Optics 074 · Polarization, anisotropy, and modulation

Polarized Reflections & Sky

Why does a polarizing filter usually need different angles to darken water glare and blue sky? The two are polarized by different processes. Glare off water is polarized by reflection, most strongly near Brewster's angle. Blue sky is polarized by Rayleigh scattering, which depends on the angle from the Sun and is strongest about 90 degrees away. Their polarization directions generally differ, so the filter angle that kills the glare is not the one that darkens the sky.

Turn one filter and see two mechanisms. Fresnel reflection, with its signed coefficients and glare near Brewster's angle, acts on polarized skylight; Rayleigh scattering polarizes the sky differently in each direction.

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 $\text{valid model regime}$

Model checked against analytic results

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. The glare on the water can disappear while another part of the sky stays bright. Line up the Sun, the water surface and the observer, then compare the same view with and without the filter at a fixed exposure. The sky is not a uniform blue lamp; its polarization direction changes across the hemisphere.

The water is an ideal flat dielectric, and the sky uses single Rayleigh scattering. The angular panoramas are synthetic, radiance is in arbitrary units, and the Sun is drawn larger than it is. Waves, clouds, noise and camera calibration are not modeled.

Physics tutorial

Audit the physics of Polarized Reflections & Sky

BackgroundGlare off water is polarized by reflection at the surface; skylight is polarized by scattering from molecules. Brewster’s condition is $\tan\theta_B=n_2/n_1$ . The angle is measured from the normal to the surface, not from the water plane. At this angle the reflected field parallel to the plane of incidence vanishes, but the rest of the reflection remains.

Why it mattersWhy does a polarizing filter usually need different angles to darken water glare and blue sky?

Start with the essentials

Focus question: Why does a polarizing filter usually need different angles to darken water glare and blue sky?
One-sentence intuition: The sky follows a different angle: $P_{\rm sky}=\frac{\sin^2\gamma}{1+\cos^2\gamma}$ . Here the scattering angle is the angle between the direction to the Sun and the direction you look in the sky; it is not the viewing angle at the water. The polarization axis changes across the sky, so one filter does not darken every direction by the same amount.

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}

The cosines are of angles measured from the normal. The last equation holds for a lossless interface and for each pure polarization separately, not for a source already changed by a photographic filter. The index of air is one, and the reference index of water is 1.333.

Scattered polarization depends on direction

\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}

The incident light is unpolarized with unit intensity, and all radiances share one arbitrary scale. Two mutually incoherent transverse modes give both the angular intensity and the degree of polarization. In exact forward or backward scattering the axis is undefined, but the intensity stays finite and unpolarized.

Both views come from one field

\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 pass through the signed reflection amplitudes and the ideal analyzer projection. The unfiltered view shows those same rays without the filter, as a reference. The blue color and the nonlinear display mapping are for illustration, not a 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 cuts unpolarized light in half, but it passes polarized light selectively. Skylight reflected in the water is already polarized before it reaches the surface. The calculation keeps that incoming polarization and the signed Fresnel amplitudes, instead of assuming every ray reaching the water starts unpolarized.

Run the experiment

01
Scene 1: Brewster-angle glare
Start with Brewster glare. Compare the horizontal and vertical axes, then remove the filter. The reflected sunlight points at the camera only when the geometry is right.
What to observe: The panoramas are synthetic maps of radiance by direction, not camera recordings. An ideal filter axis is projected onto each ray. Calibrating a wide-angle lens and filter, 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 the local polarization axis; their length and the sphere they sit on are enlarged only to show direction.
03
Scene 3: Photographic polarizer
Choose the photographic preset and rotate the same physical axis. Compare the two marked directions at the same exposure. Then change the display exposure and check that the physical ratios stay the same.
What to observe: The water is flat and semi-infinite, with no waves and no light from the bottom. The source is widened on purpose so the glare is visible; its width and arbitrary brightness are not measurements of the Sun.