T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1932-1941 (1995)

A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.

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Space Z_{5} is spanned by all polynomials from this table, subspace DZ_{5} is spanned by all polynomials that are above the diagonal dotted line, subspace UZ_{5} is spanned by all polynomials that are below the diagonal dotted line, and subspace Z_{3} ≡ Z_{5–2} is spanned by all polynomials that are above the horizontal dotted line. Note that
$c\equiv 1/\sqrt{\pi}$.

The rules separate the blocks A_{pq}. The indexing system is described in the text [formulas (49) and (52)–(56)].

Table 3

Phase Retrieval Matrix
${\tilde{A}}_{(4)}^{-1}$ for R = 1^{a}

i

j

1

2

3

4

5

6

7

8

1

2

3

4

5

6

2

0

1/2

0

0

0

0

0

0

0

7/24

0

3

1/2

0

0

0

7/24

0

5

0

0

$1/(2\sqrt{6})$

0

0

0

0

0

0

0

0

0

5/48

0

6

0

0

0

$1/(2\sqrt{6})$

0 p

0

0

0

0

0

0

0

0

5/48

9

0

0

0

0

$1/(6\sqrt{2})$

0

0

0

0

0

0

0

0

0

10

0

0

0

0

0

$1/(6\sqrt{2})$

0

0

0

0

0

0

0

0

14

0

0

0

0

0

0

0

$1/(4\sqrt{10})$

0

0

0

0

0

0

15

0

0

0

0

0

0

$1/(4\sqrt{10})$

0

0

0

0

0

0

0

4

0

0

0

0

0

0

0

0

$-1/(8\sqrt{3})$

0

0

1/16

0

0

7

0

0

0

0

0

0

0

0

0

0

$-1/(24\sqrt{2})$

0

0

0

8

0

0

0

0

0

0

0

0

0

$-1/(24\sqrt{2})$

0

0

0

0

11

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

0

0

12

0

0

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

13

0

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

0

The rules separate the blocks
${A}_{pq}^{-1}$ he first 14 nonconstant Zernike aberrations (j = 2, 3, …, 15) of a phase can be retrieved by the application of this matrix to the vector {ψ_{(4)}, f_{(2)}} of the Fourier coefficients of the boundary slope and the Zernike coefficients of the curvature of the wave front. The indexing system is described in the text [formulas (57)–(60)].

Space Z_{5} is spanned by all polynomials from this table, subspace DZ_{5} is spanned by all polynomials that are above the diagonal dotted line, subspace UZ_{5} is spanned by all polynomials that are below the diagonal dotted line, and subspace Z_{3} ≡ Z_{5–2} is spanned by all polynomials that are above the horizontal dotted line. Note that
$c\equiv 1/\sqrt{\pi}$.

The rules separate the blocks A_{pq}. The indexing system is described in the text [formulas (49) and (52)–(56)].

Table 3

Phase Retrieval Matrix
${\tilde{A}}_{(4)}^{-1}$ for R = 1^{a}

i

j

1

2

3

4

5

6

7

8

1

2

3

4

5

6

2

0

1/2

0

0

0

0

0

0

0

7/24

0

3

1/2

0

0

0

7/24

0

5

0

0

$1/(2\sqrt{6})$

0

0

0

0

0

0

0

0

0

5/48

0

6

0

0

0

$1/(2\sqrt{6})$

0 p

0

0

0

0

0

0

0

0

5/48

9

0

0

0

0

$1/(6\sqrt{2})$

0

0

0

0

0

0

0

0

0

10

0

0

0

0

0

$1/(6\sqrt{2})$

0

0

0

0

0

0

0

0

14

0

0

0

0

0

0

0

$1/(4\sqrt{10})$

0

0

0

0

0

0

15

0

0

0

0

0

0

$1/(4\sqrt{10})$

0

0

0

0

0

0

0

4

0

0

0

0

0

0

0

0

$-1/(8\sqrt{3})$

0

0

1/16

0

0

7

0

0

0

0

0

0

0

0

0

0

$-1/(24\sqrt{2})$

0

0

0

8

0

0

0

0

0

0

0

0

0

$-1/(24\sqrt{2})$

0

0

0

0

11

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

0

0

12

0

0

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

13

0

0

0

0

0

0

0

0

0

0

0

0

$-1/(16\sqrt{15})$

0

The rules separate the blocks
${A}_{pq}^{-1}$ he first 14 nonconstant Zernike aberrations (j = 2, 3, …, 15) of a phase can be retrieved by the application of this matrix to the vector {ψ_{(4)}, f_{(2)}} of the Fourier coefficients of the boundary slope and the Zernike coefficients of the curvature of the wave front. The indexing system is described in the text [formulas (57)–(60)].