Most ordinary surfaces are near Lambertian diffuse reflector, i.e. a small local radiates most strongly at norm then attenuates by cosine law when one gets to the tangentials. However this seems hard to square with the fact that it's simply an incoherent reflector. When we consider only a single wavelength, the surface would consist of point sources of random magnitude and phase, making it isotropic not Lambertian. What's the physics behind Lambertian reflectors? Small local coherences favoring norm direction?
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$\begingroup$ Does the Figure 2 on the Wikipedia answers your question? en.wikipedia.org/wiki/Lambert%27s_cosine_law $\endgroup$– NoctCommented Mar 9 at 20:20
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It seems you are worried about situation A in the drawing, where a collection of independent isotropic point sources does indeed create the same amount of outgoing radiation at a grazing angle than in the normal direction.
Case A would actually give the same luminous intensity in all directions but higher radiance in the grazing direction. That somewhat confusing distinction is explained in Wikipedia. But in any case it differs from Lambertian behavior.
The resolution is that the sources are blocking each other, like in situation B in the drawing. There you have essentially the same situation for observers in all directions, from their point of view any ray will at some point end at one of the radiating elements. This is a bit reminiscent of Olbers' paradox, where at every point in the sky we would see the surface of some distant star [Olbers].
So in case A, the observer from the normal direction would see the sources more spread out, with empty space between them. While in case B both observers see everything filled with sources, the normal observer just looks a bit deeper into the material.
Case C shows that surface roughness does the same as the randomly distributed immersed particles in case B. Only case A with its point sources will make the observer see an angle-dependent density of sources, because point sources are always sparse, while the other types fill up the viewing field regardless from which angle they are viewed. (And of course a mirror-smooth surface is excluded because we need incoherent addition of the light, otherwise a pronounced beam direction will result.)
answered Mar 10 at 11:07
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$\begingroup$ just a quick follow up: how do you then explain rough metal surfaces are also near Lambertian? This explanation makes sense for materials like paper or wall paint. What if there's an electron gas doing the reflection? $\endgroup$ Commented Mar 11 at 18:41
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$\begingroup$ Some kind of irregular hilly microscopic landscape, you mean... I played with the thought of drawing that as an example and maybe I'll add it later, I think it would work the same. But the example I chose (of randomly distributed immersed particles) is the most straightforward. $\endgroup$ Commented Mar 11 at 19:29