Optics 018 · Ray worlds, boundaries, and natural optics

Ray Cloaking Challenge

A four-lens optical bench audits paraxial ray cloaking with explicit lens kicks, free-space propagation, tilted fields, finite apertures, and a fixed off-axis hidden volume. Dashed free-space targets remain independent while the physical train must restore both output height and slope.

Free-space matrix residual

\lVert M-M_{\mathrm{free}}\rVert

$0.50$

Aperture-clipped rays

N_{\mathrm{clip}}

$50\%$

Output-ray restoration error

\varepsilon_{\mathrm{ray}}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Ray Cloaking Challenge

BackgroundIn first-order optics a ray is represented by height and slope. A successful symmetric four-lens cloak must have $M_{\mathrm{cloak}}=\begin{pmatrix}1&L\\0&1\end{pmatrix}$ , the same transfer matrix as empty space of equal length. That condition restores a background ray after the train but says nothing yet about finite apertures or aberrations.

Why it mattersCan an optical train route rays around an object and restore the background afterward?

Start with the essentials

Focus question: Can an optical train route rays around an object and restore the background afterward?
One-sentence intuition: The Rochester spacings are $d_1=f_1+f_2,\qquad d_2=\frac{2f_2(f_1+f_2)}{f_1-f_2}$ . Between the inner lenses the central ray bundle forms a narrow tunnel; the useful hidden volume lies beside that tunnel rather than directly on the axis.

Core mathematical model

Thin-lens ray update

\begin{pmatrix}y\\\theta\end{pmatrix}_{\!+}=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}\begin{pmatrix}y\\\theta\end{pmatrix}_{\!-}

A thin lens leaves ray height continuous and changes slope by an amount proportional to height.

Symmetric Rochester spacing

d_1=f_1+f_2,\qquad d_2=\frac{2f_2(f_1+f_2)}{f_1-f_2}

Positive finite separation requires the outer focal length to exceed the inner focal length.

Finite-aperture condition

|y_j|\le a_j\qquad(j=1,2,3,4)

Matrix equivalence helps only rays whose footprints remain inside all four physical clear apertures.

Common difficulties

Hiding an axial object behind hand-drawn detours

Typical misconceptionAny symmetric pair of bent polylines around a central object demonstrates a four-lens cloak.

Better mental modelReal rays must be kicked at the actual lens planes and recover empty-space height and slope. In the Rochester geometry the on-axis light tunnel remains occupied; the hidden zone is off axis.

Run the experiment

01
Scene 1: Four-lens Rochester cloak
Follow the parallel bundle through all four lens planes and compare two selected output rays with their dashed empty-space targets.
What to observe: Unclipped paraxial rays leave with the same state as equal-length free propagation even though their internal route is strongly rearranged.
02
Scene 2: Tilted-field audit
Select the tilted-field audit and inspect how every footprint shifts while the ideal output state remains restored.
What to observe: Field tilt exposes aperture demand and moves the usable off-axis hidden region; matrix equality alone does not promise a wide field of view.
03
Scene 3: Finite-aperture failure
Stop down the apertures and identify the exact lens where each red path terminates.
What to observe: Vignetting breaks the background by removing rays, even though the focal lengths and ideal matrix remain unchanged.