I am trying to understand the dispersive Fourier transform (DFT) technique for spectral characterization of pulses. In the literature, I found this far-field condition from Fraunhofer approximation in temporal domain which has to be satisfied to determine the required amount of dispersion: $$ t_0^2 << |2\beta_2 z| ~~~~~~~~~~~~~~~(1) $$ This was from 'Godin, Thomas, et al. "Recent advances on time-stretch dispersive Fourier transform and its applications." Advances in Physics: X 7.1 (2022): 2067487.'
In another article, I found this condition showing the required amount of dispersive stretching, $$ \frac{\lambda^2 t_0}{2\pi c B}<< |\beta_2 z|~~~~~~~~~~~(2) $$ This condition (2) is from the article 'Runge, Antoine FJ, et al. "Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers." Optics Letters 38.21 (2013): 4327-4330.'
Here, $t_0$ is the pulse width, $\beta_2 z$ accounts for the total group velocity dispersion experienced by the pulse, $B$ is the pulse spectral bandwidth, $\lambda$ is the central wavelength of the pulse, and $c$ is the speed of light in vacuum.
I would like to know why there is a bandwidth dependence in the equation (2). Also, can anyone help me with how the temporal analogue of the Fraunhofer condition has been derived? Any help would be appreciated. Thanks.
