Open Access
Add to BibSonomy
Abstract
A vector optical field with inhomogeneous spatial polarization distribution offers what we believe to be a new paradigm to form controllable filaments. However, it is challenging to steer multiple performances (e.g. number, orientation, and interval) of filaments in transparent nonlinear media at one time. Herein, we theoretically self-design and generate a kind of believed to be novel ellipticity and orientation co-variant vector optical field to interact with Kerr medium to solve this issue. The collapsing behaviors of such a new hybrid vector optical field reveal that, by judiciously adjusting the inherent topological charge and initial phase of incident optical field, we are able to give access to stable collapsing filamentation with tunable numbers, orientations and interval. Additionally, the collapsing patterns presented are immune nearly to the extra random noise. The relevant mechanism behind the collapse of the vector optical field is elucidated as well. The findings in this work may have huge potential in optical signal processing, laser machining, and other related applications.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The interplay between ultrashort pulse lasers and transparent nonlinear media can induce the optical field collapsing into filaments [1–4]. This process is mainly caused by the dynamic competition of the self-focusing effect and the plasma defocusing [5–10]. Owing to their unique and promising properties, the filamentations can be applied in a gamut of areas, such as cell manipulation in biomedicine [11], optical micromanipulation [12,13], laser cutting [14], and monitoring environmental pollutants [15–18]. To further satisfy diverse demands in practical applications [19–23], it is necessary to accurately control the distribution and number of filaments [24–27]. In this regard, numerous methods have been put forward as follows: modulating the laser pulse power [28], adjusting the divergence angle of the initial laser [29], launching negatively chirped ultrashort pulses [30], using the amplitude/phase mask [31], introducing wire-mesh [32] and lens array setup [33]. However, these methods only possess limited controllable ability for the filaments because their realizations are totally dictated by the amplitude and phase of scalar optical fields.
Polarization, a crucial intrinsic property of light, plays a significant role in the light–matter interaction. It is widely accepted that $\alpha$ and $\phi$ are two important parameters of the polarization ($\alpha$ represents the ellipticity angle of polarization and $\phi$ represents the orientation angle, respectively). Recently, according to the mechanism of filament formation, the cylindrical vector beams have been introduced into filament manipulation. However, it is unfeasible to obtain organized filaments in isotropic Kerr medium by only regulating the parameter $\phi$. In general, both the radially and rotationally polarized optical fields will break up disorderly in the case of random noise [34]. Thus, a novel method of filaments modulation focuses on the parameter $\alpha$ of polarization. By designing the azimuthal-variant hybrid polarized (AVHP) vector optical field, the number of the filaments can be controlled due to the axial-symmetry breaking of the optical field [35,36]. Under such a circumstance, the filaments are equal to $4m$ of the topological charge. Based on this, we have designed the elliptically symmetric hybrid polarization vector optical field (ES-HP-VOF) utilizing the eccentricity to precisely manipulate the filament distribution. It has been shown that, in higher eccentricity, the number of filaments evolves from $4m$ times to $2m$ times [37]. Overall, despite these prominent progresses in controlling the filaments [38], nearly all the approaches are dictated by engineering either the $\alpha$ or the $\phi$ in nonlinear media. To our knowledge, there is no work related to the field collapse and subsequent filaments by jointly manipulating the $\alpha$ and $\phi$ of optical field in Kerr medium. A question is raised that whether we are able to generate novel vector optical fields by resorting to the synergetic effect of the $\alpha$ and $\phi$ for inducing new collapsing behaviors. The answer is "yes".
We here theoretically design and demonstrate a novel hybrid polarization vector optical field (NHP-VOF) with co-variant ellipticity and orientation. When such a vector optical field interacts with an nonlinear optical medium, it enables us to garner tunable collapsing patterns (e.g. number, orientation, and interval) by changing the topological charges of the $\alpha$ and $\phi$ simultaneously. In addition, by changing the initial phase of ellipticity, the filaments can rotate continuously and the filament interval can be adjusted on demand. We further investigate the effect of random noise on the filament formation of the novel vector optical field. In principle, nonlinearity-related physical mechanisms are responsible for the well-defined filaments. These new findings may provide theoretical and numerical basis for optical communication [39], laser micromachining [40], and other related fields.
2. Theory
A NHP-VOF can be written as
We can deduce the states of polarization (SOP) of NHP-VOF througth the Poincaré sphere (PS) as shown in Fig. 1(a). The angles of $2\alpha$ and $2\phi$ represent the latitude and longitude angles in the spherical coordinate system, respectively. The ellipticity of SOP along the longitude can be controlled by $\alpha$, while the orientation of SOP along the latitude can be controlled by $\phi$. To guarantee an NHP-VOF with the co-variant ellipticity and orientation, we can design and construct a novel field, making its trajectory neither along the latitude nor longitude on PS. The trajectory of this field is related to Stokes parameters in Eq. (2).
Fig. 1. (a) Schematic of the PS in the spherical coordinate system represented by the traditional latitude and longitude circles, (b) The trajectory curve of NHP-VOF on the PS. The first row is the front view, and the second row is the top view.
The red curve of the PS represents the incident field, as shown in Fig. 1(b). It is worth mentioning that the polarization state distribution of the azimuthal angle in the region $[0, \pi ]$ is exactly well corresponding to the red curve on PS. When varying along $[0, 2\pi ]$ for one cycle, the polarization actually travels along the red curve twice. We can classify the incident field into three main types based on the numerical relationship between $m$ and $l$. (i) When $m=l=1$, $S_{3} =sin(2\varphi )$, the trajectory of the incident field is shaped as "8". The red curve undergoes two cycles of polarization (linear polarization-elliptical polarization-circular polarization-elliptical polarization-linear polarization) in the region $[0, \pi ]$ in this case. More specifically, the ellipticity changes from linear polarization (point A) to right circular polarization (north pole B) to linear polarization (point A) to left circular polarization (south pole C) to linear polarization (point A). On the trajectory, all polarization states are presented, including linear polarization, elliptical polarization, and circular polarization. (ii) When $m=1$, $l=2$, $S_{3} =sin(4\varphi )$, the trajectory is A-B-C-D-A-E-F-G-H-E which passes over two times in the Northern Hemisphere and two times in the Southern Hemisphere, respectively. Hence, the red curve undergoes four cycles of polarization (linear polarization-elliptical polarization-circular polarization-elliptical polarization-linear polarization) in the region $[0, \pi ]$. (iii) When $m=1$, $l=3$, $S_{3} =sin(6\varphi )$, the trajectory is also shaped as "8". The red curve undergoes six cycles of polarization (linear polarization-elliptical polarization-circular polarization-elliptical polarization-linear polarization) in the region $[0, \pi ]$. Likewise, when $m=1$, $l=n$, $S_{3} =sin(2n\varphi )$, the red curve undergoes $2n$ cycles of polarization in the region $[0, \pi ]$. In summary, when $l$ takes odd number, the trajectory is recirculated as "8"; while $l$ takes even number, the trajectory is similar to "8" looply, but no closure point at the equator, and the number of cycles is equal to $2l$.
In order to theoretically study the nonlinear propagation behaviors of NHP-VOF in Kerr medium, the nonlinear-Schrödinger (NLS) equation is used. Under the slow-varying envelope approximation, the NLS equation can be decomposed into a pair of coupled wave equations as follows:
The $P=2n_{0} \varepsilon _{0} c\iint [|E_{H} (\rho, \varphi ; \zeta )|^{2} +|E_{V} (\rho, \varphi ; \zeta )|^{2}]\rho d\rho d\varphi$ is the power and $P_{c} =\gamma \lambda ^{2} /4\pi n_{0} n_{2}$ is the critical power for self-focusing, respectively. $\rho =\frac {r}{r_{0} }$ and $\zeta =\frac {Z}{L_{d} }$ ($L_{d}=\pi r_{0} ^{2}/\lambda$) are the dimensionless cylindrical coordinates, $n_{0}$ and $n_{2}$ are the linear and nonlinear refractive indices of the Kerr medium, respectively. $\gamma$ is a parameter related to a Gaussian profile and so should be taken $\gamma$ = 1.896 [41]. $c$ is the speed of light in vacuum, $\lambda$ is the wavelength of light in vacuum. In Eqs. (3) and (4), the first term of the equation represents diffraction and the second term is related to the Kerr nonlinearity. $\nabla ^{2} _{\perp } =\partial ^{2} /\partial \rho ^{2} +\rho ^{-1} \partial /\partial \rho +\rho ^{-2}\partial ^{2}/\partial \varphi ^{2}$, which denotes the Laplace operator in the transverse direction is related to diffraction.
Obviously, the Eqs. (3) and (4) do not satisfy the principle of linear superposition, and there is no universal analytical solution. However, it can be simulated numerically using the beam propagation method (BPM) [42]. The BPM numerical algorithm divides the propagation length into equal segments, which allows the roles of the diffraction and nonlinear terms within each segment to be calculated separately, thus simplifying the computational difficulty. It is important to select a sufficient number of points when using this algorithm. Here we present research results of numerical simulation of nonlinear self-focus behaviors of NHP-VOF using BPM algorithm.
3. Numerical simulations
It is obvious that the presentation of a new vector optical field is not enough, as novel properties and useful applications are always expected. We study its self-focused collapsing behaviors in Kerr medium. When $P$=$5P_{cr}$, extensive numerical simulations are performed by the BPM in Eqs. (3) and (4). It reveals that the filaments distribute stablely, when $10{\% }$ of random amplitude and phase noise are added (multiplication of the field by 1+$A_{n}exp(i2\pi \cdot B_{n}$), where $A_{n}$ is the noise percentage, and $B_{n}$ is a random number between 0 and 1). Even with $30{\% }$ noise, the stable collapsing patterns are still possible. The detailed content is organized as follows. Section 3.1 presents the simulation for the collapsing behaviors of the NHP-VOF ($m=l$). Section 3.2 shows the collapsing behaviors of the NHP-VOF ($m\ne l$), similarly. Section 3.3 demonstrates the rotation of the filaments and the adjustment of filament interval. In additional, it also should be emphasized that the intensity normalized to its maximal value in each figure below.
3.1 $m=l$
Topological charge is an important parameter to control the field collapse and filamentation. The collapsing behaviors of NHP-VOF ($m$ = $l$) in Kerr medium are investigated in Fig. 2(c). The incident field, uniformly distributed and without singularities, is pre-focused on the input face of the Kerr medium in Figs. 2(a) and 2(b). It can be seen that the intensity distribution of the weakly focused field is inhomogeneous along the rotational direction. And the period of polarization change is consistent with that depicted in Fig. 1, undergoing a $2l$ (linear polarization-elliptical polarization-circular polarization-elliptical polarization-linear polarization) cycles in the region $[0, \pi ]$.
Fig. 2. (a) Optical intensity and polarization distribution at the focal plane of NHP-VOF ($m=1$, $l=1$), (black/green/yellow, corresponding to right, linear, left polarization, respectively), (b) $m=2$, $l=2$, (c) The collapsing behaviors of the ($m=1$, $l=1$ and $m=2$, $l=2$) vector optical field in Kerr medium, respectively. The four columns (from left to right) correspond to the four propagation distances ($\zeta =0, 0.4, 0.6.0.73$). Each image size is ($2\lambda \times 2\lambda$).
The first and second rows display the optical intensity for the case of ($m=1, l=1$) without or with noise respectively. The third and fourth rows are corresponding to the case of ($m=2, l=2$), similarly. It has been shown that (i) the NHP-VOF ($m=l$) collapse into one deterministic filament located at the center of the optical field, corresponding to the maximum optical intensity. (ii) And the collapsing patterns are insensitive to the random noise.
3.2 $m\ne l$
By controlling $m=1$ and varying $l$, the collapsing behaviors of the optical field simulated by Eqs. (3) and (4) are studied. Although all the initial input NHP-VOF have the uniform distribution, the pre-focused fields incident on the input plane of the Kerr medium are different, as simulated results shown in Fig. 3 and Fig. 4, respectively. Here we categorize into two cases depending on the value of $l$: one for even numbers and another for odd numbers. It can be seen that NHP-VOF ($m=1$, $l=2,4$) eventually collapse into four deterministic filaments located at $\varphi =0, \pi /2, \pi, 3\pi /2$ from Fig. 3. After comparing $l=2$ with $l=4$, we find that the final filament interval at $l=4$ is larger than $l=2$. It is also evident that NHP-VOF ($m=1$, $l=3, 5$) ultimately converge to two deterministic filaments positioned at $\varphi =0, \pi$ as depicted in Fig. 4. Similarly, the filament interval at $l=5$ is larger than $l=3$. We can draw following conclusions, (i) when $l$ takes even number, the filament is formed finally at $\varphi =n\pi /2$ ($n=0, 1, 2, 3$), while $l$ takes odd number, the filament is finally located at $\varphi =n\pi$ ($n=0, 1$). (ii) When $m=1$, as $l$ increases, the filament interval also grows. (iii) The collapsing behaviors of the optical field are insensitive to the random noise.
Fig. 3. (a) Optical intensity and polarization distribution at the focal plane of NHP-VOF ($m=1$, $l=2$), (black/green/yellow, corresponding to right, linear, left polarization, respectively), (b) $m=1$, $l=4$, (c) The collapsing behaviors of the vector optical field ($m=1$, $l=2$ and $m=1$, $l=4$) in Kerr medium. The four columns (from left to right) correspond to the four propagation distances ($\zeta =0, 0.1, 0.2, 0.4$). Each image size is ($2\lambda \times 2\lambda$).
Fig. 4. (a) Optical intensity and polarization distribution at the focal plane of NHP-VOF ($m=1$, $l=3$), (black/green/yellow, corresponding to right, linear, left polarization, respectively), (b) $m=1$, $l=5$, (c) The collapsing behaviors of the vector optical field ($m=1$, $l=3$ and $m=1$, $l=5$) in Kerr medium. The four columns (from left to right) correspond to the four propagation distance ($\zeta =0, 0.1, 0.2, 0.45$). Each image size is (2$\lambda \times 2\lambda$).
3.3 Adjustment of filament interval
As shown in Fig. 5(a), the collapsing behaviors of NHP-VOF ($m=1$, $l=2$) with different initial phase are displayed. By changing the initial phase of ellipticity, the filaments can achieve rotate with a period of $\pi$. As shown in Fig. 5(c), the four deterministic filaments formed by the NHP-VOF ($m=1, l=2, \alpha _{0} =\pi /4$) are rotated by $\pi /4$ with respect to the NHP-VOF ($m=1, l=2, \alpha _{0}=0$). In addition, the filament interval formed by NHP-VOF ($m=1, l=2$) are shown in Fig. 5(b). With the increasing of the initial phase, the filament interval is significantly widen as $\alpha _{0} \in (0, \pi /2 )$. While begins to decline, as $\alpha _{0} \in (\pi /2, \pi )$. Interestingly, as $\alpha _{0} \in (\pi, 2\pi ),$ the collapsing patterns are the same as the filaments formed at $(0, \pi )$. The intervals can be expressed as $L=2r_{max}$, where $r_{max}$ denotes the radius corresponding to the position of filamentation related to the initial phase. Therefore, filament interval can be regulated by changing the initial phase of ellipticity.
Fig. 5. (a) The collapsing behaviors of the vector optical field ($m=1$, $l=2$) at different $\alpha _{0}$, (b) the relationship between the filament interval and $\alpha _{0}$ ($m=1$, $l=2$), (c) the relationship among the $I$, $S_{3}$ and the $\alpha _{0}$ ($m=1$, $l=2$).
The results indicate that the $\alpha _{0}$ is a characteristic parameter of the ellipticity in the incident optical field. The orientation and interval of the collapsing patterns can be precisely controlled by $\alpha _{0}$. These discoveries may offer new ideas in the design of optical waveguide structures, optical signal processing, etc.
4. Discussion
All the results of the simulation described above can be summarized as follows: (i) when $m=l$, the NHP-VOF eventually collapses into a stable filament at the center of the optical field. (ii) When $l$ takes even number, the filaments are formed finally at $\varphi =n\pi /2$ ($n=0, 1, 2, 3$); while $l$ takes odd number, the filaments appear ultimately in $\varphi =n\pi$ ($n=0, 1$). (iii) By changing the $\alpha _{0}$, the filaments can rotate continuously and the filaments interval can be adjusted. In order to further clarify the physical nature behind the collapsing behaviors of the NHP-VOF, the nonlinear refractive index expressed for the NHP-VOF in Kerr medium is derived from reference [43]. When the vector optical field propagates along the z-axis, considering the self-focusing nonlinear process, where the refractive index of the medium is
where $n_{0}$ represents the linear refractive index of the medium, $I$ represents the weakly focused optical intensity, $n_{2}$ represents the nonlinear refractive index intensity factor of the medium, as follows $E_{xf}$, $E_{yf}$ denote the weakly focused optical intensity which obtained from the Fresnel diffraction theorem as followsFirstly, we simulate the evolution of Stocks parameter $S_{3}$ during the collapsing behaviors of NHP-VOF ($m=1$, $l=1$) in Kerr medium, as shown in Figs. 6(a) and 6(b). For the case of $\zeta =0$, $S_{3}$ of the weakly focused field obeys $S_{3}=sin(2\alpha )=sin(2\varphi )$, shown by the green curve in Fig. 6(a); as the azimuth angle changes, the $S_{3}$ is constantly equal to 0. However, the optical intensity and the $\Delta n$ remain constantly equal to 1, shown by the red curve in Fig. 6(a). For the case of $\zeta =0.73$, the curves have no change to keep the linear polarization, the both $\Delta n$ and $I$ maintain at a maximum of 1. These features are the same as the case of $\zeta =0$ in Fig. 6(a). Due to the $\Delta n$ remains constant during the the collapsing behaviors of NHP-VOF ($m=1$, $l=1$), it eventually leads to the formation of a single stable self-focusing nucleus at the center of the optical field in Fig. 2.
Fig. 6. The evolutions of $\Delta n$, $I$, and $S_{3}$ during the collapsing behaviors of NHP-VOF. The propagation distance is (a) $\zeta =0$ ($m=1$, $l=1$), (b) $\zeta =0.73$ ($m=1$, $l=1$), (c) $\zeta =0$ ($m=1$, $l=2$), (d) $\zeta =0.4$ ($m=1$, $l=2$), (e) $\zeta =0$ ($m=1$, $l=3$), (f) $\zeta =0.45$ ($m=1$, $l=3$).
Meanwhile, we continue to illuminate the physical mechanisms behind the collapsing behaviors of the NHP-VOF ($m=1$, $l=2$). The Stocks parameter $S_{3}$, $\Delta n$, and optical intensity are drawn with green, blue, and red curves, respectively, as shown in Figs. 6(c) and 6(d). For the case of $\zeta =0$, $S_{3}$ of the weakly focused field obeys $S_{3}=sin(2\alpha )=sin(4\varphi )$ in eight azimuthal locations ($\varphi =n\pi /2, n =0,1\cdot \cdot \cdot 7$) with the linear polarization. Moreover, the both $\Delta n$ and optical intensity have four maximum values located at $\varphi =n\pi /2$ ($n=0, 1, 2, 3$). For $\zeta =0.4$, the position of linear polarization remains unchanged, but the overall value of $S_{3}$ decreases. Meanwhile, the position of the maximum value of $\Delta n$ remains unchanged compared to $\zeta =0$. And the position of maximum optical intensity also corresponds to the location of the maximum nonlinear refractive index. Drawing from the findings in Fig. 3, we observe that the ultimate filaments ($m=1$, $l=2$) occur at $\varphi =n\pi /2$ ($n=0, 1, 2, 3$), aligning with the location of the maximum nonlinear refractive index, while simultaneously adhering to conditions of linear polarization and peak optical intensity. Likewise, the filamentation regulations of NHP-VOF ($m=1$, $l=3$) as above. The evolutions of $S_{3}$, $\Delta n$, and optical intensity during the collapsing behaviors of NHP-VOF ($m=1$, $l=3$) in Kerr medium are drawn with green, blue and red curves, respectively, shown in Figs. 6(e) and 6(f). By comparing the two cases, we find that $\Delta n$ have two maximum values located at $\varphi =0,\pi$, corrsponding to the position of filaments in Fig. 4. And it also follows the conditions of linear polarization and maximum optical intensity simultaneously.
Finally, we should state that both the $\alpha$ and $\phi$ act together on optical intensity. The weakly focused field have ununiform distribution optical intensity and polarization with co-variant ellipticity and orientation, which is very different from all the previously existing kinds of optical fields. The dynamic competition between the non-uniform optical field intensity and the non-linear refractive index coefficients induces changes in the self-aggregating nuclei, so that NHP-VOF with varying values of $m$ and $l$ exhibit diverse collapsing behaviors. It is imperative to comprehensively investigate the effects of optical intensity and polarization states on the self-focusing nucleus. Importantly, $\Delta n$ is related to optical intensity and polarization states. Considering the results from the above curves completely, we can summarise as follows: (i) the parameter $\Delta n$ plays a crucial role in predicting the formation of self-focusing nucleus. The field collapse originates from the spatially variant refractive index change $\bigtriangleup n$ and requires the existence of self-focusing nucleation(s) where the induced $\bigtriangleup n$ must be local maximum. (ii) And the filament formation follows the conditions of linear polarization and maximum optical intensity simultaneously. Furthermore, it should be emphasized that the optical intensity, $S_{3}$, and $\Delta n$ have all been normalized to their respective maximum values.
Experimentally, we will follow the same method as [37] for creating the NHP-VOF required for the experimental verification. The used light source is a Ti: sapphire regenerative amplifier fs laser system (Coherent Inc.), which provides a fundamental Gaussian mode with a central wavelength of 800 nm, a pulse duration of 35 fs, and a repetition rate of 1 kHz. An achromatic 1/2-wave plate and a broadband polarized beam splitter were used to control the laser fluence input on the sample. Another achromatic 1/2-wave plate was used to change the polarization direction of the input fs laser into the vector field generation unit, where all the elements are achromatic to suppress the pulse broadening as much as possible. Based on our previous experimental results, we predict that the interaction of the NHP-VOF with Kerr media will definitely yield controllable self-focusing nucleus. Importantly, the next step will focus on experiments to accomplish controllable filaments, thereby exploring additional surprising applications.
5. Conclusion
In conclusion, we have demonstrated a new approach to obtain the controllable filaments in Kerr media by designing a kind of novel ellipticity and orientation co-variant vector optical field. It is found that both the topological charges of $\alpha$ (determined by $l$) and $\phi$ (depends on $m$) of the NHP-VOF play a crucial role in controlling the number of the collapsing patterns. On the one hand, when the $m$=$l$, a single stable filament is formed at the center of the optical field. On the other hand, when $m$=1 and $l$ is taken as even number, four filaments appear together at $\varphi =0, \pi /2 , \pi, 3\pi /2$; whereas the $l$ is taken as odd number, two similar filaments are located at $\varphi =0, \pi$. It is shown that, regardless of the $\alpha$ and $\phi$, the filaments have the robust feature insensitive to the random noise. We further investigate the dependence of the filaments on the initial angle of ellipticity, revealing that the initial phase can affect the orientation and interval of the collapsing patterns. At the result, we succeed in achieving controllable (tunable number, orientation and interval) and robust (immune to noise) filaments in nonlinear media for the first time. Physically, the formation and manipulation of these superior collapsing patterns are due to the maximum optical intensity and ellipticity of the NHP-VOF. This method provides a convenient and effective approach with more degrees of freedom to flexibly steer the field collapse and multiple filaments, which may open a new window for optical signal processing, laser machining, microwave guiding, and other related applications.
Funding
National Natural Science Foundation of China (12104333, 52106024, 11974258).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69(2), 507–574 (1997). [CrossRef]
2. C. D’Amico, A. Houard, M. Franco, et al., “Conical forward thz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007). [CrossRef]
3. J. Kasparian, P. Rohwetter, K. Stelmaszczyk, et al., “Laser–induced water condensation in air,” in The European Conference on Lasers and Electro-Optics, (Optica Publishing Group, 2011), p. CG2_5.
4. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Phys. D 157(1-2), 112–146 (2001). [CrossRef]
5. A. Braun, G. Korn, and X. D. Liu, “Self-channeling of high peak-power femtosecond laser pulses in air,” Springer Ser. Chem. Phys. 60, 248–249 (1994). [CrossRef]
6. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
7. L. Bergé, S. Skupin, R. Nuter, et al., “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 71(10), 109801 (2008). [CrossRef]
8. S. Varma, Y.-H. Chen, and H. Milchberg, “Trapping and destruction of long-range high-intensity optical filaments by molecular quantum wakes in air,” Phys. Rev. Lett. 101(20), 205001 (2008). [CrossRef]
9. J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16(1), 466–493 (2008). [CrossRef]
10. J. Kasparian, M. Rodríguez, G. Méjean, et al., “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). [CrossRef]
11. H. Kawauchi, K. Yonezawa, Y. Kozawa, et al., “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007). [CrossRef]
12. K. Lou, S. X. Qian, X. L. Wang, et al., “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20(1), 120–127 (2012). [CrossRef]
13. K. Liao, W. Wang, X. Mei, et al., “Fabrication of millimeter-scale deep microchannels in fused silica by femtosecond laser filamentation effect,” Opt. Laser Technol. 142, 107201 (2021). [CrossRef]
14. X. Zhan, H. Xu, C. Li, et al., “Remote and rapid micromachining of broadband low-reflectivity black silicon surfaces by femtosecond laser filaments,” Opt. Lett. 42(3), 510 (2017). [CrossRef]
15. J. F. Gravel, Q. Luo, D. Boudreau, et al., “Sensing of halocarbons using femtosecond laser-induced fluorescence,” Anal. Chem. 76(16), 4799–4805 (2004). [CrossRef]
16. H. Xu, Y. Cheng, S. L. Chin, et al., “Femtosecond laser ionization and fragmentation of molecules for environmental sensing,” Laser Photonics Review 9(3), 275–293 (2015). [CrossRef]
17. B. Yan, H. Liu, C. Li, et al., “Laser-filamentation-assisted 1.25 gb/s video communication under harsh conditions,” Opt. Laser Technol. 131, 106391 (2020). [CrossRef]
18. D. Thul, R. Bernath, N. Bodnar, et al., “The mobile ultrafast high energy laser facility-a new facility for high-intensity atmospheric laser propagation studies,” Opt. Lasers Eng. 140, 106519 (2021). [CrossRef]
19. Z.-Q. Nie, H. Lin, X.-F. Liu, et al., “Three-dimensional super-resolution longitudinal magnetization spot arrays,” Light: Sci. Appl. 6(8), e17032 (2017). [CrossRef]
20. Y. Zhang, X. Liu, H. Lin, et al., “Ultrafast multi-target control of tightly focused light fields,” Opto-Electron. Adv. 5(3), 210026 (2022). [CrossRef]
21. Y. Zhang, Z. Zhang, H. Lin, et al., “Dual-point noncoaxial rotational doppler effect towards synthetic oam light fields for real-time rotating axis detection,” Light: Adv. Manuf. 4, 1–11 (2023). [CrossRef]
22. G. Yao, Y. Li, and R.-P. Chen, “Collapse dynamics of vortex beams in a kerr medium with refractive index modulation and pt-symmetric lattices,” in Photonics, (MDPI, 2022), p. 249.
23. A. D. Bulygin, Y. E. Geints, and I. Y. Geints, “Vortex beam in a turbulent kerr medium for atmospheric communication,” in Photonics, (MDPI, 2023), p. 856.
24. J. Li, W. Tan, J. Si, et al., “Control of the spatial characteristics of femtosecond laser filamentation in glass via feedback-based wavefront shaping with an annular phase mask,” Opt. Express 29(4), 5972–5981 (2021). [CrossRef]
25. D. Li, J. Chang, L. Xu, et al., “Free control of filaments rotating induced by vortex femtosecond laser beams interference in fused silica,” Opt. Laser Technol. 150, 107974 (2022). [CrossRef]
26. D. Li, J. Chang, T. Xi, et al., “Filament-necklace generated by femtosecond vector beams in fused silica,” Opt. Commun. 533, 129283 (2023). [CrossRef]
27. D. V. Apeksimov, A. V. Bulygin, Y. E. Geints, et al., “Statistical parameters of femtosecond laser pulse post-filament propagation on a 65 m air path with localized optical turbulence,” J. Opt. Soc. Am. B 39(12), 3237–3246 (2022). [CrossRef]
28. A. Houard, M. Franco, B. Prade, et al., “Femtosecond filamentation in turbulent air,” Phys. Rev. A 78(3), 033804 (2008). [CrossRef]
29. Z. Jin, J. Zhang, M. Xu, et al., “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005). [CrossRef]
30. P. Sprangle, J. Penano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66(4), 046418 (2002). [CrossRef]
31. H. Gao, W. Chu, G. Yu, et al., “Femtosecond laser filament array generated with step phase plate in air,” Opt. Express 21(4), 4612–4622 (2013). [CrossRef]
32. Y. E. Geints, O. V. Minina, I. Y. Geints, et al., “Nonlinear propagation and filamentation on 100 meter air path of femtosecond beam partitioned by wire mesh,” Sensors 22(17), 6322 (2022). [CrossRef]
33. L. Xu, D. Li, J. Chang, et al., “Helical filaments array generated by femtosecond vortex beams with lens array in air,” Results Phys. 26, 104334 (2021). [CrossRef]
34. A. A. Ishaaya, L. T. Vuong, T. D. Grow, et al., “Self-focusing dynamics of polarization vortices in kerr media,” Opt. Lett. 33(1), 13–15 (2008). [CrossRef]
35. S.-M. Li, Z.-C. Ren, L.-J. Kong, et al., “Unveiling stability of multiple filamentation caused by axial symmetry breaking of polarization,” Photonics Res. 4(5), B29–B34 (2016). [CrossRef]
36. R.-P. Chen, L.-X. Zhong, K.-H. Chew, et al., “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys. 25(7), 075401 (2015). [CrossRef]
37. D. Wang, Y. Pan, J.-Q. Lü, et al., “Controlling optical field collapse by elliptical symmetry hybrid polarization structure,” J. Opt. Soc. Am. B 35(10), 2373–2381 (2018). [CrossRef]
38. D. Wang, G.-G. Liu, J.-Q. Lü, et al., “Femtosecond polarization-structured optical field meets an anisotropic nonlinear medium,” Opt. Express 26(21), 27726–27747 (2018). [CrossRef]
39. X.-Z. Gao, Y. Pan, M.-D. Zhao, et al., “Focusing behavior of the fractal vector optical fields designed by fractal lattice growth model,” Opt. Express 26(2), 1597–1614 (2018). [CrossRef]
40. O. Kosareva, T. Nguyen, N. Panov, et al., “Array of femtosecond plasma channels in fused silica,” Opt. Commun. 267(2), 511–523 (2006). [CrossRef]
41. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25(5), 335–337 (2000). [CrossRef]
42. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equation and the Schrödinger Equation (John wiley & sons, 2001).
43. D. Hutchings, J. Aitchison, and J. Arnold, “Nonlinear refractive coupling and vector solitons in anisotropic cubic media,” J. Opt. Soc. Am. B 14(4), 869–879 (1997). [CrossRef]
