Optics 001 · Ray worlds, boundaries, and natural optics

Optical Boundary Explorer

What happens to light's speed, direction and power when it crosses into a new material? In the new medium light travels at the vacuum speed divided by the refractive index, and Snell's law sets its direction. The Fresnel equations split the power between reflected and transmitted beams; without absorption they add up to all the incoming power. At Brewster's angle only p-polarized light stops reflecting, and total internal reflection needs light coming from the higher-index side.

In a three-dimensional interface chamber, exact Snell geometry sets the ray directions and lossless Fresnel coefficients set the power. Eighty-one angle samples, beams that track polarization and fixed detectors show Brewster extinction and total internal reflection.

Transmitted angle

\theta_t

$0.50$

Selected reflectance

R

$50\%$

Energy-conservation error

\Delta_E

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Optical Boundary Explorer

BackgroundLight crossing the boundary between two transparent dielectrics has to satisfy phase matching and energy conservation at the same time. The ray angles follow Snell's law, $n_1\sin\theta_1=n_2\sin\theta_2$ . The Fresnel amplitudes then set how much s- and p-polarized power goes into the reflected and transmitted beams. The Lab solves both parts before it draws any beam.

Why it mattersWhat happens to light's speed, direction and power when it crosses into a new material?

Start with the essentials

Focus question: What happens to light's speed, direction and power when it crosses into a new material?
One-sentence intuition: The angle alone does not tell you how bright each beam is. At a smooth, lossless boundary the reflected and transmitted fractions always add up to one: $R+T=1$ . Brewster extinction happens only for p polarization, and total internal reflection needs light arriving from the optically denser side.

Core mathematical model

Snell's law from phase matching

n_1\sin\theta_1=n_2\sin\theta_2

Angles are measured from the surface normal. If the formula gives a sine whose magnitude exceeds one, there is no real transmitted angle.

Energy conservation at a lossless boundary

R_s+T_s=1,\qquad R_p+T_p=1

Transmitted power carries an extra factor made of the refractive indices and direction cosines. Squaring the field coefficient alone does not give it.

Brewster and critical angles

\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)

At the Brewster angle, p-polarized light is not reflected. Beyond the critical angle, the transmitted ray no longer propagates.

Common difficulties

Thinking beam width is just decoration

Typical misconceptionHow wide the reflected and transmitted beams are drawn is a matter of style, so either could be changed without changing the physics.

Better mental modelHere each beam gets wider as its Fresnel power grows, and both widths come from the same boundary calculation. For exact comparisons, read the reflectance and the energy residual.

Run the experiment

01
Scene 1: Snell and Fresnel split
Keep the second medium's refractive index fixed and sweep the incidence angle from normal toward grazing.
What to observe: Depending on the index ratio, the transmitted ray bends toward or away from the normal. The energy residual stays near zero, within numerical precision.
02
Scene 2: Brewster-angle blackout
Select p polarization and move through the predicted Brewster angle.
What to observe: The p-polarized reflection drops to zero at one angle. Unpolarized light still reflects at that angle and does not black out completely.
03
Scene 3: Critical angle and total reflection
Choose the denser-to-rarer preset and slowly raise the angle past the critical angle.
What to observe: Past the critical angle, no propagating transmitted beam remains and the reflectance reaches one.