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1、相移技术：五帧任意步距相移算法G. Stoilov & T. Dragostinov光信息存储和处理的科学院中心实验室，保加利亚，索菲亚，邮政信箱1113，保加利亚 9.5(收到1995.7.31修订版 1996.8.2；出版 1996.8.16)摘要The present paper offers a new formula for calculating the phase in interferometric measurements by the phase-stepping method. The proposed five-frame algorithm utilizes equa
2、l phase steps with an arbitrary value. Results are presented on computer simulation of the errors occurring at a phase-step different from 90o and due to vibrations in the system interferometer-object . 1997 Elsevier Science Ltd.本文提供了一种新的相移法计算在干涉测量中的相位计算公式。该算法所提出的五步运算法 则利用相移量任意的等步长相移。用计算机模拟该算法得出的结果误
3、差出现在相移步长不为 90o和系统中干涉对象的摆动。 1997 Elsevier科学有限公司1 引言It is known that the phase-stepping method (PSM), used to calculate the phase in interferometric measurements, is based on the basic interferometric equation.1-4 When two plane waves interfere, the interferometer can be adjusted to produce an interfe
4、rence pattern in the form of vertical fringes：A = I (1 + m sin kx + (P) + nA )(1)n已知的相移法(PSM)是用于计算干涉测量中的相位，是基于这个基本的干涉等式1-4.当两 个平面波的干涉，干涉结果可以被校正产生垂直条纹的干涉图案：A = I (1 + m sin kx + (P) + nA) nwhere A is the intensity of the nth picture, Z is the illuminating intensity, m is the modulation index, k if th
5、e spatial frequency, 9 (p) is the phase change, caused by the change in the measured quantity p, and A is the angle of each phase step.其中A是第n个图像的强度，Z表示背景强度，k是空间频率，机p)表示相位这个相位的 变化，由被测量量p的变化而变化，A是每个相位的变化角。The phase is calculated by making at least three measurements of the intensity distribution at di
6、fferent phases. With the evolution of the phase-stepping method, several algorithms for phase calculation have become popular.计算相位时至少要在不同相位下做三次强度分布的测量。随着相移法的发展，几种相移法的计算已经变得很普遍。5Three-frame method with a phase step 90o ：90 度相移的三帧法：(A - A(2)3)9 = arctan 02I 2 A A A102Three-frame method with a phase st
7、ep 120o : 4 120 度相移的三帧法：9 = arctan(J3a -a 212 A A A 丿0 1 2Four-frame method with a phase step 90o :590 度相移的四帧法：9 = arctanIA A )02 A A 丿13C(4)Four-frame method with a phase step90 with reduced error:减少误差90 度相移四帧法：f A - A )21 A A 丿320 = 0.5 arctanf A - A )t0 + arctan A A 丿217Four-frame method with an
8、arbitrary step:任意相移的四帧法：A = arccos0 = arctanA - A + A - A1“42(A A 丿23(A -A )GosA- 1)-(A -A )Gos2A-1) sin A(A - A )- sin 2A(A - A ) I 2 0 1 0(6)Five-frame method with a phase step 90 :8 90 度相移的5 帧法：0 = arctan2 A - A(7)132 A - A - A204A ,A ,A ,A Awhere 0123 and A4, are the measured intensities of each
9、 pixel at phase steps0,A,2A,3A ,and4A , respectively.Ao A1, A2 A3和A4分别是在A，2A，3A和4A相移下测的各像素的强度。According to the manner of motion of the phase-shifting mirror in the interferometer, four approaches are feasible:根据对干涉仪中相移反映的运动方式，四种方法都是可行的：1. All phase-steps in all pixels of the image are equal (A = 60
10、, 90 or 120 ): 1图像中所有像素的全部相移都是相等的(A = 60, 90 or 120):(x, y)= A (p, q)iA (x, y)= A (x, y)ijwhere i = 0.N, j = 0.N; N is the number of phase-steps, and x,y and p,q are indices of a pair of pixels in the picture. This is the ideal case that should give the best results. However, it is very difficult to
11、 achieve.当i = 0.N, j = O.N;N是相移的步数，x,y和p,q是图像中的一对像素指数。这是理想的情况 下，能得到最好的结果。然而在现实中很难实现。2. The consecutive phase-steps are unequal, but the phase-steps with equal serial number are equal:2. 连续的相移步步宽是不同的，但相移步步骤序号是相同的。(x, y)= A (p, q)iy)This approach has less stringent requirements to the stability of the
12、 phase-setting device. In this9 case the phase steps are among the unknowns in the set of equations.这种方法的相位设定装置的稳定性要求不太严格。在这种情况下的相位的步骤是在方程组中 的未知。3. All phase-steps at a specific pixel are equal, but the phase-steps in different pixels are different:3. 全部的相移步在特定的像素下是相等的，但在不同的像素中是不同的。i, q)A (x, y)= Ai
13、j(x, y )This is the case when the phase-shifting mirror does not move parallel to itself. 这是当相移镜不动平行与自身的情况。4. All phase-steps in all pixels may be different. However, the proportionality between the changes in phase-steps of equal numbers in different pixels is constant:4. 在所有像素的所有相移步可能是不同的。然而，在不同的像
14、素的序号相等的相移变化之间 的比例是不变的：i, q)(x, y )工 A (x,ij,q)There is no data about this case in the literature; it may attract researchersinterest in the future. The majority of the existing phase-stepping methods are based on the first approach. It demands that the phase-stepping system be calibrated prior to ea
15、ch measurement in order to eliminate the possible drift due to changes in temperature and other ambient factors. Quite often the calibration procedure takes more time than the measurement itself. The change in the transfer function of the phase-shifting device leads to a change in sensitivity and, hence, to imprecision in setting the phase step. It is much easier to set steps equal to one another than to make them all equal to a certain value. This has caused the authors to examine the
