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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. ^[ref]

The underlying phase space has four coordinates. In a planar section with constant refractive index, let $(x,y)$ locate a light ray and setThe Jacobian determinant from spherical coordinates givesThus 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.

For paraxial rays, the Lagrange optical invariant is $H=n(y_1{u}_2-y_2{u}_1)$. Lossless paraxial ray transfer preserves this quantity: in coordinates $(y,nu)$, free propagation and a thin lens act by matrices with determinant one, which preserve the oriented area of two ray vectors. Consequently a full angular optical slit width $\theta$ at an optical pupil diameter $D$ satisfies $D\theta =A_{\mathrm{cam}}p/f_{\mathrm{cam}}$ for its image width $p$ and optical camera optical pupil width $A_{\mathrm{cam}}$ in the same meridional plane.

Let $f_t$ be the optical telescope focal length, $x=f_t\theta$ the physical optical slit width, $f_{\mathrm{coll}}$ the collimator focal length, and $A_{\mathrm{coll}}=D{f}_{\mathrm{coll}}/f_t$ the incident collimated beam diameter. For the grating equation $m\lambda =d(\sin \alpha +\sin \beta )$, differentiation at fixed wavelength gives $|d\beta /d\alpha |=\cos \alpha /\cos \beta$. Thus the monochromatic optical slit image width isThe incident and emergent optical pupil widths are $A_{\mathrm{coll}}=W\cos \alpha$ and $A_{\mathrm{cam}}=W\cos \beta$. Multiplying cancels the anamorphic magnification of a grating:This is conservation of the Lagrange optical invariant in the dispersion plane. It uses paraxial optical slit angles, an unvignetted optical pupil and matching full-width conventions for $\theta ,p,D,A_{\mathrm{cam}}$.