Spatiotemporal ghost imaging and interference

Ghost imaging retrieves the image of an object by using second-order correlation, and this technique studied so far has been considered in either a spatial or temporal domain, which is hard to apply to the scenario of imaging spatiotemporal dynamic objects (STDOs). Here we propose to demonstrate spatiotemporal ghost imaging and interference (STGII) for STDOs by extending the correlation theory of optical coherence into the spatiotemporal domain. We have derived the generalized analytical formula for such STGII with partially coherent pulsed beams. Through a simple example of spatiotemporal slits at different location and time, the interesting phenomena of STGII are achieved under suitable conditions and it is observed that the image quality and interference visibility are affected by both the spatial and temporal parameters of partially coherent pulsed beams.

Figure 1

Schematic of STGII. Paths 1 and 2 are test and reference arms, respectively. Every matrix in optical paths denotes all optical elements in each part. The coordinates of $\mathit{r},\mathit{\nu },{\mathit{\rho }}_i$ ($i=1,2$) show the spatiotemporal points at source, object, and detector planes, respectively. D1 detector should have space and time resolution and D2 is a bucket detector.

Figure 2

(a) Spatiotemporal points of slits at the object plane. (b),(c) $G^{\left(2\right)}({\mathit{\rho }}_1,{\mathit{\rho }}_2=0)/G^{\left(2\right)}{({\mathit{\rho }}_1,{\mathit{\rho }}_2=0)}_{max}$ as functions of spatial and temporal coordinates of D1, with spatiotemporal ghost image (b) and interference (c) at $z_1=30$ mm and 50 mm, respectively. Other parameters are ${\sigma }_{\text{I}}=3$ mm, ${\sigma }_{\text{cs}}=0.009$ mm, ${\sigma }_{\tau }=35$ ps, ${\sigma }_{\text{ct}}=0.6$ ps, $z_{21}=30$ mm, $z_{22}=20$ mm, $\omega =2.691$ rad/fs, $n=1.00027$, and ${\beta }_2=0.023835$ ${\mathrm{ps}}^2{\mathrm{km}}^{-1}$. For the object, the pointlike shutter opens at $(x_{a1},{\tau }_{a1})=(-0.03\text{mm},-2\text{ps})$ and $(x_{a2},{\tau }_{a2})=(0.03\text{mm},2\text{ps})$.

Figure 3

Visibility of the spatiotemporal ghost image as functions of (a) ${\sigma }_{\text{I}}$, (b) ${\sigma }_{\text{cs}}$, (c) ${\sigma }_{\tau }$, and (d) ${\sigma }_{\text{ct}}$. Other unmentioned parameters are the same as in Fig. 2.

Figure 4

Dependence of $G^{\left(2\right)}({\mathit{\rho }}_1,{\mathit{\rho }}_2=0)/G^{\left(2\right)}{({\mathit{\rho }}_1,{\mathit{\rho }}_2=0)}_{max}$ on different values of (a),(b) ${\sigma }_{\text{I}}$, (c),(d) ${\sigma }_{\tau }$, (e),(f) ${\sigma }_{\text{cs}}$, and (g),(h) ${\sigma }_{\text{ct}}$ for spatiotemporal ghost interference. Other parameters are the same as in Fig. 2.