Derivation of the Vergence Equation for Mirrors
Consider Figure 1-A3. The incident ray (shown in dark green) moves from left to right and would intersect the optic axis at u if the ray did not first intersect the mirror (here drawn convex to the left, with radius of curvature r) at height h above the optic axis. The direction of the reflected ray (shown in light green) is determined by the law of reflection, so the angle γ/2 between the incident ray and the mirror’s surface normal (drawn as a dashed line) equals the angle γ/2 between the surface normal and the reflected ray. The reflected ray travels from right to left, but if extended, it would intersect the optic axis at ν. The angles between the optic axis and the reflected ray, radius of curvature to the point of reflection, and incident ray are denoted α, θ, and β, respectively.
In this case, the direction of the reflected ray is determined by the law of reflection instead of Snell’s law. We have:
Inserting Eq. (2) into Eq. (1) gives
α = 2α − 2θ + β
Figure 1-A3 Geometry for deriving the vergence equation for mirrors from the law of reflection.
(Illustration developed by Scott E. Brodie, MD, PhD.)
α = 2θ − β
Using the small-angle approximations for α, β, and θ (as in Appendix 1.1), we have
Canceling the common factor h and rearranging gives
Identifying P = −2/r as the power of the mirror, and interpreting the minus sign on the right side of the equation as indicating a vergence of “1/ν” in the “reversed” image space, yields the vergence equation for mirrors.
Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.