= Conservation of etendue
{title2=$G_{\rm out}=G_{\rm in}$}
Lossless reversible propagation preserves the <etendue> of the same bundle of <light rays>. Here the model uses stationary isotropic media, <specular reflection> and <refraction>, with every selected <light ray> reaching the output and with no <optical gain> or <scattering of light>. An <aperture> that blocks some incident <light rays> changes which bundle is counted; the full incident and transmitted <etendue> need not be equal. <Étendue (Photon Snacks) by R. John Koshel>[Étendue, pp. 1–2, lossless-system and entrance-pupil restrictions; p. 3, Figure 1, preserved generalized phase-space element]{ref}
The underlying <phase space> has four coordinates. In a planar section with constant <refractive index>, let $(x,y)$ locate a <light ray> and set
$$
p_x=n\sin\theta\cos\phi,\qquad p_y=n\sin\theta\sin\phi.
$$
The <Jacobian determinant> from <spherical coordinates> gives
$$
dp_x\,dp_y=n^2\sin\theta\cos\theta\,d\theta\,d\phi
=n^2\cos\theta\,d\Omega,
\qquad dG=dx\,dy\,dp_x\,dp_y.
$$
Thus <etendue> is a four-dimensional volume, preserved by the optical counterpart of <Liouville theorem in Hamiltonian mechanics>. Individual position and direction extents may change, and correlations between them matter. The <Lagrange optical invariant> is a related <paraxial approximation> invariant for pairs of <light rays>, rather than the full four-dimensional <measure>.
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