Abstract

Study of phase retrieval technology is quite meaningful, for its wide applications related to many domains, such as adaptive optics, detection of laser quality, precise measurement of optical surface, and so on. Here a hybrid iterative phase retrieval algorithm is proposed, based on fusion of the intensity information in three defocused planes. First the conjugate gradient algorithm is adapted to achieve a coarse solution of phase distribution in the input plane; then the iterative angular spectrum method is applied in succession for better retrieval result. This algorithm is still applicable even when the exact shape and size of the aperture in the input plane are unknown. Moreover, this algorithm always exhibits good convergence, i.e., the retrieved results are insensitive to the chosen positions of the three defocused planes and the initial guess of complex amplitude in the input plane, which has been proved by both simulations and further experiments.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006 (2)

2004 (2)

L. J. Allen, W. Mcbride, and M. P. Oxley, "Exit wave reconstruction using soft x rays," Opt. Commun. 233, 77-82 (2004).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

2003 (1)

L. Bruel, "Numerical phase retrieval from beam intensity measurements in three planes," Proc. SPIE 4932, 590-598 (2003).
[CrossRef]

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991), pp. 213-255.

1987 (1)

1982 (1)

1978 (1)

D. Malacara, Optical Shop Testing (Wiley, 1978).

1972 (1)

G. W. Gerchberg and W. O. Saxon, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Allen, L. J.

L. J. Allen, W. Mcbride, and M. P. Oxley, "Exit wave reconstruction using soft x rays," Opt. Commun. 233, 77-82 (2004).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Brady, G. R.

Bruel, L.

L. Bruel, "Numerical phase retrieval from beam intensity measurements in three planes," Proc. SPIE 4932, 590-598 (2003).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Fienup, J. R.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Gerchberg, G. W.

G. W. Gerchberg and W. O. Saxon, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Ivanov, V. Yu.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1978).

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Mcbride, W.

L. J. Allen, W. Mcbride, and M. P. Oxley, "Exit wave reconstruction using soft x rays," Opt. Commun. 233, 77-82 (2004).
[CrossRef]

Oxley, M. P.

L. J. Allen, W. Mcbride, and M. P. Oxley, "Exit wave reconstruction using soft x rays," Opt. Commun. 233, 77-82 (2004).
[CrossRef]

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Saxon, W. O.

G. W. Gerchberg and W. O. Saxon, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Sivokon, V. P.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991), pp. 213-255.

Wen-han, J.

J. Wen-han, "Adaptive optical technology," Chin. J. Nature 28, 7-13 (2006) (in Chinese).

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Appl. Opt. (2)

Chin. J. Nature (1)

J. Wen-han, "Adaptive optical technology," Chin. J. Nature 28, 7-13 (2006) (in Chinese).

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

L. J. Allen, W. Mcbride, and M. P. Oxley, "Exit wave reconstruction using soft x rays," Opt. Commun. 233, 77-82 (2004).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Opt. Express (1)

Optik (Stuttgart) (1)

G. W. Gerchberg and W. O. Saxon, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Proc. SPIE (1)

L. Bruel, "Numerical phase retrieval from beam intensity measurements in three planes," Proc. SPIE 4932, 590-598 (2003).
[CrossRef]

Other (2)

D. Malacara, Optical Shop Testing (Wiley, 1978).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991), pp. 213-255.

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Figures (11)

Fig. 1
Fig. 1

Optical setup for phase retrieval.

Fig. 2
Fig. 2

Given complex amplitude in the input plane to be retrieved, (a) amplitude and (b) phase.

Fig. 3
Fig. 3

Initial guess for (a) amplitude and (b) phase, and the retrieved (c) amplitude and (d) phase.

Fig. 4
Fig. 4

(Color online) Practical optical setup for phase retrieval.

Fig. 5
Fig. 5

Raw measured intensity patterns in planes with defocus of (a) 8 , (b) 16 , and (c) 24   mm .

Fig. 6
Fig. 6

Intensity patterns corresponding to Fig. 5 after pretreatment.

Fig. 7
Fig. 7

(Color online) Retrieved complex amplitude with defocus of 8 , 16 , and 24   mm , (a) amplitude and (b) phase.

Fig. 8
Fig. 8

(Color online) Retrieved complex amplitude with defocus of 9 , 18 , and 26   mm , (a) amplitude and (b) phase.

Fig. 9
Fig. 9

(Color online) Differences between Figs. 7 and 8, (a) amplitude and (b) phase.

Fig. 10
Fig. 10

(Color online) Retrieved phase of the DOE based on one group of intensity information.

Fig. 11
Fig. 11

(Color online) Retrieved phase of the DOE based on another group of intensity information.

Tables (1)

Tables Icon

Table 1 Phase Retrieval Accuracy under Different SNR Conditions

Equations (13)

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G k ( m , n ) = F k ( m , n ) exp [ i θ k ( m , n ) ] , k = 1 , 2 , 3 , 4 , 5 ,
F k ( m , n ) = [ I k ( m , n ) ] 1 / 2 , k = 1 , 2 , 3 , 4 , 5 ,
s m = λ f M d u , s n = λ f N d v ,
G 5 ( m , n ) = A d 1 , 5   exp { i π λ f ( 1 z f ) [ ( m s m ) 2 + ( n s n ) 2 ] } × D F T { G 1 ( u , v ) } ,
G 1 ( u , v ) = A d 5 , 1 D F T { G 5 ( m , n ) exp { i π λ f ( 1 z f ) × [ ( m s m ) 2 + ( n s n ) 2 ] } } .
G j , k ( m , n ) = A d k , j I D F T { D F T [ G k ( m , n ) ] × exp { i π λ d k , j [ ( u M s m ) 2 + ( v N s n ) 2 ] } } ,
k = 2 , 3 , 4 , 5 , j = 2 , 3 , 4 , 5.
G j , k ( m , n ) = P j , k [ G j ( m , n ) ] ,
e = j = 2 4 E j , E j = k = 2 , k j 4 m , n ( F j ( m , n ) | G j , k ( m , n ) | ) 2 .
E j θ j ( m , n ) = P j , k 1 [ G j w ( m , n ) ] ,
G j w ( p , q ) = [ F j ( m , n ) G j , k ( m , n ) | G j , k ( m , n ) | G j k ( m , n ) ] .
RMSE = m = 1 M n = 1 N ( θ 1 m , n θ 1 m , n a ) 2 M × N ,
a = m = 1 N n = 1 N ( θ 1 m , n θ 1 m , n ) M × N .

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