Abstract

High-resolution optical systems require a very accurate control of the optical paths. For the measurement of aberrations on extended objects, several iterative phase-diversity algorithms have been developed, based on aberration estimation from focal-plane intensity measurements. Here we present an analytical estimator in the case of small aberrations. Under this assumption, a quadratic criterion is derived that allows us to express the solution (phase and object) under a simple analytical form. We also compare the performance of our algorithm with the iterative phase diversity, demonstrating that the analytic estimator is appropriate for closed-loop operation.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Gonsalves, Opt. Eng. 21, 829 (1982).
  2. O. M. Bucci, A. Capozzolo, and G. D'Elia, J. Opt. Soc. Am. A 16, 1759 (1999).
    [CrossRef]
  3. A. Blanc, L. M. Mugnier, and J. Idier, J. Opt. Soc. Am. A 20, 1035 (2003).
    [CrossRef]
  4. J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
    [CrossRef] [PubMed]
  5. L. M. Mugnier, A. Blanc, and J. Idier, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1-76.
    [CrossRef]
  6. M. G. Löfdahl and A. L. Duncan, Proc. SPIE 3353, 952 (1998).
    [CrossRef]
  7. C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
    [CrossRef]
  8. G. B. Scharmer, in High Resolution Solar Physics: Theory, Observations and Techniques, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds., ASP Conferences Series (1999), Vol. 183, pp. 330-341.
  9. M. G. Löfdahl and G. B. Scharmer, Proc. SPIE 4013, 737 (2000).
    [CrossRef]
  10. J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
    [CrossRef]
  11. I. Mocœur, L. M. Mugnier, and F. Cassaing, in 21ième Colloque GRETSI (2007).

2005

J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
[CrossRef] [PubMed]

2003

2000

M. G. Löfdahl and G. B. Scharmer, Proc. SPIE 4013, 737 (2000).
[CrossRef]

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

1999

1998

M. G. Löfdahl and A. L. Duncan, Proc. SPIE 3353, 952 (1998).
[CrossRef]

C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
[CrossRef]

1982

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).

Benson, L.

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

Blanc, A.

J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
[CrossRef] [PubMed]

A. Blanc, L. M. Mugnier, and J. Idier, J. Opt. Soc. Am. A 20, 1035 (2003).
[CrossRef]

L. M. Mugnier, A. Blanc, and J. Idier, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1-76.
[CrossRef]

Bucci, O. M.

Capozzolo, A.

Cassaing, F.

I. Mocœur, L. M. Mugnier, and F. Cassaing, in 21ième Colloque GRETSI (2007).

Chan, T.

C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
[CrossRef]

D'Elia, G.

Duncan, A. L.

M. G. Löfdahl and A. L. Duncan, Proc. SPIE 3353, 952 (1998).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).

Idier, J.

J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
[CrossRef] [PubMed]

A. Blanc, L. M. Mugnier, and J. Idier, J. Opt. Soc. Am. A 20, 1035 (2003).
[CrossRef]

L. M. Mugnier, A. Blanc, and J. Idier, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1-76.
[CrossRef]

Löfdahl, M. G.

M. G. Löfdahl and G. B. Scharmer, Proc. SPIE 4013, 737 (2000).
[CrossRef]

M. G. Löfdahl and A. L. Duncan, Proc. SPIE 3353, 952 (1998).
[CrossRef]

Mocœur, I.

I. Mocœur, L. M. Mugnier, and F. Cassaing, in 21ième Colloque GRETSI (2007).

Mugnier, L. M.

J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
[CrossRef] [PubMed]

A. Blanc, L. M. Mugnier, and J. Idier, J. Opt. Soc. Am. A 20, 1035 (2003).
[CrossRef]

I. Mocœur, L. M. Mugnier, and F. Cassaing, in 21ième Colloque GRETSI (2007).

L. M. Mugnier, A. Blanc, and J. Idier, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1-76.
[CrossRef]

Paxman, R.

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

Plemmons, R.

C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
[CrossRef]

Scharmer, G. B.

M. G. Löfdahl and G. B. Scharmer, Proc. SPIE 4013, 737 (2000).
[CrossRef]

G. B. Scharmer, in High Resolution Solar Physics: Theory, Observations and Techniques, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds., ASP Conferences Series (1999), Vol. 183, pp. 330-341.

Seldin, J.

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

Stone, R.

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

Vogel, C. R.

C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
[CrossRef]

Zarifis, V.

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

IEEE Trans. Image Process.

J. Idier, L. M. Mugnier, and A. Blanc, IEEE Trans. Image Process. 14, 2107 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Eng.

R. A. Gonsalves, Opt. Eng. 21, 829 (1982).

Proc. SPIE

M. G. Löfdahl and G. B. Scharmer, Proc. SPIE 4013, 737 (2000).
[CrossRef]

J. Seldin, R. Paxman, V. Zarifis, L. Benson, and R. Stone, Proc. SPIE 401, 48 (2000).
[CrossRef]

M. G. Löfdahl and A. L. Duncan, Proc. SPIE 3353, 952 (1998).
[CrossRef]

C. R. Vogel, T. Chan, and R. Plemmons, Proc. SPIE 3353, 994 (1998).
[CrossRef]

Other

G. B. Scharmer, in High Resolution Solar Physics: Theory, Observations and Techniques, T.R.Rimmele, K.S.Balasubramaniam, and R.R.Radick, eds., ASP Conferences Series (1999), Vol. 183, pp. 330-341.

L. M. Mugnier, A. Blanc, and J. Idier, in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, pp. 1-76.
[CrossRef]

I. Mocœur, L. M. Mugnier, and F. Cassaing, in 21ième Colloque GRETSI (2007).

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.


Figures (3)

Fig. 1
Fig. 1

Focal (left) and extrafocal (right) images obtained when a piston of 1 rad is applied on a subaperture.

Fig. 2
Fig. 2

Piston linearity obtained when a piston ramp is applied on an subaperture.

Fig. 3
Fig. 3

Error estimated over the subapertures when a set of random aberrations is applied.

Tables (1)

Tables Icon

Table 1 Aberrations Applied over the Configuration (rad rms)

Equations (11)

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

J I ( o , a ) = N ν ln σ 2 + 1 2 σ 2 ν = 1 N ν d = 1 2 | i ̃ d ( ν ) h ̃ d ( a , ν ) o ̃ ( ν ) | 2 + ν = 1 N ν | o ̃ ( ν ) o ̃ m ( ν ) | 2 2 S o ( ν ) + R ( a ) ,
o ̃ ̂ ( a , ν ) = i ̃ 1 ( ν ) h ̃ 1 * ( a , ν ) + i ̃ 2 ( ν ) h ̃ 2 * ( a , ν ) | h ̃ 1 ( a ̃ , ν ) | 2 + | h ̃ 2 ( a , ν ) | 2 + σ 2 S o ( ν ) ,
J II ( a ) = 1 2 σ 2 ν = 1 N ν | i ̃ 1 ( ν ) h ̃ 2 ( a , ν ) i ̃ 2 ( ν ) h ̃ 1 ( a , ν ) | 2 | h ̃ 1 ( a , ν ) | 2 + | h ̃ 2 ( a , ν ) | 2 + ε + Cst .
h ̃ d ( a , ν ) = α d ( ν ) a + β d ( ν ) ,
J II ( a ) = 1 2 σ 2 ν = 1 N ν | A ( ν ) a B ( ν ) | 2 + Cst ,
A ( ν ) = i ̃ 2 ( ν ) α 1 ( ν ) i ̃ 1 ( ν ) α 2 ( ν ) | h ̃ 1 ( 0 , ν ) | 2 + | h ̃ 2 ( 0 , ν ) | 2 + ε ,
B ( ν ) = i ̃ 2 ( ν ) β 1 ( ν ) + i ̃ 1 ( ν ) β 2 ( ν ) | h ̃ 1 ( 0 , ν ) | 2 + | h ̃ 2 ( 0 , ν ) | 2 + ε .
J II ( a ) = 1 2 σ 2 A a B 2 + Cst .
a ̂ = [ R ( A H A ) ] R ( A H B ) ,
p n ( u ) = Π ( u ) exp [ j k = 1 k max a k n Z k ( u ) ] ,
Π ( u ) = { 1 for 0 | u | R n 0 elsewhere } . .

Metrics