Fourier optics modeling of interference microscopes

Abstract

We propose a practical theoretical model of an interference microscope that includes the imaging properties of optical systems with partially coherent illumination. We show that the effects on measured topography of a spatially extended, monochromatic light source at low numerical apertures can be approximated in a simplified model that assumes spatially coherent light and a linear, locally shift-invariant transfer function that accounts for optical aberrations and the attenuation of diffracted plane wave amplitudes with increasing spatial frequencies. Simulation of instrument response using this model agrees with methods using numerical pupil-plane integration and with an experimental measurement of surface topography.

Fourier optics modelling of coherence scanning interferometers

“…The results shown here are representative of what is observed experimentally [20,27], and have also been confirmed by direct comparison with more advanced models [8]. However, the EFO method has its limits of validity, given the many geometrical approximations built into the model.…”

“…This completes the description of the single-wavelength EFO model, consistent with prior work [9,27].…”

“…It can be nonetheless constructive to make several approximations that allow us to leverage familiar Fourier optics models. The first of these approximations is to replace the actual surface topography with a flat reflecting aperture function, with the local surface height represented by phase changes in the complex amplitude [25][26][27]. The main assumption is that the surface topography variations are small relative to the depth of field (DOF).…”

“…For a 2D pupil, the frequency x f is replaced by a radius in the frequency domain. Within the implied limits of the approximations discussed above, the TF can also incorporate overall image defocus and optical aberrations [27]. Figure 6 shows example TFs illustrating the wavelength dependence of the response.…”

“…The approximation of a 2D complex-valued aperture function in place of the 3D topography assumes that all surface points in the topography are at the same focus position [25][26][27]. This gives rise to the DOF limitation for fixed-focus systems, such as PSI microscopes, noted already in Section 3.…”