Fractional Fourier transform in temporal ghost imaging with classical light

We investigate temporal, second-order classical ghost imaging with long, incoherent, scalar plane-wave pulses. We prove that in rather general conditions, the intensity correlation function at the output of the setup is given by the fractional Fourier transform of the temporal object. In special cases, the correlation function is shown to reduce to the ordinary Fourier transform and the temporal image of the object. Effects influencing the visibility and the resolution are considered. This work extends certain known results on spatial ghost imaging into the time domain and could find applications in temporal tomography of pulses.

Figure 1

Illustration of the geometry for temporal ghost imaging. Light from a source is split into two mutually correlated beams which are directed to the reference and test arms. The reference arm contains two sections of single-mode fiber and a temporal lens characterized by the “focal length” $\gamma$. The test arm likewise contains two fiber sections and an object $m(t)$ between them. The fibers are characterized by the group-delay dispersion parameters ${\Phi }_i$, with $i=(a,b,c,d)$. The output intensities of the arms, $I_i(t_i)$, with $i=(1,2)$, are measured and correlated.