基于复振幅实验测量的散斑场特性的研究
摘要
散斑场是相干光经随机表面散射后形成的随机强度分布图样,对它的统计特性的研究是一个多年来倍受关注的课题。散斑场强度零点周围形成相位涡旋,由于其中含有丰富的光波复振幅和相位信息,对相位涡旋的研究也越来越引起人们的广泛关注并逐渐成为非线性光学、激光物理和光信息处理等领域的研究热点。利用散斑场与参考光的干涉技术和傅立叶变换法实现随机光场的实验提取后,人们逐步开始了散斑场相位涡旋的实验研究。菲涅耳极深区散斑中包含了丰富的表面信息,它比远场散斑更能体现出随机表面高度分布的统计信息,因此它必将对关于随机表面及其标定的研究方面有一定的帮助。
本学位论文利用散斑场和参考光的干涉技术和傅立叶变换法成功地实现了各种散斑场的实验提取,研究了散斑的强度、复振幅、相位和相位涡旋等的统计特性和传播规律。理论上,用Frankenthal提出的多刻度相位屏模型解释了大散射角度处散斑场的区域分形结构、利用圆型高斯散斑与直透光的叠加理论解释了菲涅耳极深区散斑的强度概率密度特征和对比度;实验上,设计散斑场与参考光的干涉装置并实现了不同散射角处散斑场复振幅和相位的实验测量,在干涉装置中又设计了一个显微系统从而实现了菲涅耳极深区散斑的实验测量和提取;数值模拟计算中,实现了菲涅耳极深区散斑场产生,并结合实验中得到的干涉图样数据完成了不同散射角处及菲涅耳极深区散斑场的数字提取。本论文共分六章。
第一章是对本课题的研究历史和现状以及论文相关的一些概念进行了综合性描述。内容包括随机表面的描述方法、主要的统计参量及其高度的统计函数;光散射的基本原理、散斑场的形成以及其在随机表面标定中的应用;相位涡旋的定义和一般性质,给出相位涡旋符号的定义方法和相关函数;讲述了干涉图样的傅立叶分析过程和散斑场复振幅和相位等参量的实验提取方法。
第二章对散斑场相位涡旋传播特性以及准孪生涡旋的性质进行了实验研究。我们首先用CCD记录散斑场和平面参考光的干涉图样,利用傅立叶变换法实现了散斑场复振幅以及相位分布的实验提取过程。发现在复振幅实部零值线和虚部零值线的切点处形成了一种新的相位奇异,与两条零值线相交形成的传统相位奇点周围单调螺旋变化的相位分布形式不同的是,若以该点为圆心做一个半径很小的圆周,在圆周上的一个最小相位值点开始沿着圆周行走一圈,我们将会看到相位的变化总是先增大到某一个最大值然后再减小到这个最小值,而且相位的取值范围也小于2π,即这里总是缺少某些相位值。结合相位涡旋的产生过程,我们称实部零值线和虚部零值线切点周围形成的为准孪生相位涡旋,即该点是一对孪生涡旋产生的临界状态。在理论研究散斑场强度的纵向自相关函数的基础上,我们在实验中研究了散斑场相位涡旋的纵向传播规律,发现在光强纵向相关长度范围内的不同传播距离处的观察面上,散斑场复振幅实部和虚部都是随机变化的,但是相位涡旋的相对位置及涡旋中心处实部零值线和虚部零值线的交角几乎不随传播距离而改变。
第三章对不同散射角度处散斑场的强度、复振幅、相位以及相位涡旋的统计特性进行了实验研究。首先介绍了在实验中利用散斑场和球面参考光的干涉图样提取出不同散射角处散斑场复振幅、相位和强度的方法,并对这些参量的统计特性进行了研究。在提取过程中我们可以通过控制傅立叶谱平面的大小而把CCD测量数据中的高频噪声滤掉,因此最终能够正确地统计出强度的分布规律,更重要的是利用这种方法我们成功地实现了不同散射角处散斑场复振幅和相位统计特性的实验研究。我们通过对不同散射角处散斑场强度的研究发现,散斑颗粒随着散射角的增大而被横向展宽,强度的自相关函数的变化趋势在各个方向上不再相同,呈各向异性,但是散斑场仍然严格服从高斯分布。根据对实验提取的不同散射角度处散斑场复振幅和相位的研究发现,由于复振幅和相位的空间分布特征随着散射角的增大发生了很大的变化,在相位奇异方面出现了一些特殊的现象和性质。如随着散射角的增大,相位奇异处光场的实部零值线和虚部零值线交角的统计分布向小角度方向趋近;相位奇异附近等光强曲线的离心率的平均值逐渐增大,有趣的是在大散射角度处出现了离心率大于1的特殊情况;相位涡旋的相关曲线在空间间隔较大的时候收敛的越来越缓慢等等。经过研究我们给出了导致这些特征产生的原因是,大散射角度处散斑场中出现了光场实部零值线和虚部零值线在某段区域相互重合而形成的相位奇异线。发现相位奇异线与传统相位奇异点周围的相位分布完全不同,在奇异线上相位不确定,它两侧的等相位曲线都平行于该奇异线分布,奇异线两端存在相位的突变。相位奇异线这个有趣的现象在传统的散斑场中很难发现。
第四章,我们用半导体泵浦激光器Verdi V-5(最大功率为5W)输出的较高功率激光做散射光源,从而使大散射角处的强度有所提高,并用电荷耦合器(CCD)对其进行测量。利用散斑与球面参考光的干涉技术和傅立叶变换的计算方法我们从实验中提取出大散射角处散斑的复振幅。根据对大散射角处散斑的强度相关函数的研究,我们发现该散斑的分形特征与空间距离ρ有关。在ρ比较小的区域内,分形指数α= 1,然而当ρ增大到某个一定数值以后分形指数α又小于1。显然该散斑的分形特征与通常的分形图样不同,它的分形指数因空间区域大小而异,我们称这种散斑为区域分形散斑。我们为这种区域分形散斑的强度相关函数整理出一个经验解析式,并用其拟合实验曲线求出分形指数α。根据区域分形散斑强度和复振幅的概率分布特征,我们证明了这种散斑仍然符合高斯分布规律。
第五章,我们首先利用基尔霍夫近似和格林函数理论对极度靠近随机表面区域(也可以称之为菲涅耳极深区)的散斑进行了理论分析,然后通过数字模拟产生了2维的菲涅耳极深区散斑。通过与产生该散斑的随机表面的高度分布特征进行对比我们发现,无论是散斑强度的分布规律还是其分形特征都与对应随机表面的高度分布有着密切的关系。我们计算出菲涅耳极深区散斑的强度概率密度,发现它并不是与远场散斑一样遵从负指数的分布规律。利用圆型高斯散斑场与直透光的叠加理论解释菲涅耳极深区散斑强度的统计规律。我们通过设计的显微系统对菲涅耳极深区散斑进行了实验测量,并分析了散斑强度分布特征和概率分布密度,为我们数字模拟得出的结论提供了实验依据。利用散斑场和参考光的干涉技术和傅立叶变换法实现了菲涅耳极深区散斑复振幅和相位的实验提取,提取的复振幅和相位的分布可以为研究散斑的形成过程和光波的传播规律提供实验依据。
第六章,总结了本学位论文所取得的主要成果和创新点,并简单地介绍了下一步要深入开展的研究工作。
Speckle pattern is the random intensity distribution produced by the coherent light scattering from a random surface. Moreover, great attention has been paid to the study of the statistical properties of speckle. There is abundant information of complex amplitude and phase in the phase vortices, forming around the zero-point of speckle intensity, so more and more papers pay attention to the investigation of phase vortices, and it is of great importance in many science and technology fields such as the nonlinear optics, the laser physics and the optical information processing, etc. After the experimental extraction of amplitude and phase distribution is realized, which is based on the interference technique of speckle fields with reference beam and the digital Fourier transform, most of the papers in the literature are focused on the experimental investigation of phase vortices. There is more abundant random surface information in the speckle pattern in the extremely deep Fresnel diffraction region than in the normal far field, so the statistical properties of speckle in the extremely deep Fresnel diffraction region would be interest in the study of the measurements of random surface.
Combining the interference technique of speckle fields and reference beam with the digital Fourier transform arithmetic, we have finished the experimental extraction of the amplitude, phase, intensity and phase vortices of speckle and studied their properties of statistical and the propagation etc. In the theoretical studies, based on the Frankenthal’s multiscale phase screens model, we have proposed a new empirical analytic expression of the intensity correlation function of the regional fractal speckle. We also have explained the intensity probability density and the contrast of the speckle in the extremely deep Fresnel diffraction region by the theory of the sum of zero-order diffraction and circular Gauss speckle. In experimental studies, we have designed an interference system of speckle and reference beam to extract the complex amplitude and the phase of random light field in far field. Moreover, we have realized the experimental measure and extraction of the speckle in the extremely deep Fresnel diffraction region with a microscope system. In the numerical simulations, we have tackled a series of problems in the extraction of random light fields, and we have calculated the speckle in the extremely deep Fresnel diffraction region based on the Kirchhoff approximation and Green’s function. The whole paper is divided into six chapters.
Chapter 1: Introduction. In this part, we give a summary and an overall review of the background and the current situation of the research on speckles as well as some conceptions about speckles. First, we introduce the characterization, the measurement, the primary statistical parameters and high statistical functions of random surfaces. Then we recite the fundamental theories of light scattering, the formation and the application of speckles; the definition and the character of the phase vortices; the Fourier-transform method of fringe-pattern and the experimental extraction of amplitude and phase of speckles.
Chapter 2: The experimental studies on phase vortices of speckles and their propagation properties. By recording the interference patterns of the speckle fields and the reference beam with the charge-coupled device (CCD), and using the digital Fourier transform technique, we realize the experimental extraction of the amplitude and the phase distribution of the speckle fields. We find that at the tangential points of the zero curves of the real and the imaginary parts, a new kind of speckle phase singularities may appear. Differing from the conventional singularities at the zero crossings of the real and imaginary parts with the monotonically spiral change of phase, this new kind of singularities has the property that the phase undergoing an increase and then a decrease around singular point and assuming nearly a symmetric distribution. Otherwise, the range of the phase around the singular point less than2π, that is to say, there always lose some phase around the tangential point. We introduce the concept of quasi twin phase vortices to explain the formation of the new kind of phase vortices. We also experimentally observe the propagation of the phase vortices of speckles based on a theoretical study of the longitudinal autocorrelation function of the speckle intensities. It is found that in planes at the different propagation distances but within a longitudinal correlation length, the real and the imaginary parts of the complex speckle fields vary considerably, but the position of the phase vortices and the angles of the zero crossings of the real and imaginary parts remain almost unchanged.
Chapter 3: The experimental studies on the statistical functions of speckle fields produced in different angle scattering based on the extraction of the complex amplitudes by use of interference beam. Using the interference patterns of speckle fields and the reference beam, we extract digitally the complex amplitudes, phases and intensities of speckles fields and then study experimentally their properties. The influences of the noises on the measurement of statistical functions, especially on that of the probability density function, appearing in the conventional method is satisfactorily eliminated. The experimental studies of the statistical properties of the complex amplitude and the phases of the speckle fields are also fulfilled. By the practical measurement of speckle fields produced in different angle scattering, we find that the speckles are laterally broadened gradually with the increase of the scattering angle and the average size of speckles become anisotropic. Such anisotropy brings no change in the probability density function of the speckle fields. The speckle fields produced in large angle scattering remain the circularly Gaussian distribution as that produced in the traditional small angle scattering. Based on the investigation of the amplitude and the phase extracted digitally, we find that there are some special properties of phase singularity when the scattering angle is large enough. With the increase of the scattering angle the spatial distribution of the amplitude and the phase have taken place great changes, and the probability of the angle between the zero-contour lines of real part and that of imaginary part tends to small value, and the average eccentricity of the intensity contours around the phase singularity is gradually increasing. Moreover, the most interesting thing is that the eccentricity is probably greater than 1 in large anger scattering, and the convergence of the correlation curve of phase vortices becomes slower with the spatial distance increasing. All of the properties originate from the appearance of the phase singularity line in larger angle scattering.
In chapter 4, we use the higher-power laser produced by semiconductor pump laser Verdi V-5 (the maximum power is 5W) to enhance the weak intensity in large angle scattering and record them with a charge-coupled device (CCD) camera. The complex amplitudes of the speckle are extracted experimentally according to the technique of the interference of the speckle and the spherical reference wave. By studying the intensity correlation function of the speckle, we find that the fractal property of the speckle is related to the spatial separate distanceρ. In the area of smallρ, the fractal exponentα= 1, but the fractal exponentαbecomes less than 1 whenρincrease to a certain number. Such speckle is named regional fractal speckle by us. We deduce an empirical analytic expression of the intensity correlation function of regional fractal speckle, and calculate the fractal exponentαby fitting the experimental curves with the empirical intensity correlation analytic expression. From the probability distribution properties of the intensities and the complex amplitudes, we demonstrate that the regional fractal speckle still obeys Gaussian distribution.
In chapter 5, we first make a theoretical analysis of the speckles in the region infinitely close to the random surface (also called the extremely deep Fresnel diffraction region) based on the Kirchhoff approximation and Green’s function theory, and then produce the 2D speckle field by numerical simulation. By comparing the 2D speckle field with the corresponding random surfaces, we find that both the intensity space distribution and the fractal property of the speckles depend on the random height distribution of the surfaces, and the intensity probability density of the speckles don’t obey negative exponential law. We explain the intensity probability density and the contrast of the speckle in the extremely deep Fresnel diffraction region by the theory of the sum of zero-order diffraction and circular Gauss speckle. We design a microscope system to detect the speckles in the extremely deep Fresnel diffraction region and find experimental evidence of the conclusions we reached in the numerical simulation. Based on the interference technique of speckle fields with reference beam and the digital Fourier transform, we realize the experimental extraction of the amplitude and phase of the speckles in the extremely deep Fresnel diffraction region.
In chapter 6, we sum up the main conclusions and the innovations of the dissertation, and briefly introduce the in-depth researches we will conduct.
本学位论文利用散斑场和参考光的干涉技术和傅立叶变换法成功地实现了各种散斑场的实验提取,研究了散斑的强度、复振幅、相位和相位涡旋等的统计特性和传播规律。理论上,用Frankenthal提出的多刻度相位屏模型解释了大散射角度处散斑场的区域分形结构、利用圆型高斯散斑与直透光的叠加理论解释了菲涅耳极深区散斑的强度概率密度特征和对比度;实验上,设计散斑场与参考光的干涉装置并实现了不同散射角处散斑场复振幅和相位的实验测量,在干涉装置中又设计了一个显微系统从而实现了菲涅耳极深区散斑的实验测量和提取;数值模拟计算中,实现了菲涅耳极深区散斑场产生,并结合实验中得到的干涉图样数据完成了不同散射角处及菲涅耳极深区散斑场的数字提取。本论文共分六章。
第一章是对本课题的研究历史和现状以及论文相关的一些概念进行了综合性描述。内容包括随机表面的描述方法、主要的统计参量及其高度的统计函数;光散射的基本原理、散斑场的形成以及其在随机表面标定中的应用;相位涡旋的定义和一般性质,给出相位涡旋符号的定义方法和相关函数;讲述了干涉图样的傅立叶分析过程和散斑场复振幅和相位等参量的实验提取方法。
第二章对散斑场相位涡旋传播特性以及准孪生涡旋的性质进行了实验研究。我们首先用CCD记录散斑场和平面参考光的干涉图样,利用傅立叶变换法实现了散斑场复振幅以及相位分布的实验提取过程。发现在复振幅实部零值线和虚部零值线的切点处形成了一种新的相位奇异,与两条零值线相交形成的传统相位奇点周围单调螺旋变化的相位分布形式不同的是,若以该点为圆心做一个半径很小的圆周,在圆周上的一个最小相位值点开始沿着圆周行走一圈,我们将会看到相位的变化总是先增大到某一个最大值然后再减小到这个最小值,而且相位的取值范围也小于2π,即这里总是缺少某些相位值。结合相位涡旋的产生过程,我们称实部零值线和虚部零值线切点周围形成的为准孪生相位涡旋,即该点是一对孪生涡旋产生的临界状态。在理论研究散斑场强度的纵向自相关函数的基础上,我们在实验中研究了散斑场相位涡旋的纵向传播规律,发现在光强纵向相关长度范围内的不同传播距离处的观察面上,散斑场复振幅实部和虚部都是随机变化的,但是相位涡旋的相对位置及涡旋中心处实部零值线和虚部零值线的交角几乎不随传播距离而改变。
第三章对不同散射角度处散斑场的强度、复振幅、相位以及相位涡旋的统计特性进行了实验研究。首先介绍了在实验中利用散斑场和球面参考光的干涉图样提取出不同散射角处散斑场复振幅、相位和强度的方法,并对这些参量的统计特性进行了研究。在提取过程中我们可以通过控制傅立叶谱平面的大小而把CCD测量数据中的高频噪声滤掉,因此最终能够正确地统计出强度的分布规律,更重要的是利用这种方法我们成功地实现了不同散射角处散斑场复振幅和相位统计特性的实验研究。我们通过对不同散射角处散斑场强度的研究发现,散斑颗粒随着散射角的增大而被横向展宽,强度的自相关函数的变化趋势在各个方向上不再相同,呈各向异性,但是散斑场仍然严格服从高斯分布。根据对实验提取的不同散射角度处散斑场复振幅和相位的研究发现,由于复振幅和相位的空间分布特征随着散射角的增大发生了很大的变化,在相位奇异方面出现了一些特殊的现象和性质。如随着散射角的增大,相位奇异处光场的实部零值线和虚部零值线交角的统计分布向小角度方向趋近;相位奇异附近等光强曲线的离心率的平均值逐渐增大,有趣的是在大散射角度处出现了离心率大于1的特殊情况;相位涡旋的相关曲线在空间间隔较大的时候收敛的越来越缓慢等等。经过研究我们给出了导致这些特征产生的原因是,大散射角度处散斑场中出现了光场实部零值线和虚部零值线在某段区域相互重合而形成的相位奇异线。发现相位奇异线与传统相位奇异点周围的相位分布完全不同,在奇异线上相位不确定,它两侧的等相位曲线都平行于该奇异线分布,奇异线两端存在相位的突变。相位奇异线这个有趣的现象在传统的散斑场中很难发现。
第四章,我们用半导体泵浦激光器Verdi V-5(最大功率为5W)输出的较高功率激光做散射光源,从而使大散射角处的强度有所提高,并用电荷耦合器(CCD)对其进行测量。利用散斑与球面参考光的干涉技术和傅立叶变换的计算方法我们从实验中提取出大散射角处散斑的复振幅。根据对大散射角处散斑的强度相关函数的研究,我们发现该散斑的分形特征与空间距离ρ有关。在ρ比较小的区域内,分形指数α= 1,然而当ρ增大到某个一定数值以后分形指数α又小于1。显然该散斑的分形特征与通常的分形图样不同,它的分形指数因空间区域大小而异,我们称这种散斑为区域分形散斑。我们为这种区域分形散斑的强度相关函数整理出一个经验解析式,并用其拟合实验曲线求出分形指数α。根据区域分形散斑强度和复振幅的概率分布特征,我们证明了这种散斑仍然符合高斯分布规律。
第五章,我们首先利用基尔霍夫近似和格林函数理论对极度靠近随机表面区域(也可以称之为菲涅耳极深区)的散斑进行了理论分析,然后通过数字模拟产生了2维的菲涅耳极深区散斑。通过与产生该散斑的随机表面的高度分布特征进行对比我们发现,无论是散斑强度的分布规律还是其分形特征都与对应随机表面的高度分布有着密切的关系。我们计算出菲涅耳极深区散斑的强度概率密度,发现它并不是与远场散斑一样遵从负指数的分布规律。利用圆型高斯散斑场与直透光的叠加理论解释菲涅耳极深区散斑强度的统计规律。我们通过设计的显微系统对菲涅耳极深区散斑进行了实验测量,并分析了散斑强度分布特征和概率分布密度,为我们数字模拟得出的结论提供了实验依据。利用散斑场和参考光的干涉技术和傅立叶变换法实现了菲涅耳极深区散斑复振幅和相位的实验提取,提取的复振幅和相位的分布可以为研究散斑的形成过程和光波的传播规律提供实验依据。
第六章,总结了本学位论文所取得的主要成果和创新点,并简单地介绍了下一步要深入开展的研究工作。
Speckle pattern is the random intensity distribution produced by the coherent light scattering from a random surface. Moreover, great attention has been paid to the study of the statistical properties of speckle. There is abundant information of complex amplitude and phase in the phase vortices, forming around the zero-point of speckle intensity, so more and more papers pay attention to the investigation of phase vortices, and it is of great importance in many science and technology fields such as the nonlinear optics, the laser physics and the optical information processing, etc. After the experimental extraction of amplitude and phase distribution is realized, which is based on the interference technique of speckle fields with reference beam and the digital Fourier transform, most of the papers in the literature are focused on the experimental investigation of phase vortices. There is more abundant random surface information in the speckle pattern in the extremely deep Fresnel diffraction region than in the normal far field, so the statistical properties of speckle in the extremely deep Fresnel diffraction region would be interest in the study of the measurements of random surface.
Combining the interference technique of speckle fields and reference beam with the digital Fourier transform arithmetic, we have finished the experimental extraction of the amplitude, phase, intensity and phase vortices of speckle and studied their properties of statistical and the propagation etc. In the theoretical studies, based on the Frankenthal’s multiscale phase screens model, we have proposed a new empirical analytic expression of the intensity correlation function of the regional fractal speckle. We also have explained the intensity probability density and the contrast of the speckle in the extremely deep Fresnel diffraction region by the theory of the sum of zero-order diffraction and circular Gauss speckle. In experimental studies, we have designed an interference system of speckle and reference beam to extract the complex amplitude and the phase of random light field in far field. Moreover, we have realized the experimental measure and extraction of the speckle in the extremely deep Fresnel diffraction region with a microscope system. In the numerical simulations, we have tackled a series of problems in the extraction of random light fields, and we have calculated the speckle in the extremely deep Fresnel diffraction region based on the Kirchhoff approximation and Green’s function. The whole paper is divided into six chapters.
Chapter 1: Introduction. In this part, we give a summary and an overall review of the background and the current situation of the research on speckles as well as some conceptions about speckles. First, we introduce the characterization, the measurement, the primary statistical parameters and high statistical functions of random surfaces. Then we recite the fundamental theories of light scattering, the formation and the application of speckles; the definition and the character of the phase vortices; the Fourier-transform method of fringe-pattern and the experimental extraction of amplitude and phase of speckles.
Chapter 2: The experimental studies on phase vortices of speckles and their propagation properties. By recording the interference patterns of the speckle fields and the reference beam with the charge-coupled device (CCD), and using the digital Fourier transform technique, we realize the experimental extraction of the amplitude and the phase distribution of the speckle fields. We find that at the tangential points of the zero curves of the real and the imaginary parts, a new kind of speckle phase singularities may appear. Differing from the conventional singularities at the zero crossings of the real and imaginary parts with the monotonically spiral change of phase, this new kind of singularities has the property that the phase undergoing an increase and then a decrease around singular point and assuming nearly a symmetric distribution. Otherwise, the range of the phase around the singular point less than2π, that is to say, there always lose some phase around the tangential point. We introduce the concept of quasi twin phase vortices to explain the formation of the new kind of phase vortices. We also experimentally observe the propagation of the phase vortices of speckles based on a theoretical study of the longitudinal autocorrelation function of the speckle intensities. It is found that in planes at the different propagation distances but within a longitudinal correlation length, the real and the imaginary parts of the complex speckle fields vary considerably, but the position of the phase vortices and the angles of the zero crossings of the real and imaginary parts remain almost unchanged.
Chapter 3: The experimental studies on the statistical functions of speckle fields produced in different angle scattering based on the extraction of the complex amplitudes by use of interference beam. Using the interference patterns of speckle fields and the reference beam, we extract digitally the complex amplitudes, phases and intensities of speckles fields and then study experimentally their properties. The influences of the noises on the measurement of statistical functions, especially on that of the probability density function, appearing in the conventional method is satisfactorily eliminated. The experimental studies of the statistical properties of the complex amplitude and the phases of the speckle fields are also fulfilled. By the practical measurement of speckle fields produced in different angle scattering, we find that the speckles are laterally broadened gradually with the increase of the scattering angle and the average size of speckles become anisotropic. Such anisotropy brings no change in the probability density function of the speckle fields. The speckle fields produced in large angle scattering remain the circularly Gaussian distribution as that produced in the traditional small angle scattering. Based on the investigation of the amplitude and the phase extracted digitally, we find that there are some special properties of phase singularity when the scattering angle is large enough. With the increase of the scattering angle the spatial distribution of the amplitude and the phase have taken place great changes, and the probability of the angle between the zero-contour lines of real part and that of imaginary part tends to small value, and the average eccentricity of the intensity contours around the phase singularity is gradually increasing. Moreover, the most interesting thing is that the eccentricity is probably greater than 1 in large anger scattering, and the convergence of the correlation curve of phase vortices becomes slower with the spatial distance increasing. All of the properties originate from the appearance of the phase singularity line in larger angle scattering.
In chapter 4, we use the higher-power laser produced by semiconductor pump laser Verdi V-5 (the maximum power is 5W) to enhance the weak intensity in large angle scattering and record them with a charge-coupled device (CCD) camera. The complex amplitudes of the speckle are extracted experimentally according to the technique of the interference of the speckle and the spherical reference wave. By studying the intensity correlation function of the speckle, we find that the fractal property of the speckle is related to the spatial separate distanceρ. In the area of smallρ, the fractal exponentα= 1, but the fractal exponentαbecomes less than 1 whenρincrease to a certain number. Such speckle is named regional fractal speckle by us. We deduce an empirical analytic expression of the intensity correlation function of regional fractal speckle, and calculate the fractal exponentαby fitting the experimental curves with the empirical intensity correlation analytic expression. From the probability distribution properties of the intensities and the complex amplitudes, we demonstrate that the regional fractal speckle still obeys Gaussian distribution.
In chapter 5, we first make a theoretical analysis of the speckles in the region infinitely close to the random surface (also called the extremely deep Fresnel diffraction region) based on the Kirchhoff approximation and Green’s function theory, and then produce the 2D speckle field by numerical simulation. By comparing the 2D speckle field with the corresponding random surfaces, we find that both the intensity space distribution and the fractal property of the speckles depend on the random height distribution of the surfaces, and the intensity probability density of the speckles don’t obey negative exponential law. We explain the intensity probability density and the contrast of the speckle in the extremely deep Fresnel diffraction region by the theory of the sum of zero-order diffraction and circular Gauss speckle. We design a microscope system to detect the speckles in the extremely deep Fresnel diffraction region and find experimental evidence of the conclusions we reached in the numerical simulation. Based on the interference technique of speckle fields with reference beam and the digital Fourier transform, we realize the experimental extraction of the amplitude and phase of the speckles in the extremely deep Fresnel diffraction region.
In chapter 6, we sum up the main conclusions and the innovations of the dissertation, and briefly introduce the in-depth researches we will conduct.
引文
[1] J. Kondev, C. L. Henley and D.G. Salinas, Nonlinear measures for characterizing rough surface morphologies[J], Phys. Rev. E, 2000, 61(1): 104-125.
[2] A. Sagy, Z. Reches and J. Fineberg, Dynamic fracture by large extraterrestrial impacts as the origin of shatter cones[J], Nature, 2002,418(6895): 310-313.
[3] Cardoso, B.Gluckmann, O. Parcollet, and P. Tabeling, Dispersion in a quasi-two- dimensional-turbulent flow: An experimental study[J], Physics of Fluids, 1996, 8(1): 209-214.
[4] G. Palasantzas and J. Barnas, Surface-roughness fractality effects in electrical conductivity of single metallic and semiconducting films[J], Phys. Rev. B, 1997, 56(12): 7726-7731.
[5] Q. Jiang, H. N. Yang, G. C. Wang, Effect of interface roughness on hysteresis loops of ultrathin Co films from 2 to 30 ML on Cu(001) surfaces[J], Surface Science, 1997, 373(2-3):181-194.
[6] P. -G. De Gennes, Wetting: Statistics and dynamics[J], Rev. Mod. Phys, 1985, 57(3): 827-863.
[7] K. A. O’Donnell, R. Torre and C. S. West, Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface[J], Opt. Lett , 1996, 21(21): 1738-1740.
[8] J. C. Dainty, Laser speckle and Related Phenomena, 2nd ed[M], Berlin: Spring-Verlag, 1984.
[9] P. H. Van Cittert, Physica, Utrecht, 1934, 1, 202; F. Zernike, Physica, Utrecht, 1938, 5, 785.
[10]朱星,近场光学与近场光学显微镜[J],北京大学学报(自然科学版),1997, 33(3):394-407.
[11] J. Levy, J. M. Kikkawa, J. M. Kikkawa, D. D. Awschalom and V. Nikitin, Femtosecond near-field spin microscopy in digital magnetic heterostructures[J], J. Applied Physics, 1996, 79(8):6095-6100.
[12] T. R. Thomas, Rough Surface[M], New York: Longman, 1982.
[13] Y.-P. Zhao, T.-M. Lu and G.-C. Wang, Diffraction from non-Gaussian rough surface[J], Phys. Rev. B, 1997, 55(20):13938-13952.
[14] S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy and O. L. Krivanek, Sruface roughness at the Si(100)-SiO2 interface[J], Phys. Rev. B, 1985, 32(12):8171-8186.
[15] G. Rasigni, M. Rasigni, J. Palmari, C. Dussert, F. Varnier and A. Llebaria, Statistical parameters for random and pseudorandom rough surfaces[J], J. Opt. Soc. Am. A, 1988, 5(1): 99-103.
[16] M. V. Berry, Diffractals[J], J. Phys. A: Math. Gen., 1979, 12(6): 781-797.
[17] H.-N. Yang, Y.–P. Zhao, A. Chan, T.-M. Lu and G.-C. Wang, Sampling–induced hidden cycles in correlated random rough surfaces[J], Phys. Rev. B, 1997, 56(7): 4224-4232.
[18] B. B. Mandelbrot. How long is the coast of Britain? Statistical self-similarity and fractional dimension[J], Science, 1967, 156: 636-638.
[19] B. B. Mandelbrot, Fractals: Form, Chance and Dimension[M], San Francisco: Freeman, 1977.
[20] F. Hausdorff, Dimension und usseres mass[J], Math. Ann., 1919, 79: 157-179.
[21] A. S. Besicovitch, On linear sets of points of fractional dimensions[J], Math. Ann., 1929, 101(1): 161-193.
[22] I. Proccacia, Fractal structures in turbulence[J], J. Stat. Phys., 1984, 36(5-6), 649-663.
[23] J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations[J], Lecture Notes in Mathematics, 1979, 730: 204-227.
[24] M. Suzuki, Phase transition and Fractal[J], Prog. Theor. Phys., 1983, 69(1): 65-76.
[25] R. S. Sayles and T. R. Thomas, Surface topography as nonstationary random process[J], Nature, 1978, 271:431-434.
[26] M. V. Berry and J. H. Hannay, Topography of random surface, Nature, 1978, 271(5644): 431-434.
[27] S. K. Sinha, E. B. Sirata, S. Garoff and H. B. Stanley, X-ray and neutron scattering from rough surfaces[J], Phys. Rev. B, 1988, 38(4): 2297-2311.
[28] J.-K. Zuo and J. F. Wendelken, Evolution of Mound Morphology in Reversible Homoepitaxy on Cu(100)[J], Phys. Rev. Lett. , 1997, 78(14): 2791-2794.
[29] J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces[M], New York: Adam Hilger, 1991.
[30] V. A. Ruiz-Cortes, J. C. Dainty, Experimental light-scattering measurements from large-scale composite randomly rough surfaces[J], J. Opt. Soc. Am. A, 2002, 19(10): 2043-2052.
[31] S. Savaidis, P. Frangos, D.L. Jaggard, K. Hizanidis, Scattering from fractally corrugated surfaces with use of the extended boundary condition method, J. Opt. Soc. Am. A, 1997, 14(2): 475-485.
[32] J. W. Goodman, Some effects of target-induced scintillation on optical radar performance[J], Proc. IEEE, 1965, 53(11): 1688-1700.
[33] B. M. Levcine, J. C. Dainty, Non-Gaussian image plane speckle: measurements from diffusers of known statistics[J], Opt. Commun., 1983, 45(4): 252-257.
[34] J. Uozumi and T. Asakura, The first order statistics of partially developed non-Gaussian speckle[J], J. Optics, 1981, 12(3): 177-186.
[35] J. Ohtsubo, Statistical properties of differentiated partially developed speckle patterns[J], J. Opt. Soc. Am., 1982, 72(9): 1249-1252.
[36] D. L. Jordan, R. C. Hollins and E. Jakeman, Experimental measurements of non-Gaussian scattering by a fractal diffuser[J], Appl. Phys. B, 1983, 31(3): 179-186.
[37] E. Jakeman, Speckle statistics with a small number of scatterers[J], Opt. Eng., 1984, 23(4):453-461.
[38] E. Jakeman and R. J. A. Tough, Generalized K-distribution—a statistical model for weak scattering[J], J. Opt. Soc. Am. A, 1987, 4(9):1764-1772.
[39] E. Jakeman and R. J. A. Tough, Non-Gaussian models for the statistics of scattered waves[J], Advances in Physics, 1988, 37(5): 471-529.
[40] P. Z. Wong and A. J. Bray. Scattering by rough surfaces[J]. Phys. Rev. B, 1988, 37(13): 7751-7758.
[41] E. L. Church, Fractal surface finish[J], Appl. Opt., 1988, 27(8): 1518-1526.
[42] Y. Y. Hung and C. Y. Liang, Image shearing camera for direct measurement of surface strains[J], Appl. Opt., 1979, 18(7):1046-1051.
[43] R. K. Mohanty, C. Joenathan and R. S. Sirohi, Speckle and speckle-shearing interferometers combined for the simultaneous determination of out-of-plane displacement and slope[J], Appl. Opt., 1985, 24(18):3106-3109.
[44] L Basano, S. Leporatti, V. Palestini and R. Rolandi, Measurements of surface roughness: use of the CCD camera to correlate doubly scattered speckle patterns[J], Appl. Opt., 1995, 34(31): 7286-7290.
[45] P. Lehmann, Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination[J], Appl. Opt., 1999, 38(7): 1144-1152.
[46] D. Léger, E. Matieu and J. C. Perrin, Optical surface roughness determination using speckle correlation technique[J], Appl. Opt., 1975, 14(4): 872-877.
[47] B. Ruffing and J. Anschutz, Surface roughness measurement of 2-D digital correlation of speckle images[C], in Photomechanics and speckle Metrology, Proc. SPIE, 1987, 814:105-112.
[48] C. J. Tay, S. L. Toh, H. M. Shang and J. Zhang, Whole-field determination of surface roughness by speckle correlation[J], Appl. Opt., 1995, 34(13):2324-2335.
[49] R. A. Sprague, Surface-roughness measurement using white light speckle[J], Appl. Opt., 1972, 11(12): 2811-2816.
[50] Berry M, Making waves in physics. Three wave singularities from the miraculous 1830s[J], Nature, 2000, 403(6765): 21.
[51] Nye J F and Berry M V, Dislocation in wave trains[J], Proc. R. Soc. Lond. A, 1974, 336(1605): 165-190.
[52] Baranova N B, Mamaev A V, Pilipetskji N F, Shkunov V. V. and Zel’dovich B. Ya., Wave-front dislocations: topological limitations for adaptive systems with phase conjugation[J], J. Opt. Soc. Am. 1983, 73(5): 525-528.
[53] Vaughan J M and Willetts D V, Temporal and interference fringe analysis of TEM01* laser modes[J], J. Opt. Soc. Am. A. 1983, 73(8): 1018-1021.
[54] Coulet P, Gil L and Rocca F, Optical vortices[J], Opt. Commun. 1989, 73(5): 403-408.
[55] Soskin M S and Vasnetsov M V, Singular optics[J], Progress in Optics, 2001, 42: 219-276.
[56] Nye J F, Natural Focusing and Fine Structure of Light[M], Bristol: Institute of Physics Publishing, 1999.
[57] Plyanskii P V, Some current views on singular optics[J], Proc. of SPIE. 2004, 5477: 31-40.
[58] J. E. Curtis, Dynamic Holographic Optical Tweezers[J], Optics Comm., 2002, 207: 169-175.
[59] Foley J T, Wolf E, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light[J], J. Opt. Soc. Am. A, 2002, 19(12): 2510-2516.
[60] Wei Wang, Tomoaki Yokozeki, Reika Ishijima, Atsushi Wada, Yoko Miyamoto, Mitsuo Takeda, and Steen G. Hanson, Optical vortex metrology for nanometric speckledisplacement measurement[J], Opt. Express, 2006, 14(1): 120-127.
[61] B Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, F. Kaiser, C. Denz and Y. S. Kivshar, Observation of Multivortex Solitons in Photonic Lattices[J], Phys. Rev. Lett. , 2008, 101(1): 013903.
[62] M. S. Soskin, M. V. Vasnetsov. Singular optics[J]. Prog. Opt. 2001, 42:219～276.
[63] A S Desyatnikov, and Yu. S. Kivshar, Optical vortices and vortex solitons[J], Prog. Opt. , 2005, 47: 291-391.
[64] J. W. Goodman, Statistical Optics[M], New York: John Wiley&Sons, inc., 1985.
[65] J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces[M], New York: Adam Hilger, 1991.
[66] I. Freund, Optical vortices in Gaussian random wavefields: Statistical probability densities[J], J. Opt. Soc. Am. A, 1994, 11(5): 1644-1652.
[67] M. V. Berry, Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves[J], J. Phys. A, 1978, 11(1): 27-37.
[68] N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetslii and V. V. Shkukov, Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment)[J]. Jetp. Lett., 1981, 33(4):195～199; N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov and B. Ya. Zel’dovich, Wave-front dislocations: topological limitations for adaptive systems with phase conjugation[J], J. Opt. Soc. Am. A, 1983, 73(5): 525-528.
[69] I. Freund, N. Shvartsman and V. Freilikher, Optical dislocation networks in highly random media[J], Opt. Commun., 1993, 101(3-4): 247-264.
[70] N. Shvartsman, I. Freund, Vortices in random wave fields: nearest neighbor anticorrelations[J], Phys. Rev. Lett., 1994, 72(7):1008-1011.
[71] F. J. Write, Structural Stability in physics[M], Berlin: Springer-Verlag 1979: 141; R. J. Mondragon, M. V. Berry, The Quantum Phase 2-Form Near Degeneracies: Two Numerical Studies[J], Proc. R. Soc. London A, 1989, 424: 263-278.
[72] M. V. Berry and M. R. Dennis. Phase singularities in isotropic random waves[J]. Proc. R. Soc. Lond. A, 2000, 456: 2059-2079.
[73] J. W. Goodman, Statistical optics[M], New York: Wiley. 2000: 347
[74] M. Harris, Light-field fluctuations in space and time[J], Contemporary Physics, 1995, 36(4): 215-233.
[75] J. W. Goodman, Some fundamental properties of speckle[J], J. Opt. Soc. Am. 1976, 66(11): 1145-1150.
[76] C. Tamm and C. O. Weiss, Bistability and optical switching of spatial patterns in a laser[J], J. Opt. Soc. Am. B, 1990, 7(6): 1034-1038.
[77] M. Brambilla and F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm and C. O. Weiss, Transverse laser patterns. I. Phase singularity crystals[J], Phys. Rev. A 1991,43(9): 5090-5113
[78] Wang L, Wang Q F, Wang X Q and Lu B D, Transversal optical vortex in the interference field of two off-axis Gaussian beams[J], Acta. Physica. Sinica, 2007, 56(1): 0201-0207 (in Chinese); [王莉,王庆峰,王喜庆,吕百达,两束离轴高斯光束干涉场中的横向光涡旋[J],物理学报, 2007 56(1): 201-207].
[79] W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields[J], Phys. Rev. Lett., 2005, 94(10): 103902.
[80] W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto and M. Takeda, Experimental Study of Coherence Vortices: Local Properties of Phase Singularities in a Spatial Coherence Function[J], Phys. Rev. Lett. 2006, 96(7): 073902.
[81] M. R. Dennis, Nodal densities of planar gaussian random waves[J], Eur. Phys. J. Special Topics, 2007,145: 191-210.
[82] S. Zhang and A. Z. Genack, Statistics of Diffusive and Localized Fields in the Vortex Core[J], Phys. Rev. Lett., 2007, 99(20): 203901.
[83] R. I. Egorow, M. S. Soskin, D. A. Kessler and I. Freund, Experimental Measurements of Topological Singularity Screening in Random Paraxial Scalar and Vector Optical Fields[J], Phys. Rev. Lett. 2008, 100: 103901.
[84] M. Takeda, H. Ina and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry[J], J. Opt. Soc. Am., 1982, 72(1): 156-160.
[85] W. W. Macy, Two-dimensional fringe-pattern analysis[J], Appl. Opt., 1983, 22(23): 3898-3901.
[86] I. Freund, Critical-point level-crossing geometry in random wave fields[J], J. Opt. Soc. Am. A, 1997, 14(8): 1911-1927.
[87] Dainty J C, Laser Speckle and Related Phenomena[M], Berlin: Springer-Verlag, 1975:15.
[88] McKechnie T S, Measurement of some second order statistical properties of speckle[J], Optik, 1974,39: 258-267.
[89] Wang N Q, Zhang H J and E Y, THE STATISTICAL PROPERTIES OF POLARIZED LASER SPECKLE[J], Acta Phys. Sin. 1983, 32(1): 124-132 (in Chinese) [王迺权,张洪钧,鄂云,偏振激光散斑的积分光强的统计特性[J],物理学报, 1983, 32(1): 124-132].
[90] Cheng Chuan-Fu, Qi Dong-Ping, Liu De-Li et al.. The Computational simulations of the gaussian correlation random surface and its light-scattering speckle field and the analysis of the intensity probability density[J]. Acta Physica Sinica, 1999, 48(9): 1635-1643 (in Chinese) [程传福,亓东平,刘德丽等.高斯相关随机表面及其光散射散斑场的模拟产生和光强概率分析[J].物理学报, 1999, 48(9): 1635-1643].
[91] Runge E and Zimmermann R, Statistical properties of speckle distributions in resonant secondary emission[J], Phys. Rev. B, 2000, 61(7): 4786-4794.
[92] Soest G V, Poelwijk F J and Lagendijk A, Speckle experiments in random lasers[J], Phys. Rev. E, 2002, 65(4): 046603.
[93] Zhu H M, Wu Y Y, Huang Z W, Wang Y G and Zhu C, First-order statistical properties of ultrasonic speckles back-scattered from a kind of weak scattering interface[J], Acta Phys. Sin., 2003, 52(6) 1438-1443 (in Chinese) [朱鸿茂,吴艳阳,黄忠文,王寅观,朱成,一类弱散射界面背向散射超声散斑一阶统计特性[J],物理学报,2003 52(6): 1438-1443].
[94] Duncan D D, Kirkpatrick S J and Wang R K, Statistics of local speckle contrast[J], J. Opt. Soc. Am. A, 2008, 25(1): 9-15.
[95] Bone D J, Bachor H A and Sandeman R J, Fringe-pattern analysis using a 2-D Fourier transform[J], Appl. Opt., 1986, 25(10): 1653-1660.
[96] Yamaguchi I and Zhang T, Phase-shifting digital holography[J], Opt. Lett., 1997, 22(16): 1268-1270.
[97] Iaconis C and Walmsley I A, Spectral phase interferometry for direct electric-fieldreconstruction of ultrashort optical pulses[J], Opt. Lett. 1998,23(10): 792-794.
[98] Cuche E, Marquet P and Depeursinge C, Spatial Filtering for Zero-Order and Twin-Image Elimination in Digital Off-Axis Holography[J], Appl. Opt., 2000, 39(23): 4070-4075.
[99] Dainty J C, Laser Speckle and Related Phenomena[M], Berlin: Springer-Verlag, 1975: 39
[100] Takayasu H, Fractals in Physical Science[M], Manchester University: Manchester, 1990.
[101] Feder J, Fractals[M], New York: Plenum,1996.
[102] Vicsek T, Fractal Growth Phenomena[M], World Scientific: Singapore, 1992.
[103] Yang J, Lai X M, Peng G, Bian B M and Lu J, The fractal measurement of aerosol equivalent volume based con counting signal[J], Acta Phys. Sin., 2009, 58(5): 3008-3013 (in Chinese) [杨娟,赖晓明,彭刚,卞保民,陆建,悬浮颗粒计数信号等效体积的分形测度[J],物理学报, 2009, 58(5): 3008-3013].
[104] Luo S H and Zeng J S, Multi-fractal identification of the fluctuation of silicon content in blast furnace hot metal based on multi-resolution analysis[J], Acta Phys. Sin., 2009, 58(1) 150-157 (in Chinese)[罗世华,曾九孙,基于多分辨分析的高炉铁水含硅量波动多重分形辨识[J],物理学报, 2009, 58(1): 150-157].
[105] Funamizu H and Uozumi J, Effects of clipping threshold on the scaling property of clipped fractal speckle intensities[J], J. Opt. A, 2008, 10(2): 025004.
[106] O’Holleran K, Dennis M R, Flossmann F and Padgett M J, Fractality of light’s darkness[J], Phys. Rev. Lett., 2008, 100(5): 053902.
[107] Uno K, Uozumi J and Asakura T, Correlation properties of speckles produced by diffractal-illuminated diffusers[J], Opt. Commun., 1996, 124(1-2): 16-22.
[108] Uozumi J, Ibrahim M and Asakura T, Fractal speckles[J], Opt. Commun., 1998, 156(4-6): 350-358.
[109] Funamizu H and Uozumi J, Generation of fractal speckles by means of a spatial light modulator[J], Opt. Express, 2007, 15(12): 7415-7422.
[110] Funamizu H and Uozumi J, Scaling reduction of the contrast of fractal speckles detected with a finite aperture[J], Opt. Commun., 2008, 281(4): 543-549.
[111] Goodman J W, speckle phenomena in optics: Theory and applications[M], Greenwood Village CO: Roberts and Company. 2007: 53.
[112] Song H S, Cheng C F, Liu M, Teng S Y and Zhang N Y, Experimental study on phasevortices of speckles and their propagation properties[J], Acta Phys. Sin., 2009, 58(6): 3887-3886 (in Chinese) [宋洪胜,程传福,刘曼,滕树云,张宁玉,散斑场相位涡旋及其传播特性的实验研究[J],物理学报, 2009, 58(6): 3887-3886].
[113] Dainty J C, Laser Speckle and Related phenomena[M], Berlin: Springer. 1975:13.
[114] Yang H N, Wang G C and Lu T M, Diffraction from rough surfaces and dynamic growth fronts[M], Singapore: World Scientific. 1993:50.
[115] Frankenthal S and Whitman A M, Scintillations due to multiscale phase screens[J], J. Opt. Soc. Am. A, 1989, 6(2): 1827-1834.
[116] Skipetrov S E, Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity[J], Opt. Lett., 2003, 28(8): 646-648.
[117] Song H S, Cheng C F, Teng S Y, Liu M, Liu G Y and Zhang N Y, Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam[J], Acta Phys. Sin., 2009 58(11): 76547661. (in Chinese) [宋洪胜,程传福,滕树云,刘曼,刘桂媛,张宁玉,参考光干涉提取复振幅的散斑统计函数的实验研究[J],物理学报, 2009 58(11): 76547661].
[118] Liu M, Cheng C F, Song H S, Teng S Y and Liu G Y, Phase singularities of speckle field produced by the scattering from Gaussian correlation random surfaces[J], Acta Phys. Sin., 2009, 58(8): 5376-5384. (in Chinese) [刘曼,程传福,宋洪胜,滕树云,刘桂媛,高斯相关随机表面光散射散斑场相位奇异及其特性的理论研究[J],物理学报, 2009, 58(8): 5376-5384].
[119] Dixon P K and Durian D J, Speckle Visibility Spectroscopy and Variable Granular Fluidization[J], Phys. Rev. Lett., 2003, 90(18): 184302.
[120] Williams G J, Pfeifer M A, Vartanyants I A and Robinson I K, Three-Dimensional Imaging of Microstructure in Au Nanocrystals[J], Phys. Rev. Lett., 2003, 90(17): 175501.
[121] Pierce M S, Moore R G, Sorensen L B, Kevan S D, Hellwig O, Fullerton E E and Kortright J B, Quasistatic X-Ray Speckle Metrology of Microscopic Magnetic Return-Point Memory[J], Phys. Rev. Lett., 2003, 90(17): 175502.
[122] Cerbino R, Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem[J], Phys. Rev. A, 2007,75(5): 053815.
[123] Gatti A, Magatti D and Ferri F, Three-dimensional coherence of light speckles: Theory[J],Phys. Rev. A, 2008, 78(6): 063806.
[124] Teng S Y, Cheng C F, Liu M, Gui W L and Xu Z Z, The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces[J], Chin. Phys. Soc., 2005, 14(10): 1990-1995.
[125] Lopez J M, Scaling Approach to Calculate Critical Exponents in Anomalous Surface Roughening[J], Phys. Rev. Lett., 1999, 83(22): 4594-4597.
[126] Qi D P, Liu D L, Teng S Y, Zhang N Y and Cheng C F, MORPHOLOGICAL ANALYSIS BY ATOMIC FORCE MICROSCOPE AND LIGHT SCATTERING STUDY FOR RANDOM SCATTERING SCREENS[J], Acta Phys. Sin., 2000 49(7): 1260-1266. (in Chinese) [亓东平,刘德丽,滕树云,张宁玉,程传福,随机散射屏的原子力显微镜形貌分析及其光散射特性[J],物理学报, 2000 49(7): 1260-1266.].
[127] Goodman J W, speckle phenomena in optics: Theory and applications[M], Greenwood Village CO: Roberts and Company. 2007:30.
[2] A. Sagy, Z. Reches and J. Fineberg, Dynamic fracture by large extraterrestrial impacts as the origin of shatter cones[J], Nature, 2002,418(6895): 310-313.
[3] Cardoso, B.Gluckmann, O. Parcollet, and P. Tabeling, Dispersion in a quasi-two- dimensional-turbulent flow: An experimental study[J], Physics of Fluids, 1996, 8(1): 209-214.
[4] G. Palasantzas and J. Barnas, Surface-roughness fractality effects in electrical conductivity of single metallic and semiconducting films[J], Phys. Rev. B, 1997, 56(12): 7726-7731.
[5] Q. Jiang, H. N. Yang, G. C. Wang, Effect of interface roughness on hysteresis loops of ultrathin Co films from 2 to 30 ML on Cu(001) surfaces[J], Surface Science, 1997, 373(2-3):181-194.
[6] P. -G. De Gennes, Wetting: Statistics and dynamics[J], Rev. Mod. Phys, 1985, 57(3): 827-863.
[7] K. A. O’Donnell, R. Torre and C. S. West, Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface[J], Opt. Lett , 1996, 21(21): 1738-1740.
[8] J. C. Dainty, Laser speckle and Related Phenomena, 2nd ed[M], Berlin: Spring-Verlag, 1984.
[9] P. H. Van Cittert, Physica, Utrecht, 1934, 1, 202; F. Zernike, Physica, Utrecht, 1938, 5, 785.
[10]朱星,近场光学与近场光学显微镜[J],北京大学学报(自然科学版),1997, 33(3):394-407.
[11] J. Levy, J. M. Kikkawa, J. M. Kikkawa, D. D. Awschalom and V. Nikitin, Femtosecond near-field spin microscopy in digital magnetic heterostructures[J], J. Applied Physics, 1996, 79(8):6095-6100.
[12] T. R. Thomas, Rough Surface[M], New York: Longman, 1982.
[13] Y.-P. Zhao, T.-M. Lu and G.-C. Wang, Diffraction from non-Gaussian rough surface[J], Phys. Rev. B, 1997, 55(20):13938-13952.
[14] S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy and O. L. Krivanek, Sruface roughness at the Si(100)-SiO2 interface[J], Phys. Rev. B, 1985, 32(12):8171-8186.
[15] G. Rasigni, M. Rasigni, J. Palmari, C. Dussert, F. Varnier and A. Llebaria, Statistical parameters for random and pseudorandom rough surfaces[J], J. Opt. Soc. Am. A, 1988, 5(1): 99-103.
[16] M. V. Berry, Diffractals[J], J. Phys. A: Math. Gen., 1979, 12(6): 781-797.
[17] H.-N. Yang, Y.–P. Zhao, A. Chan, T.-M. Lu and G.-C. Wang, Sampling–induced hidden cycles in correlated random rough surfaces[J], Phys. Rev. B, 1997, 56(7): 4224-4232.
[18] B. B. Mandelbrot. How long is the coast of Britain? Statistical self-similarity and fractional dimension[J], Science, 1967, 156: 636-638.
[19] B. B. Mandelbrot, Fractals: Form, Chance and Dimension[M], San Francisco: Freeman, 1977.
[20] F. Hausdorff, Dimension und usseres mass[J], Math. Ann., 1919, 79: 157-179.
[21] A. S. Besicovitch, On linear sets of points of fractional dimensions[J], Math. Ann., 1929, 101(1): 161-193.
[22] I. Proccacia, Fractal structures in turbulence[J], J. Stat. Phys., 1984, 36(5-6), 649-663.
[23] J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations[J], Lecture Notes in Mathematics, 1979, 730: 204-227.
[24] M. Suzuki, Phase transition and Fractal[J], Prog. Theor. Phys., 1983, 69(1): 65-76.
[25] R. S. Sayles and T. R. Thomas, Surface topography as nonstationary random process[J], Nature, 1978, 271:431-434.
[26] M. V. Berry and J. H. Hannay, Topography of random surface, Nature, 1978, 271(5644): 431-434.
[27] S. K. Sinha, E. B. Sirata, S. Garoff and H. B. Stanley, X-ray and neutron scattering from rough surfaces[J], Phys. Rev. B, 1988, 38(4): 2297-2311.
[28] J.-K. Zuo and J. F. Wendelken, Evolution of Mound Morphology in Reversible Homoepitaxy on Cu(100)[J], Phys. Rev. Lett. , 1997, 78(14): 2791-2794.
[29] J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces[M], New York: Adam Hilger, 1991.
[30] V. A. Ruiz-Cortes, J. C. Dainty, Experimental light-scattering measurements from large-scale composite randomly rough surfaces[J], J. Opt. Soc. Am. A, 2002, 19(10): 2043-2052.
[31] S. Savaidis, P. Frangos, D.L. Jaggard, K. Hizanidis, Scattering from fractally corrugated surfaces with use of the extended boundary condition method, J. Opt. Soc. Am. A, 1997, 14(2): 475-485.
[32] J. W. Goodman, Some effects of target-induced scintillation on optical radar performance[J], Proc. IEEE, 1965, 53(11): 1688-1700.
[33] B. M. Levcine, J. C. Dainty, Non-Gaussian image plane speckle: measurements from diffusers of known statistics[J], Opt. Commun., 1983, 45(4): 252-257.
[34] J. Uozumi and T. Asakura, The first order statistics of partially developed non-Gaussian speckle[J], J. Optics, 1981, 12(3): 177-186.
[35] J. Ohtsubo, Statistical properties of differentiated partially developed speckle patterns[J], J. Opt. Soc. Am., 1982, 72(9): 1249-1252.
[36] D. L. Jordan, R. C. Hollins and E. Jakeman, Experimental measurements of non-Gaussian scattering by a fractal diffuser[J], Appl. Phys. B, 1983, 31(3): 179-186.
[37] E. Jakeman, Speckle statistics with a small number of scatterers[J], Opt. Eng., 1984, 23(4):453-461.
[38] E. Jakeman and R. J. A. Tough, Generalized K-distribution—a statistical model for weak scattering[J], J. Opt. Soc. Am. A, 1987, 4(9):1764-1772.
[39] E. Jakeman and R. J. A. Tough, Non-Gaussian models for the statistics of scattered waves[J], Advances in Physics, 1988, 37(5): 471-529.
[40] P. Z. Wong and A. J. Bray. Scattering by rough surfaces[J]. Phys. Rev. B, 1988, 37(13): 7751-7758.
[41] E. L. Church, Fractal surface finish[J], Appl. Opt., 1988, 27(8): 1518-1526.
[42] Y. Y. Hung and C. Y. Liang, Image shearing camera for direct measurement of surface strains[J], Appl. Opt., 1979, 18(7):1046-1051.
[43] R. K. Mohanty, C. Joenathan and R. S. Sirohi, Speckle and speckle-shearing interferometers combined for the simultaneous determination of out-of-plane displacement and slope[J], Appl. Opt., 1985, 24(18):3106-3109.
[44] L Basano, S. Leporatti, V. Palestini and R. Rolandi, Measurements of surface roughness: use of the CCD camera to correlate doubly scattered speckle patterns[J], Appl. Opt., 1995, 34(31): 7286-7290.
[45] P. Lehmann, Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination[J], Appl. Opt., 1999, 38(7): 1144-1152.
[46] D. Léger, E. Matieu and J. C. Perrin, Optical surface roughness determination using speckle correlation technique[J], Appl. Opt., 1975, 14(4): 872-877.
[47] B. Ruffing and J. Anschutz, Surface roughness measurement of 2-D digital correlation of speckle images[C], in Photomechanics and speckle Metrology, Proc. SPIE, 1987, 814:105-112.
[48] C. J. Tay, S. L. Toh, H. M. Shang and J. Zhang, Whole-field determination of surface roughness by speckle correlation[J], Appl. Opt., 1995, 34(13):2324-2335.
[49] R. A. Sprague, Surface-roughness measurement using white light speckle[J], Appl. Opt., 1972, 11(12): 2811-2816.
[50] Berry M, Making waves in physics. Three wave singularities from the miraculous 1830s[J], Nature, 2000, 403(6765): 21.
[51] Nye J F and Berry M V, Dislocation in wave trains[J], Proc. R. Soc. Lond. A, 1974, 336(1605): 165-190.
[52] Baranova N B, Mamaev A V, Pilipetskji N F, Shkunov V. V. and Zel’dovich B. Ya., Wave-front dislocations: topological limitations for adaptive systems with phase conjugation[J], J. Opt. Soc. Am. 1983, 73(5): 525-528.
[53] Vaughan J M and Willetts D V, Temporal and interference fringe analysis of TEM01* laser modes[J], J. Opt. Soc. Am. A. 1983, 73(8): 1018-1021.
[54] Coulet P, Gil L and Rocca F, Optical vortices[J], Opt. Commun. 1989, 73(5): 403-408.
[55] Soskin M S and Vasnetsov M V, Singular optics[J], Progress in Optics, 2001, 42: 219-276.
[56] Nye J F, Natural Focusing and Fine Structure of Light[M], Bristol: Institute of Physics Publishing, 1999.
[57] Plyanskii P V, Some current views on singular optics[J], Proc. of SPIE. 2004, 5477: 31-40.
[58] J. E. Curtis, Dynamic Holographic Optical Tweezers[J], Optics Comm., 2002, 207: 169-175.
[59] Foley J T, Wolf E, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light[J], J. Opt. Soc. Am. A, 2002, 19(12): 2510-2516.
[60] Wei Wang, Tomoaki Yokozeki, Reika Ishijima, Atsushi Wada, Yoko Miyamoto, Mitsuo Takeda, and Steen G. Hanson, Optical vortex metrology for nanometric speckledisplacement measurement[J], Opt. Express, 2006, 14(1): 120-127.
[61] B Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, F. Kaiser, C. Denz and Y. S. Kivshar, Observation of Multivortex Solitons in Photonic Lattices[J], Phys. Rev. Lett. , 2008, 101(1): 013903.
[62] M. S. Soskin, M. V. Vasnetsov. Singular optics[J]. Prog. Opt. 2001, 42:219～276.
[63] A S Desyatnikov, and Yu. S. Kivshar, Optical vortices and vortex solitons[J], Prog. Opt. , 2005, 47: 291-391.
[64] J. W. Goodman, Statistical Optics[M], New York: John Wiley&Sons, inc., 1985.
[65] J. A. Ogilvy, Theory of Wave Scattering from Rough Surfaces[M], New York: Adam Hilger, 1991.
[66] I. Freund, Optical vortices in Gaussian random wavefields: Statistical probability densities[J], J. Opt. Soc. Am. A, 1994, 11(5): 1644-1652.
[67] M. V. Berry, Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves[J], J. Phys. A, 1978, 11(1): 27-37.
[68] N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetslii and V. V. Shkukov, Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment)[J]. Jetp. Lett., 1981, 33(4):195～199; N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov and B. Ya. Zel’dovich, Wave-front dislocations: topological limitations for adaptive systems with phase conjugation[J], J. Opt. Soc. Am. A, 1983, 73(5): 525-528.
[69] I. Freund, N. Shvartsman and V. Freilikher, Optical dislocation networks in highly random media[J], Opt. Commun., 1993, 101(3-4): 247-264.
[70] N. Shvartsman, I. Freund, Vortices in random wave fields: nearest neighbor anticorrelations[J], Phys. Rev. Lett., 1994, 72(7):1008-1011.
[71] F. J. Write, Structural Stability in physics[M], Berlin: Springer-Verlag 1979: 141; R. J. Mondragon, M. V. Berry, The Quantum Phase 2-Form Near Degeneracies: Two Numerical Studies[J], Proc. R. Soc. London A, 1989, 424: 263-278.
[72] M. V. Berry and M. R. Dennis. Phase singularities in isotropic random waves[J]. Proc. R. Soc. Lond. A, 2000, 456: 2059-2079.
[73] J. W. Goodman, Statistical optics[M], New York: Wiley. 2000: 347
[74] M. Harris, Light-field fluctuations in space and time[J], Contemporary Physics, 1995, 36(4): 215-233.
[75] J. W. Goodman, Some fundamental properties of speckle[J], J. Opt. Soc. Am. 1976, 66(11): 1145-1150.
[76] C. Tamm and C. O. Weiss, Bistability and optical switching of spatial patterns in a laser[J], J. Opt. Soc. Am. B, 1990, 7(6): 1034-1038.
[77] M. Brambilla and F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm and C. O. Weiss, Transverse laser patterns. I. Phase singularity crystals[J], Phys. Rev. A 1991,43(9): 5090-5113
[78] Wang L, Wang Q F, Wang X Q and Lu B D, Transversal optical vortex in the interference field of two off-axis Gaussian beams[J], Acta. Physica. Sinica, 2007, 56(1): 0201-0207 (in Chinese); [王莉,王庆峰,王喜庆,吕百达,两束离轴高斯光束干涉场中的横向光涡旋[J],物理学报, 2007 56(1): 201-207].
[79] W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields[J], Phys. Rev. Lett., 2005, 94(10): 103902.
[80] W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto and M. Takeda, Experimental Study of Coherence Vortices: Local Properties of Phase Singularities in a Spatial Coherence Function[J], Phys. Rev. Lett. 2006, 96(7): 073902.
[81] M. R. Dennis, Nodal densities of planar gaussian random waves[J], Eur. Phys. J. Special Topics, 2007,145: 191-210.
[82] S. Zhang and A. Z. Genack, Statistics of Diffusive and Localized Fields in the Vortex Core[J], Phys. Rev. Lett., 2007, 99(20): 203901.
[83] R. I. Egorow, M. S. Soskin, D. A. Kessler and I. Freund, Experimental Measurements of Topological Singularity Screening in Random Paraxial Scalar and Vector Optical Fields[J], Phys. Rev. Lett. 2008, 100: 103901.
[84] M. Takeda, H. Ina and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry[J], J. Opt. Soc. Am., 1982, 72(1): 156-160.
[85] W. W. Macy, Two-dimensional fringe-pattern analysis[J], Appl. Opt., 1983, 22(23): 3898-3901.
[86] I. Freund, Critical-point level-crossing geometry in random wave fields[J], J. Opt. Soc. Am. A, 1997, 14(8): 1911-1927.
[87] Dainty J C, Laser Speckle and Related Phenomena[M], Berlin: Springer-Verlag, 1975:15.
[88] McKechnie T S, Measurement of some second order statistical properties of speckle[J], Optik, 1974,39: 258-267.
[89] Wang N Q, Zhang H J and E Y, THE STATISTICAL PROPERTIES OF POLARIZED LASER SPECKLE[J], Acta Phys. Sin. 1983, 32(1): 124-132 (in Chinese) [王迺权,张洪钧,鄂云,偏振激光散斑的积分光强的统计特性[J],物理学报, 1983, 32(1): 124-132].
[90] Cheng Chuan-Fu, Qi Dong-Ping, Liu De-Li et al.. The Computational simulations of the gaussian correlation random surface and its light-scattering speckle field and the analysis of the intensity probability density[J]. Acta Physica Sinica, 1999, 48(9): 1635-1643 (in Chinese) [程传福,亓东平,刘德丽等.高斯相关随机表面及其光散射散斑场的模拟产生和光强概率分析[J].物理学报, 1999, 48(9): 1635-1643].
[91] Runge E and Zimmermann R, Statistical properties of speckle distributions in resonant secondary emission[J], Phys. Rev. B, 2000, 61(7): 4786-4794.
[92] Soest G V, Poelwijk F J and Lagendijk A, Speckle experiments in random lasers[J], Phys. Rev. E, 2002, 65(4): 046603.
[93] Zhu H M, Wu Y Y, Huang Z W, Wang Y G and Zhu C, First-order statistical properties of ultrasonic speckles back-scattered from a kind of weak scattering interface[J], Acta Phys. Sin., 2003, 52(6) 1438-1443 (in Chinese) [朱鸿茂,吴艳阳,黄忠文,王寅观,朱成,一类弱散射界面背向散射超声散斑一阶统计特性[J],物理学报,2003 52(6): 1438-1443].
[94] Duncan D D, Kirkpatrick S J and Wang R K, Statistics of local speckle contrast[J], J. Opt. Soc. Am. A, 2008, 25(1): 9-15.
[95] Bone D J, Bachor H A and Sandeman R J, Fringe-pattern analysis using a 2-D Fourier transform[J], Appl. Opt., 1986, 25(10): 1653-1660.
[96] Yamaguchi I and Zhang T, Phase-shifting digital holography[J], Opt. Lett., 1997, 22(16): 1268-1270.
[97] Iaconis C and Walmsley I A, Spectral phase interferometry for direct electric-fieldreconstruction of ultrashort optical pulses[J], Opt. Lett. 1998,23(10): 792-794.
[98] Cuche E, Marquet P and Depeursinge C, Spatial Filtering for Zero-Order and Twin-Image Elimination in Digital Off-Axis Holography[J], Appl. Opt., 2000, 39(23): 4070-4075.
[99] Dainty J C, Laser Speckle and Related Phenomena[M], Berlin: Springer-Verlag, 1975: 39
[100] Takayasu H, Fractals in Physical Science[M], Manchester University: Manchester, 1990.
[101] Feder J, Fractals[M], New York: Plenum,1996.
[102] Vicsek T, Fractal Growth Phenomena[M], World Scientific: Singapore, 1992.
[103] Yang J, Lai X M, Peng G, Bian B M and Lu J, The fractal measurement of aerosol equivalent volume based con counting signal[J], Acta Phys. Sin., 2009, 58(5): 3008-3013 (in Chinese) [杨娟,赖晓明,彭刚,卞保民,陆建,悬浮颗粒计数信号等效体积的分形测度[J],物理学报, 2009, 58(5): 3008-3013].
[104] Luo S H and Zeng J S, Multi-fractal identification of the fluctuation of silicon content in blast furnace hot metal based on multi-resolution analysis[J], Acta Phys. Sin., 2009, 58(1) 150-157 (in Chinese)[罗世华,曾九孙,基于多分辨分析的高炉铁水含硅量波动多重分形辨识[J],物理学报, 2009, 58(1): 150-157].
[105] Funamizu H and Uozumi J, Effects of clipping threshold on the scaling property of clipped fractal speckle intensities[J], J. Opt. A, 2008, 10(2): 025004.
[106] O’Holleran K, Dennis M R, Flossmann F and Padgett M J, Fractality of light’s darkness[J], Phys. Rev. Lett., 2008, 100(5): 053902.
[107] Uno K, Uozumi J and Asakura T, Correlation properties of speckles produced by diffractal-illuminated diffusers[J], Opt. Commun., 1996, 124(1-2): 16-22.
[108] Uozumi J, Ibrahim M and Asakura T, Fractal speckles[J], Opt. Commun., 1998, 156(4-6): 350-358.
[109] Funamizu H and Uozumi J, Generation of fractal speckles by means of a spatial light modulator[J], Opt. Express, 2007, 15(12): 7415-7422.
[110] Funamizu H and Uozumi J, Scaling reduction of the contrast of fractal speckles detected with a finite aperture[J], Opt. Commun., 2008, 281(4): 543-549.
[111] Goodman J W, speckle phenomena in optics: Theory and applications[M], Greenwood Village CO: Roberts and Company. 2007: 53.
[112] Song H S, Cheng C F, Liu M, Teng S Y and Zhang N Y, Experimental study on phasevortices of speckles and their propagation properties[J], Acta Phys. Sin., 2009, 58(6): 3887-3886 (in Chinese) [宋洪胜,程传福,刘曼,滕树云,张宁玉,散斑场相位涡旋及其传播特性的实验研究[J],物理学报, 2009, 58(6): 3887-3886].
[113] Dainty J C, Laser Speckle and Related phenomena[M], Berlin: Springer. 1975:13.
[114] Yang H N, Wang G C and Lu T M, Diffraction from rough surfaces and dynamic growth fronts[M], Singapore: World Scientific. 1993:50.
[115] Frankenthal S and Whitman A M, Scintillations due to multiscale phase screens[J], J. Opt. Soc. Am. A, 1989, 6(2): 1827-1834.
[116] Skipetrov S E, Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity[J], Opt. Lett., 2003, 28(8): 646-648.
[117] Song H S, Cheng C F, Teng S Y, Liu M, Liu G Y and Zhang N Y, Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam[J], Acta Phys. Sin., 2009 58(11): 76547661. (in Chinese) [宋洪胜,程传福,滕树云,刘曼,刘桂媛,张宁玉,参考光干涉提取复振幅的散斑统计函数的实验研究[J],物理学报, 2009 58(11): 76547661].
[118] Liu M, Cheng C F, Song H S, Teng S Y and Liu G Y, Phase singularities of speckle field produced by the scattering from Gaussian correlation random surfaces[J], Acta Phys. Sin., 2009, 58(8): 5376-5384. (in Chinese) [刘曼,程传福,宋洪胜,滕树云,刘桂媛,高斯相关随机表面光散射散斑场相位奇异及其特性的理论研究[J],物理学报, 2009, 58(8): 5376-5384].
[119] Dixon P K and Durian D J, Speckle Visibility Spectroscopy and Variable Granular Fluidization[J], Phys. Rev. Lett., 2003, 90(18): 184302.
[120] Williams G J, Pfeifer M A, Vartanyants I A and Robinson I K, Three-Dimensional Imaging of Microstructure in Au Nanocrystals[J], Phys. Rev. Lett., 2003, 90(17): 175501.
[121] Pierce M S, Moore R G, Sorensen L B, Kevan S D, Hellwig O, Fullerton E E and Kortright J B, Quasistatic X-Ray Speckle Metrology of Microscopic Magnetic Return-Point Memory[J], Phys. Rev. Lett., 2003, 90(17): 175502.
[122] Cerbino R, Correlations of light in the deep Fresnel region: An extended Van Cittert and Zernike theorem[J], Phys. Rev. A, 2007,75(5): 053815.
[123] Gatti A, Magatti D and Ferri F, Three-dimensional coherence of light speckles: Theory[J],Phys. Rev. A, 2008, 78(6): 063806.
[124] Teng S Y, Cheng C F, Liu M, Gui W L and Xu Z Z, The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces[J], Chin. Phys. Soc., 2005, 14(10): 1990-1995.
[125] Lopez J M, Scaling Approach to Calculate Critical Exponents in Anomalous Surface Roughening[J], Phys. Rev. Lett., 1999, 83(22): 4594-4597.
[126] Qi D P, Liu D L, Teng S Y, Zhang N Y and Cheng C F, MORPHOLOGICAL ANALYSIS BY ATOMIC FORCE MICROSCOPE AND LIGHT SCATTERING STUDY FOR RANDOM SCATTERING SCREENS[J], Acta Phys. Sin., 2000 49(7): 1260-1266. (in Chinese) [亓东平,刘德丽,滕树云,张宁玉,程传福,随机散射屏的原子力显微镜形貌分析及其光散射特性[J],物理学报, 2000 49(7): 1260-1266.].
[127] Goodman J W, speckle phenomena in optics: Theory and applications[M], Greenwood Village CO: Roberts and Company. 2007:30.