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Figure 4.2, it is clear that the thickness of the hyperhemisphere is −R × (n + 1)/n, or −1.625 × R. To calculate the value of R, we set up a matrix for the system. The first matrix corresponds to refraction at the planar air/glass boundary, the second to translation to the spherical surface and the final matrix to the refraction at that surface. On this occasion, translation to the original reference is not included.

equation

      From the above matrix, the focal length is −R/0.6 and hence R = −9.0 mm. The thickness, t, we know is −1.625 × R and is 14.625. In this sign convention, R is negative, as the sense of its sag is opposite to the direction of travel from object to image space.

      The (virtual) image is at (n + 1) × R from the sphere vertex or 2.6 × 9 = 23.4 mm.

      In summary:

equation

      4.2.2 Astigmatism and Field Curvature

      (4.10)equation

Geometrical illustration of field curvature for single refraction.

      (4.11)equation

      The systematic field dependent defocus can be represented as a spherical surface where the each field point is in focus. The curvature of this surface, CPETZ and equivalent to 1/RPETZ where RPETZ is the Petzval Radius, is given by:

      (4.12)equation

      It must be emphasised that the condition for perfect image formation on the Petzval surface applies specifically to the scenario where astigmatism has been removed.

Geometrical illustration of reflection from a spherical mirror. equation equation equation

      The optical path is given by:

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