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      This analysis provides a very simple introduction to the concept of third order aberrations. In the basic illustration so far considered, we have looked at the example of a simple lens focussing an on axis object located at infinity. In the more general description of monochromatic aberrations that we will come to, this simple, on-axis aberration is referred to as spherical aberration. In developing a more general treatment of aberration in the next sections we will introduce the concept of optical path difference (OPD).

      In the preceding section, we considered the impact of optical imperfections on the transverse aberration and the construction of ray fans. Unfortunately, this treatment, whilst providing a simple introduction, does not lead to a coherent, generalised description of aberration. At this point, we introduce the concept of optical path difference (OPD). For a perfect imaging system, with no aberration, if all rays converge onto the paraxial focus, then all ray paths must have the same optical path length from object to image. This is simply a statement of Fermat's principle. We now consider an aberrated system where we accurately (not relying on the paraxial approximation) trace all rays through the system from object to image. However, at the last surface, we (hypothetically) force all rays to converge onto the paraxial focus. For all rays, we compute the optical path from object to image. The OPD is the difference between the integrated optical path of a specified ray with respect to the optical path of the chief ray. Of course, if there were no aberration present, the OPD would be zero. Thus, the OPD represents a quantitative description of the violation of Fermat's principle.

Geometrical illustration of rays accurately traced from the object through the system, emerging into image space—optical path difference. equation

      Having established an additional way of describing aberrations in terms of the violation of Fermat's principle, the question is what is the particular significance and utility of this approach? The answer is that, when expressed in terms of the OPD, aberrations are additive through a system. As a consequence of this, this treatment provides an extremely powerful general description of aberrations and, in particular, third order aberrations. Broadly, aberrations can be computed for individual system elements, such as surfaces, mirrors, or lenses and applied additively to the system as a whole. This generality and flexibility is not provided by a consideration of transverse aberrations.

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      The sign convention is important, as it now concurs with the definition of OPD. As the wavefronts form surfaces of constant optical path length, there is a direct correspondence between OPD and WFE. A positive OPD indicates the optical path of the ray at the reference sphere is less than that of the chief ray. Therefore, this ray has to travel a small positive distance to ‘catch up’ with the chief ray to maintain phase equality. Hence, the WFE is also positive.

Wavefront illustration of aberration. Graphical illustration of wavefront and ray geometry.

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