Optics 011 · Ray worlds, boundaries, and natural optics

Rainbow Caustic Observatory

How can one water drop create primary and secondary bows with a dark band between them? Light piles up at the angle where the deflection stops changing from one ray to the next. One reflection inside the drop gives the primary bow, about 42 degrees from the point opposite the sun; two give the secondary bow, about 51 degrees out, with its colors reversed. Between them lies Alexander's dark band, which light reflected once or twice cannot reach.

One spherical raindrop is magnified so you can trace every refraction and internal reflection; the stationary meridional path is then rotated into a three-dimensional scattering cone. The primary and secondary rainbow rings frame Alexander's dark band, and a scalar Airy detector resolves finite caustic peaks and supernumerary bows that depend on the drop radius.

Selected scattering result

\Theta

$0.50$

Angular structure

\Delta\Theta

$50\%$

Ray or wave check

\mathcal C

$0.00\pi$

Model regime $\text{valid model regime}$

The model uses IAPWS water dispersion and exact geometric rays in a sphere. The supernumerary scene uses a scalar Airy fold, not full Mie scattering. Deformation of falling drops, blur from the Sun's disc, the spread of drop sizes, multiple scattering and absolute sky radiance are not included.

Physics tutorial

Audit the physics of Rainbow Caustic Observatory

BackgroundA rainbow is a caustic formed by a whole family of deflected rays, not by one chosen ray. For a spherical water drop, the deflection in the meridional plane is $D_p(i)=p\pi+2i-2(p+1)r,\qquad \sin i=n(\lambda)\sin r$ . The order index counts internal reflections, and every segment comes from exact intersections with the sphere and Snell refraction. Before tracing, the Lab computes the water index at each wavelength from IAPWS R9-97.

Why it mattersHow can one water drop create primary and secondary bows with a dark band between them?

Start with the essentials

Focus question: How can one water drop create primary and secondary bows with a dark band between them?
One-sentence intuition: Rays entering at neighboring impact parameters pile up only where the deflection is stationary: $\frac{\mathrm dD_p}{\mathrm di}=0,\qquad \cos^2 i_p=\frac{n^2-1}{p(p+2)}$ . One internal reflection gives the primary bow; two give the secondary bow, with its colors in reverse order. The angular gap between the lit sides of the two bows is Alexander’s dark band.

Core mathematical model

IAPWS water dispersion

\frac{n^2-1}{n^2+2}\frac{1}{\bar\rho}=a_0+a_1\bar\rho+a_2\bar T+a_3\bar\lambda^2\bar T+\frac{a_4}{\bar\lambda^2}+\frac{a_5}{\bar\lambda^2-\bar\lambda_{\mathrm{UV}}^2}+\frac{a_6}{\bar\lambda^2-\bar\lambda_{\mathrm{IR}}^2}+a_7\bar\rho^2

The Lab holds the water at 20 degrees Celsius and 998.2 kilograms per cubic metre and evaluates the published Lorentz–Lorenz formulation from 400 to 700 nanometres. The same index feeds both the drawn ray trace through the sphere and every angle readout.

Stationary rays for the primary and secondary bows

D_p(i)=p\pi+2i-2(p+1)r,\qquad \frac{\mathrm dD_p}{\mathrm di}=0,\qquad \cos^2 i_p=\frac{n^2-1}{p(p+2)}

Here p is the number of internal reflections. The analytic stationary impact parameter is checked two independent ways: with a finite-difference derivative, and with a full vector trace through the sphere.

Airy wave correction at the fold caustic

I(\Theta)\propto \operatorname{Ai}^2\!\left[-\frac{\Theta_R-\Theta}{\Delta\Theta_A}\right],\qquad \Delta\Theta_A=\left(\frac{|\Theta^{\prime\prime}(q_R)|}{2}\right)^{1/3}(ka)^{-2/3}

Ray optics puts infinite ray density at the stationary angle. The scalar Airy theory of the fold gives a finite main peak instead, followed by oscillating supernumerary bows. A larger drop or a shorter wavelength raises the size parameter and squeezes the fringes together. An exact electromagnetic prediction needs Mie theory or complex-angular-momentum theory.

Common difficulties

Sending the rainbow ray through the drop centre

Typical misconceptionA rainbow comes from one representative ray through the middle of the drop, and a larger or smaller drop should shift the geometric rainbow angle.

Better mental modelThe rainbow ray enters far from the centre, at the stationary impact parameter. With lengths scaled by the radius, every sphere has the same geometry, so the geometric angle does not depend on radius. Radius comes back in through the wave phase: it sets the spacing of the supernumerary bows, not the Descartes angle.

Run the experiment

01
Scene 1: Primary rainbow cone
In the primary-cone scene, hold the radius fixed and sweep the wavelength from violet to red. Follow one ray through the entry refraction, the internal reflection and the exit refraction, then orbit the camera to see the meridional path sweep out a cone.
What to observe: Water has a smaller index for red light, so the red primary bow has the larger angular radius. Changing the radius changes the drawn droplet, which is labeled as magnified, but not the geometric stationary angle.
02
Scene 2: Secondary bow and Alexander band
In the secondary scene, count the two internal reflections. Compare the color order of the primary and secondary bows on the same angular shell, then read the two red edges that bound Alexander’s band.
What to observe: The secondary bow has its colors reversed, with red on the inner edge. Its extra internal reflection leaves less power in this branch after the smooth surfaces. In this low-order ray model, the angular region from the primary red edge to the secondary red edge stays dark.
03
Scene 3: Airy supernumeraries
In the Airy scene, keep the wavelength fixed and sweep the radius across its full range. Watch the finite intensity curve and the supernumerary rings squeeze together, and note when the size-parameter indicator shows that the Airy theory is no longer quantitative.
What to observe: On the lit side of the fold, rays at two different impact parameters reach the same angle. Adding their waves replaces the geometric divergence with a main Airy peak and weakening supernumerary bows, spaced as the predicted size law says.