## A possible approach on optical analogues of gravitational attractors |

Optics Express, Vol. 21, Issue 7, pp. 8298-8310 (2013)

http://dx.doi.org/10.1364/OE.21.008298

Acrobat PDF (6244 KB)

### 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

## 1. Introduction

*differential geometry*. Einstein’s general theory of relativity tells us that geometry and gravity are linked by means of the metric tensor of space-time; in optics the intricacy of the relation between geometry and the physical properties of materials (permittivity and permeability) has been evinced by miscellaneous methods, most based on differential geometry [4

4. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of
light,” in *Progress in Optics*53, 69–153, E. Wolf, ed. (Elsevier, 2008) [CrossRef] .

7. 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] .

18. D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of *J*_{0}-Bessel-Gauss
beam by an heterogeneous refractive index map,” J. Opt. Soc.
A **29**, 1252–1258 (2012) [CrossRef] .

21. 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 (As_{2}S_{3}) thin fim,” J.
App. Phys. **113**, 044116 (2013) [CrossRef] .

## 2. Deriving the model

*f*, e.g. in photo-refractive media index of refraction is a real valued function

*f*of the incident light intensity

*I*; therefore

*n*(

*x⃗*) =

*f*(

*I*(

*x⃗*)), where

*x⃗*is the space-time quadrivector which corresponds to the coordinates {

*x*

^{0}:

*x*

^{1},

*x*

^{2},

*x*

^{3}}. In this representation

*x*

^{0}corresponds to the time coordinate, and the rest to the usual three-dimensional space. In this context the photon’s path is governed by the geometry of the space the light is propagating on, which in general has the metric tensor

*g*, and the infinitesimal length element

_{ij}*ds*

^{2}=

*g*. Therefore, if we parameterized the path followed by light with reference to the proper time

_{ij}dx^{i}dx^{j}*τ*, i.e.

*x⃗*(

*τ*), we can analytically derive the path equations by means of Lagrange-Euler equations, whit a Lagrangian of the form: where the derivatives are taken in respect to the affine parameter

*τ*, i.e. the

*proper time*.

*g*and the refractive index

_{ij}*n*. This can be achieved by means of conformal mappings, where it is possible to build the relation between the metric and the electromagnetic properties of a [meta]material [4

4. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of
light,” in *Progress in Optics*53, 69–153, E. Wolf, ed. (Elsevier, 2008) [CrossRef] .

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

*ds*

^{2}=

*g*

_{00}

*dx*

^{0}

*dx*

^{0}−

*g*, it is possible to map the curved space-time to an isotropic, non-dispersive and non-absorbing medium as [4

_{ii}dx^{i}dx^{i}4. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of
light,” in *Progress in Optics*53, 69–153, E. Wolf, ed. (Elsevier, 2008) [CrossRef] .

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

*g*= det(

*g*). Furthermore if we consider Minkowski’s metric of special relativity, the tensor

_{ij}*g*has the entries: then the refractive index

*n*=

*n*(

*x*

^{0},

*r⃗*) can be found using Eq. (4). And consequently, the length element

*ds*of Eq. (2) can be written as:

### 2.1. Path equations and the inverse problem

*τ*, e.g.

*ṙ*=

*∂r*/

*∂τ*. Equation (7) through Eq. (9) describe the parametrized path of a light beam traveling in a space-time with such geometry or equivalent refractive index.

*ṙ*,*ϕ̇*→ 0,*ṙ*→ 0 and*ϕ*≤_{min}*ϕ*≤*ϕ*,_{max}*ϕ̇*→ 0 and*r*≤_{min}*r*≤*r*,_{max}*r*≤_{min}*r*≤*r*and_{max}*ϕ̇*≤_{min}*ϕ̇*≤*ϕ̇*,_{max}

*τ*≥

*τ*

^{*}; where

*τ*

^{*}marks the moment at which the initial conditions for trapping are met. The first condition is only possible if the refractive index

*n*goes to infinity, or if the refractive index co-moves with the wave [16

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

17. 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] .

*stationary*refractive index [15

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

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

*a priori*conditions that describes a generalized trapping scenario. If we impose premises 1 to 4, described earlier, the path equations reduce to new set of equations where the affine parameter is time itself; whence we can rework Eq. (7) through Eq. (10) into polar coordinates:

*ṙ*≤

*ṙ*and

_{max}*ϕ̇*≤

_{min}*ϕ̇*≤

*ϕ̇*, on Eq. (11) and Eq. (12), and find a non-singular refractive index distribution,

_{max}*n*(

*r*,

*ϕ*), that satisfies them. Notice that we require the refractive index to be smoothly varying and

*C*

^{2}continuous, i.e. ∇

*n ·e⃗*≤

^{i}*M*, where

*M*is a real valued number, and

*e⃗*a vector in a give direction in space-time. The general solution to this problem, however, is complex and a unique solution is not guaranteed [18

^{i}18. D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of *J*_{0}-Bessel-Gauss
beam by an heterogeneous refractive index map,” J. Opt. Soc.
A **29**, 1252–1258 (2012) [CrossRef] .

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

*r*is bounded to a region of space, i.e.

*r*≤

_{min}*r*≤

*r*, for all

_{max}*ϕ*; whence

*ṙ*(

*r*) =

_{max}*ṫ*(

*r*) = 0, and

_{min}*r*can be written in terms of the angle

*ϕ*or the time

*t*. Hence a natural solution to the inverse problem is an autonomous refractive index that varies with the radial variable, i.e.

*n*=

*n*(

*r*). This refractive index map resembles the centro-symmetric characteristics of gravitational fields, where, as pointed out independently by Genov and Ni [15

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

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

18. D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of *J*_{0}-Bessel-Gauss
beam by an heterogeneous refractive index map,” J. Opt. Soc.
A **29**, 1252–1258 (2012) [CrossRef] .

21. 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 (As_{2}S_{3}) thin fim,” J.
App. Phys. **113**, 044116 (2013) [CrossRef] .

### 2.2. System stability and dynamic equillibrium

*ℓ*is a constant of integration. This equation is reminiscent of orbital mechanics, with the sole exception that

*μ*, the mass in celestial mechanics, varies with the radius. Substituting the latter result into Eq. (14), and rearranging yields: Equation (17) arrangement allows us to calculate the

*ratio of external energy supply*, which is a measure of the equilibrium stability of the system [25, 26

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

**5**, 687–692 (2009) [CrossRef] .

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

*ṙ*≠ 0, for all time

*t*; then from premises 3 and 4 it follows that

*ṙ*≤

_{min}*ṙ*≤

*ṙ*, where

_{max}*ṙ*< 0; and from the definition of

_{min}*n*(

*r*) as a monotonous decreasing function it follows that

*∂*ln

_{r}*n*≤ 0. Therefore, as the radius decreases, the radial velocity goes to its minimum and

*ℰ*≤ 0, therefore the system approaches the equilibrium point and the amplitude of the path decreases. On the other hand, as the radius increases, the velocity tends to a maximum, and

_{t}*ℰ*≥ 0, thus the system is driven away from the equilibrium point and the amplitude of the path increases. In Fig. 2 we show

_{t}*ℰ*for a Gaussian like refractive index described in section 3.1. Observe that in the light confinement scenario we will see an oscillatory behavior in

_{t}*ℰ*.

_{t}**5**, 687–692 (2009) [CrossRef] .

*β*=

*ϕ̇*and

*χ*(

*r*) =

*∂*ln

_{r}*n*. Then, if we introduce small perturbations

*r*=

*r*+

_{o}*δ*and ϰ = ϰ

*+*

_{o}*ε*, expand

*χ*(

*r*) in a Taylor series around

*r*, and drop all higher order terms for

_{o}*δ*and

*ε*; we can rewrite Eq. (19) as a linear system of equations for the perturbation terms: where

*χ*and

_{o}*χ*′

*are the linear terms of the Taylor expansion of*

_{o}*χ*(

*r*) around

*r*. Ensuing the eigenvalues of the matrix above are given by:

_{o}*eigenvalues*of Eq. (20), ℜ{

*λ*

_{1,2}}, are negative or equal to zero. Recall that 0 ≤

*β*≤ lim

_{r}_{→∞}

*n*(

*r*)

^{−1}, and

*χ*(

*r*) ≤ 0; and therefore the condition for Lyapunov stability is reached if:

*r*near the stability point

*r*= 0, provided the radial velocity meets

_{o}*ṙ*≤ 0. In general, since the refractive index is monotonous decrescent, the system will be Lyapunov stable at points where

_{o}*ṙ*≤ 0, see, for example, the phase-space plots in Figs. 6(b) and 8(b). In addition, it is interesting to analyze whether the system will be Lyapunov stable under the conditions necessary to sustain a circular orbit. In this case

*ϰ*= 0, and

_{o}*r*=

_{o}*a*, constant, for all time

*t*; consequently

*V*= 0,

*W*≤ 0, and the second term in the equality (22) is imaginary, provided that

*χ*(

*r*) ≤ 0; thus resulting in ℜ{

*λ*

_{1,2}} =

*V*= 0. These conditions, however, result in a different family of refractive indexes. In our study we focus on Gaussian-like refractive index distributions, feasible through photo-refraction, where

*ϰ*≠ 0, ergo the system will not describe closed circular orbits.

*ratio of external energy supply*. In the next section we will present the confinement geodesics resulting from these refractive index distributions. We will show that the geodesics in this system mimic those of a spinning black hole and other celestial objects, thus reproducing the paths of light traveling in curved space.

## 3. Results and discussion

### 3.1. Guassian distribution, single attractor

*n*≤ 3.8. Hence, given a range of values for the back-ground and maximum refractive index,

*n*and

_{c}*n*, respectively, we can evaluate different light confinement scenarios, see Fig. 3.

_{a}*n*(

*r*) and {

*r*,

_{o}*ṙ*,

_{o}*ϕ*,

_{o}*ϕ̇*}. As portrayed by Fig. 3 different initial conditions will result in new orbit manifolds with distinct characteristics, albeit maintaining a common morphology. As it can be seen from these figures the orbital paths are highly sensitive to the modifications in the refractive index and the initial conditions; a change in ∼ 0.5% in

_{o}*n*alters the orbit, as seen from the plot of radii extrema in Fig. 4.

_{c}*n*= 0.8 and

_{c}*n*= 3.0. As detailed in Fig. 6(a), the radial velocity describes an oscillatory form, which indicates that the radius follows a similar motion, swinging between two extrema, as expected in a confine and open orbital motion. The latter can be observed in greater detail by analyzing the phase-space, plotted in Fig. 6(b); notice that when the velocity

_{a}*ṙ*reaches zero the orbit radius

*r*reaches an extrema; for a closed single loop the amplitude of the velocity oscillation would be zero, and the phase space would become a horizontal line at

*ṙ*= 0. The behaviour of the angle versus the radial velocity, and consequently to the radius movement, is described by Fig. 6(c), where the absolute value of the radial velocity is plotted against the angular position, which results in a periodic motion. Observe that in the limit at every angle there would be a time at which the radial velocity would be zero, which is typical of open orbits.

### 3.2. Mexican hat distribution

*mexican hat*is:

*n*= 0.8,

_{c}*n*= 3.0, and set

_{a}*n*= 0.2. The results, shown in Figs. 7 and 8, display an orbit whose appearance resembles that of the regular Gaussian trap studied earlier. However, a closer look reveals that the trap confinement area is ∼ 2% slimmer, and the oscillatory frequency of the radial and angular velocities, Fig. 8(a), increases by ∼ 30%. Whereas the radial phase-space, Fig. 8(b), shows a smaller eccentricity compared to its Gaussian counterpart, and the absolute value of the radial velocity versus the angle, Fig. 8(c), describes the typical behaviour of an open orbit.

_{d}### 3.3. Double attractors

*n*> 0.03 modifies the trapping orbits, in some cases releasing the light beam.Yet, it is possible to confine the light beam to an orbit which mimics the behaviour of Newton’s three body problem, where, as expected, small perturbations result in sundry possible solutions to the path equation, i.e. chaos.

### 3.4. The space-time equations, trapping

### 3.5. Material realization

*J*_{0}-Bessel-Gauss
beam by an heterogeneous refractive index map,” J. Opt. Soc.
A **29**, 1252–1258 (2012) [CrossRef] .

21. 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 (As_{2}S_{3}) thin fim,” J.
App. Phys. **113**, 044116 (2013) [CrossRef] .

*As*

_{x}S_{1−x}in multiple layers. Recalling that in photo-refraction the optical properties of the material, permeability and permittivity, change as a function of the light intensity, allowing the fabrication of a continuously varying refractive index as the one required by a centro-symmetrical attractors. Moreover, since photo-refractivity is elastic,

*i.e.*reversible, it allows one to modify the refractive index of the layers combined in time, thence enabling or disabling the trap at will.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Levin and G. Perez-Giz, “A periodic table for black hole
orbits,” Phys. Rev. D |

2. | F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light
in the Schwarzchild metric,” Phys. Rev. D |

3. | T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the
classroom,” Eur. J. Phys. |

4. | U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of
light,” in |

5. | H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and
metamaterials,” Nat. Mater. |

6. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic
fields,” Science |

7. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in
transformation media,” Opt. Express |

8. | U. Leonhardt and Piwnicki, “Optics of non-uniformly moving
media,” Phys. Rev. A |

9. | U. Leonhardt and Piwnicki, “Relativistic effects of light in moving media with
extremely low group velocity,” Phys. Rev. Lett. |

10. | I. Brevik and G. Halnes, “Light rays at optical black holes in moving
media,” Phys. Rev. D |

11. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,”
Science |

12. | T. Ergin, N. Stenger, Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical
wavelengths,” Science |

13. | 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. |

14. | D.-H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection
based on coordinate transformation techniques,” Opt.
Express |

15. | D. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in
metamaterials,” Nat. Phys. |

16. | Piwnicki and U. Leonhardt, “Optics of moving media,”
App. Phys. B |

17. | 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. |

18. | D. P. San-Román-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, “Generation of |

19. | A. Zakery, “Optical properties and applications of chalcogenide
glasses: a review,” J. Non-Crystaline Solids |

20. | 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. |

21. | 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 (As |

22. | X. Ni and Y.-C. Lai, “Transient chaos in optical
metamaterials,” Chaos |

23. | B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic
waves,” J. App. Math. |

24. | M. Piana, “On uniqueness for an isotropic inhomogeneous inverse
scattering problems,” Inv. Problems |

25. | D. W. Jordan and P. Smith, |

26. | F. Verhulst, |

27. | Y. Zarmi, “The Bertrand theorem revisited,”
Am. J. Phys |

**OCIS Codes**

(230.1150) Optical devices : All-optical devices

(290.3200) Scattering : Inverse scattering

(350.4600) Other areas of optics : Optical engineering

(350.5720) Other areas of optics : Relativity

(080.5692) Geometric optics : Ray trajectories in inhomogeneous media

**History**

Original Manuscript: January 14, 2013

Revised Manuscript: March 3, 2013

Manuscript Accepted: March 4, 2013

Published: March 28, 2013

**Citation**

Damián P. San-Román-Alerigi, Ahmed B. Slimane, Tien K. Ng, Mohammad Alsunaidi, and Boon S. Ooi, "A possible approach on optical analogues of gravitational attractors," Opt. Express **21**, 8298-8310 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8298

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### References

- J. Levin and G. Perez-Giz, “A periodic table for black hole orbits,” Phys. Rev. D 77, 103005 (2008). [CrossRef]
- F. T. Hioe and D. Kuebel, “Characterizing planetary orbits and trajectories of light in the Schwarzchild metric,” Phys. Rev. D 81, 084017 (2010). [CrossRef]
- T. Müller and J. Frauendiener, “Studying null and time-like geodesics in the classroom,” Eur. J. Phys. 32, 747–759 (2011). [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]
- H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010). [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]
- U. Leonhardt and Piwnicki, “Optics of non-uniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999). [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]
- I. Brevik and G. Halnes, “Light rays at optical black holes in moving media,” Phys. Rev. D 65, 024005 (2001). [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]
- 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]
- 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]
- Piwnicki and U. Leonhardt, “Optics of moving media,” App. Phys. B 72, 51–59 (2001). [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]
- 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]
- A. Zakery, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Crystaline Solids 330, 1–12 (2003). [CrossRef]
- 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]
- 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]
- X. Ni and Y.-C. Lai, “Transient chaos in optical metamaterials,” Chaos 21, 033116 (2011). [CrossRef] [PubMed]
- B. Gangand and L. Peijun, “Inverse medium scattering problems for electromagnetic waves,” J. App. Math. 65, 2049–2066 (2005).
- M. Piana, “On uniqueness for an isotropic inhomogeneous inverse scattering problems,” Inv. Problems 14, 1565–1579 (1998). [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]
- Y. Zarmi, “The Bertrand theorem revisited,” Am. J. Phys 70, 446–449 (2002). [CrossRef]

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