The propagation of an optical discharge (OD) along hollow-core optical fibers (HCFs) is investigated experimentally. Silica-based revolver-type HCFs filled with atmospheric air were used as test samples. We observed that the average propagation velocity of an OD along the HCF (VAV) depends on the properties of the medium around the silica structure of the fiber. It is shown that the value of VAV changes by approximately a factor of three, depending on whether the optical discharge is moving along a polymer coated or uncoated fiber. The value of VAV practically does not change when the polymer is replaced by an immersion liquid (such as glycerol) or liquid gallium. By analyzing the destruction region’s patterns that appear in the fiber cladding after an OD propagation, we propose the physical picture of the phenomenon.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Any type of optical fiber can be destroyed by propagating laser radiation if some critical power level is exceeded. Moreover, the process of destruction, once started, can transform itself into a self-sustaining mode in which a destruction wave moves along the fiber towards the source of laser radiation. Similar phenomena have been previously observed and investigated in silica fibers with solid cores (the so-called fiber fuse effect [1–4]), in microstructured silica fibers [5,6], and in other types of fibers, like chalcogenide, fluoride, and polymer [7–9].
The development of hollow-core fibers (HCFs) has opened up new possibilities for transmitting high-power laser radiation through optical fibers . In these HCFs, the radiation is concentrated mainly in the hollow core, and the material of the fiber is not exposed to as high of laser radiation as what occurs on the axis of the hollow core. In addition, the thresholds for nonlinear effects in HCFs are also significantly higher than in solid-core fibers, simply because the density of the gas filling the hollow core is usually ≈3 orders of magnitude lower than the density of the solid state. As a result, HCFs are capable of transmitting radiation of much higher intensity than fully-solid-state optical fibers. At present, optical cables based on HCFs for transmitting ultrashort high-power laser pulses are already produced for industrial applications (see, for example ,). However, these fibers, of course, also can transmit only limited power and intensity of radiation, above which the process of fiber destruction begins. An understanding of the physical phenomena that occur during the destruction of HCFs is necessary for their correct use at high intensities of transported laser radiation.
The simplest picture of interactions between the laser radiation and the fiber is usually observed when a destruction wave moves through a fiber under CW laser radiation [1,2,12]. However, the propagation of the destruction wave along a HCF has not yet been observed in CW mode. Apparently, this fact can be explained by the need to use CW lasers of sufficiently high power in such experiments. Some insight into the physical processes that can take place during the destruction of HCFs under the action of CW laser radiation can be obtained using the results of . In that work, the processes of plasma propagation through glass tubes under the action of laser radiation (in other words, the propagation of an optical discharge (OD)) were studied. From the point of view of gas dynamics, these glass tubes are similar to HCFs, only their diameter is much larger at approximately 10 mm (for comparison, the diameter of a hollow fiber core is usually tens of microns). The experiments were carried out in a quasi-CW mode, since the duration of the laser (λ = 1.06 μm) pulses was approximately 5 ms, which exceeded the duration of practically all transient processes involved. When a laser radiation absorption center occurs inside a tube, an optical discharge arises and moves along the tube axis towards the laser at a relatively low intensity of CW laser radiation (≈1 MW/cm2, the total laser power was approximately 2 MW) . The movement of gas in the tube caused by its heating by the laser radiation has a significant influence on the propagation process of the OD. With increasing laser radiation intensity, the OD propagation mode is observed to change from the heat conduction mode to the light-detonation one. Accordingly, the observed velocity of the OD increases from approximately 1 km/s to 2 km/s. Similar processes should also occur in HCFs, with a correction for the significantly smaller core diameters.
The magnitude of the laser power required for the propagation of a continuous OD through a HCF can be estimated as follows. It is known that the threshold intensity of laser radiation for the propagation of an OD through a pipe is inversely proportional to the pipe diameter . This dependence appears as a result of considering energy losses sideways into the internal walls of the pipe. Therefore, based on the data of , we obtain that for HCFs with a core diameter of 20 μm, the threshold power should be on the order of 500 W (when the intensity of the laser radiation in the optical fiber is Ith≈5⋅108 W/cm2).
For lasers with a lower average power, the observation of plasma formation and propagation through a HCF is also possible with the use of a pulsed-periodic laser mode. In this case, significantly higher radiation intensities in the core of the fiber are necessary than the above estimate for the CW mode. Thus, during the propagation of 10 ns pulses of Nd: YAG laser radiation through an 8-mm-long photonic-crystal fiber with a hollow core filled with atmospheric air, an optical breakdown was observed at laser radiation intensities of approximately . In another paper , in the case of transporting nanosecond pulses (Nd: YAG, τ = 12 ns, pulse frequency of 10 Hz) by the HCF (“Kagome” type, core diameter of 50 μm), in several experiments, spontaneous initiation of the fiber destruction process was observed. The radiation intensity in the core was approximately 5⋅1010 W/cm2. After that, the wave of destruction propagated through the optical fiber towards the laser with an average velocity of ≈5 cm/s. In this case, the reflecting microstructured cladding of the fiber was completely destroyed. In the case of ultrashort pulses, the threshold intensity for the destruction of a HCF increases. For example, it was demonstrated that picosecond (~6 ps) pulses with an intensity of up to 1.5⋅1012 W/cm2 could be delivered by HCFs at a distance of 1 m .
In , the fundamental possibility of an OD propagation through a HCF under the action of laser radiation was revealed. Revolver hollow-core fibers (RFs) were used as the HCFs . It was observed that an OD propagated along an RF with an average speed of approximately 1 m/s under the action of pulsed-periodic radiation with an average power of approximately 2 W (wavelength of 1064 nm). In experiments , the intensity of laser radiation at the fiber axis reached ≈1012 W/cm2 at maxima of picosecond pulses. However, optical breakdowns were not observed in RFs in these conditions, and the OD had to be specifically initiated to observe the process of fiber destruction as in most similar experiments (see, for example ,). In addition, it should be noted that in all experiments on the effect of a sequence of short pulses on optical fibers, one should consider the following. In the case of a large (millisecond or longer) interval between pulses, the OD should actually be initiated in the fiber by each subsequent pulse for a quasi-continuous OD propagation. This imposes additional restrictions on the intensity of laser radiation necessary for the propagation of an OD.
This paper presents the first detailed study of the catastrophic destruction of HCFs under the action of pulsed laser radiation. In order to form a physical picture of the OD propagation through HCF, we used the analysis of fiber damage caused by the movement of the OD that was previously used in . The use of hollow-core revolver fibers with a reflecting cladding consisting of cylindrical silica capillaries allowed us to obtain an independent estimate of the pressure value in the core of the optical fiber during the propagation of an OD (using the threshold pressure for the capillary destruction). The variations in the medium surrounding the silica RF structure made it possible to detect an essential dependence of the average OD propagation velocity along a HCF on the properties of this medium. Note that such a phenomenon has never been previously noted when an OD propagated along ordinary, fully solid-state fibers (see, e.g., reviews [3,4]).
2. Experimental setup
The experimental scheme is shown in Fig. 1(a). It is similar to that used in . Laser radiation through the lens (1) was launched into the RF core with a length of approximately 50 cm or more and an input factor up to 80%. The experiments were carried out with RFs of two types: RF1 and RF2. In both cases, the fiber was a polymer coated silica structure, and the cross sections for RF1 and RF2 are shown in Fig. 2. In some experiments, the polymer coating on certain lengths of the fiber was removed. In particular, in a number of experiments, the RF2 polymer coating was removed (see Fig. 1(a), site 3) to place the fiber into an index-matched fluid and to observe the propagation of an OD with a microscope (4) without optical distortions. The image was recorded with a camera (5), the shutter of which was opened during the entire time of movement of the OD in the microscope’s field of view.
The adjustment quality of the launching unit for radiation into the RF was controlled by a power meter (7). In addition, the general picture of the OD propagations was recorded by a camera (6) with a frame rate of up to 240 fps. At the selected part of the RF, the glow of the OD plasma was recorded for a length of approximately 30 mm using a photodiode (Fig. 1a, PD). After initiation of an OD (9) near the output end face of a fiber (8) (by touching the output end of the fiber with a metal plate), the OD starts to move at velocity V along an air-filled HCF (2). The arrow indicates the OD propagation direction. Note that without the initiation, an OD in our experiments did not occur, and radiation without visible disturbances passed through the entire length of the RF.
The source of the single-mode radiation was a Nd:YAG laser. It operated in a combined Q-switch and mode-lock mode, generating nanosecond trains of picosecond pulses (NTPPs) with the parameters specified in Fig. 1(b). The radiation power at the maxima of the picosecond pulses (PPs) was up to 1 MW, which allowed one to observe the OD propagation in the RFs. The average laser radiation power PAV at the RF output during the OD initiations was approximately 4 W for RF1 and approximately 2 W for RF2. The maximum average power of the nanosecond pulses at the output for RF1/RF2 was 16/8 kW, while at the maxima of the picosecond pulses, it reached 2.0/1.0 MW. This corresponds to the following intensities of laser radiation on the axes of the RF1/RF2 cores: the average intensity over the NTPP was 2.4⋅109/5.2⋅109 W/cm2; the maximum picosecond pulse intensity was 3.2⋅1011/7.0⋅1011 W/cm2.
The wavelength of the Nd:YAG laser was in the transparency band for each of the optical fibers (RF1 and RF2). The optical losses of fibers at this wavelength were substantially less than 1 dB/m and did not actually affect the passage of radiation through the short lengths of the RFs used. ODs were initiated in both types of RFs.
3. Experimental results
After launching the laser radiation into the RF (4 W into RF1 and 2 W into RF2), the OD was initiated and then propagated through the fiber at an average speed VAV of ~1 m/s. The OD looked like a bright spot in the visible wavelength range moving through the fiber towards the laser radiation source. Despite the fact that the laser operated in a pulsed mode and the duration of nanosecond pulses (more precisely, the duration of NTPPs) was on the order of 10−4 of the time between them, the OD moved along the fiber with an approximately constant average speed VAV (averaged over several periods between NTPPs). Interestingly, VAV turns out to be approximately equivalent to the case of an OD moving along a standard solid core silica optical fiber under the action of CW laser radiation of the same average power PAV . Figure 3 shows a picture of the OD propagation through RF1.
The results of passing the OD through the RF1 and RF2 fibers were significantly different.
In the case of RF1, the silica structure of the optical fiber was completely destroyed after an OD propagation, and its fragments were held together only by the polymer coating. This led to a change in the equilibrium conditions of RF1, which was fixed in a suspended state during the experiment (see Fig. 3). Therefore, in the process of the OD propagation, mechanical oscillations of RF1 were excited that can be considered as a manifestation of some “optical-mechanical effect” (see Fig. 3, Visualization 1).
The RF1 fiber in Fig. 3 was completely covered with polymer except for the approximately 3 cm length without a polymer coating, which was fixed in the laser radiation launch unit. When the OD reached the unclad length of RF1, the fiber collapsed and the propagation of the OD stopped (as indicated by the RF1 fiber break at the end of the video, Fig. 3, Visualization 1). Apparently, the destruction of the RF1 support tube without a polymer coating leads to a decrease in air pressure in the core of the fiber. The absorption length of the laser radiation in the hollow core of RF1 then increases, which ultimately leads to a disruption of the OD initiation process by the next NTPP and, accordingly, to stopping the OD. It should be noted that this phenomenon can be used to protect fiber lines based on HCFs against OD propagations. For this, it is sufficient that the fiber line contains a segment of the optical fiber with a sufficiently thin unclad support tube. In such a case, an OD that appeared for any reason will stop when it reaches this segment. That will protect the rest of the fiber circuit from destruction.
In the case of RF2 having a twice as thick support tube (see Fig. 2), the fiber was outwardly preserved as a whole after the OD propagation. The preservation of the basic elements of the RF2 structure after passing the OD allowed us to investigate the damage to the fiber structure caused by the OD, and by the shape of the damage, we could obtain information on the processes taking place during the propagation of the OD. For this reason, most experiments were performed with RF2.
A comparison of the oscillograms of laser radiation and the visible emission of the OD plasma shows that in most cases, the emission of the OD plasma correlates with the laser radiation. Typical oscillograms are shown in Fig. 4. A comparison of the traces in Fig. 4(a) and (b) shows that the OD plasma in the intervals between the PPs does not extinguish, and the number of laser PPs supporting the propagation of the OD is substantially less than the number of pulses in the NTPP. The OD is supported only by PPs with a sufficiently high amplitude. PPs manifest themselves as peaks of the OD plasma glow, but the emission waveform of the OD (Fig. 4(b)) also has a CW (in the nanosecond time scale) component. Consequently, the relaxation time constant of the OD plasma exceeds 13 ns (the time interval between picosecond pulses). However, there is no plasma emission between NTPPs. Each new NTPP initiates an OD plasma in the core of the fiber. A comparison of the traces in Fig. 4(a) and (b) shows that the initiation of the OD occurs when the intensity of laser radiation is ≈20% of the maximum intensity of the PPs.
In some cases, however, the number of intensity maxima of the OD emission during one NTPP is reduced to a few units (in Fig. 4b, there are ≈17 such maxima). In some other cases, the emission of the OD plasma due to the action of the next NTPP is not observed at all; certain NTPPs do not initiate the propagation of an OD. Such variations in the behavior of the OD plasma appear randomly with a relative frequency of ~10%. The reason for this requires further investigation.
To estimate the threshold conditions for the initiation and propagation of an OD in RF2, a series of experiments were carried out with a sequential stepwise (with a step size of 0.5 W) increase in the average laser power starting with PAV = 0.5 W. A half-wave plate and a polarization beam splitter were additionally included in the experimental scheme (before the lens 1, see Fig. 1(a)) to control the level of PAV. As a result, it was found that at a PAV of less than 1 W, the initiation of an OD and its propagation through RF2 were never observed. However, at PAV ≥1.5 W, OD propagation was observed in all experiments. So, the radiation intensity thresholds at the PP maxima at the RF2 axis corresponding to these PAV levels are .
The average OD propagation velocity was measured according to the video data of camera 6 (Fig. 1(a)). It turned out that the average velocity of the OD in our experiments VAV depends significantly on whether the RF was coated with polymer or not. If RF2 was completely covered with a polymer coating, then the dependence of the OD path versus time corresponded to movement with an approximately constant average speed (see Fig. 5, line 1), and in this case, VAV = 0.91 m/s.
However, if the polymer coating was partially removed and the silica support tube directly bordered air, then the OD propagated over this unclad RF2 segment with a significantly larger VAV value. The dependence of the OD path versus time L(t) had, in this case, the form of a polyline (Fig. 5, line 2). Different sections of this line correspond to the different OD velocities: VAV = 0.92 m/s for the regions with a polymer coating, and VAV = 2.84 m/s for the region without a coating (≈3 times faster).
Notice that we measured here the average value of OD velocity. And accuracy of these measurements was not better than 5%. More detailed information on OD velocity can be obtained from direct Doppler frequency shift measurements for speed values of ~100 km/s [23,24] and from much more precise measurements using heterodyne detection .
Fig.6, Fig.7, and Fig.8 illustrate damages of RF2 after OD propagation for fiber immersed in the index-matching fluid (glycerol), for uncoated fiber in air, and for fiber in polymer coating. Figure 6 shows pictures of the same section of the RF2 before, during, and after the OD propagation through it. These pictures were taken by camera 5 (Fig. 1(a)). The fiber is uniform in length before the OD propagation (Fig. 6(a)). The time-integral photograph of the OD plasma (b) (the shutter of the camera was open the entire time that the OD was in the camera's field of view) indicates that the OD in this segment propagated continuously along the hollow core of RF2. The photograph taken after the passage of OD (c) shows the damage of RF2 reflecting cladding, which looks like a periodic structure with a period of approximately 180 μm. The pictures of fiber cross-sections of damaged and undamaged regions can be found in . Within each period (where the periodicity is observed), areas with significant destruction of the capillaries of the reflecting cladding (bright in Fig. 6(c)) alternate with less damaged regions where the capillaries remained practically free of defects (dark in Fig. 6(c)). However, the periodicity is sometimes significantly disturbed. For example, in Fig. 6(c), we can see a segment of the fiber where the damaged areas are practically not separated from each other but form one substantially longer damaged area. Figure 6 shows the results of the experiment when the RF2 fiber without a polymer coating was immersed in a fluid (index-matching fluid, glycerol). A set of experiments has shown that approximately the same pictures of RF2 damage (as in Fig. 6(c)) are also observed when the OD propagates through a fiber completely covered with a polymer coating (acrylate, Fig. 8(a)), as well as when RF2 is immersed in liquid gallium metal at a temperature above its melting point 30 °C (see below for the reasons for choosing gallium).
A significantly different picture of damage was formed in RF2 when the OD was propagated through the RF2 unclad section (without a polymer coating) located in air. Figure 7 shows images of RF2 similar to those shown in Fig. 6, but the unclad segment of the fiber was located in the field of view of the microscope 4 (Fig. 1) at position 3 without an index-matching fluid. Images of fiber regions 1, 3 and 5 in Fig. 7 are similar to the images in Fig. 6. The essential difference between Fig. 6 and Fig. 7 is the presence of regions 2 and 4 in Fig. 7 where there is no glow of the OD plasma (Fig. 7(b)) and there are significant (up to ~1 mm) lengths of the fiber without capillary damage in the cladding. It should be noted that in regions 1, 3, and 5 in Fig. 7, where periodic (with a period of approximately 180 μm) structures of destruction are observed, it is also possible to see regions with a significant violation of periodicity (as in Fig. 6(c)).
Thus, we see that there is a large dispersion of lengths of regions with damaged/undamaged capillaries in a reflecting fiber cladding. A statistical analysis of these lengths was performed in the following way. We obtained high-resolution panorama pictures of damaged fibers (length: up to 120 mm) combining sequential photos. Their 9 mm parts are shown in Fig. 8(a) for a polymer coated fiber and in Fig. 8(b) for an uncoated fiber. Using these panorama pictures, we obtained dependencies for the image brightness along the axis of the fiber core against the length of the fiber I(x). Bright and dark points correspond to damaged and undamaged regions, respectively. The dependency of I(x) for the polymer coated RF2 is given in Fig. 8(c) and for the uncoated RF2 in Fig. 8(d) (for the whole length of the fiber, 52 mm). Figure 8(d) demonstrates larger gaps between damaged regions in comparison with Fig. 8(c).
Using Fourier transformations, we computed the spatial frequency spectra of I(x) in both cases. The goal was to determine the main spatial periods of these functions. The spatial frequency spectra of the functions from Fig. 8(c) and (d) are shown in Fig. 9. The blue line corresponds to the polymer coated fiber, and the red line corresponds to the uncoated one. Both spectra contain peaks at 5.4⋅10−3 μm−1 (denoted by 1), which corresponds to the spatial period of T1≈185 μm. However, the uncoated fiber spectrum (red line) contains an additional peak at 3.3⋅10−4 μm−1 (denoted by 2), which corresponds to the spatial period of T2≈3 mm. The T2 period includes a train of T1 periods and an undamaged gap up to the next train.
The obtained results can be explained as follows. Experiments show the formation of plasma after touching the output end of an RF with a metal target. That is the process of initiating an OD. Immediately after initiation, a laser-supported detonation wave (LSDW) starts to propagate under the action of a laser PP. The LSDW after the end of a PP continues its movement as an attenuating shock wave (SW). The next PP arrives 13 ns later but before the complete relaxation of the OD plasma in the hollow core (see Fig. 4(b)). This picosecond pulse is absorbed in the front of a still powerful SW. As a result, the propagation of the LSDW and its relaxation are repeated. A similar process take place in the focal region of the lens focusing a train of PPs into a gas in the absence of any waveguide [26–28].
In those parts of the RF where the pressure pulses in the LSDW and the SW are large enough, there is a destruction of the reflective cladding capillaries. In the areas between them, destroyed capillaries are not observed (see Fig. 6, Fig. 7, and Fig. 8). After the end of the NTPP, the OD plasma in the core completely relaxes and the propagation of the OD under the action of the next NTPP begins with the new initiation of the OD. In this case, an OD can be initiated in the destroyed capillary region closest to the laser — for example, by increasing the electric field on the sharp fragments of silica glass in this area. This occurs when the OD propagates in an RF with a polymer coating.
If the OD propagates along a fiber without a polymer coating, then the OD is not initiated by subsequent NTPPs in the region of the destroyed capillaries but approximately 1 mm in front of it (see Fig. 7(b) and (c), regions 2 and 4; some of similar gaps in Fig. 8(d) are marked by green arrows and letters δ). Experiments show that this difference is due to the properties of the medium in which the silica structure of RF2 is located. In our opinion, the conditions for the reflection of elastic waves from the outer surface of the RF support tube should play an essential role here.
The SW formed in the air of the RF2 core as a result of an NTPP action partially penetrates into the silica support tube as a weak elastic shock wave diverging along the radius (Fig. 2(b), DSW). The DSW reflects from the outer surface of the support tube as a convergent wave (Fig. 2(b), RCW). Moreover, the reflection coefficient depends on a combination of the properties of the silica and the medium around the support tube. In the case of a silica-air boundary, the reflection coefficient is close to unity, and at the boundaries of a silica-polymer or silica-glycerol, and especially silica-gallium, it is significantly lower.
In the case of an OD moving in a fiber without a polymer coating, the shock wave is almost completely reflected back in the form of a rarefaction wave after reaching the outer boundary of the support tube. This wave reaching the inner surface of the support tube fills the gas (air) in the hollow core with small fragments of silica glass (including very small ones, ~10 nm in size). Fragments are moving along the hollow core following the air velocity after the shock wave front in the interval between NTPPs for approximately 1 ms. Fragments may be displaced approximately 1 mm along the optical fiber towards the laser radiation.
The air optical breakdown threshold is decreased with the presence of silica particles compared with pure air. This makes it possible to initiate an OD at a distance of approximately 1 mm from the last capillary destruction zone in a fiber without a polymer coating (as can be seen in Fig. 7(b) and (c)). Adding these undamaged parts of the fiber to damaged ones makes an essential contribution to the increase in the observed average OD velocity during an OD propagation in a fiber without a polymer coating in comparison with a coated one. Below is a more detailed description of the listed processes with necessary numerical estimates.
Laser-supported detonation wave
After initiation, an OD propagates under the action of a picosecond laser pulse with a maximum intensity of approximately 1012 W/cm2. If we assume that this propagation process is the LSDW, then its velocity and path are :
Here, is the radiation intensity of the picosecond laser pulse on the fiber axis, γ is an adiabatic index, and is the air density in the core of the fiber. For the above mentioned radiation intensity and the density of the laboratory air, we obtainand .
Attenuating shock wave
During the time between picosecond pulses, laser energy does not flow, so the LSDW turns into a usual SW with a complicated geometric shape (see, for example ,). Distances traveled together by the LSDW () and SW () along the axis of RF2 between PPs (Δ) can be measured as the distance between adjacent regions of destruction in the fiber (see Fig. 6(c)). This is approximately Δ = 180 μm.
First, we have used the point explosion theory with spherical symmetry for evaluating the distance traveled by the shock wave between PPs . The released energy was assumed to be equal to the energy of the PP (≈0.17 mJ). This approximation is closer to the experiment for smaller times when the SW passed no more than several diameters of the fiber core:30]:
Pressure value in a hollow core optical fiber during the propagation of an optical discharge
The core of a HCF after an OD propagation consists of subsequent regions with damaged and undamaged capillaries. A determination of the parameters of the pressure wave under which the capillaries are collapsing is a rather complicated task. As a first approximation, the critical pressure for capillary destruction can be estimated by using the conditions of buckling collapse of long cylindrical tubes under an external pressure . If is the critical pressure corresponding to the destruction of a capillary with radius r and thickness of the cylinder wall , thenEq. (4) gives the estimation ≈190 atm. Thus, the pressure in the hollow core in regions of destruction apparently exceeds this value. The estimated value of pressure in an LSDW essentially exceeds the critical pressure for capillaries collapse and corresponds to the presence of destruction zones :Fig. 6(c)).
Therefore, a consistent physical picture of an OD propagating along an RF (as a version of HCFs) during separate NTPPs includes the following processes:
- 1. Initiation and propagation of a LSDW under the action of a picosecond laser pulse.
- 2. Further movement of the attenuating shock wave along a HCF during the period between picosecond pulses.
- 3. The process of a new LSDW initiation in front of the shock wave by the next picosecond pulse.
The initiation of an optical discharge in the hollow core with a large time interval between the pulses of laser radiation
In all our experiments where the support tube of a fiber was not destroyed, the OD movement, once started, always continued until the OD reached the input face of the HCF. The start of the absorption of each of the next picosecond pulses in the NTPP was ensured by the absorption of the not yet relaxed (during ) plasma. However, between the NTPPs () the plasma glow in the OD went out completely. Therefore, the question of initiating an OD under these conditions is worthy of special consideration.
Indeed, in a time of almost 1 ms between NTPPs, the shock wave travels a distance of approximately 30 cm along the fiber before the first PP arrives from the next NTPP. However, it can already be considered a weak shock wave, and it does not cause air ionizations in the core. Experimentally we observed that the OD reinitiation takes place either at the region of capillary destruction nearest to the laser (in the case of a fiber coated with a polymer), or ≈1 mm in front of it (in the case of a fiber without a polymer coating).
The initiation of an OD in the region of capillary destruction is undoubtedly facilitated by the presence of small spiky splinters of silica capillaries. The concentration of the electric field of the laser radiation takes place on the edges of these fragments, which leads to a decrease in the threshold intensity of air breakdown. It is more difficult to explain the presence of millimeter breaks in the structure of RF2 destructions (Fig. 7(c)) and in the track of the OD movement (Fig. 7(b)) along the fiber without a polymer coating.
It is our opinion that this phenomenon can be explained by the influence of the OD-induced sound waves propagating in the silica across the support tube of RF2. Indeed, when the SW, which appeared during laser detonation, reaches the support tube, it partially penetrates into the silica glass of the tube, goes to its outer boundary (as a divergent shock wave, DSW), and is reflected back from it (as a reflected convergent rarefaction wave, RCW), see Fig. 2(b). The reflection coefficient of the acoustic wave from the cylindrical outer surface of the support tube depends significantly on the coating material. It is known that the reflection coefficient (by power) of a sound wave from the interface between two media is determined as follows :
where and are the density and speed of sound in each of the boundary media, respectively. is the specific acoustic impedance of the medium. Expression (6) is valid for the normal incidence of waves.
The reflection coefficient of the sound wave at the silica-air boundary is equal to one, with an accuracy of about 10−4. On the silica-polymer border, however, it is ≈35%. If the fiber without a polymer is immersed in liquid gallium at a temperature of approximately 30 °C, then the reflection coefficient at the silica-gallium interface will be approximately 3%. Thus, the formation of long (approximately 1 mm) undamaged gaps in the fiber cladding after the propagation of an OD correlates with the value of the reflection coefficient of the sound wave from the boundary of the silica-external environment. If the reflection coefficient is close to unity, then such gaps are formed. If it is significantly lower, then no gaps are formed.
The role of the rarefaction wave after reflection from the outer surface of the support tube of the fiber may be as follows. When the RCW reaches the inner surface of the support tube, the shock wave fills the air in the hollow core with small fragments of silica glass. The speed at which they acquire the velocity of the surrounding gas due to viscous forces depends on the size of the particles. For spherical particles, it can be estimated from Stokes law for the motion of a sphere in a viscous fluid :30]. Using the results of the calculations in , it can be estimated that under the conditions of our experiments, the displacement of air particles in the hollow core in the direction towards the laser will be approximately 600 μm under the action of a single NTPP (in the approximation of a spherical shock wave). Figure 10 shows the calculated values of velocities and the path of small (~10 nm) particles of silica behind the SW front. The displacement increases with the transition to an approximation of a flat shock wave, as is undoubtedly the case at longer movement times of the wave along the core of the fiber. Thus, the air in the hollow core can “capture” small particles of silica glass (formed as a result of the destruction of the capillaries of the cladding of RF2) and transport them a distance of ~1 mm in the direction of the SW propagation. The presence of small particles in the air of the core decreases the optical breakdown threshold, and as a result, one of the first pulses of the new NTPP induces an optical breakdown that ensures the next step of the quasi-continuous propagation of the OD.
Of course, these estimates do not consider all the circumstances of the process. For example, after the first NTPP, the second NTPP interacts with the perturbed gas, which was not considered here. More accurate approximations will be considered in further research.
Estimation of the average speed of OD propagation along a HCF
The picture of the OD propagation process considered above allows us to estimate the average OD velocity in our experiments.
When an OD propagates through unclad RF2 under a NTPP, its path is made up of segments with a length of approximately Δ = 180 μm. Each 180 μm segment corresponds to the OD path traveled under a single PP. If we assume that, on average, there are 10 such PPs in each NTPP, then the OD displacement during one NTPP is approximately 10⋅Δ = 1.8 mm (the length of a NTPP “fingerprint”). In addition, in fibers without a polymer coating, the distance between successive “fingerprints” of a NTPP δ is approximately δ≈1 mm (see Fig. 8). Thus, we find that one NTPP shifts an OD in uncoated fiber by a distance of 10⋅Δ + δ≈2.8 mm. Taking into account that the repetition rate of an NTPP is 1200 Hz, the average OD velocity is , which is in good agreement with the experimental value of (see Fig. 5).
If an OD propagates through a polymer coated RF2, then its path is made up practically only of ~180 µm long segments. In this case, there are almost no segments of undamaged optical fibers approximately δ≈1 mm in length between successive NTPP fingerprints (see Fig. 8). Therefore, the estimate of the average OD speed in this case is . This estimate is the same order of magnitude but is approximately two times higher than the experimental value of the OD average velocity, which in this case was 1 m/s. The reason for this discrepancy is still to be explained. It can only be noted that not all NTPPs reinitiate an OD. The relative number of “working” NTPPs is high and close to 100%. However in the case of coated fibers, the number of working NTPPs is about 10% less than in the case of uncoated ones.
In the present work, the effect of a catastrophic destruction of HCFs under the action of repetitively pulsed laser radiation was investigated. The dependence of the average OD propagation velocity on the properties of the medium around the silica structure of the HCF was discovered. The OD propagated through the fiber without the polymer coating at an average velocity approximately three times higher than through the same fiber but with polymer coating.
Based on the data that were obtained, we can propose the following physical picture of the OD motion under the action of a repetitively pulsed laser that generates NTPPs with a time interval on the order of 1 ms or more. Under the powerful picosecond pulses, a laser-supported detonation wave moves in a gas along the hollow core of the fiber. After the end of the PP, the SW continues to move along the fiber and attenuates, but when the next PP arrives, its amplitude is high enough to initiate a new LSDW. At the end of the NTPP, the OD plasma is no longer maintained by laser radiation and extinguishes. Under the action of the following NTPP, the formation of an absorbing plasma occurs, as follows.
- a) The initiation of a new OD occurs at the destroyed capillaries region closest to the laser. This happens when the OD propagates through an RF covered with a material with a specific acoustic impedance comparable with the silica. Reinitiation is not observed in approximately ten percent of NTPPs. This means that not all NTPPs contribute to the motion of the OD.
- b) The initiation of a new OD occurs at small (~10 nm) particles of SiO2 moving through the hollow core with air disturbed by the shock wave. This takes place when the RF is surrounded by an ambient material with a much lower specific acoustic impedance than silica (in our experiments, it was air). This results in the fact that the position of the repeat OD initiation is shifted in the direction of the laser by ~1 mm, compared to case a). In addition, the process of reinitiation of the OD at destroyed capillary regions by a new NTPP as in case a) described above is still possible. All of this leads to an increase in the observed average velocity of OD propagation.
Russian Foundation for Basic Research (Project No. 18-02-00324); Presidium RAS Program No. 5: Photonic technologies in probing inhomogeneous media and biological objects.
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