Introduction

During the propagation of intense femtosecond laser pulses in air filaments are generated due to an interplay between the optical Kerr effect causing self-focusing and the defocusing effect of generated plasma [1–4]. High order nonlinearities involved in the propagation dynamics of a filament complicate a thorough understanding of all its properties. Different theoretical models have evolved in the last years to describe this complex process such as the self-channeling model [1], the moving focus model [2] and the dynamic spatial replenishment model [3]. Filaments can be generated when the peak power of the driving laser pulse exceeds the critical power for self focusing P_cr of a few GW in air [5]. They consist of a plasma core with high laser intensity (about 100 µm in diameter) surrounded by an energy reservoir. Inside the filament core the laser intensity is limited to a constant value (intensity clamping [6]). Higher laser pulse powers (more than 10 times P_cr [7]) and/or irregularities of the spatial intensity distribution of the femtosecond laser pulse give rise to multifilamentation. Multiple filaments are then formed and propagate next to each other competing for the same energy resources [8]. The plasma that is generated in the core of a filament forms a channel that can act as a wave-guiding structure. This plasma channel represents a lossy waveguide for wavelengths in the visible up to the near infrared region (about 850 nm [4]). A lossless guiding behavior is expected for longer wavelengths [4]. The laser intensity inside the plasma channel is kept high enough to broaden the spectrum of the laser pulse while it can travel over long distances [1].

Femtosecond multifilaments that are generated in the atmosphere by powerful laser pulses emit a broad supercontinuum spectrum that ranges from the ultraviolet up to the far-infrared [9,10]. However, due to the high number of involved filaments the detailed information of the contribution of each single filament remains concealed.

To get a better understanding of the complex propagation dynamics of single filaments and their emission behavior we investigated the supercontinuum emission in the infrared spectral region with single-shot resolution. In previous studies it was already demonstrated that different properties of the filamentation process can be modified by spatial modulation of the driving laser pulse (pointing stability [11], spectral broadening (VIS) [12]). We used a deformable mirror as pulse shaping device and were able to generate filaments with a rich variety of different supercontinuum emission patterns.

One of the most interesting emission patterns that we obtained during the pulse shaping experiments has a spiral structure and will be the main subject of this report. The closer investigation and explanation of its relatively complex, non axial symmetric, but very regular structure is an experimental and theoretical challenge. The structural details of different IR emission patterns promise to be an ideal indicator to study the influence of small perturbations to the filamentation process in air and could provide further insight into the interaction of two or more adjacent filaments.

Setup

Figure 1 shows the experimental setup. Femtosecond laser pulses (800 nm, 50 fs) were generated by a 10 Hz Ti:Sa Laser System with an integrated DAZZLER device (Fastlite). The initial laser beam diameter of about 10 mm is reflected by a piezoelectrical deformable mirror (PDM, Flexible Optical, 2.54 cm diameter, 37 channels). The diameter of the reflected beam is then reduced to about 5 mm in a telescope. After a propagation distance of about 2 m single filaments with a length of about 2 m are generated. The input pulse energy is between 2 and 5 mJ. Depending on the phase front that is impressed by the PDM only part of the initial pulse energy is involved in the formation of a filament. The infrared radiation that is generated by the filament hits a metal screen and the intensity distribution of this IR emission is imaged by an IR-camera (CMT detector) for each laser shot in four spectral windows ranging from 1.5 µm to 5.3 µm. The average spectral distribution of the detected infrared emission corresponds qualitatively to the spectra shown in [9,10]. The data is then evaluated by an evolutionary algorithm that can optimize both pulse shaping devices (DAZZLER + PDM) in a closed feedback loop [13]. Initially, before the experiments, the parameters of the DAZZLER were adaptively optimized to achieve maximal output in the spectral region of 1.5 - 2.5 µm with a flat phase profile on the PDM. Afterwards only the PDM was modified while the DAZZLER was kept at fixed settings.

 

Fig. 1 (Color online) Setup for adaptive optimization of the infrared emission of filaments

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Experimental results

The first spiral emission patterns (see Fig. 2 ) were originally found as a byproduct during an adaptive optimization of the PDM to maximize the overall IR-emission of the filament. However, in order to specifically find a setting that favored a relative stable emission of spirals, most of the time the parameters of the PDM were adjusted manually starting from a random distribution. The stability of the setup was not high enough to guarantee day to day operation without complete readjustment of the PDM. An adaptive algorithm to optimize and/or isolate the emission of spirals automatically is in preparation.

 

Fig. 2 Snapshots of the intensity distribution of emitted spiral patterns in the infrared (1.5 - 1.8 µm). The central bright spot in the middle of the spirals is caused by the filament hitting the target screen.

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The stable emission of spirals is very easily disturbed by shot-to-shot fluctuations of the laser system and small air turbulences which was the main problem during the experiments. A detected spiral pattern can differ notably from shot-to-shot and is often completely replaced by an alternative emission pattern of the filament for consecutive laser shots. Generally, only a low percentage of laser shots resulted in the emission of a spiral pattern. Most likely rings of conical emission (CE) were detected.

To our knowledge spiral emission patterns have not been detected before. Therefore open questions are the origin of the generation and the position along the filament where the generation occurs. These questions will be addressed in the following paragraphs.

The spiral patterns presented in Fig. 2 are representative examples of this new class of emission. The snapshots were obtained by placing the target screen near the end of the filaments. Therefore an intact filament still hits the screen, resulting in a bright spot in the center of each spiral. Only Fig. 3(b) shows a short spiral with the target-screen placed after the filament has ended (no IR signal in the central region). Spirals that wind clockwise [Fig. 2(a), (b)] and counterclockwise [Fig. 2(c)] have been observed. Generally, the gap between neighboring windings of one spiral is constant in the inner region of the spiral disc. In the outer region it can increase [see Fig. 2(a), (b) and Fig. 3(a)], before the emission fades out. Alternatively the gap sometimes decreases and touches an inner winding to form an enclosing ring [see Fig. 2(c), (d) and Fig. 3(b)]. Interestingly, the position of those windings that “transform” to circles often corresponds to the position of the ring of a typical CE [Fig. 3(b), (c)].

 

Fig. 3 Further snapshots of IR emissions (1.5 - 1.8 µm): (a) A short spiral emission connected to the filament core. (b) A short spiral emission with enclosing ring structure. (c) A typical CE corresponding to the measurement of (b). (d) Emission composed of two spiral-like patterns.

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Similar to conical emissions the divergence angle of spiral emissions increases with increasing wavelength in the infrared. However, our current data set for spiral emissions exceeding wavelengths of 2.5 µm is limited preventing a detailed analysis of their divergence behavior. Nevertheless, it seems that the divergence angle of the enclosing ring of a spiral emission is roughly identical to the divergence angle of a conical emission at the same wavelength. The divergence angles of various conical emissions in the infrared were calculated by estimating the location of the generation position of the emission and the respective distance to the screen. The generation of each conical emission was localized by blocking the filament with a razor blade from below so that only the upper half of the emission could reach the screen. The upper half of the signal is then only visible if the filament is blocked after the emission process. The obtained values for the half-angle divergence range from about 5 mrad for 1.5 µm to about 12 mrad for 5.3 µm. These values correspond quantitatively to the half-angle divergence for CE given by the X-wave dispersion relation k _⊥ = (k ² − k_z ²)^1/2 used by Théberge et al. [10], with the wave vector k = ωn(ω)/c, k _⊥ as its transversal component and k_z as its longitudinal component which has to be a linear function of ω for the group velocity to be constant for all wavelengths [14]. We could not observe any emission at larger emission angles, that could be attributed to four-wave-mixing processes which are mentioned in [10].

So far, we only mentioned the generation of a single filament. This is based on our observation, that we did not detect any clear sign of interference due to overlapping emissions of neighboring sources in the infrared and in the visible spectrum that would indicate, that more than one filament has contributed to a emission. However, further investigation indicated that the emission of spiral patterns is indeed the result of the interaction of two or more filaments.

By monitoring the beam profile of a spiral-emitting filament with burn paper, we noticed next to a strong burn spot (main filament) the appearance of additional weaker spots (satellite filaments). These satellite filaments had a tendency to intersect with the main filament at a later time. We assume that the supercontinuum generation of weaker satellite filaments is inferior compared to the main filament so that no additional detectable emission above 1.5 µm can be expected. To check if the presence of these satellite filaments is a prerequisite for a spiral emission, we inserted an iris aperture around the filament and closed it to an open diameter of about 1 mm to block all satellite filaments while minimally disturbing the main filament [15].

The intensity profile of the filament right after the aperture was more compact and no satellite filaments could be recorded [Fig. 4(b) ]. However, the results of the measurements with the apertured filament remained inconclusive as both the emission behavior was slightly disturbed by the aperture and few spiral patterns could still be detected (The spiral pattern of Fig. 3(b) was obtained during this measurement). We measured the beam profile after the aperture again in various steps with increasing distance to the aperture [Fig. 4(c) - (h)]. We could observe the formation of a new satellite filament and its interaction with the apertured main filament. As is shown in Fig. 4, this satellite filament emerges from the main filament and again partly merges with it for a short moment at a later time [Fig. 4(f)]. This merging process leads to a local disturbance of the waveguiding plasma channel of the main filament and would explain an asymmetric emission behavior of the main filament.

 

Fig. 4 Burn paper patterns taken at different positions alongside the filament: (a) Pattern obtained in front of an iris aperture. (b)-(h) Pattern obtained after the filament passes the aperture (open diameter ca. 1 mm). The distance between neighboring measurements along the propagation axis is about 8 cm. The main filament splits into two filaments that travel very close to each other. (f) The two filaments merge for a short time before their intensity drops sharply in (g) and (h).

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In Fig. 5 a small wire was mounted near the end of the main filament as close to its core as possible with minimal disturbance to its plasma channel. The emitted IR radiation is partly blocked by the wire. With this setup it was possible to obtain snapshots of spiral patterns including the shadow of the wire (Fig. 5). The shadow is only visible in the outer region of each spiral emission. Therefore, the inner region of the spiral seems to have been emitted after the filament has passed the position of the wire while the outer region was emitted before this time. This is an indication that the whole emission process happens continuously alongside the filament. Furthermore, we could not record any IR intensity distribution that had both a spiral pattern in it and additionally some kind of other IR emission signal within this pattern, which was not directly connected to the spiral pattern. Therefore, we believe that after the spiral emission process the filament is too weak to generate new IR radiation within its core. This implies that during the emission of a spiral the filament loses a substantial amount of its energy and its plasma channel presumably breaks up.

 

Fig. 5 (a),(c) The spiral emission is partly blocked by a thin wire mounted near the filament core. The corresponding position of the wire is visualized in (b) and (d). The resulting intensity pattern indicates that the inner section of the spiral is emitted at a later time compared to the outer section.

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It was estimated by Nibbering et al. [4] that for infrared light with wavelengths exceeding 850 nm the index of refraction inside the plasma channel of a filament core gets higher in comparison to the air cladding that surrounds it. Under ideal conditions the infrared light can therefore travel inside the plasma channel of the filament without losses. This stands in direct contrast to the visible wavelength range, where the generated radiation is partly lost in the form of CE due to the antiguiding structure of the plasma channel for this wavelength region [4]. For a homogeneous structure of the waveguide the infrared radiation is kept inside until the plasma channel breaks up at the end of the filament. This is in accordance with various reports that no CE could be detected in the infrared region [4,16].

However, part of the SWIR and MIR radiation could escape at inhomogeneities or variations of the refractive index inside the waveguide structure. A complex interaction of two filaments due to the existence of a satellite filament leaves much room for speculation. A weak, neighboring filament will disturb the plasma channel of the main filament. A consecutive loss of radiation and energy at this point could trigger some kind of dynamic disintegration process of the whole plasma waveguide structure possibly leading to a complex non-symmetric emission pattern with a spiral pattern as a remote possibility.

Another approach is a possible change of the curvature of the plasma channel of the main filament due to the interaction of two closely spaced filaments as was proposed in [17]. Even spiral propagation of two filaments is possible as was recently demonstrated by Shim et al. [18]. In their experiment the phase of two filaments in air was coherently controlled. Depending on the relative phase of the two filaments they did demonstrate mutual fusion and repulsion of the two filaments and the onset of a spiral motion for suitable crossing angles. This kind of propagation dynamics would explain the IR emission spectrum of Fig. 3(d). A directional emission of two identical filaments that spiral around each other would indeed cause a point symmetric emission pattern of two short spirals. The absence of a point symmetry in the remaining spiral patterns of Fig. 2 and Fig. 3 suggests that only one single filament contributes to the emission in these cases. Therefore, if two filaments are really involved, they are unlikely identical.

The observed spiral propagation in [18] is still limited to a small rotation of only a few degrees. A rotation of 48° is predicted for a longer propagation distance of 10 m. To explain the emission of Fig. 2 more than two full revolutions during a propagation distance of only 1 - 2 m would be required.

This kind of motion was already obtained in previous studies of the interaction of 3D solitons [19,20]. A spiraling orbit can be obtained for coherent solitons that are exactly in phase and identical. However, slight perturbations in the phase or amplitude of the solitons would unbalance this propagation dynamics. Therefore, one can only speculate if an analogue model for the mutual propagation of two unidentical filaments in air can be applied.

Conclusion

A new class of emission from filaments was presented. The emission of spiral patterns from single filaments in the presence of a satellite filament was observed in the infrared. In air, this complex emission may be exclusive to only the infrared part of the spectrum. A detailed analysis of the visible part of the emission spectrum for similar filament formations will be informative. The mechanism of the spiral emission process is still unclear and leaves room for theoretical speculation. The application of a model of two filaments that spiral around each other was discussed but further experimental proof of some critical aspects of this model are still required. The adaptive optical setup used for the experiments has proven to be capable of producing a rich variety of different filament constellations. Pulse shaping with the PDM promises to be a helpful tool to investigate various scenarios of the interaction of multiple filaments and control their emission behavior.