## Abstract

Spatiotemporal coupling is present when the electric fields of any pulsed beams fail to separate into a product of purely spatial and temporal factors, and it affects ultimately on the propagation of the light. Here we study the effect of spatiotemporal coupling on the Hanbury Brown and Twiss (HBT) effect for the first time, with the help of our recent developed method [Opt. Express **28**, 32077 (2020) [CrossRef] ], by assuming the Gaussian statistics of partially coherent spatiotemporal pulsed sources containing the intensity and correlation coupling effect between the spatial and temporal domain. The generalized results for the spatiotemporal coupling HBT effect are investigated and through a nontrivial two-dimensional case, we numerically illustrate the influence of the spatiotemporal coupling on the HBT effect. It observes that even a very tiny coupling effect will strongly change the HBT effect at longer distances. This work will be potentially useful for the study of the HBT effect (intensity-intensity correlations) with dynamic sources having spatiotemporal coupling in both optics and other branches of physics.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The importance of spatiotemporal coupling (STC) has been largely overlooked in optics as the light field is generally assumed non-coupled. However, the electric field for an ultrashort pulse usually contains the STC term which plays an important role in physical effects [1]. One finds that the spatial properties of such ultrashort light pulse sources depends upon time and vice versa. STC is normally generated when light pulses propagate through lenses [2], prisms [3], diffraction gratings [4], apertures [5,6], nonlinear media [7] etc., so in the experiments with optical instruments it is difficult to avoid STC.

Recently, many detailed studies of STC have been introduced in which all possible types of couplings including pulse-front tilt, spatial chirp, angular dispersion, time versus angle and other have been explained [8]. It was suggested that when STC is present then the pulse must be described at least in two domains like space and time domain or other three combinations with frequency and wave vector. Different kinds of coupling effects can be found by simple Fourier transform from one pair of domains to the other. So even the two domains like space and time can be useful to study the coupling effects and the other coupling effects can be found from these [8]. Introduction and control of STC allows much better longitudinal resolution in microscopy experiments than usual techniques [9]. By utilizing the pulse front tilt, broadband nearly single-cycle terahertz pulses have been generated [10]. Attosecond light houses [11,12] and the control on the velocity of ultrashort pulses [13] have been introduced by using STC. By modulating the transverse spatiotemporal dispersion, the honeycomb beam and the picket-fence beam, are generated in the spatiotemporal domain and it is observed that STC can provide new potentials for coherent control in optics [14]. Very recently, STC induced by the misalignment of mirrors and gratings in the parallel grating pair pulse stretcher is analysed by using the ray tracing simulation method and an adjustment procedure is introduced to accurately calibrate these misalignments [15]. The coupling in space-frequency or space-time twisted beams for partially coherent light fields has been suggested recently by Hyde and their behaviors have been discussed during the propagation [16]. Matrix optics methods have also been used for the study of STC [17–20].

It is known that while passing through optical media the coupling effect can be generated and influence the propagation effect of light. However, to the best of our knowledge, there is no investigation for the influence of STC on the HBT effect specifically with partially coherent sources. Here we mentioned partially coherent sources specifically due to their coverage from fully coherent to completely non-coherent cases which makes them hot topic of research in temporal [21–26], spatial [27–34] and spatiotemporal domain [20,35–37] in recent years. Recently we have extended the phenomenon of HBT from spatial [38,39] to spatiotemporal domain [40]. After realizing the importance of the STC which can be generated even due to light propagation in optical media, in this work we extend our recent study of HBT effect in spatiotemporal domain by assuming the source containing spatiotemporal coupling and observe how it can influence the HBT effect. Here we do not go in details of the origin of the STC but only focus on the effect of STC on HBT correlations. We introduce the STC matrix in the compact form of partially coherent Gaussian Schell-model pulsed beams (GSMPBs) and used the generalized results for HBT effect in spatiotemporal domain which is suitable for any optical system with the second-order linear dispersion [40], by using matrix optics methods under the paraxial approximation [41]. By considering two-dimensional case in air, we observe the influence of STC on HBT effect with the help of numerical examples. This study actually helps to approach the maximum possible accuracy of the results for HBT correlations as it highlights the influence of STC, which can be present even due to the laboratory instruments used in the measurements of optical effects.

## 2. Partially coherent GSMPBs with spatiotemporal coupling

Partially coherent GSMPBs for the non-coupling case have been explained in the literature in the compact form with the help of matrix optics theory (for detailed derivation of compact form please see [20,40] ). Such sources could be realizable with the help of Q-switching technology or by using the modulated random ground glass plate. Here we will explain the case when such light source goes through any medium which causes coupling effects in intensity and correlation between spatial and temporal domain. It should be noted that we are not going in details of any specific medium which produces the coupling effect, rather we will treat generalized case for the study of STC effect on the HBT effect. As we know that every pulse is identical and coherent for coherent pulsed beams, while for partially coherent situations, every pulse is different and it 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. For such a partially coherent pulsed source, both the intensity and the correlation obey Gaussian statistics in spatial and temporal domain, and under the paraxial slowly varying envelope approximation, the correlation function of such pulsed sources can be expressed as [20]

*longitudinal flying reference frame*. For free space, this value $v_{g}$becomes the speed of light. The parameters $\sigma _{\textrm {I}}$, $\sigma _{\textrm {cs}}$, $\sigma _{\textrm {t}}=\sigma _{\tau }/v_{g}$ and $\sigma _{\textrm {ct}}=\sigma _{c\tau }/v_{g}$ are positive constants representing the spatial width, transverse coherence length, temporal width, and longitudinal temporal coherence length of GSMPBs, respectively. It should be noted that in the 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. Here we emphasize that the matrix $\overline {\boldsymbol {Q}}^{-1}$ is completely different from the one explained in Ref. [40], and the off diagonal elements in $\overline {\boldsymbol {Q}}^{-1}$ are non-zero, which explains the spatiotemporal coupling case. At the same time, by the characteristics of the correlation function $\Gamma ^{\ast }(\overline {\boldsymbol {r}}_{0})=\Gamma (\overline {\boldsymbol {r}}_{0})$, there exists the relations $\boldsymbol {q}_{12}=\boldsymbol {q}_{21}^{T}=\boldsymbol {q}_{34}=\boldsymbol {q}_{43}^{T}$ and $\boldsymbol {q}_{14}=\boldsymbol {q}_{41}^{T}=\boldsymbol {q}_{32}=\boldsymbol {q}_{23}^{T}$, the superscript “T” denotes the transposed operator. We call the parameters $\boldsymbol {q}_{12}$, $\boldsymbol {q}_{21}$, $\boldsymbol {q}_{34}$ and $\boldsymbol {q}_{43}$ the spatiotemporal intensity self-coupling at the same spatiotemporal point and the parameters $\boldsymbol {q}_{14}$, $\boldsymbol {q}_{41}$, $\boldsymbol {q}_{23}$, and $\boldsymbol {q}_{32}$ the spatiotemporal correlation mutual-coupling between any two different spatiotemporal points [20]. The later mutual-coupling between two different spatiotemporal points is similar to the cases considered in the recent work [16]. Actually we will see that both the spatiotemporal intensity self-coupling and correlation mutual-coupling play the important role in the propagation of the intensity correlation. Here we do not consider the coupling between spatial points, if there exist such coupling then the off-diagonal elements of matrices $\boldsymbol {\sigma }_{I}^{-2}$ and $\boldsymbol {\sigma }_{cs}^{-2}$ will also be nonzero.

## 3. Model for the HBT effect with spatiotemporal coupling

Figure 1 shows the schematic diagram for spatiotemporal HBT effect with a partially coherent source having STC. Consider a partially coherent pulsed source with STC, which has the Gaussian statistics in spatial and temporal domains and whose fluctuations are jointly Gaussian random process at any spatiotemporal point, light from the source reaches at 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 their response time 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 theoretical frame of the second-order intensity correlation function between the two detectors is already explained [42–44]. For any spatiotemporal correlated pulsed fields obeying the Gaussian statistics, the second-order intensity correlation function between the two detectors on different spatiotemporal points $\boldsymbol {\rho }_{1,2}$ at the observation planes D1 and D2 can be expressed by [42]

Here $\boldsymbol {\rho }_{i}=(x_{i},y_{i},\tau _{i})^{\textrm {T}}$. For such sources, the normalized second-order correlation function (the spatiotemporal HBT effect) of the intensity fluctuations $\Delta I_{1,2}(\boldsymbol {\rho }_{1,2})=I_{1,2}(\boldsymbol {\rho }_{1,2})-\left \langle I_{1,2}(\boldsymbol {\rho }_{1,2})\right \rangle$ at two detectors is defined as [45]

## 4. Example and numerical results

Here we will demonstrate the above generalized results numerically by considering two dimensional case for simplicity, one dimension is in spatial domain along $x$ direction and another dimension is the temporal domain. The source has Gaussian statistics in spatiotemporal domain and contains the coupling effect between spatial and temporal domain. For easy understanding we further simplify the case by considering both optical paths are in the same homogeneous medium and air is considered as homogeneous medium for light propagation. The distances from the source plane to the planes of detectors D1 and D2 are same, i.e., $z_{1}=z_{2}=z$. Now we can define the spatiotemporal transfer matrices under above mentioned assumptions 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 $b_{x}=z/n$, $b_{\tau }=-\beta _{2}c\omega z$ while $n$, $\beta _{2}$, $c$, $\omega$ representing the refractive index of the medium, group-velocity dispersion of a homogeneous medium, the speed of light in vacuum, angular frequency of light at its carrier frequency, respectively and group velocity is defined as $v_{g}=c/(n+2\beta _{2}\omega c)$. Although we have simplified the case for the numerical example, since the matrix $\overline {\boldsymbol {Q}}^{-1}$ contains non-zero elements at non diagonal places, so it is hard to get a concise analytical solution for HBT effect. According to the definitions, $\widetilde {\mathbf {A}}_{i}$, $\widetilde {\mathbf {C}}_{i}$ and $\widetilde {\mathbf {D}}_{i}$ transmission submatrices are constant for homogeneous medium, so their influence on HBT effect will not be obvious according to Eqs. (4) and (5). Since there exists spatiotemporal coupling which resides in matrix $\overline {\boldsymbol {Q}}^{-1}$, the spatial and temporal coordinates of HBT will be greatly affected due to this coupling as the matrix $\overline {\boldsymbol {Q}}^{-1}$ is multiplying with the arbitrary spatiotemporal coordinates in Eqs. (4) and (5).

For the sake of numerical results, we assume that the angular frequency of the pulsed beams is 2.355 rad/fs, the refractive index is 1.00028 and its group velocity dispersion is 0.021233 ps$^{2}$km$^{-1}$ for the medium of air taken from literature [36]. We also fix the detector D2 in the space at $x_{2}=0$ while, for the temporal part the detector D2 will be triggered with a fast response on the arrival of pulsed beams so the temporal position of it is also $t_{2}=0$. The detector D1 can move along $x$-axis spatially with temporally high resolution.

The relations $\boldsymbol {q}_{12}=\boldsymbol {q}_{21}^{T}=\boldsymbol {q}_{34}=\boldsymbol {q}_{43}^{T}$ and $\boldsymbol {q}_{14}=\boldsymbol {q}_{41}^{T}=\boldsymbol {q}_{32}=\boldsymbol {q}_{23}^{T}$ are equal respectively, so we will only consider $\boldsymbol {q}_{12}$ and $\boldsymbol {q}_{14}$ to elaborate the influence of spatiotemporal intensity self-coupling at the same spatiotemporal point and spatiotemporal correlation mutual-coupling between any two different spatiotemporal points, respectively on HBT effect. Figure 2 shows the spatiotemporal HBT effect under different parameters of the STC. It is observed that the value of HBT effect is maximum at $x_{1}=0$ and $t_{1}=0$, 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 spatiotemporal intensity self-coupling at the same point ($q_{12}$). As $q_{12}$ increases, the tilt of spatiotemporal HBT correlation increases very quickly in anti-clockwise direction up to a certain angle. It can be observed that the intensity self-coupling can affect the results of HBT effect and it can not be ignored. From Figs. 2(e) to (h), with increasing the spatiotemporal correlation mutual-coupling between any two spatiotemporal points, the increase of the tilt in the HBT correlation appears in clockwise direction and then the peaks starts expanding at a certain angle, further increase in $q_{14}$ will expand the HBT peak without any increase in tilt angle. So the spatiotemporal correlation mutual-coupling between any two spatiotemporal points of such correlated pulsed beams play an important role in the HBT effect in spatiotemporal domain. There are interesting effects of the spatiotemporal coupling parameters $q_{12}$ and $q_{14}$ on the spatiotemporal HBT effect, when both are present simultaneously as shown in Figs. 2(i) to (p). As $q_{12}$ is small as compare to $q_{14}$ the tilt appears in clockwise direction, when both $q_{12}$ and $q_{14}$ are equal then the tilt completely disappears, as the $q_{12}$ increases than $q_{14}$, the tilt appears in anti-clockwise direction. From Figs. 2(i) to (p), we can conclude that both the $q_{12}$ and the $q_{14}$ at the source plane can affect the spatiotemporal HBT effect at output plane and should be considered for the study of HBT effect.

Figure 3 shows the spatiotemporal HBT effect with different distances from the source plane to the detectors under different values of STC. Figures 3(a) to (d) show the dependence of spatiotemporal HBT effect at different values $z$ with very small values of STC. It can be observed that when the coupling effect is very small the tilt in the HBT correlations appears at far away distances from the source plane. This is in accordance with Eqs. (4) and (5). From Figs. 3(e) to (p) it shows that as the values of $q_{12}$ and $q_{14}$ increase the tilt appears near to the source plane. From Fig. 3 one can observe that the effect of STC on HBT effect can be enhanced by increasing propagation distance.

Figure 4 shows the spatiotemporal HBT effect under different spatial parameters of the source with and without STC. Figures 4(a) to (d) show the dependence of spatiotemporal HBT effect on different values of spatial beam width of the source without coupling effect. As $\sigma _{\textrm {I}}$ decreases, the spatiotemporal HBT correlation increases very quickly in spatial domain. Figures 4(e) to (h) show the dependence of spatiotemporal HBT effect on different values of spatial beam width of the source with coupling effects. Here the effect of spatiotemporal coupling is obvious on the HBT effect which appears in the form of tilt. From Figs. 4(a) to (h) we find that the effect of STC influence the HBT correlation in the form of tilt. From Figs. 4(i) to (l), with increasing the spatial coherence of the source without coupling, the increase in the HBT correlation appears in spatial domain. With the same parameters Figs. 4(m) to (p) show the HBT effect influenced by the STC. In Fig. 4 we see the tilt in the HBT effect due to STC, as expected.

Figure 5 shows the spatiotemporal HBT effect under different temporal parameters of the source with and without STC. Since $\beta _{2}$ is very small, we take the large distances from the source to two detectors in order to observe the influence of temporal and coupling parameters. Figures 5(a) to (d) show the dependence of spatiotemporal HBT effect on different values of temporal pulse width of the source without coupling effect. As $\sigma _{\textrm {t}}$ decreases, the spatiotemporal HBT correlation increases in temporal domain but in Figs. 5(a) to (c) this increase is very minor which is not very obvious in the absence of coupling. Figures 5(e) to (h) show the dependence of spatiotemporal HBT effect on different values of temporal pulse width of the source with STC effects. The effect of coupling appears as the tilt of HBT correlation as well as very prominent expansion of the HBT results, although the coupling is very small as in Fig. 4 but the expansion is mainly due to the large distance from the source plane. It can be observed that without coupling there was very minute change in the spatiotemporal HBT effect as in Figs. 5(a) to (c), but with STC this change is very obvious as can be noted in Figs. 5(e) to (g). This comparison highlights that the STC can be useful for finding minute changes in the intensity-intensity correlations. Figures 5(i) to (l) show the effect of temporal coherence of the source without coupling, with increasing the temporal coherence the increase in the HBT correlation appears in temporal domain. With the same parameters, in Figs. 5(m) to (p) the effect of STC on HBT effect is shown, there is obvious change in the results due to presence of coupling. In Fig. 5 we observe that very small amount of STC can be very sensitive for intensity-intensity correlations results, sometimes at far away distances. Here we also find that by producing STC one can observe the minute changes in the intensity-intensity correlations, so STC can be used for zooming the tiny effects in the intensity-intensity correlations. By exploring these tiny coupling effects, this work will be helpful to measure spatiotemporal parameters in intensity interferometry more precisely.

## 5. Conclusion

In conclusion, a spatiotemporal source containing Gaussian statistics with STC is considered for the detail study of the HBT effect. We used the matrix optics method and consider a generalized matrix containing the effect of coupling between spatial and temporal domain. By considering air as an example for linear dispersive medium we observe the effect of intensity and correlation coupling, between spatial and temporal domain on the HBT effect. It is observed that both type of coupling can produce the tilt in the results of HBT effect but in opposite direction. By comparing the case with and without coupling it is found that the HBT effect is very sensitive to STC and even very minute coupling can influence the results at larger distances. Our results realize the importance of considering the STC for the study of HBT effect, which can be present in the source by any means. In this work we find an interesting application of STC for enhancing the tiny effects in intensity-intensity correlations. The importance of the considering STC will be useful for detail study of intensity-intensity correlations in other branches of physics like astronomy, atomic physics, particle physics and condensed matter physics.

## Funding

National Natural Science Foundation of China (11974309); Natural Science Foundation of Zhejiang Province (LD18A040001); National Key Research and Development Program of China (2017YFA0304202).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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