Wavefront reconstruction in radial shearing interferometry with general aperture shapes is challenging because the problem may be ill-conditioned. Here we propose a Gram-Schmidt orthogonalization method to cope with off-axis wavefront reconstruction with any aperture type. The proposed method constructs a set of orthogonal basis functions and computes the corresponding expansion coefficients, which are converted into another set of expansion coefficients to reproduce the original wavefront. The method can effectively alleviate the ill-conditioning of the problem, and is numerically stable compared with the classic least-squares method, especially for non-circular apertures and in the presence of noise. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.
© 2016 Optical Society of America
Radial shearing interferometry (RSI), which is a powerful tool in optical metrology, has been widely used in many different applications [1–4], such as optical testing [5, 6], wavefront sensing [7–9] and high-power laser beam characterization . Unlike conventional interferometers, such as the Twyman-Green configuration, using a separate planar wavefront as the reference, the RSI expands the distorted wavefront under test and uses part of it as the reference. The self-reference nature enables it to be insensitive to ambient conditions, but makes acquired interferograms difficult to be interpreted because what the resultant interferogram indicates is not the original wavefront under test, but the differential wavefront between the expanded and the contracted wavefronts. This is especially true for the cases with large shear ratios. Therefore, it is necessary to employ certain types of wavefront evaluation algorithms to reproduce original wavefronts in RSI.
Early algorithms [11–13] mainly deal with the problem of wavefront reconstruction in on-axis RSI, in which the contracted and the expanded beams are coaxial and no lateral shear exists. This is the ideal situation. However, in practice, off-axis measurements are usually inevitable either due to misalignments [14, 15] or some special requirements, such as diagnosis of transient wavefront with an incomplete aperture in wind tunnel . The induced lateral shear due to off axis further complicates the problem and makes conventional on-axis algorithms incapable of accurately evaluating original wavefronts. The practical difficulty is stimulating recent developments of wavefront reconstruction techniques and drives them to be compatible with off-axis wavefront reconstruction.
Several advanced methods have been developed to cope with the wavefront reconstruction problem in off-axis RSI [10, 14–17]. Kohler et al. first proposed a reconstruction algorithm using successive iterations, which gives accurate results but is limited by sampling space and reading errors . Li et al. developed an explicit mathematical formula for precise wavefront estimation in the presence of lateral shear . Gu et al. presented a modal reconstruction technique , which employs the Zernike polynomials and its matrix formalism to estimate the expansion coefficients of the wavefront under test. However, since the Zernike polynomials are only orthogonal on a unit disk, Kewei et al. noticed that it was better to use the Legendre polynomials, which are orthogonal on a unit square, to estimate wavefronts with square apertures in inertial confinement fusion (ICF) . Both Gu’s method and Kewei’s method are intended for interferograms with special apertures, i.e. circular and square, because they take advantage of the orthogonality and aberration balancing of the employed polynomials.
However, in practice, we may need to deal with radial shearing interferograms with general apertures, such as non-circular or non-square [9, 18–20]. For such cases, advantages of orthogonality and aberration balancing of the Zernike polynomials and the Legendre polynomials no longer exist and reconstructed results may be numerically unstable when even a small perturbation or noise is present [20–22]. As a matter of fact, we found that even when the Zernike polynomials for circular apertures or the Legendre polynomials for square apertures are employed, the new basis functions (differences of two orthogonal polynomials) in off-axis wavefront reconstruction are not orthogonal any more [See Eq. (5) in Section 2.1]. This motivates us to seek a universal method to cope with radial shearing interferograms with any aperture shape encountered in experiment. As a further development of our previous work [9, 18], here we suggest that using a Gram-Schmidt process effectively improves the conditioning of the off-axis wavefront reconstruction problem and stabilizes the final solution. The proposed method gives comparable results with the classic least-squares method for well-conditioned systems, but has a more superior performance for ill-conditioned cases. It is expected to be capable of analyzing wavefronts with any aperture shape and arbitrary off-axis amount in RSI. The principle, comparisons, examples and discussions are presented below.
2. Modal wavefront reconstruction
In the following, we develop a mathematical model to describe the off-axis wavefront reconstruction problem and analyze the numerical stability of the least-squares method. Afterwards, a Gram-Schmidt procedure is proposed to ease the ill-conditioning of the problem and stabilize the wavefront reconstruction process with irregular apertures.
2.1 Mathematical model and its least-squares solution
The off-axis RSI in Fig. 1 works as follows . The incident distorted wavefront W is divided into two parts by a beam splitter. The transmitted wavefront, after being reflected by a mirror, enters a Galilean telescope and is contracted in diameter. The reflected wavefront, after being reflected again by another mirror, enters an identical Galilean telescope but with reversed direction with respect to the first one, and is expanded in diameter. The contracted wavefront Wc and the expanded wavefront We meet at a beam combiner and interfere with each other in the overlapping area. The shear ratio β = f 2 2/f 2 1, where f1 > f2 are the focal length of the lenses L1 and L2, respectively, and 0 < β < 1.
Slightly tilting the beam combiner will induce lateral displacements to the contracted beam Wc and make it decentered with respect to the expanded beam We. Define the coordinate system xOy and assume that the radii of Wc and We are 1 and r0 (r0 = 1/β > 1), respectively. The differential wavefront ∆W measured in experiment can be written as
Using finite terms of Zernike polynomials to represent the original wavefront, we have1] of the jth term, which is known and defined asEquation (5) indicates that the deduced polynomials Uj form a new set of basis functions and can be used to represent the differential wavefront ∆W. But the new basis functions Uj are not orthogonal as their mother functions Zj.
Written in discrete and matrix forms, Eq. (5) becomesEq. (7) is an overdetermined linear system and the solution can be obtained by solving the normal equation,
If the new basis functions Uj are orthogonal on predefined discrete data sets, the coefficient matrix UTU is a diagonal matrix and the least-squares solution is accurate and stable. However, as we already mentioned beforehand, Uj are not orthogonal and, therefore, the accuracy and stability of the solution depend on the conditioning of the system. When it is severely ill-conditioned, for example, on data sets with irregular apertures that seriously deviate from a unit circle, Uj will become somewhat dependent on each other and the accuracy and stability of the solution greatly degrade. For such cases, a small perturbation ϵ in the differential wavefront ∆W will cause great fluctuations δa to the coefficients a. Mathematically, this can be written asEqs. (11) and (12), we have23], which is a common process to be used to improve numerical stability.
2.2 Gram-Schmidt orthogonalization for general apertures
A. General process
Assume the basis Vj is orthogonal on the discrete data set and satisfies
Combining Eqs. (5) and (17), we haveEq. (16), the relations between bk and aj can be expressed asEqs. (17)-(19), and the expansion coefficient b can be computed by multiplying Vk to Eq. (16) and utilizing its orthogonality property [Eq. (15)], i.e.Eqs. (17)-(19).
B. A simple example with only the first three terms
To better understand the general procedure, we use a specific example with only the first three terms (N = 3) to illustrate the recursive process. For clarity, we define ∆W = ∆W(xi, yi), Vj = Vj(xi, yi), Uj = Uj(xi, yi). In this way, ∆W can be written as [Eq. (16)],Eqs. (17) - (19)]Eqs. (20), (27)-(29)]Eq. (26), we haveEqs. (27) - (29), and b is computed asEqs. (27) - (29)].
C. Error propagation and analysis
The proposed method mainly suffers from two errors, i.e. truncation error caused by fitting using finite terms of Zernike polynomials and random noise induced in experiment. Both errors can be modelled as a small perturbation ϵ to the differential wavefront ∆W. Writing Eq. (16) in matrix form, we haveEq. (34) becomesEqs. (36) and (34), we getEqs. (15) and (35)], where I is an identity matrix.
Using Eq. (24), the final coefficient error δa caused by δb can be formulated asEq. (2), the resultant wavefront reconstruction error δW can be correspondingly written as
Comparing Eq. (40) with Eq. (13), for a given amount of noise ϵ, the coefficient error δa in the proposed algorithm and the least-squares algorithm are determined by the condition number of the matrix α and UTU, i.e. κ(a) and κ(UTU), respectively. The matrix α is an upper triangular matrix and usually has a much smaller condition number than the matrix UTU. Therefore, the wavefront reconstruction problem employing the Gram-Schmidt orthogonalization is typically well-conditioned and has stable solutions compared with that using the least-squares method. This advantage is especially obvious when processing wavefronts with irregular aperture shapes.
Figure 2 shows the flow chart of the least-squares method and the Gram-Schmidt orthogonalization method.
3. Computer simulation and experimental results
To test the validity and performance of the proposed algorithm, both numerical and real experiments were carried out. The Zernike polynomials up to the 10th order (n = 10, N = 66) were used for all examples presented below.
3.1 Comparisons between the least-squares method and the Gram-Schmidt method
A simulation was first investigated to compare the numerical stability of the least-squares method and the Gram-Schmidt orthogonalization method by purposely designing a differential wavefront ∆W with an elliptic aperture, which is defined by the equation x2 + 4y2 = 1, where −1 ≤ x, y ≤ 1. The preset shear ratio, decenters are β = 0.5, x0 = 1, y0 = 0. The original wavefront W with 512 × 512 pixels was generated by assigning random coefficients to the first 21 terms of the Zernike polynomials. Figure 3 shows the original wavefront W, the contracted wavefront Wc, the expanded wavefront We and its subaperture that overlaps Wc, and the differential wavefront ∆W. To simulate real cases, additive Gaussian white noise with a mean 0 and a standard deviation 0.1 was added to ∆W.
We first used the least-squares method to reconstruct the original wavefront W from the differential wavefront ∆W. The estimated expansion coefficient aj, the recovered wavefront and the residual error are shown in Figs. 4(a) - 4(c), respectively. Due to the ill-conditioning of the problem (the condition number κ(UTU) = 9.0 × 1010), the least-squares method is very sensitive to noise and produces a large reconstruction error. As a comparison, the wavefront was also recovered using the proposed Gram-Schmidt orthogonalization method. The computed expansion coefficients bj of the orthogonal bases Vj and aj of Uj are shown in Fig. 4(d). As we can see, only the first few terms of bj have significant values and most higher-order terms are close to zero due to the orthogonality of Vj. Also the expansion coefficients bj are independent from each other and using a different fitting order n will not affect the values of bj. The finally reconstructed wavefront [Fig. 4(e)] is consistent with the true one W [Fig. 3(a)] and the residual error [Fig. 4(f)] is small. For such a non-circular aperture case, the proposed Gram-Schmidt orthogonalization method outperforms the classic least-squares method.
3.2 Experimental results
The proposed Gram-Schmidt orthogonalization algorithm has been successfully applied in wavefront reconstruction in practical applications.
Figure 5 shows an example of wavefront recovering from a single-shot off-axis RSI interferogram with a non-circular aperture by the Gram-Schmidt orthogonalization method. The shadow in the middle of the interferogram [Figs. 5(a) and 5(b)] was caused by an opaque blunt cone model placed in the light path. The phase [i.e. the differential wavefront ∆W, Fig. 5(c)] with piston and tilt removed was demodulated by use of the Fourier transform technique [24, 25]. The calibrated shear ratio β = 0.25 and decenters x0 = 0, y0 = −2.30. The computed expansion coefficients bj and aj, and the finally reconstructed wavefront are shown in Figs. 5(d) and 5(e), respectively. To verify the validity of the result, the wavefront was also evaluated using the iterative method in . The outcome of the iterative method and its difference from the result of the proposed method are shown Figs. 5(f) and 5(g), respectively. To assess the consistency, we calculated the quality index (Q index) of the two results [Figs. 5(e) and 5(f)] and the root mean square (RMS) value of the difference map [Figs. 5(g)], which are 0.897 and 0.166 rad, respectively. The Q index [26, 27] is defined as
Figure 6 shows another wavefront reconstruction example from a single-shot off-axis RSI interferogram with an even more non-circular aperture [Fig. 6(a)]. A sample, which is only transparent in a rectangular region, was placed in the beam path. A mask was created to segment the region of interest (ROI, 254 × 940 pixels) from the background [Fig. 6(b)]. The calibrated shear ratio β = 0.25 and decenters x0 = 1.20, y0 = 0.35. The encoded differential wavefront ∆W [Fig. 6(c)] with piston and tilt removed was also demodulated by the Fourier transform technique [24, 25]. The original wavefront was simultaneously reconstructed using the least-squares method, the proposed method and the iterative method in , respectively, and the results are shown in Figs. 6(d) – 6(f). It is obvious that the reconstructed wavefront by the proposed method is consistent with that by the iterative method, while the least-squares method gives erroneous result.
In summary, we presented an effective method base on the Gram-Schmidt orthogonalization for modal wavefront reconstruction in RSI with general aperture shapes. The proposed method constructs a set of orthogonal functions Vj using the Gram-Schmidt orthogonalization, and computes the independent expansion coefficients bj, which are converted into expansion coefficients aj of the non-orthogonal basis functions Uj to reproduce the original wavefront. The method has a comparable numerical stability with the least-squares method for well-conditioned systems, but reveals a more superior performance when handling ill-conditioned problems, especially for non-circular aperture cases. The method is robust to noise and does not suffer from truncation errors caused by omission of higher-order terms. It provides a universal tool for practical wavefront reconstruction from interferograms with general aperture shapes in RSI.
This work was supported by National Natural Science Foundation of China (NSFC) under Grant Nos. 60877043 and 61575061.
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