https://doi.org/10.20965/jaciii.2019.p1012 Ohkubo, T. et al. Paper: Recurrent Neural Network for Predicting Dielectric Mirror Reflectivity Tomomasa Ohkubo∗,†, Ei-ichi Matsunaga∗, Junji Kawanaka∗∗, Takahisa Jitsuno∗∗, Shinji Motokoshi∗∗∗, and Kunio Yoshida∗∗ ∗Tokyo University of Technology 1404-1 Katakuramachi, Hachioji, Tokyo 192-0982, Japan E-mail: [email protected] ∗∗Institute of Laser Engineering, Osaka University 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan ∗∗∗Institute for Laser Technology 1-8-4 Utsubo-honmachi, Nishi-ku, Osaka 550-0004, Japan †Corresponding author [Received February 19, 2019; accepted June 25, 2019] Optical devices often achieve their maximum effective- 1. Introduction ness by using dielectric mirrors; however, their de- sign techniques depend on expert knowledge in spec- Advances in optics, such as in the field of lasers, ifying the mirror properties. This expertise can also have been remarkable and dielectric multilayer films have be achieved by machine learning, although it is not played a significant role. In particular, the dielectric mir- clear what kind of neural network would be effective ror has met the demands of those ongoing developments; for learning about dielectric mirrors. In this paper, it is one of the most important components and is used we clarify that the recurrent neural network (RNN) is in many optical devices. A dielectric mirror functions by an effective approach to machine-learning for dielec- interference between light waves reflected from the inter- tric mirror properties. The relation between the thick- faces between its films, and has various characteristics de- ness distribution of the mirror’s multiple film layers pending on the interior structure. and the average reflectivity in the target wavelength A dielectric mirror is made up of multiple layers of region is used as the indicator in this study. Reflection dielectric materials having different refractive indexes. from the dielectric multilayer film results from the se- Therefore, we can control factors such as the overall re- quence of interfering reflections from the boundaries flectivity and the phase delay of individual reflected rays between film layers. Therefore, the RNN, which is usu- inside a dielectric mirror by changing the thickness of ally used for sequential data, is effective to learn the re- each film. For example, to construct a band pass filter, lationship between average reflectivity and the thick- i.e. a highly reflective mirror that passes only a particular ness of individual film layers in a dielectric mirror. We set of wavelengths, an anti-reflection coating and high- found that a RNN can predict its average reflectivity reflection coating are made of dielectric films. with a mean squared error (MSE) less than 10−4 from To design a general dielectric mirror, nonlinear opti- representative thickness distribution data (10 layers mization is used to obtain a distribution of film thick- with alternating refractive indexes 2.3 and 1.4). Fur- nesses that produces the required characteristics [1]. It thermore, we clarified that training data sets gener- is therefore necessary to determine a set of initial values ated randomly lead to over-learning. It is necessary for the thickness of each dielectric film for the iterative to generate training data sets from larger data sets so optimization process. that the histogram of reflectivity becomes a flat dis- However, determining these initial values requires an tribution. In the future, we plan to apply this knowl- expert’s know-how. If optimization were performed from edge to design dielectric mirrors using neural network inappropriate initial values (i.e., obtained without expert approaches such as generative adversarial networks, knowledge), we would obtain only a locally optimal so- which do not require the know-how of experts. lution, which is unlikely to meet requirements. This is a particularly serious problem when designing a dielec- tric mirror, which has severe requirements. For example, Keywords: recurrent neural network, dielectric mirror, a chirped mirror, which is necessary to realize an ultra- optical design short laser pulse, requires both very high reflectivity and very low group delay dispersion over a wide range of tar- get wavelengths [2]. It is impossible to design a dielectric mirror that meets the requirements for an exa watt-class 1012 Journal of Advanced Computational Intelligence Vol.23 No.6, 2019 and Intelligent Informatics © Fuji Technology Press Ltd. Creative Commons CC BY-ND: This is an Open Access article distributed under the terms of the Creative Commons Attribution-NoDerivatives 4.0 International License (http://creativecommons.org/licenses/by-nd/4.0/). RNN for Predicting Dielectric Mirror Reflectivity laser [3] by traditional design methods, and a new design from mirrors in this study. The reflectivity R of an in- process is necessary. put ray with wavelength λ is calculated in the following Machine learning is currently being applied in many paragraphs [6]. The direction of incidence of the ray is types of analysis to replace the know-how of experts. We assumed to be perpendicular to the mirror’s surface. believe that machine learning can also be used for design- The optical phase shift δi is calculated from ing dielectric mirrors; however, we are not aware of re- πd n δ = 2 i i , search to date in this area. It is not clear what kind of i λ ............. (1) neural network (NN) would be effective for this applica- tion. where the thickness and refractive index of each dielectric In this paper, as the first step in studying adapting film are denoted by di and ni, respectively. The index i is machine learning to dielectric mirror design, we com- used to number the film layers, counting from the base pared traditional fully connected simple neural networks plate. (FCNNs) and recurrent neural networks (RNNs) [4] for Assuming the base plate has a refractive index n,we this situation. The fully connected neural network does can get a vector of complex numbers α and β from not consider ordering in the input data, while the recur- Eq. (2), which will describe the characteristics of the di- rent neural network does. Because the ordering of the electric mirror. In this equation, Π denotes an infinite thin films is very important in the dielectric mirror (as de- product of matrices and j is the imaginary unit. ⎛ ⎛ ⎞⎞ scribed in Section 2), we chose the recurrent neural net- sin(δi) α (δ ) j work as suitable. = ⎝ ⎝ cos i n ⎠⎠ · 1 . β ∏ n (2) Therefore, we evaluated our RNN results in compari- i jnsin(δi) cos(δi) son to FCNN. Although curve fitting in itself is not our ob- jective, our models are evaluated using the mean squared The reflectivity R(λ) at a specific wavelength λ is then error of average reflectivity predicted by NN from the given by Eq. (3), using α and β. The refractive index of thickness distribution of films, compared to accurately the incident atmosphere is denoted by n0, and it will be calculated reflectivity. set to 1.0 in this study. When it becomes clear what kind of neural network (n α − β)2 + (n α − β)2 is suitable for learning about dielectric mirror charac- R(λ)=Re 0 Im 0 . 2 2 .. (3) teristics, the design method is expected to be drastically Re(n0α + β) + Im(n0α + β) changed. Although Eq. (3) calculates the reflectivity at a single wavelength, it is necessary to consider a range of wave- lengths for this mirror application. The average reflectiv- 2. Properties of a Dielectric Mirror Made of ity Rave for a target wavelength range λs to λe is then: Multiple Films λe R(λ)dλ A dielectric mirror is made of thin films of dielectric λs Rave = . .......... (4) materials and a base plate; the typical dielectric mirror λe has two kinds of dielectric material with different refrac- dλ λs tive indexes, alternately laminated onto a base plate. Each R boundary between thin films reflects incident light and The average reflectivity ave is therefore a function of n the interference between these reflected rays is controlled the refractive index i of each material and the thickness d by changing the thickness of each film. For example, a of each film, i. highly reflective mirror is designed so that all the rays re- flected from each boundary between films reinforce each other, while an anti-reflection coating is designed so that 3. Prepared Data for Training the reflected rays cancel each other owing to destructive interference. Furthermore, a band-pass filter can be real- In this study, we developed a neural network system R ized by balancing both reinforcing and canceling of am- that predicts the average reflectivity ave from a vector d plitudes in the target wavelength range, and a chirped mir- that is dependent on the thickness i of each film layer ror can be designed for the characteristics of both reflec- and assumes only two refractive indexes, alternating be- tivity and group delay dispersion. tween layers. We prepared training and testing data for a A dielectric mirror can have a higher damage threshold dielectric mirror to evaluate the system’s learning ability. than a metallic mirror because thin films of stable mate- The target system has 10 layers of dielectric thin film rials such as oxides and fluorides have a lower absorp- of low and high refractive indexes, alternately stacked on tion ratio and a higher damage threshold than metals [5]. a glass substrate. For simplicity, the refractive indexes for Dielectric mirrors are therefore especially important for the films are set to 2.3 and 1.4, and for the base plate, 1.45. controlling high power density light from sources such as The target wavelength range is 800–1300 nm to consider lasers. exawatt-class lasers.