1. Introduction

The combination of a negative electric permittivity ε and a negative magnetic permeability μ of an optical material leads to a negative refractive index n [1]. Natural optical materials with this strange property are not known, but recent theoretical and experimental work has brought artificial materials of this sort (“metamaterials”) into reach [2–16]. While currently available metamaterials are mostly not quite in the visible regime yet and suffer from small sample size, anisotropies, strong frequency-dependencies, and excessive losses, it is still interesting to ask: How would an isotropic material like that appear? How would a rod in a glass of “negative-index water” appear to an observer looking at the scenery? While these questions appear like trivial exercises in geometrical optics at first sight, it turns out that the resulting images reveal a number of surprises.

It is the aim of this article to present selected photorealistic images and simple movies obtained from the ray-tracing simulation software package “POV-Ray 3.6” [17] (open-source software). These images can help both scientists and laymen to get an intuition and visual understanding of negative-index materials. Beyond the specific examples presented here [18], the reader may generate own images with this software.

2. Images

What is ray tracing? It is a well-known and established rendering technique [19] that calculates an image of a scene by simulating the way rays of light travel. Naively, one would follow rays emitted from an object and its surrounding. However, the vast majority of these rays never hit an observer (or a virtual camera) and lots of computation time would be wasted. Thus, it is advantageous to take the reverse approach, i.e., follow rays that go from the observer to the object or its surrounding. For optical effects like caustics that cannot be accurately simulated with the reverse approach, ray tracing in POV-Ray 3.6 has been extended with forward-tracing techniques like photon mapping. Refraction and reflection of rays are calculated from Snell’s law and the Fresnel equations, respectively. Often, however, these formulae are formulated in a way that is restricted to the case of positive refractive indices, n > 0.

In general, the refractive index n follows n ² = εμ. For all known natural optical materials, the magnetic permeability is unity, i.e., μ = + 1 . For ε > 0, this leads to n = +√ε. Indeed, the input of a positive refractive index in POV-Ray 3.6 implies μ = + 1 . For ε < 0 and μ = -1 , the refractive index becomes n = -√εμ. Indeed, the input of a negative refractive index in POV-Ray 3.6 implies μ = -1. We have verified the formulae for refraction and reflection by carefully inspecting the source code. Furthermore, we have run several simple test cases. For example, regarding refraction, we have considered a diverging laser beam impinging onto a parallel plate with n = -1 under oblique incidence (not shown). We find negative refraction, a focus inside the plate, and a second focus behind the plate. Furthermore, we have verified that a convex-shaped negative-index lens acts like a concave-shaped positive-index lens, and vice versa (not shown). For example, regarding reflection, we have compared reflections from a material with, e.g., n = +1.3 and n = -1.3, and find identical results (not shown). Note that the Fresnel equations have to be explicitly switched on, otherwise POV-Ray assumes constant reflectance. The POV-Ray “scene files” for all figures and movies shown below are made available [18]. In this fashion, we avoid communicating the numerous image parameters in the main text.

 

Fig. 1. (a) Calculated ray-tracing image of a metal rod in an empty drinking glass. (b) Same scenery, but the glass is filled with normal water, n =1.3 , leading to ordinary refraction. (c) The water is replaced by “water” with a fictitious refractive index of n = -1.3 .

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Figure 1 exhibits a rather simple first example: (a) A metallic (i.e., unity reflectance) rod in an empty glass, (b) the same rod, but the glass is filled with water with refractive index n =1.3 , and (c) the water replaced by “negative-index water” with n = -1.3 . To enhance the contrast in Fig. 1 and all other images we have tinted the “water” bluish using the POV-Ray light attenuation functions. As expected, the rod appears broken towards the left-hand side – the “wrong” side – for a negative index when looking at the side of the glass. Note that the rod appears broken towards the rear if one directly looks at the air-”water” interface in (c) because rays traveling from the under-”water” part of the rod towards the rear are negatively refracted at this interface towards the observer. Furthermore, one can see the lower side of the “water” surface (with a part of the rod), because rays emitted from the surface and traveling downwards are negatively refracted upwards to the observer at the “water”(-glass)-air interface: “The observer can look around the corner.” For a similar reason, one can not see the bottom of the glass in (c), which can obviously be seen for normal water (compare Fig. 1(b) and (c)).

 

Fig. 2. Illustration of the depth-dependent magnification factor s´/ s of an object with size s in a material with n = -1. The observer looks straight down onto the object (arrow). The observer-interface distance is a, the interface-object distance is a’. (a) a´ < a and (b) a´ > a.

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In general, however, images tend to be less intuitive and much more complex. They sensitively depend on the precise value of the refractive index as well as on the position of the observer. To prepare ourselves for the images and movies to come below, Fig. 2 schematically shows an interface between air (unity refractive index) and a material (“water”) with negative index n = -1. An object of size s (here an arrow) at a distance a’ below the “water” surface emits rays of light (blue) that are negatively refracted at the interface. To an observer or a camera at distance a above the surface, looking straight down, the object appears magnified because of a viewing α’ angle of that is increased with respect to α Here, α is the viewing angle of the object if the rays would solely travel in air. From straightforward trigonometry one quickly gets that the object appears with size s’ given by the magnification factor s´/s = (a + a´)/(a - a´). Obviously, this magnification depends both on the observer position and on the depth of the object a’. For fixed distance a and starting from the interface at a´ = 0, the magnification factor increases with increasing a’ and eventually diverges at a´ = a. Intuitively, the optical path lengths of rays are strictly zero in this case and the object “sits right in the face of the observer”. This behavior is very closely connected to the lensing action of negative-index materials [1,20]. For a´ > a , the image appears to be its mirror image with respect to the vertical axis in Fig. 2. Here, the optical path length is negative. Eventually, the image decreases in size for further increase of a’. With increasing modulus of |n < 0| of the “water”, rays are refracted more towards the surface normal, reducing the viewing angle, hence the apparent magnification factor. In the limit |n < 0|≫1 we get the simple form s´/s = (a + a´)/a.

 

Fig. 3. (a) Scheme of the movie scenery. The observer is at the cross and looks straight down. Four characteristic snapshots are depicted. (b) (1.2 MB) The colored ball falls (i.e., moves away from the observer) with constant velocity v in a material with n =1 . (c) (1.3 MB) As (b), but n = - 1 . Note the reversed apparent velocity of the ball in (c) as well as the infinite size at a´ = a (compare Fig. 2).

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Fig. 4. (a) Scheme of the movie scenery. The observer is at the cross and looks straight down. Four characteristic snapshots are depicted. (b) (1.6 MB) The tilted arrow falls with constant velocity v in a material with n =1 . (c) (2.3 MB) As (b), but n = -1 . Note the extreme distortions and rotation of the arrow in (c) when parts of it are in or near to the “focus” at a´ = a (compare Fig. 2). It even appears to be torn into pieces.

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This overall behavior is illustrated in the movies in Fig. 3. Here, a colored ball falls into “water” and drops with a constant velocity v, i.e., it moves away from the observer (see illustration in Fig. 3(a)). Clearly, it becomes continuously smaller for the case of normal water in Fig. 3(b). In contrast, for “negative-index water”, the ball appears to move towards the observer, becomes infinitely large, flips sides, and eventually moves away from the observer (see Fig. 3(c)). Notably, in the regime 0 ≤ a´ ≤ a, the negative phase velocity of light (⇔ n < 0) translates into a negative (apparent) velocity of the ball.

A yet enriched behavior results if the object is more extended along the viewing direction. Figure 4(a) defines the scenery. In this case, different parts of the object are located in different depths a’, leading to different magnification factors, and, hence, to significant distortions of the objects shape. These distortions become extreme if the object extends over the point a’ = a , in which case parts of the object appear in normal orientation and others with reversed sides. This tears connected objects apart. Figure 4(c) illustrates this behavior with a textured arrow falling with constant velocity in “negative-index water”. Figure 4(b) is the corresponding reference for normal water.

 

Fig. 5. Scenery as Fig. 1, but the observer position is much closer to the intersection of the rod with the “water” surface. (a) Normal water with n=1.2, (b) “water” with n=-1.2, (c) n=-1, and (d) n=-1.4.

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The same behavior can be found for a rod in a glass of water – which brings us back to the scenery of Fig. 1. If we move the camera closer to the intersection of the rod and the air-water interface as compared to Fig. 1, the cylindrical metal rod acquires a “trumpet” shape in Fig. 5(b). Figure 5(a) exhibits the corresponding reference for normal water. For increasing modulus |n < 0| of the “water”, the magnification factor decreases (compare Figs. 5(b), (c), and (d)) due to refraction of rays towards the surface normal – as qualitatively discussed in the context of Fig. 2 above.

3. Conclusion

In summary, we have presented selected examples of simulated photorealistic images and of simple movies showing objects in negative-index materials. All calculations are based on a ray-tracing approach. The calculated images reveal “looks around the corner”, apparent reversal of object velocity, strong magnification, and distortions of their shape. Objects may even appear to be torn into different pieces. Broadly speaking, the visual material presented here demonstrates that even the geometrical optics of negative-index materials is full of surprises.