Lagrange optical invariant ($H=n(y_1{u}_2-y_2{u}_1)$)

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.