Focus variation microscope: linear theory and surface tilt sensitivity

Abstract

Additional Information:• This paper was accepted for publication in the journal Ap- shown that surface roughness measurements made using a focus variation microscope (FVM) are influenced by surface tilt. The effect appears to be most significant when the surface has microscale roughness (Ra ≈ 50 nm) that is sufficient to provide a diffusely scattered signal that is comparable in magnitude to the specular component. This paper explores, from first principles, image formation using the focus variation method. With the assumption of incoherent scattering, it is shown that the process is linear and the 3D point spread characteristics and transfer characteristics of the instrument are well defined. It is argued that, for the case of micro-scale roughness and through the objective illumination, the assumption of incoherence cannot be justified and more rigorous analysis is required. Using a foil model of surface scattering the images that are recorded by a FVM have been calculated. It is shown that for the case of through the objective illumination at small tilt angles, the signal quality is degraded in a systematic manner. This is attributed to the mixing of specular and diffusely reflected components and leads to an asymmetry in the k-space representation of the output signals. It is shown that by using extra-aperture illumination or at tilt angles greater than the acceptance angle of aperture (such that the specular component is lost), the incoherent assumption can be justified once again. The work highlights the importance of using ring-light illumination and/or polarizing optics, which are often available as options on commercial instruments, as a means to mitigate or prevent these effects.

Three‐dimensional transfer function of optical microscopes in reflection mode

“…More recently, a 3D TF of a focus variation microscope in reflection mode has been derived by correlating two identical spherical caps. 19 This result is totally different from those obtained by the group of Sheppard et al for microscopes operated in reflection mode. 12,17,18 Nevertheless, focus variation microscopy utilizes the depth discrimination of single point scatterers by spatially resolved analysis of a through focus image stack.…”

“…Due to the rotational symmetry, the transverse coordinates and can be substituted by the single coordinate = √ 2 + 2 , representing the radial distance. Hence, ℎ( , ) equals In contrast to Equation (19), where the phase information is considered due to the Fourier transform, Equation (20) integrates the intensity values on the spatial frequency axis . Thus, the phase information is lost and an axial focus shift Δ will result in a phase shift of ( , ), which does not affect ( ).…”

Topography of selectively laser melted surfaces: A comparison of different measurement methods

“…The sample was measured three times consecutively with each instrument, with no refixturing between measurements, constant set-up parameters, and controlled temperature environments (CM: ±0.1 °C, CSI and FV: ±1 °C, XCT: ±0.2 °C). The following set-ups were adopted:  CM: 20× objective lens (NA 0.6, field of view-FoV (0.64 × 0.64) mm); lateral resolution (pixel width) 0.625 µm; lateral resolution (optical limit): 0.12 µm; measured area: (2.9 × 2.9) mm, stitched;  CSI: 20× objective at 1× zoom (NA 0.4, FoV (0.42 × 0.42) mm) lateral resolution (pixel width) 0.409 µm; lateral resolution (optical limit): 0.71 µm; measured area: (3.4 × 3.4) mm, stitched;  FV: 20× objective lens (NA 0.4, FoV (0.81 × 0.81) mm) lateral resolution (pixel width) 0.439 µm; lateral resolution (optical limit): 0.88 µm; ring light illumination [17]; measured area: (3.7 × 3.7), stitched;  XCT: geometric magnification of 42.6× leading to a voxel size of 4.69 μm, 3142 X-ray projections (formed from averaging of two exposures per projection, each lasting 2 s); tube voltage 150 kV; current 30 μA; 1 mm copper pre-filter. Data were reconstructed in the manufacturer's proprietary software, using no beam hardening correction.…”

Sub-millimeter depth-resolved digital holography

“…To obtain a theoretical description, we neglect the details of surface scattering processes, following the assumption of Nikolaev et al (2016) in their linear theory approach to FVM [7], and consider the object surface to be a perfect plane with a reflection coefficient that has a random distribution. The object will be considered to have a reflection coefficient that has a "white" frequency distribution, i.e., all spatial frequencies have equal power.…”

“…FVM on the other hand is not an interferometric technique. It uses axial focus scanning and exploits the limited depth of focus of the objective lens to extract topology information from focus variation quantified with a focus metric, provided the surface is optically rough [7].…”