The phase information of an optical surface can be demodulated by the analysis of interference fringes in optical interferometry. The phase demodulation method often includes phase shifting and Fourier transform algorithms. The phase shift method is used to acquire several fringe patterns at a certain phase interval and then analyze them with a phase shift algorithm. Although the demodulated phase is accurate, the phase shift method is very sensitive to the measuring environment and is not suitable for measurements that only capture a single interference fringe pattern [1, 2]. The Fourier transform (FT) method only needs to analyze a single interferogram with a carrier frequency to demodulate the phase [3], but the Fourier transform also has some shortcomings. First, it is difficult for the FT to accurately locate the carrier frequency in the frequency spectrum and remove it if the magnitude and direction of the carrier frequency are unknown when analyzing the interference fringes. In other words, the uncertainty of the introduced carrier frequency makes errors in the phase demodulation. Second, FT is a linear transform that generates equal scale spatial spectrum components, which leads to lower resolution of low frequency components, but the phase information is only contained in the low frequency components. Therefore, it is very important to do research on a high precision and high resolution phase reconstruction method.

Wavelet transform, a multi-scale transform method, was developed from the idea of localization in windowed Fourier transform [4]. It has good spatial localization characteristics and multi-scale scaling capability. Wavelets can provide a time-frequency window with frequency conversion, and the window size is constant with altering frequencies, evading the unwanted spectrum aliasing that affects the accuracy of the demodulation phase. The scale conversion window can be extended and contracted effectively to eliminate noise disturbances and accurately extract the phase information.

Wavelet transforms have been applied to analyze interference fringes by many researchers. Zhong et al. proposed a discrete Gabor wavelet transform to overcome the limitation of the measurable slope of the phase caused by the height modulation of a spatial carrier-fringe pattern for 3-D shape measurement [5]. Cherbulie et al. [6] used the Morlet wavelet to show the in-plane horizontal deformation of a concrete beam and a movie of the deformation of carbon-carbon cylinder by introducing a PZT, but the phase was recovered by analysis of a phase map of the transform rather than ridges of the transform. Li et al. [7, 8] proposed to use the Mexican straw hat as the mother wavelet to limit the error introduced by edge discontinuity within the local edge area with its ability of spatial localization. However, since the Mexican straw hat is a real wavelet, it is impossible to obtain the phase information directly from the wavelet coefficients, increasing the difficulty of phase demodulation. Villa et al. [9] proposed a method for interferogram analysis with a sliding two-dimensional continuous wavelet transform, which solves the problem of phase ambiguity when analyzing closed fringe. Wang et al. [10] improved two-dimensional continuous wavelet transform by accurately, quickly and automatically analyzing fringe patterns that contain complex fringes, noises and defects by choosing suitable mother wavelet functions, a smaller number of scale factors, and applying a phase determination rule.

This paper proposes a continuous wavelet transform to analyze a single interference pattern from an interferometer that is used to measure a rotary symmetric aspheric surface. Compared with discrete wavelet transform, continuous wavelet transform can be run at various scales on a continuous-slope phase, and the maximum scale is determined by the need for the extent of detailed analysis. The wavelet function can move smoothly over the entire domain of the analyzed function. Therefore, continuous wavelet transform is more suitable for the feature extraction of interference fringes of a rotary symmetric aspheric surface. The Morlet wavelet is chosen as a mother wavelet because of its good local characteristics in both spatial and frequency domains compared with other wavelets. The phase of a single interferogram is demodulated through Morlet wavelet decomposition, reconstruction and wavelet ridge detection algorithms.

When choosing the wavelet function, the orthogonal and non-orthogonal, negative and real values, as well as the width and graph of the mother wavelet are all considered according to the factors for the mother wavelet selection proposed by Farge (1992). The orthogonal wavelet is generally used for discrete wavelet transform. Non-orthogonal wavelets can be used for both discrete and continuous wavelet transforms [11]. In general, when analyzing the continuous function, it is desirable to obtain continuous smooth wavelet amplitudes, so the non-orthogonal wavelet function is more appropriate. In addition, if both the amplitude and phase of the continuous function are necessarily retained, the negative wavelet needs to be selected. Because it is a complex function with an imaginary part that includes phase information, phases can be demodulated from the negative wavelet.

This paper constructed an analytical signal with an imaginary part, and applied a continuous Morlet wavelet as a mother wavelet for single interferogram analysis. Apart from being non-orthogonal, Morlet wavelets are also exponentially negative after they are adjusted by Gaussian functions. As a complex wavelet, the Morlet wavelet exhibits good local characteristics in both space and frequency domains, making it appropriate for the measurement of continuous wave surfaces.

The light field distribution of the interference fringe can be expressed as:

where, g₀(x, y) is the background illumination, g₁(x, y) is the amplitude modulation, and φ(x, y) is the phase distribution of the optical element surface. Along the x direction, the one-dimensional distribution of the light field is z(x). z(x) can be expressed as:

The analytical function of z(x) is described as follows:

where z(x) is the real part and z_H(x) is the imaginary part.

The Morlet wavelet function is as follows:

The corresponding wavelet basis function is as follows:

where a is the stretch factor and b is the translation factor. Then the wavelet transform of $\tilde{z}(x)$ is as follows:

where the amplitude function is $A_{a,b}(x)=A(x)A{*}_{\psi }\left(\frac{x-b}{a}\right),$, and the phase function is ${\Phi }_{a,b}(x)=\phi (x)-{\phi }_{\psi }\left(\frac{x-b}{a}\right)$.

According to the equations above, the integral value of W_z(a, b) is most dependent on the stagnation point x_z(a, b). If the component is a single function and the mother wavelet function $\tilde{\psi }(x)$ and $\tilde{z}(x)$ are both progressive, the phase function Φ_a,b (x) will have one and only one stagnation point x_z(a, b), which meets the following conditions:

The wavelet ridge is defined as follows:

The wavelet ridge is a set of points (a, b) that meet the condition x_z(a, b) = b. According to the property of the stagnation point x_z(a, b), there is

Obviously, the scale a is a function of the translation factor b, that is

If the Morlet wavelet is brought into the formula, then

A[a_r(b),b] is the amplitude of the wavelet transform. The line connecting the maximum values R[a_r(b),b] is defined as the wavelet ridge [12–15], which can be described as follows:

where a_r(b) is the scale factor representing the wavelet ridge at point b. The phase φ(x, y) is obtained by the arctangent of the ratio of the imaginary part of R[a_r(b),b] to the real part.

Here, the phase calculated is wrapped in [−π,π], so its unwrapping is required to reconstruct the three-dimensional wave surface.

The location of the wavelet ridge represents the region where the signal energy accumulates, and the signal energy reaches its maximum value around the ridge. The most important features of the signal in the time-frequency domain are extracted perfectly, the redundant component is removed, and the phase information can be recovered by reconstructing the wavelet coefficient set on the wavelet ridge. Generally, there are two kinds of extraction methods, one based on the phase information of the wavelet coefficients, and the other based on the modulo information. We will analyze the first method in detail below:

Wavelet ridges can be obtained by analyzing a single interferogram with a continuous Morlet wavelet. In order to achieve this, we need to calculate the maximum amplitude of an interference fringe. This method is called the direct maximum ridge detection algorithm.

The Morlet wavelet can be used to directly retain low-frequency wavelet coefficients that contain the phase information of the measured surface and remove high frequency disturbances by wavelet decomposition and reconstruction. Next, by running a program, the wavelet ridge can be extracted from the low-frequency wavelet coefficients by calculating the maximum values of wavelet coefficients. Finally, the wrapped phases can be accurately obtained by multiple iterative calculations on wavelet ridges.

Next, the wavelet ridge line is obtained through iterative calculation as follows:

(1) Select the appropriate initial value a = a₀, b = b₀; Based on a = ω0 / 2πf, the scale sequence a_n(n = 1, …, m) is determined, and the initial value . b₀ keeps away from the half value of the width of the wavelet to avoid the influence of edge effect of wavelet transform.

(2) Calculate the next ridge point,

the iteration step size Δb = b_n+1 − b_n is valued between one and three times Δx.

(3) Set the convergence value ε, and ε is a very small positive threshold, if |a_r (b_n+1) − a_r (b_n) / a_r (b_n)| < ε, then iteration is stopped;

(4) Repeat steps (2) and (3) until all ridge points have been calculated. The wavelet ridge is obtained by connecting the ridge points.

In order to verify the feasibility of the proposed method, we used MATLAB to simulate a fringe pattern with an arbitrary carrier frequency. The interference fringes of the circular domain are extended into the rectangular domain [16]. Morlet wavelet transform is used to obtain the wrapped phase, followed by phase unwrapping through two-dimensional discrete cosine transform to reconstruct the 3-D optical wavefront. The simulation results in Fig. 1 show that a 3-D wavefront can be reconstructed accurately from an interference fringe pattern with carrier frequency.

Figure 1. Simulation analysis with Morlet wavelet transform: (a) Original interference fringes, (b) interference fringes after extension, (c) wrapping phase, and (d) three-dimensional wavefront.

The interference experiment setup is as shown in Fig. 2 and a photo of the experiment setup is shown in Fig. 3. A He-Ne laser with a wavelength of 632.8 nm is used as the light source, a tested aspheric mirror is placed in the testing arm, and a reflective phase-only liquid crystal spatial light modulator (hspdm 512-635) is situated in the reference arm. The interference fringes are received by the charge coupled device (CCD) with a resolution of 2,448 × 2,450 at the output end. Here, the liquid crystal spatial light modulator is used as a computer-generated hologram (CGH) to modulate the wavefront phase of the incident light. The modulated light wave acts as the reference light and interferes with the test light from an aspheric element with a 52 mm aperture and −340 mm curvature radius. Then, the interfered light is received by the CCD.

Figure 2. Optical path diagram of aspheric measurement.

Figure 3. Photo of aspheric measurement experiment setup.

Interference fringes of the measured aspheric element are analyzed by the Morlet wavelet transform proposed in this paper. Due to the influence caused by stray light, air disturbance, and environmental vibrations, etc., the nonlinear relationship among interference fringes in the interferometer will lead to spectrum aliasing, which reduces measurement accuracy. This nonlinearity is mainly due to disturbances from high frequency components. Wavelet decomposition and reconstruction can preserve the low-frequency components of the spectrum, which contain phase information, and filter out the high-frequency components that contain noise. The Morlet wavelet decomposition and reconstruction is shown in Figs. 4(a) and 4(b).

Figure 4. Wavelet decomposition and reconstruction of interferograms: (a) Low frequency and high frequency information in vertical, horizontal and diagonal directions obtained by wavelet decomposition, (b) restoration images obtained by wavelet reconstruction.

Figure 5(a) shows the original interferograms from the interference experiment, and Fig. 5(b) illustrates the reconstructed Morlet wavelet coefficients. Figure 5(c) is the rectangular interferogram after the extension algorithm of Figs. 5(b) and 5(d) is the wavelet ridge extracted by the wavelet ridge extraction algorithm of Fig. 5(c). Figure 5(e) is the package phase calculated on the wavelet ridge, and Fig. 5(f) is a 3-D wavefront reconstructed by a two-dimensional discrete cosine transform. The wavefront aberration of the wavefront reconstructed compared with the theoretical wavefront is 0.4328λ in RMS. In order to verify the feasibility of the proposed method, a Zygo interferometer was used to measure the same aspheric lens, and the wavefront aberration compared with the theoretical wavefront is 0.3232λ in RMS.

Figure 5. Phase demodulation process with Morlet wavelet: (a) Original interferograms, (b) reconstructed Morlet wavelet coefficients, (c) extended rectangular interferogram, (d) extracted wavelet ridge, (e) package phase, and (f) three-dimensional wavefront.

Next, the Fourier transform method is run to analyze the same interference fringe from the interference experiment for comparison with the Morlet wavelet. Because the carrier frequency of the interference fringes is randomly generated in interferometry, both the first-order spectral center and boundary of the carrier frequency are uncertain. Therefore, these two parameters need to be determined by human observation of the frequency spectrum of interference pattern when using Fourier transform. However, it is easy introduce additional errors in locating the center of the carrier frequency by observing, resulting in defects of the wavefront in phase reconstruction and reducing the phase demodulation accuracy. As shown in Fig. 6, due to inaccurate positioning of the carrier frequency center, the package phase is obviously incomplete, making the reconstruction of 3-D wavefront incomplete. Manual acquisition of the carrier frequency requires more repeated attempts, which increases the difficulty to extract the phase and causes repeatability errors.

Figure 6. Wrapping phases and unwrapping phases demodulated by using Fourier transform: (a)–(d) Wrapping phases, (e)–(h) unwrapping phases.

The comparative results show that the Fourier transform for analyzing interference fringes exhibits errors in carrier frequency positioning, and it is difficult to develop automation programming. However, the Morlet wavelet transform does not need to locate the center of the carrier frequency, and it can automatically extract the phase from the low carrier frequency, introducing no additional error. The Morlet wavelet transform can be used to reconstruct a wavefront with high precision, solve the problem of spectrum aliasing to some extent and ensure automatic running of programs without human operation. After many operations, the wavefront aberration of the wavefront reconstructed by Fourier transform compared with the theoretical wavefront is 0.5124λ in RMS. The precision of Fourier transform is slightly lower than wavelet transform. The Morlet wavelet proposed in this paper is very feasible.

This paper proposes a method to extract a phase to reconstruct a 3-D wavefront from a single interferogram. The Morlet wavelet is used as the mother wavelet, and a single interferogram with arbitrary carrier frequencies was analyzed. The method proposed could retain the low-frequency wavelet coefficients containing phase information and eliminate the high-frequency components by wavelet decomposition and reconstruction. The modulus of the low-frequency wavelet coefficients was compared and calculated, and the wavelet ridge was extracted, which is the line connecting the positions of maximum values of the wavelet coefficient amplitude. The wrapped phase was accurately obtained through iterative calculation of the wavelet ridge, and a two-dimensional discrete cosine transform was run to unwrap the packed phase and reconstruct the 3-D wavefront. An interference fringe pattern of an aspherical lens measured by interferometry was analyzed by using the Morlet wavelet and the Fourier transform algorithms. The results showed that an extra error has to be introduced with Fourier transform because of the manual positioning of the carrier frequency center, but the Morlet wavelet does not need manual positioning. So, the Morlet wavelet is more suitable for automatic interferometry and makes programming easy.

This research is supported by the Natural Science Foundation of Heilongjiang Province of China (No. LH2021E080).

Natural Science Foundation of Heilongjiang Province of China (No. LH2021E080).

The authors declare no conflicts of interest.

All data generated or analyzed during this study are included in this published article.