<title>Transparent object analysis in a Mach-Zehnder interferometer using the multiplicative analogical moire phase-shifting method</title>

A multiplicative analogical moiré phase-shifting Twyman-Green technique for Nd:YAG rod wavefront distortion measurement

“…0 and c 3 = −1) and the calculated results, Y φ MI (x, y) and N φ MI (x, y), are put into equation (10), with the aim of obtaining the principal values φ G (x, y) of phase increment. On the other hand, the latter values are obtained directly if a set of selected phase-shifted moiré images is combined at the same time in the two-plus-two-point DPSAs, equations (11).…”

“…A WFD of the order of 0.25λ in −1 is achieved if the phase increment φ G (x, y) is calculated with the three-point GPSA. On the other hand, the unwrapped phase-increment values provided directly by the two-plustwo-point DPSAs, φ D (x, y), equations (11), present an average value similar to φ G (x, y) for the same fringe profiles, but several discontinuities appear when the numerator and denominator of the inverse trigonometric function employed take small values (figure 5). This discrepancy points out the need for a detailed study of the sensitivity of the DPSA to the various sources of phase error.…”

“…This representation provides information about the amount and characteristics of the type of error. The graphic representation (figure 6) of the calculated phase error against the phase increment, for all two-plus-two-point DPSA phase calculation formulae, equations (11), shows that the error is phase dependent. A detailed analysis of these graphical results shows that this severe error E[ φ(x, y)] exhibits a direct sinusoidal dependence on the phase increment φ(x, y) with an amplitude of the order of the miscalibration error δα, that is to say,…”

“…These multiplicative moiré images are digitized to 8 bits and lowpass filtered in a computer in such a way that two phase-shifted moiré image sets, with, Y m k (x, y), and without, N m k (x, y), a test object, are obtained. Because the influence of aberrations of the interferometer and phase-modulator (φ TG (x, y) and d g (x, y) respectively) affects both image sets in the same way, their influence on the calculation of the incrementation of the phase of the transparent object φ(x, y) can be cancelled out directly, if a DPSA is employed, for example the two-plus-twopoint DPSAs given by equations (11), or first the aberrations are calculated with a GPSA, equation ( 9), and later these stored data are subtracted, equation (10), from the results provided by the same algorithm when the transparent object is located in the measuring area. This subtraction removes, at the same time, any linear phase term related to the spatial frequencies of the grating or carrier (that is to say, for example, those terms depending on f x , g x or g y ).…”

“…In this work, as an extension of a previous paper [11] in which a similar application was carried out with a different interferometric configuration, an analysis of a Nd:YAG rod with the multiplicative analogical moiré phase-shifting Twyman-Green technique is presented. At the same time, in this paper a deeper analysis of the error sensitivities for evaluation of the DPSA is provided.…”