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.
Photochromic films enable reversible optical modulation for smart windows, adaptive optics, and reconfigurable photonic devices [1][2][3][4]. Among various photochromic architectures, hybrid systems comprising active nanoparticles dispersed within polymer matrices represent a particularly promising approach, exhibiting rapid photochromic response, high optical contrast, and ambient-condition reversibility [5][6][7][8][9]. Yet despite extensive material development [3,[6][7][8][9][10][11][12][13][14][15][16], rational design of these hybrid films remains hindered by a fundamental challenge: unlike homogeneous thin films with welldefined refractive indices and extinction coefficients, photochromic hybrid films possess no well-defined effective optical constants [17]. Solution-processed deposition inevitably produces spatially inhomogeneous particle distributions, rendering the optical response dependent on fabrication-specific microstructure rather than tabulated material properties [5,7,15]. This absence of predictive optical models forces iterative device development to rely on empirical trial-and-error.
Rigorous electromagnetic simulations offer one potential solution. Finite-difference time-domain (FDTD) and finite element methods can directly model particle distributions within films [18][19][20], but their application to micron-scale photochromic layers encounters prohibitive computational cost and a deeper limitation: accurate predictions require precise knowledge of particle arrangements that vary unpredictably between fabrication runs. Effective medium theories provide computational efficiency by homogenizing optical properties through analytical mixing formulas, yet these approaches assume quasi-static conditions and spherical geometries violated in micron-scale films and fail to capture state-dependent photochromic transitions [21][22][23]. The predictive gap between assumed simulation geometries and realized experimental structures significantly constrains reliable design of photochromic devices [24,25].
Recent progress in photonic characterization demonstrates that optical properties can be extracted directly from experimental measurements rather than calculated from microscopic structure [26][27][28][29][30][31][32][33][34]. Data-driven approaches have successfully characterized homogeneous thin films through spectrophotometric measurements, including envelope methods, Kramers-Kronig analysis, and optimization-based techniques [26][27][28][31][32][33]. Machine learning and deep learning methods have further advanced inverse characterization capabilities for photonic structures [35][36][37][38][39]. These methods work well for uniform layers but do not address photochromic hybrid films, where the challenge is extracting effective statedependent parameters that account for structural inhomogeneity across reversible optical transitions.
Here, we introduce a data-driven framework that extracts effective optical constants for photochromic hybrid films directly from minimal transmittance mea-surements. Our dual-state model approximates the inhomogeneous photochromic layer as a compressed homogeneous medium for each optical state (pristine and UV-irradiated). The model is characterized by wavelength-dependent pseudo-refractive indices and pseudo-extinction coefficients, along with compression factors that account for effective thickness reduction. These parameters are simultaneously optimized against experimental data from only a few samples, enabling accurate prediction of optical modulation across arbitrary film thicknesses. Implementation within a fully differentiable transfer matrix formulation [40] substantially reduces computational cost compared to full-wave simulations while maintaining physical interpretability [41]. Validation with WO 3-x -PVP films [5,7,16] demonstrates quantitative agreement with experimental spectra and successful interpolation to untested configurations, providing a pathway toward rational engineering of adaptive photonic devices.
In this section, we formulate the numerical methodology employed for the optical modeling of photochromic micron-scale hybrid films. To elucidate the underlying principles of the model, we consider a generalized multilayer structure containing a single photochromically active layer. This active layer sits between sets of homogeneous layers, as defined by the following sequence:
Here, M inc represents the incident medium, a semiinfinite layer where light enters the multilayer film, while M sub denotes the substrate medium where light exits the structure, also treated as semi-infinite because its thickness far exceeds that of the structure layers. P C represents a hybrid layer consisting of photochromically active particles that possess an inhomogeneous distribution within a micron-scale matrix. The top layers L T i (i = 1, . . . , N ) and bottom layers L B j (j = 1, . . . , M ) are homogeneous layers with optical constants n i,j (λ), k i,j (
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