Modified Fourier transform method for interferogram fringe pattern analysis

Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method

We propose a new concept of single-shot deflectometry for real-time measurement of three-dimensional surface profile using a single composite pattern. To retrieve an accurate phase from one-frame composite pattern, we adapt the Fourier Transform (FT) method and the spatial carrier-frequency phase-shifting (SCPS) technique to our proposed deflectometry. Based on Lissajous figure and ellipse fitting method, we also correct the phase extraction error in SCPS technique by reducing the effect of background and modulation variations. The proposed technique is verified by comparing our measurement results with phase-shifting deflectometry, and the maximum difference between two measurement results is less than 30 nm rms. We also test the robustness to vibration and the measurement capability for dynamic object.

Precise calibration of spatial phase response nonuniformity arising in liquid crystal on silicon

In order to calibrate the spatial phase response nonuniformity of liquid crystal on silicon (LCoS), we propose to use a Twyman-Green interferometer to characterize the wavefront distortion, due to the inherent curvature of the device. During the characterization, both the residual carrier frequency introduced by the Fourier transform evaluation method and the lens aberration are error sources. For the tilted phase error introduced by residual carrier frequency, the least mean square fitting method is used to obtain the tilted phase error. Meanwhile, we use Zernike polynomials fitting based on plane mirror calibration to mitigate the lens aberration. For a typical LCoS with 1×12,288 pixels after calibration, the peak-to-valley value of the inherent wavefront distortion is approximately 0.25λ at 1550 nm, leading to a half-suppression of wavefront distortion. All efforts can suppress the root mean squares value of the inherent wavefront distortion to approximately λ/34.

General spatial phase-shifting interferometry by optimizing the signal retrieving function

A general spatial phase-shifting (GSPS) interferometry method is proposed to achieve phase retrieval from one-frame spatial carrier frequency interferogram. By optimizing the internal signal retrieving function of the spatial phase-shifting (SPS) method, the accuracy, anti-noise ability and speed of phase retrieval can be significantly improved, meanwhile the corresponding local calculation property is reserved. Especially, in the case that the ratio of the spatial carrier to the phase variation rate are small, the proposed method reveals obvious advantage in the accuracy improvement relative to the conventional SPS methods, so the more details of measured sample can be effectively reserved through introducing smaller spatial carrier frequency, and this will facilitate its application in interference microscopy. The principle analysis, numerical simulation and experimental result are employed to verify the performance of the proposed GSPS method.

Hybrid algorithm for phase retrieval from a single spatial carrier fringe pattern

A hybrid algorithm is proposed in this study for demodulating a single spatial carrier fringe pattern (FP) of interferometric measurement, which essentially combines the spatial carrier phase shift (SCPS) method and Fourier transform (FT) method. It firstly extracts three phase-shifted FPs from a single spatial carrier FP, then employs the FT method and a subtraction operation to determine the phase shift of three phase-shifted FPs, and finally retrieves the phase map using a least-square phase shift algorithm. The subtraction operation could considerably mitigate the inherent edge error of the FT method, resulting in an increase of accuracy compared with the FT method. It also does not require the background and modulation amplitude of the spatial carrier FP to be constant; thus it is robust and quite suitable for engineering. The factors that may influence the performance of the proposed algorithm are analyzed, including the random and speckle noise, carrier frequency, shape of the background, and modulation amplitude. The feasibility of the proposed algorithm is validated by two experiments, comparing them with the temporal phase-shifted method. The proposed algorithm is expected to be used in interferometric measurement under adverse environments.

Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review [Invited]

The paper reviews a technique for fringe analysis referred to as Fourier fringe analysis (FFA) or the Fourier transform method, with a particular focus on its application to metrology of extreme physical phenomena. Examples include the measurement of extremely small magnetic fields with subfluxon sensitivity by electron wave interferometry, subnanometer wavefront evaluation of projection optics for extreme UV lithography, the detection of sub-Ångstrom distortion of a crystal lattice, and the measurement of ultrashort optical pulses in the femotsecond to attosecond range, which show how the advantages of FFA are exploited in these cutting edge applications.

Fringe extrapolation technique based on Fourier transform for interferogram analysis

A new fringe extrapolation method for extracting phase from fringe patterns is presented. The proposed method can be used to reduce the errors at the border of the pattern. The simulation was carried out considering the random noise. The result of the simulation shows that this method is effective for precision phase measurement. The experimental result for the fringe extrapolation method is compared with the result of the Fourier transform method without using the extrapolated interferogram patterns.