Abstract

Phase errors in multiple planes create an anisoplanatic (space-variant) blur in an image. We show how phase errors in multiple planes can be corrected with the use of a sharpness metric for heterodyne or holographic imaging. We derive the theoretical framework necessary for this anisoplanatic imaging situation. A digital simulation and results are presented. We demonstrate the success of this nonlinear optimization technique for phase errors in two planes.

© 2009 Optical Society of America

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References

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2008 (3)

2003 (1)

1997 (1)

1996 (1)

F. Berizzi and G. Corsini, IEEE Trans. Aerosp. Electron. Syst. 32, 1185 (1996).
[CrossRef]

1993 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, Waves Random Media 2, 209 (1992).
[CrossRef]

1988 (1)

R. G. Paxman and J. C. Marron, Proc. SPIE 976, 37 (1988).

1974 (1)

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Berizzi, F.

F. Berizzi and G. Corsini, IEEE Trans. Aerosp. Electron. Syst. 32, 1185 (1996).
[CrossRef]

Buffington, A.

Corsini, G.

F. Berizzi and G. Corsini, IEEE Trans. Aerosp. Electron. Syst. 32, 1185 (1996).
[CrossRef]

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, Waves Random Media 2, 209 (1992).
[CrossRef]

Fienup, J. R.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, Waves Random Media 2, 209 (1992).
[CrossRef]

Guizar-Sicairos, M.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering through Random Media (Academic, 1978).

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, Waves Random Media 2, 209 (1992).
[CrossRef]

Marron, J. C.

R. G. Paxman and J. C. Marron, Proc. SPIE 976, 37 (1988).

Miller, J. J.

Muller, R. A.

Paxman, R. G.

R. G. Paxman and J. C. Marron, Proc. SPIE 976, 37 (1988).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Thurman, S. T.

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Appl. Opt. (2)

IEEE Trans. Aerosp. Electron. Syst. (1)

F. Berizzi and G. Corsini, IEEE Trans. Aerosp. Electron. Syst. 32, 1185 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Proc. SPIE (1)

R. G. Paxman and J. C. Marron, Proc. SPIE 976, 37 (1988).

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, Waves Random Media 2, 209 (1992).
[CrossRef]

Other (3)

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering through Random Media (Academic, 1978).

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Figures (3)

Fig. 1
Fig. 1

Layout for recording a digital hologram with multiple planes of phase error present.

Fig. 2
Fig. 2

Image reconstruction results using two phase screens. (a) The ideal image. Two-screen simulation when (b) there is no aberration correction and (c) the phase estimates have been used in the reconstruction.

Fig. 3
Fig. 3

Field magnitude versus z from the image to the detector (left to right); each pair: truth and reconstructed. (a), (b) Horizontal slice; (c), (d) vertical slice; and (e)–(h) magnified slices of (a), (c), (b), and (d) near the image.

Equations (8)

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I ( ξ , η ) = 1 K k = 1 K f k ( ξ , η ) 2 ,
A [ z ; g ( x , y ) ] = F 1 { F [ g ( x , y ) ] exp [ i π λ z ( f x 2 + f y 2 ) ] } ,
P 1 p [ f k ( ξ , η ) ] = A { z p ; exp ( i φ p 1 ) A [ z 3 ; exp ( i φ 2 ) × A { z 2 ; exp ( i φ 1 ) A [ z 1 ; f k ( ξ , η ) ] } ] } .
P P + 1 p [ G k ( u , v ) ] = exp ( i φ p ) A ( z p + 1 ; exp ( i φ p + 1 ) exp ( i φ P 1 ) × A [ z P ; exp ( i φ P ) A { z P + 1 ; G k ( u , v ) } ] ) .
S = ξ , η I β ( ξ , η ) α k = 1 K f ξ , f η M ( f ξ , f η ) F k ( f ξ , f η ) 2
S φ p ( x , y ) = 2 K k = 1 K Im ( P 1 p [ β I β 1 ( ξ , η ) f k ( ξ , η ) ] × { P P + 1 p [ G k ( u , v ) ] } * ) 2 α K k = 1 K Im ( P 1 p { F 1 [ M ( f ξ , f η ) F k ( f ξ , f η ) ] } × { P P + 1 p [ G k ( u , v ) ] } * ) .
φ p ( x , y ) = j c p , j ψ j ( x , y ) ,
S c p , j = x , y ψ j ( x , y ) S φ p ( x , y ) .

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