We propose a least-squares phase-stepping algorithm (LS-PSA) consisting of only 14 steps for high-quality optical plate testing. Optical plate testing produces an infinite number of simultaneous fringe patterns due to multiple reflections. However, because of the small reflection of common optical materials, only a few simultaneous fringes have amplitudes above the measuring noise. From these fringes, only the variations of the plate’s surfaces and thicknesses are of interest. To measure these plates, one must use wavelength stepping, which corresponds to phase stepping in standard digital interferometry. The designed PSA must phase demodulate a single fringe sequence and filter out the remaining temporal fringes. In the available literature, researchers have adapted PSAs to the dimensions of particular plates. As a consequence, there are as many PSAs published as different testing plate conditions. Moreover, these PSAs are designed with too many phase steps to provide detuning robustness well above the required level. Instead, we mathematically prove that a single 14-step LS-PSA can adapt to several testing setups. As is well known, this 14-step LS-PSA has a maximum signal-to-noise ratio and the highest harmonic rejection among any other 14-step PSA. Due to optical dispersion and experimental length measuring errors, the fringes may have a slight phase detuning. Using propagation error theory, we demonstrate that measuring distances with around 1% uncertainty produces a small and acceptable detuning error for the proposed 14-step LS-PSA.
We propose a least-squares phase-stepping algorithm (LS-PSA) consisting of only 14 steps for high-quality optical plate testing. Optical plate testing produces an infinite number of simultaneous fringe patterns due to multiple reflections. However, because of the small reflection of common optical materials, only a few simultaneous fringes have amplitudes above the measuring noise. From these fringes, only the variations of the plate's surfaces and thickness are of interest. To measure these plates, one must use wavelength-stepping, which corresponds to phase-stepping in standard digital interferometry. The designed PSA must phase-demodulate a single fringe sequence and filter out the remaining temporal fringes. In the available literature, researchers have adapted PSAs to the dimensions of particular plates. As a consequence, there are as many PSAs published as different testing plate conditions. Moreover, these PSAs are designed with too many phase-steps to provide detuning robustness well above the required level. Instead, we mathematically prove that a single 14-step LS-PSA can adapt to several testing setups. As is well known, this 14-step LS-PSA has a maximum signal-to-noise ratio (SNR) and the highest harmonics rejection among any other 14-step PSA. Due to optical dispersion and experimental length measuring errors, the fringes may have a slight phase detuning. Using propagation error theory, we demonstrate that measuring distances with around 1% uncertainty produces a small and acceptable detuning error for the proposed 14-step LS-PSA.
We propose a least-squares phase-stepping algorithm (LS-PSA) consisting of only 14 steps for high-quality optical plate testing. Optical plate testing produces an infinite number of simultaneous fringe patterns due to multiple reflections. However, because of the small reflection of common optical materials, only a few simultaneous fringes have amplitudes above the measuring noise. From these fringes, only the variations of the plate's surfaces and thickness are of interest. To measure these plates, one must use wavelength-stepping, which corresponds to phase-stepping in standard digital interferometry. The designed PSA must phase-demodulate a single fringe sequence and filter out the remaining temporal fringes. In the available literature, researchers have adapted PSAs to the dimensions of particular plates. As a consequence, there are as many PSAs published as different testing plate conditions. Moreover, these PSAs are designed with too many phase-steps to provide detuning robustness well above the required level. Instead, we mathematically prove that a single 14-step LS-PSA can adapt to several testing setups. As is well known, this 14-step LS-PSA has a maximum signal-to-noise ratio (SNR) and the highest harmonics rejection among any other 14-step PSA. Due to optical dispersion and experimental length measuring errors, the fringes may have a slight phase detuning. Using propagation error theory, we demonstrate that measuring distances with around 1% uncertainty produces a small and acceptable detuning error for the proposed 14-step LS-PSA.
Fringe projection profilometry (FPP) is a well-known technique for digitizing solids. In FPP, straight fringes are projected over a digitizing solid, and a digital camera grabs the projected fringes. The sensitivity of FPP depends on the spatial frequency of the projected fringes. The projected fringes as seen by the camera are phase modulated by the surface of the digitizing object; the demodulated phase is usually wrapped. If the digitizing object has discontinuities larger than the fringe period, the phase jumps are lost. To preserve large phase discontinuities, one must use very low spatial frequency (low-sensitivity) fringes. The drawback of low-sensitivity FPP is that the demodulated phase has low signal-to-noise ratio (SNR). Much higher SNR is obtained by projecting shorter wavelength, at the cost of obtaining wrapped phase. A way out of this problem is to use dual-wavelength FPP (DW-FPP). In DW-FPP, two sets of projected fringes are used, one with long wavelength and another with shorter wavelength. Due to harmonics and gamma distortion, in DW-FPP, one usually needs four phase-shifted fringes for each sensitivity. Here we are proposing to combine the two sensitivities simultaneously, one coded in phase (PM) and the other coded in amplitude (AM), in order to obtain phase and amplitude modulated (DW-PAM) fringes. The low-sensitivity phase is coded as AM of the DW-PAM fringes. The main advantage of DW-PAM fringes is that one reduces the number of phase-shifted fringes by half: instead of using eight phase-shifted fringes (four for low and four for high sensitivities), one would need only four DW-PAM fringes. Of course, if one wants to increase the harmonic rejection of the recovered phase, one may use a higher order phase-shifting algorithm (PSA).
The signal-to-noise ratio (SNR) and Shannon’s channel capacity are two of the main figures of merit in statistical communication theory (SCT). Understanding their relation has been crucial for the exponential increase of communication speed observed during the past 70 years. However, these concepts remain essentially unexplored in the context of digital interferometry (DI). This is despite DI uses common concepts with SCT such as quadrature filtering and phase demodulation. Therefore, our purpose in this tutorial is to show the importance of Shannon information theory for DI. Particularly, we believe that the SNR gain is a very useful figure of merit when comparing phase-demodulation algorithms. We also show that dual-sensitivity phase unwrapping produces significantly higher Shannon’s entropy versus increasing only the SNR on low-sensitivity phase measurements. Our claims are supported by mathematical analysis, numerical simulations and experimental results.
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