- 11.1 INTRODUCTION. - In Sections and 11.5, techniques for continuous determination of the phase of the fringe function are described. - 11.2 INTENSITY-BASED ANALYSIS METHODS 11.2.1 Introduction. - 11.2.2 Prior Knowledge. - As shown in Figure 11.2 the flaw sites are marked with small crosses.. - 11.2.3 Fringe Tracking and Thinning. - Figure 11.1 Procedure for the analysis of Figure 11.2 (From Robinson, D. - With respect to the four directions shown in Figure 11.4(b), the peak conditions are defined as. - Figure 11.3 Holographic interferogram with a computer tracked fringe. - Figure 11.4 (a) 5 × 5 pixel matrix and (b) directions for fringe peak detection. - 11.2.4 Fringe Location by Sub-Pixel Accuracy. - I (x lo )]x lu + I (x lu ) (11.5) connecting the two points is calculated. - Figure 11.5 Illustration of the detection of the crossover points x l and x r with the mean intensity I m. - x ru (11.7) where x ro is the last pixel with intensity I (x ro ) over I m and x ru is the first pixel with intensity I (x ru ) below I m . - N 2 (11.10a). - N 2 x 1 (11.10b). - I i (11.11a). - I i (11.11b). - Figure 11.6 Intensity distribution and mean intensity I m along a TV line. - 11.3 PHASE-MEASUREMENT INTERFEROMETRY 11.3.1 Introduction. - I 1 I 2 | γ | (11.13). - 11.3.2 Principles of TPMI. - b(x, y) cos φ(x, y) (11.14). - I = a + b cos(φ + α) (11.15). - (I 2 − I 3 ) sin α 1 − (I 1 − I 3 ) sin α 2 + (I 1 − I 2 ) sin α 3 (11.17). - I 2 − I 1 (11.18). - a 0 + a 1 cos α i + a 2 sin α i (11.19a) where. - a 1 = b cos φ (11.19b). - Equation (11.19a) can be written in matrix. - (11.20). - A − 1 B (11.21). - (11.22). - (11.23). - From Equation (11.19b) we find that φ = tan −1. - a 2 a 1 (11.24). - 11.3.3 Means of Phase Modulation. - 11.3.4 Different Techniques. - Figure 11.7 Means of modulating or shifting the phase of the light in an interferometer (a) moving mirror, (b) tilted glass plate. - (11.26a) V. - (11.26b) (2) α i. - 3 I 1 − I 3 2I 1 − I 2 − I 3 (11.27a) V. - 3I 0 (11.27b). - I 1 − I 3 (11.28a) V. - (11.28b). - 4I 1 + I 2 + 6I 3 + I 4 − 4I 6 (11.29a) V. - 14I 0 (11.29b). - (I 1 − I 4 ) (11.30a) φ = tan −1. - (11.30b). - V I 0 (11.32). - 11.3.5 Errors in TPMI Measurements. - Figure 11.8 P– V phase error versus percent linear phase-shifter error. - 11.4 SPATIAL PHASE-MEASUREMENT METHODS 11.4.1 Multichannel Interferometer. - Consider Figure 11.9, which is a polarization (Michelson) interferometer. - (11.33) The resultant matrix after two passes (first through the QWP and back after being reflected from the mirror or object) therefore becomes. - (11.34) which is the same as for a halfwave plate. - (11.35). - Figure 11.9 Polarization interferometer for direct phase measurement. - (11.37a). - A three channel DPM (Bareket 1985) is illustrated in Figure 11.10. - a − b cos φ (11.40) I 3 = a + b cos φ. - a n e i2π nf x (11.42). - a n e i2π nf (x−x o ) (11.43). - a n e i2π nf (x−x o ) (11.44). - δ n = 2π nf x o (11.45). - 11.4.2 Errors in Multichannel Interferometers. - 11.4.3 Spatial-Carrier Phase-Measurement Method. - 2πf 0 x] (11.46) where f 0 is the carrier frequency in the x-direction. - Following Takeda’s method we write Equation (11.46) in the form. - c(x, y)e i2πf o x + c ∗ (x, y)e −2πf o x (11.47a) where. - By use of a filter function H (f x − f 0 , y) in the frequency plane the function C(f x − f 0 , y) can be isolated and translated by f 0 towards the origin to remove the carrier and obtain C(f x , y) as shown in Figure 11.11(b). - Figure 11.11 Separated Fourier spectra of a tilted fringe pattern. - Re[c(x, y)] (11.50). - Figure 11.12 (Takeda et al. - 11.4.4 Errors in the Fourier Transform Method. - (11.51). - (11.53). - Figure 11.12 Various stage in the Fourier transform method: (a) 1-D intensity distribution;. - Because the fringe amplitude approaches zero at the ends of the window (see Figure 11.12(b. - 11.4.5 Space Domain. - 2 cos(2ωt − φ) (11.58a) g(t) sin ωt = b. - 2 cos φ (11.59a). - 2 sin φ (11.59b). - 11.5 PHASE UNWRAPPING 11.5.1 Introduction. - I = a + b cos φ (11.61). - Figure 11.13 (a) Characteristic ‘saw-tooth’ wrapped phase function. - Figure 11.13(b) shows the data in Figure 11.13(a) after unwrapping.. - Figure 11.14 shows the effect of the addition of noise to unwrapped data. - 11.5.2 Phase-Unwrapping Techniques. - 11.5.3 Path-Dependent Methods. - The simplest of the phase-unwrapping methods involves a sequential scan through the data, line by line, see Figure 11.15. - Figure 11.15 Line by line sequential scanning path. - 11.5.4 Path-Independent Methods. - Figure 11.16 A series of iterations of the automation-unwrapping algorithm. - 11.5.5 Temporal Phase Unwrapping. - (11.64) where I ij (t. - The ϕ(t) values calculated from Equation (11.66), however, lie in the range − π to + π . - ϕ(t) (11.67). - 11.1 Suppose we have measured the intensities f 1 , f 2 , f 4 and f 5 at the pixels x 1. - 11.3 The phase in a three-frame technique is given by f = tan φ. - 11.4 Consider a four-frame technique with the following phase steps: α i
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