## 3 Results and Discussion
Accompanied by changes in the pulse width based on changes in the center wavelength for the input beam, the peak power of a femtosecond pulse is inversely proportional to the pulse width and the strength of TPA, proportional to the square of the peak power, and inversely proportional to the square of the pulse width. The sensitivity of TPA (\(S_{two-photon}\))--defined as the number of generated photoelectrons per incident photon--can be calculated as follows:  
\[S_{two-photon}(A/W^{2})=\frac{V_{out}(V)}{P_{input}{}^{2}(W^{2})\times Transimpedance\ Gain(V/A)}, \tag{1}\]  
where \(P_{input}\) denotes the input power for TPA with repetition rate, pulse width, and beam waist, and \(V_{out}\) denotes the output voltage from the detector at each preset gain dB [28]. Accordingly, we simultaneously collected the intensity AC trace at a center wavelength to fit the pulse width and amplitude of the intensity AC function, with the extracted amplitude of the AC function reflecting the sensitivity of TPA at the center wavelength of the PD detector. The fitted amplitude of the AC function at the center wavelength was corrected by the squared ratio between the corresponding fitted and shortest pulse widths--that is, at the 1300 nm wavelength for the GaAsP PD detector--as the TPA signal is proportional to the square of the peak power. Moreover, the longer the center wavelength, the shorter the pulse width. As shown in Figure 2b, the corrected signal denotes the amplitude of the AC function at an 800 nm wavelength (on a voltage scale), corrected by the squared ratio of the fitted pulse width at 800 nm to that at 1300 nm. As described in our previous study [27], we have shown that the output voltage at long wavelengths outside the linear response could be attributed to the TPA process of the GaP detector--based on the quadratic power dependence with a slope of approximately two; moreover, the threshold power at 40 dB for a complete AC trace is as low as 15 mW, where the PD detector responds to as low a pulse energy as \(\sim\)40 pJ. In photoconductive detectors, the autocorrelator sensitivity depends only on the TPA efficiency (sensitivity). In our previous study, the sensitivity (\(S_{two-photon}\)) was estimated to be \(\sim\)5.8 \(\times\) 10\({}^{-5}\) A/W\({}^{2}\) for a GaP PD [27]. For the GaP PD, the maximum sensitivity for the one-photon process is \(S_{one-photon}\)\(\sim\)0.1 A/W at 430 nm [28], which is three orders of magnitude larger than that of the TPA process. The maximum sensitivities for the one-photon process are \(S_{one-photon}\)\(\sim\)0.3 A/W at 600 nm for the GaAsP PD and \(S_{one-photon}\)\(>\) 0.7 A/W at 970 nm for the Si PD [31]. Since the photocarriers excited by the two-photon process across the bandgap result in tracking the linear response of the PD, we anticipate that the TPA signal for the Si PD will exhibit the most efficient sensitivity, with the GaAsP PD exhibiting higher TPA sensitivity than the GaP PD. To ensure high performance and a wide tuning range, we collected the intensity AC traces at the same incident input power of 300 mW and fitted them to a Gaussian-assumed AC function for GaP, GaAsP, and Si detectors at three typical wavelengths along the full wavelength range of 680-1300 nm available from the oscillator. In Figure 3, the amplitude of the AC function at a given wavelength is corrected by the squared ratio of the fitted pulse width to the shortest pulse width for each PD detector. This corrected amplitude represents the TPA spectral sensitivity of the detector at a specific wavelength. Figure 3 ashows a corrected amplitude near 1 V for the GaAsP detector held to a 30 dB gain at three wavelengths--that is, 685, 900, and 1300 nm. This is consistent with the fact that TPA can occur for wavelengths from 600 to 1360 nm, considering the linear response range of 300-680 nm. Notably, the signal strength at 1300 nm is slightly lower than the expectation based on the linear response curve.