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Spin-orbit interactions of a circularly polarized vortex beam in paraxial propagation

Xiaojin Yin, Ziyue Zhao, Pengqi Hao, and Jinhong Li

Open Access Open Access

Abstract

Spin-orbit interactions (SOIs) of circularly polarized beam and circularly polarized vortex beam during paraxial propagation in a radial gradient-index (GRIN) fiber are analyzed using the generalized Huygens–Fresnel principle and the GRIN fiber’s ABCD matrix. SAM is only associated with polarized light helicity and OAM is only associated with topological charge m. SAM and OAM do not crosstalk or convert between each other; SOIs did not occur at the GRIN fiber’s focal plane. SOIs of partially coherent circularly polarized beam and partially coherent circularly polarized vortex beam in the GRIN fiber are also studied and show the same characteristics as the perfectly polarized beam.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The angular momentum (AM) of light can be divided into two categories: orbital angular momentum (OAM) and spin angular momentum (SAM). Over recent decades, OAM and SAM have been used to form optical tweezers [1–4], chiral nanostructures [5,6], and high-capacity communication structures [7–12]. Of these two categories, OAM can also be divided into two types: (i) intrinsic orbital AM (IOAM) and (ii) extrinsic orbital AM (EOAM). IOAM originates from the vortex phase, and EOAM is generated based on the beam trajectory in a specific transmission medium [13]. SAM is associated with the helicity of polarized light, and for a circularly polarized beam, the right-hand and left-hand circular polarizations correspond to different SAMs with opposite rotation directions [14]. Collectively, these three types of AM are associated with optical vortices, the beam trajectory, and circular polarizations, respectively [13,14]. When transmission in a medium and interactions between matter and light take place, spin-orbit interactions (SOIs) of light will occur. When circularly polarized light is transmitted in an inhomogeneous medium, the spin-Hall effect, which represents the interaction between the SAM and EOAM, occurs in the transmission medium. Spin-to-orbital conversions are produced by coupling between the SAM and the IOAM in nonparaxial fields, where the light is tightly focused by high-numerical-aperture lenses and scattered by small particles. Experimental and theoretical demonstrations have shown that SAM can be converted into OAM and that the topological charge of OAM will be change when the helical phase is opposite or identical to that of the circular polarization in the tightly focused circularly polarized beam and the tightly focused circularly polarized vortex beam [15]. OAM can be converted into SAM under the condition that the beam is tightly focused, with evanescent waves or an anisotropic structure (e.g., a metasurface structure), and during transmission in a gradient refractive index fiber. The distribution of photonic skyrmions is formed with an SOIs in the evanescent waves with the Au film and an anisotropic structure [16,17]. The spin of focused cylindrical vector vortex beam can be mapped using a spin-resolved near-field scanning optical microscope (NSOM) and also be detected at the focal plane of the radial gradient-index (GRIN) fiber [18–20].

The circularly polarized vortex beam is used widely in the study of SOIs because it has both SAM and OAM. In this study, we analyze the SOIs of a focused circularly polarized vortex beam during paraxial transmission in a GRIN fiber. The field distribution, the state of polarization (SOP), the SAM and the OAM of ·the different topological charges of focused circularly polarized vortex beam at the focal plane of the GRIN fiber are studied. The SOP of the circularly polarized vortex beam remains circularly polarized and no conversion occurs between OAM and SAM. The SOIs of the focused circularly polarized vortex beam during paraxial transmission is different to the SOIs of a tightly focused circularly polarized vortex beam and the focused cylindrical vector vortex beam in the GRIN fiber [15,20]. Finally, we analyze the SOIs of a partially coherent circularly polarized vortex beam during paraxial transmission using the formula for partially coherent SAM and OAM. It is demonstrated that the findings described above are also applicable to partially coherent circularly polarized vortex beam during paraxial transmission.

2. Theoretical model

When circularly polarized beam carries the vortex phase, they become circularly polarized vortex beam. In this paper, the circularly polarized vortex beam which topological charge m = 0 is circularly polarized beam. The electric field of a circularly polarized vortex beam at the source plane can be derived from the expression for the cylindrical vector vortex beam and is expressed as [20,21]:

$${\boldsymbol E}({{x_0},{y_0},0} )= {[{x_0} + i {\mathop{\rm sgn}} (m){y_0}]^{|m|}}\exp ( - \frac{{x_0^2 + y_0^2}}{{{w^2}}})({{\boldsymbol e}_x} \pm \textrm{i}{{\boldsymbol e}_y})$$
where w is the beam width, m is the topological charge of the circularly polarized vortex beam, sgn() is a sign function, x0 and y0 denote position vectors at the source plane. In addition, i represents the phase difference between the x-component and the y-component of the circularly polarized vortex beam, which is π/2. Finally, ± denote the left-polarized and right-polarized circularly polarized vortex beam, respectively.

Huygens–Fresnel principle is generally used to study light transmission through a vacuum, and the ABCD optical system is used to characterize transmission systems. A combination of the Huygens–Fresnel principle and the ABCD system can be used to characterize the transmission of light in different media. The electric field of the beam in the ABCD optical system is given in Eq. (2) [22].

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

The x-component and y-component expressions for the circularly polarized vortex beam at the receiver plane are derived by substituting Eq. (1) into Eq. (2). The x-component and y-component expressions for a left circularly polarized vortex beam passing through an ABCD optical system are shown here as Eq. (3). In Eq. (3), x and y denote position vectors at the receiver plane, k represents wavenumber, w is the beam width of the circularly polarized vortex beam at the source plane. A, B, D are matrix elements in GRIN fiber ABCD transmission matrix respectively, and Hermite polynomials was used in the calculation process to replace some terms in Eq. (3) obtained by integration.

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

Fibers are commonly used in the study of paraxial transmission, and GRIN fibers have also been widely studied because of their self-focusing properties, which make it possible to focus automatically during paraxial transmission. The ABCD matrix of a radial GRIN fiber is as shown in Eq. (4), and the matrix parameters are expressed as follows: β=5.257 mm−1 is a GRIN coefficient, and n0 = 1.46977 and n1 = 1.45702 are the refractive indices of the center of the fiber core and the fiber cladding, respectively [23]. The fiber core radius is set at 25 µm. The periodic self-focusing property of the GRIN fiber is characterized by the fiber’s ABCD matrix. The distribution of a circularly polarized vortex beam in the GRIN fiber can be obtained by substituting Eq. (4) into Eq. (3).

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

It can be seen that the circularly polarized vortex beam shows periodicity in its transmission from Fig. 1, which shows the field distributions of the circularly polarized vortex beam in the GRIN fiber. When m = 0, the beam is circularly polarized, as shown in Fig. 1(c), and the spots at the focal plane are dots that are arranged symmetrically along the optical axis; in contrast, the spots at the focal plane are hollow when m =±1 and m =±2 (the vortex phase with OAM in the beam), as shown in Fig. 1(a), (b), (d), and (e). The radius of the spot when m =±2 is greater than the radius of that when m =±1. Because of the periodic nature and the self-focusing property of the GRIN fiber, the circularly polarized vortex beam in the GRIN fiber exhibits periodic focusing and divergence.

 figure: Fig. 1.

Fig. 1. Axial light field distribution of a circularly polarized vortex beam in a 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 performed by dividing the light intensity in each case by its maximum, and the corresponding values of the different graphs are not comparable.)

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When the circularly polarized vortex beam propagating along the z-axis, the wavevector k is perpendicular to the vector field. The circularly polarized vortex beam constrained to the xy plane and manifests a truly two-dimensional character and according to the ±π/2 phase between the transverse field components x and y resulting in longitudinal SAM [24]. The longitudinal component of the SAM (Sz) is given by Eq. (5) and Eq. (6) [18–20,25,26]. Equation (6) shows the relationship between Sz and two circularly polarized components (the left circularly polarized (LCP) and right circularly polarized (RCP) components) [18–20,25,26]. The distribution of Sz of a focused beam can be obtained by mapping its left and right components experimentally. Sz is equal to the total SAM in the paraxial transmission because the longitudinal component (Ez) of the electric field will not be generated during paraxial transmission in GRIN fibers. It should be noted that as n0 (i.e., the refractive index at the center of the GRIN fiber) increases, tight focusing will occur at the focal plane of the GRIN fiber, which means that a new theory is required to analyze the light transmission in GRIN fibers.

$${S_z} \simeq {\mathop{\rm Im}\nolimits} [{{\boldsymbol E}_x^\ast {{\boldsymbol E}_y} - {\boldsymbol E}_y^\ast {{\boldsymbol E}_x}} ]$$
$${S_z} \simeq {I_{\textrm{LCP}}} - {I_{\textrm{RCP}}}$$

The circularly polarized beam carries the vortex phase resulting in longitudinal OAM and the longitudinal component of the OAM (Oz) is shown in Eq. (7), and Oz is equal to the total OAM when the beam carries a vortex phase in the transverse electric field [27]. Only longitudinal SAM (Sz) and OAM (Oz) exist during paraxial transmission. The SOIs of the different beam in the paraxial transmission case could then be analyzed by characterizing the SAM and OAM of the beam. SOIs is an optical phenomenon which the spin (circular polarization or ellipse polarization) affects and controls the spatial of light and the vortex phase affects and controls the polarization of light. The spatial and the polarization of light affect each other, resulting in the mutual conversion of spin and orbit [13].

$${O_z} \simeq {\mathop{\rm Im}\nolimits} [{x{T_y} - y{T_x}} ]$$
$${T_x} = {\boldsymbol E}_x^ \ast \frac{{\partial {{\boldsymbol E}_x}}}{{\partial x}} + {\boldsymbol E}_y^ \ast \frac{{\partial {{\boldsymbol E}_y}}}{{\partial x}}$$
$${T_y} = {\boldsymbol E}_x^ \ast \frac{{\partial {{\boldsymbol E}_x}}}{{\partial y}} + {\boldsymbol E}_y^ \ast \frac{{\partial {{\boldsymbol E}_y}}}{{\partial y}}$$

Figure 2 shows the distributions of the SOP, the SAM, and the OAM at the focal plane of the GRIN fiber for a left-hand circularly polarized vortex beam. The horizontal and vertical coordinates axes of Fig. 2–5 range from -12.5µm-12.5µm. The polarization distribution of the left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber is shown by a black line and indicates that no changes have occurred, which is different from the case of the focused cylindrical vector vortex beam [20]. The left and right circularly polarized components of the focused left-hand circularly polarized vortex beam are shown in Fig. 2(b) and (c), respectively. The images show that the right circularly polarized component does not exist for the focused left-hand circularly polarized vortex beam. The SAM distributions of the focused left-hand circularly polarized vortex beam, which represent the differences between the left and right circularly polarized components, are shown in Fig. 2(d) and the corresponding OAM distributions are shown in Fig. 2(e). When m = 0, the SAM exists and the OAM does not exist. When m = ±1 and ±2, the distribution of the SAM is the same on both sides, but the distributions of the OAM on the two sides are opposites when the topological charge values are equal but the rotation directions are opposite.

 figure: Fig. 2.

Fig. 2. The distributions of (a) the SOP, (b) the LCP component, (c) the RCP component, (d) the SAM, and (e) the OAM of the left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber.

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 figure: Fig. 3.

Fig. 3. The fields of (a) the SOP, (b) the LCP component, (c) the RCP component, (d) the SAM, and (e) the OAM of the right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber.

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 figure: Fig. 4.

Fig. 4. The fields of (a), (d) the SOP, (b), (e) the SAM, and (c), (f) the OAM of the partially coherent left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber. (a), (b), and (c): σ = 100 m. (d), (e), and (f) σ = 15 µm.

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 figure: Fig. 5.

Fig. 5. The fields of (a), (d) the SOP, (b), (e) the SAM, and (c), (f) the OAM of the partially coherent right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber. (a), (b), and (c): σ = 100 m. (d), (e), and (f) σ = 15 µm.

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The distributions of the SOP, the SAM, and the OAM of a right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber are shown in Fig. 3. Similarly, the polarization distribution of the right-hand circularly polarized vortex beam remains circularly polarized at the focal plane of the GRIN fiber. The left circularly polarized component is zero in this case, and the right circularly polarized component is same as the light field distribution of the focused right-hand circularly polarized vortex beam (the left and right circularly polarized components are normalized by dividing their intensities by the maximum light intensity). Figure 3(d) shows the SAM distribution of the right-hand circularly polarized vortex beam. The SAM of the circularly polarized vortex beam at the focal plane of the GRIN fiber, which has the opposite polarization direction in this case, is shown to have the opposite distribution by comparing Fig. 3(d) with Fig. 2(d), and this is related to the beam polarization alone. The distribution of the OAM of the circularly polarized vortex beam at the focal plane of the GRIN fiber is related to the topological charges alone. The SOIs observed for the circularly polarized vortex beam have not occurred at the focal plane of the GRIN fiber, indicating that the SAM and the OAM are independent of each other during transmission in GRIN fibers. This is different to the phenomenon observed for the circularly polarized vortex beam, which is tightly focused [15]. The self-focusing property of GRIN fibers is unable to realize a tightly focused beam in the paraxial propagation.

Partially coherent vortex beam have added a new degree of freedom, i.e., coherence, when compared with perfectly coherent vortex beam [28]. By modulating the coherence, which is the new degree of freedom, and the other degrees of freedom of the optical field simultaneously, the partially coherent vortex beam show unique properties in terms of beam shaping [29], polarization state conversion [30], and coherent singularities [31].

The partially coherent light at the source plane is expressed as a cross-spectral density matrix, which is shown in Eq. (8) [32], where matrix elements Wij(s1,s2,0)=‹Ei∗ (s1, 0)Ej(s2, 0)› (i, j = x, y). s1 and s2 denote the position vectors at the source plane. The elements of the cross-spectral density matrix of the partially coherent circularly polarized vortex beam at the source plane can be derived using the expression from Eq. (1) for a circularly polarized vortex beam and are expressed as shown in Eq. (9). Here, $\sigma$ is the correlation length of the partially coherent circularly polarized vortex beam. Similarly, the partially circularly polarized vortex beam which topological charge m = 0 is partially circularly polarized beam.

$$\mathop {\boldsymbol W}\limits^ \leftrightarrow ({{\boldsymbol s}_1},{{\boldsymbol s}_2},0) = \left[ {\begin{array}{cc} {{W_{xx}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}&{{W_{xy}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}\\ {{W_{yx}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)}&{{W_{yy}}({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)} \end{array}} \right]$$
$${W_{xx}}({{s_1},{s_2},0} )= {[{{s_{1x}} + \textrm{i}{\mathop{\rm sgn}} (m ){s_{1y}}} ]^{|m |}}{[{{s_{2x}}\textrm{ - i}{\mathop{\rm sgn}} (m ){s_{2y}}} ]^{|m |}}\exp \left( { - \frac{{s_1^2 + s_2^2}}{{{w^2}}}} \right)\exp \left[ { - \frac{{{{({{s_1} - {s_2}} )}^2}}}{{2{\sigma^2}}}} \right]$$
$${W_{xy}}({{s_1},{s_2},0} )={-} \textrm{i}{[{{s_{1x}} + \textrm{i}{\mathop{\rm sgn}} (m ){s_{1y}}} ]^{|m |}}{[{{s_{2x}}\textrm{ - i}{\mathop{\rm sgn}} (m ){s_{2y}}} ]^{|m |}}\exp \left( { - \frac{{s_1^2 + s_2^2}}{{{w^2}}}} \right)\exp \left[ { - \frac{{{{({{s_1} - {s_2}} )}^2}}}{{2{\sigma^2}}}} \right]$$
$${W_{yx}}({{s_1},{s_2},0} )= \textrm{i}{[{{s_{1x}} + \textrm{i}{\mathop{\rm sgn}} (m ){s_{1y}}} ]^{|m |}}{[{{s_{2x}}\textrm{ - i}{\mathop{\rm sgn}} (m ){s_{2y}}} ]^{|m |}}\exp \left( { - \frac{{s_1^2 + s_2^2}}{{{w^2}}}} \right)\exp \left[ { - \frac{{{{({{s_1} - {s_2}} )}^2}}}{{2{\sigma^2}}}} \right]$$
$${W_{yy}}({{s_1},{s_2},0} )={-} {\textrm{i}^2}{[{{s_{1x}} + \textrm{i}{\mathop{\rm sgn}} (m ){s_{1y}}} ]^{|m |}}{[{{s_{2x}}\textrm{ - i}{\mathop{\rm sgn}} (m ){s_{2y}}} ]^{|m |}}\exp \left( { - \frac{{s_1^2 + s_2^2}}{{{w^2}}}} \right)\exp \left[ { - \frac{{{{({{s_1} - {s_2}} )}^2}}}{{2{\sigma^2}}}} \right]$$

Based on the generalized Huygens–Fresnel principle, which characterizes the transmission of partially coherent light in a medium, as given by Eq. (10), and the ABCD matrix of a radial GRIN fiber, the cross-spectral density matrix of a partially coherent circularly polarized vortex beam when transmitted in the GRIN fiber system is expressed as Eq. (11). The field distribution of the left-hand partially coherent circularly polarized vortex beam can then be mapped using Eq. (11).

$$\begin{aligned} {W_{ij}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z) & = {\left( {\frac{k}{{2\pi B}}} \right)^2}\int {\int {\int {\int {{W_{ij}}} } } } ({{\boldsymbol s}_1},{{\boldsymbol s}_2},0)d{s_{1x}}d{s_{1y}}d{s_{2x}}d{s_{2y}} \\ & \times \exp \left. {\left\{ { - \frac{{\textrm{i}k}}{{2B}}[{A({\boldsymbol s}_1^2 - {\boldsymbol s}_2^2) - 2({{{\boldsymbol s}_1}{{\boldsymbol \rho }_1} - {{\boldsymbol s}_2}{{\boldsymbol \rho }_2}} )+ D({\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2)} ]} \right.} \right\} \end{aligned}$$
$${\left[ {\begin{array}{cc} {{W_{xx}}}&{{W_{xy}}}\\ {{W_{yx}}}&{{W_{yy}}} \end{array}} \right]_{({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z)}} = \left[ {\begin{array}{cc} {G{G_1}{H_{{\eta_1}}}({l_1}){H_{{\eta_2}}}({l_2})}&{( - \textrm{i})G{G_1}{H_{{\eta_1}}}({l_1}){H_{{\eta_2}}}({l_2})}\\ {\textrm{i}G{G_1}{H_{{\eta_1}}}({l_1}){H_{{\eta_2}}}({l_2})}&{\textrm{(} - {\textrm{i}^2})G{G_1}{H_{{\eta_1}}}({l_1}){H_{{\eta_2}}}({l_2})} \end{array}} \right]$$
$$\begin{aligned} G & = {\left( {\frac{k}{B}} \right)^2}\exp \left( { - \frac{{\textrm{i}kD}}{{2B}}({{\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2} )} \right)\exp \left[ {{{\left( {\frac{{\textrm{i}k{{\boldsymbol \rho }_{1y}}}}{{2B\sqrt {{M_2}} }} - \frac{{\textrm{i}k{{\boldsymbol \rho }_{2y}}}}{{4B{\sigma^2}{M_1}\sqrt {{M_2}} }}} \right)}^2}} \right]\\ & \times \exp \left[ {{{\left( {\frac{{\textrm{i}k{{\boldsymbol \rho }_{1x}}}}{{2B\sqrt {{M_2}} }} - \frac{{\textrm{i}k{{\boldsymbol \rho }_{2x}}}}{{4B{\sigma^2}{M_1}\sqrt {{M_2}} }}} \right)}^2}} \right]\exp \left( { - \frac{{{k^2}{\boldsymbol \rho }_2^2}}{{4{M_1}{B^2}}}} \right){2^{ - \frac{{5|m |}}{2} - 2}}{M_1}^{ - \frac{{|m |- 2}}{2}}{\textrm{i}^{ - 2|m |}}\\ & \times \sum\limits_{{d_1} = 0}^{|m |} {\sum\limits_{{d_2} = 0}^{|m |} {\frac{{|m |!{\textrm{i}^{{d_1}}}}}{{{d_1}!({|m |- {d_1}} )!}}\frac{{|m |!{\textrm{i}^{{d_2}}}}}{{{d_2}!({|m |- {d_2}} )!}}} } {\mathop{\rm sgn}} {(m)^{{d_1}}}{[ - {\mathop{\rm sgn}} (m)]^{{d_2}}} \end{aligned}$$
$${G_1} = \sum\limits_{{r_1} = 0}^{|m |- {d_2}} {\sum\limits_{{r_2} = 0}^{{d_2}} {\left( {\begin{array}{c} {|m |- {d_2}}\\ {{r_1}} \end{array}} \right)\left( {\begin{array}{c} {{d_2}}\\ {{r_2}} \end{array}} \right)} \textrm{ }{H_{{r_1}}}({{k_1}} ){H_{{r_2}}}({{k_2}} )\sum\limits_{{n_1} = 0}^{\frac{{|m |- {d_2} - {r_1}}}{2}} {\sum\limits_{{n_2} = 0}^{\frac{{{d_2} - {r_2}}}{2}} {{{( - 1)}^{{n_1} + {n_2}}}} } } \textrm{ }q{(j )^{|m |- {r_1} - {r_2} - 2{n_1} - 2{n_2}}}M_2^p$$
$$\begin{array}{l} {\eta _1} = 2|m |- {d_1} - {d_2} - {r_1} - 2{n_1}\textrm{ }{\eta _2} = {d_1} + {d_2} - {r_2} - 2{n_2}\\ {l_1} = \frac{{k{{\boldsymbol \rho }_{2x}}}}{{4B{\sigma ^2}{M_1}\sqrt {{M_2}} }} - \frac{{k{{\boldsymbol \rho }_{1x}}}}{{2B\sqrt {{M_2}} }}\textrm{ }{l_2} = \frac{{k{{\boldsymbol \rho }_{2y}}}}{{4B{\sigma ^2}{M_1}\sqrt {{M_2}} }} - \frac{{k{{\boldsymbol \rho }_{1y}}}}{{2B\sqrt {{M_2}} }} \end{array}$$
$$\begin{array}{l} {M_1} = \frac{1}{{w_{}^2}} + \frac{1}{{2\sigma _{}^2}} - \frac{{ikA}}{{2B}}\textrm{ }{M_2} = \frac{1}{{w_{}^2}} + \frac{1}{{2\sigma _{}^2}} + \frac{{ikA}}{{2B}} - \frac{1}{{4{M_1}\sigma _{}^4}}\textrm{ }\\ {k_1} = \frac{{k{{\boldsymbol \rho }_{2x}}}}{{\sqrt {2{M_1}} B}}\textrm{ }{k_2} = \frac{{k{{\boldsymbol \rho }_{2y}}}}{{\sqrt {2{M_1}} B}}\textrm{ }q = \frac{{({|m |- {d_2} - {r_1}} )!}}{{{n_1}!({|m |- {d_2} - {r_1} - 2{n_1}} )!}}\frac{{({{d_2} - {r_2}} )!}}{{{n_2}!({{d_2} - {r_2} - 2{n_2}} )!}}\textrm{ }\\ j = \frac{1}{{\sqrt {2{M_1}} {\sigma ^2}}}\textrm{ }p ={-} \frac{{2|m |- {r_1} - {r_2} - 2{n_1} - 2{n_2} + 2}}{2} \end{array}$$

Here, $k = 2\pi /\lambda$ is the wavenumber. $H({\cdot} )$ is the Hermite polynomial and $\left( {\begin{array}{c} .\\ . \end{array}} \right)$ represents a binomial coefficient. ${{\boldsymbol \rho }_1} = ({{\rho_{1x}},{\rho_{1y}}} )$ and ${{\boldsymbol \rho }_2} = ({{\rho_{2x}},{\rho_{2y}}} )$ are two different positions of the x-y plane in the GRIN fiber system.

The relationships between the distributions of the field, the SAM, and the OAM of the partially coherent beam and the elements of the cross-spectral density matrix are shown in Eq. (12), Eq. (13), and Eq. (14), respectively. Similarly, Sz and Oz are equal to the total SAM and total OAM during paraxial transmission because the longitudinal component (Ez) of the electric field will not be generated during the paraxial transmission. The distributions of the field, the SAM, and the OAM of the partially coherent circularly polarized vortex beam at the focal plane of the GRIN fiber can then be mapped using Eq. (12), Eq. (13), and Eq. (14), respectively [33–35].

$$I = {\textrm{Re}} [{{W_{xx}} + {W_{yy}}} ]$$
$${S_z} \simeq {\mathop{\rm Im}\nolimits} [{{W_{yx}} - {W_{xy}}} ]$$
$${O_z} \simeq {\mathop{\rm Im}\nolimits} [{\rho _{1y}}{\partial _{{\rho _{1x}}}}{W_{yy}} - {\rho _{1x}}{\partial _{{\rho _{1y}}}}{W_{xx}} + {\rho _{2x}}{\partial _{{\rho _{2y}}}}{W_{yy}} - {\rho _{2y}}{\partial _{{\rho _{2x}}}}{W_{xx}}]$$

According to the polarization relationship between the left-hand circularly polarized beam and the right-hand circularly polarized beam, and using the cross-spectral density matrix of the partially coherent beam, the cross-spectral density matrix between the partially coherent right-hand circularly polarized vortex beam and the partially coherent left-hand circularly polarized vortex beam can be defined as:

$${\left[ {\begin{array}{cc} {{W_{xx}}}&{{W_{xy}}}\\ {{W_{yx}}}&{{W_{yy}}} \end{array}} \right]_{\textrm{RHC}}} = {\left[ {\begin{array}{cc} {{W_{xx}}}&{ - {W_{xy}}}\\ { - {W_{yx}}}&{{W_{yy}}} \end{array}} \right]_{\textrm{LHC}}}$$

The distributions of the SOP, the SAM, and the OAM of the partially coherent left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber were characterized and the results are shown in Fig. 4. Figure 4(a), (b), and (c) and Fig. 4(d), (e), and (f) show the field distributions of the SOP, the SAM, and the OAM of the partially coherent left-hand circularly polarized vortex beam in the cases where the correlation length $\sigma$=100 m and $\sigma$=15 µm, respectively. The distributions of the SOP, the SAM, and the OAM of the partially coherent left-hand circularly polarized vortex beam when the correlation length $\sigma$=100 m show the same distributions as the perfectly left-hand circularly polarized vortex beam because the partially coherent light is perfectly coherent light when σ = 100 m. Therefore, Eq. (12), Eq. (13), and Eq. (14), which characterize the fields for the SOP, the SAM, and the OAM of partially coherent light, have been proven to be correct based on the field distributions of the SOP, the SAM, and the OAM of the partially coherent left-hand circularly polarized vortex beam with σ = 100 m. The properties of the partially coherent light are also presented when σ = 15 µm. It can be seen that the polarization distribution has not changed and that the beam is still circularly polarized. The light-field distribution, the SAM, and the OAM have changed with the correlation length, but the pattern remains the same as that for perfectly coherent light.

Figure 5 shows the distributions of the SOP, the SAM, and the OAM of the partially coherent right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber. The hollow part which the light field intensity is zero of partially coherent circularly polarized light is reduced gradually as the coherence length decreases from both Fig. 4 and Fig. 5. The SAM and the OAM of the partially coherent circularly polarized light are dependent on the polarization and the topological charge m, respectively. For the partially coherent circularly polarized light, the SOIs also does not occur during paraxial transmission in the GRIN fiber. The SOIs of the partially coherent circularly polarized light during paraxial transmission is relevant to the coherence. The correctness of Eq. (13) and Eq. (14) is further demonstrated via characterization of both the SAM and the OAM of the partially coherent circularly polarized light.

3. Conclusion

The SOIs of circularly polarized vortex beam and partially coherent circularly polarized vortex beam during paraxial transmission in a GRIN fiber have been researched based on the generalized Huygens–Fresnel principle and the ABCD matrix for a radial GRIN fiber. The polarization of the circularly polarized vortex beam remains unchanged. The SAM and the OAM are not coupled and converted with each other at the focal plane of the GRIN fiber. The SOIs of the circularly polarized vortex beam and the partially coherent circularly polarized vortex beam do not occur during paraxial transmission, and this is different from the case in the tightly focused condition. By analyzing the transmission characteristics and the SOIs of the circularly polarized vortex beam and the partially coherent circularly polarized vortex beam in the GRIN fiber, research value is realized in the fields of optical communications and fiber-based manipulation.

Funding

Fundamental Research Project of Shanxi Province (202103021223299); Fundamental Research Project of Shanxi Province (202203021211192); Taiyuan University of Science and Technology Scientific Research Initial Funding (20222009); Fundamental Research Project of Shanxi Province (202103021223271).

Acknowledgments

X. Yin acknowledges the support given by the Fundamental Research Project of Shanxi Province (202103021223271) and the 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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Data availability

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

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Figures (5)
Fig. 1.
Fig. 1. Axial light field distribution of a circularly polarized vortex beam in a 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 performed by dividing the light intensity in each case by its maximum, and the corresponding values of the different graphs are not comparable.)

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Fig. 2.
Fig. 2. The distributions of (a) the SOP, (b) the LCP component, (c) the RCP component, (d) the SAM, and (e) the OAM of the left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber.

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Fig. 3.
Fig. 3. The fields of (a) the SOP, (b) the LCP component, (c) the RCP component, (d) the SAM, and (e) the OAM of the right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber.

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Fig. 4.
Fig. 4. The fields of (a), (d) the SOP, (b), (e) the SAM, and (c), (f) the OAM of the partially coherent left-hand circularly polarized vortex beam at the focal plane of the GRIN fiber. (a), (b), and (c): σ = 100 m. (d), (e), and (f) σ = 15 µm.

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Fig. 5.
Fig. 5. The fields of (a), (d) the SOP, (b), (e) the SAM, and (c), (f) the OAM of the partially coherent right-hand circularly polarized vortex beam at the focal plane of the GRIN fiber. (a), (b), and (c): σ = 100 m. (d), (e), and (f) σ = 15 µm.

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Equations (26)

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E ( x 0 , y 0 , 0 ) = [ x 0 + i sgn ( m ) y 0 ] | m | exp ⁡ ( − x 0 2 + y 0 2 w 2 ) ( e x ± i e y )
E ( x , y , z ) = i k 2 π B ∫ ∫ E ( x 0 , y 0 , 0 ) × exp ⁡ { − i k 2 B [ A ( x 0 2 + y 0 2 ) − 2 ( x 0 x + y 0 y ) + D ( x 2 + y 2 ) ] } d x 0 d y 0
E x = k B 2 − | m | − 1 i − | m | + 1 M − | m | 2 − 1 exp ⁡ [ − i k D 2 B ( x 2 + y 2 ) ] exp ⁡ [ − k 2 ( x 2 + y 2 ) 4 B 2 M ] × ∑ s = 0 m | m | ! i s s ! ( | m | − s ) ! sgn ( m ) s H | m | − s ( − k x 2 B M ) H s ( − k y 2 B M ) E y = k B 2 − | m | − 1 i − | m | + 2 M − | m | 2 − 1 exp ⁡ [ − i k D 2 B ( x 2 + y 2 ) ] exp ⁡ [ − k 2 ( x 2 + y 2 ) 4 B 2 M ] × ∑ s = 0 m | m | ! i s s ! ( | m | − s ) ! sgn ( m ) s H | m | − s ( − k x 2 B M ) H s ( − k y 2 B M )
M = 1 w 2 + i k A 2 B
H n ( x ) = n ! ∑ s = 0 n 2 ( − 1 ) s s ! ( n − 2 s ) ! ( 2 x ) n − 2 s
[ A B C D ] = [ cos ⁡ ( β z ) sin ⁡ ( β z ) n 0 β − n 0 β sin ⁡ ( β z ) cos ⁡ ( β z ) ]
S z ≃ Im [ E x ∗ E y − E y ∗ E x ]
S z ≃ I LCP − I RCP
O z ≃ Im [ x T y − y T x ]
T x = E x ∗ ∂ E x ∂ x + E y ∗ ∂ E y ∂ x
T y = E x ∗ ∂ E x ∂ y + E y ∗ ∂ E y ∂ y
W ↔ ⁡ ( s 1 , s 2 , 0 ) = [ W x x ( s 1 , s 2 , 0 ) W x y ( s 1 , s 2 , 0 ) W y x ( s 1 , s 2 , 0 ) W y y ( s 1 , s 2 , 0 ) ]
W x x ( s 1 , s 2 , 0 ) = [ s 1 x + i sgn ( m ) s 1 y ] | m | [ s 2 x  - i sgn ( m ) s 2 y ] | m | exp ⁡ ( − s 1 2 + s 2 2 w 2 ) exp ⁡ [ − ( s 1 − s 2 ) 2 2 σ 2 ]
W x y ( s 1 , s 2 , 0 ) = − i [ s 1 x + i sgn ( m ) s 1 y ] | m | [ s 2 x  - i sgn ( m ) s 2 y ] | m | exp ⁡ ( − s 1 2 + s 2 2 w 2 ) exp ⁡ [ − ( s 1 − s 2 ) 2 2 σ 2 ]
W y x ( s 1 , s 2 , 0 ) = i [ s 1 x + i sgn ( m ) s 1 y ] | m | [ s 2 x  - i sgn ( m ) s 2 y ] | m | exp ⁡ ( − s 1 2 + s 2 2 w 2 ) exp ⁡ [ − ( s 1 − s 2 ) 2 2 σ 2 ]
W y y ( s 1 , s 2 , 0 ) = − i 2 [ s 1 x + i sgn ( m ) s 1 y ] | m | [ s 2 x  - i sgn ( m ) s 2 y ] | m | exp ⁡ ( − s 1 2 + s 2 2 w 2 ) exp ⁡ [ − ( s 1 − s 2 ) 2 2 σ 2 ]
W i j ( ρ 1 , ρ 2 , z ) = ( k 2 π B ) 2 ∫ ∫ ∫ ∫ W i j ( s 1 , s 2 , 0 ) d s 1 x d s 1 y d s 2 x d s 2 y × exp ⁡ { − i k 2 B [ A ( s 1 2 − s 2 2 ) − 2 ( s 1 ρ 1 − s 2 ρ 2 ) + D ( ρ 1 2 − ρ 2 2 ) ] }
[ W x x W x y W y x W y y ] ( ρ 1 , ρ 2 , z ) = [ G G 1 H η 1 ( l 1 ) H η 2 ( l 2 ) ( − i ) G G 1 H η 1 ( l 1 ) H η 2 ( l 2 ) i G G 1 H η 1 ( l 1 ) H η 2 ( l 2 ) ( − i 2 ) G G 1 H η 1 ( l 1 ) H η 2 ( l 2 ) ]
G = ( k B ) 2 exp ⁡ ( − i k D 2 B ( ρ 1 2 − ρ 2 2 ) ) exp ⁡ [ ( i k ρ 1 y 2 B M 2 − i k ρ 2 y 4 B σ 2 M 1 M 2 ) 2 ] × exp ⁡ [ ( i k ρ 1 x 2 B M 2 − i k ρ 2 x 4 B σ 2 M 1 M 2 ) 2 ] exp ⁡ ( − k 2 ρ 2 2 4 M 1 B 2 ) 2 − 5 | m | 2 − 2 M 1 − | m | − 2 2 i − 2 | m | × ∑ d 1 = 0 | m | ∑ d 2 = 0 | m | | m | ! i d 1 d 1 ! ( | m | − d 1 ) ! | m | ! i d 2 d 2 ! ( | m | − d 2 ) ! sgn ( m ) d 1 [ − sgn ( m ) ] d 2
G 1 = ∑ r 1 = 0 | m | − d 2 ∑ r 2 = 0 d 2 ( | m | − d 2 r 1 ) ( d 2 r 2 )   H r 1 ( k 1 ) H r 2 ( k 2 ) ∑ n 1 = 0 | m | − d 2 − r 1 2 ∑ n 2 = 0 d 2 − r 2 2 ( − 1 ) n 1 + n 2   q ( j ) | m | − r 1 − r 2 − 2 n 1 − 2 n 2 M 2 p
η 1 = 2 | m | − d 1 − d 2 − r 1 − 2 n 1   η 2 = d 1 + d 2 − r 2 − 2 n 2 l 1 = k ρ 2 x 4 B σ 2 M 1 M 2 − k ρ 1 x 2 B M 2   l 2 = k ρ 2 y 4 B σ 2 M 1 M 2 − k ρ 1 y 2 B M 2
M 1 = 1 w 2 + 1 2 σ 2 − i k A 2 B   M 2 = 1 w 2 + 1 2 σ 2 + i k A 2 B − 1 4 M 1 σ 4   k 1 = k ρ 2 x 2 M 1 B   k 2 = k ρ 2 y 2 M 1 B   q = ( | m | − d 2 − r 1 ) ! n 1 ! ( | m | − d 2 − r 1 − 2 n 1 ) ! ( d 2 − r 2 ) ! n 2 ! ( d 2 − r 2 − 2 n 2 ) !   j = 1 2 M 1 σ 2   p = − 2 | m | − r 1 − r 2 − 2 n 1 − 2 n 2 + 2 2
I = Re [ W x x + W y y ]
S z ≃ Im [ W y x − W x y ]
O z ≃ Im [ ρ 1 y ∂ ρ 1 x W y y − ρ 1 x ∂ ρ 1 y W x x + ρ 2 x ∂ ρ 2 y W y y − ρ 2 y ∂ ρ 2 x W x x ]
[ W x x W x y W y x W y y ] RHC = [ W x x − W x y − W y x W y y ] LHC
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