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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 4612–4622

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Femtosecond laser filament array generated with step phase plate in air

Hui Gao, Wei Chu, Guoliang Yu, Bin Zeng, Jiayu Zhao, Zhi Wang, Weiwei Liu, Ya Cheng, and Zhizhan Xu  »View Author Affiliations


Optics Express, Vol. 21, Issue 4, pp. 4612-4622 (2013)
http://dx.doi.org/10.1364/OE.21.004612


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Abstract

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 OSA

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

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005). [CrossRef]

–5

J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16(1), 466–493 (2008). [CrossRef] [PubMed]

]. 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

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef] [PubMed]

, 7

P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef] [PubMed]

]. 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

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

–11

S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B 74(1), 67–76 (2002). [CrossRef]

]. 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

T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett. 96(2), 025003 (2006). [CrossRef] [PubMed]

–16

H. Cai, J. Wu, P. F. Lu, X. S. Bai, L. E. Ding, and H. P. Zeng, “Attraction and repulsion of parallel femtosecond filaments in air,” Phys. Rev. A 80(5), 051802 (2009). [CrossRef]

]. As a consequence, multiple filaments are normally distributed disorderly in space. This has been identified as ‘optical turbulence’ [17

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83(15), 2938–2941 (1999). [CrossRef]

]. 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

Z.-Q. Hao, J. Zhang, T. T. Xi, X. H. Yuan, Z. Y. Zheng, X. Lu, M. Y. Yu, Y. T. Li, Z. H. Wang, W. Zhao, and Z. Y. Wei, “Optimization of multiple filamentation of femtosecond laser pulses in air using a pinhole,” Opt. Express 15(24), 16102–16109 (2007). [CrossRef] [PubMed]

], deformable mirror [19

C. P. Hauri, J. Gautier, A. Trisorio, E. Papalazarou, and P. Zeitoun, “Two-dimensional organization of a large number of stationary optical filaments by adaptive wave front control,” Appl. Phys. B 90(3-4), 391–394 (2008). [CrossRef]

], phase plate [20

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Circular phase mask for control and stabilization of single optical filaments,” Opt. Lett. 31(15), 2326–2328 (2006). [CrossRef] [PubMed]

–22

Y. X. Fu, H. Xiong, H. Xu, J. P. Yao, B. Zeng, W. Chu, Y. Cheng, Z. Z. Xu, W. W. Liu, and S. L. Chin, “Generation of extended filaments of femtosecond pulses in air by use of a single-step phase plate,” Opt. Lett. 34(23), 3752–3754 (2009). [CrossRef] [PubMed]

], axicon [23

B. Zhou, S. Akturk, B. Prade, Y. B. André, A. Houard, Y. Liu, M. Franco, C. D’Amico, E. Salmon, Z. Q. Hao, N. Lascoux, and A. Mysyrowicz, “Revival of femtosecond laser plasma filaments in air by a nanosecond laser,” Opt. Express 17(14), 11450–11456 (2009). [CrossRef] [PubMed]

–27

H. Gao, X. Sun, B. Zeng, S. Xu, W. Chu, W. Liu, Y. Cheng, Z. Xu, and G. Mu, “Cylindrical symmetry breaking leads to multiple filamentation generation when focusing femtosecond lasers with axicons in methanol,” J. Opt. 14(6), 065203 (2012). [CrossRef]

], diffractive elements [28

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] [PubMed]

, 29

L. Guyon, K. M. Hajek, F. Courvoisier, V. Boutou, R. Nuter, A. Vincotte, S. Champeaux, L. Bergé, and J.-P. Wolf, “Control of lasing filament arrays in nonlinear liquid media,” Appl. Phys. B 90(3-4), 383–390 (2008). [CrossRef]

], mesh [30

J. Liu, H. Schroeder, S. L. Chin, R. Li, and Z. Xu, “Ultrafast control of multiple filamentation by ultrafast laser pulses,” Appl. Phys. Lett. 87(16), 161105 (2005). [CrossRef]

, 31

H. Schroeder, J. Liu, and S. L. Chin, “From random to controlled small-scale filamentation in water,” Opt. Express 12(20), 4768–4774 (2004). [CrossRef] [PubMed]

], astigmatic focusing [32

G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, “Control of multiple filamentation in air,” Opt. Lett. 29(15), 1772–1774 (2004). [CrossRef] [PubMed]

, 33

Y. Fu, H. Gao, W. Chu, J. Ni, H. Xiong, H. Xu, J. Yao, B. Zeng, W. Liu, Y. Cheng, Z. Xu, and S. L. Chin, “Control of filament branching in air by astigmatically focused femtosecond laser pulses,” Appl. Phys. B 103(2), 435–439 (2011). [CrossRef]

] or varying the beam size [34

Q. Luo, S. A. Hosseini, W. Liu, J.-F. Gravel, O. G. Kosareva, N. A. Panov, N. Aközbek, V. P. Kandidov, G. Roy, and S. L. Chin, “Effect of beam diameter on the propagation of intense femtosecond laser pulses,” Appl. Phys. B 80(1), 35–38 (2004). [CrossRef]

–36

O. G. Kosareva, N. A. Panov, N. Akozbek, V. P. Kandidov, Q. Luo, S. A. Hosseini, W. Liu, J.-F. Gravel, G. Roy, and S. L. Chin, “Controlling a bunch of multiple filaments by means of a beam diameter,” Appl. Phys. B 82(1), 111–122 (2006). [CrossRef]

] and ellipticity [37

D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010). [CrossRef]

–40

D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009). [CrossRef]

], 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

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]

–47

M. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D Appl. Phys. 45(26), 265401 (2012). [CrossRef]

].

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.

Fig. 1 Experimental setup.
Fig. 2 Schematic diagram of the phase plates: (a) SCPP; (b) QCPP; (c) EOPP.

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:
2ik ∂A ∂z+( ∂2 ∂ x2+ ∂2 ∂ y2)A+2 k2ΔnA+ik βm I m−1A=0
(1)
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

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14(3), 650–660 (1997). [CrossRef]

]) and an effective counteracting higher order nonlinear refractive index (Δnh = – γIm) [6

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef] [PubMed]

, 7

P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef] [PubMed]

]. 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

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

, 49

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti: sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999). [CrossRef]

]. 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

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71(6), 877–879 (2000). [CrossRef]

]. The absorption coefficient βm=mℏ ω0 σ (m) N0/2 is related to the ionization cross section σ(m) at 800 nm (m = 8, σ(m) = 3.7 × 10−96 cm16/W8/s) [3

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

]. N0, ω0 point to the initial density of neutral molecular in air and the laser central frequency, respectively [3

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

]. 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

G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, “Control of multiple filamentation in air,” Opt. Lett. 29(15), 1772–1774 (2004). [CrossRef] [PubMed]

, 38

A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29(10), 1126–1128 (2004). [CrossRef] [PubMed]

]. On the other hand, instead of plasma defocusing effect, HOKE has been recently proposed as an alternative mechanism to arrest the self-focusing [6

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef] [PubMed]

, 7

P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef] [PubMed]

]. Though the outcome parameter might be slightly different quantitatively in two models, key spatial aspects of the nonlinear propagation are particularly preserved [51

M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010). [CrossRef] [PubMed]

–55

O. Kosareva, J.-F. Daigle, N. Panov, T. Wang, S. Hosseini, S. Yuan, G. Roy, V. Makarov, and S. Leang Chin, “Arrest of self-focusing collapse in femtosecond air filaments: higher order Kerr or plasma defocusing?” Opt. Lett. 36(7), 1035–1037 (2011). [CrossRef] [PubMed]

].

In our simulation, the initial Gaussian beam reads:
A(x,y,z=0)=exp(− x2+ y2 w0 2)exp(−i k0 x2+ y2 2f)exp(iθ)
(2)
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

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]

]. 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

D. G. Papazoglou, S. Suntsov, D. Abdollahpour, and S. Tzortzakis, “Tunable intense Airy beams and tailored femtosecond laser filaments,” Phys. Rev. A 81(6), 061807 (2010). [CrossRef]

]. 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

A. Pasquazi, S. Stivala, G. Assanto, J. Gonzalo, J. Solis, and C. N. Afonso, “Near-infrared spatial solitons in heavy metal oxide glasses,” Opt. Lett. 32(15), 2103–2105 (2007). [CrossRef] [PubMed]

–60

E. D’Asaro, S. Heidari-Bateni, A. Pasquazi, G. Assanto, J. Gonzalo, J. Solis, and C. N. Afonso, “Interaction of self-trapped beams in high index glass,” Opt. Express 17(19), 17150–17155 (2009). [CrossRef] [PubMed]

]. 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

B. Zhou, S. Akturk, B. Prade, Y. B. André, A. Houard, Y. Liu, M. Franco, C. D’Amico, E. Salmon, Z. Q. Hao, N. Lascoux, and A. Mysyrowicz, “Revival of femtosecond laser plasma filaments in air by a nanosecond laser,” Opt. Express 17(14), 11450–11456 (2009). [CrossRef] [PubMed]

–25

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16(20), 15733–15740 (2008). [CrossRef] [PubMed]

, 61

O. G. Kosareva, A. V. Grigor’evskii, and V. P. Kandidov, “Formation of extended plasma channels in a condensed medium upon axicon focusing of a femtosecond laser pulse,” Quantum Electron. 35(11), 1013–1014 (2005). [CrossRef]

]. It is explained by the quasi-Bessel beam shape generated by focusing a Gaussian beam with an axicon [62

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44(8), 592 (1954). [CrossRef]

]. Since Bessel beam features remarkable non-diffractive central spot [63

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

], it may result in a high contrast filament along its long focal depth during nonlinear propagation [25

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16(20), 15733–15740 (2008). [CrossRef] [PubMed]

, 64

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008). [CrossRef]

]. 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

X. D. Sun, H. Gao, B. Zeng, S. Q. Xu, W. W. Liu, Y. Cheng, Z. Z. Xu, and G. G. Mu, “Multiple filamentation generated by focusing femtosecond laser with axicon,” Opt. Lett. 37(5), 857–859 (2012). [CrossRef] [PubMed]

, 27

H. Gao, X. Sun, B. Zeng, S. Xu, W. Chu, W. Liu, Y. Cheng, Z. Xu, and G. Mu, “Cylindrical symmetry breaking leads to multiple filamentation generation when focusing femtosecond lasers with axicons in methanol,” J. Opt. 14(6), 065203 (2012). [CrossRef]

]. Another benefit of using an axicon is that it may help to suppress multiple refocusing cycles [65

Z. X. Wu, H. B. Jiang, L. Luo, H. C. Guo, H. Yang, and Q. H. Gong, “Multiple foci and a long filament observed with focused femtosecond laser pulse propagation in fused silica,” Opt. Lett. 27(6), 448–450 (2002). [CrossRef] [PubMed]

, 66

W. Liu, S. L. Chin, O. Kosareva, I. S. Golubtsov, and V. P. Kandidov, “Multiple refocusing of a femtosecond laser in a dispersive liquid (methanol),” Opt. Commun. 225(1-3), 193–209 (2003). [CrossRef]

], 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

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]

, 42

M. N. Shneider, A. M. Zheltikov, and R. B. Miles, “Long-lived laser-induced microwave plasma guides in the atmosphere: self-consistent plasma-dynamic analysis and numerical simulations,” J. Appl. Phys. 108(3), 033113 (2010). [CrossRef]

, 47

M. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D Appl. Phys. 45(26), 265401 (2012). [CrossRef]

].

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).

References and links

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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005). [CrossRef]

2.

S. L. Chin, F. Théberge, and W. Liu, “Filamentation nonlinear optics,” Appl. Phys. B 86(3), 477–483 (2007). [CrossRef]

3.

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

4.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]

5.

J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16(1), 466–493 (2008). [CrossRef] [PubMed]

6.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef] [PubMed]

7.

P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef] [PubMed]

8.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

9.

G. Paunescu, G. Spindler, W. Riede, H. Schröder, and A. Giesen, “Multifilamentation of femtosecond laser pulses induced by small-scale air turbulence,” Appl. Phys. B 96(1), 175–183 (2009). [CrossRef]

10.

G. Spindler and G. Paunescu, “Multifilamentation of femtosecond laser pulses propagating in turbulent air near the ground,” Appl. Phys. B 96(1), 185–191 (2009). [CrossRef]

11.

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C. P. Hauri, J. Gautier, A. Trisorio, E. Papalazarou, and P. Zeitoun, “Two-dimensional organization of a large number of stationary optical filaments by adaptive wave front control,” Appl. Phys. B 90(3-4), 391–394 (2008). [CrossRef]

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T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Circular phase mask for control and stabilization of single optical filaments,” Opt. Lett. 31(15), 2326–2328 (2006). [CrossRef] [PubMed]

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P. Rohwetter, M. Queißer, K. Stelmaszczyk, M. Fechner, and L. Wöste, “Laser multiple filamentation control in air using a smooth phase mask,” Phys. Rev. A 77(1), 013812 (2008). [CrossRef]

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66.

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OCIS Codes
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.5950) Physical optics : Self-focusing

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 12, 2012
Revised Manuscript: February 4, 2013
Manuscript Accepted: February 6, 2013
Published: February 14, 2013

Citation
Hui Gao, Wei Chu, Guoliang Yu, Bin Zeng, Jiayu Zhao, Zhi Wang, Weiwei Liu, Ya Cheng, and Zhizhan Xu, "Femtosecond laser filament array generated with step phase plate in air," Opt. Express 21, 4612-4622 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4612


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References

  1. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys.83(9), 863–905 (2005). [CrossRef]
  2. S. L. Chin, F. Théberge, and W. Liu, “Filamentation nonlinear optics,” Appl. Phys. B86(3), 477–483 (2007). [CrossRef]
  3. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.441(2-4), 47–189 (2007). [CrossRef]
  4. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys.70(10), 1633–1713 (2007). [CrossRef]
  5. J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express16(1), 466–493 (2008). [CrossRef] [PubMed]
  6. P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett.104(10), 103903 (2010). [CrossRef] [PubMed]
  7. P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. P. Wolf, and O. Faucher, “Transition from plasma-driven to Kerr-driven laser filamentation,” Phys. Rev. Lett.106(24), 243902 (2011). [CrossRef] [PubMed]
  8. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett.3, 307–310 (1966).
  9. G. Paunescu, G. Spindler, W. Riede, H. Schröder, and A. Giesen, “Multifilamentation of femtosecond laser pulses induced by small-scale air turbulence,” Appl. Phys. B96(1), 175–183 (2009). [CrossRef]
  10. G. Spindler and G. Paunescu, “Multifilamentation of femtosecond laser pulses propagating in turbulent air near the ground,” Appl. Phys. B96(1), 185–191 (2009). [CrossRef]
  11. S. L. Chin, A. Talebpour, J. Yang, S. Petit, V. P. Kandidov, O. G. Kosareva, and M. P. Tamarov, “Filamentation of femtosecond laser pulses in turbulent air,” Appl. Phys. B74(1), 67–76 (2002). [CrossRef]
  12. T. T. Xi, X. Lu, and J. Zhang, “Interaction of light filaments generated by femtosecond laser pulses in air,” Phys. Rev. Lett.96(2), 025003 (2006). [CrossRef] [PubMed]
  13. Y.-Y. Ma, X. Lu, T.-T. Xi, Q.-H. Gong, and J. Zhang, “Filamentation of interacting femtosecond laser pulses in air,” Appl. Phys. B93(2-3), 463–468 (2008). [CrossRef]
  14. B. Shim, S. E. Schrauth, C. J. Hensley, L. T. Vuong, P. Hui, A. A. Ishaaya, and A. L. Gaeta, “Controlled interactions of femtosecond light filaments in air,” Phys. Rev. A81(6), 061803 (2010). [CrossRef]
  15. S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett.86(24), 5470–5473 (2001). [CrossRef] [PubMed]
  16. H. Cai, J. Wu, P. F. Lu, X. S. Bai, L. E. Ding, and H. P. Zeng, “Attraction and repulsion of parallel femtosecond filaments in air,” Phys. Rev. A80(5), 051802 (2009). [CrossRef]
  17. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett.83(15), 2938–2941 (1999). [CrossRef]
  18. Z.-Q. Hao, J. Zhang, T. T. Xi, X. H. Yuan, Z. Y. Zheng, X. Lu, M. Y. Yu, Y. T. Li, Z. H. Wang, W. Zhao, and Z. Y. Wei, “Optimization of multiple filamentation of femtosecond laser pulses in air using a pinhole,” Opt. Express15(24), 16102–16109 (2007). [CrossRef] [PubMed]
  19. C. P. Hauri, J. Gautier, A. Trisorio, E. Papalazarou, and P. Zeitoun, “Two-dimensional organization of a large number of stationary optical filaments by adaptive wave front control,” Appl. Phys. B90(3-4), 391–394 (2008). [CrossRef]
  20. T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Circular phase mask for control and stabilization of single optical filaments,” Opt. Lett.31(15), 2326–2328 (2006). [CrossRef] [PubMed]
  21. P. Rohwetter, M. Queißer, K. Stelmaszczyk, M. Fechner, and L. Wöste, “Laser multiple filamentation control in air using a smooth phase mask,” Phys. Rev. A77(1), 013812 (2008). [CrossRef]
  22. Y. X. Fu, H. Xiong, H. Xu, J. P. Yao, B. Zeng, W. Chu, Y. Cheng, Z. Z. Xu, W. W. Liu, and S. L. Chin, “Generation of extended filaments of femtosecond pulses in air by use of a single-step phase plate,” Opt. Lett.34(23), 3752–3754 (2009). [CrossRef] [PubMed]
  23. B. Zhou, S. Akturk, B. Prade, Y. B. André, A. Houard, Y. Liu, M. Franco, C. D’Amico, E. Salmon, Z. Q. Hao, N. Lascoux, and A. Mysyrowicz, “Revival of femtosecond laser plasma filaments in air by a nanosecond laser,” Opt. Express17(14), 11450–11456 (2009). [CrossRef] [PubMed]
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