Derivation of the Vergence Equation and the Lensmaker’s Equation from Snell’s Law
Consider Figure 1-A1. A light ray traveling from left to right (shown in dark green) is incident on a curved refracting surface with radius of curvature r and center of curvature C on the optic axis OC. Suppose the index of refraction to the left of the surface is n1, to the right of the surface, n2, (shown with n2 > n1). The incident ray strikes the axis at distance u from the vertex, O, of the refracting surface and meets the surface at a height h from the axis, making an angle, α, with the optic axis. The incident ray makes an angle θ with the surface normal and is then refracted such that it makes, say, a smaller angle, σ, with the surface normal to the right of the refracting surface. The refracted ray (shown in light green) meets the axis at a distance ν from the vertex, forming an angle β with the axis. The angle subtended by the height h at the center of the circle is denoted by γ.
In the paraxial regime, angles θ, σ, α, β, and γ are small (that is, h is substantially smaller than r). We use the “small-angle approximation” from trigonometry: for small angles ϕ,
ϕ = sin ϕ = tan ϕ
With this approximation, Snell’s law, n1 sin θ = n2 sin σ, simplifies to
Similarly, using the tangent approximation for angles α, β, and, γ, and in the same spirit ignoring the discrepancy between the foot of the perpendicular from the point of incidence and O, we have
Figure 1-A1 Geometry for deriving the vergence equation and the lensmaker’s equation from Snell’s law.
Finally, from geometry (the “exterior-angle theorem” and the fact that “opposite” angles are equal), we observe that
γ = θ + α and γ = β + σ
From here, it is only algebra: substituting Eq. (3) into Eq. (1) gives
1(γ − α) = n2(γ − β)
Substituting Eq. (2) into Eq. (4) gives
Clearing the common factor h and rearranging yields
Equating (n2 − n1)/r with P yields both the vergence equation and the lensmaker’s equation at once!
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.