## 3 Chirped Microwave Pulse Generation and Compression
### Chirped Microwave Pulse Generation  
Over the years, photonic generation of arbitrary RF waveforms, particularly chirped microwave waveforms, attracted considerable attention. A review of different approaches can be found in Reference [16]. Due to its simplicity, one of the most widespread techniques is based on spectral shaping combined with wavelength-to-time (WTM) mapping: An optical spectral shaper (OSS) is used to tailor the amplitude spectrum of a pulsed BBS (generally, the output from a mode-locked laser) and the shaped amplitude spectrum is then propagated in a dispersive medium to map the spectral content into the time domain. Generating a chirped microwave waveform requires an OSS with a constant free spectral range (FSR) and a nonlinear WTM, or an OSS with a linearly variable FSR and a linear WTM. OSSs with variable FSR were realized using a variety of fiber and integrated platforms, with reviews in References [16; 17]. One implementation uses a Sagnac loop incorporating an LCFBG [18]. By controlling the path misbalance in the loop using an optical delay line (ODL), the sign of the linear variation in the FSR can be tuned from positive to negative, which in turn can tune the sign of the chirp of the generated microwave waveform. We showed that this approach can be scaled to generate simultaneously multiple wavelength-division-multiplexed (WDM) chirped microwave waveforms by incorporating multiple LCFBGs within an arrayed waveguide grating Sagnac interferometer (AWGSI) [19]. We then combined this technique of generating multiple WDM chirped microwave waveforms with the approach for real-time interrogation of LCFBG sensors for simultaneous interrogation of multiple LCFBG temperature sensors. Figure 12 illustrates the schematic of our system used to generate multiple WDM chirped microwave waveforms. The output from a pulsed broadband source is first propagated through a dispersive medium to perform WTM (note that due to the linearity of the system, the order of performing spectral shaping and WTM does not matter). The signal is then shaped using a two-channel OSS. The OSS is based on an AWGSI, which is implemented using a Sagnac loop and two pairs of optical band-pass filters (OBPFs) that define each channel. In each channel, there is an LCFBG and an optical delay line (ODL), which is used to control the wavelength independent path length difference; adjusting this path length difference allows for tuning the sign of the RF chirp of the generated chirped microwave signal. The transfer function for the \(i\)th spectral channel is [16; 18]:  
\[T_{i}(\lambda)=\frac{1}{2}W_{i}(\lambda)\Bigg{\{}1+cos\Bigg{[}\frac{4\pi n_{ eff}}{\lambda_{C,i}^{2}}\lambda\bigg{(}\Delta L_{0,i}+\frac{\lambda-\lambda_{C,i}}{C_ {i}}\bigg{)}\Bigg{]}\Bigg{\}} \tag{3}\]  
where \(\Delta L_{0,i}\) represents the wavelength independent path length difference, \(\lambda_{C,i}\) is the center wavelength, \(C_{i}\) [nm/cm] is the grating chirp, and \(i=1,2\). The term \(\frac{\lambda-\lambda_{C,i}}{C_{i}}\) in Equation (3) defines the wavelength dependent path difference for the \(i\)th channel. \(W_{i}(\lambda)\) includes the amplitude spectrum of the BBS, the spectral response of the pair of OPBFs used to implement each channel in the AWGSI, and the reflection response of the LCFBG. After WTM, the temporal waveform is [16; 18]:  
\[y_{i}(t)=\frac{1}{2}r_{i}(t)W_{i}\bigg{(}\frac{t}{D}\bigg{)}\Bigg{\{}1+cos \Bigg{[}\frac{4\pi n_{eff}}{\lambda_{C,1}^{2}D}t\bigg{(}\Delta L_{0,1}+\frac{ \delta t}{C_{i}D}\bigg{)}\Bigg{]}\Bigg{\}} \tag{4}\]where \(r_{i}(t)\) is the envelope of the \(i\)th detected pulse and \(D\) [ps/mn] is the dispersion of the dispersive medium. The WTM parameters are given by \((\lambda-\lambda_{C,i})\rightarrow\delta t/D\), \(\lambda\to t/D\), and \(\delta t\) is the detuning from the center of the temporal waveform. If a temperature increase is applied to an LCFBG, its reflection spectrum will shift to longer wavelengths as temperature increases.