Inverse Engineering of Optical Constants in Photochromic Micron-Scale Hybrid Films

Abstract

Photochromic materials enable dynamic optical modulation through reversible transitions between distinct absorption states, with broad potential for smart windows, adaptive optics, and reconfigurable photonic devices. Micron-scale photochromic hybrid films present a particularly attractive platform for these applications, combining straightforward preparation with substantial optical modulation and scalability for high-volume fabrication. However, rational design of such films remains fundamentally constrained by the absence of well-defined optical constants. Unlike homogeneous thin films, micron-scale hybrid photochromic materials comprise active particles dispersed non-uniformly within polymer matrices. Conventional first-principles electromagnetic simulations face substantial computational costs and discrepancies between simulated and experimental particle distributions. Here, we introduce a data-driven framework that extracts effective optical constants directly from minimal experimental transmittance measurements. Our dual-state effective model approximates the complex inhomogeneous photochromic layer as a compressed homogeneous medium characterized by pseudo-refractive indices and pseudo-extinction coefficients for both pristine and UV-irradiated states. Through systematic optimization against experimental data from tungsten oxide-polyvinylpyrrolidone hybrid films, we determine wavelength-dependent pseudo-optical constants and compression ratios that enable accurate prediction of optical modulation within the tested thickness range. Our methodology establishes a framework for engineering hybrid photochromic systems and demonstrates how data-driven modeling can overcome limitations in characterizing complex nanostructured materials.

Figures (6)

• Figure 1: Effective compressed homogeneous layer approximation for photochromic hybrid films. The physical photochromic micron-scale hybrid film (left) consists of photochromically active particles dispersed non-homogeneously within a polymer matrix on a substrate. Upon UV irradiation, the particles undergo a photochromic transition, changing their optical properties. In our dual-state effective model (right), we approximate this complex inhomogeneous structure as an effectively compressed homogeneous layer for each state. The pristine state is characterized by pseudo-optical constants $\tilde{n}^P(\lambda)$, $\tilde{k}^P(\lambda)$ and compression factor $\kappa^P$, while the UV-irradiated state is described by $\tilde{n}^I(\lambda)$, $\tilde{k}^I(\lambda)$ and compression factor $\kappa^I$. This approximation enables efficient optical modeling using the transfer matrix method while capturing the essential optical response of the photochromic hybrid film through state-dependent effective optical properties.
• Figure 2: Data-driven optimization framework for dual-state effective modeling of photochromic micron-scale hybrid films. Pseudo-optical constants and compression factors are optimized through inverse engineering of experimental transmittance data, enabling optical modulation prediction across arbitrary film configurations. (a) Experimental data collection workflow: photochromic hybrid films of varying thickness are measured in both pristine and UV-irradiated states, yielding transmittance spectra pairs that constitute the training dataset. (b) Model training architecture: the transfer matrix method calculates wavelength-dependent transmittance using trainable pseudo-refractive indices ($\tilde{n}^P$, $\tilde{n}^I$), pseudo-extinction coefficients ($\tilde{k}^P$, $\tilde{k}^I$), and scalar compression ratios ($\kappa^P$, $\kappa^I$). These parameters are iteratively refined to minimize mean squared error between model predictions and experimental measurements across all samples, wavelengths, and incidence angles. (c) Predictive capability: once optimized, the model generates optical modulation spectra for film thicknesses beyond the training set, enabling rational design of photochromic devices with tailored switching characteristics.
• Figure 3: Microstructure of photochromic hybrid films. Optical microscope images of $\text{WO}_{3-x}$ hybrid photochromic single-layer film deposited at 1200 rpm, showing (a) pristine state and (b) UV-irradiated state. Scale bars, 50 $\mu\text{m}$. The images indicate that crystallized tungsten-oxide particles are embedded and non-uniformly distributed throughout the PVP matrix, highlighting the intrinsically inhomogeneous micro-scale morphology of the hybrid film.
• Figure 4: Optimized pseudo-optical constants for photochromic hybrid films. Wavelength-dependent pseudo-refractive indices (blue) and pseudo-extinction coefficients (red) extracted through inverse engineering for films deposited at (a) 1200 rpm, (b) 2000 rpm, and (c) 2500 rpm. Solid lines represent pristine state parameters ($\tilde{n}^P$, $\tilde{k}^P$), while dashed lines indicate UV-irradiated state parameters ($\tilde{n}^I$, $\tilde{k}^I$). Insets show mean squared error versus training iteration (150 iterations), demonstrating convergence of the optimization process.
• Figure 5: Experimental transmittance and transmission predictions of the dual-state effective model. Measured transmittance spectra (solid lines: blue for pristine state, green for UV-irradiated state) compared against model predictions (dashed black lines) for $\text{WO}_{3-x}$--PVP hybrid films with varying thicknesses. Red lines indicate bare glass substrate transmittance. Subplots (a--c) show 1200 rpm films, (d--f) show 2000 rpm films, and (g--i) show 2500 rpm films. Training samples include single-layer (a, d, g) and five-layer (b, e, h) configurations, while intermediate three-layer films (c, f, i) serve as test cases. Photographic insets display the visible color change of the $\text{WO}_{3-x}$--PVP hybrid films upon UV irradiation. The model accurately reproduces experimental measurements across all configurations, demonstrating reliable thickness interpolation and state-dependent optical response prediction.