# Evolution of Spatiotemporal Intensity of Partially Coherent Pulsed Beams with Spatial Cosine-Gaussian and Temporal Laguerre–Gaussian Correlations in Still, Pure Water

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Method

_{1}and τ

_{2}are two time instants, cos(•) is the cosine function, and m is the beam order parameter. Further, L

_{n}(•) denotes the Laguerre polynomial of mode order n, ω

_{0}is the pulse’s carrier frequency, while w

_{0}and δ denote the r.m.s. spatial beam width and the r.m.s. beam transverse coherence width, respectively. T

_{0}and T

_{c}represent the pulse duration and the temporal coherence length, respectively. The pulsed beam expressed by Equation (1) is therefore the spatial cosine-Gaussian and temporal Laguerre–Gaussian correlated Schell-model (SCTLGSM) pulsed beam. The SCTLGSM pulsed beam reduces to the conventional Gaussian Schell-model (GSM) pulsed beam when m = 0 and n = 0. For m ≠ 0 and n ≠ 0, the spatial and the temporal coherence parts of the mutual coherence function are modulated by the cosine function and the Laguerre function, respectively. The details of the model can be found in the Appendix A.

_{0}) = n(ω

_{0})ω

_{0}/c denotes the wave number, in which n(ω

_{0}) is the refractive index of the medium at carrier frequency ω

_{0}, c is the speed of light in vacuum,

**r**

_{1}= (x

_{1}, y

_{1}) and

**r**

_{2}= (x

_{2}, y

_{2}) are position vectors in any plane z > 0 and t

_{1}and t

_{2}are two time instants of the pulse profile. Further, A

_{S}, B

_{S}and D

_{S}are the transfer matrix elements of the optical system in the spatial domain [48], while A

_{T}, B

_{T}and D

_{T}are those in the temporal domain [45].

_{2q}is Hermite polynomial of order 2q, and

_{a}ω + n

_{b}, where n

_{a}= β

_{2}c and n

_{b}= c/v

_{g}− 2β

_{2}ω

_{0}c. Here β

_{2}denotes the group velocity dispersion and v

_{g}is the group velocity of the pulse [49]. Furthermore, the time coordinate is a retarded time with respect to a frame moving with group velocity v

_{g}and s is the chirp coefficient of the pulse.

**r**

_{1}=

**r**

_{2}=

**r**and t

_{1}= t

_{2}= t in Equations (8) and (15), respectively. Then the spatiotemporal intensity of the SCTLGSM pulsed beam becomes

## 3. Spatiotemporal Intensity Evolution of the SCTLGSM Pulsed Beams in Water

_{g}= c/v

_{g}= 1.3591 and its group velocity dispersion coefficient is β

_{2}= 58.174 ps

^{2}km

^{−1}[50]. In the following calculation, the pulses’ and medium’s parameters are chosen to be w

_{0}= 2 mm, σ = 2 mm, T

_{0}= 4 ps, T

_{c}= 2 ps, ω

_{0}= 3.667 rad/fs (λ

_{0}= 514 nm [51]), unless different values are specified. Because of still water absorption [52] the propagation distances will be limited to 100 m throughout the text.

^{1/2}m/δk of z in the exponential function increases as well. Hence, the rate of change in the spatiotemporal intensity increases with the increasing propagation distance.

^{−2}and m = 0. One can see from Figure 5 that the beam exhibits self-splitting behavior when n > 0. Specifically, the beam splits into n + 1 sub-beams with increasing propagation distance z, being in good agreement with the results of the Laguerre–Gaussian Schell-model pulsed beam in time domain [31]. While for n = 0, i.e., for the conventional Gaussian-correlated Schell-model pulsed beam, there is no self-splitting upon propagation. These results can be interpreted as follows. For n = 0, the last term of Equation (21) can be re-expressed as

_{n}

_{=1}is a product of functions $\mathrm{exp}\left(-\phi {t}^{2}\right)$ and ${\xi}_{1}{t}^{2}+{\psi}_{1}$. Thus, letting

_{n}

_{=1}will obtain two symmetric intensity maxima at $t=\pm \sqrt{({\xi}_{1}-\phi {\psi}_{1})/\phi {\xi}_{1}}$ when ${\xi}_{1}\ne \phi {\psi}_{1}$. Therefore, the beam starts splitting into two sub-beams in Figure 5b. In addition, when z is very short, the influence of term ξ

_{1}t

^{2}+Ψ

_{1}can be neglected compared with that of exp(−φt

^{2}). This is why there is only one beam when z is much closer to the source plane. Similarly, for n = 2 and n = 3, we can obtain 3 intensity maxima and 4 intensity maxima in the far field, respectively.

^{−2}, −0.8ps

^{−2}, −1.2ps

^{−2}, and for n = 0, m = 0, x = 0, y = 0. It is shown that the self-focusing phenomenon takes place upon propagation when chirp coefficient s < 0, and it becomes more noticeable with decreasing chirp coefficient s, while the position of the focus shifts towards the source plane. This result can be explained as follows. When n = 0, m = 0, x = 0, y = 0, Equation (23) can be expressed as

_{min}will decrease, which can be seen from Equation (33), because for large |s|, $4/{T}_{0}^{2}+4/{T}_{c}^{2}$ can be omitted. Equation (33) can be expressed as z

_{min}≈ β

_{2}/|s| which is why the position of focus shifts towards the source plane with increasing |s|. Here the influence of s on propagation in the temporal ABCD matrix is equivalent to a lens. Hence, the self-focusing phenomenon appears.

^{−2}, −0.8ps

^{−2}, −1.2ps

^{−2}, and for n = 1, m = 0, x = 0, y = 0. It is shown that the self-focusing and the self-splitting phenomena take place at the same time when chirp coefficient s < 0. Due to the shift of the position of self-focusing, there are three intensity peaks appearing for larger |s|.

^{−2}, n = 0, m = 0, y = 0. As can be seen, there is a circular intensity distribution in the x–t plane for z = 0. With the increasing propagation distance z, the circular intensity distribution becomes elliptical (Figure 8c–e), where the self-focusing appears in the coordinate axis t. Figure 8f shows the normalized spatiotemporal intensity I(x, y, t, z) of the SCTLGSM pulsed beam as a function of propagation distance z for different values of the chirp coefficient s = 0, −0.4ps

^{−2}, −0.8ps

^{−2}, −1.2ps

^{−2}, where n = 0, m = 0, x = 0, y = 0. One can see that the self-focusing phenomenon becomes more and more noticeable with increasing chirp coefficient |s|. Moreover, for the case of m = n = 1, shown in Figure 9a–c, the beam splits first in the coordinate axis t, and then splits in the coordinate axis x (Figure 9d). In the far field, the beam splits into four sub-beam spots in the x–t plane. Figure 9f shows the normalized spatiotemporal intensity I(x, y, t, z) of the SCTLGSM pulsed beam as a function of propagation distance z for different chirp coefficient values: s = 0, −0.4ps

^{−2}, −0.8ps

^{−2}, −1.2ps

^{−2}, where n = 1, m = 1, x = 0, y = 0, t = 0. One can find that the self-focusing phenomenon is more noticeable for s = −1.2ps

^{−2}. Comparison of Figure 8 with Figure 9 implies that the beam orders m and n play an important role in determining the spatiotemporal intensity distribution in x–t plane.

^{−2}, y = 0. It is shown that no beam splitting occurs in the far field for m = n = 0. However, for m = n = 1, the initial beam starts to split into two sub-beams in coordinate axis t and coordinate axis x, respectively. With increasing beam order, such as m = n = 2, the initial beam starts to split into three sub-beams in coordinate axis t and two sub-beams in coordinate axis x, respectively. In addition, for m = n = 3, the initial beam starts to split into four sub-beams in coordinate axis t and two sub-beams coordinate axis x, respectively. These results are consistent with those of Figure 5.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{S}and p

_{T}are non-negative weight functions, and H(

**ρ**,

**ε**) and h(t, v) are arbitrary kernels in the space domain and the time domain, respectively. We assume that

_{S}and p

_{T}as follows:

_{n}is a Hermite polynomial of order n [31]. Here, we choose the mutual coherence functions in the spatial domain and in time domain from Refs. [12,31], respectively.

_{0}, temporal coherence length T

_{c}, spectral bandwidth Ω and the spectral coherence width Ω

_{c}are related by

**Figure A1.**(

**a**) Illustrating figure of the SCTLGSM pulsed beam propagating in water. (

**b**) Spatial degree of coherence of the pulsed beam source as a function of $\left({x}_{2}^{\prime}-{x}_{1}^{\prime}\right)/\delta $. Solid black: m = 0; Dashed red: m = 1; Dash-dotted green: m = 2; Dotted blue: m = 3. (

**c**) Temporal degree of coherence of the pulsed beam source as a function of $\left({\tau}_{2}-{\tau}_{1}\right)/{T}_{c}$. Solid black: n = 0; Dashed red: n = 1; Dash-dotted green: n = 2; Dotted blue: n = 3. (

**d**) Spectral density of the pulsed beam source as a function of (ω − ω

_{0})/Ω

_{0}. Solid black: Ω

_{c}/Ω

_{0}=1; Dashed red: Ω

_{c}/Ω

_{0}= 1; Dash-dotted green: Ω

_{c}/Ω

_{0}= 0.5; Dotted blue: Ω

_{c}/Ω

_{0}= 0.1.

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**Figure 1.**Evolution of spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam as a function of propagation distance z and horizontal coordinate x with different beam orders: (

**a**) m = 0, (

**b**) m = 1, (

**c**) m = 2 and (

**d**) m = 3.

**Figure 2.**Evolution of spatiotemporal intensity I(x, y, t, z) (Equation (20)) of spatial cosine-Gaussian and temporal Laguerre–Gaussian correlated Schell-model (SCTLGSM) pulsed beam as a function of propagation distance z and horizontal coordinate x with different time (

**a**) t = 2 ps and (

**b**) t = 5 ps, and for m = 1, s = 0.

**Figure 3.**Density plots for the normalized spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam in the x–y plane at some propagation distances: (

**a**) z = 0, (

**b**) z = 30 m, (

**c**) z = 40 m, (

**d**) z = 60 m and for t = 0, s = 0, m = 1.

**Figure 4.**Density plots for the normalized spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam in the x–y plane at some propagation distances (

**a**) z = 0, (

**b**) z = 30 m, (

**c**) z = 40 m, (

**d**) z = 60 m and for t = 0, s = 0, m = 2.

**Figure 5.**Evolution of spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam as a function of propagation distance z and time t with different beam orders (

**a**) n = 0, (

**b**) n =1, (

**c**) n =2 and (

**d**) n =3.

**Figure 6.**Evolution of spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam as a function of propagation distance z and time t with different chirp coefficients (

**a**) s = 0, (

**b**) s =−0.4 ps

^{−2}, (

**c**) s =−0.8 ps

^{−2}, (

**d**) s =−1.2 ps

^{−2}and for n = 0, m = 0.

**Figure 7.**Evolution of spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam as a function of propagation distance z and time t with different chirp coefficients (

**a**) s = 0, (

**b**) s =-0.4 ps

^{−2}, (

**c**) s =-0.8 ps

^{−2}, (

**d**) s =−1.2 ps

^{−2}and for n = 1, m = 0.

**Figure 8.**Density plots for the normalized spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam in the x–t plane at some propagation distances (

**a**) z = 0, (

**b**) z =20 m, (

**c**) z =30 m, (

**d**) z =40 m, (

**e**) z =50 m and for s = −0.4ps

^{−2}, n = 0, m = 0. (

**f**) Normalized intensity I(x, y, t, z) (Equation (22)) of SCTLGSM pulsed beam as a function of propagation distance z for different chirp coefficients s = 0, −0.4ps

^{−2}, −0.8ps

^{−2}, −1.2ps

^{−2}.

**Figure 9.**Density plots for the normalized spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam in the x–t plane at some propagation distances (

**a**) z = 0, (

**b**) z =20 m, (

**c**) z = 30 m, (

**d**) z = 40 m, (

**e**) z = 50 m and for s = −0.4, n = 1, m = 1. (

**f**) Normalized intensity I(x, y, t, z) (Equation (22)) of SCTLGSM pulsed beam as a function of propagation distance z for different chirp coefficients s = 0, −0.4 ps

^{−2}, −0.8 ps

^{−2}and −1.2 ps

^{−2}.

**Figure 10.**Density plots for the normalized spatiotemporal intensity I(x, y, t, z) (Equation (20)) of SCTLGSM pulsed beam in the x–t plane for different beam orders (

**a**) m = n = 0, (

**b**) m = n = 1, (

**c**) m = n = 2, (

**d**) m = n = 3 at z = 50 m with s = −0.4ps

^{−2}.

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**MDPI and ACS Style**

Ding, C.; Korotkova, O.; Horoshko, D.; Zhao, Z.; Pan, L.
Evolution of Spatiotemporal Intensity of Partially Coherent Pulsed Beams with Spatial Cosine-Gaussian and Temporal Laguerre–Gaussian Correlations in Still, Pure Water. *Photonics* **2021**, *8*, 102.
https://doi.org/10.3390/photonics8040102

**AMA Style**

Ding C, Korotkova O, Horoshko D, Zhao Z, Pan L.
Evolution of Spatiotemporal Intensity of Partially Coherent Pulsed Beams with Spatial Cosine-Gaussian and Temporal Laguerre–Gaussian Correlations in Still, Pure Water. *Photonics*. 2021; 8(4):102.
https://doi.org/10.3390/photonics8040102

**Chicago/Turabian Style**

Ding, Chaoliang, Olga Korotkova, Dmitri Horoshko, Zhiguo Zhao, and Liuzhan Pan.
2021. "Evolution of Spatiotemporal Intensity of Partially Coherent Pulsed Beams with Spatial Cosine-Gaussian and Temporal Laguerre–Gaussian Correlations in Still, Pure Water" *Photonics* 8, no. 4: 102.
https://doi.org/10.3390/photonics8040102