Abstract
Field collapse, which occurs in various nonlinear systems, has attracted much attention, owing to its universality, complexity, and applicability. A great challenge and expectation is to achieve the controllable and designable collapsing pattern. Here we predict theoretically and demonstrate experimentally the novel collapsing behaviors of the vector optical fields in a selffocusing Kerr medium. Surprisingly, the results reveal that the collapse of the vector optical field is controllable and designable by engineering the distribution of hybrid states of polarization, and has the robust feature insensitive to the random noise. Our idea has its significance which it opens a new window for manipulating the optical field and the different kinds of field, and then facilitates to push the related researches.
Introduction
Optical field will undergo collapse at the input power above a certain critical power when an intense laser field propagates in a transparent selffocusing medium^{1,2,3,4,5}. The high intensity as the collapse exacerbates will induce other nonlinear effects, such as multiphoton and saturable absorptions, to counterbalance selffocusing, resulting ultimately in filamentation^{3,4,5,6,7,8,9,10,11,12}. Filamentation is an interesting and important topic due to its underlying physics and practical implication in many branches of physics^{13,14,15,16,17,18}. A great challenge and expectation is to tame the randomicity and then to realize the controllable and even designable filamentation. For this purpose, several methods have been proposed such as, controlling the input power and divergence angle^{19}, shaping the field profile^{20,21}, using amplitude/phase mask^{22,23,24,25}, and introducing spatial regularization^{26}.
Polarization, as an intrinsic and fundamental vectorial nature of light, plays an indispensable role in the interaction of light with matter. Trisorio and Hauri^{27} demonstrated the spatial control on two filaments using circularly polarized femtosecond laser pulses. Fibich and Ilan^{3,28} presented a theoretical prediction that under the nonparaxial condition solely, the vectorial (polarization) effect of the homogeneously linearlypolarized field can lead to two filaments. However, polarization and its diversity of spatial distribution are not yet utilized fully. For the field collapse and the multiple filamentation, besides the selffocusing effect which is a requirement for the propagation medium, a prerequisite is that the optical field should be axialsymmetry breaking in and of itself. Therefore, to achieve the active control for the field collapse and the multiple filamentation, the key points are that the axialsymmetry breaking must be controllable and even designable, and the randomicity must be effectively suppressed simultaneously. A question is raised that whether the state of polarization (SoP) of light, as a degree of freedom, can be used to break the axial symmetry of optical field and to then availably control the field collapse and multiple filamentation. Vector optical fields with spatially inhomogeneous SoP distribution^{29,30} provide indeed such an opportunity and possibility, due to the designability and diversity of SoP distribution.
Results
In this paper, we present an idea that the axialsymmetry breaking of optical field by engineering the azimuthallyvariant SoP distribution controls the collapsing filamentation. We perform, to the best of our knowledge, the first study of the collapse of the vector optical fields in a selffocusing Kerr medium. The experimental results are in excellent agreement with our theoretical prediction. Some novel features are found, including the controllable, designable and robust features. The present study can contribute further to the study of collapsing filamentation in other nonlinear systems and other kinds of fields.
An important route breaking the axial symmetry of optical field is based on the azimuthallyvariant hybridpolarized vector field (AVHPVF), which is a family of vector optical field with a “tophatlike” intensity profile excluding a central singularity caused by the polarization uncertainty^{30}. Under the weak focusing condition, the focused AVHPVF exhibits a uniformintensity focal ring instead of a focal spot^{30}. Its radially variant amplitude A(r) can be described by the generalized hypergeometric function; nevertheless, as a well approximation, A(r) can take the form of , where r_{0} is the radius of the focal ring. Inasmuch as the weak focusing does not change the SoP distribution, the focused AVHPVF in a polar coordinate system (r, ) can be written as where (m is the topological charge and is the initial phase), and and are two unit vectors indicating the horizontally and vertically polarized components, respectively. For the AVHPVF with m and , the local linear polarizations are located at a series of azimuthal positions , whereas local circular polarizations are at a series of (where n = 0, 1, ..., 4m−1). Differently from the optical vortex^{31}, the AVHPVFs carry no orbital angular momentum though they also have a strictly zero field at the central singularity^{32}. By contrast, an azimuthallyvariant linearlypolarized vector field (AVLPVF) can be written as follows Compared Eq. (2) with Eq. (1), it can be found the unique difference that the horizontally and vertically polarized components have a phase difference of −π/2 for AVHPVF, while are in phase for AVLPVF.
To theoretically explore the nonlinear propagation behavior of a vector optical field in a Kerr medium, the (2+1)dimensional vectorversion nonlinear Schrödinger (NLS) equation should be used. Under the slowly varying amplitude approximation, the vectorversion NLS equation can be divided into a pair of coupled NLS equations for two orthogonal horizontal and vertical components where is the nondimensional qcomponent normalized by the total field, . , ζ = z/L_{d} (L_{d} = πr_{0}^{2}/λ) are the nondimensional cylindrical coordinates. is the power, P_{C} = αλ^{2}/4πn_{0}n_{2} is the critical power for selffocusing, n_{0} and n_{2} are the linear and nonlinear refractive indices of the Kerr medium, α is a constant dependent on the initial field shape^{33}, c is the speed of light in vacuum, and λ is the wavelength, respectively. On the righthand side of Eq. (3), the first term represents the contribution from the diffraction described by the transverse Laplacian , and the second term is from the Kerr nonlinearity.
We performed a large number of simulations using Eq. (3) under P = 10P_{C} in a selffocusing Kerr medium. When the spatial random noise is added in simulation, the random noise with a level of ±10% amplitude is used. The first and second rows in Fig. 1 illustrate the simulation results for the collapsing behavior of the focused AVHPVF (with m = 1 and ) described by Eq. (1), without and with random noise, respectively. By comparison, the third and fourth rows in Fig. 1 show the simulation results for the AVLPVF (with m = 1 and ) described by Eq. (2), which is the wellknown radiallypolarized vector field (RPVF)^{29}, without and with random noise. Although both AVHPVF and RPVF display the same focusing ring, their nonlinear propagation behaviors are quite different. Figure 2 shows the simulated collapsing patterns of the AVHPVFs with m = 1 ( and π/4) and m = 2 ( and π/8) when the spatial random noise is added. Summarizing all the simulation results performed, we can draw following conclusions. For the AVHPVF with the given m and , (i) the azimuthallyvariant hybrid SoP distribution leads to the axialsymmetry breaking; (ii) the field undergoes the collapse to converge into the deterministic filaments; (iii) the number of filaments is 4m depending solely on m; (iv) the filaments are always located at the azimuthal positions , where their local SoPs are linearly polarized; (v) the collapsing pattern exhibits a 4mfold rotation (C_{4m}) symmetry; and (vi) the collapsing patterns persist among simulations, with and without different random noise, implying that collapsing patterns are insensitive to noise. Meanwhile, for the AVLPVFs, (i) the purely ideal AVLPVF always remains in the axially symmetric ring, indicating that the purely ideal AVLPVF cannot converge to the deterministic multiple filaments; and (ii) the collapsing filaments produced by AVLPVF with the random noise has the uncertainty, similar with the collapses of optical vortices^{31} and polarization vortices^{34}.
To experimentally confirm our theoretical prediction and numerical simulations, all the required vector fields with different m and were created by using the experimental scheme^{30}. The created AVHPVFs and AVLPVFs were weakly focused into a uniformintensity focal ring and then incident into the CS_{2} cell with a length of 10 mm, by an achromatic lens with a focal length of f = 300 mm. The intensity patterns of the field transmitted from the CS_{2} cell are imaged on a detector (Beamview, Coherent Inc.) using an achromatic lens with a focal length of f = 60 mm. The experimental results of the AVLPVFs (not shown here) reveal that the number and positions of the induced collapsing filaments have the uncertainty among the experiments, depending on the spatial random fluctuation of the input field. Any one of the measured collapsing patterns of the m = 1 charge AVHPVFs, with four different ( = 0, π/8, π/4, 3π/8), consists of four deterministic filaments and exhibits a fourfold rotation (C_{4}) symmetry (Fig. 3). The four patterns have no difference in nature, whereas their unique difference is that the patterns from Fig. 3a–d are rotated counterclockwisely in sequence by an angle of π/8. Figure 4 shows the measured collapsing patterns of AVHPVFs (with ) for m = 1, 2, 3, 4. The mcharge AVHPVF produces 4m deterministic filaments, where m = 1, 2, 3, 4, with the number of filaments being 4, 8, 12, 16, respectively. The collapsing pattern exhibits a 4mfold rotation (C_{4m}) symmetry. In particular, the positions of filaments for any AVHPVF always coincide with those of the local linear polarizations in the azimuthal dimension. The collapsing patterns of the AVHPVF with a given m and persist among experiments, implying that the collapsing patterns are indeed insensitive to the ransom noise. Moreover, the collapsing pattern of the AVHPVF is controllable and designable by engineering the spatial hybrid SoP structure.
Discussion
To understand the physics behind the collapse of AVHPVF, a crosscoupling model is presented (Fig. 5). Multiple filamentation requires the simultaneous spatial confinements in both the radial and azimuthal dimensions. The selfconfinement in the radial dimension is due to the focal ring. However, it is of great interest and of crucial importance to reveal the mechanism behind the axial symmetry breaking that leads to the multiple filamentation. Therefore, we ignore the radial terms and in the transverse Laplacian , while concentrate only on the azimuthal term in Eq. (3). The Kerr medium is divided into N equallength thin segments (each thin segment has a normalized length of Δζ = L/N), so that the nondepletion approximation is valid within any thin segment. After the focused AVLPVF described in Eq. (2) passing though the 1st segment, based on Eq. (3), and can be expressed as follows thereby obtaining where and . Based on Eqs. (4) and (5), we can find: (i) the horizontal and vertical components are always held in phase, implying that spatial SoP distribution has no change; and (ii) the intensity distribution still exhibits axial symmetry. Therefore, a purely ideal AVLPVF can never produce the deterministic multiple filamentation, which is in excellent agreement with the simulation (the third row of Fig. 1). As such, any azimuthal position is equivalent or indistinguishable. In the presence of azimuthal perturbations, axial symmetry breaking will lead to the filamentation^{1,2,3,4,5} (also see the fourth row of Fig. 1).
After the AVHPVF described in Eq. (1) passing through the 1st segment, based on Eq. (3), and can be expressed as follows thereby obtaining where Equation 7 suggests that the nonlinear crosscoupling results in the energy redistribution of AVHPVF in the azimuthal dimension (or the axial symmetry breaking). It should be pointed out that the intensity distribution still holds the axial symmetry in the absence of nonlinearity (). As an example, the AVHPVF with m = 1 and is considered (Figs. 5a–d). In the input plane (ζ = 0), the focused AVHPVF has the azimuthallyinvariant intensity distribution, while the azimuthallyvariant hybrid SoP distribution (Fig. 5a). After the 1st segment, the crosscoupling results in the energy exchange in the azimuthal dimension toward the locations at from those at (Fig. 5b). As the propagation distance increases, more energy is transferred toward the locations at after the jth segment (Fig. 5c). In the output plane (Fig. 5d), the crosscoupling leads to the energy almost converging into the azimuthal locations at . The axial symmetry of the focused AVHPVF is broken ultimately to form four deterministic filaments. From Eq. 6, the crosscoupling acquires an additional phase, which makes the phase difference between the horizontal and vertical components to not always preserve π/2, resulting in the SoP evolution in contrast to AVLPVF. However, at two series of azimuthal locations, and , the SoPs have no change. The simulated evolution of the azimuthalvariant hybrid SoPs, by the Stokes parameter S_{3} characterizing the ellipticity^{30}, during the nonlinear propagation of the AVHPVF with m = 1 and confirmed the above prediction (Fig. 5e). As propagation distance increases, the local ellipticity has a tendency to decrease (i.e, the local polarization ellipse will become flatter). Except for at the special azimuthal locations of and , the SoPs have indeed no change, like the above discussion from the crosscoupling model.
To understand the collapsing filaments, the azimuthal selffocusing model is also very beneficial (Fig. 6). The selffocusing phenomenon originates from the spatially variant refractive index change. In an isotropic Kerr medium under the same intensity, the induced refractive index change depends on the SoP of optical field (Δn^{lin} > Δn^{ell} > Δn^{cir}), where Δn^{lin}, Δn^{ell}, and Δn^{cir} are the refractive index changes induced by the linearly, elliptically, and circularly polarized light, respectively^{35}. As a result, the refractive index change induced by the AVHPVF is azimuthally variant, leading to the axialsymmetry breaking. The local linear and circular polarizations are located at and (where n = 0, 1, 2, 3) for the AVHPVF with m = 1 and (Fig. 6a). Therefore, the induced four azimuthal selffocusing lenses have their centers located at and their edges located at (Fig. 6b). The input field undergoes the azimuthal multiplenucleation selffocusing, making the energy converge toward the four locations at (Fig. 6c). As the propagation distance increases, the “focal lengths” of the induced azimuthal selffocusing lenses become shorter (Fig. 6d), and then the energy converges ultimately into the four deterministic filaments (Fig. 6e). In particular, Δn^{lin} is greater than 50% Δn^{cir}, which is the reason why AVHPVF can suppress the random noise (the robust feature) during the deterministic collapsing filamentation.
We have presented the theoretical prediction and conducted the experimental evidence on the novel collapsing behaviors of AVHPVF in the selffocusing Kerr medium. The azimuthallyvariant hybrid SoP structure is crucial in the axial symmetry breaking. The collapsing filamentation has the controllable and robust features. The mcharge AVHPVF produces 4m filaments with 4mfold rotation (C_{4m}) symmetry. The number of multiple filaments is a function solely of m, and their locations are determined by both m and . Our results suggest that engineering the hybrid SoP structure can tame the randomicity and then realize the designable collapsing filamentation. This study only concentrated on the collapsing behavior; thus, the higherorder nonlinearity, which is important in the postcollapse process, has been neglected. Of course, we also simulated the multiple filamentation when the higherorder Kerr effect is considered, the results indicate a novel feature that the relative phase difference among the multiple filaments is stable and insensitivity to the input fluctuations of the pulse energy, which is quite different from the reported fact that the relative phase difference is instable and sensitivity to the small fluctuations of the pulse energy, i.e., loss of phase of collapse^{36}. Our idea may offer an alternative route to produce the controllable and robust multiple filamentation in other nonlinear systems, thereby facilitating the development of additional surprising applications. This work has the great significance that it not only opens a new window for manipulating the optical field and facilitates to push the related researches, but also has a certain reference value for manipulating the different kinds of field.
Methods
Generation of femtosecond vector fields
We follow the same method as Ref. 30 for creating the vector fields required in this study. The used light source is a Ti:sapphire regenerative amplifier femtosecond 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 incident laser fluence on the sample surface. Another achromatic 1/2 wave plate was used to change the polarization direction of the incident femtosecond laser into the vector field generation unit, where all the elements are achromatic to suppress pulse broadening as much as possible. The created AVHPVFs and AVHPVFs have a “tophatlike” spatial profile, excluding a central singularity caused by the SoP uncertainty, with a radius of 1.5 mm, a pulse energy of 6.0 μJ, and a pulse duration of ~65 fs.
References
 1.
Robinson, P. A. Nonlinear wave collapse and strong turbulence. Rev. Mod. Phys. 69, 507–573 (1997).
 2.
Bergé, L. Wave collapse in physics: principles and applications to light and plasma waves. Phys. Reports 303, 259–370 (1998).
 3.
Fibich, G. & Ilan, B. Vectorial and random effects in selffocusing and in multiple filamentation. Physica D 157, 112–146 (2001).
 4.
Couairon, A. & Mysyrowicz, A. Femtosecond filamentation in transparent media. Phys. Reports 441, 47–189 (2007).
 5.
Bergé, L., Skupin, S., Nuter, R., Kasparian, J. & Wolf, J. P. Ultrashort filaments of light in weakly ionized, optically transparent media. Rep. Prog. Phys. 70, 1633–1713 (2007).
 6.
Chiao, R. Y., Garmire, E. & Townes, C. H. Selftrapping of optical beams. Phys. Rev. Lett. 13, 479–482 (1964).
 7.
Bespalov, V. I. & Talanov, V. I. Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3, 307–312 (1966).
 8.
Kelley, P. L. Selffocusing of optical beams. Phys. Rev. Lett. 15, 1005–1008 (1965).
 9.
Varma, S., Chen, Y. H. & Milchberg, H. M. Trapping and destruction of longrange highintensity optical filaments by molecular quantum wakes in air. Phys. Rev. Lett. 101, 205001 (2008).
 10.
Braun, A., Korn, G., Liu, X., Du, D., Squier, J. & Mourou, G. Selfchanneling of highpeakpower femtosecond laser pulses in air. Opt. Lett. 20, 73–75 (1995).
 11.
Kasparian, J. & Wolf, J. P. Physics and applications of atmospheric nonlinear optics and filamentation. Opt. Express 16, 466–493 (2008).
 12.
Kasparian, J., Rodriguez, M., Méjean, G., Yu, J., Salmon, E., Wille, H., Bourayou, R., Frey, S., André, Y. B., Mysyrowicz, A., Sauerbrey, R., Wolf, J. P. & Wöste, L. Whitelight filaments for atmospheric. Science 301, 61–64 (2003).
 13.
Kolesik, M., Wright, E. M. & Moloney, J. V. Femtosecond filamentation in air and higherorder nonlinearities. Opt. Lett. 35, 2550–2552 (2010).
 14.
D'Amico, C., Houard, A., Franco, M., Prade, B., Mysyrowicz, A., Couairon, A. & Tikhonchuk, V. T. Conical forward THz emission from femtosecondlaserbeam filamentation in air. Phys. Rev. Lett. 98, 235002 (2007).
 15.
Alfano, R. R. & Shapiro, S. L. Emission in the region 4000 to 7000 Å via fourphoton coupling in glass. Phys. Rev. Lett. 24, 584–587 (1970).
 16.
Hauri, C. P., Kornelis, W., Helbing, F. W., Heinrich, A., Couairon, A., Mysyrowicz, A., Biegert, J. & Keller, U. Generation of intense, carrierenvelope phaselocked fewcycle laser pulses through filamentation. Appl. Phys. B 79, 673–677 (2004).
 17.
Rohwetter, P., Kasparian, J., Stelmaszczyk, K., Hao, Z., Henin, S., Lascoux, N., Nakaema, W. M., Petit, Y., Queißer, M., Salamé, R., Salmon, E., Wöste, L. & Wolf, J. P. Laserinduced water condensation in air. Nature Photon 4, 451–456 (2010).
 18.
Belgiorno, F., Cacciatori, S. L., Clerici, M., Gorini, V., Ortenzi, G., Rizzi, L., Rubino, E., Sala, V. G. & Faccio, D. Hawking radiation from ultrashort laser pulse filaments. Phys. Rev. Lett. 105, 203901 (2010).
 19.
Jin, Z., Zhang, J., Xu, M. H., Lu, X., Li, Y. T., Wang, Z. H., Wei, Z. Y., Yuan, X. H. & Yu, W. Control of filamentation induced by femtosecond laser pulses propagating in air. Opt. Express 13, 10424–10430 (2005).
 20.
Dubietis, A., Tamošauskas, G., Fibich, G. & Ilan, B. Multiple filamentation induced by inputbeam ellipticity. Opt. Lett. 29, 1126–1128 (2004).
 21.
Grow, T. D. & Gaeta, A. L. Dependence of multiple filamentation on beam ellipticity. Opt. Express 13, 4594–4599 (2005).
 22.
Pfeifer, T., Gallmann, L., Abel, M. J., Neumark, D. M. & Leone, S. R. Circular phase mask for control and stabilization of single optical filaments. Opt. Lett. 31, 2326–2328 (2006).
 23.
Kandidov, V. P., Aközbek, N., Scalora, M., Kosareva, O. G., Nyakk, A. V., Luo, Q., Hosseini, S. A. & Chin, S. L. A method for spatial regularisation of a bunch of filaments in a femtosecond laser pulse. Quantum Electronics 34, 879–880 (2004).
 24.
Hao, Z. Q., Stelmaszczyk, K., Rohwetter, P., Nakaema, W. M. & Woeste, L. Femtosecond laser filamentfringes in fused silica. Opt. Express 19, 7799–7806 (2011).
 25.
Hao, Z. Q., Zhang, J., Xi, T. T., Yuan, X. H., Zheng, Z. Y., Lu, X., Yu, M. Y., Li, Y. T., Wang, Z. H., Zhao, W. & Wei, Z. Y. Optimization of multiple filamentation of femtosecond laser pulses in air using a pinhole. Opt. Express 15, 16102–16109 (2007).
 26.
Kandidov, V. P., Akozbek, N., Scalora, M., Kosareva, O. G., Nyakk, A. V., Luo, Q., Hosseini, S. A. & Chin, S. L. Towards a control of multiple filamentation by spatial regularization of a highpower femtosecond laser pulse. Appl. Phys. B 80, 267–275 (2005).
 27.
Trisorio, A. & Hauri, C. P. Control and characterization of multiple circularly polarized femtosecond filaments in argon. Opt. Lett. 32, 1650–1652 (2007).
 28.
Fibich, G. & Ilan, B. Deterministic vectorial effects lead to multiple filamentation. Opt. Lett. 26, 840–842 (2001).
 29.
Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Adv. Opt. Photon. 1, 1–57 (2009).
 30.
Wang, X. L., Li, Y. N., Chen, J., Guo, C. S., Ding, J. P. & Wang, H. T. A new type of vector fields with hybrid states of polarization. Opt. Express 18, 10786–10795 (2010).
 31.
Vuong, L. T., Grow, T. D., Ishaaya, A. A., Gaeta, A. L., 't Hooft, G. W., Eliel, E. R. & Fibich, G. Collapse of optical vortices. Phys. Rev. Lett. 96, 133901 (2006).
 32.
Wang, X. L., Chen, J., Li, Y. N., Ding, J. P., Guo, C. S. & Wang, H. T. Optical orbital angular momentum from the curl of polarization. Phys. Rev. Lett. 105, 253602 (2010).
 33.
Fibich, G. & Gaeta, A. L. Critical power for selffocusing in bulk media and in hollow waveguides. Opt. Lett. 25, 335–337 (2000).
 34.
Ishaaya, A. A., Vuong, L. T., Grow, T. D. & Gaeta, A. L. Selffocusing dynamics of polarization vortices in Kerr media. Opt. Lett. 33, 13–15 (2008).
 35.
Sutherland, R. L. Handbook of Nonlinear Optics (second Edition, Marcel Dekker, Inc., 2003).
 36.
Shim, B., Schrauth, S. E., Gaeta, A. L., Klein, M. & Fibich, G. Loss of phase of collapsing beams. Phys. Rev. Lett. 108, 043902 (2012).
Acknowledgements
This work was supported by the 973 Program of China under Grant No. 2012CB921900, and the National Natural Science Foundation of China under Grants 10934003.
Author information
Affiliations
MOE Key Laboratory of Weak Light Nonlinear Photonics and School of Physics, Nankai University, Tianjin 300071, China
 SiMin Li
 , Yongnan Li
 , XiLin Wang
 , LingJun Kong
 , Kai Lou
 , Chenghou Tu
 & HuiTian Wang
National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
 HuiTian Wang
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China
 Yongjun Tian
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Contributions
S.M. Li presented idea, performed the experiments, carried out simulations, and wrote the manuscript. Y.N. Li assisted with the experiments and the discussion of mechanism. X.L. Wang presented the method for generating the vector fields. L.J. Kong and K. Lou assisted with the experiments and the discussion. C.H. Tu provided technical support. Y.J. Tian assisted with the discussion of mechanism and wrote the manuscript. H.T. Wang planned the project, presented the idea and the mechanism, guided and supervised the simulations and experiments, and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to HuiTian Wang.
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