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\documentclass[a4paper]{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\title{Ray-rotating Windows}
\author{Michael Forret}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
METATOYs are optical components that can produce light ray fields that were thought to have been forbidden until now. They can be utilised to produce effects that were thought to be science-fiction. Invisibility cloaks, but on a larger scale than previously possible, is the major one as well other abstract effects to light rays. This report will explore what happens when two dove prism arrays are brought together and rotated and how they can produce these supposed illegal light ray fields.
\end{abstract}
\section{Introduction}
Metamaterials for light rays (METATOYs)[1] allow for interesting effects to light rays. One of such effects can be observed when small dove prisms (see figure 1) are stacked upon each other to create a ‘window’, or dove prism array which is an example of one of these METATOYs. These METATOYs are extremely exciting as they can alter light rays in ways that were thought to be impossible before. This can allow for unpresidented control over light rays and opens up a vast amount of opportunites in technological advances. When an object is looked upon through a dove prism array, the image of the object is rotated vertically. As well as this, rotating a dove prism array by \(\theta\), the light rays rotate by 2\(\theta\) dependent on the distance between the dove prism arrays and the observer. Combining these effects together using multiple dove prism arrays allow for interesting observations. The expectations for combining these two effects are as follows when the distance between the camera lens and the two dove prism arrays equals zero;
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\begin {itemize}
\item When the camera that is looking through the dove prism arrays is moved in the negative x-direction, the image of the object is expected to move in the positive y-direction,
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\item When the camera that is looking through the dove prism arrays is moved in the positive y-direction, the image of the object is expected to move in the negative x-direction.
\end {itemize}
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This can be explained by the following equation[1];
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\begin {equation}
\partial_x \partial_y \phi = \partial_y \partial_x \phi
\end {equation}
Equation (1) states that when the x component of the light-ray direction differentiated with respect to y is the same as the y component differentiated with respect to x. This means that the observed light ray field is in fact 'legal'. Where \(\phi\) is the phase of the light ray direction. When this equation holds, it is stating that the curl of the light-ray field is zero.
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The observed effects should be impossible due to the laws of the wave-optical analog making the implications of METATOYs very exciting.
\begin {figure}
\centering
\includegraphics [scale = 0.15]{dove.jpg}
\caption {Single Dove Prism}
\end {figure}
\section {Method}
To prove that these arbitrary effects are achievable, we used dove prism substitutes called confocal lenticular arrays which were pressed together and rotated relative to each other by 45 degrees using the set-up shown below using the following equipment (see figure 2);
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\(\bullet\) A camera with a 400mm lens
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\(\bullet\) Adjustable platforms to move the camera in the x and y direction
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\(\bullet\) Steel rules to measure the specified 5mm increments
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\(\bullet\) Two dove prism arrays
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\(\bullet\) The object, which in this case was a 'rubiks cube'
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\begin {figure}
\centering
\includegraphics[scale = 0.5]{Picture4.jpg}
\caption {Here is a camera mounted on a mechanical jack pointed at the object through a ray-rotating window}
\end {figure}
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Results were recorded by changing both the x and y direction of the camera independently in increments of 5mm. To provide a varied set of results, D1 and D2 were altered as follows;
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\(\bullet\) D1 = 100mm, D2 = 100mm
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\(\bullet\) D1 = 100mm, D2 = 200m
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\(\bullet\) D1 = 0mm, D2 = 100mm
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\section{Results}
After analysing the photographs that were taken in the experiment, the data collected agrees with the initial hypothesis. The photographs successfully show that when D1 equals zero and the camera looking through the dove prism arrays is moved in the negative x-direction, the image of the object moves in the positive y-direction. The same sort of effect is observed such that when the camera looking through the dove prism array is moved in the positive y-direction, the image of the object moves in the negative x-direction. However, something rather peculiar happens when D1 is assigned a value greater than zero.
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Figures 3 and 4 show the effect that the dove prism arrays have on the image of the object when D1 and D2 have a ratio of 1:1
\begin {figure}
\includegraphics [scale = 0.035]{Picture_016.jpg}
\includegraphics [scale = 0.035]{Picture_017.jpg}
\includegraphics [scale = 0.035]{Picture_018.jpg}
\includegraphics [scale = 0.035]{Picture_019.jpg}
\caption {Photos with D1:D2 ratio of 1:1 and change in the y-direction of the object }
\end {figure}
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\begin {figure}
\includegraphics [scale = 0.035]{Picture_011.jpg}
\includegraphics [scale = 0.035]{Picture_012.jpg}
\includegraphics [scale = 0.035]{Picture_013.jpg}
\includegraphics [scale = 0.035]{Picture_014.jpg}
\caption {Photos with D1:D2 ratio of 1:1 and change in the x-direction of the object }
\end {figure}
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Figures 5 and 6 also successfully show this effect. However D1 is set up at a distance of 100mm whereas D2 was set up at a distance of 200mm.
\begin {figure}
\includegraphics [scale = 0.035]{IMG_6438.JPG}
\includegraphics [scale = 0.035]{IMG_6439.JPG}
\includegraphics [scale = 0.035]{IMG_6440.JPG}
\includegraphics [scale = 0.035]{IMG_6441.JPG}
\caption {Photos with D1:D2 ratio of 1:2 and change in the y-direction of the object}
\end {figure}
\begin {figure}
\includegraphics [scale = 0.035]{IMG_6432.JPG}
\includegraphics [scale = 0.035]{IMG_6433.JPG}
\includegraphics [scale = 0.035]{IMG_6434.JPG}
\includegraphics [scale = 0.035]{IMG_6435.JPG}
\caption {Photos with D1:D2 ratio of 1:2 and change in the x-direction of the object}
\end {figure}
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\section{Analysis}
The hypothesis was that when the camera looking through the two dove prism arrays is moved in any direction along the x and y component, the effects observed should be wave-optically forbidden. When D1 is equal to zero, the relationship between the angle of rotation relative to the two dove prism arrays is equal to the angle of rotation of the projected image multiplied by two (‘\(\theta\)’ = ‘2 \(\theta\)’)
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The hypothesis also stated that when D1 is assigned a value greater than zero, then the ratio between the angle of rotation relative to the two dove prism arrays and the angle of rotation of the projected image begins to decrease.
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Looking at the photos and comparing each of the photos together, it can be said that the initial hypothesis was in-fact correct.
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\section{Conclusion}
After analysing the data, METATOYs can be expanded on to provide an array of implications in the real world. Implications such as optical-engineering which could then be expanded on to even greater things. On the other end of the spectrum, it could be used for recreational purposes.
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Through looking at the data gathered, the next logical step as far as ray-rotating windows are concerned, is to establish the mathematical relationship between the ratio of the angle of the two dove prisms relative to each other and the angle of the projected image combined with the ratio of D1 and D2. By establishing the mathematical relationship, the ray-rotating windows could be engineered in such a way to provide the desired effect allowing them to be more reliable in the real world.
\section {References}
[1] Alasdair C Hamilton and Johannes Courtial, N.J.P Volume 11, Page 013042 2009 Metamaterials for light rays
\end{document}
```