# Light-Matter Complex Interactions in Stereolithographies
## 4 Production of Free Radicals by Light
### One-Photon Process
#### 4.3.5 Two-Photon Stereolithography
Indeed, as shown schematically in Figure 21, the voxel is made of resin whose reactional progress in the course of transformation is variable. There is therefore a fragile zone that should not be degraded before washing for the measurement of the long axis [3]. In this simplified approach, the flux density is, within a multiplicative factor \(\lambda\), of the form F\({}_{0}\)/length L\({}^{2}\) in the beam direction, with T1 defined by T1 = ((2 kt/R\({}_{0}\))\({}^{1/2}\))/kq. Under the conditions of Figure 21, on the one hand, T1 is proportional to ((1/R0)\({}^{1/4}\)) since it is a two-photon absorption and, on the other hand, to the inverse of the concentration of the initiator. On this basis, all other things being equal, it is possible to relate the length of the voxel with the irradiation time. These experiments were conducted with different laser powers and are shown in Figure 22.  
Figure 21: Two-photon polymerization as a function of time. Figure 22: Voxel length vs. irradiation time (sum of the different laser pulses). Laser 1 (500 fs, 515 nm, 1 kHz): 2.3 mW; 2: 5.4 mW; 3: 10 mW; 4: 20 mW; 5: 38.5 mW. Laser 2 (500 ps, 532 nm, 1 kHz): 40 mW. In order to optimize the production of an object using this two-photon absorption method, we have tried to model the variations in the voxel length vs. the number of shots (or time) and the power of the source, knowing that the absorbed light is proportional to the number of shots (or to the concentration of the initiator (irgacure)) and to the square of the luminous flux, and that this luminous flux is approximately proportional to the inverse of the distance to the focal plane. However, it is the polymerization reaction mechanism that leads to the S-curve in Figure 2 that poses a problem for simplified and operational modeling. Indeed, it is only after the disappearance of the inhibitor (dissolved oxygen and a stabilizer based on hydroquinone or its family) that the polymerization reaction begins; in fluid media, it starts with a chain reaction whose kinetics depend on the square root of the light absorption rate, and then changes to a proportionality with this last parameter [2]. As a reminder, in zone (I), between 0 and T1 there is no reaction (T1 = Q/Ia); in zone (II), the kinetics starts at Ia1/2 then evolves towards a proportionality to Ia because of the rheological change induced by the photochemical cross-linking; in zone (III), we consider for simplicity that the reaction is blocked. In addition, in these considerations there are a number of assumptions that should be validated, such as the stability of Ia, which assumes the non-consumption of the initiator, and non-knowledge, such as the values of the rate constants, the concentrations of the inhibitors, the concentration of M over time, etc. For all these reasons, it was not possible to consider a model other than phenomenologically linking some main parameters in order to optimize the fabrication of objects by the two-photon process which is the subject of this work. Based on this, and as an illustration of the control of voxel size as a function of femtosecond laser power, Figure 23 shows the phenomenological relationship defined by a classical kinetic model of free radical polymerization adapted to the two-photon initiation process (cf. [2; 60; 92; 93]). This type of figure illustrates the possibility of controlling the relationship between the incident flux and the voxel size. Figure 24 shows the effect of this experimental law on the length of the voxel for all the measurements presented in Figure 23. Given the difficulty of measuring the value of the length (presence of \"soft\" material around the hard material), it is possible to express all the experimental data by a simple law, which is:  
\[\text{L}=\alpha\cdot\text{t}\cdot\text{C}^{1/4} \tag{16}\]  
where \(\alpha\) is a fitting parameter, t represents the irradiation time (number of shots), and C is the initiator concentration.