Abstract

A method to retrieve small-phase aberrations from a single far-field image is proposed. It is found that in a small-phase condition, the odd and even parts of a phase aberration can be obtained with a simple linear calculation method. The difference between a single measured image with aberration and a calibrated image with inherent aberration is used in the calculation process. It is proved that most of the inherent phase aberration of the imaging system must be of an even type, such as defocus, astigmatism, etc., to keep the method working. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the proposed small-phase retrieval method works well when the RMS phase error is less than 1 rad. It is also shown that the method is valid in a noise condition when the SNR>100.

© 2008 Optical Society of America

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References

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  1. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Eng. 21, 2758-2769 (1982).
  2. J. M. Wood, M. A. Fiddy, and R. E. Burge, "Phase retrieval using two intensity measurements in the complex plane," Opt. Lett. 6, 514-516 (1981).
    [CrossRef] [PubMed]
  3. J. R. Fienup, "Phase retrieval using a support constraint," presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va. October 28-30 (1985).
  4. R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 66, 961-964 (1976).
    [CrossRef]
  5. B. Ellerbroek and D. Morrison, "Linear Methods in Phase Retrieval," Proc. SPIE 351, 90-95 (1983).
  6. R. A. Gonsalves, "Small-phase solution to the phase-retrieval problem," Opt. Lett. 26, 684-685 (2001).
    [CrossRef]
  7. N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
    [CrossRef]

2001

1990

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

1983

B. Ellerbroek and D. Morrison, "Linear Methods in Phase Retrieval," Proc. SPIE 351, 90-95 (1983).

1982

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Eng. 21, 2758-2769 (1982).

1981

1976

Burge, R. E.

Ellerbroek, B.

B. Ellerbroek and D. Morrison, "Linear Methods in Phase Retrieval," Proc. SPIE 351, 90-95 (1983).

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Eng. 21, 2758-2769 (1982).

Gonsalves, R. A.

Morrison, D.

B. Ellerbroek and D. Morrison, "Linear Methods in Phase Retrieval," Proc. SPIE 351, 90-95 (1983).

Roddier, N.

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Wood, J. M.

Appl. Eng.

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Eng. 21, 2758-2769 (1982).

J. Opt. Soc. Am.

Opt. Eng.

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Opt. Lett.

Proc. SPIE

B. Ellerbroek and D. Morrison, "Linear Methods in Phase Retrieval," Proc. SPIE 351, 90-95 (1983).

Other

J. R. Fienup, "Phase retrieval using a support constraint," presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va. October 28-30 (1985).

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Figures (3)

Fig. 1.
Fig. 1.

Principal diagram of the linear phase retrieval method.

Fig. 2.
Fig. 2.

Numerical simulation results of the phase retrieval method proposed in this paper without noise. (a) Relationship between the average Strehl ratio SR of aberration and the average RMS of atmosphere disturbed aberration σ; (b) relationship between the average error coefficient η and the average RMS of atmosphere disturbed aberration σ.

Fig. 3.
Fig. 3.

Detailed retrieved wavefront comparison under the conditions of with and without noise. (a) Initial disturbed wavefront; (b) (c) even and odd parts of the initial disturbed wavefront; (d) retrieved wavefront with SNR=∞; (e) (f) even and odd parts of the retrieved wavefront with SNR=∞; (g) retrieved wavefront with SNR=120; (h) (i) even and odd parts of the retrieved wavefront with SNR=120.

Tables (1)

Tables Icon

Table 1. Results of phase retrieval with different SNRs

Equations (27)

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f ( x , y ) = f o ( x , y ) + f e ( x , y ) ,
S = A exp ( i B ) ,
B = B e + B o ,
S = A + i Z + i T .
P = ss * = ( a + x ) 2 + z 2 = a 2 + x 2 + z 2 + 2 a x ,
P e = a 2 + x 2 + z 2 ,
P o = 2 ax .
W = W e + W o ,
H B = A exp [ i ( B + W ) ] A · [ 1 + i ( B + W ) ]
= A + i A · ( B e + B o + W e + W o ) = A + i Z + i T + i V + i Q ,
h B = a + i ( z + t + v + q ) ,
h B = ( a + x + y ) + i ( z + v ) .
P B = h B h B * = ( a + x + y ) 2 + ( v + z ) 2
= a 2 + x 2 + y 2 + z 2 + v 2 + 2 xy + 2 v z + 2 ax + 2 ay ,
P Be = a 2 + x 2 + y 2 + z 2 + v 2 + 2 xy + 2 v z = P e + y 2 + v 2 + 2 xy + 2 v z ,
P Bo = 2 ax + 2 ay = P o + 2 ay .
y = 0.5 × ( P Bo P o ) a = 0.5 × ( P B P ) o a .
W ̂ o = Im { Y } A .
v 2 + 2 v z + 2 xy + ( P e + y 2 P Be ) = 0 .
v 2 + 2 z v + 2 xy + ( P e P Be ) = 0 .
v 2 + ( P e P Be ) = 0 .
v ( P Be P e ) ( 2 z ) = ( P B P ) e ( 2 z ) .
W ̂ e = Re al ( V ) A .
E ( x , y ) = W ( x , y ) W ̂ ( x , y ) .
η = RMS [ E ( x , y ) ] RMS [ W ( x , y ) ] .
SNR = P σ n ,
Threshold = 3 × σ n .

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