We report the development of a novel measurement system designed to measure bubble properties in bubble curtains (i.e. planar bubble plumes) in situ alongside acoustical measurements. Our approach is based on electrical, contact-based needle sensors in combination with an optical system. The latter is used for calibration and validation purposes. Correcting for the insensitive distance of the needle tips yields very good agreement between the two approaches in terms of the local void fraction and bubble size distributions. Finally, the system is employed to study bubble plumes evolving from three different hose types. All hoses display consistent self-similar behaviour with spreading rates increasing with increasing gas flow. The spreading is further found to be significantly higher when the bubble plumes originated from a porous hose compared to the two other hose types featuring either discrete holes or nozzle elements. Bubble curtains are widely employed across diverse engineering applications such as controlling the movement of fish  The images provide a way to independently verify the results from the electrical sensors — seeing is believing. (ii) The optical measurements allow us to obtain the input required for the sensor calibrations under exactly the same conditions. This greatly improves the robustness and accuracy of the method. (iii) The camera recordings provide a means to gauge to what extent very small bubbles, which may not be picked up by the electrical sensors, are of relevance in a particular configuration. This is an important aspect in interpreting the sensor results, that would otherwise be unaccounted for.As mentioned previously, typical void fractions (in the per cent range) and lateral dimensions (on the order of metres) do not lend themselves to the application of optical techniques if the camera is to be placed outside the bubble plume. Immersing the optical system inside the bubbly flow region can mitigate some of the issues. However, doing so introduces disturbances and typically covers but a small region of the flow per camera unit. In order to capture the bubble distribution across the full height and width of the bubble curtain, conductivity-based electrical sensors are therefore the most appealing choice. Even though each individual sensor only represents a point measurement, their low cost and simple operation render it feasible to combine many of them into a larger array to achieve a high spatial resolution. For our purpose, we opted to complement the electrical sensors with an underwater camera that simultaneously captures a part of the bubble field in the centre of the curtain. Doing so has three main benefits: The electrical probes are made of coated stainless steel acupuncture needles. The acupuncture needles have a diameter of mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimA potential  is applied between the needles and a common electrode in the water and each electrical probe is connected to a comparator circuit. Gas encapsulating the tip of the probe increases the resistance between the tip of the probe and the common electrode resulting in a switch of the output signal of the comparator. It is most straightforward to drive the circuits in direct current (DC) mode, and this is used for the results presented here. We noticed, however, that this leads to the build-up of tarnish on the tip of the probe, which significantly hampers operation and eventually results in an insulated tip. It is possible to restore functionality by cleaning between experiments, as has been done for the results reported here. Yet, a more elegant and sustainable solution is, to operate the circuits in alternating current (AC) mode. Doing so proved to effectively minimize the impact of tarnish on the measurement results. A diagram of a circuit suitable for both DC as well as for AC operation  is shown in appendix A.The complete system allows for simultaneous operation of up to 240 needles across 6 PCB’s covering a range of 1.8 m in total  at MARIN (MAritime Research Institute Netherlands). The CB measures LxWxD: 220x4x3.6 i is the average of the signal in timeT is the averaging period. Subscript n is used to indicate results from the needle measurements, and the overline denotes time averaging throughout the paper.The output signal of an individual probe, T, since for every pierced bubble a small part is not measured. Provided that i. The relevancy of the correction term is demonstrated based on our results in Sect. Due to the finite lag distance mentclass2pt{minimThe probes measure a chord length distribution i et al. . The maii-th bubble appears with a probabilitya and b denoting the major and minor axes, respectively), which is given empirically in the form a in the case of Besagni et al. (a).The algorithm assumes that the distribution of the volume equivalent bubble diameter l et al. , who shoi et al.  who empli et al. , while oN bubbles can therefore be constructed by summing the chord length contributions of the N individual bubbles  individual bubbles contained in it and dividing it by the volume associated with that image, such thatc’ denoting quantities obtained from the camera images. Note that in line with the discussion in Sect. a, b determined by the ellipse fits.The void fraction averaged across the Here, we present sample measurements to cross-validate camera and needle results and to illustrate the capabilities of the new system. To enable a meaningful comparison between camera and needle results, we consider a case with a moderate void fraction. This ensures that uncertainties in the image analysis due to overlaps remain limited, while the number of bubbles is high enough to consider the mixture locally homogeneous. This trade-off was met best for a case with an air flow rate of The void fraction is the most basic quantity extracted from the measurements. Results measured via the camera are shown in Fig.i et al. . For refi et al.  results An appropriate reference to the camera results for the void fraction is the spatial average see Fig. b. A direAs described in Sect. i et al. :12\documV, while the variation of We note, however, that the results for and Deen  proposedmentclasspt{minimaR between R could be computed. Moreover, this value for With V and heights z for both the porous hose and for the holes hose. For the other cases, no comparison to camera results was possible, either due to the high void fraction . Especially in cases exhibiting stronger motion, the bubble curtain deformed into an S-shape across the basin.Finally, we present two types of temporal averages over the entire measurement time of 120 s in Fig ase Fig. g, where ase Fig. i, the pe Milgram  already,V and for other hose types very well (not shown). In Fig. In Fig. mentclasspt{minimamentclasspt{minimaV. This trend as well as the magnitude of z is reasonably approximated by a 1/z dependence for these data, again consistent with the self-similar scaling for this quantity. The inset of Fig. V due to limited statistical convergence. It appears clear, however, that the relevance of centreline movement increases with increasing height, and presumably also with increasing V.The spreading parameter mentclasspt{minima Ditmars  for plan Ditmars  and Milg Ditmars . Finallymentclasspt{minimamentclasspt{minimaFinally, the spreading of the bubble plumes created with the three different hoses is compared in Fig. see Fig. , while dsee Fig.  who alsoV do not align very well between the two data sets. This is also reflected in the fact that the fit in Eq. The only reference data on the spreading of planar bubble plumes known to us are by Kobus . Their ain Kobus  to the te et al.  based one et al. ), which e et al.  who sugg Milgram . We compby Kobus  for threstiansen , but theof Kobus  fall intin situ. A key feature of our approach is the combination of a contact-based sensor array with an optical system. This enables the validation of the needle-based measurements and allows us to calibrate the aspect ratio dependence In summary, we report the development of a system capable of measuring the void fraction and the bubble size distribution in bubble curtains The system is then employed to measure the characteristics of bubble plumes originating from three different hose types. Our results show that especially at higher flowrates and further from the hose, it is critical to account for the meandering of the plume in order to obtain representative distributions as an input to an acoustic model. This can be achieved by conditioning on the instantaneous centre. We find that for all three hose types, such conditioned void fraction distributions evolve in a self-similar manner up to the vicinity of the free surface. The spreading rates increase with increasing gas flow rate, which is consistent with previous reports on round Milgram  and planu et al.  have shou et al.  in the rWith the proposed measurement technique a data set with the hydrodynamical properties of different bubble curtains can be constructed. Future research should extend the use of this system to longer measurement times and larger arrays and focus on combining the measurements with acoustical measurements.


We focused the generated beams using a 1 m focal length lens and we investigated their intensity profile and the polarization state in the focal plane of the focusing element. We showed that the achieved intensity profile can be controlled by changing the polarization state of the beam incident on the crystal. We observed that, for a particular configuration of the system, the initial circular symmetry of the beam is no longer preserved. The experiments were performed with Ti:Sapphire lasers that were operated in continuous wave mode, as well as in pulsed regime. The optical system presented here ensures a convenient way to generate a great variety of vector vortex beams and it is expected to be of interest for applications that use low and high-power laser sources, such as STED microscopy, light trapping, or material processing.An optical system comprising a  Particular types of structured light fields are vortex beams. They are characterized by a doughnut (or ring-like) transversal intensity profile with zero intensity in the centre and a phase that changes by an integer multiple m of 2π along a contour around that zero-intensity point. The phase structure enables the beam to carry an amount mħ of orbital angular momentum (OAM) per photon, where m is referred to as topological charge and ħ represents the reduced Planck's constant. The polarization state of such beams is homogeneous. If the polarization state is space-variant across the transversal section, the vortex beam is commonly named vector vortex beam (VVB). This is usually described as a collinear superposition of two complex conjugated vortex modes with orthogonal polarization states. On the other hand, the superposition of multiple modes that are not complex conjugated generates a so-called hybrid vector beam (HVB), which has some interesting properties not observed in the case of VVBs3. VVBs are increasingly more relevant for today’s emerging fields such as STED microscopy4, quantum comunication5, material processing7, or optical manipulation8.Structured light is referring to the configurations of the electromagnetic field with non-trivial variations in phase, amplitude, or polarization state11. Despite the advantages of these techniques, like low cost, high power, and high efficiency, the systems could be deemed rigid in terms of the assortment of the generated VVBs. Other methods make use of specially designed optical elements with spatially variant polarization properties placed outside of the laser cavities. Such elements are patterned liquid crystal retarders (q-plates)12, birefringent plates written with sub-wavelength gratings13, metasurfaces14, Brewster angle reflectors15 or anisotropic films16. Some alternative methods make use of interferometric optical systems that combine two beams with orthogonal polarization states generated either using spiral phase plates (SPP)17 or spatial light modulators (SLM)18. Method such as that described by Liu et al.18 can be also used for the generation of HVB as the phase can be independently controlled for each arm of the interferometer. However, SLM-s are characterized by low efficiency due to diffraction losses. The approaches using combinations of diffractive or geometrical optical elements are much simpler and more efficient20. Quiceno-Moreno et al.21 used two q-plates placed in line, one having tunable retardance, for the generation and the investigation of HVBs in both near and far-field.Due to the remarkable characteristics of VVBs and HVBs but also to their considerable applicability potential, plenty of generation techniques has been proposed. Some of them regard the VVBs generation by direct emission of laser beams with the desired optical vortex phase inside a laser cavityc-cut crystals was explored in references27. In the experimental works, the uniaxial crystal was illuminated either by Gaussian26 or Bessel beams29. The advantage of using Bessel beams is given by the possibility to achieve any value of polarization conversion efficiency up to 100% compared to the case of Gaussian beams where the maximum efficiency conversion is 50%. To assist practical applications we aimed to develop a simple, reliable, and efficient system that could be easily adapted to any laser source to achieve a variety of vortex beams mixtures.Polarization conversion that is due to the propagation of a beam along the anisotropy axis of a uniaxial crystal could also be an alternative to the methods mentioned above. The possibility to generate vortex beams relying on polarization conversion in 26 is the only one that describes an optical system illuminated by a divergent laser beam, but only for the generation of single- and double-charge vortex beams. It is emphasized that in this way the possibility of self-focusing is avoided and thus this method is appropriate for femtosecond (fs) high power laser applications, while also being cheaper and less sensitive to wavelength variation than using diffractive optics. Despite these advantages, the method is limited by the polarization conversion efficiency. Loussert et al.30 have demonstrated that 100% efficiency can be achieved using a single piece of uniaxial material illuminated by a nearly conical superposition of plane-waves. In addition, a collimated input beam on an optical system consisting of an SPP and a uniaxial crystal with a special shape has been theoretically proven to generate HVBs31. Using uniaxial crystals to generate VVBs and HVBs may have some advantages over the other methods, for instance, the possibility to achieve conversion without losses and also to use high power fs pulsed beams. However, in most of the aforementioned research regarding polarization conversion, the exploration of the polarization state and the intensity profile was done for a collimated beam. The effects of subsequent focusing of the beam have not been explored.In most of the studies that describe the dynamics of polarization conversion in uniaxial crystals, the techniques rely on focusing the input laser beam inside the uniaxial crystal or close to its surface. However, this method makes almost impossible the use of high-power light sources. The paper of Shvedov et al.c-cut uniaxial crystal placed between two axicons and illuminated by a Gaussian or a Laguerre–Gauss mode beam. The advantages of using this system are given by: (i) yielding a conversion efficiency of up to 100% in relation to the thickness of the uniaxial crystal and the features of the axicons; (ii) achieving of VVBs and HVBs by varying the input beam characteristics like spatial distribution and polarization; (iii) can be used for a wide range of beam power conditions. The Sapphire crystal was chosen because of its hardness and ability to withstand mechanical and thermal shocks compared to other used crystals, such as Calcite, or potassium dihydrogen phosphate (KDP) that, in addition, are hygroscopic; furthermore, the Sapphire crystal could be fabricated at large enough dimensions. We theoretically describe the phenomenon of polarization conversion in the c-cut uniaxial crystal using Jones matrices and we simulate the beam propagation through the system. One of the most intriguing phenomena that have been observed for the linearly polarized Laguerre–Gauss beam incident on the optical system is the breaking of the circular symmetry as the beam propagates to the lens’s focal plane. This behaviour is theoretically explained in one of our previous work31 as well as in Quiceno-Moreno et al.21 to be caused by the Gouy phase shift associated with the propagation of Laguerre–Gauss modes to the focal plane of a lens.In this study, we investigated, in the focal plane of a lens, the polarization state and the transverse intensity distribution of VVBs or HVBs. Such beams are produced as a result of the overlapping of the vortex modes generated by an optical system comprising a Sapphire c-cut uniaxial crystal  oriented with the crystal axis parallel to the propagation direction of the beam . It is well known that a plane-wave passing through a uniaxial material will experience: (i) an ordinary refractive index,, if the initial direction of the electric field before the crystal is parallel to the plane defined by the plane-wave's propagation direction and the crystal axis, and in this case, the polarization state could be considered radial. Otherwise, the initial polarization will not be preserved during the propagation of the beam through the uniaxial crystal. The change of polarization is determined by the phase difference acquired between the ordinary and the extraordinary waves and by the orientation of the extraordinary axis.In this section, we will briefly discuss the generation of VVBs and HVBs by polarization conversion in uniaxial crystals. Let's consider a c-cut uniaxial crystal placed between them. This part solely will be discussed in this section. The first axicon is used to produce a divergent beam with a conical wavefront incident on the crystal. The tilt angle of the rays inside the crystal , and the interface between air and the uniaxial crystal, considering a ray that is initially parallel to the optical axis. Herein nThe beam resulting after the refraction from the first axicon is a circular superposition of plane-waves having a fixed tilt with respect to the optical axis; hence, all of the waves experience the same extraordinary refractive index. In this particular configuration, the orientation of the direction of extraordinary polarization is always radially with regard to the optical axis. As a consequence of this feature, the change in polarization is accompanied by a transverse mode conversion into a vortex mode. The amount of OAM resulting from this transition balances the change of spin angular momentum (SAM) due to the polarization conversion. This phenomenon is known as optical spin-orbital angular momentum coupling or polarization conversion in uniaxial crystals. The second axicon is used to recollimate the beam.Let’s consider in the following that an optical beam with linear polarization passes through the optical system. The propagation through the uniaxial crystal changes the polarization, intensity, and phase of the beam. The ordinary and the effective extraordinary refractive indexes are very similar, which means that is not a significant spatial walk-off between the waves. In this instance, we can consider that the intensity profile of both ordinary/extraordinary waves changes in an identical manner with the propagation through the crystal. To find a simplified mathematical description of the investigated phenomenon, we should factor out the radial dependence. We will keep only the azimuthal dependence of the phase. In this case, we can associate the crystal with an axially rotated waveplate, although the transformation produced is much more complex. However, for the numerical simulations, we have considered the effect of the propagation through the crystal on the intensity, phase, and polarization of the beam.32 in uniaxial crystals since they have a well-defined refractive index; therefore they do not change their structure up to the scale transformation due to diffraction. The refractive index experienced by the radial mode is the effective extraordinary refractive index Linear polarization could be described as a linear combination of two orthogonal circularly polarized states entclass1pt{minimaWe would like to point out that the following analysis is proper for plane-waves, but it is also valid for rays. As long as the beam is convergent/divergent, we can associate a ray having the same direction as each of the plane-waves that form the beam. In the small birefringence regime and for the propagation of the beam close to the optical axis, we can trace the intensity of the beam along the ray. The phase changes will be determined by the optical path length between the input and the output planes. For the radially and azimuthally polarized modes we have well determined refractive indexes and we can easily model the propagation through the crystal as a multiplication by a phase factor, as given in Eq. .3\documeBy replacing the expression for the amplitude of the radial/ azimuthal polarized modes as given in Eq. , we findThe polarization state of the exit beam expressed using right/left circularly polarized states results by applying the matrix over Eq. . In Eq. over Eq.  we have The matrix from Eq.  can be wT in Eq. . The conT in Eq. .7\documeL, wavelength λ, and rays tilt inside the uniaxial crystal π, the crystal matches the half-waveplate condition and the polarization conversion efficiency is η = 100%. In this case, the resulting beam at the output of the crystal has an annular transversal intensity profile with a polarization state that depends on the input polarization. The initial right-circularly polarized light will be transformed into left-handed circular; conversely, if the initial polarization is left-circularly, then the generated mode will be right-circularly polarized. The conservation of the angular momenta determines that a vortex mode with a double topological charge, m =  ± 2 having an inhomogeneous polarization state. The amount of energy that is converted will be determined by the thickness of the crystal L and the tilt angle, For given values of the ordinary, mentclass2pt{minimBy adding an SPP in the system, which will change the topological charge of the composing modes by 1 or − 1, we can achieve the vortex mode composition ± 1 and 3 or ∓ 1 and − 3, respectively. In this case, the intensity profile of the resulting beam will be affected by the interference of the composing beams.The experimental setup used to generate the superposition of vortex beams and to investigate their polarization state is presented in Fig. m =  − 1. The lens L1 brings the beam to the far-field where the intensity profile has the typical doughnut-like shape. The resulting uniform polarized vortex beam is expanded and collimated by the telescope formed of L2 and L3 lenses. Then, the beam illuminates the VBS, which adds or decreases the topological charge by 2, leading to a superposition of vortex beams. The weight of each vortex beam depends on the crystal thickness and on the incoming laser beam polarization that is controlled by the PCS. To investigate the polarization state of the generated VVBs and HVBs in the far-field region, intensity distributions of the beams were acquired in the focal plane of the lens L4 with a CMOS camera. The PAS was used to determine the polarization of the beam from multiple experimentally acquired intensity profiles.Experiments were performed with and without SPPS within the system in order to generate various VVBs and HVBs. Considering the presence of the SPPS, a Gaussian laser beam first passes through the SPPS which generates a vortex beam with uniform polarization and topological charge 4. The resulting beam profiles are studied in the focal plane of the lens L4.The numerical simulations, up to the VBS, were done considering the polarization noninteracting with the physical propagation of the beam because optical elements from the PCS change the polarization in the same way on the entire cross-section of the beam. Otherwise, the polarization state of the beam right before the first axicon is decomposed in radial/azimuthal polarization states and these modes are traced through the system considering the effective extraordinary respective ordinary refractive index for the crystal. At the output of the VBS, the two modes are recomposed and propagated to the lens L33 (SPW) method for the near-field. In addition, the propagation through the optical components is evaluated numerically using geometrical optics, by tracing the intensity and phase along the ray with a correction factor for the intensity according to the change in ray density between the input and the output planes.All the wave-plates, including the VHWP, are ideal components modelled using Jones matrices. The diffraction of the beam was computed using a combination of Fresnel integral for the far-field region, and the Spectrum of Plane-Waves4, we have used a variation of the Debye-Wolf diffraction integral (DW). Our implementation is different from the conventional DW regarding the way we evaluate the wave-front aberrations, which are normally determined from the deviation of the wavefront from a reference sphere centred on the focal point. Having a plane wavefront before an ideal focusing element, all the rays converge to one point and the optical path length is equal for all the rays. If optical aberrations are present, we expect that the optical path length would not be the same for all rays, and the rays do not intersect at the same point. Every point source generates a spherical wave, however for a well-designed system the focal spot is much smaller than the curvature radius of such a wave; thus, we can approximate it as a plane-wave. We can associate a plane-wave to any point source on the plane tangent to the focusing element and a ray on which the plane-wave is centred. Our approach is to evaluate the wavefront aberration as the following quantity:k.For the propagation of the beam to the focal plane of the lens LS denotes the optical path length along the ray starting from a plane before the focusing element and the focal plane and corresponding to the plane-wave indexed by k, rs is the coordinate of intersection between the ray and the focal plane, and k0 = 2π/λ. The quantity Ψ(k) is equal to the phase acquired by a plane wave indexed by k between the plane tangent to the rear surface of the lens and the focal plane plus the phase acquired by the ray indexed k due to the propagation through the lens up to the same plane tangent to the lens.. The spherical phase is The electric field in the focal plane is given by the equation:34, where kz is the longitudinal wave-vector of the plane-wave, θ is the azimuthal angle around the optical axis.The matrix:In the experiments, we have used a Ti:Sapphire oscillator that was operated in continuous-wave, as well as a high-power laser system that generates pulses of up to 2 mJ. The intensity distribution of the beams generated by each laser source is of Gaussian type.2 and L3 lenses. On the other hand, the diameter of the laser beam delivered by the high-power laser system is 12 mm, large enough not to be aberrated by the tip of the axicons, the reason for which the lenses have been removed when this beam was used.The rounded tip of the first axicon from VBS will induce aberrations in the central part of a beam up to 1 mm from the optical axis, thus we needed a reasonably large beam incident on it. For this reason, the beam from the oscillator, which has a diameter of 1.7 mm, is magnified with the help of LThe Sapphire crystals from VBS were purchased for thickness values for which the polarization conversion efficiency reaches 50% (2.65 mm) and 100% (5.3 mm), respectively. Following the measurements, it was observed that the deviation from these values is in the thickness error of 0.1 mm given by the manufacturer, which might have a small influence on the obtained results.1, S2, and S3 for the polarization state of the beam incident on the uniaxial crystal. Orthogonal beams occupy points at opposite ends of the sphere diameter. To the north and south poles are represented the VVB or the HVB corresponding to right- and left-circularly input polarization states, respectively, while any VVB or HVB along the equator line is generated by the linear polarization before the VBS. All the elliptically input polarization states will generate VVBs or HVBs with intermediary configurations.The weight of the composing modes of the VVBs or HVBs generated using the system shown in Fig. m =  ± 2 having variable weights. Positive or negative double charge vortex beams with homogeneous orthogonal polarization states could be generated if the polarization of the incident laser beam on the VBS is left or right-handed circularly. The transition between these two vortex beams is achieved by transforming the input circular polarization into elliptical polarization and obtaining various intermediate configurations of VVBs. The energy contained in one of the vortex modes decreases, down to zero value, by the same amount as it increases in the other mode. The intensity distribution of the VVBs in the focal plane of the lens L4 is given by the sum of the intensities of the composing modes. Each VVB has an elliptical polarization state with the same handedness across the whole aperture, given by the most intense generated mode, and with the orientation of the ellipse rotating linearly with the azimuthal angle by 4π along a closed contour around the singularity. When the input laser beam is linearly polarized, the generated vortex modes have the same intensity. Hence, the superposed right- and left-handed circular polarized modes will provide an overall linear polarization state with a spatially variable direction. The intensity and polarization distribution for these cases are shown in Fig. We will start by analyzing the simplest case of the polarization conversion in our system, namely for the Sapphire crystal with a thickness of 5.44 mm illuminated by a Gaussian beam. According to Fig. 0) is shown using false colours. The angle of orientation of the major semi-axis of the polarization ellipse and the degree of ellipticity were calculated from the Stokes parameters. Ellipses were rendered for an intensity threshold above 0.3 from the maximum value. For the simulations, the polarization state is indicated using ellipses that correspond to the electric field calculated using the formula provided in Eq. , has a “doughnut” intensity distribution specific to the topological charge. The local polarization state is linear. It is not similar to that for m =  ± 2 because the orientations of ellipses differ, from a topological charge value to another, due to the phase difference between the modes with right- and left-handed circular polarization. On row (iii) is highlighted a focal spot that has a two-lobe configuration. The HVB is a mixture of four vortex modes in the context in which the initial linear polarization is decomposed into right- and left-handed circular polarization. Thus, we will have two modes with m =  ± 1 with left-handed circular polarization and two modes with m =  − 1 and − 3 with right-handed circular polarization. The remarkable behaviour of breaking the circular symmetry of the intensity distribution is caused by the interference between the mode m =  − 3 and the mode m =  − 1, as their polarization states are not orthogonal, and the m =  − 3 mode acquires an additional amount π of Gouy phase due to the propagation to the focal plane.On row (i), Fig. η = 50%. We consider an initial beam with a diagonal polarization state at 45° between vertical and horizontal directions, identified by the state m =  − 1 induces a vortex spatial phase To illustrate the effect of the Gouy phase, we present below a simplified mathematical model. We will consider that an operator n in Eq. . The effmentclasspt{minimam =  − 3 explains the breaking of the circular symmetry. A simple theoretical analysis using the Debye-Wolf integral will show that a linear polarized beam strongly focused tends to produce an elliptical spot that lacks circular symmetry despite that the initial profile of the beam is circularly symmetric; for instance, see Fig. 37. The experimental work by Kotlyar et al.38, reported a similar phenomenon of circular symmetry breaking of the focal spot generated from a Gaussian beam with a very small waist radius  as well as by reproducibility of the results.As laser sources, we used a Ti:Sapphire oscillator from FemtoLasers that operates in free-running regime at 780 nm and a high-power laser system (Legend Elite HE+) that provides laser pulses at 1 kHz repetition rate with a central wavelength λ = 800 nm and a pulse duration of 35 fs.40). A simple analysis using Jones matrices will reveal that illuminating the VHWP with circularly polarized beam results in a vortex beam with uniform polarization, orthogonal to the initial one. If the VHWP is inserted between quarter-wave plates, the components thus formed will transform a linearly polarized Gaussian beam into a linearly polarized Laguerre–Gauss beam with topological charge As we stated in the experimental setup subheading, the SPPS consists of a VHWP placed in between two QWPs. The VHWP is made from a Liquid Crystal Polymer  for which the fast axis varies azimuthally. For this reason, the VHWP introduces a constant phase difference between the components of the optical field for which the polarization has an ordinary/extraordinary orientation. For example, in the case of the VHWP plate used in experiments, the polarization state of the resulting optical beam may be radial or azimuthal if the element is illuminated with a linearly polarized laser beam. It can be considered that VHWP independently modifies the components of the beam with right-circular polarization and left-circular polarization, realizing the exchange between the angular momentum due to the circular polarization and the angular momentum due to the phase structure of the beam (spin-angular momentum couplingL4 with the camera for each 6-degree rotation of the QWP from the PAS, more exactly from 0° to 180°. The images were used to determine the Stokes parameters for each pixel as described by Schaefer et al.41.The camera used in the experiment is a Basler acA4112-20um with a spatial resolution of 4096 × 3000 pixels. For the investigation of the polarization state, we have acquired images of the intensity profile in the focal plane of the lens


Very often a manufacturing process is followed by some surface treatment. Such a process induces residual stress into the manufactured component. Compressive residual stress is desirable for enhancing the fatigue properties of the component. The residual stress is often measured only at the surface, if at all. However, residual stress is equilibrating in the whole component. Therefore, compressive residual stress at the surface induces undesirable tensile stress inside in the component. Knowledge of the residual stress distribution in a body can be very useful in engineering applications. The authors found this knowledge necessary for a proper description of fatigue crack propagation in railway axles described in the original paper •More accurate procedure for a residual stress determination in cylindrical bodies.The presented paper aims to describe a procedure developed for proper residual stress determination. The procedure can be easily used e.g. in R&D centers, where X-ray diffraction residual stress measurement is already in use. The procedure is suitable for the residual stress determination in sizable cylindrical bodies or components, e.g. railway axles. It uses X-ray diffraction residual stress surface measurement and layer removal by machining. Results experimentally obtained are corrected by a general procedure developed in MATLAB software in order to obtain the original residual stress state in the cylindrical body.  Specifications tableResidual stress is an elastic response 1.Residual stress is measured by X-ray diffraction on the surface of a cylindrical body. Both tangential and axial residual stress are recorded.2.Then, a surface material layer is removed from the cylindrical body by turning.3.Residual stress is measured again on the newly created surface .4.Steps 2 and 3 are repeated until only a small part of the original body remains.Note that thicknesses of the removed layers do not have to be uniform during the whole experiment. However, the typical thickness of the removed layer ranges from 1 to 5 mm for sizable components. It is generally better to plan a finer spacing in parts where larger gradients of residual stress can be expected.The iterative evaluation method is designed for the correction of residual stress measurement by X-ray diffraction on cylindrical components. The measurement itself must be carried out in the following steps:Technical details of the X-ray diffraction technique are out of the scope of this paper. However, it should be mentioned that a certain preparation of the surface before the X-ray diffraction measurement itself is necessary. The machining itself produces residual stress, which should be removed by etching the machined surface. Typically, a layer of tens or hundreds of micrometers is removed to avoid the influence of machining on obtained values of residual stress.1.The measurement points should be at least one diameter away from the free end of the component – according to Saint-Venant's principle 2.Residual stress distribution in the investigated part should be axisymmetric (the method described here is unsuitable for other cases).3Even the measured residual stress is zero, the residual stress measurement must be carried out in the whole cross-section and not only near the original outer surface.The following conditions should be met when performing the X-ray diffraction measurement on a sizable cylindrical component to achieve the best results:r). However, these values are not the correct values of the original residual stress distribution. They were measured after disrupting the original state of equilibrium in the part by removing a layer of the material with the presence of residual stress . These experimental results must be corrected by the numerical procedure. The correction procedure by the iterative evaluation method is described here.After the experiment, there is a set of values of residual stress  that were measured in certain positions in the cylinder , and the i subscript corresponds to the radius of the evaluated surface .The iterative procedure is based on repeatedly calculated values of an unbalance 1.inr = 0 mm.A theoretical value of 0 MPa is added to the measured values for the inner radius, assuming that after removing all material , 0 MPa, would be measured. This value is added to estimate residual stress in the region, followed by the final physically possible measurement. If there is no hole in the cylinder (solid cylinder), 0 MPa value corresponds to radius 2.The linear residual stress distribution is assumed between each measured point.Two assumptions are made at the beginning of the determination process:The whole procedure was programmed as a MATLAB function. The determination process itself runs in two nested cycles. The outer cycle determines the portion of layers for the inner cycle to work on, starting from the innermost layer and adding one layer in each step. The inner cycle evaluates the force equilibrium condition for the current portion of layers. It is important to note that the layers must correspond precisely to the removed layers when performing the measurements.1st iterationinr to r3), which is bounded by the 0 MPa at inr and the last measured point at r3. Then, the inner circle works on this interval only. It calculates the value of unbalance U, and then shifts the values of stress by the calculated unbalance and calculates the unbalance again until it reaches the point when the U < 0.1%  of the first unbalance (this condition is referred to as the equilibrium condition in further text). However, the measured value at r3 cannot be changed at this point. It is regarded as the boundary condition. The only value that can be shifted is the value of 0 MPa. The equilibrium equation for the first iteration in the case of axial residual stress evaluation is:r3 is the radius of the last measured point and inr is zero for the solid cylinder and inner radius for the hollow cylinder. To consider the linear stress distribution between the measured points, the area corresponding to the evaluated point must be equal to half of the current layer thickness. The determination starts from the innermost layer .The next layer is added to the innermost layer to form the next interval for the equilibrium search. In the 2nd iteration, the interval is from nth iterationIn every following iteration, the process of adding the next layer and searching for the equilibrium by shifting the previously calculated values is repeated until every measured depth is used and the final equilibrium state is obtained.The procedure of the iterative evaluation method is illustrated in this article using a fictive set of measured values. The values were chosen arbitrarily. The shape of the distribution is based on typical residual stress distribution in hardened railway axles. A railway axle is a typical example of a bulky cylindrical component, where it is required to describe the residual stress accurately.The showcase considers a cylindrical component with an outer diameter of 200 mm and an inner diameter of 100 mm with sufficient length according to the first experimental assumption. The measurements were considered with steps (thickness of machined layers) of 1 mm from 0 to 6 mm depth, with steps of 2 mm from 6 to 20 mm depth, and with steps of 5 mm from 20 to 45 mm depth. Together it represents 18 removed layers, i.e., measured points. The imaginary residual stress measurements are summarized in Table 1 , and Table 1  contains all the data presented in this paper. Measurement points can be used for the validation in case of own procedure development.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


The factors shaping the composition of the tree mycobiome are still under investigation. We tested the effects of host genotype, site, host phenotypic traits, and air fungal spore communities on the assembly of the fungi inhabiting Norway spruce needles. We used Norway spruce clones and spore traps within the collection sites and characterized both needle and air mycobiome communities by high‐throughput sequencing of the ITS2 region. The composition of the needle mycobiome differed between Norway spruce clones, and clones with high genetic similarity had a more similar mycobiome. The needle mycobiome also varied across sites and was associated with the composition of the local air mycobiome and climate. Phenotypic traits such as diameter at breast height or crown health influenced the needle mycobiome to a lesser extent than host genotype and air mycobiome. Altogether, our results suggest that the needle mycobiome is mainly driven by the host genotype in combination with the composition of the local air spore communities. Our work highlights the role of host intraspecific variation in shaping the mycobiome of trees and provides new insights on the ecological processes structuring fungal communities inhabiting woody plants. We hypothesized that (i) the needle mycobiome is genetically determined and that (ii) the effect of the host genotype on the needle mycobiome depends on the composition of the local air mycobiome.The aim of this study was to understand the ecological mechanisms contributing to the assembly of the tree mycobiome. For this, we determined the relative effect of host genotype, site, host phenotypic traits, and composition of the air mycobiome on the assembly of the needle mycobiome of Norway spruce trees  that belong to the Southern Swedish Norway spruce breeding programme  and tested its association with the needle mycobiome. The composition of both the needle Table  and air P = 0.66) in contrast to when all clones were considered, suggesting that differences in tree diameter at breast height (dbh) across sites was one factor underlying the interaction between clone and site. The same approach could not be taken in the North–South pair of sites because the tree diameters were not overlapping. In the subset of five clones with similar diameter across the East–West sites, both the clone (P = 0.008) and the site (P = 0.001) had a significant effect on the composition of the mycobiome. Moreover, the composition of the mycobiome was associated with the genetic similarity between the clones, i.e. clones that with higher genetic similarity also displayed more similar mycobiome communities, compared with clones with lower genetic similarity  and in the East–West (P = 0.037).We found an interactive effect between clone and site on the composition of the needle mycobiome and therefore the effect of the clone could not be interpreted alone. We explored whether differences in crown health or diameter were involved in the interaction by selecting sets of clones with the same diameter or crown health between sites and testing if the interaction ‘clone × site’ was still significant after controlling for phenotypic variation. When we selected the five clones with the same diameter in the East–West pair of sites, the interaction ‘clone × site’ was no longer significant  and location of the tree on the alpha and beta diversity of the needle mycobiome within each of the four sites. For both alpha and beta diversity, the marginal amount of variation explained by the host genotype within each individual site was consistently higher than the variation explained by phenotypic traits, which was always close to zero in most sites and indexes . RegardiThe aim of this study was to understand the ecological mechanisms contributing to the assembly of the needle mycobiome of Norway spruce. We showed that host genotype and site were the main factors shaping the composition of the needle mycobiome. The host genotypic effect was associated with the genetic similarity between the Norway spruce clones. The site effect was related to differences in the composition of local air mycobiome. Moreover, the effect of the host genotype on the composition of the mycobiome was site dependent, and it was mainly driven by differences of tree size for the same clone across sites. Altogether, our results suggest that Norway spruce trees recruit their needle mycobiome based on factors comprised of their genotype, the available air mycobiome and the tree size.et al., et al., et al., et al., The composition of the needle mycobiome differed between Norway spruce clones for both pairs of sites. This host genotype effect was consistently larger than that of the site, and that of the measured phenotypic traits, such as dbh and crown health. We found that the crown health did not influence the diversity of the needle mycobiome. This result is in line with a previous work showing that trees with contrasting physiological state as a result of the infection of a plant pathogen had similar needle mycobiome . The effect of the genotype reported here is similar to that reported for the phyllosphere microbiome of some perennial  and when separating the OTUs at different trophic modes and dominant guilds. In both analyses, the amount of variation explained by the genotype was between 8% and 10% for both pairs of sites  and gITS (primers spore traps) is that gITS have one more base pair degeneration and amplify some species that fITS do not, e.g. fungi belonging to the genera et al.  breeding programme  were established in different years with grafted clonal material. Söregärde was planted in 2009, Runesten and Larslund between 2011 and 2012, and Gåtebo in 2012. The establishment of the sites was also done in a pairwise design  shared between the two sites in the North–South pair (Larslund and Söregärde), and another 15 Norway spruce clones shared in the East–West pair of sites (Gåtebo and Runesten). From each of these clones, we sampled 10 vegetatively propagated Rosvall,  trees ‘. The Noret al., et al., et al., et al.  based on an identity‐by‐state matrix (IBS) using PLINK 1.9 , previously soaked in 4× TE buffer . Sequences with an average quality score < 10, base average quality < 2, lacking the primer sites  or containing false combinations of used primer tags, i.e. tag jumps  using a threshold value of 0.5 (plausible classification)  was tested on the alpha and beta diversity of the needle mycobiome across sites. The analyses were done separately for the two pairs of sites. To control for the fact that trees had different sizes between sites, as a result of being planted at different years, the diameter values were standardized for each clone within sites prior to the analyses. For all diversity analyses a non‐rarefied OTU table was used, and the square root of the total number of reads per sample as the first factor in all models was included. By doing this, we explicitly accounted for the potential bias resulting from different sequencing depth instead of rarefying , i.e. effect size, following the formula ω2 = (SSQeffect − dfeffect × MSerror)/ found on http://onlinestatbook.com, where SSQ is the sum of squares, df is the degrees of freedom and MS is the mean squares. The ω2 values were grouped by the effect size by trophic modes and most dominant guilds. To account for the fact that some OTUs belonged to more than one guild, the guilds and trophic modes were converted to dummy variables and assigned as 1 (presence) or 0 (absence) values of each guild and trophic mode to each OTU  of the East–West pair, the effect of ‘clone × site’ interaction was no longer significant. Only on those five clones in East–West pair of sites were further tested whether their genetic similarity was associated with the composition of their mycobiome. For this, the scores of each clone on the first and second axis of the PCA displaying their genetic similarity were obtained and fitted on the ordinations of the needle mycobiome using the envfit function in the vegan package.To test whether differences in crown health or dbh were causing the interaction clone × site, a set of clones with the same diameter or crown health status between sites were selected and tested if the interaction clone × site was still significant. We found that when considering clones with the same diameter between sites (www.smhi.se). These data were included as covariables in a global analysis of the composition of the needle mycobiome across all four sites. PERMANOVA analysis was performed for climate and weather separately.To test to which extent differences in climate and weather across sites influenced the composition of the needle mycobiome, we collected values of mean yearly air temperature and total yearly precipitation of the last 20 years  and also of the year before the sampling  from the closest active climatic station to each of the sites  using the variable ‘origin’, i.e., either needles or spore traps, as explanatory. This analysis was performed using both Bray–Curtis and Jaccard index.The air mycobiome was subdivided by month for May, June and July to identify which month had the highest similarity to the needle mycobiome. A principal coordinate analysis was performed for each site including needle samples and spore trap samples. The similarity between needle and spore trap communities was obtained for each month and site based on the et al., et al., Within each site, the contribution of host genotype, phenotype and relative location of the tree in the site  was tested based on alpha and beta diversity. The spatial descriptors, i.e. the distance‐based Moran eigenvector maps (dbMEM), were obtained using the adespatial package in the R environment (Dray Appendix S1: Supplementary InformationClick here for additional data file.Appendix S2: Table S2Click here for additional data file.


In this paper, we consider an isothermal glass tube drawing model consisting of three coupled nonlinear partial differential equations. The steady-state solution of this model is required in order to investigate its stability. With the given initial and boundary conditions, it is not possible to determine an analytical solution of this model. The difficulty lies in determining the constants of integrations while solving the second order ordinary differential equation analytically appearing in the steady-state model. To overcome this difficulty, we present a numerical based approach for the first time to develop an analytical solution of the steady-state isothermal tube drawing model. We use a numerical technique called shooting method to convert the boundary value problem into a set of initial value problems. Once the model has been converted into a system of differential equations with initial values, an integrating technique is implemented to develop the analytical solution. The computed analytical solution is then compared with the numerical solution to better understand the accuracy of obtained solution with necessary discussions. In Danner process, glass is melted in a furnace to the stage where it is soft and pliable. Molten glass is then let to fall with low feeding speed L. It is then pulled out by a drawing machine with a drawing speed 3. For the Vello process, we refer to5.In glass industry, tubes are drawn through various manufacturing processes used to achieve the continuous production of glass tubes having correct wall thickness and diameter. The most commonly used are the Danner process and the Vello processL. Inner and outer radii, inside pressure, feeding and take up speeds, axis of symmetry of the tube during the production process are shown in Figure. All other parameters involved along with their numerical values are illustrated in Table The geometry of the drawn tube is illustrated in Fig. The shaping parameters such as the wall thickness and cross-sectional area (or diameter) are the main characterizations of the drawn tube. In either of the manufacturing processes, the required shape of the tube can be maintained by the stream of the air gently blown through the mandrel. Insufficient quantity of the air blown through the mandrel can disturb the desired shape of the glass tube. As a result, this insufficient balance collapse the walls of the glass tube. Moreover, the geometry of the glass tube can be controlled by the parameters involved herein such as the glass temperature, the composition of the raw material, the pressure of the blowing air in the mandrel and the rate of draw. We have added a Fig. 4 and11. In7 numerical solutions of both the isothermal and non-isothermal models have been found and used to optimally control the geometry of the glass tube. In recent years, a variety of mathematical models representing physical and real world problems have been investigated for numerical solutions and stability analysis (for example see24). Thus, the concept of finding numerically accurate and exact solutions of real world problems has attracted attention from all over the world.A variety of mathematical models both for the isothermal and non-isothermal tube drawing processes, with different levels of descriptions and needs, are available in the literature e.g., see3 by incorporating a steady state numerical solution. To better understand the physical model and to examine the accuracy of the obtained numerical solution, our objective here is to develop an analytical solution of the steady-state isothermal model. To start with, it is not possible to determine an analytical solution of this type of model with the given initial and boundary conditions. When we solve the second order ordinary differential equation appearing in the steady-state model, the difficulty lies in determining the constants of integrations. To overcome this concern, we utilize a numerical technique called shooting method to convert the boundary value problem into a set of initial value problems. An integrating technique is implemented to develop an analytical solution of the converted initial value problems. Similar approach has been suggested in22 to construct an analytical solution of a steady-state melt-spinning model. The semi-analytical approach developed in this paper has an advantage over other numerical methods that it may be applied directly to both linear and nonlinear systems of equations without the need for discretization, linearization, or perturbation.In this paper, we have considered an isothermal glass tube drawing model consisting of three coupled nonlinear partial differential equations of first and second order. The steady-state numerical solution of this model is required in order to investigate its stability. Recently, we have analyzed the stability of an isothermal tube drawing model and performed a complete mathematical analysis inThe organization of the paper is as follows: In “9 and28 to model the tube drawing process. In this section, we briefly explain the mathematical model for an isothermal tube drawing process.In the literature, different types of models for the drawing processes with different level of demands and descriptions are available. A considerable amount of work has been carried out by different researchersTo model the tube drawing process, we consider an incompressible Newtonian flow of a molten glass between two free surfaces mentclass2pt{minim4 and are known as the standard equations showing the axi-symmetric stokes flow. The first equation is the continuity equation and the last two equations are the momentum equations respectively in r and z directions. We denote the derivatives by subscripts r and z where z denotes the distance along the axis of the glass tube and r measures the distance perpendicular to it. The velocity of the molten glass is defined to be u and v are the components of velocity z and r direction respectively. The pressure, density and the acceleration due to gravity are denoted by p, g respectively.The Eqs. –1c) are are1c) aNow, at the free surfaces On the inner and outer surfaces of the glass tube, the stress conditions are given as:The kinematic conditions are described asnditions –1e) can can1f\doW is the width of the glass tube that has a very small value than the typical length of hot-forming zone L, U is the typical drawing speed and To take benefit of the small parameters present in the problem, it is now convenient to convert the Eq.  into dimThe stress conditions –1k) rea rea1k) rThe kinematics conditions –1g) bec bec1g) b6 and27. However, we have included the remaining considerations of the modeling process in “g and R of the tube is defined asA, velocity v, and average radius R of the glass tube. Width W of the tube can be determined by the equationThe above model can be simplified by means of an asymptotic expansion, in which the inverse aspect ratio Since the temperature throughout the forming zone of the tubing process remains constant, the viscosity We introduce the following dimensionless quantitieshe model  to get tThe initial and boundary conditions are respectively transformed toThe steady state form of the model  is writt3. For details, we refer the reader to3.The system  is a sys22) where it is not possible to find integration constants with the given conditions. It is interesting to mention that integration constants can be found if boundary value problem (BVP) is transformed to an initial value problem (IVP).There are different techniques available in the literature to solve boundary value problems analytically. The obtained general solutions contains arbitrary constant which can be determined by using boundary conditions. However, there exist some problems \documentgiven in .A and v in Eq. , B(z) are given asDifferentiating  with resz in Eq. , we reac)).Using  and the )).Using , we get2 the Eq.  along wiively in  and 17)z and theively in , 16) an anz and ively in .A, v and R are negligibly small (see Table W, the width of the glass tube, by using the relation W and the corresponding error are also shown in the Fig. A, v, and R is maximum at W is maximum at The developed analytical solution is plotted and compared with the numerical solution in Figs. entclass1pt{minimaIn this work, we have presented a numerical based approach to find analytical solution of a boundary value problem where one cannot find constants of integration with the given boundary conditions. The approach is to convert a boundary value problem into initial value problems and then to develop an analytical solution of the resulting problems with the usual methods. To explain the approach, we have developed an analytical solution of a steady-state model of the isothermal tube drawing process with the help of shooting method. The obtained analytical solution is almost in agreement with the numerical solution that justifies our approach.


The rock mass around deep roadways has obvious creep characteristics in high-stress environments. Meanwhile, the cyclic impact load induced by roof fracturing also causes dynamic damage to the surrounding rock, leading to long-term large deformation. This paper examined the rock mass deformation mechanism around deep roadways based on the theory of rock creep perturbation effect considering perturbation sensitive zone. This study proposed a long-term stability control guideline for deep roadways under dynamic load. An innovative support system was developed for deep roadways, with concrete-filled steel tubular support being recommended as the main supporting body. A case study was conducted to validate the proposed supporting system. Monitoring over one year in the case study mine showed that the overall convergence deformation of the roadway was 35 mm, indicating that the roadway’s long-term large deformation induced by creep perturbation was effectively controlled by using the proposed bearing circle support system. In deep mining operations, the dynamic impact load is mainly caused by the sudden fracturing of the competent main roof units. This impact load is termed as roof-fractured impacting loading or RFIL in this paper. This type of impact load does not cause the instantaneousdamage of a roadway’s surrounding rock. However, it will cause a certain perturbation deformation of the surrounding rock. When this type of deformation is superimposed with the creep deformation of the surrounding rock, the deformation rate of the roadway increases, resulting in long-term large deformation. At the same time, the RFIL has the characteristics of cyclic impacts. With the periodic fracturing of the main roof above the face, the impact load will act on the surrounding rock repeatedly, exacerbating the long-term large deformation of the deep roadway. Rock burst in deep mining has been increasingly investigated globally. The mechanisms of rock burst, monitoring and early warning of impacting load, and pressure relief technology during mining have been thoroughly studied, with many innovative results achieved10. Nevertheless, most of the existing research work has focused on the dynamic disasters of a deep roadway under the influence of impact loads with high energy levels (impact energy > 105 J). However, the energy level induced by roof strata fracture in deep mining is much lower than that of a rock burst, being approximately 103–105 J. The long-term large deformation mechanism and quantitative support design of the surrounding rock of deep roadways under RFIL has not been investigated thoroughly, with a limited number of relevant studies.As mining goes deeper, the stress concentration caused by roadway excavation increases significantly, and the rock mass shows creep characteristics. The deformation of the roadway has pronounced time effect characteristics. The short-term equilibrium state of the rock mass in creep state could be easily disturbed due to the impact of the external perturbation load. This increases the creep deformation rate of the roadway, and could even lead to dynamic disastersTherefore, this study conducted theoretical assessment of the support system and verified them using a case study in a coal mine in China. The theory of the perturbation effect of creeping rock was tested, where the influence mechanism of RFIL on the surrounding rock in the creep state was analysed. Then, a quantitative design method for the support system of the deep roadway affected by the dynamic pressure was proposed.12 and developed a new type of equipment to test the perturbation effect of creeping rock. Based on this new testing system, the process and method of the dynamic impacting experiment of creeping rock were introduced at a laboratory scale. In addition, some preliminary research on typical creeping sandstone specimens under RFIL was conducted, considering the constitutive relationship of the rock’s perturbation deformation.The perturbation effect of creeping rock describes the mechanical phenomenon that the rock’s creep deformation will increase at a certain stress level when affected by external impact loads (such as shooting vibration etc.). It is a critical research area for the prevention and control of dynamic disasters in deep roadways impacted by the external dynamic load. However, in the existing literature of rock dynamics, the research focus on this issue is still limited. The authors’ research group previously proposed the concept of the perturbation effect of creeping rock13. This shows that the rock has already begun to become sensitive to the external impact load, and the impact has more damage on the creeping rock in this stage. The cumulative perturbation deformation and creeping deformation of the rock will cause long-term large deformation until failure. At the same time, as the intensity of the impact load increases, the curve of the rock’s cumulative perturbation deformation is transformed from the attenuation stage to the continuous development stage in advance. As a result, the rock begins to become more prone to external perturbation at lower static stress levels.The impact load acts on the creeping rock in the form of shock waves, causing the rock to generate the corresponding disturbing deformation increment. Under varying static stress conditions, rocks typically exhibit two different responses to external impact loads. At a lower static stress level, the rock is in the initial stage of creep, showing an obvious hardening characteristic under the effect of external cyclic impact load. The curve of the rock’s cumulative perturbation deformation at this stage is shown in Fig. 14:Based on different sensitivities of the rock to the external impacts under various static stress conditions, the authors propose the new mechanical concepts of “limit strength neighbourhood” and “anti-perturbation strength” of the rock. The limit strength neighbourhood of rock refers to the stress state of the rock. That is, under a simple compressive  state, the rock has a limit strength entclass1pt{minima12. According to the results, for medium to hard sandstone (siltstone and red sandstone), the anti-perturbation strength index is basically equivalent to the sandstone’s long-term strength, which is There are different mechanical behavior of the rock when the applied static stress is in the neighborhood of the limit strength or outside the neighborhood. If the static stress is about the limit strength, the rock is more sensitive to external impact perturbation and its deformation is in the stage of damage and destruction. In contrast, when the static stress is outside the neighbourhood, the rock is not sensitive to external impact perturbation, and the impact will not cause perturbation deformation or the resultant deformation will gradually decay. Under such a condition, the rock’s deformation will occur in the impact hardening stage. According to Fig. The rock mass in deep roadways is affected by both high static stress and cyclic external impact load, and the characteristics of its deformation are long term and dynamically gradual. This is much more complicated than the surrounding rock deformation in roadways on low stress environment. Therefore, the deformation theory of shallow roadways appears to be not suitable for analysing the long-term large deformation mechanism of deep roadways. The present study adopted the basic theory of rock creep perturbation effect to characterise the mechanism of deformation of deep roadways by redefining the surrounding rock’s state zone and analysing the evolution law of the dynamic stress field.17. This model is illustrated in Fig. In the traditional theory of surrounding rock deformation in shallow roadways, the rock’s state zone is divided into three zones: fractured zone, plastic zone, and elastic zone after roadway excavation. This classification is based on the static stress level and deformation characteristics of the surrounding rock. When considering the creep perturbation characteristics of the rock and comprehensively analysing it with the high static stress level and dynamic impact load, it is understood that the surrounding rock’s state zone will change after roadway excavation. Then a new classification model for the surrounding rock’s state zone was establishedIFractured zone: In this zone, the large concentrated static stress exceeds the compressive ultimate strength of the rock mass. The stress relationship is IIPerturbation sensitive zone: After the surrounding rock of the roadway is failed, the concentration of the static stress decreases. At the same time, the rock surrounding the fractured zone can provide a small lateral confining pressure to the internal surrounding rocks, making them enter a three-way compression state. This increases the various strengths of the inner surrounding rock. In the perturbation-sensitive zone, the reduced concentrated static stress is higher than the rock’s anti-perturbation strength but lower than its ultimate strength. The stress relationship is IIIPerturbation insensitive zone: The stress relationship in this zone is IVElastic zone: The surrounding rock in this zone is basically in the state of elastic deformation, which has little effect on the roadway deformation.After excavation of a deep roadway, a larger stress concentration will occur within a certain range of the surrounding rock where the maximum concentrated tangential stress at the roadway rib can reach about twice the original stress level. Going further into the surrounding rock, the concentrated stress will gradually attenuate and eventually decrease to the original stress level. This stress curve is shown by the black curve in Fig. After excavation of a deep roadway, the major deformation of the roadway are induced from the dilitant deformation and the integral extrusion. As for the rock mass in the perturbation-sensitive zone, creep deformation occurs as a result of the external dynamic loading, then the zone will extend to a fractured zone, which is gradually increasing as time. Because of the presence of the perturbation-sensitive zone, an irreversible cumulative perturbation deformation of the surrounding rock in zone II will occur after the rock is impacted by the RFIL. When the deformation continues to increase to the ultimate deformation of the surrounding rock, this zone will transform into the fractured zone, and the perturbation-sensitive zone will continue to move into the deeper surrounding rock. This dynamic process is represented in Fig. During the dynamic evolution of the perturbation-sensitive zone into the deeper surrounding rock, the concentrated static stress is further reduced, and the ultimate strength and anti-perturbation strength of the internal surrounding rock continue to increase accordingly. When the decreasing concentrated static stress peak is equal to the increasing anti-perturbation strength of the surrounding rock, the surrounding rock reaches a relatively balanced state. The distribution of stress in this process is shown in Fig. In summary, the long-term large deformation and instability mechanism of a deep roadway under the action of RFIL can be analysed and obtained. This deformation and instability are due to the continuous transfer of the perturbation-sensitive zone into the interior surrounding rock under the double effect of greater static stress and the external impact load. If the dynamic developing progress of the perturbation sensitive zone cannot be controlled, a large extent of fractured zone will eventually form, resulting in the long-term large deformation and failure of the deep roadway. This deformation mechanism based on the theory of rock creep perturbation effect can better explain the long-term large deformation and failure phenomenon of deep roadways under dynamic load.19. According to the above analysis of the dynamic stress field evolution and the deep roadway deformation mechanism, to ensure the long-term stability of a deep roadway, there are two key aspects for the ground support design. The first one is to control the short-term bulking deformation of the surrounding rock in the fractured zone, and then to control the dynamic transfer and evolution of the perturbation-sensitive zone into the interior surrounding rock. This study proposes an innovative supporting principle for deep roadways affected by RFIL, named the “bearing circle”, together with our support elements as a system.After excavation of a deep roadway, the surrounding rock will soon be damaged under the double action of dynamic and static loads if there is no artificial supportThe main principle of the “bearing circle” anti-impact supporting technology is to construct a circle within the fractured zone around the roadway which is reinforced by supporting structures, as shown in the green area in Fig. 22.The circle functions to strengthen the integrity and bearing capacity of the surrounding rock in the fractured zone, control the short-term bulking large deformation of the fractured zone, and improve the stability and impact resistance of the surrounding rock in the anchorage zone. The circle can also provide a higher lateral confining pressure to the surrounding rock in the perturbation sensitive zone, increasing the anti-perturbation strength and ultimate strength of the surrounding rock in the entire perturbation sensitive zone. The bearing circle can also artificially redistribute the high concentrated static stress in this zone from the limit strength neighborhood range of the surrounding rock, and improve the bearing capacity and impact resistance of the surrounding rock in the zone. The greater the lateral confining force provided by the “bearing circle”, the earlier the high static stress in the perturbation sensitive zone exits the rock limit strength neighbourhood range, and the better the supporting effect will beFigure Bolt, mesh and concrete spray layerThe “bearing circle” support system can resist high pressure and external impact. The mechanical properties of this structure are determined by the specific properties of the support capacity of the elements used. The “bearing circle” structure proposed in this paper uses the following four types of support elements.(2)Cable bolts supportThe supporting effect of the bolt significantly improves the cohesion and internal friction angle of the fractured surrounding rock, thereby increasing the strength of the surrounding rock in the anchorage zone. At the same time, the bolt can improve the integrity of the anchorage zone and strengthen its bearing capacity. The main function of the metal mesh and the concrete spraying layer is to provide surface support and coverage. At the same time, these two structures can spread the force of the bolts evenly to the surrounding rock and establish a stable anchorage zone around the roadway.(3)Concrete-filled steel tubular support (CSTS)As for cable bolt, one end of the cable is anchored in the deep interior of the surrounding rock, and the other end is closely attached to the roadway wall through the locking device. This suspends the anchorage zone in the more stable interior strata. It can also increase the friction between the “bearing circle” and the perturbation-sensitive zone, and prevent the layer sliding between the surrounding strata.21. The ultimate bearing capacity of the CSTS is about 3–5 times that of the same type U-type steel support. Therefore, installing it in the roadway as a passive supporting structure can provide extra radial force for the “bearing circle”. This is equivalent to applying a large amount of lateral confining pressure to the surrounding rock improving its bearing capacity. The force provided by the CSTS can also act on the surrounding rock in the perturbation-sensitive zone through the “bearing circle”, providing a substantial increase in its anti-perturbation strength and ultimate strength. Of note is that the filling material between rock surface and concrete-filled steel tubular support is an important link to achieve energy absorption capacity. It is usually filled with yielding material, such as timber plates, to allow further deformation prior to the close contact between the rock and the concrete-filled steel tubular support.(4)Surrounding rock groutingThe CSTS is a new type of high-strength structure to support the soft surrounding rock of deep roadways. The structural composition, mechanical properties and engineering application of CSTS were previously introducedC and internal friction angle of the rock mass and therefore increase the strength of the fractured rock.During the development of the surrounding rock’s deformation, the internal cracks in the rock mass will gradually develop as the number of deformation increases. When the cracks develop to a certain extent, they will cause the destruction and instability of the surrounding rock. Surrounding rock grouting technology can improve the physical and mechanical properties of the fractured rock mass from the inside the surrounding rock, increase the static and dynamic elastic modulus of the rock mass, weaken the anisotropy of the rock mass, ensure the complete unity of the surrounding rock and improve its bearing capacity. At the same time, the internal cracks are filled by grouting into the surrounding rock, which can improve the cohesion Mechanical analysis in perturbation-sensitive zoneAccording to the analysis, the condition for the long-term stability of a roadway is to redistribute the concentrated static stress in the perturbation-sensitive zone from the limit strength neighbourhood of the surrounding rock. The mechanical condition that should be satisfied is P is the original stress level; C is the cohesion of surrounding rock in the original state; and After roadway excavation, the fractured zone will form in a short time, and the peak of concentrated static stress will immediately be transferred to the boundary of the fractured zone and perturbation-sensitive zone. According to rock mechanics theory, the peak value can be calculated according to the following formula:24. After the artificial support structure is implemented, the “bearing circle” is formed. Assuming that the lateral support strength provided by the circle is (2)Mechanical analysis in the “bearing circle” zoneThe surrounding rock strength at the outer boundary of the perturbation sensitive zone can be considered as the uniaxial compressive strength assuming that the surrounding rock in the fractured zone completely loses its bearing capacity and its lateral support for the perturbation sensitive zone. This kind of safety consideration is allowed in engineering design. At the same time, because the formation of the fractured zone occurs in a relatively short period of time, it can be assumed that the surrounding rock in the perturbation-sensitive zone during this stage still follows the Coulomb–Mohr strength criterionr is the radius at any position in the anchorage zone, and In Eq. , to get A can be calculated as Solving the differential Eq.  gives: \into Eq. , the forR1; therefore, the radial stress of the surrounding rock at the outer boundary of the anchorage zone at r = R1, i.e., the lateral support strength provided by the “bearing circle” to the perturbation sensitive zone, can be expressed as:The radius of the anchorage zone is the effective length of the bolt Then the unknown parameters e in Eq.  can be eBy combining the above analysis and calculations, that is by substituting Eqs.  and 7)  7) into with Eq. , it can entclass1pt{minimaThe Yangcheng Coal Mine is located in the Jining mining area in eastern China. There are three development roadways in the southern mining district at − 650 m depth level: a rail roadway, haulage roadway, and pedestrian roadway. The planar and vertical views of the three roadway arrangements are shown in Fig. The primary mined coal seam of the 3301 working face is seam No. 3. The coal seam’s dip angle is approximately 22°, and the average thickness is 6.5 m. Due to the influence of faults, the design length of the face is 123 m in the early stage and 200 m in the later stage. Comprehensive mechanised caving coal mining method is used for coal extraction. The roof strata above the working face are mainly composed of medium sandstone, mudstone and fine sandstone, and the overall strength of the surrounding rock is medium. The three roadways are all located in the fine sandstone stratum 10.6 m thick above the No. 3 coal seam. The designed cross-section of the rail roadway is approximately circular with a shallow bottom arch. The net height of the roadway is 4200 mm, the net width is 5088 mm and the section radius is 2544 mm. The original ground support system and scheme of the roadway consist of first-stage support utilising a bolt, cable and metal mesh plus a shotcrete layer system, and the secondary reinforcement system using U-type steel support. The pre-tensioned bolt (Ф22 × 2400 mm) was installed with a spacing of 0.7 m by 0.7 m. The 21.6 mm steel strand cable has a length of 7500 mm. The row and line spacing parameters of the bolt arrangement in the rail roadway were 1.6 m by 1.6 m. The metal mesh was welded to a φ5 mm steel bar, and the mesh size was 100 mm by 100 mm. The thickness of the shotcrete was 100 mm. The distance between the adjacent U29 steel supports was 700 mm. It is of note that due to the high level of stress, the initial ground support system controls the long-term instability of the roadway.The overall occurrence of the rock strata in the location of the rail roadway is a monoclinic structure that is striking northeast and dipping southeast. The geological structure in this area is complex and there is a large horizontal tectonic stress. According to the test of the static in-situ stress at − 650 m mining level, the maximum principal stress is the horizontal stress is about 23.9 MPa. The k ratio is about 1.31 ~ 1.52. The direction of the maximum principal stress is approximately perpendicular to the strike direction of the rail roadway. According to rock mechanics theory, after the excavation of the roadway, the peak value of concentrated static stress can reach more than twice the in-situ stress level. In addition, the surrounding rock of the rail roadway is prone to large deformation and failure under high concentrated static stress.26. In the later stage of the mining face, the horizontal distance from the roadway is only about 80 m, and the dynamic impact is obvious. Yangcheng Coal Mine is a typical mine with risk of rock burst, so a real-time microseismic monitoring system has been installed29. The microseismic monitoring and analysis of the 3304 working face (mining finished) at the same elevation in the same mining district was used to obtain the impacting energy from the overlying basic roof’s periodic break as shown in Table The main source of the dynamic load that has a great impact on the rail roadway is the roof breakage after mining of the 3301 working face32. The distance between the main roof of the 3301 working face and the rail roadway is about 70–80 m. Therefore, the source energy after the roof break of the 3301 working face will be reduced by about two orders of magnitude when propagating to the rail roadway. This is also consistent with the results obtained from on-site monitoring as indicated in Table According to relevant research, the source energy generated after the roof break will gradually attenuate during the propagation within the rock medium, and its attenuation conforms to the law of power exponential decline12. The present study used similar research methods, combined with the process of rock creep perturbation experiments12, and the dynamic parameters test was also conducted on the fine sandstone of the surrounding rock in the rail roadway.Before designing the support scheme for the rail roadway, the basic rock mechanical parameters and key dynamic parameters of the surrounding rock need to be determined. The authors conducted similar rock creep perturbation experiments on siltstone at the mining level of − 1100 m in Huafeng Coal Mine in eastern China and obtained experimental results12. According to the monitoring results of the impacting energy, the average value of the impact energy during ten cycles of roof break was taken as the dynamic impact strength in the experiment, which is 2.288 × 103 J. In the experiment, it was assumed that the gravitational potential energy of weights was completely converted into the impacting energy on the surrounding rock. Through calculation, it was revealed that this impact energy could be achieved by taking a weight of 3 kg and a falling height of 11 cm. In the experiment, the MTS servo rock testing machine was used to measure the basic mechanical parameters of the rock, and the RRTS-II rock creep perturbation experimental system was used to test the rock dynamic parameters. The final experimental results are summarised in Table Our test is a type of creep perturbation experiment under a specific impact loadAccording to the analysis of the current status of the deep roadway support, the traditional support measures of bolt plus cable and U-type steel support cannot effectively control the large creep and perturbation deformation of the deep roadway in the studied coal mine. Therefore, it was decided to use the composite supporting technology with the CSTS as the main support in the rail roadway. The bolt, metal mesh and concrete spray layer and cables were used as the first step support of the roadway, while the CSTS was used as the secondary step support. Meanwhile, the surrounding rock grouting was used as an alternative technology only when intense development of rock cracks was detected. The supporting technology system described here was used to construct the “bearing circle” structure within the surrounding rock.R1 is the radius of the anchorage zone where R1 = 2.3 m. The yield strength of the bolt is not less than 335 MPa. For the row and line spacing parameters of the bolt arrangement, if the values are too small, the original integrity of the surrounding rock will be easily damaged; but if they are too large, the bolts cannot effectively improve the mechanical properties of the surrounding rock. According to the previous findings, the supporting density of the bolts is relatively reasonable between 0.6 m and 1.0 m. Therefore, in this study, the designed row and line spacing parameters of the bolt arrangement in the rail roadway were 0.8 m by 0.8 m. According to the test results of the mechanical parameters of the surrounding rock in Table Cm = 0.135 MPa was provided by the bolt to the surrounding rock, and the increased cohesion of the surrounding rock strengthened in the anchorage zone was According to the analysis of Eqs.  and 9)  9) in Seentclass1pt{minimaAccording to the analysis in Section "21, this type of CSTS can provide the ultimate bearing capacity of about 2650kN. In the roadway, the CSTS was installed at actual distances of 0.8 m. The radius of the CSTS section at about 2.5 m and of the roadway at R0 = 2.5 m were about the same. Through calculation, the supporting strength of the CSTS relative to the surrounding rock was determined to be approximately 1.33 MPa, which The CSTS structure consists of a seamless steel pipe and internal core-filling concrete. To produce the CSTS structure, an empty steel pipe is bent into the appropriate shape according to the shape of the roadway section; then concrete is poured into the empty steel tube support following the installation of the pipe in the roadway. In order to satisfy the strength needed for the support control of the surrounding rock of the rail roadway, we should have designed the CSTS with Φ194 × 10 mm steel tubing and a concrete strength grade of C40. According to the theoretical algorithm combined with the previous mechanical performance test of the supportP = 23.9 MPa. The cohesion of roadway surrounding rock (fine sandstone) under uniaxial compression was 14.2 MPa, which is C = 14.2 MPa. The testing result of the surrounding rock’s anti-perturbation strength was 45.1 MPa, which is The parameters obtained by the above calculation included: Bringing all the above parameters into Eqs.  and 7)  7) in Se3 J. Supplementing the roadway with a concrete spraying layer, surrounding rock grouting, and other auxiliary strengthening support measures further improves the bearing performance of the surrounding rock. Thus, the deep roadway can maintain long-term stability under the dual action of high static stress and cyclic external impact load.The relationship between these two parameters is entclass1pt{minimaBolt: Cable: Metal mesh and concrete spray layer: the metal mesh was a square grid, welded by φ5mm steel bars, and the grid size was 100 mm × 100 mm. The thickness of the concrete spray layer was 50 mm, and the concrete strength grade was C20.CSTS: composed of Φ194 × 10 mm steel tube and a concrete strength grade of C40. The selected steel tube’s material was 20# low-carbon steel, which had a yield strength of 245 MPa and a tensile strength of 410 MPa. The distance between two adjacent rows of CSTS was 0.8 m. A type of connecting rod was used to connect the two adjacent CSTSs. The designed shape of the CSTS cross-section was approximately circular with a shallow bottom arch, consistent with the cross-section of the roadway. A 200 mm deformational space was reserved between the roadway-surrounding rock and the CSTS to meet the deformation and stress release requirements of the surrounding rock. The deformation space was filled with flexible blocks.# ordinary Portland cement and water. The slurry was injected into the surrounding rock cracks with grouting bolts. The design of the overall supporting structure is presented in Fig. Surrounding rock grouting technology: the slurry material was ordinary single-liquid cement slurry, which was composed of 42.5The application of the supporting technology with the CSTS as the main body in the rail roadway is shown in Fig. This study established a new model of the surrounding rock’s state zone for deep roadways and proposed an improved understanding of the surrounding rock’s deformation mechanism under RFIL. Such deformation is due to the continuous transfer of the perturbation-sensitive zone into the interior surrounding rock under the double effect of greater static stress and the external impact load. The findings were applied to the rail roadway in Yangcheng Coal Mine where the roadway’s long-term creep perturbation large deformation was effectively controlled. To maintain the long-term stability of a deep roadway under the action of RFIL, the key principle is to make the concentrated static stress in the perturbation-sensitive zone redistributing away from the limit strength neighbourhood of the surrounding rock. For this purpose, the most effective measure is to provide the surrounding rock with sufficient lateral confining pressure after excavation.Then this paper proposed a new dynamic anti-perturbation strength index of rock and established the quantitative design method of support parameters applied in the deep roadway under the action of RFIL. The study proposed the “bearing circle” support system as an innovative supporting principle for a deep roadway affected by RFIL, particular considering the yielding performance of the system under dynamic loading which was limited studies previously.To construct the “bearing circle”, the supporting system was designed with the CSTS as the main body. This is the main technological innovation of this paper.34 and machine learning36 algorithms would further enrich understanding of deep roadways’ surrounding rock and improve design of support systems to prevent mining disasters. In particular, deep learning is a recent development of artificial intelligence that future research can investigate for potential applications.The in-situ application of the research findings in a deep coal mine in China verified the accuracy and reliability of the proposed theory and design method for support. The proposed method can also be used for different rock types in deep operations, considering the presence of rock discontinuities. Even though the methodology developed in this study proved both theoretically and practically trustworthy, incorporating new technologies such as digital rock technology