Correlated imaging, quantum and classical

We analytically show that it is possible to perform coherent imaging by using the classical correlation of two beams obtained by splitting incoherent thermal radiation. A formal analogy is demonstrated between two such classically correlated beams and two entangled beams produced by parametric down-conversion. Because of this analogy, the classical beams can mimic qualitatively all the imaging properties of the entangled beams, even in ways which up to now were not believed possible. A key feature is that these classical beams are spatially correlated both in the near field and in the far field. Using realistic numerical simulations the performances of a quasithermal and a parametric down-conversion source are shown to be closely similar, both for what concerns the resolution and statistical properties. The results of this paper provide a scenario for the discussion of what role the entanglement plays in correlated imaging.

Figure 1

Correlated imaging with incoherent thermal light. The thermal beam $a$ at the beam splitter BS is divided into two beams, $b_1$ and $b_2$, which travel through, respectively, a test and a reference system, described by their impulse response functions $h_1$ and $h_2$. The test arm 1 includes an object. Detector $D_1$ is either a pointlike detector or a bucket detector. $D_2$ is an array of pixel detectors. $v$ is a vacuum field.

Figure 2

Imaging scheme. $L$ denotes two identical lenses of focal length $f$. $D_1$ is a pointlike detector.The distance $z$ is either $z=f$ or $z=2f$.

Figure 3

Numerical simulation of the reconstruction of the diffraction pattern of a double slit in the scheme $z=f$ of Fig. 2. $G({\vec{x}}_1,{\vec{x}}_2)$ is plotted vs ${\vec{x}}_2$ after ${10}^4$ pump shots for $a$ entangled signal∕idler beams from PDC, $b$ classically correlated beams by splitting the signal beam. $c$ is the analytical result of Eq. (14). Parameters are those of a 4 mm $\beta$-barium-borate crystal ($=16.6\mu$m, ${\tau }_{\text{coh}}=0.97$ ps). The pump waist is $664\mu$m, and the pulse duration is $1.5$ ps. The slits are $36\mu$m wide and slit separation is $122\mu$m. $x_0$ is defined as $\Delta q\lambda f∕\left(2\pi \right)$.

Figure 4

Numerical simulation of the reconstruction of the image of a double slit in the scheme $z=2f$ of Fig. 2. $G({\vec{x}}_1,{\vec{x}}_2)$ is plotted vs ${\vec{x}}_2$ after ${10}^4$ shots for $a$ entangled signal and idler beams from PDC, $b$ classically correlated beams by splitting the idler beam. $c$ is the analytical result of Eq. (18). Parameters are as in Fig. 3.

Figure 5

Degree of spatial correlation $C$ between two identical detection regions of the beams obtained by splitting thermal light, as a function of the ratio $\delta$ between the pixel size and the coherence length. $n_{\max }$ is the mean photon number in the most intense mode. $C=1$ represents the maximum degree of correlation.

Figure 6

Visibility ${\mathcal{V}}_S$ of the spatial correlation between two identical detection regions of the beams obtained by splitting thermal light, as a function of the ratio $\delta$ between the pixel size and the coherence length.