A possible approach on optical analogues of gravitational attractors

Damián P. San-Román-Alerigi; Ahmed B. Slimane; Tien K. Ng; Mohammad Alsunaidi; Boon S. Ooi

doi:10.1364/OE.21.008298

1. Introduction

Einstein’s general relativity theory allows us to understand the dynamics of mass and massless particles as they travel through space-time. From it, we learn that matter and geometry are inextricably related; the latter transfigures the metric of space-time, and this alteration in turn modifies the geodesics, the paths of particles, resulting in a manifold of relativistic effects, v. gr. the apparent shift in the position of the stars due to the gravitational lensing effect, depicted in Fig. 1(a), where photons traveling near a massive star are deflected from their otherwise straight path due to the curvature of space-time; or trapping in multifarious orbits around black holes with respect to the initial conditions and shape of the gravitational attractor [1–3], see Figs. 1(b) and 1(c).

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.

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Keeping in mind the previous illustrations, let us momentarily examine a more mundane experiment. Imagine a photon moving in a region of space filled with some monotonically increasing refractive index. What we observe as the photon moves in the medium is that its path describes a curve like the one depicted in Fig. 1(d). If the gradient of the refractive index is tailored properly, it would be possible to build an optical analogy to gravitational bending (see Fig. 1(a)) for example. This leads us to wonder whether it is possible to fabricate an optical analogue to other sidereal geodesics by means of a differential control on the index of refraction.

The latter analogy reveals a connection between both examples, a weft where both realms [sidereal and laboratory] can be understood as manifestations of the same mathematical ground, albeit significantly different physics, 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 [4–7].

These methods can be used to show that the optical example of differential light bending can be understood in terms of geometry, essentially by bending the space on which the light propagates, hence the metric is related to the electromagnetic tensors [4, 6]. This realization paved the way for the theoretical realization of optical cloaks, super-lenses, optical illusion devices and recently black-hole analogues in moving refractive index perturbations [6–14].

Concerning trapping and light confinement in particular, different methods have been proposed [2, 15–17]. Of particular interest is Genov et al study, which presented an isotropic inhomogeneous refractive index mapping which was able to confine light mimicking a closed orbit around a black hole [15]. As the authors noted, such mapping could be implemented on a dielectric photonic crystal, composed of air-gaps in Gallium-Arsenide.

In our work, we posit theoretical traps that provide an optical analogue to celestial mechanics to realize light confinement, trapping in a region of space, by means of two-dimensional stationary refractive index mappings, which could be implemented under current technological and photo-refractive [meta]materials, constraint to optical frequencies, which could be easily integrable on all-optical-chips [18–21]. The mathematical and physical background to make these effects possible could bring forth an exciting ground to test sidereal mechanics in the laboratory; and provide the key to enable a miscellany planar optical systems that are of great interest for diverse photonic applications, ranging from time delays, to temporal optical memories, or random resonators, to name a few; thence its significance.

2. Deriving the model

We regard a model equation based in the paraxial approximation of non-linear electrodynamics, by looking into a medium where the index of refraction can be modulated in space and time by some smooth varying function 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 {x⁰ : x¹, x², x³}. In this representation x⁰ 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_ij, and the infinitesimal length element ds² = g_ijdxⁱdx^j. Therefore, if we parameterized the path followed by light with reference to the proper time τ, i.e. x⃗(τ), we can analytically derive the path equations by means of Lagrange-Euler equations, whit a Lagrangian of the form:

L = \frac{1}{2} [g_{i j} \frac{\partial x^{i}}{\partial τ} \frac{\partial x^{j}}{\partial τ}],

where the derivatives are taken in respect to the affine parameter τ, i.e. the proper time.

The physical description behind Eq. (1) is relevant to optics, as we set forth earlier in the introduction. If the metric were indeed related to the electromagnetic tensors, permittivity and permeability, then it could be possible to write the light-path in the paraxial approximation as a function of either the refractive index $n = f (x^{0}, \vec{r}) = \sqrt{ε (\vec{r}, t) μ (\vec{r}, t)}$ , or the metric ds. Hence, it could be possible to use Fermat’s principle, where Eq. (1) would take the role of an optical Lagrangian, i.e. the solution to the variational problem which states that given a refractive index n(x⁰, r⃗) the path light follows is given by the extrema of:

δ τ = δ \int_{x_{1}}^{x_{2}} d s = 0,

where the integration is done between the arbitrary points

x_{1}^{j}

to

x_{2}^{j}

, with j = 0,..., 3, and δ indicates the functional maximization. Recursively, the solution to Eq. (2) is the Lagrange-Euler Eq. (1). Meanwhile, observe that in the integral of Eq. (2) the term ds takes into account the metric by virtue of the covariant metric tensor g_ij of the curved space:

d s = \sqrt{g_{i j} d x^{i} d x^{j}} .

Yet, we are still to determine the relation between the metric tensor g_ij and the refractive index 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 [4–6]. Specifically, if the line element of space-time takes the form ds² = g₀₀dx⁰dx⁰ − g_iidxⁱdxⁱ, it is possible to map the curved space-time to an isotropic, non-dispersive and non-absorbing medium as [4, 15]:

n = ε = μ = \sqrt{\frac{- g \cdot g^{i j}}{g_{00}}},

where g = det(g_ij). Furthermore if we consider Minkowski’s metric of special relativity, the tensor g has the entries:

g = diag (g_{00}, g_{11}, g_{22}, g_{33}) = diag (1, - 1, - 1, - 1),

then the refractive index n = n(x⁰, r⃗) can be found using Eq. (4). And consequently, the length element ds of Eq. (2) can be written as:

d s^{2} = \frac{1}{n^{2}} d x^{0} d x^{0} - d x^{i} d x^{i}, i = 1, 2, 3

2.1. Path equations and the inverse problem

Accordingly, from the Lagrange-Euler (1) we can derive the light-path, which in cylindrical coordinates takes the form:

\ddot{r} = r {\dot{ϕ}}^{2} + \frac{{\dot{t}}^{2} \partial_{r} n}{n^{3}},

\ddot{ϕ} = \frac{- 2 r \dot{r} \dot{ϕ} + \frac{{\dot{t}}^{2} \partial_{ϕ} n}{n^{3}}}{r^{2}},

\ddot{ϕ} = \frac{{\dot{t}}^{2} \partial_{z} n}{n^{3}},

\ddot{t} = \frac{\dot{t} (\dot{t} \partial_{t} n + 2 (\dot{z} \partial_{z} n + \dot{ϕ} \partial_{ϕ} n + \dot{r} \partial_{r} n))}{n},

where all derivatives marked by □̇ and □̈ are with respect to the affine parameter 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.

We will describe light being trapped, when the geodesics are perpetually bounded to some region of space, for as long as the material properties remain unchanged. In this sense a light trapping geodesic is one where any of the following conditions arise:

ṙ, ϕ̇ → 0,
ṙ → 0 and ϕ_min ≤ ϕ ≤ ϕ_max,
ϕ̇ → 0 and r_min ≤ r ≤ r_max,
r_min ≤ r ≤ r_max and ϕ̇_min ≤ ϕ̇ ≤ ϕ̇_max,

for all 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, 17]. The other three cases can be realized in stationary refractive index [15, 22]. Thus, they can be imposed as general conditions to solve the inverse problem to Eq. (7) through Eq. (10).

The inverse problem begins by defining a set of 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:

\ddot{r} = \frac{n r {\dot{ϕ}}^{2} - 2 \dot{r} \dot{ϕ} \partial_{ϕ} n - {\dot{r}}^{2} \partial_{r} n + r^{2} {\dot{ϕ}}^{2} \partial_{r} n}{n},

\ddot{ϕ} = - \frac{2 n r \dot{r} \dot{ϕ} - {\dot{r}}^{2} \partial_{ϕ} n + r^{2} {\dot{ϕ}}^{2} \partial_{ϕ} n + 2 r^{2} \dot{r} \dot{ϕ} \partial_{r} n}{n r^{2}} .

Since we are interested in the general stationary case, we impose the fourth condition above, ensuing 0 ≤ ṙ ≤ ṙ_max and ϕ̇_min ≤ ϕ̇ ≤ ϕ̇_max, on Eq. (11) and Eq. (12), and find a non-singular refractive index distribution, n(r, ϕ), that satisfies them. Notice that we require the refractive index to be smoothly varying and C² continuous, i.e. ∇n ·e⃗ⁱ ≤ M, where M is a real valued number, and e⃗ⁱ 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, 23, 24].

The latter being said, the problem can be simplified further if we recall that a trapping orbit occurs when the radius r is bounded to a region of space, i.e. r_min ≤ r ≤ r_max, for all ϕ; whence ṙ(r_max) = ṫ(r_min) = 0, and 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, 22], light would behave similarly to that traveling in curved space-time. Such distribution can be constrained to the conditions listed previously.

This result is significant in terms of fabrication where novel photo-refractive [metal]materials could be used. The refractive index in these composites can be modulated by the intensity of the change-inducing beam; which can be controlled by shaping its transversal profile [18–21]. Accordingly several families of refractive index distributions can be achieved, e.g. Gaussian, Laguerre, or Bessel, which are all centro-symmetrical.

2.2. System stability and dynamic equillibrium

Since the refractive index is a function of the radius, Eq. (11) and Eq. (12) can be rewritten in the form (13):

\ddot{x} = - h (x, \dot{x}) - g (x),

\ddot{r} = (r^{2} {\dot{ϕ}}^{2} - {\dot{r}}^{2}) \partial_{r} ln n + r {\dot{ϕ}}^{2},

\ddot{ϕ} = - 2 \dot{r} \dot{ϕ} \frac{n + r \partial_{r} n}{n r},

the system’s symmetry now allows us to integrate Eq. (15), hence:

\dot{ϕ} = \frac{ℓ}{μ r^{2}}, μ = n^{2},

where 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:

\ddot{r} = (\frac{ℓ^{2}}{n^{4} r^{4}} r^{2} - {\dot{r}}^{2}) \partial_{r} ln n + \frac{ℓ^{2}}{n^{4} r^{4}} r .

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]; simplifying we get:

ℰ_{r} = \frac{d ℰ}{d t} = - \dot{r} h (r, \dot{r}) = \dot{r} (\frac{ℓ^{2}}{n^{4} r^{2}} - {\dot{r}}^{2}) \partial_{r} ln n .

By virtue of Bertrand’s theorem [15, 27], we know that ṙ ≠ 0, for all time t; then from premises 3 and 4 it follows that ṙ_min ≤ ṙ ≤ ṙ_max, where ṙ_min < 0; and from the definition of n(r) as a monotonous decreasing function it follows that ∂_r ln n ≤ 0. Therefore, as the radius decreases, the radial velocity goes to its minimum and ℰ_t ≤ 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 ℰ_t ≥ 0, thus the system is driven away from the equilibrium point and the amplitude of the path increases. In Fig. 2 we show ℰ_t 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 ℰ_t.

Fig. 2: Ratio of external energy supply, ℰ_t, for a radial symmetric system, where x = r and y = ṙ. Notice that there exists a x_c, critical radius, at wich for all phase space pairs {x, y|x ≥ x_c} ℰ_t = 0, which implies that the light geodesics do not describe an orbital motion.

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To conclude we study the stability of the orbits. by means of Lyapunov’s stability theory for the governing Eq. (17), and derive the constraints applicable to the refractive index mapping to attain stable orbital paths [15]. First, let us rewrite Eq. (17) as:

\begin{array}{l} \dot{r} = ϰ, \\ \dot{ϰ} = (r^{2} β^{2} - ϰ^{2}) χ (r) + r β^{2}, \end{array}

with β = ϕ̇ and χ(r) = ∂_r ln 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_o, and drop all higher order terms for δ and ε; we can rewrite Eq. (19) as a linear system of equations for the perturbation terms:

\begin{matrix} (\begin{matrix} \dot{δ} \\ \dot{ε} \end{matrix}) = (\begin{matrix} 0 & 1 \\ W & 2 V \end{matrix}) (\begin{matrix} δ \\ ε \end{matrix}) \\ V = - ϰ_{o} χ_{o}, W = 2 χ_{o} β^{2} r_{o} + χ_{o}^{'} (r_{o}^{2} β^{2} - ϰ_{o}^{2}), \end{matrix}

where χ_o and χ′_o are the linear terms of the Taylor expansion of χ(r) around r_o. Ensuing the eigenvalues of the matrix above are given by:

λ_{1, 2} = V \pm \sqrt{V^{2} + W} .

The geodesics will be stable if the real part of the 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:

{- \frac{\partial}{\partial r} χ (r) |}_{r = r_{o}} ϰ \leq 0, V^{2} + W \leq 0,

This constrained is met for all points r near the stability point r_o = 0, provided the radial velocity meets ṙ_o ≤ 0. In general, since the refractive index is monotonous decrescent, the system will be Lyapunov stable at points where 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_o = 0, and 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.

In summary, light confinement will occur as long as the conditions cited earlier apply. Under them the system will oscillate around the equilibrium point as depicted by the 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

In this section we explore three refractive index distributions based on Gaussian profiles. The first two mimic light trapped in the vicinity of a black hole, and the third case that of a particle moving in-between two planets.

3.1. Guassian distribution, single attractor

A Gaussian refractive index distribution has the form:

n = n_{a} e^{- r^{2} / σ^{2}} + n_{c} .

Then making it possible to constrain the maximum and minimum of the refractive index distribution to feasible values, e.g. 0.8n ≤ 3.8. Hence, given a range of values for the back-ground and maximum refractive index, n_c and n_a, respectively, we can evaluate different light confinement scenarios, see Fig. 3.

Fig. 3: Light path dynamics for Gaussian refractive index shape for different n_a and n_c. 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 r_min as n_c increases, and the orbit width reduces.

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As discussed in the previous section the orbital paths are open, while their shape and stability depend significantly on the initial conditions, n(r) and {r_o, ṙ_o, ϕ_o, ϕ̇_o}. 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 n_c alters the orbit, as seen from the plot of radii extrema in Fig. 4.

Fig. 4: Inner and outer radius as a function of the background refractive index n_c. As it increases we observe that the outer radius diminishes, while the inner increases.

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In Fig. 5 we show the 2D open orbit for a refractive index map where n_c = 0.8 and n_a = 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 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.

Fig. 5: Confinement orbit for light traveling in a Gaussian attractor where n_a = 3.0 and n_c = 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.

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Fig. 6: Phase space for Gaussian-like refractive index map with n_a = 3.0, n_c = 0.8. The plots show: (a) radial ṙ and angular ϕ̈ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle (c).

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3.2. Mexican hat distribution

We study a variation on the Gaussian trap to understand the potential modifications on the trapping orbits when large modifications ∼ 20% of the refractive index take place. The mathematical description of the mexican hat is:

n = (n_{a} - \frac{n_{d} r^{2}}{σ^{2}}) e^{- r^{2} / σ^{2}} + n_{c}

Using the same conditions as in the previous analysis we set n_c = 0.8, n_a = 3.0, and set n_d = 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.

Fig. 7: (a) Schematic of Mexican hat like refractive index map where n_a = 3.0 and n_c = 0.8 and depression is n_d = 0.2; (b) depicts the resulting trapped orbit in polar coordinates as calculated by the simulation routine.

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Fig. 8: Phase-space for a mexican hat refractive index map with n_a = 3.0, n_c = 0.8. The plots show: (a) radial ṙ and angular ϕ̇ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle (c).

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These variations in the trapping conditions could potentially be used to differentiate trapping in transient optical memories to allocated binary data.

3.3. Double attractors

Interested in observing phenomenologically trapping in a binary system we evinced a planar refractive index map resulting from the superposition of two Gaussian functions,

n = n_{a} (e^{- {(r - r_{off 1})}^{2} / σ^{2}} + e^{- {(r - r_{off})}^{2} / σ^{2}}) + n_{c} .

The resulting orbits are shown in Fig. 9. A binary system, however, presents a relatively higher sensitivity to modifications in the refractive index, where a distortion of Δn > 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.

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.

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3.4. The space-time equations, trapping

Briefly, we show that the same Gaussian solution explored in section 3.1 fulfills the trapping conditions in the full eikonal space-time, Eq. (7) through Eq. (9). The results are shown in Fig. 10.

Fig. 10: Phase space of the trapping orbit for n_a = 3.0, n_c = 0.8. The plots show: (a) radial ṙ and angular ϕ̇ velocities, (b) radial phase space, and (c) radial velocity as a function of the angle.

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In this case we observe a similar behaviour to that of the Gaussian trap in the stationary approximation, the difference in the amplitude values is due to normalization. An oscillatory behaviour in the radius and angular velocities is described in Figs. 10(b) and 10(c); as it would be expected from a centro-symmetrical refractive index mapping.

3.5. Material realization

Recent advancement in photo-refractive materials [18–21] could enable the construction of the light confinement devices discussed. For example, a prototype Gaussian trap could be fabricated by stacking different photo-refractive [meta]materials like As_xS_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

We have demonstrated that, under the paraxial description of light in heterogeneous media, it is possible to use centro-symmetrical refractive index distributions as attractors for light, which are able to perform light trapping or confinement in open orbits. These mappings are bound within fabrication constrains, and due to the paraxial approximation they are several times larger than the mean beam width and wavelength, ∼ 50 times larger. The proposed mappings have potential applications to transient optical memories, delays, concentrators, random cavities, and beam stirrers, to cite a few; which could enable the next generation of photonic integrated circuits, PICs, processors and computation systems. Due to their sensitivity to the change in the refractive index these devices could be employed as a sensor platform. Furthermore, since they are drawn in close analogy to celestial phenomena they could be used as laboratory setups to test a large variety of effects in celestial mechanics.

Acknowledgments

The authors wish to acknowledge David Ketcheson, Carlos Argaez, Pedro Guerra, and Jose Alcantara-Felix for their suggestions, corrections and valuable discussions while preparing this manuscript.

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A possible approach on optical analogues of gravitational attractors

Abstract

1. Introduction

2. Deriving the model

2.1. Path equations and the inverse problem

2.2. System stability and dynamic equillibrium

3. Results and discussion

3.1. Guassian distribution, single attractor

3.2. Mexican hat distribution

3.3. Double attractors

3.4. The space-time equations, trapping

3.5. Material realization

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Equations (25)

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