Abstract

A ray-rotation sheet consists of miniaturized optical components that function — ray optically — as a homogeneous medium that rotates the local direction of transmitted light rays around the sheet normal by an arbitrary angle [A. C. Hamilton et al., arXiv:0809.2646 (2008)]. Here we show that two or more parallel ray-rotation sheets perform imaging between two planes. The image is unscaled and un-rotated. No other planes are imaged. When seen through parallel ray-rotation sheets, planes that are not imaged appear rotated.

© 2008 Optical Society of America

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References

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  1. J. Courtial, "Ray-optical refraction with confocal lenslet arrays," New J. Phys. 10, 083033 (2008).
    [CrossRef]
  2. A. C. Hamilton and J. Courtial, "Optical properties of a Dove-prism sheet," J. Opt. A: Pure Appl. Opt. 10, 125302 (2008).
    [CrossRef]
  3. J. Courtial and J. Nelson, "Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays," New J. Phys. 10, 023028 (2008).
    [CrossRef]
  4. A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, "Local light-ray rotation," arXiv:0809.2646v2 [physics.optics] (2008).
  5. A. C. Hamilton and J. Courtial, "Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit," arXiv:0809.4370v1 [physics.optics] (2008).
  6. "POV-Ray - The Persistence of Vision Raytracer," http://www.povray.org/.
  7. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  8. W. J. Smith, "Image Formation: Geometrical and Physical Optics," in "Handbook of Optics," W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, 1978), Chap. 2.

2008 (3)

J. Courtial, "Ray-optical refraction with confocal lenslet arrays," New J. Phys. 10, 083033 (2008).
[CrossRef]

A. C. Hamilton and J. Courtial, "Optical properties of a Dove-prism sheet," J. Opt. A: Pure Appl. Opt. 10, 125302 (2008).
[CrossRef]

J. Courtial and J. Nelson, "Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays," New J. Phys. 10, 023028 (2008).
[CrossRef]

2000 (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Courtial, J.

J. Courtial and J. Nelson, "Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays," New J. Phys. 10, 023028 (2008).
[CrossRef]

J. Courtial, "Ray-optical refraction with confocal lenslet arrays," New J. Phys. 10, 083033 (2008).
[CrossRef]

A. C. Hamilton and J. Courtial, "Optical properties of a Dove-prism sheet," J. Opt. A: Pure Appl. Opt. 10, 125302 (2008).
[CrossRef]

Hamilton, A. C.

A. C. Hamilton and J. Courtial, "Optical properties of a Dove-prism sheet," J. Opt. A: Pure Appl. Opt. 10, 125302 (2008).
[CrossRef]

Nelson, J.

J. Courtial and J. Nelson, "Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays," New J. Phys. 10, 023028 (2008).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt. (1)

A. C. Hamilton and J. Courtial, "Optical properties of a Dove-prism sheet," J. Opt. A: Pure Appl. Opt. 10, 125302 (2008).
[CrossRef]

New J. Phys. (2)

J. Courtial and J. Nelson, "Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays," New J. Phys. 10, 023028 (2008).
[CrossRef]

J. Courtial, "Ray-optical refraction with confocal lenslet arrays," New J. Phys. 10, 083033 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Other (4)

W. J. Smith, "Image Formation: Geometrical and Physical Optics," in "Handbook of Optics," W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, 1978), Chap. 2.

A. C. Hamilton, B. Sundar, J. Nelson, and J. Courtial, "Local light-ray rotation," arXiv:0809.2646v2 [physics.optics] (2008).

A. C. Hamilton and J. Courtial, "Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit," arXiv:0809.4370v1 [physics.optics] (2008).

"POV-Ray - The Persistence of Vision Raytracer," http://www.povray.org/.

Supplementary Material (1)

» Media 1: MOV (2123 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Optics of a single Dove-prism array. (a) A Dove-prism array is formed by a stack of Dove prisms that form a sheet. (b) Each individual Dove prism flips one of the transverse direction components of light rays passing through it, here the vertical component. (c) If the Dove prisms are miniaturized, the overall effect of a Dove-prism sheet on the flipped direction of transmitted light rays is equivalent to that of a planar interface between two materials with opposite refractive indices, +n and -n.

Fig. 2.
Fig. 2.

Example of local light-ray rotation with Dove-prism sheets. (a) Structure of two parallel Dove-prism sheets, S 1 and S 2, that are rotated with respect to each other by α/2=45°. The two sheets individually flip the transverse direction of transmitted light rays, but with respect to different axes. (b) Successive flipping of a light-ray direction, d, first by sheet S 1 (left frame), resulting in the intermediate direction d , then by sheet S 2, resulting in direction d . Dove-prism sheets are represented by semi-transparent, light-blue, squares (seen from a 3D position from where they appear as parallelograms). Light-ray directions are represented by red arrows; the planes containing the light-ray directions and the local Dove-prism-sheet normal (dashed black lines) are indicated by semi-transparent red rectangles. The flip of the transverse light-ray direction is with respect to an axis in the sheet plane (dotted line). (c) A plot of the transverse light-ray directions (orthographic projection of the light-ray directions into the sheet plane or any other transverse plane) reveals that the two successive flips of the transverse ray direction are equivalent to rotation through an angle α.

Fig. 3.
Fig. 3.

Trajectories of light rays that originate from three point light sources, L 1 to L 3, and pass through ray-rotation sheets. All light-ray trajectories are shown in red, apart from the trajectory of one light ray from each light source, which is highlighted in blue. The ray-rotation sheets are shown in light blue; the first rotates the direction of transmitted light rays through an angle α around the sheet normal, the second through an angle β. In the case of a single ray-rotation sheet, light rays from a point light source not located in the sheet plane generally do not intersect again after passage through the sheet (a). The same is true for passage through two ray-rotation sheets (b), unless the light source is located in the one transverse plane that is imaged by the two sheets, in which case all the light rays originating from the light source, L 3, intersect again in a point L 3 (c). The points at which the highlighted trajectories intersect the first sheet are marked A 1 to A 3, those where they intersect the second sheet are B 2 and B 3. Frames (d), (e) and (f) show the orthographic projections into a transverse plane of the light-ray trajectories respectively shown in (a), (b) and (c). The figure is drawn for α=β=150°.

Fig. 4.
Fig. 4.

Geometry of a light-ray trajectory in the imaging case. The trajectory (solid darkblue arrow) is the same as that highlighted in Figs. 3(c) and (f). (a) Three-dimensional representation. The ray-rotation sheets (rotation angles α and β, respectively) are shown in light blue. θ is the angle between the ray trajectory and the normal to both ray-rotation sheets. (The angle is shown in the form of a red segment.) The dashed line is the sheet normal through the point light source, L 3, and its image, L 3. The distances o (between L 3 and the first sheet), s (between the two sheets), and i (between the second sheet and L 3) are marked by double-sided arrows. (b) Orthographic projection into a transverse plane. The angles α and β by which the direction of the projected ray changes are simply the angles by which the two ray-rotation sheets rotate the ray around the local sheet normal. The relevant angles are marked as red segments.

Fig. 5.
Fig. 5.

Focussing of light rays with different angles θ with the common sheet normal by a pair of parallel ray-rotation sheets (shown as light blue, semi-transparent, squares). The red light rays leave the point light source, L 3, with an angle θ=17° with respect to the sheet normal; they are the light rays shown in Fig. 3(c), which is calculated for the same ray-rotation angles (α=β=150°), the same separation between the ray-rotation sheets, and the same position of L 3. The blue light rays leave L 3 at angles θ=0°,5°,10°, …40° with respect to the sheet normal. All light rays intersect again in the same point, L 3, the image of L 3. To facilitate appreciation of the 3D structure of the ray trajectories, the supplementary material contains a movie (MPEG-4, 1.4MB) of the same arrangement seen from different viewing positions.

Fig. 6.
Fig. 6.

Simulated view through a pair of ray-rotation sheets. The ray-rotation angle of each sheet is 137°; in units of the floor-tile length, the sheets are separated by a distance s=5, so the object and image distance are o=i=3.42. (a) A green chess piece is placed behind the ray-rotation sheets, in the object plane. The rendering parameters are chosen such that only the image plane is in focus — the picture is a simulation of a photo taken with a camera that is focussed on the image plane and that has a finite-size aperture. As expected, a sharp image of the chess piece can be seen in the image plane. (b) Additional (upright) chess pieces are placed in different planes behind the ray-rotation sheets. As before, the green chess piece is in the object plane (at distance o), the other pieces are not (they are at distances o/2 (blue), 2o (orange), and 4o (red)). This time the rendering parameters were chosen so that all planes are rendered in focus; the frames are now simulations of photos taken with a camera with a negligibly small aperture. Therefore all chess pieces can be seen in sharp focus, but note that there is still only one object and one image plane that are being imaged into each other by the pair of ray-rotation sheets. To demonstrate this, three frames were calculated for different camera positions, namely (from left to right) right of the central sheet normal, on the central sheet normal, and left of the central sheet normal. This horizontal camera movement makes the chess pieces that are not in the object plane appear to move vertically, which cannot be reconciled with these chess pieces being imaged, as discussed in the main text. The figure was calculated using the ray-tracing software POVRay [6], which simulated transmission through the detailed structure of two ray-rotation sheets, each consisting of two Dove-prism sheets [4], each in turn consisting of 1000 Dove prisms.

Equations (4)

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Δ r Δ z = tan θ .
o = sin β sin ( α + β ) s .
i = sin α sin ( α + β ) s .
o = j s j sin ( k = j + 1 N α k ) sin α , i = j s j sin ( k = 1 j α k ) sin α

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