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We demonstrate a lensless phase-shifting point diffraction interferometer with a tiny pinhole to measure spherical mirror surface figures. The interferograms used to reconstruct the surface figure are formed without imaging optics. Fresnel diffraction calculations with a coordinate transformation are studied as a means of reconstruction method for wavefront amplitude and phase. The Radon transform is used to determine the distance from the tiny pinhole to the CCD target, which guarantees the accuracy of the diffraction calculations. Our lensless phase-shifting point diffraction interferometer not only retains the higher measurement precision by the ideal reference wavefront, but also overcome the fabrication and mounting limitation of the imaging optics. The simulations and experiments have validated the accuracy and feasibility of the spherical mirror measurement by proposed lensless phase-shifting point diffraction interferometer. [ABSTRACT FROM AUTHOR]

Abstract: A method for measuring the radius of curvature with a pinhole point diffraction interferometer (PDI) is proposed. Using the wavefront difference method and the Gaussian imaging equation, the longitudinal displacement of the converging rays passing through a standard plane-parallel-plate sample in PDI interference cavity is determined. Based on this longitudinal displacement, a precise formula for radius of curvature calculation is deduced. An experimental system for radius of curvature measurements is set up to verify the principle. With this testing system, the radius of curvature of spherical mirrors and the surface figure can be measured in a higher precision simultaneously. Some sources of uncertainty in measurement are discussed based on detailed error analysis. The experimental results indicate that the measurement accuracy is in the order of 10−4. [Copyright &y& Elsevier]

We proposed a new method for the radius of curvature (ROC) of spherical wave measurement based on vortex beam interference and Fermat's spiral fitting, which can not only provide a large measurement range but also have high precision. Theoretical analysis showed that the maximum distribution of the interference light intensity is in accordance with the Fermat's spiral, and the spiral coefficient is proportional to the ROC of spherical wave. For a concave spherical mirror with a ROC of 899.89mm and a convex spherical lens with a ROC of 59.99mm, the average measurement error of five measurements are 0.080% and 0.083%, respectively. Controlled experiments showed that, our proposed method has a higher measurement accuracy and a larger measurement range than Newton's rings method, it has almost equal measurement accuracy for large and small ROC. This research provides a theoretical basis for the development of a ROC measurement system with large measurement range and high precision. [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]

• A simple and effective strategy is proposed to measure the focal lengths. • It enables non-contact measurement and no need to measure the distance between optics components. • The entire measurement system occupies a small space that is easy to set up and operates. This paper proposes a new method to measure the focal lengths of both convex and concave lenses based on vortex beam interference. Theoretical analysis illustrates that the maximum intensity distribution of the interferogram is in accordance with the Fermat's spiral, and the spiral coefficient is proportional to the difference between the focal length of the test lens and the distance from the test lens to the camera. In order to calculate the precise focal length, we separately block the light paths to obtain the reference and test beam spots, and get the ratio of these two spot sizes. Then, the ratio and the spiral coefficient are used to determine the focal length. For a biconvex lens with a focal length of 200 mm and a biconcave lens with a focal length of −100 mm, the relative measurement errors are 0.0892 % and 0.0800 %, respectively. This research provides a novel and convenient method for measuring the focal length, and no-sophisticated displacement devices and moving the lens to be measured are required in the measurement. [ABSTRACT FROM AUTHOR]

White-light scanning interferometry is an effective and widely used technology for measuring the microscopic three-dimensional morphology of an object. Its vertical resolution can reach the sub-nanometer level, and its lateral resolution reaches submicron level. However, for the samples containing complex structure or high-density periodic distribution structural units, the measurement results are strongly restricted by magnification and numerical aperture (NA) of the microscopic objective. In this paper, we proposed a three-dimensional reconstruction algorithm for white-light interferograms after super resolution processing, using fast super-resolution convolutional neural networks (FSRCNN) to improve the detailed information of the interferograms, and then we used centroid method combined with the five-step phase-shift method to extract the zero optical path difference (ZOPD) position of the interference signal after super resolution processing. After processed by the proposed method, the interferograms collected by the 10X microscope objective (NA=0.3) recovered the 3D surface is the same as that measured by the 100X microscope objective (NA=0.7), which is proved by the experiment results. [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]

Dual-wavelength interferometry (DWI) could extend the measured range of each single-wavelength interferometry. The synchronization of the two working wavelengths in DWI is of high efficient, and the generated moiré fringe indirectly represents the information of the measured long synthetic-wavelength ( λ S ) phase. However, the extraction of the measured synthetic-wavelength phase is rather arduous from the moiré fringe. To retrieve the synthetic-wavelength phase from the moiré fringe patterns, we present a carrier squeezing dual-wavelength interferometry method (CSDI) in simultaneous DWI (SDWI). After the mathematical square of the moiré fringe patterns, the multiplicative moiré phase-shift fringe patterns with π/2 phase shift at λ S are combined into a single spatial-temporal fringe (STF). By converting the temporal phase shift into spatial carrier and the introduction of the carrier, the measured synthetic wavelength phase is retrieved by the filter and inverse Fourier transform of the STF spectrum. Compared with other methods, CSDI method could suppress the influence of the phase-shift error and only requires 4 frame phase-shift interferograms. Numerical simulations are executed to demonstrate the performance of the CSDI method in SDWI with the peak-to-valley (PV) value of 1.46 nm and the root mean square (RMS) values of 0.23 nm for the demodulated error. And the precision is better than PV of 20 nm (0.0059 λ s ) and RMS of 6 nm (0.0017 λ s ) even when the distribution range of the phase-shift error is as high as ± 10% relative to the π/2 phase shift step at λ S . Finally, our experimental results indicate that the measurement accuracy is better than 1.3% for a step with the height of 7.8 µm, and 0.5% for the step height of 6.233 µm for a Fresnel lens. [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]

A novel white-light interference signal leakage sampling points (LSPs) correction method based on Morlet wavelet transform and phase seven-step phase shift algorithm (PSA) is presented in this paper. During the correction process, the method can eliminate the LSPs error accurately without changing traditional optical system. Meanwhile, it can effectively de-noise the interference signals and calculate a relatively smooth recovery shape. • The main advantage of the LSPs correction method is that we use wavelet transform to find LSPs in white-light interference signals innovatively for its characteristic of the wavelet coefficient at the distortion points position are different from the general positions, which is different from the traditional method of using wavelet transform for noise reduction and ZOPD extraction. • Furthermore, the algorithm can be applied for any traditional optical system of white-light interferometry. It is simple, accurate and easy to optimize other algorithms. White-light scanning interferometry has become a well-established tool for measuring the microscopic three-dimensional morphology of objects. However, it is sensitive to external disturbances and suffers from leakage sampling, which compromises the accuracy of the measurements. To avoid the error, a robust correction method is proposed, based on wavelet transform, to find leakage sampling point (LSP), and the cubic spline interpolation method is used to complete the interference signal to achieve the signal correction. Further, phase evaluation at each pixel provides a high vertical resolution to characterize the microstructure. In addition, the proposed method exhibits a good denoising performance. Simulations and comparison experiments demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]