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偏振态的表征方法中,Stokes矢量法不仅可以表述任意偏振态、而且容易探测,因此,在偏振探测中得到了广泛的应用。Stokes矢量由4个光强参数S0、S1、S2、S3描述,表示如下[31]:
$$ S = \left[ {\begin{array}{*{20}{c}} {{S_0}}\\ {{S_1}}\\ {{S_2}}\\ {{S_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{I_x} + {I_y}}\\ {{I_x} - {I_y}}\\ {{I_a} - {I_b}}\\ {{I_l} - {I_r}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{I_x} + {I_y}}\\ {{I_x} - {I_y}}\\ {2 \times {I_a} - {S_0}}\\ {2 \times {I_l} - {S_0}} \end{array}} \right] $$ (1) 式中:S0为总光强;S1、 S2、 S3分别为入射光在正交基矢
$ \left( {\overrightarrow x ,\overrightarrow y } \right) $ 、$ \left( {\overrightarrow a ,\overrightarrow b } \right) = \left( {1/\sqrt 2 } \right)\left( {\overrightarrow x + \overrightarrow y ,\overrightarrow x - \overrightarrow y } \right) $ 和$ \left( {\overrightarrow r ,\overrightarrow l } \right) = \left( {1/\sqrt 2 } \right)\left( {\overrightarrow x + i\overrightarrow y ,\overrightarrow x - i\overrightarrow y } \right)$ 下测量的光强差。为了数据更加直观,通常对Stokes参数用S0进行归一化。由公式(1)可知任意一束偏振光的Stokes矢量可通过测量垂直于传播方向平面内的光强$ I_x $ 、$I_y $ 、$ I_a$ 、$ I_l $ 得到。一般而言,探测器平行于设计器件,当入射光倾斜入射时,只有部分光被分解聚焦在焦平面上,这会导致得到的Stokes参数不准确。因此,需要明确Stokes参数与入射角和获得的4个光强之间的关系。在推导过程中,建立了两个笛卡尔坐标系x′y′z′和x″y″z″,其中z′轴和z″轴分别对应实际入射光的方向和超表面的法线,探测器平行于x″y″。假设一束单色平面波沿x′z′平面以θ角入射,其电场的振幅可以由琼斯矩阵表示为下式:$$ \left[ {E_0^{'}} \right] = {\left[ {A_x^{'}\;\;\;\;A_y^{'}{e^{i\delta }}} \right]^{\rm{T}}} $$ (2) 式中:
$ A_x^{'} $ 、$ A_y^{'}$ 、δ分别为水平与垂直电场分量的振幅与相位差。分别在两个坐标系下表示预设偏振分量的光强,将两个结果进行比较,得到矫正的Stokes参数公式,通过公式揭示Stokes矢量与入射角和焦平面聚焦光强的关系。为了得到在不同偏振态下的光强值,将入射光分解为所设计的偏振分量。在x′y′z′坐标系下,得到4个偏振分量的光强为:$$ \left[ {\begin{array}{*{20}{c}} {I_x^\prime }\\ {I_y^\prime }\\ {I_a^\prime }\\ {I_l^\prime } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {A_x^{\prime 2} + A_y^{\prime 2}}\\ {A_x^{\prime 2} - A_y^{\prime 2}}\\ {\left( {A_x^{\prime 2} + A_y^{\prime 2}} \right)/2 + A_x^\prime \times A_y^\prime \times {\rm{cos}} \delta }\\ {\left( {A_x^{\prime 2} + A_y^{\prime 2}} \right)/2 + A_x^\prime \times A_y^\prime \times {\rm{sin}} \delta } \end{array}} \right] $$ (3) 因此,x′y′z′坐标系下入射光的Stokes参数可以表示为:
$$ {S^\prime } = \left[ {\begin{array}{*{20}{c}} {S_0^\prime }\\ {S_1^\prime }\\ {S_2^\prime }\\ {S_3^\prime } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {I_x^\prime + I_y^\prime }\\ {I_x^\prime - I_y^\prime }\\ {2 \times I_a^\prime - S_0^\prime }\\ {2 \times I_l^\prime - {S_0^\prime }} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {I_x^\prime + I_y^\prime }\\ {I_x^\prime - I_y^\prime }\\ {2 \times A_x^\prime \times A_y^\prime \times {\rm{cos}} \delta }\\ {2 \times A_x^\prime \times A_y^\prime \times {\rm{sin}} \delta } \end{array}} \right] $$ (4) 同样地,在探测坐标系x″y″z″下,分解可得到各个偏振态的光强为:
$$ \left[ {\begin{array}{*{20}{c}} {I_x^{\prime \prime }}\\ {I_y^{\prime \prime }}\\ {I_a^{\prime \prime }}\\ {I_l^{\prime \prime }} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {A_x^{\prime 2} \times {{\rm{{cos }}}^2}\theta + A_y^{\prime 2}}\\ {A_x^{\prime 2} \times {{\rm{{cos }}}^2}\theta - A_y^{\prime 2}}\\ {\left( {A_x^{\prime 2} \times {{\rm{{cos }}}^2}\theta + A_y^{\prime 2}} \right)/2 + A_x^\prime \times A_y^\prime \times {\rm{cos}} \delta \times {\rm{cos}} \theta }\\ {\left( {A_x^{\prime 2} \times {{\rm{{cos }}}^2}\theta + A_y^{\prime 2}} \right)/2 + A_x^\prime \times A_y^\prime \times {\rm{sin}} \delta \times {\rm{cos}} \theta } \end{array}} \right] $$ (5) 得到x″y″z″坐标系下的计算结果,表示如下:
$$ \left[ {\begin{array}{*{20}{c}} {I_x^{\prime \prime } + I_y^{\prime \prime }}\\ {I_x^{\prime \prime } - I_y^{\prime \prime }}\\ {2*I_a^{\prime \prime } - \left( {I_x^{\prime \prime } + I_y^{\prime \prime }} \right)}\\ {2*I_l^{\prime \prime } - \left( {I_x^{\prime \prime } + I_y^{\prime \prime }} \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {I_x^\prime \times {{\rm{{cos }}}^2}\theta + I_y^\prime }\\ {I_x^\prime \times {{\rm{{cos }}}^2}\theta - I_y^\prime }\\ {2 \times A_x^\prime \times A_y^\prime \times {\rm{cos}} \delta \times {\rm{cos}} \theta }\\ {2 \times A_x^\prime \times A_y^\prime \times {\rm{sin}} \delta \times {\rm{cos}} \theta } \end{array}} \right] $$ (6) 对比公式(4)和(6),可以发现通过测量的4个光强值可以重构Stokes参数,校正的Stokes参数公式如下:
$$ {S^\prime } = \left[ {\begin{array}{*{20}{c}} {S_0^\prime }\\ {S_1^\prime }\\ {S_2^\prime }\\ {S_3^\prime } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {I_x^{\prime \prime }/{{\rm{{cos }}}^2}\theta + I_y^{\prime \prime }}\\ {I_x^{\prime \prime }/{{\rm{{cos }}}^2}\theta - I_y^{\prime \prime }}\\ {\left( {2 \times I_a^{\prime \prime } - I_x^{\prime \prime } - I_y^{\prime \prime }} \right)/{\rm{cos}} \theta }\\ {\left( {2 \times I_l^{\prime \prime } - I_x^{\prime \prime } - I_y^{\prime \prime }} \right)/{\rm{cos}} \theta } \end{array}} \right] $$ (7) 式中:
$I_x^{''} $ 、$I_y^{''} $ 、$ I_a^{''} $ 、$ I_l^{''} $ 分别为x″y″z″坐标系下的光强,可以通过仿真进行测量;θ角为入射角。通过公式(7)可知要实现对大角度入射光的偏振态探测,需要准确地确定入射光的角度。二次相位具有将倾斜入射光的旋转效应转换为聚焦光束的平移对称性的性质,并且平移量与入射光角度密切相关,因此这里采用二次相位设计聚焦相位波前[32, 34]:$$ \varPhi \left( r \right) = {k_0}\dfrac{{{r^2}}}{{2f}} $$ (8) 式中:k0为自由空间中的波数,k 0=2π/ λ ,λ为10.6 μm;f为预设焦距;
$ r = \sqrt {{{\left( {x - {x_0}} \right)}^2} + {{\left( {y - {y_0}} \right)}^2}} $ 为任意像素的中心点(x,y)与焦点中心(x0,y0)的距离。前面提到的以θ角入射的入射光通过二次透镜时,出射光所携带的相位为[32]:$$ \begin{split} \varPhi (r) =& {k_0}\dfrac{{{r^2}}}{{2f}} + {k_0}x{\rm{sin}} \theta =\\ &\dfrac{{{k_0}}}{{2f}}\left( {{{(x + f{\rm{sin}} \theta )}^2} + {y^2}} \right) - \dfrac{{f{k_0}{{\rm{{\sin }}}^2}\theta }}{2} \end{split} $$ (9) 式中:
$ {k_0}x sin \theta $ 为由斜入射引入的梯度相位;等式右边最后一项与r无关,可以忽略。与正入射相比,仅在x方向上存在fsinθ的横向偏移[32]。二次相位的波前调控原理如图1(a)所示,根据焦点位置的偏移量可以得到光束的入射角。图 1 紧凑型大视场偏振探测超表面示意图。(a) 利用二次相位超表面的旋转−平移对称性变换原理图;(b) 设计的超表面。上图为超表面俯视图,下图为单元结构示意图;(c) 设计的紧凑型大角度偏振探测3D原理图
Figure 1. Schematic illustration of metasurface of the compact polarimetry for large field of view. (a) Schematic diagram of rotational-translational symmetry conversion with quadratic phase metasurface; (b) Designed metasurface. The upper one is the top view of the designed metasurface, and the lower picture is illustration of a unit cell; (c) 3D schematic of the compact polarimetry for large field of view
为了实现紧凑型大视场偏振探测,设计了如图1(b)所示的超表面。设计的超表面由4个子透镜组成,以2×2阵列形式排布,这些子阵列由硅衬底上周期排布的不同尺寸的椭圆硅柱组成,分别使一个特定的偏振态聚焦。这里预设的4个偏振态为水平线偏振(X)、垂直线偏振(Y)、+45°线偏振态(A)和左旋圆偏振态(L),如图1(b)中红色箭头所示。调控Y和A偏振的子阵列可以通过简单地旋转调控X偏振的子阵列来获得。其中单元结构如图1(b)中下图所示,硅柱位于单元结构的中心。设计的紧凑型大视场偏振探测器工作原理如图1(c)所示,入射光正入射时,光沿
$ Z^{''} $ 传播,设计的超表面将出射的不同偏振光集中在图像传感器像素的中心,由黑色虚线圆圈表示。当光以θ角斜入射时,光沿$ z^{'} $ 传播,焦点由实心红点表示,与垂直入射相比,焦点横向偏移了fsinθ。
Large field-of-view and compact full-Stokes polarimetry based on quadratic phase metasurface
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摘要: 偏振是光的固有属性之一,然而传统的光强、光谱探测技术会造成电磁波的偏振信息的丢失。同时,基于偏振测量的器件及技术不仅存在视场局限的问题,而且系统复杂。基于介质型超表面设计了一种紧凑型大视场偏振探测器件,实现了对入射光的角度及偏振态的探测。该器件由2×2的二次相位超表面组成,每个超表面可实现对特定偏振的对称性变换,即将入射角旋转对称性转变为焦平面内焦点平移对称性。二次相位的对称性变换理论使得此文可以在宽角度范围内(-40°~+40°)通过测量焦点的偏移量实现对入射角的表征。在此基础上,分析了斜入射对测量Stokes参数的影响,得到矫正的Stokes公式。利用4个焦点的强度和矫正的Stokes公式可计算出入射光的Stokes参数。在视场角为0°、20°、40°时,测量的Stokes参数与理论值吻合良好。Abstract: Polarization is one of the inherent characteristics of light, but the polarization information of electromagnetic waves is lost by traditional intensity and spectral detection technology. At the same time, not only the devices and technologies based on polarimetry have the problem of limited field of view, but also the measuring systems are complicated. In this paper, a compact large field-of-view polarimeter was designed based on dielectric metasurface, which realized the detection of the angle and polarization state of the incident light. The device was composed of 2×2 quadratic phase metasurfaces, each of which realized the symmetry transformation for a specific polarization, that is, the rotational symmetry of the incident angle was converted into the translational symmetry of focus in the focal plane. The theory of the symmetry transformation of the quadratic phase made it possible to characterize the angle of incidence by measuring the offset of the focus over a wide angle range (−40°—+40°). On this basis, the influence of oblique incidence on the measurement of Stokes parameters was elaborated, and the modified Stokes formula was obtained. The Stokes parameters of the incident light can be calculated by utilizing the intensities of the four focal points and the modified Stokes formula. The measured Stokes parameters agree well with the theoretical values, when the field-of-view is 0°, 20°, and 40°.
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Key words:
- metasurface /
- quadratic phase /
- wavefront modulation /
- polarimetry /
- compact /
- large field-of-view
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图 1 紧凑型大视场偏振探测超表面示意图。(a) 利用二次相位超表面的旋转−平移对称性变换原理图;(b) 设计的超表面。上图为超表面俯视图,下图为单元结构示意图;(c) 设计的紧凑型大角度偏振探测3D原理图
Figure 1. Schematic illustration of metasurface of the compact polarimetry for large field of view. (a) Schematic diagram of rotational-translational symmetry conversion with quadratic phase metasurface; (b) Designed metasurface. The upper one is the top view of the designed metasurface, and the lower picture is illustration of a unit cell; (c) 3D schematic of the compact polarimetry for large field of view
图 2 设计的单元结构。(a), (b) 优化的单元结构的短轴(W)和长轴(L)尺寸;(c) 0°、10°、20°、30°、40°和50°入射时前8个结构对应的相位响应;(d)单元结构在0°,10°,20°,30°和40°入射时的透射振幅
Figure 2. Designed unit cells. (a), (b) Short axis (W) and long axis (L) dimensions of the optimized unit cells; (c) Corresponding phases response of the first eight elements for incident angles of 0°, 10°, 20°, 30°, 40°, and 50°, respectively; (d) Corresponding transmitted amplitudes of all elements for incident angles of 0°, 10°, 20°, 30°, and 40°, respectively
图 4 超表面的表征结果。(a)、(b)、(c) 分别代表入射角为0°、20°和40°时的仿真结果,从左到右入射光偏振态为X、Y、A、B、L、R偏振光,每个插图的上排是不同偏振态入射的焦平面光斑分布图,下排为对应的归一化Stokes参数
Figure 4. Characterization results of the metasurface. (a), (b), and (c) Simulation results at incident angles of 0°, 20°, and 40°, respectively. The incident light from left to right corresponds to X, Y, A, B, L, and R polarization states. The upper rows of every illustration are facula distributions at the focal plane. The lower rows are calculated normalized Stokes parameters corresponding with the upper one
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