5 Noise Reduction in Amplitude-Fluctuation Electronic Speckle-Pattern Interferometry 29 Fig. 5.2 Interferometer sensitivity respectively; their intensities are assumed to be equal to each other. (In reality they are not equal to each other but the gist of the argument here is not affected by inequality. The second term in the rightmost-hand side of Eq. (5.1) with cos(ı0 Cısin!t) is called the interference term. This term is important as it contains the relative phase change information. The light intensity read by the photo-detector can be expressed as follows. Is.t/ DI.t/ 2I0 D2I0 cos.ıo Cısin!t/ (5.2) Our purpose here is to extract ıDkd fromIs(t), and thereby estimate the oscillation amplitude ı. It is apparent that the sensitivity of the signal depends on the operation-point (initial) phase ıo. Figure 5.2 illustrates the signal sensitivity for three values of ıo. It is clear that when ıo D /2, the sensitivity is the highest. The condition is that the arm length difference in the phase is a quarter wavelength, which means that the round trip phase difference between the two arms at the beam splitter is half wavelength. This condition is known as the destructive interference. Under this condition, when the amplitude dDı/kDı/(2 ) is null, the optical intensity is zero. 5.2.2 Two-Dimensional Phase Analysis Based on Amplitude-Fluctuation Interferometry Figure 5.3 illustrates a typical setup for two-dimensional phase analysis. Out-of-plane displacement of a specimen is analyzed as a full-field image with the use of a digital camera. The dashed line indicates the case when the specimen is slightly tilted from the normal direction to the laser beam path from the laser so that a carrier fringe system is introduced. Call this case the carrier fringe configuration. We first discuss the other case where the specimen is normal to laser beam path (the solid line). Call this case the normal incident configuration. In both cases, since the data acquisition time is significantly longer than the oscillation frequency, it is convenient to expand Eq. (5.1) into harmonics of the driving frequency and approximate the output signal with the time-independent components of the signal. I.t/ D2I0 f1Ccos.ıo Cısin!t/gD2I0 f1Ccosıo cos.ısin!t/ sinıo sin.ısin!t/g (5.3) We can rewrite Eq. (5.3) in terms of Bessel functions of the first kind as follows. I.t/ D2I0 cosıo f1CJ0 .ı/g C2I0 hcosıo n2J2 .ı/cos2!t C2J4 .ı/cos4!t C t o sinıo f2J1 .ı/sin!t C2J3 .ı/sin3!t C gi (5.4)
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