## 2 Dispersive Fourier Transformation Technique
### Theory of DFT  
#### 2.1.1 Mathematical Description  
It is known that a dispersive element can be modeled as a linear time-invariant (LTI) system with a transfer function given by \(H\left(\omega\right)=\left|H\left(\omega\right)\right|\exp\left[-j\Phi\left( \omega\right)\right]\), where \(\left|H\left(\omega\right)\right|\) and \(\Phi\left(\omega\right)\) are the magnitude and phase response of the dispersive element at angular frequency \(\omega\), respectively. Mathematically, the phase response \(\Phi\left(\omega\right)\) can be expanded in Taylor series. Under the second-order dispersion approximation, in which the expanded phase terms higher than the second-order are ignored within the spectral bandwidth of interest, the transfer function and the corresponding impulse response of the dispersive element can be expressed as  
\[H\left(\omega\right)\cong\left|H\left(\omega\right)\right|\exp\left(-j\Phi_{ {}_{0}}\right)\exp\left(-j\dot{\Phi}_{{}_{0}}\omega\right)\exp\left(-j\frac{1}{ 2}\ddot{\Phi}_{{}_{0}}\omega^{{}_{2}}\right) \tag{1a}\] \[h\left(t\right)=\tilde{F}^{{}_{-1}}\left[H\left(\omega\right)\right]\propto \exp\left[\ j\frac{1}{2\ddot{\Phi}_{{}_{0}}}\left(t-\dot{\Phi}_{{}_{0}}\right) ^{{}_{2}}\right] \tag{1b}\]  
where \(\dot{\Phi}_{{}_{0}}\) is the group delay and \(\ddot{\Phi}_{{}_{0}}\) denotes the second-order dispersion, also known as group velocity dispersion (GVD).  
Let \(x(t)\) and \(y(t)\) be the complex envelopes of the input and output optical pulses of a dispersion element with an impulse response \(h(t)\), respectively. If the dispersion device has a wide bandwidth covering the whole spectrum of the input optical pulse, the output can be related to the input by the convolution operation, that is, \(y(t)=x(t)*h(t)\). Considering the impulse response in Equation (1b), we can rewrite the output pulse as  
\[\begin{split} y\left(t\right)&=\int\limits_{- \infty}^{+\infty}x\left(t^{{}^{\prime}}\right)h\left(t-t^{{}^{\prime}}\right) dt^{{}^{\prime}}\propto\int\limits_{-\infty}^{+\infty}x\left(t^{{}^{\prime}} \right)\exp\left[\ j\frac{1}{2\dot{\Phi}_{{}_{0}}}\left(t-t^{{}^{\prime}} \right)^{{}_{2}}\right]dt^{{}^{\prime}}\\ &\propto\exp\left(\ j\frac{t^{{}^{\prime}}}{2\dot{\Phi}_{{}_{0}} }\right)\int\limits_{-\infty}^{+\infty}x\left(t^{{}^{\prime}}\right)\exp \left(\ j\frac{1}{2\dot{\Phi}_{{}_{0}}}t^{{}^{\prime}}^{{}_{2}}\right)\exp \left(-j\frac{1}{\dot{\Phi}_{{}_{0}}}tt^{{}^{\prime}}\right)dt^{{}^{\prime}} \end{split} \tag{2}\]  
here the constant time delay \(\dot{\Phi}_{{}_{0}}\) is ignored to allow us to focus on the pulse broadening induced by the dispersion. We assume the incident pulse has a limited pulse width of \(\Delta t_{{}_{0}}\). If the dispersion is sufficient such that \(\left|\ddot{\Phi}_{{}_{0}}\right/\left(\Delta t_{{}_{0}}\right)^{{}_{2}}\right| >>1\), which is also known as the temporal Fraunhofer approximation, the stretched pulse can be further approximated by [34]  
Figure 1: Schematic diagram illustrating dispersive Fourier transformation process in a dispersive device.  
\[y\left(t\right) \propto\exp\left(\begin{array}{c}j\frac{t^{2}}{2\dot{\Phi}_{0}} \end{array}\right)\int\limits_{-\infty}^{+\infty}x\left(t\,^{\ast}\right)\exp \left(-j\frac{t}{\dot{\Phi}_{0}}t\,^{\ast}\right)dt\,^{\ast} \tag{3}\] \[\propto\exp\left(\begin{array}{c}j\frac{t^{2}}{2\dot{\Phi}_{0}} \end{array}\right)X\left(\omega\right)\big{|}_{\omega=\frac{t}{\dot{\Phi}_{0}}}\]  
where \(X(\omega)=\widetilde{F}[x(t)]\) is the Fourier transform of the input pulse. It is clearly shown in Equation (3) that the output temporal pulse envelope is proportional to the spectrum of the input pulse with a phase factor. Therefore, DFT or dispersion-induced frequency-to-time mapping is obtained.