## 3 Results and Discussions
### Simulation Results of Brain Functional Sensing
However, the error between the theorical extinction coefficient and the simulation result is very small, not exceeding 0.06 mm\({}^{-1}\), so the result is ideal. In actual experiments, when the method described in [34] is used, the accuracy of the exponentially fitted extinction coefficient can reach 0.8%, which is about 0.72 mm\({}^{-1}\) for blood. Therefore, we set the maximum error of extinction coefficient to 1 mm\({}^{-1}\). At \(Y=70\%\), \(d\)C/C, \(d\)Y/\(Y\), and \(dS\)/\(S\) calculated by the optimized combination are 0.0859, 0.0829, and 0.0296, respectively. Accordingly, we can conclude that when imaging vascular structure, the optimization scheme shown in Figure 7 can be used to reasonably select three wavelengths to calculate the optical parameters of the tissue while ensuring high spatial resolution. It should be noted that the above optimization process was performed under the condition that \(Y\) = 0.7. The hemoglobin concentration of the blood vessel is high, and the absorption coefficient in the visible-light waveband is large, so the optimization results for different \(Y\) are approximately the same. Next, the functional sensing of the cerebral cortex was discussed, where the spatial resolution requirements are reduced compared to imaging small-sized blood vessels. In the simulation, wavelengths of 540 nm, 546 nm, and 576 nm were selected, and the window function of the STFT has a width of 6 nm in the spectral domain. Optical parameters of gray matter at these wavelengths are listed in Table 4. There were two reasons for this choice. First, the scattering coefficients of these three wavelengths were approximately the same, as shown in Figure 2; second, we made a compromise by considering the calculation errors of \(C\) and \(Y\). It should be noted that this selection of wavelengths was obtained under the condition that \(Y\) = 70% using the optimization method described in Figure 7. The \(n\) and \(g\) of tissue were 1.37 and 0.9, respectively. The scattering coefficients \(S\) in the Table 4 were obtained by dividing the scattering coefficients in Figure 2 by 10 so that they were approximately the same as the scattering coefficients in gray matter [35]. The simulated gray matter was a \(300\times 300\times 500\) (\(\upmu\)m) cube, and simulation results are shown in Figure 9. Figure 9a shows the CW fluence extracted at plane \(y=0\)\(\upmu\)m. The black data points in Figure 9b represent backscattered signals, and the interval of \(z\) is 25 \(\upmu\)m. The green curve in Figure 9b is the fitting result, and the fitting coefficients and goodness are shown in the picture. The \(R^{2}\) value of the fitting results is 0.9908. The fitting result in Figure 9b was obtained after removing the class II \begin{table}
\begin{tabular}{c c c c} \hline \hline \(A\) (nm) & \(\epsilon_{O}\) (mmol\({}^{-1}\)-mm\({}^{-1}\)) & \(\epsilon_{D}\) (mmol\({}^{-1}\)-mm\({}^{-1}\)) & \(S\) (mm\({}^{-1}\)) \\ \hline
540 & 5.89 & 4.81 & 7.014 \\
546 & 5.66 & 5.38 & 7.016 \\
576 & 5.91 & 4.45 & 7.010 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optical parameters of gray matter at selected wavelengths for functional sensing. Figure 8: Simulation results for 560 nm wavelength: (a) continuous wave (CW) fluence extracted at plane \(y=0\)\(\upmu\)m; (b) backscattered signals; and (c) fitting results when class II photons are removed. photons. If class II photons are not removed, the fitting \(\mu_{t}\) is small, as shown in Figure 9c because the optical path of the class II photon is larger. For comparison, the data in Figure 9c were normalized. The theoretical calculation values and simulation values of the three parameters are shown in Figure 9d. The points marked \"The\" in the legend are the theoretical calculation values, and the points labeled \"Rea\" are the simulation values.