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Rainbows and Other Caustics University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 12-23-2018 From Photon to Neuron Chapter 17: Rainbows and other Caustics Philip C. Nelson University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Optics Commons Recommended Citation Nelson, P. C. (2018). From Photon to Neuron Chapter 17: Rainbows and other Caustics. Retrieved from https://repository.upenn.edu/physics_papers/652 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/652 For more information, please contact [email protected]. From Photon to Neuron Chapter 17: Rainbows and other Caustics Abstract This chapter extends Part III of the book From Photon to Neuron (Princeton Univ Press 2017). This preliminary version is made freely available as-is in the hope that it will be useful. Keywords rainbow, caustic, stationary phase, optics Disciplines Optics | Physical Sciences and Mathematics | Physics This book chapter is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/652 From Photon to Neuron Princeton University Press, 2017 Supplementary Chapter Philip Nelson December 23, 2018 This chapter belongs to Part III (Advanced Topics). It is made freely available as-is in the hope that it will be useful. All reports of errata will be gratefully received: [email protected]. Copyright c 2018 Philip C. Nelson • CHAPTER 17 Rainbows and Other Caustics Don’t expect that mathematics will give you an easy answer to any physical problem. If you find that it does, that is evidence that you picked a poor problem to begin with. — Blair Kinsman 17.1 SIGNPOST: BIFURCATIONS OF STATIONARY-PHASE POINTS If any physical phenomena can be called “beautiful,” surely the rainbow is an example. Throughout human history it has played a key role in art, literature, and even religious scriptures, inspiring awe.1 And yet, for millennia the rainbow was also a goad, a rebuke to humans’ attempts to understand Nature. Practically every natural philosopher of any note had something to say about rainbows, yet very little was understood until Ren´eDescartes published an essay titled La Dioptrique in 1637.2 We now see the rainbow as an exemplar of a much broader class of optical phenomena called “caustics.” Phrased in modern language, we will see that Descartes correctly identified the origin of a caustic as a consequence of an even more broadly applicable concept called bifurcation: in this case, the merger and mutual annihilation of two stationary-phase points as a parameter is varied. Figure 17.1 shows some features of the rainbow that are directly observable with the unaided eye. The Focus Question is Question: What are the minimal requirements for the focusing of light? Can it happen naturally, or only with special apparatus? Physical idea: Focusing to a bright line of light, with a diffuse background, is generic. 17.2 PRELIMINARIES 17.2.1 Natural versus contrived focusing Chapter 6 discussed the focusing of light by a lens. Focusing to a point turned out to be a tricky affair, requiring the lens to have the right shape and distance to a focusing screen. In constrast, another form of focusing arises naturally and is ubiquitous. For example, sunlight that arrives at the bottom of a swimming pool gets concentrated into a network of bright lines (not points). The constantly changing shape of the water surface (due to wind) does not destroy these lines, but merely moves them. A partial focus of this sort is generically called a caustic; point focus is just one special case. In general, the projection of a caustic onto a viewing screen lies along a line. Because the screen need not be at any special distance from the source, the caustic itself occupies a 2D surface in 3D space. 1Awe and fear: Many folk traditions attribute grim consequences to pointing at a rainbow. 2Thomas Harriot performed similar calculations 30 years earlier, but apparently never published them. 484 17.2 Preliminaries 485 Figure 17.1: [Photograph.] Rainbow. Note (i ) the sky appears darker between the two bows; (ii ) the secondary bow is less bright than the primary; (iii ) the order of colors is opposite in the two bows; (iv ) the main bow’s color sequence is roygvgvg. The faint, extra last bands are part of a “supernumerary” bow; see Section 17.3.4. The photo was taken with a polarizing filter; this does not affect the rainbow, which is already polarized, but it does reduce background light from the sky. Inset: Color saturation was digitally enhanced to bring out the structure. [Photo courtesy Steve Nelson (Fayfoto, Boston MA).] We will soon see that a natural rainbow involves a particular kind of caustic. 17.2.2 Dispersion from a triangular prism Section 5.3.5 introduced the concept of dispersion: Because different vacuum wave- lengths travel at slightly different speeds in a transparent medium, the bending pre- dicted by the law of refraction is wavelength-dependent. Figure 17.2 shows the resulting behavior in a macroscopic setting. A bundle of parallel incoming rays of light3 gets spread in angle upon entering a prism, then spread further upon exiting. A key observation is that each spectral color exits at a specific angle relative to the incoming rays, and that this angle is the same for each of the parallel incoming rays. Thus, as we move farther away from the prism, the increasing spread with distance eventually dominates the smearing from finite beam width, and a clear spectrum emerges. 3Section 6.5.1 introduced the concept of light rays. Contents Index Notation 486 Chapter 17 Rainbows and Other Caustics reflected 3 4 1 2 5 incoming 6 Figure 17.2: [Photograph.] Colors from a prism. 1: Each incoming ray bends toward the perpendicular upon entry. 2,3: Blue is bent slightly more than red. 4: A bundle of many parallel incoming rays combine their respective spectra, leaving mostly white except at the edges. 5: Each ray bends away from the perpendicular upon exit, and again blue bends more. 6: Each color emerges at a specific angle, which is the same for each of the parallel incoming rays. [https://sciencing.com/happens-light-passes-through-prism-8557530.html ] 17.3 SPHERICAL DROPLETS: THE CARTESIAN RAINBOW 17.3.1 The rainbow as a caustic Rainbows are always seen with the Sun behind the observer. Eventually, early scientists realized that the light we see must be sunlight scattered backward to us from water droplets in the sky. Figure 17.3 shows how this is possible: Light rays are shown entering a droplet, then partially reflecting inside it. (Much of the incoming light instead exits the droplet at point Q, continuing away from the observer, but some will be internally reflected.) As with the prism, two rounds of refractive bending are involved, and each can spread incoming white light. Closer inspection of Figure 17.3 reveals a problem with this explanation, however. For the case of a spherical water droplet in air, outgoing ray directions depend on wavelength, but also on which ray we are following, so that a jumble arrives on any projection screen. How, then, can we get a rainbow? Here is where Descartes made his decisive contribution. Descartes realized that although an incoming bundle of parallel rays will emerge in many directions, nevertheless there is a minimum angle of deviation. That is, in the ray-optics approximation no light will be returned at angles less than this minimum. (Larger angles all the way up to 180 deg are all possible.) You should stop reading now and work out the details in Problem 17.1. A key conclusion is that a projection screen placed behind the droplet will catch light covering a disk-shaped region, leaving the exterior of that disk dark, and moreover the edge of the disk will be especially bright. In fact, the edge is a caustic. Descartes confirmed the predicted phenomena with an experiment in which a spherical flask of water was illuminated by a shaft of sunlight in an otherwise dark room. Contents Index Notation 17.3 Spherical Droplets: the Cartesian Rainbow 487 incoming P Q outgoing Figure 17.3: [Ray diagram.] Parallel incoming rays emerge nonparallel from a droplet. The red and blue lines depict rays that obey the laws of reflection and refraction for two different wavelengths. Not shown: Much of the incoming light exits at point Q, but we are interested in the part that internally reflects as shown. (Also, some of the incoming light reflects at P.) 17.3.2 The primary bow arises from one internal reflection When we see a rainbow, we are looking not at a single object, but at the combined effects of countless droplets. Descartes drew a diagram similar to Figure 17.4 to explain the situation: • Each droplet reflects light back in a caustic shaped like a cone, with the droplet at its apex. Even if the droplets are not all the same size, each of their cones has the same opening angle, because that angle does not depend on the size. • We can catch part of that droplet’s cone of light if our angle of view relative to the Sun’s rays equals the cone’s opening angle. • The locus of all points in space that meet this condition is also a cone, but with our eye at the apex: When we look in a direction lying on that cone, we see glints from every droplet lying along that line of sight. The combined effect is an apparent glowing arch in the sky. Descartes did not have the framework needed to complete his understanding of the rainbow, because he did not understand that white light is a mixture of different spectral colors.
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