bessel-Gauss beam by an heterogeneous refractive index map,” J. Opt. Soc. A 29, 1252–1258 (2012).
[CrossRef]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Nat. Phys. (1)

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

New J. Phys. (1)

S. L. Cacciatori, F. Belgiorno, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, and D. Faccio, “Spacetime geometries and light trapping in travelling refractive index perturbations,” New J. Phys. 12, 095021 (2010).
[CrossRef]

Opt. Express (1)

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

Opt. Express (1)

Phys. Rev. A (1)

U. Leonhardt and Piwnicki, “Optics of non-uniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Phys. Rev. D (3)

I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001).
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J. Levin and G. Perez-Giz, “A periodic table for black hole orbits,” Phys. Rev. D 77, 103005 (2008).
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F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light in the Schwarzchild metric,” Phys. Rev. D 81, 084017 (2010).
[CrossRef]

Phys. Rev. Lett. (2)

U. Leonhardt and Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102, 253902 (2009).
[CrossRef] [PubMed]

Science (3)

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

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

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

Other (3)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” in Progress in Optics53, 69–153, E. Wolf, ed. (Elsevier, 2008).
[CrossRef]

D. W. Jordan and P. Smith, Nonlinear ordinary differential equations (Oxford University Press, 2007).

F. Verhulst, Nonlinear differential equations and dynamical systems (Springer Verlag, 1990).
[CrossRef]

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Figures (10)

Fig. 1:
Fig. 1:

Comparison of sidereal and laboratory frame geodesics for light beams in (a) the vicinity of a star, (b) and (c) spinning black hole at different initial conditions, and (d) the laboratory across an heterogeneous monotonic increasing refractive index medium.

Fig. 2:
Fig. 2:

Ratio of external energy supply, t, for a radial symmetric system, where x = r and y = . Notice that there exists a xc, critical radius, at wich for all phase space pairs {x, y|x ≥ xc} t = 0, which implies that the light geodesics do not describe an orbital motion.

Fig. 3:
Fig. 3:

Light path dynamics for Gaussian refractive index shape for different na and nc. The maximum value of n in all cases is 3.8 at the center of the device. σ is constant for all results. Notice the increment in the internal radius rmin as nc increases, and the orbit width reduces.

Fig. 4:
Fig. 4:

Inner and outer radius as a function of the background refractive index nc. As it increases we observe that the outer radius diminishes, while the inner increases.

Fig. 5:
Fig. 5:

Confinement orbit for light traveling in a Gaussian attractor where na = 3.0 and nc = 0.8. (a) Shows a schematic representation of the planar trapping device where the refractive index increases towards the center, where red indicates maximum refractive index n = 3.8, and light blue n = 0.8; (b) polar-2D path as calculated by the simulation routine.

Fig. 6:
Fig. 6:

Phase space for Gaussian-like refractive index map with na = 3.0, nc = 0.8. The plots show: (a) radial and angular ϕ̈ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle (c).

Fig. 7:
Fig. 7:

(a) Schematic of Mexican hat like refractive index map where na = 3.0 and nc = 0.8 and depression is nd = 0.2; (b) depicts the resulting trapped orbit in polar coordinates as calculated by the simulation routine.

Fig. 8:
Fig. 8:

Phase-space for a mexican hat refractive index map with na = 3.0, nc = 0.8. The plots show: (a) radial and angular ϕ̇ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle (c).

Fig. 9:
Fig. 9:

Geodesics for light in a binary system. Observe that trapping also occurs for certain initial conditions at specific background and maximum refractive index.

Fig. 10:
Fig. 10:

Phase space of the trapping orbit for na = 3.0, nc = 0.8. The plots show: (a) radial and angular ϕ̇ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle.

Equations (25)

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

L = 1 2 [ g i j x i τ x j τ ] ,
δ τ = δ x 1 x 2 d s = 0 ,
d s = g i j d x i d x j .
n = ε = μ = g g i j g 00 ,
g = diag ( g 00 , g 11 , g 22 , g 33 ) = diag ( 1 , 1 , 1 , 1 ) ,
d s 2 = 1 n 2 d x 0 d x 0 d x i d x i , i = 1 , 2 , 3
r ¨ = r ϕ ˙ 2 + t ˙ 2 r n n 3 ,
ϕ ¨ = 2 r r ˙ ϕ ˙ + t ˙ 2 ϕ n n 3 r 2 ,
ϕ ¨ = t ˙ 2 z n n 3 ,
t ¨ = t ˙ ( t ˙ t n + 2 ( z ˙ z n + ϕ ˙ ϕ n + r ˙ r n ) ) n ,
r ¨ = n r ϕ ˙ 2 2 r ˙ ϕ ˙ ϕ n r ˙ 2 r n + r 2 ϕ ˙ 2 r n n ,
ϕ ¨ = 2 n r r ˙ ϕ ˙ r ˙ 2 ϕ n + r 2 ϕ ˙ 2 ϕ n + 2 r 2 r ˙ ϕ ˙ r n n r 2 .
x ¨ = h ( x , x ˙ ) g ( x ) ,
r ¨ = ( r 2 ϕ ˙ 2 r ˙ 2 ) r ln n + r ϕ ˙ 2 ,
ϕ ¨ = 2 r ˙ ϕ ˙ n + r r n n r ,
ϕ ˙ = μ r 2 , μ = n 2 ,
r ¨ = ( 2 n 4 r 4 r 2 r ˙ 2 ) r ln n + 2 n 4 r 4 r .
r = d d t = r ˙ h ( r , r ˙ ) = r ˙ ( 2 n 4 r 2 r ˙ 2 ) r ln n .
r ˙ = ϰ , ϰ ˙ = ( r 2 β 2 ϰ 2 ) χ ( r ) + r β 2 ,
( δ ˙ ε ˙ ) = ( 0 1 W 2 V ) ( δ ε ) V = ϰ o χ o , W = 2 χ o β 2 r o + χ o ' ( r o 2 β 2 ϰ o 2 ) ,
λ 1 , 2 = V ± V 2 + W .
r χ ( r ) | r = r o ϰ 0 , V 2 + W 0 ,
n = n a e r 2 / σ 2 + n c .
n = ( n a n d r 2 σ 2 ) e r 2 / σ 2 + n c
n = n a ( e ( r r off 1 ) 2 / σ 2 + e ( r r off ) 2 / σ 2 ) + n c .

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