- Email: [email protected]
High-order polarization vortex spatial solitons
Physics Letters A 376 (2012) 3051–3056
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
High-order polarization vortex spatial solitons Bingzhi Zhang a,∗ , Hongcheng Wang b , Gaoyong Luo a , Wenqing Man a , Yanhua Zheng a a b
School of Physics and Electric Engineering, Guangzhou University, Guangzhou 510006, China Department of Electronic Engineering, Dongguan University of Technology, Dongguan 523808, China
a r t i c l e
i n f o
Article history: Received 16 October 2011 Received in revised form 12 June 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop
a b s t r a c t We investigate the formation of high-order polarization vortex spatial solitons. The high-order polarization vortex solitons have novel polarization states which are different from fundamental polarization vortex solitons and have rotational symmetry only in intensity. It is proved that the polarization vortex solitons cannot carry vortex phase. The existence domain and dynamical characteristic of these highorder polarization vortex solitons in Bessel optical lattices are discussed in detail. © 2012 Elsevier B.V. All rights reserved.
Keywords: Polarization vortex Optical soliton Non-diffracting Bessel beam
1. Introduction Polarization is one of important property of light. This vector nature of light and its interactions with matter make many optical devices and optical system designs possible. Light wavefronts with spatially homogeneous polarization were most commonly studied in the past. It includes linear, circular and elliptical polarizations. In recent years, spatially inhomogeneous polarization states of light, especially, cylindrical vector (CV) beams, have attracted much of the interest [1,2], because they are beneficial for trapping and acceleration of particles [3,4], microscopy [5], and laser processing [6]. The CV beams are solutions of the Maxwell equations that obey cylindrical symmetry both in intensity and polarization (not including phase). The most basic form of polarization vortices is a purely radially polarized (RP) or azimuthally polarized (AP) beam with a Laguerre–Gaussian TEM01 ring-shaped intensity distribution and a polarization singularity at the center, where its intensity is zero. In the linear optical region, the dynamical propagation process of the CV beams can be described by full vector wave equation and the most interesting phenomenon is that the RP (AP) beams produce strong longitudinal electric (magnetic) field components at the focus [7]. This phenomenon has been applied to tighter focus, optical trapping, surface plasmon excitation and laser machining. When the paraxial approximation condition is valid, the z component of the electric (magnetic) field for RP (AP) beams is weak and can be ignored.
*
Corresponding author. Tel.: +86 13922252071; fax: +86 13922252071. E-mail address: [email protected] (B. Zhang).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.013
For the nonlinear case, the dynamical propagation characteristic of CV beams is more complicated. Thus far, only a few theoretical and experimental studies with RP and AP beams have been reported. A catastrophic result that these CV beams will undergo azimuthal modulation instability when propagating in a nonlinear Kerr medium was found by J.W. Haus et al. [8] in the theory and was subsequently verified experimentally by A.A. Ishaaya et al. [9] In addition, the spatial solitons formed by CV beams in the nonlinear medium also have been studied. For example, A. Ciattoni et al. [10] obtained the exact AP spatial dark solitons solutions (to all nonparaxial orders) in a Kerr medium, H. Wang and W.L. She [11] studied the existence properties of nonparaxial optical Kerr vortex solitons with radial polarization (to the second-order nonparaxial approximation), and L. Ge et al. [12] researched the existence properties of polarization vortex (PV) solitons with inhomogeneous polarization in nonlocal self-focusing nonlinear media. The nonlinear dynamical characteristic of these PV solitons have been not considered. In our previous work, we studied the existence properties and dynamical characteristic of PV solitons formed by cylindrical vector beams in Bessel optical lattices [13]. These PV solitons have complicated dynamical characteristic and can be stabilized in some parameter region. In that work, we found that the PV soliton solutions are similar to phase vortex solitons with topological charge m = ±1. So the field amplitude of PV solitons U (ρ ) ∼ ρ , at ρ → 0 where is polarization singular point which means that this polarization singular point is first-order singular point. However, the dynamical characteristic of these two type solitons are quite different. As is well known, phase vortex solitons can have highorder phase singular point when |m| > 1. Therefore, an interesting open question is raised: can high-order PV solitons can exist? To the best of our knowledge, the existence properties and dynamical
3052
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
where (ρ , θ) is the polar coordinate representation of transverse (η, ζ ) coordinates, and tan θ = ζ /η . For polar coordinate repre2
∂ 2 sentation, the Laplacian ∇⊥ = ρ1 ∂∂ρ (ρ ∂∂ρ ) + ρ12 ∂θ 2 . U (ρ ) is a real function describing the profile of soliton and Γ is soliton propagation constant. The term of exp(imθ) represents the vortex phase and m is the value of topological charge. Substitution of the light field in PV form into Eq. (1) yields
∂2Uη Uη 1 ∂Uη + − 2 1 + m2 − 2miU ζ − Γ U η 2 ρ ∂ρ ∂ρ ρ + σ (γ + 1)U 2 U η − p RU η = 0,
(4a)
∂ Uζ Uζ 1 ∂Uζ + − 2 1 + m2 + 2miU η − Γ U ζ 2 ρ ∂ρ ∂ρ ρ 2
Fig. 1. Schematic of the expression for polarization vector by radial and azimuthal unit vectors.
characteristic of these new high-order PV solitons have not been reported. In addition, can these polarization vortex solitons carry vortex phase? Polarization vortex solitons with vortex phase means that a polarization singular point and phase singular point coexist at the center of these solitons. If this novel soliton exist, what is the value of the order of singular point? 2. The models
∂E 2 E + f(E) − p R (η, ζ )E = 0, + ∇⊥ ∂ξ
(1)
where the longitudinal ξ and transverse (η, ζ ) coordinates are scaled to the diffraction length and input beam width, respectively. 2 The Laplacian ∇⊥ = ∂ 2 /∂ η2 + ∂ 2 /∂ζ 2 . The function f(E) describes the nonlinear response of the medium. In this Letter, we select the well-known nonlinear isotropic Kerr medium which means f = σ [|E|2 E + γ (E · E)E∗ ] (σ = ±1 stands for the focusing/defocusing nonlinearity, and γ is a constant which depends on the physical origin of the Kerr effect) [15]. Parameter p is proportional to the depth of refractive index modulation and the function R (η, ζ ) describes the lattice profile. An important conservation quantity of +∞ +∞ Eq. (1) is the total power, P = −∞ −∞ |E|2 dη dξ . Since cylindrical vector beams have O 2 rotational symmetry in polarization, we first present the expression for polarization vector in polar coordinates using corresponding radial and azimuthal unit vectors:
E = (E · eρ )eρ + (E · eθ )eθ = E cos φ eρ + E sin φ eθ ,
(2)
where eρ = (cos θ eη , sin θ eζ ), eθ = (− sin θ eη , cos θ eζ ) are radial and azimuthal unit vectors respectively, so E = E cos(θ + φ)eη + E sin(θ + φ)eζ . The angle φ is the rotation of the local polarization orientation vector from radial polarization (φ = 0) to the azimuthal polarization (φ = π /2) as shown in Fig. 1. As shown above, a cylindrical vector beams can be viewed as the coherent summation of two orthogonal linearly polarized modes oriented along the eη and eζ axes. We begin to look for soliton solutions of Eq. (1) in the PV solitons with vortex phase form for two orthogonal linearly polarized modes:
E η (ρ , θ, ξ ) = U (ρ ) cos(θ + φ) exp(i Γ ξ ) exp(imθ), E ζ (ρ , θ, ξ ) = U (ρ ) sin(θ + φ) exp(i Γ ξ ) exp(imθ),
(4b)
where U η = U (ρ ) cos(θ + φ), U ζ = U (ρ ) sin(θ + φ). According to (4a) × cos(θ + φ) + (4b) × sin(θ + φ), we can obtain
d2 U dρ
2
+
1 dU
ρ dρ
−
U
ρ
2
1 + m2 − Γ U + σ (γ + 1)U 3 − p RU = 0. (5)
It is obvious that if m = 0, Eq. (5) will reduce to Eq. (4) of Ref. [13] √ automatically. If m = 0, U (ρ ) vanishes at the center,
m +1 at ρ → 0, which means that the singular point U (ρ √) ∼ ρ 2 is m + 1 order. Now, we can obtain ‘polarization vortex soliton with vortex phase solutions’ by solving numerically Eq. (5) with a shooting method or relaxation method. However, we would like to note that Eq. (5) is not equivalent to Eqs. (4). In fact, we can easily obtain topological charge m must be equal to zero from (4a) × sin(θ + φ) − (4b) × cos(θ + φ). So the solution of Eq. (5) may not be the solution of Eqs. (4). In other words, Eq. (5) is really equivalent to Eqs. (4) only when and m = 0. This result indicates that the PV solitons cannot carry vortex phase. ‘Polarization vortex soliton with vortex phase’ does not exist. To confirm this conclusion, we look for PV soliton with vortex phase solutions of Eq. (1) for polarization vector in polar coordinates. For example, we can assume the radially polarized field E with vortex phase takes the form 2
We consider a beam propagating along the ξ -axis of a nonlinear medium in a lattice. The propagation of the beam can be described by the following generalized Schrödinger equation for the dimensionless complex vector field amplitude E under the paraxial approximation and slowly varying amplitude approximation [14]
i
+ σ (γ + 1)U 2 U ζ − p RU ζ = 0,
(3)
E = U (ρ ) exp(i Γ ξ ) exp(imθ)eρ .
(6)
Substitution of this field E into Eq. (1) yields
d2 U dρ 2
+
1 dU
ρ dρ
U − 2 1 + m2 − Γ U + σ (γ + 1)U 3 − p RU eρ
ρ
+ 2imU eθ = 0.
(7)
It is clearly that Eq. (7) is equivalent to Eq. (5) and m = 0. So the PV solitons cannot carry vortex phase. This result indicates that the PV solitons formed by cylindrical vector beams must be first-order solitons and the order of singular point is equal to one. To make an analogy to the high-order phase vortex solitons, we try to find the high-order PV solitons in the following. If we assume angular dependencies cos(nθ + φ) and sin(nθ + φ) in Eq. (3), we can obtain
d2 U dρ 2
+
1 dU
ρ dρ
−
n2
ρ2
U − Γ U + σ (γ + 1)U 3 − p RU = 0,
(8)
which is analogous to charge-n vortex solitons in form. In this case, the direction of polarization of optical fields E will recur after rotating 2π /n around the polarization singular point as shown in Fig. 2(a) (for this case, n = 2). Obviously, these solitons beams are not cylindrical vector beams and they have rotational symmetry only in intensity but not in polarization. We would like
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
3053
Fig. 2. (a) The distribution of polarization vectors of this second-order polarization vortex soliton whose polarization vectors recur after rotating π around the polarization singular point; (b) the intensity of a second-order (n = 2) polarized vortex soliton; (c), (d) the optical field of each component of (a) which is two crossed quadrupole– quadrupole vector solitons in nature.
Fig. 3. (a) Power curves of second-order polarization vortex spatial solitons in the zero-order Bessel lattice with focusing Kerr effect for different p; (b), (c) the amplitude profiles of solitons for p = −4 and p = 4 with the dashed line is the effective zero-order Bessel lattice potential.
3054
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
Fig. 4. (a), (b) The dynamical evolution of the second-order polarization vortex spatial solitons in Figs. 3(b), (c).
Fig. 5. (a) Power curves of second-order polarization vortex spatial solitons in first-order Bessel lattice with focusing Kerr effect for different p; (b), (c) the amplitude profiles of solitons for p = 4 and p = −4 with the dashed line is the effective first-order Bessel lattice potential.
to note that the PV soliton shown in Fig. 2(b) is not scalar solitons. In nature, it is composed with two crossed quadrupole– quadrupole vector solitons in nature as shown in Figs. 2(c) and (d). To the best of our knowledge, these novel high-order PV soliton have not been reported. In the following, we will inves-
tigate the existence properties and dynamical characteristic of these high-order PV solitons. We select typical Bessel lattices R (η, ζ ) = J l2 [(2blin )1/2 (η2 + ζ 2 )1/2 ] [14,16,17]. Since one can use the scaling transformation E(η, ζ, ξ, p ) → χ E(χ η, χ ζ, χ 2 ξ, χ 2 p ) to obtain the various families of solitons by a given blin , here we
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
3055
Fig. 6. (a), (b) The dynamical evolution of the second-order polarization vortex spatial solitons in Figs. 5(b), (c).
set the transverse scale blin = 1/2 and vary Γ and p. For convenience, we choose two typical Bessel lattices (R = J 02 [ρ ], J 12 [ρ ]), namely, the value of order of Bessel function l = 0, 1. In addition, we choose γ = 0 in the following calculation. Firstly, we consider the PV spatial solitons in zeroth-order Bessel lattice with focusing Kerr nonlinearity (σ = 1, R = J 02 [ρ ]). For the facility of discussion, we just investigate the second-order soliton with single ring. The power of such soliton versus the propagation constant for different p are shown in Fig. 3(a) and two typical amplitude profiles of these solitons are shown in Figs. 3(b) and (c), respectively. One can see that the lattice depth p has evident influence on such soliton from Fig. 3(a). In the case of p < 0, the second-order soliton power becomes saturated and tends to a constant corresponding to that of without lattice ( p = 0) as Γ → ∞. Interestingly, the power curve of these lattice solitons is similar to that of discrete surface solitons in a semi-infinite lattice system which exhibits a minimum at lower Γ , and also exists a cutoff on Γ (Γc ) which means that Γ cannot approach the value of zero. These results in turn imply that these solitons can exist only above a certain power threshold P th . Below this power level no solitons can be supported. This result is quite different from fundamental PV solitons as shown in Fig. 3(a) of Ref. [13]. In that case, there is no power threshold to excite the fundamental PV solitons for p < 0. In addition, according to Vakhitov–Kolokolov stability criterion, all these solitons are stable against radial perturbations for d P /dΓ > 0, but it cannot guarantee stability against azimuthal modulation perturbations.
In the case of p > 0, the power of soliton also tends to be a constant like that for p < 0. The reason for this asymptotic behavior is that the lattice depth gradually becomes ignorable when the amplitude of lattice soliton becomes larger and Γ is large enough. However, these power curves are quite different from that of case of p < 0 at lower Γ as shown in Fig. 3(a). These power curves do not have a lower cutoff on Γ , which means that the propagation constant of these PV solitons can approach Γ = 0. In addition, their power curve exhibits a maximum, and on the left and the right of this maximum, d P /dΓ > 0, d P /dΓ < 0, respectively. To study the stability of these solitons, we numerically simulate the propagation of these second-order PV solitons by using the split-step Fourier method. The accuracy of the numerical simulations checked against the total power P , an invariant during propagation. The dynamical propagation of two typical second-order PV solitons is shown in Fig. 4. Obviously, they undergo evident azimuthal modulation instability and break up into some splinters without rotation. For the solitons in the zeroth-order Bessel lattice with p < 0, they firstly break into five splinters and then attract each other rapidly. We note that these five splinters turn to not a “particle” but two splinters in the center though there are relatively deep potential well at the center of lattices. Due to the effect of this lattice potential well, the fundamental PV solitons can propagate stably up to about 40 diffraction lengths. However, these second-order PV solitons cannot be stable in several diffraction lengths. For the solitons in the zeroth-order Bessel lattice with p > 0, they are also unstable under the azimuthal modulation
3056
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
perturbations but they firstly break into seven splinters with different evidently intensity. These splinters also attract each other, and some of them fuse randomly. However, the potential barrier of lattices center cannot localize these splinters. In addition, a small amount of radiation arises due to the collision of soliton with lattice. The total energy is conserved globally. For the PV spatial solitons in first-order Bessel lattices with focusing Kerr nonlinearity (σ = 1, R = J 12 [ρ ]), the power versus Γ is shown in Fig. 5(a). It is clear that the power curves have the same asymptotic behaviors as those of solitons in the zeroth-order Bessel lattices when Γ → ∞ but they are quite different at lower Γ . The physical mechanism is the same as that for the zeroth-order Bessel lattice. For p > 0, the power curve of these lattice solitons is similar to that of solitons in the zeroth-order Bessel lattices for p < 0 as shown in Fig. 5(a). There still have lower cutoff on Γ , and have power threshold P th to excite these solitons. The origin of the comparability of these two cases is that there are relatively deep potential well at the center of lattices as shown in Fig. 5(b). We can also see that the power of these solitons in first-order Bessel lattices is higher than that of the zeroth-order Bessel lattices. The reason is that the potential well of the zeroth-order Bessel lattices center for p < 0 is deeper at the same conditions, so it is easier to excite the solitons than that of first-order Bessel lattice solitons for p > 0. In the case of p < 0, the relatively deep lattice potential well locate at the neighborhood of ρ ≈ 2, so the soliton is excited easily at this position as shown in Fig. 5(c). In addition, it is natural that the exciting power of lattice solitons for deeper lattice potential well (p = −8) will be smaller than that of p = −4. However, when the amplitude of lattice soliton becomes larger (Γ large enough), the lattice depth gradually becomes ignorable, so the exciting power will tend to a constant corresponding to that of without lattice ( p = 0) as Γ → ∞. For positive and negative p the first-order Bessel lattices, two typical soliton solutions are shown in Figs. 5(b) and (c). Their dynamical propagation is shown in Figs. 6(a) and (b). Obviously, they also undergo evident azimuthal modulation instability and break up into five splinters with about same intensity. For the solitons in first-order Bessel lattice with p > 0, one can see that they firstly break into five splinters and then “repulse” each other rapidly. These five splinters cannot rotate because they don’t have vortex phase which means that they don’t have orbit angular momentum. The reason of these five splinters expanding is the repulsion force
formed by potential barrier of lattices at ρ ≈ 2. For the solitons in first-order Bessel lattice with p < 0, the second-order PV solitons also break into five splinters at first, but these splinters can keep oscillating (or breathing) in long-term evolution at the ring orbit due to the potential well of lattices at ρ ≈ 2. In conclusion, we have proved that the polarization vortex solitons cannot carry vortex phase, and found the high-order polarization vortex solitons have novel polarization states distribution where the direction of polarization of optical fields recur after rotating 2π /n around the polarization singular point for n-order PV solitons. The existence curves of these high-order PV solitons in the zeroth- and first-order Bessel lattices are discussed and dynamical characteristic of these high-order polarization vortex solitons are discussed in detail. We do not find the strictly stable azimuthally high-order polarization vortex solitons in this Letter, and it needs be explored in the future. Acknowledgements B. Zhang and G. Luo acknowledge support from the Xin Miao Science Foundation of Guangzhou University (No. ZBZ1-1001), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM11098), and the National Natural Science Foundation of China (Grant No. 60971093). H. Wang acknowledges support from the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM10124). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Q. Zhan, Adv. Opt. Photon. 1 (2009) 1. T. Brown, Q. Zhan, Opt. Express 18 (2010) 10775. Q. Zhan, Opt. Express 12 (2004) 3377. H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, Opt. Lett. 32 (2007) 1839. R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901. V.G. Niziev, A.V. Nesterov, J. Phys. D 32 (1999) 1455. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, Opt. Commun. 179 (2000) 1. J.W. Haus, Z. Mozumder, Q. Zhan, Opt. Express 14 (2006) 4757. A.A. Ishaaya, L.T. Vuong, T.D. Grow, A.L. Gaeta, Opt. Lett. 33 (2008) 13. A. Ciattoni, B. Crosignani, P. Di Porto, A. Yariv, Phys. Rev. Lett. 94 (2005) 073902. H.C. Wang, W.L. She, Opt. Express 14 (2006) 1590. L. Ge, Q. Wang, M. Shen, J. Shi, Q. Kong, Optik 122 (2011) 749. B. Zhang, Phys. Lett. A 375 (2011) 1110. Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 94 (2005) 043902. G. Fibich, B. Ilan, Physica D 157 (2001) 112. X.S. Wang, Z.G. Chen, J.K. Yang, Opt. Lett. 31 (2006) 1887. X.S. Wang, Z.G. Chen, P.G. Kevrekidis, Phys. Rev. Lett. 96 (2006) 083904.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
High-order polarization vortex spatial solitons Bingzhi Zhang a,∗ , Hongcheng Wang b , Gaoyong Luo a , Wenqing Man a , Yanhua Zheng a a b
School of Physics and Electric Engineering, Guangzhou University, Guangzhou 510006, China Department of Electronic Engineering, Dongguan University of Technology, Dongguan 523808, China
a r t i c l e
i n f o
Article history: Received 16 October 2011 Received in revised form 12 June 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by A.R. Bishop
a b s t r a c t We investigate the formation of high-order polarization vortex spatial solitons. The high-order polarization vortex solitons have novel polarization states which are different from fundamental polarization vortex solitons and have rotational symmetry only in intensity. It is proved that the polarization vortex solitons cannot carry vortex phase. The existence domain and dynamical characteristic of these highorder polarization vortex solitons in Bessel optical lattices are discussed in detail. © 2012 Elsevier B.V. All rights reserved.
Keywords: Polarization vortex Optical soliton Non-diffracting Bessel beam
1. Introduction Polarization is one of important property of light. This vector nature of light and its interactions with matter make many optical devices and optical system designs possible. Light wavefronts with spatially homogeneous polarization were most commonly studied in the past. It includes linear, circular and elliptical polarizations. In recent years, spatially inhomogeneous polarization states of light, especially, cylindrical vector (CV) beams, have attracted much of the interest [1,2], because they are beneficial for trapping and acceleration of particles [3,4], microscopy [5], and laser processing [6]. The CV beams are solutions of the Maxwell equations that obey cylindrical symmetry both in intensity and polarization (not including phase). The most basic form of polarization vortices is a purely radially polarized (RP) or azimuthally polarized (AP) beam with a Laguerre–Gaussian TEM01 ring-shaped intensity distribution and a polarization singularity at the center, where its intensity is zero. In the linear optical region, the dynamical propagation process of the CV beams can be described by full vector wave equation and the most interesting phenomenon is that the RP (AP) beams produce strong longitudinal electric (magnetic) field components at the focus [7]. This phenomenon has been applied to tighter focus, optical trapping, surface plasmon excitation and laser machining. When the paraxial approximation condition is valid, the z component of the electric (magnetic) field for RP (AP) beams is weak and can be ignored.
*
Corresponding author. Tel.: +86 13922252071; fax: +86 13922252071. E-mail address: [email protected] (B. Zhang).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.013
For the nonlinear case, the dynamical propagation characteristic of CV beams is more complicated. Thus far, only a few theoretical and experimental studies with RP and AP beams have been reported. A catastrophic result that these CV beams will undergo azimuthal modulation instability when propagating in a nonlinear Kerr medium was found by J.W. Haus et al. [8] in the theory and was subsequently verified experimentally by A.A. Ishaaya et al. [9] In addition, the spatial solitons formed by CV beams in the nonlinear medium also have been studied. For example, A. Ciattoni et al. [10] obtained the exact AP spatial dark solitons solutions (to all nonparaxial orders) in a Kerr medium, H. Wang and W.L. She [11] studied the existence properties of nonparaxial optical Kerr vortex solitons with radial polarization (to the second-order nonparaxial approximation), and L. Ge et al. [12] researched the existence properties of polarization vortex (PV) solitons with inhomogeneous polarization in nonlocal self-focusing nonlinear media. The nonlinear dynamical characteristic of these PV solitons have been not considered. In our previous work, we studied the existence properties and dynamical characteristic of PV solitons formed by cylindrical vector beams in Bessel optical lattices [13]. These PV solitons have complicated dynamical characteristic and can be stabilized in some parameter region. In that work, we found that the PV soliton solutions are similar to phase vortex solitons with topological charge m = ±1. So the field amplitude of PV solitons U (ρ ) ∼ ρ , at ρ → 0 where is polarization singular point which means that this polarization singular point is first-order singular point. However, the dynamical characteristic of these two type solitons are quite different. As is well known, phase vortex solitons can have highorder phase singular point when |m| > 1. Therefore, an interesting open question is raised: can high-order PV solitons can exist? To the best of our knowledge, the existence properties and dynamical
3052
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
where (ρ , θ) is the polar coordinate representation of transverse (η, ζ ) coordinates, and tan θ = ζ /η . For polar coordinate repre2
∂ 2 sentation, the Laplacian ∇⊥ = ρ1 ∂∂ρ (ρ ∂∂ρ ) + ρ12 ∂θ 2 . U (ρ ) is a real function describing the profile of soliton and Γ is soliton propagation constant. The term of exp(imθ) represents the vortex phase and m is the value of topological charge. Substitution of the light field in PV form into Eq. (1) yields
∂2Uη Uη 1 ∂Uη + − 2 1 + m2 − 2miU ζ − Γ U η 2 ρ ∂ρ ∂ρ ρ + σ (γ + 1)U 2 U η − p RU η = 0,
(4a)
∂ Uζ Uζ 1 ∂Uζ + − 2 1 + m2 + 2miU η − Γ U ζ 2 ρ ∂ρ ∂ρ ρ 2
Fig. 1. Schematic of the expression for polarization vector by radial and azimuthal unit vectors.
characteristic of these new high-order PV solitons have not been reported. In addition, can these polarization vortex solitons carry vortex phase? Polarization vortex solitons with vortex phase means that a polarization singular point and phase singular point coexist at the center of these solitons. If this novel soliton exist, what is the value of the order of singular point? 2. The models
∂E 2 E + f(E) − p R (η, ζ )E = 0, + ∇⊥ ∂ξ
(1)
where the longitudinal ξ and transverse (η, ζ ) coordinates are scaled to the diffraction length and input beam width, respectively. 2 The Laplacian ∇⊥ = ∂ 2 /∂ η2 + ∂ 2 /∂ζ 2 . The function f(E) describes the nonlinear response of the medium. In this Letter, we select the well-known nonlinear isotropic Kerr medium which means f = σ [|E|2 E + γ (E · E)E∗ ] (σ = ±1 stands for the focusing/defocusing nonlinearity, and γ is a constant which depends on the physical origin of the Kerr effect) [15]. Parameter p is proportional to the depth of refractive index modulation and the function R (η, ζ ) describes the lattice profile. An important conservation quantity of +∞ +∞ Eq. (1) is the total power, P = −∞ −∞ |E|2 dη dξ . Since cylindrical vector beams have O 2 rotational symmetry in polarization, we first present the expression for polarization vector in polar coordinates using corresponding radial and azimuthal unit vectors:
E = (E · eρ )eρ + (E · eθ )eθ = E cos φ eρ + E sin φ eθ ,
(2)
where eρ = (cos θ eη , sin θ eζ ), eθ = (− sin θ eη , cos θ eζ ) are radial and azimuthal unit vectors respectively, so E = E cos(θ + φ)eη + E sin(θ + φ)eζ . The angle φ is the rotation of the local polarization orientation vector from radial polarization (φ = 0) to the azimuthal polarization (φ = π /2) as shown in Fig. 1. As shown above, a cylindrical vector beams can be viewed as the coherent summation of two orthogonal linearly polarized modes oriented along the eη and eζ axes. We begin to look for soliton solutions of Eq. (1) in the PV solitons with vortex phase form for two orthogonal linearly polarized modes:
E η (ρ , θ, ξ ) = U (ρ ) cos(θ + φ) exp(i Γ ξ ) exp(imθ), E ζ (ρ , θ, ξ ) = U (ρ ) sin(θ + φ) exp(i Γ ξ ) exp(imθ),
(4b)
where U η = U (ρ ) cos(θ + φ), U ζ = U (ρ ) sin(θ + φ). According to (4a) × cos(θ + φ) + (4b) × sin(θ + φ), we can obtain
d2 U dρ
2
+
1 dU
ρ dρ
−
U
ρ
2
1 + m2 − Γ U + σ (γ + 1)U 3 − p RU = 0. (5)
It is obvious that if m = 0, Eq. (5) will reduce to Eq. (4) of Ref. [13] √ automatically. If m = 0, U (ρ ) vanishes at the center,
m +1 at ρ → 0, which means that the singular point U (ρ √) ∼ ρ 2 is m + 1 order. Now, we can obtain ‘polarization vortex soliton with vortex phase solutions’ by solving numerically Eq. (5) with a shooting method or relaxation method. However, we would like to note that Eq. (5) is not equivalent to Eqs. (4). In fact, we can easily obtain topological charge m must be equal to zero from (4a) × sin(θ + φ) − (4b) × cos(θ + φ). So the solution of Eq. (5) may not be the solution of Eqs. (4). In other words, Eq. (5) is really equivalent to Eqs. (4) only when and m = 0. This result indicates that the PV solitons cannot carry vortex phase. ‘Polarization vortex soliton with vortex phase’ does not exist. To confirm this conclusion, we look for PV soliton with vortex phase solutions of Eq. (1) for polarization vector in polar coordinates. For example, we can assume the radially polarized field E with vortex phase takes the form 2
We consider a beam propagating along the ξ -axis of a nonlinear medium in a lattice. The propagation of the beam can be described by the following generalized Schrödinger equation for the dimensionless complex vector field amplitude E under the paraxial approximation and slowly varying amplitude approximation [14]
i
+ σ (γ + 1)U 2 U ζ − p RU ζ = 0,
(3)
E = U (ρ ) exp(i Γ ξ ) exp(imθ)eρ .
(6)
Substitution of this field E into Eq. (1) yields
d2 U dρ 2
+
1 dU
ρ dρ
U − 2 1 + m2 − Γ U + σ (γ + 1)U 3 − p RU eρ
ρ
+ 2imU eθ = 0.
(7)
It is clearly that Eq. (7) is equivalent to Eq. (5) and m = 0. So the PV solitons cannot carry vortex phase. This result indicates that the PV solitons formed by cylindrical vector beams must be first-order solitons and the order of singular point is equal to one. To make an analogy to the high-order phase vortex solitons, we try to find the high-order PV solitons in the following. If we assume angular dependencies cos(nθ + φ) and sin(nθ + φ) in Eq. (3), we can obtain
d2 U dρ 2
+
1 dU
ρ dρ
−
n2
ρ2
U − Γ U + σ (γ + 1)U 3 − p RU = 0,
(8)
which is analogous to charge-n vortex solitons in form. In this case, the direction of polarization of optical fields E will recur after rotating 2π /n around the polarization singular point as shown in Fig. 2(a) (for this case, n = 2). Obviously, these solitons beams are not cylindrical vector beams and they have rotational symmetry only in intensity but not in polarization. We would like
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
3053
Fig. 2. (a) The distribution of polarization vectors of this second-order polarization vortex soliton whose polarization vectors recur after rotating π around the polarization singular point; (b) the intensity of a second-order (n = 2) polarized vortex soliton; (c), (d) the optical field of each component of (a) which is two crossed quadrupole– quadrupole vector solitons in nature.
Fig. 3. (a) Power curves of second-order polarization vortex spatial solitons in the zero-order Bessel lattice with focusing Kerr effect for different p; (b), (c) the amplitude profiles of solitons for p = −4 and p = 4 with the dashed line is the effective zero-order Bessel lattice potential.
3054
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
Fig. 4. (a), (b) The dynamical evolution of the second-order polarization vortex spatial solitons in Figs. 3(b), (c).
Fig. 5. (a) Power curves of second-order polarization vortex spatial solitons in first-order Bessel lattice with focusing Kerr effect for different p; (b), (c) the amplitude profiles of solitons for p = 4 and p = −4 with the dashed line is the effective first-order Bessel lattice potential.
to note that the PV soliton shown in Fig. 2(b) is not scalar solitons. In nature, it is composed with two crossed quadrupole– quadrupole vector solitons in nature as shown in Figs. 2(c) and (d). To the best of our knowledge, these novel high-order PV soliton have not been reported. In the following, we will inves-
tigate the existence properties and dynamical characteristic of these high-order PV solitons. We select typical Bessel lattices R (η, ζ ) = J l2 [(2blin )1/2 (η2 + ζ 2 )1/2 ] [14,16,17]. Since one can use the scaling transformation E(η, ζ, ξ, p ) → χ E(χ η, χ ζ, χ 2 ξ, χ 2 p ) to obtain the various families of solitons by a given blin , here we
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
3055
Fig. 6. (a), (b) The dynamical evolution of the second-order polarization vortex spatial solitons in Figs. 5(b), (c).
set the transverse scale blin = 1/2 and vary Γ and p. For convenience, we choose two typical Bessel lattices (R = J 02 [ρ ], J 12 [ρ ]), namely, the value of order of Bessel function l = 0, 1. In addition, we choose γ = 0 in the following calculation. Firstly, we consider the PV spatial solitons in zeroth-order Bessel lattice with focusing Kerr nonlinearity (σ = 1, R = J 02 [ρ ]). For the facility of discussion, we just investigate the second-order soliton with single ring. The power of such soliton versus the propagation constant for different p are shown in Fig. 3(a) and two typical amplitude profiles of these solitons are shown in Figs. 3(b) and (c), respectively. One can see that the lattice depth p has evident influence on such soliton from Fig. 3(a). In the case of p < 0, the second-order soliton power becomes saturated and tends to a constant corresponding to that of without lattice ( p = 0) as Γ → ∞. Interestingly, the power curve of these lattice solitons is similar to that of discrete surface solitons in a semi-infinite lattice system which exhibits a minimum at lower Γ , and also exists a cutoff on Γ (Γc ) which means that Γ cannot approach the value of zero. These results in turn imply that these solitons can exist only above a certain power threshold P th . Below this power level no solitons can be supported. This result is quite different from fundamental PV solitons as shown in Fig. 3(a) of Ref. [13]. In that case, there is no power threshold to excite the fundamental PV solitons for p < 0. In addition, according to Vakhitov–Kolokolov stability criterion, all these solitons are stable against radial perturbations for d P /dΓ > 0, but it cannot guarantee stability against azimuthal modulation perturbations.
In the case of p > 0, the power of soliton also tends to be a constant like that for p < 0. The reason for this asymptotic behavior is that the lattice depth gradually becomes ignorable when the amplitude of lattice soliton becomes larger and Γ is large enough. However, these power curves are quite different from that of case of p < 0 at lower Γ as shown in Fig. 3(a). These power curves do not have a lower cutoff on Γ , which means that the propagation constant of these PV solitons can approach Γ = 0. In addition, their power curve exhibits a maximum, and on the left and the right of this maximum, d P /dΓ > 0, d P /dΓ < 0, respectively. To study the stability of these solitons, we numerically simulate the propagation of these second-order PV solitons by using the split-step Fourier method. The accuracy of the numerical simulations checked against the total power P , an invariant during propagation. The dynamical propagation of two typical second-order PV solitons is shown in Fig. 4. Obviously, they undergo evident azimuthal modulation instability and break up into some splinters without rotation. For the solitons in the zeroth-order Bessel lattice with p < 0, they firstly break into five splinters and then attract each other rapidly. We note that these five splinters turn to not a “particle” but two splinters in the center though there are relatively deep potential well at the center of lattices. Due to the effect of this lattice potential well, the fundamental PV solitons can propagate stably up to about 40 diffraction lengths. However, these second-order PV solitons cannot be stable in several diffraction lengths. For the solitons in the zeroth-order Bessel lattice with p > 0, they are also unstable under the azimuthal modulation
3056
B. Zhang et al. / Physics Letters A 376 (2012) 3051–3056
perturbations but they firstly break into seven splinters with different evidently intensity. These splinters also attract each other, and some of them fuse randomly. However, the potential barrier of lattices center cannot localize these splinters. In addition, a small amount of radiation arises due to the collision of soliton with lattice. The total energy is conserved globally. For the PV spatial solitons in first-order Bessel lattices with focusing Kerr nonlinearity (σ = 1, R = J 12 [ρ ]), the power versus Γ is shown in Fig. 5(a). It is clear that the power curves have the same asymptotic behaviors as those of solitons in the zeroth-order Bessel lattices when Γ → ∞ but they are quite different at lower Γ . The physical mechanism is the same as that for the zeroth-order Bessel lattice. For p > 0, the power curve of these lattice solitons is similar to that of solitons in the zeroth-order Bessel lattices for p < 0 as shown in Fig. 5(a). There still have lower cutoff on Γ , and have power threshold P th to excite these solitons. The origin of the comparability of these two cases is that there are relatively deep potential well at the center of lattices as shown in Fig. 5(b). We can also see that the power of these solitons in first-order Bessel lattices is higher than that of the zeroth-order Bessel lattices. The reason is that the potential well of the zeroth-order Bessel lattices center for p < 0 is deeper at the same conditions, so it is easier to excite the solitons than that of first-order Bessel lattice solitons for p > 0. In the case of p < 0, the relatively deep lattice potential well locate at the neighborhood of ρ ≈ 2, so the soliton is excited easily at this position as shown in Fig. 5(c). In addition, it is natural that the exciting power of lattice solitons for deeper lattice potential well (p = −8) will be smaller than that of p = −4. However, when the amplitude of lattice soliton becomes larger (Γ large enough), the lattice depth gradually becomes ignorable, so the exciting power will tend to a constant corresponding to that of without lattice ( p = 0) as Γ → ∞. For positive and negative p the first-order Bessel lattices, two typical soliton solutions are shown in Figs. 5(b) and (c). Their dynamical propagation is shown in Figs. 6(a) and (b). Obviously, they also undergo evident azimuthal modulation instability and break up into five splinters with about same intensity. For the solitons in first-order Bessel lattice with p > 0, one can see that they firstly break into five splinters and then “repulse” each other rapidly. These five splinters cannot rotate because they don’t have vortex phase which means that they don’t have orbit angular momentum. The reason of these five splinters expanding is the repulsion force
formed by potential barrier of lattices at ρ ≈ 2. For the solitons in first-order Bessel lattice with p < 0, the second-order PV solitons also break into five splinters at first, but these splinters can keep oscillating (or breathing) in long-term evolution at the ring orbit due to the potential well of lattices at ρ ≈ 2. In conclusion, we have proved that the polarization vortex solitons cannot carry vortex phase, and found the high-order polarization vortex solitons have novel polarization states distribution where the direction of polarization of optical fields recur after rotating 2π /n around the polarization singular point for n-order PV solitons. The existence curves of these high-order PV solitons in the zeroth- and first-order Bessel lattices are discussed and dynamical characteristic of these high-order polarization vortex solitons are discussed in detail. We do not find the strictly stable azimuthally high-order polarization vortex solitons in this Letter, and it needs be explored in the future. Acknowledgements B. Zhang and G. Luo acknowledge support from the Xin Miao Science Foundation of Guangzhou University (No. ZBZ1-1001), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM11098), and the National Natural Science Foundation of China (Grant No. 60971093). H. Wang acknowledges support from the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM10124). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Q. Zhan, Adv. Opt. Photon. 1 (2009) 1. T. Brown, Q. Zhan, Opt. Express 18 (2010) 10775. Q. Zhan, Opt. Express 12 (2004) 3377. H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, Opt. Lett. 32 (2007) 1839. R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901. V.G. Niziev, A.V. Nesterov, J. Phys. D 32 (1999) 1455. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, Opt. Commun. 179 (2000) 1. J.W. Haus, Z. Mozumder, Q. Zhan, Opt. Express 14 (2006) 4757. A.A. Ishaaya, L.T. Vuong, T.D. Grow, A.L. Gaeta, Opt. Lett. 33 (2008) 13. A. Ciattoni, B. Crosignani, P. Di Porto, A. Yariv, Phys. Rev. Lett. 94 (2005) 073902. H.C. Wang, W.L. She, Opt. Express 14 (2006) 1590. L. Ge, Q. Wang, M. Shen, J. Shi, Q. Kong, Optik 122 (2011) 749. B. Zhang, Phys. Lett. A 375 (2011) 1110. Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 94 (2005) 043902. G. Fibich, B. Ilan, Physica D 157 (2001) 112. X.S. Wang, Z.G. Chen, J.K. Yang, Opt. Lett. 31 (2006) 1887. X.S. Wang, Z.G. Chen, P.G. Kevrekidis, Phys. Rev. Lett. 96 (2006) 083904.