Optical invariants and étendue

Take two paraxial rays at the same plane, with coordinates $(y_1,u_1)$ and $(y_2,u_2)$ where $u=n\theta$. Their signed parallelogram area in reduced ray coordinates is $H=y_1u_2-y_2u_1$. Any first-order optical map with determinant one preserves this quantity because the determinant of the pair of output columns equals the map determinant times the input determinant. H has units of length because reduced angle is dimensionless.

With a marginal and a chief ray chosen appropriately, H becomes the familiar Lagrange invariant relating object height and numerical aperture to corresponding image-side quantities. Its sign depends on ray ordering, but its constancy does not. The conclusion requires admitted rays and centered paraxial transport. It does not imply that an aperture transmits every possible pair, and it does not convert a first-order determinant identity into a universal theorem for arbitrary scattering or nonlinear media.

For a three-dimensional bundle define étendue by integrating projected area and solid angle with an index-squared weight. The projection cosine accounts for a surface being seen obliquely by a direction. In a uniform medium, a planar area A accepting a full circular cone of half-angle alpha has étendue $\pi A n^2\sin^2\alpha$. This exact cone integral follows by integrating cosine times sine over polar angle and full azimuth.

Writing numerical aperture as $n\sin\alpha$ makes the same result $G=\pi A\,\mathrm{NA}^2$. It assumes every point of the selected area admits that same cone; arbitrary position-direction correlations require the integral instead. Solid angle is dimensionless in SI but conventionally labeled steradians, so étendue is reported in square metres steradians. A smaller spot can preserve G by admitting a larger cone. When the available numerical aperture is exhausted, further area reduction must reject part of the original bundle.

Radiance describes power per projected area per solid angle. Irradiance describes power per receiving area after integration over incident directions. A focusing lens can increase irradiance by delivering a wider accepted angular distribution at the spot while leaving reduced radiance unchanged. Across a refractive-index change, ordinary radiance need not remain numerically constant; the conserved quantity is $L/n^2$. Omitting that factor incorrectly rules out familiar immersion-optics benefits or falsely claims a brightness violation.

The passive brightness statement applies to a specified frequency band, polarization channels, and geometrical-optics transport without gain or internal emission. Absorption reduces transported radiance; clipping can reduce the étendue of the surviving subset only by rejecting power. Scattering generally broadens occupied phase space rather than losslessly compressing it. Combining separate wavelengths or orthogonal polarizations changes the channels being counted. Coherent interference and wavelength-scale structures require wave descriptions rather than an unqualified application of an incoherent ray-bundle formula.