The preceding discussion largely focused on perfect imaging in specific and restricted circumstances. However, even where perfect imaging is not theoretically possible, aspheric surfaces are extremely useful in the correction of system aberrations with a minimum number of surfaces. For more general design problems, therefore, even asphere terms may be added to the surface prescription. With the stop located at a specific surface, adding aspheric terms to the form of that surface can only control the spherical aberration at that surface. One perspective on the form of a surface is that second order terms only add to the power of that surface, whereas fourth order terms control the third order (in transverse aberration) aberrations. The reasoning behind this assertion may be viewed a little more clearly by expanding the sag of a conic surface in terms of even polynomial terms:
Adding a conic term to the surface, in addition to defining the curvature of the surface by its base radius, effectively adds an independent term to Eq. (5.9), effectively controlling two polynomial orders in Eq. (5.9). To this extent, adding separate additional second order and fourth order terms to the even asphere expansion in Eq. (5.1) is redundant. From the perspective of controlling third order aberrations, Eq. (5.9) confirms the utility of a conic surface in adding a controlled amount of fourth order optical path difference (OPD) to the system. In fact, the amount of OPD added to the system, to fourth order, is simply given by the change in sag produced by the conic surface multiplied by the difference in refractive indices. If the refractive index of the first medium is n0, and that of the second medium, n1, then the change in OPD produced by introducing a conic parameter of k is given by:
Equation (5.10) allows estimation of the spherical aberration produced by a conic surface introduced at the stop position. However, by virtue of the stop shift equations introduced in the previous chapter, providing fourth order sag terms at a surface remote from the stop not only influences spherical aberration, but also the other third order aberrations as well. In principle, therefore, by using aspheric surfaces, it is possible to eliminate all third order aberrations with fewer surfaces that would be possible with using just spherical surfaces alone. In fact, assuming that a system has been designed with zero Petzval curvature, it is only necessary to eliminate spherical aberration, coma, and astigmatism. Therefore, only three surfaces are strictly necessary. This represents a considerable improvement over a system employing only spherical surfaces. Notwithstanding the difficulties in manufacturing aspheric surfaces, some commercial camera systems are designed with this principal in mind.
Having introduced the underlying principles, it must be stated that design using aspheric surfaces is not especially amenable to analytical solution. In principle, of course, Eq. (5.10) could be used together with the relevant stop shift equations to compute analytically all third order aberrations. However, in practice, this is a rather cumbersome procedure and design of such systems proceeds largely by computer optimisation. Nevertheless, a clear understanding of the underlying principles is of invaluable help in the design process. An example, a simple two lens system, employing aspheric surfaces is sketched in Figure 5.3. This lens system replicates the performance of a three lens Cooke triplet with an aperture of f#5 and a field of view of 40°. Figure 5.3 is not intended to present a realistic and competitive design, but it merely illustrates the flexibility introduced by the incorporation of aspheric surfaces. In particular, it offers the potential to achieve the same performance with fewer surfaces.
Whilst aspheric components represent a significant enhancement to the toolkit of an optical designer, they represent something of a headache to the component manufacturer. As will be revealed later, in general, aspheric components are more difficult to manufacture and test and hence more costly. As such, their use is restricted to those situations where the advantage provided is especially salient. At the same time, advanced manufacturing techniques have facilitated the production of aspheric surfaces and their application in relatively commonplace designs, such as digital cameras, is becoming a little more widespread. Of course, the presence of conic and aspheric surfaces in large reflecting telescope designs is, by comparison, relatively well established.
Figure 5.3 Simple two lens system employing aspheric components.
5.3 Zernike Polynomials
5.3.1 Introduction
In describing wavefront aberrations at any surface in a system, it is convenient to do so by expressing their value in terms of the two components of normalised pupil functions Px and Py. Where the magnitude of the pupil function is equal to unity, this describes the position of a ray at the edge of the pupil. With this description in mind, we now proceed to describe the normalised pupil position in terms of the polar co-ordinates, ρ and θ. This is illustrated in Figure 5.4.
Figure 5.4 Polar pupil coordinates.
The wavefront error across the pupil can now be expressed in terms of ρ and θ. What we are seeking is a set of polynomials that is orthonormal across the circular pupil described. Any continuous function may be represented in terms of this set of polynomials as follows:
The individual polynomials are described by the term fi(ρ,θ), and their magnitude by the coefficient, Ai. The property of orthonormality is significant and may be represented in the following way:
The symbol, δij is the Kronecker delta. That is to say, when i and j are identical, i.e. the two polynomials in the integral are identical, then the integral is exactly one. Otherwise, if the two polynomials in the integral are different, then the integral is zero. The first property is that of normality, i.e. the polynomials have been normalised to one and the second is that of orthogonality, hence their designation as an orthonormal polynomial set.
