The influence of the numerical aperture of a beam probing an object on the determination of the thickness of a layered object in confocal microscopy

Depth-dependent scaling of axial distances in light microscopy

“…In fact, both re-scaling factors from Lyakin and Stallinga give better agreement with the measured data [15,16] and in practical terms seems to provide a better value for 𝜁 crit . At extreme depths of 𝑧 𝐴 = 10 mm the depth-dependent re-scaling factor approaches the 𝑛 2 /𝑛 1 = 1.34 limit set by Carlsson et al (not shown in Figure 7) [13].…”

“…The various re-scaling theories found in the literature can be understood as a result of different (overt or covert) assumptions on which maximum constructive interference contribution is leading. For example, Lyakin et al used an analysis similar to ours, but explicitly sets the critical point 𝜃 * such that 𝑘 ′ 𝑧 (𝜃 * ) = 1 2 × (𝑘 ′ 𝑧 (0) + 𝑘 ′ 𝑧 (𝜃 max )), and then from Equation 8 follows [16]:…”

“…For example, Lyakin et al . used an analysis similar to ours, but explicitly sets the critical point θ ∗ such that, and then from Equation 8 follows [16]: The same value was proposed earlier by Stallinga while posing different criteria of minimal variation of the phase function: θ ∗ : min ⟨ (Φ − Φ( θ ∗ )) 2 ⟩ [15]. For Diel et al .…”

Three-dimensional confocal reflectance microscopy for surface metrology

“…The optical sectioning capability of confocal microscopy enables not only 3D surface measurement, but also tomographic depth profiling, multilayer imaging, and defect inspection of optically transparent, semi-transparent, or weakly scattering materials [143,[145][146][147][148]. When inspecting glass material using conventional optical microscopy, surface and subsurface data will be combined, making the detection of subsurface defects difficult [143].…”

“…When measuring the subsurface area of transparent materials or determining the thickness of a layered object, the direct result of thickness measurement using a confocal microscope has a nonlinear dependence on various parameters, including refractive index and geometrical thickness of a sample and the numerical aperture of an objective lens. Thus, correction methods for precise measurement of material thickness have been proposed by theoretical analysis and experimental verification [146,147,149]. A calibration curve can be used to correct errors induced by different refractive indexes and material curvature [145].…”