## 2 Methods  
### Structural and Functional Imaging Based on OCT  
The spectroscopic OCT includes the spectroscopic time domain OCT (TD-OCT) and spectroscopic Fourier domain OCT (FD-OCT), both of which are implemented by windowed Fourier transform [21; 25]. Here we mainly introduce the principle of structural imaging and functional sensing based on FD-OCT. In FD-OCT, it is only necessary to perform a Fourier transform on the interference signal collected by the spectrometer or detectors to obtain the structural image, i.e.,  
\[I(k)=S(k)\Big{|}aR+\int_{0}^{\infty}a(z)\exp\{i2k\mathbf{n}(k)z\}dz\Big{|}^{2} \tag{1}\]  
and  
\[H(z)=F[I(k)]=F[S(k)]\otimes\Big{\{}a_{R}^{2}\delta(z)+a_{R}[a(z)+a^{*}(-z)]+AC( I_{au}(k)\Big{\}}, \tag{2}\]  
where \(S(k)\) is the power spectrum of the light source; \(a_{R}\) is the amplitude of the reflected reference light; \(k\) is the wavenumber; \(\mathbf{n}(k)\) is the refractive index of the sample at \(k\); and \(a(z)\) is the amplitude of the backscattered light at the \(z\) position of the sample. \(AC\) represents the self-coherent signal of the sample, which can usually be ignored because it is small. During the calculation above, \(\mathbf{n}(k)\) is treated as a real number. However, it is in fact a complex valued function \(\mathbf{n}(k)=n(k)+i\kappa(k)\), defining both the group delay (\(n(k)\)) and absorption (\(\kappa(k)\)) within a medium [26]. Therefore, ignoring the conjugate and direct-current terms, Equation (2) should be expressed as  
\[H(z)=F[I(k)]=F[S(k)]\otimes\big{\{}a_{R}a(z)\exp(-2 nz\mu_{t}(z))\big{\}} \tag{3}\]  
in which \(\mu_{t}\) is the extinction coefficient of tissue; and \(n\) is the average refractive index of tissue. When a STFT or wavelet transform is applied to Equation (1), we can obtain the depth-resolved spectrum \(H\left(z,k_{n}\right)\), shown as  
\[H(z,k_{n})=F[I(k)g(k_{n};\Delta k)]=A\big{[}\mu_{b}(z,k_{n})\exp(-2 nz\mu_{t}(z,k_{n})\big{\}} \tag{4}\]  
where \(A\) represents the system parameter and \(\mu_{b}\) is the backscattered coefficient [7] and \(g(k_{n};\Delta k)\) is a window function having a width \(\Delta k\) centered at a mean scattering wave number \(k_{n}\). Thus, the spectra of backscattering coefficient \(\mu_{b}\) and extinction coefficient \(\mu_{t}\) as a function of \(z\) and \(k_{n}\) are obtained. From Equation (4) we can see that there is an inherent trade-off between spectral resolution and spatial resolution. For Gaussian windows, the resolution in \(z\)-space and \(k\)-space is related by Equation \(\Delta k\Delta z=1/4\pi\), i.e., \(\Delta z=\lambda^{2}/(2\Delta\lambda)\)[21]. For brain tissue, the main absorption comes from water and hemoglobin. For simplicity, chromophores other than hemoglobin such as fat, melanin, and lipids are neglected [19]. Consequently, the extinction coefficient of the sample can be defined as  
\[\mu_{t}(z,\lambda)=C(z)\cdot\ln(10)\cdot[Y\cdot\varepsilon_{O}(z,\lambda)+( 1-Y)\cdot\varepsilon_{D}(z,\lambda)]+\mu_{s}(z,\lambda). \tag{5}\]  
\(\varepsilon_{O}\) and \(\varepsilon_{D}\) are the molar extinction coefficients--the extinction coefficient divided by \(\ln(10)\) per unit molar concentration--of oxyhemoglobin and deoxyhemoglobin, respectively [27]. \(\mu_{s}\) is the scattering coefficient. The unit of \(C\) is mmol, the unit of \(\varepsilon_{O}\) and \(\varepsilon_{D}\) is mmol\({}^{-1}\) mm\({}^{-1}\), and the unit of \(\mu_{s}\) and \(\mu_{t}\) is mm\({}^{-1}\). Functional sensing involves the measurement of \(C\), \(Y\), and \(\mu_{s}\) based on the absorption and scattering spectra (Figure 2). Figure 2 shows the curves of \(\mu_{a}\) and \(\mu_{s}\) as a function of wavelength, where \(\mu_{a}=C\ln(10)\varepsilon_{o}\) and \(C=2\) mmol [22]. For the visible-light waveband (500-600 nm), the magnitude of the absorption and scattering coefficients are of the same order, and it can be approximately considered that the scattering coefficient is not affected by the wavelength when using the method introduced in [6].