1. Introduction

The outlook for application of laser beams with nonzero orbital angular momentum (OAM) for increasing the effectiveness of laser energy transfer through the atmosphere and probing the atmosphere determines the need of study of propagation of beams carrying OAM [1,2] in the free space. The equiphase surface of the optical field $U=A\exp \left(\pm jm\theta \right)$ having nonzero orbital angular momentum ($\pm m\ne 0$) in the plane transverse to the direction of propagation looks like a wound helix. The helix shape is determined by the azimuth angle $\theta$ and the value of the topological charge $\pm m$, where the sign determines the energy winding direction [1,2]. Presence of OAM can increase the resistance of laser beams to the turbulent distortions [3,4]. The vortex beam fields corresponding to different values of the parameter $m$ have the properties of orthogonality [2,5]. This, in fact, allows vortex beams with different values of the topological charge to be used for increasing the number of communication channels operating simultaneously at the same frequency [5–11].

Refractive turbulence is one of the main obstacles in the practical application of vortex beams in atmospheric optical systems because it disrupts the spatial coherence of the field of propagating beam. This leads to a decrease of the signal level due to the turbulent broadening of laser beams and appearance of intensity fluctuations in the beam cross-section.

In [12,13] with the Gaussian laser beam taken as an example, it is shown that the pulsed beams is less prone to the distorting effect of turbulence than the continuous-wave (cw) beams, and, consequently, they are more noise-resistant when used in atmospheric optical systems. In this study we consider turbulent fluctuations of energy density of pulsed Laguerre–Gaussian beam as a beam with nonzero OAM. Based on the algorithm for numerical simulation of propagation of broadband optical radiation [12,13], we calculate the relative variance and the coefficient of spatial correlation of energy density fluctuations of Laguerrian beams with different topological charges in dependence on refractive turbulence strength.

2. Basic equations

Let the pulsed laser beam propagates in the turbulent atmosphere along the axis $x\ge 0$, and the electric field strength of the wave at the initial plane x = 0 can be represented in the form

The parameter a determines the limitedness of the Laguerre beam in transverse plane,$j=\sqrt{-1}$, $\rho =\{y,z\}$ is the radius vector in the plane perpendicular to the axis х, $\theta =arctg(y/z)$, $L_n^m=x^{-1}\exp (x)\frac{d^n}{d{x}^n}[x^{n+1}\exp (-x)]$ is the Laguerre polynomial, integer numbers m and n determine the Laguerre–Gaussian modes $E_{nm}^0$, ${\psi }_0$ is the wave phase independent of $\rho$ and t, t is time, $f_0$ is the frequency at the point of maximum of the beam temporal spectrum, ${\tau }_0$ is the initial pulse duration determined from the decrease of ${\left|E(0,0,t)\right|}^2$ down to the level $e^{-1}{\left|E_{nm}^0(0)\right|}^2$. The pulse duration ${\tau }_P(x)$ determined as a pulse full width at half maximum of the power ${\displaystyle {\int }_{-\infty }^{+\infty }{d}^2\rho |E(x,\rho ,t){|}^2}$, in the plane x = 0 is related to ${\tau }_0$ as ${\tau }_P(0)=2\sqrt{\ln 2}{\tau }_0$. At m = n = 0, Eq. (2) reduces to the Gaussian beam

In neglect of the nonlinear effects and absorption of the radiation by air and aerosol particles, the complex spectral amplitude of the field strength of optical wave

3. Procedure of numerical simulation

Statistical characteristics of the energy density of a short pulse were studied through the simulation of the propagation of a Laguerre–Gaussian laser beam based on numerical solution of Eq. (5) for the complex spectral amplitude by the split-step method [12,13]. The essence of the method consists in replacement of a continuous medium with an N-set of thin phase screens ${\Psi }_i(\rho ,f)$ imitating turbulent distortions of an optical wave in the process of propagation. Between the screens, only the diffraction of the propagating wave is taken into account.

The frequency spectrum of short-pulse radiation has a finite width, which increases with a decrease of the pulse duration. Therefore, the wave parabolic Eq. (5) for the complex spectral amplitude of the field $U(x,\rho ,f)$ is solved consecutively for every individual component of the spectrum f, where $f=f_0+(k-K/2)\Delta f$, $k=0,1,\mathrm{...},K$, and $\Delta f$ is the frequency step.

For simulation of random phase screens ${\Psi }_i(\rho ,f)$, it is necessary to set their statistical properties. We assume that in the i-th layer of the propagation path the probability density of phase fluctuations $p({\Psi }_i)$ has the normal distribution [16]. It is commonly accepted that the spatial structure of turbulent inhomogeneities of the refractive index of air obeys the fundamental Kolmogorov-Obukhov law [16]. Therefore, in the equation for the structure function of the wave phase

The application of the two-dimensional fast Fourier transform (FFT) to the array of complex spectral phase amplitudes simulated in accordance with spectrum (8) [17–19] allows us to obtain independent random realizations of phase ${\Psi }_i(\rho ,f_0)$ ($<{\Psi }_i{\Psi }_{i^{\prime }\ne i}>$ = 0) for the given frequency $f_0$ on the $(M\times M)$ computational grid with the step h. To obtain realizations of a random phase screen at other frequencies f, the equation ${\Psi }_i(\rho ,f)=(f/f_0){\Psi }_i(\rho ,f_0)$ is used.

The two-dimensional (2D) distributions of spectral amplitudes $U(x,\rho ,f)$ at different frequencies obtained from numerical solution of Eq. (5) are used for calculation of the beam energy density distributions integral over the pulse duration [20] (see also [12,13])

Independent random realizations of 2D distributions of $U(x,\rho ,f)$ in the transverse plane were used for estimation of the average value $〈W(x,\rho )〉$, the relative variance

4. Results of numerical simulation

The numerical simulation of random distributions of $W(x,\rho )$ in the turbulent atmosphere was carried out for the modes $E_{0m}^0$, where m = 0, 2, 8, of the Laguerre–Gaussian beam. The complex spectral amplitude $U(x,\rho ,f)$ was calculated for 60 spectral channels at ${\tau }_P(0)$ = 3 fs (${\tau }_0$ = 1.8 fs) and for 41 (${\tau }_P(0)$ = 5 fs (${\tau }_0$ = 3 fs)) spectral channels with the width $\Delta f$ = 10 THz for frequencies $f=f_1+k\Delta f$, where k = 0, 1,..., K. The initial frequency$f_1$ = 10 THz at K = 59 (${\tau }_P(0)$ = 3 fs) and $f_1$ = 100 THz at K = 40 (${\tau }_P(0)$ = 5 fs).

The method of subharmonics was applied for simulation of random phase screens for narrow beams [18, 19]. The number of subharmonics used in the simulation of a phase screen was equal to eight. The averaging was performed over 5000 independent random realizations of 2D distributions of the beam energy density in the transverse plane. Independence of random realizations of the beams energy distribution means that simulation was performed under the assumption that pulse repetition rate does not exceed frequencies of temporal variations of refractive turbulence in the atmosphere in the range $10-{10}^4$Hz.

The turbulent conditions of optical wave propagation in the atmosphere correspond to the regime of strong scintillations, when the parameter

The number of nodes of the $M\times M$ computational grid and the discretization step h on the grid were specified with regard for the parameter ${\beta }_0^2$. At ${\beta }_0^2$< 36, the $M\times M$ = 512$\times$512 grid with the step h = 3 mm was used. At $36\le {\beta }_0^2\le 124$, the computations were performed at the $M\times M$ = 1024 $\times$1024 grid with the step of 1.5 mm. At $124<{\beta }_0^2\le 185$, the grid was 1540$\times$1540 with $h$ = 1.2 mm. The path length was taken equal to 1 km and 3 km. The number of layers, N the model propagation path was divided into, varied from 20 to 30 depending on the parameter ${\beta }_0^2$.

Figures 1–4 show the random distributions of the energy density for the Gaussian beam (Figs. 1, 2) and the $E_{08}^0$ mode of the Laguerre beam (Figs. 3, 4) at the end of the 1-km long path. In Figs. 1–4, the energy density in every distribution is normalized to the maximal value. For cw Laguerrian beams distributions of the intensity $I(x,\rho )={\left|U(x,\rho ,f_0\right|}^2$, normalized to the maximum, are depicted in Figs. 1-4. Since the spectral intensity of the cw radiation $S_I(x,\rho ,f_0)$ is independent of the frequency f, the energy density $W(x,\rho ){|}_{{\tau }_0\to \infty }$ (Eq. (9)) in the limiting case of cw radiation is related to the intensity $I(x,\rho )$ by the formula $W(x,\rho ){|}_{{\tau }_0\to \infty }/{\displaystyle {\int }_{-\infty }^{\infty }df}=I(x,\rho )$.

 

Fig. 1 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity $I_N$ of cw radiation (a, c, e, g) and normalized energy density $W_N$ of pulsed ${\tau }_P(0)$ = 3 fs (b, d, f, h) Gaussian beam with а = 5 cm at $C_n^2$ = 10⁻¹³ m^-2/3 (a)–(d) and $C_n^2$ = 10⁻¹² m^-2/3 (e)–(h).

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Fig. 2 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity $I_N$ of cw radiation (a, c, e, g) and normalized energy density $W_N$ of pulsed ${\tau }_P(0)$ = 3 fs (b, d, f, h) Gaussian beam with а = 2 cm at $C_n^2$ = 10⁻¹³ m^-2/3 (a)–(d) and $C_n^2$ = 10⁻¹² m^-2/3 (e)–(h).

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Fig. 3 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity $I_N$ of cw radiation (a, c, e, g) and normalized energy density $W_N$ of pulsed ${\tau }_P(0)$ = 3 fs (b, d, f, h) Laguerre beam $E_{08}^0$, а = 5 cm at $C_n^2$ = 10⁻¹³ m^-2/3 (a)–(d) and $C_n^2$ = 10⁻¹² m^-2/3 (e)–(h).

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Fig. 4 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of normalized intensity $I_N$ of cw radiation (a, c, e, g) and normalized energy density $W_N$ of pulsed ${\tau }_P(0)$ = 3 fs (b, d, f, h) Laguerre beam $E_{08}^0$, а = 2 cm at $C_n^2$ = 10⁻¹³ m^-2/3 (a)–(d) and$C_n^2$ = 10⁻¹² m^-2/3 (e)–(h).

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From the comparison of Figs. 1–4 (a, c, e, g) and 1–4 (b, d, f, h), one can see that the random distributions of the energy density of pulsed radiation are less speckled than the intensity distributions of cw radiation. With an increase of fluctuations of the refractive index, when $C_n^2$ increases by an order of magnitude, the distributions of the energy density of pulsed radiation become even more uniform in comparison with the intensity d istributions of cw radiation.

Figures 3 and 4 demonstrate that the decrease of the parameter a leads to blurring of the initial ring structure of the Laguerre beam due to the turbulent broadening. The distributions of energy density in Fig. 4 (e, h) for the Laguerre beam do not differ qualitatively from the energy density distributions of the Gaussian beam in Fig. 2 (e, h).

To estimate quantitatively turbulent fluctuations of energy density of pulsed radiation, we consider statistical characteristics of energy density. Figure 5 shows the standard deviation ${\sigma }_W$ of relative fluctuations of energy density of pulsed radiation calculated by Eq. (10) at the pulse duration ${\tau }_P(0)$ = 5 fs (${\tau }_0$ = 3 fs, curves 1'–3′), 3 fs (${\tau }_0$ = 1.8 fs, curves 1”–3”) and cw radiation ${\tau }_0$→∞ (curves 1–3) of Laguerre–Gaussian beams. The calculations of ${\sigma }_W$ were carried out for the point of maximum of the average energy density distribution of the corresponding mode of the Laguerre–Gaussian beam as a function of the parameter ${\beta }_0$ at a = 2 cm.

 

Fig. 5 Standard deviation of relative fluctuations of the energy density of pulsed (curves 1'–3′, 1”–3”) and cw (curves 1–3) Laguerre–Gaussian modes $E_{00}^0$ (curves 1, 1', 1”), $E_{02}^0$ (curves 2, 2', 2”), $E_{08}^0$ (curves 3, 3′, 3”) as a function of the parameter ${\beta }_0$. Pulse duration ${\tau }_P(0)$ = 5 fs (curves 1', 2', 3′) and 3 fs (curves 1”, 2”, 3”).

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It is seen from Fig. 5 that at the weak refractive turbulence (${\beta }_0$ < 1) there is no almost difference in energy density fluctuations of the modes of the Laguerre–Gaussian beam. As the parameter ${\beta }_0$ increases, the level of fluctuations begins to differ for the cw and pulsed beams. The higher the mode order of the Laguerre–Gaussian beam and the shorter the pulse duration ${\tau }_P(0)$, the larger the difference. Standard deviation of strong (${\beta }_0^2$ > 1) intensity fluctuations of the cw Laguerre–Gaussian beams tends to unity very slowly with an increase of ${\beta }_0^2$, which is in consistent with the known results for cw optical radiation [3, 21]. The energy density of pulsed Laguerre–Gaussian beams under the regime of strong scintillations ${\beta }_0^2$ > 1, as follows from Fig. 5, fluctuates much less than the intensity of cw beams. In contrast to the cw radiation, ${\sigma }_W$ of pulsed beams does not saturate at some level, but becomes smaller and smaller with an increase of ${\beta }_0^2$. The shorter the pulse, the smaller ${\sigma }_W$ at the same values of ${\beta }_0^2$. Starting from approximately ${\beta }_0$ = 6, ${\sigma }_W$ in pulsed beams becomes smaller than unity. An increase in the mode order (topological charge) of the Laguerre beam leads to a small decrease of energy density fluctuations due to an increase of the initial size of the beam.

Qualitatively a decrease of turbulent fluctuations of pulsed beams energy density can be explained by the decrease of a magnitude of the parameter ${\beta }_0^2$, averaged over all the wavelengths corresponding to the pulse bandwidth, as compared to ${\beta }_0^2$(${\lambda }_0$).

The issues of resistance of different modes of Laguerre–Gaussian beams to the distorting effect of refractive turbulence are discussed [3, 4]. It follows from the results of [3, 4] that the higher resistance of higher modes of vortex beams to the impact of atmospheric turbulence is not explained by the phase topology, but by the increase of the initial size of vortex beams with an increase of the topological charge. If for Laguerre–Gaussian beam we choose such values of the parameter a in Eq. (2) that the initial size of the beam is the same for modes of different orders, than the impact of turbulence on the vortex Laguerre–Gaussian beams is identical regardless of the mode.

To check the validity of this conclusion for energy density fluctuations of pulsed Laguerre–Gaussian beams, the calculations have been performed, and their results are depicted in Fig. 6. The values of parameter а = 4, 2.9 and 2 cm for the Laguerre beam modes $E_{0m}^0$ with $m$ = 0, 2, 8 in the calculations of ${\sigma }_W$ shown in Fig. 6 were chosen so that the energy density distributions of all modes in the initial plane to be fitted in a circle of a preset diameter. The total energy of every Laguerre mode $E_{0m}^0$ within the circle [22] is the same for chosen values of а. In Fig. 6, the standard deviation ${\sigma }_W$ for the $E_{02}^0$ mode of the Laguerre–Gaussian beam with the parameter a = 2.9 cm (curves 2, 2'), for the $E_{08}^0$ mode with the parameter a = 2 cm (curves 3, 3′), and for the Gaussian beam $E_{00}^0$ with the radius a = 4 cm (curves 1, 1') was calculated as a function of ${\beta }_0^2$ at the point of maximum of the average energy density distribution of the corresponding mode. Dots in Fig. 6 show the confidence interval. Curves 1', 2', 3′ correspond to the pulsed beams with ${\tau }_P(0)$ = 5 fs (${\tau }_0$ = 3 fs), while curves 1, 2, 3 are for the cw beams.

 

Fig. 6 Standard deviation of relative energy density fluctuations of pulsed (${\tau }_P(0)$ = 5 fs, curves 1'–3′) and cw (curves 1–3) Laguerre–Gaussian modes $E_{00}^0$ (curves 1, 1'), $E_{02}^0$ (curves 2, 2'), $E_{08}^0$ (curves 3, 3”) as a function of the parameter ${\beta }_0$.

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It is seen from Fig. 6 that for the Laguerre–Gaussian modes $E_{00}^0$, $E_{02}^0$ and $E_{08}^0$ at the parameter а values providing the same initial transverse size of the beam modes, the level of energy density fluctuations within the statistical uncertainty is identical both for the pulsed radiation (curves 1', 2', 3′) and for the cw radiation (curves 1, 2, 3). That is, the phase singularity does not affect the level of energy density (intensity) fluctuations of Laguerre–Gaussian beams. Insignificant decrease of energy density fluctuations with an increase of the Laguerre mode order in Fig. 5 is a consequence of the change of diffraction conditions on the transmitting aperture due to an increase of the initial size of the Laguerre beams with an increase of m at constant a [3,4].

Figures 7–9 show the results of calculation of the spatial correlation coefficient of strong (${\beta }_0^2$ > 1) energy density fluctuations of cw (curves 1, 2, 3) and pulsed ${\tau }_P(0)$ = 5 fs (curves 1', 2', 3′), 3 fs (curves 1”, 2”, 3”) Laguerre–Gaussian beams for the modes $E_{00}^0$ (curves 1, 1', 1”), $E_{02}^0$ (curves 2, 2', 2”), and $E_{08}^0$ (curves 3, 3′, 3”) in the turbulent atmosphere at a = 2 cm. The distance from the beam axis is shown as an abscissa in dimensionless units $\rho /a_g$ and $\rho /{\rho }_c$, where $a_g=a\left[1+{\left({\lambda }_{0}x/2\pi a\right)}^2\right]$ is the diffractive radius of the collimated Gaussian beam (3) [21], and ${\rho }_c={(1.45C_n^2{k}^{2}L)}^{-3/5}$ is the radius of spatial coherence of a plane wave in a turbulent atmosphere [21].

 

Fig. 7 Coefficient of spatial correlation of energy density fluctuations of pulsed (curves 1'–3′, 1”–3”) and cw (curves 1–3) Laguerre–Gaussian modes $E_{00}^0$ (curves 1, 1', 1”), $E_{02}^0$ (curves 2, 2', 2”), $E_{08}^0$ (curves 3, 3′, 3”) at ${\beta }_0^2\approx 124$and ${\tau }_P(0)$ = 5 fs (curves 1'–3′), 3 fs (curves 1”–3”).

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Fig. 8 Coefficient of spatial correlation of energy density fluctuations of pulsed (curves 1'–3′, 1”–3”) and cw (curves 1–3) Laguerre–Gaussian modes $E_{00}^0$ (curves 1, 1', 1”), $E_{02}^0$ (curves 2, 2', 2”), $E_{08}^0$ (curves 3, 3′, 3”) at ${\beta }_0^2\approx 185$, ${\tau }_P(0)$ = 5 fs (curves 1'– 3′), 3 fs (curves 1”– 3”).

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Fig. 9 Coefficient of spatial correlation of energy density fluctuations of pulsed ${\tau }_P(0)$ = 5 fs (curves 1'–6') and cw (curves 1–6) Laguerre–Gaussian modes $E_{00}^0$ (curves 1, 1', 4, 4'), $E_{02}^0$ (curves 2, 2', 5, 5′), $E_{08}^0$ (curves 3, 3′, 6, 6') at ${\beta }_0^2\approx 124$ (curves 1–3, 1'–3′) and ${\beta }_0^2\approx 185$ (curves 4–6, 4'–6').

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It follows from Fig. 7 that the spatial correlation of intensity fluctuations for all modes of the cw Laguerre–Gaussian beam at ${\beta }_0^2\approx 124$ is two-scale in accordance with the results for strong intensity fluctuations [21]. The first scale ${\rho }_1$ calculated by the level $C_W(x,{\rho }_1)=e^{-1}$ determines the area of high spatial correlation, whose dimensions are specified by the coherence radius ${\rho }_c$ and decrease with increase of the refractive turbulence strength [21].

The second scale ${\rho }_2\sim x{\lambda }_0\text{/}{\rho }_c$ [21] determines the area of weak (residual) spatial correlation within the range of the correlation coefficient from $C_W(x,\rho )=0.2$ to $C_W(x,{\rho }_2)=0.2e^{-1}$, whose dimensions increase with increase of the turbulence strength [21]. In the pulsed beams, the level of residual spatial correlation of density fluctuations increases nearly threefold at the pulse duration ${\tau }_P(0)=$ 5 fs and nearly fourfold at ${\tau }_P(0)$ = 3 fs in comparison with the level of residual intensity correlation of cw beams for all the considered Laguerrian modes. That is, the speckle structure arising due to turbulence in the cross section of pulsed Laguerre–Gaussian beams is more smoothed as that of cw beams [see Figs. 1–4].

Figure 8 shows $C_W(x,\rho )$ calculated at ${\beta }_0^2$ = 185 for the same other parameters as in Fig. 7.

From the comparison of Figs. 7 and 8, it follows that the increase of refractive turbulence strength leads to a decrease in the first correlation scale and an increase in the level of residual correlation. Figure 9 illustrates this conclusion.

It follows from Figs. (7)–(9) that at a = 2 cm the spatial correlation of the energy density of cw and pulsed Laguerre–Gaussian beams for the turbulent conditions of propagation taken in the calculations is independent of the topological charge within the calculation accuracy.

5. Conclusion

In this paper, based on the numerical solution of the parabolic wave equation for the complex spectral amplitude of the wave field by the split-step method, we have simulated the propagation of broadband pulsed vortex beams in the turbulent atmosphere using, as an example, the second and eighth Laguerre–Gaussian modes. Energy density fluctuations of pulsed Laguerre–Gaussian beams have been studied in comparison with intensity fluctuations of cw Laguerre–Gaussian beams.

It is shown that, as the refractive turbulence strength increases, the relative variance of energy density fluctuations of pulsed beams becomes much smaller than the variance of intensity fluctuations of cw beams and, in contrast to the latter, in the regime of strong scintillations can take values smaller than unity. The conclusion is valid for both the Gaussian and the Laguerre–Gaussian beams. For the Laguerre–Gaussian beams having the same value of the parameter а of the initial field distribution (2), the higher the mode order, the smaller the energy density fluctuations. In the regime of strong scintillations the spatial structure of energy density correlation in the cross section of the Laguerre–Gaussian beam is a two-scale one for all the considered modes. It is characterized by the presence of small-scale areas of high correlation, whose size is determined by the radius of spatial coherence of the wave field in the turbulent atmosphere [21] and decreases with the increase of turbulence strength. Along with the areas of high spatial correlation of the energy density, there are large-scale areas of low (residual) correlation, whose size, to the contrary, increases with an increase of the refractive turbulence strength. For pulsed beams with the pulse duration ${\tau }_P(0)$ = 3 fs and 5 fs, the level of residual spatial correlation exceeds that for cw beams by several times.

The decrease of the relative variance and the increase of the level of residual spatial correlation of energy density fluctuations of the pulsed beams as compared to those of cw beams is similar to the behavior of these characteristics for the partially coherent beam as compared to the fully coherent one [21,23].

The results of numerical simulation of the propagation of Laguerre–Gaussian beams with the same initial transverse dimensions, but different topological charges show that the statistics of energy density fluctuations in these beams under the same refractive turbulence strength is independent of the value of topological charge within the accuracy of calculation.