## Abstract

Fluctuations of energy density of short-pulse optical radiation in the turbulent atmosphere have been studied based on numerical solution of the parabolic wave equation for the complex spectral amplitude of the wave field by the split-step method. It has been shown that under conditions of strong optical turbulence, the relative variance of energy density fluctuations of pulsed radiation of femtosecond duration becomes much less than the relative variance of intensity fluctuations of continuous-wave radiation. The spatial structure of fluctuations of the energy density with a decrease of the pulse duration becomes more large-scale and homogeneous. For shorter pulses the maximal value of the probability density distribution of energy density fluctuations tends to the mean value of the energy density.

© 2014 Optical Society of America

## 1. Introduction

It is widely known that optical waves propagating in the atmosphere experience random distortions owing to turbulent fluctuations of the refractive index of air [1–5]. One of the manifestations of these distortions are intensity fluctuations of the optical wave field, which reduce the efficiency of atmospheric free space communication and navigation systems. Estimation possible distortions of optical waves caused by the atmospheric turbulence and, in particular, random energy redistribution in the cross section of laser beams is important step in designing atmospheric optical systems.

The problems of propagation of narrow-band continuous-wave (cw) and pulsed optical beams in the turbulent atmosphere are solved on the basis of equations for statistical moments of the complex amplitude of the wave field obtained in the Markov’s approximation from the stationary parabolic wave equation [2–5]. The rigorous solution of these equations at arbitrary turbulent conditions of propagation is possible only for the second-order coherence function. For higher-order statistical moments equations only asymptotic solutions for the regimes of weak and strong intensity fluctuations are known [2–5]. The development of femtosecond lasers and its applications in atmospheric problems [6] determines the urgency of investigation of the propagation of short-pulse laser radiation in the turbulent atmosphere [7–9]. The calculation of time broadening, time of pulse arrival, intensity fluctuations, and other characteristics of pulsed radiation in the turbulent atmosphere already requires the solution of equations for statistical moments of the complex amplitude of the wave field at different frequencies [5, 10–16]. However, in this case, even the equation for the second statistical moment has no rigorous solution.

To study the propagation of broadband pulsed radiation, when the pulse duration can be equal to only few wave periods, it is necessary to invoke the nonstationary wave equation [17–19]. Diffraction broadening and coherent properties of broadband pulsed partially coherent optical beams in the absence of turbulence were considered on the basis of the nonstationary wave equation in the paraxial approximation [19–24]. In [25] the results on spatio-temporal couplings in short pulses are reviewed. The nonstationary wave parabolic equation also allows the derivation of equations for coherence functions (statistical moments) of the complex spectral amplitude of the field of optical wave propagating in random media, in particular, in the turbulent atmosphere. However, the rigorous solution of these equations, similar to the equations for the complex amplitude of the field [2–5], can be obtained only for the second-order coherence function.

For solution of statistical problems of propagation of narrow-band optical radiation in the turbulent atmosphere, numerical methods are widely used, in particular, the split-step method for solution of the parabolic equation for the complex amplitude of the wave field [26]. In this study, the numerical methods are applied to study fluctuations of energy density of the broadband pulsed optical radiation in the turbulent atmosphere. For this purpose, an algorithm is developed for the numerical simulation of propagation of short-pulse radiation through solution of the wave parabolic equation for the complex spectral amplitude of the field by the SPF method. The algorithm is described, and the results of calculation of the statistical characteristics of energy density fluctuations of pulsed radiation with femtosecond pulse durations are presented.

## 2. Formulation of the problem

The pulsed laser radiation propagates in the turbulent atmosphere along the axis $z\ge 0$. The complex strength of the electric field of the wave at the point $(z,\rho )$ ($\rho =\{x,y\}$is the radius vector in the plane perpendicular to the optical axis) at the time $t$ is denoted as $E(z,\rho ,t).$ The laser radiation is assumed to be fully coherent, and the strength $E(0,\rho ,t)$ in the initial plane can be represented in the form:

The wave equation for the complex spectral amplitude of the wave field

We assume that the refractive index of air in the atmosphere is a statistically homogeneous field and use the model of dry air for it [2]

In the absence of turbulent fluctuations of the refractive index $n(z,\rho ,f)\equiv \text{\hspace{0.17em}}<n(f)>.$ Then for the regime of plane wave, when we can neglect diffraction and throw away the second term in Eq. (3), the solution of Eq. (3) has the form [17]

Using, according to Eq. (2), the relationshipAccording to the calculation by Eqs. (13) and (14), at ${\tau}_{0}$ = 3 fs (${\tau}_{P}(0)$ = 5 fs), ${\lambda}_{0}$ = 1 µm (${f}_{0}$ = 300 THz), ${P}_{a}$ = 1013 mb and $<T>$ = 288° K, the pulse duration at the distance $z$ = 1 km exceeds the initial pulse duration in 2358 times ($c{\tau}_{P}(z)$ = 1.5 µm at $z$ = 0 and $c{\tau}_{P}(z)$ = 3.5 mm at $z$ = 1 km).

By analogy with Eq. (10), we represent the spectral amplitude of the wave field $\tilde{E}(z,\rho ,f)$ in the form

## 3. Numerical solution of the wave parabolic equation for the complex spectral amplitude

For numerical solution of Eq. (16), we use the split-step method [26, 27]. The essence of the method as applied to our case consists in the following. The entire propagation path of the length $L$ is divided into $N$ layers every with thickness$\Delta z=L/N.$ The complex spectral amplitude $U({z}_{j},\rho ,f)$ for every frequency$f$, where ${z}_{j}=j\Delta z$ and $j=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}N$, is calculated consecutively at transition from one layer to another. Equation (16) for the spectral amplitude at the end of the $j$th layer ($z\in [{z}_{j-1}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{z}_{j-1}+\Delta z]$), by analogy with the parabolic equation for the complex amplitude of the field [26, 27], is solved in two stages.

Stage 1. Only phase distortions acquired by the wave with the frequency $f$ upon its propagation through turbulent inhomogeneities of the refractive index inside the layer are taken into account. Then the spectral amplitude of the wave field denoted as ${U}_{1}({z}_{j},\rho ,f)$ is described by Eq. (16), in which the second term is put equal to zero, $U({z}_{j},\rho ,f)$ is replaced by ${U}_{1}({z}_{j},\rho ,f),$ and the boundary condition

With allowance for Eq. (17), the solution of this equation has the formwhereStage 2. Only the wave diffraction is taken into account. Then the spectral amplitude of the wave field denoted as ${U}_{2}({z}_{j},\rho ,f)$ is described by Eq. (16) with omitted third term and $U({z}_{j},\rho ,f)$ replaced with ${U}_{2}({z}_{j},\rho ,f)$ with the boundary condition

Upon the application of the direct two-dimensional Fourier transformFor simulation of random phase screens ${\Psi}_{j}(\rho ,f)$, it is necessary to set their statistical properties. We assume that in the $j$th layer of the propagation path the probability density of phase fluctuations $p({\Psi}_{j})$ has the normal distribution [1–5]. It is commonly accepted that the spatial structure of turbulent inhomogeneities of the refractive index of air obeys the fundamental Kolmogorov-Obukhov law [1]. 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 the spectrum (26) [27–30] allows us to obtain independent random realizations of phase ${\Psi}_{j}(\rho ,{f}_{0})$ ($<{\Psi}_{j}{\Psi}_{{j}^{\prime}\ne j}>\text{\hspace{0.17em}}$ = 0) for 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 ${\Psi}_{j}(\rho ,f)=(f/{f}_{0}){\Psi}_{j}(\rho ,{f}_{0})$ is used.

Upon the substitution of Eq. (26) into Eq. (25) and integration, we obtain the well-known equation [2]

The spectral amplitude $U({z}_{j},\rho ,f)$ is calculated with the use of the split-step method and the algorithm for simulation of random phase screens in every layer, the path is divided into, for $K+1$ beams on the frequencies $f={f}_{1}+k\Delta f$ ($k=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}K$ and $\Delta f$ is the frequency step) at nodes of the uniform grid $\rho =\{({m}_{x}-M/2)h\text{\hspace{0.17em}},\text{\hspace{0.17em}}({m}_{y}-M/2)h\}$ (${m}_{x,y}=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}M-1$). For calculation of ${\tilde{U}}_{2}({z}_{j-1},\kappa ,f)$ at nodes of the computational grid $\kappa =\{{k}_{x}/(Mh)\text{\hspace{0.17em}},\text{\hspace{0.17em}}{k}_{y}/(Mh)\}$ (${k}_{x,y}=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}M-1$) and $U({z}_{j},\rho ,f)$, the direct and inverse two-dimensional FFT is applied, respectively, in place of integration in accord with Eqs. (23) and (24). The parameters $K$ and $M$ determine the dimension of 3D $(K\times M\times M)$ arrays of calculated complex variables and the needed RAM volume.

The simulation parameters ${f}_{1}$, $\Delta f$ and $K$can be selected empirically for given ${f}_{0}$ and ${\tau}_{0}$. We used the described above algorithm, assuming the condition: $A(0)<0.01,$ where $A(f)=\text{\hspace{0.17em}}|\tilde{E}(0,0,f)|/|\tilde{E}(0,0,{f}_{0})|.$ Then from Eq. (4) we have: ${\tau}_{0}>\sqrt{\mathrm{ln}10}/(\pi {f}_{0})$ or ${\tau}_{P}(0)>0.8/{f}_{0}.$ According to this condition, for example, in the case of ${f}_{0}$ = 300 THz the pulse duration ${\tau}_{P}(0)$ must exceed 2.67 fs (or ${\tau}_{0}$ > 1.6 fs).

From simulated values of the complex spectral amplitude $U(L,\rho ,f),$ we can calculate the spectral intensity

spectral powerWhen the absorption of the radiation energy on the propagation path is neglected, the spectral power of radiation ${S}_{P}(L,f),$ in contrast to the spectral intensity ${S}_{I}(L,\rho ,f),$ is independent of the path length $L$, that is, ${S}_{P}(L,f)={S}_{P}(0,f)$ [17]. To take the absorption into account, we should multiply the calculated values of $U({z}_{j},\rho ,f)$ by $\mathrm{exp}[-{\alpha}_{j}(f)\Delta z/2]$ in every layer, where ${\alpha}_{j}(f)$ is the absorption coefficient of the radiation energy by atmospheric air and aerosol in the *j*th layer.

The numerically simulated independent realizations of $W(L,\rho )$ allow calculation of statistical characteristics of the radiation energy density, in particular, the average value $<W(L,\rho )>,$ relative variance

In calculation of the energy density $W(L,\rho )$ by Eqs. (28) and (31), the integration of $\text{\hspace{0.17em}}|U(L,\rho ,f){|}^{2}$ with respect to $f$ is replaced with summation over all indices $k$.

## 4. Results of numerical simulation

For numerical simulation of random realizations $W(L,\rho )$, we set the following parameters: ${\lambda}_{0}$ = 1 µm (${f}_{0}$ = 300 THz), ${\tau}_{P}(0)$ = 5 fs (${\tau}_{0}$ = 3 fs, ${\sigma}_{f}$ = 53 THz), ${a}_{0}$ = 5 cm, and $L$ = 1 km. For these values of ${\lambda}_{0}$ and ${\tau}_{0}$, the radiation spectrum is nonzero in the frequency range 100-500 THz, which corresponds to the wavelength range 0.6-3 µm . At every node of the computational grid with $h$ = 1 mm and $M$ = 512, the values of the complex spectral amplitude were calculated for 41 spectral channels with the width $\Delta f$ = 10 THz, that is, in the discrete representation the frequency takes values $f={f}_{1}+k\Delta f$ where ${f}_{1}$ = 100 THz, $k=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}K$ and $K$ = 40.

In parallel with $W(L,\rho )$, we calculated the spectral intensity ${S}_{I}(L,\rho ,{f}_{0})$ and the normalized intensity ${I}_{N}(L,\rho )={S}_{I}(L,\rho ,{f}_{0})/{S}_{I}(0,0,{f}_{0})$ of the narrow-band (continuous-wave) laser radiation. During the measurement time $\Delta t$ satisfying the condition ${f}_{P}^{-1}>>\Delta t>>{\tau}_{P}(L)$ (${f}_{P}$ is the pulse repetition frequency), the intensity $I(L,\rho )$ of cw radiation with the frequency ${f}_{0}$ does not change. Consequently, the energy density $W(L,\rho ){|}_{{\tau}_{0}=\infty}=\Delta t\text{\hspace{0.17em}}I(L,\rho )$ measured in a time $\Delta t$ after normalization to $W(0,0){|}_{{\tau}_{0}=\infty}$ is ${I}_{N}(L,\rho )$. The calculation of statistical characteristics of the energy density of pulsed radiation ${W}_{N}(L,\rho )=W(L,\rho )/W(0,0)$ and the intensity ${I}_{N}(L,\rho )$ of continuous-wave radiation from the same realizations of arrays of random phase screens allows us to perform the comparative analysis of turbulent fluctuations of the pulsed and cw radiation.

Figure 1 shows an example of simulated random realizations
of normalized energy densities ${I}_{N}(L,\rho )$ and ${W}_{N}(L,\rho )$. The simulation was conducted at $N$ = 20, $\Delta z$ = 50 m, and ${C}_{n}^{2}$ = 10^{−12} m^{-2/3}. One can see that the
distribution of ${W}_{N}(L,\rho )$ is smoother than that of ${I}_{N}(L,\rho )$.

From the array of simulated random realizations of ${I}_{N}({z}_{j},\rho )$ and ${W}_{N}({z}_{j},\rho )$ we calculated the standard deviations of relative fluctuations of the radiation energy density ${\sigma}_{W}({z}_{j})$ at different durations of the pulse ${\tau}_{P}(0)$. For the cw radiation, ${\sigma}_{W}$ is nothing else than the standard (rms) deviation of relative intensity fluctuations ${\sigma}_{I}$. Calculated ${\sigma}_{W}$ as functions of the parameter ${\beta}_{0}=\sqrt{1.23{C}_{n}^{2}{(2\pi /{\lambda}_{0})}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{z}_{j}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$ characterizing the strength of optical turbulence on the propagation path of the length ${z}_{j}$ [1–5] are shown in Fig. 2. The calculations were carried out for the regime of plane wave (when we can take ${a}_{0}=\infty $ in numerical simulation), $N$ = 100, $\Delta z$ = 10 m, and the same values of ${\lambda}_{0}$, $L$, and ${C}_{n}^{2}$ as taken in calculations shown in Fig. 1.

The simulation parameters ${f}_{1}$, $\Delta f$, $K$, ${f}_{K}={f}_{1}+K\Delta f$, ${\lambda}_{\mathrm{min}}=c/{f}_{K}$ and ${\lambda}_{\mathrm{max}}=c/{f}_{1}$ are given in Table 1.At the same ${f}_{1}$ and ${f}_{K}$ we decreased $\Delta f$ and increased $K$by factor 2 and obtained similar result that is sown in Fig. 2. Therefore we can assume that the simulation parameters given in Table 1 are quite acceptable.

It is seen from Fig. 2 that the values of ${\sigma}_{W}$ for cw and pulsed radiation at ${\beta}_{0}\le 1$ differ only slightly. With an increase of ${\beta}_{0}$, the standard deviation of energy density fluctuations in the both cases achieves the maximum in the vicinity of ${\beta}_{0}$ = 2 and then decreases. Under the condition ${\beta}_{0}$ = 2 there is a focusing of cw plane wave intensity fluctuations in the atmosphere [2–5]. This effect occurs, as is seen from Fig. 2, for energy density fluctuations of pulsed plane wave as well. The standard deviation ${\sigma}_{W}$ for the pulsed radiation (curves 2-6) decreases more rapidly than that for cw radiation (curve 1) and, starting from certain values of ${\beta}_{0}$, depending on the pulse duration, becomes smaller than unity. At the same time, ${\sigma}_{W}$ for cw radiation at ${\beta}_{0}>1$ exceeds unity and tends to unity in the limit ${\beta}_{0}\to \infty $ [2–5].

For the plane wave, the average value of the spectral intensity ${S}_{I}({x}_{i},\rho ,f)$ is independent of spatial coordinates $({x}_{i},\rho )$ and determined as $<{S}_{I}(f)>\text{\hspace{0.17em}}=\text{\hspace{0.17em}}|\tilde{E}(0,\rho ,f){|}^{2}$ at ${a}_{0}=\infty $ (see Eq. (4)), while the relative variance of the energy density ${\sigma}_{W}^{2}({x}_{i},\rho )$ depends only on ${x}_{i}$. Then, from Eqs. (31) and (32) we have

In the process of numerical simulation imitating the propagation of a femtosecond pulses in the turbulent atmosphere, one can see how the form of the spectrum ${S}_{I}({z}_{j},\rho ,f)$ changes with an increase of the index $j$ at fixed $\rho $. With an increase of the distance of pulse propagation in the turbulent atmosphere under the condition ${\beta}_{0}>1$, the characteristic frequency scale ${f}_{T}$ of spectral fluctuations decreases. As a result, after integration with respect to $f$ in Eq. (31), the partial averaging of radiation energy density fluctuations occurs. In the limit of ${\beta}_{0}\to \infty $, the ratio ${f}_{T}/{\sigma}_{f}\to 0$ and, correspondingly, ${\sigma}_{W}\to 0$ in contrast to the cw radiation, for which ${\sigma}_{W}\to 1$. Much earlier, the effect of decorrelation of strong intensity fluctuations of frequency-separated optical waves and the effect of averaging of intensity fluctuations of the spatially partially coherent cw radiation in the turbulent atmosphere were described in [16, 32, 33].

Figure 3 shows the calculated normalized correlation
function of the radiation energy density ${C}_{W}(r)\text{}\text{\hspace{0.17em}}/<W{>}^{2}=\text{\hspace{0.17em}}<W(L,\rho +r)W(L,\rho )>/<W{>}^{2}-1,$ where $r=\text{\hspace{0.17em}}\left|r\right|$, for $L$ = 1 km and ${C}_{n}^{2}$ = 10^{−12} m^{-2/3}, when the parameter
${\beta}_{0}^{2}$ takes the value ${\beta}_{0}^{2}\approx $33.

It is well-known [2–5] that the correlation function of strong intensity fluctuations of the narrow-band radiation (curve 1 in Fig. 3) at ${\beta}_{0}>>1$ is characterized by two scales. The first scale determining the size of area of strong correlation of intensity fluctuations is roughly equal to the radius of spatial coherence of the wave field in the turbulent atmosphere ${\rho}_{c}$. The second scale determining the distance of weak correlation is proportional to $\lambda L/{\rho}_{c}$. With increasing the optical turbulence strength (increase of ${\beta}_{0}$), the first scale decreases, while the second one increases. It follows from Fig. 3 that the decrease of the pulse duration ${\tau}_{P}(0)$ leads to a significant decrease of the correlation function of energy density fluctuations at the distances determined by the first scale. Thus, the frequency averaging mostly affects the small-scale strongly correlated fluctuations of radiation energy. The analogous effect takes place in the temporal averaging of the spatially partially coherent cw radiation [32, 33].

Figure 4 shows dimensionless distributions of the probability density of radiation energy density fluctuations $<W>p(W)$ at different pulse durations ${\tau}_{P}(0)$ calculated from simulated random realizations of the complex spectral amplitude of the wave field. For comparison, the dashed curve shows the exponential distribution of the normalized probability density $\mathrm{exp}(-W/<W>)$, which corresponds to the normal distribution of the probability density of fluctuations of the complex amplitude of wave field for the cw radiation. It can be seen that even at rather large values of ${\beta}_{0}^{2}\approx $33 the curve 1 for the probability density of the distribution of intensity fluctuations of the cw radiation differs significantly from the dashed curve.

It is seen from Fig. 4 that for cw plane wave (curve 1) the maximum of probability density is at $W/<W>\text{\hspace{0.17em}}=0$. For pulsed plane waves (curves 2-6) the probability density has maxima at points $W/<W>\text{\hspace{0.17em}}\ne 0$. A decrease of the pulse duration ${\tau}_{P}(0)$ leads to a decrease of $p(W)$ in the ranges $0\le (W/<W>)\text{\hspace{0.17em}}<<1\text{\hspace{0.17em}}$and $(W/<W>)\text{\hspace{0.17em}}>2\text{\hspace{0.17em}}$and approaching the maximal value of the distribution of energy density fluctuations ${W}_{\mathrm{max}}$ to the mean value of the energy density $<W>$.

As it follows from Eq. (31), due to the decorrelation of intensity fluctuations of waves with different frequencies (wavelengths), the averaging over the frequencies takes place. Taking into account the calculation results shown in Fig. 3, such averaging is equivalent to the spatial averaging of small-scale energy density fluctuations (compare Fig. 1(b) and Fig. 1(d)). As a result for strong turbulence conditions ${\beta}_{0}^{2}\approx $33 and extremely short pulse durations the probabilities of small ($W/<W>\text{\hspace{0.17em}}<<1$) and large ($W/<W>\text{\hspace{0.17em}}\text{\hspace{0.17em}}>>1$) values of the energy density become very small (see curve 6 in Fig. 4).

## 5. Conclusions

Fluctuations of energy density of short-pulse optical radiation propagating in the turbulent atmosphere in absence of nonlinear effects and absorption of the radiation energy have been studied. The dependence of the variance ${\sigma}_{W}^{2}$ of energy density fluctuations of the femtosecond pulsed radiation (at ${f}_{0}$ = 300 THz and ${\tau}_{P}\ge $ 3 fs) propagating in the atmosphere in the regime of plane wave on the strength of optical turbulence (parameter ${\beta}_{0}$) has been calculated. It is shown that with increase of the optical turbulence strength the dependence ${\sigma}_{W}({\beta}_{0})$ for the pulsed radiation begins to differ increasingly widely from the well-known dependence of the standard deviation of plane wave intensity ${\sigma}_{I}({\beta}_{0})$ for the cw radiation. In the limit ${\beta}_{0}^{2}\to \infty ,$ ${\sigma}_{W}({\beta}_{0})\to 0,$ whereas ${\sigma}_{I}({\beta}_{0})\to 1.$ Under conditions of strong optical turbulence, as the pulse duration decreases, the averaging of small-scale fluctuations of radiation energy density occurs and the spatial structure of energy density fluctuations becomes more large-scale and homogeneous. The maximum of the probability density of energy density distribution fluctuations $p(W)$ tends to the mean value of the radiation energy density.

The described algorithm for numerical solution of the parabolic wave equation for the spectral complex amplitude of the wave field can be generalized to the case of arbitrary spatiotemporal initial distribution of the amplitude and phase of a pulsed radiation beam. It can also be used for numerical investigation of laser detection and ranging [34, 35] based on broad band pulsed radiation.

## Acknowledgments

This study was supported by the Russian Scientific Foundation for Maintenance and Development (Project 14-17-00386).

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