Optics 003 · Ray worlds, boundaries, and natural optics

Mirror Engineering Challenge

Can a few mirrors steer a beam to a fixed target just by getting the angles right? Not by angles alone. Every bounce must land on an actual mirror, and the beam must end on the fixed detector. A ray can meet a mirror's extended line where there is no glass, or be blocked before the target, and either way the design fails.

Route a beam through a periscope, a retroreflector and a mirror maze with obstacles, all with fixed hardware. Exact intersections with finite mirror segments and specular reflection set both the glowing path and the miss distance.

Output direction

\theta_{\mathrm{out}}

$0.50$

Fixed-target miss

d_{\mathrm{miss}}

$50\%$

Direction norm error

\Delta_{|\hat{\mathbf k}|}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Mirror Engineering Challenge

BackgroundDesigning a mirror path comes down to finding what each ray hits first. At every mirror hit, the ray leaves along the unit direction $\hat{\mathbf k}_{r}=\hat{\mathbf k}_{i}-2(\hat{\mathbf k}_{i}\cdot\hat{\mathbf n})\hat{\mathbf n}$ . That reflection counts only if the ray actually meets the finite mirror segment before it reaches an obstacle or the target.

Why it mattersCan a few mirrors steer a beam to a fixed target just by getting the angles right?

Start with the essentials

Focus question: Can a few mirrors steer a beam to a fixed target just by getting the angles right?
One-sentence intuition: The target detector stays fixed, and the Lab reports the miss distance $d_{\mathrm{miss}}$ . If the target moved to wherever the computed beam ended, every design would look like a success.

Core mathematical model

Direction after a mirror reflection

\hat{\mathbf k}_{r}=\hat{\mathbf k}_{i}-2(\hat{\mathbf k}_{i}\cdot\hat{\mathbf n})\hat{\mathbf n}

Reflection flips the part of the direction along the mirror normal and keeps the part along the mirror. The direction stays a unit vector.

Miss distance from a fixed target

d_{\mathrm{miss}}=\min_{t\ge 0}\left\lVert\mathbf r_0+t\hat{\mathbf k}-\mathbf r_{\mathrm{target}}\right\rVert

The detector ring does not move with the computed ray, so a miss distance of zero means the design really works.

Common difficulties

Adding up angles without checking the hits

Typical misconceptionIf the output angle comes out right, the beam must have hit every mirror it was meant to.

Better mental modelThe line a mirror lies on extends forever, but the mirror has edges. A ray can cross that line where there is no mirror, and the Lab rejects such hits before it applies any reflection.

Run the experiment

01
Scene 1: Periscope build
Start from the aligned periscope and rotate one mirror away from its reference angle.
What to observe: The beam visibly drifts off the fixed target, and the miss-distance readout rises.
02
Scene 2: Corner retroreflector
Test the retroreflector with equal and unequal mirror rotations.
What to observe: In the symmetric reference setup, the beam leaves in exactly the opposite direction to the way it came in. Unequal rotations break this.
03
Scene 3: Occluded ray maze
Guide the beam through the occluded maze and watch for a ray that misses a finite mirror segment before it reaches the target.
What to observe: When a mirror hit is rejected, the computed path ends there. The Lab does not draw a reflected segment that could not exist.