Laser inertial confinement nuclear fusion is significant for explosion simulation, astrophysics, and other research. Meanwhile, the target pellet as a fuel container in fusion requires extremely high precision in surface morphology, and since any small morphological defect on its surface may cause asymmetric compression and experimental failure, the measurement of target surface morphology is essential. However, during the actual measurement of the surface morphology of the target pellet, the measurement results are susceptible to vibration, such as ambient light changes and optical platform vibration, which will introduce random errors to cause inaccurate measurement results. Therefore, it is of practical significance to correct the random errors during the measurement and improve the vibration resistance of the target shot. The non-uniform fast Fourier transform (NUFFT) algorithm can correct non-uniform interference signals, which is characterized by high accuracy and low hardware cost. Thus, based on the NUFFT algorithm, we propose an anti-vibration white light interferometry method. Specifically, the white light interferometry optical path adopts dual imaging channels and the main channel collects the white light interferogram. The secondary channel collects the quasi-monochromatic optical interferogram, calculates the phase shift interval of the vibration according to the quasi-monochromatic light interferogram, and corrects the white light interference signals collected in the vibration environment according to the obtained phase-shifting interval combined with the NUFFT algorithm to obtain a more accurate white light interference signal. According to the corrected white light interference signal combined with the seven-step phase-shifting method, the three-dimensional topography information of the object to be measured is restored. Additionally, the algorithm can be adopted for correcting non-uniform interference signals with random phase shift interval in random vibration conditions.

First, the Fourier transform algorithm is employed to extract the phase information of the quasi-monochromatic light interference signal, and the phase information is expanded into continuous phases by unwrapping to obtain the non-uniform phase-shifting interval of each pixel position in the interferogram. Meanwhile, the non-uniform phase-shifting interval is sorted from small to large, and then the interferogram corresponding to the non-uniform phase-shifting interval is also sorted accordingly. The sampling interval is normalized and oversampled into uniform grid coordinates, and the NUFFT algorithm is utilized to convolute the sorted white light interference signal according to the phase-shifting interval after sorting. The convoluted interference signal is transformed by the Fourier transform, the influence of the Gaussian kernel function in the spectrum is removed, and the uniform interference signal is obtained by the inverse Fourier transform. Finally, the topographic distribution of the step surface is acquired by calculating the phase of the uniform interference signal and the peak position of the modulation system.

Figure 7 shows that in the vibration environment, the step surface morphology directly restored by the white light interferogram before correction has a large distortion, and its morphology information cannot be restored correctly. The average height measured in Table 1 is 0.1419 μm, the relative error between the nominal value of 0.139 μm and the step plate is 2.13%, and the restored step surface shape is close to the reference surface shape measured by the Veeco interferometer. In Table 1, the corrected peak-to-veally (PV) and root-mean-square (RMS) values of the corrected step surface are 0.2011 μm, which are significantly higher than those of 0.3417 μm and 0.0735 μm before correction, and are close to the Veeco reference value. The results show that the surface shape of the step measured by this method is in good agreement with the actual measured surface shape, with high measurement accuracy.

A white light interferometry anti-vibration measurement method based on a non-uniform fast Fourier transform algorithm is studied to solve the problem of white light interferometry in a vibrating environment. We employ a dual-channel optical path system to calculate the actual phase-shifting interval by adopting the quasi-monochromatic optical interferogram collected by the secondary channel camera and correcting the white light interferogram collected by the main channel camera according to the obtained phase-shifting interferogram. The simulation and experimental measurement results show that the NUFFT algorithm can accurately correct the non-uniform white light interference signal, and the morphological information of the object to be measured can be well recovered from the corrected uniform white light interference signal. The results show that our method can restore the surface morphology of the measured object in the vibration environment.