# Spectroscopic Studies of Laser-Based Far-Ultraviolet Plasma Light Source
## 3 Computational Model
Figure 2b presents the spectral radiance (in units of W cm\({}^{-2}\) nm\({}^{-1}\) sr\({}^{-1}\)) of an artificial homogeneous Si plasma at \(\rho\approx 10^{-4}\) and \(\approx\) 4\(\times 10^{-4}\) g cm\({}^{-3}\), kT \(\approx\) 5 eV, and the photon path length (e.g., the plasma radius) of d \(\approx\) 350 \(\upmu\)m. An inspection of Figures 1 and 2b reveal the calibrated Si spectra is more and less in agreement with the calculated spectral radiance, especially at \(\rho\approx 10^{-4}\) g cm\({}^{-3}\). Note that in the FAC code, the accuracy of the wavelengths and transition rates for VUV and visible lines that usually result from the transitions within the same configurations from near-neutral ions can be limited, especially for transition metals with incomplete d-shell. Although it is possible to adjust the energy levels, however, the detailed investigation on the spectral line positions was beyond the scope of the present study. The calculations in Figure 2b demonstrate the spectral radiations from two Si III and Si IV charge states are main contributors for measured calibrated FUV emission in Figure 1. Note that the time-space integrated spectral irradiance (Figure 1) can be estimated using the CR solver coupled to hydrodynamics code. However, a very crude approximation may be made as follows. The spectral irradiance (\(M_{\lambda}\), in units of W cm\({}^{-2}\) nm\({}^{-1}\)) is linked to the spectral radiance (\(L_{\lambda}\), in units of W cm\({}^{-2}\) nm\({}^{-1}\) sr\({}^{-1}\)) by the [PERSON]'s cosine law [23], i.e., \(M_{\lambda}\) = \(\pi\)\(L_{\lambda}\). This irradiance is inversely proportional to the square of the distance from the plasma, that is, the inverse square law. Suppose a spherical plasma with a radius d \(\approx\) 350 \(\upmu\)m (somewhat larger than the laser spot size), a plasma emission duration \(\Delta\tau\approx\) 20 ns (i.e., 2 \(\times\) FWHM laser pulse), and the laser repetition-rates \(v\) = 100 Hz. With these assumptions, Figure 2b gives approximately the measured irradiance in Figure 1. For example, for strong spectral lines about the wavelength 140 nm (with the radiance of \(\approx\)4 \(\times\) 10\({}^{4}\) W cm\({}^{-2}\) nm\({}^{-1}\) in Figure 2b), the irradiance at 100 cm from the target is \(\approx\)(4 \(\times\) 10\({}^{4}\)\(\times\) 10\({}^{6}\)) \(\pi\) A\({}_{\rm S}\)\(\Delta\tau\)\(v/100^{2}\)\(\approx\) 0.4 \(\upmu\)W cm\({}^{-2}\) nm\({}^{-1}\). Here, the A\({}_{\rm S}\) is the plasma area. Indeed, in transient, non-LTE plasmas with strong temperature and density gradients generated in LPP sources, the plasma parameters such as the temperature and density, plasma size, and the integrated spectral emission are space- and time-dependent and cannot be simulated under the assumption of the perfect homogeneity of the plasma (e.g., Figure 2b). Consequently, there are a few discrepancies between the calculated spectral radiance (Figure 2b) and measured spectral irradiance (Figure 1). For example, the calculation cannot reproduce the measured intensity ratio of two strong Si IV lines around 140 nm. In addition, a comparison among the corresponding spectral emissivities and opacities of these lines (Figure 2a), the spectral radiance (Figure 2b) and the measured spectral irradiance reveals that the ratio approaches the black-body limit due to the opacity effect. Thus, the calculation may reproduce the measured Si IV line intensity ratio at different plasma conditions, e.g., from an optically thinner plasma.