Emergence in classical optics: caustics and rainbows

                                                                Photo by Chris Lawton on Unsplash

I love seeing patterns such as those above in bodies of water. I did not know that they are an example of emergence, according to Michael Berry, who states:

“A caustic is a collective phenomena, a property of a family of rays that is not present in any individual ray. Probably the most familiar example is the rainbow.”

Caustics are envelopes of families of rays on which the intensity diverges. They occur in media where the refractive index is inhomogeneous. In the image above, there is an interplay of the uneven air-water interface and the difference in the refractive index between air and water. For rainbows, key parameters are the refractive index of the water droplets and the size of the droplets. The caustic is not the "rainbow", i.e., the spectrum of colours, but rather the large light intensity associated with the bow. The spectrum of colours arises because of dispersion (i.e., the refractive index of water depends on the wavelength of the light).

Caustics illustrate several characteristics of emergence properties: novelty, singularities, hierarchies, new scales, effective theories, and universality. 

Novelty. The whole system (a family of light rays) has a property (infinity intensity) that individual light rays do not.

Discontinuities. A caustic defines a spatial boundary across which there are discontinuities in properties.  

Irreducibility and singular limits. Caustics only occur in the theory of geometrical optics which corresponds to the limit where the wavelength of light goes to zero in a wave theory of light. Caustics (singularities) are not present in the wave theory.

Hierarchies. 

a. Light can be treated at the level of rays, scalar waves, and vector waves. At each level, there are qualitatively different singularities: caustics, phase singularities (vortices, wavefront dislocations, nodal lines), and polarisation singularities. 

b. Treating caustics at the level of wave theory, as pioneered by George Bidell Airy, reveals a hierarchy of non-analyticities, and an interference pattern, reflected in the supernumerary part of a rainbow.

New (emergent) scales. An example, is the universal angle of 42 degrees subtended by the rainbow, that was first calculated by Rene Descartes. Airy's wave theory showed that the spacing of the interference fringes shrinks as lambda^2/3.

Effective theories. At each level of the hierarchy, one can define and investigate effective theories. For ray theory, the effective theory is defined by the spatially dependent refractive index n(R)  and the ray action.

Universality. Caustics exist for any kind of waters: light, sound, and matter. They exhibit "structural stability". They fall into equivalence (universality) classes that are defined by the elementary catastrophes enumerated by Rene Thom and Vladimir Arnold and listed in the Table below. Any two members of a class can be smoothly deformed into one another.

The first column in the Table below is the name of the class given by Thom, and the second is the symbol used by Arnold. K is the number of parameters needed to define the class and the associated polynomial, which is given in the last column. 

For this post, I have drawn on several beautiful articles by Michael Berry.  A good place to start may be 

Nature's optics and our understanding of light (2015), which contains the figure I used above of the rainbow.

There is a beautiful description of some of the history and basic physics of the rainbow in Rainbows, Snowflakes, and Quarks: Physics and the World Around Us by Hans Christian Von Baeyer