## 2 Basic Mechanism  
The mathematical models of the pulse propagation and carrier generation in semiconductors are as follows: [8,9]  
\[\frac{dI(r,z)}{dz}=-\alpha I(r,z)-\beta_{2}I^{2}(r,z)-\sigma_{ex}NI(r,z) \tag{1}\]  
\[\frac{d\Phi(r,z)}{dz}=\beta_{1}I(r,z)-\gamma_{1}N(r,z) \tag{2}\]  
\[\frac{dN(r,z)}{dt}=\frac{\alpha I(r,z)}{h\omega}+\frac{\beta_{2}I^{2}(r,z)}{2 h\omega} \tag{3}\]  
where \(I\) is the pulse laser irradiance, \(N\) is the free carrier density, and \(\varphi\) is the phase. In addition, \(\alpha\) is the single-photon absorption coefficient, \(\beta_{2}\) is the two-photon absorption coefficient, which is proportional to the imaginary part of third-order nonlinear optical susceptibility (\(\chi\)), \(\beta_{1}\) is proportional to the real part of \(\chi\), \(\sigma_{EX}\) is the absorption rate of laser-induced carriers, and \(\gamma_{1}\) represents the refraction of the free carriers. In Formula (3), \(h\omega\) is the photon energy, and the coefficient 2 in the denominator of the second term on the righthand side of the formula means that the simultaneous absorption of two photons is required to produce an electron hole pair. In summary, Formula (1) describes the light absorption process as the pulsed laser propagates through the medium, Formula (2) describes the phase change caused by the free carrier and the pulsed laser, and Formula (3) describes the change in the carrier concentration with time due to single photon absorption and two-photon absorption, as shown in Figure 1. If the selected pulse laser wavelength is in the sub-band gap, the \(\alpha\) in the above equation should be equal to 0, in essence, which means that the contribution of single-photon-absorption-induced carriers can be ignored. Then, Formula (3) can be simplified as:  
\[\frac{dN_{TPA}(z,r)}{dt}=\frac{\beta_{2}l^{2}(z,r,t)}{2h\omega} \tag{4}\]  
Focusing the laser beam on a small spot enables the formation of a high density of photons in a small area, which can induce the two-photon absorption process. Figure 2, below, shows the absorption spectra of silicon materials at room temperature for different wavelengths of light (visible and near infrared) [1]. The common wavelengths of the light used in the pulsed laser single-event effect simulation tests are shown in Figure 2, above. When the wavelength of the light is more than 1200 nm, the linear absorption coefficient can be ignored, which means that, when the wavelength of the light is more than 1200 nm, most carriers are generated by two-photon absorption, and the corresponding energy of one photon is about 1 eV. This TPA SEE technique has two characteristics that combine to make it particularly useful in SEE testing. Firstly, the pulse test equipment avoids the influences of the top metal layer and device packaging material during the process of the charging of the carriers through the wafer, and there is no exponential attenuation of the light irradiance. Secondly, the quadratic dependence on the irradiance of the pulsed laser restricts the charge carrier to  
Figure 1: Mechanisms of SPA and TPA. ((**a**): single photon absorption, (**b**): two photon absorption). Figure 2: Absorption coefficients of light at different wavelengths in silicon. ensure that the maximum light intensity is injected into the confocal region, which makes the charge injection possible at any depth of the material (device). The two-photon absorption process is different from the single-photon absorption process, which produces a pair of electron hole pairs by absorbing a photon. It is a nonlinear absorption process, and the energy, pulse width, and beam spot diameter of the pulse laser all affect the characteristic parameters of the single-particle effect. When the pulse is prevented from hitting the top of the device, it should hit the inner wafer from the back of the device. For incident from the backside, either the single-photon absorption test method or the two-photon absorption test method can be used.