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The matrix for the system is simply as below – note the order:

We have two translations. The first translation represents the thickness of the lens and the second translation, by convention, traces the refracted rays back to the origin in z. This is so that, in interpreting the formulae for Cardinal points, we can be sure that they are all referenced to a common origin, located as in Figure 1.19. Positive axial displacement (z) is to the right and a positive radius, R, is where the centre of curvature lies to the right of the vertex. The final matrix is as below:

As both object and image space are in the same media, there is a common focal length, f, i.e. f₁ = f₂ = f. All relevant parameters are calculated from the above matrix using the formulae tabulated in Section 1.6.2.

The focal length, f, is given by:

The formula above is similar to the simple, ‘Lensmaker’ formula for a thin lens. In addition there is another term, linear in thickness, t, which accounts for the lens thickness.

The focal positions are as follows:

The principal points are as follows:

Figure 1.20 Hubble space telescope schematic.

Of course, since the refractive indices of the object and image spaces are identical, the nodal points are located in the same place as the principal points. If we take the example of a biconvex lens where R₂ = −R₁, then:

So, for a biconvex lens with a refractive index of 1.5, then the principal points lie about one third of the thickness from their respective vertices.