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Femtosecond laser filament arrays are generated in air by using three kinds of step phase plates with π phase lag, namely, the semicircular phase plate (SCPP), the quarter-circle phase plate (QCPP) and eight-octant phase plate (EOPP). Experimental results and simulations show that filament arrays consisting of two, four and eight filaments, respectively, are produced by three phase plates. The transverse patterns of the filament arrays are determined by the geometrical shapes of the phase plates. At the same time, the separation distances are found to vary with the focal lengths of the used lenses. We further propose that by using an axicon, filament array in the form of ring shape could be realized while the lengths of the filaments could be significantly elongated at the same time. Our study has suggested a realistic method to generate filament array by the step phase plate with π phase lag.
©2013 Optical Society of America
1. Introduction
Filamentation is an attractive nonlinear optical phenomenon which takes place during the propagation of high power femtosecond laser in transparent optical media [1–5]. Its dynamic is generally described by the counteraction of the optics Kerr effect induced self-focusing and the de-focusing effect caused by either plasma diffraction or higher-order-Kerr-effect (HOKE) [6, 7]. The occurring of filamentation is characterized by the so-called critical power for self-focusing. When the laser power is higher than the critical power for self-focusing, multi-filamentation can be frequently observed in practice due to the perturbation in the intensity distribution of the initial beam pattern or the refractive index perturbation of the optical media [8–11]. Depending on the phase differences, crossing angles or distances among them, multiple filaments will interact with each other, manifesting as repelling, attraction, fusion or energy exchange etc [12–16]. As a consequence, multiple filaments are normally distributed disorderly in space. This has been identified as ‘optical turbulence’ [17]. Hence, the general idea of multi-filamentation control is to impose strong modulation onto the initial beam transverse intensity distribution or phase front to overcome the inhomogeneity of the initial laser intensity or the optical media density. Extensive efforts, such as using pinhole [18], deformable mirror [19], phase plate [20–22], axicon [23–27], diffractive elements [28, 29], mesh [30, 31], astigmatic focusing [32, 33] or varying the beam size [34–36] and ellipticity [37–40], have been made to achieve regulative distributions of multiple filaments.
In our present work, filament arrays have been experimentally created in air by the step phase plate with π phase lag, such as the semicircular phase plate, the quarter-circle phase plate and eight-octant phase plate. Our results have indicated that with additional focusing lens, geometric features of the filament array, mainly including the number of filaments and the distance among the individual filament, could be controlled. We have further proposed that combined with the usage of an axicon, a filament array in the form of a hollow cylinder could be easily generated. Meanwhile, the longitudinal filament lengths could be significantly prolonged in such a way. The generated filament array would be potentially favorable for the application of manipulating electromagnetic wave in air [41–47].
2. Experimental setup
The experimental setup is shown in Fig. 1 . The femtosecond pulses with duration of ~40fs and central wavelength of 800nm at the repetition rate of 1 kHz are provided by a Ti: Sapphire laser (Legend Elite-Duo, Coherent, Inc.). The pulse energy is 6 mJ/pulse. Phase plates were inserted to control the phase of the incident beam. Three kinds of step phase plates, namely, the semicircular phase plate (SCPP), the quarter-circle phase plate (QCPP) and eight-octant phase plate (EOPP) as schematically shown in Fig. 2 , were employed in our experiment. They are all approximately 9 mm in diameter and 1.6 mm in thickness. The phase plates were divided into two, four and eight parts, respectively. In each phase plate π phase lag at 800 nm wavelength was induced between two adjacent parts. For example, for the semicircular phase plate, a half part of the phase plate was thinned by wet etching to produce a π-phase shift of 800 nm wave. After the phase plate, various focusing lens would be used. The focal lengths are 0.5 m, 1.0 m and 2.0 m, respectively. It is necessary to point out that no white light was observed from the lens during the experiment. It implies that additional nonlinear phase distortion inside the lens would not play crucial role in our cases.
The spatial distributions of the generated filament arrays were characterized by two methods. First, the longitudinal distributions were investigated by side imaging of the scattered white light from the filaments. Images of the filament arrays were captured by a 10 × microscopy objective and a CCD camera. Secondly, the transverse patterns of the filament arrays were recorded straightforwardly by burn paper. Note that single shot laser burn spot was obtained during the experiment by quickly sweeping the burn paper across the propagation direction. A blue band pass filter with a high transmissivity ranging from 400 nm to 500 nm was positioned in front of the objective to eliminate the scattered fundamental wavelength.
3. Experimental results
Figures 3(a) -3(d) illustrate the side images of the filament array created by using QCPP with the focal length of f = 50 cm. The images are taken at distances of (a) z = 46.5 cm, (b) z = 49.5 cm, (c) z = 50.5 cm and (d) z = 51 cm from the lens, respectively. Three bright lines could be clearly distinguished in Figs. 3(a)-3(d). The bottom one and the upper one each represent one filament. However, the signal of the middle one is higher since it corresponds to a superposition of two filaments located at the same horizontal level. This could be easily interpreted by Figs. 3(e)-3(h), which are the filament array cross-section profiles registered by burn papers at corresponding distances. Four-leaf clover like graph could be seen in Figs. 3(e)-3(h). Figure 3 also hints that the filaments are curved in the propagation direction, i.e. the distance among them become closer when approaching the geometrical focus.
Fig. 3 (a)-(d) Side images of multiple filaments using QCPP with focal length of f = 50 cm: (a) z = 46.5 cm, (b) z = 49.5 cm, (c) z = 50.5 cm, and (d) z = 51 cm. (e)-(f) The cross-section beam profiles captured by burn paper at corresponding distances of panels (a)-(d).
Similar to Fig. 3, representative experimental results obtained by different experimental configurations are present in Fig. 4 . The experimental parameters are: (a) using SCPP with the focal length f = 50 cm and at the distance z = 46.5 cm; (b) using SCPP, f = 100 cm and z = 95 cm; (c) using EOPP, f = 50 cm and z = 49.5 cm, respectively. It is worth mentioning that similar to Fig. 2, each bright line in Fig. 4(c) is given rise by the overlapping images of two filaments. As seen in Fig. 4, the number of filaments is determined by the number of portions that the phase plate has been divided into. In our cases, two, four and eight filaments were obtained by using SCPP, QCPP and EOPP respectively. The spatial patterns formed by multiple filaments closely follow the geometric characteristics of the phase plates. For example, when SCPP was used, two filaments are aligned vertically since the phase plate was divided into two portions from the top to bottom (see Figs. 4(a) and 4(b)). While in Figs. 4(c) and 4(f) multiple filaments are circumferentially distributed leaving low intensity at the center of the beam. Figure 4 also reveals that the filament separation distance is varied with the focal length. Specifically, in the case of SCPP, the displacement of the two filaments are varied from d = 0.37 mm (Fig. 4(a), f = 50 cm) to d = 0.48 mm (Fig. 4(b), f = 100 cm). The minimum separation distances of two diagonal filaments as a function of the focal length are depicted in Fig. 5 for three phase plates. The outcome implies that the separation distance mainly relies on the focal length and has weak dependence on the type of the phase plate used.
Fig. 4 (a)-(c) Side images of multiple filaments: (a) using SCPP, f = 50 cm, z = 46.5 cm; (b) using SCPP, f = 100 cm, z = 95 cm; (c) using EOPP, f = 50 cm, z = 49.5 cm. (d)-(f) corresponding cross-section beam profiles inside the filaments.
Fig. 5 The minimum separation distances of two diagonal filaments for three phase plates as a function of the focal length. (a) Experimental results; (b) Simulation results.
4. Numerical simulation
In order to shed further light onto the dynamic of filament array formation by step phase plates, numerical simulations based on a 2D + 1 (A(x,y,z)) nonlinear wave equation have been carried out as well:
Where A represents the amplitude of the light field, k denotes the wave number of the beam. Δn corresponds to the intensity dependent refractive index Δn = n2I – γIm, which includes the optical Kerr effect induced nonlinear refractive index (Δnk = n2I, n2 = 3.9 × 10−19 cm2/W [48]) and an effective counteracting higher order nonlinear refractive index (Δnh = – γIm) [6, 7]. The last term at the left hand side of Eq. (1) accounts for energy loss due to ionization. It is easy to see that Eq. (1) describes the propagation of CW beam in a medium with saturable nonlinearity. Therefore, without losing generality, the plasma defocusing has been considered as the major balancing effect to the self-focusing in our simulation. Hence m is chosen to be equal to 8 which is approximately the reported effective nonlinearity order of air ionization by near infrared femtosecond laser [3, 49]. γ denotes an empirical parameter determined from our preparatory simulation of the propagation of a Gaussian pulse based on Eq. (1). It gives rise to approximately a clamped intensity of 5 × 1013 W/cm2 [50]. The absorption coefficient is related to the ionization cross section σ(m) at 800 nm (m = 8, σ(m) = 3.7 × 10−96 cm16/W8/s) [3]. N0, ω0 point to the initial density of neutral molecular in air and the laser central frequency, respectively [3]. Since we focus mainly on the spatial distribution of the multiple filaments, the temporal aspects of the nonlinear propagation are not considered in Eq. (1). The validity of this kind simplification has been demonstrated by previous studies on the multi-filamentation [32, 38]. On the other hand, instead of plasma defocusing effect, HOKE has been recently proposed as an alternative mechanism to arrest the self-focusing [6, 7]. Though the outcome parameter might be slightly different quantitatively in two models, key spatial aspects of the nonlinear propagation are particularly preserved [51–55].In our simulation, the initial Gaussian beam reads:
where w0 = 2 mm is the radius of the input beam at 1/e2 level, f represents the focal length and θ is referred to the phase introduced by the phase plates. Both linear propagation and nonlinear propagation have been studied for different parameters. The results are selectively shown in Fig. 6 for the cases of linear propagation and in Fig. 7 for the cases of nonlinear propagation. In Fig. 7, the input power is 2.5, 6 and 15 times critical power for self-focusing for SCPP, QCPP and EOPP, respectively. As one could be seen from Fig. 7, the major spatial features of the filament arrays observed in experiments have been reproduced by the numerical simulations. Multiple filaments are generated by using phase plates. The number of filaments, the spatial pattern of the filament array and the separation distance between filaments have strong dependence on the type of the phase plate. Just like Fig. 5(a), Fig. 5(b) highlights the variation of separation distance as a function of the focal length for three phase plates. Figures 5(a) and 5(b) agree with each other qualitatively. The quantitative difference could be due to the different input beam diameters used in experiments and simulations. Figures 7(a)-7(d) even confirm that multiple filaments could be curved when combining a step phase plate with focusing lens. Curved filamentation has been reported previously by using Airy beams [56]. It has been explained by the strong energy confinement along a bent trajectory caused by a net cubic phase modulation introduced into the initial beam [57]. We conclude that similar statement holds valid also in our experiment with the difference that we have added abrupt π-shifted phase modulation by step phase plates.Fig. 6 (a)-(d) Simulated laser intensity distribution as a function of propagation distance during linear propagation: (a) using SCPP, f = 50 cm; (b) using QCPP, f = 50 cm; (c) using QCPP, f = 100 cm; (d) using EOPP, f = 100 cm. (e)-(f) the corresponding cross-section beam profiles at the focus. The intensity is normalized.
Fig. 7 (a)-(d) Simulated laser intensity distribution during the nonlinear propagation: (a) using SCPP, f = 50 cm; (b) using QCPP, f = 50 cm; (c) using QCPP, f = 100 cm; (d) using EOPP, f = 100 cm. (e)-(f) Corresponding cross-section beam profiles inside the filaments. For (e) z = 48 cm; For (f) z = 50 cm; For (g) z = 96 cm; For (h) z = 85 cm.
Comparing Fig. 6, one is also impressed that the transverse distributions of filament arrays shown in Fig. 7 closely follow the laser beam patterns formed during linear propagation. It suggests that the experimentally observed transversal and side view images are mainly due to spatial distribution of a laser field in a linear case with some small nonlinear correction. The understanding is the following. When focused by a lens, the laser beams passing through two diagonally opposite phase petals will both converge towards the propagation axis and interfere with each other. The π phase difference imposed by the phase plate leads to destructive interference on the propagation axis, giving rise to an intensity minimum at the beam center. For the similar reason, the laser beams passing through two adjacent phase plate parts will have π phase shift. The interference within their overlapping region would be destructive and reflected as azimuth intensity minima as shown in Figs. 6(e)-6(h). Therefore, the laser beam would be divided into multiple parts having abrupt phase jump. Each part will propagate almost independently leading to multiple foci in the case of linear propagation (Fig. 6) or multiple filaments in the case of nonlinear propagation (see Fig. 3, Fig. 4 and Fig. 7).
Furthermore, similar simulation model has also been successfully adapted to study the subtle propagation dynamic of a picosecond laser pulse in the heavy-metal-oxide glass. It is deduced that two-photon or three-photon absorption play crucial roles in balancing the self-focusing [58–60]. In our cases, the energy loss due to multi-photon absorption has been found not to exceed a few percents. In order to clarify the effect of multi-photon absorption, simulations by neglecting the multi-photon absorption has been performed. The results indicate that the multi-photon absorption does not influence the propagation dynamic significantly in our studies.
On the basis of the previously discussion, we have attempted to extend our work to create ring shaped filament array. The idea is to replace spherical focusing lenses used in the above discussion with an axicon. The axicon has recently appeared as a promising optical component in elongating the length of a single filament [23–25, 61]. It is explained by the quasi-Bessel beam shape generated by focusing a Gaussian beam with an axicon [62]. Since Bessel beam features remarkable non-diffractive central spot [63], it may result in a high contrast filament along its long focal depth during nonlinear propagation [25, 64]. We have recently demonstrated that multi-filamentation could also be generated by using an axicon and the lengths of the multiple filaments are significantly elongated [26, 27]. Another benefit of using an axicon is that it may help to suppress multiple refocusing cycles [65, 66], achieving uniform long filaments.
The numerical simulation by using an axicon has been performed according to Eq. (1). The top angle of the axicon is 179°. The input power is set as 20 times critical power for self-focusing. Figure 8(a) illustrates the distributions of the filaments array generated by the optical configuration consisting of an eight-octant phase plate as shown in Fig. 2(c) and the axicon. The longitudinal intensity distribution plotted in Fig. 8(a) indicates that the length of the filament array is about 30 cm, which is approximately equal to the focal depth obtained in linear propagation as displayed in Fig. 8(b). Moreover, the multiple filaments are spread in the form of hollow cylinder which is particularly interesting for electromagnetic wave guiding in air [41, 42, 47].
Fig. 8 (a) and (b) The longitudinal intensity distribution with the eight-octant phase plate using the axicon during the nonlinear propagation and linear propagation, respectively. (c) and (d) the cross-section beam profiles during the nonlinear and linear propagation at the distance of z = 25 cm. For (b) and (d), the intensity is normalized.
5 Conclusions
In conclusion, we have demonstrated both experimentally and numerically that femtosecond laser filament array could be generated in air by using step-phase-plate with π phase lag. The spatial arrangement of the filaments, including pattern, separation distance and length, could be controlled by pre-designed optical configurations. Thus, our results imply a simple method to generate configurable filament array. It could be foreseen that filament array with more complicatedly structured could be obtained in air by using subtly designed phase plate. It may find diverse applications in manipulating electromagnetic wave in air.
Acknowledgment
This work is financially supported by National Basic Research Program of China (2011CB808100), and National Natural Science Foundation of China (11174156, 10974213, 60825406). WL acknowledges the support of the open research funds of State Key Laboratory of High field Laser Physics (SIOM).
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