**A.2.2 Indirect measurement**

*Nonlinear Optics - Novel Results in Theory and Applications*

**A.2.1 Direct measurement**

which is readily transformed into:

where *IS*

sample [36, 39, 42].

*SRS Gain* = 10∗ *log*10(

intensity of the output Stokes radiation, *IP*(0) <sup>=</sup> \_\_*<sup>P</sup>*

**A.2 Experiment: measurements of Raman gain**

The theory developed in previous paragraph is a theory of Raman amplification. This means that to measure Raman gain, we should perform experiments on Raman amplifiers [47, 48]. Several materials, such as silicon, allow a direct measurement of SRS [49, 50]. In this case the Raman gain can be evaluated by measuring the Stokes amplification in a Raman amplifier having as active medium the material under test [2].

In the steady-state (no pump depletion) regime of SRS, assuming no losses at the Stokes frequency, the value of the gain coefficient *g* can be obtained by fitting Eq. (8),

*IS*(*L*) \_\_\_\_\_

radiation, *P* is the power incident onto the sample, *A* as the effective area of pump beam and *L* is the effective length. Since the sample is transparent to the incident light, *L* is taken to be equal to the thickness of the sample along the path of the incident light. As an example, in **Figure 5** a typical trend of the maxima of the signal power plotted as a function of the effective pump power at the exit of Raman amplifier is reported. As the laser power increases, the SRS gain is first constant and then grows approximately linear when the power is greater than the threshold value and so stimulated scattering begins to prevail. The threshold is usually defined as the power at which the linear behavior starts, while the slope of the line is proportional to the Raman gain coefficient *g*. The estimation of the Raman gain coefficient *g* is not straightforward due to the uncertainty in the effective focal volume inside the

(0) is the intensity of the input Stokes radiation (Stokes seed), *IS*(*L*) is the

*A*

*IS*(0)) <sup>=</sup> 4.34 <sup>∗</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>L</sup>* <sup>∗</sup> *IP*(0) (8)

is the intensity of the input pump

**138**

**Figure 5.**

*A typical trend of SRS signal plotted as a function of the effective pump power.*

If Raman gain is weak and the length of sample is small, Raman amplification is difficult to measure and an indirect measurement should be implemented. Frequently, this happens for glasses, for which the spontaneous Raman spectra is firstly measured by a standard Raman set up, then the Raman gain is estimated by a numerical procedure [2].

In order to eliminate in the measured Stokes Raman intensity *I*(*ω*) its dependence on both the temperature and the frequency of the vibrational modes [51, 52], the following relation can be used: *<sup>R</sup>*(ω*S*) <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>ω</sup>*S* [*<sup>N</sup>*(ω*S*,*T*) <sup>+</sup> 1] (ω*<sup>P</sup>* <sup>−</sup> <sup>ω</sup>)

$$R\text{(co}\_{\text{S}}) \quad = \frac{1}{\left[N\text{(co}\_{\text{S}},T) + 1\right] \text{(co}\_{P} - \text{o)}^{4}} I\text{(co}\_{\text{S}}\text{)}\tag{9}$$

where *ωS* is the Stokes Raman shift (in cm<sup>−</sup><sup>1</sup> units), *ωP* is the laser excitation frequency, *N*(*ωS*,*T*) is the Bose-Einstein mean occupation number and *T* is the temperature [31]. Afterwards, with the aim to properly relate the Raman spectra of investigated glasses with the standard silica glass, the measured Raman spectra can be adjusted also for the differences in reflection and angle of collection [26, 27, 31]. The relation between Raman gain spectrum and spectral and differential Raman cross section is expressed by the following equation:

$$\mathbf{g}\{\mathbf{u}\mathbf{s}\}\quad = \frac{\lambda\_{\mathcal{S}}^{\frac{3}{2}}}{c^{2}\hbar\,n^{2}}\left(\frac{\partial^{2}\sigma}{\partial\Omega\,\partial\,\mathbf{u}\_{\mathcal{S}}}\right)\_{\mathbf{0}}\tag{10}$$

where ( <sup>∂</sup><sup>2</sup> \_\_\_\_\_\_ <sup>σ</sup> ∂*Ω*∂ω*S*)<sup>0</sup> is the Raman cross section at *T* = 0 K (i.e., corrected considering the thermal population factor), *λS* is the Stokes wavelength (in m), *c* is the velocity of light in vacuum and *n* is the refraction index at the excitation wavelength [24–27, 31].
