INVESTIGATION OF INVERSE DESIGN OF MULTILAYER THIN -FILMS WITH CONDITIONAL INVERTIBLE NEURAL NETWORKS

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INVESTIGATION OF INVERSE DESIGN OF MULTILAYER

THIN-FILMS WITH CONDITIONAL INVERTIBLE NEURAL

NETWORKS

A PREPRINT

Alexander Luce

University Erlangen-Nürnberg

ams OSRAM

Regensburg

Ali Mahdavi

ams OSRAM

Regensburg

Heribert Wankerl

ams OSRAM

Regensburg

Florian Marquardt

University Erlangen-Nürnberg

Max Planck Institute for the Science of Light

Erlangen

October 11, 2022

ABSTRACT

The task of designing optical multilayer thin-ﬁlms regarding a given target is currently solved using

gradient-based optimization in conjunction with methods that can introduce additional thin-ﬁlm

layers. Recently, Deep Learning and Reinforcement Learning have been been introduced to the task

of designing thin-ﬁlms with great success, however a trained network is usually only able to become

proﬁcient for a single target and must be retrained if the optical targets are varied. In this work, we

apply conditional Invertible Neural Networks (cINN) to inversely designing multilayer thin-ﬁlms

given an optical target. Since the cINN learns the energy landscape of all thin-ﬁlm conﬁgurations

within the training dataset, we show that cINNs can generate a stochastic ensemble of proposals for

thin-ﬁlm conﬁgurations that that are reasonably close to the desired target depending only on random

variables. By reﬁning the proposed conﬁgurations further by a local optimization, we show that the

generated thin-ﬁlms reach the target with signiﬁcantly greater precision than comparable state-of-the

art approaches. Furthermore, we tested the generative capabilities on samples which are outside the

training data distribution and found that the cINN was able to predict thin-ﬁlms for out-of-distribution

targets, too. The results suggest that in order to improve the generative design of thin-ﬁlms, it is

instructive to use established and new machine learning methods in conjunction in order to obtain the

most favorable results.

Keywords Invertible Neural Networks ·optical multilayer thin-ﬁlms ·latent space

1 Introduction

In optics, being able to develop devices which manipulate light in a desired way is a key aspect for all applications

within the ﬁeld such as illumination [

] or integrated photonics [

]. Recent developments in machine learning, deep

learning and inverse design offer new possibilities to engineer such optical and photonic devices [

Nanophotonics in particular beneﬁts from the recent advancements in optimization and design algorithms [

]. For

example the development of meta optics or the design of scattering nano particles was greatly improved by employing

gradient-based inverse design and deep learning [

]. Multilayer thin-ﬁlms are another instance of nanophotonic

devices which are employed to fulﬁll a variety of different functionalities. Application examples are vertical-cavity

surface-emitting lasers [

], anti-reﬂection coatings [

] and wavelength demultiplexers [

]. Recently, they were

arXiv:2210.04629v1 [physics.comp-ph] 10 Oct 2022

APREPRINT - OCTOBER 11, 2022

successfully employed to enhance the directionality of a white LED while maintaining the desired color temperature

[

]. Designing multilayer thin-ﬁlms [

] has been a task in the nanophotonics community for a long time

and sophisticated techniques for the synthesis of thin-ﬁlms, which exhibit desired optical characteristics have been

developed in open-source or commercially available software [

]. Methods such as the Fourier method

[

] or the needle method [

] compute the position inside the thin-ﬁlm where the introduction of a new

layer is most beneﬁcial. Then the software will continue with a reﬁnement process, often based on a gradient-based

optimization such as the Levenberg-Marquardt algorithm [

], until it reaches a local minimum where it will

then introduce another layer. Although the software will often converge to a satisfying solution with respect to the

given target, the presented solutions often use excessive amounts of layers and the optimization is still limited by the

selected parameters in the beginning of the optimization. The problem of converging to local optima was tackled in

the past by the development of numerous global optimization techniques which have been introduced and tested in

the ﬁeld of thin-ﬁlm optimization [

]. Recently, the innovations of machine learning attracted much

interest in the thin-ﬁlm community and resulted in interesting new ways to create thin ﬁlms [5, 38]. Particularly, deep

reinforcement learning or Q-learning showed promising results in designing new and efﬁcient multilayer thin-ﬁlms

while punishing complicated designs, which employ many layers [

] and require targets that are difﬁcult to achieve

with conventional optimization.

In this work we employ so called conditional Invertible Neural Networks (cINNs) [

] to directly infer the loss

landscape of all thin-ﬁlm conﬁgurations with a ﬁxed number of layers and material choice. The cINN learns to map the

thin-ﬁlm conﬁguration to a latent space, conditional on the optical properties, ie. the reﬂectivity of a thin-ﬁlm. During

inference, due to the invertibility of the architecture, the cINN maps selected points from the latent space to their most

likely thin-ﬁlm conﬁgurations, conditional on a chosen target. This results in requiring only a single application of the

cINN to obtain the most likely thin-ﬁlm conﬁguration given an optical target. Additionally, the log-likelihood training

makes the occurrence of mode-collapse [

] almost impossible. For thin-ﬁlms, many different conﬁgurations lead to

similar optical properties. For conventional optimization, this leads to the convergence of the optimization to possibly

unfavorable local minima. A cINN circumvents this due to the properties of the latent space - by varying the points

in the latent space, a perfectly trained cINN is able to predict any possible thin-ﬁlm conﬁguration that satisﬁes the

desired optical properties. In this work, we investigated how good the generative capabilities of a cINN are for ﬁnding

suitable thin-ﬁlm conﬁgurations in a real-world application. We present an optimization algorithm, which is suitable to

improve the thin-ﬁlm predictions of the cINN. Then, we compared the optimization results of the presented algorithm to

state-of-the-art software. Finally, we discuss the limitations of the approach and give a guideline when the application

of a cINN is advantageous.

2 Normalizing ﬂows and conditional invertible neural networks

Invertible neural networks are closely related to normalizing ﬂows, which were ﬁrst popularized by Dinh et. al. [

]. A

normalizing ﬂow is an architecture that connects two probability distributions by a series of invertible transformations.

The idea is to map a complex probability distribution to a known and simple distribution such as a Gaussian distribution.

This can be used both for density estimation, but also for sampling since points can easily be sampled with a Gaussian

distribution and mapped to the complex distribution via the normalizing ﬂow. The architecture of a normalizing ﬂow is

constructed from the following. Assume two probability distributions,

which is known and for which

z∼π(z)

holds

and the complex, unknown distribution p. The mapping between both is given by the change-of-variables formula

p(x) = π(z)

det ∂z

∂x 

.(1)

Consider a transformation fwhich maps f(x) = z. Then the change-of-variables formula can be written as

p(x) = π(z)

det ∂f(x)

∂x 

.(2)

The transformation

can be given by a series of invertible transformations

f=fK◦fK−1. . . ◦f0

with

x=zK=

f(z0)=(fK◦. . . ◦f0)(z0)

. Then, the probability density at any intermediate point is given by

pi(xi) = zi=fi(zi−1)

By rewriting the change-of-variables formula and taking the logarithm one obtains

log (p(x)) = log π(z0)

i=1 

det ∂fi(zi−1)

∂zi−1

−1!= log (π(z0)) −

i=1

log 

det ∂fi(zi−1)

∂zi−1

.(3)

To be practical, a key component of any transformation of a normalizing ﬂow is that the Jacobian determinant of

the individual transformations must be easy to compute. A suitable invertible transformation, which is sufﬁciently

APREPRINT - OCTOBER 11, 2022

expressive is the so called RNVP block [

]. The input

is split into two separate vectors

and

and is processed

with the help of two transformation functions, s2and t2, while the other input u2is kept ﬁxed

fi(u1⊗u2) = {u1exp(s2(u2)) ⊕t2(u2)⊗u2}={v1⊗u2}.(4)

,⊕,

and



denote element wise computation while

⊗

denotes a concatenation. The inverse of this transformation

is given by

f−1

i(v1⊗u2) = {(v1t2(u2)) exp(s2(u2)) ⊗u2}.(5)

To invert the entire transformation, no inversion of the transformations

and

is required. Therefore, a neural network

can be used as a transformation to make the normalizing ﬂow expressive. The Jacobian of the transformation is given

by an upper triangular matrix. Therefore, the Jacobian determinant is easy to compute since only the diagonal elements

contribute.

det ∂fi

∂z =det 1d0

∂v1

∂u2diag(exp(s2(u2) + t2(u2))).(6)

Since only one part of the input is transformed by a neural network, the RNVP block is repeated and applied to the yet

untransformed part of the input

with two additional transformation functions

and

. Other transformations, which

can be advantageous depending on the speciﬁc application are the GLOW transformation [

], which utilizes invertible

1x1 convolutions or the masked autoregressive ﬂow with a generalized implementation of the RNVP block [46].

Utilizing normalizing ﬂows for the task of a generative neural network for the proposition of thin-ﬁlms requires some

modiﬁcation of the normalizing ﬂow. Ardizzone et. al. [

] propose a so called conditional invertible neural network

(cINN) which extends the change-of-variables formula to conditional probability densities with condition c

p(x|c) = π(z)

det ∂f(x;c)

∂x 

.(7)

By assuming a Gaussian probability distribution for

and taking the logarithm, the conditional maximum likelihood

loss function is derived by Ardizzone et. al. for training of a cINN as

LcML =E"kf(x;c)k2

2−log 

det ∂f(x;c)

∂x #.(8)

The condition

can be the result of another neural network

ϕ(c) = s

, which extracts features from the given condition

. The features are then passed to the RNVP transformations by appending the features to the input vectors

and

. By jointly training the cINN via the conditional maximum likelihood loss, Ardizzone et. al. showed that the

feature extraction network learns to extract useful information for the invertible transformation. The condition

can be

thought of as the target of the cINN. During inference, the cINN transforms gaussian samples to the learned distribution,

conditional on the target

. Ardizzone et. al. also provide a Python implementation FrEIA

for conditional and regular

invertible neural network based on the Pytorch2Deep Learning library.

3 Application of conditional invertible neural networks to generating multilayer thin-ﬁlms

Multilayer thin-ﬁlms are an optical component that consists of a sequence of planar layers with different materials

stacked on top of each other with a varying layer thickness. Light, which irradiates the thin-ﬁlm, can be transmitted and

reﬂected at the layer interfaces and absorbed within the layer, as depicted in Figure 2. This interaction of reﬂection,

transmission and absorption is different for different wavelengths of light and angles of incidence

. A fast and

convenient way to compute the optical response of a thin-ﬁlm is to employ the transfer matrix method (TMM) [

]. In

a previous work, we developed the Python package TMM-Fast

[

] which allows to compute the optical response of a

thin-ﬁlm. The package also implements convenience functionality for thin-ﬁlm-dataset generation, which is especially

important for machine learning. By changing the layer thicknesses, the behavior of the thin-ﬁlm can be modiﬁed, which

are the parameters pthat are up to optimization.

Finding a thin-ﬁlm design, which fulﬁlls the optical criteria while employing only a very limited number of layers is a

challenging task since the loss landscape is highly non-convex and high-dimensional. The loss landscape is given by a

mapping of the optical characteristics

of the thin-ﬁlms by a loss function

L(M(Θ, λ),Mtarget(Θ, λ)) = E

with

1https://github.com/VLL-HD/FrEIA#papers

2https://pytorch.org/

3https://github.com/MLResearchAtOSRAM/tmm_fast

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INVESTIGATIONOFINVERSEDESIGNOFMULTILAYERTHIN-FILMSWITHCONDITIONALINVERTIBLENEURALNETWORKSAPREPRINTAlexanderLuceUniversityErlangen-NürnbergamsOSRAMRegensburgAliMahdaviamsOSRAMRegensburgHeribertWankerlamsOSRAMRegensburgFlorianMarquardtUniversityErlangen-NürnbergMaxPlanckInstitutefortheScienceofLightEr...

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INVESTIGATION OF INVERSE DESIGN OF MULTILAYER THIN -FILMS WITH CONDITIONAL INVERTIBLE NEURAL NETWORKS

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