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在光学旋转多普勒效应研究中,常用的是拉盖尔-高斯(LG)光束,其复振幅在柱坐标系下可以表示为
${{E}}(r,\varphi ,{\textit{z}}) = {{E}}(r,{\textit{z}})\exp (il\varphi )$ ,其中${{E}}$ 为电场强度,$\exp (il\varphi )$ 为螺旋相位,$r$ 为极径,${\textit{z}}$ 为传播距离,$l$ 为拓扑荷数。光束的能流密度矢量由坡印廷矢量表示,由于存在螺旋波前,涡旋光束的坡印廷矢量与光束传播方向存在夹角$\alpha $ ,且$\sin \alpha = l/kr$ ,其中$k$ 为角波数。涡旋光与球面波干涉会出现螺旋状条纹,可用于微小位移的测量[13]。在速度投影模型中,光束照射到物体表面发生反射,物体对出射光的频率进行调制,产生频移,大小可表示为
$\Delta f = {f_0}v/c$ ,其中${f_0}$ 为入射光频率,$c$ 为光速,$v$ 为物体速度在光束坡印廷矢量方向上的投影。当涡旋光垂直入射到物体的旋转中心时,如图1(a)所示。涡旋光束的坡印廷矢量分布可以近似为半顶角为$\alpha $ 的圆锥,取物体表面任意散射点为研究对象,该处的坡印廷矢量方向如向量$p$ 所示。将散射点速度沿着坡印廷矢量方向进行投影,如图1(b)所示,可得光束在该散射点上产生的多普勒频移为:图 1 (a)涡旋光垂直入射到旋转物体表面;(b)涡旋光坡印廷矢量分布示意图
Figure 1. (a) A vortex beam illuminates on the surface of a rotating object vertically; (b) Pointing vector distribution of vortex beam
$$\Delta f = \cos \;\beta \frac{{v{f_0}}}{c}$$ (1) 其中,
$\;\beta $ 满足$\alpha {\rm{ + }}\;\beta {\rm{ = }}{\pi / 2}$ 。在速度投影模型中,光轴与物体转轴重合,涡旋光斑上的每个散射点旋转运动的半径为$r$ ,速度$v = \varOmega r$ 。可得旋转多普勒频移:$$\Delta f = \sin \alpha \frac{{v{f_0}}}{c} = \frac{{l\lambda }}{{2\pi r}}×\frac{{{f_0}\varOmega r}}{c} = \frac{{l\varOmega }}{{2\pi }}$$ (2) 式中:
$\varOmega $ 表示物体旋转角速度;$c$ 为光速。 -
考虑涡旋光任意入射条件下的光学旋转多普勒效应,即分析光斑在平板物体表面的相对位置,约束要素包括三个:(1)光斑中心与物体旋转中心的距离;(2)光斑的偏心率;(3)光斑中心、物体旋转中心的连线与椭圆形光斑长轴的夹角。其中(2)和(3)是由于光束倾斜入射引起的。
由于倾斜照射,物体表面的涡旋光斑将由圆环状变为椭圆环状,光束传播轴与物体转轴之间的夹角为
$\gamma $ ,最小为0;光斑中心偏离物体旋转中心,横向位移为$d$ ;光斑中心、物体旋转中心的连线与椭圆形光斑长轴的夹角为$\varphi $ ,以$Oy'$ 轴为基准,逆时针为正,顺时针为负,如图2(a)所示。建立两个坐标系,分别为光束坐标系$O - xy{\textit{z}}$ 与光斑坐标系$O' - xy{\textit{z}}$ ,如图2(b)所示。则光束坐标系到光斑坐标系的坐标转换矩阵为${{\boldsymbol{M}}_x}( - \gamma )$ 。图 2 (a)涡旋光任意入射到旋转物体表面;(b)物体表面光束坐标系与光斑坐标系
Figure 2. (a) A vortex beam illuminates on the surface of a rotating object at general incidence; (b) Coordinate systems of the vortex beam and beam spot on the object surface
在光束坐标系
$O - xy{\textit{z}}$ 中,坡印廷矢量在$(r,0,0)$ 处的方向可以表示为:$$\overrightarrow {{p_{{x_0}}}} = \frac{1}{{2\pi r}}{(0, - l\lambda , - \sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} )^{\rm T}}$$ (3) 在光束横截面上,光场中任意点
$A$ 的坡印廷矢量方向为:$$ \begin{split} \overrightarrow {{p_A}} = \overrightarrow {{p_{{x_0}}}} \times {{\boldsymbol{M}}_{\textit{z}}}(\theta ) = \frac{1}{{2\pi r}}( - l\lambda \sin \theta ,l\lambda \cos \theta ,- \sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} )^{\rm T} \end{split} $$ (4) 式中:
$\theta $ 为光束横截面上位置矢量$\overrightarrow {OA} $ 与$O{\rm{ - }}x$ 轴的夹角;矩阵${{\boldsymbol{M}}_{\textit{z}}}(\theta )$ 表示空间坐标系内绕${\textit{z}}$ 轴旋转$\theta $ 的坐标转换矩阵。在光斑坐标系中,物体旋转中心
$Q$ 的坐标为$( - \rm d\sin \varphi , - d\cos \varphi ,0)$ ,$A$ 点在光斑上的投影点为$A'$ ,$\overrightarrow {O'A'} $ 与$O'x$ 轴的夹角为$\theta '$ ,且$r$ 与$r'$ 满足$r'\cos \theta ' = r\cos \theta $ ,$r'\sin \theta ' = r\sin \theta \sec \gamma $ ,则:$$\overrightarrow {QA'} = {(r'\cos \theta ' + {\rm{d}}\sin \varphi ,r'\sin \theta ' + {\rm{d}}\cos \varphi ,0)^{\rm T}}$$ (5) $A'$ 点的速度矢量为:$$ \begin{split} \overrightarrow {{v_{A'}}} =& \overrightarrow \varOmega \times \overrightarrow {QA'} = \varOmega ( - r'\sin \theta ' - {\rm{d}}\cos \varphi , \\ & r'\cos \theta ' + {\rm{d}}\sin \varphi ,0{)^{\rm T} } \end{split} $$ (6) $A'$ 点的坡印廷矢量方向与$A$ 点相同,在光斑坐标系中,$A'$ 点的坡印廷矢量为:$$ \begin{split} \overrightarrow {{P_{A'}}} =& {M_x}( - \gamma ) \cdot \overrightarrow {{P_A}} + {(0,r\sin \gamma ,r\tan \gamma \sin \gamma )^{\rm T}}{\rm{ = }}\\ &\dfrac{1}{{2\pi r}}( - l\lambda \sin \theta ,l\lambda \cos \theta \cos \gamma + \sin \gamma \sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} + \\ & r\sin \gamma \lambda \cos \theta \sin \gamma - \cos \gamma \sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} + \\ &{r\tan \gamma \sin \gamma )^{\rm T}} \end{split} $$ (7) 此时,光斑上
$A'$ 点发生的多普勒频移为:$$ \begin{split} \Delta f =& \dfrac{f}{c}\overrightarrow {{v_{A'}}} \cdot \overrightarrow {{P_{A'}}} {\rm{ = }}\frac{{l\varOmega }}{{2\pi }}\Bigg[\dfrac{{\sin \theta \sin \theta '}}{{\cos \gamma }} + \cos \gamma \cos \theta \cos \theta ' + \\ &\dfrac{d}{r}(\sin \theta \cos \varphi - \cos \theta \sin \varphi \cos \gamma )\Bigg] + \\ &\dfrac{{f\sin \gamma (r\cos \theta ' - d\sin \varphi )}}{{c2\pi r}}\sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} \end{split} $$ (8) 同时,
$\theta '$ 与$\theta $ 满足投影关系,当$\theta $ 为$\pi /2$ 的整数倍时,其数值等于$\theta '$ ,将公式(8)改写为:$$ \begin{split} \Delta f =& \dfrac{f}{c}\overrightarrow {{v_{A'}}} \cdot \overrightarrow {{P_{A'}}} {\rm{ = }}\frac{{l\varOmega }}{{2\pi }}\Bigg[\dfrac{{{{\sin }^2}\theta }}{{\cos \gamma }} + \cos \gamma {\cos ^2}\theta +\\ & \dfrac{d}{r}(\sin \theta \cos \varphi - \cos \theta \sin \varphi \cos \gamma )\Bigg]+ \\ &\dfrac{{f\sin \gamma (r\cos \theta - d\sin \varphi )}}{{c2\pi r}}\sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} \end{split} $$ (9) 其中,中括号内的部分表示
$A'$ 点发生的旋转多普勒频移,后一项为线性多普勒频移。当采用拓扑荷数为$ \pm l$ 的叠加态涡旋光作为探测光时,${\rm{ + }}l$ 与$ - l$ 的坡印廷矢量在散射点速度方向上的投影大小相同,方向相反,产生的旋转多普勒频移大小相同符号相反,而线性多普勒效应完全相同。由于叠加态涡旋光的两种成分相互干涉,检测信号是两成分的频率差。则线性频移相减抵消,而旋转多普勒频移加倍。因此,出射光的旋转多普勒频移为:$$ \begin{split} \Delta f{\rm{ = }}&\dfrac{{l\varOmega }}{\pi }\Bigg[\frac{{{{\sin }^2}\theta }}{{\cos \gamma }} + \cos \gamma {\cos ^2}\theta + \\ & \dfrac{d}{r}(\sin \theta \cos \varphi - \cos \theta \sin \varphi \cos \gamma )\Bigg] \end{split} $$ (10) 当
$\gamma = 0$ 且$\varphi {\rm{ = }}0$ 时,即只存在横向位移$d$ ,频移为:$\Delta f = {{l\varOmega } / \pi }(1 + {d / r}\sin \theta )$ ;当$d = 0$ 且$\varphi {\rm{ = }}0$ 时,即只存在倾斜角$\gamma $ ,频移为:$\Delta f = {{l\varOmega } / \pi }({{{{\sin }^2}\theta } / {\cos \gamma }} + \cos \gamma {\cos ^2}\theta )$ ;当$\gamma $ 、$d$ 、$\varphi $ 都为0时,即涡旋光传播轴与物体转轴重合,旋转多普勒频移$\Delta f = {{l\varOmega } / \pi }$ ,这与已有研究得出的理论公式完全吻合,同时也证明了这一理论结果的普适性。由公式(10)可得,涡旋光任意入射条件下,旋转多普勒频移会受到
$\gamma 、d、\varphi$ 等因素的影响,同时光斑上每一点对应一个频移值。为了提取出物体的旋转频率,选取$\theta $ 的特殊值点:$$\begin{array}{l} \theta = 0,\;\Delta f = \dfrac{{l\varOmega }}{\pi }\Bigg(\cos \gamma - \dfrac{d}{r}\sin \varphi \cos \gamma \Bigg) \\ \theta = \dfrac{\pi }{2},\Delta f = \dfrac{{l\varOmega }}{\pi }\Bigg(\dfrac{1}{{\cos \gamma }} + \dfrac{d}{r}\cos \varphi \Bigg) \\ \theta = \pi ,\;\Delta f = \dfrac{{l\varOmega }}{\pi }\Bigg(\cos \gamma + \dfrac{d}{r}\sin \varphi \cos \gamma \Bigg) \\ \theta = \dfrac{{3\pi }}{2},\Delta f = \dfrac{{l\varOmega }}{\pi }\Bigg(\dfrac{1}{{\cos \gamma }} - \dfrac{d}{r}\cos \varphi \Bigg) \end{array} $$ (11) 通过观察公式(11)可以得到:
$$ {f}_{\rm{mod}}=\sqrt{\frac{\Delta {f}_{0}+\Delta {f}_{\pi }}{2}\cdot \frac{\Delta {f}_{\pi /2}+\Delta {f}_{3\pi /2}}{2}}=\frac{l\varOmega }{\pi }$$ (12) 该结果不包含
$\gamma $ 、$d$ 、$\varphi $ 等因素,可分析得到物体的旋转频率$f = {f_{\boldsymbolod }}/2l$ 。为了直观地观察涡旋光在任意入射条件下的旋转多普勒频移分布规律,根据公式(10),设定参数
$\gamma = {30^\circ },\varphi {\rm{ = }}{30^\circ },d = 1\;\rm mm$ ,涡旋光半径$r = 4\;\rm mm$ ,拓扑荷数$l = \pm 18$ ,物体旋转频率$f = 50\;\rm Hz$ ,可得仿真的频率曲线如图3所示。
Rotational frequency detection of spinning objects at general incidence using vortex beam (Invited)
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摘要: 涡旋光是一种携带轨道角动量的空间结构光束,照射到旋转的平板物体表面时频率会发生移动,这一现象被称为光学旋转多普勒效应,通过测量光束频移可获得平板物体的旋转速率。频移受光束入射条件的影响,通过揭示入射条件影响规律,可实现任意入射条件下的旋转物体转速测量。首先,建立了速度投影模型,分析了光学旋转多普勒效应的产生机理。其次,通过理论推导得出了涡旋光任意入射条件下的旋转多普勒频移分布规律,并提出了提取物体旋转频率的理论方法。最后,搭建了旋转多普勒效应的实验装置,采用拓扑荷数为
$ \pm 18$ 的叠加态拉盖尔-高斯光束在四种不同入射条件下测量旋转多普勒频移谱,将实验频谱与理论频移曲线结合,测得物体的旋转频率,相对误差低于1%。Abstract: The vortex beam is a kind of spatially structured optical beam carrying orbital angular momentum, whose frequency shifts when it illuminates the surface of a rotating object. This phenomenon, known as the optical rotational Doppler effect (RDE), can be used to obtain the rotation frequency of a flat object by measuring the frequency shift. While the frequency shift is influenced by the incident condition, by revealing the influencing law of incident condition, the rotational frequency of the object can be measured directly. Firstly, a method of velocity projection was used to analyze the mechanism of optical RDE. Then, the rotational Doppler frequency shift distribution law at general incidence of vortex beam was obtained, and the theoretical method of extracting the rotational frequency was proposed. In the end, an experiment of RDE using a superimposed Laguerre-Gaussian beam with topological charge$l = \pm 18$ was set up, and rotational Doppler frequency shift spectrum at 4 incident conditions was obtained. The experimental spectrum and the theoretical result were combined, then rotational frequency of the object could be extracted with an error less than 1%. -
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