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5 Aspheric Surfaces and Zernike Polynomials

      5.1 Introduction

      The previous chapters have provided a substantial grounding in geometrical optics and aberration theory that will provide the understanding required to tackle many design problems. However, there are two significant omissions.

      Firstly all previous analysis, particularly with regard to aberration theory, has assumed the use of spherical surfaces. This, in part, forms part of a historical perspective, in that spherical surfaces are exceptionally easy to manufacture when compared to other forms and enjoy the most widespread use in practical applications. Modern design and manufacturing techniques have permitted the use of more exotic shapes. In particular, conic surfaces are used in a wide variety of modern designs.

      The second significant omission is the use of Zernike circle polynomials in describing the mathematical form of wavefront error across a pupil. Zernike polynomials are an orthonormal set of polynomials that are bounded by a circular aperture and, as such, are closely matched to the geometry of a circular pupil. There are, of course, many different sets of orthonormal functions, the most well known being the Fourier series, which, in two dimensions, might be applied to a rectangular aperture. As the wavefront pattern associated with defocus forms one specific Zernike polynomial, the orthonormal property of the series means that all other terms are effectively optimised with respect to defocus. This topic was touched on in Chapter 3 when seeking to minimise the wavefront error associated with spherical aberration by providing balancing defocus. The optimised form that was derived effectively represents a Zernike polynomial.

5.2 Aspheric Surfaces

      5.2.1 General Form of Aspheric Surfaces

      In this discussion, we will restrict ourselves to surfaces that are symmetric about a central axis. Although more exotic surfaces are used, such symmetric surfaces predominate in practical applications. The most general embodiment of this type of surface is the so-called even asphere. Its general form is specified by its surface sag, z, which represents the axial displacement of the surface with respect to the axial position of the vertex, located at the axis of symmetry. The surface sag of an even asphere is given by the following formula:

(5.1)

      c = 1/R is the surface curvature (R is the radius); k is the conic constant; α_n is the even polynomial coefficient.

The curvature parameter, c, essentially describes the spherical radius of the surface. The conic constant, k, is a parameter that describes the shape of a conic surface. For k = 0, the surface is a sphere. More generally, the conic shapes are as set out in Table 5.1.

Table 5.1 Form of conic surfaces.

Without the further addition of the even polynomial coefficients, α_n, the surfaces are pure conics. Historically, the paraboloid, as a parabolic mirror shape, has found application as an objective in reflective telescopes. As will be seen subsequently, use of a parabolic mirror shape entirely eliminates spherical aberration for the infinite conjugate. The introduction of the even aspheric terms add further useful variables in optimisation of a design. However, this flexibility comes at the cost of an increase in manufacturing complexity and cost. Strictly speaking, at the first approximation, the terms, α₁ and α₂ are redundant for a general conic shape. Adding the conic term, k, to the surface prescription and optimising effectively allows local correction of the wavefront to the fourth order in r. In this context, the first two even polynomial terms are, to a significant degree, redundant.

      5.2.2 Attributes of Conic Mirrors

There is one important attribute of conic surfaces that lies in their mathematical definition. To illustrate this, a section of an ellipsoid, i.e. an ellipse, is shown in Figure 5.1. An ellipse is defined by its two foci and has the property that a line drawn from one focus to any point on the ellipse and thence to the other focus has the same total length regardless of which point on the ellipse was included.

The ellipsoid is defined by its two foci, F₁ and F₂. In this instance, the shape of the ellipsoid is defined by its semi-major distance, a, and its semi-minor distance, b. As suggested, the key point about the ellipsoid shape sketched

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