In geometric optics, a caustic is a bright curve of light that appears when many incoming light rays are focused in the same outgoing direction. The most famous example of a caustic is a rainbow, which occurs when light interacts with spherical water droplets. Consider a spherical liquid droplet of radius r with index of refraction 1 < n < 2, suspended in air with index of refraction n = 1.
a. Consider a light ray that enters the droplet with impact parameter b, reflects once off the inside surface of the droplet, then exits, as shown at left below. Give your answers in terms of the dimensionless impact parameter x = b/r. (Hint: the four marked angles are congruent.)
[Figure: Left diagram shows light ray path through droplet with impact parameter b and radius r. Right diagram shows droplet at center of spherical screen of radius R with intensity I₀ and angle φ₀.]
i. Find the angle by which the light ray is deflected at the first refraction.
ii. Find the angle by which the light ray is deflected at the reflection.
iii. Find the angle by which the light ray is deflected at the second refraction.
The sum of these three quantities is the net deflection angle φ(x). (The light can also reflect inside more than once, or never enter at all, but for simplicity we will ignore these other paths.)
b. Next, we consider when caustics form in general. Suppose the droplet is uniformly illuminated by parallel light rays of intensity I₀, and sits at the center of a spherical screen of radius R ≫ r, as shown at right above. Consider the light that enters near dimensionless impact parameter x₀, and exits near angle φ₀ = φ(x₀).
i. What is the power incident on the droplet at x₀ ≤ x ≤ x₀+dx?
ii. What is the area on the screen illuminated by the outgoing rays, at φ₀ ≤ φ ≤ φ₀+dφ?
iii. A caustic occurs when the intensity of light on the screen diverges. Assume that φ₀ ≠ 0 and φ₀ ≠ π. Under what conditions does light incident at x₀ lead to a caustic at φ₀? Express your answer as a condition on the function φ(x).
c. Find the angle φ₀ of the rainbow in terms of n. (Hint: the derivative of arcsin(x) is 1/1−x2.)
d. For water, the index of refraction of red light is 1.331, and the index of refraction of blue light is 1.340. Find the angular width of the rainbow on the screen and give your answer in degrees.
e. A glory is an optical phenomenon which involves light scattered directly backward, at φ = π, leading to an apparent halo around the shadow of an observer's head. For what values of n is there a caustic at φ = π? Can glories from water droplets be explained in terms of caustics?