Generation of perfect vectorial vortex beams by employing coherent beam combining

Pei Ju; Pei Ju; Wenhui Fan; Wenhui Fan; Wenhui Fan; Wenhui Fan; Wei Gao; Wei Gao; Zhe Li; Zhe Li; Qi Gao; Qi Gao; Gang Li; Gang Li; Xiaoqiang Jiang; Xiaoqiang Jiang; Tongyi Zhang; Tongyi Zhang; Tongyi Zhang

doi:10.1364/OE.485396

1. Introduction

With the continuous improvement of laser technology and the continuous expansion of its application scope, the complex modulation of phase, polarization, and even multi-dimensional beam parameters emerge as the time require. The beams with specially tailored intensity, phase, or polarization state in the spatial domain, such as vortex beams [1], vector beams [2], and other beams have been proposed successively. As a special beam, vectorial vortex beams (VVBs) possess spiral phase distribution and spatial inhomogeneous polarization state [3–6]. Owing to the unique optical properties, VVBs have earned a wide variety of applications in high resolution microscopy [7,8], optical trapping [9,10], classical and quantum communications [11,12], and nonlinear optics [13], and so on.

In general, VVBs can be considered the superposition of right-handed (RH) and left-handed (LH) circularly polarized vortices. However, the beam radius of the vortex beam is closely related to its value of phase topological charge [1,14]. Therefore, the RH and LH circularly polarized vortices with different phase topological charges are confronted with two unavoidable issues. On the one hand, the intensity distribution cannot completely coincide with each other. On the other hand, the propagation characteristics of synthesized VVBs are unstable [15], and the intensity distribution will collapse during the propagation [16]. To address on this issue, the concept of perfect vortex beams has been proposed by Ostrovsky et al. [17], whose radii are independent of phase topological charge. This concept is also introduced into the research of VVBs, and the perfect VVBs have been come into being [18–23]. Recently, various methods have been proposed to generate perfect VVBs, including Sagnac interferometer [18,19], cascaded liquid-crystal display devices [22], specially designed metasurface [23], and so on. However, these reported methods mostly involve devices such as spatial light modulator or metasurface, resulting in making the system expensive and inefficient. More importantly, the limitation of related devices makes it impossible to realize high power perfect VVBs.

Compared with other optical devices, the photon sieves composed of a large number of independent pinholes have better optical and processing characteristics [24]. By designing a special arrangement of pinholes, various types of structured beams can be obtained, such as vortex beams [25–27], Airy beams [28], and OAM comb [29], etc. However, low diffraction efficiency may lead to challenges in achieving high power structured beams. Similarly, coherent beam combining using signals from multiple parallel lasers can break through the power limitation of a single laser, greatly scale up the output power and maintain good beam quality [30,31]. Since the intensity, polarization, and phase of each beamlet in the laser array can be arranged independently, various structured beams based on coherent beam combining, such as vortex beams [32–35], vector beams [36,37], airy beams [38], and Bessel-Gaussian beam [39], have been proposed theoretically and verified experimentally. In addition, high-speed photoelectric phase modulation devices can be used in the laser array, so rapidly switchable structured beams can be realized based on coherent beam combining [40].

In this work, we propose an approach to generate perfect VVBs based on coherent beam combining by using a specially designed radial phase-locked Gaussian laser array, which is composed of two discrete vortex arrays with RH and LH circularly polarized states and in turn adjacent to each other. By analyzing different parameter parameters of generated beams, including intensity, normalization Stokes parameters, polarization orientation, Pancharatnam phases, diameter, and thickness, we have verified whether the generated beams are perfect VVBs or not. Further, the influence of different constant phases φ₀ between the RH and LH circularly polarized laser arrays on the perfect VVBs’ characteristics has also been numerically investigated. Utilizing the Fresnel diffraction theory, the propagation characteristics of the generated perfect VVBs in the free space are numerically simulated. Finally, the generation and propagation properties of perfect VVBs with elliptically polarized states are numerically investigated. This proposed method enables further applications of a high power perfect VVBs, such as laser manufacture and nonlinear frequency conversion.

2. Generation method of perfect VVBs

The generalized VVBs can be decomposed into two orthogonal circularly polarized vortices with coefficients $\psi _R^m$ and $\psi _L^n$, and its expression can be expressed as [41]

(1)$$|{{\psi_{m,n}}} \rangle = \psi _R^m|{{R_m}} \rangle + \psi _L^n|{{L_n}} \rangle$$

Here, |R_m〉 and |L_n〉 represent the RH and LH circularly polarized vortices carrying topological charges of m and n, respectively, such as [42]

(2)$$\begin{array}{l} |{{R_m}} \rangle = \exp ({im\varphi - i{\varphi_0}/2} )({{{\boldsymbol e}_x} + i{{\boldsymbol e}_y}} )/\sqrt 2 \\ |{{L_n}} \rangle = \exp ({in\varphi + i{\varphi_0}/2} )({{{\boldsymbol e}_x} - i{{\boldsymbol e}_y}} )/\sqrt 2 \end{array}$$

where e_x and e_y are the unit vectors along x and y directions of Cartesian coordinate system, respectively, φ is the azimuthal angle of polar coordinate system, and φ₀ is the constant phase. From Eq. (2), it can be found that there exists the constant phase term exp(-iφ₀) between RH and LH circularly polarized vortices.

In order to characterize the inhomogeneous polarization states of generalized VVBs, we adopt the arbitrary-order Poincaré spheres (PS) [42], which represents the Stokes parameters in sphere’s Cartesian coordinates. According to Eqs. (1) and (2), the Stokes parameters can be written as [42]

(3)$$\begin{array}{l} {S_0} = {|{\psi_R^m} |^2} + {|{\psi_L^n} |^2},{S_1} = 2|{\psi_R^m} ||{\psi_L^n} |\cos \phi \\ {S_2} = 2|{\psi_R^m} ||{\psi_L^n} |\sin \phi ,{S_3} = {|{\psi_R^m} |^2} - {|{\psi_L^n} |^2} \end{array}$$

where $\phi = \arg ({\psi_R^m} )- \arg ({\psi_L^n} )$, $|\psi _R^m{|^2}$ and $|\psi _L^n{|^2}$ are the intensities of |R_m〉 and |L_n〉, respectively. Furthermore, by neglecting the constant phase exp(iφ₀/2), and extracting a common phase factor exp [i(m + n)φ/2], Eq. (1) can be transformed as

(4)$$|{{\psi_{m,n}}} \rangle = \exp \left( {i\frac{{m + n}}{2}\varphi } \right)\left[ {\psi_R^m|R\rangle \exp \left[ {i\frac{{m - n}}{2}\varphi } \right] + \psi_L^n|L\rangle \exp \left[ { - i\frac{{m - n}}{2}\varphi } \right]} \right]$$

where |R〉 = (e_x + ie_y)/$\sqrt {2} $, |L〉 = (e_x - ie_y)/$\sqrt {2} $, representing RH and LH circularly polarized, respectively. According to Eq. (4), The state of a point on the equator of arbitrary-order PS can be characterized by the polarization order P = (m - n)/2 and the topological Pancharatnam charge of l₀= (m + n)/2, respectively [16]. Once the polarization order and the topological Pancharatnam phase is changed, the intensity profile and states of polarization of the VVB will also be changed.

Utilizing S₁, S₂, S₃ as the sphere’s Cartesian coordinates, a hybrid-order PS with S₀ the unit radius can be constructed, as shown in Fig. 1, where 2α ∈ (0, 2π), 2β ∈ (-π/2, π/2). It is clear that all the possible sates of VVBs can be mapped on the arbitrary-order PS. The north pole A (2α, π/2) and the south pole B (2α, -π/2) of arbitrary-order PS correspond to the eigenstates |R_m〉 and |L_n〉, respectively. The equatorial points of the arbitrary-order PS represent a superposition of the two orthogonal states with equal intensity. The points C (0, 0), D (π, 0), E (π/2, 0) and F (3π/2, 0) on the equator, denoted by |H_m,_n〉, |V_m,_n〉,|D_m,_n〉 and|A_m,_n〉, are the horizontal, vertical, diagonal and antidiagonal polarization basis, and their expressions can be expressed as [36]: |H_m,_n〉 = (|R_m〉 + |L_n〉)/2, |V_m,_n〉 = -i(|R_m〉 - |L_n〉)/2, |D_m,_n〉 = (|H_m,_n〉 + |V_m,_n〉)/2 and |A_m,_n〉 = (|H_m,_n〉 - |V_m,_n〉)/2. Figure 1(b) and (c) schematically display the intensity and polarization distributions of states A–F on an arbitrary-order PS under the conditions of Laguerre–Gaussian (LG) modes – m = n = 1 and m = 5, n = 1, respectively. For the case of – m = n = 1, the north and south poles represent LH and RH circularly polarized vortices with topological charges of ∓1, and the equatorial points represent the generalized cylindrical vector sates, especially polarization basis of |H_-1,₁〉 and |V_-1,₁〉 correspond to radially and azimuthally polarized cylindrical vector sates. It is worth noting that when – m = n = constant, the arbitrary-order PS degenerates into a higher-order PS [41,43]. While for the case of |m| ≠ |n|, for example, m = 5, n = 1, the north and south poles represent RH and LH circularly polarized vortices with topological charges of 5 and 1, respectively. According to Eq. (4), the polarization order of equatorial points is 2. However, due to the difference in intensity distribution between |R₅〉 and |L₁〉, the states of polarization of equatorial points present anisotropic linear polarization distribution. The arbitrary-order PS can be transformed into a hybrid order PS [42].

Fig. 1. (a) Schematic illustration of the arbitrary-order PS. The intensity and polarization distributions at points A – F for the cases of – m = n = 1 (b) and m = 5, n = 1 (c), respectively.

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The radial phase-locked discrete vortex beam is composed of M linearly polarized Gaussian beamlets, which have same waist width. The electric field distribution of radial phase-locked Gaussian laser array with discrete vortex can be written as

(5)$$E(r,\theta ) = \sum\limits_{m = 1}^M {{E_m}} (r,\theta )$$

with

(6)$${E_m}(r,\theta ) = \exp \left[ { - \frac{{({{r^2} + R_1^2} )}}{{\omega_0^2}}} \right]\exp \left[ {\frac{{2[{{R_1}r\cos ({\theta - {\phi_m}} )} ]}}{{\omega_0^2}}} \right]\exp ({i({{l_0}{\phi_m} + \sigma {\varphi_0}/2} )} )$$

where ω₀ is the waist width of the laser beam, R₁ is the radius of the array, ϕ_m= 2π(m-1)/M and m is the number of beamlets (m = 1,2…M), l₀ is the topological charge, σ = 1 or -1, which correspond to RH and LH circularly polarized sates, The phase difference of adjacent beamlet Δϕ =Δ (2πm/M) = 2πΔm/M. When the number of beamlets increases to enough large, the phase difference Δϕ tends to be infinitesimal, and the summation formula of Eq. (3) on m will approximate the integral formula on ϕ

(7)$$E(r,\theta ) = \frac{M}{{2\pi }}\exp \left[ { - \frac{{({{r^2} + R_1^2} )}}{{\omega_0^2}}} \right]\exp ({i\sigma {\varphi_0}/2} )\int_0^{2\pi } {\exp } \left[ {\frac{{2[{{R_1}r\cos (\theta - \phi )} ]}}{{\omega_0^2}}} \right]\exp ({i{l_0}\phi } )\textrm{d}\phi$$

According to the integral formula [44]

(8)$$\int_0^{2\pi } {\exp } (x\cos \varphi - il\varphi )\textrm{d}\varphi = 2\pi {I_l}(x)$$

where I_l is the lth order modified Bessel function of the first kind. Thus, Eq. (7) can be simplified as [39]

(9)$$E(r,\theta ) = M \exp \left[ { - \frac{{({{r^2} + R_1^2} )}}{{\omega_0^2}}} \right]{I_{{l_0}}}\left( {\frac{{2{R_1}r}}{{\omega_0^2}}} \right)\exp ({i({{l_0}\theta + \sigma {\varphi_0}/2} )} )$$

Referring to the literature [45], it can be found that Eq. (9) is basically consistent with Eq. (3) in the literature [45], indicating the beam generated by a radial phase-locked Gaussian laser array with discrete vortex is a quasi-perfect vortex beam.

In practical application, each beamlet will output laser after passing through a hard aperture, thus Eq. (4) will be rewritten as

(10)$$\scalebox{0.82}{$\displaystyle{E_m}(r,\theta ) = circ\left( {\frac{{[{{r^2} + R_1^2 - 2{R_1}r\cos ({\theta - {\phi_m}} )} ]}}{{{\rho_0}/2}}} \right)\exp \left[ { - \frac{{({{r^2} + R_1^2} )}}{{\omega_0^2}}} \right]\exp \left[ {\frac{{2[{{R_1}r\cos ({\theta - {\phi_m}} )} ]}}{{\omega_0^2}}} \right]\exp ({i({{l_0}{\phi_m} + \sigma {\varphi_0}/2} )} )$}$$

where circ (r₀) accounts for the transmittance function of the hard aperture. when r₀< 1, circ(r₀) = 1, when r₀= 1, circ(r₀) = 0.5, when r₀> 1, circ(r₀) = 0 [40]. Furthermore, considering the limited number of beamlets, we adopt the power in the bucket (PIB) to accurately reflect the performance of the radial phase-locked Gaussian laser array with discrete vortex. PIB describes the degree of energy concentration, and its expression can be expressed as the proportion of between the power contained in the bucket and the total power [46]

(11)$$\textrm{PI}{\textrm{B}_M} = \frac{{\int_0^{d/2} {\int_0^{2\pi } {{I_M}} } (r,\theta )rdrd\theta }}{{\int_0^{D/2} {\int_0^{2\pi } {{I_M}} } (r,\theta )rdrd\theta }}$$

where d and D represent the width of the main ring and the receiving plane on the focal plane with focal length f, respectively. I_m(r,θ) denotes the intensity distribution of the radial phase-locked Gaussian laser array with discrete vortex with beamlets number of M on the focal plane. Based on the Eq. (10), the PIB under different beamlets number of M is numerically calculated, as shown in Fig. 2(a). In the numerical calculation, the parameters are set as: λ = 1064 nm, R₁ = 3.2 mm, ω₀ = ρ₀= 0.35 mm, l₀ = 2, and f = 500 mm. When beamlets number M is small, the radial phase-locked Gaussian laser array with discrete vortex has not yet formed a complete annular structure, thus the PIB is relatively small. With the increasing of beamlets number M, the energy of sidelobes will decrease significantly and transform to the inner annular hollow-shaped spot, resulting in that the PIB increases rapidly. When the number of beamlets reaches a certain valve, the PIB will not increase. Notably, the maximum PIB of radial phase-locked Gaussian laser array with discrete vortex is smaller than that of Bessel-Gaussian vortex beam, which is attributed to that part of the energy of the radial phase-locked Gaussian laser array with discrete vortex in the peripheral sidelobes on the focal plane. Therefore, the beamlets number M is set as 26 in the subsequent simulation. Then, we simulate and analyze the relationship between the diameter of the beam formed by a radial phase-locked Gaussian laser array with discrete vortex passing through an optical 4f system and the topological charge, as shown in Fig. 2(b). Since generated beam is circularly symmetric, that is ω = ω_x= ω_y, the diameter ω can be computed numerically by following expression in the Cartesian coordinates [45]

(12)$${\omega ^2} = \omega _x^2 \approx 4\frac{{\sum\limits_{i = 1}^H {\sum\limits_{j = 1}^V {x_i^2} } I({{x_i},{y_j}} )\varDelta x\varDelta y}}{{\sum\limits_{i = 1}^H {\sum\limits_{j = 1}^V I } ({{x_i},{y_j}} )\varDelta x\varDelta y}}$$

where I (x_i, y_j) is the intensity value at pixel coordinates (i, j), (Δx,Δy) are the dimensions of a single pixel, x_i can be obtained via the relation x_i = (-x₀+ i)Δx, x₀ presents so called the “center of mass” of image in the x direction. Clearly, Bessel-Gaussian vortex beams have a stability intensity profile regardless of topological charge. While for radial phase-locked Gaussian laser array with discrete vortex, although the diameter fluctuates somewhat with the alteration of the topological charge, the amplitude is relatively small. It can be considered that the diameter is independent of the topological charge. Therefore, we can draw the following conclusion that a radial phase-locked Gaussian laser array with discrete vortex composed of a certain number of beamlets can generate a perfect vortex.

Fig. 2. The relationship between the PIB and beamlets number M (a), and between the diameter and topological charge (b) for a diameter phase-locked Gaussian laser array with discrete vortex and a Bessel-Gaussian vortex beam.

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Based on above discussion and in combination with coherent beam combining, we proposed a method to generate the perfect VVBs, as schematically depicted in Fig. 3. The Gaussian laser array is distributed side by side as close to each other as possible in a circular array arrangement. The polarization of adjacent beamlets are perpendicular to each other, which are RH and LH circular polarization, respectively. For RH or LH circularly polarized laser array, the vortex phase distribution with variable topological charge is constructed by adjusting the phase of each beamlet. Therefore, the beam from the designed Gaussian laser array can be considered as the coaxial superposition of RH and LH circularly polarized perfect vertex beams with different topological charges, which realizes the generation of perfect VVBs.

Fig. 3. Schematic illustration for the generation perfect VVBs with a radial phase-locked Gaussian laser array.

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Comparing with other methods, the proposed method has three advantages: (1) The spatial light modulators and other diffractive optical elements are not needed, and the utilization of the laser can be improved. (2) Based on coherent beam combining technology, high power perfect VVBs can be realized. (3) The whole optical system involves less optical elements and does not require complex optical path adjustment technology.

Notably, in order to filter unwanted sidelobes, the radial phase-lock Gaussian laser array is placed at the front focal plane of a 4f (f = 500 mm) imaging system with a circular aperture pinhole. The Stokes parameters can be obtained by placing a polarizer and a quarter wave plate in front of the CCD.

3. Results and discussions

Firstly, under the conditions of constant phase φ₀ = 0 and all the beamlets with same amplitude, we numerically investigate the intensity distribution, normalization Stokes parameters, polarization orientation, and Pancharatnam phases of different |H_m_,n〉 states, as shown in Fig. 4. Since the number of beamlets constituting RH and LH circularly polarized laser arrays is the same, resulting in the same intensity of RH and LH circularly polarized vortex beams, that is, S₃ ≈ 0. This is basically consistent with the simulation results depicted in the fourth row of Fig. 4. Due to the difference of topological charge, namely, |m| ≠ |n|, the intensity distribution of RH and LH circularly polarized vortex light is not completely consistent, so that S₃ is not completely zero. When m = n, for instance m = n = 1, the VVBs degenerate into the vortex beams, so resulting in only S₁ component, which equals to S₀, while S₂ and S₃ components are zero. For the cases of m ≠ n, the maximum and minimum values of S₁/S₀ and S₂/S₀ are arranged in turn along the azimuthal direction, and the number of them is exactly equal to the absolute value of the polarization order |P| = |m-n|/2. But for VVBs with opposite polarization orders, for example, the cases of m = -5, n = 1 and m = 4, n = -2, the positions of maximum value of S₂/S₀ are opposite to that of the minimum value. According to Eq. (4), the polarization orientation can be defined as $\theta \textrm{ = }\arctan ({{{S_2}} / {{S_1}}})/2$. We select the x component of VVBs to interfere with a plane wave, the phase information can be extracted from the interference patterns via the method of four-step phase-shifting. The Pancharatnam phases can be obtained after removing the polarization phases. For different |H_m_,n〉 states depicted in Fig. 4, the polarization orientations separately rotate -2π, 8π, 4π, -4π, 6π, 0, and π rad in a full circle, which correspond to the polarization order P = (m-n)/2 of -1, 4, 2, -2, 3, 0, and 0.5. While the Pancharatnam phases spirally vary 0, 0, 6π, -4π, 2π, 2π, and 3π, which means that the topological Pancharatnam charges l₀= (m + n)/2 exactly equal to 0, 0, 3, -1, 1 and 1.5. From the above results, we can draw the conclusions: (1) The VVBs with right polarization order and topological Pancharatnam charge can be successfully generated; (2) The polarization orders and topological Pancharatnam charges are no longer limited to integers, as a result, the VVBs carrying half-integer orbital angular momentum and possess unique polarization states with half-integer order can also be realized. It is worth noting that the polarization distributions of different |H_m_,n〉 states shown in first row of Fig. 4 also confirm the correctness of the calculated polarization order.

Fig. 4. Intensity distribution (top row), S₁/S₀ (second row), S₂/S₀ (third row), S₃/S₀ (fourth row), polarization orientation (fifth row), and Pancharatnam phase (bottom row) of different |H_m_,n〉 state. (a) -m = n = 1; (b) -m = n = 4; (c) m = 5, n = 1; (d) m = -5, n = 1; (e) m = 4, n = -2; (f) m = 1, n = 1; (g) m = 2, n = 1.

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More importantly, by comparing the intensity distributions of different |H_m_,n〉 states, the intensity distributions are basically consistent regardless of the polarization order and topological Pancharatnam charge. In order to quantitatively characterize the intensity distributions of different |H_m_,n〉 states, according to Eq. (12), the diameter ω are calculated and shown in Fig. 5(b). Obviously, the diameter is independent of polarization order and topological Pancharatnam charge of VVBs. This indicates that the VVBs generated from a specially designed radial phase-locked Gaussian laser array are perfect VVBs. In addition, as shown in Fig. 5(a), by defining thickness T as the distance when the maximum value of intensity in a certain direction drops to 1/2, the thickness is calculated and plotted in Fig. 5(c). It is found that, the fluctuation amplitude of thickness is small, which can be approximately recognized as consistent. Notably, since high-order ring of synthetic beams has been filtered by a circular aperture pinhole, the perfect VVBs have a wider annular profile relative to the designed thickness after the inverse Fourier transformation.

Fig. 5. (a) Intensity profile, (b) calculated diameter ω, and (c) calculated thickness T for different |H_m_,n〉 states.

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Further, note that Eq. (10) also contains constant phase φ₀, so we numerically investigate the impact of different constant phases on the perfect VVBs’ characteristics. Here, we chose |H_-1,1〉 state as the research objectives, and the simulation results are shown in Fig. 6. It is noteworthy that the case of φ₀ = 0 has been shown in Fig. 4(a). By comparison, when the constant phase φ₀ varies, S₃/S₀ is still zero, the number of maximum and minimum valves of S₁/S₀ and S₂/S₀ remain unchanged, but their relative positions rotate by φ₀/2. After calculation, for different constant phase φ₀, polarization order P and topological Pancharatnam charge l₀ are still 1 and 0. However, it can be seen from the fourth column of Fig. 6 that there are some differences in the polarization orientation. To characterize these differences, the polarization distribution has been calculated and depicted by white arrow upon the intensity distribution. When φ₀ = 0, the perfect VVBs degenerate into the perfect radially polarized beams, when φ₀ = π, the perfect VVBs degenerate into the perfect azimuthally polarized beams, and when φ₀ is set other angles, the perfect VVBs become the perfect first order generalized vector beams.

Fig. 6. Intensity distribution (first column), S₁/S₀ (second column), S₂/S₀ (third column), S₃/S₀ (fourth column), polarization orientation (fifth column), and Pancharatnam phase (last column) of |H_-1,1〉 state with different constant phase φ₀. (a) φ₀ = π/2; (b) φ₀ = π; (c) φ₀ = 3π/2.

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Based on the Fresnel diffraction theory, the propagation characteristics of the generated perfect VVBs in the free space are numerically simulated, as shown in Fig. 7. Figure 7(a)-(d) display the intensity profiles in x-z plane propagating in a distance of 200 mm and the intensity distributions, polarization orientation, Pancharatnam phase, and polarization distribution at the propagating distance of 5 mm,100 mm, and 195 mm (corresponding to the planes of A, B, and C, respectively) for |H_-1,1〉, |H_-8,8〉, |H_2,1〉, and |H_5,1〉 states, respectively. By comparing with intensity distributions at different propagation distances, it is clear that the generated perfect VVBs maintain a stable ring structure within 100 mm of propagation distance. When propagation distance exceeds 100 mm, the ring structure of |H_-1,1〉, |H_2,1〉, and |H_5,1〉 states will deteriorate, while that of |H_-8,8〉 state is mostly invariant. For |H_-1,1〉, |H_-8,8〉 states, although the color of Pancharatnam phase has some differences at different propagation distances, but Pancharatnam phases still spirally vary 0 in a full circle, and their polarization distribution and polarization orientation almost remain unchanged during the propagation process. While for |H_2,1〉, |H_5,1〉 states composed of two different phase vortex beams, the polarization distributions are elliptical during the propagation process. With the increase of propagation distance, the polarization ellipticity gradually increases, and the polarization orientation also appears a certain distortion, but their Pancharatnam phase remain unchanged. In addition, the influence of propagation distance on the diameter of generated perfect VVBs is also numerically studied, as depicted in Fig. 7(e). With the increase of propagation distance, the diameter starts to shrink, but the reduction rate is relatively small when the propagation distance is less than 100 mm. For the propagation distance over 100 mm, the reduction rate of diameter for is |H_-8,8〉 state still small, while the reduction rate of diameter size for |H_-1,1〉, |H_2,1〉, and |H_5,1〉 states suddenly increase. Based on the previous discussion, the generated perfect VVBs have three features: (1) The generated perfect VVBs can stably propagate for a certain distance, and 100 mm stable propagation has been achieved here. (2) The generated higher order perfect VVBs have a longer stable propagation distance. (3) The generated perfect VVBs with half-integer orbital angular momentum can also realize stable propagation in free space [47]. Li et al. [18] pointed out that Bessel-Gaussian beams can maintain their shape when the propagation distance is far less than the Rayleigh length $L = {{\pi \omega _0^2} / \lambda }$. The simulated results of this paper also confirm this conclusion. In addition, by selecting appropriate parameters, such as waist width of beamlets and diameter of the array, the stable propagation distance of generated perfect VVBs can be effectively improved.

Fig. 7. Intensity profiles propagating in x-z plane and intensity distributions, polarization orientation, Pancharatnam phase at z = 5 mm, 100 mm, and 195 mm for |H_-1,1〉 (a), |H_-8,8〉 (b), |H_2,1〉 (c), and |H_5,1〉 (d) states. The relationship between the propagation distance and on the diameter for selected |H_m_,n〉 states (e).

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It is mentioned that $\psi _R^m$ and $\psi _L^n$ are amplitude coefficients of RH and LH circularly polarized vortices in Eq. (1). For our generated generation scheme, by adjusting the intensity ratio between the RH and

LH circularly polarized laser arrays, the amplitude coefficient of $\psi _R^m$ and $\psi _L^n$ can be flexibly modulated. For the convenience of expressing on the hybrid-order PS, $\psi _R^m$ and $\psi _L^n$ can be represented as $\psi _R^m\textrm{ = }\cos ({\mathrm{\pi} / 4} - \beta )$, $\psi _L^n\textrm{ = sin}({\mathrm{\pi} / 4} - \beta )$. When β ≠ 0 and ±π/2, the generated perfect VVBs carry elliptically polarized states, which correspond to the points between the north/south poles and the equator on the hybrid-order PS. Here, we built a hybrid-order PS composed of |R₅〉 and |L₁〉 eigenstates, as shown in Fig. 8(a). For universality, we chose four different points A (0, π/3), B (0, -π/3), C (π/3, π/4), and D (-π/3, -π/4). Figure 8(b) depicts the intensity distributions, S₃/S₀, polarization orientation, and Pancharatnam phase for points A, B, and C. Obviously, the polarization orientation remains unchanged regardless of the intensity ration between |R₅〉 and |L₁〉 eigenstates. Different from the points on the equator, S₃ of the generated perfect VVBs at points on upper and lower hemisphere is no longer zero, and the value ranges of S₃ for them are opposite. Most notably, for points on the upper hemisphere, the topological Pancharatnam charge is 5, while for points on the lower hemisphere, topological Pancharatnam charge is 1. The reason is that there is topological competition between |R₅〉 and |L₁〉 eigenstates, resulting in the valve of topological Pancharatnam charge l₀ = 5 if $\psi _R^m > \psi _L^n$ and l₀ = 1 if $\psi _R^m < \psi _L^n$[48].

Fig. 8. Schematic illustration of the hybrid-order PS constructed by |R₅〉 and |L₁〉 eigenstates (a). The intensity distributions (first column), S₃/S₀ (second column), polarization orientation (third column), and Pancharatnam phase (last column) at points A, B, and C (b). Intensity profiles propagating in x-z plane (D-1) and intensity distributions, S₃/S₀, polarization orientation, and Pancharatnam phase at z = 0 mm (D-2) and z = 100 mm for point D (c).

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Furthermore, we numerically simulate the propagation characteristics of perfect VVBs corresponding to point D, as shown in Fig. 8(c). Top row of Fig. 8(c) shows the intensity profiles of a beam at point D propagating in a distance of 100 mm. Clearly, the annular structure of the generated perfect VVBs is mostly invariant during propagation. The second row and bottom row in Fig. 8(c) separately depict intensity, S₃/S₀, polarization orientation, and phase properties of generated perfect VVBs at z = 0 mm and z = 100 mm. As can be seen, except for a little difference in the polarization orientation, the intensity distribution, S₃/S₀, and Pancharatnam phase remain nearly constant. This indicates generated perfect VVBs with elliptical polarization states are stable on beam propagation.

4. Conclusions

In conclusions, based on coherent beam combining, a novel approach to generate a perfect VVBs with a specially designed radial phase-locked Gaussian laser array composed of two groups of discrete vortex array with RH and LH circularly polarized states and in turn adjacent to each other is presented. By analyzing intensity distributions, normalization Stokes parameters, polarization orientation, and Pancharatnam phases for different |H_mn〉 states, we successfully generate VVBs with correct polarization order and topological Pancharatnam charge. In addition, the VVBs with half-integer orbital angular momentum and possess unique polarization states with half-integer order can also be realized. More importantly, we found the diameter and thickness of generated VVBs is independent of the polarization orders and topological Pancharatnam charges, which indicates that the generated VVBs are perfect. Further, the influence of different constant phases φ₀ on the perfect VVBs’ characteristics are numerically investigated. The results show that the constant phases φ₀ has no effect on polarization order and topological Pancharatnam charge, but cause polarization orientation to rotate φ₀/2. Utilizing the Fresnel diffraction theory, the propagation characteristics of the generated perfect VVBs in the free space are numerically simulated. It has been found that the generated perfect VVBs can stably propagate for a certain distance, even with half-integer orbital angular momentum. Moreover, the generated higher order perfect VVBs have a longer stable propagation distance. Notably, one can improve the stable propagation distance for generated perfect VVBs with selecting appropriate parameters. For the proposed scheme, by adjusting the intensity ratio between the RH and LH circularly polarized laser arrays, the perfect VVBs with elliptically polarized states can be flexibly generated, and such perfect VVBs are also stable on beam propagation. This scheme improves the utilization of laser and can realize high power perfect VVBs, which will have potential in the fields of laser manufacture, nonlinear frequency conversion, and so on.

Funding

Youth Innovation Promotion Association XIOPM-CAS (XIOPMQCH2021003); National Natural Science Foundation of China (62005310, 62171443); Key Research and Development Projects of Shaanxi Province (2021GY-298).

Acknowledgments

The authors appreciate Peng Li for valuable discussions.

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.

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Generation of perfect vectorial vortex beams by employing coherent beam combining

Abstract

1. Introduction

2. Generation method of perfect VVBs

3. Results and discussions

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (12)

Optics Express