Validity of the perturbation model for the propagation of MSF structure in 2D

Abstract

Assessment of the performance degradation caused by the mid-spatial frequency (MSF) structure on optical surfaces often relies on a perturbation method that dovetails with the familiar sequence of models based on geometrical and physical optics. In the case of imaging systems, the perturbative step yields estimates of wavefronts in the exit pupil which are, in turn, used to extract performance measures such as MTF, PSF, and Strehl ratio. To date, the validity of that perturbation appears to be poorly understood. We present methods to estimate the errors of this approach and thereby arrive at a rule of thumb for its accuracy: the error is approximately equal to the RMS of the MSF structure at its source multiplied by the square of the ratio between a particular Fresnel zone size and a characteristic length of the MSF structure.

Rapidly decaying Fourier-like bases

“…We were motivated to pursue the ideas reported above by the fact that the fourth moment of an RDF-spectrum gives intuitive access to a physically significant characteristic length in investigations of MSF structure on optical surfaces [9,15]. As indicated in Eqs.…”

“…1, this basis is also a natural choice for the assessment of the validity of the perturbation model for the propagation of one-dimensional MSF structures in imaging systems and -as given in Eq. (12) -it has a valuable fourth spectral moment in that context [9].…”

“…(15), the solutions given by 2 V m n (r, φ) happen to describe the elastic modes of unclamped thin disks [11][12][13], and have been used to study deformable mirrors [14]. Furthermore, in two-dimensional paraxial imaging systems with MSF structures, these solutions are a natural basis for the assessment of the perturbation model [15]; the basis required for analogous studies of non-paraxial imaging systems is obtained by solving Eq. (35) on a sphere rather than on flat space, and this is considered in Sec.…”

“…1, and can provide a useful connection between these two limits. In this way, we finally resolve the nature of the spectra referred to in [9] that gives access to an intuitive characteristic scale of a function's variations. Interestingly, some of these RDF bases are useful in a range of other applications [11][12][13][14][15].…”

“…For most of these purposes, an orthogonal polynomial basis has been shown to be well matched [8]. The second of these steps, however, is typically performed by using a perturbation approximation and we recently showed that the accuracy of that approximation can be estimated in terms of the mean squared Laplacian of the MSF [9]. In a simple one-dimensional model, it is the mean squared second derivative that is required, of course.…”