Эротические рассказы

Optical Engineering Science. Stephen RoltЧитать онлайн книгу.

Optical Engineering Science - Stephen Rolt


Скачать книгу
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 Ai. To calculate an individual term, one simply takes the cross integral of the function, Φ(ρ,θ), with respect to an individual polynomial, fi(ρ, θ):

equation

      By definition we have:

      5.3.2 Form of Zernike Polynomials

      Following this general discussion about the useful properties of orthonormal functions, we can move on to a description of the Zernike circle polynomials themselves. They were initially investigated and described by Fritz Zernike in 1934 and are admirably suited to a solution space defined by a circular pupil. We will suppose initially, that the polynomial may be described by a component, R(ρ), that is dependent exclusively upon the normalised pupil radius and a component G(φ) that is dependent upon the polar angle, φ. That is to say:

      (5.15)equation

      We can make the further assumption that R(ρ) may be represented by a polynomial series in ρ. The form of G(φ) is easy to deduce. For physically realistic solutions, G(φ) must repeat identically every 2π radians. Therefore G(φ) must be represented by a periodic function of the form:

       where m is an integer

      Having dealt with the polar part of the Zernike polynomial, we turn to the radial portion, R(ρ). The radial part of the Zernike polynomial, R(ρ), comprises of a series of polynomials in ρ. The form of these polynomials, R(ρ), depends upon the angular parameter, m, and the maximum radial order of the polynomial, n. Furthermore, considerations of symmetry dictate that the Zernike polynomials must either be wholly symmetric or anti-symmetric about the centre. That is to say, the operation r → −r is equivalent to φ → φ + π. For the Zernike polynomial to be equivalent for both (identical) transformations, for even values of m, only even polynomials terms can be accepted for R(ρ). Similarly, exclusively odd polynomial terms are associated with odd values of m.

      Overall, the entirety of the set of Zernike polynomials are continuous and may be represented in powers of Px and Py or ρcos(φ) and ρsin(φ). It is not possible to construct trigonometric expressions of order, m, i.e. cos(mφ) and ρsin(mφ) where the order of the corresponding polynomial is less than m. Therefore, the polynomial, R(ρ), cannot contain terms in ρ that are of lower order than the angular parameter, m.

      Cn,m,i represents the value of a specific coefficient

      The parameter, Nn,m, is a normalisation factor. Of course, any arbitrary scaling factor may be applied to the coefficients, Cn,m,i, provided it is compensated by the normalisation factor. By convention, the base polynomial has a value of unity for ρ = 1. Of course, with this in mind, the purpose of the normalisation factor is to ensure that, in all cases, the rms value of the polynomial is normalised to one. It now remains only to calculate the values of the coefficients, Cn,m,i. These are determined from the condition of orthogonality which applies separately for Rn,m(ρ) and may be set out as follows:

      (5.19)equation

      More completely we can express the entire polynomial:

      (5.21a)equation

      (5.21b)equation


Скачать книгу
Яндекс.Метрика