A multi-surface phase demodulation algorithm is proposed to measure surfaces of the transparent plate simultaneously. A non-linear least-squares method based on the trust region is utilized to fit the window function derived from the characteristic polynomials. The fitted window function requires fewer parameters and has a flexible form, which means the window length can be stretched freely and used flexibly. Then, a multi-surface phase demodulation technique combining the discrete Fourier principle and the fitted window function is investigated, allowing simultaneous reconstruction of the measured front surface, rear surface, and thickness variation. During the implementation of our algorithm, the processing time for the interferograms with 850×850 pixels is around 5.1s. Measurement errors caused by several factors are analyzed, including wavelength-tuning error, frequency order error, cavity length error, and average thickness error. The experimental result on a rectangular transparent plate further verifies the effectiveness of our algorithm.

To simultaneously measure topographies of both surfaces of a transparent plate through wavelength-tuning interferometry at arbitrary cavity lengths, a flexible phase shift matching algorithm (FPSMA) is proposed. The FPSMA includes two parts: 1) taking the four-term third-order Nuttall window as an example, a phase demodulation approach is designed and analyzed for extracting the harmonic phases of the front surface, rear surface, and thickness variation of the measured plate and 2) based on the residual error analysis of the developed $5N$ -Nuttall algorithm for different cavity length coefficients $M$ and the phase-shifting parameters $N$ , a parameter-change technique that allows the developed phase demodulation approach is applied to arbitrary cavity lengths by adaptively changing the phase shifts and frames of the captured interferograms. By this technique, the algorithm failures due to the spectral aliasing of harmonics can be avoided effectively and conveniently, and then the correct phase demodulation can be ensured. Besides, using the developed method, when the values of $N$ or $M$ are chosen in advance, the corresponding available measurement ranges can be given. Besides, the relationship between the window length and the measurement limitation of the measurable thinnest optical thickness is investigated. Considering the common linear and nonlinear errors in the wavelength-shifting process, the error analysis of FPSMA is performed and discussed for situations close to reality. The comparative studies and the experimental results of two transparent parallel plates also support the performance of the developed FPSMA.