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A concave spherical mirror reflects at the inside of a spherical surface of radius of curvature $R$. For incident light rays near its optical axis, the law of reflection gives paraxial focal length $f=R/2$. Rays farther from the optical axis have different axial crossing positions, causing spherical aberration; one must specify a criterion to define the best full-aperture focus.

Use the unfolded propagation convention after reflection. Free-space propagation and a concave spherical mirror have matricesThe mirror is equivalent in this paraxial convention to a thin lens of focal length $R/2$. The excursion from the reference plane to mirror 1 and back isLikewise $S_2=T(L-L_1)F(R_2)T(L-L_1)$. Multiplying $S_2{S}_1$, or using cyclic invariance of the trace to move the reference plane, givesConsequently the strict criterion of a isThe trace is independent of $L_1$, as a change of reference plane only conjugates the round-trip matrix. For the concrete confocal counterexample $R_1=R_2=L$, $L_1=L/2$, each excursion is $\left(\begin{array}{cc}0& L/2\\ -2/L& 0\end{array}\right)$ and its square is $-I$. This gives $g_1{g}_2=0$ and the degenerate boundary exception already explained.

A distant satellite supplies an approximately collimated incident beam. A concave spherical mirror of radius of curvature $R_c=1.5\mathrm{m}$ has paraxial focal length $f_m=R_c/2$. Put the receiving feed at the principal focus, on the optical axis in front of the dish:The radius of curvature relation is a paraxial approximation. A practical dish is usually a parabolic reflector, which focuses an on-axis collimated beam without spherical aberration; a literal spherical dish has spherical aberration, so an exact optimum over its full aperture would need a specified focusing criterion.