Optics 074 · Polarization, anisotropy, and modulation

Polarized Reflections & Sky

One rotating filter, two mechanisms: signed Fresnel reflection of polarized skylight, Brewster glare, and a directional Rayleigh sky.

Water polarization / transmission

P_{\rm water},T

$0.50$

Sky scattering / polarization

\gamma,P_{\rm sky}

$50\%$

Brewster angle / parallel reflectance

\theta_B,R_p

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Polarized Reflections & Sky

Water viewing angle from normal

\theta_i

53.12^{\circ}

Transmission axis from image horizontal

\alpha

90^{\circ}

Solar elevation

h_\odot

36.88^{\circ}

Solar azimuth

A_\odot

0^{\circ}

Viewing azimuth

A_{\rm view}

0^{\circ}

Lower-medium index (water: 1.333)

n_2

1.33

Filter: zero removed, one inserted

j_F

1

Display exposure stops (not source power)

e

1.50

Turn the gold transmission axis. Water glare can vanish while a different part of the sky survives. Align the Sun, surface and observer; then compare the same angular scene without and with a filter at fixed exposure. The sky is not a uniform blue lamp: its polarization direction changes across the hemisphere.

Ideal flat dielectric and single Rayleigh scattering. Synthetic angular panoramas, arbitrary radiance and enlarged solar source; no waves, clouds, noise or calibrated camera.

Physics tutorial

Audit the physics of Polarized Reflections & Sky

BackgroundWater glare is polarized by a boundary. Sky light is polarized by molecular scattering. Brewster’s condition is $\tan\theta_B=n_2/n_1$ . This angle is measured from the surface normal, not the water plane. At that angle the reflected parallel field vanishes; not all reflected light disappears.

Why it mattersWhy can one rotating filter darken water glare and blue sky in different directions?

Start with the essentials

Focus question: Why can one rotating filter darken water glare and blue sky in different directions?
One-sentence intuition: The sky follows a different angle: $P_{\rm sky}=\frac{\sin^2\gamma}{1+\cos^2\gamma}$ . The scattering angle is between Sun and sky directions, not the viewing angle at water. Its polarization axis changes across the sky, so one filter does not darken every direction equally.

Core mathematical model

Boundary reflection and energy

\begin{aligned}r_s&=\frac{n_1c_i-n_2c_t}{n_1c_i+n_2c_t}\\r_p&=\frac{n_2c_i-n_1c_t}{n_2c_i+n_1c_t}\\n_1\sin\theta_i&=n_2\sin\theta_t\\R_j&=r_j^2\\T_j&=\frac{n_2c_t}{n_1c_i}t_j^2\\R_j+T_j&=1\end{aligned}

Cosines refer to angles from the normal. The last equality is for the lossless interface and each pure polarization, not for a source modified by a photographic filter. Air index is one; water reference is 1.333.

A directional scattering state

\begin{aligned}\cos\gamma&=\widehat{\mathbf s}\cdot\widehat{\mathbf d}\\\mathbf e_\perp&\parallel\widehat{\mathbf s}\times\widehat{\mathbf d}\\I_\perp&=\tfrac12\\I_\parallel&=\tfrac12\cos^2\gamma\end{aligned}

Unit incident unpolarized illumination, arbitrary common radiance scale. Two mutually incoherent transverse modes produce the angular intensity and polarization degree together. At forward/back scattering the axis is undefined but intensity stays finite and unpolarized.

One field feeds both views

\begin{aligned}\mathbf C&=\sum_m w_m\mathbf e_m\mathbf e_m^{\mathsf T}\\I&=\operatorname{tr}\mathbf C\\I_\alpha&=\mathbf a_\alpha^{\mathsf T}\mathbf C\mathbf a_\alpha\\0&\le I_\alpha\le I\end{aligned}

The same incoherent modes propagate through the signed reflection amplitudes and the ideal analyzer projection. The unfiltered view is a reference of those same rays, not a separately invented image. Blue color and nonlinear display mapping are illustrative, not spectral prediction.

Common difficulties

A polarizer is just a dark filter?

Typical misconceptionRotate it and the whole landscape should become darker by the same amount.

Better mental modelAn ideal polarizer always halves unpolarized light, but selectively transmits polarized light. Reflected skylight is already polarized before meeting the surface. The calculation preserves that input and the signed Fresnel amplitudes; it does not assume every water ray begins unpolarized.

Run the experiment

01
Scene 1: Brewster-angle glare
Begin at Brewster glare. Compare horizontal and vertical axes, then remove the filter. The reflected solar path aligns with the camera only for the correct geometry.
What to observe: The panoramas are synthetic angular radiance views, not recorded camera images. An ideal axis is projected into each ray. Wide-angle lens/filter calibration and real atmospheric photometry are outside the model.
02
Scene 2: Rayleigh sky map
Choose the sky preset. Move the solar azimuth and watch the sky map, scattering-angle readout and sky transmission change together.
What to observe: The sky map is a separate ideal reference. Its short lines show local physical polarization axes; their lengths and angular sphere are enlarged direction guides.
03
Scene 3: Photographic polarizer
Choose the photographic preset. Rotate the same physical axis. Compare the two marked directions at identical exposure; then change display exposure without changing the physical ratios.
What to observe: Water is flat and semi-infinite, with no bottom illumination or waves. The source is deliberately broadened to make glare visible; its width and arbitrary brightness are not measurements of the Sun.