Optical Thickness

As we've seen in the last section, the amount of scattering matters quite a bit for visuals. Based on the probable fate of a single photon, one can distinguish two regimes:

If a single photon is unlikely to scatter while passing through a medium, most of a ray will pass unaltered. In that case the medium is optically thin - which in essence means one can see through it, albeit with some distortion.

If on the other hand a single photon has a high probability of scattering more than once on its path through the medium, the medium is optically thick - which is to say one can not see through it, it is opaque and generally a ray will not pass through.

Diffuse scattering and absorption

In an optically thick medium, multiple scattering occurs. Consider hundreds of Mie scattering processes for a photon - each single one will be predominantly into the direction of the previous photon, but after a sufficient number of such processes, the resulting photon will no longer remember the direction of the original one.

In essence, the scattering is diffuse - it sends light into all possible directions like Rayleigh scattering, but the process is not dependent on wavelength.

An example for such an optically thick scatterer is a thick Stratus cloud. One can not see through it, it looks solid (in fact one can only see into it for the average distance to the first photon scattering - one optical depth). The cloud is highly reflective - which is the result of an asymmetric process - any light that is scattered out of the cloud is gone for good, but any photons scattered into the cloud remain there and gradually diffuse through the layer through a large number of scatterings - but if the light again happens to emerge on top, again it is gone. And the light that reaches the other end of the cloud in this diffusion process is highly attenuated - the cloud casts a shadow. the light in overcast conditions is dimmed by a factor of about 100 in intensity compared to direct sunlight. Whatever light emerges beneath a cloud layer is directionless - one can no longer point to the position of the sun, the sky looks more or less uniformly bright.

Light can also be absorbed by an optically thick medium - in this case the photon is gone and used to increase the temperature of the absorber. Clouds can absorb incoming sunlight, heat up in this way and ultimately dissolve.

Optical thickness and cloud illumination

For the illumination of clouds, understanding the optical thickness is rather crucial. Some cloud types - especially the high Cirrus clouds, are always translucent. Others like Cumulus clouds are more opaque, but despite appearing so, even very compact clouds like strongly developing Cumulonimbus are not solid objects, they have 'fuzzy' edges, and that means that there is always a region at the fringe of the cloud where it is optically thin.

Consider the following scene - what do we see?

Silver lining - Mie scattering at cloud edges.

The Sun has gone behind a low cloud bank at the horizon and the cloud bank blocks the sunlight - which is why it appears very dark. But the upper fringe is glowing brightly - that is Mie forward scattering in the region where the cloud is optically thin. During the day this phenomenon appears a a white glow and is hence known as silver lining, but with the illuminating ray colored by Rayleigh scattering, here it would more properly appear as 'gold lining'.

The sudden change from optically thin to optically thick makes for a strinking contrast, the brightly glowing region is bordered by a very dark shaded cloud. Note also how the bright glow really only occurs inside the Mie scattering halo region around the sun, the light on clouds outside this region is much more muted (for the moment, we do not consider the cloudsin the upper part of the picture, their illumination works differently).

Annotated version of the above.

The strength of the transition from bright glow to dark shadow is determined by how opaque the cloud really is - for thick clouds, silver lining occurs when the Sun moves behind the cloud, for thinner cloud types the pattern can be more complex. Still, note how the strong glow is confined to the Mie halo region around the sun.

Complex pattern for less opaque clouds.

When the clouds are even less opaque, the pattern is not perceived as a contrast between 'light and shadow' but rather as a variation in intensity in the Mie glow - the clouds are always lit, but the strong bright glow occurs only when they are truly optically thin. Note also how the overall strength of Rayleigh scattering on the illuminating ray sets the color of the scene.

Mie halo for even less opaque clouds and more white light.

Continue with Different Altitude, different Light.

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