Beam wandering of femtosecond laser filament in air

Jing Yang; Tao Zeng; Lie Lin; Weiwei Liu

doi:10.1364/OE.23.025628

1. Introduction

During recent years, the filamentation of femtosecond laser pulses has stimulated extensive research interests [1–8 ]. Particularly, pulse compression by the filamentation has become one of the highly expected applications of this unique nonlinear optical phenomenon [9–14 ]. Significant studies in this field have focused on the limit of the pulse duration obtained by this pulse compression method. The maximum extractable pulse energy is another major concern. At the same time, the spatial mode of the self-compressed pulse after the filamentation process has been found to be quite satisfying. It has been attributed to the self-cleaning phenomenon accompanying the filamentation [15–17 ]. Eventually, the beam pointing stability is another critical factor in subtle experiment such as high order harmonic generation (HHG) by using the filamentational laser pulse [18]. Besides, a new concept of “remote air lasing” has become a quite active topic recently due to its great potential in atmospheric remote-sensing [19–22 ]. The beam pointing stability will be also a crucial issue for achieving a high performance remote air laser.

In order to study the effect of turbulence on the filament wondering, artificial air turbulence has been introduced before or after the onset of filamentation [23]. It has been found in the later case the beam pointing fluctuation is weaker [23]. The present work reports a recent study on the stability of the beam pointing of a filament in air when no artificial turbulence is introduced. The results have revealed that the transverse position of the filament will fluctuates inside a cone angle of approximately 15 μrad. The observed spatial beam wandering could be associated with the air turbulence resulted from the filament heating effect [24]. By using numerical simulation, the strength of the filamentation induced turbulence could be characterized by a turbulent structure constant of $C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{- 2 / 3}$ in framework of classical Karman model describing the atmospheric turbulence.

2. Experiment results

In our experiment, femtosecond laser pulses with repetition rate of 1 kHz were sent in ambient air. In order to produce a long filament, the initial laser beam was squeezed by an inverse telescope consisting of a concave mirror (f = 40 cm) and a plan-concave lens (f = −20 cm). The displacement between the mirror and the lens was 20 cm. At the output of the lens, the laser beam diameter was about 3 mm at 1/e² level. The pulse duration was 45 fs (full width at half maximum, FWHM). The central wavelength of the laser pulse was 800 nm, and the laser energy was 2 mJ/pulse. In this case, a filament was produced starting from a distance 95 cm with respect to the plan-concave lens. Perpendicular to the propagation axis, the emitted fluorescence light was collected by a 1-inch-diameter fused silica lens, whose focal length was 1 inch, onto a photomultiplier tube (PMT). In front of PMT, an interference filter was inserted to isolate one particular nitrogen fluorescence spectral line, namely, 337 nm, from the scattered light of the pump beam. Then the PMT signal was sent to an oscilloscope for analysis. The longitudinal fluorescence distribution is recorded in Fig. 1(a) , in which two signal peaks manifest the re-focusing process [25].

Fig. 1 (a) Fluorescence distribution perpendicular to the propagation axis of a filament. (b) Beam full width at half maximum (FWHM) along the horizontal and vertical axises versus the propagation distance, tree insets show beam pattern at various distance.

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The beam profiles inside the filament have also been investigated in our experiment. Two parallel fused silica wedges were inserted in the laser beam path, both at grazing angles, yielding a reflectivity of about 10% at each front surface. Therefore, after two surface reflections the laser intensity was reduced to approximate 1%. The cross sections of the laser beam were then registered by a CCD camera through a calibrated 1:4 image setup [26]. The exposure time of the CCD was set as 1 ms in order to capture single shot laser for each picture. Various neutral density filters were put in front of the CCD camera to further attenuate the laser intensity. The evolution of the laser beam pattern as a function of the propagation distance was investigated by moving the wedges and the CCD together longitudinally. Note that at each position, the laser beam patterns were recorded for 20 times. The insets of Fig. 1(b) demonstrate the representative recorded beam pattern at various distance, namely, z = 80 cm, 120 cm and 160 cm, respectively. And Fig. 1(b) indicates the calculated beam FWHM along the horizontal and the vertical axises versus the propagation distance. Two pronounce minima in Fig. 1(b) are consistent with the results shown in Fig. 1(a) that re-focusing takes place in our experiment. More interesting, the beam pointing wandering has been quantified by calculating the standard deviation of the beam center position along both the horizontal direction and the vertical direction. The beam center coordinates (x_c, y_c) are calculated according to:

x_{c} = \sum_{x} \sum_{y} x \cdot S (x, y) / \sum_{x} \sum_{y} S (x, y), y_{c} = \sum_{x} \sum_{y} y \cdot S (x, y) / \sum_{x} \sum_{y} S (x, y),

where S indicates the intensity of the pixel. The obtained results are depicted in Fig. 2 . As indicated in Fig. 2, the beam pointing is very stable up to roughly 85 cm, where the plasma filament starts to emerge. It hints that the nature turbulence in our lab is not strong enough to cause significant deviation of the beam pointing. On the further propagation, once the filament is created, the center of the laser beam wanders transversely. The longer the filament length is, the more severe the beam wanders. According to the results shown in Fig. 2, the beam center during the filamentation scatters mainly within a cone angle of 15 μrad.

Fig. 2 Standard deviation of the beam center position along both the horizontal and vertical direction versus the propagation distance and three fitting curves with different turbulent structure constants $C_{n}^{2} = 1.9 \times 10^{- 12} {cm}^{- 2 / 3}$ (orange dashed line), $C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{- 2 / 3}$ (green short dotted line) and $C_{n}^{2} = 4.9 \times 10^{- 13} {cm}^{- 2 / 3}$ (blue dotted line).

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3. Simulation results

The origin of the beam wandering occurred during the filamentation could be attributed to the air turbulence induced by the laser filament heating [24]. In order to describe the spatial fluctuations of the refractive index resulted from the filament heating, the classical model describing the atmospheric turbulence, namely the modified Karman spectrum, has been considered [23,27 ]:

F_{n} (κ_{x}, κ_{y}, κ_{z}) = 0.033 C_{n}^{2} {(κ^{2} + κ_{0}^{2})}^{- 11 / 6} \exp (- {(κ / κ_{m})}^{2})

where

C_{n}^{2}

refers to the refractive index structure constant, while

κ_{0} = 2 π / L_{0}

and

κ_{m} = 5.92 / l_{0}

. L₀ and l₀ are outer and inner scales of turbulence, respectively. The scale of the turbulent air refractive index fluctuations varies from the inner scale l₀ which is about 0.1~1cm to the outer scale L₀ which can be tens of meters [27].

Then the phase screen, which represents the cumulative phase shift during the propagation, is used to model the refractive index fluctuations. The beam propagation range is divided into several segments and the phase fluctuations spectral density of each segment Δz has the form:

F_{ϕ} (κ_{x}, κ_{y}) = 2 π k_{0}^{2} Δ z F_{n} (κ_{x}, κ_{y}, 0)

The outer scale of turbulence is chosen to be 1 m and the inner scale equals to 1 mm in our simulation. Different phase screens have been created with different series of random number via the same computation method as Ref. 28. According to the method described in [28], the spatial phase fluctuations ϕ(x, y) are reconstructed by the summation of the Fourier harmonics of the spatial spectrum indicated by Eq. (3) and taking into account the contribution of the low-frequency spectral component to the resulting phase screen. Then, the refractive index fluctuations are taken into account in the numerical simulation of the 2D + 1 (A(x,y,z)) nonlinear wave equation [14, 29 ]:

2 i k_{0} \frac{\partial A}{\partial z} + (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) A + 2 k_{0}^{2} Δ n A + 2 k_{0}^{2} \tilde{n} (x, y, z) A = 0

where A represents the amplitude of the light field and k₀ is the wave number of the beam with the central wavelength of 800 nm in our simulation. Δn includes the nonlinear refractive index induced by the optical Kerr effect (Δn = n ₂ I) and the effective counteracting higher order nonlinear refractive index of plasma defocusing effect (Δn _plasma = -σI ^m). The coefficient n₂ is 2.0 × 10⁻¹⁹ cm²/W and m is chosen to be 8 [30]. δ is an empirical parameter which gives rise to a clamped intensity of 5.0 × 10¹³ W/cm² [29]. The fluctuations of refractive index

\tilde{n} (x, y, z)

within the beam propagation distance of Δz can be obtained by:

\tilde{n} (x, y, z) = ϕ (x, y) / k_{0} Δ z .

As an illustration, Fig. 3(a) shows a representative phase screen with the structure constant $C_{n}^{2} = 1.0 \times 10^{- 13} {cm}^{- 2 / 3}$ and the distance Δz = 1 m. The validity of the simulated phase screens through the above method has been tested with reference to the theoretical phase structure function, which is defined as [26]:

D (r) = 〈 {[φ (r^{'}) - φ (r^{'} + r)]}^{2} 〉 = 6.88 {(r / r_{0})}^{5 / 3}

where the brackets mean the average over an ensemble phase screens, r denotes the distance between any two points in the phase screen and r₀ represents the Fried parameter [31]. The testing result is shown in Fig. 3(b). The structure function of phase fluctuations shown as the black solid line in Fig. 3(b) is obtained by averaging over 100 statistically independent screens [28]. The red dotted line represents the theoretical result calculated on the basis of Eq. (6). The slight difference in Fig. 3(b) could be attributed to the limited screen dimension. It is worth mentioning that previous study has found that though the free electron density varies significantly, the electron temperature is rather constant inside the filament [32]. Since the electron temperature is the crucial factor causing filament heating [33], the structure constant is set to be invariable during each simulation.

Fig. 3 (a) A representative phase screen with the structure constant $C_{n}^{2} = 1.0 \times 10^{- 13} {cm}^{- 2 / 3}$ and the distance Δz = 1 m. (b) Simulative and theoretical result of the structure function of phase fluctuations. (c) Spatial displacement of filament in turbulent air with the turbulence parameter $C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{- 2 / 3}$ , which is introduced after filament formation. (d) Relationship between the standard deviation of the single beam wandering and filament length in turbulent air with different turbulent structure constants and the fitting (solid line).

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In the simulation of the filamentation process, the radius of the initial CW laser beam with a Gaussian profile was 1 mm, while the initial laser power was chosen to be 5 times the critical power for self-focusing. The turbulence is introduced after the beam collapse distance where the plasma starts to be formed. The turbulence structure constant is 1.0 × 10¹³ cm^-2/3. The filament center position has been recorded at a distance of 6 m away from the beginning of the turbulence. This process has been repeated for 30 times by using different phase screens with the same turbulence parameters. Figure 3(c) shows the result of the statistical characteristics of the filament wandering when the turbulence ( $C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{-2 / 3}$ ) occurs in the middle of the filament. The standard deviation of the transverse displacement of the filament center under this condition is about 210 μm.

We further looked into the relationship between the standard deviation of the single filament transverse displacement as a function of the propagation distance when the turbulent $C_{n}^{2}$ varies from $1.0 \times 10^{- 14} {cm}^{-2 / 3}$ to $1.0 \times 10^{- 12} {cm}^{-2 / 3}$ . The results are shown in Fig. 3(d). Interestingly, all the data indicated in Fig. 3(d), can be well fitted by an empirical formula:

δ = \sqrt{C_{n}^{2} / 10^{- 14}} z^{3 / 2}

where z = 0 corresponds to the beginning point of the plasma filament. In Eq. (7), the units of

C_{n}^{2}

and z are cm^-2/3 and m, respectively. It expected that by using Eq. (7), one may be able to estimate quantitatively the displacement of a filament over long distance propagation. Besides, it is feasible to fit the spatial displacement of the filament center shown in Fig. 2 by Eq. (7). As illustrated in Fig. 2, three fitting curves are obtained for the upper limit (dashed line), down limit (dotted line) and the best fitting (short dotted line). The resulted structure constants are

C_{n}^{2} = 1.9 \times 10^{- 12} {cm}^{- 2 / 3}

,

C_{n}^{2} = 4.9 \times 10^{- 13} {cm}^{- 2 / 3}

and

C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{- 2 / 3}

, respectively. Therefore, a structure constant of

C_{n}^{2} = 1.0 \times 10^{- 12} {cm}^{- 2 / 3}

might be a good indication for the strength of the turbulence induced by the filament, particularly in the cases of the loose focusing and the free propagation.

4. Conclusion

In conclusion, the spatial wandering of a femtosecond laser filament in air has been studied. The displacement of the filament center as a function of the propagation distance could be fitted by an empirical formula deduced from the classical Karman model describing the atmospheric turbulence. The determined effective structure constant associated with a single filament generated by 1kHz repetition rate laser system is about 1.0 × 10⁻¹² cm^-2/3. The study with higher or lower repetition rate laser has been planned for future study. With the deduced effective turbulence structure constant, one may be able to estimate the spatial wondering of the filament as a function of the propagation distance and interpret the practical performance of the experiments assisted by the femtosecond laser filamentation, such as the remote air lasing, pulse compression and HHG.

Acknowledgments

This work is financially supported by National Basic Research Program of China (2014CB339802, 2011CB808100), National Natural Science Foundation of China (11174156) and Tianjin Research Program of Application Foundation and Advanced Technology (15JCZDJC31700).

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Beam wandering of femtosecond laser filament in air

Abstract

1. Introduction

2. Experiment results

3. Simulation results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (7)

Optics Express