## Abstract

The influence of air turbulence on the long-range filamentation of femtosecond laser pulses has been numerically investigated. Simulations are performed for different parameters of air turbulence and laser pulses. Simulation results indicate that the diameter of filaments formed by free propagated fs laser pulse can be widened to mm level under air turbulence. However, the widening effect can be suppressed if the propagating distance before the on-set position of filamentation becomes shorter. The reduction of non-linear focal length can be realized by pre-focusing of the laser pulse or increasing of the laser intensity. The effect of the inner scale of air turbulence on the filamentation has also been studied.

©2008 Optical Society of America

## 1. Introduction

The filamentation of intense femtosecond (fs) laser pulses in the atmosphere [1, 2, 3, 4, 5] has attracted many interests of scientists during the last few years due to its potential applications such as atmospheric remote sensing and lighting control [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For these practical applications, the laser pulse has to be launched into space of high altitude. Hence, a deeper understanding of filamentation in natural air is of great importance.

Most of the previous theoretical and numerical studies were devoted to femtosecond pulse propagation in uniform air. Only a few works considered the perturbation in the laser field induced by atmospheric turbulence and aerosols [17, 18, 19, 20]. The self-focusing distance is found to be averagely smaller than that in uniform atmosphere under turbulent conditions. The random moving focus in the transverse plane outlines a ‘curved’ filament along the propagation distance. Recent experiments [21, 22, 23] investigated the light filaments generated by the free propagated ps and fs laser beams and demonstrated that not only the filaments drifted in transverse plane, but also their diameters expanded to mm size in comparison with numerical simulation, which is usually simulated the ideal Gaussian pulse in uniform air. As a result, such differences should be related to the feature of natural air and the initial intensity perturbation of the laser pulse.

For deep investigation of the widening filamentation, we present in this paper the systematic simulation of filamentation in turbulent atmosphere. The widening of filament diameter to mm level was induced for freely propagated fs pulse under air turbulence. The effect of nonlinear focal length on the filamentation under air turbulence was also studied by varying the initial focusing condition and initial pulse duration.

## 2. Numerical simulation model

The classical equations describing the propagation of ultra-short laser pulses consist of a (3D+1)-dimensional extended nonlinear Schrödinger (NLS) equation for the electric field envelope, and the evolution equation for the local plasma density [24]:

where *n*′ (*x*,*y*,*z*) describes random fluctuations of refractive index induced by air turbulence. These equations are expressed in the reference frame moving with the pulse group velocity (*t*→*t*-*z*/*ν _{g}*), characterized by the central wave number

*k*=2

*π*/λ

_{0}, where λ

_{0}=800 nm is the central wavelength of the laser beam in air. The terms on the right-hand side of Eq. (1) account for the transverse diffraction with Δ

_{⊥}=∂

^{2}/∂

*x*

^{2}+∂

^{2}/∂

*y*

^{2}, the group velocity dispersion, the Kerr response of air, the turbulent effect, the defocusing induced by multiphoton ionization (MPI) and the power dissipation caused by multiphoton absorption (MPA) with coefficient

*k*″=0.2 fs

^{2}/cm,

*n*

_{2}=3.2×10

^{-19}cm

^{2}/W,

*β*

^{(K=8})=4.25×10

^{-98}cm

^{-13}/W

^{7}at 800 nm. The number of photons needed to extract electrons from neutral atoms is

*K*=8. The plasma frequency is ${\omega}_{pe}=\sqrt{\frac{{q}_{e}^{2}\rho}{{m}_{e}{\epsilon}_{0}}}$ (

*q*,

_{e}*m*, and

^{e}*ρ*, are the electron charge, mass, and density, respectively), and the density of neutral atoms is

*ρ*=2.7×10

_{at}^{19}cm

^{-3}. We ignore the effect of self-steepening and space-time focusing. These effects mainly influence the temporal characteristics of the pulses which are not the main interest of our work. The previous study also reported that pulse spatial dynamics was little affected by self-steepening and space-time focusing when the input power was moderate, even for the very short pulse duration as 50fs [25].

In the simulation, equation (1) is solved using full-step (not split-step) Fast Fourier Transformation in transverse space and time dimensions. We use a square grid (256×256) with the step 0.117mm in the transverse section. The whole time interval was discretized into 64 points. The spatial resolution is enough to describe the mm size filaments and basically indicate the characteristics of thin filaments, although it is a little bit lower than that in Ref. [19], where the size of spatial grid is 0.08mm. We also performed testing simulation by increasing the number of spatial grids to 512. However, we found no significant change with higher resolution.

The effects of the air turbulence on the laser propagation are usually simulated using thin phase screens which perturb the phase of a propagating wavefront [26]. The chain of phase screens, located along the propagation direction, reproduces adequately the properties of a continuous medium. The laser pulse will propagate freely between the two neighboring phase screens. To describe a wide range of refractive-index fluctuation we use the von Kármán model spectrum [27]. The detail of the atmospheric turbulence model is presented in the Appendix. There are three important parameters: *C*
^{2}
_{n} is the structure constant which represents the strength of the turbulence; *L*
_{0} and *l _{m}* are outer and inner scales of turbulence. The outer scale was set to be 1 m [18, 19, 27] in our simulation. We varied the inner scale of turbulence

*l*in the range of 0.5~1.5 mm and

_{m}*C*

^{2}

_{n}in the range of 2.0×10

^{-17~}2.75×10

^{-16}m

^{-2/3}. The separation distance between two neighbouring phase screens is 1 m. A sample phase screen with inner scale

*l*=1mm is shown in figure 1. The size of the screen is 30 mm×30 mm, 256×256 sample points, and the structure constant

_{m}*C*

^{2}

_{n}=2.75×10

^{-16}m

^{-2/3}.

## 3. Numerical simulations and discussions

In order to reveal the effect of air turbulence on filamentation of fs laser pulse, we performed numerical simulations of propagation in uniform air without turbulence for comparison. In our simulation, a negative chirped pulse with 270 fs duration is used to generate long range filament. The initial beam waist is 10 mm and the pulse energy is 10 mJ. The laser pulse has a Gaussian shape in both spatial and temporal domains.

Figure 2(a) shows the isosurface of energy fluence of laser filament in unperturbed air. The energy fluence is normalized by the maximum in transverse plane for every propagation distance. From Fig. 2(a), we can see the beam smoothly self-focuses and form a thin filament at the distance of 45 m. The transverse intensity distribution of filament with no turbulence for different propagation distances is shown in Fig. 3 (a). The filament diameter, which is defined as full width at the half maximum of the energy fluence on the transverse cross section, is about 146 *µm* along the propagation distance, as shown in Fig. 4 (solid line). The 146 *µm* is the average filament diameter we obtained using Gaussian approximation of normalized transverse energy fluence distribution. Figure 2(b) and 2(c) show the energy fluence distribution in weak (*C*
^{2}
_{n}=2.7×10^{-17} m^{-2/3}) and moderate (*C*
^{2}
_{n}=2.75×10^{-16} m^{-2/3}) turbulent atmosphere respectively with the inner scale *l _{m}*=1 mm. The beam profile was disturbed in the weak turbulent air (Fig. 2(b)) compared with the beam profile without turbulence in Fig. 2(a). However, the energy still fuses into one thin filament in Fig. 2(b). In figure 2(c), the beam is randomly nucleated and forming a bath of spiky filaments over shorter distances (<30 m) in moderate turbulence. The center of the beam wanders before the beam collapsing [18,19] and the energy fuses into one thin filament with widened diameter. Many small random local intensity peaks emerge after the propagation distance of 60 m. These small intensity peaks are formed due to the random dynamics of laser field and they disappear quickly. Only the continued and long channel is optical filament, whose length is much longer than the natural diffraction (Rayleigh) length for the beam waist of this optical filament. Figure 3(b) shows the transverse intensity distribution of filament with turbulence for different propagation distances. The beam diameter is about 1

^{~}2 mm in most part of the propagation track (Fig. 4 dashed line). The inner scale of turbulence in our simulation is 1 mm, which is relatively small to the transverse size of filament and much smaller than the background energy reservoir, which is extended to cm distance from the filament center for such long filament. As a result, the air turbulence causes phase perturbations on the energy background and this perturbation can be accumulated with propagation. The distortion of wave front can partly break the process of energy replenishment from background reservoir to the filament core [28]. Thus, this effect can lead the widening of filament size on large distance propagation and decrease of the filament intensity to 10

^{12}W/cm

^{2}which is around the ionization threshold of air. Fig. 5 shows peak electron densities for the femtosecond laser pulse in unperturbed air (a) and in moderate turbulent air (b) as a function of propagation distance z. We can find that the electron density generated in the widened filament is only at 10

^{0~}10

^{1}level and the plasma is inefficient in saturating the self-focusing. The electron plasma generation does not play a critical role in the long distance propagation of free filaments. The beam size of the filament keeps stable if the parameters of the turbulence are the same. However, the trajectory of the filament depends on the random number generator used to define the phase screen. Thin filament with higher intensity generates a large spectral broadening due to self phase modulation. The spectral broadening which occurs during filamentation in unperturbed air (a) and in moderate turbulent air (b) is shown in Figure 6. The spectrum of the pulse is broadened in the case of unperturbed air despite the limited simulating spectral range (720 nm

^{~}880 nm). However, the pulse in strong turbulent air undergoes a much smaller spectral broadening due to the weak intensity of the filament (Fig. 6(b)).

For convenience of indoor experiments, a converging lens is always required to reduce the propagation distance before the on-set position of filaments is formed. Figure 7 shows the energy fluence distribution when a laser pulse with 10mJ energy is prefocused by lenses of focal length 4 m (a), 6 m (b), and 8 m (c). Although the turbulent air is the same as the free propagation (Fig. 2(b)), it can be seen from Fig. 7 that the transverse widths of the filaments are very small for all focal conditions, which agrees well with both numerical and experimental results [29, 30]. Earlier experiment [31] shows that for filamentation of prefocused laser pulse, the size of energy background is about 1mm, which is comparable to the inner scale of air turbulence. In this case, the energy background is slightly influenced by the turbulent air. The dynamic energy exchange between the filament and background is successfully taking place to form a thin filament, as that happens in unperturbed air. The influence of the air turbulence reduces with the decrease of the on-set position of filamentation. This is consistent with the experiment of Ref. [17] which reported that a thin filament can be generated by a prefocusing beam even in strong turbulent air.

Another method to reduce the on-set position of filamentation is to increase the peak power of the laser pulses, which is related to the pulse duration. Figure 8 (a, b) shows the energy fluence distribution of unchirped pulse (50 fs) and chirped pulse (100 fs) in turbulent air, with the structure constant *C*
^{2}
_{n}=2.75×10^{-16} m^{-2/3}. We can see in Fig. 8 that the filamentation starts at different propagation distance depending on the pulse duration. For shorter laser pulse duration (Fig. 8 (a) and (b)), the peak power of laser pulse becomes higher, and the distance required for generation of filaments becomes shorter. This effect can also reduce the accumulating influence of turbulent air. So thin filament with diameter 0.15 mm ^{~} 0.2 mm can be obtained in this case, which is similar to the case of prefocused laser pulses. It has been confirmed by earlier experiments that thin filament can also be formed by free propagated fs laser pulses, when the nonlinear focal length was in several meters range [25]. However, without the initial focusing, the multifilamentation takes place due to the perturbation of laser wave front.

The role of inner scale of the turbulence has also been studied. Figure 9 shows the energy fluence distribution of the laser propagation through turbulent atmosphere with different inner scale *l _{m}*=0.7 mm (a), 1.5 mm (b). The structure constant is fixed on 2.75×10

^{-16}m

^{-2/3}. Turbulent atmosphere is comprised of many small turbulent “lenses” with diameters comparable to the inner scale of the turbulence. These lenses can decrease the overall spatial coherence of the laser beam. However, the laser field can keep the local coherence in the spatial range of inner scale. When the inner scale is small (Fig. 9(a)), i.e.,

*l*=0.7 mm, background energy can not effectively replenish the filament core. Then the diameter of the filament keeps wide. When the inner scale is large, (Fig. 9(b)), i.e.,

_{m}*l*=1.5 mm, more laser energy can be self-focused into filament. As a result, thin filament can be formed.

_{m}## 4. Conclusion

In conclusion, we have numerically investigated the influence of the air turbulence on the long-range fs laser filamentation in detail. Air turbulence causes random fluctuations of refractive index of the atmosphere. Widening filament of hundred meters is induced by the perturbation of air turbulence. Thin filament in turbulent air can be formed by prefocused laser pulse, due to the much shorter distance required for generation of filament. In addition, our simulation indicates that the perturbation of the air turbulence is increased with the decrease of the inner scale *l _{m}*.

## Appendix

The von Kármán models of atmospheric turbulence, as used in the present calculation, is adopted from Refs. [26,27]. In this model, the power spectral density Φ_{n} (*k*, *z*) of the refractive index fluctuation of the atmosphere [27] is:

where *C*
^{2}
_{n} is the structure constant which represents the strength of the turbulence and *k* is the three-dimensional spatial wavenumber,
$k=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2}}$
. The critical wavenumbers correspond to turbulence scale lengths *L*
_{0}=2*π*/*k*
_{0} and *l _{m}*=2

*π*/

*k*, where

_{m}*L*

_{0}and

*l*are outer and inner scales of turbulence respectively. For

_{m}*k*<

*k*

_{0}, Φ

_{n}(

*k*,

*z*) is limited by

*k*

_{0}, and for

*k*>

*k*, Φ

_{m}_{n}(

*k*,

*z*) would be quickly forced to zero.

The generation of phase screen *θ*′ (*x*,*y*) can be realized by filtering a Gaussian white noise process with the square root of the power spectral density, followed by an inverse Fourier transform, using

where *a*(*k _{x}*,

*k*) is a zero-mean unit-variance Hermitian complex Gaussian white noise process.

_{y}The discrete formulation of (A2) is

where Δ*x* and Δ*y* are the desired intervals. *N _{x}* and

*N*are integer indices.

_{y}*a*(

*n*,

*m*) and

*b*(

*n*,

*m*) are discrete zero-mean Gaussian variables. The variances are

The minimum spatial frequency of the phase screen generated by the FFT-method is 1/Δ*xN _{x}*. The lower frequency information of the phase screen is given by

where *N _{p}* is the subharmonic levels.

*a*(

*n*,

*m*,

*p*) and

*b*(

*n*,

*m*,

*p*) are discrete zero-mean Gaussian variables, whose variances are

where Δ*k _{xp}*=Δ

*k*/3

_{x}^{p}, Δ

*k*=Δ

_{yp}*k*/3

_{y}^{p}.

The total phase screen is the sum of (A3) and (A4)

And the random fluctuations of refractive index induced by air turbulence is

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos 60621063, 10634020, 60478047, 10734130 and 10521002), the National Key Basic Research Special Foundation of China under Grant No 2007CB815101, No 2006CB806007, and the National Hi-tech ICF Programme.

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