Rays in a room: crepuscular rays, sun glitter or caustics?

 

Rays in a room: crepuscular rays, sun glitter or caustics?

We went for a holiday to Cornwall, and stayed a couple of days in a hotel in Falmouth, named Green Banks that overlooks Falmouth Harbour.

One morning I observed some flickering rays of light just behind the bedroom, which is just metres away from the shore (see the picture below and the attached video). I’d never seen this before so I was fascinated.  But I wanted an explanation.

Fig.1

After back tracking a bit I saw the rays were coming through the French window on the extreme right:

Fig.2

I figured the culprit must be outside the room:

Fig.3

But I was still puzzled because there was no wind, so the clouds were almost static and the water of the harbour mostly unruffled, other than by a few ripples due to some passing boat. And, anyway, most of the boats were anchored and pretty far away.

I would describe the flickering rays as scintillating – but they would not be covered by the term ‘scintillation’ which means something completely different.

The photos were taken early in the morning (7:55 am – 8:00 am on 26^th Aug.2023). So, one possible explanation could be crepuscular rays. The word crepuscular originates in the Greek word for twilight – but people often refer to pictures taken at dawn or dusk. But usually these are sun rays that are partly obstructed by clouds. And this photo doesn’t seem to match that description.

Much later I came across the phrase ‘sun glitter’. This seems to match perfectly: the glitter path – the collection of sun glints reflected by the water – is quite clearly visible in the photo.

But I was conflicted.

At this point I decided to ask an expert, so I e-mailed Dr.Christopher S.Baird of West Texas A & M University and exchanged some e-mails – which I have reproduced in the Appendix for completeness. To summarise: he says that what I photographed is sun glitter and that he has seen this several times in sun rays reflected from a body of water. He rules out crepuscular rays as an explanation and states that they aren’t rays at all:

“Rather, it is the image of the sun, which has been greatly elongated by the glitter-reflection process and by your ceiling intercepting the light at a deep angle.”

But what is to distinguish an extremely elongated image of the sun at a deep angle from a ray?

I’m still conflicted, but I know more now!

While I agree completely with the sun glitter explanation, I’m also inclined to ask if they could not be crepuscular rays as well: an overlap of categories – since the term crepuscular rays is often used quite loosely. This is what I would like to discuss. But, I admit that a strict definition would probably rule out crepuscular rays, as Dr.Baird states.

I Crepuscular Rays:

I think there are 4 conditions to be satisfied for crepuscular rays [1]:

a)       The word ‘crepusculum’ in Latin refers to ‘twilight’, but crepuscular rays also refer to near dawn or near dusk.  “Loosely, the term crepuscular rays is sometimes extended to the general phenomenon of rays of sunlight that appear to converge at a point in the sky, irrespective of time of day” [1]. However [2]: “When the sun is closer to the horizon, the rays tend to be longer and more pronounced, creating a dramatic effect.”

b)      Obstructions: by clouds, mountains or even the leaves of trees, which cause dark areas between rays. “Crepuscular rays are alternating light and dark sun rays streaming through gaps in clouds [3].”

c)       Scattering [3]: ”The beams of sunlight are made visible by haze in the atmosphere. They are most noticeable when considerable scattering is present, caused by airborne dust, inorganic salts, organic aerosol particles, small water droplets and the air molecules themselves.”

d)      Rays Apparently converge [3]: “Crepuscular rays are alternating light and dark sun rays streaming through gaps in clouds. They appear to fan out from the Sun's position, although they are actually parallel (like the apparent converging of railroad tracks when you look down a long straight track).” 

Fig.4: Crepuscular rays from [2]: close to sunrise/sunset time

Fig.5: Crepuscular rays: from [4]: not so close to sunrise or sunset

Fig.6: crepuscular rays obstructed by mountainous topography from [5]

Fig.7:  crepuscular rays through the leaves of trees, called komorebi in Japanese [6], an integral part of ‘forest bathing’ (shinrin-yoku in Japanese). You can also see more detailed descriptions of komorebi by Simon Wilkes [7].

Fig.8: crepuscular rays, through windows, into a dusty room [8].

Fig.9: crepuscular rays underwater from [9]

Lunar crepuscular rays:

Fig.10: lunar crepuscular rays [10]

Yes, you can see crepuscular rays at night due to moon rays. I agree that night is not twilight…but, as mentioned above,  the term ‘crepuscular’ is rather loose.

Anticrepuscular rays:

Crepuscular rays, as you can see above (and even more so, online) are all over the place. But anticrepuscular rays [11] are rare. Only once have I ever seen anticrepuscular rays, and, sadly,  I didn’t have a camera! To tell the truth, I didn’t even know what I was seeing.

Anticrepuscular rays [11] appear to converge to a point, the antisolar point (the point on the horizon exactly opposite the sun). “Anticrepuscular rays are most frequently visible around dawn or dusk. This is because the atmospheric light scattering that makes them visible (backscattering) is larger for low angles to the horizon than most other angles. Anticrepuscular rays are dimmer than crepuscular rays because backscattering is less than forward scattering.”   The weakness of backscattering explains why anticrepuscular rays are not seen that often. But look at Fig.10 below: the rays diverge from the solar point (on the left) and appear to converge at the antisolar point (180 degrees away). To be clear: that’s not exactly how they look in 3D (although Fig.11 gives you a better idea): the rays pass overhead in a curving fan, and converge at both the solar and ant-solar point. Quite a sight!

Fig.11: Anti-crepuscular rays from [5]

Fig.12: Crepuscular and anticrepuscular rays [12b]

II Sun glints, sun glitter and glitter paths:

“A glitter path is a vertical reflection of a very bright light source on water, extending from the horizon straight down to the water near the viewer. Typically the source is the sun or moon, so sometimes it's called a "moon-path." The glitter path widens where the water is disturbed, and it narrows in the areas where the water is calmer. [12].”

Fig.13: A glitter path of the Sun (a), from [13], and a moon path [13].

“Wavelets present many small reflecting surfaces at a variety of angles. Wherever those surfaces are just the right angle to reflect the sun, a spot or dash of light appears [13].”

One of the most iconic paintings that depicts glitter paths is Van Gogh’s “Starry Night over the Rhone.”

If the surface of the sea is perfectly still and flat, the glitter is just a specular (mirror) reflection of the source corresponding to a single point P (shown in Fig.14). The width of the reflection equals the width of the light source (sun, moon, or other light source). This bright reflection is referred to as a glint e.g. a sun glint of a moon glint. If the Surface is rippled or wavy, many glints will be seen by the observer – as seen in Fig.15. The collection of glints is the glitter path, and the large number of ripples give the eye an impression of a continuously lit area. The glitter path is a collection of a large number of reflections of a light source by suitably tilted water waves.

Fig.14: Reflections from different points on ripples [14a]: at the point P, the reflection is from the top of the ripple; to the left of P, reflection is from the right-hand side of the ripple; for ripples on the right of P, the left-hand side of the ripple reflects the light to the observer. If the water were perfectly still, only the point P would reflect light to the observer. The farther the ripple is from P, the greater the inclination at the reflecting position. Since there is a maximum slope for the ripples, there is a finite region around P that will glitter.

Fig.15:  from [14b] by Shaw

Shaw [14b] points out that:

“For a high light source, the angular length of a glitter pattern is equal to four times the angle of the maximum wave slope. Waves inclined both toward and away from the observer create glints, resulting in a factor of two times the maximum wave slope; the additional factor of two is a result of angular doubling on reflection.” That is, if the maximum wave slope is α, the angle over which the glitter path is visible is 4α. For people like me who forget that a mirror tilted by angle α results in the image shifting by 2α, please see [14c].

Fig.16: The point that reflects light to the observer is also shown here [15].

 Mariners have noted that the higher the waves, the broader the glitter path [16a]. This observation is used at sea when closing on land like islands to spot rough water over shallower ground close to the shore – as can be seen in Fig.16.

“The glitter path is most beautiful and clear if the water is almost still and the sun is about 5 degrees above the horizon. Large waves will spread out the glitter path over such a large angle that it becomes less noticeable [16b].”  The point about 5° above the horizon also applies to the moon path [16c].  In addition, O’Meara [16c] raises an interesting point: will the glitter path of the full moon look different from that of the crescent moon? Similarly, if one were to observe the glitter path of the sun during a solar eclipse, would it, too, look different? Of course, observing the crescent moon would be easier.

A more detailed explanation of glitter paths is given in [14b, 17].

Fig.17: Note how the width of the glitter path varies depending on the local surface roughness (from [14b])

Fig.18: A glitter path is visible not just on water, but also on ice and snow [18a]; a glitter path on blue ice in the Antarctic [18b].

Sun glint is a phenomenon that occurs when sunlight reflects off the surface of the ocean at the same angle that a satellite or other sensor is viewing the surface. In the affected area of the image, smooth ocean water becomes a silvery mirror, while rougher surface waters appear dark [19].

Sun glitter can be caused by the “natural movement of water, or the movement of birds or animals in the water…or even momentarily by a rock thrown in the water [20].”

“A rippled but locally smooth surface such as water with waves will reflect the sun at different angles at each point on the surface of the waves [21].”

Further: “The colour and the length of the glitter depend on the altitude of the Sun. The lower the sun appears, the longer and more reddish the glitter is. When the sun is really low above the horizon, the glitter breaks because of the waves, which obstruct the Sun [21].”

Shaw [14b] describes the work of Cox and Munk [21a], where the observer of Figs.14 and 15 has literally taken off from the ground:

“The maximum wave slope can be determined from the geometry of glitter patterns. In 1951, Charles Cox and Walter Munk found a more quantitative way of using glitter patterns to derive a statistical model for the complete wave-slope distribution. They used cameras in the bomb bay of a World War II surplus B-17G aircraft to photograph sun glitter on the Pacific Ocean near Hawaii. By relating the photographic density to the probability of a sun-glint wave slope, Cox and Munk derived a wave-slope probability density function.” What did they find? The effect of wind on wave slope distribution: “The mean-square slope increases approximately linearly with wind speed, indicating that the surface gets steadily rougher as the wind blows harder.”

Nowadays, sensors are mounted even further up, on satellites. For example, Zhang et al [22] modelled sea surface roughness (SSR) from the sun glitter hi-res images taken by the ASTER satellite sensor from multiple angles (Advanced Spaceborne Thermal Emission and Reflection Radiometer). The modelled SSR was converted to wind speed and compared with meteorological, with good agreement. 

Another example of observation of sun glitter from space is of the shoreline of Marin in California [23], as reported by Adam Voiland:

Fig.19: the picture was taken by the MODIS (Moderate Resolution Imaging Spectroradiometer) on the Terra satellite in 2015 [23]

I couldn’t resist this beautiful image of sun glints [24]:

Fig.20: sun glitter on the surface [24] (image from Shutterstock): “These glints appear as the sunlight reflects off slopes of the waves. As the waves move, the slopes of the waves move, and the sunlight continues to reflect light off the moving slopes. Voila! You will see the magical dancing flickers of light known as sun glitter!”

 

 

III. Caustic reflections

At the end of his 2^nd email, while I was going on about crepuscular rays, Baird suddenly brought up caustic reflections [25, 26]:

“Note that if the ripples on the surface of the water are very large, the pattern of light changes from an elongated image of the sun surrounded by glitter, to a network of wave-shaped light streaks called water-surface-reflection caustics.”

Fig.21: reflection caustics: a) under the arch of a bridge b) on the hulls of boats in the water [25].

Essentially, sun rays are reflected by undulating waves which act like lenses and mirrors simultaneously, as in Fig.22a and Fig.22b:

Fig.22: reflection and refraction caustics from [25] and [26b]

Reflection caustics are called cata-caustics and refraction caustics are called dia-caustics [26b]. Note that reflections occur from the concave portions in between waves in Fig.22, while refractions occur from near the peaks.

One of the first to draw reflection caustics from a circular mirror was Leonardo da Vinci [26b], but the Greek word ‘kaustikos’ (‘to burn’) refers to the famous mirrors Archimedes used to burn the sails of enemy ships.

Two examples of refraction caustics:

Fig.23: Refraction caustics in a swimming pool [27a] and on a lakebed [27b].

“Waves and ripples on the surface of the shallow water refract the sun's rays. When the surface curvature happens to be right the rays cluster on the lakebed to form a sharply illuminated line - a ‘caustic’ [27].” Of course, most of us are familiar with these dia-caustics: we have all seen them on swimming-pool floors.

Math bump: if you are allergic to maths, go straight to “Concluding remarks”

“A fascinating result stemming from catastrophe theory is that there are only seven stable elementary forms of caustics. These forms include cusp and fold caustics, which exhibit distinct patterns and interactions [28].” These 7 types of cata-caustics are explained [26a]: ellipse, secant, square root, circle, parabola, inverse power law and Cauchy. However, Keith Beven [27b] states that there are just 5 types in up to 3D: fold, cusp, swallowtail, elliptic umbilical, and hyperbolic umbilical. The Wikipedia page lists these 5 as well as 2 more catastrophes: butterfly and parabolic umbilic [27c]. The two catastrophes that are excluded have a codimension K of 4, while the ones included have codimensions of K between 1 and 3. Beven seems to be an outlier, since he’s the only one that I know of who claims that Thom proposed just 5 elementary catastrophes.

However, anyone who wants to know about real-world caustics in terms of the 7 types should read the book by J.F.Nye [29a] (accessible on Google Books) or Michael Berry [29b]. A codimension refers to a subspace within a space and the number of dimensions it does not use: a point in a 3D space has K = 3, a line has K = 2 and a plane K = 1 [29c]. The complexity of the caustic curves (see Table below) increases as the codimension K increases [30d]. (And the names in [27a] seem to be different!).

 

Type of catastrophe	Codimension K	Potential function
Fold	1	V = x3 + ax
Cusp	2	V = x4 + ax2 + bx
swallowtail	3	V = x5 + ax3 + bx2 + cx
Elliptic umbilic	3	V = (x3)/3 – xy2 + a(x2 +y2) + bx + cy
Hyperbolic umbilic	3	V = x3 + y3 + axy + bx + cy
Butterfly	4	V = x6 + ax4 + bx3 + cx2 + dx
Parabolic umbilic	4	V = y4 + x2y + ax2 + by2 + cx + dy

Npte: in each potential function, the ‘germ’ (the non-singular part) is in bold letters. The number of control parameters in each potential function equals its codimension K; with the codimension representing all the different ‘directions’ required to completely unfold the singularity [29e].

How are caustics defined? According to Beven [26b]: “Caustics may be defined as the envelope of light rays that have been reflected or refracted from a curved surface and projected onto a surface where they can be visualised.”  Also [26b]: “The caustic surface or caustic sheet separates the region where rays of light cross and the light is concentrated, from those where they spread giving a darker region.” (as in Figs.21 and 22).

And another definition [26a]: “A caustic is understood mathematically as the envelope function of multiple rays that converge in the Fourier domain (angular deflection measured at far distances).  These are points of mathematical stationarity, in which the ray density is invariant to first order in deviations in the refracting surface.”  So, the regions of bright light in Fig.21 and Fig.22 are bounded by these envelopes.

The connection of caustics to catastrophes [26b]: “A caustic surface is effectively a form of separating surface analogous to the catastrophe surfaces in the theory.”.

Catastrophe theory has been connected with abrupt changes away from a stable equilibrium, leading to tipping points or even hysteresis [26c].

One might well ask: what is so catastrophic about some reflected ripples of light? In catastrophe theory, caustics are singular points or lines formed by the intersection of light rays reflected (or refracted) from a curved surface. These are points or lines where the light intensity changes dramatically due to focusing (very high spatial gradients, but not infinite). In ray tracing, caustics are the ‘edges’ where multiple rays converge at a single point (this paragraph was generated by Google AI Overview!).

Fig.24: a coffee cup caustic [26h] (it could even be a tea-cup, or any old mug!)

A confirmed sceptic might well ask: these 7 elementary catastrophes are wonderful mathematics but in optics have they been experimentally observed? The quickest answer is that we can all check out is the cusp catastrophe that can be seen when sun rays fall on a coffee cup as in Fig.24 [26d, 26g]. Pretty irrefutable, that. But there is more: Zannotti et al [26d] point out that Airy beams had already been identified with fold catastrophes and the Pearcey beam with the cusp catastrophe, so they designed structures to create beams corresponding to the swallowtail and butterfly beams. And Nye [26e] had shown in 1997 that liquid drop lenses are associated with higher umbilic catastrophes. However, I must add that often the pictures are just described as ‘caustic reflections’, without specifying which type.

IV: Concluding remarks:

The Sun elevation was 23.63° and azimuth was 102.35° at 8 am at Falmouth on 26^th Aug.2023 [30].

Shaw [14b] notes that:

“As the sun or moon drops lower in the sky, the glitter pattern gets progressively narrower until the width-to-length ratio reaches a minimum when the source (Sun, moon, etc) elevation angle is twice the maximum wave slope. Beyond this angle, as the sun or moon approaches the horizon, the glitter pattern becomes shorter because of shadowing and eventually disappears.”

The fact that the glitter path is visible when the sun elevation is 23° tells us something about the maximum wave slope (that it is more than 12°). But not enough: I should have woken up a bit earlier that morning to note the time that the glitter path emerged.

“When the sun is high, glitter is elliptical and grows gradually more elongated and narrower as the sun elevation decreases [32]”. Lynch et al [31] mention that the solution of simultaneous equations involving the phase function they derive for the position of sun glints as a function of time “in a different context” gives the location of glints whose coalescence “produces the associated diffraction caustic [33].” One might have expected a connection between glints and caustics, and Marston [33] indicates that it does exist. Lynch [31] also compares the number of sun glints produced by capillary waves and gravity waves: “a raft of capillary waves typically has 5-10 waves and thus produces 5-10X as many glints as the gravity waves upon which it rides.” Gravity waves (typically have longer wavelengths) produce longer glitter paths; capillary waves (wavelengths typically less than 1.7 cms) exhibit much denser, more continuous, reflections. Further, the schematic of ripples in Fig.15 would have to be modified in the more complicated case of capillary waves riding on gravity waves. The curvatures of capillary and gravity waves were determined by Cox and Munk from their measurements and analysis of glitter [21b].

 

At the end of this long collection of figures, what can one conclude?

Obviously, the shifting rays I saw were connected with sun glitter, since the glitter path is clearly visible in Fig.3. Also, the window is static so it cannot introduce any time dependence into the (putative) crepuscular rays (unlike clouds or leaves that move with the wind).

I initially believed the rays could be explained by crepuscular rays, simply because they look similar. I have tried to show various different types of crepuscular rays generated by different kinds of obstructions, at times of the day that may not be twilight or even close to twilight. The obstructions could be clouds, mountains, the leaves of trees and even windows. Scattering is also needed by different types of scatterers: water droplets, aerosols, dust motes and even air molecules.

That’s why the rays could also be described as crepuscular rays (a rather capacious term). Baird argued that these are merely ‘a greatly elongated image of the sun’ due to the glitter-path geometry ‘at a deep angle’. However, that sounds to me as an alternate description of a ray – although, that may be a matter of opinion. However, one should note that crepuscular rays usually occur when sun rays are directly incident on the obstructions (clouds, mountains, leaves etc). Whereas the rays that hit the obstructing window in the Green Banks hotel room are already reflected by the sea water.

The third possibility raised by Dr.Baird is the one of caustic surface reflections. This is entirely possible, since caustic surfaces are practically everywhere. Even in rainbows [26]. Just looking at Fig.21, I doubt that the rays I saw are caustics, because they caustics mostly look like streaks and striations (that do intersect at places) that are pretty well-separated, and are most likely examples of the fold catastrophe. But the distance between the water and the arch is a few metres at most, while the rays I observed were between 10-100 metres away from the water.  I’m not inclined to be caustic, but I can’t rule it out, since Lynch states that there is a connection between glints and caustics [31,32].

I would tentatively conclude that I could explain these rays by any one of these three explanations, but with a descending order of plausibility. Or they could all be valid!

Fig.25: ‘God rays” are often associated with the Buddha [33].

Of course, there is a last description: in mythology crepuscular rays are described as ‘god rays’, also the ‘fingers of god’ [34], or Buddha’s fingers (twilight rays that resemble elongated, finger-like projections extending from the sun's position) and Jacob’s ladder [35] (a specific type of twilight ray formation where the rays appear as vertical columns or curtains stretching from the horizon towards the sky).

But it isn’t just crepuscular rays: research by Ilario Cristofaro [36] mentions that the Incas were much taken by the interaction of Sun and Water, and that the glitter path features in multiple artistic depictions of seascapes of the Aegean Sea (dotted by islands) by Greeks from the Late Bronze Age to the Archaic Period.

References:

1)      https://en.wikipedia.org/wiki/Crepuscular_rays

 

2)      https://www.atoptics.co.uk/blog/crepuscular-rays/

 

 

3)      https://www.weatheronline.co.uk/reports/wxfacts/Crepuscular-rays.htm

 

4)      https://atoptics.co.uk/blog/crepuscular-rays-2/

 

 

 

5)      https://earthsky.org/earth/crepuscular-rays-sunrays-photos-around-world/

 

6)      https://shoreacres.wordpress.com/tag/komorebi/

 

 

7)      https://www.simonwilkes.co.uk/blog/komorebi/

 

8)      https://stockcake.com/i/sunlit-dusty-interior_148683_19575

 

9)      https://www.dreamstime.com/illustration/underwater-light-rays.html

 

 

10)   https://epod.usra.edu/blog/2008/11/lunar-crepuscular-rays.html

 

11) https://en.wikipedia.org/wiki/Anticrepuscular_rays

 

 

12)   a) https://science.nasa.gov/anticrepuscular-rays-converge-opposite-sun

b) https://epod.usra.edu/blog/2019/12/anticrepuscular-rays-observed-over-miami-beach-florida.html

13)   https://gurneyjourney.blogspot.com/2019/03/glitter-path.html

 

14)   a) Joachim Schlichting, “The Glitter path: an everyday life phenomenon relating physics to other disciplines”

 

b) Joseph A.Shaw “Glittering light on water” Optics and Photonics News (!999) 43

c)  https://testbook.com/question-answer/when-a-mirror-is-rotated-by-an-angle--    5f69d400cb1e1a2a6fff6059

 

 

 

15)   https://sky-lights.org/2019/02/18/glitter-paths/

 

16)    a) https://www.naturalnavigator.com/news/2020/01/the-glitter-path/

 

b) https://www.weatherscapes.com/album.php?cat=optics&subcat=glitter_path

c) https://www.astronomy.com/observing/the-long-track-of-the-moon/

Stephen James O’Meara

 

17)   https://atoptics.co.uk/blog/glitter-paths/

 

 

18)   a) https://photosynthesis-in-nature.com/optics_glitter-path.html

 

b) http://optics.kulgun.net/GlitterPath/

 

 

19)   https://en.wikipedia.org/wiki/Sunglint

 

20)   https://en.wikipedia.org/wiki/Sun_glitter

 

21)   a) C.Cox and W.Munk J.Opt.Soc. of America 44(1954) 838-50

 

b) C.Cox and W.Munk “Slopes of Sea Surface Deduced from Photographs of Sun Glitter” (Univ. of California Press, Berkeley and Los Angeles, 1956) p.459

 

22)   H.Zhang et al Remote Sensing Environment 218 (2018) 97-108

 

23)   https://earthobservatory.nasa.gov/blogs/earthmatters/2016/11/09/ground-to-space-a-glittering-path-of-san-francisco-sunglint

 

24)   https://www.willyswilderness.org/post/when-conditions-are-right-glitter-can-appear-on-the-water

 

 

 

25)  https://gurneyjourney.blogspot.com/2010/07/caustic-reflections.html

 

 

26)   a):

https://galileo-unbound.blog/2021/02/28/casual-caustics-and-the-optics-of-rays/

b) Keith Beven:  

 https://www.onlandscape.co.uk/2019/01/physics-of-caustic-light-in-water/   

c):

https://en.wikipedia.org/wiki/Catastrophe_theory

d)      B.Guilfoyle and W.Klingenberg “Symmetries of the Coffee-cup caustic” arXiv 0411189v1 math.DG 9Nov2004

e)      A.Zannotti et al “Optical catastrophes of the swallowtail and butterfly beams” arXiv 1703.07716v1 22 Mar2017

f)        J.Nye “Catastrophic optics of liquid drop lenses” Proc.Roy.Soc. A403 (1986) 1-26

g)       https://www.youtube.com/watch?app=desktop&v=kA3Q_Z591qU

h)      C.Ucke and C.Engelhardt “Playing with caustic phenomena” published in: “New ways in physics teaching” GIRE/ICPE Conference in Ljubljana (!996) 440-444

27)    

a)       https://www.alamy.com/pure-caustic-refractions-on-the-bottom-of-a-swimming-pool-caused-by-the-sun-passing-through-the-rippling-surface-of-water-the-picture-was-made-underw-image334854362.html

 

b)      https://atoptics.co.uk/blog/lake-caustics/

 

28)   https://atoptics.co.uk/blog/opod-trout-optical-catastrophes/

29)    a) “Natural Focusing and fine structure of light: caustics and wave dislocations” by J.F.Nye  (Institute of Physics, Bristol, 1999)

 

 

b)       https://michaelberryphysics.files.wordpress.com/2013/07/berry089.pdf

or: “Singularities in waves and rays” Course 7

c)        https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry105.pdf

 

d) https://en.wikipedia.org/wiki/Codimension

e) Michael Berry and C.Upstil “Catastrophe optics: morphologies of caustics and their diffraction patterns” Progress in Optics Vol.18 (North Holland, 1980) Ed.E.Wolf

f) http://home.swipnet.se/~w-48087/faglar/materialmapp/teorimapp/ekt1.html

 

 

30)   https://www.omnicalculator.com/physics/sun-angle

 

31)   David K.Lynch et al “Glitter and glints on water” Applied Optics 50 (2011) F39-49

 

32)   P.L.Marston, “Geometrical and catastrophe optics methods in scattering” in “Physical Acoustics (Academic, 1992) eds. A.D.Pierce and R.N.Thurston Vol.21 pp.1-234.

 

 

33)   https://www.craiyon.com/image/A45IlM8JRialevEgtr10hQ

 

34)   https://weather.com/news/news/fingers-of-god-crepuscular-rays-20130220

 

35)   https://atoptics.co.uk/blog/twilight-rays-2/

 

36)   https://doi.org/10.1558/jsa.39053 :”When the Sun Meets Okeanos” Ilario Cristofaro (2020)