Abstract

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.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  2. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Sect. 9.10.
  3. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  4. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  5. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  6. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  7. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  8. J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  9. J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
    [CrossRef]
  10. 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]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sect. 9.2.
  12. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  14. W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
    [CrossRef]
  15. J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Applied Opt. 19, 1510–1518 (1980).
    [CrossRef]
  16. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes 17, S21–S24 (1994).
  17. C. Schwartz, E. Ribak, S. G. Lipson, “Bimorph adaptive mirrors and curvature sensing,” J. Opt. Soc. Am. A 11, 895–902 (1994).
    [CrossRef]
  18. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981); J. Opt. Soc. Am. 71, 1408 (1981); J. Opt. Soc. Am. A 1, 685 (1984).
    [CrossRef]
  19. W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832–1837 (1994).
    [CrossRef] [PubMed]
  20. P. Hickson, “Wave-front curvature sensing from a single defocused image,” J. Opt. Soc. Am. A 11, 1667–1673 (1994).
    [CrossRef]
  21. S. G. Mikhlin, Mathematical Physics, An Advanced Course (North-Holland, Amsterdam, 1970).
  22. 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]
  23. F. Gori, M. Santasiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  24. S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Value Problems (Akademie-Verlag, Berlin, 1982).

1995 (1)

1994 (5)

1993 (2)

1992 (1)

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

1990 (2)

1988 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1982 (1)

1981 (1)

1980 (1)

J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Applied Opt. 19, 1510–1518 (1980).
[CrossRef]

1977 (1)

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

1976 (1)

1974 (1)

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

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), Sect. 9.2.

Chow, W. W.

Goelz, S.

Gori, F.

Grimm, B.

Guattari, G.

Gureyev, T. E.

Hickson, P.

Jagourel, P.

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

Liang, J.

Lipson, S. G.

Mahajan, V. N.

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes 17, S21–S24 (1994).

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981); J. Opt. Soc. Am. 71, 1408 (1981); J. Opt. Soc. Am. A 1, 685 (1984).
[CrossRef]

Marot, G.

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

Mikhlin, S. G.

S. G. Mikhlin, Mathematical Physics, An Advanced Course (North-Holland, Amsterdam, 1970).

Millane, R. P.

Nieto-Vesperinas, M.

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

Noll, R. J.

Nugent, K. A.

Ravelet, R.

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

Rempel, S.

S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Value Problems (Akademie-Verlag, Berlin, 1982).

Ribak, E.

Roberts, A.

Roddier, C.

Roddier, F.

Santasiero, M.

Schulze, B.-W.

S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Value Problems (Akademie-Verlag, Berlin, 1982).

Schwartz, C.

Silva, D. E.

J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Applied Opt. 19, 1510–1518 (1980).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Susini, J.

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

Swantner, W.

Tango, W. J.

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

Teague, M. R.

Wang, J. Y.

J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Applied Opt. 19, 1510–1518 (1980).
[CrossRef]

Wolf, E.

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

Zhang, L.

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

W. J. Tango, “The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).
[CrossRef]

Applied Opt. (1)

J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Applied Opt. 19, 1510–1518 (1980).
[CrossRef]

Eng. Lab. Notes (1)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Eng. Lab. Notes 17, S21–S24 (1994).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Rev. Sci. Instrum. (1)

J. Susini, G. Marot, L. Zhang, R. Ravelet, P. Jagourel, “Conceptual design of an adaptive x-ray mirror prototype for the ESRF,” Rev. Sci. Instrum. 63, 489–492 (1992).
[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 (4)

S. G. Mikhlin, Mathematical Physics, An Advanced Course (North-Holland, Amsterdam, 1970).

S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Value Problems (Akademie-Verlag, Berlin, 1982).

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (3)

Tables Icon

Table 1 Spaces of Zernike Polynomialsa

Tables Icon

Table 2 Matrix Ã(4) for R = 1a

Tables Icon

Table 3 Phase Retrieval Matrix A ˜ ( 4 ) - 1 for R = 1a

Equations (68)

Equations on this page are rendered with MathJax. Learn more.

exp ( i k z ) u ( r ) = I 1 / 2 ( r ) exp [ i k z + i ϕ ( r ) ] ,
( 2 i k z + Δ ) u ( x , y , z ) = 0 ,
2 k z ϕ = - ϕ 2 + D ( I ) ,
k z I = - I · ϕ - I Δ ϕ ,
I ( r , θ ) = I 0 H ( R - r ) ,             I 0 = constant , H ( t ) = { 1 t > 0 0 t 0 .
- H ( R - r ) Δ ϕ ( r , θ ) + δ ( R - r ) r ϕ ( R , θ ) = k I 0 - 1 z I ( r , θ ) ,
k I 0 - 1 z I ( r , θ ) = f ( r , θ ) + δ ( R - r ) ψ ( θ ) ,
- Δ ϕ = f
r ϕ = ψ
Ω f ( r , θ ) r d r d θ f ( r , θ ) r d r d θ + Γ ψ ( θ ) R d θ = 0.
k 0 2 π 0 R z I ( r , θ ) r d r d θ = - I 0 R 0 2 π r ϕ ( R , θ ) d θ ,
ψ ( θ ) - k z I ( R , θ ) r I ( R , θ ) .
f ( r , θ ) k I 0 - 1 z I ( r , θ ) ,             r < R .
Z j ( r , θ ) = { c n m R n m ( r ) cos ( m θ ) j even , m 0 c n m R n m ( r ) sin ( m θ ) j odd , m 0 c n 0 R n 0 ( r ) m = 0 ,
c n m = [ ( 2 - δ m 0 ) ( n + 1 ) / π ] 1 / 2
R n m ( r ) = s = 0 ( n - m ) / 2 γ n , m s r n - 2 s , γ n , m s = ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] !
j = j ( m , n ) = n ( n + 1 ) 2 + m ,
j N = j ( N , N ) + 1 = ( N + 1 ) ( N + 2 ) / 2.
Z i , Z j = 0 2 π 0 1 Z i ( r , θ ) Z j ( r , θ ) r d r d θ = δ i j .
Z N = { j J N a j Z j ,             a j real numbers }
D Z N = { a j Z j :             Z j Z N , m = n }
dim ( Z N ) - dim ( D Z N ) = dim ( Z N - 2 ) ,
R ˜ n m ( r ) = s = 0 ( n - m ) / 2 γ ˜ n , m s r n - 2 s - 2 , γ ˜ m , n s = γ m , n s [ m 2 - ( n - 2 s ) 2 ] .
Ker [ ( - Δ ) N ] = D Z N .
Im [ ( - Δ ) N ] = Z N - 2 .
dim { Im [ ( - Δ ) N ] } = dim ( Z N ) - dim { Ker [ ( - Δ ) N ] } = dim ( Z N ) - dim ( D Z N ) = dim ( Z N - 2 ) ,
ϕ = ϕ ( 0 ) + ϕ ( 1 ) ,             ϕ ( 0 ) D Z N , ϕ ( 1 ) U Z N .
( - Δ ) N Z N = [ 0 0 0 - Δ ] ( D Z N U Z N ) = ( 0 Z N - 2 ) ,             or ( - Δ ) N ( ϕ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = ( 0 f ( N - 2 ) ) ,
ϕ ( N ) = ϕ ( N ) ( 0 ) + ϕ ( N ) ( 1 ) ,             ϕ ( N ) ( 0 ) = j D J N ϕ j Z j , ϕ ( N ) ( 1 ) = j U J N ϕ j Z j ,
f ( N - 2 ) ( R ρ , θ ) = i J N - 2 f i Z i ( ρ , θ ) ,
i J N - 2 f i Z i = f ( N - 2 ) = ( - Δ ) N ϕ ( N ) ( 1 ) = ( - Δ ) N j U J N ϕ j Z j = R - 2 i J N - 2 j U J N ϕ j Λ i j Z i .
j U J N Λ i j ϕ j = R 2 f i ,             i J N - 2 .
ϕ j = R 2 i J N - 2 Λ j i - 1 f i ,             j U J N ,
F N = { η ( θ ) :             η ( θ ) = η 0 + m = 1 N [ η m sin ( m θ ) + η m cos ( m θ ) ] } .
ψ ( N ) ( θ ) = ψ 0 + m = 1 N [ ψ m sin ( m θ ) + ψ m cos ( m θ ) ] .
r ϕ ( N ) ( R , θ ) = ψ ( N ) ( θ ) .
ψ ˜ ( N ) ( θ ) = ψ ˜ 0 + m = 1 N [ ψ ˜ m sin ( m θ ) + ψ ˜ m cos ( m θ ) ] .
ϕ ( N ) ( 0 ) ( R ρ , θ ) = c 0 0 ϕ 0 + m = 1 N c m m ρ m [ ϕ m sin ( m θ ) + ϕ m cos ( m θ ) ] ;
r ϕ ( N ) ( 0 ) ( R , θ ) = R - 1 m = 1 N c m m m [ ϕ m sin ( m θ ) + ϕ m cos ( m θ ) ] .
ϕ m = R ψ ˜ m m c m m ,             ϕ m = R ψ ˜ m m c m m ,             m = 1 , 2 , , N .
ψ ˜ 0 = 1 2 π 0 2 π [ ψ ( N ) ( θ ) - r ϕ ( N ) ( 1 ) ( R , θ ) ] d θ = 0.
0 2 π ψ ( N ) ( θ ) d θ = - R 0 2 π 0 1 f ( N - 2 ) ( R ρ , θ ) ρ d ρ d θ .
ψ 0 = - R 2 π f 0 ,
A N Z N = [ B B 0 - Δ ] ( D Z N U Z N ) = ( F N Z N - 2 ) ,
A N ( ϕ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = [ B ( N ) ( 0 ) B ( N ) ( 1 ) 0 - Δ ( N ) ] ( ϕ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = ( B ( N ) ( 0 ) ϕ ( N ) ( 0 ) + B ( N ) ( 1 ) ϕ ( N ) ( 1 ) - Δ ϕ ( N ) ( 1 ) ) = ( ψ ( N ) f ( N - 2 ) ) ,
A ˜ N ( ϕ ˜ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = [ B ˜ ( N ) ( 0 ) B ˜ ( N ) ( 1 ) 0 - Δ ( N ) ] ( ϕ ˜ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = ( B ˜ ( N ) ( 0 ) ϕ ˜ ( N ) ( 0 ) + B ˜ ( N ) ( 1 ) ϕ ( N ) ( 1 ) - Δ ϕ ( N ) ( 1 ) ) = ( ψ ˜ ( N ) f ( N - 2 ) ) .
A ˜ N - 1 = [ [ B ˜ ( N ) ( 0 ) ] - 1 - [ B ˜ ( N ) ( 0 ) ] - 1 B ˜ ( N ) ( 1 ) [ - Δ ( N ) ] - 1 0 [ - Δ ( N ) ] - 1 ] , A ˜ N - 1 ( F ˜ N Z N - 2 ) = ( D Z ˜ N U Z N ) ,
( ϕ ˜ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = A ˜ N - 1 ( ψ ˜ ( N ) f ( N - 2 ) ) = ( [ B ˜ ( N ) ( 0 ) ] - 1 ψ ˜ ( N ) - [ B ˜ ( N ) ( 0 ) ] - 1 B ˜ ( N ) ( 1 ) [ - Δ ( N ) ] - 1 f ( N - 2 ) [ - Δ ( N ) ] - 1 f ( N - 2 ) ) .
A ˜ N = [ A 11 A 12 0 A 22 ] = [ R - 1 B i j ( 0 ) R - 1 B i j ( 1 ) 0 R - 2 Λ i j ] .
R - 1 B i j = B Z j , F i Γ = R - 1 0 2 π ( ρ Z j ) ( 1 , θ ) F i ( θ ) d θ ,
F i = { c cos ( i 2 θ ) i even c sin ( i + 1 2 θ ) i odd ,             c = 1 / π ,
R - 1 B i j ( 0 ) = R - 1 m c m m δ i k ,             i = 1 , 2 , , 2 N ,             j D J N \ { 1 } ,             m = m ( j ) ,
R - 1 B i j ( 1 ) = R - 1 σ j δ i k ,             i = 1 , 2 , , 2 N , j U J N ,
k = { 2 m ( j ) - 1 j odd 2 m ( j ) j even ,
R - 2 Λ i j = - Δ Z j , Z i = 0 2 π 0 R ( - Δ ) Z j ( r / R , θ ) Z i ( r / R , θ ) r d r d θ ,
Λ i j = δ m m 2 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 s = 0 ( n - m ) / 2 s = 0 ( n - m ) / 2 γ n , m s γ n , m s × m 2 - ( n - 2 s ) 2 n + n - 2 s - 2 s ,
A ˜ N - 1 = [ A 11 - 1 A 12 - 1 0 A 22 - 1 ] = [ R [ B i j ( 0 ) ] - 1 - R 2 [ B i j ( 0 ) ] - 1 B i j ( 1 ) Λ j i - 1 0 R 2 Λ j i - 1 ] .
( A 11 - 1 ) j i = R δ i k / m c m m ,             i = 1 , 2 , , 2 N , j D J N \ { 1 } , m = m ( j ) ,
( A 22 - 1 ) j i = R 2 ( - 1 ) i + j M i j / det Λ ,             i J N - 2 , j U J N ,
( A 12 - 1 ) j i = - R 2 k l ( A 11 - 1 ) j k B k l ( 1 ) ( A 22 - 1 ) l i ,             j D J N \ { 1 } , i J N - 2 .
( ϕ ˜ ( N ) ( 0 ) ϕ ( N ) ( 1 ) ) = A ˜ N - 1 ( ψ ˜ ( N ) f ( N - 2 ) ) = [ A 11 - 1 A 12 - 1 0 A 22 - 1 ] ( ψ ˜ ( N ) f ( N - 2 ) ) .
Ker [ ( - Δ ) N ] D Z N ,
ϕ = j J N ϕ j Z j = n m r n [ ϕ n m sin ( m θ ) + ϕ n m cos ( m θ ) ] ,
- Δ ϕ = n m ( m 2 - n 2 ) r n - 2 [ ϕ n m sin ( m θ ) + ϕ n m cos ( m θ ) ] .
( m 2 - n 2 ) ϕ n m = 0 ,             ( m 2 - n 2 ) ϕ n m = 0.
Im [ ( - Δ ) N ] Z N - 2 .
r l = n l σ n R n m ( r )
r m + 2 k = [ R m + 2 k m - s = 1 k γ m + 2 k , m s r m + 2 ( k - s ) ] / γ m + 2 k , m 0 .

Metrics