Objective White-light interferometry is widely employed in precision manufacturing sectors, including semiconductor processing, thin-film thickness measurement, and metal defect detection, owing to its merits of high precision, high measurement speed, and non-contact nature. The frequency domain analysis (FDA) technique has been applied to surface profile measurement by utilizing the slope of the phase distribution of the white-light scanning interference signal in the wavenumber domain. Through simulations and experiments, this study demonstrates that the accuracy of the phase obtained in the wavenumber domain is related to the amplitude distribution, whether in the presence of additive noise or phase noise, and quantitatively provides the optimal phase selection range for the FDA method. When the sample has strong spectral absorption, parts of the amplitude distribution in the wavenumber domain decrease significantly, rendering the corresponding phase information invalid and unreliable. To address this limitation, an improved algorithm is proposed in this paper, which incorporates the wavenumber-domain amplitude as a weighting factor into the least-squares fitting procedure. This method avoids the selection of the phase range for the FDA method and ensures high measurement accuracy.

Methods In the simulation, the zero-mean Gaussian noise with a standard deviation (STD) of 10 nm is used as the phase noise ε_n(z) to generate a white-light interference signal. The Fourier transform is performed on the signal to obtain the amplitude I_F(σ) and phase (σ) in the wavenumber domain. For the FDA method, the selected phase range Rσ is defined as the wavenumber range corresponding to amplitudes in I_F(σ) that are greater than CImax, where the coefficient C∈(0, 1) and I_max is the maximum value of I_F(σ). The STD of z_s represents the measurement accuracy of the FDA approach. The influence of phase noise on the measurement accuracy and its relationship with amplitude distribution I_F(σ) are investigated by calculating the STD of z_s across different R_σ. Furthermore, additive noise is added to the white-light interference signal with phase noise, and the STD of z_s is evaluated under varying R_σ to determine the optimal phase selection range. Since conventional phase selection is not suitable for interference signals with complicated spectral distributions, this paper proposes a special frequency domain algorithm based on amplitude optimization in the wavenumber domain (amplitude optimization for frequency domain analysis, AOFDA). In this improved strategy, the amplitude distribution I_F(σ) is introduced as a weighting factor in the phase linear fitting process to obtain the optimized surface measurement parameter z_w.

Results and Discussions The simulation results indicate that when only phase noise exists, the accuracy of phase extraction improves with the increase in the amplitude corresponding to the selected phase, thereby enhancing the overall measurement precision. However, the phase extraction accuracy has an upper limit. When the range R_σ corresponding to C≥0.5 is selected in the FDA method, the measurement results exhibit higher accuracy. Additive noise with different signal-to-noise ratio (SNR) is introduced into the white-light interference signal containing phase noise. For interference signals with the same SNR, the STD of z_s obtained with C≥0.5 is stable and relatively low. For signals with stronger additive noise, the STD of z_s shows a slight increase when C>0.5. In summary, if only phase noise exists, selecting the range R_σ corresponding to C≥0.5 yields high measurement precision. When both phase noise and additive noise exist, C=0.5 provides higher measurement precision. Therefore, the Rσ corresponding to C=0.5 is the optimal phase selection range for the FDA method.

To verify the feasibility of the AOFDA method for interference signals with complex spectral distributions, a band-stop filter spectral distribution was used as the absorption spectrum of the measured object in the simulation. In the band-stop range, simulations with different reflectivities were conducted to investigate the measurement accuracy. When the reflectivity of the sample is ≥20%, it exerts little influence on the measurement accuracy of the FDA method. However, the reflectivity of ＜20% moves the measurement results away from their true values. When the reflectivity of the sample is ＜20%, the measurement results obtained by the AOFDA method always exhibit better performance. This indicates that the measurement accuracy of the AOFDA method is largely unaffected by the absorption spectrum of the sample.

The flat surface of wedged window and the step surface were measured in the experiment. When the different value of C is selected to obtain the surface profile of the wedged window, the C=0.5 has the better performance that agrees with simulation result. The S_z of the wedged window surface calculated by the AOFDA method and the FDA method (C=0.5) are 34.6 nm and 50.2 nm, and the S_q are 5.5 nm and 6.7 nm, respectively. The average step heights, whose value is regarded as 3.0 μm, of 10 measurements calculated by the AOFDA method and the FDA method (C=0.5) are 3.0054 μm and 2.9566 μm, and the repeatability values are 17.9 nm and 19.4 nm, respectively. The experimental results demonstrate that the AOFDA method can achieve the measurement accuracy comparable to that of the FDA method when selecting the optimal wavenumber-domain phase range. Since the selection of an appropriate wavenumber-domain phase range is no longer required for the AOFDA method, it enables the white-light interferometer to better adapt to complex measurement environments.

Conclusions The measurement accuracy of the FDA method is degraded by environmental vibration and background noise. This work confirms that the measurement precision of the FDA method is correlated with the amplitude distribution in the wavenumber domain through the selection of diverse phase ranges, and proposes a feasible phase selection strategy to enhance its measurement performance. It is verified that the FDA method is difficult to adapt to the situation where the sample has a strong absorption spectrum. To address these limitations, the AOFDA method is developed in this paper, which incorporates the wavenumber-domain amplitude distribution as a weighting function into the phase fitting procedure. The AOFDA method can achieve a measurement accuracy comparable to that of the FDA method processed with the optimal phase range, while eliminating the need for determination of the optimal phase range.