Page 1 of 32 EllipsoNet Deep -learning -enabled optical ellipsometry for complex thin films

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EllipsoNet: Deep-learning-enabled optical

ellipsometry for complex thin films

Ziyang Wang†, Yuxuan Cosmi Lin‡, 1, *, Kunyan Zhang₴, Wenjing Wu† ,%, Shengxi Huang†,*

†. Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005,

USA

‡. Department of Electrical Engineering and Computer Sciences, University of California,

Berkeley, CA 94720, USA

₴. Department of Electrical Engineering, The Pennsylvania State University, University Park, PA

16802, USA

%. Applied Physics Graduate Program, Smalley-Curl Institute, Rice University, Houston, TX

77005, USA

1. Z.W. and Y.C.L. should be considered joint first author

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ABSTRACT

Optical spectroscopy is indispensable for research and development in nanoscience and

nanotechnology, microelectronics, energy, and advanced manufacturing. Advanced optical

spectroscopy tools often require both specifically designed high-end instrumentation and intricate

data analysis techniques. Beyond the common analytical tools, deep learning methods are well

suited for interpreting high-dimensional and complicated spectroscopy data. They offer great

opportunities to extract subtle and deep information about optical properties of materials with

simpler optical setups, which would otherwise require sophisticated instrumentation. In this work,

we propose a computational ellipsometry approach based on a conventional tabletop optical

microscope and a deep learning model called EllipsoNet. Without any prior knowledge about the

multilayer substrates, EllipsoNet can predict the complex refractive indices of thin films on top of

these nontrivial substrates from experimentally measured optical reflectance spectra with high

accuracies. This task was not feasible previously with traditional reflectometry or ellipsometry

methods. Fundamental physical principles, such as the Kramers-Kronig relations, are

spontaneously learned by the model without any further training. This approach enables in-

operando optical characterization of functional materials within complex photonic structures or

optoelectronic devices.

KEYWORDS: Convolutional neural network, refractive index, ellipsometry, Kramers-Kronig

relation, optical thin films

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INTRODUCTION

Complex refractive indices are among the most fundamental properties of materials. They act as

optical “fingerprints” of materials and contain rich information about light-matter interaction, such

as optical interband/intraband transitions, quantum confined state transitions, phonon polaritons,

and exciton polaritons. They are also essential material properties in designing photonic and

optoelectronic devices.[1–6] Ellipsometry is a widely used technique to measure the refractive index

of thin films.[7–9] It first measures the changes in polarization in terms of the amplitude ratio Ψ and

phase difference Δ (Figure S1). The measured Ψ and Δ are related to the optical reflectance ratio

between p and s polorizations, 𝑟𝑝

𝑟𝑠=tan(𝛹)𝑒𝑖𝛥, which is a function of the thickness and complex

refractive index. Ψ and Δ will be fitted by a model that describes the multilayer sample, where the

refractive index of the target layer consists of multiple oscillators. Therefore, the refractive indices

of the target layer can be obtained (Figure S2). Despite its wide use, ellipsometry technique still

faces three main challenges. First, selecting a proper model requires intervention by human experts

due to intensive parameter tuning, including the selection of type and number of oscillators .[10]

Further, the substrate structures have to be simple and known. However, in many practical

scenarios, the substrate structures are inevitably more complicated with partially unknown

information. Finally, the optical setup for ellipsometry demands a large incident angle which

requires special instrumentation and is not directly implementable on a common reflection optical

microscope setup. Previous studies have made efforts to simplify the process of parameter

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selections but were still limited by the requirements of simple and well-defined substrate structures,

and unconventional optical setups.[11–13]

In this work, we developed a deep learning method that extracts complex refractive indices of

thin films placed on unknown and arbitrary multilayer substrates from optical reflectance spectra

measured on an optical microscope. Unlike traditional reflectometry or ellipsometry, our approach

does not require extensive fitting and is able to tackle all the three challenges mentioned above.

First, our framework obtains refractive indices of thin films without solving inverse functions or

tuning parameters. Second, our method can be used in complex substrate structures without

knowing the structure parameters or materials of substrates. Moreover, our model only takes

reflectance spectra as the inputs, which can be easily integrated with optical microscopes (Figure

1a). Specifically, we designed an encoder-decoder convolutional neural network named

EllipsoNet that takes reflectance spectra as the inputs and predicts the corresponding refractive

indices (Figure 1a, b, c). To train the neural network, we generated a dataset using over 400

materials with density functional theory (DFT) simulated refreactive indeices and 450,000

multilayer stack structures. With an independently generated dataset of testing materials and

multilayer stack structures, the predictions made by EllipsoNet reach an overall median Pearson’s

correlation coefficient of 0.88. We further validated our method using experimentally measured

reflectance of real 2D materials on different substates. We also showed that a more complex

version of EllipsoNet, called C-EllipsoNet, can deal with even more complicated substrate

structures. Finally, we found that both EllipsoNet and C-EllipsoNet spontaneously learn the

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Kramers-Kronig (KK) relations, a fundamental physical principle governing the light-matter

interaction, without any extra training.[14] EllipsoNet will enable in-operando characterization of

unknown materials in complex photonic structures. Our deep neural network approach can also be

extended to extract other material properties and be applied to various spectroscopic data.

Figure 1. Overall workflow of refractive index prediction. (a) Input of 5 pairs of reflectance

spectra measured with and without the target layers. (b) Schematic illustration of the encoder-

decoder convolutional neural network, EllipsoNet, with multiple convolutional layers (blue

rectangles), four down-sampling layers (red arrows), and four up-sampling layers (green arrows).

The inset shows a schematic of the multilayer stack structure used in this study. (c) Prediction of

the real part, n, and the imaginary part, k, of refractive indices in dash curves and their

corresponding ground truths in solid curves.

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Page1of32EllipsoNet:Deep-learning-enabledopticalellipsometryforcomplexthinfilmsZiyangWang†,YuxuanCosmiLin‡,1,*,KunyanZhang₴,WenjingWu†,%,ShengxiHuang†,*†.DepartmentofElectricalandComputerEngineering,RiceUniversity,Houston,TX77005,USA‡.DepartmentofElectricalEngineeringandComputerSciences,Universityof...

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