Wenhan Sun, Shuai Wang, Xing He, and Bing Xu, "Jacobi circle and annular polynomials: modal wavefront reconstruction from wavefront gradient," J. Opt. Soc. Am. A 35, 1140-1148 (2018)

Jacobi circle polynomials, which are orthogonal on the unit circle with orthogonal radial derivatives, have been developed previously. As the classical Zernike mode can be represented as a linear combination of Jacobi modes, Zernike wavefront modes can be reconstructed using Jacobi modes. Comparison of the Jacobi and Zernike modes for the modal approach indicates that a modal approach incorporating the Gram matrix with the Jacobi modes has potential application in high-sampling wavefront gradient sensors. The Gram matrix method using the Jacobi modes can be extended to annular pupils.

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Coefficients of Zernike Modes Reconstructed by Gram Matrix for Truncation Numbers $J$ of Order $n$, Where Actual Wavefront of Zernike Modes [Normalized as Eq. (1)] Is ${Z}_{1}^{1}+{Z}_{3}^{1}+{Z}_{5}^{1}+{Z}_{7}^{1}+{Z}_{9}^{1}$

$J$ of Order $n$

${Z}_{1}^{1}$

${Z}_{3}^{1}$

${Z}_{5}^{1}$

${Z}_{7}^{1}$

${Z}_{9}^{1}$

${Z}_{11}^{1}$

${Z}_{13}^{1}$

1

15.9312

3

−8.3481

6.8682

5

6.8541

−0.6558

3.9504

7

−4.3646

2.5173

0.2036

2.4047

9

1.0000

1.0000

1.0000

1.0000

1.0000

11

1.0000

1.0000

1.0000

1.0000

1.0000

$-6.5781\times {10}^{-15}$

13

1.0000

1.0000

1.0000

1.0000

1.0000

$-9.5342\times {10}^{-15}$

$2.3437\times {10}^{-15}$

Table 2.

Coefficients of Jacobi Modes Reconstructed by Gram Matrix for $J$ of Order $n$, Where Actual Wavefront of Jacobi Modes Is ${G}_{1}^{1}(2,2;\rho ,\theta )+{G}_{2}^{1}(2,2;\rho ,\theta )+{G}_{3}^{1}(2,2;\rho ,\theta )+{G}_{4}^{1}(2,2;\rho ,\theta )+{G}_{5}^{1}(2,2;\rho ,\theta )+{G}_{6}^{1}(2,2;\rho ,\theta )+{G}_{7}^{1}(2,2;\rho ,\theta )+{G}_{8}^{1}(2,2;\rho ,\theta )+{G}_{9}^{1}(2,2;\rho ,\theta )$^{
a
}

$J$ of Order $n$

${G}_{1}^{1}$

${G}_{2}^{1}$

${G}_{3}^{1}$

${G}_{4}^{1}$

${G}_{5}^{1}$

${G}_{6}^{1}$

${G}_{7}^{1}$

${G}_{8}^{1}$

${G}_{9}^{1}$

${G}_{10}^{1}$

${G}_{11}^{1}$

1

1.0000

2

1.0000

1.0000

3

1.0000

1.0000

1.0000

4

1.0000

1.0000

1.0000

1.0000

5

1.0000

1.0000

1.0000

1.0000

1.0000

6

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

7

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

8

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

9

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

10

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

$1.8679\times {10}^{-5}$

11

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

$1.8679\times {10}^{-5}$

$9.5867\times {10}^{-6}$

The weight function of the radial derivatives of the Jacobi modes is $w(\rho )={\rho}^{2}(1-\rho )$ [17]. If the inner product is taken through numerical integration on the Cartesian coordinate system, considering the Jacobian determinant, the weight function is taken as $w(\rho )=\rho (1-\rho )$.

Table 3.

Coefficients of Jacobi Modes Reconstructed by Gram Matrix for $J$ of Order $n$, Where Actual Wavefront of Zernike Modes Is ${Z}_{1}^{1}+{Z}_{3}^{1}+{Z}_{5}^{1}+{Z}_{7}^{1}+{Z}_{9}^{1}$^{
a
}

$J$ of Order $n$

${G}_{1}^{1}$

${G}_{2}^{1}$

${G}_{3}^{1}$

${G}_{4}^{1}$

${G}_{5}^{1}$

${G}_{6}^{1}$

${G}_{7}^{1}$

${G}_{8}^{1}$

${G}_{9}^{1}$

${G}_{10}^{1}$

${G}_{11}^{1}$

1

−0.7650

2

−0.7650

0.9655

3

−0.7650

0.9655

−0.7045

4

−0.7650

0.9655

−0.7045

0.8463

5

−0.7650

0.9655

−0.7045

0.8463

−1.0869

6

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

7

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

8

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

9

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

10

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

$-9.0010\times {10}^{-6}$

11

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

$-9.0010\times {10}^{-6}$

$9.3693\times {10}^{-6}$

The weight function indices are also taken as ${p}_{1}={q}_{1}=2$.

The $n=0$ orders are neglected, as discussed in Section 5.

Tables (4)

Table 1.

Coefficients of Zernike Modes Reconstructed by Gram Matrix for Truncation Numbers $J$ of Order $n$, Where Actual Wavefront of Zernike Modes [Normalized as Eq. (1)] Is ${Z}_{1}^{1}+{Z}_{3}^{1}+{Z}_{5}^{1}+{Z}_{7}^{1}+{Z}_{9}^{1}$

$J$ of Order $n$

${Z}_{1}^{1}$

${Z}_{3}^{1}$

${Z}_{5}^{1}$

${Z}_{7}^{1}$

${Z}_{9}^{1}$

${Z}_{11}^{1}$

${Z}_{13}^{1}$

1

15.9312

3

−8.3481

6.8682

5

6.8541

−0.6558

3.9504

7

−4.3646

2.5173

0.2036

2.4047

9

1.0000

1.0000

1.0000

1.0000

1.0000

11

1.0000

1.0000

1.0000

1.0000

1.0000

$-6.5781\times {10}^{-15}$

13

1.0000

1.0000

1.0000

1.0000

1.0000

$-9.5342\times {10}^{-15}$

$2.3437\times {10}^{-15}$

Table 2.

Coefficients of Jacobi Modes Reconstructed by Gram Matrix for $J$ of Order $n$, Where Actual Wavefront of Jacobi Modes Is ${G}_{1}^{1}(2,2;\rho ,\theta )+{G}_{2}^{1}(2,2;\rho ,\theta )+{G}_{3}^{1}(2,2;\rho ,\theta )+{G}_{4}^{1}(2,2;\rho ,\theta )+{G}_{5}^{1}(2,2;\rho ,\theta )+{G}_{6}^{1}(2,2;\rho ,\theta )+{G}_{7}^{1}(2,2;\rho ,\theta )+{G}_{8}^{1}(2,2;\rho ,\theta )+{G}_{9}^{1}(2,2;\rho ,\theta )$^{
a
}

$J$ of Order $n$

${G}_{1}^{1}$

${G}_{2}^{1}$

${G}_{3}^{1}$

${G}_{4}^{1}$

${G}_{5}^{1}$

${G}_{6}^{1}$

${G}_{7}^{1}$

${G}_{8}^{1}$

${G}_{9}^{1}$

${G}_{10}^{1}$

${G}_{11}^{1}$

1

1.0000

2

1.0000

1.0000

3

1.0000

1.0000

1.0000

4

1.0000

1.0000

1.0000

1.0000

5

1.0000

1.0000

1.0000

1.0000

1.0000

6

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

7

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

8

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

9

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

10

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

$1.8679\times {10}^{-5}$

11

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

$1.8679\times {10}^{-5}$

$9.5867\times {10}^{-6}$

The weight function of the radial derivatives of the Jacobi modes is $w(\rho )={\rho}^{2}(1-\rho )$ [17]. If the inner product is taken through numerical integration on the Cartesian coordinate system, considering the Jacobian determinant, the weight function is taken as $w(\rho )=\rho (1-\rho )$.

Table 3.

Coefficients of Jacobi Modes Reconstructed by Gram Matrix for $J$ of Order $n$, Where Actual Wavefront of Zernike Modes Is ${Z}_{1}^{1}+{Z}_{3}^{1}+{Z}_{5}^{1}+{Z}_{7}^{1}+{Z}_{9}^{1}$^{
a
}

$J$ of Order $n$

${G}_{1}^{1}$

${G}_{2}^{1}$

${G}_{3}^{1}$

${G}_{4}^{1}$

${G}_{5}^{1}$

${G}_{6}^{1}$

${G}_{7}^{1}$

${G}_{8}^{1}$

${G}_{9}^{1}$

${G}_{10}^{1}$

${G}_{11}^{1}$

1

−0.7650

2

−0.7650

0.9655

3

−0.7650

0.9655

−0.7045

4

−0.7650

0.9655

−0.7045

0.8463

5

−0.7650

0.9655

−0.7045

0.8463

−1.0869

6

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

7

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

8

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

9

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

10

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

$-9.0010\times {10}^{-6}$

11

−0.7650

0.9655

−0.7045

0.8463

−1.0869

0.6542

−0.1886

0.0259

−0.0014

$-9.0010\times {10}^{-6}$

$9.3693\times {10}^{-6}$

The weight function indices are also taken as ${p}_{1}={q}_{1}=2$.