Spin&#x2013;orbit periodic conversion in a gradient-index fiber

Xiaojin Yin; Xiaojin Yin; Xiaojin Yin; Chungang Zhao; Chungang Zhao; Chen Yang; Jinhong Li; Jinhong Li

doi:10.1364/OE.457375

1. Introduction

Angular momentum (AM) as an important characteristic of light is divided into two categories: orbital AM (OAM) and spin AM (SAM). The OAM originates from the vortex phase and nonuniform polarization distribution, whereas the SAM is associated with the helicity of polarized light [1–5]. In the last few decades, the OAM and SAM have been used to trap particles in the optical tweezer field [6–9]. The orbiting motion of particles is observed when the OAM is transferred to the particles, and the SAM induces the rotation of the particles around their axis [8]. The OAM is also employed to increase the communication capacity based on the orthogonality of different topological charges [10–15] and construct chiral nanostructures [16,17]. Spin–orbit conversions occur in tight focusing [18,19], scattering, and imaging optical systems [19] as well as surface plasmon polaritons [20,21]. Numerous physical phenomena occur owing to spin–orbit conversions, such as spin–orbit interactions [19,22,23] and the spin Hall effect of topological photonics [24–26]. The complexity of the SAM increases when the confined field exhibits a vortex phase; for example, the distribution of photonic skyrmions is achieved because of spin–orbit interactions [20,21]. Spin-orbit conversion also occurs in the transmission process of the fiber, the spin-orbit interaction is studied in the multihelicoidal fiber which possesses a multihelical refractive index profile [27], few-mode fiber [28], ring-core optical fibers [29] etc. Especially, SAM-IOAM, SAM-EOAM and IOAM-EOAM interaction is implemented with the gradient-index fiber [30] and a new single-shot Müller matrix polarimeter is constructed though GRIN cascades which generates a vectorial beam [31]. However, spin-orbit interaction in gradient-index fibers is concentrated in the case of shorter lengths such as GRIN lens. It is necessary to study spin-orbit interaction in long-distance transmission of gradient-index fibers.

Partially coherent beams can be generated by appropriately reducing the beam coherence, which also affords strong directionality [32] and can address the shortcomings of the inhomogeneous intensity distribution and low anti-interference ability of highly coherent light in the far field [33]. Tuning the phase, polarization, and coherent structure of beams can yield new optical fields with increased complexity, such as partially coherent cylindrical vortex vector beams (CVVBs). Compared with fully coherent vortex beams, partially coherent vortex beams exhibit unique properties; in particular, the combined regulation of coherence and topological charges can promote beam shaping [34], polarization state conversion [35], and coherent singularity [36].

In this study, we analyzed the transmission characteristics of a cylindrical vector beams (CVB) and a CVVB in a GRIN fiber. The field distribution and state of polarization (SOP) of the CVB and CVVB propagating in the GRIN fiber are studied. The SOP of CVVB becomes elliptical from the radial and angular, and the OAM is converted to SAM at the focal plane of the GRIN fibers. Furthermore, spin–orbit periodic conversions occur in the GRIN fibers. Moreover, a SAM equation of partially coherent beams is established and the spin–orbit periodic conversion of partially coherent beams in the GRIN fibers is analyzed.

2. Theoretical model

CVB contains two different polarized light beams, radially polarized light (RB) and angularly polarized light (AB). When RB and AB carry the vortex phase, they become radially polarized vortex light (RVB) and angularly polarized vortex light (AVB). In this paper, we set RVB and AVB which topological charge m = 0 are RB and AB.

The electric field of a radially polarized vortex beam (RVB) at the source plane is expressed as [37]

(1)$${\boldsymbol E}({x_0},{y_0} \,0) = {[{x_0} + {\mathop{\rm {isgn}}} (m){y_0}]^{|m|}}\exp ( - \frac{{x_0^2 + y_0^2}}{{{w^2}}})(\frac{{{x_0}}}{w}{\boldsymbol{e}_x} + \frac{{{y_0}}}{w}{\boldsymbol{e}_y})$$

where w denotes the beam width, m represents a topological charge, sgn() is a sign function, and x₀ and y₀ denote position vectors at the source plane. The electric field of the beam in an ABCD optical system can be characterized using the generalized Huygens–Fresnel principle [38].

(2)$$\begin{aligned}{\boldsymbol E}(x,y,z) &= \frac{{i k}}{{2\pi B}}\int {\int {\boldsymbol E} } ({x_0},{y_0},0)\\&\times \exp \left. {\left\{ { - \frac{{\textrm{i}k}}{{2B}}[{A(x_0^2{\boldsymbol + }y_0^2) - 2({{x_0}x + {y_0}y} )+ D(x_{}^2{\boldsymbol + }y_{}^2)} ]} \right.} \right\}d{x_0}d{y_0}\end{aligned}$$

Substituting Eq. (1) into Eq. (2) yields the electric field expression of the RVB at the receiver plane.

(3)$$\begin{aligned}{{\boldsymbol E}_x} &= \frac{k}{{Bw}}{\sqrt M ^{ - |m\textrm{|} - 3}}{2^{ - |m\textrm{|} - 2}}{\textrm{i}^{ - |m\textrm{|}}}\exp \left[ { - \frac{{{k^2}({{x^2} + {y^2}} )}}{{4{B^2}M}}} \right]\exp \left[ { - \frac{{\textrm{i}kD}}{{2B}}(x_{}^2{\boldsymbol + }y_{}^2)} \right]\\&\times \sum\limits_{d = 0}^{\textrm{|}m\textrm{|}} {\frac{{\textrm{|}m\textrm{|}!{\textrm{i}^d}}}{{d!(\textrm{|}m\textrm{|} - d)!}}} {\mathop{\rm sgn}} {(m)^d}{H_{|m\textrm{|} - d + 1}}\left( { - \frac{{kx}}{{2B\sqrt M }}} \right){H_d}\left( { - \frac{{ky}}{{2B\sqrt M }}} \right)\\{{\boldsymbol E}_y} &= \frac{k}{{Bw}}{\sqrt M ^{ - |m\textrm{|} - 3}}{2^{ - |m\textrm{|} - 2}}{\textrm{i}^{ - |m\textrm{|}}}\exp \left[ { - \frac{{{k^2}({{x^2} + {y^2}} )}}{{4{B^2}M}}} \right]\exp \left[ { - \frac{{\textrm{i}kD}}{{2B}}(x_{}^2{\boldsymbol + }y_{}^2)} \right]\\&\times \sum\limits_{d = 0}^{\textrm{|}m\textrm{|}} {\frac{{\textrm{|}m\textrm{|}!{\textrm{i}^d}}}{{d!(\textrm{|}m\textrm{|} - d)!}}} {\mathop{\rm sgn}} {(m)^d}{H_{|m\textrm{|} - d}}\left( { - \frac{{kx}}{{2B\sqrt M }}} \right){H_{d + 1}}\left( { - \frac{{ky}}{{2B\sqrt M }}} \right)\end{aligned}$$

M = \frac{1}{{{w^2}}} + \frac{{\textrm{i}kA}}{{2B}}

where x and y denote position vectors at the receiver plane and k = 2π/λ denotes the wavenumber. For paraxial beam propagation, the ABCD matrix of a radial GRIN fiber is expressed as [39]

(4)$$\left[ {\begin{array}{cl} A&B\\ C&D \end{array}} \right] = \left[ {\begin{array}{cl} {\cos (\beta z)}&{\frac{{\sin (\beta z)}}{{{n_0}\beta }}}\\ { - {n_0}\beta \sin (\beta z)}&{\cos (\beta z)} \end{array}} \right]$$

which shows periodicity. In Eq. (4), β=5.257mm^-1 is a GRIN coefficient, n₀= 1.46977 denotes the refractive index at the center of the fiber core, and n₁= 1.45702 represents the refractive index of the fiber cladding. The fiber core radius is set as 25µm. Consequently, the ABCD matrix of the GRIN fiber is obtained. The light field distribution is obtained by substituting Eq. (4) into Eq. (3).

Figure 1 shows the axial light field distribution of the RVB in the GRIN fiber, which exhibits periodicity. The light distribution is symmetrical at the focal plane. After focusing the beam using the GRIN fiber, a hollow light spot is observed when m = -2, 0 and 2 (without and with the presence of the vortex phase in the beam, respectively). When m =±1, the light spot is a dot that is symmetrical along the optical axis (with the presence of the vortex phase in the beam).

Fig. 1. Axial light field distribution of the RVB in the GRIN fiber. (a) m = -2. (b) m = -1 (c) m = 0. (d) m = 1. (e) m = 2. (The normalization of all graphs in this study is for the same topological charge m, and the corresponding values of different topological charges m are not comparable.)

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The SAM distribution is estimated using the following equation [5,40]

(5)$${S_E} \sim {\mathop{\rm Im}\nolimits} ({{\boldsymbol E}_{}^\ast{\times} {\boldsymbol E}} )\textrm{ = }\left( {\begin{array}{c} {{\boldsymbol E}_y^\ast {{\boldsymbol E}_z} - {\boldsymbol E}_z^\ast {{\boldsymbol E}_y}}\\ {{\boldsymbol E}_z^\ast {{\boldsymbol E}_x} - {\boldsymbol E}_x^\ast {{\boldsymbol E}_z}}\\ {{\boldsymbol E}_x^\ast {{\boldsymbol E}_y} - {\boldsymbol E}_y^\ast {{\boldsymbol E}_x}} \end{array}} \right)$$

The longitudinal component of the SAM is related to the transverse field, expressed as [5,40]

(6)$${S_z} \simeq {\mathop{\rm Im}\nolimits} ({{\boldsymbol E}_x^\ast {{\boldsymbol E}_y} - {\boldsymbol E}_y^\ast {{\boldsymbol E}_x}} )$$

The relation between the longitudinal component of the SAM (S_Z) and the intensities of two circularly polarized components (namely, left circularly polarized (LCP) component, right circularly polarized (RCP)) of the transverse field can be obtained as follows [5,40].

(7)$${S_z} \simeq {I_{LCP}} - {I_{RCP}}$$

Figure 2 presents the field, SOP, and SAM density distributions at the focal plane when the RVB propagates in the GRIN fiber. In Fig. 2(a), the black line represents the polarization distribution. This figure shows that the RVB is focused using the GRIN fiber; moreover, the polarization is unchanged when m =0 and changed when m = ±1and ±2, respectively (without and with the vortex phase in the beam, respectively). The polarization of the RVB in the GRIN fiber changes from a radial to an elliptical distribution. The vortex phase induces changes in the polarization, and the SAM appears at the focal plane (Fig. 2(d)). The left and right circularly polarized components of the RVB at the focal plane of the GRIN fiber are estimated using Eq. (7) and shown in Fig. 2(b) and (c). The SAM is expressed as the difference between the left and right circularly polarized components. At m = 0, the left and right circularly polarized components of the RVB show the same distribution and the SAM density does not exist. When the beam carries a vortex phase (m = ±1and ±2), the left and right circularly polarized components of the beam exhibit different distributions. In this case, the SAM appears at the focal plane of the GRIN fiber and the SAM density is mapped as the difference between the left and right circularly polarized components. The OAM is converted to the SAM at the focal plane of the GRIN fiber. By comparing Fig. 2(d1) and (d5), Fig. 2(d2) and (d4), It can be seen that SAM at the focal plane showing the opposite distribution when the beam carries the OAM of opposite topological charges.

Fig. 2. (a)The field, SOP, (b) LCP component, (c)RCP component, and (d)SAM of the RVB at the focal plane of the GRIN fiber.

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Equation (8) presents the polarization relation between angularly and radially polarized lights. Figure 3 shows the field, SOP, and SAM density distributions of angularly polarized vortex light (AVB) at the focal plane in the GRIN fiber obtained using Eq. (4). The AVB at the focal plane without a vortex phase (m = 0) continues to exhibit angular polarization, and the AVB with a vortex phase (m = ±1and ±2) shows nonuniform elliptical polarization (Fig. 3(a)).

(8)$${\left[ {\begin{array}{c} {{{\boldsymbol E}_x}}\\ {{{\boldsymbol E}_y}} \end{array}} \right]_{\textrm{azimuthally}}}\textrm{ = }{\left[ {\begin{array}{c} { - {{\boldsymbol E}_y}}\\ {{{\boldsymbol E}_x}} \end{array}} \right]_{\textrm{radially}}}$$

Figures 3(b)–(d) show the distribution of the left and right circularly polarized components and the SAM of the AVB when m = 0, ±1, and ±2. At m = 0, the left and right circularly polarized components of the AVB show the same distribution and the SAM distribution of the AVB does not exist. At m = ±1 and ±2, the left and right circularly polarized components of the AVB show different distributions and the SAM density of the AVB is expressed as the difference between the left and right circularly polarized components. The spin–orbit conversion characteristics of the CVB and CVVB in GRIN fibers is analyzed based on the Huygens–Fresnel principle. This research is suitable for the study of spin-orbit conversion under paraxial conditional transport. Since no longitudinal component of the electric field (E_z) is generated under paraxial conditions, the radial component of SAM (S_r) is 0 and the longitudinal component of SAM (S_z) can be considered equal to the total SAM (S = S_z). OAM converted to SAM at the focal plane (z = (2n+1)l/2) and SAM converted to OAM at the period plane (z = nl). The spin–orbit periodic conversion in a GRIN fiber is proved by simulating the AM distribution at different locations.

Fig. 3. (a)The field, SOP, (b)LCP, (c)RCP, and (d)SAM of the AVB at the focal plane of the GRIN fiber.

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Partially coherent light at the source plane can be represented using a cross-spectral density matrix [41]

(9)$$\mathop {\boldsymbol W}\limits^ \leftrightarrow ({{\boldsymbol s}_1},{{\boldsymbol s}_2},0) = \left[ {\begin{array}{cc} {{W_{xx}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}&{{W_{xy}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}\\ {{W_{yx}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}&{{W_{yy}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)} \end{array}} \right]$$

where matrix elements ${W_{ij}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0) = \langle {\boldsymbol E}_i^\ast ({{\boldsymbol s}_1},0){{\boldsymbol E}_j}({{\boldsymbol s}_2},0)\rangle$ $({i,j = x,y} )$. ${\boldsymbol{s}_1}$ and ${\boldsymbol{s}_2}$ denote position vectors at the source plane.

The elements of the cross-spectral density matrix of a partially coherent radially polarized vortex beam (PRVB) at the source plane can be expressed as [37]

(10)$$\begin{aligned}{W_{ij}}({\boldsymbol{s}_1},{\boldsymbol{s}_2},0) &= \frac{{{s_{1i}}{s_{2j}}}}{{{w^2}}}{[{s_{1x}} + {\mathop{\rm {isgn}}} (m){s_{1y}}]^{|m|}}{[{s_{2x}} - {\mathop{\rm {isgn}}} (m){s_{2y}}]^{|m|}}\\&\times \exp \left( { - \frac{{\boldsymbol{s}_1^2 + \boldsymbol{s}_2^2}}{{{w^2}}}} \right)\exp \left[ { - \frac{{{{(\boldsymbol{s}_1^{} - \boldsymbol{s}_2^{})}^2}}}{{2{\sigma^2}}}} \right]\end{aligned}$$

where $\sigma$ denotes the correlation length. Based on the generalized Huygens–Fresnel principle [38],

\begin{aligned}{W_{ij}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z) &= {\left( {\frac{k}{{2\pi B}}} \right)^2}\int {\int {\int {\int {{W_{ij}}} } } } ({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)d{s_{1x}}d{s_{1y}}d{s_{2x}}d{s_{2y}}\\&\times \exp \left. {\left\{ { - \frac{{\textrm{i}k}}{{2B}}[{A({\boldsymbol s}_1^2 - {\boldsymbol s}_2^2) - 2({{{\boldsymbol s}_1}{{\boldsymbol \rho }_1} - {{\boldsymbol s}_2}{{\boldsymbol \rho }_2}} )+ D({\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2)} ]} \right.} \right\}\end{aligned}

The cross-spectral density matrix of a PRVB transmitted in the ABCD optical system is shown in Eq. (12), using which the light field distribution of the PRVB can be estimated.

(12)$${\left[ {\begin{array}{cc} {{W_{xx}}}&{{W_{xy}}}\\ {{W_{yx}}}&{{W_{yy}}} \end{array}} \right]_{({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z)}} = \left[ {\begin{array}{cc} {G{G_1}{H_{\delta + 2}}({{l_1}} ){H_\eta }({{l_2}} )}&{G{G_2}{H_{\delta + 1}}({{l_1}} ){H_{\eta + 1}}({{l_2}} )}\\ {G{G_1}{H_{\delta + 1}}({{l_1}} ){H_{\eta + 1}}({{l_2}} )}&{G{G_2}{H_\delta }({{l_1}} ){H_{\eta + 2}}({{l_2}} )} \end{array}} \right]$$

\begin{aligned}G &= {\left( {\frac{k}{{Bw}}} \right)^2}\exp \left( { - \frac{{\textrm{i}kD}}{{2B}}({\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2)} \right)\exp \left[ {{{\left( {\frac{{\textrm{i}k{\rho_{1x}}}}{{2B\sqrt {{M_2}} }} - \frac{{\textrm{i}k{\rho_{2x}}}}{{4{\sigma^2}B{M_1}\sqrt {{M_2}} }}} \right)}^2}} \right]\\&\times \exp \left[ {{{\left( {\frac{{\textrm{i}k{\rho_{1y}}}}{{2B\sqrt {{M_2}} }} - \frac{{\textrm{i}k{\rho_{2y}}}}{{4{\sigma^2}B{M_1}\sqrt {{M_2}} }}} \right)}^2}} \right]\exp \left( { - \frac{{{k^2}\rho_2^2}}{{4{B^2}{M_1}}}} \right){2^{ - \frac{{5|m |+ 9}}{2}}}{\textrm{i}^{ - 2|m |- 2}}M_1^{ - \frac{{|m |+ 3}}{2}}\\&\times \sum\limits_{{d_1} = 0}^{|m\textrm{|}} {\sum\limits_{{d_2} = 0}^{|m\textrm{|}} {\frac{{|m |!{\textrm{i}^{{d_1}}}}}{{{d_1}!(|m |- {d_1})!}}} \frac{{|m |!{\textrm{i}^{{d_2}}}}}{{{d_2}!(|m |- {d_2})!}}} {\mathop{\rm sgn}} {(m)^{{d_1}}}{[{ - {\mathop{\rm sgn}} (m)} ]^{{d_2}}}\end{aligned}

\begin{aligned}&{G_1}\textrm{ = }\sum\limits_{{r_1} = 0}^{|m |-{d_2} + 1} {\sum\limits_{{r_2} = 0}^{{d_2}} {\left( {\begin{array}{c} {|m |- {d_2} + 1}\\ {{r_1}} \end{array}} \right)\left( {\begin{array}{c} {{d_2}}\\ {{r_2}} \end{array}} \right){H_{{r_1}}}({{k_1}} ){H_{{r_2}}}({{k_2}} )} } \sum\limits_{{n_1} = 0}^{\frac{{|m |- {d_2} + 1 - {r_1}}}{2}} {\sum\limits_{{n_2} = 0}^{\frac{{{d_2} - {r_2}}}{2}} {{{( - 1)}^{{n_1} + {n_2}}}} } M_2^p{(j )^{|m |+ 1 - {r_1} - 2{n_1} - {r_2} - 2{n_2}}}{q_1}\\&{G_2}\textrm{ = }\sum\limits_{{r_1} = 0}^{|m |- {d_2}} {\sum\limits_{{r_2} = 0}^{{d_2} + 1} {\left( {\begin{array}{c} {m - {d_2}}\\ {{r_1}} \end{array}} \right)} \left( {\begin{array}{c} {{d_2} + 1}\\ {{r_2}} \end{array}} \right){H_{{r_1}}}({{k_1}} )} {H_{{r_2}}}({{k_2}} )\sum\limits_{{n_1} = 0}^{\frac{{|m |- {d_2} - {r_1}}}{2}} {\sum\limits_{{n_2} = 0}^{\frac{{{d_2} + 1 - {r_2}}}{2}} {{{( - 1)}^{{n_1} + {n_2}}}} } M_2^p{(j )^{|m |+ 1 - {r_1} - 2{n_1} - {r_2} - 2{n_2}}}{q_2}\\&\begin{array}{lll}{l_1} = \frac{{k{\rho _{2x}}}}{{4{\sigma ^2}B{M_1}\sqrt {{M_2}} }} - \frac{{k{\rho _{1x}}}}{{2B\sqrt {{M_2}} }}&{l_2} = \frac{{k{\rho _{2y}}}}{{4{\sigma ^2}B{M_1}\sqrt {{M_2}} }} - \frac{{k{\rho _{1y}}}}{{2B\sqrt {{M_2}} }}&\delta \textrm{ = }2\left| m \right| - {d_1} - {d_2} - {r_1} - 2{n_1}\end{array}\\& \eta \textrm{ = }{d_1} + {d_2} - {r_2} - 2{n_2}\end{aligned}

$H({\cdot} )$ represents the Hermite polynomial and $\left( {\begin{array}{c} .\\ . \end{array}} \right)$ is a binomial coefficient. ${{\boldsymbol \rho }_1} = ({{\rho_{1x}},{\rho_{1y}}} )$ and ${{\boldsymbol \rho }_2} = ({{\rho_{2x}},{\rho_{2y}}} )$ are any two points at the output plane, and $k = 2\pi /\lambda$ is the wavenumber.

{M_1} = \frac{1}{{w_{}^2}} + \frac{1}{{2\sigma _{}^2}} - \frac{{ikA}}{{2B}}{\kern 5pt}{M_2} = \frac{1}{{w_{}^2}} + \frac{1}{{2\sigma _{}^2}} + \frac{{ikA}}{{2B}} - \frac{1}{{4{M_1}\sigma _{}^4}}{\kern 5pt}j = \frac{\textrm{1}}{{\sqrt 2 {\sigma ^2}\sqrt {{M_1}} }}

\begin{array}{c}{k_1} = \frac{{k{\rho _{2x}}}}{{\sqrt 2 B\sqrt {{M_1}} }}{\kern 10pt}{k_2} = \frac{{k{\rho _{2y}}}}{{\sqrt 2 B\sqrt {{M_1}} }}{\kern 10pt}{q_1} = \frac{{(|m |- {d_2} + 1 - {r_1})!}}{{{n_1}!(|m |- {d_2} + 1 - {r_1} - 2{n_1})!}}\frac{{({d_2} - {r_2})!}}{{{n_2}!({d_2} - {r_2} - 2{n_2})!}}\\{\kern -30pt}{q_2} = \frac{{(|m |- {d_2} - {r_1})!}}{{{n_1}!(|m |- {d_2} - {r_1} - 2{n_1})!}}\frac{{({d_2} + 1 - {r_2})!}}{{{n_2}!({d_2} - {r_2} + 1 - 2{n_2})!}}{\kern 5pt} p= - \frac{{2|m |+ 4 - {r_1} - 2{n_1} - {r_2} - 2{n_2}}}{2}.\end{array}

The SAM density of partially coherent light can be obtained using that of fully coherent light [5,40,42] and the cross-spectral density matrix of partially coherent light:

(13)$${S_z} \simeq {\mathop{\rm Im}\nolimits} ({{W_{yx}} - {W_{xy}}} )$$

Figure 4 shows the distribution of a PRVB with different coherences at the focal plane of the GRIN fiber for σ = 100m and 15µm. The light intensity, SOP, and SAM density distributions of the PRVB remain unchanged compared with those of the RVB (Figs. 4(a) and (b)) when σ = 100m for partially coherent light can be considered perfectly coherent light at σ = 100m. Thus, Eq. (13) is proven correct based on the light intensity, SOP, and SAM density distributions of the PRVB with σ = 100m. When σ = 15µm, the light intensity, SOP, and SAM density distributions of the PRVB change with the low σ value (Figs. 4(c) and (d)).

Fig. 4. (a)The field, SOP, (b)LCP, (c)RCP, and (d)SAM of the PRVB at the focal plane of the GRIN fiber. (a and b) σ = 100m. (c and d) σ = 15µm.

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On the basis of the polarization relation between angularly and radially polarized lights, the cross-spectral density matrix between partially coherent angularly polarized and partially coherent radially polarized beam can be defined as

(14)$${\left[ {\begin{array}{cc} {{W_{xx}}}&{{W_{xy}}}\\ {{W_{yx}}}&{{W_{yy}}} \end{array}} \right]_{\textrm{azimuthally}}}\textrm{ = }{\left[ {\begin{array}{cc} {{W_{yy}}}&{ - {W_{yx}}}\\ { - {W_{xy}}}&{{W_{xx}}} \end{array}} \right]_{\textrm{radially}}}$$

The light intensity, SOP, and SAM distributions of a partially coherent angularly polarized vortex beam (PAVB) is shown in Fig. 5 for σ = 100 m (Figs. 5(a) and (b)) and 15µm (Figs. 5(c) and (d)). The vortex phase causes the polarization of the optical field to change from angular to elliptical and the OAM change to the SAM at the focal plane of the GRIN fiber. The spin–orbit periodic transition occurs in the GRIN fiber.

Fig. 5. (a)The field, SOP, (b)LCP, (c)RCP, and (d)SAM of the PAVB at the focal plane of the GRIN fiber. (a and b) σ = 100 m. (c and d) σ = 15µm.

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The SAM is mapped as the difference between the left and right circularly polarized components of the light field using Eq. (7). For perfectly coherent light, the SAM mapped based on the aforementioned difference is the same as that obtained using the original Eq. (6). Figure 6 shows the SAM distributions of the PRVB (σ = 100 m) based on the difference between the left and right circularly polarized components. The change is occurred in the SAM distributions of the PRVB (σ = 100 m) compared with Fig. 4(b) for the light field of a partially coherent beam is the sum of W_xx and W_yy which are elements of the cross-spectral density matrix. Using the difference between left and right circularly polarized component of light field is just for W_xx+W_yy.

Fig. 6. The (a)LCP, (b)RCP, and (c)SAM obtained based on the difference between the left and right circularly polarized components of the PRVB (σ = 100 m) at the focal plane of the GRIN fiber.

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3. Conclusion

Based on the generalized Huygens–Fresnel principle, the periodic transmission characterization and spin–orbit periodic conversions of the RVB, AVB, PRVB, and PAVB in a radial GRIN fiber are presented. Changes in the polarization occur and SAM is appearing at the focal plane of the GRIN fiber in the presence of a vortex phase. Moreover, the SAM distributions of partially coherent light cannot be achieved by characterizing the left and right circularly polarized components. The analysis of the transmission characteristics and SAM of the CVB and CVVB in the GRIN fiber has application significance in the field of optical communication and manipulation.

Funding

Taiyuan University of Science and Technology Scientific Research Initial Funding (20222009); Fundamental Research Program of Shanxi Province (202103021223271, 202103021223299); Shanxi Provincial Central Government Guides Local Science and Technology Development Fund Project (YDZX20201400001386).

Acknowledgments

Supported by Fundamental Research Program of Shanxi Province (202103021223271) and Taiyuan University of Science and Technology Scientific Research Initial Funding (20222009).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Spin–orbit periodic conversion in a gradient-index fiber

Abstract

1. Introduction

2. Theoretical model

3. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (19)

Optics Express