Influence of Spectral Bandwidth on the Working Curve in Vat Photopolymerization

In vat photopolymerization, 3-dimensional parts are fabricated by using patterned light to spatially cure a liquid resin. One of the foundational measurements for vat photopolymerization is known as the working curve whereby the depth (i.e. thickness) of cured resin is measured as a function of radiant exposure. The commonly applied mathematical model for the working curve - known widely as the Jacobs model - assumes a monochromatic light source. The Jacobs model has been widely used, but in many cases significant deviations between the Jacobs model and the data have been observed. Herein, we extend the Jacobs model by deriving a polychromatic model that accounts for broadband light sources (e.g. light emitting diodes, LEDs). We demonstrate through experiment and theory that in certain cases the deviations from Jacobs' original model can be explained and understood as an optical 'inner filter' effect. The ability of the Jacobs model to capture the working curve behavior is shown to be dependent on the bandwidth of the light source in conjunction with the gradient in the absorption spectrum of the resin in the vicinity of the light source spectrum. Additionally, we offer an empirical model function that better fits experimental data and allows for an improved estimate of model parameters. Broadly, this work aims to strengthen the conceptual link between the working curve measurement and the photophysical parameters that are intrinsic to vat photopolymerization printing.