Monday, July 26, 2010


(Click on the images to enlarge)

These images formed a part of a study and exhibition on prismatic colors and rainbow phenomena back in 1999. A phenomenological approach, following Goethe's method of science was used. Drawings were made by my colleaque Klaus Salomaa and myself. You are invited to comment the texts and images! The idea of this blog is to evoke discussion on the premises of modern physics, light theory, Goethe, Newton, Land and others, additive and subtractive color mixing, meaning of phenomenology in physics and what ever.

Since the aim of this blog is mainly to present our project, I have no intention of adding new material to it at the moment. However, I do wish to continue discussions on the subject with you in the comment fields provided by the program.

In modern physics one consideres the theory of rainbows to be a settled field of optics. All conventional optical theories (geometrical optics, wave optics and quantum physics) provide an unified understanding of rainbow phenomena.

Although an unfamiliar approach in physics, a phenomenological method can widen our understanding of nature phenomena, just as the different approaches mentioned above are considered to be useful in their own fields. They are complementary in the sense that they agree with each other but are not derivable from each other alone.

The optical entities studied by conventional mathematical physics are either geometrical "rays" of light, electromagnetic "waves", or "packages" of light - the photons. These entities are highly theoretical. They are problematic in the sense that they all exist outside the field of direct human observation (sense perception). This is not to say that these entities are not useful in physics, on the contrary. It is the use of such entities that has made physical science so triumphant. The problem is, that as a consequence, we have been compelled to pronounce certain sense qualities as "non existing", as merely "apparent". These include colors, sounds, odors and tastes as such (the so called subjective or secondary sense perceptions).

Phenomenology, on the other hand, is concerned precisely with that which appears. In optics these are images and colors as such. By studying the images, which appear in raindrops, we can find such aspects of rainbow optics, which are mostly left unattended in conventional physics. These findings do not contradict optical physics. In that respect we can say that, by providing an alternative view, a phenomenological study of rainbow phenomena may add to our knowledge concerning the secrets of nature and it may therefore be called a complementary method of scientific investigation.

Today we acknowlege phenomenology as a relevant scientific method in humanistic sciences. Recent studies have, however, revealed that a phenomenological method in connection with the study of nature was already used by Goethe. Although still unfamiliar in scientific circles, Goethe's phenomenological approach to the study of nature phenomena begins to be a fairly well commented field of philosophical study.


(Click on the images to enlarge)

Instead of using unobservable theoretical entities, such as refracting and reflecting rays of light, we observe images in "drops of water". The following pictures are photographed from pastel drawings (50cm x 70cm and 70cm x 100cm format), forming an exhibition on the topic: Phenomenological rainbow study. The drawings also make up for the observational material for the rainbow study mentioned earlier.

Picture 1.
Inverted image of a landscape seen through a suspended "drop of water". A falling raindrop in a rainstorm takes the form of a perfect sphere. Therefore spherical glass vessels, filled with water, have been used for rainbow study by most researchers in history.

Picture 2.
Same landscape seen through a larger "drop", a glass vessel filled with water.

Picture 3.
Here the "drop" is in a darkened room. The bright doorway is seen as a reflection on the drop's surface.

Picture 4.
We place ourself between the drop and the open doorway, facing the drop. Two images of the doorway are seen in the drop. One is inverted and the other is upright.

Picture 5.
We move to the side (left from the line, drop - doorway). The two images on the drop move away from each other. The inverted image on the left moves towards the left edge of the drop. As it reaches the edge, its geometry is broken and the image is deformed and bent in a peculiar way. Bluish colors also appear on its edges.


(Click on the images to enlarge)

Picture 6.
We follow the formation of candle images in a "rain drop" as we move to the side, left from the base line: light source - center of the drop.
Two images of a candle flame are seen in the drop, a glass vessel filled with water. The one on the right is erect, the other one is an inverted image of the flame.

Picture 7.
By moving to the left, away from the base line, the two candle images move away from each other. The inverted image on the left moves towards the left edge of the drop.

Picture 8.
When moving more to the left, we see the inverted image on the left appear with colored borders, red towards the center of the drop and blue towards the edge of the drop. A streak of white light appears on the extreme left edge of the drop.

Picture 9.
By moving more to the left, we see the streak of white light change into a third image of the candle flame! It has also colored borders, but in reverse order compared to the image next to it. This leaves the blue edges of the candle images facing each other. These two images we call primary and secondary images, the last one being the one cosest to the left edge of the drop. Also a faint ring or oval of light is seen around these two candle images. The oval has a red colored outer border.

Picture 10.
As we move yet to the left, the two candle images on the left merge in a red spot of light. The red bordered oval also shrinks together and seems to merge in the same spot of red light. This angle of observation is called the Cartesian angle. It is the limit value of the deflected Sunlight in a spherical raindrop, being some 42 degrees in a real rainbow. By moving still to the left the red spot disappears and this ends the phenomenon.

Picture 11.
Here we have pictured a close up of the merging of the two inverted images of the candle flame seen in the drop. The secuence is formed as a consequence of moving our eye to the left. In the empty dark space between the candle flames there appears at first a broad band of violet color. After that the violet disappears and a bluish color is left in the middle, then green, yellow and finally red. These colors conform to the prismatic edge or boundary colors explained in Goethe's theory of colours.
A smaller "ghost" image is seen at the side of the primary image (right). This results from the fact that there are two reflecting surfaces in our glass sphere - one between air and glass, one between glass and water. In a rain drop there is only one - between air and water.

Picture 12.
Here we have presented the merging of the two candle images and the colored oval of light. In order to make the oval appear more vividly, we have placed a soft paper tissue in contact with the back surface of the drop. It serves as a screen on which the passing light forms a more distinct trace of the oval.

As so many rainbow investigators before him, also Goethe had studied these phenomena observable in glass vessels filled with water. He was not, however, completely satisfied with his own results (he died before he could publish his latest findings). In the next sections we shall try to go further to see what Goethe might have had in mind.


(Click on the images to enlarge)

Picture 13.
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).

Picture 14.
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.

Picture 15.
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.

Picture 16.
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.

Picture 17.
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.

Picture 18.
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?

Picture 19.
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.

Picture 20.
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!

Picture 21
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?


(Click on the images to enlarge)

Picture 22.
Light passing through a spherical glass vessel filled with water producing an image of the light source on the screen at the image point (above). When the light source is brought closer to the vessel the light on the screen grows larger and forms a circular bright area with red -yellow outer borders. Immediately inside this red -yellow border one can observe a bluish or violet colored area. The center of the circle is dully illuminated white or gray (middle).
Bringing the light source right next to the vessel produces an even larger light filled circular area on the screen, with colored borders as discribed above.

Picture 23.
Here is a "smoke-chamber" view of the same situation as in the previous Picture 22. The light and dark areas behind the glass vessel illuminated by a candle reveal themselves to be caustical phenomena found in many optical instances.
The first case (above) shows a curved edge caustic, which fades in its apex. This is called the "Newton's zero order caustic", as Newton is said to have studied it and interpreted it to be a rainbow phenomenon in a single raindrop where no internal reflection takes place. That Newton should have taken such a view has been disputed by some authors, however.
The second case (middle) shows a straight edge caustic with parallel borders. The apex has moved to infinity and the curved borders have straightened out. Here the light source is at a distance of one radius from the surface of the spherical glass vessel.
In the third case (below) the light source is brought into contact with the glass vessel. Now the caustic borders open at their widest, about 21 degrees (half the amount of the actual rainbow angle in the sky).

Picture 24.
By placing a large sheet of paper with a circular hole in the middle, betwen the light source and the glass vessel, one can observe a similar caustic phenomenon on the paper as in Picture 22 above. Here, however, the caustic shines from the glass vessel towards the light source. The passing light has suffered two refractions, once while entering the glass vessel and once while exiting it. In addition, it has been reflected once from the back wall between the two refractions. The colored border of this "raincircle" shows the familiar rainbow colors.
The connection between these caustical phenomena, known very well in conventional physics, and the double images explained in previous pictures is that they are presentations of the same overall phenomenon. The caustics are observations of an objective nature and the double images of a subjective nature ("subjective" meaning here observation "towards the incoming light" and "objective" meaning observation "from the side"). Objective study of caustics provides a means of using geometrical optics and mathematics. Subjective study of double images provides a means of recognizing the visual entities for what they are: refracting and reflecting images of the light source as such. Thus we have two complementary ways of investigating the rainbow phenomenon.


(Click on the images to enlarge)

Images in the previous pages were seen in a single drop of water. In order to understand how a complete rainbow is formed in the sky we must produce the same necessary conditions as in a real rainbow phenomenon. These include that the light source (the Sun) is behind the observer's back, and in front of him a multitude of falling raindrops, a rainfront, upon which the light shines.

Picture 25.
Here we have an observer sitting in front of a wall with a bright spot light behind him. An assistant holds the galss sphere near the wall, letting it "drop" along side it. The sitting observer calls "mark" each time he sees the two light source images merge into a red spot of light on the glass sphere. A red tape is fastened to that spot on the wall. After a strenious while of drop "dropping", there appears, as a consequence, an arc of red tapes on the wall. We thus understand that in these directions the observer can see the red spots of light in the falling raindrops as they pass through that region in the sky. Some raindrops would be nearer some further. The direction is substantial! As there are millions of raindrops falling in the sky following close to each other, the appearance is that of a stationary rainbow arc in the sky.

Picture 26.
Here we have presented a falling drop of water as it passes a part of the rainbow arc, illuminated by the Sun. Images of the Sun, as we have seen earlier, can be observed in the drop. When above the rainbow arc, one image of the Sun is visible in the upper left edge of the drop (the 'direct glare' from the front surface of the drop). As the drop reaches the red area of the rainbow arc, a red spot of light appears in its lower right edge. As the drop descends, the red is changed into yellow, then green, blue and finally violet. After that, two, mostly colorless images of the Sun are seen in the lower right edge of the drop. As the drop descends more, the image that is situated at the extreme right edge of the drop disappears leaving one Sun image on the right and the direct glare on the left visible as long as the drop falls.
This is the reason why the sky is darker above the primary rainbow (Alexander's dark belt), as there is only one tiny image of the Sun visible in each drop situated in that sky area. Under the rainbow arc there are three images of the Sun visible. One of these images disappears as the drop moves down and away from the rainbow arc. That is why the sky is brighter directly underneath the rainbow and why it gradually grows dim towards the center of the rainbow.

Picture 27.
As we observe the rainbow in a landscape, the Sun is always behind our back and the rain drops in front of us. Here we see a perfect 'half circle' -rainbow. The Sun is exactly at the opposite horizon, rising or setting. The line from the center of the Sun to the center of the rainbow arc runs through the head of the observer.

Picture 28.
Here we have presented the position of the Sun and the rainbow arc in the vault of the sky. When the Sun is at the far horizon, there appears a full half circle rainbow at the opposite horizon (presupposing, of course, that there is a rainfront fully illumunated by the Sun). The highest point of the rainbow arc is then at 42 degrees. As the Sun rises the rainbow arc sinks lower. When the Sun reaches a hight of 42 degrees in the sky, then the rainbow arc disappears at the opposite horizon. Observer is pictured in the middle.

We have thus given a phenomenological explanation for the understanding of the rainbow phenomenon. In this we have followed Goethe's way of "taking part" in the phenomena by remaining within the sphere of the observable. We have chosen images and colors as such as the objects of our study (instead of theoretical entities such as "rays" or "waves" of light). Yet, we have come to the same conclusions as mathematical physics. The phenomenological way, of course, lacks the exactness of a mathematical study, yet it is based on well grounded philosophical premises in its exact presentation of that which appears. Both ways are possible and combined they give a more wholistic and comprehensive view of reality.