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光的一阶相干性表征了光场与光场之间的干涉能力,这里用杨氏双缝干涉实验来进一步说明。图2(a)为杨氏双缝干涉实验的原理图,光经过小孔
$ {S}_{1} $ 和$ {S}_{2} $ 后在干涉屏表面P点的振幅分布为:图 2 (a)杨氏双缝干涉实验原理图;(b) Hanbury Brown-Twiss星体干涉仪的原理图
Figure 2. (a) Schematic diagram of Young's double slit interference experiment; (b) Schematic diagram of the Hanbury Brown-Twiss stellar interferometer
$$ \begin{array}{c}E\left(\overrightarrow{r},{t}\right)={K}_{1}E\left(\overrightarrow{{r}_{1}},{t}_{1}\right)+{K}_{2}E\left(\overrightarrow{{r}_{2}},{t}_{2}\right)\end{array} $$ (1) 式中:
$ E\left(\overrightarrow{{r}_{1}},{t}_{1}\right) $ 表示在$ {t}_{1} $ 时刻通过小孔$ {S}_{1} $ 的光在P点的振幅分布;$ E\left(\overrightarrow{{r}_{2}},{t}_{2}\right) $ 表示在$ {t}_{2} $ 时刻通过小孔$ {S}_{2} $ 的光在P点的振幅分布。P点的强度分布为:$$ \begin{split} {I}\left(\overrightarrow{r},{t}\right)=&{\left|{K}_{1}\right|}^{2}\left\langle{{\left|E(\overrightarrow{{r}_{1}},{t}_{1})\right|}^{2}}\right\rangle+{\left|{K}_{2}\right|}^{2}\left\langle{{\left|E(\overrightarrow{{r}_{2}},{t}_{2})\right|}^{2}}\right\rangle+\\ &2Re\left[{K}_{1}^{\mathrm{*}}{K}_{2}\left\langle{{E}^{\mathrm{*}}\left(\overrightarrow{{r}_{1}},{t}_{1}\right)E\left(\overrightarrow{{r}_{2}},{t}_{2}\right)}\right\rangle\right] \end{split} $$ (2) 其中,
$ {(\cdot )}^{\mathrm{*}} $ 表示求复共轭,等号右边第三项为干涉项,为了更方便地描述干涉特性,定义一阶相干函数为:$$ \begin{array}{c}{G}^{\left(1\right)}\left(\overrightarrow{{r}_{1}},\overrightarrow{{r}_{2}};{t}_{1},{t}_{2}\right)=\left\langle{{E}^{\mathrm{*}}\left(\overrightarrow{{r}_{1}},{t}_{1}\right)E\left(\overrightarrow{{r}_{2}},{t}_{2}\right)}\right\rangle\end{array} $$ (3) 二阶相干函数表征的是光场强度之间的干涉特性,其可以表示为:
$$ {G}^{\left(2\right)}\left(\overrightarrow{{r}_{1}},\overrightarrow{{r}_{2}};{t}_{1},{t}_{2}\right) =\left\langle{{E}^{\mathrm{*}}\left(\overrightarrow{{r}_{1}},{t}_{1}\right)E\left(\overrightarrow{{r}_{1}},{t}_{1}\right){E}^{\mathrm{*}}\left(\overrightarrow{{r}_{2}},{t}_{2}\right)E\left(\overrightarrow{{r}_{2}},{t}_{2}\right)}\right\rangle$$ (4) 20世纪50年代中期,曼彻斯特大学的Hanbury Brwon和Twiss为了测量双星的角半径,设计了一个光强干涉实验[64],如图2(b)所示,该实验就是著名的HBT实验。在HBT实验之前,所有的干涉仪都是基于相干光场的干涉现象,也即光场的一阶相干性。而HBT星体干涉仪利用了二阶相干性,其测量方式是对两个光强信号进行关联运算,从关联运算结果获取所要测量的星体角半径。
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关联成像对于参考光与单像素探测器在信号光路收集的一系列光强做关联运算,得到物体的振幅及相位的空间分布信息,这是一种利用光场的二阶相干性对目标物体实现重构的过程。基于赝热光调制和空间光调制器调制的关联成像原理分别如图3(a)和图3 (b)所示。
在赝热光关联成像系统中,如图3(a)所示,激光经旋转毛玻璃调制产生的赝热光被分束器分成两束,一束经过距离L被CCD探测,得到赝热光强度的空间分布I(x,y),这一路被称为参考光路;另一束经过相同的距离L照射在物体表面,物体的透射光由收集透镜会聚后被没有空间分辨能力的桶探测器接收。桶探测器探测到的光强可表示为:
$$ \begin{array}{c}y=\displaystyle\iint I\left(x,y\right)\cdot O\left(x,y\right){\rm{d}}x{\rm{d}}y\end{array} $$ (5) 式中:
${O}({x},{y})$ 为物体的透过率空间分布。当毛玻璃旋转时,其调制的结构光场也在不断变化,不同时刻CCD拍摄到的结构光场强度分布可表示为
$ {I}_{t}(x,y) $ , 与每一结构光场对应的光强测值为:$$ \begin{array}{c}{y}_{t}=\iint {I}_{t}\left(x,y\right)\cdot O\left(x,y\right){\rm{d}}x{\rm{d}}y\end{array} $$ (6) 故,参考光路结构光场与信号光路单像素探测器收集到的光强值的二阶关联为:
$$ \begin{array}{c}{G}^{\left(2\right)}\left(x,y\right)=\left\langle{{I}_{t}\left(x,y\right){y}_{t}}\right\rangle\end{array} $$ (7) 一次完整的关联成像过程,需要CCD多次拍摄结构光场,多个结构光场的强度分布可用向量进行表示,记为
$ \overrightarrow{{I}_{t}}(x,y) $ ,同样地,桶探测器探测的一系列探测值也可记为$ \overrightarrow{{y}_{t}} $ 。则二阶关联为:$$ \begin{array}{c}\overrightarrow{{G}^{\left(2\right)}}\left(x,y\right)=\left\langle{\overrightarrow{{I}_{t}}\left(x,y\right)\overrightarrow{{y}_{t}}}\right\rangle\end{array} $$ (8) 对结构光场
$ \overrightarrow{{I}_{t}}(x,y) $ 和探测值$ \overrightarrow{{y}_{t}} $ 进行零均值化处理,可以降低串扰项的影响,故物体${O}({x},{y})$ 的重构图像$ \overrightarrow{{G}^{\left(2\right)}}\left(x,y\right) $ 可以表示为:$$ \overrightarrow{{G}^{\left(2\right)}}\left(x,y\right)=\left\langle{\overrightarrow{{[I}_{t}}\left(x,y\right)-\left\langle{\overrightarrow{{I}_{t}}\left(x,y\right)}\right\rangle]\cdot [\overrightarrow{{y}_{t}}-\left\langle{\overrightarrow{{y}_{t}}}\right\rangle]}\right\rangle $$ (9) 相较于传统的成像技术,关联成像作为一种新型的成像技术,利用一个单像素探测器即可完成图像的重构,在特殊波段、弱光成像等领域有着显著的优势。
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从文中可以看出,关联成像的必要条件之一是实现光场强度的涨落。为了实现这种涨落,研究人员设计了利用旋转毛玻璃产生赝热光场或利用调制器件产生变化的结构照明光场等多种方法。
由于热光场具有强度随机涨落的性质,满足经典关联成像的需要,因此在最开始时,科研人员考虑使用热光源实现关联成像。但是,热光是由原子或分子自发辐射产生,其相干时间远小于光电探测器的响应时间,探测器一般无法探测到热光的涨落。随着研究深入,Martienssen和Spiller发明了赝热光源,其既可以模拟热光场的涨落性质,又具有较长的相干时间,可以被探测器测量。常用的产生赝热光场的办法之一是用激光光束照射旋转的毛玻璃。国防科技大学陈平形课题组证明了激光照射旋转毛玻璃产生的赝热光场符合热光场强度随机涨落的统计特性,也具有满足探测器测量要求的相干时间。由于旋转毛玻璃产生的赝热光场是未知的,因此在进行关联运算时,仍需设置参考光路来探测赝热光场的强度空间分布信息,这使得整个系统光路较为复杂,且成像信噪比较低。
随着科学技术的不断进步,数字微镜阵列(DMD)、液晶空间光调制器和LED阵列问世,为关联成像提供了更多思路。将激光光束入射至DMD、SLM等调制器件表面可以产生特定的、已知的结构光场,从而省略了参考光路,提高了重构图像的信噪比。由关联成像的成像机制可知,空间光调制器件的刷新速率决定了成像速度,但目前,此类器件中最快的刷新速率也只有22.7 kHz,远达不到对于高速运动物体的成像需求。为了进一步提升成像速度,破除现有光场调制器件对关联成像速度限制的问题,北京航空航天大学孙鸣捷课题组提出了一种基于LED阵列的高速空间结构光场调制方法及成像系统。LED阵列的调制速率达到了2.5 MHz,成像速度达到了5 kHz[66-67]。但基于LED阵列的关联成像也有其固有缺点,LED的发光波长已然是确定的,失去了关联成像与任何光源的兼容优势。
虽然LED阵列及DMD等空间光调制器的使用在很大程度上优化了关联成像系统,但是上述器件自身的缺陷仍局限着关联成像的适用范围,因此,人们也在不断地探究其他结构光场调制方式。早在20世纪70年代,基于哈达玛变换光学就利用掩膜版实现了空间复用成像。之后随着光电调制器件的诞生,掩膜版由于调制精度与调制速率的限制逐渐被放弃。如今,由于掩膜版加工工艺的进步,掩膜版已经具有了相当良好的调制精度,并且,掩膜版在光谱响应方面的优势一直不能被其他调制器件替代。利用掩膜版与待成像物体之间的相对运动就可以实现关联成像所必需的“涨落”条件,所以,基于掩膜版调制的关联成像近些年来又正在兴起,包括笔者课题组在内的许多研究人员针对基于掩膜版调制的关联成像作了进一步的研究。
Development and application of mask modulated correlated imaging (Invited)
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摘要: 关联成像作为一种新型的计算成像技术,使用不具备空间分辨能力的单像素探测器,结合空间光场调制技术,运用关联算法重构出目标的二维空间信息,成为近二十多年来广泛关注的研究课题。单像素探测器和结构光调制器作为关联成像中的两个核心要素,其性能直接决定了关联成像的各项指标。单像素探测器往往具有极高的光谱响应范围和工作带宽,但结构光调制器却少有与之匹配的性能。因此在一定程度上,结构光调制器的更新历程决定了关联成像技术的发展史。到目前为止,在关联成像中常用的结构光调制器有毛玻璃、空间光调制器、LED阵列以及掩膜版。其中,掩膜版作为一款有着悠久历史的结构光调制器,目前依旧是关联成像中空间调控的重要手段,并发挥着不可替代的作用。以关联成像的基本概念和发展历程为铺垫,着重介绍了一些基于掩膜版调制关联成像技术的工作原理及应用前景,并对非光学波段的掩膜版关联成像工作进行了简要的总结。Abstract: Correlated imaging, as a novel computational imaging technology, uses a single pixel detector without spatial resolution capability and combines with spatial light modulation technology to reconstruct two-dimensional spatial information of targets by correlation algorithm. It has been a research topic of widespread concern for two decades. Single pixel detector and structured light modulator are two core elements in correlated imaging, and their performance directly determines the property of correlated imaging. Single pixel detectors often have a very high spectral response range and working bandwidth. In these respects, structured light modulators rarely match the performance of detectors. Therefore, to some extent, the renewal process of the structured light modulator determines the development of the correlated imaging technology. So far, the common structured light modulators used in correlated imaging include ground-glass, spatial light modulator, LED array and masks. Among them, masks, which have been used as a structured light modulator with a long history, are still an important choice for spatial modulation in correlated imaging and play an irreplaceable role. This paper started with the basic concept and development process of correlated imaging, analysed the working principles and application prospects of some existing correlated imaging technologies based on mask modulation, and briefly summarized the work of mask-modulated correlation imaging in non-optical wavebands.
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Key words:
- correlated imaging /
- computational imaging /
- structured light modulator /
- mask
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