J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026) 437
approaches combine these mechanisms to further tailor the spectral response [ 23, 24 ]. These developments have enabled large-area color printing, high-resolution patterning, and the fabrication of functional optical surfaces for applications in decoration, optical encoding, and anti-counterfeiting. In particular, laser-induced multiplexed image printing, where different images are revealed under different observation conditions, provides a powerful approach for secure document personalization [ 25 – 27 ].
Despite these advances, laser processing remains limited in its ability to fully control the optical response. The physical parameters governing color generation cannot be tuned independently, and the resulting color gamuts are generally restricted compared to standard color spaces such as sRGB. This results in missing hues and reduced chromaticity. While gamut mapping strategies can be used to adapt images to the available color palette [ 28 ], they do not compensate for the intrinsic limitations of the process. Maximizing the accessible color gamut therefore remains a key objective. Achieving this requires identifying optimal laser processing parameters, such as power, scan speed, or repetition rate, within a high-dimensional parameter space. However, the relationship between these parameters and the resulting colors is highly nonlinear and sensitive to material variability. Advances in ultrafast laser processing have highlighted the critical role of precise parameter control in achieving reproducible and high-quality material structuring [ 29 ]. For industrial applications, the ability to rapidly adapt to new materials or to variations between samples is therefore essential. In this work, we address this challenge by developing an efficient method to identify, with a minimal number of experiments, the optimal laser parameters that maximize the achievable color gamut for a given material. While similar optimization problems have been addressed using genetic algorithms [ 30, 31 ], we show that Bayesian optimization provides a more efficient framework forthistask.
While Genetic Algorithms( GA) and Bayesian Optimization( BO) are well-established global optimization techniques, their application to laser material processing has gained increasing attention in recent years. This is largely due to their ability to work on high-dimensional spaces and to efficiently explore parameter spaces without relying on gradient information, making them well suited to the nonlinear nature of laser-matter interactions. Recent studies have demonstrated the efficacy of surrogate-based optimization methods for complex laser processing tasks. For instance, hybrid frameworks combining physical models, such as the two-temperature model, and machine learning have been proposed to predict ultrafast laser ablation depths under data-scarce conditions [ 32 ]. Similarly, Gaussian Process-based surrogate models have been successfully employed for the inverse design of complex metasurfaces, significantly reducing the number of computationally expensive electromagnetic simulations [ 33, 34 ]. Beyond simulation-driven studies, surrogate-based models are increasingly applied to laser manufacturing processes, including the identification of process windows for laser-cutting [ 35 ], the exploration of parameter spaces for laser-induced graphene synthesis [ 36 ] or the optimization of photonic surfaces texturing [ 37 ]. However, maximizing the structural color gamut presents a unique challenge: the mapping between laser parameters and the resulting colors is highly nonlinear and strongly affected by microscopic variability, which prevents reliable analytical or predictive modeling. This limitation is particularly pronounced in random plasmonic metasurfaces, where the optical response from statistical distributions of nanostructures. The optimization process must therefore rely on an iterative experimental feedback, where each laser inscription provides information to guide the exploration of the parameter space. In this article, we introduce a grey-box Bayesian optimization framework tailored for rapid gamut expansion. In contrast to conventional black box formulations, where objective function is modeled directly, our approach uses a different structure. The goal here is not to optimize a single color, but to maximize the hypervolume of a set of printed colors in a perceptual space. To address this, we propose a grey-box, set-based Bayesian optimization framework. Instead of modeling the volume directly, a Gaussian Process is used as a surrogate model for intermediate physical quantities, namely the color coordinates. The acquisition function then analytically evaluates the expected contribution of a candidate point in terms of its ability to expand the volume of the current color gamut. This formulation simplifies the optimization problem and enables a more efficient exploration of the color space compared to genetic algorithms.
2 Material and methods
2.1 Material
The first type of sample is a thin mesoporous film of titanium dioxide elaborated by sol-gel process containing silver nanoparticles. The fabrication process and the laserinduced optical properties have been described in several articles or our group so far [ 38 – 40 ]. This type of sample is semi-transparent and will be observed in two observation modes transmission and backside reflection. The second type of sample is a metal-dielectric-metal-dielectric thin film stack elaborated by magnetron sputtering. For more detailed descriptions, the reader can refer to the reference [ 41 ]. This sample does not transmit light and will be observed in the following two observation modes in frontside reflection: diffuse and specular. These two types of samples contain silver nanoparticles in thin films, whose shape and spatial arrangement can be modified by laser treatment. The observed colors arise from the coupled interaction between localized plasmonic resonances of the silver nanoparticles, strongly influenced by their morphology, spatial distribution and near- or far field coupling, and the interference and propagation effects in thin films, which vary under different viewing conditions [ 17, 42 ]. The material transformation at the nanometer scale and the color variations induced by changing the laser parameters are not predictable using current physical models, as these variations do not follow a simple or linear trend in any of the observation modes. Deep neural networks have successfully modeled the complex relationships between the laser