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A while ago I was wondering how we could use mathematics to increase the efficiency of solar panels. The kind of mathematics I was thinking about in particular was Dynamical Billiards. Though I think it is improbable that it is currently technologically feasible to create the solar panel I am thinking about, I guess the following question could be interesting from a mathematical point of view.

We first need some definitions:

An $(k+l)$-setting is a collection of two sets of points in $\mathbb{R^d}$, $\{n_1,...,n_k \}, \{m_1,...,m_l \}$ such that each of the points $m_i \in \{m_1,...,m_{l-1} \}$ is connected with the point $m_{i+1}$, by some continuous function (the continuous function may differ for each pair of points $(m_i,m_{i+1})$).

For example, this is an $(k+l)$-setting in $\mathbb{R^2}$:

enter image description here

I hope it's somewhat readable. In this case, we have $k=2$ and $l=5$.

Furthermore, we say that a $(k+l)$-setting is good, if it is possible create a circle, such that the points $m_1,...,m_l$ are in the circle, but the points $n_1,...,n_k$ are not in the circle. If a $(k+l)$-setting isn't good, it's bad. For example:

enter image description here

Please notice that the first example of a $(k+l)$-setting is bad. No matter how you draw the circle, the point $n_2$ is always contained in it.

Now, the point of these definitions is that I would like to launch a light rays from the the points $n_1,...,n_k$, that bounces against the continuous functions. These continuous functions act as a mirror, causing the light ray to reflect according to the laws of reflection. We assume that the light ray loses no energy with each reflection, thereby maintaining its intensity on it whole course. The continuous functions that act as a boundary of the "solar panel" aren't affected by a reflection either.

Question 1: Does there exist a good $(1+l)$-setting in $\mathbb{R^2}$, such that we can send a light ray from point $n_1$ into the circle that encloses the points $m_1,...,m_l$, in such a way that the light ray never leaves the circle?

If you can answer the question in the affirmative, I have a number of follow-up questions:

Question 2.1: If there exists such a setting, how is it visualised?

Question 2.2: What about the case $k>1$ ?

Question 2.3: What about the case $d>2$ ?

If you answer the question in the negative, I also have some follow-up questions:

Question 3.1: Why does such a setting not exist? Can you prove it cannot exist?

Question 3.2: What if $d>2$ ?

Please notice that, when $d=3$, the continuous functions between the points become continuous surfaces, and when $d=4$, they become volumes, etc.

By the way, it would be great if someone told me how I could center the images, or places them in the middle themselves.

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    This is related: http://www.nature.com/ncomms/2015/150908/ncomms9092/full/ncomms9092.html . Maybe you'll find it interesting.2015-09-08

2 Answers 2

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This has been studied, under various assumptions. For the case of a parallel beam of light rays, there is a nice description in Serge Tabachnikov's 2005 book Geometry and Billiards, pp. 52-53:


enter image description here
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enter image description here


Going deeper is a paper by Daswon et al., "Light Traps" (PDF link). They show it is possible to trap a pencil of rays (rays through a point), but it is impossible to trap diffuse light. Their main negative theorem

states that if the initial light has nonzero measure in the phase space, it cannot be trapped.

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    Thank you! I may come back with more questions, based on the answers you have given.2012-07-10
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OK - I believe the answer is yes:

Take an ellipse, make a small hole, and send in a ray through one of the ellipses focai. You can see that this leads to a series of rays ever converging towards the major axis.

enter image description here

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    Two remarks. Firs$t$, the hole doesn't need to be small; two portions of ellipse near the major axis would have the same effect. Second, if the ray doesn't pass exactly through a focus, then it will leave the system, even if the hole is very small.2012-07-09