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.

R. S. Biesheuvel, A. J. E. M. Janssen, P. Pozzi, and S. F. Pereira OSA Continuum 1(2) 581-603 (2018)

References

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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 ${p}_{1}={q}_{1}=2$.