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 [15]. 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 [25]. 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 [25]. 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 [79]. 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, 1016]. 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 [1719]. 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 [1924]. 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 [25], 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 z0. The complex strength of the electric field of the wave at the point (z,ρ) (ρ={x,y}is the radius vector in the plane perpendicular to the optical axis) at the time t is denoted as E(z,ρ,t). The laser radiation is assumed to be fully coherent, and the strength E(0,ρ,t) in the initial plane can be represented in the form:

E(0,ρ,t)=E0exp(ρ22a02t22τ022πif0t+jψ0),
where E0=E(0,0,0) is the amplitude at the axis, a0 and τ0 are the initial beam radius and pulse duration determined from decrease of, respectively, |E(0,ρ,0)|2 and |E(0,0,t)|2 to the level E02e1, i=1, f0 is the frequency at the point of maximum of the radiation spectrum, and ψ0 is the wave phase independent of ρ and t. The pulse duration τP(z) determined from decrease of the radiation power P(z,t)=+d2ρI(z,ρ,t) = +d2ρ|E(z,ρ,t)|2, where I(z,ρ,t)=|E(z,ρ,t)|2 is the intensity (power density) of radiation, down to the level (1/2)P(z,tmax) to the right and to the left from the point of maximum t=tmax, in the plane z = 0 is related with τ0 by the equation: τP(0)=2ln2τ0.

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

E˜(z,ρ,f)=+dtE(z,ρ,t)exp(2πift)
in the absence of radiation absorption by air and aerosol particles and nonlinear effects has the form [17]
2E˜(z,ρ,f)z2+ΔE˜(z,ρ,f)+(2πfc)2n2(z,ρ,f)E˜(z,ρ,f)=0.
In Eq. (3), Δ=2/x2+2/y2 is the transverse Laplace operator, c is the speed of light in vacuum, and n(z,ρ,f) is the refractive index of air depending on the radiation wavelength λ=c/f. The boundary condition for Eq. (3), according to Eqs. (1) and (2), is
E˜(0,ρ,f)=2πτ0E0exp[ρ22a02(ff0)22σf2+iψ0],
where σf=(2πτ0)1 is the width of the radiation spectrum determined from decrease of the spectral power SP = |E˜(0,0,f)|2 from the point of maximum to the level 2πτ02E02e1. Laser pulse peak power threshold of nonlinear optical Kerr effect, self-focusing, filamentation in the atmosphere depends on wavelength and nonlinear refractive index [17]. For considered below conditions the peak power should be less than ~3 GW to avoid the nonlinear effects.

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]

n(z,ρ,f)=1+106PaT(z,ρ)[77.6+0.584λ02(ff0)2],
where Pa is the atmospheric pressure in mb, T(x,ρ) is the air temperature in K, and λ0=c/f0 in µm. With allowance for the fact that the temperature <T> averaged over an ensemble of realizations in the atmosphere is much larger, in the absolute value, than turbulent fluctuations of temperature T(z,ρ)=T(z,ρ)<T> [2], the refractive index can be written as
n(z,ρ,f)=<n(f)>+n(z,ρ,f),
where
<n(f)>=1+106Pa<T>[77.6+0.584λ02(ff0)2]
is the average value and
n(z,ρ,f)=[1<n(f)>]T(z,ρ)/<T>
are fluctuations of the refractive index caused by turbulent variations of the air temperature. Taking into account that in the atmosphere |n(z,ρ,f)|<<1 [1], in Eq. (3) we use the approximate equality

n2(z,ρ,f)=<n(f)>2[1+2n(z,ρ,f)/<n(f)>].

In the absence of turbulent fluctuations of the refractive index n(z,ρ,f)<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]

E˜(z,ρ,f)=E˜(0,ρ,f)exp[2πif<n(f)>z/c].
Using, according to Eq. (2), the relationship
E(z,ρ,t)=+dfE˜(z,ρ,f)exp(2πitf).
and Eqs. (4) and (10), we obtain the following equation for the radiation power P(z,t):
P(z,t)=2(πτ0a0E0)2|+dfexp{(ff0)22σf22πi[t<n(f)>z/c]f}|2.
The integral in Eq. (12) can be calculated analytically, if we use Eq. (7) for<n(f)> and expand the function Ω(f)=f<n(f)> into the Taylor series in the vicinity of the point f=f0, restricting our consideration to the three first terms of this series [17]. As a result, for the pulse duration as a function of the distance z we obtain
τP(z)=2ln2τ01+(3πμzcτ02f0)2,
where the dimensionless parameter μ is determined as
μ=106Pa<T>0.584λ02.
In Eq. (14), Pa, T, and λ0 are given in the same units as in Eq. (5).

According to the calculation by Eqs. (13) and (14), at τ0 = 3 fs (τP(0) = 5 fs), λ0 = 1 µm (f0 = 300 THz), Pa = 1013 mb and <T> = 288° K, the pulse duration at the distance z = 1 km exceeds the initial pulse duration in 2358 times (cτP(z) = 1.5 µm at z = 0 and cτP(z) = 3.5 mm at z = 1 km).

By analogy with Eq. (10), we represent the spectral amplitude of the wave field E˜(z,ρ,f) in the form

E˜(z,ρ,f)=U(z,ρ,f)exp(2πif<n(f)>z/c).
Upon the substitution of Eq. (15) into Eq. (3), we obtain the equation for the complex spectral amplitude U(z,ρ,f), in which we can neglect the term 2U(z,ρ,f)/z2 [25]. Ultimately, replacing <n(f)> with unity (because the refractive index of air differs from unity only in the fourth decimal place [1]) and n(z,ρ,f) with n(z,ρ) (in the dispersive medium, the dependence of turbulent fluctuations of the refractive index on the frequency f can be neglected), with allowance for Eq. (9) we come to the parabolic wave equation for the complex spectral amplitude in the form
i4πfcU(z,ρ,f)z+ΔU(z,ρ,f)+2(2πfc)2n(z,ρ)U(z,ρ,f)=0.
Since U(0,ρ,f)=E˜(0,ρ,f), the boundary condition of Eq. (16) is determined by Eq. (4).

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 thicknessΔz=L/N. The complex spectral amplitude U(zj,ρ,f) for every frequencyf, where zj=jΔz and j=0,1,...,N, is calculated consecutively at transition from one layer to another. Equation (16) for the spectral amplitude at the end of the jth layer (z[zj1,zj1+Δz]), 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 U1(zj,ρ,f) is described by Eq. (16), in which the second term is put equal to zero, U(zj,ρ,f) is replaced by U1(zj,ρ,f), and the boundary condition

U1(zj1,ρ,f)=U(zj1,ρ,f).
With allowance for Eq. (17), the solution of this equation has the form
U1(zj,ρ,f)=U(zj1,ρ,f)exp[iΨj(ρ,f)],
where
Ψj(ρ,f)=(2πf/c)0Δzdzn˜(zj1+z,ρ)
is a random phase screen.

Stage 2. Only the wave diffraction is taken into account. Then the spectral amplitude of the wave field denoted as U2(zj,ρ,f) is described by Eq. (16) with omitted third term and U(zj,ρ,f) replaced with U2(zj,ρ,f) with the boundary condition

U2(zj1,ρ,f)=U1(zj,ρ,f).
Upon the application of the direct two-dimensional Fourier transform
U˜2(z,κ,f)=+d2ρU2(z,ρ,f)exp(2πiκρ),
where κ={κx,κy} is the vector of spatial frequencies, from the equation for U2 we obtain the ordinary differential equation for U˜2,whose solution with allowance for Eqs. (20) and (21) has the form
U˜2(zj,κ,f)=U˜2(zj1,κ,f)exp(iπκ2Δzc/f),
where
U˜2(zj1,κ,f)=+d2ρU1(zj,ρ,f)exp(2πiκρ).
Finally, the spectral amplitude U(zj,ρ,f)=U2(zj,ρ,f) is calculated based on the inverse two-dimensional Fourier transform:

U(zj,ρ,f)=+d2κU˜2(zj,κ,f)exp(2πiκρ).

For simulation of random phase screens Ψj(ρ,f), it is necessary to set their statistical properties. We assume that in the jth layer of the propagation path the probability density of phase fluctuations p(Ψj) has the normal distribution [15]. 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

DΨ(r,f)=<[Ψj(ρ+r,f)Ψj(ρ,f)]2>=2+d2κSΨ(κ,f)[1exp(2πiκρ)],
we use the model spectrum of phase fluctuations SΨ(κ,f) in the form [15, 27]
SΨ(κ,f)=0.382Cn2Δz(f/c)2|κ|11/3,
where Cn2 is the structure characteristic of turbulent fluctuations of the refractive index of air.

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) [2730] allows us to obtain independent random realizations of phase Ψj(ρ,f0) (<ΨjΨjj> = 0) for given frequency f0 on the (M×M) computational grid with the step h. To obtain realizations of a random phase screen at other frequencies f, the equation Ψj(ρ,f)=(f/f0)Ψj(ρ,f0) is used.

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

DΨ(r,f)=2.92Cn2Δz(2πf/c)2|r|53.
Since in computational grid h>0 and Mh<, the values h and Mh/2 can be considered as analogs of, respectively, the inner and outer scales of turbulence. In this case, the structure function of simulated random phase screens differs from the result of calculation by Eq. (27). For the structure function of the simulated phase screen at h<<|r|Mh to coincide with the results of calculation by Eq. (27), the method of subharmonics [29, 30] can be used.

The spectral amplitude U(zj,ρ,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=f1+kΔf (k=0,1,...,K and Δf is the frequency step) at nodes of the uniform grid ρ={(mxM/2)h,(myM/2)h} (mx,y=0,1,...,M1). For calculation of U˜2(zj1,κ,f) at nodes of the computational grid κ={kx/(Mh),ky/(Mh)} (kx,y=0,1,...,M1) and U(zj,ρ,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×M×M) arrays of calculated complex variables and the needed RAM volume.

The simulation parameters f1, Δf and Kcan be selected empirically for given f0 and τ0. We used the described above algorithm, assuming the condition: A(0)<0.01, where A(f)=|E˜(0,0,f)|/|E˜(0,0,f0)|. Then from Eq. (4) we have: τ0>ln10/(πf0) or τP(0)>0.8/f0. According to this condition, for example, in the case of f0 = 300 THz the pulse duration τP(0) must exceed 2.67 fs (or τ0 > 1.6 fs).

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

SI(L,ρ,f)=|U(L,ρ,f)|2,
spectral power
SP(L,f)=+d2ρSI(L,ρ,f),
and radiation energy density
W(L,ρ)=+dtI(L,ρ,t)=+dt|E(L,ρ,t)|2.
From Eqs. (11), (15), (28), and (30), we obtain

W(L,ρ)=+dfSI(L,ρ,f).

When the absorption of the radiation energy on the propagation path is neglected, the spectral power of radiation SP(L,f), in contrast to the spectral intensity SI(L,ρ,f), is independent of the path length L, that is, SP(L,f)=SP(0,f) [17]. To take the absorption into account, we should multiply the calculated values of U(zj,ρ,f) by exp[αj(f)Δz/2] in every layer, where αj(f) is the absorption coefficient of the radiation energy by atmospheric air and aerosol in the jth layer.

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

σW2(L,ρ)=<W2(L,ρ)>/<W(L,ρ)>21,
correlation function CW(L,ρ1,ρ2)=<W(L,ρ1)W(L,ρ2)><W(L,ρ1)><W(L,ρ1)>, and probability density function p(W).

In calculation of the energy density W(L,ρ) by Eqs. (28) and (31), the integration of |U(L,ρ,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,ρ), we set the following parameters: λ0 = 1 µm (f0 = 300 THz), τP(0) = 5 fs (τ0 = 3 fs, σf = 53 THz), a0 = 5 cm, and L = 1 km. For these values of λ0 and τ0, 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 Δf = 10 THz, that is, in the discrete representation the frequency takes values f=f1+kΔf where f1 = 100 THz, k=0,1,...,K and K = 40.

In parallel with W(L,ρ), we calculated the spectral intensity SI(L,ρ,f0) and the normalized intensity IN(L,ρ)=SI(L,ρ,f0)/SI(0,0,f0) of the narrow-band (continuous-wave) laser radiation. During the measurement time Δt satisfying the condition fP1>>Δt>>τP(L) (fP is the pulse repetition frequency), the intensity I(L,ρ) of cw radiation with the frequency f0 does not change. Consequently, the energy density W(L,ρ)|τ0==ΔtI(L,ρ) measured in a time Δt after normalization to W(0,0)|τ0= is IN(L,ρ). The calculation of statistical characteristics of the energy density of pulsed radiation WN(L,ρ)=W(L,ρ)/W(0,0) and the intensity IN(L,ρ) 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 IN(L,ρ) and WN(L,ρ). The simulation was conducted at N = 20, Δz = 50 m, and Cn2 = 10−12 m-2/3. One can see that the distribution of WN(L,ρ) is smoother than that of IN(L,ρ).

 

Fig. 1 Two-dimensional (а, c) and one-dimensional (b, d) random realizations of the normalized energy density of cw (а, b) and pulsed (c, d) radiation at Cn2 = 10−12 m-2/3.

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From the array of simulated random realizations of IN(zj,ρ) and WN(zj,ρ) we calculated the standard deviations of relative fluctuations of the radiation energy density σW(zj) at different durations of the pulse τP(0). For the cw radiation, σW is nothing else than the standard (rms) deviation of relative intensity fluctuations σI. Calculated σW as functions of the parameter β0=1.23Cn2(2π/λ0)76zj116 characterizing the strength of optical turbulence on the propagation path of the length zj [15] are shown in Fig. 2. The calculations were carried out for the regime of plane wave (when we can take a0= in numerical simulation), N = 100, Δz = 10 m, and the same values of λ0, L, and Cn2 as taken in calculations shown in Fig. 1.

 

Fig. 2 Standard deviation of relative energy density fluctuations of cw (curve 1)and pulsed (curves 2-6) radiation as a function of the parameter β0.Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs

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The simulation parameters f1, Δf, K, fK=f1+KΔf, λmin=c/fK and λmax=c/f1 are given in Table 1.At the same f1 and fK we decreased Δf and increased Kby 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.

Tables Icon

Table 1. Simulation Parameters

It is seen from Fig. 2 that the values of σW for cw and pulsed radiation at β01 differ only slightly. With an increase of β0, the standard deviation of energy density fluctuations in the both cases achieves the maximum in the vicinity of β0 = 2 and then decreases. Under the condition β0 = 2 there is a focusing of cw plane wave intensity fluctuations in the atmosphere [25]. This effect occurs, as is seen from Fig. 2, for energy density fluctuations of pulsed plane wave as well. The standard deviation σW for the pulsed radiation (curves 2-6) decreases more rapidly than that for cw radiation (curve 1) and, starting from certain values of β0, depending on the pulse duration, becomes smaller than unity. At the same time, σW for cw radiation at β0>1 exceeds unity and tends to unity in the limit β0 [25].

For the plane wave, the average value of the spectral intensity SI(xi,ρ,f) is independent of spatial coordinates (xi,ρ) and determined as <SI(f)>=|E˜(0,ρ,f)|2 at a0= (see Eq. (4)), while the relative variance of the energy density σW2(xi,ρ) depends only on xi. Then, from Eqs. (31) and (32) we have

σW2(zj)=(πτ0E02)2+df1+df2<SI(f1)><SI(f2)>Ks(zj,f1,f2)
where Ks(zj,f1,f2)=<s(zj,ρ,f1)s(zj,ρ,f2)> is the frequency correlation function of normalized fluctuations of the spectral intensity of radiation s(zj,ρ,f)=SI(zj,ρ,f)/<SI(f)>1. It follows from the experiment [31] that the ratio Ks(zj,f1,f2)/Ks(zj,f,f), where f=(f1+f2)/2 and f1f2, in the range 0β01 remains close to unity and decreases with an increase of β0 in the range β0>1.Therefore, with allowance for Eq. (33), for the broadband radiation in the regime of strong intensity fluctuations, when β0>>1, the variance of energy density fluctuations is significantly less than that for the narrow-band radiation (τ0).

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 SI(zj,ρ,f) changes with an increase of the index j at fixed ρ. With an increase of the distance of pulse propagation in the turbulent atmosphere under the condition β0>1, the characteristic frequency scale fT 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 β0, the ratio fT/σf0 and, correspondingly, σW0 in contrast to the cw radiation, for which σW1. 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 CW(r)/<W>2=<W(L,ρ+r)W(L,ρ)>/<W>21, where r=|r|, for L = 1 km and Cn2 = 10−12 m-2/3, when the parameter β02 takes the value β0233.

 

Fig. 3 Normalized correlation function of energy density fluctuations of cw (curve 1) and pulsed (curves 2-6) radiation at β0233. Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs (6).

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It is well-known [25] that the correlation function of strong intensity fluctuations of the narrow-band radiation (curve 1 in Fig. 3) at β0>>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 ρc. The second scale determining the distance of weak correlation is proportional to λL/ρc. With increasing the optical turbulence strength (increase of β0), the first scale decreases, while the second one increases. It follows from Fig. 3 that the decrease of the pulse duration τP(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 τP(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 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 β0233 the curve 1 for the probability density of the distribution of intensity fluctuations of the cw radiation differs significantly from the dashed curve.

 

Fig. 4 Normalized probability density distribution of energy density fluctuations of cw (curve 1) and pulsed (curves 2-6) radiation at β0233. Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs (6). Dashed curve shows the exponential distribution of the probability density.

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It is seen from Fig. 4 that for cw plane wave (curve 1) the maximum of probability density is at W/<W>=0. For pulsed plane waves (curves 2-6) the probability density has maxima at points W/<W>0. A decrease of the pulse duration τP(0) leads to a decrease of p(W) in the ranges 0(W/<W>)<<1and (W/<W>)>2and approaching the maximal value of the distribution of energy density fluctuations Wmax 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 β0233 and extremely short pulse durations the probabilities of small (W/<W><<1) and large (W/<W>>>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 σW2 of energy density fluctuations of the femtosecond pulsed radiation (at f0 = 300 THz and τP 3 fs) propagating in the atmosphere in the regime of plane wave on the strength of optical turbulence (parameter β0) has been calculated. It is shown that with increase of the optical turbulence strength the dependence σW(β0) for the pulsed radiation begins to differ increasingly widely from the well-known dependence of the standard deviation of plane wave intensity σI(β0) for the cw radiation. In the limit β02, σW(β0)0, whereas σI(β0)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).

References and links

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7. V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999). [CrossRef]  

8. S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

9. V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012). [CrossRef]  

10. C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).

11. C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977). [CrossRef]  

12. C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]  

13. C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980). [CrossRef]  

14. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Broadening of a space-time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37, 7655–7660 (1998). [CrossRef]   [PubMed]  

15. D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999). [CrossRef]  

16. V. U. Zavorotnyi, “Frequency correlation of strong intensity fluctuations in a turbulent medium,” Radiophys. Quant. Electr. 24(5), 601–608 (1981). [CrossRef]  

17. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses, (Nauka, Moscow: 1988), pp. 1–310. [in Russian]

18. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985). [CrossRef]  

19. I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta (Lond.) 33(1), 63–72 (1986). [CrossRef]  

20. I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009). [CrossRef]  

21. V. A. Banakh, “Diffraction-free propagation of a focused delta-pulsed beam,” Opt. Lett. 36(23), 4539–4541 (2011). [CrossRef]   [PubMed]  

22. L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).

23. V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013). [CrossRef]  

24. V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).

25. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010). [CrossRef]  

26. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976). [CrossRef]  

27. V. P. Kandidov, “Monte Carlo method in nonlinear statistical optics,” Phys. Usp. 39(12), 1243–1272 (1996). [CrossRef]  

28. V. A. Banakh and I. N. Smalikho, Coherent Doppler Wind Lidars in a Turbulent Atmosphere (Artech House, Boston-London, 2013), Chap. 2.

29. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39(3), 393–397 (2000). [CrossRef]   [PubMed]  

30. V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012). [CrossRef]  

31. A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).

32. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

33. V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).

34. V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011). [CrossRef]  

35. I. N. Smalikho, “Calculation of the backscatter amplification coefficient of laser radiation propagating in a turbulent atmosphere using numerical simulation,” Atmos. Oceanic Opt. 26(2), 135–139 (2013). [CrossRef]  

References

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  1. V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), pp. 1–312.
  2. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, Moscow, 1976), pp. 1–280. [in Russian]
  3. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation Through Random Media (Springer, 1989), pp. 1–246.
  4. V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere, (Artech House, Boston & London, 1987), pp. 1–185.
  5. V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of Turbulent Atmosphere (Gidrometeoizdat, Leningrad, 1988), pp. 1–270. [in Russian]
  6. Femtosecond Atmospheric Optics. S. N. Bagaev and G. G. Matvienko, eds., (Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2010), (1–238 pp.). [in Russian]
  7. V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
    [Crossref]
  8. S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).
  9. V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012).
    [Crossref]
  10. C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).
  11. C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977).
    [Crossref]
  12. C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
    [Crossref]
  13. C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980).
    [Crossref]
  14. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Broadening of a space-time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37, 7655–7660 (1998).
    [Crossref] [PubMed]
  15. D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
    [Crossref]
  16. V. U. Zavorotnyi, “Frequency correlation of strong intensity fluctuations in a turbulent medium,” Radiophys. Quant. Electr. 24(5), 601–608 (1981).
    [Crossref]
  17. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses, (Nauka, Moscow: 1988), pp. 1–310. [in Russian]
  18. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
    [Crossref]
  19. I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta (Lond.) 33(1), 63–72 (1986).
    [Crossref]
  20. I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009).
    [Crossref]
  21. V. A. Banakh, “Diffraction-free propagation of a focused delta-pulsed beam,” Opt. Lett. 36(23), 4539–4541 (2011).
    [Crossref] [PubMed]
  22. L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).
  23. V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
    [Crossref]
  24. V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).
  25. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
    [Crossref]
  26. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
    [Crossref]
  27. V. P. Kandidov, “Monte Carlo method in nonlinear statistical optics,” Phys. Usp. 39(12), 1243–1272 (1996).
    [Crossref]
  28. V. A. Banakh and I. N. Smalikho, Coherent Doppler Wind Lidars in a Turbulent Atmosphere (Artech House, Boston-London, 2013), Chap. 2.
  29. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39(3), 393–397 (2000).
    [Crossref] [PubMed]
  30. V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
    [Crossref]
  31. A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).
  32. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).
  33. V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).
  34. V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011).
    [Crossref]
  35. I. N. Smalikho, “Calculation of the backscatter amplification coefficient of laser radiation propagating in a turbulent atmosphere using numerical simulation,” Atmos. Oceanic Opt. 26(2), 135–139 (2013).
    [Crossref]

2013 (3)

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).

I. N. Smalikho, “Calculation of the backscatter amplification coefficient of laser radiation propagating in a turbulent atmosphere using numerical simulation,” Atmos. Oceanic Opt. 26(2), 135–139 (2013).
[Crossref]

2012 (2)

V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
[Crossref]

V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012).
[Crossref]

2011 (3)

V. A. Banakh, “Diffraction-free propagation of a focused delta-pulsed beam,” Opt. Lett. 36(23), 4539–4541 (2011).
[Crossref] [PubMed]

L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).

V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011).
[Crossref]

2010 (1)

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

2009 (1)

I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009).
[Crossref]

2007 (1)

S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

2000 (1)

1999 (2)

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
[Crossref]

1998 (1)

1996 (1)

V. P. Kandidov, “Monte Carlo method in nonlinear statistical optics,” Phys. Usp. 39(12), 1243–1272 (1996).
[Crossref]

1986 (1)

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta (Lond.) 33(1), 63–72 (1986).
[Crossref]

1985 (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
[Crossref]

1983 (2)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).

1981 (1)

V. U. Zavorotnyi, “Frequency correlation of strong intensity fluctuations in a turbulent medium,” Radiophys. Quant. Electr. 24(5), 601–608 (1981).
[Crossref]

1980 (1)

C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980).
[Crossref]

1979 (2)

C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).

1977 (2)

C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).

C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977).
[Crossref]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Akturk, S.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

Andrews, L. C.

D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
[Crossref]

C. Y. Young, L. C. Andrews, and A. Ishimaru, “Broadening of a space-time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37, 7655–7660 (1998).
[Crossref] [PubMed]

Banakh, V. A.

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).

V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011).
[Crossref]

V. A. Banakh, “Diffraction-free propagation of a focused delta-pulsed beam,” Opt. Lett. 36(23), 4539–4541 (2011).
[Crossref] [PubMed]

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).

Bowlan, P.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

Brodeur, A.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

Buldakov, V. M.

V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Chin, S. L.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

Christov, I. P.

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta (Lond.) 33(1), 63–72 (1986).
[Crossref]

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
[Crossref]

Falits, A. V.

V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
[Crossref]

I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009).
[Crossref]

Fedorov, V. Yu.

S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Frehlich, R.

Gerasimova, L. O.

V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).

Gu, X.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

Gurvich, A. S.

A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).

Ishimaru, A.

Kan, V.

A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).

Kandidov, V. P.

V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012).
[Crossref]

S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

V. P. Kandidov, “Monte Carlo method in nonlinear statistical optics,” Phys. Usp. 39(12), 1243–1272 (1996).
[Crossref]

Kelly, T. S.

D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
[Crossref]

Kosareva, O. G.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

Lui, C.-H.

C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980).
[Crossref]

C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).

C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977).
[Crossref]

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Shlyonov, S. A.

V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012).
[Crossref]

S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

Smalikho, I. N.

I. N. Smalikho, “Calculation of the backscatter amplification coefficient of laser radiation propagating in a turbulent atmosphere using numerical simulation,” Atmos. Oceanic Opt. 26(2), 135–139 (2013).
[Crossref]

V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011).
[Crossref]

Tamarov, M. P.

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

Tikhomirova, O. V.

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

Tjin, D. E.

D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
[Crossref]

Trebino, R.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

Yeh, K. C.

C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980).
[Crossref]

C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977).
[Crossref]

C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).

Young, C. Y.

Zaloznaya, I. V.

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).

I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009).
[Crossref]

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Frequency correlation of strong intensity fluctuations in a turbulent medium,” Radiophys. Quant. Electr. 24(5), 601–608 (1981).
[Crossref]

Appl. Opt. (2)

Appl. Phys. (Berl.) (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976).
[Crossref]

Atmos. Oceanic Opt. (9)

I. V. Zaloznaya and A. V. Falits, “Diffraction contraction of short pulses,” Atmos. Oceanic Opt. 22(6), 590–594 (2009).
[Crossref]

L. O. Gerasimova and I. V. Zaloznaya, “Spatial and temporal coherence of short pulses (in Russian),” Atmos. Oceanic Opt. 24(3), 185–189 (2011).

V. A. Banakh, L. O. Gerasimova, I. V. Zaloznaya, and O. V. Tikhomirova, “Diffraction of broadband pulsed light beams,” Atmos. Oceanic Opt. 26(3), 178–184 (2013).
[Crossref]

V. A. Banakh and L. O. Gerasimova, “Propagation of broadband pulsed optical beams (in Russian),” Atmos. Oceanic Opt. 26(1), 5–10 (2013).

V. A. Banakh, I. N. Smalikho, and A. V. Falits, “Effectiveness of the subharmonic method in problems of computer simulation of laser beam propagation in a turbulent atmosphere,” Atmos. Oceanic Opt. 25(2), 106–109 (2012).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Determination of optical turbulence intensity by atmospheric backscattering of laser radiation,” Atmos. Oceanic Opt. 24(5), 457–465 (2011).
[Crossref]

I. N. Smalikho, “Calculation of the backscatter amplification coefficient of laser radiation propagating in a turbulent atmosphere using numerical simulation,” Atmos. Oceanic Opt. 26(2), 135–139 (2013).
[Crossref]

S. A. Shlyonov, V. Yu. Fedorov, and V. P. Kandidov, “Filamentation of phase-modulated femtosecond laser pulse on kilometer-long paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 20(4), 275–283 (2007).

V. P. Kandidov and S. A. Shlyonov, “Thermal self-action of laser beams and filamentation of pulses in turbulent atmosphere,” Atmos. Oceanic Opt. 25(3), 192–198 (2012).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

C.-H. Lui and K. C. Yeh, “Pulse propagation in random media,” IEEE Trans. Antenn. Propag. 26, 561–566 (1977).

J. Opt. (1)

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 1–20 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

C.-H. Lui and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain or fog,” J. Opt. Soc. Am. A 67(9), 1261–1266 (1977).
[Crossref]

C.-H. Lui and K. C. Yeh, “Statistics of pulse arrival time in turbulent media,” J. Opt. Soc. Am. A 70(2), 168–172 (1980).
[Crossref]

Opt. Acta (Lond.) (1)

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta (Lond.) 33(1), 63–72 (1986).
[Crossref]

Opt. Commun. (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53(6), 364–366 (1985).
[Crossref]

Opt. Lett. (1)

Opt. Spectrosc. (2)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 55(4), 423–426 (1983).

Phys. Usp. (1)

V. P. Kandidov, “Monte Carlo method in nonlinear statistical optics,” Phys. Usp. 39(12), 1243–1272 (1996).
[Crossref]

Quantum Electron. (1)

V. P. Kandidov, O. G. Kosareva, M. P. Tamarov, A. Brodeur, and S. L. Chin, “Nucleation and random movement of filaments in the propagation of high-power laser radiation in a turbulent atmosphere,” Quantum Electron. 29(10), 911–915 (1999).
[Crossref]

Radio Sci. (1)

C.-H. Lui and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

Radiophys. Quant. Electr. (2)

V. U. Zavorotnyi, “Frequency correlation of strong intensity fluctuations in a turbulent medium,” Radiophys. Quant. Electr. 24(5), 601–608 (1981).
[Crossref]

A. S. Gurvich and V. Kan, “Fluctuations of the radiation intensity from two wave sources in a turbulent medium,” Radiophys. Quant. Electr. 22(7), 843–847 (1979).

Waves Random Media (1)

D. E. Tjin, T. S. Kelly, and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999).
[Crossref]

Other (8)

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses, (Nauka, Moscow: 1988), pp. 1–310. [in Russian]

V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), pp. 1–312.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, Moscow, 1976), pp. 1–280. [in Russian]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics: Wave Propagation Through Random Media (Springer, 1989), pp. 1–246.

V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmosphere, (Artech House, Boston & London, 1987), pp. 1–185.

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of Turbulent Atmosphere (Gidrometeoizdat, Leningrad, 1988), pp. 1–270. [in Russian]

Femtosecond Atmospheric Optics. S. N. Bagaev and G. G. Matvienko, eds., (Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2010), (1–238 pp.). [in Russian]

V. A. Banakh and I. N. Smalikho, Coherent Doppler Wind Lidars in a Turbulent Atmosphere (Artech House, Boston-London, 2013), Chap. 2.

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Figures (4)

Fig. 1
Fig. 1 Two-dimensional (а, c) and one-dimensional (b, d) random realizations of the normalized energy density of cw (а, b) and pulsed (c, d) radiation at C n 2 = 10−12 m-2/3.
Fig. 2
Fig. 2 Standard deviation of relative energy density fluctuations of cw (curve 1)and pulsed (curves 2-6) radiation as a function of the parameter β 0 .Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs
Fig. 3
Fig. 3 Normalized correlation function of energy density fluctuations of cw (curve 1) and pulsed (curves 2-6) radiation at β 0 2 33. Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs (6).
Fig. 4
Fig. 4 Normalized probability density distribution of energy density fluctuations of cw (curve 1) and pulsed (curves 2-6) radiation at β 0 2 33. Curves 2–6 correspond to pulse durations of 50 (2), 20 (3), 10 (4), 5 (5), and 3 fs (6). Dashed curve shows the exponential distribution of the probability density.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (33)

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E ( 0 , ρ , t ) = E 0 exp ( ρ 2 2 a 0 2 t 2 2 τ 0 2 2 π i f 0 t + j ψ 0 ) ,
E ˜ ( z , ρ , f ) = + d t E ( z , ρ , t ) exp ( 2 π i f t )
2 E ˜ ( z , ρ , f ) z 2 + Δ E ˜ ( z , ρ , f ) + ( 2 π f c ) 2 n 2 ( z , ρ , f ) E ˜ ( z , ρ , f ) = 0.
E ˜ ( 0 , ρ , f ) = 2 π τ 0 E 0 exp [ ρ 2 2 a 0 2 ( f f 0 ) 2 2 σ f 2 + i ψ 0 ] ,
n ( z , ρ , f ) = 1 + 10 6 P a T ( z , ρ ) [ 77.6 + 0.584 λ 0 2 ( f f 0 ) 2 ] ,
n ( z , ρ , f ) = < n ( f ) > + n ( z , ρ , f ) ,
< n ( f ) > = 1 + 10 6 P a < T > [ 77.6 + 0.584 λ 0 2 ( f f 0 ) 2 ]
n ( z , ρ , f ) = [ 1 < n ( f ) > ] T ( z , ρ ) / < T >
n 2 ( z , ρ , f ) = < n ( f ) > 2 [ 1 + 2 n ( z , ρ , f ) / < n ( f ) > ] .
E ˜ ( z , ρ , f ) = E ˜ ( 0 , ρ , f ) exp [ 2 π i f < n ( f ) > z / c ] .
E ( z , ρ , t ) = + d f E ˜ ( z , ρ , f ) exp ( 2 π i t f ) .
P ( z , t ) = 2 ( π τ 0 a 0 E 0 ) 2 | + d f exp { ( f f 0 ) 2 2 σ f 2 2 π i [ t < n ( f ) > z / c ] f } | 2 .
τ P ( z ) = 2 ln 2 τ 0 1 + ( 3 π μ z c τ 0 2 f 0 ) 2 ,
μ = 10 6 P a < T > 0.584 λ 0 2 .
E ˜ ( z , ρ , f ) = U ( z , ρ , f ) exp ( 2 π i f < n ( f ) > z / c ) .
i 4 π f c U ( z , ρ , f ) z + Δ U ( z , ρ , f ) + 2 ( 2 π f c ) 2 n ( z , ρ ) U ( z , ρ , f ) = 0.
U 1 ( z j 1 , ρ , f ) = U ( z j 1 , ρ , f ) .
U 1 ( z j , ρ , f ) = U ( z j 1 , ρ , f ) exp [ i Ψ j ( ρ , f ) ] ,
Ψ j ( ρ , f ) = ( 2 π f / c ) 0 Δ z d z n ˜ ( z j 1 + z , ρ )
U 2 ( z j 1 , ρ , f ) = U 1 ( z j , ρ , f ) .
U ˜ 2 ( z , κ , f ) = + d 2 ρ U 2 ( z , ρ , f ) exp ( 2 π i κ ρ ) ,
U ˜ 2 ( z j , κ , f ) = U ˜ 2 ( z j 1 , κ , f ) exp ( i π κ 2 Δ z c / f ) ,
U ˜ 2 ( z j 1 , κ , f ) = + d 2 ρ U 1 ( z j , ρ , f ) exp ( 2 π i κ ρ ) .
U ( z j , ρ , f ) = + d 2 κ U ˜ 2 ( z j , κ , f ) exp ( 2 π i κ ρ ) .
D Ψ ( r , f ) = < [ Ψ j ( ρ + r , f ) Ψ j ( ρ , f ) ] 2 > = 2 + d 2 κ S Ψ ( κ , f ) [ 1 exp ( 2 π i κ ρ ) ] ,
S Ψ ( κ , f ) = 0.382 C n 2 Δ z ( f / c ) 2 | κ | 11 / 3 ,
D Ψ ( r , f ) = 2.92 C n 2 Δ z ( 2 π f / c ) 2 | r | 5 3 .
S I ( L , ρ , f ) = | U ( L , ρ , f ) | 2 ,
S P ( L , f ) = + d 2 ρ S I ( L , ρ , f ) ,
W ( L , ρ ) = + d t I ( L , ρ , t ) = + d t | E ( L , ρ , t ) | 2 .
W ( L , ρ ) = + d f S I ( L , ρ , f ) .
σ W 2 ( L , ρ ) = < W 2 ( L , ρ ) > / < W ( L , ρ ) > 2 1 ,
σ W 2 ( z j ) = ( π τ 0 E 0 2 ) 2 + d f 1 + d f 2 < S I ( f 1 ) > < S I ( f 2 ) > K s ( z j , f 1 , f 2 )

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