Abstract

Circular symmetry singular light beams (CS-SLBs) possessing spatially variant field distributions have drawn extensive attention because of their unique optical properties. However, the extraction of spatial phase and polarization distributions is always a significant but difficult issue in CS-SLB applications. Here, we propose and experimentally investigate an orthogonal polarization separation (OPS) method to retrieve the spatial phase and polarization distributions of arbitrary CS-SLBs. Theoretically, the CS-SLB, including the vortex beam (VB), cylindrical vector beam (CVB), and cylindrical vector vortex beam (CVVB), can be decomposed into two orthogonal circularly polarized sub-VBs. Therefore, once the spatial phase distributions and initial phase difference of the two components are obtained, the phase and polarization distributions of the CS-SLB can be retrieved, and its type can also be identified. Based on this analysis relationship, we first separated the CS-SLB into two circularly polarized sub-VBs and designed an astigmatic phase iterative algorithm to restore their spatial phase information. After retrieving the phases of the two components, we have experimentally obtained the spatial phase and polarization distributions of three typical CS-SLBs, including VBs, CVBs, and CVVBs. These results demonstrate that this method provides a feasible way to retrieve the variant field distributions of CS-SLBs and may have great application prospects in optical imaging, optical communication, etc.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Circular symmetry singular light beams (CS-SLBs) have a zero amplitude at the center of beam cross-section and show “doughnut” intensity distributions due to their spatially non-uniform distributions of phase or polarization [18]. The vortex beam (VB) is a typical CS-SLB possessing phase singularity and carrying orbital angular momentum (OAM). The helical phase wave-front of VBs can be characterized by a phase factor of $\exp (il\theta )$, where $\theta $ is the azimuthal angle, and l is the topological charge (TC) representing OAM modes [9,10]. Theoretically, the number of OAM eigenstates is infinite, providing an additional degree of freedom compared with conventional light beams. Except for the phase singularity, polarization singularity is also a factor leading to CS-SLBs. The cylindrical vector beam (CVB) [11] is another CS-SLB possessing spatial polarization uncertainty and mainly include radial and azimuthal polarization. Researchers found that CVBs have stronger longitudinal fields and smaller spot sizes after focused by a high numerical aperture objective [1214]. The cylindrical vector vortex beam (CVVB) is also a kind of CS-SLB, which have both the helical-phase wave-front carrying OAM (phase singularity) and spatially inhomogeneous polarization profile (polarization singularity). These variant field distributions reward CS-SLBs unique optical properties and have attracted extensive attention in many fields, such as optical trapping [1518], optical multiplexing [1922], optical super-resolution [23], quantum information processing [24], and tight focusing [25], etc. In those applications, retrieving the spatial phase and polarization distributions is always one of the most significant issues.

Considerable research efforts have been devoted to extracting the polarization and phase distributions of CS-SLBs [2630]. By interfering the VB with a planar or spherical wave [31], or diffracting the VB with a cylindrical lens [32], the TC of OAM modes can be confirmed from the obtained intensity pattern. To extract the phase distributions of VBs, algorithms are normally used to deal with the intensity information. The most common way to obtain the spatial phase is using the algorithm to deal with the interfered intensity images. Further, iterative algorithms are developed to retrieve the phase from intensity image of a single beam. For example, the Gerchberg-Saxton (G-S) algorithms [33,34] have been employed to retrieve the phase distribution by setting known correct constraints for each iteration, but the uniqueness of the reconstruction using these algorithms cannot be guaranteed. To solve this problem, some researchers have proposed the astigmatic diffraction imaging methods [35,36]. The Stokes parameter method is the most common way for measuring the polarization distributions of beams [37,38], which has been also demonstrated that the Stokes parameter method is available for extracting phase information [39]. Unfortunately, the Stokes parameter method can only retrieve the relative phase difference of the two orthogonal polarized components, which is hard to obtain the integral phase of the two components. Based on the Pancharatnam-Berry phase theory, a method for simultaneously measuring polarization and the phase of arbitrarily polarized beams by using digital holography was proposed [29]. It only requires a single-exposure imaging containing interference fringes without changing any optical elements. However, the extracted phase of the beam is imperfect, and the singularity phase in the center of the cross-section will lose. Therefore, it is still necessary to find a method that can accurately retrieve the complete phase and polarization distributions of arbitrarily CS-SLBs.

In this paper, we proposed an orthogonal polarization separation (OPS) method for retrieving the spatial phase and polarization distributions of arbitrary CS-SLBs. Theoretically, the CS-SLB can be decomposed into two orthogonal circularly polarized sub-VBs. Therefore, once the spatial phase distributions and initial phase difference of the two components are obtained, both the phase and polarization distributions of the CS-SLB can be retrieved. Combined with our previous works [40], we used Q-plates to produce CS-SLBs. By manipulating the incident beam to circularly polarized or linearly polarized, a VB or CVB can be obtained respectively. To produce CVVBs, we adopted the combination of two Q-plates and a quarter-wave plate. After decomposing the CS-SLB, we introduced a spherical wave to interfere with the sub-VBs and designed an astigmatic phase iterative (API) algorithm to calculate the phase distributions of the two components. The experimental results show that by superimposing the phase distributions of the two decomposed sub-VBs, the spatial phase and polarization distributions of the VBs (with the TCs of l = 1∼3), CVBs (with the polarization orders of m = 1∼3), and CVVBs (with the TCs and polarization orders of (l, m) = (-1, 2) and (2, -1)) were all successfully retrieved.

2. Principles and Methods

CS-SLBs mainly include VBs, CVBs, and CVVBs. The VB can be written as:

$$\begin{aligned} {E_{vortex}} &= {E_0}\exp [{i({l\theta + {\varphi_0}} )} ]\left[ {\begin{array}{{c}} 1\\ 0 \end{array}} \right]\nonumber \\ &= \frac{1}{2}{E_0}\exp [{i({l\theta + {\varphi_0}} )} ]\left[ {\begin{array}{{c}} 1\\ i \end{array}} \right] + \frac{1}{2}{E_0}\exp [{i({l\theta + {\varphi_0}} )} ]\left[ {\begin{array}{{c}} 1\\ { - i} \end{array}} \right], \end{aligned}$$
where ${E_0}$ is the simplified amplitude, l is the TC, $\theta$ is the azimuthal angle, and ${\varphi _0}$ is the initial phase. The Jones matrix is used to represent the polarization states of VBs. From the Jones matrices, we find that the VB can be decomposed into two orthogonal circularly polarized sub-VBs (a left-handed circularly polarized (LHCP) VB and a right-handed circularly polarized (RHCP) VB).

Theoretically, the CVB and CVVB can also be decomposed into two orthogonal circularly polarized sub-VBs. The Jones matrices of them can be expressed as:

$$\begin{aligned} {E_{vector}} &= {E_0}\left[ {\begin{array}{{c}} {\textrm{cos(m}\theta + {\varphi_0})}\\ {\textrm{sin(m}\theta + {\varphi_0})} \end{array}} \right]\nonumber \\ &= \frac{1}{2}{E_0}\exp (i{\varphi _0})\exp ( - im\theta )\left[ {\begin{array}{{c}} 1\\ i \end{array}} \right] + \frac{1}{2}{E_0}\exp (i{\varphi _0})\exp ({i\textrm{m}\theta } )\left[ {\begin{array}{{c}} 1\\ { - i} \end{array}} \right], \end{aligned}$$
$$\begin{aligned} {E_{vector\_vortex}} &= {E_0}\exp (il\theta )\left[ {\begin{array}{{c}} {\textrm{cos(m}\theta + {\varphi_0})}\\ {\textrm{sin(m}\theta + {\varphi_0})} \end{array}} \right] \nonumber \\ &= \frac{1}{2}{E_0}\exp (i{\varphi _0})\exp [i(l - m)\theta ]\left[ {\begin{array}{{c}} 1\\ i \end{array}} \right]\nonumber \\ &+ \frac{1}{2}{E_0}\exp (i{\varphi _0})\exp [i(l + m)\theta ]\left[ {\begin{array}{{c}} 1\\ { - i} \end{array}} \right], \end{aligned}$$
where m is the polarization order. As shown in Eq. (2), the CVB can be decomposed into an LHCP and an RHCP VB with opposite TCs. However, the TCs of the two sub-VBs of CVVBs are different. The TC of the LHCP component is $l + m$, and the RHCP component is $l - m$. Therefore, once the phase distributions of the two components are obtained, the phase and polarization distributions of CS-SLBs, including VBs, CVBs, and CVVBs, can be retrieved.

In this work, we use an API algorithm to reconstruct the diffracted wave-function of VBs, which is achieved by iterating wave-functions between the lens plane and the detector plane. This method is analogous to an iterative method of G-S algorithms. For the traditional G-S algorithms, a priori knowledge of the sample must be obtained in advance. These priori information are introduced to the algorithm and constrain the reconstruction to a solution. Particularly, the size of the sample is the most important constraint of G-S algorithm, and the information regarding the size of the sample can be obtained by a pupil function [33]. However, in the API algorithm, by measuring the intensity of diffracted wave-functions at the detector plane 36 times, the algorithm becomes over-constrained, and we do not require the explicit imposition of sample-plane constraints. Hence, there is no need to apply some numerical image analysis on the registered beam images. The schematic diagram of the algorithm is illustrated in Fig. 1. The algorithm begins with an initial light field ${U_{in}}$ at the lens plane, which could be completely random. ${A_n}$ is the Fourier transform of the light field U incorporating a C-lens phase (${\Phi _n}$). ${\Phi _n}$ is determined by the optical axis orientation ($\theta $) and the focal length ($f$) of C-lens. To improve the accuracy of the extracted phase information, we average the phase results by changing the optical axis direction of the C-lens [36]. Here, we make $\theta = n\pi /18,(n = 1,2,\ldots ,36)$ to improve the convergence speed. The phase of the C-lens is shown in Eq. (4), where k is the wavenumber, and f is the focal length. The focal length of the C-lens in the simulation is consistent with the experiment.

$${\Phi _n} = \exp ( - ik{(x \cdot \cos (\theta ) - y \cdot \sin (\theta ))^2}/2/f).$$
The intensity distribution of ${A_n}$ is then replaced by ${D_n}$, and the phase maintains to get a new light filed of ${U_n}$, where ${D_n}$ is the square root of the experimentally recorded intensity of the VB passing through the C-lens with the same $\Phi $. The ${U_n}$ is then Fourier transformed ($F{T^{ - 1}}$) back to the lens plane to get ${U_n}^{\prime}$. The ${U_n}^{\prime}$ contains a phase aberration brought by C-lens in the experiment. This aberration can be corrected by multiplying the conjugate phase of C-lens (${\Phi _n}^{ - 1}$). And an updated light field ($U$) is obtained via a simple arithmetic average. Finally, the updated field U is taken for the next iteration, where p is the iteration time, and $\varepsilon $ is the error metric of $p \cdot th$ iteration. $\varepsilon (p)$ is defined by [36]
$$\varepsilon (p) = \frac{1}{N}\sum\limits_j {{{({I_j} - I_j^p)}^2}} ,$$
where N is the total pixel of the picture, ${I_j}$ is the measured intensity in the $j \cdot th$ pixel, and $I_j^p$ is the intensity in $j \cdot th$ pixel at the $p \cdot th$ iteration. The iteration continues until the error metric (${\varepsilon }$) reaches the preset value ($i$). With different values of i or n, the required time of retrieve algorithm will be different. The smaller the value of i, the longer time it takes. The larger the value of n, the faster the convergence. Moreover, the required iteration time also related to the quality of the input image and the computing ability of computers. In this work, we set i = 0.001, for n = 36 and the iteration times is 154, the required time of the algorithm is 315.01 s with an i5-6500 CPU.

 figure: Fig. 1.

Fig. 1. The schematic diagram of the API algorithm.

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However, the diffraction patterns of the VBs with the same TC but different initial phase are identical for symmetric intensity distributions, making the retrieved phases the same. Thus, we have to know the relative phase difference between the RHCP and LHCP components to get accurate polarization distributions. In the experiment, we introduced a spherical wave to interfere with the decomposed sub-VBs and used the interference patterns as the input of API algorithm. Since the phase of the reference light is fixed, the interference intensity will be rotated with the initial phase of VBs. Based on this property, the complete phase distributions, including the initial phase of RHCP and LHCP components, can be obtained.

Figure 2 describes the intensity and phase distributions of the interfered light fields with different energy ratios. The first and second rows of Fig. 2 are the interference patterns of the VBs with the same TC but different initial phases (${\varphi _0}$), and the intensity ratios of the VB to spherical wave from left to right are 1:0; 1:1; 2:1; 5:1; 10:1. As shown in the figures, the interference patterns are rotated with the initial phase difference, providing a way to get the relative phase difference of the decomposed VBs. The third and fourth rows present the phase distributions obtained from the interferogram by using the API algorithm. If the VBs with different initial phases have not interfered with the reference light, the retrieved phases are the same. However, if the energy ratio of the VB to spherical waves is below 2:1, the retrieved phase distributions are seriously destroyed because the spherical wave dominates the interference. As this ratio increases to 10:1, the phase distributions of the two sub-VBs are well retrieved. Thus, in order to get the phase distributions, including the initial phase difference of the RHCP and LHCP components, we choose the energy ratio (VB to spherical waves) of 10:1 in the next experiment.

 figure: Fig. 2.

Fig. 2. The interference intensity distributions (first and second rows) of VBs and the spherical wave with different ratios, and the restored phase distributions (third and fourth rows), ${\varphi _0}$ is the initial phase.

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After orthogonal polarization separation, we retrieve the phase distributions of LCHP and RCHP components by the API algorithm (${E_1}$ and ${E_2}$). According to the theorem of Stokes polarization, the Stokes parameter can also be obtained from the recovered field (${E_1}$ and ${E_2}$). The Stokes parameters (${S_0},{S_1},{S_2},{S_3}$) can be described by (${E_1},{E_2}$):

$$S0 = {\langle{{E_1}} \rangle ^2} + {\langle{{E_2}} \rangle ^2},$$
$$S1 = {\langle{{E_1} + {E_2}} \rangle ^2} + {\langle{{E_1} - {E_2}} \rangle ^2},$$
$$S2 = {\langle{(1 - i){E_1} + (1 + i){E_2}} \rangle ^2} + {\langle{( - 1 + i){E_1} + ( - 1 - i){E_2}} \rangle ^2} - 2{\langle{(1 + i){E_1} + (1 - i){E_2}} \rangle ^2},$$
$$S3 = {\langle{(1 + i){E_1}} \rangle ^2} - {((1 + i){E_2})^2},$$
where ${\langle{{E_1}} \rangle ^2} = {E_1} \cdot {E_1}^ \ast $, ${E_1}^ \ast $ is the conjugate value of ${E_1}$. The azimuth ($\chi $) and ellipsometry ($\varepsilon $) of the polarization can be calculated from $\chi = 0.5\arctan (S2/S1)$ and $\varepsilon = \tan (0.5\arcsin (S3/S0))$. Thus, after retrieving the accurate phases of the two components, we can obtain the polarization distribution with the Stokes polarization calculation method. According to Eqs. (1)–(3), the spatial phase distribution of CS-SLBs can also be obtained by subtracting the polarization azimuth with the phase azimuth of RHCP VB.

3. Experimental results and analysis

The experimental setup for retrieving the spatial phase and polarization distributions of arbitrary CS-SLBs is illustrated in Fig. 3. A laser beam with the working wavelength of 633 nm was emitted from a He-Ne laser (Thorlabs, HNL210LB) and filtered by a linear polarizer (LP1) to ensure linear polarization. By properly adjusting the transmission axis of LP, we can achieve the wanted energy ratio of the spherical wave to CS-SLB. Then the linearly polarized incident light was divided by a polarization beam splitter (PBS1) into two sub-beams: one transmitted through the CS-SLB generating system (CS-SLBGS) to produce target CS-SLBs, and the other was converted to spherical wave via a spherical lens. The focal length of the double convex lens is 250.0 mm, and the focal point of the lens is considered as the point source of the spherical wave. The distance between the focal point and the interfere plane, as well as the radius of the spherical wavefront, is 300.0 mm. The spot radius of the spherical beam at the interfere plane is 6.2 mm, which is approximately equal to the singular beam. Before the interference, the LHCP and RHCP components of CS-SLBs were converted to x- and y-polarizations by a quarter-wave plate (QWP). And the interference field of the LHCP and RHCP components can be obtained after split by PBS2. After passing through the cylindrical lens (C-lens) with 10 cm focal length, the two interference fields experienced astigmatic diffraction. And a CCD camera (Ophir, SP928) was placed at the focal plane of C-lens to record far-field diffraction patterns. These recorded patterns are used as the input of API algorithm for calculation.

 figure: Fig. 3.

Fig. 3. The experimental setup for phase and polarization retrieval of arbitrary CS-SLBs. (a) and (b) are the cross- and parallel-polarized measured images of the Q-plate. (c) and (d) are the theoretical and measured optical axis distributions of the Q-plate with q = 0.5. LP: linear polarizer; PBS: polarization beam splitter; CS-SLBGS: CS-SLBs generating system; HWP: half-wave plate; QWP: quarter-wave plate; BS: beam splitter; M: mirror. The inserted pictures are the corresponded optical intensity distributions.

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Referring to our previous works [40], we use a Q-plate to produce CS-SLBs, which is an optical phase-shifting device and can be regarded as the combination of multiple linear birefringent half-wave plates. According to the characteristics of Q-plate, VBs and CVBs can be directly generated by manipulating the polarization state of the incident beams. The measured cross- and parallel-polarized images are shown in Figs. 3(a) and 3(b), ${P_{in}}$ and ${P_{out}}$ denote the polarization states of the input and output beam, respectively. The theoretical and measured optical axis distributions retrieved by the cross-polarized images are shown in Figs. 3(c) and 3(d) [41]. Here, q is a constant related to the spatial rotation of the optical axis. The Q-plate can perform the transformation of $|L \rangle \to {e^{i({2q} )\theta }}|R \rangle $ and $|R \rangle \to {e^{ - i({2q} )\theta }}|L \rangle $, where the left ($|L \rangle $) and right ($|R \rangle $) circular polarizations are converted to opposite spin states and carry $\pm 2q\hbar $ OAM. By this way, CVBs and CVVBs can also be generated with the Q-plates.

a. The spatial phase and polarization retrieval of VBs

The mechanism of generating VBs by using Q-plate is illustrated in Fig. 4(a). If the incident light is an LHCP Gaussian beam, an RHCP VB with positive TC will be produced. Conversely, if the incident light is an RHCP Gaussian beam, the output is an LHCP VB with a negative sign. When the VBs pass through the C-lens, it will experience strong astigmatism diffraction at the focal position. The intensity distributions of the VBs with TCs ranging from −3 to 3 at the focal position are presented in the bottom row of Fig. 4(b). As shown in the figures, the diffraction patterns of the VBs with different TCs have different dark fringes. The number of fringes is equal to TCs, and the oblique direction determines the sign of TC.

 figure: Fig. 4.

Fig. 4. (a) The schematic diagram of generating VBs with Q-plates. (b) The intensity distributions of VBs (upper) and the diffraction patterns of the VBs pass through C-lens (bottom) with $l = - 3, - 2, - 1,1,2,3$.

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To test the accuracy and efficiency of the retrieval method, we first generated linearly polarized VBs, whose polarization angle is $-{\pi \mathord{\left/ {\vphantom {\pi 4}} \right.} 4}$. Figure 5 shows the experimentally retrieved phase and polarization distributions of the VBs with l = 1∼3. The interferograms of the two separated sub-VBs and spherical waves are shown in the first and third column of Fig. 5(a). Owing to the low proportion of spherical wave, the interferograms remain a “doughnut” shape. The phase distributions of the separated LHCP and RHCP components are shown in the second and fourth column of Fig. 5(a). Both the phase distributions of LHCP and RHCP components have the same spiral phase but different initial phase, which is related to the polarization state. Through a series of calculations as previously introduced, we obtained the phase and polarization distributions of the VBs, which are shown in Fig. 5(b). The phase distributions are spirally distributed, and the spiral periods correspond to the TCs of l = 1∼3. The polarization states are all linearly polarized, which is determined by the initial phase difference between the LHCP and RHCP components. Because the incident VBs are $-{\pi \mathord{\left/ {\vphantom {\pi 4}} \right.} 4}$ linearly polarized, the initial phase of the LHCP component is ${\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}$ more than RHCP. These results agree well with the theoretical calculations by the Jones matrix.

 figure: Fig. 5.

Fig. 5. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of VBs with l = 1∼3 (from the first row to the end row). The blue ellipses represent the measured polarization states.

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b. The spatial phase and polarization retrieval of CVBs

Based on the characteristics of Q-plate, a CVB will be produced when the incident beam is linearly polarized. The schematic diagram of generating CVBs is shown in Fig. 6(a). After filtered by LP, the CVB was decomposed into several side lobes, which can be used to identify CVBs. The number of side lobes depends on the polarization order, and the side lobes will rotate with the optical axis of the LP. When the rotation direction of the side lobes is consistent with the transmission axis of LP, the sign of polarization order is positive. Conversely, the sign of polarization order is negative. In this experiment, we used the Q-plates (q = 0.5, 1 and 1.5) to generate the CVBs with the polarization orders of 1∼3. Their intensity distributions are shown in the first column of Fig. 6(b). From the figures, these CVBs present “doughnut” intensity distributions. Based on the measurement of side lobes, the polarization orders of the produced CVBs are m = 1∼3, which is consistent with the theory (m = 2q).

 figure: Fig. 6.

Fig. 6. (a) The schematic diagram of generating CVBs with Q-plates. (b)The intensity distributions of the CVBs (m = 1∼3) filtered by the LP. The white arrow indicates the transmission axis of the polarizer.

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Then, we used the API algorithm to retrieve the phase distributions of the LHCP and RHCP components of CVBs. The interferograms are shown in the first and third column of Fig. 7(a). The beam-size of LHCP and RHCP components are almost the same because the signs of VBs have no impact on the spot size. As shown in Fig. 7(a), the second and fourth columns are the retrieved helical phase distributions of the LHCP and RHCP components, respectively. The TCs of LHCP and RHCP components are opposite, which is consistent with the Jones matrix theory. According to the Stokes Parameters ${S_0}\sim {S_3}$ obtained from the LHCP and RHCP fields, the intensity and retrieved spatial polarization distributions of CVBs are shown in the second column of Fig. 7(b), where the blue ellipses represent the measured polarization states. Due to the influence of Q-plate quality, the proportion of LHCP and RHCP components in the cross-section has a certain deviation, which leads to some elliptical polarization. We can also see that the polarization orientation periods of these CVBs along the direction of azimuthal are 1∼3, which are consistent with the practical polarization order. The retrieved phases of CVBs present in the first column of Fig. 7(b). The phase distributions of these CVBs are just like a spherical wavefront because the TCs of CVBs are zero.

 figure: Fig. 7.

Fig. 7. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of the CVBs (m = 1∼3). The blue ellipses represent the measured polarization states.

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c. The spatial phase and polarization retrieval of CVVBs

Figure 8(a) describes the principle of generating CVVBs by cascading two Q-plates. The first Q-plate (q = 0.5) is used to convert the circularly polarized Gaussian beam into a circularly polarized VB and then transformed into a linearly polarized VB with a QWP. Then, the linearly polarized VB is converted to a CVVB, which possesses both the helical phase and non-uniform spatial polarization after passing another Q-plate (q = 1). By swapping the position of the two Q-plates, we got two completely different CVVBs with $m = - 1,l = 2$ and $m = 2,l = - 1$. As shown in Fig. 8(b), although the intensities distributions of the two CVVBs are the same, the intensity distributions of CVVBs filtered by the polarizer are different because they have different polarization and phase distributions. As illustrated in [40], the TC’s sign of CVVBs can be identified through observing the typical “s”-shape or anti-“s”-shape formed by two light spots on the same diameter. And we can get the polarization order of CVVBs from the intensity distributions filtered by LP. From the first row of Fig. 8(b), the polarization order of the CVVB is −1, and the sign of TC is positive. The second CVVB’s polarization order is 2, and the sign of TC is negative, which are shown in the second row of Fig. 8(b).

 figure: Fig. 8.

Fig. 8. (a) The schematic diagram of generating CVVBs by cascading Q-plates. (b) The intensity distributions of CVVBs filtered by the polarizer, m= −1, l = 2 (upper); m = 2, l= −1 (lower). The white arrow indicates the transmission axis of the polarizer.

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Since CVB is a special type of CVVB, we can obtain the phase and polarization distributions of CVVBs with the above-mentioned extraction method. The retrieved phase distributions of the LHCP and RHCP components of CVVBs are shown in the second and fourth column in Fig. 9(a), respectively. The corresponding interference patterns are presented in the first and third column. As shown that a CVVB with $m = - 1,l = 2$ can be decomposed into an LHCP VB with $l = 1$ and an RHCP VB with $l = 3$. And the CVVB with $m = 2,l = - 1$ can be decomposed into an LHCP VB with $l = 1$ and an RHCP VB with $l = - 3$. The experimentally retrieved phase and polarization distributions of the CVVBs are illustrated in the first and second column of Fig. 9(b). The spatial phases of the two CVVBs have a spiral phase distribution with the TCs of 2 and −1, which are consistent with the predicted values. The polarization orientation periods of these two CVVBs along the direction of azimuthal are 1 and 2. The sign of the polarization order is determined by the direction of polarization rotation. When the polarization rotates counter-clockwise, the polarization order is positive and vice versa. These experimental results match the anticipations well, and the polarization and phase structures of the CVVBs are retrieved. Furthermore, this method can also analyze the complex vector vortex beam since it can also be decomposed into two orthogonal circularly polarized components [42,43].

 figure: Fig. 9.

Fig. 9. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of CVVBs with m= −1, l = 2 (upper); m = 2, l= −1 (lower). The blue ellipses represent the measured polarization states.

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4. Discussion

The above experimental results demonstrate that the OPS method can effectively retrieve the phase and polarization distributions of CS-SLBs. To measure the phase and polarization simultaneously, we have to obtain the spatial phase distributions and initial phase difference of the two decomposed orthogonal circularly polarized sub-VBs. A designed API algorithm has been used to retrieve the phase of incoming beams. However, owing to the diffraction patterns of the VBs with the same TC but different initial phases are identical for the symmetric intensity distributions, the conventional API algorithm is difficult for the initial phase measuring. To solve this problem, we introduced a reference beam (spherical wave) to interfere with the VB, and the interference field after passing through a C-lens is used as the input of API algorithm. By properly selecting the energy ratio of the VB to spherical wave, we have successfully obtained the spatial phase distribution of VBs, including the initial phase. It is worth mentioning that the energy ratios of the VB to the spherical wave have a great impact on the extracted phases. With the increase in the proportion of spherical wave, the helical phase will lose. In order to ensure that the main information contained in the output phase comes from the target beam, the ratio of the VB and spherical wave is set as 10:1. Meanwhile, we found that although the proportion of spherical wave was restricted to small, the recovered phase still has a certain distortion. Further studies should focus on reducing the phase distortion by optimizing the ratio of the target and reference beam or put forward some other initial phase extraction schemes. In addition, we can try to add an attenuation factor in the algorithm, so that after iteration, the effect of the spherical wave phase on the whole phase can be reduced, and a more accurate phase distribution containing the initial phase can be achieved.

After retrieving the phases of the two orthogonal circularly polarized sub-VBs, we need to confirm the relationship between the phase and polarization state. Here, we build up the relationship by combining the Jones matrix theory. The spatial phase and polarization distributions of CS-SLBs were then retrieved by using the API algorithm and the Stokes polarization calculation method. Compared with conventional detection methods, this method can simultaneously retrieve both of the phase and polarization distributions of CS-SLBs without needing to know its type in advance.

5. Conclusion

In conclusion, we proposed and experimentally investigated an OPS method to reconstruct the phase and polarization distributions of arbitrary CS-SLBs. Decomposition of CS-SLB into two orthogonal circularly polarized sub-VBs and then, retrieval of spatial phase distribution, together with the initial phase difference between them, leads to final retrieval of phase and polarization distributions of the CS-SLB. The phase distributions of the two components were restored with the designed API algorithm, and the initial phase difference was obtained by interfering the target beam with a spherical wave. In the experiment, the phase and polarization distributions of the VBs (with the TCs l = 1∼3), CVBs (with the polarization orders m = 1∼3), and CVVBs (with the TCs and polarization orders of (l, m) = (−1, 2) and (2, −1)) were successfully retrieved. This work provides an effective way for reconstructing the spatial phase and polarization distributions of arbitrarily CS-SLBs. Due to the ability to detect polarization and phase information of complex objects, it may have great potentials in polarization or phase imaging. And this method can also be used to demodulate complex CS-SLB shift-keying signals in SLB modulation communications.

Funding

Program of Fundamental Research of Shenzhen Science and Technology Plan (JCYJ20180507182035270); National Natural Science Foundation of China (61490713, 61575127, 61805149); Natural Science Foundation of Guangdong Province (2016A030310065); Science and Technology Planning Project of Guangdong Province (2016B050501005); Science and Technology Planning Project of Shenzhen Municipality (ZDSYS201707271014468); Department of Education of Guangdong Province (2016KCXTD006).

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2. P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989). [CrossRef]  

3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]  

4. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]  

5. Z. Zhao, J. Wang, S. Li, and A. E. Willner, “Metamaterials-based broadband generation of orbital angular momentum carrying vector beams,” Opt. Lett. 38(6), 932–934 (2013). [CrossRef]  

6. Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017). [CrossRef]  

7. X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014). [CrossRef]  

8. Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017). [CrossRef]  

9. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]  

10. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

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

12. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]  

13. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [CrossRef]  

14. R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]  

15. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]  

16. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]  

17. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef]  

18. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010). [CrossRef]  

19. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]  

20. W. Qiao, T. Lei, Z. Wu, S. Gao, Z. Li, and X. C. Yuan, “Approach to multiplexing fiber communication with cylindrical vector beams,” Opt. Lett. 42(13), 2579–2582 (2017). [CrossRef]  

21. J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. 40(21), 4827–4830 (2015). [CrossRef]  

22. Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free- space optical communication,” Opt. Express 19(2), 482–492 (2011). [CrossRef]  

23. R. Chen, K. Agarwal, C. J. Sheppard, and X. Chen, “Imaging using cylindrical vector beams in a high-numerical-aperture microscopy system,” Opt. Lett. 38(16), 3111–3114 (2013). [CrossRef]  

24. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef]  

25. C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]  

26. F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19(25), 25143–25150 (2011). [CrossRef]  

27. D. J. Lee and A. M. Weiner, “Optical phase imaging using a synthetic aperture phase retrieval technique,” Opt. Express 22(8), 9380–9394 (2014). [CrossRef]  

28. S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015). [CrossRef]  

29. S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017). [CrossRef]  

30. S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express 17(24), 21396–21407 (2009). [CrossRef]  

31. Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017). [CrossRef]  

32. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017). [CrossRef]  

33. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758 (1982). [CrossRef]  

34. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994). [CrossRef]  

35. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003). [CrossRef]  

36. C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17(14), 11905–11915 (2009). [CrossRef]  

37. N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019). [CrossRef]  

38. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011). [CrossRef]  

39. V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” J. Opt. Soc. Am. B 31(6), A40–A45 (2014). [CrossRef]  

40. Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017). [CrossRef]  

41. R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018). [CrossRef]  

42. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019). [CrossRef]  

43. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex spectrum in a vortex-beam array without cuts and gluing of the wavefront,” Opt. Lett. 43(22), 5635–5638 (2018). [CrossRef]  

References

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  • |
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  • |

  1. M. Padgett, “Light’s twist,” Proc. R. Soc. London, Ser. A 470(2172), 20140633 (2014).
    [Crossref]
  2. P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
    [Crossref]
  3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
    [Crossref]
  4. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref]
  5. Z. Zhao, J. Wang, S. Li, and A. E. Willner, “Metamaterials-based broadband generation of orbital angular momentum carrying vector beams,” Opt. Lett. 38(6), 932–934 (2013).
    [Crossref]
  6. Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
    [Crossref]
  7. X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
    [Crossref]
  8. Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
    [Crossref]
  9. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
    [Crossref]
  10. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  11. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  12. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7(2), 77–87 (2000).
    [Crossref]
  13. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref]
  14. R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref]
  15. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
    [Crossref]
  16. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  17. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [Crossref]
  18. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
    [Crossref]
  19. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  20. W. Qiao, T. Lei, Z. Wu, S. Gao, Z. Li, and X. C. Yuan, “Approach to multiplexing fiber communication with cylindrical vector beams,” Opt. Lett. 42(13), 2579–2582 (2017).
    [Crossref]
  21. J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. 40(21), 4827–4830 (2015).
    [Crossref]
  22. Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free- space optical communication,” Opt. Express 19(2), 482–492 (2011).
    [Crossref]
  23. R. Chen, K. Agarwal, C. J. Sheppard, and X. Chen, “Imaging using cylindrical vector beams in a high-numerical-aperture microscopy system,” Opt. Lett. 38(16), 3111–3114 (2013).
    [Crossref]
  24. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
    [Crossref]
  25. C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
    [Crossref]
  26. F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19(25), 25143–25150 (2011).
    [Crossref]
  27. D. J. Lee and A. M. Weiner, “Optical phase imaging using a synthetic aperture phase retrieval technique,” Opt. Express 22(8), 9380–9394 (2014).
    [Crossref]
  28. S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
    [Crossref]
  29. S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
    [Crossref]
  30. S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express 17(24), 21396–21407 (2009).
    [Crossref]
  31. Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
    [Crossref]
  32. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
    [Crossref]
  33. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758 (1982).
    [Crossref]
  34. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994).
    [Crossref]
  35. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
    [Crossref]
  36. C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17(14), 11905–11915 (2009).
    [Crossref]
  37. N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
    [Crossref]
  38. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
    [Crossref]
  39. V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” J. Opt. Soc. Am. B 31(6), A40–A45 (2014).
    [Crossref]
  40. Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
    [Crossref]
  41. R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
    [Crossref]
  42. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
    [Crossref]
  43. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex spectrum in a vortex-beam array without cuts and gluing of the wavefront,” Opt. Lett. 43(22), 5635–5638 (2018).
    [Crossref]

2019 (2)

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
[Crossref]

2018 (2)

2017 (8)

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

W. Qiao, T. Lei, Z. Wu, S. Gao, Z. Li, and X. C. Yuan, “Approach to multiplexing fiber communication with cylindrical vector beams,” Opt. Lett. 42(13), 2579–2582 (2017).
[Crossref]

C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref]

2015 (2)

2014 (4)

2013 (2)

2012 (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

2011 (5)

Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free- space optical communication,” Opt. Express 19(2), 482–492 (2011).
[Crossref]

F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19(25), 25143–25150 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref]

2010 (2)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
[Crossref]

2009 (3)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

2004 (1)

2003 (2)

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
[Crossref]

2002 (2)

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

2000 (1)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1994 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1989 (1)

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

1982 (1)

Agarwal, K.

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Akimova, Y.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Barnett, S. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Boyd, R. W.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Bretsko, M.

Brown, T. G.

Cai, Y.

C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
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R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
[Crossref]

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

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S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
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Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
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Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
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Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
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Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
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S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
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R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Figures (9)

Fig. 1.
Fig. 1. The schematic diagram of the API algorithm.
Fig. 2.
Fig. 2. The interference intensity distributions (first and second rows) of VBs and the spherical wave with different ratios, and the restored phase distributions (third and fourth rows), ${\varphi _0}$ is the initial phase.
Fig. 3.
Fig. 3. The experimental setup for phase and polarization retrieval of arbitrary CS-SLBs. (a) and (b) are the cross- and parallel-polarized measured images of the Q-plate. (c) and (d) are the theoretical and measured optical axis distributions of the Q-plate with q = 0.5. LP: linear polarizer; PBS: polarization beam splitter; CS-SLBGS: CS-SLBs generating system; HWP: half-wave plate; QWP: quarter-wave plate; BS: beam splitter; M: mirror. The inserted pictures are the corresponded optical intensity distributions.
Fig. 4.
Fig. 4. (a) The schematic diagram of generating VBs with Q-plates. (b) The intensity distributions of VBs (upper) and the diffraction patterns of the VBs pass through C-lens (bottom) with $l = - 3, - 2, - 1,1,2,3$.
Fig. 5.
Fig. 5. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of VBs with l = 1∼3 (from the first row to the end row). The blue ellipses represent the measured polarization states.
Fig. 6.
Fig. 6. (a) The schematic diagram of generating CVBs with Q-plates. (b)The intensity distributions of the CVBs (m = 1∼3) filtered by the LP. The white arrow indicates the transmission axis of the polarizer.
Fig. 7.
Fig. 7. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of the CVBs (m = 1∼3). The blue ellipses represent the measured polarization states.
Fig. 8.
Fig. 8. (a) The schematic diagram of generating CVVBs by cascading Q-plates. (b) The intensity distributions of CVVBs filtered by the polarizer, m= −1, l = 2 (upper); m = 2, l= −1 (lower). The white arrow indicates the transmission axis of the polarizer.
Fig. 9.
Fig. 9. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of CVVBs with m= −1, l = 2 (upper); m = 2, l= −1 (lower). The blue ellipses represent the measured polarization states.

Equations (9)

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E v o r t e x = E 0 exp [ i ( l θ + φ 0 ) ] [ 1 0 ] = 1 2 E 0 exp [ i ( l θ + φ 0 ) ] [ 1 i ] + 1 2 E 0 exp [ i ( l θ + φ 0 ) ] [ 1 i ] ,
E v e c t o r = E 0 [ cos(m θ + φ 0 ) sin(m θ + φ 0 ) ] = 1 2 E 0 exp ( i φ 0 ) exp ( i m θ ) [ 1 i ] + 1 2 E 0 exp ( i φ 0 ) exp ( i m θ ) [ 1 i ] ,
E v e c t o r _ v o r t e x = E 0 exp ( i l θ ) [ cos(m θ + φ 0 ) sin(m θ + φ 0 ) ] = 1 2 E 0 exp ( i φ 0 ) exp [ i ( l m ) θ ] [ 1 i ] + 1 2 E 0 exp ( i φ 0 ) exp [ i ( l + m ) θ ] [ 1 i ] ,
Φ n = exp ( i k ( x cos ( θ ) y sin ( θ ) ) 2 / 2 / f ) .
ε ( p ) = 1 N j ( I j I j p ) 2 ,
S 0 = E 1 2 + E 2 2 ,
S 1 = E 1 + E 2 2 + E 1 E 2 2 ,
S 2 = ( 1 i ) E 1 + ( 1 + i ) E 2 2 + ( 1 + i ) E 1 + ( 1 i ) E 2 2 2 ( 1 + i ) E 1 + ( 1 i ) E 2 2 ,
S 3 = ( 1 + i ) E 1 2 ( ( 1 + i ) E 2 ) 2 ,

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