## 4 Discussion  
The simulated transmission spectra of a close-packed monolayer of PS beads on a glass substrate in comparison with the experimental data are shown in Figure 5. The simulated curves have dips corresponding to the resonance frequencies and lay under the experimental curves. As mentioned earlier, the samples are not ideal 2D photonic crystals; therefore, they have smaller fill factors which make the measured transmittance higher. Moreover, due to imperfection in the experimental samples, most of the dips are also smeared out in the measured transmission curves. Only the first dip (\(Z=0.7\)) is the most noticeable. Figure 4: Simulated transmission spectra of a close-packed monolayer of 701 nm PS beads on a dielectric substrate (**a**) fixing \(\varepsilon_{\text{sphere}}=2.25\) while the dielectric constant of the substrate varies in the range of 1.5 to 3 and (**b**) keeping \(\varepsilon_{\text{sub}}=2.5\) while the dielectric constant of the dielectric spheres changes from 2 to 4. Figure 3: Measured transmission spectra of (**a**) 574 nm PS beads suspended in solution, as a sub-monolayer, as a multilayer, and as a close-packed monolayer (dotted blue, dashed brown, dash–dotted red, and solid black lines, respectively) and of (**b**) self-assembled films of PS beads with diameters of 474, 574, 684, and 812 nm. The transmission dips in the spectra shift to longer wavelengths as the size of the beads increases. The spectral position of this dip is used to estimate the diameter of the PS sub-microspheres by a simple formula:  
\[d=\frac{2\times Z\times\lambda_{1^{\mathrm{g}}}\ \mathrm{dip}}{\sqrt{3}} \tag{1}\]  
where \(d\) and \(\lambda_{1^{\mathrm{g}}}\ \mathrm{dip}\) (corresponding to \(Z\) = 0.7) are the diameter of the PS beads and the spectral position of the first dip in the transmission spectrum, respectively. In Figure 6, the diameters of the PS beads are plotted against the spectral positions of the first dips appearing in the transmission spectra of close-packed monolayers of the PS beads. These data points were well fitted by a simple linear function with a slope of 0.799. The \(Z\) parameter calculated from the slope of the plot was \(Z\) = 0.69, which agrees very well with the theoretical value \(Z\) = 0.7. This result, first of all, confirms the validity of the physical model. Hence, it allows us to use Formula (1) for the size estimation of PS sub-microspheres. This technique works well with PS beads of diameter as small as 200 nm. It further extends the capability of the optical method for the size determination of PS beads to the sub-micrometer range [32]. Figure 5: Transmittance spectra of a close-packed monolayer of PS beads; solid lines indicate experimental results and dotted lines indicate simulated results. The parameters for the simulations were \(\varepsilon_{\mathrm{sphere}}=2.25\) (\(n_{\mathrm{sphere}}=1.5\)) and \(\varepsilon_{\mathrm{sub}}=2.5\) (\(n_{\mathrm{sphere}}=1.58\)), diameters of the PS microspheres: 507, 525, 701, and 820 nm. Figure 6: The diameters of the PS beads plotted against the spectral positions of the first dips. The \(Z\) parameter retrieved from the slope of the plot was \(Z\) = 0.69, which agrees well with the theoretical value \(Z\) = 0.7 used for finite-difference time-domain (FDTD) simulation. Unfortunately, when the size of the PS beads reaches the nanometer region, the spectral position of the first dip in the transmission spectrum falls in the deep UV range and cannot be measured by the detectors of common commercial UV-vis spectrometers. Glass substrates can be used to determine the sizes of PS beads with diameters in the upper sub-micrometer range. However, quartz substrates are necessary for smaller PS bead size determination. The dielectric constant of the substrate can affect the spectra in general and the spectral position of the first dip specifically. While the latter directly defines the estimated size of the PS beads, the simulation showed that the first peak (\(Z\) = 0.7) is insensitive to the refractive index of the substrate (Figure 4a).