Rays emerging from the drop

To find out from which directions the drop is seen, and how it is seen, it is interesting to study several issues.

First of all, we must determine which directions are traveled by each of the four emerging rays when the impact point parameter runs through its range of variation. And it is also interesting to know if there are, in these "direction arcs", and for any of the four categories covered, areas in which the rays emerging from the drop are concentrated with greater intensity, thus giving rise to directions with greater luminous intensity. Furthermore, if everything that we have seen so far for a ray of a single color, corresponding to a specific frequency, with a well-defined index of refraction, is repeated for another single color, what relationship is there between the previous arcs of the directions and those of the new emerging rays? Having reached conclusions regarding these questions, we will have to apply them to the real situation that interests us: that of a solar ray, which is a mixture of rays of different colors, with refraction indices different from each other.

The following figure shows points on horizon-circles representing directions of the rays emerging from the drop.

For the reflected rays \(C_{1}\), nothing depends on the color and there is a distribution of directions along the horizon-circumference, with a greater rarefaction on the right side, along the horizontal axis of symmetry of the drop, which is natural, because this region corresponds to grazing reflections from the top and bottom of the drop. As for the first refracted rays (\(C_{2}\)), they are directed to the right, especially closer to the aforementioned axis, therefore not far from the direction of the rays hitting the drop. There remain the directions of the rays \(C_{3}\) and \(C_{4}\), represented by points located on two arcs, respectively, on the left side with a semi-amplitude of about \(42.5^ {\circ}\) and on the right with a semi-amplitude of \(129.9^{\circ}\). There are still two small arcs left, between the semi-amplitudes of \(42.5^{\circ}\) and \(50,1^{\circ}(=180^{\circ}-129 ,9^{\circ})\), representing directions that are not obtained by any radii of classes \(C_{3}\) or \(C_{4}\).

The conclusions and values indicated refer to red. What changes if the colour is violet is that, to the directions of the radii of classes \(C_{3}\) or \(C_{4}\) correspond arcs with semi-amplitudes, respectively of \(40.9^{\circ}\) and \(127^{\circ}\).

For the other colors, something intermediate will occur between what is indicated for red and for violet (see figure below with an enlargement of the area close to the extreme directions for red). In particular, the small arcs between the red ends of the directions of \(C_{3}\) and of \(C_{4}\), above and below the horizontal axis, represent directions that are not obtained for any ray of class \(C_{3}\) or \(C_{4}\), whatever the color.

In the next interactive application, you can vary the color of the incident ray and analyse its influence on the directions of the rays emerging from the drop.