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Most, if not all, curved optical surfaces are at least approximately spherical and are widely employed in the fabrication of lens components. Figure 1.13 illustrates refraction at a spherical surface.

As before, the special case of refraction at a spherical surface may be described by Snell's law:

If we now wish to calculate the angle φ in terms of θ, this process is, in principle, straightforward. We need also to take into account the angle the surface normal makes with the optical axis, Δ, and the radius of curvature, R, of the spherical surface. However, calculation is a little unwieldy, so therefore we make the simplifying assumption that all angles are small and:

Figure 1.13 Refraction at a spherical surface.

Hence:

We can finally calculate φ in terms of θ:

(1.14)

There are two terms on the RHS of Eq. (1.14). The first term, depending on the input angle θ is of the same form as Snell's law (for small angles) for a plane surface. The second term, which gives an angular deflection proportional to the height, h, and inversely proportional to the radius of curvature R, provides a focusing effect. That is to say, rays further from the optic axis are bent inward to a greater extent and have a tendency to converge on a common point. The sign convention used here assumes that positive height is vertically upward, as displayed in Figure 1.13 and a positive spherical radius corresponds to a scenario in which the centre of the sphere lies to the right of the point where the surface intersects the optical axis. Finally, a positive angle is consistent with an increase in ray height as it propagates from left to right in (1.13).

Equation (1.14) can be used to trace any ray that is incident upon a spherical refractive surface. If this surface is deemed to comprise ‘the optical system’ in its entirety, then one can use Eq. (1.14) to calculate the location of all Cardinal Points, expressed as a displacement, z along the optical axis. Positive z is to the right and the origin lies at the intersection of the optical axis and the surface. The Cardinal points are listed below. Cardinal points for a spherical refractive surface

Both Principal Points: z = 0
Both Nodal Points: z = R

In this instance, the two focal lengths, f₁ and f₂ are different since the object and image spaces are in different media. If we take the first focal length as the distance from the first focal point to the first principal point, then the first focal length is positive. Similarly, the second focal length, the distance from the second principal point to the second focal point, is also positive. The principal points are both located at the surface vertex and the nodal points at the centre of curvature of the sphere. It is important to note that, in this instance, the principal and nodal points do not coincide. Again, this is because the refractive indices of object and image space differ.