Visual Defects When Extending Two-Dimensional Invisible Lenses with Circular Symmetry Into the Third Dimension

Visual defects when extending two-dimensional invisible lenses with circular symmetry into the third dimension

Aaron J. Danner1 and TomáˇsTyc2 1Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117583 Singapore 2Department of Theoretical Physics and Astrophysics, Masaryk University, Kotláˇrská 2, 61137 Brno, Czech Republic E-mail: [email protected]

November 26, 2015

Abstract. A common practice in the design of three-dimensional objects with transformation optics is to ﬁrst design a two-dimensional object and then to simply extend the refractive index proﬁle along the z-axis. For lenses that are transformations of free space, this technique works perfectly, but for many lenses that seem like they would work, this technique causes optical distortions. In this paper, we analyze two such cases, the invisible sphere and Eaton lens, and show photorealistically how serious such optical distortions might appear in practice.

1. Introduction

Since the first invisible cloaks were introduced by Leonhardt [1] and Pendry [2], the excitement generated has led to a large number of investigations of both cloaking in general as well as other interesting transformation optics devices. The term ”transformation optics” has now traditionally come to refer specifically to the act of changing the metric tensor in electromagnetic space, corresponding to a specific type of change in the refractive index profile in physical space (for example, a requirement that the tensors µ = ). Since this requirement can rarely be met in practice, various sacrifices to device performance must be made in order to achieve desired functionality. This includes, but is not limited to, sacrificing one polarization’s performance to implement a device in dielectrics, or limiting performance of a device designed with transformation optics to only two dimensions, which can significantly ease design constraints. For example, a Pendry cloak [2], originally presented as a spherically symmetric device, can instead be implemented with cylindrical symmetry, and remains invisible for all polarizations with such symmetry. There exist other types of invisible structures that do not cloak. For example, various types of invisible spheres [3, 4, 5] and other refractive index profiles have been 2 shown to exist which would still be interesting to investigate but nominally easier to fabricate in practice compared to those profiles which actually cloak objects. For example, Fig. 1 shows an invisible sphere which cannot cloak anything. Although the profile contains a singularity where the refractive index n approaches infinity at the origin, this singularity can be removed by introducing material anisotropy [6], bringing such a device within the realm of possible fabrication. A spherically symmetric graded refractive index profile would be very challenging to fabricate. A much easier type of refractive index profile to fabricate would be one with cylindrical symmetry, since then a layer-by-layer fabrication method could be used to construct a device, and such fabricaton would be largely compatible with approaches commonly used in semiconductor device fabrication. Unfortunately, as will be shown below, devices such as the invisible sphere which are not transformations of free space, but instead are non-Euclidean in nature [13], cannot be extended into the third dimension without introducing unwanted optical distortions. We now consider the optical distortions introduced and give a physical explanation and visualization for two example devices, the invisible sphere of Fig. 1 and the Eaton lens [7] of Fig. 2.

2 Figure 1. (Color online) Invisible lens of radius r0, with n(r) = [Q − 1/(3Q)] , where q 3 p2 2 2 Q = −r0/r + r0/r + 1/27.

2. The spherically symmetric case

Similarly as in [8], we make use of an analogy between classical mechanics and geometrical optics [9] for analyzing these lenses. This analogy comes from similarity between Fermat’s principle in optics [10] and Maupertuis’s principle in mechanics [11] and its consequence is that the trajectories of a particle with the Lagrangian L = v2/2 − U(~r) (we set the mass to unity) and energy E have the same geometrical shapes as light rays in a medium with refractive index q n(~r) = 2[E − U(~r)] . (1) 3

q 2r0 Figure 2. (Color online) Eaton lens of radius r0, with n(r) = r − 1.

First, we will consider the spherically symmetric potential U = U(r), where r = √ x2 + y2 + z2, and then proceed to the cylindrical case.

Let us first consider a ray impinging upon a spherical lens of radius r0. This ray will be confined to the plane of incidence containing both the incoming ray and the center of the lens. On this plane we introduce polar coordinates (r, φ), and define β as the interior angle between the ray and a radial line drawn from the sphere’s center to the point of incidence. (For example, in Fig. 2, β ranges between a small positive value to slightly less than π/2 for the rays shown.) As the ray travels through the lens, let it sweep out a polar angle ∆φ before exiting the lens. The polar angle swept by the ray can be determined by the following equation, which has been derived in mulitple references, for example see [12, 13, 14],

Z r0 dr ∆φ = 2L , (2) q 2 rt 2 2 L r n(r) − r2

where L is the angular momentum (also called the impact parameter) of the ray, and rt is the radial turning point, i.e., the shortest distance from the sphere’s center to the ray along the ray’s path. Since the angular momentum is conserved, it is constant along

the ray’s path, so L(r) = L(r0) = n(r0)r0 sin β. This can also be used to determine

rt using the relation L = rtn(rt). The scattering angle χ corresponding to ∆φ is χ(β) = 2β − π + ∆φ. For an Eaton lens, χ = π for all rays, while for an invisible sphere χ = 2π.

3. The cylindrically symmetric case

Let us now consider the case of cylindrical symmetry, with a lens of refractive index n(r), √ where now r = x2 + y2, symmetric about the z-axis. Now it is no longer true that the ray trajectory is conﬁned to one plane as was the case in the spherically symmetric case, and we can expect that optical distortions will be introduced for rays with z- 4

motion passing through the lens. This can be seen easily using again the mechanical analogy. Due to the translational symmetry of the problem, the total energy E of the

particle can be divided into a part Ez corresponding to the z-motion and a part Exy corresponding to the xy-motion, E = Exy + Ez. The potential does not depend on z,

so Ez is just the kinetic energy of the z-motion. Therefore Exy < E for particles with z-motion, so we see that these particles will have less energy available to escape the potential well U(r), so the potential will influence them more and the total scattering angle will therefore increase. If we denote by α the angle between the impinging ray 2 and the xy plane, then Ez = [n(r0) sin α] /2, and we can re-evaluate Eq. (2) with the 0 0 new index n (r) that corresponds to E = E − Ez via Eq. (1) instead of E. This way 0 2 2 2 2 we get n (r) = n(r) − n(r0) sin α. The scattering angle χ(α, β), now also a function of α, is plotted in Fig. 3 for the invisible cylinder. It is clear that the scattering angle increases by up to 50% with increasing ray tilt towards the z-axis, which will cause the “invisible cylinder” to become visible when looking at it at steeper angles. Another visual distortion is also present. Since each ray experiences a greater (integrated) refractive index on its trip through the cylinder than it would through free space, simple refraction causes the total z-distance traversed by the ray to be less than what it would be if it had traversed free space. This effect is also present in an everyday situation; when looking through any glass window, tall objects appear slightly taller because of such a shift. The shift magnitude depends on the glass thickness. In the cylinder, the total vertical distance traversed by any ray can be determined by Z r0 dr ∆z = 2 sin α . (3) q 2 rt 0 2 L n (r) − r2 A single ray traversing an invisible cylinder is shown in Fig. 4, and both effects can be seen.

3 Π Π Χ Α, Β 2

2 Π H L 0 Β

Α Π 0 2

Figure 3. (Color online) Scattering angle χ as a function of α and β. For rays parallel to the xy plane (β = 0), the scattering angle χ = 2π, which is the desired behavior; any value other than χ = 2π will cause unwanted optical distortions. 5

Figure 4. (Color online) For a ray entering an invisible cylinder at an angle not parallel to the xy plane (here α = π/5), the scattering angle χ > 2π. It is evident in the image that the rays entering and exiting the cylinder are not parallel. The portion of the ray inside the cylinder is shown in blue (curved segment) and the portion outside the cylinder is shown in red (straight segment).

4. Photorealistic images

Photorealistic raytracing simulations have been carried out to illustrate the severity of such visual eﬀects. First, a panoramic image extending in all directions was created by taking multiple photographs in the VFW Park, Illinois, USA, and combining them together with free Autostitch software [15]. An outdoor park where the nearest objects are several meters away was chosen so that parallax can be ignored; the entire image of the park is treated as a 2-dimensional spherical shell lying at inﬁnity. Rays are traced from the origin outwards. Let the unit of distance be the radius of an invisible cylinder. An invisible cylinder of unit radius is placed 7 units from the ray origin, and an 8 unit square image is captured 8 units behind the origin, centered on a line connecting the origin, an invisible cylinder 7 units from the origin, and a point along the horizon of the park shell image. Fig. 5 shows a portion of the background image with no cylinder, and Fig. 6 shows the same background with the cylinder present. It is clear that the cylinder behaves more as a gathering lens when looking upwards or downwards from the horizon. Fig. 7 shows the view through an Eaton cylinder, with the background rotated by 180 degrees 6

Figure 5. (Color online) Scene viewed through the invisible cylinder. The view is perfect along the horizon (parallel to the xy plane).

Figure 6. (Color online) Scene viewed through the Eaton cylinder. The camera and ray source are invisible. The view is a perfect corner reﬂection along the horizon (parallel to the xy plane). so that the distortion caused by the lens can be easily compared to that of the invisible cylinder. Note that because the invisible cylinder is a stronger lens overall (with a greater refractive index at every point), the distortions are also greater. In another simulation, we placed an inﬁnite plane painted with a black and white checkerboard pattern, each square 12 units in height and width, at a distance of 91 units from the ray origin, and raytracing was carried out with other parts of the arrangement as before. Unlike in the simulations of the park, in this setup the vertical shift of Eq. (3) is 7

Figure 7. (Color online) Background scene, or view through a nominally perfect invisible sphere.

Figure 8. An inﬁnite plane checkerboard viewed through the invisible cylinder. The view is perfect along the horizon (parallel to the xy plane). present, although it is still a small eﬀect. Fig. 8 shows the view through the invisible cylinder, and Fig. 9 shows the view through the Eaton lens (with the checkerboard plane positioned behind the viewer).

5. Conclusion

As our simulations show, lenses with isotropic spherically symmetric refractive index proﬁles can be extended into their three-dimensional cylindrical counterparts only at the expense of degrading their functionality. The performance will be satisfactory only 8

Figure 9. An inﬁnite plane checkerboard viewed through the invisible cylinder. The view is perfect along the horizon (parallel to the xy plane). for rays that are almost perpendicular to the cylinder’s axis while for other rays the imaging can be severely distorting. Photorealistic simulations give a qualitative feeling for these optical distortions, and show that care must be taken when extending two dimensional lenses into the third.

6. Acknowledgments

T. Tyc acknowledges support of the grant P201/12/G028 of the Czech Science Foundation.

References

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