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众所周知，由于涡旋光束的独有特性，人们对偏振涡旋光束通过大数值孔径的紧聚焦特性进行了广泛的研究[1-7]。特别是紧聚焦的涡旋光束会产生波长尺度的光斑或暗核，这些结果可以应用于许多领域。近年来，部分相干光通过大数值孔径的紧聚焦特性也引起了极大的研究兴趣[8-20]。例如，FISCHER和VISSER[10]研究了部分相干紧聚焦光场的光谱相干特性；GUO等人[11]研究了部分相干径向偏振光涡旋光束紧聚焦的空间相干特性；还有一些研究人员[15-17]分析了J0相关偏振光束紧聚焦的传输特性等等。然而，通过大数值孔径的J0相关径向偏振光束的空间相干特性鲜见相关研究报道。本文中，将基于矢量德拜衍射理论，研究J0相关径向偏振光束的紧聚焦场，分析焦点区域光场的光谱相干度。结果表明，焦点区域相干度不仅与数值孔径大小和相干参量有关，而且还与拓扑电荷数有关。特别是，相干度还具有相位异常点。
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一个完全相干且径向偏振的光束，通过大数值孔径聚焦后，在焦点区域的电场可以表示为[21]:
$ \begin{array}{l} \mathit{\boldsymbol{E}}\left( {r,\psi ,z} \right) = \left[ {\begin{array}{*{20}{l}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right] = \frac{{ - {\rm{i}}A}}{{\rm{ \mathsf{ π} }}}\int_0^\alpha {\int_0^{2{\rm{ \mathsf{ π} }}} {\sqrt {\cos \theta } \sin \theta \times \;} } \;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;B\left( {\theta ,\phi } \right)\left[ {\begin{array}{*{20}{l}} {\sin \theta \cos \phi }\\ {\cos \theta \sin \phi }\\ { - sin\theta } \end{array}} \right] \times \;\\ \exp \left[ {{\rm{i}}k\left( {z\cos \theta + r\sin \theta \cos \left( {\phi - \psi } \right)} \right)} \right]{\rm{d}}\phi {\rm{d}}\theta \end{array} $
(1) 式中，r, ψ和z是焦点区域中的径向、角向和纵向坐标, φ为入射光束的方位角, 波数k=2π/λ, A是常数，α=arcsin(dNA)是由透镜的数值孔径(numerical aperture, NA)决定的最大角度，θ是入射光束的会聚角, 并从0变化到α，B(θ, φ)是入射场的切趾函数。
部分相干涡旋光束的相干特性可以用交叉谱密度矩阵来描述，矩阵元素的表达式如下[22]:
$ \begin{array}{l} {W_{ij}}\left( {{r_1}, {r_2}} \right) = \langle E_i^*\left( {{r_1}, {\psi _1}, {z_1}} \right) \times \\ \;\;{E_j}\left( {{r_2}, {\psi _2}, {z_2}} \right)\rangle , \left( {i, j = x, y, z} \right) \end{array} $
(2) 式中，Ei*(r1, ψ1, z1)和Ej(r2, ψ2, z1)表示两点处电场的笛卡尔分量，*表示复共轭，〈〉表示系综平均值。将(1)式代入(2)式，焦点区域交叉密度矩阵的元素就可以表示为:
$ \begin{array}{l} {W_{ij}}\left( {{r_1}, {\psi _1}, {z_1};{r_2}, {\psi _2}, {z_2}} \right) = {\left( {\frac{A}{{\rm{ \mathit{ π} }}}} \right)^2}\int_0^\alpha {\int_0^\alpha {\int_0^{2{\rm{ \mathit{ π} }}} {\int_0^{2{\rm{ \mathit{ π} }}} {\sqrt {\cos {\theta _1}} \times } } } } \\ \;\;\;\;\;\;\;\;\;\sqrt {\cos {\theta _2}} \sin {\theta _1}\sin {\theta _2}{B_0}\left( {{\theta _1}, {\phi _1}, {\theta _2}, {\phi _2}} \right) \times \\ \;\;\exp \left[ { - {\rm{i}}k{r_1}\sin {\theta _1}cos\left( {{\phi _1} - {\psi _1}} \right)} \right]\exp \left[ {ik{r_2}\sin {\theta _2} \times } \right.\\ \;\;\;\;\;\;\;\cos \left. {\left( {{\phi _2} - {\psi _2}} \right)} \right]{P_i}\left( {{\theta _1}, {\phi _1}} \right){P_j}\left( {{\theta _2}, {\phi _2}} \right) \times \\ \exp \left[ {{\rm{i}}k\left( {{z_2}\cos {\theta _2} - {z_1}\cos {\theta _1}} \right)} \right]d{\theta _1}d{\theta _2}d{\phi _1}d{\phi _2}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {i, j = x, y, z} \right) \end{array} $
(3) $ {P_{i, j}}\left( {\theta , \phi } \right) = \left\{ \begin{array}{l} \sin \theta \cos \phi , \left( {i, j = x} \right)\\ \cos \theta \sin \phi , \left( {i, j = y} \right)\\ - \sin \theta , \left( {i, j = z} \right) \end{array} \right. $
(4) 式中, B0(θ1, φ1, θ2, φ2)=B*(θ1, φ1)B(θ2, φ2)称为入射光束的切趾相干函数。假设通过大数值孔径聚焦的光束是一个J0相关高斯-谢尔模型涡旋光束，这种光束的切趾相干函数可以表示为[17, 23]:
$ \begin{array}{l} {B_0}\left( {{\theta _1}, {\phi _1}, {\theta _2}, {\phi _2}} \right) = {B^*}\left( {{\theta _1}, {\phi _1}} \right)B\left( {{\theta _2}, {\phi _2}} \right) = \\ \sum\limits_{m = - \infty }^{ + \infty } {\exp \left( { - \frac{{{{\sin }^2}{\theta _1} + {{\sin }^2}{\theta _2}}}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right){J_m}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _1}} \right)} \times \\ {J_m}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _2}} \right)\exp \left[ {i\left( {n - m} \right)\left( {{\phi _2} - {\phi _1}} \right)} \right] \end{array} $
(5) 式中, σ=w0/d表示光束的相对光斑大小，w0为束腰半径, d为大数值孔径的半径，δ=βd为相干参量, n和m为序数，n>m。
光谱相干度是描述紧聚焦光束特性的重要指标，其定义式[10, 17]可以表示为:
$ \mu \left( {{r_1}, {r_2}} \right) = \frac{{{T_{\rm{r}}}W\left( {{r_1}, {\psi _1}, {z_1};{r_2}, {\psi _2}, {z_2}} \right)}}{{\sqrt {{T_{\rm{r}}}W\left( {{r_1}, {\psi _1}, {z_1}} \right)} \sqrt {{T_{\rm{r}}}W\left( {{r_2}, {\psi _2}, {z_2}} \right)} }} $
(6) 式中, Tr表示矩阵对角元素之和。将(4)式和(5)式代入(3)式，即可得到焦区焦点区域的交叉谱密度Wij(r1, ψ1, z1; r2, ψ2, z2)。通过设置r1=r2=r，ψ1=ψ2=ψ，z1=z2=z，就可得到焦点区域的光强分布:
$ \begin{array}{l} I\left( {r, \psi , z} \right) = {T_{\rm{r}}}W\left( {r, \psi , z} \right) = \; = \;{W_{xx}}\left( {r, \psi , z} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;{W_{yy}}\left( {r, \psi , z} \right) + {W_{zz}}\left( {r, \psi , z} \right) \end{array} $
(7) 下面通过(3)式~(6)式，进一步分析J0相关径向偏振涡旋光束在光轴和焦平面上的光谱相干度。
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现在考虑一对点在光轴上，即r1=(0, 0, z1), r2=(0, 0, z2)。通过使用以下公式:
$ \begin{array}{l} \int_0^{{\rm{2 \mathit{ π} }}} {\sin \phi \exp \left[ {{\rm{i}}\left( {n - m} \right)\phi + {\rm{i}}kr\sin \theta \cos \left( {\phi - \psi } \right)} \right]} {\rm{d}}\phi = \\ {\rm{ \mathit{ π} }}{{\rm{i}}^{n - m}}\left[ {\exp \left[ {{\rm{i}}\left( {n - m + 1} \right)\psi } \right]} \right.{{\rm{J}}_{n - m + 1}}\left( {kr\sin \theta } \right) + \\ \exp \left[ {{\rm{i}}\left( {n - m - 1} \right)\psi } \right]{{\rm{J}}_{n - m - 1}}\left. {\left( {kr\sin \theta } \right)} \right] \end{array} $
(8) $ \begin{array}{l} \int_0^{{\rm{2 \mathit{ π} }}} {\cos \phi \exp \left[ {i\left( {n - m} \right)\phi + {\rm{i}}kr\sin \theta \cos \left( {\phi - \psi } \right)} \right]} {\rm{d}}\phi = \\ {\rm{ \mathit{ π} }}{{\rm{i}}^{n - m + 1}}\left[ {\exp \left[ {{\rm{i}}\left( {n - m + 1} \right)\psi } \right]{{\rm{J}}_{n - m + 1}}\left( {kr\sin \theta } \right) - } \right.\\ \exp \left[ {{\rm{i}}\left( {n - m - 1} \right)\psi } \right]{{\rm{J}}_{n - m - 1}}\left. {\left( {kr\sin \theta } \right)} \right] \end{array} $
(9) 由(3)式可得到交叉谱密度的表达式:
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ \begin{array}{l} {W_{xx}}\left( {0, 0, {z_1};0, 0, {z_2}} \right)\\ {W_{yy}}\left( {0, 0, {z_1};0, 0, {z_2}} \right)\\ {W_{zz}}\left( {0, 0, {z_1};0, 0, {z_2}} \right) \end{array} \right] = \\ {A^2}\left[ \begin{array}{l} {H_{n + 1, x}}^*\left( {{z_1}} \right){H_{n{\rm{ + 1}}, x}}\left( {{z_2}} \right) + {H_{n - 1, x}}^*\left( {{z_1}} \right){H_{n - 1, x}}\left( {{z_2}} \right)\\ {H_{n + 1, y}}^*\left( {{z_1}} \right){H_{n{\rm{ + 1}}, y}}\left( {{z_2}} \right) + {H_{n - 1, y}}^*\left( {{z_1}} \right){H_{n - 1, y}}\left( {{z_2}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;4{H_{n + 1, x}}^*\left( {{z_1}} \right){H_{n{\rm{ + 1}}, x}}\left( {{z_2}} \right) \end{array} \right] \end{array} $
(10) 其中,
$ \begin{array}{l} \left[ \begin{array}{l} {H_{n + 1, x}}\left( z \right)\\ {H_{n + 1, y}}\left( z \right) \end{array} \right]\; = \int_0^\alpha {\sqrt {\cos \theta } \sin \theta \left[ \begin{array}{l} \sin \theta \\ \cos \theta \end{array} \right]} \exp \left( { - \frac{{{{\sin }^2}\theta }}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\;{{\rm{J}}_{n + 1}}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin \theta } \right)\exp \left( {{\rm{i}}kz\cos \theta } \right){\rm{d}}\theta \end{array} $
(11) $ \begin{array}{l} \left[ \begin{array}{l} {H_{n - 1, x}}\left( z \right)\\ {H_{n - 1, y}}\left( z \right) \end{array} \right]\; = \int_0^\alpha {\sqrt {\cos \theta } \sin \theta \left[ \begin{array}{l} \sin \theta \\ \cos \theta \end{array} \right]} \exp \left( { - \frac{{{{\sin }^2}\theta }}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\;{{\rm{J}}_{n - 1}}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin \theta } \right)\exp \left( {{\rm{i}}kz\cos \theta } \right){\rm{d}}\theta \end{array} $
(12) 将z1=z2=z代入(7)式和(10)式，可得z轴上的总光强表达式为:
$ \begin{array}{l} I\left( {0, 0, z} \right) = {A^2}\left[ {5\left| {{H_{n + 1, x}}\left( z \right)\left| {^2 + \left| {{H_{n + 1, y}}\left( z \right)} \right.} \right.} \right.\left| {^2} \right. + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left| {{H_{n - 1, x}}\left( z \right)\left| {^2} \right.} \right. + \left. {\left| {{H_{n - 1, y}}} \right.\left( z \right)\left| {^2} \right.} \right] \end{array} $
(13) 从上式中可以注意到，对于空间完全相干的涡旋光束(即δ=0)，只有拓扑荷数n=1时在光轴上有光强，也就是说，拓扑荷数n≠1时, 在光轴上是零光强。然而，对于部分相干涡旋光束，则不存在这种零光强。通过分析(11)式和(12)式中的贝塞尔函数Jn+1和Jn－1，就很容易理解这些现象。而从(10)式、(13)式也很容易领悟一些对称性质，即:
$ \begin{array}{l} \;\;\;\;{W_{ij}}\left( {0, 0, - {z_1};0, 0, - {z_2}} \right) = \\ {W_{ij}}{\left( {0, 0, {z_1};0, 0, {z_2}} \right)^*}, \left( {i, j = x, y, z} \right) \end{array} $
(14) $ \;I\left( {0, 0, - z} \right) = I\left( {0, 0, z} \right) $
(15) 应用(14)式、(15)式和光谱相干度定义(6)式，可以获得以下对称关系:
$ \mu \left( {0, 0, {z_1};0, 0, {z_2}} \right) = \mu {\left( {0, 0, - {z_1};0, 0, - {z_2}} \right)^*} $
(16) 因此，应用(10)式、(13)式和(6)式，就可以确定z轴上的光强分布和纵向相干度。从(10)式可以看出，如果z1保持不变，μ(0, 0, z1; 0, 0, z2)是z2的振荡函数，而振荡周期会随拓扑荷数n和相干参量δ变化。例如由图 1所示，给出纵向相干度μ(0, 0, 0;0, 0, z)实数部分和模随拓扑荷数n和相干参量δ的变化曲线。可以看到，在图 1a~图 1b中的μ(0, 0, 0;0, 0, z)实数部分，其曲线的振荡周期略大于波长，约等于1.2λ~1.35λ。随拓扑荷数n的增加振荡周期会相应增大，但随相干参量δ的增加振荡周期只略有变化。从图 1c~图 1d中可知，在焦点区域的纵向相干度μ(0, 0, 0;0, 0, z)的模随拓扑荷数n的增加或随相干参量δ的减小而增大。图 1中其余参量σ=2, dNA=0.8。
Figure 1. Fig. 1 Real parts and modulus of μ(0, 0, 0;0, 0, z)for different values of topological charge n and coherence parameter δ
图 2为不同拓扑荷数n的纵向相干度μ(0, 0, z1; 0, 0, z2)分布图, 以及z1(λ)与z2(λ)的关系图。从图中可以清楚地看出，纵向相干度呈对称分布，其最小值出现在一对轴对称点z1，z2上，并且这个最小值点将随着拓扑荷数的增加而逐渐离开焦点。其余参量同图 1。
Figure 2. Modulus of μ(0, 0, z1; 0, 0, z2)for different values of topological charge n and coherence parameterδ
图 3中给出了μ(0, 0, －z; 0, 0, z)随拓扑荷数n、相干参量δ和数值孔径dNA变化的曲线图。从图中可见，在焦点区域的纵向相干度μ(0, 0, －z; 0, 0, z)会随拓扑荷数n的增加而增大，但是也会随δ或dNA的减小而增加。特别是，当n≥5 (见图 3a)或dNA≤0.4(见图 3c)时，纵向相干度在焦点附近约等于1。其余参量同图 1。
Figure 3. Modulus of μ(0, 0, -z; 0, 0, z) for different values of topological charge n and coherence parameter δ
总之，在焦点区域纵向相干度随拓扑荷数n的增大而增大，也会随相干参量δ或数值孔径dNA的减小而增大, 也就是说拓扑荷数越大，或入射光束的相干性越好(即δ越小)，或数值孔径越小(即dNA值越小)，都会使焦点附近光束的相干性越好。类似的研究见参考文献[12]和参考文献
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接下来研究位于焦平面上的一对点的光谱相干度。其中，一个点位于几何焦点上, 而另一个点位于焦平面上并与z轴有一定的距离。由于聚焦场的旋转对称性，可以假设这另一个点位于x轴上，研究相干度也不会失去一般性, 即r1=(0, 0, 0), r2=(x, 0, 0)。将(4)式和(5)式代入(3)式，可得在焦平面上聚焦场的交叉谱密度:
$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{l}} {{W_{xx}}\left( {0, 0, 0;x, 0, 0} \right)}\\ {{W_{yy}}\left( {0, 0, 0;x, 0, 0} \right)}\\ {{W_{zz}}\left( {0, 0, 0;x, 0, 0} \right)} \end{array}} \right] = }\\ {{A^2}\left[ {\begin{array}{*{20}{l}} {{P_{n + 1, x}}\left( x \right) + {P_{n - {\rm{1}}, x}}\left( x \right) - {D_{n + 1, x}}\left( x \right) - {D_{n - 1, x}}\left( x \right)}\\ {{P_{n + 1, y}}\left( x \right) + {P_{n - {\rm{1}}, y}}\left( x \right) + {D_{n + 1, y}}\left( x \right) + {D_{n - 1, y}}\left( x \right)}\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;4{P_{n + 1, x}}\left( x \right)} \end{array}} \right]} \end{array}\; $
(17) 其中,
$ \begin{array}{l} \left[ \begin{array}{l} {P_{i, x}}\left( x \right)\\ {P_{i, y}}\left( x \right) \end{array} \right] = \int_0^\alpha {\int_0^\alpha {\sqrt {\cos {\theta _1}} } } \sqrt {\cos {\theta _2}} \sin {\theta _1}\sin {\theta _2} \times \\ \;\;\;\;\left[ \begin{array}{l} \sin {\theta _1}\sin {\theta _2}\\ \cos {\theta _1}cos{\theta _2} \end{array} \right]\exp \left( { - \frac{{{{\sin }^2}{\theta _1} + {{\sin }^2}{\theta _2}}}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ \;\;\;\;\;\;{{\rm{J}}_i}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _1}} \right){{\rm{J}}_i}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _2}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\rm{J}}_0}\left( {kx\sin {\theta _2}} \right){\rm{d}}{\theta _1}{\rm{d}}{\theta _2} \end{array} $
(18) $ \begin{array}{l} \left[ \begin{array}{l} {P_{i, x}}\left( x \right)\\ {P_{i, y}}\left( x \right) \end{array} \right] = \int_0^\alpha {\int_0^\alpha {\sqrt {\cos {\theta _1}} } } \sqrt {\cos {\theta _2}} \sin {\theta _1}\sin {\theta _2} \times \\ \;\;\;\;\left[ \begin{array}{l} \sin {\theta _1}\sin {\theta _2}\\ \cos {\theta _1}cos{\theta _2} \end{array} \right]\exp \left( { - \frac{{{{\sin }^2}{\theta _1} + {{\sin }^2}{\theta _2}}}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ \;\;\;\;\;\;{{\rm{J}}_i}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _1}} \right){{\rm{J}}_i}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin {\theta _2}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\rm{J}}_2}\left( {kx\sin {\theta _2}} \right){\rm{d}}{\theta _1}{\rm{d}}{\theta _2} \end{array} $
(19) 同时，可获得焦平面上的总光强表达式为:
$ \begin{array}{l} \;\;\;\;I\left( {x, 0, 0} \right) = {A^2}\sum\limits_{m = - \infty }^{ + \infty } {\;\left\{ {\left[ {{Q_{m + 1, x}}\left( x \right) + } \right.} \right.} \\ {\left. {{Q_{m - 1, x}}\left( x \right)} \right]^2} + \left. {{{\left[ {{Q_{m + 1, y}}\left( x \right)\; + \;{Q_{m - 1, y}}\left( x \right)} \right]}^2}} \right\} \end{array} $
(20) 其中,
$ \begin{array}{l} \left[ \begin{array}{l} {Q_{m + 1, x}}\left( x \right)\\ {Q_{m + 1, y}}\left( x \right) \end{array} \right] = \int_0^\alpha {\sqrt {\cos \theta } } \sin \theta \left[ \begin{array}{l} \sin \theta \\ \cos \theta \end{array} \right]\exp \left( { - \frac{{{{\sin }^2}\theta }}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ {{\rm{J}}_m}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin \theta } \right){{\rm{J}}_{n - m + 1}}\left( {kx\sin \theta } \right){\rm{d}}\theta \end{array} $
(21) $ \begin{array}{l} \left[ \begin{array}{l} {Q_{m - 1, x}}\left( x \right)\\ {Q_{m - 1, y}}\left( x \right) \end{array} \right] = \int_0^\alpha {\sqrt {\cos \theta } } \sin \theta \left[ \begin{array}{l} \sin \theta \\ \cos \theta \end{array} \right]\exp \left( { - \frac{{{{\sin }^2}\theta }}{{{d_{{\rm{NA}}}}^2{\sigma ^2}}}} \right) \times \\ {{\rm{J}}_m}\left( {{d_{{\rm{NA}}}}^{ - 1}\delta \sin \theta } \right){{\rm{J}}_{n - m - 1}}\left( {kx\sin \theta } \right){\rm{d}}\theta \end{array} $
(22) 因此，应用(17)式、(20)式和(6)式，就可以确定焦平面的光强分布和横向相干度。从(17)式中也可看出，交叉谱密度和横向相干度都是实数；而从(10)式中也可看出，纵向相干度是复数。
在图 3中，给出了聚焦场的光强和横向相干度μ(0, 0, 0;x, 0, 0)随拓扑荷数n、相干参量δ和数值孔径dNA变化的曲线图。从图 4a可以清楚地看出，当拓扑荷数n=1时，焦平面上有一小光斑；而当拓扑荷数n>1时，焦平面上有一小暗核，可作为光镊囚禁粒子。从图 4b~图 4f可以观察到，横向相干度会随着n, δ或者dNA的增大而减小。也就是说拓扑荷数越大，或入射光束的相干性越差(即δ越大)，或数值孔径越大(即dNA值越大)，都会使焦平面上光束的相干性越差，横向相干度衰减越快。此外，当n≥2或者δ≥3时(如图 4b、图 4d~图 4f所示)，对于图中描绘的曲线与线μ(0, 0, 0;x, 0, 0)=0相交情况，横向相干度至少有两个零点，这种点也被称为光谱相干度的相位奇点或相干奇点。在这些点上，J0相关径向偏振涡旋聚焦场与点r1=(0, 0, 0)是完全不相干的。这说明部分相干矢量场也拥有相干奇点[12-17]。图 4中其余参量同图 1。
Figure 4. a—intensity distribution b~f—the degree of coherence μ(0, 0, 0;x, 0, 0) in the focal plane
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基于矢量德拜理论和相干理论，研究了J0相关径向偏振涡旋光束通过大数值孔径透镜聚焦后，在光轴和焦平面上的空间相干特性。推导出在光轴和焦平面上的相干度表达式。数值模拟结果表明，在焦点区域纵向相干度随拓扑荷数n的增大而增大，随相干参量δ或数值孔径dNA的增大而减小。特别是，当n≥5或者dNA≤0.4时，纵向相干度在焦点附近约等于1。研究还发现，焦平面的横向相干度也会随拓扑荷数n、相干参量δ或者数值孔径dNA的增大而减小。此外，焦平面的横向相干度还具有相位奇点，这意味着在这种点上J0相关矢量涡旋聚焦场与几何焦点是完全不相干的。这些研究结果对J0相关矢量涡旋光束在许多领域的应用具有重要意义。
J0相关径向偏振涡旋光束紧聚焦的空间相干特性
Spatial correlation properties of tightly focused J0-correlated
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摘要: 为了研究J0相关径向偏振涡旋光束深聚焦的相干特性，采用矢量德拜理论和相干理论，对聚焦场在光轴和焦平面上的相干特性进行了理论分析和数值仿真，取得相干度表达式和模拟数据。结果表明，在焦点区域的纵向相干度不但随着拓扑荷数n的增加而增大，而且还随着相干参量或者数值孔径dNA的增大而减小；在焦平面的横向相干度也随着拓扑荷数或者相干参量或者数值孔径的增大而减小; 当n≥5或者dNA≤0.4时，纵向相干度在焦点附近约为1;这种相干度还存在相位异常点。这些研究结果对J0相关矢量涡旋光束在许多领域的应用具有一定意义。Abstract: In order to study the spatial correlation properties of tightly focused J0-correlated radially polarized vortex beam through a high numerical aperture dNA, the vectorial Debye diffraction theory and the coherence theory were adopted. After theoretical analysis and numerical simulation of the correlation characteristics of focused field on the optical axis and the focal plane, the expression and simulation data of coherence degrees were obtained.The results show that, the longitudinal coherence degrees in the focal region increase as the topological charge n increases, but it decrease as the coherence parameter or the numerical aperture dNA; The transverse coherence degrees in the focal plane decrease with the increase of the topological charge, coherence parameter or numerical aperture. When n≥5 or dNA≤0.4, the longitudinal coherence near the focal point approximately equals to 1. Further, the coherence degrees are shown to exhibit phase singularities. The results are of great significance to the application of J0-correlated vector vortex beams in many fields.
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Key words:
- laser physics /
- correlation properties /
- J0-correlated /
- vortex beam /
- polarization
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Figure 1. Fig. 1 Real parts and modulus of μ(0, 0, 0;0, 0, z)for different values of topological charge n and coherence parameter δ
Figure 2. Modulus of μ(0, 0, z1; 0, 0, z2)for different values of topological charge n and coherence parameterδ
Figure 3. Modulus of μ(0, 0, -z; 0, 0, z) for different values of topological charge n and coherence parameter δ
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