## 3 Results and Discussion
### Four-Wave Mixing  
The efficiency in FWM is determined by parameters such as the intensities of the pump pulses, phase mismatch, and the interaction length. Under the assumption that no depletion occurs for pump pulses, the intensity of the first-order anti-Stokes emission, \(I_{\mathrm{AS1}}\), can be described as [29, 30]:  
\[I_{\mathrm{AS1}}=\frac{\epsilon_{0}n\omega_{\mathrm{AS1}}{}^{2}n_{2}{}^{2}z^{2} }{2c}\,|\epsilon_{\mathrm{P1}}|^{2}|\epsilon_{\mathrm{P2}}|^{4}\mathrm{sin}c^{ 2}(-\frac{\Delta\beta z}{2}) \tag{1}\]  
where \(\epsilon_{0}\), \(c\), \(n\), \(n_{2}\), \(z\), and \(\omega_{\mathrm{AS1}}\), are the vacuum permittivity, the velocity of light in vacuum, the linear and nonlinear refractive indices of the gas, the interaction length, and the angular frequency of AS1, respectively. The intensities of P1 (\(I_{\mathrm{P1}}\)) and P2 (\(I_{\mathrm{P2}}\)) are proportional to the square of the corresponding electric field amplitudes, \(\varepsilon_{\mathrm{P1}}\) and \(\varepsilon_{\mathrm{P2}}\), respectively. The phase mismatch \(\Delta\beta\) in the equation is expressed as \(\beta_{\mathrm{P1}}+\beta_{\mathrm{AS1}}-2\beta_{\mathrm{P2}}\), where \(\beta_{\mathrm{P1}}\), \(\beta_{\mathrm{AS1}}\), and \(\beta_{\mathrm{P2}}\) stand for the propagation constants for P1, AS1, and P2. The phase mismatch in the hollow fiber can be calculated by taking into account the contributions of the gas and the waveguide [27, 29, 30].  
In the generation of multicolor beams through cascaded FWM, the anti-Stokes emission is generated in the first step and higher-order anti-Stokes emission is then generated during the propagation in the gas. The energy of the first anti-Stokes emission is higher than that of the high-order anti-Stokes emission and it is a good measure of the efficiency of the multicolor generation.  
Equation (1) can be used to calculate the gas pressure in the hollow fiber, which leads to a similar efficiency of multicolor generation as that in the gas cell with a pressure of 2 atm. The value of \(\Delta\beta\)z/2 in Equation (1) is calculated to be 0.66 rad for the gas cell and hence the value of \(\mathrm{sinc}^{2}(-\Delta\beta\)z/2) can be approximated as unity. From Equation (1) and the fact that the nonlinear refractive index is proportional to the gas pressure, the conversion efficiency from the pump to the first-order anti-Stokes emission, \(I_{\mathrm{AS1}}\)/\(I_{\mathrm{P2}}\), is proportional to \(I_{\mathrm{P1}}I_{\mathrm{P2}}p^{2}z^{2}\), where \(p\) is the gas pressure. The output energies of P1 and P2 in the case of the hollow fiber were 0.59 and 0.76 times smaller than those for the gas cell, respectively, as indicated in Table 1. The value of \(I_{\mathrm{P1}}I_{\mathrm{P2}}\) for the hollow fiber is hence considered to be 0.43 times smaller than that for the gas cell. From this fact, together with the ratio of the interaction  
Figure 7: Spectra of the output beam from the hollow fiber at 0.4 atm (black solid line) and 0.5 atm (blue solid line) and the gas cell (red solid line) at 2 atm. The spectrum is normalized by the intensity of the highest peak.  
lengths in the hollow fiber and the gas cell (600 mm/80 mm), the gas pressure in the hollow fiber, which provides the same value of \(I_{\rm{AS1}}\)/\(I_{\rm{P2}}\) as for the gas cell, is calculated to be 0.4 atm. In Figure 7, we show the spectra of the output beams from the hollow fiber at 0.4 and 0.5 atm and the gas cell at 2 atm for comparison. Excellent agreement is observed between the spectra for the hollow fiber at 0.4-0.5 atm and for the gas cell at 2 atm, although no pump depletion is assumed to occur in the model used in Equation (1).