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Abstract
Wavelength-tuning interferometry is commonly employed to measure the optical homogeneity of parallel plates. However, the nonlinearity of phase shifts caused by wavelength tuning errors and environmental vibration leads to a spatially uniform error in the calculated phase distribution. Herein, a wavelength-tuning interferometry method based on nonuniform fast Fourier transform (WTI-NUFFT) was developed, which solves the spectral aliasing resulting from the spatially uniform error. The characteristics of the WTI-NUFFT method were estimated through comparison with the FFT method. Both the simulated and experimental results showed that the WTI-NUFFT method can improve the accuracy of the optical homogeneity measurement of parallel plates.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Parallel plates are widely used in optical systems, and their optical homogeneity, which is an important parameter of optical materials that indicates the inconsistency of refractive index in the optical material, directly affects the imaging quality of the system. Among the various methods reported for the measurement of optical homogeneity, interferometry is the most precise. However, in traditional interference measurement methods, the interference between the front and rear surfaces of the parallel plates produces significant difficulties during homogeneity measurements. To prevent such interference between the front and rear surfaces, Vaseline or other refractive index liquids can be coated on the plate surface to prevent unwanted beams from interfering, with one example including the window-flipping method proposed by Wyant et al [1]. In addition, special optical structures can also be employed to prevent such interference. More specifically, the holographic method reported by Tentori et al. records and reforms light waves from the front and rear reflectors of the tested sample prior to calculating the homogeneity distribution [2]. However, this method is time-consuming and its operation is complex. Furthermore, the grating interferometer proposed by Groot et al. uses two diffraction gratings to locate the coherent beams on the surface to be measured, avoiding interference from other surfaces, and yielding the homogeneity distribution of the tested sample [3]. However, the equivalent wavelength of the interference fringe pattern obtained by this method is too long to achieve the high-precision measurement of surface profiles. Moreover, although Novak's short coherence polarized light method can also measure the optical homogeneity of parallel plates and avoid interference from other surfaces [4], it is difficult to eliminate the influence of background light and the internal stress of materials in such a common-path interferometer. Jungjae P. et al. improved the original Roberts-Langenbeck 4-step method [5] by simplifying the measurement steps using a tunable infrared Fizeau interferometer, and obtained the optical homogeneity of a 300 mm wafer [6]. In addition, Luan Zhu proposed a five-step method to measure homogeneity over the interferometer aperture [7]. In this case, the variation of sample homogeneity was calculated directly using five interferograms measured at oblique incidence. Furthermore, Rui Zhang et al. proposed an absolute measurement method for the optical homogeneity of parallel plates based on the principle of phase shifting by lateral displacement of the point source [8]. This method employed a dynamic Fizeau interferometer to complete the absolute detection of the optical homogeneity of parallel plates through a three-step measurement process. Dual-wavelength interferometry [9,10] and optical combs [11–13] have also been examined in the detection of optical homogeneity.
To simplify the measurement steps, wavelength-tuning phase shifting interferometry has been applied to the measurement of optical homogeneity in parallel plates [14–20]. When carrying out measurements using a wavelength-tuning interferometer, interference fringes formed by multiple surfaces can be separated by a fast Fourier transform (FFT) algorithm, which requires the collection of large numbers of interferograms. However, upon increasing the number of captured interferograms, the wavelength-tuning degree is increased, and the wavelength-tuning nonlinearity becomes more pronounced. In addition, owing to the long testing time, environmental vibrations and other factors will also cause nonlinearity of the phase shifts. When the interferometric intensities are transformed into the frequency domain, the spectrum will spread, and the spectra corresponding to different frequencies are aliased. Thus, if the FFT method is used directly in the calculations, this will result in errors in the final wavefront.
Thus, to improve the measurement precision, we herein propose a wavelength-tuning interferometry method based on nonuniform fast Fourier transform (WTI-NUFFT) for determining the optical homogeneity of parallel plates. The NUFFT algorithm will be used to precisely recover the multi-surface wavefronts through uniform grid sampling and FFT. Initially, the correctness of the algorithm will be validated by simulations, after which a wavelength-tuning interferometer will be constructed to test the homogeneity of parallel plates and verify the accuracy of the test method. We also wish to determine whether this method can solve the problem of spectral aliasing caused by wavelength tuning errors and environmental vibrations, which leads to deviation of the final wavefronts calculation. Ultimately, we wish to develop a method that allows the optical homogeneity of parallel plates to be determined with high precision.
2. Principle
The interferometric cavity of the parallel plate measured by the Fizeau wavelength-tuning interferometer is shown in Fig. 1(a) (TF, transmission flat; RF, reflection flat). The thickness of the plate is set to t, the refractive index is n0, and the optical homogeneity is Δn. The distance from TF to the front surface of the plate is L1, and the distance from RF to the rear surface of the plate is L2. T, A, B, and R are used to represent the surface profiles of TF, the front and rear surfaces of the plate, and RF, respectively. If multiple reflections are not considered, the interferograms are composed of fringes interfered by T and A, A and B, T and B, T and R, A and R, and B and R. As shown in Fig. 1(b), the plate is removed, and the distance between TF and RF is maintained for the empty cavity measurement. The interferograms of TF and RF are then obtained in the cavity. The optical path differences and interference cavity lengths of each group of fringes are outlined in Table 1.
Fig. 1 Measurement procedure. (a) Cavity of the Fizeau interferometer employed for testing a parallel plate. (b) The empty cavity.
Table 1. Optical path distances and cavity lengths of each interference fringe
From the optical path differences outlined in Table 1, the optical homogeneity of the plate was calculated as follows:
As such, W1, W2, and W6 must be obtained when the plate is placed in the interference cavity and W7 should be obtained when measuring the empty cavity.When the plate is in the cavity, the interference fringes formed should be the sum of the intensities of six groups of interference fringes. Each group of interference intensities can be expressed as follows:
where ai represents the background intensity, bi represents the intensity modulation, represents the initial phase, δi represents the phase shift, and i = 1–6. where hi is the cavity length of interference, λ0 is the initial wavelength, Δλ is the wavelength variation at each phase shift, t is the number of phase shifts, and.It can be seen from h1–h6 in Table 1 that the hi value of each interference fringe can vary upon selection of the appropriate L1 and L2 values. From the expression of fi, we can see that the fringe frequencies of different hi values differ under the same wavelength shift. They can therefore be separated theoretically in the frequency domain. However, during measurement of the optical homogeneity of the parallel plate by the wavelength-tuning interferometer, the wavelength-tuning error and environmental vibrations can result in phase shift errors, resulting in spectrum aliasing, which ultimately causes difficulties in separating the spectra. We therefore propose a novel algorithm to solve this problem. This algorithm is composed of two key steps.
Firstly, Step 1 involves the phase-shifting interferograms being obtained upon testing of the parallel plate. More specifically, calculation of the nonlinear sampling intervals is initially carried out [21]; the background area of interferograms with only a single interference fringe around the measured area is selected, N intensity values are extracted at a certain position, a Fourier transform is performed to obtain the corresponding spectral information. A filtered window is used to pick up one of the sidelobes and an inverse Fourier transform is then performed, unwrapping is carried out to obtain the phase shift intervals, and normalization to [−π,π] is carried out as the input parameters of nonuniform Fourier transform x.
Subsequently, calculation of the multiple fringes is carried out. As mentioned in Section 2.1, the interference intensity of the multi-group fringes superposition region is the sum of six groups of interference intensity. Since the sampling sequence has been normalized to , according to Eq. (2), N sampling intensity values I (x) of a single pixel in the region can be expressed as:
Owing to the nonuniformity of the normalized sampling interval x, I (x) is a sequence of nonuniform interference data. To smooth these data, the Gaussian pulse function gτ(x) is used for convolution. The Gaussian function is also known as the kernel function and is expressed as:
where the exponential decay rate of the kernel function is determined by parameter τ, x is the phase shift intervals in the range of [−π,π]. The interference signal Iτ(x) obtained by the convolution operation can then be given by:where * denotes convolution, and I(x) is the sum of the 6 groups of interference intensity, i.e., .In this case, Iτ(x) is the smoothing and infinite integral function of variable x in the range of [−π,π], which can be sampled by equispaced grid points. This process can be described as shown in Fig. 2.
If the number of uniform interference data is Mr, then the discrete values at equispaced grid points can be expressed as:
where N is the number of interferograms, Δx is the sampling interval of oversampling. Δx = 2π/Mr, and m is the grid sampling point. According to Eq. (7), the interference intensity at any uniform grid point mΔx can therefore be estimated.According to the Fourier transform relationship, the discrete transform spectrum of convolutional interference data Iτ(mΔx) can be obtained by fast Fourier transform, which can be expressed as:
where Bτ(k) is the spectrum estimation corresponding to Iτ(mΔx).Deconvolution in the frequency domain is necessary to reduce the smoothing effect of the Gaussian kernel function. The spectral intensity corresponding to the unequal intervals sampled interference data Iτ(mΔx) is:
where G(k) is the discrete transform spectrum of the Gaussian kernel function, which can be expressed as:It should be noted that B(k) contains six groups of spectral information F1–F6,which correspond to the interference fringes in Eq. (2) and can be extracted by adding a window to B(k). Subsequent inverse Fourier transformation yields the required phase information ɸi:
where, represents the inverse Fourier transform. After converting ɸi to OPD from Eq. (3), W1, W2, and W6 in Table 1 can be obtained:Subsequently, Step 2 involves measurement of the empty cavity. In this case, when the plate is removed to give the empty cavity, is obtained by the wavelength-tuning phase shifting algorithm. Finally, the optical path differences W1, W2, W6, and W7 are obtained from Eq. (3), and the optical homogeneity of the plate is calculated using Eq. (1).
According to this described principle, the measurement steps involved in the WTI-NUFFT process could be outlined as shown in Fig. 3.
3. Simulation and analysis
3.1 Simulation process and results
The described algorithm is simulated and compared using an FFT algorithm to validate its correctness. Firstly, the measured wavefront data are used to simulate the surface profiles of T, A, B, R, and the optical homogeneity distribution of the plate. The phase difference of each group of interference fringes can be calculated from W1–W6 in Table 1 and Eq. (3). Assuming that L1 = 60 mm, t = 60 mm, L2 = 120 mm, and n0 = 1.5, then the maximum interference cavity length is 270 mm, in which 4 points are guaranteed per period. According to the nonlinear relationship between the wavelength of the wavelength-tuning laser and the control voltage, a quadratic coefficient of 0.003 nm/V was set to obtain the nonuniform phase shifts. The phase shift errors vary nonlinearly upon increasing the sampling times, as shown in Fig. 4. The sampling intervals required for the NUFFT method are obtained by normalizing the nonuniform phase shifts to [−π, π]. Six groups of interference fringes are generated by simulation, and the total interference intensity is then obtained by the superposition of these six groups of intensity. As such, 128 phase-shifting interferograms are obtained.
The 128 intensity values of each point on the interferograms are then calculated, and the spectral distribution of each point is obtained by applying the FFT and NUFFT algorithms, as shown in Fig. 5. Subsequently, the abscissa frequency values are converted into the cavity length values by the relationship between the frequency and the cavity length. Consistent with the theory, peaks exist at the corresponding cavity lengths of 60, 90, 120, 150, 210, and 270 mm. Owing to the nonlinearity of the sampling intervals, the spectrum calculated using FFT is broadened, resulting in aliasing, as shown by the red line in Fig. 5, while the spectrum calculated using the NUFFT method is significantly less broadened (blue line). The Hamming window is then added to the spectra of different frequencies to extract the corresponding spectra. Through the inverted Fourier transform of the extracted spectra, the phases corresponding to each group of fringes are obtained. Finally, the phases are unwrapped and de-tilted.
According to Eq. (1), the optical homogeneity of the plate can be calculated from the phases obtained above (see Fig. 6). The simulated optical homogeneity distribution is shown in Fig. 6(a), the optical homogeneity distribution calculated using FFT is shown in Fig. 6(b), and the optical homogeneity distribution calculated using NUFFT is shown in Fig. 6(c). The residual distribution can then be obtained by subtracting the simulated optical homogeneity distribution from that calculated by FFT and NUFFT, as shown in Fig. 7. The PV and RMS values of the optical homogeneity and residual distribution are given in Table 2.
Fig. 6 Simulated and calculated phases of the optical homogeneity of a parallel plate. (a) Simulated optical homogeneity. (b) Optical homogeneity calculated using FFT. (c) Optical homogeneity calculated using NUFFT.
Fig. 7 Residual phases of optical homogeneity with the simulated system. (a) Residual phase calculated using FFT. (b) Residual phase calculated using NUFFT.
Table 2. Simulation results
As shown in Figs. 6 and 7 and in Table 2, the calculated phase distribution was achieved using the proposed WTI-NUFFT method. A good agreement was observed between the optical homogeneity distribution calculated using the NUFFT method and the simulated distribution, with deviations of the residual phase in the PV and RMS values of 3.2 × 10−7 and 6.7 × 10−8 being calculated. In contrast, the optical homogeneity distribution calculated using the FFT method differs considerably from the simulated distribution. From the information presented in Table 2, it is apparent that the PV value of the residual distribution calculated using the FFT method is 2.55 × 10−6, whereas the RMS value is 3.35 × 10−7. These results indicate that the NUFFT method offers greater precision than the FFT method.
3.2 Effect of the nonlinearity of wavelength tuning
Compared with the FFT method, our proposed WTI-NUFFT method is advantageous in that there is a nonlinear relationship between the output wavelength and the input voltage of the wavelength-tuning laser. The nonlinearity is particularly noticeable when acquiring a large number of interferograms. If the FFT algorithm is used, the phase shifts must be calibrated prior to measurement, otherwise, the spectrum will be aliased and separation will be difficult. Furthermore, the NUFFT algorithm does not require equal sampling intervals, and so even if the wavelength varies nonlinearly with voltage, no calibration is required. Indeed, the test process is simple and can improve the measurement efficiency and accuracy.
Examination of the effect of the nonlinearity of wavelength tuning on the test results was then carried out by altering the nonlinearity error of nonuniform sampling for several simulations, thereby allowing the adaptability of the NUFFT and FFT algorithms to different nonlinearity errors to be studied. The scale of the nonlinear error is determined by the ratio of the maximum difference between the actual sampling value and the ideal sampling value to the sampling range, i.e., the “linearity.” The initial linearity is set to 3.21%, and the nonlinear error is found to gradually increase. The PV and RMS values of the residual phase distribution of the calculated optical homogeneity phase and simulated phase (see Section 3.1) are shown in Figs. 8 and 9, respectively.
The obtained results show that the results calculated using FFT are worse than those using NUFFT in any value of linearity. Particularly, when the linearity gets higher (the linearity in the Figs is close to 20%), it is impossible to calculate the residual wavefront results due to the aliasing in the spectrum result using FFT algorithm. The NUFFT algorithm can solve this problem successfully, which means upon increasing the nonlinear error, the NUFFT algorithm described herein exhibits a good robustness. The PV and RMS values of the residual phase calculated under different linearity values are near to the simulations. When the linearity is close to 25%, the NUFFT calculation results show a distinct jump. However, the linearity of a wavelength tuning laser does not reach such high levels, so this method can be widely applicable in wavelength phase shifting interferometers.
4. Experiments and results
Experiments were carried out using the wavelength phase shifting interferometer developed by our research team. The WTI-NUFFT method and the traditional flipping method were used to measure the same sample, and the results of the two methods were compared.
The hardware configuration of the experimental wavelength phase shifting interferometer is shown in Fig. 10(a), and the experimental interferometer is shown in Fig. 10(b), where a testing caliber of 100 mm was employed. TheTLB6800 laser (central wavelength = 632.8 nm) was developed by the NewFocus Company (USA). The measured object is a square brick with a thickness of 60 mm and a refractive index of 1.5163. The values of L1 and L2 were set as 30 and 60 mm, respectively, and four points of interference fringes formed by a maximum cavity length were captured per period. Figure 11 shows four consecutive interferograms selected arbitrarily from the 256 captured interferograms. The interferograms are then calculated using FFT and NUFFT and the corresponding optical homogeneity phases are obtained. The spectral distributions obtained by the two methods are shown in Fig. 12, and the calculation results of optical homogeneity are given in Fig. 13(b) and 13(c). The PV and RMS values of each phase are listed in Table 3.
Fig. 10 Photographic image of a square brick in the wavelength-tuning Fizeau interferometer. (a) Hardware configuration. (b) The experimental system.
Fig. 13 Optical homogeneities of the square brick calculated using different methods. (a) Optical homogeneity obtained using the flipping method and a Zygo interferometer. (b) Optical homogeneity calculated using FFT. (c) Optical homogeneity calculated using NUFFT.
Table 3. Results obtained through calculations using the three different methods
To validate the method correctness, a Zygo GPI phase shifting interferometer was used to test the same square brick. We employed the flipping method because the interference fringes formed by the front and back surfaces cannot be separated due to the parallelism of the brick. The profiles of the front and rear surfaces of the square brick, the transmission wavefront when the square brick is placed in the interference cavity, and the empty cavity wavefront are measured, and the optical homogeneity is then calculated from these four groups of wavefront data, as shown in Fig. 13(a).
From Fig. 13 and Table 3, it can be seen that the phase distribution of the NUFFT method is consistent with that of the flipping method, with deviations in the PV and RMS values of 9 × 10−8 and 2 × 10−9 being calculated. In contrast, a larger deviation was observed in the case of the FFT method, with differences in the PV and RMS values of 3.3 × 10−7 and 3.3 × 10−8 being obtained. These results indicate that the NUFFT method is more accurate than the FFT method. In addition, it should be noted that the six peaks in the frequency spectrum obtained using the FFT algorithm cannot be easily separated without using the spectral distribution of NUFFT to indicate the peak positions.
5. Conclusion
We herein reported the development of a wavelength-tuning interferometry method based on nonuniform fast Fourier transform (WTI-NUFFT) for the measurement of the optical homogeneity of parallel plates. The reported WTI-NUFFT method compensates for up to 25% nonlinearity of the phase shift error, which solves the spectral aliasing issue resulting from phase shifting errors, which is in turn caused by wavelength-tuning errors and environmental vibrations during measurement. The characteristics of the WTI-NUFFT method were estimated by comparison with the FFT method, and we found through the simulated and experimental results that the WTI-NUFFT method can indeed improve the accuracy of the optical homogeneity measurement of parallel plates. We therefore expect that this method will applicable widely in the context of wavelength-tuning interferometers. Future work in our group will focus on research into the NUFFT algorithm to solve the nonlinearity of phase shifts.
Funding
The National Natural Science Foundation of China (NSFC: 61405092). The Fundamental Research Funds for the Central Universities (30918014115).
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