With the naked eye, the wind turbines were hardly visible (Wide angle image, medium focal length); they were “zoomed in”. Assuming straight light rays and a straight background, this image would not be possible. But you can clearly see the mirages downwards, so the light rays that produce this mirror image are curved.
The picture was taken on March 18, 2018; the air layer directly above the water surface must have been slightly warmer than the layers above, so that the rays bend upwards. The adjacent sketch shows this schematically (and exaggeratedly). At the bottom the air is warmer, the refractive index smaller and therefore the wavelength longer than at the top.
The jagged wavy borderline between sky and water is not the horizon, but the lower edge of the mirror image. The horizon is at the height where the rotor blades and their mirror image seem to touch in the rounded bend.
A slight curvature of the light rays in the atmosphere is always present, since the density of the air and thus its refractive index depends on the altitude and the temperature. This is particularly noticeable in the case of almost horizontal lines of view. Since the density of the air generally decreases with increasing height, the rays are usually curved very weakly downwards. As a result, range of vision is larger than it would be without refraction, and this can be taken into account by using an apparent curvature slightly smaller than the curvature of the Earth's surface (larger apparent Earth radius) in the calculations. (According to Wikipedia the mean observed apparent earth radius is Rapparent ≈ 7680 km.) This will be explained in more detail below.
However, the temperature distribution plays an important role. If the temperature increases upward, the visibility can be increased strongly if the light-rays follow the earth-curvature, and there can occur even mirages upward (Fata Morgana).
According to a map of the offshore wind turbines in the German Bight, these wind turbines should belong to the wind farm Borkum-Riffgrund II.
The picture was taken at the end of June, when the weather was very warm. The water was colder than the air, so that the lower air layers were probably cooler than the slightly higher ones (inversion) and the normal trend in temperature was only reached at higher altitudes.
How much do light rays bend in the atmosphere? READ MORE …
The low autumn sun shines on the glass entrance door of the Lower Saxony State Museum in Hanover, Germany. Light hitting the bevelled edge is refracted and split into a spectrum, as schematically sketched in the middle. To the right the enlarged detail shows that at larger distance from the prism, the spectrum shows only red, green, and blue. (This has already been noticed by Goethe, see e.g. one of the plates added to his Theory of Colour, Tafel V, which, apart from the faded colours, has much in common with the present sketch.)
The reflection of sunlight by structures exhibiting some regularity may produce light columns or curved paths, depending on the geometrical conditions.
When looking through a crisscross of glistening silk towards the sun, the glosses seem to form circular arcs around the sun. Likewise light rings are seen when looking through the wet twigs of a bare-branched tree towards a street lamp, or when a light is reflected by a scratched glossy plane. Ireggularly oriented strands or scratches seemingly always produce circles.
If, however, the scratched glossy surface is not plane, the apparent curves are no circles anymore. I noticed this when I recently saw a spider web glistening. The sheet web – a tangle of strands within a thin nearly horizontal layer – was visible only in its glistening parts.
To figure out which patterns might occur under which conditions strains spatial imagination. To see some simple examples may help.
Instead of a spider web we consider an invisible cambered surface to which glistening needles (tangents) are attached at random points with random orientation.
The surface is given by the equation
For the sake of reproducibility, the other parameters are given: the length of the needles is 8 mm; drawn are only their glistening parts for which the difference of the angles between incoming ray and needle and needle and direction towards the eye is less than 0.25°. (The apparent diameter of the sun is 0.5°)
This accidental observation is completely different from all my previous ones. I was surprised (but not amused) by an otherwise interesting video on YouTube (in German). There the topic is that sometimes it is controversial whether a given bluish green (or greenish blue) is green or blue, and some weird hypotheses on colour vision. But in the introduction, the colour yellow is discussed, and a distinction is made between “real yellow” and the “illusion of yellow” produced by additive mixing of red and green on the computer screen. According to this video – and to that one too –
Apparently it is a popular fallacy that “every colour has a certain wavelength”. And the colour yellow is the best example to show that this in not true. Only a prism is needed.
Indeed, according to Helmholtz most colours can be characterized by a wavelength of the same hue (the exceptions are the purple hues between red and violet). But to see colour on a surface, it must remit a non-negligible fraction of the incoming (white) light, certainly not only “one wavelength”.
For the following pictures a nice flower had to sacrifice one of its ray florets. From that, a narrow streak was separated and photographed on black paper, in the second image below through a glass prism. (The experimental arrangement is the same as described earlier.)
By the prism, the light coming from the yellow streak is spread out to a spectrum, showing that the red and the green portion of the light is remitted, and only the blue light is absorbed. Yellow is always “white minus blue” and thus the sum of red and green. Yellow or any other colour consisting of only one wavelength (one spectral line) is only possible for light sources like lasers or other special light emitting devices.
The right pair of images shows photographs of the first picture in the row, taken from the LCD flat screen, the last one again through the prism. While the third photo is hardly distinguishable from the first one, the fourth is quite different from the second one. In contrast to daylight, the white light backlighting the screen consists of only few narrow lines or bands and a broad continuum only in the short-wavelength (blue-violet) region.
The last picture differs, however, from the visual impression when looking through the prism at the leftmost one, in that the weak blue line cannot be seen (at least, I can't see it), and the green line is stronger – the sensibility curves of the camera clearly differ from those of my eyes. Moreover, remember that it is impossible to render a spectrum faithfully on the screen or in print, see the sections on white light and the prism experiments.
The spectral composition of the yellow light from the screen is different from that remitted by the yellow flower – but this is generally the case for every colour and every possible method of colour reproduction.
Glass which is buried in the soil slowly decomposes. The alkali is leached out from the surface layer and the silica rich remnant then partially crystallizes and is likely to get fine cracks and to separate from the bulk, thus making the layers beneath accessible to weathering. The result of this decomposition is often a patina consisting of thin stacked layers which exhibits vivid iridescence.
Now I got a cullet found when gardening, probably a fragment of the rim of a thick-walled wide dish. Its size is about 70 × 50 mm, measured along the diagonals. How old it might be?
The outermost brownish layer flakes off when touched, thus exposing the layers beneath and their blurry iridescence. In transmitted light, only the green colour of the bulk is seen.
A moderate thunderstorm was over, in the west it was still raining as the sky glowed bright red. From my position the sun could not be seen, its height was only 0.16° (computed from data given in Wikipedia), part of it was already below the horizon.
In connection with the classification of rainbows, this phenomenon has been called “zero order glow”. Order in this case means the number of internal reflections in the raindrop. The light seen here thus has not been reflected but only fanned out by refraction. I have already photographed this once (“Behind the rainbow”), then the sun was slightly higher (about 2°) and the glow wasn't red but orange-yellow.
To the right in the left half of the image the incoming rays from the sun and the reflected ones which form the rainbow are shown; in the right half the rays refracted by the drop acting like a thick convex lens.
Each one of the falling raindrops in the field of view contributes a tiny speck of light in the colour of the setting sun.
Recently my attention was caught by a spider's horizontally extended sheet web which showed light rings and paths of light glistening in purple and green colours. The images below show the same area viewed from slightly different directions; to the right a detail of the left image.
The web – a tangle of lines within a thin, slightly undulating sheet – extends over the whole area of the image, but only the glinting parts of the threads are visible. Light rings formed by the glints are not circular, but oval – the imagined lines resemble contour lines on a map. This is due to the camber of the sheet containing the strands and shall not be the topic now.
Besides white, i.e. colourless reflections there are purple ones as well as green ones. What causes these colours?
Colours seen on spider webs in most cases occur if the web is looked at against the light when there is a dark background. Light scattered by small angles conveys the colours. But here now the scattering angles are large. If the sheet web were a membrane, the sun would be reflected at the brightest spots, similar to the reflections in the image to the right which shows a membrane produced by a snail. It is suggestive to speak of reflection instead of scattering in this case.
How the reflection of the sun in a transparent cylinder looks like depends on the viewing direction. To the right there are three macro-photographs of a nylon wire of 1 mm thickness. (a): only one reflection at the front side – the angle between illumination and observation (projected onto the plane perpendicular to the line) exceede the rainbow angle –, (b): to the left of the narrow reflection at the front there is the double reflection at the back, and at the right rim there is some light which has been reflected twice. (The double reflection from the back arises because there are two different rays which are deflected by the same angle, see the sketch below and the discussion of the rainbow.) In image (c) the direction of the strand, the sun, and the observer's eye are in the same plane. The reflections from front and back side overlap in the middle, and near both rims there is light which has been reflected twice at the back.
However, the silk strands are so thin that geometrical optics is no good approximation any more. The exact solution of light scattering by a transparent cylinder is available and may be found in the standard work of van de Hulst . But presumably the cross section of s silk strand is not exactly circular. Wrinkles are the reason for the specific silky gloss.
Therefore, we content ourselves with a rough estimate. Part of the light is reflected at the upper side of the strand, another part at the back side. The reflected waves interfere; the path difference of both rays is somewhat less or equal to twice the thickness of the strand. This has to be multiplied with the refractive index of the material to get the optical path difference.
Comparing the colours with computed interference colours of two rays, one can deduce that the optical path differences should be between 1000 nm and 1300 nm, or (what I think is less probable) between 1500 and 1800 nm. The refractive index of the silk is assumed to be 1.5, from that we obtain slightly more than 330 nm for the thickness of the purple strands and 430 nm for the greenish ones (or 500 and 600 nm respectively). The peculiar feature of this web was that the diameters of the strands didn't change noticeably along the glistening lengths.
“Typically, a spider's silk line is only about 0.001 mm - 0.004 mm thick”, but here the lines seem to be considerably thinner.
In the centre of the light rings the reflections are brightest, but white or only very faintly coloured. This may partially be due to over-exposition. In addition, if the threads run in east-west direction, the reflection is of type (a), thus there is no interfering second ray. (In the images, south is at the top. The height of the sun was 60.5° in the south, the web approximately horizontal, thus the angle between illumination and observation about 59°; for n=1.5 the rainbow angle is only 26°.)
 H.C. van de Hulst (1957): Light Scattering by Small Particles. Dover Publications, New York, Inc. 1981, ISBN 0-486-64228-3
Colours in spider webs are treated also here:
The purple emperor (Apatura iris) is very rare in the region where I live. Recently I could spot one and even take photographs, but I could not see its iridescence, as it opened the wings only a few times for a very short moment (left image below). On the right image which I got some years ago, a smidge of the blue iridescence is seen.
Two layers of tiny scales cover the wings of butterflies like tiles on a roof. The lamellar scales are hollow; their underside is smooth, the upper lamina is finely sculptured.
Red, orange, yellow, brown and black colour of the wings is due to pigmentation of the scales, blue, purple, and green with few exceptions are structural colours.
The scales of the purple emperor have been investigated by Pantelić et al., the following images are from that paper:
Comparing the images one can see that the scales of Apatura iris follow the same general blueprint as those of the peacock butterfly, but the proportions are very different. The cross section of the ridges on the cover scales reminds of the shape of fir trees. Depending on the angles of viewing and illumination, iridescent multilayer interference is seen or simply remitted light, brown due to the pigmentation of the scales. But as the ridges are so closely spaced, the rays reflected interfere with their neighbours, next neighbours and so on, reducing the angular distribution of the reflected light. According to Pantelić et al. with fixed illumination iridescence is seen only within a narrow angle (18°) and the spectral distribution of the reflected light has a maximum at 380 nm and width (FWHM) of 50 nm. Thus, most of the reflected radiation is in the ultraviolet region which the butterflies can see, but we can't.
The iridescent scales of the Morpho butterflies are very similar to those of Apatura. But why is their appearance so different? One reason is, of course, that the maximum reflectivity of Apatura is in the UV, and what we see is only the tail of the spectral distribution. Apparently, this is not the case with Morpho. (In Morpho species, not the cover scales, but the ground scales are responsible for the iridescence. M. rhetenor, however, has no cover scales.)
Secondly, the ribbons at the sides of the ridges are not opposed to each other, but stacked, see the image to the right, and adjacent ridges slightly and randomly differ in their height, so that interference between neighbours is averaged out with the consequence that reflection goes into a wider solid angle . Cover scales are transparent with coarser sculpture, or missing.
Two images to compare Apatura iris with Morpho rhetenor:
Many species of butterflies have scales of similar shape, but in almost all cases the iridescence occurs in the ultraviolet region. The clouded yellow (Colias croceus) is an example.
 Dejan Pantelić, Srećko Ćurčić, Svetlana Savić-Šević, Aleksandra Korać, Aleksander Kovačević, Božidar Ćurčić, and Bojana Bokić, “High angular and spectral selectivity of purple emperor (Lepidoptera: Apatura iris and A. ilia) butterfly wings”. Opt. Express 19, 5817–5826 (2011). doi:10.1364/OE.19.005817
 Shuichi Kinoshita, “Structural Colors in the Realm of Nature”, World Scientific Publishing Co. Pte. Ltd., ISBN-13 978-981-270-783-3
“I was skating on a random bog pond when I noticed these colorful arcs on both sides of the bright reflection of the sun. As you can see the sun was fairly low, and the ice was incredibly reflective, but even though perfect for skating it was of course full of small cracks and all kinds of features […] The phenomenon was visible while moving, without a camera, sunglasses or polarizing filters, and I even noticed it showing on an iphone video. … ”
This is what Mika-Pekka Markkanen wrote when he asked if I knew this phenomenon. But I never have seen anything like that, and I don't think that many people have.
Mika-Pekka is a keen photographer, see his instagram or fb-galleries, and had his camera and also a tripod ready when skating.
The colours can be explained by the interference of light reflected by very peculiar structures of the ice which probably have been caused by snow falling on the water and floating just before congelation. As the floating snow crystals protrude only slightly from the water surface, this may produce a fine sculpture of the ice, consisting of tiny ridges, bumps and pits. How the light is reflected by these will be discussed in some detail.
Click on the following thumbnails to see zoomed-in details and a frame of the video clip!
Photos © Mika-Pekka Markkanen, shown with permission.
The most prominent effect seen on the images is the bright glitter path running from the far shore towards the observer. If the ice were perfectly flat one should see the mirror image of the sun. Glitter paths are a quite common sight on water when the sun is low, and their shape depends on the size of the waves. The first image in the row below shows where the sun is reflected by surface elements inclined by 0° (one point), 3.5° (smaller contour), and 8°. (The height of the sun is 10° as is approximately the case in the photos). Comparing this with the photographs, we can conclude that the majority of glints was due to facets inclined by less than 3.5°, but up to 8° inclination occurred. However, randomly oriented facets with up to 8° inclination only produce glints within the 8° contour, even if second reflections of the rays are accounted for, and no rays come from the regions where the vivid colours occur. There must be surface elements so steep that they don't contribute to the glitter path, but can scatter rays already reflected once to larger angles. This is shown by the second figure below, assuming facets with 3.5° and with 60° inclination. Glints from light rays reflected first by gently inclined and then by steep facets are possible in the grey region. Note that the apex of this region is lower than the horizon. This compares well with the region where the coloured fringes occur in the photos. But as yet there is no colour.
Colours appear if there are glints due to interfering rays which slightly differ in the path lengths which the light has to travel. Interference may suppress parts of the spectrum while enhancing other parts. This occurs if there are smooth transitions between flat and steep regions, as shown in the third sketch in the row above. Two light paths are shown where the light comes from the left side after an assumed first reflection on a flat facet of the surface. Until they reach the yellow dashed line, the path lengths are equal, and after passing the orange dashed line, the distance to the observer's eye is also the same. But the distance AD is not the same as BC, and therefore destructive interference is possible.
The fourth sketch in the row shows the result of a model computation which reproduces some of the features seen in the photographs. (Click to enlarge!) It has been assumed that the light is first reflected by a slightly inclined facet and then hits the steep flank of a step as shown in the third image above.
While in the photographs the colours fade and vanish in the foreground, this is not the case in the simulation. The reason is simply that apart from the inclination of the reflecting facets, all geometrical details have been ignored, as well as the dependence of the reflectivity on the angle of the light's incidence.
The following parameters have been used: the shallow facets are inclined by an angle of θ = 6°, their orientation φ is random. The maximum inclination angle of the steep flanks relative to the shallow ones is Θmax = 60° (thus Θ is ranging from 0° to 60° due to the smoothed transitions), the cross section is sinusoidal as in the above sketch. The orientation of the flanks is related to the shallow facets in the following way: introducing a coordinate system fixed to the shallow facet, the z'-axis being orthogonal to the facet and pointing upwards, the x'-axis pointing in the direction of steepest descent, then the surface normal to the flank has polar coordinates Θ and Φ, with Φ restricted to Φ = ±150° only. The total height of the step is H = 10 μm.
The parameters used do not reflect the hexagonal symmetry of snow crystals, this is a serious flaw. But anyhow, this simulation cannot be the final answer to the problem, because the postulated height H of the steps is so small, about 0.01 mm. The widths of the interference fringes is determined by H; the smaller H, the wider the fringes. But if the light is reflected by such narrow facets, diffraction effects cannot be ignored and simple geometric optics (as used here) is only a crude approximation at best. A correct calculation is, however, not a simple task …
In 1818 the French Academy of Sciences launched a competition to explain the properties of light, where Siméon Denis Poisson was one of the members of the judging committee. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new wave theory of light. Poisson, being a supporter of the particle theory of light, thought that he had found a flaw when he argued that a consequence of Fresnel's theory was that there would exist an on-axis bright spot in the shadow of a circular obstacle (see Wikipedia).
However, the head of the committee, Dominique-François-Jean Arago decided to do the experiment and found the bright spot (wich therefore is also called Arago spot), with the consequence that Fresnel won the competition. Arago later pointed out that the phenomenon had already been described in 1715 by Delisle  and in 1723 by Maraldi .
In view of the experimental possibilities of that time – dependent on the sun as light source – this achievement is admirable. With today's artificial light sources and digital cameras, the proof is much easier, as seen from numerous images found in the web. This encouraged me to try the experiment with as simple means as possible, to see the effect with my own eyes.
As light sources a small LED torch and a small laser pointer in combination with a convex lens with 10 cm focal length have been used; shadow giving objects were a neodymium-magnet sphere of 5 mm diameter and pins with spherical heads of about 3.5 mm thickness. Small wooden blocks and a cork served as supports, a sheet of paper taped to the wall was the screen. Two tables. Photos were taken with a amall digital camera mounted on a tripod.
The distance from the light source to the object was 3.40 m, that from the object to the screen was 1.30 m. Using only the torch without diaphragm, the shadows were too diffuse to see the central light spot, therefore a pinhole diaphragm (ca 1 mmø) was put in front of it.
The press key of the laser pointer was fixed with adhesive film. The lens in front of it makes a light cone from the laser beam.
The spots are not easily seen with bare eyes, but on the photographs they are obvious. (Click to enlarge!)
There are other possibilities to produce bright spots within shadows. Browsing images and videos in the web I found the following example, which is too beautiful to be true, namely a CD with central hole and transparent part covered.
The illumination of the centre is not due to Fresnel diffraction but rather to refraction and reflection. The rim of the disc is transparent. The image to the right shows the transformation of a beam of light which passes the outermost part of the disk. The beam of a laser pointer was directed through the rim on the left side “at 9 o'clock” in approximately the same direction as the rays from the torch. The major part is refracted slightly outwards – note the bright ring surrounding the shadow in the left image above –, but another part is reflected at the slightly cambered outer surface and lightens the shadow region along a radial line. In the centre all these lines meet, producing a brighter spot.
Even with today's means it is not easy to demonstrate the Poisson spot, a small bright spot in the middle of the shadow of a sphere or a circular disk. Arago reportedly has discovered the spot using a tiny disk of 2 mm diameter.
With a steel ball of 24 mmø and an LED torch or a small laser pointer I could not make the spot visible. If, however, one only wants to see that light is diffracted into the shadow region, there is an easier way. If your eye is in the shadow of the ball, you will see part of its rim brightened by the diffracted light. And if the eye (or the lens of the camera) is in the centre of the shadow, the whole rim is bright.
Left pair of pictures: an LED torch behind a steel ball of 24 mm, right pair: the beam of a laser pointer is widened by a magnifying glass and lightens the ball from behind.
Similar pictures can be obtained by a coin with unruffled rim like the Euro five cent (with 21.2 mm diameter).
Fresnel diffraction can be seen occasionally on photographs when bright “points” are out of focus , . Such patterns are tiny on the sensor of the camera. But larger patterns are possible: the paradoxical counterpart of the Poisson spot, a dark speck in the middle of the bright patch of light behind a circular aperture, is easily demonstrated with a small laser pointer and a magnifying glass and can be clearly seen with bare eyes.
Left pair: diffraction patterns of a small, not exactly circular hole (1 mmø) pierced into an aluminium foil. The centre can be dark or bright, depending on the distances of light source and screen.
Right pair: theoretical diffraction patterns for a circular pinhole; image widths 2 cm. Parameters: aperture 1 mm, wave length λ=650 nm, distance from diaphragm to screen 1.5 m; distance from the point-source to the diaphragm 22.5 cm and 15 cm. (The point source is the focus of the magnifying glass in front of the laser.)
The glasses used for viewing 3D movies have already been discussed here . I use one pair as a handy and cheap analyzer of circular  and, turned backwards, linear polarisation. When, at sunset, I was looking for the polarisation of the skylight in different directions and, inadvertently, held the goggles not reverted, I was surprised to see stripes on the glass of my garret window.
Left: the sun is setting in the south-west, in south-east direction the skylight is polarised vertically. Turned around, the goggles in this position let pass only horizontally polarised light, the glasses therefore appear dark.
Middle: Held in normal position, the goggles show stripes on the glass of the inclined garret window. (Only left-handed light passes the left glass, only right-handed the right one.)
Right: the left lens held immediately in front of the camera. As the window is double-glazed, there is some moiré from the superposition of the patterns of both panes. In addition, there are wide stripes perpendicular to the narrow ones.
I noticed that the stripes remain stationary when changing the viewing angle; this rules out interferences like those discussed earlier  as a reason.
Thus, the stripes must be due to the properties of the panes, presumably periodically changing compressive stresses. Thermally toughened safety glass! At present for the inner panes of garret windows laminated safety glass is used, but here in this older one, obviously, both panes are of the toughened type.
From internal stress glass becomes birefringent. The incoming linearly polarised light is split into parts oscillating in directions orthogonal to each other, and behind the pane there is a phase shift between these parts. If this shift is exactly λ/4, the light is circularly polarised, as has been described earlier , in general elliptical polarisation results which may be considered as superposition of circular and linear polarisation.
Goethe's essays on colour science contain many interesting and accurately described observations. Here an addendum from 1820 to his treatise on chromatics (“Zur Farbenlehre”) is considered (digitized by Google), an essay on “entoptic” colours, colours arising from double refraction and polarisation of light as we understand it now. Goethe repeats and extends experiments of Seebeck and of Brewster with considerable effort. Specially prepared small plates of glass are used to mirror the light of the blue sky, showing black or white crosses. Blackened glass plates are used as mirrors to enhance the visibility of the effects in various ways.
What did Goethe see?
The simplest experiment – the glass square lies horizontally on a dark ground and reflects the blue sky. Depending on the direction ov viewing, dark or bright crosses can be observed. This simplest observation is more difficult to explain than the more sophisticated experiments which we therefore consider first.
The black mirrors polarise the light or serve to detect polarisation. Goethe was obviously not aware of that, writing that they serve to reduce brightness. Today we have more convenient methods: the computer's LED monitor screen is a source of linearly polarised light, to analyse this, polarising sunglasses, goggles for 3D-movies (held in reversed position) or, of course, a camera polarising filter may be used.
The treatment of the glass squares produces strong internal tensions, making the glass birefringent in a pattern which under most circumstances is invisible for the naked eye, but can be seen if the plate is placed between polarising filters. For the more elaborate experiments, Goethe used an “entoptic” glass cube. Will a glass die work as well?
Today, to see this effect, no specially prepared glass specimens are necessary, instead, objects made of acrylic glass or other transparent synthetic resins made by injection moulding may be used. Beakers or round lids are very suitable as their inner tensions and orientation of the filiform molecules are nearly rotationally symmetric.
A plastic beaker in front of the white screen of my computer, photographed with a polarising filter. To the left the polarising filter blocks the light from the screen (“crossed polarisers”) and a dark cross is seen; to the right the polarising direction of the filter is the same as that of the screen and a white cross appears.
Due to the rotational symmetry of the pattern of birefringence, light that enters the material is split into one part polarised in the radial direction and another part polarised tangentially. After passing the polarising filter, what remains interferes and thus parts of the spectrum are suppressed, others enhanced. But if the impinging light oscillates in radial or tangential direction already before entering the material, there is no splitting, as the other component is zero, and there is no interference. This happens along two diameters orthogonal to each other which form a dark or a bright cross, depending on the orientation of the analysing filter. (The colours arising from interference have been dealt with in a special section.)
Of course, the pattern of internal tensions of Goethe's glass squares was not rotationally symmetric, but the symmetries of the square produce quite similar patterns.
Goethe describes his experiments correctly, but in his interpretations and attempted explanations he is ensnared by his ideas on the origin of colours.
Now consider the simplest experiment where no black mirror is used. The glass square lies horizontally on dark ground and mirrors the blue sky. The blue sky is sunlight scattered by air molecules (Rayleigh scattering), and is polarised with the electric vector at right angle to the sun's rays. We consider the case that the electric field oscillates in the plane spanned by the incoming ray and the perpendicular to the glass (p-polarisation), and the incoming ray's angle is close to the Brewster angle. In this case, there is little reflection at the surface, the light refracted into the glass is split into two rays (ordinary and extraordinary ray). The reflection at the bottom surface of the glass acts like a polarising filter because of the near Brewster angle, and so the situation is that of a birefringent medium between crossed polarisers. The first polariser is Rayleigh scattering, the second one the reflection near Brewster's angle. Colours and the black cross appear. If the incoming light is polarised horizontally (s-polarisation) the bright cross can be seen. The following two photos of a plastic lid reflecting the white computer screen have been taken without polarising filter; the colours are the same as seen with the bare eye.
With a bit of good will in the left picture the black cross can be seen, and in the right one the white cross. As the light from the screen is polarised obliquely, I had to position my “entoptic” plastic lid obliquely too to demonstrate this effect.
We see the dark cross if p-polarisation prevails, the bright cross in the opposite case.
Goethe was fascinated by the colours arising in crystals, mica sheets, or glass between polarising black mirrors, as expressed in a poem (no translation). Here is another plastic beaker:
The coloured reflections on compact discs are well known and an everyday experience. But similar colours could also be seen on the old vinyl discs, though only at grazing incidence of the light. To the right there is a photograph taken in about 1960.
When I was going to abuse an old vinyl (which I couldn't play any more) to serve as disc of a Wimshurst machine, I was surprised to see faint colours on it in the spotlight. Not at all grazing incidence, instead lit and viewed from almost perpendicular directions.
The colours bear some slight resemblance to Quetelet fringes as well as to those of thin films. The fringes seem to be circular. My first idea therefore was to look for light paths with slightly differing lengths, possibly with two reflections in the grooves, in analogy to the Quetelet mechanism. There are such paths as shown in the sketches at the end of this short section. The two rays coming from the same groove cannot be resolved by the eye, so the waves are superposed on the same spot of the retina (or the camera's sensor) and interfere to produce the colours well known from soap bubbles, oil films on wet pavements, generally from two-beam interference.
Using a small LED torch I tried to reproduce the effect with other old vinyls. It turned out that under similar conditions the results with different discs were quite different, presumably because of the shapes of the grooves' cross sections. Below are some examples.
The last picture in the row shows the record where I first saw the effect (which survived the unsuccessful test as a Wimshuest machine disc with minor injuries). Click on the images for larger versions! The sequence of colours is quite different from that seen in soap films or other examples of two-beam interference which means that the physics is not so simple: the grooves are so narrow that diffraction effects must not be ignored. Instead of accounting only for the paths where the light is deflected by specular reflection, we should, in the spirit of Huygens and Fresnel, consider the elementary waves scattered from all points along the cross section of the groove, scattered once more and finally forming the outgoing wave.
We have seen that small surface elements of black paper glisten in different colours, and as the surface structure there is irregular, the pattern of coloured specks is irregular too. Here now the surface is structured by slightly wiggly grooves with constant cross section. If at a certain point some colour is reflected at given illumination and observation angles, along the groove the colour changes only slowly, and the neighbouring grooves, having the same cross section and viewed and illuminated at almost the same angles, are likely to show nearly the same colour too. Therefore there are “regular” fringes instead of chaotic speckles.
A sheet of black cardboard in the sunlight. It looks inconspicuously, black or, more precisely, dark grey. But when looking at it closely in the direction to the sun, a pattern of colourful tiny specks can be seen. The image to the right is an enlarged detail of the left one. At the bottom the millimetre-scale of a set square.
I remember that once I have read about this phenomenen in Goethe's “Colour Theory”. This book, scanned and digitized, is now accessible in the web in the “Deutsches Text-Archiv”, here is the citation:
Of course, a correct explanation is not to be expected from Goethe. Diffraction and interference produce this effect, as has been discussed in detail in the corresponding section.
The dark brown fur of the dog in the next image is shimmering. The effect is much the same. Colours are seen where the gloss of the fur is slightly out of focus, and pink and green prevail.
A similar observation has also been described accurately by Goethe:
(The text images have been derived from the scans of the Deutsches Textarchiv. Goethe, Johann Wolfgang von: Zur Farbenlehre. Bd. 1. Tübingen, 1810.)