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Abstract
Optical skyrmions provide a new idea and approach to endow structured light and spatial-temporal light with topological properties. In this paper, the longitudinal and transversal components of the focused light field are decoupled and can be controlled independently by modulating the polarizations and phases of two pairs of counter-propagating incident cylindrical vector beams under 4π focal configuration. Under this condition, Néel-type and Bloch-type optical skyrmions formed by electromagnetic field vectors are constructed in the focal plane. When one pair of the incident beams is radially polarized with a phase difference of π and the other pair is radially polarized in phase, a Néel-type optical skyrmion formed by electric field vectors can be constructed in the focal plane of the 4π focal system. The corresponding focal magnetic field is purely azimuthally polarized. If we substitute the other pair of the incident beams with azimuthally polarized beams, Bloch-type optical skyrmions formed by electromagnetic field vectors can be constructed in the focal plane, where the one formed by magnetic field vectors has a π/2 phase lead compared with the that formed by electric field vectors. This work provides a theoretical basis for further research on the interactions between matter and optical skyrmions formed by electromagnetic field vectors at micro and nano scales in free space.
Keywords
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1. 引 言
4π聚焦系统为解决此问题提供了可行的技术手段。4π聚焦系统由两个共焦且焦距相等的具有相同高数值孔径的透镜组成，常应用于对称性结构光场的构造，例如光球、光链、光学暗通道等[24-29]。基于Richards-Wolf矢量衍射理论[30]，在4π聚焦系统中，通过控制聚焦光场各偏振分量的相长相消干涉，可以在焦场中心产生高纯度均匀偏振且可被独立调控的横向和纵向场分量，即实现聚焦光场偏振分量间的解耦合。基于此原理，目前人们已实现基于反法拉第效应的高纯度均匀面内光致磁化场[31]以及亚波长空间尺度的三维任意取向光场能流分布[32]。
在本文中，自由空间中两对相向传播的柱矢量入射光在4π聚焦系统的焦面上相干产生了电磁矢量光学斯格明子。当其中一对相向传播的入射光是相位差为π的径向偏振光，另一对为同相的径向偏振光时，系统焦平面上将产生Néel型电场斯格明子分布，而磁场分布由于缺少纵向场分量，仅表现为角向分布的纯横向场。进一步，将另一对相向传播的入射光替换为对称分布的角向偏振光，此时系统焦面上将同时产生Bloch型的电场矢量斯格明子与相位超前π/2的磁场矢量斯格明子。本工作将为进一步研究自由空间中微纳尺度电磁矢量光学斯格明子与物质的相互作用提供理论基础。
光学斯格明子最为重要的潜在应用之一是实现微纳尺度的光与物质相互作用并产生新颖的物理现象。然而，在传统的自由空间紧聚焦系统[21-23]中，聚焦光场的各偏振分量很难实现完全的同相或反相，其纵向分量往往与某一横向分量或全部横向分量具有π/2的相位差。因此，如何在自由空间中产生微纳尺度的电磁矢量光学斯格明子仍然是本领域亟待解决的问题。
光学斯格明子是近年来随着拓扑光学的发展而快速兴起的研究领域，为实现时空光、结构光和光与物质相互作用的拓扑属性提供了新的方法与思路[1-3]，相关的应用范围涵盖了自旋光学、成像和计量学、光力、结构光、拓扑和量子技术等[4]。斯格明子最早由英国物理学家Tony Skyrme提出[5]，是一种已在量子领域、固体物理学和磁性材料中研究证实的具有拓扑稳定性的准粒子。磁性斯格明子在过去几十年中得到了广泛的研究，是实现高密度磁性信息存储、传输和自旋电子器件的有效途径[6-12]，而光学斯格明子直到最近五年才进入人们的视野。光学斯格明子最初是通过入射光场的倏逝波在金属表面激发的表面等离子体激元的干涉来实现。2018年，Shai Tsesses等人利用对称的六边形谐振腔干涉将斯格明子扩展到了电磁场领域，引发了跨不同光谱范围的拓展研究[13]。与此同时，Luping Du等人基于紧聚焦矢量光束自旋角动量与轨道角动量的耦合，利用紧聚焦柱矢量涡旋光束在金属表面的倏逝波实现了自旋矢量光学斯格明子，其光场局部自旋取向在λ/60的空间范围内实现了反转，这种深亚波长范围内的自旋反转可为高分辨率成像、量子技术和光学存储等领域带来深远的影响[14]。随后，不同类型的光学斯格明子也被相继提出[15-20]。
2. 理论模型
$$ {{\boldsymbol{E}}_{3{\text{f}}}}(r,\varphi , {\textit{z}}) = {\rm{i}}{{\mu}_0 }c{{\boldsymbol{H}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}), $$ 由E1、E1′、E3和E3′产生的总聚焦电场以及由H1、H1′、H3和H3′产生的总聚焦磁场可以分别表示为
$$\begin{split} {{\boldsymbol{E}}_{\text{I}}}(r,\varphi , {\textit{z}}) =& {A}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} {({W_1} - {\rm{i}}{W_2}){{\rm{J}}_1}(kr\sin \theta )\cos \theta } \\ 0 \\ {({W_1} + {\rm{i}}{W_2}){{\rm{J}}_0}(kr\sin \theta )\sin \theta } \end{array}} \right]} \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \\& \cdot \sqrt {\cos \theta } \sin \theta d\theta , \end{split} $$ 在柱坐标系下分析斯格明子矢量场，可以将其分解为径向场分量、角向场分量与纵向场分量。本文在4π聚焦系统中，通过灵活调控相向传播的径向偏振光与角向偏振光，在焦面上构造了Néel型和Bloch型的电磁矢量光学斯格明子。图1为两对相向传播的柱矢量光束通过4π系统聚焦形成电磁矢量光学斯格明子的示意图。当其中一对相向传播的入射光是相位差为π的径向偏振光E1和E1′，且它们分别经过具有π/2相位延迟的相位板PP1和PP2，另一对相向传播的入射光为同相的径向偏振光E2和E2′时，4π聚焦系统的焦面上将构造出Néel型电场矢量斯格明子。在此基础上，若将E2和E2′替换为对称分布的角向偏振光E3和E3′，焦面上将构造出Bloch型电场矢量斯格明子。焦点放大图表示焦面上相应的斯格明子结构。如图1所示，E1和E1′的振动方向相反，E2和E2′以及E3和E3′的振动方向相同，即焦面上电场的纵向分量由E1和E1′独立调控，横向分量由E2和E2′或E3和E3′独立调控。这说明4π聚焦系统的引入使聚焦光场在焦面上实现了偏振分量间的解耦合。此外，在入射光场中，电场矢量、磁场矢量和它们的传播方向，即入射波矢方向，满足右手螺旋关系，即入射的径向偏振光携带了角向偏振的磁场，入射的角向偏振光携带有径向偏振的磁场。因此我们将同时分析焦场内的磁场矢量分布。
其中，
$$ {{\boldsymbol{H}}_{2{\text{f}}}}(r,\varphi , {\textit{z}}) = {\rm{i}}{{\boldsymbol{H}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}), $$ $$\begin{split} {{\boldsymbol{E}}_{{\text{II}}}}(r,\varphi , {\textit{z}}) = & {A}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} { - {\rm{i}}{W_2}{{\rm{J}}_1}(kr\sin \theta )\cos \theta } \\ {{W_1}{{\rm{J}}_1}(kr\sin \theta )} \\ {{W_1}{{\rm{J}}_0}(kr\sin \theta )\sin \theta } \end{array}} \right] } \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \cdot \sqrt {\cos \theta } \sin \theta {\rm{d}}\theta , \end{split} $$ $$ \left\{ {\begin{array}{*{20}{c}} {{W_1} = {{\rm{e}}^{{\rm{i}}k {\textit{z}}\cos \theta }} + {{\rm{e}}^{ - {\rm{i}}k {\textit{z}}\cos \theta }}} \\ {{W_2} = {{\rm{e}}^{{\rm{i}}k {\textit{z}}\cos \theta }} - {{\rm{e}}^{ - {\rm{i}}k {\textit{z}}\cos \theta }}} \end{array}} \right.. $$ 其中，(r, φ, z)为聚焦空间的柱坐标，μ0为真空磁导率，c为真空中光速，θ为聚焦会聚角，最大聚焦会聚角θmax = arcsin(NA)，NA为聚焦系统的数值孔径，k为真空中波数，er、eφ和ez为聚焦空间的柱单位矢量，J0和J1分别表示0阶和1阶第一类贝塞尔函数。假设两对相向传播的柱矢量入射光具有均匀且相等的场强分布，则A为常数且入射光场E2和E3及其相应的磁场H2和H3的聚焦场可以分别表示为
根据Richards-Wolf矢量衍射理论[33]，图1中入射光场E1及其相应磁场H1的聚焦场可以分别表示为
$$ {{\boldsymbol{H}}_{3{\text{f}}}}(r,\varphi , {\textit{z}}) = - {\rm{i}}/({{\mu }_0}{c}){{\boldsymbol{E}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}). $$ ![Figure 1. Schematic of the generation of optical skyrmions formed by electromagnetic field vectors in the focal region of two pairs of counter-propagating cylindrical vector beams under 4π focal condition. PP1, PP2: homogeneous phase plate with a phase delay of π/2; BS1, BS2: beam splitter; L1, L2: objectives]()
Schematic of the generation of optical skyrmions formed by electromagnetic field vectors in the focal region of two pairs of counter-propagating cylindrical vector beams under 4π focal condition. PP1, PP2: homogeneous phase plate with a phase delay of π/2; BS1, BS2: beam splitter; L1, L2: objectives
基于4π聚焦系统的场叠加特性[28, 31, 34]，由E1、E1′、E2和E2′产生的总聚焦电场以及由H1、H1′、H2和H2′产生的总聚焦磁场可以分别表示为
$$ {{\boldsymbol{E}}_{2{\text{f}}}}(r,\varphi , {\textit{z}}) = {\rm{i}}{{\boldsymbol{E}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}), $$ $$ {{\boldsymbol{H}}_{{\text{II}}}}(r,\varphi , {\textit{z}}) = \frac{{A}}{{{{\mu}_0}{c}}}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} { - {W_2}{{\rm{J}}_1}(kr\sin \theta )\cos \theta } \\ { - {\rm{i}}{W_1}{{\rm{J}}_1}(kr\sin \theta )} \\ { - {\rm{i}}{W_1}{{\rm{J}}_0}(kr\sin \theta )\sin \theta } \end{array}} \right]} \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \cdot \sqrt {\cos \theta } \sin \theta {\rm{d}}\theta . $$ $$ \begin{split} {{\boldsymbol{H}}_{\text{I}}}(r,\varphi , {\textit{z}}) =& \frac{{A}}{{{{{\mu_0 }}{{}}}{\text{c}}}}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} 0 \\ {({W_2} - {\rm{i}}{W_1}){{\rm{J}}_1}(kr\sin \theta )} \\ 0 \end{array}} \right]} \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \\& \cdot \sqrt {\cos \theta } \sin \theta {\rm{d}}\theta . \end{split}$$ $$\begin{split} \\ {{\boldsymbol{H}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}) =& \frac{{A}}{{{\mu _0}{c}}}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} 0 \\ { - {\rm{i}}{{\rm{J}}_1}(kr\sin \theta )} \\ 0 \end{array}} \right]} \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \\& \cdot {{\rm{e}}^{{\rm{i}}k {\textit{z}}\cos \theta }}\sqrt {\cos \theta } \sin \theta {\rm{d}}\theta . \end{split}$$ $$ \begin{split} {{\boldsymbol{E}}_{1{\text{f}}}}(r,\varphi , {\textit{z}}) = & {A}\int\limits_0^{{\theta _{\max }}} {\left[ {\begin{array}{*{20}{c}} { - {\rm{i}}{{\rm{J}}_1}(kr\sin \theta )\cos \theta } \\ 0 \\ {{{\rm{J}}_0}(kr\sin \theta )\sin \theta } \end{array}} \right]} \begin{array}{*{20}{c}} {{{\boldsymbol{e}}_r}} \\ {{{\boldsymbol{e}}_\varphi }} \\ {{{\boldsymbol{e}}_ {\textit{z}}}} \end{array} \\ & \cdot {{\rm{e}}^{{\rm{i}}k {\textit{z}}\cos \theta }}\sqrt {\cos \theta } \sin \theta {\rm{d}}\theta , \end{split}$$ 特别地，在焦平面上，即z = 0时，W1 = 2，W2 = 0。此时，EI仅具有径向分量与纵向分量且两分量同相，HI仅具有角向分量且与EI间具有−π/2的相位差。EII与HII均仅具有同相的角向分量与纵向分量，且HII与EII间具有−π/2的相位差。
3. 数值结果
3.1 Néel型电场矢量斯格明子
由两对径向偏振光推导得出的式(7)与式(8)表明聚焦平面内电场矢量只存在径向分量和纵向分量，而磁场矢量只有角向分量。取NA=0.95。图2(a)~2(b)所示为焦平面内的聚焦电场Er和Ez分量的振幅实部分布(Er和Ez分量的振幅虚部为0，忽略常数A的影响)，图2(c)所示为Hφ分量的振幅虚部分布(Hφ分量的振幅实部为0)，其中白色箭头表示电磁场相应的振动方向。相较于Er和Ez分量，Hφ分量的相位要超前π/2。图2(d)是电磁场全部分量的归一化截面曲线分布。
$$ {{N}}_{\text{sk}}=\frac{1}{4\pi }{\displaystyle \underset{S}{\iint }n{\rm{d}}S}=\frac{1}{4\pi }{\displaystyle \iint {{\boldsymbol{e}}}_{0}} \cdot \left(\frac{\partial {{\boldsymbol{e}}}_{0}}{\partial x}\times \frac{\partial {{\boldsymbol{e}}}_{0}}{\partial y}\right){\rm{d}}x{\rm{d}}y. $$ ![Figure 3. Vectorial distributions of the Néel-type optical skyrmion formed by electric field vectors in the focal plane. (a)~(b) Normalized energy density distributions of the focused electric field and magnetic field in the focal plane. The black arrows represent the projections of the electric or magnetic field unit vectors on the focal plane; (c) Three-dimensional vectorial structure of the optical skyrmion formed by electric field vectors within the red circle in (a); (d) Two-dimensional vectorial structure of the focal magnetic field in the central region of (b); (e) Orientations of the electric field unit vectors along the radial direction of the constructed Néel-type optical skyrmion; (f) Variation of θxy versus r; (g) Skyrmion numberdensity distribution of the constructed optical skyrmion in the focal plane]()
Vectorial distributions of the Néel-type optical skyrmion formed by electric field vectors in the focal plane. (a)~(b) Normalized energy density distributions of the focused electric field and magnetic field in the focal plane. The black arrows represent the projections of the electric or magnetic field unit vectors on the focal plane; (c) Three-dimensional vectorial structure of the optical skyrmion formed by electric field vectors within the red circle in (a); (d) Two-dimensional vectorial structure of the focal magnetic field in the central region of (b); (e) Orientations of the electric field unit vectors along the radial direction of the constructed Néel-type optical skyrmion; (f) Variation of θxy versus r; (g) Skyrmion numberdensity distribution of the constructed optical skyrmion in the focal plane
图3(a)~3(b)所示为焦平面上电场和磁场的归一化能量密度分布及电磁场单位矢量在焦平面上的投影(如黑色箭头所示)。在图3(a)中的红色圆圈内，电场矢量形成了一个完整的斯格明子结构，其由电场单位矢量构成的三维矢量结构如图3(c)所示，可以看出，电场矢量以Néel型方式从中心的方向“向上”连续变化反转至外围的方向“向下”，而焦平面处的磁场矢量由于缺少纵向分量无法形成斯格明子，仅在焦平面内呈现角向分布，如图3(d)所示。图3(e)所示为电场单位矢量沿斯格明子结构的径向变化情况，可以看出电场矢量连续并重复地将其方向由“向上”状态改变至“向下”状态。其中，θxy是电场矢量相对于x-y平面的方向角。图3(f)所示为θxy沿径向从中心到外围的变化，它表明θxy从r = 0时的π/2单调递减为r = 1.06λ时的−π/2，即电场矢量指向由z轴正方向连续变化为z轴负方向，如粉色阴影区域所示。
斯格明子数(Nsk)和斯格明子数密度是用于表征斯格明子结构的重要参数。斯格明子数可以表示为[9, 14, 35]：
![Figure 2. Normalized distributions of the components of the Néel-type optical skyrmion formed by electric field vectors in the focal plane. (a)~(b) Radial and longitudinal components of the focused electric field in the focal plane; (c) Angular component of the focused magnetic field in the focal plane; (d) The cross sections of the individual components of the focused electromagnetic field]()
Normalized distributions of the components of the Néel-type optical skyrmion formed by electric field vectors in the focal plane. (a)~(b) Radial and longitudinal components of the focused electric field in the focal plane; (c) Angular component of the focused magnetic field in the focal plane; (d) The cross sections of the individual components of the focused electromagnetic field
其中，面积S覆盖完整的斯格明子，n是斯格明子数密度。e0表示局部电场或磁场的单位矢量。图3(g)表示焦平面内电场矢量斯格明子的斯格明子数密度分布，计算得到的Nsk为0.9992，其与1的偏差源于数值误差。斯格明子数密度的大小在一定程度上反映了相应区域内矢量取向变化的快慢。
3.2 Bloch型电磁矢量斯格明子
由一对角向偏振光与一对径向偏振光推导得出的式(10)与式(11)表明聚焦平面内电场矢量只存在角向分量和纵向分量，聚焦磁场具有超前于聚焦电场π/2的归一化场分布。取NA=0.95。图4(a)~4(b)所示为焦平面内的聚焦电场Eφ和Ez分量的振幅实部分布(Eφ和Ez分量的振幅虚部为0，忽略常数A的影响)，图4(c)~4(d)所示为Hφ和Hz分量的振幅虚部分布(Hφ和Hz分量的振幅实部为0)，其中白色箭头表示电磁场相应的振动方向。相较于Eφ和Ez分量，Hφ和Hz分量的相位要超前π/2。图4(e)是电磁场全部分量的归一化截面曲线分布。
![Figure 4. Normalized distributions of the components of the Bloch-type optical skyrmions formed by electromagnetic field vectors in the focal plane. (a)~(b) Angular and the longitudinal components of the focused electric field in the focal plane; (c)~(d) Angular and longitudinal components of the focused magnetic field in the focal plane; (e) Cross sections of the individual components of the focused electromagnetic field]()
Normalized distributions of the components of the Bloch-type optical skyrmions formed by electromagnetic field vectors in the focal plane. (a)~(b) Angular and the longitudinal components of the focused electric field in the focal plane; (c)~(d) Angular and longitudinal components of the focused magnetic field in the focal plane; (e) Cross sections of the individual components of the focused electromagnetic field
![Figure 5. Vectorial distributions of the Bloch-type optical skyrmions formed by electromagnetic field vectors in the focal plane. (a)~(b) Normalized energy density distributions of the focused electric field and magnetic field in the focal plane. The black arrows represent the projections of the electric or magnetic field unit vectors on the focal plane; (c)~(d) Three-dimensional vectorial structures of the optical skyrmions formed by electromagnetic field vectors within the red circles in (a) and (b); (e)~(f) Orientations of the electric field unit vectors and the magnetic field unit vectors along the radial directions of the constructed Bloch-type optical skyrmions; (g)~(h) Variations of θxy versus r; (i)~(j) Skyrmion number density distributions of the constructed optical skyrmions in the focal plane]()
Vectorial distributions of the Bloch-type optical skyrmions formed by electromagnetic field vectors in the focal plane. (a)~(b) Normalized energy density distributions of the focused electric field and magnetic field in the focal plane. The black arrows represent the projections of the electric or magnetic field unit vectors on the focal plane; (c)~(d) Three-dimensional vectorial structures of the optical skyrmions formed by electromagnetic field vectors within the red circles in (a) and (b); (e)~(f) Orientations of the electric field unit vectors and the magnetic field unit vectors along the radial directions of the constructed Bloch-type optical skyrmions; (g)~(h) Variations of θxy versus r; (i)~(j) Skyrmion number density distributions of the constructed optical skyrmions in the focal plane
图5(a)~5(b)所示为焦平面上电场和磁场的归一化能量密度分布及电磁场单位矢量在焦平面上的投影(如黑色箭头所示)。在红色圆圈内，电场矢量和磁场矢量分别形成了一个完整的斯格明子结构。其由电磁场单位矢量构成的三维矢量结构如图5(c)~5(d)所示。可以看出，电场矢量以Bloch型方式从中心的方向“向上”连续变化反转至外围的方向“向下”，而磁场矢量的变化情况超前电场矢量π/2。图5(e)~5(f)所示为电磁场单位矢量沿斯格明子结构的径向变化情况，可以看出电场与磁场矢量连续并重复地将其方向由“向上”状态改变至“向下”状态。θxy是电磁场矢量相对于x-y平面的方向角，图5(g)~5(h)所示为θxy沿径向从中心到外围的变化。图5(g)表明θxy从r = 0时的π/2单调递减为r = 0.92λ时的−π/2，即电场矢量指向由z轴正方向连续变化为z轴负方向，如粉色阴影区域所示。图5(h)所示的磁场矢量的θxy从r = 0时的−π/2单调递增为r = 0.92λ时的π/2，即磁场矢量指向由z轴负方向连续变化为z轴正方向。图5(i)~5(j)分别表示焦平面内电场矢量和磁场矢量斯格明子的斯格明子数密度分布，相应的斯格明子数分别为0.9989和−0.9989，其与±1的偏差源于数值误差。
4. 结 论
本文在自由空间中，通过调控两对相向传播的柱矢量入射光的相位与偏振态，在4π聚焦系统中实现了电磁场纵向偏振分量与横向偏振分量的独立控制，并在4π聚焦系统的焦面上相干产生了微纳空间尺度的Néel型与Bloch型的电磁矢量光学斯格明子。当其中一对相向传播的入射光是相位差为π的径向偏振光，另一对为同相的径向偏振光时，系统焦面上产生了Néel型电场矢量斯格明子，而磁场分布由于缺少纵向场分量，仅表现为角向分布的纯横向场。进一步，将另一对相向传播的入射光替换为对称分布的角向偏振光，此时系统焦面上同时产生了Bloch型的电场矢量斯格明子与相位超前π/2的磁场矢量斯格明子。本工作将为进一步研究自由空间中微纳尺度电磁矢量光学斯格明子与物质的相互作用提供理论基础。
利益冲突
李向平是期刊的编委，除此之外，所有作者声明无利益冲突。
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1. 周志凯，王思聪，李向平. 光学斯格明子的产生与调控（特邀）. 光学学报. 2024(10): 85-99 . 
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Sun Jialin, Wang Sicong, Zhou Zhikai, Zheng Zecan, Jiang Meiling, Song Shichao, Deng ZiLan, Qin Fei, Cao Yaoyu, Li Xiangping. Generation of optical skyrmions formed by electromagnetic field vectors under 4π focal configurations. Opto-Electronic Engineering 50, 230059 (2023). DOI: 10.12086/oee.2023.230059Download CitationArticle History
- Received Date March 13, 2023
- Revised Date May 06, 2023
- Accepted Date May 08, 2023
- Available Online June 01, 2023
- Published Date June 24, 2023
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