Abstract

Wavefront aberrations can be represented accurately by a number of Zernike polynomials. We develop a method to retrieve small-phase aberrations from a single far-field image with a Zernike modal-based approach. The difference between a single measured image with aberration and a calibrated image with inherent aberration is used in the calculation process. In this paper, the principle of linear phase retrieval is introduced in a vector–matrix format, which is a kind of linear calculation and is suitable for real-time calculation. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the proposed Zernike modal-based linear phase retrieval method works well when the rms of phase error is less than 1 rad, and it is valid in a noise condition when the signal-to-noise ratio (SNR)>3.

© 2009 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).
    [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).
  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).
  7. M. Li, X.-Y. Li, and W.-H. Jiang. "Small-phase retrieval with a single far-field image," Opt. Express 16, 8190-8197 (2008).
    [PubMed]
  8. L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).
  9. X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang, "A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system," in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212-218.
  10. N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).

2008 (1)

2002 (1)

L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).

2001 (1)

1990 (1)

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

1983 (1)

B. Ellerbroek and D. Morrison, "Linear methods in phase retrieval," Proc. SPIE 351, 90-95 (1983).

1982 (1)

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

1981 (1)

1976 (1)

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.

Jiang, W.-H.

Li, M.

Li, X.-Y.

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).

Wenhan, J.

L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).

Wood, J. M.

Xinyang, L.

L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).

Appl. Eng. (1)

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

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Optica Sinica (1)

L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).

Proc. SPIE (1)

B. Ellerbroek and D. Morrison, "Linear methods in phase retrieval," Proc. SPIE 351, 90-95 (1983).

Other (2)

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).

X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang, "A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system," in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212-218.

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Figures (3)

Fig. 1.
Fig. 1.

Principal diagram of the linear phase retrieval method.

Fig. 2.
Fig. 2.

Relationship between the error coefficient η and the proportion of coma to the total system aberration; the defocus is set as σ=4 rad.

Fig. 3.
Fig. 3.

Numerical simulation results of the linear phase retrieval method proposed in this paper without noise. (a) Relationship between the average Strehl ratio SRˉ of aberration and the average RMS of atmosphere disturbed aberration σ̄ ; (b) relationship between the average error coefficient η̄ and the average rms of atmosphere-disturbed aberration σ̄.

Tables (1)

Tables Icon

Table 1. Results of phase retrieval with different SNRs

Equations (17)

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W=We+Wo,
f(x,y)=fo(x,y)+fe(x,y),
f=fJxyf2+f+Jxyf2=fo+fe,
Ŵo PBo Po.
Ŵe PBe Pe,
ŵo=Ro·Δo,
ŴeRe·Δe.
ŵ=ŵo+ŵeRoΔo+ReΔe=(RoRoJxy2+Re+ReJxy2)Δ=R·Δ,
Δ=R+·ŵ,
W(x,y)=Σi=1pciMi(x,y),
w=D · c ,
Δ=R+D·c=Z · c .
c=Z+·Δ,
E(x,y)=W (x,y)Ŵ(x,y).
η=RMS [E(x,y)]RMS[W(x,y)].
SNR=P1σn,
Threshold=3×σn.

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