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In this report, a method is proposed to demodulate a single-shot interferogram based on deep learning using a convolutional neural network (CNN). The time required to demodulate a single interferogram is on the order of milliseconds. The root mean square (RMS) of the residual wavefront between the original wavefront and the estimated wavefront reached the order of 10−2 λ. Moreover, the cost was lower than using commercial dynamic and phase-shifting interferometers. This method yielded superior real-time performance and high precision. The results indicated that this approach based on deep learning is robust and feasible for demodulating single-shot interferograms. • A method based on deep learning is proposed to demodulate a single shot interferogram. • The single shot interferogram is used as the input of the convolutional neural network, and the Zernike polynomials coefficient is output. • Compared with other methods of demodulating a single shot interferogram, the method is faster, and the speed of demodulating an image is in the order of microseconds. The RMS (Root-Mean-Square) of the residual wavefront between the original wavefront and the estimated wavefront approached 10 − 2 λ. • The cost of the method is lower than commercial dynamic and phase-shifting interferometers. In addition, we compare the speed and accuracy of the method with the Zygo interferometer, and find that the method leads to better or comparable results yet is orders of magnitudes faster.it will be great helpful for the dynamic fringe demodulation or other real-time applications. • The process of demodulation of the convolutional neural network is robust. [ABSTRACT FROM AUTHOR]

• In this research, we proposed a novel optical surface shapes measurement method based on conformal mapping-angular phase-shifting interferometry. It paves the way for the application of vortex beams in the field of optical surface shape measurement. It is of great significance to the miniaturization design of high-precision interferometers, and can significantly reduce the experimental cost. • We use a Q-plate in the reference optical path to introduce a 90-degree phase shift, and conformal mapping is used to demodulate the surface shape. • This method not only ensures the accuracy of surface shape measurement; but also avoids the use of expensive precision translation stages; and has specific anti-vibration performance. • The measuring device is simple, compact, easy to operate and low cost. We proposed a novel optical surface shapes measurement method based on conformal mapping-angular phase-shifting interferometry (APSI). This method rotates a Q-plate in the reference light path to introduce a 90-degree phase shift. The mathematical tool of conformal mapping is used for surface shape demodulation. After removing the helical phase in the compound phase, high-precision surface shape measurement results are obtained, and the measurement surface shape deviations are 3.33 nm (PV) and 0.26 nm (RMS). This work paves the way for surface shape measurement and compound phase demodulation using vortex beams. [ABSTRACT FROM AUTHOR]

• STP method can retrieve the synthetic-wavelength phase from moiré fringe patterns in simultaneous dual-wavelength interferometry. • In STP method, by the additive-to-multiplicative moiré fringe transition, the measured synthetic-wavelength interferometry information exists directly and is demodulated conveniently. • STP method is implemented with π/(2k) phase shift at the synthetic wavelength to be retrieved by the conventional phase-shift algorithm. • STP method releases the restriction of spatial carrier for spectrum separation and the requirement of quadratic polar coordinate transform for the circle carrier. • The separation distance of the phase lobes in spectrum for STP method is longer than 9.56 times of the distance for traditional temporal FT method. Simultaneous dual-wavelength interferometry (SDWI) could extend the measured range of each single-wavelength interferometry, and has higher efficiency when the two single wavelengths work simultaneously. However, the directly measured phase demodulation at the long synthetic wavelength (λ S) from the generated moiré fringe patterns in SDWI is rather arduous. To reduce the difficulty of the processing for the moiré fringe patterns, a synthetic wavelength temporal phase retrieval method (STP) with the π/(2k) phase shift at λ S was proposed in this paper. By the additive-to-multiplicative moiré fringe transition, the synthetic-wavelength interferometry information exists in the form of the superposition with other wavelength information. And then for the longer separation distance of the phase lobes in spectrum, the synthetic-wavelength spectrum in STP method could be filter out more easily than traditional temporal Fourier transform (FT) method. After a low-pass filtering in the spectrum of the multiplicative moiré fringe intensity and adopting the π/(2k) phase shift at λ S , the synthetic-wavelength phase is retrieved by the conventional phase-shift algorithm from the obtained synthetic-wavelength phase-shift interferograms with the interval of π/2 phase shift. Compared with the spatial FT method with the introduction of spatial carrier, STP method releases the restriction of spatial carrier for spectrum separation and the requirement of quadratic polar coordinate transform for the circle carrier. Finally, the feasibility and applicability of the STP method are verified using simulation and experimental results. [ABSTRACT FROM AUTHOR]

• The CCFC method could extract the synthetic-wavelength interferogram from moiré fringe patterns in simultaneous dual-wavelength interferometry without the introduction of other wavelength interferometric information. • Coupled by the multiplication for the conjugate complex functions, only the lower beat-frequency synthetic-wavelength interferometric fringe pattern is derived directly. • The half-band filter in used in CCFC method is appropriate for the different spectral width or carrier spatial frequency. • The necessary introduced linear carrier in CCFC method is merely about 0.0947 times of the carrier in the traditional spatial domain fourier transform demodulation theory numerically. • The introduced carrier is only used to pursue the separation between the ±1 order phase lobes instead of the separation between the different wavelength phase lobes in CCFC method. Simultaneous dual-wavelength interferometry (SDWI) supplies a better solution to extend the measurement range of traditional single-wavelength interferometry with higher efficiency. To decompose the needed longer beat-frequency synthetic-wavelength (λ S) information from the generated moiré fringe pattern in SDWI, a conjugate complex function coupling strategy (CCFC) of interferometric field is proposed in this study. By the Fourier transform and the half-band filter in one side of the spectrum, the positive half spectrum is obtained. The inverse Fourier transform and conjugate operation are performed on the half spectrums to obtain the two conjugate complex functions of the interferometric field. Coupled by the multiplication for the conjugate complex functions, the lower beat-frequency synthetic-wavelength interferometric fringe pattern is derived directly. And if the SDWI is implemented with the π/2 phase shift at λ S , the obtained phase-shift interferogram for λ S could be demodulated by the conventional phase-shift algorithm. Compared with other spatial-domain Fourier transform demodulation theory, the half-band filter used in CCFC method is appropriate for the different spectral width and carrier spatial frequency. The necessary introduced linear carrier is merely about 0.0947 times of the carrier in the former numerically. And it is only used to pursue the separation between the ±1 order phase lobes instead of the separation between the different wavelength phase lobes. The feasibility and applicability of the CCFC method are verified using simulation and experimental results. [ABSTRACT FROM AUTHOR]