Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions

Abstract

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

Bilinear solution to the phase diversity problem for extended objects based on the Born approximation

“…The choice of Zernike polynomials for the representation of the wavefront is motivated by the fact that they have analytical expressions and can be easily included in the computation of the PSFs. The derivations here follow the lines of [19][20][21] . We consider a point source of monochromatic light in the object plane of a centered optical system.…”

“…(16) and Eq. (19). It is stated in 3 that the computation time of the algorithm is completely dominated by the time required to obtain these gradients.…”

Image reconstruction with analytical point spread functions

“…For small aberrations (a j 1), the previous series can be safely cut at first order, something that was successfully exploited by Janssen (2002) and Braat et al (2002) to empirically estimate point spread functions of microscope-type optical systems. This approximation may be useful if one is interested in describing aberrations introduced by well adjusted optical systems, but it is hardly acceptable for aberrations introduced by atmospheric turbulence.…”

“…The resulting integral expressions were very complicated and for a long time remained impossible to evaluate. What is presently known as the extended Nijboer-Zernike theory (ENZ, Janssen 2002;Braat et al 2002) presents analytical expressions for the complex field at the focal volume that depend on the aberration described by the coefficients of the Zernike expansion of the wavefront. The expressions are closed in the sense that they depend only on computable functions and the coefficients of the Zernike expansion.…”

Through-Focus Point-Spread Function Evaluation for Lens Metrology using the Extended Nijboer-Zernike Theory

“…It has turned out that for nonzero values of f, as large as ±2π, a well-converging series expression for Eq. (2.2) can be found [6] and this solution has proven to be very useful and effective [7] once we are confronted with the inversion problem described in the introduction.…”