广义相移数字全息干涉术相移提取及波前再现算法的理论及实验研究
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摘要
数字全息是近年来信息光学领域发展迅速的一个研究分支。它利用电荷耦合器件(Charge Coupled Devices,简称CCD)等光电记录器件以数字化的方式记录全息干涉图的强度信息并输入计算机,使用相应算法进行数据处理,获得原始物波信息。数字全息既拥有传统全息技术可以同时记录、再现物波振幅及相位信息的优点,又避免了传统全息的湿处理及难以精确复位等不便之处。同时结合了迅速发展的现代CCD技术及计算机图像处理技术,可以方便准确地实施数据及图像的量化记录、处理、存储、变换、加工、再现,实现光机算一体化。该技术已广泛应用于三维物体识别、振动分析、微小位移测量、形变检测、表面干涉检测、流体流速测量、粒子场测定、显微成像、生物医学诊断等方面。
使用离轴数字全息虽然可以将再现的物像与零级项和孪生像分开,但由于记录器件CCD分辨率比传统全息使用的干版低一到二个数量级,要求物光与参考光的夹角很小,此条件对数字全息的实际应用带来很大限制。将相移方法引入数字全息,利用相移干涉术(Phase-Shifting Interferometry,简称PSI),可消除同轴数字全息的零级项和孪生像,很好地解决这一矛盾。在相移数字全息干涉术中,首先通过改变参考光(一般为轴向平面波)的相位对同一物波记录多幅全息图,然后利用对不同相移设计所发展的不同算法对全息图数据进行计算,可得到记录面上物光场的复振幅分布。该技术结合了相移干涉和数字全息的优点,具有测量精度高、易于操作、能够实现全场测量、数字处理能力强大方便等优点,大大推进了全息技术的发展和应用。
传统的相移干涉术采用定步长标准相移算法,即每步的相移量均为特定的特殊值(多为2π/K,K为正整数)。一些研究者将其推广到等步长相移算法,即相移量不一定取2π/K这种特殊值,但每步相移量仍需严格相等。这两种算法对相移器的要求都特别苛刻。实践中由于各种因素,相移器的实际相移量与其标称值或多或少都存在一定偏差,而且这种偏差往往是难以预测和控制的,它会对波前再现或测量结果带来误差。另外,传统PSI需要至少三幅干涉图,实际上基于减小误差等考虑,往往要用到多幅甚至10幅以上干涉图来满足算法的需要,使记录的数据量庞大,给随后的存储、传输及处理带来很大负担。
针对以上问题,我们提出了可以应用任意未知相移的广义相移干涉术(Generalized Phase-Shifting Interferometry,简称GPSI)的概念。在GPSI中,每步相移量不仅可以是不等的和任意的,而且一般来说可以是未知的。利用我们发展的一套算法,可以从干涉图中盲取(blindly extract)相移量;然后根据这些相移量可依特定的GPSI计算公式恢复波场复振幅。这就取消了相等相移及精确相移这两个苛刻的限制,提高了这一技术的误差免疫能力以及普适性和方便性,减少了这一技术对相移器精度的依赖性,大大提高了测量精度。同时,由于GPSI可以相当精确地求得真实相移量,也使应用者不必再采用针对相移误差而设计的各种复杂的波前误差校正算法。最后,我们发展的GPSI可以应用于两幅及两幅以上的任何算法,最低只需两幅干涉图,这可以大大降低计算量和存储量,对某些应用具有重要意义。
由于真正意义上的广义相移干涉术提出较晚,国内外与此相关的报道甚少,该技术涉及的许多问题需要进一步研究探讨。概括地说它应包含以下几个方面。首先,在未知相移的提取方面,还需对已提出的几种算法进行改进和完善,并探索新的更为方便可靠的算法。其次,需建立物光波前误差评估体系,研究各种误差尤其是相移误差对最终再现物光波面振幅和相位(不仅仅是记录面恢复相位)误差的传递作用,并对最终再现的物光波面的复振幅误差与算法和相移的设计关系进行定量分析,这方面在国内外几乎还是空白。再次,如何针对实验中的各种实际问题,寻求压低各种实验误差从而提高最终测量精度的途径,也需要多方面的研究和反复验证。
本论文的主要目的是在对国内外数字全息及相移干涉术进行深入考察及分析的基础上,发展对相移器精度依赖较小甚至可使用未知相移量的广义相移数字全息干涉术,包括寻找可精确和方便地抽取未知相移量并进而恢复物波复波场的算法与技术,并从模拟和光学实验两方面探索减小误差、提高测量精度的方法和手段。为此论文中给出了利用迭代方法在三步以上GPSI中快速抽取未知相移量的精确算法和两步(或两步以上)GPSI的相移量抽取和物光恢复算法;随后发展了两步(或两步以上)GPSI中相移量提取的非迭代精确算法和快速简单算法;通过研究GPSI中三种相移量误差对物光恢复误差进行传递的关系找到了对相移误差源具有较强免疫性的相移量区间,其计算结果对实际工作中的相移量选择具有重要的理论指导意义;针对光源强度稳定性问题,提出了以衍射相位场统计特性为基础的光源强度误差校正方法。
论文的主要内容如下,其中除第一点为综合评述外,其他各点都是我们的创新研究成果,已在多种国际著名学术刊物发表,并受到广泛关注:
a.分别对相移数字全息干涉术的发展及应用和广义相移数字全息干涉术的产生及研究现状做了概貌性的介绍和总结,包括相移数字全息中记录和再现的基本理论:物光波前数字恢复的各种方法;相移技术的基本原理,相移的引入方法和实验装置,以及物波恢复公式的设计方法;已有的三步以上相移数字全息干涉术的定步长、等步长相移算法公式;任意已知相移量广义相移数字全息干涉术算法的物光波前恢复公式。以上描述和结论是进一步研究广义相移数字全息干涉术的基础,同时对实际工作具有理论指导意义和参考价值。
b.在对已有的任意步长广义相移算法进行分析比较的基础上,给出了一种基于切线法迭代原理的任意步长相移量精确快速抽取的新方法,再利用已有的适合于任意步长的三步和四步GPSI波前计算公式实现物光波前恢复。该方法对相移器的精度无严格要求,甚至无需标定,计算机模拟证明了这种方法的准确性和高效性。
c.将物光再现公式结合最小二乘迭代方法实现了使用两幅干涉图进行相移量抽取的算法设计。该算法可以仅用两幅干涉图实现相移量的盲寻;然后利用相应公式恢复物波波前并进而重建原始物像。此方法可应用于干涉图数K≥2的任意情况。利用计算机模拟验证了在光滑物面和漫射物面两种情况下相移抽取和原始物光重建的准确性和有效性。
d.由于未知相移盲取的迭代算法一般计算量较大,计算时间较长,我们进而提出了两种非迭代算法。其一是通过求解一元二次方程,推导出相邻两幅干涉图之间相移量的计算式,由相邻两幅干涉图和物光及参考光强度计算任意未知相移量。其二是利用物光衍射相位场的随机统计特性,导出一个公式,可在不需测量物光强度的情况下抽取出任意相移量。这两种算法将现有GPSI中需要的干涉图由三幅以上减少到最少的极限情况——只需一次相移,即两幅干涉图,而且该相移可以是(0,π)区间内的任意值;它们也同样适用于更多幅干涉图的情况。它们可大大提高计算效率和测量精度,计算机模拟和光学实验结果已非常令人满意地证明了这一点。
e.分析了PSI和GPSI中波前再现误差的各种来源,特点和计算方法。在回顾已有的误差计算和校正方法的基础上,针对光源强度涨落造成的物光波前误差,以物光衍射相位场的统计特性为基础,通过分析光源不稳定造成物光恢复误差的机理提出了光源强度涨落引起的物光重建误差的校正算法。该算法可在不需要任何辅助测量的条件下,仅用得到的干涉图数据进行校正,将再现的物光波前振幅和相位误差都减小两个数量级以上。计算机模拟和实验结果都证明了其正确性与有效性。
f.系统定量地研究了三步广义相移全息术中相移设计和相移误差对波前恢复误差的影响。导出了一个在不同相移设置和不同相移误差情况下均方波前恢复误差的一般表达式,同时给出了对应三种重要的相移误差,即等值、线性及随机相移误差的均方波前恢复误差的具体表达式。作为实际应用中常用的特例,详细讨论了三步等值相移GPSI中的波前恢复误差,得到了一系列对实际应用有指导意义的结论,为GPSI的算法设计和最佳相移量选取提供了有用的依据。利用多达100,000套相移误差样本的计算机模拟实验结果证实了我们的理论分析。这里使用的分析方法也可以进一步应用到其他GPSI算法,如四步或五步相移算法设计,对一般的相移算法设计具有重要参考价值。相移干涉术已提出多年,它对昂贵的精密相移器(一般用进口的压电微平移器PZT)的依赖已成为限制其普及应用及测量精度的一个瓶颈。我们提出的广义相移干涉术有望克服这一瓶颈,大大拓展其应用并改善其精度。以上研究成果不仅具有理论意义,也具有重要的实用价值。
Recently, digital holography (DH) is a fast-developing area in the field of information optics. It records the intensity information of interferograms digitally by using optoelectronic recording device such as Charge Coupled Devices (CCD), stores it in a computer, and then processes the data by corresponding algorithms to gain the information of the original object. Digital holography can not only retrieve both the amplitude and phase distributions of object wave at the same time as traditional holography does, but also avoid the inconvenience of wet processing and the difficulty of repositioning for the latter. Combining the modern CCD and computer image processing technology together, DH can effectively realize the quantitatively recording of interferogram, and the dada storage, processing, transform, and reconstruction. This technology has been widely used in 3D object recognition, vibration analysis, small displacement measurement, deformation detection, surface inspection, flow measurement, particle field analysis, microscopic imaging, biomedical diagnosis and so on.
Although the object image can be separated from the twin image and zero order term by using off-line digital holography, but it requires a small angle between object wave and reference wave in recording process because the resolution of a CCD is one or two orders lower than the traditional silver halide plates. This requirement exerts a serious limitation upon the applications of digital holography in practice. This problem had been satisfactorily solved by the introduction of phase-shifting interferometry (PSI). In PSI, the interferograms are firstly recorded with different relative phases of a reference wave (generally on-line plane wave) for the same objective wave, then the recorded data is processed by the algorithms designed by different phase-shifting methods, and lastly the objective complex field distribution on the recording plane is retrieved. This technique combines the advantages of both the PSI and DH such as high measuremental precision, easy operation, whole field observation, and high ability and versatility of digital data processing, therefore it gave a great impetus to the development and applications of digital holography.
The traditional PSI uses fixed standard phase shift algorithms, in which each step has a special constant phase shift value (mostly 2π/K, K is a positive integer). Some researchers extended them to equal-step phase-shifting algorithms, that is to say, the phase shifts are not definitely the special values of 2π/K, but they still must be strictly equal. All the two kinds of methods have very strict requirements for the precision of a phase shifter. However, because of many factors in practice, the real phase shift value introduced by a phase shifter is often more or less different from its nominal value, and this difference is always difficult to predict and control. And more, traditional PSI needs at least three interferograms (in fact, often even more than 10 frames are needed for some algorithms specially designed for error suppression), which yields a heavy burden on the recording, storage, transportation, and processing of the data.
To solve the problems herein, we propose the concept of generalized phase-shifting interferometry (GPSI), in which the unknown arbitrary phase shifts can be used. In GPSI, all phase shifts can be not only unequal but also arbitrary, and can even be unknown in general. Using a set of algorithms developed by us, the phase shifts can be blindly extracted from interferograms, and then the object wave field can be retrieved by these extracted phase shifts through specific GPSI computation formulae. Now the two strict limitations of equal phase shifts and precise phase shifts in PSI are removed, and the convenience and measurement precision of this technology are greatly improved. At the same time, because GPSI can calculate the actual phase shifts with considerable precision, the users may need no longer the error correction algorithms. Finally, the GPSI developed by us can be used to any GPSI algorithms with two or more frames. The usability of only two frames in GPSI decreases the computation load, and it is important for some applications.
Because the GPSI with the use of unknown phase shifts was introduced only recently, the related studies and reports are comparatively few, and many problems on this area need to be further investigated. Generally it may include the following aspects. First, on the extraction of unknown phase shifts, the several algorithms suggested early need improvement and perfection, and new reliable and convenient algorithms should be explored. Second, the error evaluation system for object wavefront retrieval should be established to estimate the effect of different kinds of errors especially phase shift errors on the reconstruction of object wave including its amplitude and phase, and quantitatively analyze the relationship between the complex amplitude error of retrieved object wave and phase shift design in corresponding algorithms, there is few work in this aspect at home and aboard now. Last, the approaches to suppress experimental errors and then improve the final measurement precision in practice should be investigated.
Based on the detailed investigation and analysis of PSI at home and broad, this dissertation aim to develop the GPSI methods and techniques which has less dependance on the precision of phase shifter or even can use arbitrary unknown phase shifts, including finding the algorithm and technology for the phase shift extraction and object wave retrieval with high precision and convenience and exploring the method and approaches to decrease the errors and improve the measurement precision in both the computer simulations and optical experiments. Therefore both the iterative and the non-iterative algorithms to extract arbitrary unknown phase shifts and retrieve object wave with high precision in GPSI with frame number equal to or greater than 2 are proposed. The phase shift range more immune to phase shift errors is found by the study of the relationship between three kinds of phase shift errors and the object retrieval errors in GPSI, and the results can serve as a guide for phase shift selection in practice. Specifically, for the problem of light source unstability, an effective error correction method is suggested based on the statistical character of phase distributions of a diffraction field.
The main contents of this dissertation are as follows. Except the first point which is a systematic review on the previous works and some basic principles, all the other points are our initial works published in different famous international journals.
a) A systematic and comprehensive review on the development and applications of PSI and the formation and current research situation of GPSI is provided, including the basic theory of recording and reconstruction of an object wave in digital holography, the basic principle of phase shift method and the techniques to introduce the phase shifts, the method of phase shift design in PSI, and the existent object wavefront retrieval formulae with fixed, equal, or arbitrary known phase shifts in three or more step PSI. These explanations are the basic background for the further study of GPSI and useful for practical work.
b) Based on the analysis and comparison of the existing GPSI with arbitrary phase shifts, a new fast convergent algorithm to extract arbitrary unknown phase shifts mainly by tangential iterative principle has been proposed. By using the wavefront calculation formulae in three and four-step GPSI, the object wavefront can be further retrieved. This method has no strict requirement for the precision of the phase shifter, or even needs no calibration of the phase shifter. Its effectiveness and accuracy has been verified by a series of computer simulations.
c) Combining the object wave retrieval formulae with the least square iterative method, we proposed a new algorithm to extract the phase shift in two frame GPSI, and then extended it to any GPSI of frame number K≥2. This algorithm can find blindly the unknown phase shift with only two interferograms and then retrieve the object wave through corresponding formulae developed by us. A series of computer simulations have verified its effectiveness for both the smooth and diffusing objects in phase shift extraction and original object wave retrieval with high accuracy.
d) Because the unknown phase shift finding algorithms with iteration are time consuming, two non-iterative algorithms are proposed. In the first one of them, the equation to calculate the phase shift between two adjacent frames is deduced by solving a monadic quadratic equation and then the arbitrary unknown phase shift can be calculated by using the two interferograms and the intensities of object and reference waves. In the other one, based on the statistical character of the diffractive object phase field, we deduce a formula to extract arbitrary phase shift but without knowing the object intensities. These two algorithms decrease the frame number needed in GPSI to the least—only two interferograms and one phase shift, no iteration is necessary, and the phase shift can be any value in the range of (0,π). And naturally it is also applicable to the case with more interferograms. It can improve the calculation efficiency and measurement precision. The results from computer simulations and optical experiments have verified all our conclusions satisfactorily.
e) We have reviewed and analyzed different error sources in wavefront reconstruction and their characters and calculation methods in PSI and GPSI. On this basis we proposed an algorithm to correct the object wavefront retrieval error caused by the instability of light source intensity by considering the statistical property of diffraction phase field. This algorithm can work by using only interferogram data without any auxiliary measurement and decrease the amplitude and phase errors of object wave by more than two magnitude orders. The results from computer simulations and optical experiments have convincingly verified its feasibility and effectiveness.
f) We have systematically and quantitatively analyzed the effects of phase shifts selection and phase shift errors on the wavefront retrieval errors in three step GPSI. A general expression of mean square wavefront retrieval error is deduced and then the specific expressions corresponding to the three important phase shift errors, the equal, linear, and random phase shift errors, are given. As a special case used frequently in practice, the wavefront retrieval error in three equal step GPSI is discussed in details and a series of conclusions useful for practical applications are obtained, which are valuable for GPSI algorithm design and the selection of optimized phase shifts. The results from computer simulations with samples up to 100,000 sets of errors have verified our theoretic analysis. The analytical method used herein can also be applied to other GPSI algorithms, such as four-step and five-step phase shift design, and can be a guide for the design of general phase shifting algorithm.
PSI technology has been proposed for many years, and its dependence on the expensive phase shifter has become an obstacle for its wide applications and measurement precision. The GPSI suggested here may overcome this obstacle and expand its application scope and improve the corresponding precision. The research work above is not only of theoretical significance, but also of important value in practice.
使用离轴数字全息虽然可以将再现的物像与零级项和孪生像分开,但由于记录器件CCD分辨率比传统全息使用的干版低一到二个数量级,要求物光与参考光的夹角很小,此条件对数字全息的实际应用带来很大限制。将相移方法引入数字全息,利用相移干涉术(Phase-Shifting Interferometry,简称PSI),可消除同轴数字全息的零级项和孪生像,很好地解决这一矛盾。在相移数字全息干涉术中,首先通过改变参考光(一般为轴向平面波)的相位对同一物波记录多幅全息图,然后利用对不同相移设计所发展的不同算法对全息图数据进行计算,可得到记录面上物光场的复振幅分布。该技术结合了相移干涉和数字全息的优点,具有测量精度高、易于操作、能够实现全场测量、数字处理能力强大方便等优点,大大推进了全息技术的发展和应用。
传统的相移干涉术采用定步长标准相移算法,即每步的相移量均为特定的特殊值(多为2π/K,K为正整数)。一些研究者将其推广到等步长相移算法,即相移量不一定取2π/K这种特殊值,但每步相移量仍需严格相等。这两种算法对相移器的要求都特别苛刻。实践中由于各种因素,相移器的实际相移量与其标称值或多或少都存在一定偏差,而且这种偏差往往是难以预测和控制的,它会对波前再现或测量结果带来误差。另外,传统PSI需要至少三幅干涉图,实际上基于减小误差等考虑,往往要用到多幅甚至10幅以上干涉图来满足算法的需要,使记录的数据量庞大,给随后的存储、传输及处理带来很大负担。
针对以上问题,我们提出了可以应用任意未知相移的广义相移干涉术(Generalized Phase-Shifting Interferometry,简称GPSI)的概念。在GPSI中,每步相移量不仅可以是不等的和任意的,而且一般来说可以是未知的。利用我们发展的一套算法,可以从干涉图中盲取(blindly extract)相移量;然后根据这些相移量可依特定的GPSI计算公式恢复波场复振幅。这就取消了相等相移及精确相移这两个苛刻的限制,提高了这一技术的误差免疫能力以及普适性和方便性,减少了这一技术对相移器精度的依赖性,大大提高了测量精度。同时,由于GPSI可以相当精确地求得真实相移量,也使应用者不必再采用针对相移误差而设计的各种复杂的波前误差校正算法。最后,我们发展的GPSI可以应用于两幅及两幅以上的任何算法,最低只需两幅干涉图,这可以大大降低计算量和存储量,对某些应用具有重要意义。
由于真正意义上的广义相移干涉术提出较晚,国内外与此相关的报道甚少,该技术涉及的许多问题需要进一步研究探讨。概括地说它应包含以下几个方面。首先,在未知相移的提取方面,还需对已提出的几种算法进行改进和完善,并探索新的更为方便可靠的算法。其次,需建立物光波前误差评估体系,研究各种误差尤其是相移误差对最终再现物光波面振幅和相位(不仅仅是记录面恢复相位)误差的传递作用,并对最终再现的物光波面的复振幅误差与算法和相移的设计关系进行定量分析,这方面在国内外几乎还是空白。再次,如何针对实验中的各种实际问题,寻求压低各种实验误差从而提高最终测量精度的途径,也需要多方面的研究和反复验证。
本论文的主要目的是在对国内外数字全息及相移干涉术进行深入考察及分析的基础上,发展对相移器精度依赖较小甚至可使用未知相移量的广义相移数字全息干涉术,包括寻找可精确和方便地抽取未知相移量并进而恢复物波复波场的算法与技术,并从模拟和光学实验两方面探索减小误差、提高测量精度的方法和手段。为此论文中给出了利用迭代方法在三步以上GPSI中快速抽取未知相移量的精确算法和两步(或两步以上)GPSI的相移量抽取和物光恢复算法;随后发展了两步(或两步以上)GPSI中相移量提取的非迭代精确算法和快速简单算法;通过研究GPSI中三种相移量误差对物光恢复误差进行传递的关系找到了对相移误差源具有较强免疫性的相移量区间,其计算结果对实际工作中的相移量选择具有重要的理论指导意义;针对光源强度稳定性问题,提出了以衍射相位场统计特性为基础的光源强度误差校正方法。
论文的主要内容如下,其中除第一点为综合评述外,其他各点都是我们的创新研究成果,已在多种国际著名学术刊物发表,并受到广泛关注:
a.分别对相移数字全息干涉术的发展及应用和广义相移数字全息干涉术的产生及研究现状做了概貌性的介绍和总结,包括相移数字全息中记录和再现的基本理论:物光波前数字恢复的各种方法;相移技术的基本原理,相移的引入方法和实验装置,以及物波恢复公式的设计方法;已有的三步以上相移数字全息干涉术的定步长、等步长相移算法公式;任意已知相移量广义相移数字全息干涉术算法的物光波前恢复公式。以上描述和结论是进一步研究广义相移数字全息干涉术的基础,同时对实际工作具有理论指导意义和参考价值。
b.在对已有的任意步长广义相移算法进行分析比较的基础上,给出了一种基于切线法迭代原理的任意步长相移量精确快速抽取的新方法,再利用已有的适合于任意步长的三步和四步GPSI波前计算公式实现物光波前恢复。该方法对相移器的精度无严格要求,甚至无需标定,计算机模拟证明了这种方法的准确性和高效性。
c.将物光再现公式结合最小二乘迭代方法实现了使用两幅干涉图进行相移量抽取的算法设计。该算法可以仅用两幅干涉图实现相移量的盲寻;然后利用相应公式恢复物波波前并进而重建原始物像。此方法可应用于干涉图数K≥2的任意情况。利用计算机模拟验证了在光滑物面和漫射物面两种情况下相移抽取和原始物光重建的准确性和有效性。
d.由于未知相移盲取的迭代算法一般计算量较大,计算时间较长,我们进而提出了两种非迭代算法。其一是通过求解一元二次方程,推导出相邻两幅干涉图之间相移量的计算式,由相邻两幅干涉图和物光及参考光强度计算任意未知相移量。其二是利用物光衍射相位场的随机统计特性,导出一个公式,可在不需测量物光强度的情况下抽取出任意相移量。这两种算法将现有GPSI中需要的干涉图由三幅以上减少到最少的极限情况——只需一次相移,即两幅干涉图,而且该相移可以是(0,π)区间内的任意值;它们也同样适用于更多幅干涉图的情况。它们可大大提高计算效率和测量精度,计算机模拟和光学实验结果已非常令人满意地证明了这一点。
e.分析了PSI和GPSI中波前再现误差的各种来源,特点和计算方法。在回顾已有的误差计算和校正方法的基础上,针对光源强度涨落造成的物光波前误差,以物光衍射相位场的统计特性为基础,通过分析光源不稳定造成物光恢复误差的机理提出了光源强度涨落引起的物光重建误差的校正算法。该算法可在不需要任何辅助测量的条件下,仅用得到的干涉图数据进行校正,将再现的物光波前振幅和相位误差都减小两个数量级以上。计算机模拟和实验结果都证明了其正确性与有效性。
f.系统定量地研究了三步广义相移全息术中相移设计和相移误差对波前恢复误差的影响。导出了一个在不同相移设置和不同相移误差情况下均方波前恢复误差的一般表达式,同时给出了对应三种重要的相移误差,即等值、线性及随机相移误差的均方波前恢复误差的具体表达式。作为实际应用中常用的特例,详细讨论了三步等值相移GPSI中的波前恢复误差,得到了一系列对实际应用有指导意义的结论,为GPSI的算法设计和最佳相移量选取提供了有用的依据。利用多达100,000套相移误差样本的计算机模拟实验结果证实了我们的理论分析。这里使用的分析方法也可以进一步应用到其他GPSI算法,如四步或五步相移算法设计,对一般的相移算法设计具有重要参考价值。相移干涉术已提出多年,它对昂贵的精密相移器(一般用进口的压电微平移器PZT)的依赖已成为限制其普及应用及测量精度的一个瓶颈。我们提出的广义相移干涉术有望克服这一瓶颈,大大拓展其应用并改善其精度。以上研究成果不仅具有理论意义,也具有重要的实用价值。
Recently, digital holography (DH) is a fast-developing area in the field of information optics. It records the intensity information of interferograms digitally by using optoelectronic recording device such as Charge Coupled Devices (CCD), stores it in a computer, and then processes the data by corresponding algorithms to gain the information of the original object. Digital holography can not only retrieve both the amplitude and phase distributions of object wave at the same time as traditional holography does, but also avoid the inconvenience of wet processing and the difficulty of repositioning for the latter. Combining the modern CCD and computer image processing technology together, DH can effectively realize the quantitatively recording of interferogram, and the dada storage, processing, transform, and reconstruction. This technology has been widely used in 3D object recognition, vibration analysis, small displacement measurement, deformation detection, surface inspection, flow measurement, particle field analysis, microscopic imaging, biomedical diagnosis and so on.
Although the object image can be separated from the twin image and zero order term by using off-line digital holography, but it requires a small angle between object wave and reference wave in recording process because the resolution of a CCD is one or two orders lower than the traditional silver halide plates. This requirement exerts a serious limitation upon the applications of digital holography in practice. This problem had been satisfactorily solved by the introduction of phase-shifting interferometry (PSI). In PSI, the interferograms are firstly recorded with different relative phases of a reference wave (generally on-line plane wave) for the same objective wave, then the recorded data is processed by the algorithms designed by different phase-shifting methods, and lastly the objective complex field distribution on the recording plane is retrieved. This technique combines the advantages of both the PSI and DH such as high measuremental precision, easy operation, whole field observation, and high ability and versatility of digital data processing, therefore it gave a great impetus to the development and applications of digital holography.
The traditional PSI uses fixed standard phase shift algorithms, in which each step has a special constant phase shift value (mostly 2π/K, K is a positive integer). Some researchers extended them to equal-step phase-shifting algorithms, that is to say, the phase shifts are not definitely the special values of 2π/K, but they still must be strictly equal. All the two kinds of methods have very strict requirements for the precision of a phase shifter. However, because of many factors in practice, the real phase shift value introduced by a phase shifter is often more or less different from its nominal value, and this difference is always difficult to predict and control. And more, traditional PSI needs at least three interferograms (in fact, often even more than 10 frames are needed for some algorithms specially designed for error suppression), which yields a heavy burden on the recording, storage, transportation, and processing of the data.
To solve the problems herein, we propose the concept of generalized phase-shifting interferometry (GPSI), in which the unknown arbitrary phase shifts can be used. In GPSI, all phase shifts can be not only unequal but also arbitrary, and can even be unknown in general. Using a set of algorithms developed by us, the phase shifts can be blindly extracted from interferograms, and then the object wave field can be retrieved by these extracted phase shifts through specific GPSI computation formulae. Now the two strict limitations of equal phase shifts and precise phase shifts in PSI are removed, and the convenience and measurement precision of this technology are greatly improved. At the same time, because GPSI can calculate the actual phase shifts with considerable precision, the users may need no longer the error correction algorithms. Finally, the GPSI developed by us can be used to any GPSI algorithms with two or more frames. The usability of only two frames in GPSI decreases the computation load, and it is important for some applications.
Because the GPSI with the use of unknown phase shifts was introduced only recently, the related studies and reports are comparatively few, and many problems on this area need to be further investigated. Generally it may include the following aspects. First, on the extraction of unknown phase shifts, the several algorithms suggested early need improvement and perfection, and new reliable and convenient algorithms should be explored. Second, the error evaluation system for object wavefront retrieval should be established to estimate the effect of different kinds of errors especially phase shift errors on the reconstruction of object wave including its amplitude and phase, and quantitatively analyze the relationship between the complex amplitude error of retrieved object wave and phase shift design in corresponding algorithms, there is few work in this aspect at home and aboard now. Last, the approaches to suppress experimental errors and then improve the final measurement precision in practice should be investigated.
Based on the detailed investigation and analysis of PSI at home and broad, this dissertation aim to develop the GPSI methods and techniques which has less dependance on the precision of phase shifter or even can use arbitrary unknown phase shifts, including finding the algorithm and technology for the phase shift extraction and object wave retrieval with high precision and convenience and exploring the method and approaches to decrease the errors and improve the measurement precision in both the computer simulations and optical experiments. Therefore both the iterative and the non-iterative algorithms to extract arbitrary unknown phase shifts and retrieve object wave with high precision in GPSI with frame number equal to or greater than 2 are proposed. The phase shift range more immune to phase shift errors is found by the study of the relationship between three kinds of phase shift errors and the object retrieval errors in GPSI, and the results can serve as a guide for phase shift selection in practice. Specifically, for the problem of light source unstability, an effective error correction method is suggested based on the statistical character of phase distributions of a diffraction field.
The main contents of this dissertation are as follows. Except the first point which is a systematic review on the previous works and some basic principles, all the other points are our initial works published in different famous international journals.
a) A systematic and comprehensive review on the development and applications of PSI and the formation and current research situation of GPSI is provided, including the basic theory of recording and reconstruction of an object wave in digital holography, the basic principle of phase shift method and the techniques to introduce the phase shifts, the method of phase shift design in PSI, and the existent object wavefront retrieval formulae with fixed, equal, or arbitrary known phase shifts in three or more step PSI. These explanations are the basic background for the further study of GPSI and useful for practical work.
b) Based on the analysis and comparison of the existing GPSI with arbitrary phase shifts, a new fast convergent algorithm to extract arbitrary unknown phase shifts mainly by tangential iterative principle has been proposed. By using the wavefront calculation formulae in three and four-step GPSI, the object wavefront can be further retrieved. This method has no strict requirement for the precision of the phase shifter, or even needs no calibration of the phase shifter. Its effectiveness and accuracy has been verified by a series of computer simulations.
c) Combining the object wave retrieval formulae with the least square iterative method, we proposed a new algorithm to extract the phase shift in two frame GPSI, and then extended it to any GPSI of frame number K≥2. This algorithm can find blindly the unknown phase shift with only two interferograms and then retrieve the object wave through corresponding formulae developed by us. A series of computer simulations have verified its effectiveness for both the smooth and diffusing objects in phase shift extraction and original object wave retrieval with high accuracy.
d) Because the unknown phase shift finding algorithms with iteration are time consuming, two non-iterative algorithms are proposed. In the first one of them, the equation to calculate the phase shift between two adjacent frames is deduced by solving a monadic quadratic equation and then the arbitrary unknown phase shift can be calculated by using the two interferograms and the intensities of object and reference waves. In the other one, based on the statistical character of the diffractive object phase field, we deduce a formula to extract arbitrary phase shift but without knowing the object intensities. These two algorithms decrease the frame number needed in GPSI to the least—only two interferograms and one phase shift, no iteration is necessary, and the phase shift can be any value in the range of (0,π). And naturally it is also applicable to the case with more interferograms. It can improve the calculation efficiency and measurement precision. The results from computer simulations and optical experiments have verified all our conclusions satisfactorily.
e) We have reviewed and analyzed different error sources in wavefront reconstruction and their characters and calculation methods in PSI and GPSI. On this basis we proposed an algorithm to correct the object wavefront retrieval error caused by the instability of light source intensity by considering the statistical property of diffraction phase field. This algorithm can work by using only interferogram data without any auxiliary measurement and decrease the amplitude and phase errors of object wave by more than two magnitude orders. The results from computer simulations and optical experiments have convincingly verified its feasibility and effectiveness.
f) We have systematically and quantitatively analyzed the effects of phase shifts selection and phase shift errors on the wavefront retrieval errors in three step GPSI. A general expression of mean square wavefront retrieval error is deduced and then the specific expressions corresponding to the three important phase shift errors, the equal, linear, and random phase shift errors, are given. As a special case used frequently in practice, the wavefront retrieval error in three equal step GPSI is discussed in details and a series of conclusions useful for practical applications are obtained, which are valuable for GPSI algorithm design and the selection of optimized phase shifts. The results from computer simulations with samples up to 100,000 sets of errors have verified our theoretic analysis. The analytical method used herein can also be applied to other GPSI algorithms, such as four-step and five-step phase shift design, and can be a guide for the design of general phase shifting algorithm.
PSI technology has been proposed for many years, and its dependence on the expensive phase shifter has become an obstacle for its wide applications and measurement precision. The GPSI suggested here may overcome this obstacle and expand its application scope and improve the corresponding precision. The research work above is not only of theoretical significance, but also of important value in practice.
引文
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