---
abstract: 'Fermat’s principle and variational analysis is used to analyze the trajectories of light propagating in a radially inhomogeneous medium with a singularity in the center. It is found that the light trajectories are similar to those around a black hole, in the sense that there exists a critical radius within which the light cannot escape, but spirals into the singularity.'
author:
- |
    M. Marklund, D. Anderson, F. Cattani, M. Lisak and L. Lundgren\
    *Department of Electromagnetics, Chalmers University of Technology,*\
    *SE–412  96 Göteborg, Sweden*
date: '()'
title: 'Fermat’s principle and variational analysis of an optical model for light propagation exhibiting a critical radius'
---

Introduction
============

There has recently, within the scientific community, been much interest focused on the extra-ordinary properties associated with Bose-Einstein condensates - clouds of atoms cooled down to nano-Kelvin temperatures where all atoms are in the same quantum state and macroscopic quantum conditions prevail [@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00; @Hau-etal-99].

One of the most dramatic experiments which has been performed in Bose-Einstein condensates is the demonstration of optical light pulses traveling at extremely small group velocities, e.g. velocities as small as 17 m/s have been reported [@Hau-etal-99].

In a physically suggestive application of the optical properties of Bose-Einstein condensates, it was recently suggested that they could be used to create, in the laboratory, dielectric analogues of relativistic astronomical phenomena like those associated with a black hole [@Leonhardt-Piwnicki-00].

The astrophysical concept of a black hole is one of the most fascinating and intriguing phenomena related to the interaction between light and matter through the curvature of space-time within the framework of general relativity. Black holes are believed to form when compact stars undergo complete gravitational collapse due to, e.g., accretion of matter from its surroundings (in general relativity, the pressure contributes to the gravitational mass of a fluid, and an increased pressure will, after a certain point, therefore only help to accelerate the collapse phase).

In ordinary dielectric media, unrealistic physical conditions would be required in order to demonstrate any of the spectacular effects of general relativity. However, in dielectric media characterized by very small light velocities, new possibilities appear. In particular, it was recently suggested that by creating a vortex structure in such a dielectric medium it is possible to mimic the properties of an optical black hole [@Leonhardt-Piwnicki-00]. What this study in fact suggested was that one could construct, by using the aforementioned vortex, an unstable photon orbit, much like the orbit at a radius $r = 3M$ in the Schwarschild geometry [@Misner-Thorne-Wheeler]. In order to construct an event horizon, one would need to supplement the vortex flow by a radial motion of the fluid [@Visser-00; @Leonhardt-Piwnicki-00b].

The vortex, which involves a rotating cylindrical velocity field, tends to attract light propagating perpendicular to its axis of rotation and to make it deviate from its straight path. In the model considered in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the increasing, in fact diverging, rotational velocity field of the vortex core attracts light by the optical Aharanov–Bohm effect and causes a bending of the light ray similar to that of the gravitational field in general relativity. It is also shown that there exist a critical radius, $r_{crit}$, from the vortex core with the properties that if light rays come closer to the core than $r_{crit}$,the light will fall towards the singularity of the vortex core. In this sense, the critical radius, $r_{crit}$, plays the role of an optical unstable photon orbit, analogous to the unstable orbit in the Schwarzschild geometry.

The basic physical effect involved in the analysis of the ray propagation in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00] is the fact that the refractive index, $ n$, of a medium changes when the medium is moving. In Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the medium is assumed to have a cylindrical vortex velocity field $ \bf V= V(\bf r)$ given by $${\bf V(r)} = \frac{W}{r}{\bf \widehat{\varphi}}$$ where $r$ is the radius from the vortex core centre, $\widehat{\varphi }$ is the azimuthal unit vector and $2\pi W$ is the vorticity.

Advanced physical concepts like Bose-Einstein condensates, vortex velocity fields and black holes are fascinating physical effects, which however may be difficult to present in a simplified manner for undergraduate students. Nevertheless, it is important to try to convey inspiration for and arouse curiosity in such phenomena. An interesting example of such an effort was recently made in Ref. [@McDonald-00], where the basic physical mechanism, in the form of a classical two level system was used to demonstrate the possibility of the very low light velocities observed in Bose- Einstein condensates. In a later work by the same author, Ref.  [@McDonald-01], an inspiring investigation is given of a simple mechanical model that exhibits a gravitational critical radius. The purpose of the present work is similar to that of Refs. [@McDonald-00; @McDonald-01]. We will try to discuss a simple classical example of a medium where the refractive index has a divergence at $r = 0$ and which exhibits some of the dramatic properties of the light behaviour around a black hole as discussed in Refs.. In addition we want to illustrate the power and beauty of the classical principle of Fermat and of variational methods in connection with this new fascinating concepts.

Fermat’s principle {#sec:Fermat}
==================

It is well known that light tends to be deflected towards regions with higher refractive index. Consider light propagating in a cylindrically symmetric medium where the refractive index increases towards the centre. In such a situation we expect the light path to look qualitatively as in Fig. 1. The actual light path is determined by Fermat’s principle i.e.$$\delta\int n({\bf r})ds=0$$ where$ ds$ is an infinitesimal element along the light ray. We will consider light propagation in a medium where the refractive index is of the qualitative form, cf Eq. (1) $$n(r)\simeq \left\{
  \begin{array} {r@{\quad as\quad}l}1 & r\gg r_{0}\\
  r_0/r & r\ll r_{0}\end{array} \right.$$

One possible realization of $n(r)$ with the desired properties is given by $$n^{2}(r) = 1 + \left(\frac{r_{0}}{r}\right)^2$$

Using Fermat’s principle and expressing the light path as the relation $\theta=\theta(r)$ where $\theta$ is the polar angle, Eq. (2) implies $$\delta\int{n(r)\sqrt{1+r^{2}\left(\frac{d\theta}{dr}\right)^2}}dr
  = 0 \ .$$

The Euler-Lagrange variational equation corresponding to Eq. (5) reads $$\frac{d}{dr}\left[ n(r)\frac{r^2 \, d\theta/dr}%
  {\sqrt{1 + r^2(d\theta/dr)^2}}\right]=0$$ which determines the trajectory of the light.

Solution of the light trajectory
================================

Equation (6) directly implies that $$\frac{r^2n(r) \, d\theta/dr }%
  {\sqrt{1 + r^2 (d\theta/dr)^2}} = r_{i}$$ where the constant $r_{i}$ is determined by initial conditions, i.e. the properties of the incident light.

Equation (7) is easily inverted to read $$\frac{d\theta}{dr}=\pm\frac{r_{i}}{\sqrt{r^4n^2(r) - r_{i}^2r^2}}$$

For a light ray incident as shown in Fig. 1, we clearly have $d\theta/dr <0$ (at least up to some minimum radius $ r=r_{min}$. It is illustrative to first consider the trivial case of a homogeneous medium with $n(r)\equiv1$. In this case Eq. (9) becomes $$\frac{d\theta}{dr}=-\frac{r_{i}}{\sqrt{r^4 - r_{i}^2r^2}}$$ which can easily be interpreted to yield the light path in the form $$\theta=\mathrm{arccot}\sqrt{\frac{r^2}{r_{i}^2} - 1}$$ or simpler $$r=\frac{r_{i}}{\sin \theta}$$ Clearly this is the straight line solution $$y=r_{i}$$ where the parameter $r_{i}$ plays the role of “impact parameter” or minimum distance from the centre.

Let us now consider the model variation for $n(r)$ as given by Eq. (5). Within this model Eq. (10) becomes $$\frac{d\theta}{dr} =
  - \frac{r_i}{\sqrt{r^4 - (r_i^2 - r_0^2)r^2}}$$ Clearly the solution of Eq. (13) will depend crucially on the relative magnitude of $r_i$ and $r_0$, i.e. the impact parameter relative to the characteristic radial extension of the inhomogeneity. Consider first the case when $r_0 < r_i$. Equation (13) can then be rewritten as $$\begin{aligned}
  \frac{b_1}{r_i}\frac{d\theta}{dr}
  &=& -\frac{b_1}{\sqrt{r^4 -b_1^2r^2}} \ , \\
  b_1 &\equiv& \sqrt{r_{i}^{2}-r_{0}^{2}} \ .\end{aligned}$$ Equation (15) is of the same form as Eq. (9) and we directly infer the following solution $$r = \frac{\sqrt{r_{i}^{2} - r_{0}^{2}}}%
  {\sin\left[\theta\sqrt{1-r_{0}^{2}/r_{i}^{2}}\,\right]}$$

As $\theta\rightarrow0$, we still have asymptotically $$r\simeq\frac{r_{i}}{\sin \theta}$$ However, the trajectory is now bending towards the origin and the minimum distance occurs at the polar angle $\theta=\theta_{m}$ given by $$\theta_m = \frac{\pi}{2}\frac{1}{\sqrt{1-r_0^2/r_i^2}}$$

The corresponding minimum distance, $r_m$ is $$r_m \equiv r{\theta_m} = \sqrt{r_i^2 - r_0^2}$$

We also note that the trajectory is symmetrical around the angle $\theta_{m}$ and that the asymptotic angle of the outgoing light ray is $$\theta_{\infty}\equiv \lim_{r\rightarrow\infty} \theta(r)
  =\frac{\pi}{\sqrt{1 - r_0^2/r_i^2}} = 2 \theta_m$$

The solution given by eq.(10) describes a trajectory which is bent towards the centre of attraction at $ r=0$. Depending on the ratio $r_{0}/r_{i}$, the trajectory is either more or less bent or may even perform a number of spirals towards the centre before again turning outwards and escaping, cf. Fig. 2. The number of turns, $N$,which the light ray does around the origin before escaping is simply $$N = \left\lfloor \frac{2\theta_{m}}{2\pi} \right\rfloor
    = \left\lfloor \frac{1}{2\sqrt{1-r_{0}^{2}/r_{i}^{2}}} \right\rfloor$$ where $\left\lfloor x \right\rfloor$ denotes the largest integer less than $x$.

Let us now consider the special case when the impact parameter equals the characteristic width of the refractive index core i.e.$r_{i}=r_{0}$. The equation for the trajectory now simplifies to $$\frac{d\theta}{dr} = -\frac{r_{i}}{r^{2}}$$ with the simple solution $$\theta=\frac{r_{i}}{r}$$ i.e. the trajectory describes a path in the form of Arkimede’s spiral as the light falls towards the origin, cf. Fig. 3. The form of the light trajectory in the situation when $ r_{i}< r_{0}$ is now obvious, it will spiral into the singularity more or less directly depending on the magnitude of the ratio $r_{0}/r_{i}>1$.

The actual trajectory in this case is given by a slight generalization of Eq. (16) viz $$r=\frac{\sqrt{r_{0}^{2}-r_{i}^{2}}}{\sinh
  [\theta\sqrt{1-r_{i}^{2}/r_{0}^{2}}]}$$

This solution does indeed convey the expected behaviour, the trajectory spirals monotonously into the singularity of the refractive index.

Final comments
==============

The present analysis is inspired by recent discoveries and discussions about light propagation in Bose-Einstein condensates where extremely low light velocities can be obtained. This has triggered speculations about possible laboratory demonstrations of effects, which normally are associated with general relativistic conditions. In particular, it has been suggested that it may be possible to create the analogue of a black hole using a divergent in-spiral of a Bose-Einstein condensate.

In the present work we have analyzed a simple classical example of light propagation as determined by Fermat’s principle in a medium characterized by a radially symmetric refractive index. In analogy with the variation of the vortex velocity field suggested in [@Leonhardt-Piwnicki-00], the refractive index is here assumed to diverge towards the centre. This classical example exhibit some of the characteristic properties of light propagating around a black hole where the gravitational attraction deflects the light and where, under certain conditions, the light may be “swallowed” by the black hole.

In the example analyzed here, the unstable photon orbit of the Schwarzschild black hole is similar to the characteristic radius, $r_0$, of the refractive index variation, which together with the impact parameter, $r_i$, of the incident light completely determines the light trajectory. If $r_0 < r_i$, the light is more or less deflected, but ultimately escapes. However, if $r_0 \geq r_i$, the light spirals into the singularity.It should be cautioned that this result depends crucially on the presence of the singularity in the refractive index (4). If this is removed the in-spiraling photon orbit will eventually turn and start spiraling outwards. In the recent discussion about the possibility of generating analogues of optical black holes in Bose-Einstein condensates, it has been suggested that the vortex motion must also have a velocity component in the radial direction.

In the classical model considered here, a number of additional physical effects will obviously affect the light path when it comes close to the axis and will in fact remove the mathematical singularity. Nevertheless, the model provides a simple example of light dynamics, which resembles some of the properties of light propagation around a black hole. Furthermore,the investigation is based on Fermat’s principle and variational analysis, in this way illustrating the use of classical methods in connection with very new and fascinating concepts at the front line of modern research.

Figure Captions. {#figure-captions. .unnumbered}
================

Fig.1 Qualitative plot of a light ray trajectory in a cylindrically symmetric medium with a refractive index, which increases towards the centre.\
Fig.2 Light trajectories for the refractive index model of eq.() for different impact radii $r_{i}>r_{0}$\
Fig.3 Light trajectory in the case of $r_{i}=r_{0}$, the spiral of Arkimede.\

[99]{}

U. Leonhardt and P. Piwnicki, Phys. Rev. A **60**, 4301 (1999)

U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **84**, 822 (2000)

L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, Nature (London) **397**, 594 (1999)

C. Misner, K. S. Thorne and J. A. Wheeler *Gravitation* (Freeman, 1973) M. Visser, Phys. Rev. Letters **85**, 5252 (2000)

U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **85**, 5253 (2000)

K. T. Mc Donald, Am. J. Phys. **68**, 293 (2000) K. T. Mc Donald, Am. J. Phys. **69**, 617 (2001)
