We mainly investigate the polarization singularities of the focused azimuthally polarized (AP) beams modulated by spiral phase and sector obstacles. The results reveal that either the spiral phase or sector obstacle can convert the central V-point to C-points, C-point dipoles, or even double V-points under certain conditions. The conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.
©2013 Optical Society of America
Cylindrical vector (CV) beam, a kind of spatially non-homogeneous polarization beams, has recently attracted increasing research interests due to its particular tight-focusing properties [1–8]. There are two typical CV beams that people concern most: the radially polarized (RP) and azimuthally polarized (AP) beams, respectively with the instantaneously radial and azimuthal electric vector field. When focused by a high numerical aperture (NA) lens, the RP beam can be transformed into a sub-wavelength spot with stronger longitudinally polarized component, while the AP beam is focused to a hollow ring with only azimuthal component existed . Especially under the modulation of diffractive optical elements (DOEs), the tight-focusing of CV beams can give rise to many special fields, such as optical needle , optical bubble , optical cage , and optical chain . Furthermore, by being obstructed with sector-shaped  or odd-fold symmetric  obstacles, the energy flow redistribution takes place in the focal region of the CV beam. These unique focusing properties of the CV beams are expected to explore many applications in optical, biological, and atmospheric sciences
On the other hand, as an essential part of optical phenomena, the singular optics has been paid many attentions. In past several years, polarization singularities (PSs) of optical fields, in addition to the phase singularities, have been theoretically and experimentally investigated, including the classification and description [9–11], the evolution [11,12], as well as the observation and measurement [13–17] of PSs in the context of crystal optics, skylights or speckle fields. Especially in the focal field of the RP beam, several types of PSs are found . Recently, we presented that when a CV beam is blocked by a rotationally symmetric sector-shaped obstacle, the polarization of the focal field redistributes and several PSs (V-points) emerge . In this paper, we investigate the conversion and control of PSs of the focused AP beam modulated by spiral phase and sector obstacles. The results show that the conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.
2. Polarizations and singularities
An arbitrary polarization state can be also described by using the polarization ellipse, as depicted in Fig. 1(a) , where azimuthal angle Φ and tanχ denote the orientation of the major axis of the ellipse and the ellipticity, and can be expressed with Stokes parameters as: tan2Φ = s2/s1, and sin2χ = s3/s0, respectively. The ellipticity tanχ = 0, + 1 and −1 mean linear polarization, right- and left-hand circular polarizations, respectively.
In addition to phase singularities, some light fields can exhibit a variety of polarization singularities (PSs) [9–18], as depicted by the points on Poincare sphere shown in Fig. 1(b): (1) C-point (point of circular polarization), where the orientation of the major (or minor) axis of the polarization ellipse is undefined and the Stokes parameters satisfy the condition of circular polarization: s1 = s2 = 0, and s3 = ± 1, as depicted by the two points of the north and south poles, respectively; (2) L-line (line of linear polarization), where the handedness of the polarization ellipse is undefined and the Stokes parameters satisfy the condition of linear polarization: s3 = 0, as depicted by the equatorial line; (3) V-point (vector point singularities), which is an isolated and stationary point with the orientation of the linear polarization undefined and the Stokes parameters satisfy s1 = s2 = s3 = 0. The V-point dose not locate on the Poincare sphere, and the full Stokes parameter S0 = 0. To manifest the polarization singularities of a light field, the crossing curves of the zero contours of Stokes parameters can be employed [9, 10, 14].
3. Singularity of the focused azimuthally polarized (AP) beams
For an AP beam focused by a high NA lens, the focal field distribution can be described with Richards-Wolf theory [2, 20]. Ignoring the integral constant, the focal field of an AP beam can be derived as21], which is just used as an example. The relevant conclusions on the PSs drawn from the Bessel-Gaussian beam are essentially applicable to some other vector-vortex beams, although with some different minor details.
To analyze the polarization state of the focused AP beam, the focal field and the corresponding Stokes parameters are calculated by Eq. (2), and the results are displayed in Fig. 2 . Figure 2(a) shows the intensity distribution, where the dotted and dashed lines denote the zero contours of s1 and s2, respectively. Figure 2(b) shows the ellipticity (tanχ) and the orientation (Φ) of the major axis (marked by short lines) of polarization ellipse. In Fig. 2(a), there is a crossing point (point A) of the zero contours of s1 and s2, where the orientation of the major axis of polarization ellipse is undefined. It is easily seen from Fig. 2(b) that s3≡0, i.e. the focal field of the AP beam is linearly polarized. As a result, point A denotes the V-point, which represents the PS with undefined direction of linear polarization. It is the sole V-point in the focal field of the AP beam. Actually, the focal field of the AP beam still keeps azimuthal polarization [see Fig. 2(b)]. Furthermore, S0 at the V-point is zero, namely, the intensity at point A is zero, as also shown as the hollow ring distribution of the focal field in Fig. 2(a).
When an additional spiral phase is attached to the AP beam, i.e. ϑ(φ,θ) = lφ, the polarization state in the focal plane changes dramatically. Figure 3 represents the calculated results of the intensity (top) and polarization (bottom) distributions of the focused AP beams carrying spiral phases with different topological charges. Where, Figs. 3(a)-3(c) correspond to l = + 1, + 2, and + 3, respectively; the dotted, dashed and solid lines in the top row denote the zero contours of s1, s2, and s3, respectively; the background and short lines in the bottom row denote the ellipticity (tanχ) and the orientation (Φ) of polarization ellipse, respectively.
From the bottom of Fig. 3, it obviously reveals that with the introduction of spiral phase, the focal field of the AP beam would no longer maintain the linear polarization, but translate into circularly (tanχ = ± 1) or elliptically (1>|tanχ|>0) polarized field, similar to the case of AP beam obstructed by odd-fold symmetric obstacles . When l = + 1, as shown in Fig. 3(a), the AP beam is focused into a solid light spot with a smaller size (about 0.5λ) . There is a crossing spot (point A) of the zero contours of s1 and s2 in the center of the spot (where s3 = + 1≠0), presenting as a C-point with right-hand circular polarization. Namely, the spiral phase exp(iφ) converts the central V-point of the focal field of the AP beam to a C-point. And due to this conversion, the ring-like focal field of the AP beam is shrunk to a solid spot. In this case, the orbital angular momentum (OAM) carried by the spiral phase is partially converted into the spin angular momentum (SAM). Moreover, around point A, the solid spot presents right-hand elliptical polarization (tanχ>0) with radial-ward major axis, resembling the RP beam.
When l≠ + 1, the focal field of AP beam keeps ring-like profile, although elliptical polarizations are introduced into the center region. From the crossed Stokes contours [see the top of Figs. 3(b) and 3(c)] and ellipticity distributions of polarization [see the bottom of Fig. 3(b) and 3(c)], it seems that a C-point is also formed at the crossing of zero contours of s1 and s2 [see points B and C in Figs. 3(b) and 3(c), respectively]. However, at points B and C, s3 is calculated as zero, i.e. points B and C are both V-points. Actually, it can be seen that the ellipticity distributions are discontinuous at the center points, as shown in the insets (the zoom views of the center regions) of the bottom in Figs. 3(b) and 3(c). The values of these two discontinuous points have no physical meaning because the ellipticity of the polarization ellipse of V-point is undefined. In a word, the spiral phase exp(ilφ) never changes the central V-point in the focal field of the AP beam when l≠ + 1. Therefore, the type of the central singularity can be selectively converted to another one by merely changing the topological charge l of the spiral phase.
When the topological charge l of the spiral phase is negative, the intensity and polarization of the focal field of AP beam are also calculated. We find that the sign of l does not influence the intensity of the focal field at all, but changes the handedness of the polarization, as shown in Fig. 4 . Figures 4(a)-4(c) correspond to l = −1, −2, and −3, and the background and short lines denote the distribution of tanχ and orientation of Φ, respectively. Comparing with the bottom of Fig. 3, it reveals that the ellipticity of the polarization ellipse corresponding to l = −1 (−2, −3) is just opposite in sign to that of l = + 1 ( + 2, + 3). The center points (A, B and C) in Figs. 4(a)-4(c) are C-point (left-hand circularly polarized), V-point, and V-point, respectively, same as that in Fig. 3.
The conversions of the PSs of the focused AP beam with spiral phase can be seen from the focal fields of the vector-vortex beams. For the incident beam A = l0(θ)exp(ilφ) [where l0(θ) denotes the amplitude], the focal field can be expressed asEq. (4), the focal field can be considered as the composing of the two circularly polarized components. To make the central intensity (ρ = 0) of the focal plane nonzero, at least one of the Bessel functions in Eq. (4) needs to be 0th order, i.e., l = ± 1. When l = 1, for example, the central intensity of the left-hand polarized component turns to zero, and the right-hand circular polarization manifests itself at the central point. When l≠ ± 1, the central point with the zero-intensity presents as a V-point.
In addition, we investigated the PSs in the focal plane of the high-order AP beams, of which the polarization can be expressed as [-sin(nφ)ex, cos(nφ)ey, 0ez]T. The nonzero integer n denotes the topological charge of the polarization. The influence of the spiral phase on the focused high-order AP beam is similar to that on the single-charge one (n = 1). We find that the focal field can be also decomposed into eL and eR components, containing four Bessel function terms Jl ± n, and Jl ± (n-2) in the integral formula. That is to say, when l = ± n or ± (n-2), the central intensity is nonzero. Thus, the central V-point would be converted into a C-point, and the ring-like focal field transforms to a solid spot. By performing a series of calculations, we prove that this conclusion works except in the case of l = 0. This is because that the incident beam with l = 0 carry no angular momentum, and the circular or elliptical polarization would not arise due to the conservation of angular momentum. Above all, when l = ± n or ± (n-2), and l≠0, the V-point is converted into C-point; when l≠ ± n and l≠ ± (n-2), the V-point remains unchanged. This conclusion can expressed as:
4. Singularity transition associated with sector amplitude obstacle
Figure 5 depicts the schematic focusing system for the incident beam blocked by a sector obstacle. The sector angle is denoted as Ψ. For an amplitude obstacle, the transmission function of the obstacle can be expressed as
The intensity and polarization states of the focal field of AP beam blocked by different sector obstacles are calculated as shown in Fig. 6 , where Figs. 6(a)-6(f) correspond to Ψ = 45°, 90°, 135°, 180°, 225°, and 270°, respectively. The results reveal that by introducing the sector obstacle, the cylindrical symmetry of the focal field of the AP beam is broken, and elliptical and circular polarizations are introduced to the focal field. As the sector angle increasing, the focal field splits vertically into two spots firstly [Ψ<180°, as shown in Figs. 6(a)-6(c)], and then combines gradually to an elliptical spot [Ψ≥180°, as shown in Figs. 6(d)-6(f)]. From the crossed zero contours of s1, s2, and s3, it can be seen that the V-point in Fig. 2 is divided into two C-points (see the two white circle points in the top of Fig. 6), which are separated by a horizontal L-line (s3 = 0, see the solid lines in Fig. 6). It is specially noticed that the L-line depicts the boundary of elliptical polarizations with opposite handedness. Thus, the two separated C-points have opposite handedness, i.e. the upper and lower points are right- (tanχ>0) and left-hand (tanχ<0) circularly polarized, respectively. The two C-points can be considered as a dipole, around which the orientations of the major axis of the polarization ellipse are much similar to the electric field lines of the electric dipole, as shown in the bottom of Fig. 6. Moreover, with increasing the sector angle, the two C-points depart away from each other gradually, and finally locate beyond the focal spot [see Fig. 6(f)].
It is specially noted that more C-points emerge in the focal field, just as the crossing points of zero contours of s1 and s2 in Fig. 6. However, these points locate away from the focal field, where the intensity is too weak to be taken into consideration. Here, we only consider these two crucial C-points.
The effect of the spiral phase on the C-point dipole is also investigated, as shown in Fig. 7 , where the sector angle Ψ = 180°, and the topological charges in Figs. 7(a)-7(d) are l = −1, + 1, + 2, and + 3, respectively. It is clearly seen that due to the OAM carried by spiral phase, the focal spot of the half-blocked AP beam is dramatically moved away from the focus (see the top of Fig. 7). The sign and value of the OAM can be characterized separately by that of the topological charge. Thus the moving direction and displacement of the focal spot are determined by l: when l<0 (>0), focal spot moves upward (downward) and the displacement increases with |l|. Meanwhile, the spiral phase affects the C-point dipole little. Especially with the variation of l, the C-point dipole just moves with the focal spot, but the dipole spacing is almost unchanged.
Actually, the focal field of the half-blocked AP beam with spiral phase can be also decomposed to two circular polarizations with opposite handedness , i.e., Ef ∝ ei(l-1)π/2U-eL-ei(l+1)π/2U+eR = ei(l-1)π/2[U+eR + U-eL], where U+ and U- are given by Eq. (11) in Ref , with their index l-1 and l + 1 instead of l and –l in Ref , respectively. If only considered the dominant effect, U+ and U- are respectively related to S1-l and S-1-l, which are defined as the Hankel transform. The locations of the circularly polarized components can be approximated by the subscript of S, namely, the smaller absolute value of subscript corresponds to the closer spot to the central. This can qualitatively explain the motions of the C-point dipole. When l = 0, the right- and left-hand components (corresponds to S1 and S-1, respectively) locate symmetrically to the center (see Fig. 6). When l>0, the C-point dipole (corresponds to S1-l and S-1-l) moves away from the center, with the location increasing with l. While the case of l<0 goes the opposite way.
5. Singularity transition associated with sector phase obstacle
In Fig. 5, the amplitude obstacle in the focusing system is replaced by a π-shift phase obstacle, of which the transmission function expressed as
With the modulation of the phase obstacle, the intensities and polarization states of the focal field of AP beam are calculated with Eq. (2). The results are shown in Fig. 8 , where Figs. 8(a)-8(f) correspond to Ψ = 30°, 60°, 83.5°, 90°, 110° and 180°, respectively. It reveals that with the increasing of the sector region of phase modulation, the intensity profile of the focal field gradually changes to petal-like distribution, with a pair of dark cores horizontally locating on the both side of the focus [see the top of Fig. 8(f)]. From the bottom of Fig. 8, it can be seen that the sector phase obstacle also introduces elliptical and circular polarizations into the focal field. However, when the phase is modulated on half plane (Ψ = 180°), the focal field maintains locally linear polarizations, resembling the case with even-fold symmetric amplitude obstacles . From the crossed zero contours of s1, s2, and s3 in Fig. 8, it reveals that the polarization singularities are dramatically changed. With the modulation of the sector phase obstacle, the central V-point in Fig. 2 is converted into a C-point dipole, composed by two C-points with opposite handedness [see Fig. 8(a)]. With increasing the sector angle Ψ, the two C-points depart away from each other gradually [see Figs. 8(b) and 8(c)], along the directions marked by the arrowheads in Fig. 8(a). The angle Ψ = 83.5° is a critical angle for the singularity transition, at which the zero contour of s1 just intersects the two self-crossing points of the zero contour of s2, as shown in Fig. 8(c). When Ψ>83.5°, each point of the C-point dipole splits into three C-points with unchanged handedness, as shown in Fig. 8(d). With the further increase of Ψ, the six C-points move along the arrowheads in Fig. 8(d). Especially for the C-points on the sides, they can be considered two C-point dipoles, and gradually join together on each side as Ψ increasing. Interestingly, the six C-points are merged and converted into four V-points when Ψ increases to 180°.
It is worth mentioning that for the case of Ψ = 180°, due to the modulation of the phase obstacle, the ring-like focal field of the AP beam splits into a double-dark-core distribution with the maximum intensity residing the central point; meanwhile, the central V-point of the focused AP beam splits into two V-points, each of which occupies a dark core of the focal field.
Finally, we investigated the effect of spiral phase on the PSs of the focused AP beam blocked by a half-plane phase obstacle (Ψ = 180°), and the corresponding results are depicted in Fig. 9 . In Figs. 9(a)-9(d), the topological charges of the attached spiral phases are l = −1, + 1, + 2, and + 3, respectively. Comparing with Fig. 8(f), it can be seen that when |l| = 1, the central dark core and V-point of the focal field are recalled by the spiral phase. Instead, the two dark cores and the V-points in Fig. 8(f) are vanished. In addition, two left-hand (right-hand) C-points emerge when l = −1 ( + 1), as shown in Figs. 9(a) and 9(b). While for |l|≠1, the focal field becomes confusing, and more C-points arise. Furthermore, when Ψ≠180°, the focal field of the AP beam modulated with phase obstacle and spiral phase becomes also complicated as Figs. 9(c) and 9(d).
To summarize, we have investigated the influence of obstacle and spiral phase on the intensity and polarization state distributions of the focused AP beam, and discussed the conversion and control of the PSs in detail. The results reveal that the spiral phase can convert the V-point to a C-point, just if the topological charges of the spiral phase and azimuthal polarization are equal in absolute value, i.e. |l| = n. The sector obstacles, including the amplitude and phase ones, can also convert the V-point to one or more C-point dipoles, with the position of the singularities changed with the sector angle Ψ. In the case of the amplitude obstacle, the spiral phase moves the C-point dipole totally. While in the case of the phase obstacle, especially when the phase of the AP beam is modulated on half plane (Ψ = 180°), the central V-point of the focal field splits into double V-points, and then can be recalled by the single charged spiral phase (l = ± 1). Based on these conclusions, the PSs of the focused AP beam can be selectively converted and controlled. We hope these results may have some useful implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused CV beams.
This work was supported by the 973 Program (2012CB921900), the National Natural Science Foundations of China (61205001), the Natural Science Basic Research Plan in Shaanxi Province of China (2012JQ1017), the Northwestern Polytechnical University (NPU) Foundation for Fundamental Research (JC20120251), and the Technology Innovation Foundation of NPU (2011KJ01011).
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