= Radiant flux through an oriented surface
{title2=$\Phi_{\rm net}=\Phi_+-\Phi_-$}
In <geometrical optics>, suppose radiation has a nonnegative <radiance> $L(\mathbf x,\mathbf s)$ whose directional contributions add as <power>. Here $\mathbf s$ is the actual propagation <unit vector>, $d\Omega$ is a <solid angle> element, and the collecting <oriented surface> $A$ has chosen <unit normal> $\mathbf n(\mathbf x)$. Assume the following <integrals> exist. The outward and inward <radiant fluxes> are
$$
\Phi_+=\int_A\!\int_{\mathbf n\cdot\mathbf s>0}
L(\mathbf x,\mathbf s)(\mathbf n\cdot\mathbf s)\,d\Omega\,dA,
\qquad
\Phi_-=\int_A\!\int_{\mathbf n\cdot\mathbf s<0}
L(\mathbf x,\mathbf s)(-\mathbf n\cdot\mathbf s)\,d\Omega\,dA.
$$
Both quantities are nonnegative; their difference is the signed net <radiant flux>. The <dot product> supplies the <cosine> of the <angle> with the <unit normal>, hence the <projected area>. For incoming radiation the outward <unit normal> gives a negative <dot product>; the minus sign makes the collected <power> positive. Measuring the incoming <angle> from a source-facing <unit normal> instead gives the familiar positive $\cos\theta$. The units check is $\mathrm{W\,m^{-2}\,sr^{-1}}\times\mathrm{m^2}\times\mathrm{sr}=\mathrm W$.
Equivalently, the incoming <surface integral> is $\Phi_-=\int_A E\,dA$ with local <irradiance> $E$; the emitted part of $\Phi_+$ is $\int_A M\,dA$ with <radiant exitance> $M$. For a small flat patch and a narrow accepted cone, nearly constant <radiance> and <angle> give $\Phi\simeq LA\cos\theta\,\Delta\Omega$. This approximation requires uniformity across the patch and cone. Direction-independent outgoing <radiance> over a whole <hemisphere> gives the different exact result $\Phi_+=\pi LA$, by <Lambert's cosine law> and the <spherical coordinates> calculation in <radiant exitance>. Replacing the <cosine>-weighted <integral> by the <hemisphere>'s $2\pi$ <solid angle> would double the answer.<Radiometry and Photometry>{ref}
These <radiance> formulas presume a description in additive directional <power>; they do not reconstruct arbitrary coherent <wave interference> from separate ray contributions. For general <electromagnetic fields>, the signed instantaneous <power> crossing an <oriented surface> is $\int_A\mathbf S\cdot\mathbf n\,dA$, using the <Poynting vector> $\mathbf S$; optical measurements commonly average over many <wave periods>. A zero net <radiant flux> can conceal equal nonzero incoming and outgoing <radiant fluxes>.
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