Optics 009 · Ray worlds, boundaries, and natural optics

Window Ghost Observatory

An independently initialized three-dimensional apparatus connects Parallel but not collinear, Four-surface night ghosts, Bare versus AR-coated ghosts. Two dimensional physical controls, direct probe dragging, a detector trace, and three quantitative checks are recalculated from the stated equation.

Primary prediction

\mathcal P_1

$0.50$

Physical scale

\mathcal P_2

$50\%$

Limit check

\mathcal V

$0.00\pi$

Model regime $\text{valid model regime}$

Every Lab declares its approximation boundary and executable analytic or numerical benchmark. Geometry and glow are presentation encodings; use the equation, dimensional readouts, benchmark check, and validity indicator as evidence.

Physics tutorial

Audit the physics of Window Ghost Observatory

BackgroundA lossless plane-parallel window performs two Snell refractions with equal and opposite angular deviations. Its exact perpendicular beam translation is $d=t\,\frac{\sin(\theta_i-\theta_t)}{\cos\theta_t}$ . The same two physical surfaces also split power through Fresnel reflection. A double-pane window has four such surfaces, so a night lamp can return along four complete, slightly different source-to-window-to-eye paths.

Why it mattersWhy is the emerging ray parallel yet displaced, and why do windows make ghost images at night?

Start with the essentials

Focus question: Why is the emerging ray parallel yet displaced, and why do windows make ghost images at night?
One-sentence intuition: Parallel does not mean collinear. For parallel entrance and exit faces, $\theta_{\mathrm{out}}=\theta_i\quad\text{but}\quad d\ne0$ . Thickness changes the separation but not the final direction. Every rendered ghost must include the reflection that created it and every transmission it crossed.

Core mathematical model

Exact plane-parallel translation

n_{\mathrm a}\sin\theta_i=n_{\mathrm g}\sin\theta_t,\qquad d=t\,\frac{\sin(\theta_i-\theta_t)}{\cos\theta_t}

The displacement is measured perpendicular to the two parallel air rays. It vanishes at normal incidence and grows with both physical thickness and incidence angle.

Four primary surface reflections

P_j^{(q)}=R_q\left(1-R_q\right)^{2j},\qquad q\in\{s,p\},\quad j=0,1,2,3

A reflection from a deeper surface must transmit through every preceding surface on the way in and again on the way out. The displayed unpolarized power is the average of the two polarization channels.

Transmitted internal-round-trip ghosts

P_k^{(q)}=\left(1-R_q\right)^2R_q^{2k},\qquad \sum_{k=0}^{\infty}P_k^{(q)}=\frac{1-R_q}{1+R_q}

Each additional round trip adds two reflections and shifts the emerging beam by another two internal lateral traversals. The infinite incoherent sum closes with the reflected family to conserve energy.

Common difficulties

Drawing a ghost without its parent reflections

Typical misconceptionA faint duplicate can be placed beside the main transmitted ray without tracing how it entered, reflected, and left the glass.

Better mental modelThat picture has no optical history and cannot predict position or power. Here every colored ghost is one continuous piecewise-Snell path, and its width is a nonlinear display of a separately reported Fresnel power.

Run the experiment

01
Scene 1: Parallel but not collinear
In the first scene, set the plate to minimum thickness, sweep incidence from normal toward the upper limit, then increase thickness. Compare the green output with the dashed no-glass line and read the pink perpendicular bracket.
What to observe: The two external rays retain exactly the same angle because the second refraction cancels the first angular change. The displacement itself is nonzero because the connecting segment inside glass followed a different slope.
02
Scene 2: Four-surface night ghosts
Enter the night-window scene and orbit until all four paneside reflection paths are visible. Sweep incidence and thickness while watching the four angular peaks separate and the deeper paths lose power through preceding surfaces.
What to observe: At normal incidence the four angular peaks coincide even though four power paths still exist. Oblique viewing separates them, and the rear surface of one pane can remain almost as bright as its front surface because two Fresnel transmissions are still close to unity.
03
Scene 3: Bare versus AR-coated ghosts
Compare the vertically stacked bare and coated lanes. Follow the main ray, the one-round-trip ghost, and the two-round-trip ghost all the way through both faces, then compare the detector profiles and suppression readout.
What to observe: The first transmitted ghost is separated by twice the internal lateral traverse. Reducing the effective surface reflectance suppresses a round-trip ghost quadratically, which is why the coated detector trace nearly loses its secondary peaks.