href="#ulink_2a352f46-73da-5442-9253-5476c238f1fe">(3.44)p is the pupil function and h is the object height (proportional to field angle θ); φ is the ray fan angle.
In the general term, Wabc, ‘a’ describes the order of the object height (field angle dependence), ‘b’ describes the order of the pupil function dependence and ‘c’ describes the dependence on the ray fan angle. The defocus and tilt, are of course paraxial terms. Overall, the dependence of each coefficient is given by Eq. (3.45):
It should be noted that this convention incorporates powers of cosφ, so the astigmatism term contains some average field curvature. Describing each of the aberration coefficients introduced earlier in terms of these coefficients gives the following:
(3.46)
(3.47)
(3.49)
(3.50)
Another convention exists of which the reader should be aware. These are the so called Seidel coefficients, named after the nineteenth century mathematician, Phillip Ludwig von Seidel, who first elucidated the five monochromatic aberrations. The coefficients are usually denominated, SI, SII, SIII, SIV, and SV, referring to spherical aberration, coma, astigmatism, field curvature, and distortion. They nominally quantify the WFE, as the other coefficients do, but their magnitude is determined by the size of the blur spot that the aberration creates. The correspondence of these terms is as follows:
(3.51)
(3.52)
(3.53)
(3.55)
The form of Eq. (3.54) is interesting. When compared to the definition of W220 in Eq. (3.48), an additional amount of astigmatism has been compounded with the field curvature. As such, this new representation of field curvature, SIV represents a fundamental and important property of an aberrated optical system and is referred to as the Petzval curvature. Its significance will be discussed more fully in the next chapter.
The treatment of aberrations, thus far, has been entirely generic. We have introduced the five Gauss-Seidel aberrations without specific reference to how they are generated at specific optical surfaces and by individual optical components. This will be discussed in detail in the next chapter. The most important feature of this treatment is that the third order aberrations are additive through a system when described in terms of OPD. That is to say, the five aberrations may be calculated independently at each optical surface and summed over the entire optical system. This analysis is an extremely powerful tool for characterisation of aberration in a complex system.
Further Reading
1 Born, M. and Wolf, E. (1999). Principles of Optics, 7e. Cambridge: Cambridge University Press. ISBN: 0-521-642221.
2 Hecht, E. (2017). Optics, 5e. Harlow: Pearson Education. ISBN: 978-0-1339-7722-6.
3 Kidger, M.J. (2001). Fundamental Optical Design. Bellingham: SPIE. ISBN: 0-81943915-0.
4 Kidger, M.J. (2004). Intermediate Optical Design. Bellingham: SPIE. ISBN: 978-0-8194-5217-7.
5 Longhurst, R.S. (1973). Geometrical and Physical Optics, 3e. London: Longmans. ISBN: 0-582-44099-8.
6 Mahajan, V.N. (1991). Aberration Theory Made Simple. Bellingham: SPIE. ISBN: 0-819-40536-1.
7 Mahajan, V.N. (1998). Optical Imaging and Aberrations: Part I. Ray Geometrical Optics. Bellingham: SPIE. ISBN: 0-8194-2515-X.
8 Mahajan, V.N. (2001). Optical Imaging and Aberrations: Part II. Wave Diffraction Optics. Bellingham: SPIE. ISBN: 0-8194-4135-X.
9 Slyusarev, G.G. (1984). Aberration and Optical Design Theory. Boca Raton: CRC Press. ISBN: 978-0852743577.
10 Smith, F.G. and Thompson, J.H. (1989). Optics, 2e. New York: Wiley. ISBN: 0-471-91538-1.
4 Aberration Theory and Chromatic Aberration
4.1 General Points
In the previous chapter, we developed a generalised description of third order aberration, introducing the five Gauss-Seidel aberrations. The motivation for this is to give the reader a fundamental understanding and a feel for the underlying principles. At the same time, it is fully appreciated that optical system design and detailed analysis of aberrations is underpinned by powerful optical software tools. Nevertheless, a grasp of the underlying principles, including an appreciation of the form of ray fans and optical path difference (OPD) fans, greatly facilitates the application of these sophisticated tools.
The treatment presented here is restricted to consideration of third order aberrations. Before the advent of powerful software analysis tools, the designer was compelled to resort to a much more elaborate and complex analysis, in particular introducing an analytical treatment of higher order aberrations. For all the labour that this would involve, the reader would gain little in terms of a useful understanding that could be applied to current design tools. As the third order aberrations are third order in transverse aberration and fourth order in OPD, so succeeding higher order aberrations are fifth, seventh etc. order in transverse aberration, but sixth, eighth order in OPD. That is to say, aberrations, whose order is expressed conventionally in terms of the transverse aberration, can only be odd. One can re-iterate the analysis of Section 3.4 to generate the form and number of terms involved in the higher order aberrations. This is left to the reader, but it is straightforward to derive the number of distinct terms Nn as a function of aberration order, n:
(4.1)
