# Topological and Fractal Analysis of Nanostructured Metal-Dielectric Films
## 3 Results and Discussion
Figure 2:5 Variation in the metal phase filling (packing) factor with the mass thickness of Ag films on a glass substrate. However, according to (2), the shape of the probe p(x) can affect the surface correlation function's anisotropy. To test this, the shape profiles of the probe tip were determined by scanning a 17 nm high rectangular grating. Figure 12 shows the profiles obtained when scanning at different angles, with a difference of 90 degrees. The difference between the profiles cannot affect the anisotropy of the correlation function, as it is not essential. Figure 2:6 The two-dimensional autocorrelation of the surface morphology of the studied Au films at 1.0 nm (**a**), 2.0 nm (**b**), and 3.0 nm (**c**) mass thicknesses. Additionally, vertical cross-sections along the x-direction (black line), with corresponding Gaussian fits (red line) displayed, are included. Figure 2:7 The two-dimensional autocorrelation of the surface morphology of the studied Ag films at 1.1 nm (**a**), 3.6 nm (**b**), and 5.7 nm (**c**) mass thicknesses. Additionally, vertical cross-sections along the x-direction (black line), with corresponding Gaussian fits (blue line) displayed, are included. Consequently, the anisotropy of the correlation function leads to the conclusion of the anisotropic morphology of nanoparticles. The dependence of roughness (\(\delta\)) and correlation length (\(\sigma\)) on mass thickness (\(d_{m}\)) demonstrates similar trends for both Ag and Au films. An increase in \(d_{m}\) leads to an expansion of the correlation length paired with a reduction in roughness. Notably, the \(\sigma\) and \(\delta\) values for Au films are consistently 2 to 2.5 times smaller compared to those of Ag films. This disparity can be related to the intrinsically smaller dimension of the Au particles. The explanation for the decrease in roughness remains consistent, while the increase in correlation length with the increase in mass thickness arises from the growth and merger of metal nanoparticles. Evaluating roughness and conducting correlation analysis enable the assessment of surface statistical characteristics. However, understanding the size of the structural elements is also crucial. On the other hand, information about the probe's shape (Figure 12) makes it possible to use the deconvolution method [51; 52; 53] to partially eliminate the effect that occurs due to the probe's finite size (Figure 4). This method, of course, does not make it possible to completely eliminate the influence of the probe on the shape of the AFM surface, especially in the case of Figure 5a. However, the deconvolution method effectively reduces the error in determining the horizontal dimensions of objects. A change in the appearance of the surface is presented for demonstration: the surface of an ideal sphere if the probe size were infinitely small (Figure 13a), the sphere after the scanning process as in Figure 4 (Figure 13b), and the surface after using deconvolution (Figure 13c). The object size still has an error after using horizontal deconvolution, but it is much closer to the values obtained by other, often more accurate methods [56]. Therefore, the determination of particle sizes according to the following algorithms was carried out after using the de-convolution method for the corresponding AFM images. Figure 2:8 Probe tip profiles obtained by scanning 17 nm high rectangular grating at different angles. Figure 2:9 The model of an ideal AFM surface of a sphere (**a**), the surface of a sphere modeled according to the algorithm Figure 4 (**b**), and the appearance of the surface after using deconvolution (**c**). The data underwent several processing stages to identify regions in the AFM image corresponding to metal particles. The primary objective was to delineate the contours of these particles. Initially, the image was segmented using the watershed method [57], followed by automatic image thresholding using [PERSON]'s method [58], which was performed on the resulting segments. Subsequently, contours were detected on the binary images obtained.