# Design of Nano-Porous Multilayer Antireflective Coatings

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{ave}which is defined as

_{min}and λ

_{max}represent the spectral range of interest, and R

_{TE}(λ) and R

_{TM}(λ) are the TE (Transverse Electric) and TM (Transverse Magnetic) mode spectral reflectivities. After evaluating a series of multi-layer systems and thus creating a population of fitness values, the systems were selected for breeding via a proportionate selection process, where the normalized fitness value of each system would have a proportionately finite probability of being selected. The fitter individual system has a high chance of pairing with another fit individual system. The parent systems are replicated as their children, where crossover between the parent systems’ chromosomes has a high chance of occurring. Finally, random mutations of 1% are introduced for diversity and to decrease the chance of missing a global optimal solution. After this application of selection, crossover and mutation, a new generation of optical parameters are fed into the Abeles matrix to generate another population of fitness values. It is found that convergence to the highest fit is achieved in 23 generations while the algorithm was run for a total of 100 generations.

_{j}is the admittance at the j-th layer. The point (1,0) on the complex plane denotes zero reflectance, and the greater the distance the loci end-point is from (1,0) the greater the reflectance will be.

_{2}or 10 nm SnO

_{2}colloidal particle solutions, spin coat the mixture on a cleaned substrate and anneal it at high temperature to pyrolyse the polystyrene. The resulting layer is one of a porous network of SiO

_{2}or SnO

_{2}nanoparticles. More details on the fabrication method are provided in Appendix B.

## 3. Results

#### 3.1. Optimized Anti-Reflection Multi-Layer

#### 3.2. Admittance Loci Visualization

## 4. Discussion

_{2}nanoparticle solutions, the porosity induced by the removal of polystyrene is significant as seen by the steep drop in the refractive index. With increasing polystyrene, however, it becomes more difficult to achieve a very low refractive index as the change in index with further increase in porosity is gradual. The limit at which the internal framework of nanoparticles (hosting the induced porosity) can mechanically support itself becomes apparent as the index actually increases slightly at the largest polystyrene concentration. In Figure 5 inset, we determined the particle ratio of the SiO

_{2}nanoparticles to the polystyrene nanoparticles from the density of the colloidal solution (1.22 g/mL, 30 wt % in H

_{2}O), density of the PS solution (0.025 g/mL, 2.5 wt % in H

_{2}O) and the dilution factor to arrive at ~10

^{13}silica particles and ~10

^{9}PS particles in the prepared solution. In sum, it is relatively easy to induce an index drop of more than 0.3 for a silica nanoparticle film, but achieving an index of less than 1.1, which requires a packing fraction of only 10%, is difficult. However, it is conceivable that by improving the chemical adhesion between nanoparticles that a highly porous and mechanically robust scaffolding could be fabricated. For tin dioxide nanoparticle films, the trend is largely a linear decrease from an index of 3.5, and hence a combination of tin-dioxide and silica nanoparticle layered stack may be suitable for high index substrates such as silicon or ITO. From SEM cross sectional images, the PS generated voids tend to conglomerate into bigger and more elliptical voids within SnO

_{2}particle films, whereas SiO

_{2}particle films show an open network of SiO

_{2}agglomerations.

_{2}(diluted 1:15):PS = 1:5. It is clear that some variability in the index can be achieved by the spin coating speed. It is possible that with increasing RPM (Rotations Per Minute) the larger PS particles are more tightly compressed with the host silica film, and the rate of removal of PS versus silica solution is less when the rotational speed is high. The tighter compression may lead to a more interlinked porous network throughout the film. The final ratio of PS to silica after spin coating may also be greater. One characterization method for determining the characteristics of a porous film is the environmentally controlled ellipsometry, where the introduction of water moisture into the film and its removal can be studied through its change in the effective refractive index. Figure 8 shows the hysteresis curves [14] of the change in porosity as a function of increasing polystyrene (PS). If the film exhibits a large surface area due to an extensive porous network, its moisture content will be removed more rapidly as compared to a film with isolated porosity (for example, in a close-packed film of nano-spheres).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

TE Polarization | TM Polarization |
---|---|

Matrix of a single layer M, with thickness d, refractive index n at an incident angle $\psi $ | |

$\left(\begin{array}{cc}\mathrm{cos}\left({k}_{0}nd\mathrm{cos}\psi \right)& -\frac{i}{n\mathrm{cos}\psi}\mathrm{sin}\left({k}_{0}nd\mathrm{cos}\psi \right)\\ -in\mathrm{cos}\psi \mathrm{sin}\left({k}_{0}nd\mathrm{cos}\psi \right)& \mathrm{cos}\left({k}_{0}nd\mathrm{cos}\psi \right)\end{array}\right)$ | $\left(\begin{array}{cc}\mathrm{cos}\left({k}_{0}nd\mathrm{cos}\psi \right)& -\frac{in}{\mathrm{cos}\psi}\mathrm{sin}\left({k}_{0}nd\mathrm{cos}\psi \right)\\ -i\frac{\mathrm{cos}\psi}{n}\mathrm{sin}\left({k}_{0}nd\mathrm{cos}\psi \right)& \mathrm{cos}\left({k}_{0}nd\mathrm{cos}\psi \right)\end{array}\right)$ |

Matrics of a stack, M for the product sum of the matrics of _{stack}j sub-layers of d thickness | |

${M}_{stack}=\left(\begin{array}{cc}{m}_{11}& {m}_{12}\\ {m}_{21}& {m}_{22}\end{array}\right)={\displaystyle \prod _{j=1}^{N}{M}_{j}\left({d}_{j}\right)}$ | |

Expressions for field transmission and reflection coefficients | |

$t=\frac{2{n}_{i}\mathrm{cos}\phi}{\left({m}_{11}+{m}_{12}{n}_{s}\mathrm{cos}{\phi}_{s}\right){n}_{1}\mathrm{cos}\phi +{m}_{21}+{m}_{22}{n}_{s}\mathrm{cos}{\phi}_{s}}$ $r=\frac{\left({m}_{11}+{m}_{12}{n}_{s}\mathrm{cos}{\phi}_{s}\right){n}_{i}\mathrm{cos}\phi -\left({m}_{21}+{m}_{22}{n}_{s}\mathrm{cos}{\phi}_{s}\right)}{\left({m}_{11}+{m}_{12}{n}_{s}\mathrm{cos}{\phi}_{s}\right){n}_{1}\mathrm{cos}\phi +{m}_{21}+{m}_{22}{n}_{s}\mathrm{cos}{\phi}_{s}}$ | $t=\frac{2\frac{\mathrm{cos}\phi}{{n}_{1}}}{\left({m}_{11}+{m}_{12}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}\right)\frac{\mathrm{cos}\phi}{{n}_{1}}+{m}_{21}+{m}_{22}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}}$ $r=\frac{\left({m}_{11}+{m}_{12}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}\right)\frac{\mathrm{cos}\phi}{{n}_{1}}-\left({m}_{11}+{m}_{12}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}\right)}{\left({m}_{11}+{m}_{12}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}\right)\frac{\mathrm{cos}\phi}{{n}_{1}}+{m}_{21}+{m}_{22}\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}}$ |

Intensity coefficients for a film on a semi-infinite substrate | |

$T=\frac{\mathrm{Re}\left({n}_{s}\mathrm{cos}{\phi}_{s}\right)}{\mathrm{Re}\left({n}_{1}\mathrm{cos}{\phi}_{1}\right)}{\left|t\right|}^{2}$; $R={\left|r\right|}^{2}$ | $T=\frac{\mathrm{Re}\left(\frac{\mathrm{cos}{\phi}_{s}}{{n}_{s}}\right)}{\mathrm{Re}\left(\frac{\mathrm{cos}{\phi}_{1}}{{n}_{1}}\right)}{\left|t\right|}^{2};R={\left|r\right|}^{2}$ |

**Figure A1.**Admittance loci for the normal incidence optimized and angular incidence optimized ARC designs for TM mode. The TM transmittance spectra for these designs on a semi-infinite glass substrate are also shown. The admittance loci representations are shown for increasing angles of incidences from 0° to 80°.

## Appendix B

_{2}colloidal solution is first diluted with H

_{2}O: For example, 1 g of SiO

_{2}solution is diluted with 1 g of deionized H

_{2}O to prepare SiO

_{2}(1:1 dilution). In the next step, 0.015 g of SiO

_{2}solution is extracted from the diluted solution and mixed with varying mass of polystyrene (PS) colloidal solution. The mass ratio indicated in the following data refers to the ratio of the mass of PS solution with the 0.015 g mass of diluted SiO

_{2}solution. As the volume of the solutions of PS/SiO

_{2}is too small to enable uniform coating of a 1 inch by 1 inch substrate, or for multiple samples, it is necessary to dilute the preceding solutions with 100 mg H

_{2}O to increase the overall volume. For SnO

_{2}, 0.015 g of SnO

_{2}solution is extracted and mixed with varying mass of PS solution. Hence the mass ratio indicated in the following data refers to the ratio of the mass of PS solution with the 0.015 g mass of SnO

_{2}solution which is not yet diluted. The PS/SnO

_{2}solutions are then diluted with varying mass of deionized H

_{2}O.

_{2}and SnO

_{2}with different polystyrene (PS) mass ratios were spin coated at an RPM of 3500 for 40 s, and sintering at 450 °C for 40 min. The refractive index (at 633 nm wavelength) and thickness of a single layer is obtained by spin coating on a silicon substrate and characterized via spectral ellipsometry. Regression analysis using a Cauchy law dispersion formula with a single Lorentz peak was conducted using WINELLI 2.

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**Figure 1.**Enhanced transmittance spectrum of a PMMA substrate coated with a single layer of silica nanoparticles in contrast to an uncoated PMMA substrate. The silica particles are FCC close packed. The calculated transmittance of a single layer with physical thickness of 150 nm and refractive index 1.33 shows a match with the experimental results.

**Figure 2.**Two designs generated by G.A optimiz for 0° angle incidence (

**A**), and 0–80° angle incidence (

**B**). The first design follows a conventional index profile, while the second design has a thick low index third layer and a higher index fourth layer.

**Figure 3.**The TE mode and TM mode reflectance profiles of both anti-reflection coating (ARC) designs. The angular incidence optimized design has significantly lower reflectance than the normal incidence optimized design at 70° and 80° incidence.

**Figure 4.**Admittance loci for the normal incidence optimized and angular incidence optimized ARC designs for TE mode. The TE transmittance spectra for these designs on a semi-infinite glass substrate are also shown. The black and orange lines represent angular and normal incidence optimized designs respectively. The admittance loci representations are shown for increasing angles of incidences from 0° to 70°.

**Figure 5.**The index trend of SiO

_{2}nanoparticle films with increasing polystyrene solution mass ratio. There are two distinct trends, a sharp decrease in index with minimal introduction of PS, and a gradual decrease in index with significant introduction of PS. Inset: the refractive indices as a function of the estimated PS–SiO

_{2}particle ratio. Reproduced from [13] with permisson (Copyright Royal Society of Chemistry 2014). The other inset shows the cross-sectional SEM image of the void filled silica particle film.

**Figure 6.**The linear index trend of SnO

_{2}nanoparticle films at increasing water solvent dilution with increasing polystyrene solution mass ratio. The inset shows a cross sectional SEM image of the void filled SnO

_{2}particle film. The ellipses indicate the conglomeration of the voids.

**Figure 8.**Porosimetry isotherms for 1:3 diluted SiO

_{2}nanoparticles with increasing polystyrene (PS) ratio.

**Table 1.**The thickness and refractive index as measured by ellipsometry as a function of spin rate in RPM. As the spin rate decreases, the thickness decreases and the refractive index increases.

Max Rotational Speed/RPM | Thickness/nm | Refractive Index |
---|---|---|

3500 | 47.9 | 1.093 |

1000 | 58.2 | 1.121 |

500 | 64.9 | 1.141 |

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Loh, J.Y.Y.; Kherani, N. Design of Nano-Porous Multilayer Antireflective Coatings. *Coatings* **2017**, *7*, 134.
https://doi.org/10.3390/coatings7090134

**AMA Style**

Loh JYY, Kherani N. Design of Nano-Porous Multilayer Antireflective Coatings. *Coatings*. 2017; 7(9):134.
https://doi.org/10.3390/coatings7090134

**Chicago/Turabian Style**

Loh, Joel Yi Yang, and Nazir Kherani. 2017. "Design of Nano-Porous Multilayer Antireflective Coatings" *Coatings* 7, no. 9: 134.
https://doi.org/10.3390/coatings7090134