## 3 Results and Discussions
### Human Skin without Malignancies  
In this section, the influence of the pulse width \(t_{p}\) on the normalized reflectance and the transmittance for normal human skin is analyzed according to the properties described below as well as without malignancies (Figures 4 and 5). Four pulse widths are considered for a single pulse: \(t_{p}=1\) ns, 100 ps, 10 ps and 1 ps. In addition, signals reported by [PERSON] et al. [21] are plotted for a one-dimensional case so that we may compare our present three-dimensional results using the FTn FVM to those given by the discrete ordinates method utilized in [21]. In order to analyze the signals properly, we depict the results using the CLAM scheme only. In fact, for fine spatial grids, the results obtained using the STEP scheme, which are not shown here, closely match those reported using the CLAM scheme. Figure 4: Temporal signals of normalized transmittance from the normal skin subjected to the laser pulse at different pulse widths of (**a**) \(t_{p}=1\) ns, (**b**) \(t_{p}=100\) ps, (**c**) \(t_{p}=10\) ps and, (**d**) \(t_{p}=1\) ps. Figure 3: Temporal profile of the incident short-pulse (ultrafast) laser. Figures 4 and 5 show that the results of both normalized transmittance and reflectance reproduce the incident ultrafast laser, although their peak magnitudes are different. Also, the present three-dimensional signals have the same shapes as those for the one-dimensional case of [21]. For the same pulse width, the transmittance is greatly decreased (Figure 4) by comparison to the reflectance (Figure 5). In fact, for \(t_{p}=1\) ns, the peak values of the normalized transmittance and reflectance are \(1.8\times 10^{-2}\) and \(0.41\), respectively. It is shown that the decrease of the pulse width decreases the peak magnitude of the transmittance signals (Figure 4). This can be explained by the fact that large pulse widths require more thermal energy (heat) input to the skin, thus creating the trend obtained at higher peak amplitudes for a larger pulse width. Both peak magnitudes and temporal differences of the reflectance are small for the lower value of \(t_{p}\) (Figure 5). The cascading effect becomes visible only for the lowest laser pulse width (i.e., \(t_{p}=1\) ps) (Figure 5). The consecutive augmentation in the reflectance's peak magnitude is more important for \(t_{p}=1\) ps. The photons travel the skin at the speed of light within this medium, \(c_{md}=c/n\), and thus those that are not scattered in the tissue require a time of \(t=L_{x}/c_{md}=37.33\) ps to reach the boundary opposite to the laser source (i.e., the boundary at x = \(L_{x}\)). Thus, the transmittance is different from zero only after this time (Figure 4); however, the reflectance, at boundary x = 0, reveals much before this point (Figure 5). In fact, when the laser light enters the skin at the boundary x = 0, the attenuation and diffusion of the intensity start immediately, and so an increase of the reflectance signal at this collimated boundary is obtained. Figure 4 shows that the transmittance signals are very weak. In addition, in real medical applications, any malignancy in human skin is taken into account except for some cells of the affected tissues that are surgically removed and analyzed due to an amplified opacity of bone and skin. These bons and skin are obstacles to laser light and thus prevent transmittance signals from reaching the other side of the body. While transmittances are negligible in the case of malignancy, the reflectance  
Figure 5: Temporal signals of reflectance from the normal skin subjected to the laser pulse at different pulse widths of (**a**) \(t_{p}=1\) ns, (**b**) \(t_{p}=100\) ps, (**c**) \(t_{p}=10\) ps and, (**d**) \(t_{p}=1\) ps. profiles will remain present. In the next sections, only reflectance signals for malignancies in human skin are studied. To examine the transmittance and reflectance profiles correctly, we illustrate the predictions of a laser pulse with a width of \(t_{p}=1\) ps only.