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Sandbox Physics

Optics 013 · Ray worlds, boundaries, and natural optics

Gradient-Index Worlds

An independently initialized three-dimensional apparatus connects GRIN slab, Luneburg lens, Maxwell fish-eye orbit. Two dimensional physical controls, direct probe dragging, a detector trace, and three quantitative checks are recalculated from the stated equation.

Interactive modelGradient-Index Worlds
Primary prediction P1\mathcal P_10.500.50
Physical scale P2\mathcal P_250%50\%
Limit check V\mathcal V0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

How to investigate Gradient-Index Worlds

BackgroundGradient-Index Worlds is one independently initialized apparatus with three linked investigations: GRIN slab, Luneburg lens, Maxwell fish-eye orbit. Its two controls—Gradient constant per metre and Launch height—feed the governing relation dds ⁣(ndrds)=∇n\frac{\mathrm d}{\mathrm ds}\!\left(n\frac{\mathrm d\mathbf r}{\mathrm ds}\right)=\nabla n. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Why it mattersCan a drawn refractive-index landscape steer rays without any sharp interface?

Start with the essentials

Focus question
Can a drawn refractive-index landscape steer rays without any sharp interface?
One-sentence intuition
The detector curve and all three numerical readouts are recomputed from dds ⁣(ndrds)=∇n\frac{\mathrm d}{\mathrm ds}\!\left(n\frac{\mathrm d\mathbf r}{\mathrm ds}\right)=\nabla n. Geometry and glow are presentation encodings; the equation, units, conservation or limit check, and validity indicator are the quantitative evidence.

Core mathematical model

Governing relation

dds ⁣(ndrds)=∇n\frac{\mathrm d}{\mathrm ds}\!\left(n\frac{\mathrm d\mathbf r}{\mathrm ds}\right)=\nabla n

The implementation evaluates this relation with dimensional inputs and an executable analytic or numerical benchmark. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Common difficulties

Mistaking glow for measured power

Typical misconceptionA brighter cinematic trail must represent proportionally more optical power.

Better mental modelUse the detector and normalized readouts for comparison. Glow is deliberately nonlinear so weak structure stays visible.

Run the experiment

  1. 01

    Scene 1: GRIN slab

    Select GRIN slab. Sweep Gradient constant per metre, hold Launch height fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  2. 02

    Scene 2: Luneburg lens

    Select Luneburg lens. Sweep Gradient constant per metre, hold Launch height fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  3. 03

    Scene 3: Maxwell fish-eye orbit

    Select Maxwell fish-eye orbit. Sweep Gradient constant per metre, hold Launch height fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.