## Abstract

The application of the Hamilton-Jacobi equation to isotropic optical materials leads to the well-known eikonal equation which provides the surfaces normal to the ray trajectories. The symmetry between the coordinates x=(x 1,x 2,x 3) and the momenta p=(p 1,p 2,p 3) in the Hamiltonian formulation of Geometrical Optics establishes a dual Hamilton-Jacobi equation for “wavefronts” in the momentum space. This equation is also an eikonal equation when the refractive index distribution has spherical symmetry. In this case, another spherical symmetric refractive index distribution may exist such that the ray trajectories in the coordinates and momentum space are exchanged (examples of this case are given: Maxwell fish-eye, Eaton lens and Luneburg lens). The relationship between the wavefronts in the coordinate and momentum space is also analyzed. Curved orthogonal coordinates are considered as well.

© 2006 Optical Society of America

1. Introduction

In the Hamiltonian formulation of geometrical optics, rays are the solutions to a Hamiltonian system of equations. There are different ways of establishing such a Hamiltonian system. Consider this one [1]:

$dxidt=∂H∂pidpidt=∂H∂xii=1,2,3$

xi and pi are the unknown functions of the parameter t, and they define a ray-trajectory. The xi variables are called the configuration variables (or coordinates) and the pi are called the conjugate variables (or momenta). A ray-trajectory in the phase space is a curve given by the functions x 1(t), x 2(t), x 3(t), p 1(t), p 2(t), p 3(t). We are usually only interested in the projection of this phase space trajectory in the coordinate space x 1-x 2-x 3. In the Hamiltonian formulation used here, not all of the solutions to Eq. (1) are possible ray trajectories, rather, only those consistent with H(xi ,pi )=0. H is a function of the six variables xi , pi called the Hamiltonian function. It can be proven [2, 3] that the solutions to Eq. (1) fulfil that H=constant along their trajectories, i.e., the function H is a first integral (also called a constant of motion) of the Hamiltonian system. The rays are merely those solutions to Eq. (1) contained in the subspace H=0. The Hamiltonian function for an isotropic medium can be written as

$H(x1,x2,x3,p1,p2,p3)=(n2(x1,x2,x3)−p12−p22−p32)⁄2$

where x 1,x 2,x 3 are Cartesian coordinates the conjugate variables p 1, p 2, p 3 of which are the optical direction cosines, and n(x 1,x 2,x 3) is the refractive-index distribution. Hereinafter, x=(x 1, x 2, x 3) and p=(p 1, p 2, p 3). Different forms of the Hamiltonian function are related with different parameterizations along the ray trajectories. References [1, 4] give more details about this subject.

#### 1.1 Hamilton-Jacobi equation

The Hamilton-Jacobi equation is found in the Canonical Transformation theory within the Hamiltonian formulation [3]. Let Xi , Pi be the coordinates after a canonical transformation and let V(xi , Xi ) be its generating function. This type of generating function, which depends only on the old and new coordinates, is called a Type 1 generating function [5]. Henceforth, we will assume that xi , Xi form a set of independent variables, which is needed for the existence the Type 1 generating function. There are also Type 2, Type 3, and Type 4 generating functions, for which we need to assume respectively that xi , Pi ; pi , Xi and pi , Pi , are sets of independent variables. According to the Canonical Transformation theory, Type 1 transformations from x i, p i to X i, P i are derived from the equations

$pi=∂V∂xiPi=−∂V∂Xi$

The Hamilton-Jacobi equation is found when we seek a canonical transformation such that the Hamiltonian function is identical to a constant in the new (transformed) coordinates. With such transformation Eq. (1) becomes trivial. When the canonical transformation is given by a Type 1 generating function, the resulting Hamilton-Jacobi equation is

$H(x1,x2,x3,∂V∂x1,∂V∂x2,∂V∂x3)=0$

This is a non linear partial differential equation the complete integral of which is a function V dependent on x i and on 3 constants that are the X i of the generating function theory [3, 6]. There are many possibilities to choose the integration constants some of which lead to the many possible characteristic functions of an optical system [7].

We are only going to consider isotropic media, for which the Hamilton-Jacobi Eq. (4) is the eikonal equation (∇V)2=n 2(x) (where ∇ is the gradient operator ∇V=(∂V/∂x 1, ∂V/∂x 2,∂V/∂x 3)), as can be easily checked by combining Eqs. (4) and (2).

## 2. Hamilton-Jacobi equation in the momentum space

The same procedure to obtain the classic Hamilton-Jacobi equation can be applied by using a Type 3 generating function W(p i, X i) instead of Type 1. In this case the transformations from x i, p i to X i, P i are given by [5]

$xi=−∂W∂piPi=−∂W∂Xi$

and the resulting dual Hamilton-Jacobi equation is

$H(−∂W∂p1,−∂W∂p2,−∂W∂p3,p1,p2,p3)=0$

Equation (6) can also be obtained by first applying the canonical transformation given by X i=p i, P i=-x i and then using a Type 1 generating function to establish the classic Hamilton-Jacobi equation. Let’s call it a Hamilton-Jacobi equation in the momentum space.

If n(x) is the refractive-index distribution, then the dual Hamilton-Jacobi equation is n 2(-∇ p W)-p 2=0; where ∇p W=(∂W/∂p 1, ∂W/∂p 2, ∂W/∂p 3). The Hamilton-Jacobi equation is also an eikonal equation in Cartesian coordinates if the refractive index distribution has spherical symmetry (see Section 3) and is invertible, i.e., if n is only a function of |x|, nn(|x|), and it is a one-to-one correspondence between n and |x|, which is fulfilled if n(|x|) is a monotonic function. The existence of the inverse function of n guarantees that it is possible to extract the value of |∇ p W| with uniqueness from the dual Hamilton-Jacobi equation n 2(-∇ p W)-p 2=0, and thus write it as the eikonal (∇p W)2={n -1(|p|)}2, where n -1 is the inverse function of n, i.e., n -1(n(r))=r.

The functions V(x) and -W(p) (we are now omitting their dependence on the variables X i) are a Legendre transform pair, since they are different generating function types of the same canonical transformation [5]. This means (see for instance Ref. [8]) that

$V(x)−W(p)=x·p$

Eq. (7) must be fulfilled when we substitute in Eq. (7), either x as a function of p with x=-∇p W(p) [which is the left-hand side equation in Eq. (5)], or p as a function of x with p=∇V(x) [which is the left-hand side equation in Eq. (3)]. This is a well-known result when functions V and W are, respectively, Hamilton’s point and mixed characteristic functions [9].

Equation (7), together with the left-hand side equation in Eqs. (3) and (5), relate the wavefronts in the x space (i.e., V(x)=constant surfaces) to the “wavefronts” in the p space (W(p)=constant surfaces). For instance, a wavefront given by W(p)=W 0 in the p space is represented by the equation V(x)-x·∇V(x)=W 0 in the x space.

A Type 4 generating function T(p i, P i) can also be used to get the dual Hamilton-Jacobi equation, for which the coordinate transformations from x i, p i to X i, P i are

$xi=−∂T∂piXi=∂T∂Pi$

and the resulting Hamilton-Jacobi equation in the momentum space restates Eq. (6)

$H(−∂T∂p1,−∂T∂p2,−∂T∂p3,p1,p2,p3)=0$

The functions V(x, X) and -T(p, P), where X=(X i) and P=(P i), are a Legendre transform pair since they are different generating function types of the same canonical transformation [5], but now also involving the variables X i and P i in the Legendre transformation.

$V(x,X)−T(p,P)=x·p−X·P$

As Type 3 and 4 generated the same dual Hamilton-Jacobi equation, Type 2 generating functions arrive at the usual Hamilton-Jacobi Eq. (4) as Type 1 generating functions do. Since in this paper we are only interested in the Hamilton-Jacobi equation, then only Type 1 and Type 3 generating functions need be examined.

Because the solutions of Hamilton-Jacobi Eqs. (4) and (6) are related by a Legendre transform, any of them can be solved through the other. As an example, consider the refractive index distribution given by n 2=1-$x12$ . The eikonal equation and its corresponding dual Hamilton-Jacobi equation in the momentum space are:

$(∂V∂x1)2+(∂V∂x2)2+(∂V∂x3)2=1−x12(∂W∂p1)2=1−p12−p22−p32$

The eikonal equation involves the 3 partial derivatives of V, while only ∂W/∂p 1 appears in the dual Hamilton-Jacobi equation in the momentum space. This equation can be easily solved, giving

$±W=12(1−p22−p32)arcsin(p11−p22−p32)+12p11−p12−p22−p32+f(p2,p3)$

where f(p 2,p 3) is an arbitrary function. Now V can be obtained with a Legendre transformation using Eq. (7) (V(x)=x·p+W(p)) and x=-∇p W(p). Note that these 2 equations give the definition of the function V(x) in parametric form (the parameters being the components of p).

## 3. Isotropic media with spherical symmetry

Both Hamilton-Jacobi Eqs. (4) and (6) lead to eikonal-type equations when the media has spherical symmetry. This result establishes obviously important symmetries between the media.

Fig. 1. Phase space trajectories of rays in the Maxwell fish-eye lens projected on the coordinate plane x 1,x 2 (left) and projected on the momentum plane p 1,p 2 (right). These trajectories coincide with the ones of the Eaton lens if the coordinates and momenta are exchanged (xp). The same color of the dots indicates corresponding points of the trajectories. The bold arrow tip is the vector p and the hollow arrow tip is the vector x.

For instance, consider the ray trajectories in the phase space x-p for the Maxwell fish-eye lens (refractive index distribution n(|x|)=2/(1+x 2)). The projection of these trajectories in the coordinate space x are what we usually call the ray trajectories. These projections in the x space are the curves orthogonal to surfaces V(x)=constant, where V is a solution of the eikonal equation (∇V)2=n 2(|x|)=4/(1+ x 2)2. The projection of the trajectories in the momentum space p can also be calculated with an eikonal equation. In particular, the projection of the ray trajectories in the p space are orthogonal to surfaces W(p)=constant, satisfying the eikonal equation (∇p W)2={n -1(|p|)}2, where n -1 is the inverse function of n, i.e., n -1(n(r))=r. For the Maxwell fish-eye case this inverse function is n -1(|p|)=(2/|p|-1)1/2. This is the same function as the refractive index distribution of the Eaton lens, which is n E(|x|)=(2/|x|-1)1/2 [10]. This means that the phase space trajectories of rays in the Maxwell fish eye, when projected into the p space, follow the same tracks as the x space projection of rays of the Eaton lens and vice versa, i.e., the p space projection of phase space trajectories of the Eaton lens coincides with the x space projection of the rays in the Maxwell fish-eye lens. We will say that the Maxwell fish-eye and the Eaton lenses are a pair of dual lenses. Incidentally, the Hamiltonian function associated with the Eaton lens is identical to the Hamiltonian system of the Kepler problem in Mechanics, which establishes the connection between the Kepler problem in Mechanics and the Maxwell fish eye found by Buchdahl [11].

In a system with spherical symmetry we can represent a generic ray by choosing x 3=0 and p 3=0 without loss of generality. Fig. 1(left) shows the trajectories of rays on the x 1,x 2 plane and the trajectories of the same rays on the p 1,p 2 plane (right) for the Maxwell fish-eye lens. These trajectories coincide with the Eaton lens ray trajectories in the coordinate space (right) and in the momentum space (left).

To be precise, the Eaton lens refractive index distribution is defined as n(|x|)=(2/|x|-1)1/2 inside the sphere x 2≤1 and equal to 1 outside it, while the dual lens of the Maxwell fish eye is the distribution n(|x|)=(2/|x|-1)1/2 in the sphere x 2≤4 and it is undefined elsewhere. The Eaton lens has the property of retro-reflecting any ray in the incoming direction (Fig. 2).

Fig. 2. Retro reflected rays in a Eaton lens with index n=1 outside the sphere x 2=1.

Cornbleet noted the existence of these pairs of dual spherical symmetric lenses, the refractive index distribution of one being the inverse function of the other, so that they can be obtained by exchanging n and the radius (n↔|x|) [10]. Cornbleet also found that the eikonal solutions of both lenses are related by means of a Legendre transformation. What is shown now is that the trajectories in the momentum space for one of the lenses are the trajectories in the coordinate space for the other.

Since these isotropic media have spherical symmetry in the x space then a 2 axis skew conservation must hold, and since they are also spherical symmetric in the p space, another equivalent 2-axis skew conservation must also hold for the trajectories in the momentum space. When calculated it is found that both conservations are the same. It can be written as x×p=constant along the ray trajectories (× denotes vector product), which is a symmetric expression with respect to x and p.

Ray trajectories in the coordinate space are always tangent to the vector p, as can be easily seen in with the Eq. (1) and Eq. (2), which establishes dx/dt=p. Spherical symmetric systems also fulfil that ray trajectories in the momentum space are tangent to the vector x as can be seen with the same equations and noting that ∇n(|x|) is parallel to x. Moreover, as well as the vector p is always n(x) times the unit tangent to the ray trajectory in coordinate space, in spherical symmetric systems, the vector x is also n -1(p) times the unit tangent to the ray trajectory in the momentum space.

#### 3.1 Luneburg lens

Another remarkable example is the Luneburg lens [12]. Its refractive index distribution n(|x|)=(2- x 2)1/2 (inside the sphere x 2=1) is such that its inverse function is itself, i.e., n -1(|x|)=n(|x|). As a consequence, the phase space trajectories of rays in the Luneburg lens follow the same tracks when projected on the p space as when projected on the x space because both eikonal equations (∇p W)2={n -1(|p|)}2 and (∇V)2=n 2(|x|) are the same equation. (see Fig. 3). Even more remarkable is that the projection of a ray on a plane x i-x j or a plane p i-p j follows the same ellipse, although not in the same part of the ellipse, as can be easily understood by noting that p 2>1 when x2<1 because n(|x|)>1 for x 2<1. The vector x and the vector p at a given point of the trajectory in one of these planes are conjugate diameters of the same ellipse. One property of the ellipse (due to Apollonius of Perga) establishes that the sum of the squares of conjugate diameters is constant. For the case of the Luneburg lens this is x 2+p 2=2, which is simply the eikonal equation for the Luneburg lens.

Fig. 3. Phase-space trajectories of rays with x 3=p 3=0 in the Luneburg lens; projected on the x 1, x 2 plane (left) and projected on the momentum plane p 1, p 2 (right). The bold arrow tip is vector p and the hollow arrow tip is vector x.

Figure 3(left) shows the trajectories of rays on the x 1,x 2 plane and the trajectories of the same rays on the p 1,p 2 plane (right). The rays leave from point x 1=-1, x 2=0 on the x 1,x 2 plane and follow an elliptical trajectory. They cross the circle (x 1)2+(x 2)2=1 parallel to the x 1 axis, i.e., with p 2=0. In the p 1,p 2 plane, the rays are the same ellipses, and thus they are parallel to the abscissa axis at the circle of radius 1. All these trajectories converge at the point p 1=1, p 2=0.

Fig. 4. Trajectories of rays in the Luneburg lens in the plane x 3=0 (n=1 for x 2 >1).

The refractive index of the media outside the sphere x 2=1 is n=1 for the Luneburg lens, so that the ray trajectories are straight lines outside that sphere (Fig. 4). If the medium outside this sphere also follows the law n(|x|)=(2-x 2)1/2, then the trajectories of the rays are closed ellipses in both spaces. The ray trajectories can be easily calculated with Eq. (1), when H2-x 2-p 2:

$dxidt=pidpidt=−xii=1,2,3$
$d2xidt2+xi=0d2pidt2+p1=0i=1,2,3$

These solutions are valid for x 2≤1 in the Luneburg lens and valid everywhere else when the law n(|x|)=(2-x 2)1/2 extends to all points. The solutions, i.e, the phase space ray trajectories in parametric form can be written as

$(x1x2x3p1p2p3)=(costsint−sintcost)(x10x20x30p10p20p30)$

where t is a parameter along the ray trajectories and x i0, p i0 are the initial conditions. Equation (15) can be written shorter using complex vector notation as x+i p=(x 0+i p 0)exp(-it), where i2=-1 (thus, the ray trajectories in coordinate space are simply Re{(x 0+ip 0)exp(-it)}). The 6 initial conditions are not completely free, since they must fulfil that H=0, i.e., they must fulfil 2-(x 10)2-(x 20)2-(x 30)2-(p 10)2-(p 20)2-(p 30)2=0. Rays in Fig. 4 depart (t=0) from the point x 0=(-1,0,0) with p 0 ranging from (0,-1,0) to (0,1,0) and fulfilling (p 0)2=1. Equation (15) for t=π/2 establishes x(π/2)=p 0 and p(π/2)=-x 0. Then, x(π/2) is in the circle x 2=1 ranging from (0,-1,0) to (0,1,0) and p(π/2)=(1,0,0). These are the coordinates of the rays in Fig. 4 when they leave the circle x 2=1.

By eliminating t in a pair of Eq. (15) we can easily find the integral invariants in the Luneburg ray trajectories [13]. In particular, we can check that the projections of the ray trajectories onto any x i-p i plane are circles:

$xi2+pi2=xi02+pi02i=1,2,3$

The projection on any plane of configuration variables x i-x j or conjugate variables p i-p j or a combination of variables x i-p j, ij are the ellipses given by

$(xixj0+pjpi0)2+(xipj0−pjxi0)2=(xi0xj0+pj0pi0)2$
$(xipj0−xjpi0)2+(−xixj0−xjxi0)2=(xi0pj0+xj0pi0)2$
$(pipj0−pjpi0)2+(−pixj0+pjxi0)2=(xi0pj0−xj0pi0)2$

Note that Eqs. (18) and (19) represent the same ellipse but on different planes [x i-x j for Eq. (18) and p i-p j for Eq. (19)], i.e., the projection of the phase space trajectory of a ray is same curve in the planes x i-x j or in the planes p i-p j.

## 4. Non Cartesian orthogonal coordinates

Consider now a generalized orthogonal coordinate system ξ=(ξ 1,ξ 2,ξ 3) given by 3 functions ξ i(x). For any coordinate transformation from x to ξ, it is possible to find a set of conjugate momenta ζ=(ζ 1, ζ 2, ζ 3) such that the phase space transformation from (x,p) to variables (ξ, ζ) is canonical [3]. Unlike the new coordinates ξ, which only depend on the old ones x, the new momenta ζ depend in general of both old coordinates and momenta. The Hamiltonian function in the new phase space coordinates is given by [1]

$H(ξ1,ξ2,ξ3,ζ1,ζ2,ζ3)=n2(ξ1,ξ2,ξ3)−ζ12h12(ξ1,ξ2,ξ3)−ζ22h22(ξ1,ξ2,ξ3)−ζ32h32(ξ1,ξ2,ξ3)$

where h 1, h 2, h 3 are the three scale factors of the coordinate transformation, i.e., h i=1/|∇ξi|. The function n is the refractive index distribution in the new coordinates n(ξ)=n(x(ξ)). In general h i are functions of configuration coordinates ξ. We are going to assume that the scale factors are equal for at least one point, let’s say the point ξ102030, i.e., h 1102030)=h 2102030)=h 3102030). Note that if there is no point ξ102030, with this property we can do an additional canonical transformation scaling up or down each one of the coordinates ξ123 (since we want the transformation to be canonical, the corresponding conjugate momenta would result inversely scaled with respect to ζ1, ζ 2, ζ 3.)

The eikonal equation is derived similarly as Eq. (4). Since the transformation from (x,p) to variables (ξ, ζ) is canonical, the eikonal equation is obtained by replacing (ζ1, ζ2, ζ 3) by (∂V/∂ξ 1, ∂V/∂ξ 2, ∂V/∂ξ 3) in the expression H(ξ, ζ)=0, i.e.,

$1h12(ξ1,ξ2,ξ3)(∂V∂ξ1)2+1h22(ξ1,ξ2,ξ3)(∂V∂ξ2)2+1h32(ξ1,ξ2,ξ3)(∂V∂ξ3)2=n2(ξ1,ξ2,ξ3)$

Of course, the result can again be written as (∇V)2=n 2(ξ), because the gradient operator in ξ-coordinates is ∇V=((1/h 1)∂V/∂ξ 1, (1/h 2)∂V/∂ξ 2, (1/h 3)∂V/∂ξ 3) [14].

The Hamilton-Jacobi equation in the momentum space with the new coordinate system is obtained by replacing (ξ123) by (-∂W/∂ζ 1, -∂W/∂ζ 2, -∂W/∂ζ 3) in the expression H(ξ, ζ)=0:

$∑i=13ζi2hi2(−∂W∂ζ1,−∂W∂ζ2,−∂W∂ζ3)=n2(−∂W∂ζ1,−∂W∂ζ2,−∂W∂ζ3)$

The symmetry between the x and -p variables found in spherical symmetric systems seems to be dependent on the particular coordinate system (Cartesian). It seems also that other refractive index distributions may have a coordinate system in which both Hamilton-Jacobi equations are eikonal equations (similarly as for given coordinate system there are refractive index distributions which make the eikonal equation separable [15]). As in the Cartesian coordinate case, we are interested in cases in which Eq. (22) is an eikonal-type equation as Eq.(21), that is, when Eq. (22) can be written as

$1h12(ζ1,ζ2,ζ3)(∂W∂ζ1)2+1h22(ζ1,ζ2,ζ3)(∂W∂ζ2)2+1h32(ζ1,ζ2,ζ3)(∂W∂ζ3)2=f2(ζ1,ζ2,ζ3)$

being f an arbitrary function. Is this possible?. Many solutions can be found if the refractive index is anisotropic, but this is not our case of interest. For the isotropic case, at least one solution occurs when the scale factors are such that h 1=h 2/k 2=h 3/k 3 where k 2 and k 3 are 2 constants. These constants k 2 and k 3 must be equal to 1, because h 1102030)=h 2102030)=h 3102030), i.e., the 3 scale factors are the same function. Then the Hamilton-Jacobi Eq. (21) can be written as

$(∂V∂ξ1)2+(∂V∂ξ2)2+(∂V∂ξ3)2=η2(ξ1,ξ2,ξ3)$

where η is the function η123)=n123h 1123). The Hamilton-Jacobi equation in the momentum space Eq. (22) is

$ζ12+ζ22+ζ32=η2(∂W∂ζ1,∂W∂ζ2,∂W∂ζ3)$

which is an equation. of similar type as Eq. (24) if η123) is a function solely of the variable ξ2=$ξ12$+$ξ22$+$ξ32$, and is invertible.

Because h 1=h 2=h 3 the coordinate transformation from x to ξ, is conformal [16]. Inversions and similarities are the only possible conformal transformations in 3D Euclidean space [17]. We conclude that the Hamilton-Jacobi equation in momentum space can also be an eikonal equation in a curved orthogonal coordinate system, when this coordinate system can be obtained from Cartesian coordinates by means of an inversion, a similarity or a combination of both with an additional constant scaling factor for each configuration variable if necessary. Moreover, the refractive index distribution times the scale factor function h 1(ξ) must be a function solely of the variable ξ2, and must be invertible.

## 5. Conclusions

Within the framework of Geometrical Optics the ray trajectories can be obtained as the curves orthogonal to the wavefront surfaces, which are the solutions of the eikonal equation. These are the trajectories in the coordinate space x 1, x 2, x 3. The eikonal equation is a particular Hamilton-Jacobi equation found in Geometrical Optics when isotropic media are analyzed. Using the symmetry between the coordinates and the momenta of the Hamiltonian formulation we have established the equivalent Hamilton-Jacobi equation for the ray trajectories in the momentum space p 1, p 2, p 3. The “wavefronts” in this momentum space are related to the conventional wavefronts by means of a Legendre transformation.

For spherically symmetric refractive-index distributions, both Hamilton-Jacobi equations are eikonal equations, which means that there are couples of refractive indices such that the ray trajectories in the coordinate space in one of them are ray trajectories in the momentum space for the other, and vice versa. One of these couples is the Maxwell Fish-eye lens and the Eaton lens. The pair of Luneburg lens is itself, i.e., the ray trajectories in the coordinate and the momentum spaces are the same.

When orthogonal curved coordinate systems other than Cartesian are used, both Hamilton-Jacobi equations have different form due to the scale factors functions, but the duality remains the same, i.e., any of these equations can be solved through the other plus a Legendre transformation. The Hamilton-Jacobi equation in the momentum space is also an eikonal equation in the new variables, whenever the curved coordinate system can be obtained by inversions and similarities from the Cartesian coordinate system plus an arbitrary constant factor for each of the new variables.

The fact that sometimes both Hamilton-Jacobi equations are eikonal equations is a mathematical curiosity that simplifies solving one eikonal equation, once the other has been solved. As far as we know there is no physical implication of this fact.

## Acknowledgments

This study was supported by the European Union project TST3-CT-2003-506316 and the TEC2004-04316 and PCI2005-A9-0350 projects of the Spanish Ministerio de Educación y Ciencia. We thank William A. Parkyn, Jr. for his help in editing the manuscript. We also thank the reviewers for their comments, which helped very much to improve this paper.

1. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005).

2. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).

3. J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach, (Cambridge University Press, 1998).

4. G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. 59, 1130–1140 (1991). [CrossRef]

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6. O. N. Stravoudis, The Optics of Rays, Wavefronts and Caustics, (Academic, New York, 1972).

7. M. A. Alonso and G. W. Forbes, “Generalization of Hamilton’s formalism for geometrical optics,” J. Opt. Soc. Am. A 12, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744. [CrossRef]

8. Wikipedia contributors, “Legendre transformation,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Legendre_transformation&oldid=66180751 (accessed May7, 2006).

9. M. Born and E. Wolf, Principles of Optics, 5th ed, (Pergamon, Oxford, 1975).

10. S. Cornbleet, Microwave and Geometrical Optics, (Academic, 1994).

11. H. A. Buchdahl, “Kepler problem and Maxwell fish-eye,” Am. J. Phys. , 46, 840–843 (1978). [CrossRef]

12. R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles1964).

13. H. A. Buchdahl, “Luneburg lens: unitary invariance and point characteristic,” J. Opt. Soc. Am. 73, 490- (1983), http://www.opticsinfobase.org/abstract.cfm?URI=josa-73-4-490. [CrossRef]

15. H. A. Buchdahl, “Rays in gradient-index media: separable systems,” J. Opt. Soc. Am. 63, 46- (1973), http://www.opticsinfobase.org/abstract.cfm?URI=josa-63-1-46. [CrossRef]

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17. D. E. Blair, Inversion Theory and Conformal Mapping, (American Mathematical Society, 2000).

### References

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1. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005).
2. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).
3. J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach, (Cambridge University Press, 1998).
4. G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]
5. Wikipedia contributors, "Canonical transformation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June 17, 2006).
6. O. N. Stravoudis, The Optics of Rays, Wavefronts and Caustics, (Academic, New York, 1972).
7. M. A. Alonso and G. W. Forbes, "Generalization of Hamilton's formalism for geometrical optics," J. Opt. Soc. Am. A 12, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744.
[CrossRef]
8. Wikipedia contributors, "Legendre transformation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Legendre_transformation&oldid=66180751 (accessed May 7, 2006).
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10. S. Cornbleet, Microwave and Geometrical Optics, (Academic, 1994).
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[CrossRef]
12. R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964).
13. H. A. Buchdahl, "Luneburg lens: unitary invariance and point characteristic," J. Opt. Soc. Am. 73, 490- (1983).
[CrossRef]
15. H. A. Buchdahl, "Rays in gradient-index media: separable systems," J. Opt. Soc. Am. 63, 46- (1973).
[CrossRef]
16. D. J. Struik, Lectures on Classical Differential Geometry, (Dover, New York, 1988).
17. D. E. Blair, Inversion Theory and Conformal Mapping, (American Mathematical Society, 2000).

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G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

#### 1978 (1)

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

#### Buchdahl, H. A.

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

#### Forbes, G. W.

G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

#### Am. J. Phys. (2)

G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

#### Other (15)

R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964).

H. A. Buchdahl, "Luneburg lens: unitary invariance and point characteristic," J. Opt. Soc. Am. 73, 490- (1983).
[CrossRef]

H. A. Buchdahl, "Rays in gradient-index media: separable systems," J. Opt. Soc. Am. 63, 46- (1973).
[CrossRef]

D. J. Struik, Lectures on Classical Differential Geometry, (Dover, New York, 1988).

D. E. Blair, Inversion Theory and Conformal Mapping, (American Mathematical Society, 2000).

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O. N. Stravoudis, The Optics of Rays, Wavefronts and Caustics, (Academic, New York, 1972).

M. A. Alonso and G. W. Forbes, "Generalization of Hamilton's formalism for geometrical optics," J. Opt. Soc. Am. A 12, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744.
[CrossRef]

Wikipedia contributors, "Legendre transformation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Legendre_transformation&oldid=66180751 (accessed May 7, 2006).

M. Born, and E. Wolf, Principles of Optics, 5th ed, (Pergamon, Oxford, 1975).

S. Cornbleet, Microwave and Geometrical Optics, (Academic, 1994).

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005).

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).

J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach, (Cambridge University Press, 1998).

### Cited By

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### Figures (4)

Fig. 1.

Phase space trajectories of rays in the Maxwell fish-eye lens projected on the coordinate plane x 1,x 2 (left) and projected on the momentum plane p 1,p 2 (right). These trajectories coincide with the ones of the Eaton lens if the coordinates and momenta are exchanged ( x p ). The same color of the dots indicates corresponding points of the trajectories. The bold arrow tip is the vector p and the hollow arrow tip is the vector x .

Fig. 2.

Retro reflected rays in a Eaton lens with index n=1 outside the sphere x 2=1.

Fig. 3.

Phase-space trajectories of rays with x 3=p 3=0 in the Luneburg lens; projected on the x 1, x 2 plane (left) and projected on the momentum plane p 1, p 2 (right). The bold arrow tip is vector p and the hollow arrow tip is vector x .

Fig. 4.

Trajectories of rays in the Luneburg lens in the plane x 3=0 (n=1 for x 2 >1).

### Equations (25)

$d x i dt = ∂ H ∂ p i d p i dt = ∂ H ∂ x i i = 1 , 2 , 3$
$H ( x 1 , x 2 , x 3 , p 1 , p 2 , p 3 ) = ( n 2 ( x 1 , x 2 , x 3 ) − p 1 2 − p 2 2 − p 3 2 ) ⁄ 2$
$p i = ∂ V ∂ x i P i = − ∂ V ∂ X i$
$H ( x 1 , x 2 , x 3 , ∂ V ∂ x 1 , ∂ V ∂ x 2 , ∂ V ∂ x 3 ) = 0$
$x i = − ∂ W ∂ p i P i = − ∂ W ∂ X i$
$H ( − ∂ W ∂ p 1 , − ∂ W ∂ p 2 , − ∂ W ∂ p 3 , p 1 , p 2 , p 3 ) = 0$
$V ( x ) − W ( p ) = x · p$
$x i = − ∂ T ∂ p i X i = ∂ T ∂ P i$
$H ( − ∂ T ∂ p 1 , − ∂ T ∂ p 2 , − ∂ T ∂ p 3 , p 1 , p 2 , p 3 ) = 0$
$V ( x , X ) − T ( p , P ) = x · p − X · P$
$( ∂ V ∂ x 1 ) 2 + ( ∂ V ∂ x 2 ) 2 + ( ∂ V ∂ x 3 ) 2 = 1 − x 1 2 ( ∂ W ∂ p 1 ) 2 = 1 − p 1 2 − p 2 2 − p 3 2$
$± W = 1 2 ( 1 − p 2 2 − p 3 2 ) arcsin ( p 1 1 − p 2 2 − p 3 2 ) + 1 2 p 1 1 − p 1 2 − p 2 2 − p 3 2 + f ( p 2 , p 3 )$
$d x i d t = p i dp i d t = − x i i = 1 , 2 , 3$
$d 2 x i d t 2 + x i = 0 d 2 p i d t 2 + p 1 = 0 i = 1 , 2 , 3$
$( x 1 x 2 x 3 p 1 p 2 p 3 ) = ( cos t sin t − sin t cos t ) ( x 10 x 20 x 30 p 10 p 20 p 30 )$
$x i 2 + p i 2 = x i 0 2 + p i 0 2 i = 1 , 2 , 3$
$( x i x j 0 + p j p i 0 ) 2 + ( x i p j 0 − p j x i 0 ) 2 = ( x i 0 x j 0 + p j 0 p i 0 ) 2$
$( x i p j 0 − x j p i 0 ) 2 + ( − x i x j 0 − x j x i 0 ) 2 = ( x i 0 p j 0 + x j 0 p i 0 ) 2$
$( p i p j 0 − p j p i 0 ) 2 + ( − p i x j 0 + p j x i 0 ) 2 = ( x i 0 p j 0 − x j 0 p i 0 ) 2$
$H ( ξ 1 , ξ 2 , ξ 3 , ζ 1 , ζ 2 , ζ 3 ) = n 2 ( ξ 1 , ξ 2 , ξ 3 ) − ζ 1 2 h 1 2 ( ξ 1 , ξ 2 , ξ 3 ) − ζ 2 2 h 2 2 ( ξ 1 , ξ 2 , ξ 3 ) − ζ 3 2 h 3 2 ( ξ 1 , ξ 2 , ξ 3 )$
$1 h 1 2 ( ξ 1 , ξ 2 , ξ 3 ) ( ∂ V ∂ ξ 1 ) 2 + 1 h 2 2 ( ξ 1 , ξ 2 , ξ 3 ) ( ∂ V ∂ ξ 2 ) 2 + 1 h 3 2 ( ξ 1 , ξ 2 , ξ 3 ) ( ∂ V ∂ ξ 3 ) 2 = n 2 ( ξ 1 , ξ 2 , ξ 3 )$
$∑ i = 1 3 ζ i 2 h i 2 ( − ∂ W ∂ ζ 1 , − ∂ W ∂ ζ 2 , − ∂ W ∂ ζ 3 ) = n 2 ( − ∂ W ∂ ζ 1 , − ∂ W ∂ ζ 2 , − ∂ W ∂ ζ 3 )$
$1 h 1 2 ( ζ 1 , ζ 2 , ζ 3 ) ( ∂ W ∂ ζ 1 ) 2 + 1 h 2 2 ( ζ 1 , ζ 2 , ζ 3 ) ( ∂ W ∂ ζ 2 ) 2 + 1 h 3 2 ( ζ 1 , ζ 2 , ζ 3 ) ( ∂ W ∂ ζ 3 ) 2 = f 2 ( ζ 1 , ζ 2 , ζ 3 )$
$( ∂ V ∂ ξ 1 ) 2 + ( ∂ V ∂ ξ 2 ) 2 + ( ∂ V ∂ ξ 3 ) 2 = η 2 ( ξ 1 , ξ 2 , ξ 3 )$
$ζ 1 2 + ζ 2 2 + ζ 3 2 = η 2 ( ∂ W ∂ ζ 1 , ∂ W ∂ ζ 2 , ∂ W ∂ ζ 3 )$