## Imaging with parallel ray-rotation sheets

Optics Express, Vol. 16, Issue 25, pp. 20826-20833 (2008)

http://dx.doi.org/10.1364/OE.16.020826

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### 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

## 1. Introduction

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]

## 2. Ray-rotation sheets

*α*, the two Dove-prism sheets therefore have to be rotated with respect to each other by

*α*/2.

## 3. Geometric imaging with ray-rotation sheets

*L*such that all the light rays that have passed through the optical system intersect again in another point,

*L*

^{′}, then the optical system images

*L*into

*L*

^{′}.

*L*is called the object,

*L*

^{′}its image. Both

*L*and

*L*

^{′}can be real or virtual: in the former case the actual light rays intersect, in the latter case their continuations.

*L*

_{1}, and passing through a ray-rotation sheet with rotation angle

*α*. After passage through the sheet the rays form a twisted bundle; no two rays in the bundle intersect, so the sheet does not image

*L*. This is in fact typical of individual ray-rotation sheets. Except in special cases (ray-rotation angles

*α*=0° and

*α*=180°), a ray-rotation sheet does not image point light sources in any plane other than the sheet plane, which is imaged again into the sheet plane, in the sense discussed in section 1.

*L*

_{2}, passing through two parallel ray-rotation sheets with rotation angles

*α*and

*β*, respectively. Like the typical single-ray-rotation-sheet case, passage through two ray-rotation sheets results in a twisted bundle of non-intersecting rays. The sheets do not produce an image of the light source, and this situation is again typical.

*L*

_{3}, is positioned at a distance from the first ray-rotation sheet such that, after passage through both sheets, all light rays originating from

*L*

_{3}intersect again in the same point,

*L*

^{′}

_{3}, the image of

*L*

_{3}.

*all*light rays intersect at the point where the projections of the ray and the point light source intersect.

*L*

_{3}

*A*

_{3}

*B*

_{3}(where

*A*

_{3}and

*B*

_{3}are the points where the trajectory intersects the first and second sheet, respectively — see Figs 3 and 4) are the product of the corresponding

*z*distance and tan

*θ*. We call the distance between the point light source,

*L*

_{3}, and the first sheet the object distance,

*o*, we call the separation between the first and second sheet

*s*, and we call the distance between the second sheet and the image of

*L*

_{3},

*L*

^{′}

_{3}, the image distance,

*i*. Basic trigonometry then leads to the following object distance, given two sheets with ray-rotation angles

*α*and

*β*, separated by s:

*θ*. This means that rays that leave the light source at any angle

*θ*with the common sheet normal are imaged at the point

*L*

^{′}

_{3}(provided, of course, they pass through both ray-rotation sheets) (Fig. 5). So

*L*

^{′}

_{3}is the geometrical image of

*L*

_{3}. The independence from the angle

*θ*, together with the fact that no approximations were used in finding that

*L*

_{3}is imaged into

*L*

^{′}

_{3}, implies that idealized ray-rotation sheets perform ray-optically perfect imaging. Specifically, an increase in aperture size leads to no loss of ray-optical imaging quality. Obviously, combinations of particular implementations of ray-rotation sheets are limited by how closely the particular implementations approximate idealized ray-rotation sheets.

*L*

_{3}, but for any light-source position in the plane a distance

*o*in front of the first ray-rotation sheet, which is then imaged to a corresponding position in the plane a distance

*i*behind the second ray-rotation sheet. This means the entire plane is imaged. (Similarly, the fact that any point light source that is a distance other than

*o*in front of the first ray-rotation sheet results in a twisted bundle of non-intersecting rays means that no other plane is imaged.) As the orthographic projections into a transverse plane of any light source and its image coincide, the image of the plane is the same size as the object and upright, so the magnification of the imaging process is

*M*=+1.

*M*=+1) or any rotation axis (contrary to what might be expected from transmission through a combination of ray-rotation sheets, the image is not rotated with respect to the object) implies that there is no optical axis. This also follows from the lack of any preferred axis in each of the constituent ray-rotation sheets.

7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

*α*+

*β*(Fig. 4(b)) and principal planes in the object and image planes. In this way, we therefore recover the result that a single ray-rotation sheet trivially images the sheet plane into itself. This argument can also easily be extended to N ray-rotation sheets with rotation angles

*α*

_{1},

*α*

_{2}, …,

*α*

*and separations*

_{N}*s*

_{1},

*s*

_{2}, …,

*s*

_{N-1}: they act like a single ray-rotation sheet with a ray-rotation angle

*α*=Σ

_{j}*α*

*and principal planes (that is, object and image planes with magnification*

_{j}*M*=+1) at distances

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. |

2. | A. C. Hamilton and J. Courtial, “Optical properties of a Dove-prism sheet,” J. Opt. A: Pure Appl. Opt. |

3. | J. Courtial and J. Nelson, “Ray-optical negative refraction and pseudoscopic imaging with Dove-prism arrays,” New J. Phys. |

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. |

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. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.2990) Imaging systems : Image formation theory

(160.1245) Materials : Artificially engineered materials

(240.3990) Optics at surfaces : Micro-optical devices

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 26, 2008

Revised Manuscript: November 17, 2008

Manuscript Accepted: November 23, 2008

Published: December 2, 2008

**Citation**

Alasdair C. Hamilton and Johannes Courtial, "Imaging with parallel ray-rotation sheets," Opt. Express **16**, 20826-20833 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20826

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### References

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

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