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以热光关联成像理论为基础,探测面上的二阶关联函数有如下形式[15]:
$$ \Delta G^{(2)}=\left\langle I_{1}\left(r_{1}\right) I_{2}\left(r_{2}\right)\right\rangle \\ =\left\langle I_{1}\left(r_{1}\right)\right\rangle\left\langle I_{2}\left(r_{2}\right)\right\rangle+\left|\left\langle E_{1}^{*}\left(r_{1}\right) E_{2}\left(r_{2}\right)\right\rangle\right|^{2} $$ (1) 式中:
$E_{1}^{*}\left(r_{1}\right)、 I_{1}\left(r_{1}\right)、 E_{2}\left(r_{2}\right)、 I_{2}\left(r_{2}\right)$ 分别表示探测器D1和D2的光场复共轭函数和强度分布函数;$\langle\;\rangle$ 为平均值算符;$|\;|^{2}$ 为模方算符。根据光场传递函数,热光非定域边缘增强成像方案光路末端的探测面上的光场表示为:
$$ \begin{split} E_{i}\left(r_{i}\right)=&E_{0}\left(r_{0}\right) {\rm{exp }}\left\{i k\left[8 f+1 / 2 f\left(r_{i}^{2}+r_{0}^{2}\right)-i \pi\right]\right\}\times \\ & F_{i}\left(r_{i}\right) \delta\left(r_{i}-r_{0}\right) \end{split} $$ (2) 式中:
$E_{0}\left(r_{0}\right)$ 表示赝热光光场函数。当i=1时,$F_{i}\left(r_{i}\right)$ 表示相位物体的传递函数,$F_{1}=\exp \left[i \varphi\left(r_{1}\right)\right]$ ;当i=2时,$F_{i}\left(r_{i}\right)$ 表示分数阶涡旋滤波器的传递函数,$F_{2}= $ $ \exp \left[i Q \varphi\left(r_{2}\right)\right]$ ,Q表示分数阶滤波器的OAM拓扑荷值。将两个探测面的光场代入热光场二阶关联函数中即可得到相位物体的非定域边缘增强函数:$$ \Delta G^{(2)}\left(l_{d}, l_{r}\right)=I_{0}^{2}\left\{1+\sum_{l_{d}} \sum_{l_{r}} A_{l_{d}}^{2} A_{l_{r}}^{2} \delta\left(l_{d}-l_{r}\right)\right\} $$ (3) 式中:I0表示赝热光光场强度;ld、lr分别表示物体和涡旋滤波器的轨道角动量谱的本征值;
$A_{l_{d}}、A_{l_{r}}$ 分别表示物体和分数阶涡旋滤波器在轨道角动量谱上的各个分量的权重,如图6所示。由此可见,分数阶热光非定域螺旋相衬成像的二阶关联值可以看成是单独的整数阶涡旋滤波的相干叠加。在没有相位变化的部分,OAM谱只有l=0的低频成分,因此二阶关联值在l=0时最高,随着分数阶OAM拓扑荷值Q从0~1变化,在整数阶拓扑荷l=0的分量逐渐减小,因此,其背景关联值逐渐降低;相反,在存在相位阶跃的边缘,二阶关联值在l=0时最小,在l=1时达到最大。因此出现了边缘信号与背景亮度的反转。
Non-local spiral phase contrast imaging with thermal light (Invited)
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摘要: 边缘增强作为一种图像处理技术,对于研究物体边界信息有着重要的应用。根据热光鬼成像的关联机制,将鬼成像与螺旋相衬成像技术相结合,把相位物体与涡旋滤光器非定域地放在热光鬼成像系统的探测光路和参考光路中,建立了相位物体边缘增强识别系统。实验结果表明,采用具有分数阶轨道角动量拓扑荷的涡旋滤波器可以实现轨道角动量值在0~1范围内的相位物体的边缘渐变性增强效果。随着轨道角动量拓扑荷数的增加,相位物体的边缘增强效果会越明显。非定域螺旋相衬成像相较于传统的螺旋相衬成像方案,突破了被测物体和涡旋滤波器在同一光路,并且滤波器必须放置在频谱面的空间局域性限制,解除了对主动照明光源相干性的要求,增强了涡旋滤波相衬成像系统的泛化能力。Abstract: As an image processing technology, edge enhancement has important applications for studying the boundary information of objects. According to the correlation theory of thermal light ghost imaging, a phase object edge enhancement recognition system was established by combining the ghost imaging with the spiral phase contrast imaging technology. The phase object and the vortex filter were placed in the signal and idle optical path of the ghost imaging system non-locally. The results prove that by using the vortex filter with the fractional orbital angular momentum (OAM) from 0 to 1, the gradual edge enhancement of the phase object can be realized. The higher the OAM topological charge, the more obvious edge enhancement effect will be. Compared with the traditional spiral phase contrast imaging scheme, the limitation of the filter in the spatial frequency spectrum in the spectral plane is broken by the non-local spiral phase contrast imaging scheme. In addition, the coherence requirement of the active illumination light source is released, and the generalization ability of the spiral phase contrast imaging system is enhanced.
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图 3 传统边缘增强成像实验结果。(a)和(c)分别为振幅物体与相位物体成像结果;(b)和(d)为涡旋光轨道角动量拓扑荷为1的傅里叶频谱滤波的边缘增强实验结果
Figure 3. Experimental results of the traditional edge-enhanced imaging. (a) and (c) are the traditional images of the amplitude object and phase object, respectively; (b) and (d) are the Fourier spectrum edge enhanced experimental results with orbital angular momentum topological charge of 1
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