(Click on the images to enlarge)
An "off-axis" view of the "oval" seen from behind the drop. (The brightness of the ring has been exaggerated. It is visible only when the surface of the glass vessel is covered with substance illuminated by the passing light, for instance grease and dirt. The effect of the smeared glass surface is that of a screen).
This ring is formed on the back surface of the drop by the coloured border of a light-cone, called the "zero-order spherical aberration caustic". The ring forms a "horizon", inside of which the light source is always seen, when viewed at distances more than one radius away from the drop.
This picture also tells us that divergent light is coming from the drop in the direction from which we are looking. Divergent, because the candle is seen as an image (a virtual image).
By viewing directly through the raindrop, very close to the surface of the drop back towards the light source, we see an erect image of the candle flame in the center and a bright circular ring of light near the edge of the drop. This ring has colored borders so that its outer edge is red, then towards the center yellow, then a white area, then cyan blue and finally the inner edge is violet.
In the lower part of the image we have moved our eye behind the drop a bit to the left so as to place it "off axis". The bright ring of light divides into two arcs of light. The erect image of the candle flame moves towards the left as we move our eye to the left and it approaches the arc on the left.
Here we have pictured in greater detail what happens to the candle image in the center of Picture 14 and the arc of light on its left. As mentioned above the arc was formed as the ring of light divided itself as we moved to the left. This secuence of images shows how the arc on the left is formed into another mirror image of the light source, the candle flame. By moving left we observe a similar merging of these images as was found in Picture 11 with the exception that no green light is formed here.
We have thus seen that what appeared as a ring of light in Picture 14, was in fact another image of the light source, deformed by the optical properties of the spherical drop and bent into a ring, when observed from that direction. The spherical drop offered, so to speak, two different views of the light source.
A similar phenomenon is shown in this beautiful picture taken by the Hubble Space Telescope. It is a complete circular gravitational lens, galaxy B1938+666. Here we have a cosmic case of a double image. The image of the distant galaxy at center is also seen as a ring of light, bent by a massive invisible calagtical object between the galaxy and earth. The double image is thus caused by gravitational forces. It is a yet new type of double images, not belonging to the common refractive or reflective types.
Here we have pictured a more familiar optical horizon, the "horizon" of a fish, through which it sees objects that are situated above the surface of the water. Everything above the surface of the water is visible to the fish inside the circle of this horizon, but images near the edges of the horizon are strongly flattened. The fisherman tries to take advantage of this flattening, to hide himself from the fish.
This is a phenomenon that explains the refraction of images when looking from a denser medium (water) into a less dense one (air). The part of the water surface that is left outside the "horizon" is nontransparent and mirrors the images of the under water world (total internal reflection). So for instance, the rock, which is seen on the shore in the water at left can also be viewed through the "fish horizon" above the water surface.
Another familiar optical phenomenon taught to us by nature, is the reflection of images. In this picture we have the reflected image of the scenery on a tranquil surface of water. What is reflected? Rays of light or an image of the fisherman? By asking such a question we are referring to a complementary way of looking at the phenomenal reality.
The reflections in water include yet another interesting phenomenon that carries us closer to the understanding of what happens in a raindrop. To this end we ask another question: how is the reflected image of the fishingpole changed by the effect of the waves?
Candle images on a metallic surface of "waves".
Images reflected from a wave-like metallic surface come in pairs - as double images. These images are mirror images in respect to each other. There, where the convexity of the surface changes into concaveness, is a borderline of symmetry that separates the erect images from the inverted ones and forms a basis for the mirroring.
When we see a vertically situated object, such as the fishingpole, being reflected from the more or less horizontal waves of water, we do not see a continuous image being twisted sideways back and forth, but a series of images that are fragments of the original pole and appearing in pairs, as an image and its mirror image.
We thus have double images as result of reflection from a surface with borders of reflection symmetries, i.e. points of inversion. Double images are quite common phenomena in optics and appear in reflective as well as in refractive instances. Examples of double images of reflective (plus refractive) origin can be found in lenses of different types by viewing a light source at a suitable angle "off axis". Mirage phenomena at open sea, or upon the hot surface of an asphalt road are refractive phenomena producing double images. Even triple images or multiple images are possible.
Looking through the drop in a "subjective" observation.
Here we see a general principle of image formation in a spherical raindrop. We have written the letter "E" on a piece of paper, which we hold in contact with the back surface of the drop. At first we look through the middle and notice that the drop does not hinder the sighting of the "E". We then move the paper to the left and the "E" with it. Another "E" meets the first one close to the edge of the drop, but this new one is a mirror image of the former. They both have coloured borders and since they are dark images on white background, the reds are now in the middle between the two "E":s and the violets on the opposite sides. The "E":s merge and finally disappear. The original "E" is still on the paper, but inaccessible for sight from this angle. We have thus hidden the "E" behind a nontransparent edge of the drop!
The Ur-phenomenon in a raindrop.
There is a circular "blind area" or a nontransparent area, that runs along the edge of the drop, through which it is impossible to see an object that is situated at a suitable position behind (or inside) the drop. In this blind area however, it is possible to see a "mirror image" of an object that is simultaneously seen through the centre part of the drop, near the border of the nontransparent area. The border between the center, which is transparent and the impenetrable edge forms a line of symmetry between the image and its "mirror image", the primary and secondary images.
This means that what we referred to as the primary and secondary images are mirror images, however, not by reflection (as in simple mirroring) but in the case of the "letter E" by refraction and in the case of the rainbow phenomenon, by refraction and reflection!. A view through the drop is possible only through its transparent centre, the borders of which form a "horizon" comparable to that of a fish. The circular, nontransparent edge of the drop "mirrors" everything inside the transparent centre just as the water surface outside the fish horizon mirrors (by total internal reflection) everything that is underwater.
This is the reason why the boundary colours in the primary and secondary images are reversed. It also explains why the images do not just simply pass by each other (as one of my students wondered, a very good observation by the way), mixing their colours as they do so, but that in fact the colours that join in the middle, as the images merge, are one by one extinguished.
It is always a question of the primary image being completely or only partly seen inside the transparent centre of the drop. When the left, violet edge of the primary image has moved behind the nontransparent border this edge ceases to be seen from that angle of observation. As the secondary image mirrors the primary, it will reveal the same portion of the image of the light source as the primary image, but in a reverse order, as a "mirror image".
Similarly, when we observed the oval near the Cartesian angle, what we saw was not the complete circle, that can be seen on the back surface of the drop, but only half of it. The right hand side of the oval is the half still visible through the transparent centre of the drop. The left side of the oval is a "mirror image" of the former. This is also the reason why the oval contracts into a red spot. As less and less of it is seen in the transparent middle, the last part that will disappear behind the nontransparent edge is the red border of the oval facing the centre of the drop, which grows vertically shorter and shorter. A fact worth mentioning here is that the double images and the oval seem to merge and disappear in the same red spot of light, when seen in a solid glass drop, without the double surfaces of a glass vessel filled with water.
This Ur-phenomenon is the phenomenological explanation of the rainbow phenomenon within a single raindrop. It serves to explain all "subjective" observations that can be made in a single drop of water. As the actual rainbow phenomenon in the sky is also a truly "subjective" observation (consisting of multiple such phenomena as described above), we have thus given a subjective interpretation of how one can understand the rainbow phenomenon from a Goethean point of view. Also the images to be found in connection with the secondary rainbow are modifications of this Ur-phenomenon.
The "objective" method of physics is to observe "from outside" how light traverses through a raindrop, how it refracts and reflects and is split into colors. The "subjective" method of phenomenology is to "take part" in the phenomenon, to look "from within". In this case it means to look though the optical medium of the rain drop at the light source. Exactly the same happens to sight as does to light going the opposite way. In a phenomenological sense we could speak of the reflection and refraction of sight as we speak of light. They say there is a maximum speed of light. What is the maximum speed of sight? :-)
This drawing shows the paths of light rays through a raindrop, indicating some of the associated wavefronts and caustics. From C. Boyer: The Rainbow, from Myth to Mathematics.
A conventional physical explanation of the same phenomenon follows from the mathematical laws of refraction and reflection. Depending on the positions of the incident rays of white light on the front surface of the raindrop, they are refracted inside the drop at slightly different angles. As in a prism, white light is also here thought of being separated in its composite entities - colors. As this refracted light reaches the back wall of the drop some of it escapes from the drop while some is reflected back inside. When reaching the wall of the drop for the second time, this reflected light is again divided so that some of it again escapes outside according to the law of refraction and some is reflected inside the drop - and so on. Thus light is trapped, as it were, within the drop, some of it being deflected outside while the rest continues its journey within and weakening in intensity.
The result is identical to that of a phenomenological explanation: light exiting the drop after two refractions and a reflection in between is "composed" of two separate families of caustical rays, having a mutual asymptote - the cartesian ray. The angles of the family of rays composing these two caustics coincide at the cartesian ray. This is indicated in Boyer's drawing by the two wave fronts (lower left) propagating from the drop and having a mutual point on the cartesian ray. In physics, there is no difference in rank between these two causical light-paths. They appear simply as a result of the position where the incident light first entered the drop. The border between the transparent and the non-transparent part of the drop near its edge marks the point of division into two separate light-paths - the two caustics.
However, while in conventional physics one perhaps consideres these caustics "merely" as a collection of the possible routes a ray of light can travel within a raindrop, a fact simply following from the mathematical laws - a phenomenological study reveals a new ontological aspect to these caustics! They may be seen in connection with double image formation in optical instances. As we have shown, the inner caustic may be identifed with the primary image and the outer caustic with the secondary image of the original light source. The light source in the case of a rainbow phenomenon is the Sun. We can therefore conclude that it is tiny images of the Sun, which we see in drops of rain in the sky - not rays of light. Mathematics in physics works splendedly! The ontological nature of what is being confronted in a mathematícal study may not be revealed by this alone. Here physical concept formation comes into play. Our question is: how much is the ontological concept formation in physics affected by concepts borrowed from mathematics itself?