# Nano- and Micropatterning on Optical Fibers by Bottom-Up Approach: The Importance of Being Ordered
## 4 Methods for Order Quantification
### Order Quantification by Fourier Transform  
When dealing with 2D ordered structures, and specifically with hexagonally arranged patterns as those obtainable by self-assembly, Fourier transform can provide both qualitative and quantitative information. By providing information about the spatial frequencies  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Main Methods** & **Source** & **Information Provided** & **Main Fields of Application** \\ \hline
2D Fourier transform (2 DFT) [94; 95; 96; 97; 98] & 2D image acquired by optical microscopy, SEM, TEM, AFM & Symmetry, periodicity and orientation. & Block-copolymer self-assembly, breath figures, colloidal assembly \\ \hline Pair correlation function (PCF) [9  
of an image, in fact, the 2D Fourier transform (2 DFT) is a useful tool for extracting from the original image evidence about the symmetry, periodicity and orientation of specific patterns. For this reason, it has been extensively used for analyzing the order of colloidal assemblies [94; 95; 96; 97; 98]. For a triangular lattice with a hexagonal spot arrangement, whether they be particles or pores, the first order peak of the 2 DFT consists of six distinct spots on the edges of a hexagon (Figure 6(a-i)). When the arrangement is not perfectly hexagonal and defects are present, the 2 DFT pattern turns into a ring or disc-shaped form (Figure 6(a-ii)) [96]. The presence of a narrow ring indicates a preserved short-distance periodicity of the network and slightly disturbed long-range order. As the ring gets larger, the periodicity is more and more disturbed. When the 2 DFT pattern has a disc shape, the degree of order of the structure is very low. Besides such a qualitative assessment of the hexagonal order, 2 DFT can also provide a quantification of the order level. In highly ordered 2D structures, in fact, the reciprocal of the distance of the six distinct spots from the center of the 2 DFT provides the periodicity of the structure in the real image. The profile analysis along the center and two opposite distinct  
Figure 6: (**a**) Optical microscopy pictures of diblock copolymer honeycomb films obtained by breath figures with a blowup in the inset and corresponding 2D Fourier transforms. Reproduced from Ref. [96] with permission from the 2010 Royal Society of Chemistry. (**b**) FE-SEM top-view of anodic porous titania together with its 2D Fourier transform in the inset (i); profile analysis (iii) of the 2D Fourier transform intensity of a similar sample along the marked solid line in (ii). Reproduced from Ref. [97] with permission from 2010 Elsevier Ltd. (**c**) AFM micrographs of diblock copolymer reverse micelles of PS-b-P2 VP assembled at different spin-coating speed (i) showing highly ordered organization (top) and very low density (bottom); graphical representation of pair correlation function of the corresponding AFM images (ii) and plotted values of pair correlation function (iii) of the highly ordered system (red), disordered system (green) and calculated hexagonal lattice (black). Reproduced from Ref. [99]. (**d**) Voronoi tessellation performed on SEM image of a block-copolymer thin film prepared by spin-coating with legend highlighting the correspondences between colors and number of the sides of the polygons and evaluation of the corresponding degree of hexagonal order (i); SEM images and corresponding Voronoi polygons of breath figure patterns of CAB prepared by different methods: pores with five- and seven-nearest neighbors are colored red and green, respectively (ii). Reproduced from Ref. [100] with permission from the 2014 Royal Society of Chemistry and from Ref. [101] with permission from the 2004 American Chemical Society, respectively.