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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 4 — Feb. 15, 2010
  • pp: 3332–3337
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Visualizing invisibility: Metamaterials-based optical devices in natural environments

Aaron J. Danner  »View Author Affiliations


Optics Express, Vol. 18, Issue 4, pp. 3332-3337 (2010)
http://dx.doi.org/10.1364/OE.18.003332


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Abstract

Photorealistic ray tracing methods have been developed that allow us to see how devices such as imperfect invisible spheres and invisibility cloaks would appear if actually constructed and placed in outdoor environments. The methods developed allow photorealistic depiction of devices with gradient indices of refraction and birefringence or trirefringence in non-Cartesian coordinate systems (and hence accurately handle ray splitting/beam walkoff). The resulting images, which can be rendered in real time to produce animations as will be shown, allow subjective assessment of the performance of optical instruments such as invisibility devices in environments in which they are intended to ultimately be used.

© 2010 OSA

1. Introduction

When many people envision an invisibility device such as an optical cloak in action, outdoor use is often imagined. Futuristic vehicles with stealth, invisible aircraft, or even personal camouflage to be used on the ground are concepts that fire the imagination. Current metamaterial technology is far from these goals, but it is a worthwhile endeavor to show just how far current technology is from such scenarios by depicting how such devices would look if they could actually be built in the future versus how they would look if built with the best of today’s materials. In this letter, photorealistic depictions of such metamaterial-based optical instruments in action will be presented. In this case however, it is not just ideal operation with ideal (and not-yet-invented) materials that will be depicted, but imperfect, realistic materials that could potentially be fabricated that will be shown. To the extent of the author’s knowledge, this is the first report of photorealistic ray tracing that cannot only handle anisotropic (birefringent) materials in non-Cartesian coordinate systems with spatially gradient permittivity profiles, but can also incorporate real camera footage/scenery. Fast ray tracing methods will also be discussed which allow real time ray tracing and creation of animations that incorporate the above-mentioned effects.

This letter is organized into two sections that describe the ray tracing methods. The first describes ray tracing in an outdoor scene involving constant index (with n < 1) materials and 3 dimensional (3D) wireframe object models—this is essentially background material but with stunning and pedagogically useful results. The second section extends the technique to gradient index materials with anisotropy and beam walkoff in order to properly depict a spherically birefringent invisible object designed with use of transformation optics.

2. 3D ray tracing with n < 1 objects in a natural environment

Figure 1
Fig. 1 (a) Swimming pool with no water, (b) Swimming pool with normal water (n = 1.33), (c) Swimming pool demonstrating total external reflection (n = 0.9), and (d) Swimming pool with negative index water (n = −1.33) with dark lines added showing bottom inner edges of the pool.
illustrates the case of a swimming pool filled up with water of various “indices”. It is pedagogically useful to see what “negative-index water” would look like, and dark lines have been added to Fig. 1(d) to show the location of the bottom inside edges of the swimming pool that would never be visible without a negative index material. Indeed, it is clear that if a person were swimming in such a pool, the body of the swimmer would appear to glide above the plane of the pool in an illusion. Figure 1(c) shows the situation of total external reflection. The artificial sky is visible in the reflection in the pool’s surface. In this case, light was assumed to be unpolarized so an average of the Fresnel curves for both polarizations was used to calculate the reflectivity at the water/air interface as a function of incidence angle. Pseudorandom bump maps create the waves and ripples on the surface of the water.

3. Photorealistic ray tracing in anisotropic, gradient-permittivity materials

To create a device more amenable to potential fabrication at optical wavelengths, correct behavior for one of the two incident polarizations can be sacrificed to create an all-dielectric device. This is because one polarization (TM) along any plane passing through the sphere’s origin is unaffected by nonzero μr, μφ, and εθ, which can be defined at will. The other three components of permeability and permittivity are then rescaled so that μθ=1 with the same procedure described in ref [8

8. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]

]. but for the opposite polarization and without any resulting surface reflections. (They experimentally demonstrated operation of an Eaton lens at microwave frequencies so selected a different polarization.) After the above transmutation, optimization, and rescaling operations are carried out, the result is that μr=μφ=μθ=1 andεr, εφ, and εθare simply the squares of the corresponding components of the following modified refractive index tensor in spherical coordinates as a function of radius R, where n(r) is defined in Eq. (1).
nModified(R)=n(r(R))[rR000rR000rR]Rφθ
(2)
The function R(r) or its inverse r(R) appearing in Eq. (2) represents the transmuting function, the construction of which is described in ref [6

6. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys. 10, 1–8 (2009).

,7

7. A. J. Danner, and U. Leonhardt, “Lossless design of an Eaton lens and invisible sphere by transformation optics with no bandwidth limitation,” presented at the Conference on Lasers and Electro-optics (CLEO), Baltimore, Maryland, 1–5 June 2009.

]. It is possible to choose an appropriate function R(r) such that the singularity is transmuted with the resulting index always finite and greater than 1 in all components. For the invisible sphere, a possible such function is as follows.
R(r)=12r1/3+13r4/3+16r7/3
(3)
Since all components of the relative magnetic permeability μr=μθ=μφ=1, the resulting device is a gradient birefringent dielectric with all components greater than one and finite [7

7. A. J. Danner, and U. Leonhardt, “Lossless design of an Eaton lens and invisible sphere by transformation optics with no bandwidth limitation,” presented at the Conference on Lasers and Electro-optics (CLEO), Baltimore, Maryland, 1–5 June 2009.

]. This is a device that could potentially be fabricated with existing materials, the unavoidable (but finite) anisotropy at the origin notwithstanding. Forcing the use of dielectric materials as described disturbs ray trajectories for one polarization. This is illustrated in Fig. 2
Fig. 2 (a) Ray trajectories of standard invisible sphere with no birefringence; definition of impact parameter b is shown, (b) Ray trajectories for the correct polarization of the modified dielectric invisible sphere, (c) Ray trajectories for the other polarization of the modified dielectric invisible sphere.
.

The ray trajectories shown in Fig. 2 were calculated by integrating Hamilton’s equations of motion [9

9. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]

] for each polarized ray. To place this object in a natural environment with photorealistic quality, a large sample of ray trajectories for the disturbed polarization corresponding to Fig. 2(c) was pre-calculated with software capable of numerical integration, and an empirical function was created that relates the impact parameter of any ray impinging on the surface of the plane with its appropriate turning angle. The “impact parameter” b refers to the shortest distance of any impinging ray to a parallel ray that passes through the origin of the sphere. The “turning angle” refers to the amount each ray is turned about the origin. In Fig. 2(b), all turning angles are 360° for every ray shown, all with different impact parameters. Once the empirical relationship is established for Fig. 2(c), internal Hamiltonian ray calculations are no longer needed, since the macroscopic behavior (ray input vs. ray output) is established and sufficient data can offer reasonable interpolation for uncalculated impact parameters.

Once the empirical relationship between impact parameter and turning angle is established, ray tracing takes place as follows: Rays are traced in a time-reversed way from a virtual camera fanning outwards through a view frustrum. Those rays that intersect the background scene need no further action and the pixel is assigned the appropriate color. In cases where rays intersect the curved surface of the invisible sphere, the impact parameter is calculated and then translated to turning angle. The turning angle is applied along the plane connecting the three points corresponding to the camera, the impact point, and the sphere center, in order to find the ray exit point. The exit ray then intersects some point of the background which then determines the pixel color to assign at the view frustrum. This process is repeated for each polarization and the per-pixel results averaged. Because numerical integration is only required to establish the empirical relationship once, ray tracing is very fast and in fact can be done in real time to create on-the-fly animations for the invisible sphere discussed.

Figure 3
Fig. 3 (a) Background image of Chinese Garden, Singapore, (b) Single frame of an animation of the polarization-dependent invisible sphere designed through transformation optics (Media 1).
shows a background outdoor scene (formed into a panorama that surrounds the virtual camera) in Singapore’s Chinese Garden and the same scene with the modified invisible sphere. It is evident that the sphere is really only “half” invisible because the trajectories are correct for only one polarization in the dielectric-only device. The resulting animation/image capture in Fig. 3(b) is convincingly photorealistic. It is conceivable that it could actually be built and be broadband as shown (1 < n < 9.46 for all components). Further optimization of the transmuting function would not be able to yield anything better than 1 < n < 3 for all components, but even so this appears within practical reach. The scene in Fig. 3 consists only of the invisible sphere and the background panorama. No 3D object models are included, unlike the scene in Fig. 1.

4. How would these figures compare to real photographs?

The camera is assumed to be a pinhole camera, infinitely small, with no diffraction effects. This gives an infinite depth of focus in Fig. 1 and Fig. 3. There is also a parallax error in Fig. 3(b). It is caused by the way the panorama was created. A large number (about 90) photographs were taken from a camera mounted on a tripod and digitally stitched together to create the panorama that surrounds the virtual camera [10

10. M. Brown and D. Lowe, “Automatic panoramic image stitching using invariant features,” Int. J. Comput. Vis. 74(1), 59–73 (2007). [CrossRef]

]. Depth information was not recorded. As long as the separation between the virtual camera and the invisible sphere is small relative to the distance separating either the sphere or the camera from the nearest object in the background, the parallax error is negligible. This is why a scene with faraway objects was chosen, and not an interior room view with protruding features, for example. Since the objects in Fig. 1 are truly 3D objects in the model, such parallax errors would not occur, but the additional computational load caused by the 3D vertex calculations (instead of a pseudo-2D surface lookup as in Fig. 3) would probably preclude real time rendering on current processors. In addition, if objects such as mirrors were included near the invisible sphere or if realistic lighting models or dispersion effects are to be included, photons must be traced in the opposite direction (from source to camera) so that brightness is properly determined. Since many photons would not reach the camera, significantly higher computational resources are required to capture enough at the view frustrum to achieve photorealism. The real time rendering method described here can be used to handle many interesting devices such as proposed optical cloaking devices [11

11. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]

,12

12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

] though. Once materials feasible enough to fabricate such structures are designed or discovered, how they would behave in a natural environment could be characterized with this software.

5. Conclusion

Real time, photorealistic ray tracing of metamaterial-based devices in natural environments has been demonstrated. When assessing the capabilities of invisibility devices such as the modified invisible sphere here, i.e. devices that can potentially be constructed with materials that are within practical reach, it would be worthwhile to see how the devices actually look and behave in realistic environments.

Acknowledgments

Artwork contribution by D. S. Hanggoro and T. Ngo and a grant from the Singapore Ministry of Education/National University of SingaporeR263000414112/133 are acknowledged.

References and links

1.

G. Dolling, M. Wegener, S. Linden, and C. Hormann, “Photorealistic images of objects in effective negative-index materials,” Opt. Express 14(5), 1842–1849 (2006). [CrossRef] [PubMed]

2.

A. J. Danner, “Photorealistic rendering of metamaterials, gradient index devices, and polarization-dependent invisibility cloaks,” presented at Frontiers in Optics (Optical Society of America Annual Meeting), San Jose, California, 11–15 October, 2009.

3.

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009). [CrossRef] [PubMed]

4.

J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14(21), 9627–9635 (2006). [CrossRef] [PubMed]

5.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97(7), 073902 (2006). [CrossRef] [PubMed]

6.

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys. 10, 1–8 (2009).

7.

A. J. Danner, and U. Leonhardt, “Lossless design of an Eaton lens and invisible sphere by transformation optics with no bandwidth limitation,” presented at the Conference on Lasers and Electro-optics (CLEO), Baltimore, Maryland, 1–5 June 2009.

8.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]

9.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]

10.

M. Brown and D. Lowe, “Automatic panoramic image stitching using invariant features,” Int. J. Comput. Vis. 74(1), 59–73 (2007). [CrossRef]

11.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]

12.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(080.5692) Geometric optics : Ray trajectories in inhomogeneous media

History
Original Manuscript: December 24, 2009
Revised Manuscript: January 12, 2010
Manuscript Accepted: January 12, 2010
Published: February 1, 2010

Citation
Aaron J. Danner, "Visualizing invisibility: Metamaterials-based optical devices in natural environments," Opt. Express 18, 3332-3337 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3332


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References

  1. G. Dolling, M. Wegener, S. Linden, and C. Hormann, “Photorealistic images of objects in effective negative-index materials,” Opt. Express 14(5), 1842–1849 (2006). [CrossRef] [PubMed]
  2. A. J. Danner, “Photorealistic rendering of metamaterials, gradient index devices, and polarization-dependent invisibility cloaks,” presented at Frontiers in Optics (Optical Society of America Annual Meeting), San Jose, California, 11–15 October, 2009.
  3. J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009). [CrossRef] [PubMed]
  4. J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14(21), 9627–9635 (2006). [CrossRef] [PubMed]
  5. A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97(7), 073902 (2006). [CrossRef] [PubMed]
  6. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys. 10, 1–8 (2009).
  7. A. J. Danner, and U. Leonhardt, “Lossless design of an Eaton lens and invisible sphere by transformation optics with no bandwidth limitation,” presented at the Conference on Lasers and Electro-optics (CLEO), Baltimore, Maryland, 1–5 June 2009.
  8. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]
  9. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
  10. M. Brown and D. Lowe, “Automatic panoramic image stitching using invariant features,” Int. J. Comput. Vis. 74(1), 59–73 (2007). [CrossRef]
  11. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]
  12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

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