Hanbury Brown and Twiss effect in spatiotemporal domain

Adeel Abbas; Li-Gang Wang

doi:10.1364/OE.405726

1. Introduction

Hanbury Brown and Twiss (HBT) effect, named after its two discoverers, was introduced for the measurement of the angular size of stars with the help of second-order correlation [1,2]. By using HBT effect one can find the information of a source from the intensity correlation between two spatial points. The HBT work was a great development in the field of astronomy and other branches of physics, including optics (specifically in the establishment of quantum optics) [3], atomic physics [4], particle physics [5] and condensed matter physics [6].

Recently, partially coherent sources are the topic of interest due to vast applications. Among partially coherent sources, Gaussian Schell-model sources are the pioneering light sources in spatial [7,8], temporal [9,10] and spatiotemporal [11] domains. Besides, other partially coherent beams [12–17], pulses [18–20], and pulsed beams [21,22] have also been introduced with emerging properties. By introducing the study of HBT effect with partially coherent sources, unique and interesting applications emerge, such as measuring the curvature [23], mode indices [24] and recovering the size of the source [25]. Furthermore, it was observed that the random fluctuations of twisted light can correspond to the HBT effect [26]. HBT effect has also been studied by considering the degree of electromagnetic coherence [27], the influence of different source parameters [28,29], and the state of polarization [30]. A unified approach was introduced to explain the physics behind HBT effect and ghost imaging, and it was shown that both effects arise from intensity fluctuations [31]. In 2019, by using partially coherent Gaussian Schell-model sources, HBT effect and notion of scintillation were generalized through the concept of Stokes fluctuations [32], followed by the studies under the influence of the source parameters [33] at the focal plane and it was found that the strength of HBT correlations can be increased by a lens [34].

Currently, HBT effect has been explored for some partially coherent light sources in the pure spatial [2] or temporal cases [35]. It is known that the propagation effect of light may provide both spatial and temporal information, and the spatiotemporal properties of the correlated light have been paid attention [11,21,22]. However, to the best of our knowledge, there still lacks of such investigations on the HBT effect in spatiotemporal domain, which is very important for the potential applications in the correlated pulsed sources, like for partially coherent Gaussian Schell-model pulsed beams (GSMPBs) [11]. In this work, we consider the HBT effect in spatiotemporal domain and develop the theoretical method for calculating the spatiotemporal HBT effect by using the matrix-optics method under the paraxial approximation [36]. By using the partially coherent GSMPBs, we find the generalized analytical results of spatiotemporal HBT effect, which is suitable for any optical system with the second-order linear dispersion. Then as an example, we consider a simple two-dimensional case of the spatiotemporal HBT effect in air, which shows the influence of both the spatial and temporal source’s parameters on the spatiotemporal HBT effect. We believe that this study will be very useful for the study of HBT effect with spatial and temporal coordinates simultaneously and promote the applications of HBT effect in the spatiotemporal domain, like for the dynamic pulsed sources.

2. Theory and equations for spatiotemporal HBT effect

Figure 1 shows the schematic diagram for spatiotemporal HBT effect. Consider a partially coherent spatiotemporal pulsed source, which has the Gaussian statistics in spatial and temporal domains and whose fluctuations are jointly Gaussian random process at any spatiotemporal point, passing through the beam splitter, and it reaches at two detectors. Note that the beam splitter is not necessary here and it only provides two optical paths from the source to two detectors. We also assume that both detectors have the ability to collect the spatial and temporal information. The detectors can move in transverse planes and have the fast response in time. Here we only give out the requirements of possible detectors which can be helpful to realize this technique experimentally. From the fast response time, it means that the response time is comparable to (or shorter than) the characteristic time of intensity fluctuation so that it can collect the information of intensity fluctuation. The strength of intensity fluctuation in time is related with the source’s temporal coherence, so in that sense we used the term fast response time. It should be noted that the detectors response time usually should be much shorter than the source coherence time. From both detectors the output is correlated in the second-order correlator, $G^{(2)}(\boldsymbol {\rho }_{1}, \boldsymbol {\rho }_{2})$. The general theoretical frame of the second-order intensity correlation function between the two detectors is already explained for space-time case [8], and for example, it is valid for the pure spatial [26] or temporal case [37]. Here for any spatiotemporal correlated pulsed fields obeying the Gaussian statistics, the second-order intensity correlation function between the two detectors at different spatiotemporal points can be expressed by [8]

(1)$$G^{(2)}(\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})=\left\langle I_{1}(\boldsymbol{\rho }_{1})\right\rangle \left\langle I_{2}(\boldsymbol{\rho }_{2})\right\rangle +\left\vert \Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})\right\vert ^{2}.$$

Here $\boldsymbol {\rho }_{i}=(x_{i},y_{i},\tau _{i})^{\textrm {T }}$ with $i=1,2$ represent respectively the two spatiotemporal points at the observation planes D1 and D2, and the superscript "T" denotes the transposed operator. Here the temporal variable $\tau _{i}$ is defined by $\tau _{i}\equiv v_{g}t_{i}$ with a length dimension, and $t_{i}$ is the delay time coordinate given by $\tau _{i}=t_{s_{i}}-z_{i}/v_{g}$, where $t_{s_{i}}$ is the instant time and $z_{i}$ is the position of the observation plane. From these relations, it is well know that the origin of $\tau _{i}$ is moving as the position of $z_{i}$ changes, therefore it seems like a flying reference frame as light propagates with the group velocity $v_{g}$ and is called as the longitudinal flying reference frame. For the pulsed beams, one usually use the delay time in experiment, and it means that the origin of $t_{i}$ is the arrival time of the pulse center at $z_{i}$. For free space, this value of $v_{g}$ becomes the speed of light. For such sources, the spatiotemporal HBT effect can be connected with the normalized second-order correlation function of the intensity fluctuations $\Delta I_{1,2}(\boldsymbol {\rho }_{1,2})$ at two detectors, which is defined as [31]

(2)$$HBT(\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})=\frac{\left\langle \Delta I_{1}(\boldsymbol{\rho }_{1})\Delta I_{2}(\boldsymbol{\rho } _{2})\right\rangle }{\left\langle I_{1}(\boldsymbol{\rho }_{1})\right\rangle \left\langle I_{2}(\boldsymbol{\rho }_{2})\right\rangle }=\frac{\left\vert \Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})\right\vert ^{2}}{ \left\langle I_{1}(\boldsymbol{\rho }_{1})\right\rangle \left\langle I_{2}( \boldsymbol{\rho }_{2})\right\rangle },$$

where $\Gamma (\boldsymbol {\rho }_{1},\boldsymbol {\rho }_{2})$ in Eqs. (1) and (2) is the first-order spatiotemporal correlation function that depends on the spatiotemporal coordinates of both detectors, $I_{i}(\boldsymbol {\rho }_{i})$ is the instantaneous intensity of the pulsed beam arriving at the $i$th detector (with $i=1,2$), and $\left \langle \cdot \right \rangle$ denotes the ensemble average that is averaged over different realizations of the pulsed fields. Note that the normalized background constant term in the HBT correlation is unimportant, thus we only concern the correlation of the intensity fluctuation in Eq. (2). By taking into account the spatiotemporal integral for getting the information about spatial and temporal correlations of both detectors, the first-order spatiotemporal correlation between two detectors is given by

(3)$$\Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})=\iiint \iiint \Gamma _{0}(\boldsymbol{r}_{10},\boldsymbol{r}_{20})h_{1}(\boldsymbol{r}_{10}, \boldsymbol{\rho }_{1})h_{2}^{\ast }(\boldsymbol{r}_{20},\boldsymbol{\rho } _{2})d\boldsymbol{r}_{10}d\boldsymbol{r}_{20},$$

where $\Gamma _{0}\left ( \boldsymbol {r}_{10},\boldsymbol {r}_{20}\right )$ is the initial first-order correlation function of pulsed light fields at the source plane, the symbol “*” denotes complex conjugate, $h_{i}(\boldsymbol {r},\boldsymbol {\rho }_{i})$ with $i=1,2$, respectively, describe the spatiotemporal point spread functions (PSFs) of the linear optical systems on the paths 1 and 2, which can include any linear second-order dispersion optical systems [11], and $\boldsymbol {r}_{i0}=(x_{i0},y_{i0},\tau _{i0})^{\textrm {T}}$ represents arbitrary spatiotemporal point on the source plane. Similarly, the ensemble averaged intensities at D1 and D2 are given by

(4)$$\left\langle I_{i}(\boldsymbol{\rho }_{i})\right\rangle =\iiint \iiint \Gamma _{0}(\boldsymbol{r}_{10},\boldsymbol{r}_{20})h_{i}(\boldsymbol{r} _{10},\boldsymbol{\rho }_{i})h_{i}^{\ast }(\boldsymbol{r}_{20},\boldsymbol{ \rho }_{i})d\boldsymbol{r}_{10}d\boldsymbol{r}_{20},$$

and the spatiotemporal PSFs $h_{i}(\boldsymbol {r},\boldsymbol {\rho }_{i})$ for two paths in Fig. 1 are given by

(5)$$h_{i}(\boldsymbol{r},\boldsymbol{\rho }_{i})=\left( \frac{ik}{2\pi }\right) ^{\frac{m}{2}}[\det (\widetilde{\mathbf{B}}_{i})]^{-\frac{1}{2}}\exp \left[ - \frac{ik}{2}(\boldsymbol{r}^{\textrm{T}}\widetilde{\mathbf{B}}_{i}^{-1} \widetilde{\mathbf{A}}_{i}\boldsymbol{r}-2\boldsymbol{r}^{\textrm{T}} \widetilde{\mathbf{B}}_{i}^{-1}\boldsymbol{\rho }_{i}+\boldsymbol{\rho } _{i}^{\textrm{T}}\widetilde{\mathbf{D}}_{i}\widetilde{\mathbf{B}}_{i}^{-1} \boldsymbol{\rho }_{i})\right] .$$

Here $k$ is the wave number of the pulsed beam at its carrier frequency in the system, $m=$2, or 3 is the number of the dimensions for spatiotemporal points $\boldsymbol {r}_{i0}$ and $\boldsymbol {\rho }_{i}$. For $m=3$, it corresponds to the three-dimensional case: the two transverse spatial coordinates ($x$ and $y$) and another temporal dimension $\tau$. When $m=2$, it corresponds to the two-dimensional case: one transverse spatial coordinate ($x$ or $y$) and another temporal dimension $\tau$. Meanwhile $\widetilde {\mathbf {A}}_{i},\widetilde {\mathbf {B}}_{i},\widetilde {\mathbf {C}}_{i}$ and $\widetilde {\mathbf {D}}_{i}$ are $m{\times}m$ spatiotemporal ray characteristic matrices between the input and output planes of the optical systems under the paraxial approximation [11,22].

Fig. 1. Model for spatiotemporal HBT effect. Spatiotemporal light source splits into two light fields among which, one is detected by detector D1 and other is measured by detector D2. Matrices $\widetilde {\mathbf {A}}_{i}, \widetilde {\mathbf {B}}_{i},\widetilde {\mathbf {C}}_{i}$ and $\widetilde {\mathbf {D}}_{i}$ contain the information of optical path $i$ ($i = 1, 2$).

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By using Eqs. (3) and (5), we can write the first-order correlation in the most compact form as follows,

(6)$$\begin{array}{c} \Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})=\left( \frac{k}{2\pi } \right) ^{m}[(-1)^{m}\det \left( \overline{\mathbf{B}}\right) ]^{-\frac{1}{2} }\iiint \iiint d\boldsymbol{r}_{10}d\boldsymbol{r}_{20}\Gamma _{0}(\overline{ \boldsymbol{r}}_{0})\\ \times \exp \left[ -i\frac{k}{2}\binom{\overline{\boldsymbol{r}}_{0}}{ \overline{\boldsymbol{\delta }}_{12}}^{\textrm{T}}\left( \begin{array}{cc} \overline{\mathbf{B}}^{-1}\overline{\mathbf{A}} & -\overline{\mathbf{B}}^{-1} \\ \overline{\mathbf{C}}-\overline{\mathbf{D}}\overline{\mathbf{B}}^{-1} \overline{\mathbf{A}} & \overline{\mathbf{D}}\overline{\mathbf{B}}^{-1} \end{array} \right) \binom{\overline{\boldsymbol{r}}_{0}}{\overline{\boldsymbol{\delta }} _{12}}\right] , \end{array}$$

where $\overline {\boldsymbol {r}}_{0}\equiv \binom {\boldsymbol {r}_{10}}{ \boldsymbol {r}_{20}}$ and $\overline {\boldsymbol {\delta }}_{12}\equiv \binom { \boldsymbol {\rho }_{1}}{\boldsymbol {\rho }_{2}}$ are $2m\times 1$ matrices representing the two spatiotemporal points at the source plane, and the points on the planes of the detectors D1 and D2, respectively, and $\overline {\mathbf {A}}=\left ( \begin {smallmatrix} \widetilde {\mathbf {A}}_{1} & 0 \\ 0 & \widetilde {\mathbf {A}}_{2} \end {smallmatrix}\right )$, $\overline {\mathbf {B}}=\left ( \begin {smallmatrix} \widetilde {\mathbf {B}}_{1} & 0 \\ 0 & -\widetilde {\mathbf {B}}_{2} \end {smallmatrix}\right )$, $\overline {\mathbf {C}}=\left ( \begin {smallmatrix} \widetilde {\mathbf {C}}_{1} & 0 \\ 0 & -\widetilde {\mathbf {C}}_{2} \end {smallmatrix}\right )$, $\overline {\mathbf {D}}=\left ( \begin {smallmatrix} \widetilde {\mathbf {D}}_{1} & 0 \\ 0 & \widetilde {\mathbf {D}}_{2} \end {smallmatrix}\right )$ are $2m\times 2m$ matrices. Due to the scalar property of the exponential kernel, there are the following relations $( \overline {\mathbf {B}}^{-1}\overline {\mathbf {A}})^{\textrm {T}}=\overline { \mathbf {B}}^{-1}\overline {\mathbf {A}}$, $(\overline {\mathbf {D}}\overline { \mathbf {B}}^{-1})^{\textrm {T}}=\overline {\mathbf {D}}\overline {\mathbf {B}}^{-1}$, and $(\overline {\mathbf {C}}-\overline {\mathbf {D}}\overline {\mathbf {B}}^{-1} \overline {\mathbf {A}})^{\textrm {T}}=-(\overline {\mathbf {B}}^{-1})^{\textrm {T}}$. Similarly, from Eqs. (4) and (5), we can find the averaged intensities $\left \langle I_{i}(\boldsymbol {\rho }_{i})\right \rangle$ at D1 and D2, which can also be obtained via Eq. (6) by replacing $\overline {\mathbf {A}}$ with $\overline {\mathbf {A}}_{i}=\left ( \begin {smallmatrix} \widetilde {\mathbf {A}}_{i} & 0 \\ 0 & \widetilde {\mathbf {A}}_{i} \end {smallmatrix}\right )$, $\overline {\mathbf {B}}$ with $\overline {\mathbf {B }}_{i}=\left ( \begin {smallmatrix} \widetilde {\mathbf {B}}_{i} & 0 \\ 0 & -\widetilde {\mathbf {B}}_{i} \end {smallmatrix}\right )$, $\overline {\mathbf {C}}$ with $\overline {\mathbf {C }}_{i}=\left ( \begin {smallmatrix} \widetilde {\mathbf {C}}_{i} & 0 \\ 0 & -\widetilde {\mathbf {C}}_{i} \end {smallmatrix}\right )$, $\overline {\mathbf {D}}$ with $\overline {\mathbf {D }}_{i}=\left ( \begin {smallmatrix} \widetilde {\mathbf {D}}_{i} & 0 \\ 0 & \widetilde {\mathbf {D}}_{i} \end {smallmatrix}\right )$, and $\overline {\boldsymbol {\delta }}_{12}$ with $\overline {\boldsymbol {\rho }}_{ii}=\binom {\boldsymbol {\rho }_{i}}{ \boldsymbol {\rho }_{i}}$.

3. Generalized analytic results for partially coherent GSMPBs

In order to investigate the spatiotemporal HBT effect analytically, here the spatiotemporal correlated light source is a kind of partially coherent GSMPBs. In such sources both the intensity and the correlation lengths are Gaussian in spatial and temporal domain [11]. Furthermore, it is assumed that the source fields of GSMPBs are jointly Gaussian random process in spatial and temporal domain at any spatiotemporal point. Such pulsed beams could be possibly realized by modulating the polychromatic partially coherent beams with the Q-switching technology; or, they may be realized by using Gaussian pulsed beams passing through the random ground glass plate. The GSMPBs can be expressed by

(7)$$\begin{aligned} \Gamma (x_{10},y_{10},t_{10};x_{20},y_{20},t_{20}) &=\exp [i\omega _{0}(t_{10}-t_{20})]\exp \left[ -\frac{ (x_{10}^{2}+y_{10}^{2})+(x_{20}^{2}+y_{20}^{2})}{4\sigma _{\textrm{I}}^{2}}- \frac{t_{10}^{2}+t_{20}^{2}}{2\sigma _{\textrm{t}}^{2}}\right]\\ & \times \exp \left[ -\frac{(x_{10}-x_{20})^{2}+(y_{10}-y_{20})^{2}}{2\sigma _{\textrm{cs}}^{2}}-\frac{(t_{10}-t_{20})^{2}}{2\sigma _{\textrm{ct}}^{2}}\right] , \end{aligned}$$

where $\sigma _{\textrm {I}}$, $\sigma _{\textrm {cs}}$, $\sigma _{\textrm {t}}$ and $\sigma _{\textrm {ct}}$ are positive constants representing the spatial width, transverse coherence length, temporal width, and longitudinal temporal coherence length of GSMPBs, respectively. Physically speaking, for coherent pulsed beams, every pulse is identical and coherent, while for partially coherent situations, every pulse contains the random fluctuations in space and time, which can be characterized by the transverse spatial coherence and longitudinal temporal coherence in the correlation function. There is an experimental realization of temporal ghost imaging [38], which involves the ensemble average of a large number of temporal intensity fluctuations and may be helpful to realize this spatotemporal HBT effect proposed here. Equation (7) can also be written in the compact form [11]

(8)$$\Gamma (\overline{\boldsymbol{r}}_{0})=\exp \left[ -\frac{ik}{2}(\overline{ \boldsymbol{r}}_{0}^{\textrm{T}}\overline{\boldsymbol{Q}}^{-1}\overline{ \boldsymbol{r}}_{0})\right] ,$$

where $\overline {\boldsymbol {Q}}^{-1}=\left ( \begin {smallmatrix} \widetilde {\boldsymbol {\sigma }}_{1}, & \widetilde {\boldsymbol {\sigma }}_{2} \\ \widetilde {\boldsymbol {\sigma }}_{2}, & \widetilde {\boldsymbol {\sigma }}_{1}\end {smallmatrix}\right )$ is a $2m\:{\times}\:2m$ matrix. $\tilde{\boldsymbol{\sigma}}_{1}=\left(\begin{array}{cc}-\frac{i}{2 k} \boldsymbol{\sigma}_{\mathrm{I}}^{-2}-\frac{i}{k} \boldsymbol{\sigma}_{\mathrm{cs}}^{-2}, & \mathbf{0}_{2} \\ \boldsymbol{0}_{1}, & -\frac{i}{k} \sigma_{\tau}^{-2}-\frac{i}{k} \sigma_{c \tau}^{-2}\end{array}\right)$ and $\tilde{\boldsymbol{\sigma}}_{2}=\left(\begin{array}{cc}\frac{i}{k} \boldsymbol{\sigma}_{\mathrm{cs}}^{-2}, & \boldsymbol{0}_{2} \\ \boldsymbol{0}_{1}, & \frac{i}{k} \sigma_{c \tau}^{-2}\end{array}\right)$ are $m \times m$ submatrices. Here $\boldsymbol {\sigma }_{I}^{-2}=\left ( \begin {smallmatrix} \sigma _{I}^{-2} & 0 \\ 0 & \sigma _{I}^{-2}\end {smallmatrix}\right )$ and $\boldsymbol {\sigma }_{cs}^{-2}=\left ( \begin {smallmatrix} \sigma _{cs}^{-2} & 0 \\ 0 & \sigma _{cs}^{-2}\end {smallmatrix}\right )$ are the spatial properties of such pulsed beams, and $\boldsymbol {0}_{2}$ and $\boldsymbol {0}_{1},$ are $(m-1)\times$1 and 1$\times (m-1)$ zero matrices, respectively. It should be noted that in matrix $\overline {\boldsymbol {Q}}^{-1}$ the subscript $\tau$ is representing the temporal part with the same unit as the spatial parts, which ensures all units in the elements of $\overline {\boldsymbol {Q}}^{-1}$ to be the same. If the light source has the coupling effect between spatial and temporal coordinates, then the off-diagonal elements of $\overline {\boldsymbol {Q}}^{-1}$ will be modified to describes the coupling characteristics [11]. In this work, we have not considered the coupling effect between spatial and temporal coordinates [11], thus all the off-diagonal elements of $\overline {\boldsymbol {Q}}^{-1}$ are zero.

By putting Eq. (8) into Eq. (6), one can find the generalized solution for the first-order correlation,

(9)$$|\Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})|^{2}=\left\vert [\det (\overline{\mathbf{A}}+\overline{\mathbf{B}}\overline{\boldsymbol{Q}} ^{-1})]^{-\frac{1}{2}}\exp \left\{ -i\frac{k}{2}[\overline{\boldsymbol{ \delta }}_{12}^{\textrm{T}}(\overline{\mathbf{C}}+\overline{\mathbf{D}} \overline{\boldsymbol{Q}}^{-1})(\overline{\mathbf{A}}+\overline{\mathbf{B}} \overline{\boldsymbol{Q}}^{-1})^{-1}\overline{\boldsymbol{\delta }} _{12}]\right\} \right\vert ^{2}.$$

Similarly, the ensemble averaged intensities at detectors D1 and D2 can be finally given by

(10)$$\left\langle I_{i}(\boldsymbol{\rho }_{i})\right\rangle =\boldsymbol{[\det }( \overline{\mathbf{A}}_{i}+\overline{\mathbf{B}}_{i}\overline{\boldsymbol{Q}} ^{-1})]^{-\frac{1}{2}}\exp \left\{ -i\frac{k}{2}[-\overline{\boldsymbol{\rho }}_{ii}^{\textrm{T}}\overline{\mathbf{B}}_{i}^{-1}(\overline{\mathbf{A}}_{i}+ \overline{\mathbf{B}}_{i}\overline{\boldsymbol{Q}}^{-1})^{-1}\overline{ \boldsymbol{\rho }}_{ii}]\right\} .$$

The above results in Eqs. (9) and (10) are the main results of this work, which are the generalized results for considering the HBT effect in the spatiotemporal domain by using GSMPBs as a source, and these equations are suitable for any linear optical systems including the second-order dispersion optical systems. By using the definitions of ABCD matrices for linear optical systems [36], one can readily obtain the properties of spatiotemporal HBT effect under different parameters of sources.

Mainly, there are two definitions for finding the visibility of the second-order correlations, which are introduced by Cao et al. [39] and Gatti et al. [40], respectively. Although the differences in the numerical values of visibility from both definitions are possible, the increasing or decreasing trends are similar for both cases. Therefore we will follow the definition introduced in [40]. When the source follows the Gaussian statistics in both spatial and temporal coordinates we can get the definition of visibility of spatiotemporal HBT effect, given by [40]

(11)$$V=\frac{|\Gamma (\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})|_{\max }^{2} }{G^{(2)}(\boldsymbol{\rho }_{1},\boldsymbol{\rho }_{2})_{\max }},$$

where the numerator and denominator in Eq. (11) represent the maximal value of the modular square of the first-order correlation and the maximal value of the second-order correlation, respectively. The first term in Eq. (1) is not a major factor for controlling the visibility, which is known as the background term, and the second term in Eq. (1) (i.e., the numerator) provides the main characteristics of spatiotemporal HBT effect.

4. Example and numerical results

In order to have a deep insight of these equations, here we consider a simple example in which the source has the spatiotemporal Gaussian statistics, and both optical paths are in the same homogeneous medium and the distances $z_{i}$ from the source plane to the detector planes are same, i.e., $z_{1}=z_{2}=z$. For simplicity, we consider the case with two dimensions only. One is the temporal coordinate and the other is the transversal spatial coordinate in the $x$ direction. Since $x$ and $y$ directions are independent, reducing the effect in the $y$ direction does not affect the physics in the results. Under these considerations and assumptions, by putting Eqs. (9)–(10) into Eq. (2), after tedious but straightforward calculations, one can find

(12)$$HBT(x_{1},t_{1};x_{2},t_{2})=\exp \left[ -\frac{(x_{1}-x_{2})^{2}}{\Delta _{s}^{2}}-\frac{(t_{1}-t_{2})^{2}}{\Delta _{t}^{2}}\right] ,$$

where $\Delta _{s}^{2}=\sigma _{\textrm {cs}}^{2}+(b_{x}^{2}/k^{2}\sigma _{ \textrm {I}}^{2})(1+\sigma _{\textrm {cs}}^{2}/4\sigma _{\textrm {I}}^{2})$, $\Delta _{t}^{2}=\sigma _{\textrm {ct}}^{2}+(2b_{\tau }^{2}/k^{2}v_{g}^{4}\sigma _{ \textrm {t}}^{2})(1+\sigma _{\textrm {ct}}^{2}/2\sigma _{\textrm {t}}^{2})$, and we have used the two-dimensional spatiotemporal transfer matrices as $\widetilde {\mathbf {A}}_{1}=\widetilde {\mathbf {A}}_{2}=\left ( \begin {smallmatrix} 1 & 0 \\ 0 & 1 \end {smallmatrix}\right )$, $\widetilde {\mathbf {B}}_{1}=\widetilde {\mathbf {B} }_{2}=\left ( \begin {smallmatrix} b_{x} & 0 \\ 0 & b_{\tau } \end {smallmatrix}\right )$, $\widetilde {\mathbf {C}}_{1}=\widetilde {\mathbf {C} }_{2}=\left ( \begin {smallmatrix} 0 & 0 \\ 0 & 0 \end {smallmatrix}\right )$, $\widetilde {\mathbf {D}}_{1}=\widetilde {\mathbf {D} }_{2}=\left ( \begin {smallmatrix} 1 & 0 \\ 0 & 1 \end {smallmatrix}\right )$, with $b_{x}=z/n$ and $b_{\tau }=-\beta _{2}c\omega z$, and $v_{g}=c/(n+2\beta _{2}\omega c)$ in this example. Here $\beta _{2}$ is the group-velocity dispersion of a homogeneous medium, $\omega$ is the angular frequency of light at its carrier frequency, $c$ is the speed of light in vacuum, and $n$ is the refractive index of the medium. From Eq. (12) we can observe that the HBT effect have the Gaussian distribution in spatiotemporal domain, which depends on the properties of the source and the medium. The value of HBT is maximal when both detectors have the same coordinates along the space and time axes of the pulsed beams. Further, the HBT effects in spatial and temporal domain are independent for such partially coherent pulsed beams as there is no coupling effect between spatial and temporal coordinates, see Eq. (7). When the ratios of $b_{x}^{2}/k^{2}\sigma _{\textrm {I}}^{2}$ and $2b_{\tau }^{2}/k^{2}v_{g}^{4}\sigma _{\textrm {t}}^{4}$ are small, the HBT effects are purely determined by the spatial and temporal coherence lengths $\sigma _{ \textrm {cs}}$ and $\sigma _{\textrm {ct}}$. However, as long as the values $b_{x}^{2}/k^{2}\sigma _{\textrm {I}}^{2}$ and $2b_{\tau }^{2}/k^{2}v_{g}^{4}\sigma _{\textrm {t}}^{4}$ are large enough even for the incoherent pulsed beams (with $\sigma _{\textrm {cs}}<<\sigma _{\textrm {I}}$ and $\sigma _{\textrm {ct}}<<\sigma _{\textrm {t}}$), one can still observe the prominent HBT effect in both space and time domains.

For demonstrating the influence of both the spatial and temporal parameters of the source on the HBT effect, we assume that the pulsed source with $\omega$ = 2.355 rad/fs, propagates in air with $n=1.00028$ and $\beta _{2}=0.021233$ ps$^{2}$km$^{-1}$. The detector D2 is fixed at $(x_{2}=0,t_{2}=0)$, which requires that the detector D2 should be fixed in space, and it should also be triggered with a fast response exactly when the pulsed beam arrives at D2. It means that when the center of the pulse arrives at the position of the detector D2, the detector D2 should record the intensity information instantaneously. In practice the detector D2 can be triggered by a series of signal in time sequence, which can be generated from a digital signal generator. For the detector D1, it can move in the $x_{1}$ direction with high resolution in time domain. Figure 2 shows the spatiotemporal HBT effect under different parameters of the source. It is observed that the value of HBT effect is maximum at $x_{1}=0$, and the larger area with the bigger values of HBT makes it easy to realize this effect experimentally. Figures 2(a) to (d) show the dependence of spatiotemporal HBT effect on different spatial beam widths $\sigma _{\textrm {I}}$. As $\left \vert x_{1}\right \vert$ increases, the HBT correlation decreases very quickly for large $\sigma _{ \textrm {I}}$. One can find that by decreasing $\sigma _{\textrm {I}}$, the HBT effect increases in spatial domain. Thus, it is better to observe the HBT effect in the far-field region, corresponding to the large value of $b_{x}^{2}/k^{2}\sigma _{\textrm {I}}^{2}$. This property is in agreement with the HBT effect studied in the spatial domain [31]. From Figs. 2(e) to (h), with increasing the transverse coherence length, the increase in the HBT correlation is very prominent along the spatial coordinate under a fixed propagation distance. So the spatial parameters of such correlated pulsed beams still play an important role in the HBT effect in spatiotemporal domain.

Fig. 2. Spatiotemporal HBT effect under different values of (a)-(d) $\sigma _{\textrm {I}}$, (e)-(h) $\sigma _{\textrm {cs}}$, (i)-(l) $\sigma _{\textrm {t}}$ and (m)-(p) $\sigma _{\textrm {ct}}$ . Other parameters are (a)-(h) $\sigma _{\textrm {I}} = 15$ mm, $\sigma _{\textrm {cs}} = 0.1$ mm, $\sigma _{\textrm {t}} = 10$ ps, $\sigma _{\textrm {ct}} = 10$ ps, $z = 100$ m, and (i)-(p) $\sigma _{\textrm {I}} = 1000$ mm, $\sigma _{\textrm {cs}} = 100$ mm, $\sigma _{\textrm {t}} = 5$ ps, $\sigma _{\textrm {ct}} = 5$ ps, $z = 10^{6}$ m.

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There are similar effects of the temporal parameters $\sigma _{\textrm {t}}$ and $\sigma _{\textrm {ct}}$ on the spatiotemporal HBT effect, as shown in Figs. 2(i) to (p). Since $\beta _{2}$ is very small, we take the large distances from the source to two detectors in order to observe the second-order dispersion effect on the pulse duration. As $\sigma _{\textrm {t}}$ decreases, the value $2b_{\tau }^{2}/k^{2}v_{g}^{4}\sigma _{\textrm {t}}^{4}$ gradually becomes dominant. Thus the spatiotemporal HBT effect along the longitudinal direction (temporal domain) becomes more and more prominent. Of course, the HBT effect in temporal domain can also be enhanced by increasing the temporal coherence length $\sigma _{\textrm {ct}}$. From Figs. 2(i) to (p), we can conclude that both the small $\sigma _{\textrm {t}}$ and the large $\sigma _{\textrm {ct}}$ of the source are good for realizing spatiotemporal HBT effect.

When $\widetilde {\mathbf {B}}_{1}\neq \widetilde {\mathbf {B}}_{2}$ with $z_{1}\neq z_{2}$, one can analytically derive the visibility of spatiotemporal HBT effect as follows

(13)$$V=\frac{\xi ^{-\frac{1}{2}}}{(\alpha _{bx11}\alpha _{b\tau 11}\alpha _{bx22}\alpha _{b\tau 22})^{-\frac{1}{2}}+\xi ^{-\frac{1}{2}}},$$

where $\xi =(\alpha _{bx12}^{2}+\gamma _{bx12})(\alpha _{b\tau 12}^{2}+\gamma _{b\tau 12})$, and $\alpha _{bxpq}=1+(b_{xp}b_{xq}/4k^{2}\sigma _{\textrm {I}}^{4}\sigma _{\textrm {cs} }^{2})(4\sigma _{\textrm {I}}^{2}+\sigma _{\textrm {cs}}^{2})$, $\alpha _{b\tau pq}=1+(b_{\tau p}b_{\tau q}/k^{2}v_{g}^{4}\sigma _{\textrm {t}}^{4}\sigma _{ \textrm {ct}}^{2})(2\sigma _{\textrm {t}}^{2}+\sigma _{\textrm {ct}}^{2})$, $\gamma _{bx12}=(b_{x2}-b_{x1})^{2}(2\sigma _{\textrm {I}}^{2}+\sigma _{\textrm {cs} }^{2})^{2}/4k^{2}\sigma _{\textrm {I}}^{4}\sigma _{\textrm {cs}}^{4}$ and $\gamma _{b\tau 12}=(b_{\tau 2}-b_{\tau 1})^{2}(\sigma _{\textrm {t}}^{2}+\sigma _{ \textrm {ct}}^{2})^{2}/k^{2}v_{g}^{4}\sigma _{\textrm {t}}^{4}\sigma _{\textrm {ct} }^{4}$ ($p$ = 1, 2, and $q$ = 1, 2). From Eq. (13), one can find the dependence of visibility on the source parameters and the distances $z_{1}$ and $z_{2}$, and there is the largest visibility (50%) at $z_{1}=z_{2}$, i.e., $b_{x2}=b_{x1}$ and $b_{\tau 2}=b_{\tau 1}$.

Figure 3 shows the visibility of spatiotemporal HBT effect as a function of $z_{1}/z_{2}$ under different parameters. It is observed that the large spatial or temporal widths may decrease the visibility at $z_{1}\neq z_{2}$, while the large values of transverse spatial or longitudinal temporal coherence lengths will increase the visibility. Thus the choice of source parameters is very important for better observation of spatiotemporal HBT effect.

Fig. 3. Visibility of Spatiotemporal HBT effect for different values of (a) $\sigma _{\textrm {I}}$, (b) $\sigma _{\textrm {cs}}$, (c) $\sigma _{\textrm {t}}$, (d) $\sigma _{\textrm {ct}}$, with $\sigma _{\textrm {I}}$ = 900 mm and 150 mm in (b) and (c)-(d), respectively. Other parameters are $\sigma _{\textrm {cs}}$ = 100 mm, $\sigma _{\textrm {t}}$ = 20 ps, $\sigma _{\textrm {ct}} = 5$ ps and $z = 10^{6}$ m.

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5. Conclusion

In conclusion, using a spatiotemporal source with Gaussian statistics and considering the spatiotemporal integral we have derived the generalized analytical result for spatiotemporal HBT effect, which can be used for any second-order linear dispersive optical system. With examples, we show that the HBT effect and its visibility can be increased by increasing the coherence length or by decreasing the source width in spatial or temporal domain. Our results realize the importance of the spatiotemporal information which can be considered for the study of HBT effect for getting deep insight over the source information. The spatiotemporal HBT effect may provide the possibility to investigate the dynamic pulsed sources in astronomy, atomic physics, particle physics and condensed matter physics, and it can help people to obtain the spatial and temporal information of the sources simultaneously.

Funding

National Natural Science Foundation of China (11674284, 11974309); Natural Science Foundation of Zhejiang Province (LD18A040001); National Key Research and Development Program of China (2017YFA0304202); Fundamental Research Funds for the Central Universities (2019FZA3005).

Disclosures

The authors declare no conflicts of interest.

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Hanbury Brown and Twiss effect in spatiotemporal domain

Abstract

1. Introduction

2. Theory and equations for spatiotemporal HBT effect

3. Generalized analytic results for partially coherent GSMPBs

4. Example and numerical results

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (3)

Equations (13)

Optics Express