Abstract

Manipulation of femtosecond laser filamentation is essential for many potential applications. We report the simulations of the manipulation of femtosecond laser filamentation by introducing a novel gaseous lattice medium with the alternating positive and negative refractive index distribution at different stages of filamentation. The results show that the filament length has greatly been extended and a multi-filament array can be formed by the gas lattice medium. It has been found that additional focusing and discrete diffraction provided by the gas lattice medium contribute to a new dynamic equilibrium in the filamentation. As a result, a varied cross-section pattern, higher field intensity, and electron density along the filamentation are obtained. Our approach provides a new way to manipulate filamentation for many practical photonic applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The filamentation phenomenon is induced by high power femtosecond laser pulse propagating in transparent media, which has received profound research attention over decades. Filamentation is the result of dynamic equilibrium among the pulse self-focusing and the diffraction and the plasma defocusing produced by photoionization in the transparent medium [1]. The plasma filament sustains a distance with tightly radially confined high intensity region which is much greater than the Rayleigh length [2]. Many applications have been developed based on the properties of femtosecond filaments. Usually, a manipulation of femtosecond filamentation is required for the potential applications such as white light LIDAR [3], triggering HV discharges [4], THz generation [5,6], pulse compression [7], written waveguide in silica [8] and high spectral power femtosecond supercontinuum source [9], high-intensity filament-induced fluorescence sensing [10] and filament-induced fabrication of materials [11,12]. On one side, a control of filamentation properties can be realized via the adjustment of energy, duration, chirp, polarization, spatiotemporal distribution of the femtosecond laser and amplitude and phase modulation of femtosecond laser pulse [2,1317]. On the other side, the transient refractive index change along the femtosecond laser pulse transmission path of the medium is found to have a remarkable effect on the femtosecond laser filamentation. It has been demonstrated that the gas molecular alignment [18,19] and the plasma lattice structure [2022] excited by the previous pulse in the medium can significantly influence the filamentation of the successive laser pulse. A pre-written waveguide structure in solid medium is reported [23] to have a large influence on the filamentation. These examples with a pre-arranged special refractive index distribution of the medium provide an opportunity to regulate femtosecond filaments. However, the above pre-arranged refractive index change applied to particularly cases such as a requirement of a molecule gas or permanent change of medium, or only negative refractive index provided by plasma lattice. Thus, these methods have essential limitations and meet great challenge in the manipulation of filamentation. Therefore, a new method with the refractive index change that can be applied to manipulate filamentation with less restriction is strongly desired.

Recently, reports showed that the refractive index shift $\delta n$ of the femtosecond laser pulse produced density hole in air can reach a range from ${10^{\textrm{ - }6}}$ to ${10^{\textrm{ - 5}}}$ [24]; their work proved that the refractive index shift of gas medium would have a significant effect on the successive laser pulse propagation [25,26]. The same group also reported a long-lived and stable air waveguide is left after symmetric four-filament arrays; this fiber-like structure is formed with positive index change due to the enhancement of air molecules density in the center of filament array, surrounded the lower density at the location of each filament [27]. It has been demonstrated the feasibility that the injected laser beam is guided in this air waveguide to improve the guiding efficiency. In these experimental demonstrations, the air waveguide generated by four filaments was used to improve the linear transmission process of subsequent optical signals [27,28], but the effect on the nonlinearity of the ultra-fast intense femtosecond laser has not been studied until now.

In this work, inspired by the air waveguide concept based multi-filament array [29,30], a novel gaseous lattice with alternating positive and negative refractive index change to regulate intensive femtosecond laser filamentation is proposed. We present the effect of gaseous lattice medium on the filamentation. Filamentation properties on the gas lattice parameters and introducing position, the cross-section patterns of the filamentation and field intensity as well as electron density along the beam propagation are investigated. The underlying physics responsible for the manipulation of the filamentation by the gaseous lattice is discussed. Filamentation in the lattice with alternating positive and negative index change and only negative index change are also compared. In contrast to producing one direction change (negative/positive) of the refractive index [22,23], increasing the modulation direction of the refractive index shift would introduce more influences on the nonlinear process of filaments, thus it is expected that the lattice would provide the more control freedom to manipulation of filamentation.

2. Calculations

The numerical model of the ultrafast-laser propagation in the gaseous lattice medium composed of air is depicted by the (3D+1) dimensional nonlinear Schrödinger (NLS) equation with the plasma density evolution equation [31]:

$$\begin{aligned} \frac{{\partial E}}{{\partial z}} &= i\frac{1}{{2{k_0}}}{\Delta _ \bot }E - i\frac{{k^{\prime\prime}}}{2}\frac{{{\partial ^2}E}}{{\partial {t^2}}}\\ &+ i\frac{{{k_0}{n_2}}}{2}{|E |^2}E + i\frac{{{k_0}{n_2}}}{{2{\tau _k}}}\left\{ {\int_{ - \infty }^t {\textrm{exp} [{{{ - ({t - t^{\prime}} )} / {{\tau_k}}}} ]} {{|{E(t^{\prime})} |}^2}dt^{\prime}} \right\}E\\ &- i{k_0}\frac{\rho }{{2{\rho _c}}}E - \frac{{{\beta ^{(K)}}}}{2}{|E |^{2K - 2}}E + i{k_0}\delta nE\end{aligned}$$
$$\frac{{\partial \rho }}{{\partial t}} = \frac{{{\beta ^{(K )}}}}{{K{\hbar}{\omega _0}}}{|E |^{2K}}\left( {1 - \frac{\rho }{{{\rho_{at}}}}} \right)$$
here the coupled envelope of electric field moves in a local reference frame, $t \to t - {z / {{v_g}}}$, and ${v_g}$ represents the group-velocity of femtosecond laser beam with central wavelength ${\lambda _0}\textrm{ = }800\,nm$, the z is the propagation distance. The ${k_0} = {{2\pi } / {{\lambda _0}}}$ and ${\omega _0}\textrm{ = }{{2\pi c} / {{\lambda _0}}}$ are the central wave number and the central angular frequency, respectively. The first term of Eq. (1) refers to beam transverse diffraction with ${\Delta _ \bot }\textrm{ = }{{{\partial ^2}} / {\partial {x^2} + }}{{{\partial ^2}} / {\partial {y^2}}}$. The term of second-order temporal derivation is the group velocity dispersion (GVD) effect with coefficient $k^{\prime\prime} = {{\partial k} / {\partial \omega = 0.2\,{{f{s^2}} / {cm}}}}$. There are several nonlinear effects in the NLS equation, including the optical Kerr responsibility and multiphoton ionization (MPI). The Kerr term is consisted of the instantaneous and the delayed Raman effect with nonlinear coefficient ${n_2} = 3.2 \times {10^{ - 19}}\,{{c{m^2}} / W}$, corresponding to the self-focusing critical power ${P_{cr}} = {{3.77{\lambda ^2}} / {8\pi {n_2}}} \approx 3GW$ [32]. The electron density ρ is induced by MPI with coefficient ${\beta ^K} = 3.1 \times {10^{ - 98}}\,{{c{m^{13}}} / {{W^7}}}$ for K=8 photons, which causes the defocusing effect. In addition, critical electron density and the neutral oxygen molecules density are ${\rho _c} = 1.7 \times {10^{21}}\,c{m^{ - 3}}$ and ${\rho _{at}} = 5 \times {10^{18}}\,c{m^{ - 3}}$, respectively. The initial pulse envelope is defined Gaussian profile E with temporal duration (FWHM) $\Delta T = \sqrt {2\ln 2} {\tau _0} = 30fs$ and beam waist (FWHM) ${d_0} = \sqrt {2\ln 2} {w_0} = 1\,mm$, here $E = {E_0}\textrm{exp} [{ - ({{{{x^2}} / {w_0^2}} + {{{y^2}} / {w_0^2}}} )} ]\times \textrm{exp}({{{ - {t^2}} / {r_t^2}}} )\times \textrm{exp} [{{{ - ik({{x^2} + {y^2}} )} / {2f}}} ]$, the ${E_0}$ is the electric field amplitude. The initial energy of the input femtosecond laser pulse is set to 0.3 mJ and the pulse is focused by a lens with 2 m focal length.

The crucial term of the refractive index shift $\delta n$ in the gaseous lattice simulates the nonlinear effect caused by the fluctuation of the refraction index. The $\delta n$ of the gaseous lattice with Gaussian distribution can be written as:

$$\delta n = {n_0} \times \sum\nolimits_{i = 0}^k {{{({ - 1} )}^k}\textrm{exp} ({ - {{\{{{{[{2x \pm {{({ - 1} )}^k}k\Lambda } ]}^\textrm{2}}\textrm{ + }{{[{2y \pm {{({ - 1} )}^k}k\Lambda } ]}^\textrm{2}}} \}} / {4{w^2}}}} )} \textrm{ , }{z_1} \le z \le {z_2}\textrm{ }$$
where the maximal change of the refractive index induced by the gaseous lattice, namely modulation depth ${n_0}$. A set of coordinates in the center of the circle of the cross-section in Fig. 1(a) is $[{{{ \pm {{({ - 1} )}^k}k\Lambda } / 2},{{ \pm {{({ - 1} )}^k}k\Lambda } / 2}} ]$, which refers to the central axis of each cylindrical waveguide of the lattice in the z direction. The density cylinder of waveguide in the 9×9 lattices is numbered k, which has a waist w with the period of the lattice structure Λ. The ${z_1}$ and ${z_2}$ represent the onset and termination of the gas lattice along the z-axis direction.

 

Fig. 1. Refractive index shift $\delta n$ of the $9 \times 9$ gaseous lattice of (a) the transverse section distribution and its corresponding (b) 2D structure profile.

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In order to facilitate clear description, Fig. 1(b) depicts a two-dimensional profile of refractive index that corresponds to the Fig. 1(a), in which shows the positive (red part) and negative (blue part) alternating refractive index distribution. In our situations, the central optical axis of the lattice corresponds to the waveguide of positive $\delta n$. Specifically, a modification of wave-front of the femtosecond laser pulse, e.g. induced by micro lens array [29] or metal mesh [30], can generate gaseous lattice that forms a waveguide like the Fig. 1 shows. In our simulation, the boundary conditions are as follows, the size of the gaseous lattice is about 1.7 mm × 1.7 mm and the boundary of spatial volume is about 5 mm × 5 mm, indicating that the simulation volume includes empty space and allows light to leak. We used a square grid 256 × 256 with spatial resolution 0.02 mm in the transverse section. The whole time interval was discretized into 128 points with 1.19fs temporal resolution.

3. Results and discussions

It is known that in a waveguiding structure, the important elements are the transverse (the cross-sectional) characteristics. Therefore, firstly we changed the transverse parameters of the lattice such as period Λ, modulation depth n0 and a waist w of each density cylinder of the lattice in our simulation. The simulations of laser pulse propagating in the lattices with the different parameters are depicted in the Fig. 2. The iso-fluence surface of the filamentation is defined by $F({x,y,z} )= \int_{ - \infty }^{ + \infty } {|{A{{({x,y,z,t} )}^2}} |dt}$ along the z-axis. Obviously, as shown in Figs. 2(a)–2(c), the spatial distribution of multifilament is regulated by these parameters of the gaseous lattice. Specifically, the multi-filaments densely concentrate around the central axial of the lattice at period Λ=400 µm, while they become sparsely populated with the increasing of the period Λ. On the other hand, the lengths of those intense filaments increase with the increasing of refractive index modulation. More uniform and long filaments can be seen for waist w = 50 µm than the case of other two waist sizes. Furthermore, Figs. 2(d)–2(f) display effect of these three parameters on optical intensities along the central axis of the lattice, it can be seen that the period Λ has a much strong influence on the intensity while the other two parameters have less effect. Based on Fig. 2, we select the period Λ=400 µm, w=50 µm, and modulation depth n0=1×10−5 in the following simulation.

 

Fig. 2. The 3D iso-fluence (a.u.) evolution of femtosecond laser propagating with different parameters of the 3m-long gaseous lattice (at z=0 m) for (a) period Λ1=400 µm, Λ2=500 µm, Λ3=600 µm; (b) modulation depth n0(1) = 6×10−6, n0(2) = 8×10−6, n0(3) = 1×10−5; (c) each density cylinder waist w1=40 µm, w2=50 µm, w3=60 µm. The peak intensity on the optical axis as a function of propagation distance z with different parameters of the gaseous lattice are given in (d-f).

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Figures 3(a) and 3(b) show the effects of the gaseous lattice on the laser pulse propagation on the central axial of the lattice with the incident laser power higher than filamentation threshold (∼3Pcr) and lower than the threshold (∼0.001Pcr), respectively. Figure 3(a) depicts high power pulses propagating in air without lattice and with lattice of different onsets. When the laser pulse propagates in the air without a gaseous lattice introduced (the black solid line), the typical filamentation starts about z=1 m and terminates at z=1.65 m, whose peak intensity approaches $7.65 \times {10^{13}}\,{W / {c{m^2}}}$. When the gaseous lattice is added at z=0 m (the red solid curve), the optical intensity of the femtosecond laser beam quickly reaches the order of ${10^{13}}\,{W / {c{m^2}}}$ at the position ∼0.09 m and start to filamentation. The peak intensity in the whole filament path is only $3 \times {10^{13}}\,{W / {c{m^2}}}$, which is less than half of the optical intensity of the filamentation without lattice, but filamentation maintains a much longer distance. The introduction of the gaseous lattice structure at the initial position rapidly increases the intensity as the small peaks appeared at the position close to z=0 m, which enhances the energy dissipation via ionization of the air in the initial stage of filamentation, then clamping the optical intensity over a longer distance due to the decrease of the energy dissipation. When the lattice is set at z=1 m, i.e., where filamentation in air has begun to form, as seen in the curve of the blue solid line, the peak intensity is about $8.25 \times {10^{13}}\,{W / {c{m^2}}}$, and the termination point of optical filament is extended to about z=3.96 m. As a result, the filament is ∼4.5 times longer than that without gaseous lattice medium. In this case, the beam is coupled into the gaseous lattice at position (z=1 m) where self-focusing of filament has already overcome the diffraction effect and a dynamic balance has been reached, therefore maintaining a much higher intensity over longer distance than the case of introducing the lattice at z=0 m position. Figure 3(a) show that a filament length is greatly extended by the introduction of the gaseous lattice and the clamping intensity of the filamentation largely depends on the introducing point in the beam path. It is worthwhile pointing out that the synergy of focusing effect by central waveguide and the optical Kerr effect of the laser beam overcome diffraction in the case of the introduction of the gas lattice medium, thereby reaching a new dynamic balance between focusing and defocusing effect that leads to an extended length of the filament.

 

Fig. 3. Comparison of the intensities on the optical axis when input laser power Pin (a) above critical power about ∼3Pcr and (b) below critical power ∼0.001Pcr without (the black solid and dotted lines) or with the 3-meter-long lattice (red lines added at z=0 m, blue lines added at z=1 m). For Pin=3Pcr, the 3D iso-fluence representation of femtosecond pulse propagation in the gaseous lattice: (c) the onset of lattice at z=0 m; (d) the onset of lattice at z=1 m; the insets depict the gaseous lattice transverse distribution and the laser beam cross-sectional profile before coupling into the lattice. The green lines in (a) and (b) correspond to the lattice with only negative refractive index perturbations, and the yellow lines correspond to the collimated beam cases.

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In order to study the underlying physical mechanism, we simulated femtosecond laser transmission in the linear regime (∼0.001Pcr) with the same parameters as the Fig. 3(a) excepting the incident laser intensity, and the results were shown in the Fig. 3(b). As seen in both red and blue dotted lines, the beam is strongly regulated by the gaseous lattice as well, and the energy on the central axis cylinder of the lattice converges inward because of focusing effect resulting from the waveguide of positive $\delta n$ . Consequently, it can be seen that peak intensity of the propagating light beam is about 1 order of magnitude higher than that without gaseous lattice (the black dotted line). At about z=1 m in the blue dotted curve, the optical intensity of the beam increases sharply when it is coupled into the medium of gaseous lattice due to additional focusing effects introduced. As a result of the waveguide effect, the optical intensity remains stable at a high level over an extended distance. For comparison, in the simulation we also included the case of only negative refractive index perturbations lattice that has a similar refractive index profile to the work [20], it is found that, for the both cases of incident laser intensities of higher than filamentation threshold (∼3Pcr) and lower than the threshold (∼0.001Pcr), the optical intensities [green lines in Figs. 3(a) and 3(b)] on the axis for lattice with only negative index perturbation one are far lower than the cases with alternating positive and negative index perturbations lattice. Therefore, the alternating positive and negative index perturbations produces a better waveguiding results than the lattice with only negative perturbation. In addition, we also simulated the collimated beam to propagate in the gas lattice, the results are also given in the Figs. 3(a) and 3(b) (see the yellow curves). The result show that there is no big difference in the behavior of the beam in the lattice between the collimated beam and the focused one with a focusing lens of f=2 m. Our result is quite different from the work in Ref. [20] where the wavefront curvature of the input laser beam strongly influences the beam in the lattice [33]. The reason for no big difference in our case can be attributed to the following. In our case, the alternating positive and negative refractive index change of the gaseous lattice plays an important role, especially the positive part of this gaseous lattice provides the additional focusing effect, which weakens the effect of laser beam wavefront curvature by a loose focusing lens of 2 m.

The visual 3D iso-fluence pictures of filamentation propagation in the 3m-long gaseous lattice, corresponding to the cases of Fig. 3(a), are depicted in Figs. 3(c) and 3(d), in which the insets show the beam cross section profile before coupling into the lattice and the distribution of gaseous lattice. It is shown in Fig. 3(c) that many filaments with marginal length and relatively weak intensity are regularly surrounded the main one on the central optical axis of the lattice when the lattice starts at z=0. It is worth noting that the size of the beam is comparable to the cross-section of the lattice as seen in the inset. In this case, laser pulse itself does not start to filamentation when it enters the lattice medium with alternating $\delta n$, and the negative refractive index portion of the gaseous lattice scatters the beam and the positive one traps the beam, which leads to redistribution of the energy background reservoir. As a result, laser beam fuse into regularly arranged multi-filaments. Figure 3(d) shows the fluence of filamentation that is generated in the lattice medium starting from 1 m. It can be seen from the inset that the beam has collapsed to a very small diameter when propagates to z=1 m, which limits interaction region between the gaseous lattice and the collapsed laser beam to a waveguide consisting of the several gaseous density cylinders along the central axis. For the high-power pulses, the peak intensity is confined in this waveguide core due to self-focusing effect of the femtosecond laser. Besides, the additional focusing effect of the higher density cylinder with Gaussian profile on optical axis of the gaseous lattice provides further focusing effect. Therefore, a longer and more uniform filament on the optical axis with the higher field intensity is formed, which is surrounded by some weak optical filaments due to laser energy diffracted to the periphery by the negative refractive index part. The results in Figs. 3(c)–3(d) demonstrate that a multi-filament array with various length and intensity distribution of a dozen of individual filament can be realized by arranging gaseous lattice medium at different positions in the path of laser propagation.

The filamentation evolution by introducing the gaseous lattices of various lengths ($\xi = |{{z_2} - {z_1}} |$) is visualized in Fig. 4. Here, the introduced different length gaseous lattice structures have the same start position ${z_1} = 1\,m$ but different end positions at ${z_2}$. Figure 4(a) depicts the 3D profiles of the iso-fluence surface of filamentation and the 2D projection profiles in the $yz$ planes for $\xi = 0$ (without lattice), $\xi = 1\,m$, $\xi = 2\,m$, and $\xi = 3\,m$. The lattice structure has a distinct modulation effect on the beam due to its refractive index with an alternating positive and negative $\delta n$ distribution. Obviously, positive part of the lattice converges the beam, and the negative part makes the energy spread out, in consequence energy background of the pulse is re-distributed and the filament array is strongly affected by the lattice. Furthermore, results show that the length and uniformity of filaments in the array are improved with the increasing of the gaseous lattice length. Because the beam diameter has collapsed to hundreds of microns before it enters the gas lattice as seen in the inset of Fig. 3(d), and the corresponding several density cylinders near the optical axis actually constitute the air waveguide structure, providing a stronger waveguide effect and limits the outflow of the injected laser pulse. It can be seen from Fig. 4(a) that the longer the interaction length of the laser beam with the gas lattice medium, the more obvious regulation of the filamentation by the lattice is achieved.

 

Fig. 4. The evolution of femtosecond laser propagation in air and in the lattice dielectric structure. (a)The 3D Iso-fluence (a.u.) of the femtosecond laser pulse propagation and the 2D projection profiles in the yz planes for different gaseous lattice lengths ξ=0 (without lattice), ξ=1 m, ξ=2 m, and ξ=3 m. (b)the cross-sectional intensity profiles of without lattice (i, i’, i”) and with lattice(ii, ii’, ii”) at z=1.5 m. The (i) and (ii) are spatiotemporal intensity distributions of pulses; the (i’) and (ii’) are the cross-sectional instantaneous intensity distributions of the central part of the pulses (at 0fs); the (i”) and (ii”) are the cross-section hotspot profiles. (c) For ξ=3 m, the cross-sectional hotspot pattens of temporal integrated intensities at distances of z=2.1 m, 2.7 m, 3.3 m, 3.9 m.

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In order to analyze the mode of the filamentation when the beam is coupled into the gaseous lattice, we extracted the spatiotemporal intensity and the cross-sectional distribution of the filament array from Fig. 4(a), and the results are displayed in Fig. 4(b) and Fig. 4(c), respectively. The intensity profiles marked in Fig. 4(b) show the intensity distribution at distance of 1.5 m both for the ξ=0 m [see (i), (i’) and (i”)] and for the cases of ξ=1 m, 2 m, 3 m [see (ii), (ii’)and (ii”)]. Figure 4(b)(i) show that the spatiotemporal intensity distribution of the filamentation pulse in the absence of the lattice is significantly different from that with the lattice medium as shown in Fig. 4(b)(ii). The spatiotemporal intensity distribution of filamentation pulse in gaseous lattice (ii) is smaller and most of the optical intensity is tightly bound in the central waveguide cylinder compared with the case without lattice (i) at the same distance. Figure 4(b)(i’) and Fig. 4(b)(ii’) represent the section instantaneous intensity distributions of the central part of the pulses (at 0fs), corresponds to the location of the dotted line mark in (i) and (ii), respectively. Figure 4(b)(i’) shows that the beam diameter at central part of the pulse (at 0fs) diverges to about 2 mm in diameter and the instantaneous intensity is attenuated to ∼1011 W/cm2. However, the laser pulse beam is well re-regulated when it is coupled through the lattice medium as seen in Fig. 4(b)(ii’). In particularly, at z=1.5 m, the light beam is clearly divided into multiple optical filaments, in which filament positions are consistent with the those of the lattices [see Fig. 1(a)]. It is also worth mentioning that an off-axis ring maximum is formed in (i’) due to the central part of the pulse (t=0fs) is dissipated for ionization and defocused by the plasma, the defocused pulse forms a ring around the central axis. The ring in the beam profile has been experimentally observed with femtosecond laser pulse propagating in air [34,35]. The temporal integrated transverse intensity profiles at z=1.5 m without and with gaseous lattices are plotted in Fig. 4(b)(i”) and Fig. 4(b)(ii”), respectively. For overall energy of pulse, more laser energy concentrate in a smaller central range in lattice medium (ii”) than the case without lattice(i”). The laser energy defocused by plasma in lattice filamentation still concentrate in several waveguides around the central optical axis and formed multiple optical filaments with a high intensity as shown in Fig. 4(b)(ii”).

Cross-section pictures of the hotspot distribution for $\xi = 3\,m$ as a function of transmission distance z=2.1 m, 2.7 m, 3.3 m and 3.9 m are plotted in Fig. 4(c), respectively. It is observed that the energy of the collapsed laser beam is easily coupled into the central waveguide structure of the lattice, as there has always been a hotspot in the center with the higher intensity in Fig. 4(c). The phenomena are due to the waveguide effect and additional focusing by the positive cylinder with Gaussian refractive index gradient distribution. In addition to the hotspot on the central axis, there are four small hotspots around z=2.1 m. As the laser pulse is transmitted to the position of 2.7 m, 8 hotspots have evolved into the periphery surrounding the central one. Finally, the energy flows out from the inner square to the outer one, and more hotspots are formed in the periphery at z=3.9 m, while the number of original hotspots in the inner layer as well as their energies are reduced. We interpret this peripheral energy flow as consequence of the discrete diffraction [36] nature of the gas lattice. It is well known that diffraction in discrete systems is fundamentally different from that in continuous systems. Experimental results have demonstrated that energy distribution of discrete diffraction preferentially flows into the periphery and forms maximum of the beam in 1D waveguide arrays [33,37] and 2D photonic lattices [38] respectively, which is completely opposite to the diffraction in a continuous medium where the most energy remains at the center of the beam. The peripheral hotspot evolution in our result is similar with the energy flow behavior of the discrete diffraction in 2D photonic lattices. This indicates that discrete diffraction is involved in the equilibrium process of the filament formation along its propagation in gaseous lattices. It can be expected that a regulation and optimization of a femtosecond laser filament array can be realized via a further control of the discrete diffraction in gaseous lattice system.

The intensity and plasma density of the prolonged filamentation by gaseous lattice are quantitatively studied as well, and the results are shown in Fig. 5. And the Figs. 5(a) and 5(b) show the peak intensity and electron density on the optical axis as a function of transmission distance z for various $\xi$, respectively. With the lattice ($\xi \ne 0$), the peak intensity and maximum electron density is $8.25 \times {10^{^{13}}}\,{W / {c{m^2}}}$ and $1.09 \times {10^{^{17}}}\,c{m^{ - 3}}$ at z=1.05 m, respectively, which are slightly higher than those of the previously-reported typical filament. Figures 5(c)–5(f) show the spatial intensity profiles of the filaments for the four interaction lengths $\xi = 0\,m,1\,m,2\,m,3\,m$. In Fig. 5(c), the diameter of filament is kept constant ∼100 µm along the transmission axis when there is no lattice introduced ($\xi = 0$). Meanwhile, the energy fuses into a thinner filament with reduced diameter to 62 µm in the lattice medium ($\xi \ne 0$). In this case, the structure of the central positive density cylinder surrounded by four negative density cylinders acting as an optical fiber provides the waveguide effect, which limits focused light from spreading out. Additionally, a positive density cylinder with a Gaussian refractive index distribution furtherly converge light toward its center. As a result, a thinner and longer filament is formed.

 

Fig. 5. The on-axis (a) intensity and (b) maximal electron density of the femtosecond laser pulse in gaseous lattice along the propagation axis z, for the four situations of the gaseous lattice length ξ=0 (without lattice), ξ=1 m, ξ=2 m, ξ=3 m. The filament intensity profiles in the xz plane (y=0) for (c) ξ=0; (d) ξ=1 m; (e) ξ=2 m; (f) ξ=3 m.

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Noted that, in the previous research of lattice tailoring filamentation, it has been demonstrated that the longer and more uniform filaments can be obtained by using plasma lattice or direct writing lattice waveguide in solid, but the thicker filament is formed at the expense of largely reducing intensity and electron density of the filament [2023]. Differently, in our case, the filament regulated by the gas lattice medium reduce its diameter and keeps a high intensity and electron density along the beam propagation. This indicates that a new dynamic competition and equilibrium has been reached due to additional focusing and discrete diffraction effect introduced by the lattice medium.

Lastly, we need to mention that our previous work in Ref. [29] present a generation method of femtosecond multi-filament array in air experimentally by using a micro lens array. It is essentially a wavefront modulation control of laser beam for multi-filament array formation. In this work, the femtosecond multifilament array is generated by a pre-arranged gaseous lattice medium, and the ultimate goal is to manipulate filaments by using this special gaseous pre-lattice transmission medium. In fact, the two works are fundamentally different.

4. Conclusion

In summary, we have proposed a new type of gaseous lattice medium consisting of waveguide array with alternating positive and negative refractive index, to regulate femtosecond laser filamentation, which is totally different from the previously explored lattice medium. This gaseous lattice is characterized by its extraordinary refractive index distribution, which provides the additional focusing and discrete diffraction effects involved in the dynamic balance process of filamentation, and is responsible for laser energy distribution as well as other emerging properties of the filament array. Depending on the parameters of the gaseous lattice, the cross-section pattern, peak intensity, and beam width as well as other properties of the filament array can be optionally adjusted. It is found that a new dynamic competition and equilibrium that has been reached due to additional focusing effect introduced by the lattice medium can maintain a high intensity and electron density along the beam propagation. Therefore, the gas lattice-regulated filament is particularly useful to the applications where a higher electron filament is expected such as in trigging high voltage discharge in air [4,39] or guiding of radiofrequency energy with femtosecond laser filament [40,41]. The findings in this work also paves new way for many other potential applications based on the controlled and optimized the femtosecond laser filaments.

Funding

National Natural Science Foundation of China (No.11074027); Department of Science and Technology of Jilin Province (20200401052GX).

Acknowledgments

Y.J. Guo and J.J. Wang would like to thank Prof. Zuoqiang Hao at Shandong Normal University for his guidance in the initial stage of their research. Y.J. Guo would like to express her sincere thanks to Prof. Tingting Xi at University of Chinese Academy of Science for her assistance in the filamentation simulation.

Disclosures

The authors declare no conflicts of interest.

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4. K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012). [CrossRef]  

5. C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007). [CrossRef]  

6. L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013). [CrossRef]  

7. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005). [CrossRef]  

8. L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, “Stress induced birefringence tuning in femtosecond laser fabricated waveguides in fused silica,” Opt. Express 20(22), 24103–24114 (2012). [CrossRef]  

9. A. Camino, Z. Q. Hao, X. Liu, and J. Q. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014). [CrossRef]  

10. H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007). [CrossRef]  

11. H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012). [CrossRef]  

12. X. P. Zhang, H. L. Xu, C. H. Li, H. W. Zang, C. Liu, J. H. Zhao, and H. B. Sun, “Remote and rapid micromachining of broadband low-reflectivity black silicon surfaces by femtosecond laser filaments,” Opt. Lett. 42(3), 510–513 (2017). [CrossRef]  

13. W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016). [CrossRef]  

14. H. L. Li, H. W. Zhang, Q. L. Huang, C. Liu, Y. Su, Y. Fu, M. Y. Hou, A. W. Li, H. Chen, S.-L. Chin, and H. L. Xu, “Polarization-orthogonal filament array induced by birefringent crystals in air,” Opt. Express 26(7), 8515–8521 (2018). [CrossRef]  

15. R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005). [CrossRef]  

16. S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016). [CrossRef]  

17. G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004). [CrossRef]  

18. J. Wu, H. Cai, H. P. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008). [CrossRef]  

19. H. Cai, J. Wu, H. Li, X. S. Bai, and H. P. Zeng, “Elongation of femtosecond filament by molecular alignment in air,” Opt. Express 17(23), 21060–21065 (2009). [CrossRef]  

20. P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010). [CrossRef]  

21. P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011). [CrossRef]  

22. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013). [CrossRef]  

23. M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012). [CrossRef]  

24. 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]  

25. 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]  

26. 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]  

27. N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of Long-Lived High-Power Optical Waveguides in Air,” Phys. Rev. X 4(1), 011027 (2014). [CrossRef]  

28. E. W. Rosenthal, N. Jhajj, J. K. Wahlstrand, and H. M. Milchberg, “Collection of remote optical signals by air waveguides,” Optica 1(1), 5–9 (2014). [CrossRef]  

29. A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015). [CrossRef]  

30. D. E. Shipilo, N. A. Panov, E. S. Sunchugasheva, D. V. Mokrousova, A. V. Shutov, V. D. Zvorykin, N. N. Ustinovskii, L. V. Seleznev, A. B. Savel’ev, O. G. Kosareva, S. L. Chin, and A. A. Ionin, “Fifteen meter long uninterrupted filaments from sub-terawatt ultraviolet pulse in air,” Opt. Express 25(21), 25386–25391 (2017). [CrossRef]  

31. Y. Y. Ma and X. Lu, “1 T. T. Xi, Q. H. Gong, and Jie Zhang, “Widening of Long-range femtosecond laser filaments in turbulent air,” Opt. Express 16(12), 8332–8341 (2008). [CrossRef]  

32. W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005). [CrossRef]  

33. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002). [CrossRef]  

34. S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001). [CrossRef]  

35. S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002). [CrossRef]  

36. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008). [CrossRef]  

37. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef]  

38. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004). [CrossRef]  

39. M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002). [CrossRef]  

40. M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008). [CrossRef]  

41. Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013). [CrossRef]  

References

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  1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
    [Crossref]
  2. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007).
    [Crossref]
  3. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
    [Crossref]
  4. K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
    [Crossref]
  5. C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
    [Crossref]
  6. L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
    [Crossref]
  7. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005).
    [Crossref]
  8. L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, “Stress induced birefringence tuning in femtosecond laser fabricated waveguides in fused silica,” Opt. Express 20(22), 24103–24114 (2012).
    [Crossref]
  9. A. Camino, Z. Q. Hao, X. Liu, and J. Q. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014).
    [Crossref]
  10. H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
    [Crossref]
  11. H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
    [Crossref]
  12. X. P. Zhang, H. L. Xu, C. H. Li, H. W. Zang, C. Liu, J. H. Zhao, and H. B. Sun, “Remote and rapid micromachining of broadband low-reflectivity black silicon surfaces by femtosecond laser filaments,” Opt. Lett. 42(3), 510–513 (2017).
    [Crossref]
  13. W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016).
    [Crossref]
  14. H. L. Li, H. W. Zhang, Q. L. Huang, C. Liu, Y. Su, Y. Fu, M. Y. Hou, A. W. Li, H. Chen, S.-L. Chin, and H. L. Xu, “Polarization-orthogonal filament array induced by birefringent crystals in air,” Opt. Express 26(7), 8515–8521 (2018).
    [Crossref]
  15. R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005).
    [Crossref]
  16. S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
    [Crossref]
  17. G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
    [Crossref]
  18. J. Wu, H. Cai, H. P. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008).
    [Crossref]
  19. H. Cai, J. Wu, H. Li, X. S. Bai, and H. P. Zeng, “Elongation of femtosecond filament by molecular alignment in air,” Opt. Express 17(23), 21060–21065 (2009).
    [Crossref]
  20. P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
    [Crossref]
  21. P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
    [Crossref]
  22. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
    [Crossref]
  23. M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
    [Crossref]
  24. 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]
  25. 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]
  26. 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]
  27. N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of Long-Lived High-Power Optical Waveguides in Air,” Phys. Rev. X 4(1), 011027 (2014).
    [Crossref]
  28. E. W. Rosenthal, N. Jhajj, J. K. Wahlstrand, and H. M. Milchberg, “Collection of remote optical signals by air waveguides,” Optica 1(1), 5–9 (2014).
    [Crossref]
  29. A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
    [Crossref]
  30. D. E. Shipilo, N. A. Panov, E. S. Sunchugasheva, D. V. Mokrousova, A. V. Shutov, V. D. Zvorykin, N. N. Ustinovskii, L. V. Seleznev, A. B. Savel’ev, O. G. Kosareva, S. L. Chin, and A. A. Ionin, “Fifteen meter long uninterrupted filaments from sub-terawatt ultraviolet pulse in air,” Opt. Express 25(21), 25386–25391 (2017).
    [Crossref]
  31. Y. Y. Ma and X. Lu, “1 T. T. Xi, Q. H. Gong, and Jie Zhang, “Widening of Long-range femtosecond laser filaments in turbulent air,” Opt. Express 16(12), 8332–8341 (2008).
    [Crossref]
  32. W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005).
    [Crossref]
  33. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
    [Crossref]
  34. S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
    [Crossref]
  35. S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
    [Crossref]
  36. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
    [Crossref]
  37. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
    [Crossref]
  38. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004).
    [Crossref]
  39. M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
    [Crossref]
  40. M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
    [Crossref]
  41. Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
    [Crossref]

2019 (1)

2018 (1)

2017 (2)

2016 (2)

W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016).
[Crossref]

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

2015 (1)

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

2014 (3)

2013 (5)

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

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]

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]

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

2012 (4)

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, “Stress induced birefringence tuning in femtosecond laser fabricated waveguides in fused silica,” Opt. Express 20(22), 24103–24114 (2012).
[Crossref]

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

2011 (1)

2010 (1)

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

2009 (1)

2008 (4)

J. Wu, H. Cai, H. P. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008).
[Crossref]

Y. Y. Ma and X. Lu, “1 T. T. Xi, Q. H. Gong, and Jie Zhang, “Widening of Long-range femtosecond laser filaments in turbulent air,” Opt. Express 16(12), 8332–8341 (2008).
[Crossref]

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

2007 (3)

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

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

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

2005 (3)

2004 (2)

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004).
[Crossref]

2003 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

2002 (3)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

2001 (1)

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

1995 (1)

Abdollahpour, D.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

Aitchison, J. S.

Aközbek, N.

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

Alshershby, M.

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

André, Y.-B.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Andrianov, K. Y.

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Babushkin, I.

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

Bai, X. S.

Bellec, M.

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

Bergé, L.

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005).
[Crossref]

Bernhardt, J.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Biegert, J.

Birnbaum, R.

N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of Long-Lived High-Power Optical Waveguides in Air,” Phys. Rev. X 4(1), 011027 (2014).
[Crossref]

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

Bowden, C. M.

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

Braun, A.

Burghoff, J.

Cai, H.

Camino, A.

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

A. Camino, Z. Q. Hao, X. Liu, and J. Q. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014).
[Crossref]

Châteauneuf, M.

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

Chen, H.

Cheng, Y.-H.

Chin, S. L.

D. E. Shipilo, N. A. Panov, E. S. Sunchugasheva, D. V. Mokrousova, A. V. Shutov, V. D. Zvorykin, N. N. Ustinovskii, L. V. Seleznev, A. B. Savel’ev, O. G. Kosareva, S. L. Chin, and A. A. Ionin, “Fifteen meter long uninterrupted filaments from sub-terawatt ultraviolet pulse in air,” Opt. Express 25(21), 25386–25391 (2017).
[Crossref]

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005).
[Crossref]

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

Chin, S.-L.

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref]

Couairon, A.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

J. Wu, H. Cai, H. P. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008).
[Crossref]

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

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005).
[Crossref]

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

D’Amico, C.

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

Du, D.

Dubois, J.

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

Efremidis, N. K.

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

Fernandes, L. A.

Franco, M.

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005).
[Crossref]

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

Frey, S.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

Fu, Y.

Fujii, T.

Gao, X.

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

Grenier, J. R.

Guo, K. M.

Hao, Z. Q.

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

A. Camino, Z. Q. Hao, X. Liu, and J. Q. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014).
[Crossref]

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

Herman, P. R.

Herrmann, J.

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

Hou, M. Y.

Houard, A.

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

Huang, Q. L.

Ionin, A. A.

Iwasaki, A.

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Jhajj, N.

Kalkner, W.

Kamali, Y.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Kandidov, V. P.

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Kasparian, J.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Keller, U.

Kieffer, J.-C.

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

Klingbeil, L.

Köhler, C.

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

Kong, L. J.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Korn, G.

Kosareva, O. G.

D. E. Shipilo, N. A. Panov, E. S. Sunchugasheva, D. V. Mokrousova, A. V. Shutov, V. D. Zvorykin, N. N. Ustinovskii, L. V. Seleznev, A. B. Savel’ev, O. G. Kosareva, S. L. Chin, and A. A. Ionin, “Fifteen meter long uninterrupted filaments from sub-terawatt ultraviolet pulse in air,” Opt. Express 25(21), 25386–25391 (2017).
[Crossref]

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

Li, A. W.

Li, B. Z.

Li, C. H.

Li, H.

Li, H. L.

Li, S. M.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Li, Y. N.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Lin, J. Q.

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

A. Camino, Z. Q. Hao, X. Liu, and J. Q. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014).
[Crossref]

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

Litchinitser, N. M.

W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016).
[Crossref]

Liu, C.

Liu, W.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005).
[Crossref]

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Liu, X.

Lu, X.

Ma, Y. Y.

Marceau, C.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Marques, P. V. S.

Mathieu, P.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Méchain, G.

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

Méjean, G.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

Milchberg, H. M.

Mokrousova, D. V.

Mourou, G.

Mysyrowicz, A.

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

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005).
[Crossref]

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Nadeau, M.-C.

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Nolte, S.

Nuter, R.

Panagiotopoulos, P.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

Panov, N. A.

Papazoglou, D. G.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

Payeur, S.

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

Pertsch, T.

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004).
[Crossref]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

Peschel, U.

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29(5), 468–470 (2004).
[Crossref]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

Petit, S.

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

Prade, B.

C. D’Amico, A. Houard, M. Franco, B. Prade, and A. Mysyrowicz, “Conical Forward THz Emission from Femtosecond-Laser-Beam Filamentation in Air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. 93(3), 035003 (2004).
[Crossref]

Proulx, A.

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

Qian, S. X.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Ren, Y.

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

Ren, Z. C.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Rethmeier, K.

Rodriguez, M.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Rosenthal, E. W.

N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of Long-Lived High-Power Optical Waveguides in Air,” Phys. Rev. X 4(1), 011027 (2014).
[Crossref]

E. W. Rosenthal, N. Jhajj, J. K. Wahlstrand, and H. M. Milchberg, “Collection of remote optical signals by air waveguides,” Optica 1(1), 5–9 (2014).
[Crossref]

Roy, G.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Salmon, E.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Sauerbrey, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Savel’ev, A. B.

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Seleznev, L. V.

Shipilo, D. E.

Shutov, A. V.

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref]

Simard, J.-R.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Simard, P. T.

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

Skupin, S.

L. Bergé, S. Skupin, C. Köhler, I. Babushkin, and J. Herrmann, “3D Numerical Simulations of THz Generation by Two-Color Laser Filaments,” Phys. Rev. Lett. 110(7), 073901 (2013).
[Crossref]

R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005).
[Crossref]

Song, X. W.

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

Squier, J.

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Su, Y.

Sun, C. K.

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

Sun, H. B.

Sunchugasheva, E. S.

Suntsov, S.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

Tan, X.

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

Tao, H. Y.

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

Tu, C. H.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Tünnermann, A.

Tzortzakis, S.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
[Crossref]

Ustinovskii, N. N.

Wahlstrand, J. K.

Walasik, W.

W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016).
[Crossref]

Wang, H. T.

S. M. Li, Z. C. Ren, L. J. Kong, S. X. Qian, C. H. Tu, Y. N. Li, and H. T. Wang, “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016).
[Crossref]

Will, M.

Wille, H.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Wolf, J.-P.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Wöste, L.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Wu, J.

Xi, T. T.

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

Xu, H. L.

Yu, J.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, and J.-P. Wolf, “Triggering and guiding megavolt discharges by use of laser-induced ionized filaments,” Opt. Lett. 27(9), 772–774 (2002).
[Crossref]

Zang, H. W.

Zeng, H. P.

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous Refraction and Diffraction in Discrete Optical Systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[Crossref]

Zhang, H. W.

Zhang, X. P.

Zhao, J. H.

Zhao, Z. M.

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
[Crossref]

K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
[Crossref]

Zvorykin, V. D.

ACS Photonics (1)

W. Walasik and N. M. Litchinitser, “Dynamics of Large Femtosecond Filament Arrays: Possibilities, Limitations, and Trade-Offs,” ACS Photonics 3(4), 640–646 (2016).
[Crossref]

Appl. Phys. B (1)

A. Camino, T. T. Xi, Z. Q. Hao, and J. Q. Lin, “Femtosecond filament array generated in air,” Appl. Phys. B 121(3), 363–368 (2015).
[Crossref]

Appl. Phys. Lett. (4)

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, P. Panagiotopoulos, A. Couairon, and S. Tzortzakis, “Tailoring femtosecond laser pulse filamentation using plasma photonic lattices,” Appl. Phys. Lett. 103(2), 021106 (2013).
[Crossref]

H. L. Xu, Y. Kamali, C. Marceau, P. T. Simard, W. Liu, J. Bernhardt, G. Méjean, P. Mathieu, G. Roy, J.-R. Simard, and S. L. Chin, “Simultaneous detection and identification of multigas pollutants using filament-induced nonlinear spectroscopy,” Appl. Phys. Lett. 90(10), 101106 (2007).
[Crossref]

H. Y. Tao, J. Q. Lin, Z. Q. Hao, X. Gao, X. W. Song, C. K. Sun, and X. Tan, “Formation of strong light-trapping nano- and microscale structures on a spherical metal surface by femtosecond laser filament,” Appl. Phys. Lett. 100(20), 201111 (2012).
[Crossref]

M. Châteauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92(9), 091104 (2008).
[Crossref]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref]

Opt. Commun. (2)

S. L. Chin, N. Aközbek, A. Proulx, S. Petit, and C. M. Bowden, “Transverse ring formation of a focused femtosecond laser pulse propagating in air,” Opt. Commun. 188(1-4), 181–186 (2001).
[Crossref]

S. L. Chin, S. Petit, W. Liu, A. Iwasaki, M.-C. Nadeau, V. P. Kandidov, O. G. Kosareva, and K. Y. Andrianov, “Interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air,” Opt. Commun. 210(3-6), 329–341 (2002).
[Crossref]

Opt. Express (9)

D. E. Shipilo, N. A. Panov, E. S. Sunchugasheva, D. V. Mokrousova, A. V. Shutov, V. D. Zvorykin, N. N. Ustinovskii, L. V. Seleznev, A. B. Savel’ev, O. G. Kosareva, S. L. Chin, and A. A. Ionin, “Fifteen meter long uninterrupted filaments from sub-terawatt ultraviolet pulse in air,” Opt. Express 25(21), 25386–25391 (2017).
[Crossref]

Y. Y. Ma and X. Lu, “1 T. T. Xi, Q. H. Gong, and Jie Zhang, “Widening of Long-range femtosecond laser filaments in turbulent air,” Opt. Express 16(12), 8332–8341 (2008).
[Crossref]

W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005).
[Crossref]

P. Panagiotopoulos, A. Couairon, N. K. Efremidis, D. G. Papazoglou, and S. Tzortzakis, “Intense dynamic bullets in a periodic lattice,” Opt. Express 19(11), 10057–10062 (2011).
[Crossref]

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]

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]

L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, “Stress induced birefringence tuning in femtosecond laser fabricated waveguides in fused silica,” Opt. Express 20(22), 24103–24114 (2012).
[Crossref]

H. L. Li, H. W. Zhang, Q. L. Huang, C. Liu, Y. Su, Y. Fu, M. Y. Hou, A. W. Li, H. Chen, S.-L. Chin, and H. L. Xu, “Polarization-orthogonal filament array induced by birefringent crystals in air,” Opt. Express 26(7), 8515–8521 (2018).
[Crossref]

H. Cai, J. Wu, H. Li, X. S. Bai, and H. P. Zeng, “Elongation of femtosecond filament by molecular alignment in air,” Opt. Express 17(23), 21060–21065 (2009).
[Crossref]

Opt. Lett. (10)

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005).
[Crossref]

J. Wu, H. Cai, H. P. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008).
[Crossref]

R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005).
[Crossref]

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X. P. Zhang, H. L. Xu, C. H. Li, H. W. Zang, C. Liu, J. H. Zhao, and H. B. Sun, “Remote and rapid micromachining of broadband low-reflectivity black silicon surfaces by femtosecond laser filaments,” Opt. Lett. 42(3), 510–513 (2017).
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K. M. Guo, J. Q. Lin, Z. Q. Hao, X. Gao, Z. M. Zhao, C. K. Sun, and B. Z. Li, “Triggering and guiding high-voltage discharge in air by single and multiple femtosecond filaments,” Opt. Lett. 37(2), 259–261 (2012).
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Optica (1)

Photonics Res. (1)

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Phys. Rep. (2)

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Phys. Rev. A (1)

P. Panagiotopoulos, N. K. Efremidis, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Tailoring the filamentation of intense femtosecond laser pulses with periodic lattices,” Phys. Rev. A 82(6), 061803 (2010).
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Phys. Rev. E (1)

Y. Ren, M. Alshershby, Z. Q. Hao, Z. M. Zhao, and J. Q. Lin, “Microwave guiding along double femtosecond filaments in air,” Phys. Rev. E 88(1), 013104 (2013).
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Phys. Rev. Lett. (5)

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M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K. Efremidis, A. Couairon, and S. Tzortzakis, “Observation and Optical Tailoring of Photonic Lattice Filaments,” Phys. Rev. Lett. 109(11), 113905 (2012).
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Phys. Rev. X (1)

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Science (1)

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-Light Filaments for Atmospheric Analysis,” Science 301(5629), 61–64 (2003).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Refractive index shift $\delta n$ of the $9 \times 9$ gaseous lattice of (a) the transverse section distribution and its corresponding (b) 2D structure profile.
Fig. 2.
Fig. 2. The 3D iso-fluence (a.u.) evolution of femtosecond laser propagating with different parameters of the 3m-long gaseous lattice (at z=0 m) for (a) period Λ1=400 µm, Λ2=500 µm, Λ3=600 µm; (b) modulation depth n0(1) = 6×10−6, n0(2) = 8×10−6, n0(3) = 1×10−5; (c) each density cylinder waist w1=40 µm, w2=50 µm, w3=60 µm. The peak intensity on the optical axis as a function of propagation distance z with different parameters of the gaseous lattice are given in (d-f).
Fig. 3.
Fig. 3. Comparison of the intensities on the optical axis when input laser power Pin (a) above critical power about ∼3Pcr and (b) below critical power ∼0.001Pcr without (the black solid and dotted lines) or with the 3-meter-long lattice (red lines added at z=0 m, blue lines added at z=1 m). For Pin=3Pcr, the 3D iso-fluence representation of femtosecond pulse propagation in the gaseous lattice: (c) the onset of lattice at z=0 m; (d) the onset of lattice at z=1 m; the insets depict the gaseous lattice transverse distribution and the laser beam cross-sectional profile before coupling into the lattice. The green lines in (a) and (b) correspond to the lattice with only negative refractive index perturbations, and the yellow lines correspond to the collimated beam cases.
Fig. 4.
Fig. 4. The evolution of femtosecond laser propagation in air and in the lattice dielectric structure. (a)The 3D Iso-fluence (a.u.) of the femtosecond laser pulse propagation and the 2D projection profiles in the yz planes for different gaseous lattice lengths ξ=0 (without lattice), ξ=1 m, ξ=2 m, and ξ=3 m. (b)the cross-sectional intensity profiles of without lattice (i, i’, i”) and with lattice(ii, ii’, ii”) at z=1.5 m. The (i) and (ii) are spatiotemporal intensity distributions of pulses; the (i’) and (ii’) are the cross-sectional instantaneous intensity distributions of the central part of the pulses (at 0fs); the (i”) and (ii”) are the cross-section hotspot profiles. (c) For ξ=3 m, the cross-sectional hotspot pattens of temporal integrated intensities at distances of z=2.1 m, 2.7 m, 3.3 m, 3.9 m.
Fig. 5.
Fig. 5. The on-axis (a) intensity and (b) maximal electron density of the femtosecond laser pulse in gaseous lattice along the propagation axis z, for the four situations of the gaseous lattice length ξ=0 (without lattice), ξ=1 m, ξ=2 m, ξ=3 m. The filament intensity profiles in the xz plane (y=0) for (c) ξ=0; (d) ξ=1 m; (e) ξ=2 m; (f) ξ=3 m.

Equations (3)

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E z = i 1 2 k 0 Δ E i k 2 2 E t 2 + i k 0 n 2 2 | E | 2 E + i k 0 n 2 2 τ k { t exp [ ( t t ) / τ k ] | E ( t ) | 2 d t } E i k 0 ρ 2 ρ c E β ( K ) 2 | E | 2 K 2 E + i k 0 δ n E
ρ t = β ( K ) K ω 0 | E | 2 K ( 1 ρ ρ a t )
δ n = n 0 × i = 0 k ( 1 ) k exp ( { [ 2 x ± ( 1 ) k k Λ ] 2  +  [ 2 y ± ( 1 ) k k Λ ] 2 } / 4 w 2 )  ,  z 1 z z 2  

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