Elongation of filamentation and enhancement of supercontinuum generation by a preformed air density hole

Junwei Chang; Dongwei Li; Litong Xu; Lanzhi Zhang; Tingting Xi; Tingting Xi; Zuoqiang Hao; Zuoqiang Hao

doi:10.1364/OE.458128

1. Introduction

When intense femtosecond laser pulses propagate in air, filamentation occurs due to the dynamic equilibrium between Kerr self-focusing and electron defocusing effects. Several interesting phenomena are observed during the filamentation, such as intensity clamping [1,2], supercontinuum (SC) generation [3,4], long distance plasma channel [5,6], and so on. Based on these unique characteristics, filamentation has many important applications, such as single-cycle pulse generation [7,8], remote sensing [9–11], weather control [9,12], THz generation [13], and virtual antennas [14]. For remote applications, it is required to generate long-distance filaments. In addition, other characteristics, such as intensity of SC, spectral region of SC, and lifetime of plasma channel, also need to be improved. To date, various methods to control the filamentation have been reported, such as using shaped pulse [15–18], changing focusing conditions [19–22], and adding co-propagating dressing beam [23–25]. In addition, the characteristics of filamentation can also be effectively controlled by using pulse sequence with different delay times of picosecond or nanosecond [26–29]. Surprisingly, the characteristics of the filamentation can be also influenced by the pulse sequence with millisecond delay time [30,31]. In this case, the thermal diffusion of air, which is caused by the recombination of electron and ion generated by a pre-pulse, results in the formation of a “density hole” [30–34] where the refractive index distribution in air is changed. The density hole can cause defocusing effect of the subsequent pulse, and may provide an effective method to control the filamentation. However, it has not been reported.

In this paper, we investigate numerically the propagation of the femtosecond laser pulse in air with a preformed density hole. The filamentation length and the SC intensity can be considerably enhanced by modulating the delay time after the formation of the density hole and the length of the density hole.

2. Calculation

The propagation of the femtosecond laser pulse in air with a preformed density hole can be described by the nonlinear Schrödinger equation (NLSE) coupled with the electron generation equation, which can be written as [35]:

(1)$$\begin{array}{l} \frac{{\partial E}}{{\partial z}} = i\frac{1}{{2{k_0}}}{\Delta _ \bot }E - i\frac{{k^{\prime\prime}}}{2}\frac{{{\partial ^2}E}}{{\partial {t^2}}} + i\frac{{{k_0}{n_2}}}{2}R(t)E\\ - i{k_0}\frac{\rho }{{2{\rho _c}}}E - \frac{{{\beta ^{(K)}}}}{2}{|E |^{2K - 2}}E + i{k_0}\varDelta nE, \end{array}$$

(2)$$\frac{{\partial \rho }}{{\partial t}} = \frac{{{\beta ^{(K)}}}}{{K\hbar {\omega _0}}}{|E |^{2K}}(1 - \frac{\rho }{{{\rho _{at}}}}),$$

(3)$$R(t) = {|E |^2} + \frac{1}{{{\tau _k}}}(\int_{ - \infty }^t {\exp ( - \frac{{t - t^{\prime}}}{{{\tau _k}}}){{|{E(t^{\prime})} |}^2}dt^{\prime}} ).$$

The right terms of Eq. (1) account for the effects of beam diffraction, group velocity dispersion, Kerr effect, electron defocusing, multiphoton absorption, and the refractive index change induced by the preformed density hole. E is the electric field envelope and z is the propagation distance. k₀ = 2π/λ₀ is the central wavenumber for the central wavelength λ₀ = 800 nm. k^″ = 0.2 fs²/cm represents the coefficient of the group-velocity dispersion. n₂ = 1 × 10⁻¹⁹ cm²/W is the nonlinear coefficient of Kerr effect. Both the instantaneous and delayed responsibilities (Raman effect) of the Kerr effect are considered with the response function R(t) and the response time of τ_k = 70 fs. ρ, ρ_c, and ρ_at are the electron density, the critical plasma density, and the neutral oxygen molecules density, respectively. ρ_c = 1.7 × 10²¹ cm^-3 and ρ_at = 5.4 × 10¹⁸ cm^-3. β^(K) = 3.1 × 10⁻⁹⁸ cm¹³/W⁷ is the coefficient of multiple photon ionization for the photon number K = 8. Δn represents the refractive index change induced by the density hole formed by a pre-pulse.

The initial electric field envelope of the femtosecond laser pulse is written as:

(4)$$E = {E_0}\exp ( - \frac{{{x^2} + {y^2}}}{{{w^2}}})\exp ( - \frac{{{t^2}}}{{0.72{\tau ^2}}})\exp ( - \frac{{i{k_0}{r^2}}}{{2f}}).$$

where $\tau = 30\textrm{ fs}$ and $w = 0.85\textrm{ mm}$ are the pulse duration (FWHM) and beam radius, respectively. The input laser energy is 0.9 mJ, corresponding to three times of the critical power for self-focusing. These laser parameters are chosen to form single filament in the simulation. f = 2 m is the focal length of the lens. This focusing condition results in the propagation distance in meter range, which is long enough to investigate the influence of density hole on filamentation and is not a challenge for time-consuming simulation. The refractive index change induced by the density hole is written as [36]

(5)$$\varDelta n ={-} \varDelta {n_m}\exp ( - \frac{{{x^2} + {y^2}}}{{{R^2}}})$$

with the maximal refractive index change given by

(6)$$\varDelta {n_m} = ({n_0} - 1)\frac{{\varDelta {T_{peak}}}}{{{T_a}}}\frac{{R_0^2}}{{{R^2}}}.$$

where n₀ = 1.000275 and T_a = 300 K are the refractive index and temperature of the ambient air, respectively. ΔT_peak =50 K is the peak temperature change induced by the energy deposition in air [36]. For this temperature, the thermal diffusivity of air is given by α = 0.19 cm²/s [30,36]. The radius of the density hole is given by $R = {(R_0^2 + 4\alpha \varDelta t)^{1/2}}$, R₀ = 50 µm is the initial radius of the density hole, and Δt is the delay time between the formation of the density hole and the subsequent pulse.

3. Results and discussion

Figure 1(a) shows the maximal on-axis intensity of the femtosecond laser pulse as a function of the propagation distance when no density hole is introduced. Filamentation starts at z = 0.89 m and terminates at z = 1.85 m. The filamentation has a length of 0.96 m with a maximal intensity of 62 TW/cm². Here, both the filamentation onset and termination are defined as the position with the half value of the maximal intensity.

Fig. 1. (a) Maximal on-axis intensity of the femtosecond laser pulse and (c) electron density in air for the length of introduced density hole is 0 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m, respectively. (b) Filamentation range by defining the filamentation onset and termination as the intensity reaches a half value of the maximum in the L_hole = 0 m case (62 TW/cm²). The onset of the density hole is 0.9 m. The delay time between the formation of the density hole and the femtosecond laser pulse is 1 ms.

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To control the filamentation evolution, we introduce the density hole near the filamentation onset position. If the density hole is introduced in the self-focusing regime (for example, z = 0.8 m), the defocusing effect of the density hole is too strong to elongate filamentation. If the density hole is introduced at the filamentation stage (for example, z = 1.1m), the clamping intensity, which has already been formed by the dynamic balance of Kerr self-focusing and electron defocusing, is hardly influenced by the introduction of density hole. Therefore, we introduce the density hole near the filamentation onset z = 0.9 m. The delay time is fixed at 1 ms. In this case, the density hole has a radius of 280 µm, and the induced refractive change has a maximum of 1.5 × 10⁻⁶ which is calculated by Eq. (6). When the length of the density hole varies from 0.2 m to 0.5 m, the filamentation intensity and length shows greatly different. When the length of density hole L_hole is 0.2 m, the maximal intensity of filamentation is 60 TW/cm², slightly lower than that in the case of no density hole. In this case, the filamentation length is 1.5 m, much longer than that without density hole, as shown in Fig. 1(b). For the density hole length L_hole = 0.3 m, the maximal intensity is 49 TW/cm², much lower than those for L_hole = 0 m and L_hole = 0.2 m. In this case, filamentation length approach that in the case of L_hole = 0.2 m. These results show that the filamentation can be greatly prolonged by using the density hole. But if the density hole is too long, such as L_hole = 0.4 m or 0.5 m, filamentation will become shorter, and the intensity will be lower. Correspondingly, the maximal electron density becomes lower with the increase of the density hole length, as shown in Fig. 1(c).

To investigate the mechanism of filamentation elongation by density hole, we plot the spatiotemporal intensity distribution of the laser pulse at several typical propagation distances. When no density hole is introduced, as shown in Fig. 2(a), the laser pulse self-focuses firstly, and then the pulse tail is defocused by the generated electrons. With the increase of the propagation distance, the pulse tail self-focuses again. Subsequently, the combined influence of self-phase modulation and dispersion leads to the formation of two sub-pulses. With the spread of the two sub-pulses, the filamentation terminates. In Fig. 2(b), a short density hole is introduced from z = 0.9 m to z = 1.1 m. The defocusing effect of the density hole is stronger than the self-focusing effect of the leading and trailing pulse edges, and causes an obvious spread of the two parts. But it is still weaker than the self-focusing effect of the pulse peak under this condition. It only relieves the self-focusing of the central part of the pulse. As a consequence, a new balance is established between the self-focusing, defocusing of density hole, and electron defocusing, which results in relatively low intensity of laser pulse. Although the pulse peak is defocused slightly at z = 1.1 m, it then self-focuses again (at z = 1.5 m). The short density hole does not influence the maximal intensity of the second self-focusing cycle. Then the pulse splits into two sub-pulses, and the subsequent evolution is similar to that of Fig. 2(a). The new balance introduced by the density hole leads to the elongation of the filamentation. Furthermore, as shown in Fig. 2(c), a longer density hole is introduced from z = 0.9 m to z = 1.2 m. Compared with the shorter case shown in Fig. 2(b), more energy is defocused by the longer density hole, as shown in the second image of Fig. 2(c). It causes more energy spread and influences the maximal intensity of the second self-focus cycle. Therefore, when the longer density hole with L_hole = 0.3 m is introduced, the filamentation is elongated with a lower intensity, resulting from the new balance induced by the density hole. When a much longer density hole with L_hole = 0.4 m is introduced (Fig. 2(d)), too much energy of the laser pulse is defocused, and the second self-focus cycle cannot start. Finally, a very short filamentation is formed.

Fig. 2. The spatiotemporal intensity distribution of the laser pulse at several typical propagation distances for different hole lengths of (a) L_hole= 0 m, (b) 0.2 m, (c) 0.3 m, (d) 0.4 m, respectively. All subgraphs have the same size.

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Here, the Raman effect, which is responsible for the red shift of spectrum, is considered in our calculation. But the self-frequence shift caused by Raman effect, which requires longer pulse, higher energy, and longer filamentation length [37], is not obvious in our case. Moreover, we focus on the blue shift of the spectrum. Thus, we will not discuss the contribution of the Raman effect to the SC generation below. The four-wave mixing process does not contribute to the SC generation either, because it needs a strong nonlinear interaction between two-color laser pulses [38–40], or between a pump pulse with intense blueshift components [41]. In our calculations, SC is generated by self-phase modulation and electron generation. Therefore, the shape of SC spectrum depends on the length and intensity of filamentation [42]. So it is reasonable to expect an enhanced SC emission when the filamentation is elongated. Figure 3(a) shows the SC spectra generated by the femtosecond laser pulse are influenced by density holes with different lengths. It shows the spectral energy density can be greatly enhanced by the introduction of density holes. But for the different L_hole, the enhanced spectral region is different, and the cut-off wavelength in the blue side is also different. Here, the cut-off wavelength is defined as the value 10⁻⁴ of the maximal energy density. As shown in Fig. 3(b), for L_hole = 0.2 m, the cut-off wavelength approaches that without density hole. But the spectral energy density in the range from 435 nm to 634 nm is enhanced greatly, even more than one order of magnitude for some regions. For L_hole = 0.3 m, although the cut-off wavelength is longer than that without density hole, the spectral energy density is greatly enhanced in the range from 457 nm to 681 nm. The enhanced spectral region moves toward long wavelength, and the intensity amplification is less than that for L_hole = 0.2 m. The enhancement mechanism of spectral energy density should be attributed to the elongation of the filamentation. The different enhanced spectral regions and cut-off wavelength may result from the different intensity of the filamentation. Because the maximal frequency shift in the blue side depends on the maximal intensity of filamentation due to the electron generation [2]. For L_hole = 0.2 m, the filamentation is elongated with a high intensity approaching that without density hole. The maximal frequency shift in the blue side should approach that without density hole. Therefore, the spectral enhancement occurs at a shorter wavelength region. Moreover, the cut-off wavelength is approaching that without density hole. While for L_hole = 0.3 m, the filamentation intensity is lower, the maximal frequency shift is smaller, and the cut-off wavelength in the blue side is longer. Correspondingly, the spectral enhancement occurs at a relatively longer wavelength region. When the density hole is much longer (0.4 m and 0.5 m), the filamentation is shorter with low intensity. Therefore, the spectral broaden is limited, and the cut-off wavelength is around 600 nm.

Fig. 3. (a) Spectral energy density and (b) cut-off wavelength of the laser pulse at z = 3 m for L_hole = 0 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m.

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We further investigate the filamentation in a preformed density hole with different delay times of 0.5 ms, 1 ms, and 2 ms, respectively, as shown in Fig. 4. For the three cases, the maximal refractive index changes are 2.8 × 10⁻⁶, 1.5 × 10⁻⁶, and 7.4 × 10⁻⁷, respectively. The density hole is introduced from z = 0.9 m to z = 1.2 m. When the delay time is small (Δt = 0.5 ms), the induced refractive index is very large, leading to a strong defocus effect. Thus, the filamentation is terminated quickly. In this case, the electron is lower and the spectrum does not get broadening obviously. With the increase of the delay time, the induced refractive index becomes smaller. The relatively weak defocus leads to a longer filamentation. For Δt = 1 ms, the filamentation has a length of 1.5 m, much larger than that without density hole. For Δt = 2 ms, the filamentation length is 1.3 m, also larger than that without density hole. In the two cases, although the filamentation intensity is still lower than that without density hole, it becomes higher with the increase of the delay time. As a result, the spectral energy density is enhanced by the elongation of filamentation, as shown in Fig. 4(c). Moreover, enhanced spectral region moves towards blue side for the large delay time (Δt = 2 ms) due to the relatively high intensity.

Fig. 4. (a) Peak intensity of the laser pulse, (b) electron density, and (c) spectra energy density in air for the different delay times. The density hole length is introduced from 0.9 m to 1.2 m. The black solid curve in all figures is corresponding to the case without density hole in air.

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4. Conclusion

In this paper, we simulate the propagation of the femtosecond laser pulse in air with a preformed density hole. We investigate the influence of density hole on the filamentation evolution and SC generation by changing the length of the density hole and the delay time between the formation of the density hole and the subsequent femtosecond laser pulse. The filamentation can be greatly elongated when a short density hole is introduced. The elongation is due to the newly established balance of Kerr self-focusing, electron defocusing, and the defocusing induced by the density hole. Correspondingly, the enhancement of SC intensity in short wavelength region is obtained with more than one order of magnitude. The filamentation length, intensity, and enhanced spectral energy density region can be effectively controlled by changing the density hole length and delay time.

In addition, the filamentation-induced shockwave can expel the fog droplets and drastically improve subsequent beam transmission [43], and facilitate the free space laser telecommunication through fogs and clouds [44,45]. It is also reasonable to expect that the shockwave can also reduce the influence of air turbulence on filamentation of subsequent laser pulses, although there is no related report yet. Therefore, by using pulse sequence with a proper interval, one can get an elongated filamentation and enhanced SC emission, but also can mitigate the influence of adverse air conditions, which could be a promising method for long-distance filamentation in field applications. However, in the present work, we study the effect of the density hole produced by only the first pulse on the second pulse, which is the first step in the evolution and interaction of pulses in the pulse sequence. The elongated filamentation of the second pulse will leave a different density hole behind, which will influence the subsequent pulses propagating. Further studies with more pulses are needed.

Funding

National Natural Science Foundation of China (12074228, 11874056, 11774038); Natural Science Foundation of Shandong Province (ZR2021MA023); Taishan Scholar Project of Shandong Province (tsqn201812043); Innovation Group of Jinan (2020GXRC039).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]

2. W. Liu, S. Petit, A. Becker, N. Aközbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002). [CrossRef]

3. A. Dubietis, G. Tamošauskas, R. Šuminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57(3), 112–157 (2017). [CrossRef]

4. Y. Petit, S. Henin, W. M. Nakaema, P. Béjot, A. Jochmann, S. D. Kraft, S. Bock, U. Schramm, K. Stelmaszczyk, P. Rohwetter, J. Kasparian, R. Sauerbrey, L. Wöste, and J.-P. Wolf, “1-J white-light continuum from 100-TW laser pulses,” Phys. Rev. A 83(1), 013805 (2011). [CrossRef]

5. M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, “Kilometer-range nonlinear propagation of femtosecond laser pulses,” Phys. Rev. E 69(3), 036607 (2004). [CrossRef]

6. M. Durand, A. Houard, B. Prade, A. Mysyrowicz, A. Durécu, B. Moreau, D. Fleury, O. Vasseur, H. Borchert, K. Diener, R. Schmitt, F. Théberge, M. Chateauneuf, J.-F. Daigle, and J. Dubois, “Kilometer range filamentation,” Opt. Express 21(22), 26836–26845 (2013). [CrossRef]

7. L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. 100(11), 113902 (2008). [CrossRef]

8. F. Hagemann, O. Gause, L. Wöste, and T. Siebert, “Supercontinuum pulse shaping in the few-cycle regime,” Opt. Express 21(5), 5536–5549 (2013). [CrossRef]

9. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). [CrossRef]

10. H. L. Xu, J. F. Daigle, Q. Luo, and S. L. Chin, “Femtosecond laser-induced nonlinear spectroscopy for remote sensing of methane,” Appl. Phys. B 82(4), 655–658 (2006). [CrossRef]

11. S. L. Chin, H. L. Xu, Q. Luo, F. Théberge, W. Liu, J. F. Daigle, Y. Kamali, P. T. Simard, J. Bernhardt, S. A. Hosseini, M. Sharifi, G. Méjean, A. Azarm, C. Marceau, O. Kosareva, V. P. Kandidov, N. Aközbek, A. Becker, G. Roy, P. Mathieu, J. R. Simard, M. Châteauneuf, and J. Dubois, “Filamentation “remote” sensing of chemical and biological agents/pollutants using only one femtosecond laser source,” Appl. Phys. B 95(1), 1–12 (2009). [CrossRef]

12. J. P. Wolf, “Short-pulse lasers for weather control,” Rep. Prog. Phys. 81(2), 026001 (2018). [CrossRef]

13. C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007). [CrossRef]

14. Y. Brelet, A. Houard, G. Point, B. Prade, L. Arantchouk, J. Carbonnel, Y. B. André, M. Pellet, and A. Mysyrowicz, “Radiofrequency plasma antenna generated by femtosecond laser filaments in air,” Appl. Phys. Lett. 101(26), 264106 (2012). [CrossRef]

15. G. Heck, J. Sloss, and R. J. Levis, “Adaptive control of the spatial position of white light filaments in an aqueous solution,” Opt. Commun. 259(1), 216–222 (2006). [CrossRef]

16. R. Ackermann, E. Salmon, N. Lascoux, J. Kasparian, P. Rohwetter, K. Stelmaszczyk, S. Li, A. Lindinger, L. Wöste, P. Béjot, L. Bonacina, and J. P. Wolf, “Optimal control of filamentation in air,” Appl. Phys. Lett. 89(17), 171117 (2006). [CrossRef]

17. A. Chen, S. Li, H. Qi, Y. Jiang, Z. Hu, X. Huang, and M. Jin, “Elongation of plasma channel generated by temporally shaped femtosecond laser pulse,” Opt. Commun. 383, 144–147 (2017). [CrossRef]

18. L. Zhan, M. Xu, T. Xi, and Z. Hao, “Contributions of leading and tailing pulse edges to filamentation and supercontinuum generation of femtosecond pulses in air,” Phys. Plasmas 25(10), 103102 (2018). [CrossRef]

19. A. Camino, Z. Hao, X. Liu, and J. Lin, “Control of laser filamentation in fused silica by a periodic microlens array,” Opt. Express 21(7), 7908–7915 (2013). [CrossRef]

20. Y. E. Geints and A. A. Zemlyanov, “Effect of high-power laser divergence on the plasma structural parameters during multiple filamentation in air,” Phys. Rev. A 93(6), 063833 (2016). [CrossRef]

21. Z. Hong, Q. Zhang, S. Ali Rezvani, P. Lan, and P. Lu, “Extending plasma channel of filamentation with a multi-focal-length beam,” Opt. Express 24(4), 4029–4041 (2016). [CrossRef]

22. Z. Feng, W. Li, C. Yu, X. Liu, Y. Liu, J. Liu, and L. Fu, “Influence of the external focusing and the pulse parameters on the propagation of femtosecond annular Gaussian filaments in air,” Opt. Express 24(6), 6381–6390 (2016). [CrossRef]

23. M. Scheller, M. S. Mills, M. A. Miri, W. Cheng, J. V. Moloney, M. Kolesik, P. Polynkin, and D. N. Christodoulides, “Externally refuelled optical filaments,” Nat. Photonics 8(4), 297–301 (2014). [CrossRef]

24. Y. E. Geints and A. A. Zemlyanov, “Ring-Gaussian laser pulse filamentation in a self-induced diffraction waveguide,” J. Opt. 19(10), 105502 (2017). [CrossRef]

25. J. Q. Lü, P. P. Li, D. Wang, C. Tu, Y. Li, and H. T. Wang, “Extending optical filaments with phase-nested laser beams,” Photon. Res. 6(12), 1130 (2018). [CrossRef]

26. Y. E. Geints and A. A. Zemlyanov, “Long-range filaments producing upon ultrashort mid-IR laser pulses train filamentation in air: numerical simulations,” Proc. SPIE 10833, 108330W2018). [CrossRef]

27. D. Reyes, H. Kerrigan, J. Peña, N. Bodnar, R. Bernath, M. Richardson, and S. R. Fairchild, “Temporal stitching in burst-mode filamentation,” J. Opt. Soc. Am. B 36(10), G52–G56 (2019). [CrossRef]

28. X. L. Liu, X. Lu, J. L. Ma, L. B. Feng, X. L. Ge, Y. Zheng, Y. T. Li, L. M. Chen, Q. L. Dong, W. M. Wang, Z. H. Wang, H. Teng, Z. Y. Wei, and J. Zhang, “Long lifetime air plasma channel generated by femtosecond laser pulse sequence,” Opt. Express 20(6), 5968–5973 (2012). [CrossRef]

29. X. Lu, S.-Y. Chen, J.-L. Ma, L. Hou, G.-Q. Liao, J.-G. Wang, Y.-J. Han, X.-L. Liu, H. Teng, H.-N. Han, Y.-T. Li, L.-M. Chen, Z.-Y. Wei, and J. Zhang, “Quasi-steady-state air plasma channel produced by a femtosecond laser pulse sequence,” Sci. Rep. 5(1), 15515 (2015). [CrossRef]

30. Y. H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef]

31. N. Jhajj, Y. H. Cheng, J. K. Wahlstrand, and H. M. Milchberg, “Optical beam dynamics in a gas repetitively heated by femtosecond filaments,” Opt. Express 21(23), 28980–28986 (2013). [CrossRef]

32. J. K. Wahlstrand, N. Jhajj, E. W. Rosenthal, S. Zahedpour, and H. M. Milchberg, “Direct imaging of the acoustic waves generated by femtosecond filaments in air,” Opt. Lett. 39(5), 1290–1293 (2014). [CrossRef]

33. O. Lahav, L. Levi, I. Orr, R. A. Nemirovsky, J. Nemirovsky, I. Kaminer, M. Segev, and O. Cohen, “Long-lived waveguides and sound-wave generation by laser filamentation,” Phys. Rev. A 90(2), 021801 (2014). [CrossRef]

34. V. V. Pankratov, D. E. Shipilo, M. M. Yandulsky, N. A. Panov, and O. G Kosareva, “The optical waveguide generated by acoustic waves emitted from femtoseconds filaments,” Proc. SPIE 9990, 99900N (2016). [CrossRef]

35. T. Xi, Z. Zhao, and Z. Hao, “Filamentation of femtosecond laser pulses with spatial chirp in air,” J. Opt. Soc. Am. B 31(2), 321–324 (2014). [CrossRef]

36. J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “Controlling femtosecond filament propagation using externally driven gas motion,” Opt. Lett. 44(2), 199–202 (2019). [CrossRef]

37. Y. Chen, F. Théberge, C. Marceau, H. Xu, N. Aközbek, O. Kosareva, and S. L. Chin, “Observation of filamentation-induced continuous self-frequency down shift in air,” Appl. Phys. B 91(2), 219–222 (2008). [CrossRef]

38. F. Théberge, N. Aközbek, W. Liu, A. Becker, and S. L. Chin, “Tunable ultrashort laser pulses generated through filamentation in gases,” Phys. Rev. Lett. 97(2), 023904 (2006). [CrossRef]

39. T. Fuji, T. Horio, and T. Suzuki, “Generation of 12 fs deep-ultraviolet pulses by four-wave mixing through filamentation in neon gas,” Opt. Lett. 32(17), 2481–2483 (2007). [CrossRef]

40. F. Théberge, M. Châteauneuf, G. Roy, P. Mathieu, and J. Dubois, “Generation of tunable and broadband far-infrared laser pulses during two-color filamentation,” Phys. Rev. A 81(3), 033821 (2010). [CrossRef]

41. F. Théberge, M. Châteauneuf, V. Ross, P. Mathieu, and J. Dubois, “Ultrabroadband conical emission generated from the ultraviolet up to the far-infrared during the optical filamentation in air,” Opt. Lett. 33(21), 2515–2517 (2008). [CrossRef]

42. L. Zhang, T. Xi, Z. Hao, and J. Lin, “Supercontinuum accumulation along a single femtosecond filament in fused silica,” J. Phys. D: Appl. Phys. 49(11), 115201 (2016). [CrossRef]

43. L. de La Cruz, E. Schubert, D. Mongin, S. Klingebiel, M. Schultze, T. Metzger, K. Michel, J. Kasparian, and J.-P. Wolf, “High repetition rate ultrashort laser cuts a path through fog,” Appl. Phys. Lett. 109(25), 251105 (2016). [CrossRef]

44. G. Schimmel, T. Produit, D. Mongin, J. Kasparian, and J.-P. Wolf, “Free space laser telecommunication through fog,” Optica 5(10), 1338–1341 (2018). [CrossRef]

45. B. Yan, H. Liu, C. Li, X. Jiang, X. Li, J. Hou, H. Zhang, W. Lin, B. Liu, and J. Liu, “Laser-filamentation-assisted 1.25 Gb/s video communication under harsh conditions,” Opt. Laser Technol. 131(1), 106391 (2020). [CrossRef]

Elongation of filamentation and enhancement of supercontinuum generation by a preformed air density hole

Abstract

1. Introduction

2. Calculation

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (6)

Optics Express