Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can be obtained.
Optical microscopes are probably the most wide-spread optical instruments in science and technology. While in biological applications microscopes are mostly operated in transmission mode, reflection-type microscopes dominate in materials science. A further important field of application of reflection microscopy is the reconstruction of the three-dimensional (3D) surface topography of an object. This kind of 3D microscopy obtains 3D image stacks by axially scanning through the focus. While in a conventional bright-field microscope point scatterers are necessary to enable height discrimination, interference microscopy even works on specularly reflecting surfaces. In both cases, according to optical systems theory the physical behaviour of the microscope is fully represented by a 3D point spread function in the object space or, equivalently, by the corresponding 3D transfer function in the spatial frequency domain. In order to be consistent with microscopic imaging theory, integrating the 3D transfer function along the axial spatial frequency is required to result in the well-known modulation transfer function of a diffraction limited optical imaging system. Since to the best of our knowledge no formula for the calculation of the 3D transfer function of a microscope operated in reflection mode has been published, we derive a mostly analytical formula, which is perfectly consistent with the modulation transfer function of a diffraction limited system. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can be obtained. Utilizing the rotational symmetry of the 3D transfer function we present a computationally efficient formula for the numerical calculation of the 3D point spread function.
Keywords: 3D point spread function; 3D spatial frequency characterization; microscopy; reflection mode; transfer function.
© 2021 The Authors. Journal of Microscopy published by John Wiley & Sons Ltd on behalf of Royal Microscopical Society.