Two-dimensional wavefront reconstruction from lateral multi-shear interferograms

We propose and demonstrate multiple shearing interferometry for measuring two-dimensional phase object. Multi-shear interference can effectively eliminate the problem of spectral leakage that results from the single-shear interference. The Fourier coefficients of a two-dimensional wavefront are computed from phase differences obtained from multiple shearing interferograms, which are acquired by a shearing interferometer, and the desired phase is then reconstructed. Numerical and optical tests have confirmed that the multiple shearing interferometry has a higher recovery accuracy than single-shear interferometry and the reconstruction precision increases as the number of shear steps increases. ©2012 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.5050) Phase measurement; (070.2465) Finite analogs of Fourier transforms. References and links 1. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59(6), 940–950-2 (1947). 2. M. P. Rimmer and J. C. 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Introduction
Lateral shearing interferometry is a useful technique for evaluating optical wavefronts and is a promising measurement tool for many applications [1][2][3][4][5][6][7].Compared to conventional interferometry, it has the advantage of self-referencing, which makes it highly resistant to vibration and more suitable when a reference beam is not available.However, the measured shearing interferogram only contains information about the phase differences of the wavefront.The measured phase differences must be postprocessed in order to reconstruct the desired phase distribution.Many algorithms have been proposed to reconstruct the wavefront from these phase differences [8][9][10][11][12][13][14][15][16][17][18][19][20][21].One of the frequently adopted evaluation methods is the polynomial modal expansion method, which estimates the underlying wavefront by a set of polynomials whose coefficients are determined by least squares fitting.The Fourier modal expansion was proposed by Freischlad and Koliopoulos [8], and exact reconstructions are achieved for the special case in which the shear amount equals the spacing of the measurement points.An improved algorithm is proposed based on Fourier modal expansion for reconstructing a two-dimensional wavefront from phase differences with large shear [9].However, this algorithm has a disadvantage, i.e., the signal's Fourier spectra at frequency periods corresponding to multiples of the shear amount are lost, and thus the reconstruction accuracy is impaired.Elster and Weingärtner proposed a two-shear algorithm to combine the information of two sheared phase differences to avoid spectral leaking [16,17].Their method requires that the product of two shear parameters must equal the number of sampling points, which means that one or both of the two shears must be relatively large.However, in general, the precision of reconstruction decreases substantially when the shear amount increases, because the decreased phase difference along the shear direction can result in a greater amount of missing data from within the desired phase.
In this paper, we propose a new, to the best of our knowledge, lateral multiple-shearing scheme to resolve the spectral leakage problem.Multi-shear interference is accomplished in a three-wave shearing interferometer based on a spatial light modulator (SLM) [22].The influence of the amount of shear on the reconstruction accuracy is tested, and robustness against noise is also evaluated.Numerical and optical tests have confirmed that this multiple shearing interferometry method results in higher recovery accuracy than single-shear interference, especially for objects with discontinuous phase profiles (e.g., step-like optic surface)

Principle
A wavefront ϕ(m,n) can be expanded in a set of basis functions z pq (m,n) as follows: where N denotes the number of sampling points along the x or y direction.The original phase distribution ϕ(m,n) can be reconstructed if the coefficients α pq are determined from optical interferometry.If we choose the following complex exponential as the basis function: then Eq. ( 1) represents a two-dimensional discrete Fourier transform; thus, the coefficients α pq are the Fourier expansion coefficients of the desired wavefront.
In a lateral shearing interferometer, the object wavefront interferes with its sheared copy and forms a shearing interferogram.Using proper extraction techniques such as phase shifting, Fourier fringe analysis, and phase-unwrapping [23][24][25][26] we obtain from the interferogram the phase difference along the shear direction.Let us denote the phase difference in the x and y directions in the discrete form by D x (m,n) and D y (m,n), respectively.
They are related to the original phase ϕ(m,n) as follows: where s is the shear amount.Substituting Eq. ( 1) into the two expressions, we obtain the following expressions.
The measured phase differences in the x and y directions are denoted by ' ( , ) x D m n and ' ( , ) y D m n , respectively.The squared sum of the deviation between the measured and original phase differences can be defined as follows: The minimization of this residual error yields the optimal phase estimate.By performing a least-squares procedure, we derive the following condition: 0 .
Substituting Eq. (4) into Eq.( 5) and solving Eq. ( 6), we get the Fourier coefficients of the wavefront as follows: # where FT pq represents the 2-D Fourier transform and p and q are the corresponding spatial frequencies.If p and q simultaneously reach a multiple of N/s, the denominator of Eq. ( 7) becomes zero, which leads to undetermined results; i.e., the Fourier spectral components of the wavefront at these points are lost.To circumvent this problem, one can replace the frequency components at these leakage points with the average value of the adjacent points [25].However, such processing may result in reconstruction errors.Our interferometry experience thus far has shown that replacing the leaking frequency components with the average value may severely impair the reconstruction accuracy especially for object wavefronts with a discontinuous phase profile.
With the aim of resolving the abovementioned frequency-leaking problem in shearing interferometry, we propose a multi-shear interference scheme based on Fourier expansion.First, we obtain a series of phase differences from the multiple shearing interferograms with different shears.Denoting the number of shears by K (i.e., K groups of phase differences), we reconstitute the error function as follows: We then derive the following optimal solution of the Fourier coefficients after carrying out some mathematical manipulations based on least-squares fitting: ) 4sin ( ) where s j represents the j-th shear amount in the x or y direction.From Eq. ( 9), it follows that the denominator cannot vanish so long as any two of the shear amounts have no common divisors, and thus the spectral leaking problem can be removed.The wavefront phase ϕ(m,n) can be reconstructed exactly over all frequencies by performing the inverse Fourier transform of the Fourier coefficients α pq .

Error propagation
In lateral shearing interferometry, some noise may exist in the measured phase difference, which may be due to speckle, circumstance disturbance, detector noise, or other noise sources.It is valuable to give some theoretical analysis of the error propagation that indicates the sensitivity of the noise in the reconstructed phase to the noise in the phase difference measurements.The error propagation is characterized in terms of noise coefficients, which quantifies the reconstruction error resulting from the noise in the phase difference.Previous numerical and theoretical works show that the noise coefficient in single-shear process is a logarithmic function of the number of measurement points [27,28].Now, we extend the analysis to multi-shear process.For clarity in the following derivations, Eq. ( 9) is rewritten as where For the following analysis it is assumed that the measured phase difference are perturbed by the uncorrelated zero-mean additive random noises, ( , ) n m n and ( , ) j y n m n , which have the following statistics, where ﹤﹤﹤ stands for the ensemble average.From Eq. ( 10) and using the Fourier transform and autocorrelation theorem, we get the mean variance of the error in the coefficients a(p, q) ( ) Since a(p, q) is the Fourier coefficient of the desired phase, the mean variance of the error ( , ) n m n ϕ in the reconstructed phase ϕ(m, n) can be derived as follows, ( ) ( ) , .
where the second equal sign is due to Parseval's theorem.Finally, we obtain the noise coefficient: ( ) Substituting Eqs. ( 11) and ( 12) into Eq.( 16) gives a simple form of the noise coefficient: Figure 1 shows the plots of the noise coefficients for different shear amounts and various numbers of sample points.The data on the noise coefficient can be fitted to a logarithmic curve.The result is summarized in Table 1.It may be questioned that, since the number of shear steps contributes to both the numerator and the denominator in Eq. ( 9) at the same time, the noise coefficient should not change as dramatically as in the plots in Fig. 1.Our answer is that the denominator is a sum of absolute values while the numerator consists of a sum of complex terms, and accordingly, the denominator increases faster with the number of shear steps than the numerator when more shearing data are added.The simulations and experiments presented in the following sections will reaffirm our argument that the noise coefficient decreases significantly as the number of shear steps increases.0.03407 0.00113ln N +

Computer simulation
To evaluate the reliability of the proposed algorithm, we use the following phase function ϕ(x,y) for a reconstruction simulation: This function is a combination of Zernike polynomials and is sampled across a 420 × 420 grid.The Zernike function is selected because it is most often used in imaging system evaluations, and thus represents a typical phase distribution.The phase distribution is plotted in Fig. 2(a).The multi-shears s j are chosen to be 4, 5, and 6 pixels in this case.The phase differences with a shear of 4 pixels in two orthogonal directions are shown in Fig. 2(b) and 2(c).Figure 2(d) shows the reconstructed phase for the multi-shears s j of 4, 5, and 6 pixels.The deviation of the reconstructed phase from the original phase is computed using the root mean square (RMS) and peak-to-valley (PV) value of the deviations as error measures.The unit of the numerical errors is converted to wavelength, λ, by considering a phase of 2π as being equivalent to an optical length of λ.In this simulation, the RMS is 4.59 × 10 −9 λ and the PV value is 9.54 × 10 −8 λ.The relative error is expressed as a ratio of the PV or RMS value to the average value of the original phase.The relative RMS value is 3.06 × 10 -6 %, and the PV value is 6.36 × 10 -5 %.To study the reconstruction accuracy of the multi-shear method and the influence of the number of shears on the accuracy, we perform simulations with different number of shears, in which the reconstruction accuracies of single-shear and multi-shear interferences are compared.The numerical results for the two methods are presented in Tables 2 and 3, respectively.Table 2 shows the dependence of reconstruction precision of singe-shear interferometry on the shear amount s.Table 3 displays the dependence of reconstruction precision on the number of shearing steps.From Tables 2 and 3 we can see that the reconstruction accuracy of the multi-shear method is higher than that of the single-shear method, especially for larger shear amounts.In addition, from Table 3, we know that both the RMS and PV decrease as the number of shears increases, which agrees with the theoretical analysis presented in the last section.Objects with discontinuous phase distributions are also tested.Owing to its step-like phase map, we choose the letter B for use in reconstruction simulations with 420 × 420 grid points.The original phase is plotted in Fig. 3(a).The obtained RMS of the reconstruction phases are 1.3 × 10 −3 λ for the single-shearing interferograms with shear of 4 pixels and 3.1 × 10 −14 λ for multiple shears of 4, 5, and 6 pixels, respectively.The multi-shear algorithm demonstrates superior performance in recovering the discontinuous phase distribution.To visualize the performance of the two algorithms from the plot, we use the difference between the original and reconstructed phases.The difference between the original and reconstructed phases using the single-shear algorithm is plotted in Fig. 3(b), and the difference using the multi-shear algorithm is shown in Fig. 3(c).Figures 3(b) and 3(c) show that the multi-shear algorithm has a much higher reconstruction precision than the singe-shear algorithm.In addition, we use different numbers of shears in the simulation and find that the precision increases as the number of shear amounts increase, which is similar to the trend observed in Table 3.We believe that the superior performance of the multi-shear algorithm in recovering the discontinuous phase is due to the fact that a discontinuous function has a broader Fourier frequency range than does a smooth function.A step-like phase, which has more highfrequency components in the Fourier domain, is more likely susceptible to frequency-leaking damage, which occurs when the single-shear algorithm is used but can be avoided completely by using the multi-shear algorithm.To validate our viewpoint, we compare the recovery accuracy of the above two phase (one with continuous distribution and the other with discontinuous distribution) as they suffer from the spectral leaking.For the two phases, their Fourier spectra at the period of N/s are blocked (set to zero), and then inversely Fourier transformed to an erroneous phase.The recovery error due to the spectral leaking is listed in Table 4, which clearly suggests that a discontinuous phase is more susceptible to frequency-leaking than a discontinuous phase is.Consequently, our multi-shear interferometry can achieve a significant improvement over a single shear in the reconstruction of the discontinuous phase.In order to numerically investigate the performance of the algorithm in noisy environments and make a comparison with the theoretical analysis of the error propagation described in the previous section, we calculate the reconstruction error from the noisy phase differences to which a uniformly distributed random noise is added.Table 5 presents the simulation result.The RMSs of the reconstructed phases -ϕ(x, y) and the letter "B" and of the noise in the phase differences are denoted by ϕ σ , B σ and n σ , respectively.In the second row of Table 4 the noise coefficient, C, is computed according to Eq. ( 17), and the data in the third row represent the expected RMS of the reconstructed phase.The fourth and fifth rows present the actual RMS of the two phases.The numerical simulation is in agreement with the theoretical analysis in the sense of the noise propagation, and justifies the conclusion that the reconstruction error decreases significantly as the number of shear steps increases

Optical experiment
In most shearing interferometers, mechanically moving devices are required to adjust the phase shifts and wavefront shears.The accuracy and adaptability of shearing interferometers are therefore limited by mechanical elements.Because our proposed algorithm requires multiple shears, and thus both the measured phase differences and reconstruction precision are easily affected by any error in the moving parts.Therefore, we carry out optical experiments in a three-wave lateral shearing interferometer based on a SLM [22], as schematically shown in Fig. 4. A He-Ne laser provides light at a wavelength of 632.8 nm.A cosine grating is displayed on the SLM with 1024 × 768 pixels (each pixel is 18 µm × 18 µm in size) controlled by a computer.The cosine grating is used to diffract the incident light into the + 1, −1 and zero diffraction orders so that an object wave is split into three sheared waves.The common path set-up and zero optical path difference between the interfering diffraction orders makes this device easy to align.The flexibility of the SLM can be fully utilized in the sense that the phase shift, as well as the direction and amount of shear can be dynamically controlled.The interferograms are captured using a CCD camera with a cell size of 8.6 µm × 8.6 µm.The eight interferograms required for an 8-step phase-shifting algorithm are recorded [22], and then the phase differences are extracted and unwrapped.The phase difference in the perpendicular direction is obtained by rotating the grating through 90 degrees.Multiple phase differences are acquired for different shear amounts, and are then used to reconstruct the desired phase.The test object is a quartz glass with the letter "B" etched into the surface to form a surface relief.The object is placed in the object plane.The shearing interferograms are captured by the CCD camera and sampled within a 420 × 420 pixel domain.We choose the shear amount for the single-shear algorithm to be 4 pixels.The reconstructed phase is shown in Fig. 5(a).Shear amounts of 1, 2, 3, 4, 5, 6, and 7 pixels are chosen for the multipleshearing interference test, and the reconstructed phase is shown in Fig. 5(b).For comparison, the one-dimensional relief profile along the 210-th row of the measured surface map is also presented in the bottom of Fig. 5(a) and 5(b).This measurement confirms that the multi-shear reconstruction method performs better than the single-shear reconstruction method.The height d between the top and bottom of the binary relief is related to the measured phase ϕ by ( 2) , where ∆n is the difference between the refractive indexes of the glass and air.The refractive index of the glass is 1.457 at a wavelength of 632.8 nm.Therefore, the corresponding relief height of the etched pattern is around 500 nm.For comparison, the relief height is also measured with the Dektek 150 profilometer from Veeco Instruments Inc.The measured value from the profiler is around 470 nm, which corresponds to a discrepancy of less than 6% between two kinds of measurement.

Conclusion
We have developed a multiple shearing interferometry for measuring two-dimensional object phase.Multiple phase differences are acquired using different shears in the shearing interferometer.The algorithm can perform a fast phase reconstruction based on least-square estimate and the two-dimensional Fourier transform of the multiple phase differences.The multiple-shearing scheme demonstrates higher phase recovery accuracy than single-shear interference, especially for discontinuous phase distributions, because it is not subject to spectral leakage.The influence of shear steps and noise levels on reconstruction accuracy is evaluated.Multi-shear interference measurements are accomplished in a three-wave shearing interferometer based on SLM, showing a good stability and reliability in wavefront phase reconstruction.

Fig. 1 .
Fig. 1.Plot of the noise coefficient for different shear amounts versus the number of sample points.Table 1. Noise coefficient versus the shear amount s shear amount noise coefficient s = 1 0.04878 0.15915ln N +

Fig. 3 .
Fig. 3. Comparison between reconstruction accuracy of single-shear and multi-shear methods: (a) original phase map, (b) difference between original and reconstructed phases obtained using single-shear algorithm, and (c) difference between original and reconstructed phases obtained using multi-shear algorithm.

Fig. 4 .Fig. 5 .
Fig. 4. Three-wave lateral shearing interferometer based on SLM with four lenses (L1, L2, L3 and L4) and a rotating ground glass that can lower the spatial coherence of light in order to reduce speckle noise.