Quantitative detection of high-order Poincar&#x00E9; sphere beams and their polarization evolution

Junna Yao; Xinhua Jiang; Jialang Zhang; Anting Wang; Qiwen zhan

doi:10.1364/OE.479386

1. Introduction

Vector beams (VBs) are drawing increasing attention due to their unique polarization and amplitude distributions. In 1961, E. Snitzer theoretically proposed the basic concept of VBs [1]. Light fields with spatially variant polarization distribution on the same wavefront are called VBs. Compared to conventional scalar beams, VBs have shown remarkable application value and been exploited in fields like optical trapping [2,3], material processing [4], surface plasmon excitation [5,6] and optical communication [7].

In 1892, Poincaré proposed a sphere to depict the state of polarization (SOP) of light beams, which is the so-called standard Poincaré sphere (PS) [8]. Every point on the spherical surface can uniquely represent a specific SOP. The PS not only provides considerable insight into the polarization phenomena but also tremendously simplifies the handling of polarization problems. Nevertheless, the standard PS is not sufficient to represent the high-order solutions of the Maxwell wave equation including the VBs. To address the need for the diverse representation of the PS, in 2010, G. Milione demonstrated the high-order PS for the VBs [9]. In 2022, J. Zhong proposed a PS analogue for optical vortex knots [10]. Methods have been proposed to generate the high-order PS beams utilizing spatial light modulator [11,12], liquid crystal based spatially varied retarder [13,14] and polarization selection in laser cavity [15,16]. The SOP of the VBs is usually verified by a single linear polarizer based detection or point-by-point detection. However, the former can only qualitatively detect the SOP of the VBs, and the latter are complicated because point-by-point computation of multiple images is required. Few researchers have put forward the quantitative detection of the high-order Stokes parameters (SPs) for high-order PS beams even if it is necessary for identifying a particular VB. Furthermore, the polarization evolution (PE) on the traditional Poincaré sphere based on the linear birefrigent medium has been widely researched and used in the past, which makes understanding and designing the various polarization devices and experiments extremely easy. There should be a similar property for the HOPs to take full advantage of HOPs representation. But so far no one has studied this property. The PE of the PS beams on the same order and between different orders lacks a systematic description.

In this work, we propose a method to quantitatively detect the SPs of high-order PSs via the S-plate based on the Pancharatnam-Berry geometrical phase [17,18]. We start from the high-order SPs and realize the overall polarization detection by directly measuring the respective intensity of different nonuniform polarization bases. The SOP of the high-order PS beams can be determined by measuring the values of only one set of high-order SP. S-plates with spatially variant optical axes are represented by diagonal points in the equatorial plane of the sphere. The PE of VBs after passing through the S-plates can be obtained ingeniously by the trajectory rotating on the high-order PS around the diameter. The PE from the conventional PS to the high-order PS is also presented, thus arbitrary high-order PS beams can also be generated with ease. The S-plate is micromachined by femtosecond laser writing [19,20]. In contrast to the traditional q-plate [13] and recently proposed J-plate [21], the S-plate possesses the merits of a quasi-continuously varied optical axis, higher damage threshold and flexible processing.

2. Representation of high-order PS

The high-order PS is a mapping between high-order VBs and the points on the spherical surface. Here we introduce the high-order PS in a totally new manner for the purpose of reader-friendly and contextual correspondence. Two poles on the high-order PS denote the orthogonal circular polarization basis with opposite topological charges as shown

(1)$$\left\{ \begin{array}{*{20}{l}} {|{{R_m}} \rangle = {\rm exp}) ({ - im\phi } ){{({\vec{x} + i\vec{y}} )} / {\sqrt 2 }}}\\ {|{{L_m}} \rangle = {\rm exp} ({im\phi } ){{({\vec{x} - i\vec{y}} )} / {\sqrt 2 }}} \end{array}\right.,$$

where φ=arctan(y/x), m is the topological charge, i.e. the order of the PS. The SOP of any points on the high-order PS can be represented by the linear superposition of the SOPs of two poles

(2)$$\left| {\psi _m} \right\rangle = E_R^m \left| {R_m} \right\rangle + E_L^m \left| {L_m} \right\rangle ,$$

(3)$$\left[\begin{matrix} {E_R^m } \\ {E_L^m } \end{matrix}\right] = \left[ \begin{matrix} \sin \left( {\varepsilon + {\pi / 4}} \right) \\ \exp \left( {2i\theta } \right)\cos \left( {\varepsilon + {\pi / 4}} \right) \end{matrix}\right],$$

where ε controls the local ellipticity and θ represents the initial orientation of the principle axes of the ellipse. The SOPs on the sphere can also be expressed by the localized linear polarization basis with initial polarization along the x and y directions, respectively

(4)$$\left[ {\begin{array}{*{20}{c}} {E_0^m}\\ {E_{90}^m} \end{array}} \right] = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&1\\ i&{ - i} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {E_R^m}\\ {E_L^m} \end{array}} \right].$$

Actually, any two centrosymmetric points can be used as the orthogonal polarization basis. To be consistent with the standard PS, we also choose the localized linear polarization basis whose initial polarization is oriented at 45°and -45°with respect to the x axis. The relationship of the complex amplitudes with the former is

(5)$$\left[ {\begin{array}{*{20}{c}} {E_{45}^m}\\ {E_{ - 45}^m} \end{array}} \right] = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {E_0^m}\\ {E_{90}^m} \end{array}} \right].$$

Then we can define the high-order SPs

(6)$$\left\{ {\begin{array}{*{20}{l}} {{S_0} = {{|{E_R^m} |}^2} + {{|{E_L^m} |}^2}}\\ {{S_1} = {{|{E_0^m} |}^2} - {{|{E_{90}^m} |}^2}}\\ {{S_2} = {{|{E_{45}^m} |}^2} - {{|{E_{ - 45}^m} |}^2}}\\ {{S_\textrm{3}} = {{|{E_R^m} |}^2} - {{|{E_L^m} |}^2}} \end{array}} \right..$$

Figure 1 shows the representation of the high-order PS when m=±1. The two poles are the circularly polarized optical vortex beams as aforementioned, which carry an angular momentum of -(m+1)ℏ for the north pole and (m+1)ℏ for the south pole. The points on the equator are linearly polarized VBs containing azimuthally and radially polarized beams. Other points on the sphere are elliptically polarized VBs. All the SOPs possess a polarization topological charge of m.

Fig. 1. Representation of the high-order PS (a) m=1 (b) m=-1.

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3. PE between the conventional PS and high-order PS

The S-plate is a superstructured vortex retarder with spatially variant optical axis that can give rise to space-dependent phase modulation to the incident light beams, which is also the manifestation of the Pancharatnam-Berry geometrical phase. The S-plate used in this paper is micromachined by femtosecond laser writing under moderate laser intensity in silica glass. For the m/2-order S-plate (phase retardance is π) with optical axis distribution of

(7)$$\alpha = {{m\phi } / 2},$$

the Jones matrix can be written as

(8)$$M = \left( {\begin{array}{*{20}{c}} {\cos m\phi }&{\sin m\phi }\\ {\sin m\phi }&{ - \cos m\phi } \end{array}} \right).$$

For an arbitrary incident beam on the conventional PS

(9)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over P} _0} = \sin ({\varepsilon^{\prime} + {\pi / 4}} )\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} + \exp (2i\theta ^{\prime})\cos ({\varepsilon^{\prime} + {\pi / 4}} )\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over L} ,$$

where ε′ and θ′ are the angular coordinates on the conventional PS. The output beams from the S-plate can be calculated by

(10)$$\begin{aligned} {M{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over P} }_0}}& = {\exp \left( { - im\phi } \right)\cos \left( {\varepsilon ' + {\pi / 4}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over R} + \exp \left( {im\phi - \textrm{2}i\theta '} \right)\sin \left( {\varepsilon ' + {\pi / 4}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over L} }\\ {}& = {\sin \left( {{\pi / 4} - \varepsilon '} \right)\left| {{R_m}} \right\rangle + \exp \left( {2i\left( {\pi - \theta '} \right)} \right)\cos \left( {{\pi / 4} - \varepsilon '} \right)\left| {{L_m}} \right\rangle .} \end{aligned}$$

Compared with Eq. (3), a one-to-one correspondence is established between the conventional PS and the high-order PS through the S-plate, and their angular coordinates have the following relation

(11)$$\theta = \pi - \theta ^{\prime},$$

(12)$$\varepsilon ={-} \varepsilon ^{\prime}.$$

Figure 2 shows the mapping relationship between two PSs at some special points. Points A-H in Fig. 2(a) are the input SOPs on the conventional PS, and points A1-H1 in Fig. 2(b) are the corresponding output SOPs on the m-order PS. The input beams and output beams are on the conventional PS and the m-order PS respectively, and are symmetrical along the S₁ axis.

Fig. 2. The PE between (a) the conventional PS and (b) high-order PS.

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The experiments are performed to verify the above rules as shown in part 1 of Fig. 3. LP1 and QP1 are combined to covert the laser beams output from the laser diode (532 nm) into arbitrary homogeneous polarized beams on the conventional PS. Arbitrary high-order PS beams can be generated after passing through the S-plate by changing the SOP of the input beams. The experimental results of the first-order PS beams when m=1 are presented in Fig. 4. A1 to D1 are the SOPs on the equator. The two-lobe patterns of the beams after passing through the linear polarizer prove that the absolute value of the order of the PS beams is equal to 1. What’s more, the two-lobe pattern rotates counterclockwise when rotating the linear polarizer counterclockwise. The same direction of the rotations indicates the order of the PS beams is positive. So the order of the generated beams is determined to be 1. Points A1 and B1 are radially and azimuthally polarized beams. C1 and D1 are both slanted cylindrical polarized beams. E1 and F1 are circularly polarized optical vortices. G1 and H1 are localized elliptically polarized VBs. Each row in Fig. 4 shows the beam emerging from the S-plate without any analyzer and after passing through different polarizers (linear polarizer at different orientations, right circular polarizer and left circular polarizer). The intensity patterns recorded by the CCD are in line with the expectations. Thus we can conclude that the m/2-order S-plate can achieve the PE between the conventional PS and the m-order PS.

Fig. 3. The experimental setup for generating high-order PS beams (Part 1) and measuring high-order SPs (Part 2). LD, laser diode; LP, linear polarizer; QP, quarter waveplate; SP, S-plate; BS, polarization independent beam splitter; CCD, charge-coupled device; D, detector; PM, power meter.

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Fig. 4. Generation of the first-order PS beams when m=1. The polarization analyzers are indicated on the top of the figure.

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4. PE on the high-order PS

The SOPs on the high-order PS can also be interconverted via the S-plate. For the first time, we denote the S-plate as the diameter FS in the equatorial plane of the high-order PS as shown in Fig. 5. Points F and S represent the orientation of the fast axis and slow axis of the S-plate, respectively. The orientation of the fast axis of the S-plate in the xy plane satisfies the following equation

(13)$$\alpha = m\phi + \theta ,$$

where θ is the initial orientation of the fast axis of the S-plate. The Jones matrix of the S-plate with a homogeneous birefringent phase retardation of σ is described as

(14)$$\begin{array}{*{20}{l}} T&{ = {R^{ - 1}}\left( {\begin{array}{*{20}{c}} 1&0\\ 0&{{e^{ - i\sigma }}} \end{array}} \right)R}\\ {}&{ = \cos \frac{\sigma }{2}\left( {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right) + i\sin \frac{\sigma }{2}\left( {\begin{array}{*{20}{c}} {\cos 2\alpha }&{\sin 2\alpha }\\ {\sin 2\alpha }&{ - \cos 2\alpha } \end{array}} \right),} \end{array}$$

where R is the rotation matrix. For the convenience of discussion, we take $|{{R_m}} \rangle$ as the input beam, the output beams after passing through the S-plate can be given by

(15)$$\begin{array}{*{20}{l}} {{E_{out}}}& = &{T|{{R_m}} \rangle }\\ {}& = &{\sin \left( {\frac{\pi }{2} - \frac{\sigma }{2}} \right)|{{R_m}} \rangle + \exp \left( {2i\theta + i\frac{\pi }{2}} \right)\sin \left( {\frac{\pi }{2} - \frac{\sigma }{2}} \right)|{{L_m}} \rangle .} \end{array}$$

Fig. 5. S-plate representation on the high-order PS and the PE.

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Comparing Eqs. (15) and (3), it is easy to obtain the relationship

(16)$$2\varepsilon = {\pi / 2} - \sigma .$$

The SOPs of the input beams and output beams are both denoted in Fig. 5. Thus far, we note that the SOPs of the output beams can be obtained by rotating the input state clockwise (while looking from F to S) on the sphere around the diameter FS by an angle σ. Actually, the foregoing rule is generic for any other input SOPs on the PS through the similar derivation. This is an extension of the evolutionary nature of the traditional PS [8]. Figure 6 enumerates some special cases. The birefringent phase retardation of the S-plate is π/2 in Figs. 6(a) and 6(b) and π in Figs. 6(c) and 6(d). All the output beams are obtained by rotating the input state clockwise around the diameter FS by an angle (numerically equal to the phase retardance of the S-plate). The experimental verification for the PE of Figs. 6(c) and 6(d) are depicted in Fig. 7 (m=1). The first two rows of Fig. 7 illustrate the experimental results of Fig. 6(c), the VB with initial polarization oriented at 45°with respect to the x axis is passed through the S-plate. The output beam corresponding to this input will be another cylindrical VB with initial polarization oriented at -45°with respect to the x axis and can be obtained by rotating as discussed earlier. The input beam and output beam in Fig. 6(d) are circularly polarized optical vortices with opposite topological charges as shown in the last two rows of Fig. 7, which fits well the trajectory indicated in Fig. 6(d). Therefore, the m-order S-plate can implement the PE between different points on the m-order PS. The beauty of this property lies in the fact that one can visualize a geometric picture of the evolution of the HOPs beams through a vortex plate without complex matrix calculation.

Fig. 6. The PE of VBs on the high-order PS after passing through the S-plate (the initial orientation of the fast axis of the S-plate is represented by α₀ here to distinguish it from the coordinates of the beams): (a) α₀=π/8,σ=π/2;(b) α₀=0,σ=π/2;(c) α₀=0,σ=π;(d) α₀=0,σ=π.

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Fig. 7. The experimental results for cases of Fig. 6(c) (first two rows) and Fig. 6(d) (last two rows) when m=1.

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5. Measurement of the SPs

Compared to the uniform polarization bases of the traditional PS, the polarization bases of the high-order PS is nonuniform. So a high-order SP can uniquely represent a SOP of vector beams on the high-order PS. To obtain the values of the SPs of VBs, three sets of polarization bases as described in section 2 are separated and detected. The SPs in Eq. (6) can be rewritten in the following form

(17)$$\left\{ {\begin{array}{*{20}{l}} {{S_0} = 1}\\ {{S_1} = \frac{{{I_{A1}} - {I_{B1}}}}{{{I_{A1}} + {I_{B1}}}}}\\ {{S_2} = \frac{{{I_{C\textrm{1}}} - {I_{D\textrm{1}}}}}{{{I_{C\textrm{1}}} + {I_{D\textrm{1}}}}}}\\ {{S_\textrm{3}} = \frac{{{I_{E\textrm{1}}} - {I_{F\textrm{1}}}}}{{{I_{E\textrm{1}}} + {I_{F\textrm{1}}}}}} \end{array}} \right.,$$

where S₀ is normalized to 1, I_A₁ and I_B₁ represent the intensities of the A1 and B1 SOP components of the VBs respectively, as shown in Fig. 2(b). I_C₁ and I_D₁, I_E₁ and I_F₁ have the analogous meaning. Part 2 in Fig. 3 is designed to measure the SPs, where the portion of the horizontal light path is used to measure I_A₁, I_B₁, I_C₁ and I_D₁, and the vertical light path is used to measure I_E₁ and I_F₁. The m/2-order S-plate can convert the spatially variant polarization components into measurably and uniformly polarized light as the m/2-order S-plate can implement the PE between the m-order PS and conventional PS. To test our conjecture, we take the first-order PS as an example, and the 1/2-order SP2 is used to measure the SPs. The SPs are denoted by the following equations

(18)$$\left\{ {\begin{array}{*{20}{l}} {{S_1} = \frac{{I({{0^ \circ },{0^ \circ }} )- I({{0^ \circ },{{90}^ \circ }} )}}{{I({{0^ \circ },{0^ \circ }} )+ I({{0^ \circ },{{90}^ \circ }} )}}}\\ {{S_2} = \frac{{I({{0^ \circ }, - {{45}^ \circ }} )- I({{0^ \circ },{{45}^ \circ }} )}}{{I({{0^ \circ }, - {{45}^ \circ }} )+ I({{0^ \circ },{{45}^ \circ }} )}}}\\ {{S_3} = \frac{{I^{\prime}({{0^ \circ },{{45}^ \circ }} )- I^{\prime}({{0^ \circ }, - {{45}^ \circ }} )}}{{I^{\prime}({{0^ \circ },{{45}^ \circ }} )+ I^{\prime}({{0^ \circ }, - {{45}^ \circ }} )}}} \end{array}} \right.,$$

where I(i, j) stands for the intensity detected by the detector 2 in Fig. 3, (i) and (j) are the initial optical axis directions of SP2 and the transmission axis of LP2. I′(k, q) represent the intensity recorded by the detector 1, and k and q are the optical axis direction of QP2 and the transmission axis of LP3 respectively. Six cardinal points are generated and detected. The measured values are shown in Table 1. The comparison of the theoretical points and the measured points is depicted in Fig. 8. Red and blue points represent the theoretical and measured results, respectively. The results show that the two are in good agreement. The detection accuracy is greater than 91%. The definition of the accuracy is given by

(19)$$\delta = \textrm{1 - }\sqrt {{{({{S_1}^\prime - {S_1}} )}^2} + {{({{S_2}^\prime - {S_2}} )}^2} + {{({{S_3}^\prime - {S_3}} )}^2}} ,$$

where S₁ ∼ S₃ are the ideal value, S₁′ ∼ S₃′ are the measured values. The actual detection accuracy should be higher because the parameters of the generated vector beams have a certain deviation from the theoretical value (commercial quarter-waveplate and polarizer both have small errors). The impaction of the setup to the accuracy mainly depends on the misalignment of the center of the detected beam and the center of the S-plate, which can be optimized by observing the intensity profile of the beam after passing through the S-plate. Patterns with no hollow center should be observed because the high-order PS beam will be converted to a scalar beam due to the inverse conversion of the S-plate, which can be used as a principle to optimize the position of the S-plate. What’s more, a sophisticated displacement platform can be employed to improve the accuracy of the setup. The conventional SP method requires the pixel-to-pixel computation of multiple images to calculate the SPs of every point on the cross section of the beam. But it is difficult to make all the intensity profiles located at the same pixel, which is complicated and also leads to inaccuracy of calculation. Compared to conventional method, our method greatly increases the convenience of detection.

Fig. 8. The detection of the SPs of first order PS. Red and blue points represent the theoretical and measured results, respectively.

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6. Conclusion

In conclusion, we propose the quantitative detection of the high-order PS beams based on the S-plate. Overall polarization detection is realized by directly measuring the values of the high-order PSs. The polarization evolution of the PS beams on the high-order PS and between the conventional and the high-order PS are discussed in detail. The results prove that the m/2-order S-plate can realize the PE between the conventional PS and m-order PS, which are applied to the generation and detection of the m-order PS beams. The m-order S-plate can implement the PE on the same m-order PS. The PE on the m-order PS is described by the trajectory rotating around the diameter FS. The proposed methods are valid for high-order PS of any order as long as one chooses the S-plate with appropriate order. The results provide new insights for the generation, evolution and detection of arbitrary beams on the high-order PSs that may find potential applications in laser processing, multimode multiplexing, information coding and so on.

Table 1. The measurement results of the SPs.

View Table

Funding

National Key Research and Development Program of China (2020YFB2205802).

Acknowledgments

The authors gratefully acknowledge funding by the China National Key R&D Program (No. 2020YFB2205802).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Measured points	Theoretical values	Measured values
A1	(1,0,0)	(0.924,0.037,0.017)
B1	(-1,0,0)	(-0.930,-0.022,0.012)
C1	(0,1,0)	(-0.003,0.921,0.006)
D1	(0,-1,0)	(-0.014,-0.921,0.032)
E1	(0,0,1)	(-0.001,-0.009,0.974)
F1	(0,0,-1)	(0.067,-0.052,-0.977)

Quantitative detection of high-order Poincaré sphere beams and their polarization evolution

Abstract

1. Introduction

2. Representation of high-order PS

3. PE between the conventional PS and high-order PS

4. PE on the high-order PS

5. Measurement of the SPs

6. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (19)

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