Theoretical And Experimental Study Of The Algorithms Of Phase Shift Extraction And Wave-front Reconstruction In Digital Generalized Phase-shifting Interferometry

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.