## 2 Concept
To theoretically explain the processes of adding the delay and compensating for the chromatic dispersion, let us assume we start from the Kerr comb in the angular frequency domain [30] (for the remaining part of this section, we will switch from the frequency domain (\(f\)) to angular frequency domain (\(\omega=2\pi f\)) notation to make the subsequent analysis consistent with the literature):  
\[E_{comb}(\omega)\simeq\sum_{m=-\infty}^{\infty}A_{m}\cdot\delta(\omega-m \Omega_{R})\cdot\text{sech}\left(\frac{\pi T_{0}(\omega-m\Omega_{R})}{2}\right) \tag{3}\]  
where \(\Omega_{R}=\frac{2\pi}{T_{R}}\) is the repetition rate in the angular frequency domain, corresponding to the free spectral range (FSR) of the comb, and \(A_{m}\) represents the amplitude of the \(mth\) spectral line. Let us assume that the signal passes through the optical Mach-Zehnder modulator to amplitude modulate the RF-LFM \(V_{in}(\omega)\) pulse signal onto all the Kerr comb carriers, where:  
\[V_{in}(\omega)\longrightarrow\exp\Bigl{[}-jT(\omega-\omega_{0})^{2}/2B_{ \omega}\Bigr{]}\cdot\text{rect}[(\omega-\omega_{0})/B_{\omega}] \tag{4}\]  
such that \(B_{\omega}\) is the bandwidth in terms of the angular frequency. This will produce the modulated signal:  
\[E_{V_{in}}(\omega)\simeq E_{comb}(\omega)+k_{a}.E_{comb}(\omega)*V_{in}(\omega) \tag{5}\]  
where \(k_{a}\) is an amplitude modulation coefficient, and \(*\) is the convolution operator. When \(E_{V_{in}}(\omega)\) passes through the concatenation of FBGs while ignoring frequency-independent loss, we obtain the FBGs output signal as:  
\[E_{V_{FBGs}}(\omega)\simeq E_{V_{in}}(\omega).exp(-j\phi(\omega))\simeq E_{V_ {in}}(\omega).exp(-j\beta(\omega)L) \tag{6}\]  
where the medium's optical properties determine the functional dependence of \(\beta(\omega)\) on \(\omega\). Since, normally, \(\delta\omega<<\omega_{0}\), it is valid to describe \(\beta(\omega)\) by the first few terms of its Taylor expansion around \(\omega\)[31]:  
\[\beta(\omega)=\beta_{0}(\omega_{0})+\beta_{1}(\omega_{0})(\omega-\omega_{0})+ \frac{1}{2}\beta_{2}(\omega_{0})(\omega-\omega_{0})^{2}+\cdots \tag{7}\]  
Therefore:  
\[E_{V_{FBGs}}(\omega)|_{\omega_{0}}\simeq E_{V_{in}}(\omega)|_{\omega_{0}}. \Bigl{\{}h(\omega_{0})\cdot\exp[-j\beta_{1}(\omega_{0})L\omega]\cdot\exp \Bigl{[}-j\beta_{2}(\omega_{0})(\omega-\omega_{0})^{2}L\Bigr{]}\Bigr{\}} \tag{8}\]  
The first term, \(h(\omega_{0})=\exp\{-j[\beta_{0}(\omega_{0})+\beta_{1}(\omega_{0})\omega_{0}]L\}\), represents a phase term of no importance to intensity-based systems, such as ours. The second term, having an exponent that is linearly dependent on \(\omega\), stands for a pure delay:  
\[\tau(\omega_{0})=\beta_{1}(\omega_{0})L=\frac{L}{v_{g}(\omega_{0})};\text{ and }\quad v_{g}(\omega_{0})=\frac{1}{\beta_{1}(\omega_{0})}=\left.\frac{d\omega}{ \beta}\right|_{\omega_{0}} \tag{9}\]  
where \(\tau(\omega_{0})\) and \(v_{g}(\omega_{0})\) are the group delay and group velocity at the carrier frequency \(\omega_{0}\), respectively. The third term of Equation (8) shows a quadratic dependence on \(\omega\), resulting in pulse distortion. A linearly chirped FBG (LC-FBG) comprises a continuously written fiber Bragg grating of length (\(L_{LC-FBG}\)), with a continuous linearly increasing (or decreasing) period, which covers a spectral segment, \(\left[\omega_{0,Low},\omega_{0,High}\right]\).