Multiplicative moiré two-beam phase-stepping and Fourier-transform methods for the evaluation of multiple-beam Fizeau patterns: a comparison

Abstract

A phase-evaluation method of multiple-beam Fizeau patterns that combines two-beam phase-stepping algorithms with the moiré effect was previously reported [Appl. Opt. 34, 3639-3643(1995)]. The method is based on a multiplicative moiréimage-formation process obtained by the direct superposition of high-frequency multiple-beam Fizeau carrier fringes upon a transmission grating (working as a phase modulator). We present a comparison between this multiplicative moiré two-beam phase-stepping method and the well-known Fourier-transform method for the topographic measurement of an undoped silicon wafer. The discrepancy between the two methods yields a rms phase-difference value of the order of(~2pi/90).

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

“…The direct superimposition of the phase-modulated Twyman-Green spatial carrier and the transmission grating provides a multiplicative moiré pattern with an intensity distribution given by the product of the intensity of the former, equation ( 3), and the transmission function of the latter, equation (6). Suitable low-pass filtering retains the fringe pattern of lowest frequency, the moiré image [9]:…”

“…On the other hand, the phase increment due to the transparent object φ(x, y) can be directly calculated if a DPSA [2] is employed. These algorithms are widely employed during the phase-evaluation stage in electronic speckle-pattern interferometry (ESPI) techniques and provide directly the phase difference between two fringe patterns, without it being necessary to perform the aforementioned partial phase calculations by using equation (9). For example, one of the more attractive DPSAs [12] (since only two couples of phase-shifted fringe patterns are combined in an algebraical configuration that resembles a GPSA) can be rewritten, after some trigonometric calculus, as…”

“…At present, there are many demodulation methods [2][3][4] that can be employed to provide the information codified in the Twyman-Green fringe pattern. Especially attractive are those methods that employ the moiré effect [5] as an important element of the evaluation [6][7][8][9][10]. In our case, the aforementioned information on a transparent object is obtained using the multiplicative analogical moiré phaseshifting method.…”

“…In this way, a set of moiré images with different values of the additional phase term can be combined in a generic phase-shifting algorithm (GPSA) [2]. This method of evaluation can be a suitable choice when any of the following considerations apply [9]: (i) cost is an important factor; (ii) the interfering wavefronts have large dimensions; (iii) there is the need to add an evaluation module to a closed interferometer; and (iv) high spatial resolution and/or measurement range are not relevant factors. Finally, the phase variation introduced by the transparent object is obtained by subtraction of the evaluation results with and without it located in the measuring area [5].…”

“…In this case, the error sensitivities for evaluation of DPSAs are not well known and further analysis is required. During both measuring procedures, additionally aberrations of the interferometer and phase modulator [9] are cancelled, rendering an extra calibration stage unnecessary.…”