Читать книгу Optical Engineering Science - Stephen Rolt - Страница 105

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 P_x and P_y. 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:

(5.11)

The individual polynomials are described by the term f_i(ρ,θ), and their magnitude by the coefficient, A_i. The property of orthonormality is significant and may be represented in the following way:

(5.12)

The symbol, δ_ij 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.

Equations (5.11) and (5.12) give rise to a number of important properties of these polynomials. Initially we might be presented with a problem as to how to represent a known but arbitrary wavefront error, Φ(ρ,θ) in terms of the orthonormal series presented in Eq. (5.11). For example, this arbitrary wavefront error may have been computed as part of the design and analysis of a complex optical system. The question that remains is how to calculate the individual polynomial coefficients A_i. To calculate an individual term, one simply takes the cross integral of the function, Φ(ρ,θ), with respect to an individual polynomial, f_i(ρ, θ):

By definition we have:

(5.13)

So, any coefficient may be determined from the integral presented in Eq. (5.13). The coefficients, A_i, clearly express, in some way, the magnitude of the contribution of each polynomial term to the general wavefront error. In fact, the magnitude of each component, A_i, represents the root mean square (rms) contribution of that component. More specifically, the total rms wavefront error is given by the square root of the sum of the squares of the individual coefficients. That this is so is clearly evident from the orthonormal property of the series:

(5.14)