Three?dimensional transfer function of optical microscopes in reflection mode

Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in spatial frequency domain. Utilizing projection-slice theorem, 2D modulation function an incoherent system can be derived from 3D by integration with respect to axial frequency. For diffraction limited microscope homogeneous pupil illumination, equals autocorrelation circular disc. However, until now best our knowledge no has been published, which exactly leads reflection mode. In this article, we derive formula, after coordinate perfectly fits function. The inverse three-dimensional Fourier transform results complex-valued point spread function, depth field, lateral resolution and, addition, corresponding both, conventional and interference microscope, obtained. Optical microscopes are probably most wide-spread instruments science technology. While biological applications mostly operated transmission mode, reflection-type dominate materials science. A further important field application microscopy is reconstruction surface topography object. This kind obtains image stacks axially scanning through focus. bright-field scatterers necessary enable height discrimination, even works on specularly reflecting surfaces. both cases, according theory physical behaviour fully represented object space or, equivalently, order consistent microscopic theory, integrating along required result well-known system. Since formula calculation mode analytical rotational symmetry present computationally efficient numerical About 150 years ago, Ernst Abbe demonstrated that if aberrations introduced manufacturing tolerances maladjustment components occur.1 considered grating certain period as measuring found minimum resolved , where wavelength light NA aperture objective lens. different approach followed Lord Rayleigh, assumes source or scatterer plane calculates intensity distribution plane, so-called (PSF).2 systems, PSF Airy disc context, limit Rayleigh criterion separation two sources microscope. corresponds apertures domain leading expression (MTF) system.3, 4 objective's uniformly illuminated obeying sine condition.5 contrast paraxial approximation it not small apertures. More recently, characteristics analysed (3D TFs). These go back work McCutchen, who adapted Ewald sphere concept context pattern single source.6 these TFs transmission- microscopes.7-13) follows Figure 1. 1(A) shows wave vector incident illuminating located -plane under incidence angle . If object's flat, reflected characterized reflection. consists will scattered, resulting scattered described numerous waves, each scattering given full there waves angles up shown 1(B). set vectors Note simplicity schematics 1 cross sections related - -plane, respectively. Further note situation -components have same sign.10, 12, 13 maximum contributing again According 1(B), all end spherical caps. construction McCutchen sphere,6, 7, 14 connected -space then represents directly TF correlating caps -value, correlation integral zero, need considered. projection-slice-theorem,15 MTF depending only obtained vertical axis, axis Several approaches obtain spatially illumination published. should noted confocal (CM) known coherent due CM. CM closely microscope.16 Because process, refers amplitude (electric field), whereas multiplied domain.16 Therefore, Sheppard et al. 1990s12, 17, 18 CMs apply microscopes, they instead amplitude. al.17 expression, holds unity value over nearly complete range, differs zero They show Hopkins.3 Another al.12 considers high autoconvolution infinitesimal thickness proportional range their volume overlap. method elliptic integrals TFs, differ MTF. based Kirchhoff optics, also takes into account, TF, valid rough imaging.18 much higher values compared imaging. focus variation identical caps.19 totally those group mode.12, Nevertheless, utilizes discrimination analysis stack. principle requires some recorded images thus does investigation perfect mirror. crucial since process surface.20 Mirau-type obscuration reference mirror neglected. Thus, builds too. on-axis belonging specular reflection, transverse zero. Hence, square root intensities domain, finally microscope.20 As concequence, constant theoretical was recently used coherence interferometry (CSI) correct characteristics,21, 22 analyse defocus effects23 compensate lens aberration.24 Furthermore, modelling representation explains frequency, at CSI signals analysed, affects achieved measurements.20, 25, 26 noticed cannot observed bandwidth limitation consequence fact intensities, determined phase modulated signals, appear interferometric such holographic microscopy. following section, introduce differential calculate We use term ‘'incoherent’ here, assume Kohler superimpose electric fields. Although major intention NA, reducing partially because number process. projection slice theorem15 agrees aplanatic condition. turns out quite useful working well profilometry. coordinates agree al.18 represent component All three normalized wavenumber light, radius. assumption side view limiting representing cut-off portion radius 2, cut plane.7 Our overlapping area intersecting apertures.4 distance symmetry. general explained Figures 2(A) 3(A). Both, form cap. extension inverted other propagation direction (see 1). shift Contributions expected long solely depends 3(A) demonstrates -axis must intersection laterally shifted apertures, top 3(B). case, contributions, Area integrate Under assumption, lines circles tilted 2(B), ellipse contrast, attributed 3A) 3(C) subdivided four parts equal length. After integration, sum subareas difference between 5 examples monochromatic Equation (16) 0.5 0.9, 6 7 contribution (13) (14) including (15) (13). negative contributions positive sign. comparison exhibits decreasing expressed more 10(C) 11(C) dependence PSFs assuming comparison, additionally 1D PSFs, absolute reveal -function approximation, low Wilson.16 4(C) -dependent half width. 5(C), ramp shape thus, exact -function. broadened computation runs whole generally smaller than It noted, however, value, suffers loss information. real part displayed. Here, information fringes Su al.14 NA-effect appears wider fringe spacing NA.5 care taken, independent other. example, flat mirror-like uniform reflectance, concentrated -like longer ). -shaped PSF, plane.16 previous sections, caps, axis. projected cross-sectional area. amount agreement something like gold standard above generalization distribution. becomes relevant out-of-focus examined. incoherently measurement capabilities PSF. CM, blocked remaining squared surfaces better resolution. With microscopy, example CSI, frequencies provided proper appropriate setting parameters signal processing algorithms. Open access funding enabled organized Projekt DEAL.