Why light falls off exponentially
Beer–Lambert, and what happens when the medium is not uniform
Each slice of medium removes a fixed fraction of whatever light reaches it — never a fixed amount. Constant fractions compound, and compounding fractions are an exponential. Drag the density profile at the top into any shape you like and watch the other three panels follow: the optical depth accumulates it, the transmittance is what survives, and the strip at the bottom is that number as light.
The ray, from entry to the eye
Draw a density profile
σt(x) — extinction along the rayat the marker: —
τ(x) = ∫ σt — optical depth, the shaded area aboveτ(d) = —
T(x) = exp(−τ) — the fraction that survivesT(d) = —
the beam, drawn at its actual brightness—
Controls
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What the eye receives
entering, L(0) = 1.000
arriving
Optical depth τ(d)
—
Transmittance T(d)
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Light removed
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Depth where half the light is left
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Optical depth is the number that actually matters. Notice that τ is dimensionless, and that only τ reaches the exponential — a thin dense medium and a thick thin one are indistinguishable if their integrals agree. Try it: set the profile to Thin wall and then to Uniform, and adjust the density until the two give the same τ. The eye receives exactly the same light in both cases, because the ray has no way to report where along its path the extinction happened. That is also the practical reading of the numbers: τ = 1 leaves 37% of the light, τ = 3 leaves 5%, and τ = 5 leaves under 1% — which is why volumetric renderers stop marching once τ passes about 5 and call the rest black. You will meet that shortcut again as early‑ray termination.