Development and Validation of a Novel Fresnel Integral Based Method to Model MSF Errors in Optical Imaging

TL;DR

This work develops a Fresnel-diffraction–based framework to model mid-spatial frequency (MSF) errors on the PSF in optical imaging, using a hybrid approach that treats the optical system with geometrical optics while computing diffraction effects for MSF perturbations in the image plane. The MSF influence is captured through a transmission function $\tau$ and a convolution with the unperturbed PSF via $\widetilde{\text{PSF}}(x,y)=\mathcal{P}_{z_2z_3}(\tau\,\mathcal{P}_{z_1z_2}(U_{z_1}))$, which reduces to a PSF convolved with $\mathcal{F}(\tau)$ up to quadratic phase terms. Analytic results are obtained for a sinusoidal MSF, yielding a sum of translated Airy disks weighted by Bessel functions, while arbitrary MSF spectra are treated numerically through Fourier-domain convolution. Experimental validation using a spatial light modulator to imprint MSF-like phase patterns demonstrates good agreement with the model at low numerical aperture, supporting the approach and outlining paths for extending to higher NA and more complex optical systems.

Abstract

Mid-spatial frequency errors of optical components cause degradation of images which are rather difficult to quantify. In this work, we present a model for calculating the point-spread function in the presence of mid-spatial frequency errors which is based on diffraction integrals.The results of the model are compared with experiments.

Figures (6)

• Figure 1: A schematic view of imaging a point source by an optical system represented as a black box. The resulting image is a PSF of the optical system in the image plane. Note that the entrance pupil and the exit pupil can be at arbitrary positions along the optical axis.
• Figure 2: The field in the exit pupil is modelled as unaffected by the MSF structures. It may however contain low-spatial frequency errors, which can be represented by Zernike polynomials. The MSF errors are incorporated in the focussed spherical wave from the exit pupil when this wave passes through the image plane of the MSF structure.
• Figure 3: Experimental setup where specific MSF errros are realized in the setup with an SLM. The Resulting PSF is directly imaged onto the camera.
• Figure 4: Four examples of simulation results. In (a) a control image is shown for when the perturbation is set to 0, in (b) a perturbation is shown for $h=\pi/4$, we can see the interference between multiple spots. In (c) and (d) the perturbation is set such that we obtain a zero for $J_0(h)$ and $J_1(h)$ respectively. The simulations are accompanied by the corresponding simulation, indicated with an accent.
• Figure 5: Overlay of the measurement results through the cross-section of the simulated and measured spots. The pixel size of the detector is $3.45\mu$m