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Optics 001 · Ray worlds, boundaries, and natural optics

Optical Boundary Explorer

A three-dimensional interface chamber couples exact Snell geometry to lossless Fresnel power. Eighty-one angular samples, polarization-aware beams, and fixed detectors reveal Brewster extinction and total internal reflection without moving the evidence to fit the answer.

Interactive modelOptical Boundary Explorer
Transmitted angle θt\theta_t0.500.50
Selected reflectance RR50%50\%
Energy closure ΔE\Delta_E0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

Audit the physics of Optical Boundary Explorer

BackgroundA dielectric boundary must satisfy phase matching and energy conservation at the same time. Snell geometry follows n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1=n_2\sin\theta_2, while the Fresnel amplitudes decide how much s- and p-polarized power enters each branch. The apparatus solves both parts before it draws any beam.

Why it mattersHow do angle, polarization, speed, and energy change together at one boundary?

Start with the essentials

Focus question
How do angle, polarization, speed, and energy change together at one boundary?
One-sentence intuition
Angle alone does not determine brightness. For a smooth lossless boundary the check is R+T=1R+T=1; Brewster extinction applies only to p polarization, and total internal reflection requires incidence from the optically denser side.

Core mathematical model

Snell phase matching

n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1=n_2\sin\theta_2

Angles are measured from the surface normal. A real transmitted angle exists only when the computed sine magnitude does not exceed one.

Lossless Fresnel closure

Rs+Ts=1,Rp+Tp=1R_s+T_s=1,\qquad R_p+T_p=1

The transmitted power includes the refractive-index and direction-cosine factor; squaring the field coefficient alone is not enough.

Brewster and critical angles

θB=arctan⁡ ⁣(n2n1),θc=arcsin⁡ ⁣(n2n1) (n1>n2)\theta_B=\arctan\!\left(\frac{n_2}{n_1}\right),\qquad \theta_c=\arcsin\!\left(\frac{n_2}{n_1}\right)\ (n_1>n_2)

The first suppresses p-polarized reflection; the second marks the boundary beyond which a propagating transmitted ray disappears.

Common difficulties

Treating ray width as arbitrary decoration

Typical misconceptionThe reflected and transmitted trails can be resized independently without changing the physical claim.

Better mental modelHere both widths are monotone encodings of the Fresnel power from the same boundary solve. Read the numerical reflectance and energy residual for quantitative comparison.

Run the experiment

  1. 01

    Scene 1: Snell and Fresnel split

    Sweep incidence from normal toward grazing while keeping the transmitted index fixed.

    What to observe: Verify that the transmitted ray bends toward or away from the normal as the index ratio changes, while the energy residual stays near numerical precision.
  2. 02

    Scene 2: Brewster-angle blackout

    Select p polarization and move through the predicted Brewster angle.

    What to observe: The p-polarized reflected branch reaches zero at one angle; the unpolarized scene does not completely black out there.
  3. 03

    Scene 3: Critical angle and total reflection

    Enter the denser-to-rarer preset and cross the critical angle slowly.

    What to observe: Past the critical angle, the propagating transmitted branch vanishes while reflectance reaches unity.