Surface requirements for unaberrated images

Next we consider non-paraxial rays. We first consider what surface is required to make an unaberrated image.

We can derive the surface using Fermat's principle. Fermat's principle states that light travels in the path such that infinitessimally small variations in the path doesn't change the travel time to first order: d(time)/d(length) is a minimum. For a single surface, this reduces to the statement that light travels the minimum path length. An alternate way of stating Fermat's principle is that the optical path length is unchanged to first order for a small change in path. The OPL is given by:

$$OPL = \int c dt = \int {c\over v} v dt = \int n ds$$

Fermat's principle has a physical interpretation when one considers the wave nature of light. It is clear that around a stationary point of the optical path light, the maximum amount of light can be accumulated over different paths with a minimum of destructive interference. By the wave theory, light travels over all possible paths, but the light coming over the ``wrong'' paths destructively interferes, and only the light coming over the ``right'' path constructively interferes.

Fermat's principle can be used to derive the basic laws of reflection and refraction (Snell's law):

(do the derivation)

Now consider a perfect imaging system that takes all rays from an object and makes them all converge to an object. Since Fermat's principle says the only paths taken will be those for which the OPL is minimally changed for small changes in path, the only way a perfect image will be formed is when all optical path lengths along a surface between an image and object point are the same - otherwise the light doesn't get to this point!

Instead of using Fermat's principle, we could solve for the parameters of a perfect surface using analytic geometry, but this would require an inspired guess for the correct functional form of the surface.

We find that the perfect surface depends on the situation: whether the light comes from a source at finite or infinite distance, and whether the mirror is concave or convex. We consider the various cases now, quoting the results without actually doing the geometry.

Consider a concave mirror with one conjugate at infinity. Fermat's principle gives:

$$y^2 = 2 R z$$

where $R = 2 f$, the radius of curvature at the mirror vertex. This equation is that of a parabola. Note, however, that a parabola makes a perfect image only for on axis images (field angle=0).

For a concave mirror with both conjugates finite, we get an ellipse. Again, this is perfect only for field angle = 0.

$$(z-a)^2/a^2 + y^2/b^2 = 1$$

$$y^2 - 2 z b^2/a + z^2 b^2/a^2 = 0$$

where

$$a = (s + s')/2.$$

$$b = \sqrt(s s')$$

$$R = s s' / (s + s') = 2 b^2 / a$$

For a convex mirror with both conjugates finite, we get a hyperbola:

$$(z-a)^2/a^2 - y^2/b^2 = 1$$

$$y^2 + 2zb^2/a - z^2 b^2/a^2 = 0$$

where

$$a = (s + s')/2$$

$$b^2 = - s s'$$

(s is negative)

$$R = -2 b^2 / a$$

For a convex mirror with one conjugate at infinity, we get a parabola.

Note that in all cases we've considered a one-dimension surface. We can generalize to 2D surfaces by rotating around the z-axis; for the equations, simply replace $y^2$ with $(x^2 + y^2)$.

As you may recall from analytic geometry, all of these figures are conic sections, and it is possible to describe all of these figures with a single equation:

$$\rho^2 - 2 R z + (1+K) z^2 = 0$$

where

$$\rho^2 = x^2 + y^2$$

and $R$ is the radius of curvature at the mirror vertex, $K$ is called the conic constant ($K=-e^2$, where $e$ is the eccentricity for an ellipse, $e(b,a)$).

$K>0$ gives a prolate ellipsoid

$K=0$ gives a sphere

$-1<K<0$ gives a oblate ellipsoid

$K=-1$ gives a paraboloid

$K<-1$ gives a hyperboloid