Optics 012 · Ray worlds, boundaries, and natural optics

Mirage & Green-Flash Observatory

A long-baseline horizon observatory couples NIST dry-air dispersion to analytic hot, cold, and multilayer temperature profiles. Fourth-order continuous-index ray tracing reveals upward-turning inferior-mirage paths, downward-bent looming, a ten-path Fata Morgana preset, and arcsecond solar-limb color separation while a live profile curtain and two instruments expose the atmosphere that caused them.

Bending or path count

\mathcal B

$0.50$

Apparent mapping or spectral separation

\mathcal A

$50\%$

Stratified-medium invariant error

\varepsilon_{p_x}

$0.00\pi$

Model regime $\text{valid model regime}$

This is two-dimensional dry-air geometric optics over a flat surface with prescribed profiles and ideal-gas density scaling. Humidity, Earth curvature, turbulence, aerosols, radiative transfer, finite solar diameter, diffraction, and perceived green-flash brightness are explicit omissions.

Physics tutorial

Audit the physics of Mirage & Green-Flash Observatory

BackgroundA mirage is refraction through a continuous air-density field, not reflection from a road or ocean. A scalar ray in that field obeys $\frac{\mathrm d}{\mathrm ds}\!\left(n\hat{\mathbf t}\right)=\nabla n$ . The Lab first evaluates wavelength-dependent standard dry-air refractivity, then rescales it with hydrostatic pressure and the prescribed temperature profile before integrating the path.

Why it mattersHow do measured air dispersion and a prescribed thermal profile bend one horizon into depressed, loomed, and multiply imaged views?

Start with the essentials

Focus question: How do measured air dispersion and a prescribed thermal profile bend one horizon into depressed, loomed, and multiply imaged views?
One-sentence intuition: Horizontal translational symmetry supplies an independent invariant: $n(z)\cos\alpha=\mathrm{const.}$ . A ray therefore bends toward larger refractive index. The invariant residual checks every rendered path instead of trusting its shape by eye.

Core mathematical model

NIST standard dry-air dispersion

10^8(n_{\mathrm{as}}-1)=\frac{5792105}{238.0185-\sigma^2}+\frac{167917}{57.362-\sigma^2},\qquad \sigma=\lambda_{\mathrm{vac}}^{-1}

Vacuum wavelength is expressed in micrometres. Shorter visible wavelengths receive the larger refractive index, so the red and blue solar-limb paths separate even in the same thermal profile.

Prescribed dry-atmosphere density scaling

n(z,\lambda)-1=\bigl[n_{\mathrm{as}}(\lambda)-1\bigr]\frac{P(z)}{P_0}\frac{T_0}{T(z)},\qquad P(z)=P_0\exp\!\left(-\frac{z}{H_P}\right)

The hot, cold, and layered presets change the analytic temperature anomaly while the pressure scale height stays fixed. This ideal-gas rescaling is declared and is not a full humidity-aware Ciddor weather reduction.

Ray curvature and numerical audit

\frac{\mathrm d\alpha}{\mathrm ds}=\frac{1}{n}\frac{\mathrm dn}{\mathrm dz}\cos\alpha,\qquad \varepsilon_{p_x}=\max_s\left|\frac{n\cos\alpha-(n\cos\alpha)_0}{(n\cos\alpha)_0}\right|

Fourth-order Runge-Kutta advances position, direction, and optical path. The conserved horizontal optical momentum is computed separately along the resulting trajectory and shown as the validation readout.

Common difficulties

Treating a mirage as a reflecting surface

Typical misconceptionThe road acts like a mirror, so the ray can bounce at one hidden interface and the duplicate object can emit the dashed ray.

Better mental modelThere is no bounce in this apparatus. Every solid path changes direction continuously throughout the air; the dashed line is only the final tangent extended backward to locate an apparent source.

Run the experiment

01
Scene 1: Hot-surface inferior mirage
In the hot-surface scene, follow all three direct-and-turning ray pairs from the lighthouse to the observer. Compare their dashed backward tangents: the near-ground branch has negative apparent vertical magnification, so the lighthouse ordering is inverted below the horizon.
What to observe: Heating the surface lowers its density and index. The positive vertical index gradient turns a downward-launched ray upward without touching the ground, while the backward tangent places its source lower than the real lighthouse.
02
Scene 2: Cold-layer looming
Enter the cold-layer scene and increase boundary-layer height. Read the orange temperature profile and cyan refractivity profile together, then measure how far the ship is loomed above its geometric position.
What to observe: A cold dense surface layer reverses the gradient. The path arches above the direct chord and then bends down into the observer, so the reverse tangent places the ship above its actual horizon height.
03
Scene 3: Layered Fata Morgana and green-flash dispersion
Choose the multilayer preset. Count the distinct green paths from the same solar reference, orbit the scene to separate them in depth, and compare the red and blue selected branches in arcseconds.
What to observe: Alternating shelves fold the launch-angle map repeatedly, allowing the same source and observer to be connected by several rays. Normal air dispersion shifts the blue branch more than the red; visibility of a green flash still requires radiative-transfer effects excluded here.