Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes

Chao Tian; Xuefeng Chen; Shengchun Liu

doi:10.1364/OE.24.003572

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 W_c and the expanded wavefront W_e 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.

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 W_c and make it decentered with respect to the expanded beam W_e. Define the coordinate system xOy and assume that the radii of W_c and W_e are 1 and r₀ (r₀ = 1/β > 1), respectively. The differential wavefront ∆W measured in experiment can be written as

Δ W (x, y) = W_{c} (x, y) - W_{e} (x, y) = W (x, y) - W (β x + x_{0}, β y + y_{0}),

where −1 ≤ x, y ≤ 1, W_c(x, y) = W(x, y), W is the original wavefront. The expanded wavefront in the overlapping area W_e(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) = \sum_{j = 1}^{N} a_{j} Z_{j} (x, y),

where N is the total terms of the Zernike polynomials, a_j is the expansion coefficient of the jth term, and Z_j is the orthonormal Zernike polynomial [1] of the jth term, which is known and defined as

Z_{j} = {\begin{cases} \sqrt{(n + 1)} R_{n}^{m} (ρ), m = 0, \\ \sqrt{2 (n + 1)} R_{n}^{m} (ρ) \cos m θ, m \neq 0 and even j, \\ \sqrt{2 (n + 1)} R_{n}^{m} (ρ) \sin m θ, m \neq 0 and odd j, \end{cases}

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

R_{n}^{m} (ρ) = \sum_{s = 0}^{(n - m) / 2} \frac{{(- 1)}^{s} (n - s)!}{s! [(n + m) / 2 - s]! [(n - m) / 2 - s]!} ρ^{n - 2 s},

where n and m are non-negative integers, which mean the radial degree and the azimuthal frequency, respectively, and satisfy n – m = even.

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

Δ W (x, y) = \sum_{j = 1}^{N} a_{j} Z_{j} (x, y) - \sum_{j = 1}^{N} a_{j} Z_{j} (β x + x_{0}, β y + y_{0}) = \sum_{j = 1}^{N} a_{j} U_{j} (x, y),

where

U_{j} (x, y) = Z_{j} (x, y) - Z_{j} (β x + x_{0}, β y + y_{0}),

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 U_j form a new set of basis functions and can be used to represent the differential wavefront ∆W. But the new basis functions U_j are not orthogonal as their mother functions Z_j.

Written in discrete and matrix forms, Eq. (5) becomes

U a = Δ W,

where

U = [\begin{matrix} U_{1} (x_{1}, y_{1}) & U_{2} (x_{1}, y_{1}) & \dots & U_{N} (x_{1}, y_{1}) \\ U_{1} (x_{2}, y_{2}) & U_{2} (x_{2}, y_{2}) & \dots & U_{N} (x_{2}, y_{2}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ U_{1} (x_{M}, y_{M}) & U_{2} (x_{M}, y_{M}) & \dots & U_{N} (x_{M}, y_{M}) \end{matrix}],

a = [\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{N} \end{matrix}], Δ W = [\begin{matrix} Δ W (x_{1}, y_{1}) \\ Δ W (x_{2}, y_{2}) \\ ⋮ \\ Δ W (x_{M}, y_{M}) \end{matrix}],

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,

U^{T} U a = U^{T} Δ W .

where T means matrix transpose. The solution vector can be obtained by matrix inverse, i.e.

a = {(U^{T} U)}^{- 1} U^{T} Δ W .

If the new basis functions U_j are orthogonal on predefined discrete data sets, the coefficient matrix U^TU is a diagonal matrix and the least-squares solution is accurate and stable. However, as we already mentioned beforehand, U_j 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, U_j 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 + δ a = {(U^{T} U)}^{- 1} U^{T} (Δ W + ϵ) .

Combining Eqs. (11) and (12), we have

δ a = {(U^{T} U)}^{- 1} U^{T} ϵ = (S Λ^{- 1} S^{T}) U^{T} ϵ,

where (U^TU)⁻¹ = SΛ⁻¹S^T, S is an eigenvector matrix and Λ is an eigenvalue matrix. Assuming that λ₁, λ₂, λ₃, …, λ_N are eigenvalues of the matrix (U^TU)⁻¹ and are arranged in an decreasing order λ₁ ≥ λ₂ ≥ λ₃ ≥ … ≥ λ_N, the induced error of the solution can be approximately estimated as

Δ a_{j} ≅ \sum_{i = 1}^{M} U_{j i} ε_{i} / λ_{j},

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 V_j is orthogonal on the discrete data set and satisfies

\sum_{i = 1}^{M} V_{p} (x_{i}, y_{i}) V_{q} (x_{i}, y_{i}) = {\begin{cases} 0, if p \neq q, \\ 1, if p = q, \end{cases}

The wavefront ∆W can be decomposed into a linear combination of the orthogonal basis V_k

Δ W (x_{i}, y_{i}) = \sum_{k = 1}^{N} b_{k} V_{k} (x_{i}, y_{i}),

where b_k is the expansion coefficient of the kth term. Following the Gram-Schmidt orthogonalization procedure, each U_j can be expanded in terms of V_j up to the term j, i.e.

U_{j} (x_{i}, y_{i}) = \sum_{k = 1}^{j} α_{k j} V_{k} (x_{i}, y_{i}) .

The orthogonal basis V_j can also be represented by U_j through the following expression

V_{j} (x_{i}, y_{i}) = \frac{1}{α_{j j}} [U_{j} (x_{i}, y_{i}) - \sum_{k = 1}^{j - 1} α_{k j} V_{k} (x_{i}, y_{i})],

where α is the conversion coefficient, and can be derived as

α_{k j} = {\begin{cases} \sum_{i = 1}^{M} [U_{j} (x_{i}, y_{i}) V_{k} (x_{i}, y_{i})], k < j, \\ {\sum_{i = 1}^{M} {[U_{j} (x_{i}, y_{i})]}^{2} - \sum_{k = 1}^{j - 1} α_{k j}^{2}}^{1 / 2}, k = j . \end{cases}

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

Δ W (x_{i}, y_{i}) = \sum_{j = 1}^{N} a_{j} U_{j} (x_{i}, y_{i}) = \sum_{j = 1}^{N} a_{j} [\sum_{k = 1}^{j} α_{k j} V_{k} (x_{i}, y_{i})] = \sum_{k = 1}^{N} \sum_{j = k}^{N} a_{j} α_{k j} V_{k} (x_{i}, y_{i}) .

Comparing the above equation with Eq. (16), the relations between b_k and a_j can be expressed as

b_{k} = \sum_{j = k}^{N} a_{j} α_{k j} .

Written in matrix form, it becomes

[\begin{matrix} α_{11} & α_{12} & α_{13} & \dots & α_{1 N} \\ 0 & α_{22} & α_{23} & \dots & α_{2 N} \\ 0 & 0 & α_{33} & \dots & α_{3 N} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & 0 & α_{N N} \end{matrix}] [\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ ⋮ \\ a_{N} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ ⋮ \\ b_{N} \end{matrix}],

or in the following form

α a = b .

The expansion coefficient a can be found using matrix inverse as

a = α^{- 1} b,

where the conversion coefficient α can be recursively calculated through Eqs. (17)-(19), and the expansion coefficient b can be computed by multiplying V_k to Eq. (16) and utilizing its orthogonality property [Eq. (15)], i.e.

b_{k} = \sum_{i = 1}^{M} [Δ W (x_{i}, y_{i}) V_{k} (x_{i}, y_{i})],

where ∆W is the differential wavefront measured in experiment and V_k 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(x_i, y_i), V_j = V_j(x_i, y_i), U_j = U_j(x_i, y_i). In this way, ∆W can be written as [Eq. (16)],

Δ W = b_{1} V_{1} + b_{2} V_{2} + b_{3} V_{3} .

The first orthogonal basis V₁ is generated as [Eqs. (17) - (19)]

\begin{array}{l} U_{1} = α_{11} V_{1}, \\ α_{11} = {(\sum_{i = 1}^{M} U_{1}^{2})}^{1 / 2}, \\ V_{1} = U_{1} / α_{11} . \end{array}

The second orthogonal basis V₂ is generated as

\begin{array}{l} U_{2} = α_{12} V_{1} + α_{22} V_{2}, \\ α_{12} = \sum_{i = 1}^{M} (U_{2} V_{1}), \\ α_{22} = {(\sum_{i = 1}^{M} U_{2}^{2} - α_{12}^{2})}^{1 / 2}, \\ V_{2} = (U_{2} - α_{12} V_{1}) / α_{22} . \end{array}

The third orthogonal basis V₃ is generated as

\begin{array}{l} U_{3} = α_{13} V_{1} + α_{23} V_{2} + α_{33} V_{3}, \\ α_{13} = \sum_{i = 1}^{M} (U_{3} V_{1}), \\ α_{23} = \sum_{i = 1}^{M} (U_{3} V_{2}), \\ α_{33} = {[\sum_{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} . \end{array}

The wavefront ∆W can also be written as [Eqs. (20), (27)-(29)]

\begin{matrix} Δ W = 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} . \end{matrix}

Comparing it with Eq. (26), we have

[\begin{matrix} α_{11} & α_{12} & α_{13} \\ 0 & α_{22} & α_{23} \\ 0 & 0 & α_{33} \end{matrix}] [\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}] .

The expansion coefficients

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},

where α has already been calculated in Eqs. (27) - (29), and b is computed as

b_{3} = \sum_{i = 1}^{M} (Δ W \cdot V_{3}), b_{2} = \sum_{i = 1}^{M} (Δ W \cdot V_{2}), b_{1} = \sum_{i = 1}^{M} (Δ W \cdot V_{1}),

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

V b = Δ W,

where the orthogonal matrix

V = [\begin{matrix} V_{1} (x_{1}, y_{1}) & V_{2} (x_{1}, y_{1}) & \dots & V_{N} (x_{1}, y_{1}) \\ V_{1} (x_{2}, y_{2}) & V_{2} (x_{2}, y_{2}) & \dots & V_{N} (x_{2}, y_{2}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ V_{1} (x_{M}, y_{M}) & V_{2} (x_{M}, y_{M}) & \dots & V_{N} (x_{M}, y_{M}) \end{matrix}], b = [\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{N} \end{matrix}] .

Considering the perturbation ϵ, Eq. (34) becomes

V (b + δ b) = (Δ W + ϵ) .

where δb is the error of b caused by ϵ. Combing Eqs. (36) and (34), we get

V \cdot δ b = ϵ,

and, therefore,

δ b = V^{T} ϵ,

using the relationship V^TV = 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 + δ a = α^{- 1} (b + δ b) .

Further, we have

δ a = α^{- 1} \cdot δ b = α^{- 1} V^{T} ϵ .

According to Eq. (2), the resultant wavefront reconstruction error δW can be correspondingly written as

δ W = Z \cdot δ a = Z α^{- 1} V^{T} ϵ .

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 U^TU, i.e. κ(a) and κ(U^TU), respectively. The matrix α is an upper triangular matrix and usually has a much smaller condition number than the matrix U^TU. 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.

Fig. 2 Flow chart of the least-squares method (a) and the Gram-Schmidt orthogonalization method (b). Z_j 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 W_c, the expanded wavefront W_e and its subaperture that overlaps W_c, 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.

Fig. 3 Generation of a differential wavefront with an elliptic aperture. (a) Original wavefront W, contracted wavefront W_c, expanded wavefront W_e 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 a_j, 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(U^TU) = 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 b_j of the orthogonal bases V_j and a_j of U_j are shown in Fig. 4(d). As we can see, only the first few terms of b_j have significant values and most higher-order terms are close to zero due to the orthogonality of V_j. Also the expansion coefficients b_j are independent from each other and using a different fitting order n will not affect the values of b_j. 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.

Fig. 4 Reconstruction results by the least-squares method (1st row) and the proposed method (2nd row). (a) - (c) Computed coefficients a_j, reconstructed wavefront and residual error by least squares; (d) - (f) computed coefficients a_j and b_j, 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 b_j and a_j, 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 = \frac{4 σ_{AB} \cdot μ_{A} μ_{B}}{(σ_{A}^{2} + σ_{B}^{2}) (μ_{A}^{2} + μ_{B}^{2})},

where A and B represent the results of the proposed method and the iterative method, respectively; μ_A and μ_B are the mean values; 𝜎_A and 𝜎_B are the standard deviations; and 𝜎_AB 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.

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 a_j and b_j, (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.

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 V_j using the Gram-Schmidt orthogonalization, and computes the independent expansion coefficients b_j, which are converted into expansion coefficients a_j of the non-orthogonal basis functions U_j 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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Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes

Abstract

1. Introduction

2. Modal wavefront reconstruction

2.1 Mathematical model and its least-squares solution

2.2 Gram-Schmidt orthogonalization for general apertures

A. General process

B. A simple example with only the first three terms

C. Error propagation and analysis

3. Computer simulation and experimental results

3.1 Comparisons between the least-squares method and the Gram-Schmidt method

3.2 Experimental results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (42)

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