1. Introduction

Propagation characteristics of various laser beams have attracted considerable attention in the past decade. Knowledge of these subjects would contribute to applications in laser communication, remote sensing, and directed energy systems [1]. Numerous types of laser beams have been investigated [1–5]. Among them, there is a growing interest in beams with flower petal structure, which hold great promise in rotational Doppler shift based remote sensing [6,7]. Recently, a laser beam with four petals called the four-petal Gaussian beam has been introduced [8]. Researches on the propagation factor M², the nonparaxial propagation properties, and the far field vectorial structure of four-petal Gaussian beams have been reported [9–11]. Studies have also been extended to cases with phase singularity or with partial coherence, namely the four-petal Gaussian vortex beam and the partially coherent four-petal Gaussian vortex beam, respectively [12,13]. Special attention has been paid to propagation behavior of these beams in atmospheric and oceanic turbulence [14,15].

It has been shown that vortex beams with fractional orbital angular momentum (OAM) are more resistant to atmospheric thermal blooming [16]. A practical way to generate fractional-OAM optical vortices is to generate an elliptical optical vortex [17]. Elliptical optical vortex beams are also demonstrated to be capable of suppressing turbulence induced scintillations [18]. Researches on the generation, transformation, propagation, and detection of elliptical vortex beams have been carried out [19–22]. However, currently reported works mainly concentrate on Gaussian-like or Bessel-like elliptical vortex beams. To the best of our knowledge, propagating behavior of partially coherent petal-like beams with elliptical optical vortex has not been reported.

In this paper, propagation of partially coherent four-petal elliptic Gaussian vortex beams in atmospheric turbulence is investigated. Analytical formulas for the cross-spectral density function and root mean square (rms) beam width, and method for studying the position of coherence vortices are presented in Sec. 2. Propagation behavior, beam spreading character, and evolution of coherence vortices are analyzed in Sec. 3. Conclusions are drawn in Sec. 4.

2. Propagation theory

According to Gori [23], a sufficient condition for the mathematical form of cross-spectral density functions for optical fields to be genuine is that it can be expressed as:

Equations (10)-(17) are the analytical expressions for the cross-spectral density function of partially coherent four-petal elliptic Gaussian vortex beams propagating through atmospheric turbulence. The expression for the average intensity can be obtained by further setting $r_1=r_2=r$ [2], so that:

The expression for the root mean square (rms) beam width [27], defined as

The position of coherence vortices is determined by [28]:

The conservation distance of coherence vortices topological charge is defined to be the maximum propagation distance for the detected total coherence vortices topological charge keeps unchanged [29]. The total coherence vortices topological charge at the detector is calculated by evaluating $\left[{\displaystyle \oint \nabla \varphi \left(r_1\right)}\cdot \text{d}l\right]/\left(2\pi \right)$ around the perimeter of the detector aperture, where $\varphi \left(r_1\right)$ denotes the phase of $\mu \left(r_1,r_2,z\right)$ with $r_2$ and $z$ fixed. The conservation distance of coherence vortices topological charge provides a measure about how distant the detected beam could maintain its spatial two-point correlation singularity.

To compare the analytical result with those given in Ref [14], it should be noted that the choice of expressions for initial field are not exactly the same. The form of Eqs. (3) and (4) follows the customs in Refs [30]. and [31]. However, by substituting $m$ into $-M$, $2{\sigma }^2$ into ${\sigma }^2$, setting $\alpha =1$, and multiplied by ${\omega }_0^{2\left|m\right|}$, Eq. (4) in the current work are found to be the same as Eq. (4) in Ref [14]. And correspondingly, by substituting $m$ into $-M$, $2{\sigma }^2$ into ${\sigma }^2$, setting $r_1=r_2=r,$ multiplied by ${\omega }_0^{2\left|m\right|}$, and changing the name of some local index variables, the analytical result for average intensity for the special case of $\alpha =1$ in this work is found to be the same as the result in Ref [14]. According to this correspondence, we set ${\omega }_0=20\text{mm}$, $\lambda =1.064\mu m$, $C_n^2={10}^{-13}{\text{m}}^{\text{-2/3}}$, $\alpha =1$, $\sigma =7.072\text{mm}$, $m=-1$ and $n=1$, the calculated normalized intensity profiles at $z=$100m, 500m, 1000m, 5000m are found to be the same as Fig. 1 in Ref [14].

 

Fig. 1 Evolution of normalized intensity distribution of a partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere for (a) z = 200m, (b) z = 300m, (c) z = 500m, (d) z = 1000m, (e) z = 2000m, (f) z = 50000m, respectively. The calculation parameters are seen in the text. All subfigures share the same coordinate axis direction. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.754, 0.564, 0.385, 0.188, 0.060, and 1.61 × 10⁻⁵ for (a)-(f), respectively.

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3. Numerical examples and analysis

The average intensity, beam spreading, and evolution of coherence vortices of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere are numerically studied based on formulas in Sec. 2. The calculation parameters were set as ${\omega }_0=20mm$, $\lambda =1.315\mu m$, $C_n^2={10}^{-14}{m}^{-2/3}$, $\alpha =1.5$, $\sigma =40mm$unless otherwise specified.

The normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere with $m=1,$ $n=1$ is shown in Fig. 1. The average intensity experiences several stages of evolution. At the beginning, the initial four-petal profile gradually deformed into a six-petal profile in the short propagation distance. This is different from the partially coherent four-petal Gaussian vortex beam, whose number of petals and sidelobes would always be an integer multiple of four in this stage [14]. With the propagation distance increasing, the dark hollow center of the partially coherent four-petal elliptic Gaussian vortex beam will shrink due to the expansion of each petal. And at last, the beam will evolve into a Gaussian-like beam in the far field. The minimal distance at which the beam center intensity equals to its maximum intensity, denoted by $z_{flat},$ is 2.51km. This parameter can be considered as an indicator of how fast the beam evolves.

Figure 2 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with $n=1$ and different $m$. It can be found that the partially coherent four-petal elliptic Gaussian vortex beams with different $m$ have similar evolution behaviors, but as the topological charge $m$ is getting larger, the intensity profile rotation around the z axis is getting faster. For $m=2$, the transition from four-petal to six-petal profile is slower than $m=1$. For $m>2$, there is no obvious stage with six-petal pattern within the whole propagation range. All beams with different m will evolve into Gaussian-like beams in the far field. $z_{flat}$ are 5.28km, 5.49km, 3.81km, 7.10km for $m=2,$ 3, 4, and 5, respectively. Noticing that in Eq. (3) the amplitude profile is of reflective symmetry with respect to the x axis and the y axis, while the phase profile is of rotational symmetry of order $m,$ $z_{flat}$ exhibits more complex trends with the increase of $m$ rather than to be monotonic.

 

Fig. 2 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with n = 1 and different m. (a-c) m = 2, (d-f) m = 3, (g-i) m = 4, (j-l) m = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.350, 0.201, 1.73 × 10⁻⁵, 0.373, 0.183, 1.82 × 10⁻⁵, 0.389, 0.165, 1.88 × 10⁻⁵, 0.401, 0.153, and 1.92 × 10⁻⁵ for (a)-(l), respectively.

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Figure 3 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with $m=1$ and different $n$. It can be seen that as the four-petal Gaussian beam order $n$ is getting larger, the transition from four-petal to six-petal profile is getting slower, and the intensity profile rotation speed around the z axis keeps unchanged. All beams with different $n$ will evolve into Gaussian-like beams in the far field. $z_{flat}$ are 2.24km, 2.28km, 2.41km, 2.58km for $n=2,$ 3, 4, and 5, respectively.

 

Fig. 3 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1 and different n. (a-c) n = 2, (d-f) n = 3, (g-i) n = 4, (j-l) n = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.291, 0.147, 1.57 × 10⁻⁵, 0.255, 0.113, 1.55 × 10⁻⁵, 0.260, 0.0936, 1.55 × 10⁻⁵, 0.255, 0.0865, and 1.54 × 10⁻⁵ for (a)-(l), respectively.

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Figure 4 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with $m=1,$$n=1,$ and different $\alpha .$ It can be seen that while $\alpha <1$, the four-petal profile would morph into six-petal profile with three along x direction and two along y direction, and while $\alpha >1$, would morph into six-petal profile with two along x direction and three along y direction. For $\alpha >1$, the transition from four-petal to six-petal profile is getting faster with the increasing of $\alpha$. All beams with different $\alpha$ will evolve into Gaussian-like beams in the far field. Further computation shows that $z_{flat}$ are 3.41km, 2.88km, 2.27km, 1.86km, 1.68km, 1.83km, 2.23km, 2.91km, 3.30km for $\alpha =$0.5, 0.6, 0.75, 0.9, 1, 1.1, 1.33, 1.75, and 2, respectively. It is found that for $\alpha <1,$ $z_{flat}$ will decrease as $\alpha$ increases, and for $\alpha >1,$ $z_{flat}$ will increase as $\alpha$ increases.

 

Fig. 4 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different α. (a-c) α = 0.5, (d-f) α = 0.75, (g-i) α = 1.33, (j-l) α = 2, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.708, 0.373, 5.17 × 10⁻⁵, 0.630, 0.270, 3.41 × 10⁻⁵, 0.377, 0.200, 1.84 × 10⁻⁵, 0.407, 0.138, and 1.15 × 10⁻⁵ for (a)-(l), respectively.

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Figure 5 shows the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. By comparing Figs. 1-4 with Fig. 5, it can be found that the beam profiles for short propagation distances (e.g. less than 1000m) in free space are similar with those propagating in turbulence. This is because in the considered case, the turbulence is relatively weak, and for short propagation distances, the phase distortion caused by the turbulence is small, and thus have little effect on the propagated beam. For $m=1$ and $n=1,$ there is transition from four petal profile to six petal profile. For $m=5$ and $n=1,$ there is no obvious stage with six-petal pattern within the whole propagation range. Often, the transition from four-petal profile to six-petal profile takes place at short propagation distances, thus this phenomenon will be similar no matter the beam is propagated in free space or weak turbulence. Nevertheless, for far field propagation the beam profiles in free space are different from those in turbulence. The far field of partially coherent four-petal elliptic Gaussian vortex beam propagating in free space exhibits petal or bud-like pattern with a dip in the center, while those propagating in turbulence will always evolve into Gaussian-like patterns. Both beams in Fig. 5 do not have valid $z_{flat}$ value within the whole considered range ($0<z<50000\text{m}$).

 

Fig. 5 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. (a-c) m = 1, n = 1, (d-f) m = 5, n = 1, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.406, 0.218, 1.62 × 10⁻⁴, 0.414, 0.176, and 1.04 × 10⁻⁴ for (a)-(f), respectively.

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Figure 6 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with $m=1,$ $n=1,$ and different $\sigma .$ $z_{flat}$ are 1.60km and 0.90km for $\sigma =$20mm and 10mm, respectively. Associating with Fig. 1, it can be seen that as $\sigma$ is getting smaller, $z_{flat}$ is getting smaller, and each of the six-petals would getting expanded and connected to its neighbors earlier. For $\sigma$ much smaller than ${\omega }_0$, the petals would merge into two, and there is no obvious stage with six-petal pattern within the whole propagation range. The petal numbers of these beams around $z=$500m are different from those of the partially coherent four-petal Gaussian vortex beams [14].

 

Fig. 6 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different $\sigma$. (a-c) $\sigma$ = 20mm, (d-f) $\sigma$ = 10mm, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values $I_m$ of each subfigure are 0.296, 0.139, 1.57 × 10⁻⁵, 0.191, 0.0977, 1.44 × 10⁻⁵ for (a)-(f), respectively.

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Figures 7(a)-7(c) plot the relative normalized rms beam width $\omega \left(z\right)/{\omega }_{free}\left(z\right)$ of partially coherent four-petal elliptic Gaussian vortex beams with different $m,$ different $n,$ and different $\alpha ,$ respectively. Here ${\omega }_{free}\left(z\right)$ denotes the rms beam width while propagating in free space. We see that all curves increase monotonically with the propagation distance z, meaning the beam spreading in turbulence is always faster than in free space. In the far field, the relative normalized rms beam width decreases with increasing $m,$ decreasing $n,$ and increasing $\alpha .$ This means beams with larger $m,$ smaller $n,$ and larger $\alpha$ will be less influenced by atmospheric turbulence. Note that in Figs. 6(b) and 6(c), there are intersections between curves. We attempted to explain this by analysing a similar but simpler example. Consider the case of $m=0,$ and $n=0,$ the expression for $\omega \left(z\right)$ can be simplified as:

 

Fig. 7 relative normalized rms beam width of partially coherent four-petal elliptic Gaussian vortex beams with (a) n = 1, α = 1.5, and different m, (b) m = 1, α = 1.5, and different n, and (c) m = 1, n = 1, and different α. Insets show enlarged view of curves for z<6000m.

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Figure 8 shows typical curves corresponding to solutions of Eqs. (34) and (35). In the current case $m=1,$ $n=1,$ and$r_1=\left(x_1,y_1\right)$ is set to be (3cm, 6cm). One can find that as the propagation distance increases, the number of coherence vortices will change. The conservation distance d of partially coherent four-petal elliptic Gaussian vortex beams propagating through atmospheric turbulence for different values of $\alpha$ is summarized in Table 1, where the other calculation parameters are the same as those in Fig. 7. From Table 1, it follows that two local maxima occur at about $\alpha =0.5$ and $\alpha =\mathrm{1.7.}$ Thus, the ellipticity factor can be utilized as an additional degree of freedom in controlling the conservation distance of coherence vortices topological charge.

 

Fig. 8 Curves of Re μ = 0 and Im μ = 0 of a partially coherent four-petal elliptic Gaussian vortex beam with m = 1, n = 1 and (a) z = 200m, (b) z = 500m and (c) z = 1000m.

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Table 1. Conservation distance d for different values of α

View Table

The change of detected total topological charge during propagation is caused by the evolution and movement of the vortices. According to Gbur [29], as the beam propagates, the vortex “wanders”. If it wanders outside the perimeter of the detector aperture, the topological charge will not be measured by the detector. For $0.1\le \alpha \le 0.5$ and $1.7\le \alpha \le 2$, the vortex getting out of the aperture comes from the pair created within $\left(-0.5cm,0.5cm\right)\times \left(-2cm,0\right)$, while for $0.6\le \alpha \le 1.6$, comes from the pair created within $\left(0,2cm\right)\times \left(-0.5cm,0\right)$. That is to say, different vortex movements are responsible for the conservation distance for different range of $\alpha .$ This corresponds to the fact that $\alpha =0.5$ and 1.7 are extreme points. Note that a square aperture with side length of 4cm is used in Table 1. If we change the aperture to be circular with diameter of 4cm, the trend of the conservation distance with respect to $\alpha$ keeps unchanged, while the two local maxima still at about $\alpha =0.5$ and 1.7.

4. Conclusions

In this paper, taking the partially coherent four-petal elliptic Gaussian vortex beam as an example of partially coherent petal-like beams with elliptical optical vortex, the evolution of average intensity and coherence vortices, and the beam spreading behavior have been studied.

It is found that, the initial four-petal profile of partially coherent four-petal elliptic Gaussian vortex beams could deform into six-petal profiles in short propagation distance, and this is different from the partially coherent four-petal Gaussian vortex beam. The occurence, the appearance, and the transition speed from four-petal to six-petal profile depend on m, n, and $\alpha$. The rotation speed of the near field intensity profile around the z axis depends mainly on m.

The far field distributions of partially coherent four-petal elliptic Gaussian vortex beams through atmospheric turbulence are always Gaussian-like patterns. Quite differently, those propagating in free space will exhibit petal or bud-like patterns with dip in the center.

In the far field, the relative normalized rms beam width $\omega \left(z\right)/{\omega }_{free}\left(z\right)$ decreases with increasing m, decreasing n, and increasing $\alpha$. This means beams with larger m, smaller n, and larger $\alpha$ will be less influenced by atmospheric turbulence. In short propagation distances, however, the relative size order of $\omega \left(z\right)/{\omega }_{free}\left(z\right)$ with different n and $\alpha$ might alter.

As the propagation distance increases, the number of coherence vortices will change. It is found that the conservation distance of coherence vortices topological charge depends on $\alpha$, so that the ellipticity factor can be utilized as an additional degree of freedom in controlling the conservation distance.