Vectorial effect of hybrid polarization states on the collapse dynamics of a structured optical field

Rui-Pin Chen; Khian-Hooi Chew; Guoquan Zhou; Chao-Qing Dai; Sailing He

doi:10.1364/OE.24.028143

1. Introduction

Nonlinear wave collapse has been investigated in many areas of physics, including optics, fluidics, plasma physics, and Bose-Einstein condensates [1–3]. The beams collapse in a transparent, self-focusing medium when the input power surpasses a certain critical power. Alternatively, when the initial power is too high, other higher-order processes such as the multiple photon absorption attenuating the core of the beam, plasma generation, or higher-order defocusing effects may halt the collapse [1, 2]. The beam undergoes a self-guided propagation with the focusing-defocusing cycles in the form of a filament over a long range because its power is still above the critical power [3]. The collapse dynamics of a scalar field with solely homogeneous polarization such as linear polarization, circular polarization, or elliptical polarization in the field cross-section have been theoretically and experimentally investigated in numerous contexts [4–8], especially for some novel optical beams such as Airy beams, vortices, and ultra-short laser pulses.

Compared to a scalar beam with the spatially uniform state of polarization (SoP) such as linear, circular, or elliptical polarization in the cross-section of the optical field, the vector beam involves spatially variant SoP [9–11]. Many novel and interesting phenomena have been found [12–16] in the vector optical field with spatially variant linear polarization [12, 13, 16] (e.g. the azimuthal and radial polarization) and the hybrid SoP [14] (i. e. the linear, circular and elliptical polarizations are located at different positions in the cross section of the optical field). The vectorial effect of the distribution of different SoP on linear and nonlinear dynamics has attracted a lot of interest due to the potential applications and fundamental relevance in many branches of physics [15], such as flattop generation, focus engineering, optical tweezing and surface plasmons. A. A. Ishaaya et al. [16] demonstrated the collapse dynamics and polarization stability of radially and azimuthally polarized beams in a pure Kerr medium. In addition, vortices have been a subject of study in many branches of physics [5, 6, 15, 17]. The structured beam with vortices and polarization singularities has led to many interesting phenomena such as polarization rotation, the interaction of the singularities of polarization and vortex, and manipulation of orbital angular momentum [10, 14]. The structured optical field having complex wavefront structures enables novel properties and applications such as zero diffraction, self-healing and curve propagation, single-molecule spectroscopy, nanoscale focusing, and even particle acceleration and manipulation [15]. Recently, the propagation dynamics of a structured optical field that combine the spiral phase (i.e, vortex) and hybrid SoP in the cross-section of the field have been demonstrated [10]. The SoP in the field cross section rotates along the propagation axis due to the existence of a vortex.

The collapse of optical vortices in nonlinear optical media has attracted immense attention. Li et al. studied the collapse of vector optical fields in a Kerr medium [18]. Their work suggests that the axial symmetry breaking of optical field by engineering the spatial SoP structure can be an efficient approach for controlling the optical field collapse in Kerr media. Recently, we investigate the collapse dynamics of a vector optical field with hybrid SoP [19]. Our study reveals the evolution of the SoP during self-focusing process and the optical field tends to collapse at the position with linearly polarized component in Kerr media. Despite a number of studies on the collapse of optical fields in nonlinear media [5–8, 16–19], less work has been performed on the collapse dynamics in singular points of polarization and vortex in a structured optical field. This motivates the present study in which the collapse dynamics of a structured optical field with hybrid SoP and a spiral phase in the field cross section are studied. In particular, the two collapse regions associated with an on-axis collapse and a multi-part off-axis collapse of a structured optical field with the same polarization and vortex topological charge number are examined in detail. We found that the coupling between the vortex field and the hybrid SoP strongly affects the nonlinear dynamic properties of the structured beam in both the Kerr and saturated media. In particular, the interaction between the central phase singular point (i.e. dislocation) and the polarization singular point (i.e. disclination) leads to the optical field annihilation or creation in the beam center, depending on the numbers of topological charges of vortex and polarization. The linearly, circularly, or elliptically polarized component may exist in the beam center when the number of topological charge of vortex equals to that of topological charge of polarization. Therefore, the collapse may occur in the center of the optical field when apse may occur in the center of the optical field when the numbers of topological charges vortex and polarization are equal. The collapse competition between different SoPs gives rise to a single on-axis collapse or multiple off-axis partial collapses. Furthermore, the collapsing beams can rotate along the propagation axis during propagation in the Kerr medium or the saturated medium due to the existence of the vortex. Numerical results further confirm that the SoP in the cross-section of field also propagates with a spiral trajectory. The cores of collapsing beams are dominated by the linearly-polarized component during propagation.

2. Theoretical formulation

In order to gain insight into the behavior of collapsing beams of the vector vortex optical field, the two-dimensional coupled nonlinear Schrӧdinger (NLS) equations are employed to investigate the nonlinear dynamics behavior of a vector optical field [20, 21]:

\nabla_{⊥}^{2} E_{\pm} - 2 i k \frac{\partial E_{\pm}}{\partial z} + \frac{4 n_{2} k^{2}}{3 n_{0}} ({| E_{\pm} |}^{2} + 2 {| E_{\mp} |}^{2}) E_{\pm} - \frac{2 n_{4} k^{2}}{5 n_{0}} ({| E_{\pm} |}^{4} + 6 {| E_{\pm} |}^{2} {| E_{\mp} |}^{2} + 3 {| E_{\mp} |}^{4}) E_{\pm} = 0,

where the transverse Laplacian

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

, k is the linear wave number. z is the longitudinal coordinate, n₀ is the linear refraction index of the medium, n₂ is the third order nonlinear coefficient, and n₄ is the fifth order nonlinear coefficient. Here, the fifth-order nonlinearity as the nonlinear saturation term has been added in the NLS equations to capture the collapse of optical field and to allow the propagation of beam across the collapse point [20–22]. In Eq. (1),

E_{\pm} = (E_{x} \pm i E_{y}) / \sqrt{2}

with E_x, E_y are the linearly-polarized components in the x- and y - direction, respectively, whereas E₊ and E_- represent the left and right circular polarization states, respectively. The coupled NLS equations have been extensively studied [23, 24]. Two important invariants of the NLS equation are the beam power

P (z) = \iint_{s} ({| E_{+} |}^{2} + {| E_{-} |}^{2}) d x d y,

and the Hamiltonian

\begin{array}{l} H (z) = \frac{1}{4 k^{2}} \iint_{s} [{| \nabla E_{+} |}^{2} + {| \nabla E_{-} |}^{2} - \frac{2 k^{2} n_{2}}{3 n_{0}} ({| E_{+} |}^{4} + {| E_{-} |}^{4} + 4 {| E_{+} |}^{2} {| E_{-} |}^{2}) \\ + \frac{k^{2} n_{4}}{15 n_{0}} (2 {| E_{+} |}^{6} + 2 {| E_{-} |}^{6} + 6 {| E_{+} |}^{4} {| E_{-} |}^{2} + 6 {| E_{+} |}^{2} {| E_{-} |}^{4})] d x d y, \end{array}

The beam width in the root-mean-square (rms) sense can be defined as

I (z) = \iint_{s} (x^{2} + y^{2}) ({| E_{+} |}^{2} + {| E_{-} |}^{2}) d x d y,

The solutions of the NLS equation satisfy the relation [7, 24]:

I (z) = I (z = 0) + \frac{d I (z = 0)}{d z} z + 4 H (z = 0) z^{2},

Equation (5) shows the evolution of the rms beam width of a vector vortex optical field in a nonlinear medium. For a structured optical field with azimuthally inhomogeneous SoP distribution in the field cross-section as an initial field distribution [9–12], we have:

\begin{array}{l} E (x, y; z = 0) = A_{0} r^{n} \exp (- r^{2} / w_{0}^{2}) (\cos (m φ + φ_{0}) e_{x} + \sin (m φ + φ_{0}) \exp (i Δ θ) e_{y}) \exp (i n φ) \\ = A_{0} r^{n} \exp (- r^{2} / w_{0}^{2}) ([\cos (m φ + φ_{0}) + \exp (i π / 2 + i Δ θ) \sin (m φ + φ_{0})] e_{+} \\ + [\cos (m φ + φ_{0}) - \exp (i π / 2 + i Δ θ) \sin (m φ + φ_{0})] e_{-}) \exp (i n φ) / \sqrt{2}, \end{array}

where A₀ is a constant, w₀ is the beam waist associated with the Gaussian part, r and φ are the polar and azimuthal coordinates, and φ₀ is the initial phase, respectively; m and n denote the topological charges related to polarization and vortex, respectively. e_x and e_y denote the x- and y- direction unit vectors, e₊ and e_- denote left-handed (LH) and right-handed (RH) unit vectors, respectively. If Δθ = 0, it can be seen from Eq. (6) that the x- component has the same phase as that of y- component. It is always linearly polarized at any position in the cross section of the vector optical field. The directions of linearly polarization depend only on the azimuthal angle φ. When n = 0, m = 1, and φ₀ = 0 or π/2, Eq. (6) gives the radially- or azimuthally-polarized optical fields [9–12]. When Δθ ≠ 0 (Eq. (6)), the x- and y- components exhibit different phases. This indicates a hybrid-polarized optical field with the linear, elliptical and circular SoP located at different positions in the cross-section of the vector field. When H(z) = 0 (see Eq. (5)) the rms beam width becomes constant, implying that a total balance is formed between the self-focusing effect, the defocusing effect of diffraction, and the higher (fifth) order effect. For a Kerr medium (i.e. n₄ = 0), a competition exists between the self-focusing effect of Kerr nonlinearity and the defocusing effect of diffraction to determine the critical power of the structured optical field. The critical power of an optical field in a Kerr medium indicates a total balance between the self-focusing effect and the defocusing effect of diffraction. The critical power of the vector vortex beam in a Kerr medium, P_cr, can be obtained from Eqs. (2)-(6) by setting H(z = 0) = 0 and n₄ = 0 [2,7]. The corresponding critical amplitude A₀ is calculated using Eq. (3) with H(z = 0) = 0 and n₄ = 0, and then the critical power is obtained analytically by integrating the Eq. (2) [19].

P_{c r} = \frac{6}{5} \frac{4^{n} (m^{2} + n) n! Γ [n]}{(2 n)!} P_{G},

where m ≠ 0 and Γ[.] denotes the Gamma function. P_G = 2πn₀/(n₂k²) is the critical power of a Gaussian beam. Equation (7) clearly indicates that the critical power of a vector vortex beam depends mainly on the topological charges of both the polarization and the vortex. It should be noted that the critical power of Eq. (7) is obtained with m ≠ 0 and the integral in Eq. (3)

\int_{0}^{2 π} s i n^{2} (m φ + φ_{0}) = 1 / 2

(φ₀ is any certain value). However, if m = 0 and φ₀ = 0, the integral in Eq. (3)

\int_{0}^{2 π} s i n^{2} (m φ + φ_{0}) = 0

, then the critical power of a structured optical field degenerates to the critical power of a vortex optical field with linear polarization, as demonstrated in [17, 25]. When m = 0 and n = 0, the critical power of a linearly polarized Gaussian beam can be obtained from Eqs. (2)-(6) P_G = 2πn₀/(n₂k²).

When the initial power of the optical field equals the critical power, the beam rms width of the optical field remains invariant as recognized from Eq. (5). As the initial power exceeds the critical power, the beam rms width decreases during propagation and goes to zero at a certain propagation distance, leading to a global collapse [26]. However, the critical power P_cr can predict correctly the occurrence of the actual collapse only for the Townes profile [4, 26]. In general, the partial collapses occur at certain positions in the field cross-section before the rms beam width remains constant [5, 7, 26]. Thus, the critical power P_cr corresponds to the upper bound for the actual collapse. Therefore, the evolution of the beam and the occurrence of the partial collapse in the field cross-section during propagation in the Kerr medium can only be explored numerically and experimentally. In practice, the beam parameters such as the field distribution, SoP and phase should be considered carefully to determine the occurrence of collapse.

3. On-axis versus off-axis collapses of a structured optical field with a hybrid SoP and a spiral phase in a Kerr medium

We first examine the evolution and collapse of a structured optical field in a Kerr medium. Let us take λ = 0.53μm and w₀ = 10μm. The results are obtained numerically using the split-step Crank-Nicolson finite-difference method. The intensity distributions of the structured beams with different initial powers in the Kerr medium at different propagation distances are shown in Figs. 1-2. Diffraction, vortex, and the spatially-variant nonlinear refractive index change resulted from the different self-focusing effect between the linearly, elliptically, and circularly polarized components, leading to a redistribution of the intensity in the structured optical field. If a hybrid SoP exists, the energy tends to accumulate at the local position of linearly polarized component rather than the circularly polarized component because of the different nonlinear refractive index change of the linear and circular polarizations, as given by $Δ n_{\pm} = n_{2} ({| E_{\pm} |}^{2} + 2 {| E_{\mp} |}^{2}) / 3 n_{0}$ . Thus, the field-induced nonlinear refractive index change can be obtained: Δn^lin > Δn^eli >Δn^cir [2] where Δn^lin, Δn^eli and Δn^cir represent the change of the refractive index of the linearly, elliptically, and circularly polarized components, respectively.

Fig. 1 (a) The polarization states distribution in the field cross-section of the initial field with n = 1, m = 1, Δθ = π/2 and φ₀ = 0. The intensity distribution with the propagation distance for different initial powers: (b) P_in = 2.3P_G, (c) P_in = 5.3P_G. (d) The corresponding normalized peak intensity as a function of propagation distance for different initial powers. The Stokes polarization parameters for input powers (e) P_in = 2.3P_G and (f) P_in = 5.3P_G at different propagation distances. Positive and negative S1 (or S2) Stokes values represent horizontal (or 45°) and vertical (or 135°) linear polarization components, respectively, whereas positive and negative S3 values represent opposite circular components. Black circles indicate the corresponding locations of the collapse. Note here that P_in = 2.3P_G and 5.3P_G are the threshold power for the occurrence of an on-axis and off-axis partial collapses.

Download Full Size | PDF

Fig. 2 the evolution of on-axis intensity in the structured optical field as a function of propagation distances for a pure linear propagation or a nonlinear propagation with different initial powers. (a) n = m = 1, P_in = 1.15P_G; (b) n = m = 1, P_in = 2.3P_G; (c) n = m = 2, P_in = 1.45P_G; (d) n = m = 2, P_in = 2.9P_G.

Download Full Size | PDF

One interesting phenomenon is the interaction between the central phase singular point and the polarization singular point. The structured optical field described by Eq. (6) can be rewritten as:

\begin{array}{l} A_{0} r^{n} \exp (- r^{2} / w_{0}^{2}) [\cos (m φ + φ_{0}) e_{x} + \sin (m φ + φ_{0}) \exp (i Δ θ) e_{y}] \exp (i n φ) \\ = A_{0} r^{n} \exp (- r^{2} / w_{0}^{2}) [\exp (i (n + m) φ + i φ_{0}) + \exp (i (n - m) φ - i φ_{0})] e_{x} / 2 \\ - i A_{0} r^{n} \exp (- r^{2} / w_{0}^{2}) [\exp (i (n + m) φ + i φ_{0}) - \exp (i (n - m) φ - i φ_{0})] \exp (i Δ θ) e_{y} / 2. \end{array}

As seen from Eq. (8), when

n \neq \pm m

, there is always optical singularity. However, the optical field can be regarded as the superposition of a vortex component and a non-vortex component with different linear polarizations if

n = \pm m

. Therefore, the energy accumulate at the center of the optical field due to the linear diffraction and self-focusing effects. Single collapse can occur at the optical field center if the acculumated on-axis power reaches a threshold value, whereas the locally initial power in the outside ring is not strong enough to induce the partial collapses at the periphery. For example, the single collapse occurs at the field center when the initial power P_in = 2.3P_G with the hybrid SoP (n = m = 1, Δθ = π/2), as shown in Fig. 1(b). However, as the initial power increases to P_in = 5.3P_G four partial collapses occur at the positions in the ring of the optical field with linearly-polarized components, whereas no collapse occurs at the center of the field. This is because the intensity distribution in the outside ring is stronger than the accumulated intensity in the beam center, which leads to the partial collapses at the ring preceding the collapse in the center of the optical field, as shown in Fig. 1(c). However, if the initial power is 2.3P_G < P_in < 5.3P_G, the local power in the ring is not strong enough to induce the 4 off-axis partial collapse at the periphery. In this case, the energy of the non-vortex component accumulates at the center of the optical field due to the linear diffraction and self-focusing effects, leading to an on-axis collapse with a propagation distance longer than that of the off-axis collapse. The numerical results of the critical powers for the occurrence of single on-axial collapse or multiple off-axial collapse when n =± m and Δθ = π/2 are shown in Table 1.

Table 1. Critical powers for single on-axial collapse and multiple off-axial collapse when n =± m, Δθ = π/2 and φ₀ = 0

View Table

As recognized from Eqs. (6) and (8), the initial intensity at the center of the optical field is zero. The non-vortex component of the ring-shaped beam will accumulate power to on-axis position because of the linear diffraction effect. In order to clearly understand the contributions of the linear diffraction and nonlinear self-focusing effects to the on-axis collapse, the evolutions of on-axis intensity are calculated for the beam propagating in free-space versus the Kerr medium as shown in Fig. 2. Here, the intensities in Fig. 1(d) and Fig. 2 were normalized to the peak intensity of the initial peak intensity. As recognized from Fig. 2, the contribution of the linear diffraction to the on-axis intensity dominates during the initial propagation. When the accumulated on-axis intensity reaches a certain value, the nonlinear self-focusing effect will become dominant over the linear diffraction effect. Although the critical powers obtained by Eq. (1) include the effect of linear diffraction and nonlinear self-focusing effect. Actually the two effects always exist, and the on-axis collapse can be divided into two stages according to the contribution to the evolution of on-axis intensity: the initial linear diffraction and the subsequent self-focusing effect. On the other hand, the off-axis collapse occurs as a result from the nonlinear self-focusing effect because the linear diffraction cannot accumulate energy at the ring positions.

It is important to note here that the collapse is assumed to occur when the peak intensity approaches 100 times the initial peak intensity during the numerical calculation for the self-focusing process [7]. The collapse powers increase with increasing topological charge number as demonstrated in [17] and [25], as shown in Table 1. When n ≠ m, no collapse occurs in the central position of the optical field due to the existence of the phase and polarization singular points in the field center. The partial collapses always occur in outer ring with the collapse number 4m as shown in Fig. 3. The partial collapses always tend to appear at the positions of linearly polarized components, which can be understood based on the relationship between the collapse powers of linear and circular polarizations P_cr^lin = 3/2 P_cr^cir, i.e. the occurrence of the collapse of the linearly polarized component precedes that of the circularly polarized component with otherwise identical beam parameters.

Fig. 3 (a) The polarization states distribution in the cross-section of the initial field. (b) The intensity distribution of the vortex vector field (n = 1, m = 2, Δθ = π/2 and φ₀ = 0) for P_in = 8.2 P_G at different propagation distances. (c) The Stokes polarization parameters at different propagation distances.

Download Full Size | PDF

The numerical calculations indicate that the collapse power of a structured field with n = 1, m = 2 and Δθ = π/2 is 8.2P_G. There are a total of 8 partial collapses that occur at the ring of the optical field. If the initial power is lower than the collapse power, the energy of the optical field also initially tends to accumulate at the positions where the SoP is linearly polarized, e.g. 4 accumulated points for n = 1 and m = 1 and 8 accumulated points for n = 1 and m = 2. As the optical field propagates in a Kerr medium, it rotates and spreads out, and no collapse occurs. The normalized peak intensities (normalized to the initial peak intensity) as a function of the propagation distance with different initial powers confirm the phenomenon of deformation and redistribution in the optical field during propagation, as well as the different collapse distance, as shown in Fig. 1. Furthermore, it is seen that the collapsing beams rotate during propagation due the existence of the vortex optical field, as shown in Fig. 1 for m = 1 and Fig. 3 for m = 2.

4. The spiral collapsing beams of a structured optical field in a saturated medium

In practice, the other higher nonlinear optical effects would halt the collapse when the initial power is high enough [1,2]. It is important to gain further insight into the behavior of collapsing beams of the structured optical field in a saturated medium by considering the fifth-order nonlinear effect [20, 22]. In particular, when the topological numbers of the polarization and the vortex are equal n = m, it is of fundamental interest and for potential applications to study whether the collapsing beams in the center and the periphery of the structured optical field simultaneously exist and propagate in a saturated medium. In order to obtain a good understanding on the collapsing beam, we simulate the nonlinear propagation of the optical field in a saturated medium and allow the wave to propagate across the collapse point [20–22] using Eq. (1) by taking n₄ = 0.002n₂/P_G. Figures 4 and 5 show the collapsing beams of structured optical fields in the saturated medium with different vortex topological charges. It is seen that when n = m, the collapsing beams simultaneously appear in the center and the periphery of the optical field, and that the peak intensity of the collapsing beams oscillates and the beams rotate during propagation in the saturated medium. The peak intensities in Figs. 4 and 5 have been normalized to the peak intensity of the initial optical field (i. e. z = 0). The four neighboring collapsing beams oscillate simultaneously with the same amplitude and period when m = 1, but the oscillation period and amplitude are different from that of the central collapsing beam, as shown in Fig. 4(b). Furthermore, the maximum amplitudes of the central collapsing beams are greater than those of the neighboring collapsing beams, and the maximum amplitudes occur at different propagation distances as shown in Fig. 4. When n ≠ m, no collapse occurs at the optical field center because the optical singular point (both the phase and polarization singular points) is located at the center (e.g. Figure 5 for n = 2 and m = 1). The intensity of the collapsing beams oscillates and that the beams rotate during propagation. More interesting features can be found in the evolution of the SoP in the field cross-section. The SoP of the collapsing beams of the structured optical field rotates and redistributes as it propagates in the saturated medium in a spiral trajectory [Fig. 4 (c)].

Fig. 4 (a) Iso-surface I = 3 of the data in Stokes parameter S0 for n = 1, m = 1 with initial power P_in = 8P_cr. (b) The corresponding central and neighboring peak intensity as a function of the propagation distance. Vertical dashed lines mark corresponding propagation distances of the maximum amplitudes, and (c) The corresponding Stokes polarization parameters at different propagation distances as marked by the vertical dashed lines.

Download Full Size | PDF

Fig. 5 (a) Iso-surface I = 2.7 of the intensity distribution data (Stokes parameter S0) for n = 2, m = 1 with initial power P_in = 8P_cr; (b) The corresponding peak intensity as a function of the propagation distance.

Download Full Size | PDF

5. Conclusions

The collapse dynamics of a structured optical field with hybrid SoP and a spiral phase in the field cross section have been studied. The collapsing beams in the structured optical fields are demonstrated based on the coupled NLS equations. Compared to a non-vortex vector optical field, the presence of vortex field leads to the rotation of collapsing beams along the propagation axis. The collapse can occur at the center of an optical field or in the outside ring, depending on whether the numbers of the topological charge of the vortex and polarization are the same. The evolutions of SoP in the field cross section during propagation are numerically analyzed using the Stokes polarization parameters. The results indicate that the component of linear polarization always dominates the core of a collapsing beam due to the difference of the self-focusing effect between linear and circular polarizations. The initial power, the vortex order and the distribution of hybrid SoP are the key parameters for controlling and manipulating the collapsing beams in a structured optical field. The structured collapsing beams expand our present understanding of the interaction of light with matter and provide potential applications in corresponding fields.

Funding

National Natural Science Foundation of China (NSFC) (11574271, 11247014); the Zhejiang Provincial Natural Science Foundation (LZ17A040001, LQ13A040006); the Science Research Foundation of Zhejiang Sci-Tech University (14062078-Y); the University of Malaya (UMRG RP001B-13ICT); the Malaysian Ministry of Higher Education HIR Grant (UM.C/625/1/HIR/MOHE/05).

References and links

1. P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69(2), 507–574 (1997). [CrossRef]

2. L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176(3-4), 181–211 (2003). [CrossRef]

3. M. Scheller, M. S. Mills, M. A. Miri, W. Cheng, J. V. Moloney, M. Kolesik, P. Polynkin, and D. N. Christodoulides, “Externally refueled optical filaments,” Nat. Photonics 8(4), 297–301 (2014). [CrossRef]

4. R. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964). [CrossRef]

5. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006). [CrossRef] [PubMed]

6. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010). [CrossRef] [PubMed]

7. R. P. Chen, K. H. Chew, and S. He, “Dynamic control of collapse in a vortex Airy beam,” Sci. Rep. 3, 1406 (2013). [CrossRef] [PubMed]

8. A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29(10), 1126–1128 (2004). [CrossRef] [PubMed]

9. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]

10. R. P. Chen, Z. Chen, K. H. Chew, P. G. Li, Z. Yu, J. Ding, and S. He, “Structured caustic vector vortex optical field: manipulating optical angular momentum flux and polarization rotation,” Sci. Rep. 5, 10628 (2015). [CrossRef] [PubMed]

11. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010). [CrossRef] [PubMed]

12. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007). [CrossRef]

13. R. P. Chen and G. Li, “The evanescent wavefield part of a cylindrical vector beam,” Opt. Express 21(19), 22246–22254 (2013). [CrossRef] [PubMed]

14. R.-P. Chen, L.-X. Zhong, K.-H. Chew, B. Gu, G. Zhou, and T. Zhao, “Effect of a spiral phase on a vector beam with hybrid polarization states,” J. Opt. 17(6), 065605 (2015). [CrossRef]

15. N. M. Litchinitser, “Applied physics. Structured light meets structured matter,” Science 337(6098), 1054–1055 (2012). [CrossRef] [PubMed]

16. A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in Kerr media,” Opt. Lett. 33(1), 13–15 (2008). [CrossRef] [PubMed]

17. G. Fibich and N. Gavish, “Critical power of collapsing vortices,” Phys. Rev. A 77(4), 045803 (2008). [CrossRef]

18. S. M. Li, Y. Li, X. L. Wang, L. J. Kong, K. Lou, C. Tu, Y. Tian, and H. T. Wang, “Taming the collapse of optical fields,” Sci. Rep. 2, 1007 (2012). [PubMed]

19. R. P. Chen, L. X. Zhong, K. H. Chew, T. Y. Zhao, and X. Zhang, “X, Zhang, “Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization,” Laser Phys. 25(7), 075401 (2015). [CrossRef]

20. V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 38, 537 (1993).

21. N. A. Panov, V. A. Makarov, V. Y. Fedorov, and O. G. Kosareva, “Filamentation of arbitrary polarized femtosecond laser pulses in case of high-order Kerr effect,” Opt. Lett. 38(4), 537–539 (2013). [CrossRef] [PubMed]

22. B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012). [CrossRef] [PubMed]

23. A. S. Desyatnikov, D. E. Pelinovsky, and J. Yang, “Multi-component vortex solutions in symmetric coupled nonlinear Schrodinger equations,” J. Math. Sci. 151(4), 3091–3111 (2008). [CrossRef]

24. V. Kruglov, Y. Logvin, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39(11), 2277–2291 (1992). [CrossRef]

25. V. Prytula, V. Vekslerchik, and V. M. Pérez-García, “Collapse in coupled nonlinear Schrodinger equations: Suffcient conditions and applications,” Physica D 238(15), 1462–1467 (2009). [CrossRef]

26. 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] [PubMed]

Topological charge numbers	Critical powers for single on-axial collapse	Critical powers for multiple off-axial collapse
	2.3PG	5.3PG
	2.9PG	8.4PG
	3.2PG	12.4PG
	3.6PG	16.5PG

Vectorial effect of hybrid polarization states on the collapse dynamics of a structured optical field

Abstract

1. Introduction

2. Theoretical formulation

3. On-axis versus off-axis collapses of a structured optical field with a hybrid SoP and a spiral phase in a Kerr medium

4. The spiral collapsing beams of a structured optical field in a saturated medium

5. Conclusions

Funding

References and links

Cited By

Figures (5)

Tables (1)

Equations (8)

Optics Express