## Abstract

We propose a theoretical model to semi-quantitatively describe the modulation mechanism of multi-azimuthal masks on the focal fields of azimuthally polarized (AP) beam. With this model, we cannot only explain the redistributions of the polarization and intensity at the focal plane, but also consciously manage the focal fields by designing the mask structure parameters, such as the symmetry, area, and phase retardation of the sector photic regions. Our results may supply a guideline to realize the manipulation on the polarizations, angular momenta, and the distribution of focused fields.

© 2015 Optical Society of America

## 1. Introduction

The unique focusing properties of cylindrical vector (CV) beams have been extensively studied over the past years [1], owing to their wide applications in optical trapping, super-resolution imaging, as well as superfine processing [2–4]. It has been demonstrated that radially polarized (RP) beams can be focused into tighter spots with stronger longitudinal components than those of spatially homogeneous polarized beams [5, 6]. The RP beams enable also to generate special focal spots, such as optical needle, optical chain, and optical cage [7–9], under the combined modulation of a high-NA objective and diffraction optical element (DOE). Recently, more research interests focus on the tight focusing of azimuthally polarized (AP) beams, the focal field of which contains only donut shape azimuthal components. Beyond the extraordinary intensity distribution in focal field, the polarization and transverse energy flow distribution are much richer and have displayed very interesting issues [10]. We have been recently reported that the AP beams obstructed by rotationally symmetric obstacles take place energy flow redistribution in the focal region [11], and the AP beams modulated by spiral phase and sector obstacle exhibit controllable polarization singularities conversion [12]. However, the mechanism of focal fields of the AP beams modulated by the sector-shaped obstacles, which is essential to explore more novel focusing properties, has yet to be revealed.

In this paper, we propose a theoretical model to semi-quantitatively describe the modulation of multi-azimuthal masks on the focal fields of AP beam. It is shown that the angular diffraction [13–15] induces the azimuthally spin-dependent splitting of transmitted beam, and the interference of spin components gives rise to the abundant distribution of focal field. With this model, we detailedly explore the modulation rules of mask structure parameters, such as symmetry, area, and phase retardation of the sector photic regions.

## 2. Theoretical model

A multi-azimuthal mask can be considered to be composed of several sector photic regions. Assuming there are *N* photic regions in the mask, for each of which the transmission function can be expressed as

*r′*,

*φ′*) is the polar coordinates at the pupil plane;

*t*is the transmission coefficient;

_{j}*α*and

_{j}*β*denote the angular position and the angle width of the photic regions, respectively. Then the wavefunction of a cylindrically polarized beam after passing through the multi-azimuthal mask can be written as

_{j}**E**denotes the vector amplitude of the input AP beam.

_{0}The relationships between angular diffraction and OAM sidebands have been well explored, and each individual photic region is analogue to a single slit, the mask is analogue to a grating of *N* slits with 2π periodic nature [13, 14]. Consequently, the complex transmission function of the angle distribution can be expressed in terms of the distribution of angular momentum exp(i*mφ*′) with Fourier coefficients as following

*C*depicts the amplitudes of the OAM sidebands of the

_{mj}*j*th photic region and is expressed as

*E*

_{0}(

*r′*) is the amplitude profile;

*l*is the topological charge of the polarization;

*φ′*

_{0}is a constant phase;

**e**

_{R}and

**e**

_{L}stand for the unit vector of the RH and LH spin components, respectively.

For a vector beam, when the constant phase *φ*_{0}*'* = π/2, the wavefunction can be represented as **E**_{0}(*r′*, *φ′*) = *E*_{0}(*r′*)[exp(-i*lφ′*)**e**_{R}-exp(i*lφ′*)**e**_{L}] by removing the constant phase. Substituting the wavefunction into Eq. (3), we can obtain the transmitted vector beam as

*lφ′*), Eq. (7) suggests that the RH and LH spin components are both composed of an infinite series of helical phase exp(i

*mφ′*).

For simplicity, we assume the profile of the incident beam has an finite-aperture function as *E*_{0}(*r′*) *=* circ (*r′*/*r*_{0}) [10], where *r*_{0} is the beam size. As we known, the focal field distribution of a finite-aperture vortex field can be described as a Hankel transform field with finite aperture [18]. In principle, the focal field of incident beam described by Eq. (6) can be expressed as

*r*,

*φ*) is the polar coordinates at the focal plane,

*H*(

_{n}*r*) denotes the

*n*th-order Hankel transform of

*E*

_{0}(

*r′*). As can be seen, the transmitted beam from

*j*th photic region is split into a pair of spin components

*U*

_{R}

*and*

_{j}*U*

_{L}

*, with opposite spin angular momenta (SAMs) at the focal plane. Each spin component has its own set of OAM sidebands (SAM-OAM combined sidebands, which present SAM-OAM combined state [19]), and occupy the angular positions*

_{j}**[10], respectively.**

*α*π/2_{j}±As an illustrating example, we investigate the focusing behaviors of AP beam (*l =* 1) and higher-order vector beams (*l =* 5, 10) transmitted from a single sector photic mask, by comparing with the numerical results obtained by Richards-Wolf theory [11, 20]. In our numerical simulation, we choose the parameters NA = 0.95 and *λ =* 633nm. The mask is schematically depicted in Fig. 1(a), with the transmission coefficient *t*_{1} = 1 in the photic region -π/4≤*φ′*≤π/4. Figure 1(b) schematically displays the theoretically predicted polarization distribution at the focal plane from Eq. (8). According to Eqs. (8) and (9), the focused single-segmented AP beam is split into a pair of spin components *U _{R}* and

*U*with opposite handed polarizations, which occupy the angular positions

_{L}*±*π/2, respectively. Figures 1(c)-1(e) illustrate the calculated transverse distributions of intensity (top) and Stokes parameter s

_{3}(bottom) at the focal plane of the vector beams with azimuthal orders

*l*= 1, 5 and 10, respectively. Quite clearly, in agreement with our theoretical prediction, the focused field is split into a pair of

*C*-points [point of circular polarization, see the two white points in Figs. 1(c2)-(e2), where s

_{3}= ± 1], which occupy the angular positions ± π/2, respectively.

It is known that the OAM spectra have a sinc^{2} distribution [13] and the OAM component of incident light acquiring shift *m =* 0 dominate the intensity. As a result, the positions of the peak intensities of the two spin components *U*_{R}* _{j}* and

*U*

_{L}

*are determined by the terms of*

_{j}*m*= 0 in Eq. (9), i.e. exp(-i

*lφ*)

*H*

_{-}

*(*

_{l}*r*) and exp(i

*lφ*)

*H*

_{-}

*(*

_{l}*r*), respectively. For the AP beam (

*l =*1), these two components approximate the first-order Hankel transform of incident background, and their peak intensities locate close to the center coordinate. Therefore, these two spin components are too closed to be separated, and the field at the superposed region is linearly polarized, as depicted in Fig. 1 (c

_{2}). As

*l*grows, the terms of higher-order Hankel transform lead the two components away from the center, and result in obvious separation of the

*C*-point pair.

## 3. Modulation of multi-azimuthal masks on focal fields

The above semiquantitative model reveals the modulation mechanism of individual sector photic region in the focal field of vector beams. For a multi-azimuthal mask composed of several sector photic regions, it is visible that the mask structure significantly influences the characteristics of focal field, due to the interference of SAM-OAM combined sidebands from multiple photic regions. In this section, we detailedly explore the modulation rules of multi-azimuthal masks on the focal fields of AP beam, by analyzing the influences of mask structure, including the symmetry, area, and even phase retardation of the sector photic regions.

#### 3.1 Influences of symmetry

It has been shown that the symmetry of photic regions significantly influences the distribution characteristics of focal fields [11]. Therefore, we respectively explore the modulation rule of the even- and odd-fold masks. Generally, for an *N*-fold mask, each photic region has the same angle width (*β _{j}* =

*β*,

*j*= 1, 2, …,

*N*) and the same transmission coefficient

*t*=

_{j}*t*, and the corresponding angular position is

*α*= 2π(

_{j}*j*-1)/

*N*. Firstly, we take the

*N*= 2 mask as an example of even-fold mask, which has a transmission profile schematically depicted in Fig. 2(a

_{1}), with

*β =*π/2,

*α*

_{1}

*=*0,

*α*

_{2}

*=*π, and

*t*= 1. According to the theoretical model, the focused field generated from the first (

*j =*1) photic region [

*-*π/4

*≤φ′≤*π/4] is split into two opposite spin components

*U*

_{R1}and

*U*

_{L1}, occupying the angular positions π/2 and 3π/2, respectively, as schematically shown in Fig. 2(a

_{2}). Likewise, the spin components

*U*

_{R2}and

*U*

_{L2}arising from the second (

*j =*2) photic region [3π/4

*≤φ′≤*5π/4] occur similar focusing behaviors [as schematically shown in Fig. 2(a

_{3})], and occupy the angular position 3π/2 and π/2, respectively. It should be noted that the spin components

*U*

_{R1}and

*U*

_{L2}(

*U*

_{L1}and

*U*

_{R2}) overlap each other, with the same amplitude and a constant phase difference

*l*π [as described in Eq. (8) and Eq. (9)]. Therefore, the superposed field of the spin components

*U*

_{R1}and

*U*

_{L2}(

*U*

_{L1}and

*U*

_{R2}) present locally linear polarization.

Figures 2(b) and 2(c) display the calculated intensity distribution at the pupil plane (top), and the distributions of the intensity (middle) and the Stokes parameter s_{3} (bottom) at the focal plane, where Figs. 2(b) and 2(c) correspond to the AP beam and the vector beam with *l* = 5, respectively, with their locally polarized directions denoted by arrows. In agreement with our theoretical prediction, the superposed field present locally linear polarization at the focal plane, as shown in Figs. 2(b_{3}) and 2(c_{3}). In addition, the interferogram at the focal plane is closely related to the azimuthal order *l*. For the case of *l* = 1, the interference field presents an intensity profile like the TEM_{01} mode; for the case of *l =* 5, the interference of higher-order SAM-OAM combined sidebands represents four lobes in the upper/lower half planes [21].

Figure 3 provides a schematic of the *N =* 3 mask (as an example of odd-fold mask), from which an AP beam is transmitted. The mask is depicted in Fig. 3(a), which has three photic regions with angle width *β =* π/3, and angular positions *α _{j} =* 2π(

*j*-1)/3. The focusing behaviors could also be deduced from Eq. (9), that the spin components

*U*

_{R}

*occupy the angular positions π/2, 7π/6, 11π/6, while the spin components*

_{j}*U*

_{L}

*occupy the angular positions π/6, 5π/6, 3π/2, respectively, as schematically shown in Fig. 3(b). It is clear that the spin components*

_{j}*U*

_{R}

*and*

_{j}*U*

_{L}

*alternately appear along the azimuthal direction, and the focused field exhibit complex polarization distribution. Figures 3(c) and 3(d) display the numerical results of the distributions of intensity and Stokes parameter s*

_{j}_{3}at the focal plane, respectively. Six

*C*-points (point of circular polarization) with opposite rotation direction azimuthally appear at the angular positions 2π(

*j*-1)/

*N ±*π/2.

The above presented results reveal that the distributions of the focal fields of vector beams closely dependent on the symmetry of photic regions. For an even-fold symmetric mask, the focal field will be locally polarized because the destructive interference of two spin components with opposite angular momenta, and no longer carry angular momenta (including SAM and OAM). For the odd-fold one, in spite of that the total angular momentum is conserved, the local SAM and OAM are not zero, and accordingly 2*N C*-points and phase singularities are present at the focal fields.

#### 3.2 Influences of transmission coefficient

In the following we investigate the modulation rule of multi-azimuthal masks composed of multiple photic regions with different transmission coefficients *t _{j}* = exp(i

*ϕ*), where

_{j}*ϕ*is the phase retardation of the

_{j}*j*th photic region.

Figure 4 displays the schematic of *N* = 4 masks with different transmission coefficients (Top), as well as the calculated total intensity (middle) and Stokes parameter s_{3} (bottom) of focal field of AP beams passing though such masks. For the mask shown in Fig. 4(a_{1}), the parameters *ϕ*_{1} = *ϕ*_{3} = 0, *ϕ*_{2} *= ϕ*_{4} = π, the total focal field consists of two superposed fields, which generated from the photic regions with different transmission coefficients. The field (marked as **E**_{I}) passing through the photic regions *ϕ*_{1} and *ϕ*_{3} (*−*π/4*≤φ′≤*π/4 and 3π/4*≤φ′≤*5π/4) forms the same focal field as that shown in Fig. 2(b_{2}), while the other field (marked as **E**_{II}) passing through the photic regions *ϕ*_{2} and *ϕ*_{4} (π/4*≤φ′≤*3π/4 and 5π/4*≤φ′≤*7π/4) forms a linear polarized focused field similar to Fig. 2(b_{2}), with both the intensity and polarization distributions totally rotated by π/2. Note that these two linearly polarized fields from **E**_{I} and **E**_{II} are out of phase, so the superposition of such two fields is similar to that of the horizontally and vertically polarized TEM_{01} modes, and the total field exhibits the intensity and polarization characteristics resemble to the HE_{21} mode, as shown in Figs. 4(a_{2}) and 4(a_{3}). For the mask with parameters *ϕ*_{1} = *ϕ*_{3} = 0, *ϕ*_{2} *= ϕ*_{4} = π/2, as shown in Fig. 4(b_{1}), the constant phase difference between two orthogonally polarized fields (from **E**_{I} and **E**_{II}) is ∆*ϕ* = π/2, so the total field presents four *C*-points [white points in Fig. 4(b_{3})] and a *V*-point [vector polarized point, black point in Fig. 4(b_{3})].

Figure 4(c_{1}) shows an *N* = 4 mask with discrete vortex-type phase retardation, where the phase difference between two adjacent photic regions is π/2. For such a mask, the phase differences between the superposed spin components *U*_{R} and *U*_{L} at the same angular positions, is *l*π + π. As a result, their interferograms transversally shift half period, and the bright fringes located in the center of the focal field, keep the linearly polarizations. It is easy to check that the superposed field generated from regions with *ϕ*_{1} = 0 and *ϕ*_{3} *=* π is vertically polarized, the other one from regions with *ϕ*_{2} = π/2 and *ϕ*_{4} *=* 3π/2 is horizontally polarized, and the phase difference between these two parts is π/2. As a result, the focal field shows a central *C*-point, as represented in Fig. 4(c_{3}).

Figure 5 shows another case for an *N =* 6 mask, with parameters *ϕ*_{2}_{j}_{-1} = 0, *ϕ*_{2}* _{j} =* π. From Eq. (8), we can see that the total focal field [see Fig. 5(b)] correspondingly consists of six parts: three pairs of spin components

*U*

_{R}and

*U*

_{L}generated from opposite photic regions with

*ϕ*= 0 and

*ϕ =*π, respectively. It is obvious that the superposed field of a pair of spin components with opposite handedness is locally linearly polarized, and the phase differences between the three pairs of spin components are zero. As a result, the total focal field is locally linearly polarized, as depicted in Figs. 5(b) and 5(c), and the polarized direction is analogous to radial polarization.

From the results above, we can see that the focal field is totally changed by merely adjusting the transmission coefficients rather than the symmetry. By adjusting the phase difference of opposite photic regions, we can realize the manipulation of intensity pattern, especially the intensity in the center of focal field. We can also manipulate the position and handedness of singularities by engineering the azimuthal phase distribution.

#### 3.3 Influences of area

We note that our description can easily be generalized to more complicated masks, such as anisometric masks, where the areas of sector photic regions are no longer equal. Figures 6(a) and 6(b) show the focusing properties of transmitted AP beam passing through the masks [as shown in top row] with *t*_{1} = *t*_{3} = *t*_{5} = 1, *t*_{2} = *t*_{4} = *t*_{6} = −1. For the mask descripted in Fig. 6(a_{1}), the angle widths *β*_{1} = *β*_{3} = *β*_{5} = π/2, *β*_{2} = *β*_{4} = *β*_{6} = π/6, and the angular positions *α _{j}* = π(

*j*-1)/3. Equation (9) shows that the angular position of the spin components

*U*

_{R}and

*U*

_{L}are depended on the angular positions

*α*. Therefore, the pair of spin components generated from the opposite photic regions occupy the same angular positions (

_{j}*j*-1)π/3 + π/2. But, the angle width of photic region also can significantly affect the intensity and polarization distributions, those two

*C*-points with opposite handedness depart away from each other as angle width decreases [12]. As a consequence, the focused fields of light transmitted from three bigger photic regions dominate the intensity distribution, and three pairs of

*C*-points arise alternately in the azimuthal direction. Meanwhile, elliptical and circular polarizations emerge in the focal fields, similar to the results of three-fold symmetric amplitude-type mask. For the mask of descripted in Fig. 6(b

_{1}), its transmission function can be expressed by Fig. 6(a

_{1}), i.e.,

*P*

_{1}(

*φ'*)

*=*-

*P*

_{2}(

*φ'*-π/

*N*). Consequently, the focused fields exhibit identical intensity distribution, but opposite polarization, as shown in Figs. 6(b

_{2}) and 6(b

_{3}).

Figure 6(c_{1}) displays a special anisometric mask, which includes three pairs of isometric photic region, that *β*_{1} = *β*_{4} = π/6, *β*_{2} = *β*_{5} = π/3, and *β*_{3} = *β*_{6} = π/2, and the transmission coefficients are alternatively arranged as *t*_{1} = 1 and *t*_{2} = exp(iπ/2). For this structure, the opposite photic regions are isometric, while has a constant phase difference π/2. According to the theoretical model, once the two opposite photic regions are isometric, the superposed focal field from those two photic regions will be always locally linearly polarized, but the interferogram closely depends on their phase difference. For this mask, the phase difference between the superposed spin components *U*_{R} and *U*_{L} increases to *l*π + π/2, and the interferograms transversally shifts a quarter of period. As totally superposed by three linearly polarized fields, which are respectively from the three pair of isometric photic regions, the total focal field is also locally linearly polarized. These results support the conscious modulation on intensity and polarization distributions of focal field by appropriately adjusting the angle widths and phase differences of the mask.

## 4. Conclusions

To summarize, we have revealed the formation mechanism of the focal fields of AP beams passing through multi-azimuthal masks by using a Fourier transform model. With this model, we analyzed the modulation rules of mask with different structure parameters, such as the symmetry, area, and phase retardation of the sector photic regions. On the other hand, we have demonstrated that it is feasible to consciously manage the intensity distribution, polarization, and even the angular momentum of the focal fields by designing the mask structure. We hope our results may supply a guideline to realize the manipulation on the polarizations, angular momenta, and the distribution of focused fields of AP beams with multi-azimuthal masks, as well as the diffractive optical elements.

## Acknowledgments

This work was supported by the 973 Program (2012CB921900), the National Natural Science Foundations of China (11404262, 61205001, and 61377035), the Natural Science Basic Research Plan in Shaanxi Province of China (2012JQ1017), and the Fundamental Research Funds for the Central Universities (3102014JCQ01084).

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