Optics 011 · Ray worlds, boundaries, and natural optics

Rainbow Caustic Observatory

A dark rain observatory magnifies one physical spherical drop, traces every refraction and internal reflection, and rotates the stationary meridional path into a three-dimensional scattering cone. Primary and secondary spectral rings bracket an explicit Alexander band, while a scalar Airy detector resolves finite caustic peaks and radius-dependent supernumeraries.

Selected scattering result

\Theta

$0.50$

Angular structure

\Delta\Theta

$50\%$

Ray or wave check

\mathcal C

$0.00\pi$

Model regime $\text{valid model regime}$

IAPWS water dispersion and exact spherical geometric rays are used. The supernumerary scene is a scalar Airy fold, not full Mie scattering; falling-drop deformation, solar-disc blur, droplet-size distributions, multiple scattering, and absolute sky radiance are explicit omissions.

Physics tutorial

Audit the physics of Rainbow Caustic Observatory

BackgroundA rainbow is a caustic of a deflection family, not one chosen cartoon ray. For a spherical water drop, the meridional deflection family is $D_p(i)=p\pi+2i-2(p+1)r,\qquad \sin i=n(\lambda)\sin r$ . Here the order index counts internal reflections; exact sphere intersections and Snell refraction determine every segment. The water index is evaluated from IAPWS R9-97 at each wavelength before the path is traced.

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: 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 produces the primary maximum; two produce the secondary maximum with reversed color order. The angular gap between their illuminated sides 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 fixes water at 20 degrees Celsius and 998.2 kilograms per cubic metre, then evaluates the published Lorentz–Lorenz formulation from 400 to 700 nanometres. This shared material value drives both the rendered sphere trace and every angular readout.

Stationary primary and secondary rays

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 internal-reflection count. The analytic stationary impact is independently checked by a finite-difference derivative and by a complete vector trace through the physical sphere.

Airy regularization of 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}

Geometric ray density diverges at the stationary angle. The scalar Airy fold produces a finite main peak and oscillatory supernumeraries. Increasing radius or decreasing wavelength raises the size parameter and compresses the fringes; exact electromagnetic prediction requires Mie or complex-angular-momentum theory.

Common difficulties

Sending the rainbow ray through the drop centre

Typical misconceptionA rainbow is made by a representative ray through the middle of a drop, and changing drop radius should move the geometric rainbow angle.

Better mental modelThe rainbow ray enters far from the centre at the stationary impact parameter. Normalized spherical geometry makes its geometric angle independent of radius; radius re-enters through wave phase and therefore controls supernumerary spacing, not the Descartes angle.

Run the experiment

01
Scene 1: Primary rainbow cone
In the primary-cone scene, sweep wavelength from violet to red while holding radius fixed. Follow one ray through the entry refraction, internal reflection, and exit refraction, then orbit the camera to see the meridional path become a cone.
What to observe: Red water has the smaller refractive index and therefore the larger primary angular radius. Changing radius changes the declared magnified droplet size but not the geometric stationary angle.
02
Scene 2: Secondary bow and Alexander band
Enter the secondary scene and count both internal reflections. Compare the primary and secondary spectral order on the same angular shell, then read the two red-edge boundaries of Alexander’s band.
What to observe: The secondary spectrum is reversed: red is on its inner edge. Its extra internal reflection lowers the smooth-interface branch throughput, while the angular region from the primary red edge to the secondary red edge remains dark in this low-order ray model.
03
Scene 3: Airy supernumeraries
In the Airy scene, sweep radius across its full range at fixed wavelength. Watch the finite intensity curve and supernumerary rings compress, and note when the size-parameter indicator leaves the quantitative Airy regime.
What to observe: Two different impact parameters reach the same angle on the illuminated side of the fold. Their wave contributions replace the geometric divergence with a main Airy peak and decreasing supernumeraries whose spacing follows the predicted size law.