= Beam collection at fixed radiant flux
{title2=$\Phi_{\rm collected}=P\min(1,a^2/b^2)$}
Consider an ideal parallel circular beam with uniform <irradiance> inside radius $b$, zero outside, and total <radiant flux> $P=2\,\mathrm W$. A centred circular collector of radius $a$, perpendicular to the <light rays>, intercepts
$$
E_b=\frac{P}{\pi b^2},\qquad
\Phi_{\rm collected}=E_b\pi\min(a,b)^2
=P\min\!\left(1,\frac{a^2}{b^2}\right).
$$
For $b=1\,\mathrm{cm}$ and $a=0.5\,\mathrm{cm}$, $E_b=20000/\pi\,\mathrm{W\,m^{-2}}\simeq6366.20\,\mathrm{W\,m^{-2}}$ and the collected <radiant flux> is $0.50\,\mathrm W$. Expanding the beam to $b=2\,\mathrm{cm}$ while keeping $P$ fixed divides <irradiance> and the same collector's <radiant flux> by four: $E_b\simeq1591.55\,\mathrm{W\,m^{-2}}$ and $\Phi_{\rm collected}=0.125\,\mathrm W$. A collector covering either entire beam still receives $2\,\mathrm W$.
Tilting a flat collector of actual <area> $A$ by an <angle> $\theta$ to the beam direction gives $\Phi_{\rm collected}=E_b A\cos\theta$ if its entire <projected area> remains inside the uniform beam. At $60^\circ$ the small collector's <radiant flux> halves. A collector large enough to intercept the whole beam after tilting still receives $P$: the footprint on its surface grows as $1/\cos\theta$ and cancels the <irradiance> factor. This model ignores <diffraction>, beam-edge structure, and changes in absorption with incidence <angle>; it counts incident rather than absorbed <power>.
\Image[/radiant-flux-beam-collection-geometry.png]
{title=Fixed <radiant flux>, different beam widths and collecting <projected areas>}
{id=radiant-flux-beam-collection-geometry}
{description=Original ideal geometry, not measurements. The beam cross-sections use the same centimetre scale and total <radiant flux> of $2\,\mathrm W$; both collectors have radius $0.5\,\mathrm{cm}$. The side view shows a collector's cross-section at $60^\circ$: its projection perpendicular to the <light rays> has half the width, with depth unchanged. The stated half-power result requires the projected collector to stay inside the uniform beam.}
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