1. Introduction

1.1 Absolute instruments: necessary conditions and classification

An optical system is said to produce sharp imaging of an object-point x _o onto an image point x _i when any ray trajectory emitted from x _o through the optical system will pass through x _i in an exact way. Points x _o and x _i are said to be perfect conjugates [1]. The prolate ellipsoidal reflector is well known example of perfect imaging optical system, but only for points x _o and x _i coincident with the foci of the ellipsoid.

A device is called an Absolute Instrument if it produces sharp imaging not just of a single object point, but of all points in a three-dimensional domain (i.e., with non-null volume) [1]. That domain is called object space R_obj and the corresponding three-dimensional domain of conjugated image points will be referred to as R_img.

In this paper we are interested in Absolute Instruments for Homogeneous Regions (AIHR), that is when the refractive index of both regions R_obj and R_img is constant (the refractive index distribution may be inhomogeneous outside those regions). It is well known that unit magnification (either positive or negative) is a necessary condition for the existence of Absolute Instruments for which both sets R_obj and R_img have the same constant refractive index [1]. Therefore, for any pair of object points x _o and x _o ’, their images fulfil |x _i-x _i’|=|x _o-x _o’|. Such a result has been obtained by several approaches, for instance, from Maxwell’s theorem [1] or by proving that the impossibility of sharply imaging two object planes with non-unit magnification holds regardless of distortion requirements or of whether the object and image surfaces are real or virtual [2, 3]. Imaging with unit magnification is sometimes called trivial imaging [1, 2]. Henceforth we will not use the word trivial with this sense but with the usual one.

When the magnification is constant, every curve in the object space is geometrically similar to its image. Such image formation is called perfect [1]. Therefore, AIHR are perfect. Before this paper, only one AIHR was known: the plane mirror (or combinations of plane mirrors), the image formation of which is trivial.

In this paper, new AIHR based on spherical symmetric refractive index distributions are found. With this symmetry, any pair x_o, x_i must be in a straight line passing through the origin. The AIHR are classified with two criteria, depending whether the magnification m in the expression x _i-x _i’=m(x _o-x _o’) is positive or negative, and whether the object and image are real or virtual. This will lead to six possible systems, which will be labelled as ±RR, ±VV, ±RV (the latter is equivalent to ±VR, just by exchanging object and image) where the sign indicates +1 or -1 magnification and the letters indicate whether the Object and Images are real (R) or virtual (V). The trivial case is subsumed under the case +RV.

The paper is organised as follows: Section 1.2 reviews the Generalised Maxwell fish-eye lens. Section 1.3 introduces two AIHRs based on known devices the imaging properties of which were previously unknown. Section 1.4 reviews some results of spherically symmetric refractive-index distributions which together with the Generalized Maxwell fish-eye will help to calculate the ray trajectories of the new AIHRs introduced in Section 2. Section 3 gives an alternative derivation of these AIHRs, and Section 4 gives the conclusions.

1.2 Generalized Maxwell fish-eye

Let x=(x ₁, x ₂, x ₃) be the coordinates of a point in a Cartesian system. A ray trajectory is given in parametric form by three functions x(t)=(x ₁(t), x ₂(t), x ₃(t)), where t is a parameter throughout the trajectory. Let p=(p ₁, p ₂, p ₃) be the three optical direction cosines of a ray at a point x, i.e., p=n(x) x’/|x’|, where n(x) is the refractive index distribution and x’=dx/dt (a different definition of t would change only the sign of p but this is worthless for our purposes.) We will relate the trajectory of the ray in the momentum space to the vector function p(t). The generalized Maxwell fish-eye lens is given by the following refractive index distribution [4]:

For γ=1 Eq. (1) gives the classical Maxwell fish-eye distribution n=2/(1+x ²), where any point x _o is imaged in a point x _i such that x _o=-x _i/$x_{\mathrm{i}}^2$. There is sharp image formation of any point of the space. The Maxwell fish-eye is thus an Absolute Instrument [1] although not an AIHR (its magnification is not constant). This property is shared among all the lenses of the family defined by Eq. (1) when γ or 1/γ are integers. A better understanding of this imaging property can be achieved mapping the ray trajectories with the great circles of a sphere, as explained next.

Any ray trajectory in a spherically symmetric optical system is contained in a meridian plane, i.e., a plane containing the point x=0. Then we can do all the ray trajectory calculations in a plane x ₁–x ₂ without loss of generality. For the case of the generalized Maxwell fish-eye, the ray trajectories x(t) can be easily calculated with the aid of a conformal mapping between the points of the plane x ₁–x ₂ and the points of a sphere. In this conformal mapping, the optical path length differential in the plane x ₁–x ₂ calculated with the distribution of Eq. (1) equals the length differential mapped on the sphere. Consequently, the geodesics on the sphere (i.e., the great circles of the sphere) are mapped to the ray trajectories of the plane x ₁–x ₂ with refractive index given by Eq. (1). Thus, all we have to do to calculate the ray trajectories in x ₁–x ₂ is to write the equation of the great circles of the sphere and then map them onto the x ₁–x ₂ plane. For spherical coordinates θ,φ (θ=0 defines the equator of the sphere and θ=±π/2 are the poles), the mapping from the sphere to the plane x ₁–x ₂ is given by

Using Eq. (1), (2) and taking into account that the sphere has radius 1/γ, it is simple to verify the fulfillment of Eq. (3), which is a sufficient condition for the mapping to be conformal [6, 7].

This conformal mapping is a composition of the mapping defined by the complex function w=z^1/γ (w and z are complex numbers) from plane x ₁–x ₂ into another plane and of a sterographic projection from this plane to the sphere [5].

Since the great circles through any point of a sphere intersect each other at its corresponding antipodal point, the rays in the plane x ₁–x ₂ are such that any point is imaged to its corresponding mapped antipode point (see [4] for more details).

1.3 Absolute Instruments for Homogeneous Regions based on known devices.

An Eaton lens [4, 8] is defined by the distribution η(|x|)=(2/|x|-1)^½ for |x|≤1 and η(|x|)=1 for |x|>1. This distribution forms a perfect retro-reflector (see left of Fig. 1).

 

Fig. 1. The Eaton lens is an AIHR of type –RV, i.e., magnification m=-1 and virtual image. This lens forms a virtual image of any point x _o on the point x _i=-x _o.

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The trajectories inside the sphere |x|=1 are half-ellipses with semiminor axis vertices at points of the sphere. It seems to have passed unnoticed that this lens is a nontrivial example of AIHR (see right of Fig. 1) of type –RV (or –VR), i.e., magnification m=-1 and virtual image for real object. Note that part of any ray trajectory (in the plane x ₁–x ₂) is contained in two parallel straight lines given by x ₂=a x ₁±b. If one of the parallel straight lines passes through the point x _1o, x _2o then x _2o=a x _1o+b and thus -x _2o=-a x _1o-b, which means that the other straight line passes through the point -x _1o, -x _2o. This proves the image formation.

Figure 2 shows a retro-reflector made of a Luneburg lens [5, 9, 10] with a mirror on part of the sphere surface |x|=1 (thus this is not a spherically symmetric device). For the rays that hit the mirror from the inside of the lens, the device also behaves as a perfect retro-reflector, the same as the Eaton lens and for the same reasons it is an AIHR of type -RV (or -VR). Nevertheless, in this case the object and image spaces as well as the field of the instrument are restricted, since the rays being retro-reflected are only those that are reflected by the mirror from the inside of the Luneburg lens.

 

Fig. 2. Partially mirror-coated Luneburg lens is another AIHR of type -RV, i.e., magnification m=-1 and a virtual image. This lens forms a virtual image of any point x _o on the point x _i=-x _o.

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1.4 Momentum space trajectories in spherically symmetric distributions

We have recently proved [11] that when the refractive-index distribution has spherical symmetry, i.e., when the function n depends only on |x|, and when this function is invertible, then the trajectories of the rays p(t) in the momentum space p ₁-p ₂-p ₃ behave like conventional ray trajectories, just with a new refractive index distribution η(|p|).

The function η(|p|) can be calculated by solving |x| in the expression |p|=n(|x|). The resulting expression is |x|=η(|p|) (and thus η(n(x))=x). This result will be used to calculate the ray trajectories in the new AIHRs of section 3. Note that we can easily calculate the trajectories of rays in a medium with refractive index distribution η(|x|) once we know the trajectories in n(|x|), since all we have to do is to calculate the trajectories p(t) and then relabel the axes p ₁-p ₂-p ₃ as x ₁-x ₂-x ₃. The vector p(t) is just n(|x|) times the unit tangent to the ray trajectories x(t). The converse is also true for invertible spherical symmetric refractive index distributions, i.e., the vector x(t) is η(|p|) times the unit tangent to the ray trajectories in momentum space p(t), which also means that the momentum-space trajectories for the rays in a medium η(|x|) are the rays corresponding to the refractive-index distribution n(|p|).

We will introduce two examples that will be used later in Section 3 for the derivation of new AIHRs. The first is the Maxwell fish-eye lens, shown in Fig. 3 (see [1, 4, 5] for more details). Its refractive index distribution n=2/(1+x ²) is invertible, giving η(|p|)=(2/|p|-1)^½. The left hand side of Fig. 3 shows the ray trajectories and the right hand side shows the ray trajectories in momentum space. The dotted circles are the circles |x|=1 and |p|=1 respectively.

 

Fig. 3. Trajectories of rays in the meridian plane (x3=0 and so p3=0) for the Maxwell fish-eye lens (γ=1) (left); and the trajectories of the same rays (same color indicates the same ray) in the momentum space (right). The bold arrow tip is vector p and the hollow arrow tip is vector x.

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The distribution η(|x|)=(2/|x|-1)^½ for |x|≤1 and η(|x|)=1 for |x|>1 is the Eaton lens previously mentioned, so that the trajectories in momentum space coincide partially with those in the Eaton lens.

 

Fig. 4. Trajectories of rays in the meridian plane (x ₃=p ₃=0) for the generalized Maxwell fish-eye lens (γ=0.5) (left); and the trajectories of the same rays in the momentum space (right).The bold arrow tip is vector p and the hollow arrow tip is vector x.

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Figure 4 shows another example of generalized Maxwell fish-eye, the one with γ=0.5, in which the ray issuing from any point x _o are imaged at the image point x _i such that x _o=x _i/$x_{\mathrm{i}}^2$ [12]. If the point x _o is on the sphere |x|=1, then it is imaged into itself. Fig. 4 shows rays issuing and reaching point x=(1, 0, 0). The function η(|p|) can also be explicitly calculated in this case, giving η(|p|)=[1/(3u)-u]² where u=[-1/|p|+(1/|p|²+1/27)^½]^⅓.

2. Novel Absolute Instruments for Homogeneous Regions.

All the rays shown in the left hand sides of Fig. 3 and 4 cross the point x=(1, 0, 0) with different values of p that fulfill |p|=1 (note that n(1)=1 for any value of γ in Eq. (1)). Consequently, the momentum-space trajectories of these rays cross the circle |p|=1 with parallel direction x=(1, 0, 0). Independently of the value of γ in Eq. (1)), the point x=(1, 0, 0) is always imaged onto points x _i of the circle |x|=1. This is because the circle |x|=1 is the mapped curve of the sphere’s equator (according to the conformal mapping of Eq. (2)), and the antipodal points of the equator are also the in the equator. Then the ray trajectories in momentum space will cross again the circle |p|=1 with parallel direction x _i. This property is the basis of Eaton lens, and of a family of beam-splitting lenses with afocal properties [4].

 

Fig. 5. A refractive-index distribution η(|x|)=(2/|x|-1)^½ for 1≤|x|≤2 and η(|x|)=1 for |x|≤1 gives sharp imaging of any point x _o (at the point x _i=-x _o) including the points in the n=1 region.

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Next we introduce a family of lenses based on this same property but configured in a new way: The first of this lenses has the refractive-index distribution η(|x|)=(2/|x|-1)^½ for 1≤|x|≤2 and η(|x|)=1 for |x|≤1, which is the complementary case to the Eaton lens (η(|x|) is not defined outside the sphere |x|=2). Some ray trajectories are shown in Fig. 5. Of course, the trajectories inside the sphere |x|=1 are straight lines. This lens gives sharp imaging of any real point x _o, including the points in the constant refractive index region. Since the ray trajectories for the refractive index distribution of Fig. 5 (left) are symmetric with respect to the origin, it is easy to check that any real object point x _o is imaged onto its symmetric point x _i=-x _o (right), so that the magnification is m=-1. Therefore, this lens is an example of AIHR of type -RR, considering R_obj and R_img equal to |x|≤1. If we consider the region |x|>1 as virtual homogeneous object and image spaces R_obj and R_img, this lens is still performing perfect image formation, as can be proved similarly to the case of the Eaton lens. Therefore, this lens is also an AIHR, of type -VV.

 

Fig. 6. A refractive-index distribution η(|x|)=[1/(3u)-u]² being u=[-1/|x|+(1/|x|²+1/27)^½]^⅓ for |x|>1 and η(|x|)=1 for |x|≤1, gives perfect self-imaging of any point, including the n=1 points.

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A similar analysis of the trajectories in momentum space of the lens in Fig. 4 (γ=0.5) leads to the lens shown in Fig. 6. In this case any point xo is imaged into itself (x _i=x _o, and m=1). Remember that all ray trajectories shown in Fig. 4 cross the circle |p|=1 with parallel directions x=(1, 0, 0). The refractive index is η(|x|)=[1/(3u)-u]², with u=[-1/|x|+(1/|x|²+1/27)^½]^⅓ for |x|>1 and η(|x|)=1 for |x|≤1. This refractive-index distribution gives sharp self-imaging of any point including the points in the constant refractive index region. Therefore, this lens is an example of AIHR of type +RR, considering R_obj and R_img equal to |x|≤1.

Analogously to the previous case, if we consider the region |x|>1 as virtual object and image spaces R_obj and R_img, this lens still performs perfect image formation, as can be easily checked. Therefore, this lens is also an AIHR of type +VV.

The complementary lens to that of Fig. 6 leads to the lens of Fig. 7, in which η(|x|)=[1/(3u)-u]², with u=[-1/|x|+(1/|x|²+1/27)^½]^⅓ for |x|≤1 and η(|x|)=1 for |x|>1. Any ray exits the lens at the same point and with the same direction as it would have if there were no lens, but with an extra optical path length equal to 2π and after doing a loop around the origin. This lens also forms a perfect image of any point x _o in the constant refractive-index region, but this image x _i is virtual and located at the object point x _o (as free-space propagation does). Notice that this lens is invisible!

 

Fig. 7. The trajectories of rays in the invisible lens form a virtual image of any point on itself x _i=x _o.

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Some invisible lenses are known in 2D geometry within the Geometrical Optics theory [13, 14]. These lenses have some differences with respect to the one shown in Fig. 7 (besides the important fact that the invisible lens of Fig. 7 works in the 3D space): in those 2D invisible lenses (a) there is no homogeneous region (the refractive index is graded everywhere) although n tends to be constant as |x|→∞; (b) the refractive-index distribution is asymmetric and (c) there is a hidden region near |x|=0, (a region not accessible to rays coming from the “outside”, i.e., rays coming from |x|→∞).

Within the Wave Optics theory, invisible lenses for a finite number of directions are also known [15]. The device of Fig. 7 can also be considered a “nonscattering scatterer” [16] or an “invisible body” for all directions of incidence in the Geometrical Optics approximation. Wolf and Habashy proved that such devices do not exist in the Wave Optics theory [17].

Other AIHRs are obtained when γ or 1/γ is an integer. For instance when 1/γ=2k-1 (k is a positive integer), lenses with similar properties to Eaton or the one in Fig. 5 are obtained, and for 1/γ=2k they are similar to the ones in Fig. 6 and Fig. 7. The rays in all these lenses do k loops around the origin.

3. Alternative derivation

The trajectories shown in the left hand side of Fig. 5 coincide with the momentum space trajectories of a Maxwell fish-eye distribution in |x|≤1 surrounded by a reflector at the surface |x|=1 (see Fig. 8).

 

Fig. 8. Trajectories of rays in the meridian plane (x ₃=0, p ₃=0) for the Maxwell fish-eye lens distribution in |x|≤1 surrounded by a mirror at |x|=1 (left); and the trajectories of the same rays in the momentum space (right) for which corresponds a refractive index distribution η=(2/|p|-1)^½ when |p|>1 and η=1 otherwise.

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In this interpretation, the vector p does not change discontinuously at a reflection but in a continuous way, i.e., as if the refractive index distribution had a sharp but continuous step down at the surface |x|=1 that causes the reflection of the ray (see Fig. 9). In this case the refractive-index distribution is still invertible and the vector p fulfils p×N=constant throughout the reflection, (N is parallel to the gradient of n, i.e., the normal to the reflector surface) which is the equation of a straight line in momentum space. In the limit case this transition from p ₁ to p ₅ occurs at a single point x. Thus the reflection appears as a straight line trajectory in momentum space.

 

Fig. 9. Continuous transition of the vectors x and p in a reflection caused by a sharp but continuous decay of the refractive index n. The vector N is parallel to the gradient of n.

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In a similar way the trajectories shown in the Eaton lens (Fig. 1) coincide with the momentum-space trajectories of a Maxwell fish-eye distribution in |x|≥1 with a surface of refractive index n→∞ at the sphere |x|=1. In this interpretation, the refractive-index distribution had a sharp but continuous increase at the surface |x|=1 (the refractive-index function is thus invertible) that causes the vector p to increase its modulus to infinity at a constant point x. Again in this case, the vector p fulfils p×N=constant throughout the refraction, (N is parallel to the gradient of n, i.e., the normal to the sphere |x|=1) which is again the equation of a straight line in momentum space.

4. Conclusions

The first non-trivial devices forming perfect image in homogeneous three-dimensional regions of space (AIHR) have been introduced. Some of them (Fig. 1, Fig. 2) are devices already known but whose imaging properties had not been noticed. The remaining ones are new (Fig. 5, 6 and 7). None of them are trivial. No optical device except a plane mirror (or combinations of them) was previously known to form a perfect image of a homogeneous three-dimensional domain. All together, they cover all the cases considered in the introduction (±RR, ±VV and ±RV) referring to real/virtual object and image spaces and positive or negative unit magnification. One of these cases is an invisible lens, i.e., a lens in which the rays exit it through the point and with the direction that they would have if no lens were there. Such devices, however, do not exist in the Wave Optics theory. With the help of the analysis of the trajectories of rays in momentum space for the Generalized Maxwell fish-eye refractive index distributions, we have easily calculated and proved these image formation properties.