Optics 013 · Ray worlds, boundaries, and natural optics

GRIN Geometry Observatory

A volumetric GRIN observatory solves the normalized vector ray equation rather than joining decorative splines. An exact parabolic slab exposes sinusoidal pitch and relay planes, a three-dimensional Luneburg sphere focuses a collimated bundle on its opposite surface, and a Maxwell fisheye maps point-source rays to a stereographic antipode. Analytic trajectories, public profiles, focus residuals, and Noether invariants audit the same rendered paths.

Pitch or index contrast

\mathcal G

$0.50$

Relay or image residual

\mathcal F

$50\%$

Exact-solution or invariant audit

\varepsilon_{\mathrm{audit}}

$0.00\pi$

Model regime $\text{valid model regime}$

Two-dimensional scalar, isotropic, lossless geometric optics with smooth prescribed profiles. Diffraction, polarization, absorption, dispersion, fabrication microstructure, impedance mismatch, and finite-wave resolution are explicit omissions; geometrically perfect focus is not a super-resolution claim.

Physics tutorial

Audit the physics of GRIN Geometry Observatory

BackgroundA gradient-index element bends a ray everywhere, even when no material interface is crossed. With arclength as the path parameter, the governing equation is $\frac{\mathrm d}{\mathrm ds}\!\left(n\hat{\mathbf t}\right)=\nabla n$ . The apparatus integrates its normalized vector form, so position and unit direction evolve together in the exact index landscape shown by the translucent layers.

Why it mattersHow can one continuous index field relay a ray, focus every direction, or turn plane curves into great circles without a hidden interface?

Start with the essentials

Focus question: How can one continuous index field relay a ray, focus every direction, or turn plane curves into great circles without a hidden interface?
One-sentence intuition: The field symmetry supplies a second line of evidence. Translation symmetry in the slab and rotational symmetry in the radial lenses require $p_x=n t_x=\mathrm{const.},\qquad L_z=n(\mathbf r\times\hat{\mathbf t})_z=\mathrm{const.}$ . These constants are evaluated along every numerical ray; a visually convincing curve that violates them is rejected.

Core mathematical model

Normalized continuous-index ray equation

\frac{\mathrm d\mathbf r}{\mathrm ds}=\hat{\mathbf t},\qquad \frac{\mathrm d\hat{\mathbf t}}{\mathrm ds}=\frac{\nabla n-\hat{\mathbf t}(\hat{\mathbf t}\!\cdot\!\nabla n)}{n},\qquad \lVert\hat{\mathbf t}\rVert=1

Only the component of the index gradient perpendicular to the current direction curves the ray. Fourth-order Runge-Kutta advances both vectors and the implementation renormalizes direction after each step.

Exact parabolic relay

n^2(y)=n_0^2\!\left(1-g^2y^2\right),\qquad y(x)=y_0\cos\!\left(\frac{n_0g}{p_x}x\right),\qquad P=\frac{2\pi p_x}{n_0g}

This square-root profile has an exact sinusoidal ray, not merely a paraxial sketch. Its analytic height and pitch independently audit the numerical trajectory and the quarter-pitch relay plane.

Absolute-instrument profiles

n_{\mathrm L}(r)=\sqrt{2-\frac{r^2}{R^2}},\qquad n_{\mathrm M}(r)=\frac{2}{1+r^2/R^2},\qquad \mathbf r_{\mathrm{image}}=-\frac{R^2}{\lVert\mathbf r_{\mathrm{source}}\rVert^2}\mathbf r_{\mathrm{source}}

The canonical Luneburg sphere maps a plane wave to the opposite surface. The Maxwell profile is the stereographic plane image of a uniform sphere, so great circles map to curved rays that meet at the projected antipode.

Common difficulties

Reading the colored shells as hidden interfaces

Typical misconceptionEach visible band is a separate glass surface, so the ray is really a polygon with many Snell refractions and the smooth curve is cosmetic.

Better mental modelThe bands are samples of one analytic field used only to reveal its value. The solver evaluates the continuous index and gradient at every RK4 substep; no internal surface or discrete refraction event exists.

Run the experiment

01
Scene 1: Exact parabolic GRIN relay
In the parabolic relay, drag the launch offset and change profile strength. Follow the five independently integrated rays, compare the pitch and quarter-pitch readouts, and confirm that the analytic-versus-RK4 residual stays small.
What to observe: Increasing profile strength shortens the oscillation pitch because the transverse restoring curvature is larger. Translating the launch bundle changes its phase-space orbit but does not invent a new surface.
02
Scene 2: Luneburg opposite-surface focus
Switch to the Luneburg sphere at canonical strength. Orbit the nested iso-index shells and watch seven parallel rays meet the opposite-surface target; then move strength away from the canonical value and measure the honest spot residual.
What to observe: At canonical strength the surface index is exactly one and the center index is the square root of two. Every sampled plane ray reaches the same opposite-surface point; deforming the profile visibly destroys that absolute-instrument condition.
03
Scene 3: Maxwell stereographic antipode
Enter the Maxwell plane, move the source, and identify the target produced by stereographic inversion. Separate the fan rays in depth and verify that their physical curves—not the dashed source-to-target chord—reunite at the antipode.
What to observe: The Maxwell source and image are generally not Euclidean mirror points. They are stereographic antipodes, and their inverse-radius relation moves the target nonlinearly when the source is dragged.