# Nonlocal Effective Medium (NLEM) for Quantitative Modelling of Nanoroughness in Spectroscopic Reflectance

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology and Results

_{2}substrate with roughness modelled by pyramids of $H=32\mathrm{nm}$, along with three effective refractive indices obtained by using ${\tilde{n}}_{0}$, ${\tilde{n}}_{\infty}$ and (${\tilde{n}}_{0}+{\tilde{n}}_{\infty}$)/2. A remarkable agreement is achieved with the average index, which validates our approach of combining the two extreme cases.

_{2}), ${n}_{f}=2$ (e.g., Si

_{3}N

_{4}), ${n}_{f}=2.6$ (e.g., TiO

_{2}), and ${n}_{f}=3.42$ (e.g., Si). The aim of the NLEM model is to be used on-the-fly during the numerical fitting of SR data. Thus, the FDTD-extracted effective indices need to be properly parametrized by an analytical formula. To this end we adopt the MG model [9]:

_{2}/Si, SiN/Si, TiO

_{2}/Si, and Si/SiO

_{2}.

_{2}/Si system and three values of $H$ is shown in Figure 4a. A different value of $w$ produces the minimum error between the FDTD and NLEM reflectance for different $H$ values. Specifically, $w\approx 0$ (maximal weight for ${\tilde{n}}_{\infty}$) is needed for the larger $H$ (tall columnar-like roughness) and $w\to 1$ (maximal weight for ${\tilde{n}}_{0}$) is needed for the smaller $H$ (short wide-angle roughness). This finding aligns well with our intuition regarding shape-dependent behavior. Thus, it calls for a nonlocal approach, which we will develop by parametrization, using the basis of the two distinct topological representations (${\tilde{n}}_{0}$ and ${\tilde{n}}_{\infty}$) for the effective medium.

_{2}/Si case for $H=32\mathrm{nm}$. The ${R}_{NLEM}$ displays a remarkably accurate reproduction of ${R}_{FDTD}$. This NLEM accuracy also ensures a strong predictive capability. We demonstrate this in Figure 5b, where the ${R}_{FDTD}$ spectra from a series of FDTD run with different $H$ values are used as the “measured” signal, and optimization runs of various models (NLEM vs. other EMT models, all at $h=1\mathrm{nm}$ discretization) try to fit ${R}_{FDTD}$ by using $H$ as the free parameter. The relative error in the predicted $H$ value is shown in Figure 5b as a function of the real $H$ value used in the FDTD simulation. It is impressive that the NLEM error is close to zero for all $H$ values (i.e., accurate prediction of $H$), while it smoothly varies between overestimation and underestimation for the other EMTs, crossing at some H value the zero-error line. The latter means that there is always one specific height H for each EMT where it will make a correct prediction, but as evident from Figure 5b, this is accidental. This is clear proof that the effective medium of each slice does not depend on its geometrical features only, but on the overall slope it creates with its neighbors, i.e., it is nonlocal.

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Giles, C.L.; Wild, W.J. Fresnel reflection and transmission at a planar boundary from media of equal refractive indices. Appl. Phys. Lett.
**1982**, 40, 210–212. [Google Scholar] [CrossRef] - Skaar, J. Fresnel equations and the refractive index of active media. Phys. Rev. E
**2006**, 73, 026605. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ruíz-Pérez, J.J.; González-Leal, J.M.; Minkov, D.A.; Márquez, E. Method for determining the optical constants of thin dielectric films with variable thickness using only their shrunk reflection spectra. J. Phys. D Appl. Phys.
**2001**, 34, 2489–2496. [Google Scholar] [CrossRef] - Swanepoel, R. Determination of the thickness and optical constants of amorphous silicon. J. Phys. E Sci. Instrum.
**1983**, 16, 1214–1222. [Google Scholar] [CrossRef] - Djurišić, A.B.; Fritz, T.; Leo, K. Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements. Opt. Commun.
**1999**, 166, 35–42. [Google Scholar] [CrossRef] - Panagiotopoulos, N.T.; Patsalas, P.; Prouskas, C.; Dimitrakopulos, G.P.; Komninou, P.; Karakostas, T.; Tighe, A.P.; Lidorikis, E. Bare-Eye View at the Nanoscale: New Visual Interferometric Multi-Indicator (VIMI). ACS Appl. Mater. Interfaces
**2010**, 2, 3052–3058. [Google Scholar] [CrossRef] [PubMed] - Koukouvinos, G.; Petrou, P.; Misiakos, K.; Drygiannakis, D.; Raptis, I.; Stefanitsis, G.; Martini, S.; Nikita, D.; Goustouridis, D.; Moser, I.; et al. Simultaneous determination of CRP and D-dimer in human blood plasma samples with White Light Reflectance Spectroscopy. Biosens. Bioelectron.
**2016**, 84, 89–96. [Google Scholar] [CrossRef] [PubMed] - Aspnes, D.E.; Theeten, J.B.; Hottier, F. Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry. Phys. Rev. B
**1979**, 20, 3292–3302. [Google Scholar] [CrossRef] - Garnett, J.C.M. XII. Colours in metal glasses and in metallic films. Phil. Trans. R. Soc. Lond. A
**1904**, 203, 385–420. [Google Scholar] [CrossRef] - Bruggeman, D.A.G. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann. Phys.
**1935**, 416, 636–664. [Google Scholar] [CrossRef] - Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles, 1st ed.; Wiley: Hoboken, NJ, USA, 1998; ISBN 978-0-471-29340-8. [Google Scholar]
- Aspnes, D.E. Plasmonics and effective-medium theories. Thin Solid Film.
**2011**, 519, 2571–2574. [Google Scholar] [CrossRef] - Kunz, K.S.; Luebbers, R.J. The Finite Difference Time Domain Method for Electromagnetics, 1st ed.; CRC Press: Boca Raton, FL, USA, 2018; ISBN 978-0-203-73670-8. [Google Scholar]
- Bellas, D.V.; Toliopoulos, D.; Kalfagiannis, N.; Siozios, A.; Nikolaou, P.; Kelires, P.C.; Koutsogeorgis, D.C.; Patsalas, P.; Lidorikis, E. Simulating the opto-thermal processes involved in laser induced self-assembly of surface and sub-surface plasmonic nano-structuring. Thin Solid Films
**2017**, 630, 7–24. [Google Scholar] [CrossRef] - Fodor, B.; Kozma, P.; Burger, S.; Fried, M.; Petrik, P. Effective medium approximation of ellipsometric response from random surface roughness simulated by finite-element method. Thin Solid Films
**2016**, 617, 20–24. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Qiu, J.; Liu, L. Applicability of the effective medium approximation in the ellipsometry of randomly micro-rough solid surfaces. Opt. Express
**2018**, 26, 16560. [Google Scholar] [CrossRef] [PubMed] - Ohlídal, I.; Vohánka, J.; Mistrík, J.; Čermák, M.; Franta, D. Different theoretical approaches at optical characterization of randomly rough silicon surfaces covered with native oxide layers: Theoretical approaches at optical characterization of rough surfaces. Surf. Interface Anal.
**2018**, 50, 1230–1233. [Google Scholar] [CrossRef] - Ohlídal, I.; Vohánka, J.; Čermák, M.; Franta, D. Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory. Appl. Surf. Sci.
**2017**, 419, 942–956. [Google Scholar] [CrossRef] - Lidorikis, E.; Egusa, S.; Joannopoulos, J.D. Effective medium properties and photonic crystal superstructures of metallic nanoparticle arrays. J. Appl. Phys.
**2007**, 101, 054304. [Google Scholar] [CrossRef] - Bérenger, J.-P. Perfectly Matched Layer (PML) for Computational Electromagnetics. Synth. Lect. Comput. Electromagn.
**2007**, 2, 1–117. [Google Scholar] [CrossRef] [Green Version] - Gedney, S.D.; Zhao, B. An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML. IEEE Trans. Antennas Propagat.
**2010**, 58, 838–847. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic of the system under study, where roughness is modelled as a periodic array of linear square-based pyramids. (

**b**) Roughness is discretized in a series of slices. (

**c**) Obtaining an effective medium for each slice, roughness becomes just a multi-layer film easily solved by a transfer matrix method.

**Figure 2.**(

**a**) The FDTD computational cell for obtaining the exact effective index ${\tilde{n}}_{0}$ of a thin square disk suspended in air. (

**b**) The cell for obtaining the exact effective index ${\tilde{n}}_{\infty}$ of a semi-infinite square rod. (

**c**) The reflectance of an array of square-based pyramids of SiO

_{2}on top of a semi-infinite substrate of the same material (see inset). The FDTD result is compared to the effective index model of the multi-sliced system using the ${\tilde{n}}_{0}$, the ${\tilde{n}}_{\infty}$, and the average effective indices.

**Figure 3.**The parameters defining the MG depolarization factor p that best fits ${\tilde{n}}_{0}$, as a function of film index. The scaling parameters for the dashed curves are shown in Table 1.

**Figure 4.**(

**a**) The error function between FDTD and NLEM as a function of the weight parameter w for 3 different roughness heights. (

**b**) The linear scaling of the weight factor w with pyramid half-angle (see inset) for the 4 different film indices considered.

**Figure 5.**(

**a**) FDTD and NLEM reflectance for the SiO

_{2}/Si system with roughness height $H=32\text{}\mathrm{nm}$ and bulk film thickness of 2 μm, showing excellent agreement. (

**b**) Different effective medium models are tested for predicting the roughness height $H$, by fitting to corresponding FDTD results. The NLEM model is the only one offering zero-error prediction for all roughness height values.

**Figure 6.**Different effective medium models are tested for predicting the roughness shape $s$, by fitting to corresponding FDTD results (for H = 32 nm). The NLEM model is the only one offering error-free prediction for all roughness shape values.

**Figure 7.**Free fit results of the NLEM model assuming both height H and shape s as free parameters. In these threshold images, color-coded is the error function. The white space represents ($H$, $s$) points for which the error is higher than 5 × 10

^{−3}. The dashed lines point to the FDTD system they try to fit: (

**a**) $H=32\mathrm{nm}$, $s=0.5$, (

**b**) $H=12\mathrm{nm}$, $s=2.27$, (

**c**) $H=54\mathrm{nm}$, $s=0.28$. A continuous band of small errors traces through optically similar combinations of roughness height and shape. This is proved in (

**d**), where the overlayed error lines fall almost perfectly on each other.

**Figure 8.**Different effective medium models are tested for predicting (

**a**) the roughness height $H$ and (

**b**) the roughness shape $s$, by fitting to corresponding FDTD results.

**Figure 9.**FDTD and NLEM reflectance for a system of three material layers with refractive indices $n=3,1.7,3$ showing excellent agreement.

${\mathit{c}}_{0}$ | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | |
---|---|---|---|

Depolarization factor parametrization | |||

${p}_{0}$ | 5.50 | 1.822 | −0.2326 |

$\gamma $ | −18.5 | −1.062 | - |

$\delta $ | 0.57 | 0.039 | −0.0053 |

${p}_{\infty}$ | 1.25 | - | - |

Weighting factor parametrization | |||

a | −0.4762 | 0.091 | - |

b | 0.0166 | 0.00041 | - |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lampadariou, E.; Kaklamanis, K.; Goustouridis, D.; Raptis, I.; Lidorikis, E.
Nonlocal Effective Medium (NLEM) for Quantitative Modelling of Nanoroughness in Spectroscopic Reflectance. *Photonics* **2022**, *9*, 499.
https://doi.org/10.3390/photonics9070499

**AMA Style**

Lampadariou E, Kaklamanis K, Goustouridis D, Raptis I, Lidorikis E.
Nonlocal Effective Medium (NLEM) for Quantitative Modelling of Nanoroughness in Spectroscopic Reflectance. *Photonics*. 2022; 9(7):499.
https://doi.org/10.3390/photonics9070499

**Chicago/Turabian Style**

Lampadariou, Eleftheria, Konstantinos Kaklamanis, Dimitrios Goustouridis, Ioannis Raptis, and Elefterios Lidorikis.
2022. "Nonlocal Effective Medium (NLEM) for Quantitative Modelling of Nanoroughness in Spectroscopic Reflectance" *Photonics* 9, no. 7: 499.
https://doi.org/10.3390/photonics9070499