## 3 Near-Infrared Silicon Absorption Physics
### Two-Photon Absorption (TPA)  
In the process of two-photon absorption, an electron absorbs two photons at approximately the same time, achieving an excited state that corresponds to the sum of the energy of the incident photons. In order to reach the final state, no intermediate state is necessary--only a \"virtual state\" which doesn't correspond to any electronic or vibrational energy eigenstate (Figure 4). The absorption cross section \(\sigma\) describing this process increases linearly with laser intensity according to the following relation:  
\[\sigma=\sigma^{(2)}I \tag{5}\]  
where \(\sigma^{(2)}\) is a coefficient taking into account the probability that the two-photon absorption process occurs. It is clear that in conventional linear optics, the absorption cross section \(\sigma\) is a constant. Atomic transition rate R, due to two-photon absorption scales as the square of the laser intensity, is shown in the following formula:  
\[R=\frac{\sigma^{(2)}I^{2}}{\hbar\omega} \tag{6}\]In the absence of significant recombination and diffusion, the differential equation describing the beam propagation in presence of linear and two-photon absorption can be written as [36]:  
\[dI/dz=-\alpha I-\beta I^{2} \tag{7}\]  
where \(\alpha\) is the linear absorption coefficient, \(\beta\) is the two-photon absorption coefficient, z is the longitudinal propagation direction, and where the instantaneous irradiance I = I(r, z, t) is a function of time t and transverse position r as well as z. By assuming \(\alpha\) as constant, without taking into account the possible influence of lattice heating, band filling, and bandgap renormalization on \(\alpha\), Equation 7 can be solved exactly to give the transmitted irradiance I(r, L, t) in terms of the incident irradiance I(r, 0, t):  
\[I(r,L,t)=T_{0}\frac{I(r,0,t)}{1+\frac{I(r,0,t)}{I_{c2}}} \tag{8}\]  
where:  
\[T_{0}=(1-R_{1})(1-R_{2})e^{-aL} \tag{9}\]  
is the linear transmission of a sample of thickness L and front (rear) surface reflectivity R\({}_{1}\)(R\({}_{2}\)). When the irradiance approaches the critical value I\({}_{c2}\) defined as:  
\[I_{c2}=\frac{(\alpha/\beta)}{(1-R_{1})(1-e^{-aL})} \tag{10}\]  
the transmission deviates from the linear value and TPA becomes comparable to linear absorption. It is worth noting that when I(r,0,t) = \(\alpha/\beta\), the two-photon coefficient near the surface equals the linear coefficient. Of course, (1 - R\({}_{1}\)) represents the front surface transmission while (1 - e\({}^{-aL}\)) is the linear absorption of a sample of thickness L. From a physics point of view, I(r,0,t)/I\({}_{c2}\) compares the linear absorption coefficient \(\alpha\) to the two-photon coefficient \(\beta\)I. The quantity typically measured to derive \(\beta\) is the transmitted (T) energy, not the irradiance. For a Gaussian spatial and temporal irradiance profile, it can be shown that [36]:  
\[T^{-1}=T_{0}^{-1}\left[1+\frac{I_{0}}{2\sqrt{2}I_{c2}}\right] \tag{11}\]  
It should be noted that this formula has been obtained without taking into account free-carrier absorption (FCA) generated by TPA. When FCA cannot be neglected, it can be included, obtaining a modified and more complex formula for the transmittance [36]. If front and back surface reflectivities  
Figure 4: Two-photon absorption mechanism. are known, it is possible to show that the intercept of an inverse transmission _versus_ irradiance plot corresponds to the linear absorption coefficient, while the initial slope is related to \(\beta\). [PERSON] and [PERSON] [37] isolated TPA in silicon for first time, obtaining a value of \(\beta=1.5\) cm/GW at 100 K for picosecond pulses at a wavelength \(\lambda=1,\)060 nm, and, in 1986, [PERSON] _et al_. measured a similar value at 300 K [36]. In 1990, [PERSON] _et al_. reported \(5<\beta<36\) cm/GW by measurements across the direct gap of silicon in the range \(550<\lambda<620\) nm [38]. In recent years, TPA at telecommunications wavelengths of \(\lambda=1,\)300 or \(\lambda=1,\)500 nm has been reported, with values of \(\beta\sim 0.8\) cm/GW [39-41].