Abstract

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

1. Introduction

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 [10]. 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 [9]. 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 [16]. Li et al. developed an explicit mathematical formula for precise wavefront estimation in the presence of lateral shear [14]. Gu et al. presented a modal reconstruction technique [15], 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) [10]. 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 [9]. 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.

 figure: Fig. 1

Fig. 1 Schematic diagram of one type of off-axis RSI using two identical Galilean telescopes and the coordinate system.

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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

ΔW(x,y)=Wc(x,y)We(x,y)=W(x,y)W(βx+x0,βy+y0),
where −1 ≤ x, y ≤ 1, Wc(x, y) = W(x, y), W is the original wavefront. The expanded wavefront in the overlapping area We(x, y) = W(βx + x0, βy + y0), which is obtained by first scaling the coordinate system of W by a factor β and then shifting it by x0 and y0, which are decenters along the x axis and the y axis, respectively, and satisfy x2 0 + y2 0 ≤ (r0 - 1)2.

Using finite terms of Zernike polynomials to represent the original wavefront, we have

W(x,y)=j=1NajZj(x,y),
where N is the total terms of the Zernike polynomials, aj is the expansion coefficient of the jth term, and Zj is the orthonormal Zernike polynomial [1] of the jth term, which is known and defined as
Zj={(n+1)Rnm(ρ),m=0,2(n+1)Rnm(ρ)cosmθ,m0andevenj,2(n+1)Rnm(ρ)sinmθ,m0andoddj,
where 0 ≤ ρ = (x2 + y2)1/2 ≤ 1 and 0 ≤ θ = tan−1(y/x) ≤ 2π are the normalized radial coordinate and the angular coordinate, respectively; j = (n + 1)(n + 2)/2 is the index, and the radial polynomials
Rnm(ρ)=s=0(nm)/2(1)s(ns)!s![(n+m)/2s]![(nm)/2s]!ρn2s,
where n and m are non-negative integers, which mean the radial degree and the azimuthal frequency, respectively, and satisfy nm = even.

Substituting Eq. (2) into Eq. (1), we obtain

ΔW(x,y)=j=1NajZj(x,y)j=1NajZj(βx+x0,βy+y0)=j=1NajUj(x,y),
where
Uj(x,y)=Zj(x,y)Zj(βx+x0,βy+y0),
where j = 2, 3, …, N, and N is the total term of the Zernike polynomials. For j = 1, U1(x, y) = Z1(x, y) – Z1(βx + x0, βy + y0) = 0, which will lead to a singular solution of a1. To avoid the singularity, we redefine U1(x, y) = 1. Equation (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) becomes

Ua=ΔW,
where
U=[U1(x1,y1)U2(x1,y1)UN(x1,y1)U1(x2,y2)U2(x2,y2)UN(x2,y2)U1(xM,yM)U2(xM,yM)UN(xM,yM)],
a=[a1a2aN],ΔW=[ΔW(x1,y1)ΔW(x2,y2)ΔW(xM,yM)],
where M is the total number of valid data points. Since M > N is generally true, Eq. (7) is an overdetermined linear system and the solution can be obtained by solving the normal equation,
UTUa=UTΔW.
where T means matrix transpose. The solution vector can be obtained by matrix inverse, i.e.

a=(UTU)1UTΔW.

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 as

a+δa=(UTU)1UT(ΔW+ϵ).
Combining Eqs. (11) and (12), we have
δa=(UTU)1UTϵ=(SΛ1ST)UTϵ,
where (UTU)−1 = −1ST, S is an eigenvector matrix and Λ is an eigenvalue matrix. Assuming that λ1, λ2, λ3, …, λN are eigenvalues of the matrix (UTU)−1 and are arranged in an decreasing order λ1λ2λ3 ≥ … ≥ λN, the induced error of the solution can be approximately estimated as
Δaji=1MUjiεi/λj,
where j = 1, 2, …, N. For seriously ill-conditioned systems, the eigenvalues will drop quickly in the order of magnitude, which may cause great errors to the final solution. This can be cured by use of the Gram-Schmidt orthogonalization [23], 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

i=1MVp(xi,yi)Vq(xi,yi)={0,ifpq,1,ifp=q,
The wavefront ∆W can be decomposed into a linear combination of the orthogonal basis Vk
ΔW(xi,yi)=k=1NbkVk(xi,yi),
where bk is the expansion coefficient of the kth term. Following the Gram-Schmidt orthogonalization procedure, each Uj can be expanded in terms of Vj up to the term j, i.e.
Uj(xi,yi)=k=1jαkjVk(xi,yi).
The orthogonal basis Vj can also be represented by Uj through the following expression
Vj(xi,yi)=1αjj[Uj(xi,yi)k=1j1αkjVk(xi,yi)],
where α is the conversion coefficient, and can be derived as

αkj={i=1M[Uj(xi,yi)Vk(xi,yi)],k<j,{i=1M[Uj(xi,yi)]2k=1j1αkj2}1/2,k=j.

Combining Eqs. (5) and (17), we have

ΔW(xi,yi)=j=1NajUj(xi,yi)=j=1Naj[k=1jαkjVk(xi,yi)]=k=1Nj=kNajαkjVk(xi,yi).
Comparing the above equation with Eq. (16), the relations between bk and aj can be expressed as
bk=j=kNajαkj.
Written in matrix form, it becomes
[α11α12α13α1N0α22α23α2N00α33α3N0000αNN][a1a2a3aN]=[b1b2b3bN],
or in the following form
αa=b.
The expansion coefficient a can be found using matrix inverse as
a=α1b,
where the conversion coefficient α can be recursively calculated through Eqs. (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.
bk=i=1M[ΔW(xi,yi)Vk(xi,yi)],
where ∆W is the differential wavefront measured in experiment and Vk is recursively calculated through 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)],

ΔW=b1V1+b2V2+b3V3.
The first orthogonal basis V1 is generated as [Eqs. (17) - (19)]
U1=α11V1,α11=(i=1MU12)1/2,V1=U1/α11.
The second orthogonal basis V2 is generated as
U2=α12V1+α22V2,α12=i=1M(U2V1),α22=(i=1MU22α122)1/2,V2=(U2α12V1)/α22.
The third orthogonal basis V3 is generated as
U3=α13V1+α23V2+α33V3,α13=i=1M(U3V1),α23=i=1M(U3V2),α33=[i=1MU32(α132+α232)]1/2,V3=(U3α13V1α23V2)/α33.
The wavefront ∆W can also be written as [Eqs. (20), (27)-(29)]
ΔW=a1U1+a2U2+a3U3=(α11a1+α12a2+α13a3)V1+(α22a2+α23a3)V2+α33a3V3.
Comparing it with Eq. (26), we have
[α11α12α130α22α2300α33][a1a2a3]=[b1b2b3].
The expansion coefficients
a3=b3/α33,a2=(b2a23b3)/α22,a1=(b1α12a2α13a3)/α11,
where α has already been calculated in Eqs. (27) - (29), and b is computed as
b3=i=1M(ΔWV3),b2=i=1M(ΔWV2),b1=i=1M(ΔWV1),
where ∆W is the differential wavefront measured in experiment and V1, V2, V3 are calculated beforehand [Eqs. (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 have

Vb=ΔW,
where the orthogonal matrix
V=[V1(x1,y1)V2(x1,y1)VN(x1,y1)V1(x2,y2)V2(x2,y2)VN(x2,y2)V1(xM,yM)V2(xM,yM)VN(xM,yM)],b=[b1b2bN].
Considering the perturbation ϵ, Eq. (34) becomes
V(b+δb)=(ΔW+ϵ).
where δb is the error of b caused by ϵ. Combing Eqs. (36) and (34), we get
Vδb=ϵ,
and, therefore,
δb=VTϵ,
using the relationship VTV = I [Eqs. (15) and (35)], where I is an identity matrix.

Using Eq. (24), the final coefficient error δa caused by δb can be formulated as

a+δa=α1(b+δb).
Further, we have
δa=α1δb=α1VTϵ.
According to Eq. (2), the resultant wavefront reconstruction error δW can be correspondingly written as

δW=Zδa=Zα1VTϵ.

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.

 figure: Fig. 2

Fig. 2 Flow chart of the least-squares method (a) and the Gram-Schmidt orthogonalization method (b). Zj and ∆W are known, and W is the unknown to be determined.

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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.

 figure: Fig. 3

Fig. 3 Generation of a differential wavefront with an elliptic aperture. (a) Original wavefront W, contracted wavefront Wc, expanded wavefront We and their overlapping area (red curve), (b) three-dimensional (3D) and two-dimensional (2D) differential wavefront ∆W with a small additive noise. Colorbar unit: rad.

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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.

 figure: Fig. 4

Fig. 4 Reconstruction results by the least-squares method (1st row) and the proposed method (2nd row). (a) - (c) Computed coefficients aj, reconstructed wavefront and residual error by least squares; (d) - (f) computed coefficients aj and bj, reconstructed wavefront and residual error by Gram-Schmidt orthogonalization. Colorbar unit: rad.

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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 [14]. 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

Q=4σABμAμB(σA2+σB2)(μA2+μB2),
where A and B represent the results of the proposed method and the iterative method, respectively; μA and μB are the mean values; 𝜎A and 𝜎B are the standard deviations; and 𝜎AB is the covariance of A and B. The Q index measures the correlation of the two results and has a dynamic range [-1, 1], where 1 means perfect match. It is clear that the result of the proposed method has a high Q index and the RMS value of the difference map is small.

 figure: Fig. 5

Fig. 5 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and its enlarged view, (c) demodulated phase, (d) computed coefficients aj and bj, (e) and (f) reconstructed wavefronts using the proposed method and the iterative method in [14], respectively, and (g) difference map. Colorbar unit: rad.

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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 [14], 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.

 figure: Fig. 6

Fig. 6 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and the ROI, (c) demodulated phase, (d) - (f) reconstructed wavefronts using the least-squares method, the proposed method and the iterative method in [14], respectively. Colorbar unit: rad.

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4. Conclusion

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.

Acknowledgments

This work was supported by National Natural Science Foundation of China (NSFC) under Grant Nos. 60877043 and 61575061.

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References

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  1. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).
  2. W. Zhu, L. Chen, C. Gu, J. Wan, and D. Zheng, “Single-shot reflective shearing point diffraction interferometer for wavefront measurements,” Appl. Opt. 54(20), 6155–6161 (2015).
    [Crossref] [PubMed]
  3. N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
    [Crossref]
  4. R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
    [Crossref]
  5. M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” in Photonics Asia 2002 (SPIE, 2002), 673–676.
  6. C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
    [Crossref]
  7. T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25(11), 773–775 (2000).
    [Crossref] [PubMed]
  8. N. Gu, L. Huang, Z. Yang, and C. Rao, “A single-shot common-path phase-stepping radial shearing interferometer for wavefront measurements,” Opt. Express 19(5), 4703–4713 (2011).
    [Crossref] [PubMed]
  9. T. Ling, D. Liu, Y. Yang, L. Sun, C. Tian, and Y. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. 38(14), 2493–2495 (2013).
    [Crossref] [PubMed]
  10. E. Kewei, C. Zhang, M. Li, Z. Xiong, and D. Li, “Wavefront reconstruction algorithm based on Legendre polynomials for radial shearing interferometry over a square area and error analysis,” Opt. Express 23(16), 20267–20279 (2015).
    [Crossref] [PubMed]
  11. T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
    [Crossref]
  12. D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
    [Crossref]
  13. T. M. Jeong, D.-K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. 32(3), 232–234 (2007).
    [Crossref] [PubMed]
  14. D. Li, F. Wen, Q. Wang, Y. Zhao, F. Li, and B. Bao, “Improved formula of wavefront reconstruction from a radial shearing interferogram,” Opt. Lett. 33(3), 210–212 (2008).
    [Crossref] [PubMed]
  15. N. Gu, L. Huang, Z. Yang, Q. Luo, and C. Rao, “Modal wavefront reconstruction for radial shearing interferometer with lateral shear,” Opt. Lett. 36(18), 3693–3695 (2011).
    [Crossref] [PubMed]
  16. D. R. Kohler and V. L. Gamiz, “Interferogram reduction for radial-shear and local-reference-holographic interferograms,” Appl. Opt. 25(10), 1650–1652 (1986).
    [Crossref] [PubMed]
  17. E. López Lago and R. de la Fuente, “Amplitude and phase reconstruction by radial shearing interferometry,” Appl. Opt. 47(3), 372–376 (2008).
    [Crossref] [PubMed]
  18. C. Tian, Y. Yang, Y. Zhuo, T. Wei, and T. Ling, “Tomographic reconstruction of three-dimensional refractive index fields by use of a regularized phase-tracking technique and a polynomial approximation method,” Appl. Opt. 50(35), 6495–6504 (2011).
    [Crossref] [PubMed]
  19. R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29(24), 2840–2842 (2004).
    [Crossref] [PubMed]
  20. W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33(10), 1832–1837 (1994).
    [Crossref] [PubMed]
  21. F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express 20(2), 1530–1544 (2012).
    [Crossref] [PubMed]
  22. D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
    [Crossref]
  23. C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000).
  24. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  25. C. Tian, Y. Yang, T. Wei, T. Ling, and Y. Zhuo, “Demodulation of a single-image interferogram using a Zernike-polynomial-based phase-fitting technique with a differential evolution algorithm,” Opt. Lett. 36(12), 2318–2320 (2011).
    [Crossref] [PubMed]
  26. Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
    [Crossref]
  27. C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express (To be published).

2015 (3)

2013 (1)

2012 (1)

2011 (4)

2009 (1)

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

2008 (2)

2007 (1)

2004 (1)

2002 (2)

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

2000 (2)

T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25(11), 773–775 (2000).
[Crossref] [PubMed]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

1994 (1)

1990 (1)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

1986 (1)

1982 (1)

Bao, B.

Barnes, T. H.

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Carpio-Valadez, M.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Chen, H.

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Chen, L.

Chen, Z.

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Chow, W. W.

Dai, F.

de la Fuente, R.

Ellerbroek, B.

Feng, P.

Gamiz, V. L.

Gu, C.

Gu, N.

Haskell, T. G.

Huang, L.

Ina, H.

Jeong, T. M.

Kantun-Montiel, R.

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

Kewei, E.

Ko, D.-K.

Kobayashi, S.

Kohler, D. R.

Kohno, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Lee, J.

Li, D.

Li, F.

Li, M.

Ling, T.

Liu, D.

T. Ling, D. Liu, Y. Yang, L. Sun, C. Tian, and Y. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. 38(14), 2493–2495 (2013).
[Crossref] [PubMed]

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

Liu, S.

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express (To be published).

López Lago, E.

Luo, Q.

Luo, Y.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

Malacara-Hernandez, D.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Matsumoto, D.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Meneses-Fabian, C.

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Rao, C.

Rodriguez-Zurita, G.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Sanchez-Mondragon, J. J.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Sasaki, O.

Shen, Y.

Shirai, T.

Sun, L.

Swantner, W.

Takeda, M.

Tang, F.

Tian, C.

Toto-Arellano, N. I.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Uda, Y.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Upton, R.

Vázquez-Castillo, J. F.

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

Wan, J.

Wang, Q.

Wang, X.

Wang, Z.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

Wei, T.

Wen, F.

Xiong, Z.

Yang, Y.

Yang, Z.

Yazawa, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

Zhang, C.

Zhao, Y.

Zheng, D.

Zhu, W.

Zhuo, Y.

Appl. Opt. (5)

IEEE Signal Process. Lett. (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[Crossref]

J. Opt. (1)

R. Kantun-Montiel and C. Meneses-Fabian, “Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer,” J. Opt. 17(4), 045602 (2015).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

N. I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “A single-shot phase-shifting radial-shearing interferometer,” J. Opt. A, Pure Appl. Opt. 11(4), 045704 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (3)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39(10), 2696–2699 (2000).
[Crossref]

D. Li, H. Chen, and Z. Chen, “Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer,” Opt. Eng. 41(8), 1893–1898 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Other (5)

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007).

M. Wang, B. Zhang, S. Nie, and J. Fu, “Radial shearing interferometer for aspheric surface testing,” in Photonics Asia 2002 (SPIE, 2002), 673–676.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, (SPIE, 2010), 765612.
[Crossref]

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express (To be published).

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000).

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of one type of off-axis RSI using two identical Galilean telescopes and the coordinate system.
Fig. 2
Fig. 2 Flow chart of the least-squares method (a) and the Gram-Schmidt orthogonalization method (b). Zj and ∆W are known, and W is the unknown to be determined.
Fig. 3
Fig. 3 Generation of a differential wavefront with an elliptic aperture. (a) Original wavefront W, contracted wavefront Wc, expanded wavefront We and their overlapping area (red curve), (b) three-dimensional (3D) and two-dimensional (2D) differential wavefront ∆W with a small additive noise. Colorbar unit: rad.
Fig. 4
Fig. 4 Reconstruction results by the least-squares method (1st row) and the proposed method (2nd row). (a) - (c) Computed coefficients aj, reconstructed wavefront and residual error by least squares; (d) - (f) computed coefficients aj and bj, reconstructed wavefront and residual error by Gram-Schmidt orthogonalization. Colorbar unit: rad.
Fig. 5
Fig. 5 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and its enlarged view, (c) demodulated phase, (d) computed coefficients aj and bj, (e) and (f) reconstructed wavefronts using the proposed method and the iterative method in [14], respectively, and (g) difference map. Colorbar unit: rad.
Fig. 6
Fig. 6 Wavefront reconstruction from a single-shot off-axis RSI interferogram. (a) and (b) An experimental interferogram and the ROI, (c) demodulated phase, (d) - (f) reconstructed wavefronts using the least-squares method, the proposed method and the iterative method in [14], respectively. Colorbar unit: rad.

Equations (42)

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ΔW(x,y)= W c (x,y) W e (x,y)=W(x,y)W(βx+ x 0 ,βy+ y 0 ),
W(x,y)= j=1 N a j Z j (x,y),
Z j ={ (n+1) R n m (ρ),m=0, 2(n+1) R n m (ρ)cosmθ,m0andevenj, 2(n+1) R n m (ρ)sinmθ,m0andoddj,
R n m (ρ)= s=0 (nm)/2 (1) s (ns)! s![(n+m)/2s]![(nm)/2s]! ρ n2s ,
ΔW(x,y)= j=1 N a j Z j (x,y) j=1 N a j Z j (βx+ x 0 ,βy+ y 0 )= j=1 N a j U j (x,y),
U j (x,y)= Z j (x,y) Z j (βx+ x 0 ,βy+ y 0 ),
Ua=ΔW,
U=[ U 1 ( x 1 , y 1 ) U 2 ( x 1 , y 1 ) U N ( x 1 , y 1 ) U 1 ( x 2 , y 2 ) U 2 ( x 2 , y 2 ) U N ( x 2 , y 2 ) U 1 ( x M , y M ) U 2 ( x M , y M ) U N ( x M , y M ) ],
a=[ a 1 a 2 a N ],ΔW=[ ΔW( x 1 , y 1 ) ΔW( x 2 , y 2 ) ΔW( x M , y M ) ],
U T Ua= U T ΔW.
a= ( U T U) 1 U T ΔW.
a+δa= ( U T U) 1 U T (ΔW+ϵ).
δa= ( U T U) 1 U T ϵ=(S Λ 1 S T ) U T ϵ,
Δ a j i=1 M U ji ε i / λ j ,
i=1 M V p ( x i , y i ) V q ( x i , y i )={ 0,ifpq, 1,ifp=q,
ΔW( x i , y i )= k=1 N b k V k ( x i , y i ),
U j ( x i , y i )= k=1 j α kj V k ( x i , y i ).
V j ( x i , y i )= 1 α jj [ U j ( x i , y i ) k=1 j1 α kj V k ( x i , y i ) ],
α kj ={ i=1 M [ U j ( x i , y i ) V k ( x i , y i ) ],k<j, { i=1 M [ U j ( x i , y i ) ] 2 k=1 j1 α kj 2 } 1/2 ,k=j.
ΔW( x i , y i )= j=1 N a j U j ( x i , y i )= j=1 N a j [ k=1 j α kj V k ( x i , y i ) ]= k=1 N j=k N a j α kj V k ( x i , y i ).
b k = j=k N a j α kj .
[ α 11 α 12 α 13 α 1N 0 α 22 α 23 α 2N 0 0 α 33 α 3N 0 0 0 0 α NN ][ a 1 a 2 a 3 a N ]=[ b 1 b 2 b 3 b N ],
αa=b.
a= α 1 b,
b k = i=1 M [ ΔW( x i , y i ) V k ( x i , y i ) ],
ΔW= b 1 V 1 + b 2 V 2 + b 3 V 3 .
U 1 = α 11 V 1 , α 11 = ( i=1 M U 1 2 ) 1/2 , V 1 = U 1 / α 11 .
U 2 = α 12 V 1 + α 22 V 2 , α 12 = i=1 M ( U 2 V 1 ), α 22 = ( i=1 M U 2 2 α 12 2 ) 1/2 , V 2 =( U 2 α 12 V 1 )/ α 22 .
U 3 = α 13 V 1 + α 23 V 2 + α 33 V 3 , α 13 = i=1 M ( U 3 V 1 ), α 23 = i=1 M ( U 3 V 2 ), α 33 = [ i=1 M U 3 2 ( α 13 2 + α 23 2 ) ] 1/2 , V 3 =( U 3 α 13 V 1 α 23 V 2 )/ α 33 .
ΔW= a 1 U 1 + a 2 U 2 + a 3 U 3 =( α 11 a 1 + α 12 a 2 + α 13 a 3 ) V 1 +( α 22 a 2 + α 23 a 3 ) V 2 + α 33 a 3 V 3 .
[ α 11 α 12 α 13 0 α 22 α 23 0 0 α 33 ][ a 1 a 2 a 3 ]=[ b 1 b 2 b 3 ].
a 3 = b 3 / α 33 , a 2 =( b 2 a 23 b 3 )/ α 22 , a 1 =( b 1 α 12 a 2 α 13 a 3 )/ α 11 ,
b 3 = i=1 M (ΔW V 3 ), b 2 = i=1 M (ΔW V 2 ), b 1 = i=1 M (ΔW V 1 ),
Vb=ΔW,
V=[ V 1 ( x 1 , y 1 ) V 2 ( x 1 , y 1 ) V N ( x 1 , y 1 ) V 1 ( x 2 , y 2 ) V 2 ( x 2 , y 2 ) V N ( x 2 , y 2 ) V 1 ( x M , y M ) V 2 ( x M , y M ) V N ( x M , y M ) ],b=[ b 1 b 2 b N ].
V(b+δb)=(ΔW+ϵ).
Vδb=ϵ,
δb= V T ϵ,
a+δa= α 1 (b+δb).
δa= α 1 δb= α 1 V T ϵ.
δW=Zδa=Z α 1 V T ϵ.
Q= 4 σ AB μ A μ B ( σ A 2 + σ B 2 )( μ A 2 + μ B 2 ) ,

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