Abstract

In a previous paper [ J. Opt. Soc. Am. A 12, 1932 ( 1995)] we presented a method for phase recovery with the transport-of-intensity equation by use of a series expansion. Here we develop a different method for the solution of this equation, which allows recovery of the phase in the case of nonuniform illumination. Though also based on the orthogonal series expansion, the new method does not require any separate boundary conditions and can be more easily adjusted for apertures of various shapes. The discussion is primarily for the case of a circular aperture and Zernike polynomials, but we also outline the solution for a rectangular aperture and Fourier harmonics. The latter example may have some substantial advantages, given the availability of the fast Fourier transform.

© 1996 Optical Society of America

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References

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  1. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 9.10.
  2. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  3. T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
    [CrossRef]
  4. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).
  5. J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
    [CrossRef]
  6. A. W. Dreher, J. F. Bille, R. N. Weinreb, “Active optical depth resolution improvement of the laser tomographic scanner,” Appl. Opt. 28, 804–808 (1989).
    [CrossRef] [PubMed]
  7. M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
    [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,”J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,”J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  10. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  11. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  12. C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
    [CrossRef]
  13. T. E. Gureyev, A. Roberts, 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).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 9.2.
  15. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  17. P. Hickson, “Wave-front curvature sensing from a single defocused image,” J. Opt. Soc. Am. A 11, 1667–1673 (1994).
    [CrossRef]
  18. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  19. O. A. Oleinik, E. V. Radkevič, Second Order Equations with Non-negative Characteristic Form (Plenum, New York, 1973).
    [CrossRef]
  20. L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).
  21. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  22. P. C. Sabatier, ed., Inverse Methods in Action (Springer, Berlin, 1990).
    [CrossRef]

1995 (5)

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, 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).
[CrossRef]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

1994 (1)

1993 (1)

1990 (2)

1989 (1)

1988 (1)

1983 (1)

1982 (1)

1976 (1)

1974 (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Baker, R.

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Bille, J. F.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 9.2.

Dreher, A. W.

Gureyev, T. E.

Hickson, P.

Kantorovich, L. V.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Klibanov, M. V.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

Krumrey, M.

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Krylov, V. I.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Kvick, Å.

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Millane, R. P.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 9.10.

Noll, R. J.

Nugent, K. A.

Oleinik, O. A.

O. A. Oleinik, E. V. Radkevič, Second Order Equations with Non-negative Characteristic Form (Plenum, New York, 1973).
[CrossRef]

Radkevic, E. V.

O. A. Oleinik, E. V. Radkevič, Second Order Equations with Non-negative Characteristic Form (Plenum, New York, 1973).
[CrossRef]

Roberts, A.

Roddier, C.

Roddier, F.

Sacks, P. E.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

Schwegle, W.

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Stamnes, J. J.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Susini, J.

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Teague, M. R.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Tikhonravov, A. V.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

Wedberg, T. C.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Weinreb, R. N.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 9.2.

Appl. Opt. (3)

Inverse Probl. (1)

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Pure Appl. Opt. (1)

T. C. Wedberg, J. J. Stamnes, “Comparison of phase retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Rev. Sci. Instrum. (1)

J. Susini, R. Baker, M. Krumrey, W. Schwegle, Å. Kvick, “Adaptive x-ray mirror prototype: first results,” Rev. Sci. Instrum. 66, 2048–2052 (1995).
[CrossRef]

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (7)

O. A. Oleinik, E. V. Radkevič, Second Order Equations with Non-negative Characteristic Form (Plenum, New York, 1973).
[CrossRef]

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

P. C. Sabatier, ed., Inverse Methods in Action (Springer, Berlin, 1990).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sec. 9.10.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 9.2.

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Figures (6)

Fig. 1
Fig. 1

Phase retrieval can be performed by using the two intensity measurements I(x, y, 0) and I(x, y, δz) on adjacent planes z = 0 and z = δz orthogonal to the optical axis z.

Fig. 2
Fig. 2

Original distribution of phase in the plane z = 0 with the Zernike coefficients from the second column of Table 5.

Fig. 3
Fig. 3

Reconstructed phase in the plane z = 0 with the Zernike coefficients from the fifth column of Table 5.

Fig. 4
Fig. 4

Difference between the original phase from Fig. 2 and the reconstructed phase from Fig. 3.

Fig. 5
Fig. 5

Original phase distribution φ(r, θ, z = 0) used in the second example.

Fig. 6
Fig. 6

Projection of the original phase distribution φ(r, θ, z = 0) used in the second example onto the linear space spanned by the first 21 Zernike polynomials.

Tables (6)

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Table 1 Matrix M(21) in the Case of Uniform Intensity Distribution

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Table 2 Matrix M ( 21 ) 1 Which is the Inverse of the Matrix M(21) in Table 1

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Table 3 Matrix M ( 21 ) 1 ( I v a r )

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Table 4 Eigenvalues of Matrix M ( 21 ) 1 ( I v a r ) from Table 3

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Table 5 Original and Retrieved Zernike Coefficients of the Phase Obtained by the Method Developed in the Present Paper

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Table 6 Original and Retrieved Zernike Coefficients of the Phase from the Second Example

Equations (44)

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( 2 i k z + Δ ) u ( x , y , z ) = 0 ,
2 k z φ = | φ | 2 + D ( I ) ,
k z I = · ( I φ ) ,
I ( x , y ) > 0 inside Ω ,
I ( x , y ) 0 outside Ω and on Γ .
· ( I φ ˜ ) = · ( I φ 2 ) · ( I φ 1 ) = k z I k z I = 0 , 0 = Ω φ ˜ · ( I φ ˜ ) d x d y = Ω I | φ ˜ | 2 d x d y Γ I φ ˜ n φ ˜ d s .
Ω Z I d x d y = 0.
Ω k z I d x d y = Ω E k z I d x d y = Ω E · ( I φ ) d x d y = Ω I φ · E d x d y = 0.
f , g = R 2 0 2 π 0 R f ( r , θ ) g ( r , θ ) r d r d θ .
R 2 0 2 π 0 R · ( I φ ) Z j r d r d θ = R 2 0 2 π 0 R F Z j r d r d θ ,
φ ( r , θ ) = i = 1 φ i Z i ( r / R , θ )
i = 1 φ i R 2 0 2 π 0 R I Z i · Z j r d r d θ = F j .
M i j = 0 2 π 0 R I ( r , θ ) Z i ( r / R , θ ) · Z j ( r / R , θ ) r d r d θ , i , j = 1,2,3 , .
i = 1 M i j φ i = R 2 F j , j = 1,2,3 , , or M φ = R 2 F .
g ( N ) ( r , θ ) = j = 2 N g j Z j ( r / R , θ ) .
i = 2 N M i j φ i = R 2 F j , j = 2 , , N , or M ( N ) φ ( N ) = R 2 F ( N ) .
j = 2 N i = 2 N M i j φ i φ j = j = 2 N i = 2 N φ i φ j Ω I Z i · Z j r d r d θ = Ω I | φ ( N ) | 2 r d r d θ .
φ i = R 2 j = 2 N M i j 1 F j , i = 2 , , N , or φ ( N ) = R 2 M ( N ) 1 F ( N ) .
W m n ( x , y ) = exp ( i 2 π m x / a ) exp ( i 2 π n y / b ) .
f , g a b = 1 a b 0 b 0 a f ( x , y ) g * ( x , y ) d x d y ,
i j φ i j M m n i j = a b F m n ,
M m n i j ( 2 π ) 2 ( i m b / a + j n a / b ) I ˆ m i , n j ,
i , m = 0 , ± 1 , , ± M , j , n = 0 , ± 1 , , ± N , m 2 + n 2 > 0 , i 2 + j 2 > 0
φ i j = a b m , n [ M m n i j ] 1 F m n ,
M m n i j = ( 2 π ) 2 I 0 ( m 2 b / a + n 2 a / b ) δ m i δ n j ,
φ m n = ( a b ) 2 ( 2 π ) 2 ( m 2 b 2 + n 2 a 2 ) I 0 F m n ,
N F = 2 π R 2 λ δ z .
R 2 F j = N F R 2 0 2 π 0 R [ I ( r , θ , δ z ) I ( r , θ , 0 ) ] Z j r d r d θ .
I ( δ z ) I ( 0 ) z I ( 0 ) δ z + z 2 I ( 0 ) ( δ z ) 2 / 2 + σ ( I ) ,
φ ( N ) = N F M ( N ) 1 δ z I ( N ) ,
Δ 2 = ( j = 2 21 | φ j orig . φ j reconst . | 2 / j = 2 21 | φ j orig . | 2 ) 1 / 2
I ( r , θ ) = I 0 H ( R r ) , I 0 = const ., H ( t ) = { 1 , t > 0 0 , t < 0 .
I n ( r , θ ) = I 0 H n ( R r ) , I 0 = const .
k I 0 1 z I n = H n ( R r ) Δ φ + δ n ( R r ) r φ ( R , θ ) ,
R n 2 0 2 π 0 R n H n ( R r ) Δ φ Z j r d r d θ + R 0 2 0 2 π 0 R n δ n ( R r ) r φ ( R , θ ) Z j r d r d θ = F j n ,
F j n = k I 0 1 R n 2 0 2 π 0 R n z I n Z j r d r d θ .
R n 1 0 2 π H n ( R R n ) r φ Z j d θ + R n 2 × 0 2 π 0 R n H n ( R r ) φ · Z j r d r d θ = F j n .
i = 2 N M i j n φ i = R n 2 F j n , j = 2 , , N ,
M i j n = 0 2 π 0 R n H n ( R r ) Z i ( r / R , θ ) · Z j ( r / R , θ ) r d r d θ , i , j = 2 , , N .
M i j = 0 2 π 0 R Z i ( r / R , θ ) · Z j ( r / R , θ ) r d r d θ ,
i = 2 N M i j φ i = R 2 F j , j = 2 , , N ,
F j = k I 0 1 R 2 0 2 π 0 R Z I ( r , θ ) Z j r d r d θ .
φ i = R 2 j = 2 N [ M i j ] 1 F j , i = 2 , , N , or φ ( N ) = R 2 M ( N ) 1 F ( N ) .
k I 0 1 z I = f ( r , θ ) + δ ( R r ) ψ ( θ ) ,

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