Optical Conformal Mapping

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Science  23 Jun 2006:
Vol. 312, Issue 5781, pp. 1777-1780
DOI: 10.1126/science.1126493


An invisibility device should guide light around an object as if nothing were there, regardless of where the light comes from. Ideal invisibility devices are impossible, owing to the wave nature of light. This study develops a general recipe for the design of media that create perfect invisibility within the accuracy of geometrical optics. The imperfections of invisibility can be made arbitrarily small to hide objects that are much larger than the wavelength. With the use of modern metamaterials, practical demonstrations of such devices may be possible. The method developed here can also be applied to escape detection by other electromagnetic waves or sound.

According to Fermat's principle (1), light rays take the shortest optical paths in dielectric media, where the refractive index n integrated along the ray trajectory defines the path length. When n is spatially varying, the shortest optical paths are not straight lines, but are curved. This light bending is the cause of many optical illusions. Imagine a situation where a medium guides light around a hole in it. Suppose that all parallel bundles of incident rays are bent around the hole and recombined in precisely the same direction as they entered the medium. An observer would not see the difference between light passing through the medium or propagating across empty space (or, equivalently, in a uniform medium). Any object placed in the hole would be hidden from sight. The medium would create the ultimate optical illusion: invisibility (2).

However, it has been proved (3, 4) that perfect invisibility is unachievable, except in a finite set of discrete directions where the object appears to be squashed to infinite thinness and for certain objects that are small as compared with the wavelength (5, 6). In order to carry images, though, light should propagate with a continuous range of spatial Fourier components, i.e., in a range of directions. The mathematical reason for the impossibility of perfect invisibility is the uniqueness of the inverse-scattering problem for waves (3): the scattering data, i.e., the directions and amplitudes of the transmitted plane-wave components determine the spatial profile of the refractive index (3). Therefore, the scattering data of light in empty space are only consistent with the propagation through empty space. Perfect illusions are thus thought to be impossible due to the wave nature of light.

On the other hand, the theorem (3) does not limit the imperfections of invisibility—they may be very small—nor does it apply to light rays, i.e., to light propagation within the regime of geometrical optics (1). This study develops a general recipe for the design of media that create perfect invisibility for light rays over a continuous range of directions. Because this method is based on geometrical optics (1), the inevitable imperfections of invisibility can be made exponentially small for objects that are much larger than the wavelength of light.

To manufacture a dielectric invisibility device, media are needed that possess a wide range of the refractive index in the spectral domain where the device should operate. In particular, Fermat's Principle (1) seems to imply that n < 1 in some spatial regions, because only in this case the shortest optical paths may go around the object without causing phase distortions. In our example, n varies from 0 to about 36. In practice, one could probably accept a certain degree of visibility that substantially reduces the demands on the range of the refractive index.

Extreme values of n occur when the material is close to resonance with the electromagnetic field. Metamaterials (7) with man-made resonances can be manufactured with appropriately designed circuit boards, similar to the ones used for demonstrating negative refraction (8). The quest for the perfect lens (9) has led to recent improvements (7, 1013) mainly focused on tuning the magnetic susceptibilities. In such metamaterials, each individual circuit plays the role of an artificial atom with tunable resonances. With these artificial dielectrics, invisibility could be reached for frequencies in the microwave-to-terahertz range. In contrast, stealth technology is designed to make objects of military interest as black as possible to radar where, using impedance matching (14), electromagnetic waves are absorbed without reflection, i.e., without any echo detectable by radar. Recently, nanofabricated metamaterials with custom-made plasmon resonances have been demonstrated (13) that operate in the visible range of the spectrum and may be modified to reach invisibility.

The method used here is general and also applicable to other forms of wave propagation—for example, to sound waves, where the index n describes the ratio of the local phase velocity of the wave to the bulk value, or to quantum-mechanical matter waves, where external potentials act like refractive-index profiles (1). For instance, one could use the profiles of n described here to protect an enclosed space from any form of sonic tomography. This study examines the simplest nontrivial case of invisibility, an effectively two-dimensional situation, by applying conformal mapping (15) to solve the problem—an elegant technique used in research areas as diverse as electrostatics (14), fluid mechanics (16), classical mechanics (1720), and quantum chaos (21, 22).

Consider an idealized situation: a dielectric medium that is uniform in one direction and light of wave number k that propagates orthogonal to that direction. In practice, the medium will have a finite extension and the propagation direction of light may be slightly tilted without causing an appreciable difference to the ideal case. The medium is characterized by the refractive-index profile n(x,y). To satisfy the validity condition of geometrical optics, n(x,y) must not vary by much over the scale of an optical wavelength 2π/k (1). To describe the spatial coordinates in the propagation plane, complex numbers z = x iy are used with the partial derivatives ∂x = ∂z + ∂z* and ∂y = iziz*, where the asterisk symbolizes complex conjugation. In the case of a gradually varying refractive-index profile, both amplitudes ψ of the two polarizations of light obey the Helmholtz equation (1) Embedded Image(1) written here in complex notation with the Laplace operator ∂x2 + ∂y2 = 4∂z*∂z. Suppose we introduce new coordinates w described by an analytic function w(z) that does not depend on z*. Such functions define conformal maps (15) that preserve the angles between the coordinate lines. Because ∂z*∂z = |dw/dz|2w*∂w, we obtain in w space a Helmholtz equation with the transformed refractive-index profile n′ that is related to the original one as Embedded Image(2) Suppose that the medium is designed such that n(z) is the modulus of an analytic function g(z). The integral of g(z) defines a map w(z) to new coordinates where, according to Eq. 2, the transformed index n′ is unity. Consequently, in w coordinates, the wave propagation is indistinguishable from empty space where light rays propagate along straight lines. The medium performs an optical conformal mapping to empty space. If w(z) approaches z for w → ∞, all incident waves appear at infinity as if they have traveled through empty space, regardless of what has happened in the medium. However, as a consequence of the Riemann Mapping Theorem (15), nontrivial w coordinates occupy Riemann sheets with several ∞, one on each sheet. Consider, for example, the simple map Embedded Image(3) illustrated in Fig. 1, that is realized by the refractive-index profile n = |1 – a2/z2|. The constant a characterizes the spatial extension of the medium. The function (3) maps the exterior of a circle of radius a on the z plane onto one Riemann sheet and the interior onto another. Light rays traveling on the exterior w sheet may have the misfortune of passing the branch cut between the two branch points ±2a. In continuing their propagation, the rays approach ∞ on the interior w sheet. Seen on the physical z plane, they cross the circle of radius a and approach the singularity of the refractive index at the origin. For general w(z), only one ∞ on the Riemann structure in w space corresponds to the true ∞ of physical z space and the others to singularities of w(z). Instead of traversing space, light rays may cross the branch cut to another Riemann sheet where they approach ∞. Seen in physical space, the rays are irresistibly attracted toward some singularities of the refractive index. Instead of becoming invisible, the medium casts a shadow that is as wide as the apparent size of the branch cut. Nevertheless, the optics on Riemann sheets turns out to serve as a powerful theoretical tool for developing the design of dielectric invisibility devices.

Fig. 1.

Optical conformal map. A dielectric medium conformally maps physical space described by the points z = x iy of the complex plane onto Riemann sheets if+the refractive-index profile is |dw/dz| with some analytic function w(z). The figure illustrates the simple map (3) where the exterior of a circle in the picture above is transformed into the upper sheet in the lower picture, and the interior of the circle is mapped onto the lower sheet. The curved coordinate grid of the upper picture is the inverse map z(w) of the w coordinates, approaching a straight rectangular grid at infinity. As a feature of conformal maps, the right angles between the coordinate lines are preserved. The circle line in the figure above corresponds to the branch cut between the sheets below indicated by the curly black line. The figure also illustrates the typical fates of light rays in such media. On the w sheets, rays propagate along straight lines. The rays shown in blue and green avoid the branch cut and hence the interior of the device. The ray shown in red crosses the cut and passes onto the lower sheet where it approaches ∞. However, this ∞ corresponds to a singularity of the refractive index and not to the ∞ of physical space. Rays like this one would be absorbed, unless they are guided back to the exterior sheet.

All we need to achieve is to guide light back from the interior to the exterior sheet, i.e., seen in physical space, from the exterior to the interior layer of the device. To find the required refractive-index profile, we interpret the Helmholtz equation in w space as the Schrödinger equation (1) of a quantum particle of effective mass k2 moving in the potential U with energy E such that UE = –n2/2 (1). We wish to send all rays that have passed through the branch cut onto the interior sheet back to the cut at precisely the same location and in the same direction in which they entered. This implies that we need a potential for which all trajectories are closed. Assuming radial symmetry for U(w) around one branch point w1, say +2a in our example, only two potentials have this property: the harmonic oscillator and the Kepler potential (17). In both cases the trajectories are ellipses (17) that are related to each other by a transmutation of force according to the Arnol'd-Kasner theorem (1820). The harmonic oscillator corresponds to a Luneburg lens (23) on the Riemann sheet with the transformed refractive-index profile Embedded Image(4) where r0 is a constant radius. The Kepler potential with negative energy E corresponds to an Eaton lens (23) with the profile Embedded Image(5) Note that the singularity of the Kepler profile in w space is compensated by the zero of |dw/dz| at a branch point in physical space such that the total refractive index (2) is never singular. In both cases (4) and (5), r0 defines the radius of the circle on the interior w sheet beyond which n2 would be negative and hence inaccessible to light propagation. This circle should be large enough to cover the branch cut. The inverse map z(w) turns the outside of the circle into the inside of a region bounded by the image z(w) of the circle line in w space. No light can enter this region. Everything inside is invisible.

Yet there is one more complication: Light is refracted (1) at the boundary between the exterior and the interior layer. Seen in w space, light rays encounter here a transition from the refractive index 1 to n′. Fortunately, refraction is reversible. After the cycles on the interior sheets, light rays are refracted back to their original directions (Fig. 2). The invisibility is not affected, unless the rays are totally reflected. According to Snell's Law (1), rays with angles of incidence θ with respect to the branch cut enter the lower sheet with angles θ′ such that n′sinθ′ = sinθ. If n′ < 1, this equation may not have real solutions for θ larger than a critical angle Θ. Instead of entering the interior layer of the device, the light is totally reflected (1). The angle Θ defines the acceptance angle of the dielectric invisibility device, because beyond Θ, the device appears silvery instead of invisible. The transformed refractive-index profiles (4) and (5) at the boundary between the layers are lowest at the other branch point w2 that limits the branch cut, w2 = –2a, in our example. In the case of the harmonic-oscillator profile (4), n′ lies always below 1, and we obtain the acceptance angle Embedded Image(6) For all-round invisibility, the radius r0 should approach infinity, which implies that the entire interior sheet is used for guiding the light back to the exterior layer. Fortunately, the Kepler profile (5) does not lead to total reflection if r0 Θ 2|w2w1|. In this case, the invisible area is largest for Embedded Image(7) Figure 3 illustrates the light propagation in a dielectric invisibility device based on the simple map (3) and the Kepler profile (5) with r0 = 8a. Here n ranges from 0 to about 36, but this example is probably not the optimal choice. One can choose from infinitely many conformal maps w(z) that possess the required properties for achieving invisibility: w(z) ∼ z for z → ∞ and two branch points w1 and w2. The invisible region may be deformed to any simply connected domain by a conformal map that is the numerical solution of a Riemann-Hilbert problem (16). We can also relax the tacit assumption that w1 connects the exterior to only one interior sheet, but to m sheets where light rays return after m cycles. If we construct w(z) as af(z/a) with some analytic function f(z) of the required properties and a constant length scale a, the refractive-index profile |dw/dz| is identical for all scales a. Finding the most practical design is an engineering problem that depends on practical demands. This problem may also inspire further mathematical research on conformal maps in order to find the optimal design and to extend our approach to three dimensions.

Fig. 2.

Light guiding. The device guides light that has entered its interior layer back to the exterior, represented here using two Riemann sheets that correspond to the two layers, seen from above. Light on the exterior sheet is shown in blue and light in the interior, in red. At the branch cut, the thick line between the two points in the figure (the branch points), light passes from the exterior to the interior sheet. Here light is refracted according to Snell's law. On the lower sheet, the refractive-index profile (5) guides the rays to the exterior sheet in elliptic orbits with one branch point as focal point. Finally, the rays are refracted back to their original directions and leave on the exterior sheet as if nothing has happened. The circle in the figure indicates the maximal elongations of the ellipses. This circle limits the region in the interior of the device that light does not enter. The outside of the circle corresponds to the inside of the device. Anything beyond this circle is invisible.

Fig. 3.

Ray propagation in the dielectric invisibility device. The light rays are shown in yellow. The brightness of the green background indicates the refractive-index profile taken from the simple map (3) and the Kepler profile (5) with r0 = 8a in the interior layer of the device. The invisible region is shown in black. The upper panel illustrates how light is refracted at the boundary between the two layers and guided around the invisible region, where it leaves the device as if nothing were there. In the lower panel, light simply flows around the interior layer.

Finally, we ask why our scheme does not violate the mathematical theorem (3) that perfect invisibility is unattainable. The answer is that waves are not only refracted at the boundary between the exterior and the interior layer, but also are reflected, and that the device causes a time delay. However, the reflection can be substantially reduced by making the transition between the layers gradual over a length scale much larger than the wavelength 2π/k or by using anti-reflection coatings. In this way, the imperfections of invisibility can be made as small as the accuracy limit of geometrical optics (1), i.e., exponentially small. One can never completely hide from waves, but can from rays.

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