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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
where, (r′, φ′) is the polar coordinates at the pupil plane; tj is the transmission coefficient; αj and βj 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 aswhere E0 denotes the vector amplitude of the input AP beam.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(imφ′) with Fourier coefficients as following
where Cmj depicts the amplitudes of the OAM sidebands of the jth photic region and is expressed asMore generally, the cylindrically polarized beams can be considered as the synthesizing of right-handed (RH) and left-handed (LH) spin components carrying opposite OAM [16, 17], i.e.where, E0(r′) is the amplitude profile; l is the topological charge of the polarization; φ′0 is a constant phase; eR and eL stand for the unit vector of the RH and LH spin components, respectively.For a vector beam, when the constant phase φ0' = π/2, the wavefunction can be represented as E0(r′, φ′) = E0(r′)[exp(-ilφ′)eR-exp(ilφ′)eL] by removing the constant phase. Substituting the wavefunction into Eq. (3), we can obtain the transmitted vector beam as
withEvidently, in addition to the common helical phase term of exp( ± ilφ′), Eq. (7) suggests that the RH and LH spin components are both composed of an infinite series of helical phase exp(imφ′).For simplicity, we assume the profile of the incident beam has an finite-aperture function as E0(r′) = circ (r′/r0) [10], where r0 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
withwhere, (r, φ) is the polar coordinates at the focal plane, Hn(r) denotes the nth-order Hankel transform of E0(r′). As can be seen, the transmitted beam from jth photic region is split into a pair of spin components URj and ULj, 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 αj ± π/2 [10], respectively.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 t1 = 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 UR and UL with opposite handed polarizations, which occupy the angular positions ± π/2, respectively. Figures 1(c)-1(e) illustrate the calculated transverse distributions of intensity (top) and Stokes parameter s3 (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 s3 = ± 1], which occupy the angular positions ± π/2, respectively.
Fig. 1 (a) Schematic structure of the sector photic mask. (b) Theoretical map of polarizations at the focal plane. (c)-(e) Calculated transverse distributions of intensity (top) and Stokes parameter s3 (bottom) at the focal plane of the vector beams with l = 1, 5 and 10, respectively. Red and blue dashed ellipses correspond to the spin components UR and UL, respectively. The dimension of the focal plane is 6λ × 6λ.
It is known that the OAM spectra have a sinc2 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 URj and ULj are determined by the terms of m = 0 in Eq. (9), i.e. exp(-ilφ)H-l(r) and exp(ilφ)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 (c2). 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 (βj = β, j = 1, 2, …, N) and the same transmission coefficient tj = t, and the corresponding angular position is αj = 2π(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(a1), with β = π/2, α1 = 0, α2 = 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 UR1 and UL1, occupying the angular positions π/2 and 3π/2, respectively, as schematically shown in Fig. 2(a2). Likewise, the spin components UR2 and UL2 arising from the second (j = 2) photic region [3π/4≤φ′≤5π/4] occur similar focusing behaviors [as schematically shown in Fig. 2(a3)], and occupy the angular position 3π/2 and π/2, respectively. It should be noted that the spin components UR1 and UL2 (UL1 and UR2) overlap each other, with the same amplitude and a constant phase difference lπ [as described in Eq. (8) and Eq. (9)]. Therefore, the superposed field of the spin components UR1 and UL2 (UL1 and UR2) present locally linear polarization.
Fig. 2 (a1) Schematic structure of the N = 2 mask and theoretical maps of polarization for vector beams transmitted from photic regions (a2) -π/4≤φ′≤π/4, and (a3) 3π/4≤φ′≤5π/4, with the schematics of inserted. (b), (c) Intensity distributions at the pupil plane (top), transverse distributions of intensity (middle) and Stokes parameter s3 (bottom) at the focal plane of the vector beams with l = 1 (AP beam) and 5, respectively. Arrows donate the polarized direction. The dimension of the focal plane is 4λ × 4λ.
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 s3 (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(b3) and 2(c3). 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 TEM01 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 αj = 2π(j-1)/3. The focusing behaviors could also be deduced from Eq. (9), that the spin components URj occupy the angular positions π/2, 7π/6, 11π/6, while the spin components ULj 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 URj and ULj 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 s3 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.
Fig. 3 (a) Schematic structure of an N = 3 mask and (b) its theoretical focusing model. Transverse distributions of (c) intensity and (d) Stokes parameter s3 of the focal field. Red and blue dashed ellipses correspond to the spin components URj and ULj, respectively. The dimension of the focal plane is 4λ × 4λ.
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 2N 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 tj = exp(iϕj), where ϕj is the phase retardation of the jth 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 s3 (bottom) of focal field of AP beams passing though such masks. For the mask shown in Fig. 4(a1), the parameters ϕ1 = ϕ3 = 0, ϕ2 = ϕ4 = π, the total focal field consists of two superposed fields, which generated from the photic regions with different transmission coefficients. The field (marked as EI) passing through the photic regions ϕ1 and ϕ3 (−π/4≤φ′≤π/4 and 3π/4≤φ′≤5π/4) forms the same focal field as that shown in Fig. 2(b2), while the other field (marked as EII) passing through the photic regions ϕ2 and ϕ4 (π/4≤φ′≤3π/4 and 5π/4≤φ′≤7π/4) forms a linear polarized focused field similar to Fig. 2(b2), with both the intensity and polarization distributions totally rotated by π/2. Note that these two linearly polarized fields from EI and EII are out of phase, so the superposition of such two fields is similar to that of the horizontally and vertically polarized TEM01 modes, and the total field exhibits the intensity and polarization characteristics resemble to the HE21 mode, as shown in Figs. 4(a2) and 4(a3). For the mask with parameters ϕ1 = ϕ3 = 0, ϕ2 = ϕ4 = π/2, as shown in Fig. 4(b1), the constant phase difference between two orthogonally polarized fields (from EI and EII) is ∆ϕ = π/2, so the total field presents four C-points [white points in Fig. 4(b3)] and a V-point [vector polarized point, black point in Fig. 4(b3)].
Fig. 4 Schematic structures of N = 4 masks with different transmission coefficients (top), calculated intensity (middle) and Stokes parameter s3 (bottom) of focal fields. Arrows: polarization orientation. Lines: orientations of the long axis of local polarization ellipses. The dimension of the focal plane is 4λ × 4λ.
Figure 4(c1) 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 UR and UL at the same angular positions, is lπ + 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 ϕ1 = 0 and ϕ3 = π is vertically polarized, the other one from regions with ϕ2 = π/2 and ϕ4 = 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(c3).
Figure 5 shows another case for an N = 6 mask, with parameters ϕ2j-1 = 0, ϕ2j = 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 UR and UL 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.
Fig. 5 (a) Schematic structure of an N = 6 mask with two kind of transmission coefficients. (b) Calculated intensity and (c) Stokes parameter s3 of focal fields. Arrows: polarization orientation. The dimension of the focal plane is 4λ × 4λ.
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 t1 = t3 = t5 = 1, t2 = t4 = t6 = −1. For the mask descripted in Fig. 6(a1), the angle widths β1 = β3 = β5 = π/2, β2 = β4 = β6 = π/6, and the angular positions αj = π(j-1)/3. Equation (9) shows that the angular position of the spin components UR and UL are depended on the angular positions αj. Therefore, the pair of spin components generated from the opposite photic regions occupy the same angular positions (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(b1), its transmission function can be expressed by Fig. 6(a1), i.e., P1(φ') = -P2(φ'-π/N). Consequently, the focused fields exhibit identical intensity distribution, but opposite polarization, as shown in Figs. 6(b2) and 6(b3).
Fig. 6 Schematics structures of anisometric masks (top) and correspondingly calculated intensity (middle) and Stokes parameter s3 (bottom) distributions of focal fields. Arrows: polarization orientation. Lines: orientations of the long axis of local polarization ellipses. The dimension of the focal plane is 4λ × 4λ.
Figure 6(c1) displays a special anisometric mask, which includes three pairs of isometric photic region, that β1 = β4 = π/6, β2 = β5 = π/3, and β3 = β6 = π/2, and the transmission coefficients are alternatively arranged as t1 = 1 and t2 = 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 UR and UL increases to lπ + 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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