The path of a light ray in a semicircular cavity

It is well known that the path of a light ray incident on the inner reflecting wall of a circular cavity can be constructed by geometric means. Analogous to what is done in this more common geometric configuration, trajectories of light rays in a semicircular closed region with perfectly reflecting inner walls are found. A ruler-and-compass method for obtaining these trajectories is illustrated.


Introduction
The study of trajectories of point particles in billiards with perfectly reflecting walls is a known topic in the literature [1][2][3].Since the properties of mechanical reflection are similar to those of reflection in optics, we might argue that the trajectories of light rays trapped inside a cavity with perfectly reflecting inner walls are similar to those of a point particle in the same cavity.The study of these trajectories may be useful when trying to conceive light trapping devices in some solar energy harvesting applications [4,5].In fact, concentration of light in a small region of space S may be used to trap solar radiation inside a cavity in which a small opening coincides with the region S [6,7].Also laboratory devices, such as Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
integrating spheres [8], whispering gallery solids [9], confocal laser resonators [10] use optical cavities.The notion of the inner trajectory of light rays for various geometries may be of help in maximizing the energy transfer from solar radiation to an absorbing body eventually placed inside the cavity.These applications may be of some importance in the effort to construct a sustainable future, which young students are now righteously asking for.In this respect, the geometric construction of light trajectories may be seen not just as a mere mathematical exercise, but also as a useful way of looking at future applications in the realm of alternative energy devices based on solar energy.
In the present work we consider the case of a semicircular cavity with perfectly reflecting inner walls.In this respect, we may say that even though the straight portion of the semicircle or its semicircular part are taken to be partially absorbing regions, the reflection properties would not be altered if the inner surface of the cavity is smooth.In the following section, the geometric properties of circular cavities are recalled, and a graphical R De Luca approach is devised to allow students to trace trajectories by a ruler-and-compass method.In the third section, generalization of the reflection patterns to semicircular cavities is sought: We first give a recipe for a geometric construction; then, the graphical method developed for the circular cavity is adopted also for this case.Conclusions are drawn in the last section.

Circular cavities
The study of the light trajectory in circular cavities can be carried out by means of basic geometric concepts.In this way, the problem can be proposed to high-school student.
In figure 1, we start by analysing the main features of the trajectory of a light ray trapped inside a circular cavity C R of radius R. The light ray enters the cavity through a small opening in P 0 at an angle θ 0 with respect to the x-axis.The same ray is then reflected in P 1 , where the incident and reflected angles are equal to θ 0 .After reflection in P 1 , the ray is reflected in P 2 .Again, the incident and reflected angles in P 2 are equal to θ 0 .The same pattern is repeated over and over.After a certain number of reflections, light might escape the cavity in a given time T, which depends both on θ 0 and on the relative size of the opening S [11].
The points P 0 , P 1 , P 2 , … can be represented by means of the graphical construction in figure 1.Because of the law of reflection, in fact, the inner concentric circle, of radius r = R sin θ 0 , plays the role, except for its borders, of a forbidden area for the light ray, so that all points of the trajectories lie outside the open circle of radius r.Furthermore, all rays must be tangent to this inner concentric circle because all triangles OP k P k+1 (k = 0, 1, 2, 3, . ..) must be isosceles.
Therefore, once the P 0 P 1 portion of the trajectory is drawn, we may trace the inner concentric circumference C r of radius r < R having centre in O with a compass.Point P 2 will thus be obtained by starting from point P 1 and by tracing with a ruler a line tangent to C r .The intersection between this tangent and the outer circumference C R will then define point P 2 .The whole trajectory is found by iterating this procedure, i.e., we may find point P k+1 by tracing a line tangent to the inner circumference, starting from 2 , respectively.The graphs have been drawn by an analytic procedure with the aid of the software Mathematica.Details of this procedure is given in the appendix.From the same analysis given in the appendix, by setting P N = P 0 , the following condition for a closed trajectory consisting of N − 1 reflections, can be found: for N 2 < k < N. In this way, one finds the roots of unity, who also solve the well-known problem of the number of images of a point-like object placed on the bisectrix of two mirrors glued together at an angle 2π N [12].By joining together these points one finds that closed trajectories of the light ray can be given by regular polygons with N sides  or by star polygons with N points, as shown in figure 3 for N = 8 as an example.In this case, we let the integer values of k run over the interval 4 < k < 8, where only the values of k = 5 and k = 7 are used, since the value k = 6 gives a partial list of roots, which refer to the case θ 0 = π 4 .

Semicircular cavities
Having seen how the trajectories of light rays in circular cavities can be found, we can ask the following question: what type of trajectories are realized in semicircular cavities?In these types of cavities, a reflecting vertical septum at x = 0 prevents propagation of light in the right portion of C R , where 0 < x ⩽ R. A typical trajectory is represented in figure 4, where we also reproduce the second half of the whole circular cavity to draw prolongations of the light rays in this region.By starting from P 0 , the light ray propagates until it hits the vertical septum in P 1 .Its prolongations, however, intersects C R at the virtual point Q 2 representing the mirror image, with respect to the vertical septum, of the point P 2 , which is where the light ray, reflected in P 1 , is incident on C R .In this way, the portion P 1 P 2 of the real trajectory of the light ray is the mirror image of the virtual trajectory Q 1 Q 2 .The prolongation Q 1 Q 2 of the segment P 0 P 1 will follow the same path of light rays in a circular cavity, so that the reflected virtual ray One then realizes that the segment P 2 P 3 is the mirror image of Q 2 Q 3 .This construction can be iterated by now prolonging the segment Q 2 Q 3 and obtaining the portion P 3 P 4 of the real trajectory, to which the specular virtual image Q 3 Q 4 corresponds.We may thus argue that it is possible to construct the real trajectory by first Here θ 0 is the angle that the entering light ray makes with the x-axis.applying the graphical rules learned in the previous section and by then reflecting, with respect to the vertical septum, the virtual trajectories in the right region of the circular cavity (0 < x ⩽ R), into the left region, for which −R < x ⩽ 0.
The result of this construction can be seen in figures 5(a)-(c), for the three cases shown in figures 2(a)-(c).In the numerical algorithm adopted to generate figures 5(a)-(c), we first let the light ray propagate in a circular cavity.Then, for all points of the trajectory for which x > 0, we operated a reflection with respect to the y-axis.As it can be noted from figures 5(a)-(c), the lefthand side of C r is still to be considered a forbidden region for light rays.In this way, if a partially absorbing thin rod is placed along the y-axis, in place of the linear mirror, to intercept the light rays propagating inside the cavity, it will absorb most of the radiation in the vicinity of the point at radial distance r = R sin θ 0 from the centre of the cavity.Those will thus be the 'hot spots' of the thin rod.

Conclusions
In the present work the propagation of light rays inside a semicircular cavity is studied by means of simple geometric concepts.A ruler-and-compass method is developed to obtain these trajectories.In fact, it is noted that all segments composing the piecewise continuous trajectory of the light ray can be obtained by drawing tangent lines to a concentric circle of radius r = R sin θ 0 , where θ 0 is the angle that the light ray makes with the xaxis at its entrance in the circumference at point P 0 = (−R, 0).
In the field of solar energy applications, the reflection pattern can be used to give a description, by means of geometrical optics, the regions where the trapped solar radiation concentrates the most.This pattern can be useful for detecting hot spots on an absorbing thin rod placed inside the cavity.
As for didactical use of the present work, it may be considered as a first approach to the interesting topic of circular billiards and ray tracing activities.A demonstration of the reflection pattern can be given in the laboratory, carefully using a laser pointer and a circular mirror.In this way, the same impressing pattern as in figures 2(a)-(c) would be seen.Placing a linear mirror along the y-axis of the circular cavity, as in figure 4, a semicircular cavity is obtained and the patterns in figures 5(a) and (b) would be detected.Generalization of the present work to circular cavities with arbitrary inner angles can be sought.

Figure 1 . 6 √ 2 , 8 √ 2 ,
Figure 1.Trajectory of a light ray inside a circumference C R of radius R and centre O.All segments making up the piecewise continuous trajectory are tangent to a smaller concentric circumference Cr of radius R = 1, where θ 0 is the angle that the light ray makes with the x-axis.

Figure 3 . 8 (
Figure 3. Reflections of a light ray inside a circular cavity (in blue) of radius R = 1, in arbitrary units, for θ 0 = π 8 (star polygon with eight points) and for θ 0 = 3π 8 (octagon).The red dashed lines represent the borders of the two forbidden regions for the two light trajectories.

Figure 4 .
Figure 4. Trajectory of a light ray inside the left semicircular cavity H R of radius R and centre O.The vertical reflecting septum at x = 0 prevents light to propagate in the right portion of the whole circular cavity C R .Prolongments of the light rays (full lines) are represented as dotted lines in the right portion of C R .The real and virtual piecewise continuous trajectories are both tangent to the smaller concentric circumference Cr of radius r.Here θ 0 is the angle that the entering light ray makes with the x-axis.