# Nano- and Micropatterning on Optical Fibers by Bottom-Up Approach: The Importance of Being Ordered
## 4 Methods for Order Quantification
### Laser Diffraction  
Sometimes, a fast and qualitative method for order evaluation is necessary. This is especially desirable during the process optimization attempts, to have a fast response of the modification made to the process, as well as during the preparation of a series of samples, to have quick quality control. In this view, laser diffraction can help to recognize immediately whether the obtained pattern is completely disordered or it possesses some periodicity.  
The technique is suitable for studying 2D distributions of objects (particles or pores) on a transparent substrate, and consists of illuminating the sample at normal incidence with a laser beam, so that the diffracted light is projected to a screen on the other side of the sample. As shown in Figure 6(e-i), when the sample is hit by the laser beam a ring is generated on the screen, which becomes six symmetrical spots within a diffuse ring when the irradiated portion of the sample is a perfect hexagonal array, and the laser spot is smaller than the illuminated crystallite [102]. These images are called Debye rings and are 2D optical analogues of the common X-ray diffraction from 3D samples of crystalline powders (Debye-Scherrer method).  
Laser diffraction is very useful, for example, to evaluate the order of 2 DPCs [103]. In fact, by monitoring the Debye ring diffraction we can quantitatively determine the 2 DPC ordering without perturbing the sample and without using a microscope. A laser pointer is normally sufficient to analyzing this kind of samples, obtaining a quick comeback on what is going on. As 2 DPCs are often obtained by assembling microspheres at the air/water interface, it is even possible to analyze the Debye ring directly on the monolayer colloidal crystal floating on the water surface. This kind of monitoring can be particularly advantageous by the moment that, once the 2 DPC is transferred from water to a substrate, the self-assembly process is frozen and it is practically impossible to modify it.  
Besides the qualitative information about periodicity and crystallite size, laser diffraction is able to provide also information about the spacing. As depicted in Figure 6(e-ii), the diameter of the Debye ring projected onto the screen in fact depends on both the 2 DPC particle spacing, \(a\) and the distance, \(h\) between the 2 DPC and the screen, according to the following formulas:  
\[\theta =tan^{-1}\frac{d}{2h}\] \[a =\frac{2\lambda}{\sqrt{3}\text{sin}\theta}\]  
Hence, by measuring the ring diameter and its distance from the sample (knowing the wavelength of the laser, \(\lambda\)), the particle spacing can be calculated. When the parameters of the Debye ring are used to calculate the spacing of a 2 DPC on the water surface, one should take into account that the diffraction of the 2 DPC is shifted due to refraction of the light by the air-water interfaces. Consequently, the Debye ring diameter of a 2 DPC on the water surface will be smaller than that from an identical 2 DPC on a glass slide. Anyway, this deviation is related to the thickness of the water layer and can be calculated [103].