Abstract

Turbulent fluctuations of the energy density of broadband pulsed Laguerre–Gaussian beams are 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 is shown that in the regime of strong scintillations, the relative variance of energy density of the pulsed beams can take values smaller than unity, in contrast to the strong scintillation index of the continuous-wave beams, which tends to unity with increasing the turbulence strength. The level of residual spatial correlation of the energy density of pulsed beams exceeds that for the continuous-wave beams. It increases with shortening of the pulse duration and increasing of the refractive turbulence strength.

© 2016 Optical Society of America

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=Aexp(±jmθ) having nonzero orbital angular momentum (±m0) in the plane transverse to the direction of propagation looks like a wound helix. The helix shape is determined by the azimuth angle θ and the value of the topological charge ±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 x0, and the electric field strength of the wave at the initial plane x = 0 can be represented in the form

E(0,ρ,t)=Enm0(ρ)exp(t22τ022πjf0t),
where the complex amplitude Enm0(ρ) at the point (x = 0,ρ) is specified as the Laguerre–Gaussian beam [12–14]

Enm0(ρ)=Enm0(0)(j)m12(ρa)mexp[ρ22a2+jψ0+jmθ]Lnm(ρ2a2).

The parameter a determines the limitedness of the Laguerre beam in transverse plane,j=1, ρ={y,z} is the radius vector in the plane perpendicular to the axis х, θ=arctg(y/z), Lnm=x1exp(x)dndxn[xn+1exp(x)] is the Laguerre polynomial, integer numbers m and n determine the Laguerre–Gaussian modes Enm0, ψ0 is the wave phase independent of ρ and t, t is time, f0 is the frequency at the point of maximum of the beam temporal spectrum, τ0 is the initial pulse duration determined from the decrease of |E(0,0,t)|2 down to the level e1|Enm0(0)|2. The pulse duration τP(x) determined as a pulse full width at half maximum of the power +d2ρ|E(x,ρ,t)|2, in the plane x = 0 is related to τ0 as τP(0)=2ln2τ0. At m = n = 0, Eq. (2) reduces to the Gaussian beam

E000(ρ)=E0(0)exp(ρ22a2+jΨ0),
where a determines the radius of the Gaussian beam as a distance from the beam axis, at which the intensity decreases down to the level e1|E0(0)|2.

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

E˜(x,ρ,f)=+dtE(x,ρ,t)exp(2πjft)=U(x,ρ,f)exp(2πjfn(f)xc),
is described by the wave equation in the parabolic approximation [12, 13, 15]
j4πfcU(x,ρ,f)x+ΔU(x,ρ,f)+2(2πfc)2n(x,ρ)U(x,ρ,f)=0,
with the boundary condition, according to Eqs. (1) and (4),
E˜(0,ρ,f)=2πτ0Enm0(ρ)exp[(ff0)2(2πτ0)22].
where f is the linear frequency, с is the speed of light in vacuum, Δ=2/y2+2/z2 is the transverse Laplasian operator, n(f) and n(x,ρ) are the average value and fluctuations of the refractive index caused by turbulent variations of the air temperature. Equation (5) is written with regard for the fact that |n(x,ρ,f)|<<1 in the atmosphere and the dependence of turbulent fluctuations of the refractive index on the frequency f in a dispersive medium can be neglected.

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 Ψi(ρ,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,ρ,f) is solved consecutively for every individual component of the spectrum f, where f=f0+(kK/2)Δf, k=0,1,...,K, and Δf is the frequency step.

For simulation of random phase screens Ψi(ρ,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(Ψi) 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

DΨ(r,f)=<[Ψi(ρ+r,f)Ψi(ρ,f)]2>=2+d2κSΨ(κ,f)[1e2πjκρ],
we use the model spectrum of phase fluctuations SΨ(κ,f) in the form [12, 13, 16, 17]
SΨ(κ,f)=0.382Cn2Δx(f/c)2|κ|11/3,
where Cn2 is the structure constant of turbulent fluctuations of the refractive index of air in the atmosphere, κ={κy,κz} is the spatial frequency vector.

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 Ψi(ρ,f0) (<ΨiΨii> = 0) for the 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 Ψi(ρ,f)=(f/f0)Ψi(ρ,f0) is used.

The two-dimensional (2D) distributions of spectral amplitudes U(x,ρ,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])

W(x,ρ)=+dfS1(x,ρ,f)=+df|U(x,ρ,f)|2
in the plane transverse to the propagation direction at the distance x. In Eq. (9), SI(x,ρ,f)=|U(x,ρ,f)|2 is the spectral intensity. In the calculations of W(x,ρ), integration in Eq. (9) was replaced with summation of |U(x,ρ,f)|2 over all components of the frequency spectrum (over all k).

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

σW2(x,ρ)=W2(x,ρ)/W(x,ρ)21,
and the spatial correlation coefficient
CW(x,ρ1,ρ2)=W(x,ρ1)W(x,ρ2)W(x,ρ1)W(x,ρ2)((W2(x,ρ1)W(x,ρ1)2)(W2(x,ρ2)W(x,ρ2)2))1/2
of energy density of the pulsed Laguerre–Gaussian beams.

4. Results of numerical simulation

The numerical simulation of random distributions of W(x,ρ) in the turbulent atmosphere was carried out for the modes E0m0, where m = 0, 2, 8, of the Laguerre–Gaussian beam. The complex spectral amplitude U(x,ρ,f) was calculated for 60 spectral channels at τP(0) = 3 fs (τ0 = 1.8 fs) and for 41 (τP(0) = 5 fs (τ0 = 3 fs)) spectral channels with the width Δf = 10 THz for frequencies f=f1+kΔf, where k = 0, 1,..., K. The initial frequencyf1 = 10 THz at K = 59 (τP(0) = 3 fs) and f1 = 100 THz at K = 40 (τP(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 10104Hz.

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

β02=1.23Cn2(2π/λ0)76x116,
characterizing the refractive turbulence strength on the propagation path of length x [21], takes values exceeding unity. In Eq. (12), λ0=c/f0=1μm is the wavelength corresponding to the maximum of the frequency spectrum of pulsed radiation and the frequency of cw radiation.

The number of nodes of the M×M computational grid and the discretization step h on the grid were specified with regard for the parameter β02. At β02< 36, the M×M = 512×512 grid with the step h = 3 mm was used. At 36β02124, the computations were performed at the M×M = 1024 ×1024 grid with the step of 1.5 mm. At 124<β02185, the grid was 1540×1540 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 β02.

Figures 1–4 show the random distributions of the energy density for the Gaussian beam (Figs. 1, 2) and the E080 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,ρ)=|U(x,ρ,f0|2, normalized to the maximum, are depicted in Figs. 1-4. Since the spectral intensity of the cw radiation SI(x,ρ,f0) is independent of the frequency f, the energy density W(x,ρ)|τ0 (Eq. (9)) in the limiting case of cw radiation is related to the intensity I(x,ρ) by the formula W(x,ρ)|τ0/df=I(x,ρ).

 

Fig. 1 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity IN of cw radiation (a, c, e, g) and normalized energy density WN of pulsed τP(0) = 3 fs (b, d, f, h) Gaussian beam with а = 5 cm at Cn2 = 10−13 m-2/3 (a)–(d) and Cn2 = 10−12 m-2/3 (e)–(h).

Download Full Size | PPT Slide | PDF

 

Fig. 2 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity IN of cw radiation (a, c, e, g) and normalized energy density WN of pulsed τP(0) = 3 fs (b, d, f, h) Gaussian beam with а = 2 cm at Cn2 = 10−13 m-2/3 (a)–(d) and Cn2 = 10−12 m-2/3 (e)–(h).

Download Full Size | PPT Slide | PDF

 

Fig. 3 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of the normalized intensity IN of cw radiation (a, c, e, g) and normalized energy density WN of pulsed τP(0) = 3 fs (b, d, f, h) Laguerre beam E080, а = 5 cm at Cn2 = 10−13 m-2/3 (a)–(d) and Cn2 = 10−12 m-2/3 (e)–(h).

Download Full Size | PPT Slide | PDF

 

Fig. 4 Two-dimensional (a, b, e, f) and one-dimensional (c, d, g, h) distributions of normalized intensity IN of cw radiation (a, c, e, g) and normalized energy density WN of pulsed τP(0) = 3 fs (b, d, f, h) Laguerre beam E080, а = 2 cm at Cn2 = 10−13 m-2/3 (a)–(d) andCn2 = 10−12 m-2/3 (e)–(h).

Download Full Size | PPT Slide | PDF

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 Cn2 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 σW of relative fluctuations of energy density of pulsed radiation calculated by Eq. (10) at the pulse duration τP(0) = 5 fs (τ0 = 3 fs, curves 1'–3′), 3 fs (τ0 = 1.8 fs, curves 1”–3”) and cw radiation τ0→∞ (curves 1–3) of Laguerre–Gaussian beams. The calculations of σW 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 β0 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 E000 (curves 1, 1', 1”), E020 (curves 2, 2', 2”), E080 (curves 3, 3′, 3”) as a function of the parameter β0. Pulse duration τP(0) = 5 fs (curves 1', 2', 3′) and 3 fs (curves 1”, 2”, 3”).

Download Full Size | PPT Slide | PDF

It is seen from Fig. 5 that at the weak refractive turbulence (β0 < 1) there is no almost difference in energy density fluctuations of the modes of the Laguerre–Gaussian beam. As the parameter β0 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 τP(0), the larger the difference. Standard deviation of strong (β02 > 1) intensity fluctuations of the cw Laguerre–Gaussian beams tends to unity very slowly with an increase of β02, 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 β02 > 1, as follows from Fig. 5, fluctuates much less than the intensity of cw beams. In contrast to the cw radiation, σW of pulsed beams does not saturate at some level, but becomes smaller and smaller with an increase of β02. The shorter the pulse, the smaller σW at the same values of β02. Starting from approximately β0 = 6, σW 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 β02, averaged over all the wavelengths corresponding to the pulse bandwidth, as compared to β02(λ0).

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 E0m0 with m = 0, 2, 8 in the calculations of σW 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 E0m0 within the circle [22] is the same for chosen values of а. In Fig. 6, the standard deviation σW for the E020 mode of the Laguerre–Gaussian beam with the parameter a = 2.9 cm (curves 2, 2'), for the E080 mode with the parameter a = 2 cm (curves 3, 3′), and for the Gaussian beam E000 with the radius a = 4 cm (curves 1, 1') was calculated as a function of β02 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 τP(0) = 5 fs (τ0 = 3 fs), while curves 1, 2, 3 are for the cw beams.

 

Fig. 6 Standard deviation of relative energy density fluctuations of pulsed (τP(0) = 5 fs, curves 1'–3′) and cw (curves 1–3) Laguerre–Gaussian modes E000 (curves 1, 1'), E020 (curves 2, 2'), E080 (curves 3, 3”) as a function of the parameter β0.

Download Full Size | PPT Slide | PDF

It is seen from Fig. 6 that for the Laguerre–Gaussian modes E000, E020 and E080 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 (β02 > 1) energy density fluctuations of cw (curves 1, 2, 3) and pulsed τP(0) = 5 fs (curves 1', 2', 3′), 3 fs (curves 1”, 2”, 3”) Laguerre–Gaussian beams for the modes E000 (curves 1, 1', 1”), E020 (curves 2, 2', 2”), and E080 (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 ρ/ag and ρ/ρc, where ag=a[1+(λ0x/2πa)2] is the diffractive radius of the collimated Gaussian beam (3) [21], and ρc=(1.45Cn2k2L)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 E000 (curves 1, 1', 1”), E020 (curves 2, 2', 2”), E080 (curves 3, 3′, 3”) at β02124and τP(0) = 5 fs (curves 1'–3′), 3 fs (curves 1”–3”).

Download Full Size | PPT Slide | PDF

 

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 E000 (curves 1, 1', 1”), E020 (curves 2, 2', 2”), E080 (curves 3, 3′, 3”) at β02185, τP(0) = 5 fs (curves 1'– 3′), 3 fs (curves 1”– 3”).

Download Full Size | PPT Slide | PDF

 

Fig. 9 Coefficient of spatial correlation of energy density fluctuations of pulsed τP(0) = 5 fs (curves 1'–6') and cw (curves 1–6) Laguerre–Gaussian modes E000 (curves 1, 1', 4, 4'), E020 (curves 2, 2', 5, 5′), E080 (curves 3, 3′, 6, 6') at β02124 (curves 1–3, 1'–3′) and β02185 (curves 4–6, 4'–6').

Download Full Size | PPT Slide | PDF

It follows from Fig. 7 that the spatial correlation of intensity fluctuations for all modes of the cw Laguerre–Gaussian beam at β02124 is two-scale in accordance with the results for strong intensity fluctuations [21]. The first scale ρ1 calculated by the level CW(x,ρ1)=e1 determines the area of high spatial correlation, whose dimensions are specified by the coherence radius ρc and decrease with increase of the refractive turbulence strength [21].

The second scale ρ2xλ0/ρc [21] determines the area of weak (residual) spatial correlation within the range of the correlation coefficient from CW(x,ρ)=0.2 to CW(x,ρ2)=0.2e1, 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 τP(0)= 5 fs and nearly fourfold at τP(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 CW(x,ρ) calculated at β02 = 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 τP(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.

Funding

This work is supported by the Russian Science Foundation (Project No. 14-17-00386).

Acknowledgments

The authors are thankful to A.V. Falits for consultations in computer simulations.

References and links

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]   [PubMed]  

2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]  

3. A. V. Falits, “Wandering and intensity fluctuations of focused Laguerre–Gaussian beam in the turbulent atmosphere,” Opt. Atmos. Okeana 28(9), 763–771 (2015).

4. V. A. Banakh and A. V. Falits, “Turbulent broadening of Laguerre–Gaussian beam in the atmosphere,” Opt. Spectrosc. 117(6), 942–948 (2014). [CrossRef]  

5. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef]   [PubMed]  

6. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]  

7. Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014). [CrossRef]   [PubMed]  

8. Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014). [CrossRef]  

9. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014). [CrossRef]   [PubMed]  

10. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

11. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef]   [PubMed]  

12. V. A. Banakh and I. N. Smalikho, “Fluctuations of energy density of short-pulse optical radiation in the turbulent atmosphere,” Opt. Express 22(19), 22285–22297 (2014). [CrossRef]   [PubMed]  

13. V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015). [CrossRef]  

14. Yu. A. Anan’ev, Optical Cavities and Laser Beams. (Nauka, 1990) [in Russian].

15. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Nauka, 1988) [in Russian]

16. V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

17. V. A. Banakh and I. N. Smalikho, Coherent Doppler wind lidars in a turbulent atmosphere (Artech House, 2013)

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

19. 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]  

20. M. V. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1979) [in Russian].

21. V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Current Problems of Atmospheric Optics Vol.5 Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988) [in Russian].

22. A. E. Siegman, "How to (Maybe) Measure Laser Beam Quality," in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

23. 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(6), 626–629 (1983).

References

  • View by:
  • |
  • |
  • |

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
    [Crossref]
  3. A. V. Falits, “Wandering and intensity fluctuations of focused Laguerre–Gaussian beam in the turbulent atmosphere,” Opt. Atmos. Okeana 28(9), 763–771 (2015).
  4. V. A. Banakh and A. V. Falits, “Turbulent broadening of Laguerre–Gaussian beam in the atmosphere,” Opt. Spectrosc. 117(6), 942–948 (2014).
    [Crossref]
  5. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref] [PubMed]
  6. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  7. Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
    [Crossref] [PubMed]
  8. Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
    [Crossref]
  9. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
    [Crossref] [PubMed]
  10. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).
  11. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref] [PubMed]
  12. V. A. Banakh and I. N. Smalikho, “Fluctuations of energy density of short-pulse optical radiation in the turbulent atmosphere,” Opt. Express 22(19), 22285–22297 (2014).
    [Crossref] [PubMed]
  13. V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
    [Crossref]
  14. Yu. A. Anan’ev, Optical Cavities and Laser Beams. (Nauka, 1990) [in Russian].
  15. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Nauka, 1988) [in Russian]
  16. V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).
  17. V. A. Banakh and I. N. Smalikho, Coherent Doppler wind lidars in a turbulent atmosphere (Artech House, 2013)
  18. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39(3), 393–397 (2000).
    [Crossref] [PubMed]
  19. 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]
  20. M. V. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1979) [in Russian].
  21. V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Current Problems of Atmospheric Optics Vol.5 Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988) [in Russian].
  22. A. E. Siegman, "How to (Maybe) Measure Laser Beam Quality," in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  23. 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(6), 626–629 (1983).

2015 (3)

A. V. Falits, “Wandering and intensity fluctuations of focused Laguerre–Gaussian beam in the turbulent atmosphere,” Opt. Atmos. Okeana 28(9), 763–771 (2015).

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
[Crossref]

2014 (5)

2012 (2)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[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]

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

2008 (1)

2004 (1)

2000 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1983 (1)

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(6), 626–629 (1983).

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Anguita, J. A.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Banakh, V. A.

V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Fluctuations of energy density of short-pulse optical radiation in the turbulent atmosphere,” Opt. Express 22(19), 22285–22297 (2014).
[Crossref] [PubMed]

V. A. Banakh and A. V. Falits, “Turbulent broadening of Laguerre–Gaussian beam in the atmosphere,” Opt. Spectrosc. 117(6), 942–948 (2014).
[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, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54(6), 626–629 (1983).

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Barnett, S.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Boyd, R. W.

Buldakov, V. M.

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(6), 626–629 (1983).

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Courtial, J.

Dolinar, S.

Erkmen, B. I.

Falits, A. V.

A. V. Falits, “Wandering and intensity fluctuations of focused Laguerre–Gaussian beam in the turbulent atmosphere,” Opt. Atmos. Okeana 28(9), 763–771 (2015).

V. A. Banakh and A. V. Falits, “Turbulent broadening of Laguerre–Gaussian beam in the atmosphere,” Opt. Spectrosc. 117(6), 942–948 (2014).
[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]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Franke-Arnold, S.

Frehlich, R.

Gerasimova, L. O.

V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
[Crossref]

Gibson, G.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

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(6), 626–629 (1983).

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Neifeld, M. A.

Padgett, M.

Padgett, M. J.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Pas’ko, V.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Shapiro, J. H.

Smalikho, I. N.

V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
[Crossref]

V. A. Banakh and I. N. Smalikho, “Fluctuations of energy density of short-pulse optical radiation in the turbulent atmosphere,” Opt. Express 22(19), 22285–22297 (2014).
[Crossref] [PubMed]

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]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Vasic, B. V.

Vasnetsov, M.

Wang, J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Adv. Opt. Photonics (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7, 66–106 (2015).

Appl. Opt. (2)

Atmos. Oceanic Opt. (1)

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]

Nat. Commun. (1)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Opt. Atmos. Okeana (1)

A. V. Falits, “Wandering and intensity fluctuations of focused Laguerre–Gaussian beam in the turbulent atmosphere,” Opt. Atmos. Okeana 28(9), 763–771 (2015).

Opt. Express (2)

Opt. Lett. (1)

Opt. Spectrosc. (2)

V. A. Banakh and A. V. Falits, “Turbulent broadening of Laguerre–Gaussian beam in the atmosphere,” Opt. Spectrosc. 117(6), 942–948 (2014).
[Crossref]

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(6), 626–629 (1983).

Optica (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Quantum Electron. (1)

V. A. Banakh, L. O. Gerasimova, and I. N. Smalikho, “Numerical investigation of short-pulse laser radiation propagation in a turbulent atmosphere,” Quantum Electron. 45(3), 258–264 (2015).
[Crossref]

Other (7)

Yu. A. Anan’ev, Optical Cavities and Laser Beams. (Nauka, 1990) [in Russian].

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Nauka, 1988) [in Russian]

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

V. A. Banakh and I. N. Smalikho, Coherent Doppler wind lidars in a turbulent atmosphere (Artech House, 2013)

M. V. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, Wave Theory (Nauka, 1979) [in Russian].

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Current Problems of Atmospheric Optics Vol.5 Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988) [in Russian].

A. E. Siegman, "How to (Maybe) Measure Laser Beam Quality," in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
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 τ 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).
Fig. 2
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 τ 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).
Fig. 3
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 τ 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).
Fig. 4
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 τ 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).
Fig. 5
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 β 0 . Pulse duration τ P (0) = 5 fs (curves 1', 2', 3′) and 3 fs (curves 1”, 2”, 3”).
Fig. 6
Fig. 6 Standard deviation of relative energy density fluctuations of pulsed ( τ 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 β 0 .
Fig. 7
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 β 0 2 124 and τ P (0) = 5 fs (curves 1'–3′), 3 fs (curves 1”–3”).
Fig. 8
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 β 0 2 185 , τ P (0) = 5 fs (curves 1'– 3′), 3 fs (curves 1”– 3”).
Fig. 9
Fig. 9 Coefficient of spatial correlation of energy density fluctuations of pulsed τ 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 β 0 2 124 (curves 1–3, 1'–3′) and β 0 2 185 (curves 4–6, 4'–6').

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

E(0,ρ,t)= E nm 0 (ρ)exp( t 2 2 τ 0 2 2πj f 0 t ),
E nm 0 (ρ)= E nm 0 (0) (j) m 1 2 ( ρ a ) m exp[ ρ 2 2 a 2 +j ψ 0 +jmθ ] L n m ( ρ 2 a 2 ).
E 00 0 (ρ)= E 0 (0)exp( ρ 2 2 a 2 +j Ψ 0 ),
E ˜ (x,ρ,f)= + dtE(x,ρ,t)exp(2πjft)=U(x,ρ,f)exp( 2πjf n(f) x c ) ,
j4πf c U(x,ρ,f) x + Δ U(x,ρ,f)+2 ( 2πf c ) 2 n (x,ρ)U(x,ρ,f)=0,
E ˜ (0,ρ,f)= 2π τ 0 E nm 0 (ρ)exp[ (f f 0 ) 2 (2π τ 0 ) 2 2 ].
D Ψ (r,f)=< [ Ψ i (ρ+r,f) Ψ i (ρ,f)] 2 >=2 + d 2 κ S Ψ (κ,f)[1 e 2πjκρ ] ,
S Ψ (κ,f)=0.382 C n 2 Δx (f/c) 2 |κ | 11/3 ,
W(x,ρ)= + df S 1 (x,ρ,f)= + df | U(x,ρ,f) | 2
σ W 2 (x,ρ)= W 2 (x,ρ) / W(x,ρ) 2 1,
C W (x, ρ 1 , ρ 2 )= W(x, ρ 1 )W(x, ρ 2 ) W(x, ρ 1 ) W(x, ρ 2 ) ( ( W 2 (x, ρ 1 ) W(x, ρ 1 ) 2 )( W 2 (x, ρ 2 ) W(x, ρ 2 ) 2 ) ) 1/2
β 0 2 =1.23 C n 2 ( 2π/ λ 0 ) 7 6 x 11 6 ,

Metrics