Abstract

The quality of images computed from digital holograms or heterodyne array imaging is degraded by phase errors in the object and/or reference beams at the time of measurement. This paper describes computer simulations used to compare the performance of digital shearing laser interferometry and various sharpness metrics for the correction of such phase errors when imaging a diffuse object. These algorithms are intended for scenarios in which multiple holograms can be recorded with independent object speckle realizations and a static phase error. Algorithm performance is explored as a function of the number of available speckle realizations and signal-to-noise ratio (SNR). The performance of various sharpness metrics is examined in detail and is shown to vary widely. Under ideal conditions with >15 speckle realizations and high SNR, phase corrections better than λ50 root-mean-square (RMS) were obtained. Corrections better than λ10 RMS were obtained in the high SNR regime with as few as two speckle realizations and at object beam signal levels as low as 2.5 photons/speckle with six speckle realizations.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).
  2. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123-1130 (1962).
    [CrossRef]
  3. F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photodetector arrays,” Opt. Lett. 25, 716-718 (2000).
    [CrossRef]
  4. J. R. Fienup, J. N. Cederquist, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Heterodyne array phasing by digital shearing laser interferometry,” in IRIS Specialty Group on Active Systems Meeting Digest, October 16-18, 1990.
  5. J. N. Cederquist, J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Digital shearing laser interferometry for heterodyne array phasing,” Proc. SPIE 1416, 266-277 (1991).
    [CrossRef]
  6. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200-1210 (1974).
    [CrossRef]
  7. R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” Proc. SPIE 976, 37-47 (1988).
  8. J. R. Fienup, A. M. Kowalczyk, and J. E. Van Buhler, “Phasing sparse arrays of heterodyne receivers,” Proc. SPIE 2241, 127-131 (1994).
    [CrossRef]
  9. J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A 20, 609-620 (2003).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics (Wiley, 2000), Sec. 5.6, pp. 207-211.
  11. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  12. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  13. H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416-425 (1988).
    [CrossRef]
  14. E. Acosta, S. Bará, M. A. Rama, and S. Rios, “Determination of phase mode components in terms of local wave-front slopes: an analytical approach,” Opt. Lett. 20, 1083-1085 (1995).
    [CrossRef] [PubMed]
  15. Provided courtesy of Jet Propulsion Laboratories (J. B. Breckinridge).
  16. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209-224 (1992).
    [CrossRef]
  17. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372-1379 (1966).
    [CrossRef]

2003 (1)

2000 (1)

1995 (1)

1994 (1)

J. R. Fienup, A. M. Kowalczyk, and J. E. Van Buhler, “Phasing sparse arrays of heterodyne receivers,” Proc. SPIE 2241, 127-131 (1994).
[CrossRef]

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

1991 (1)

J. N. Cederquist, J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Digital shearing laser interferometry for heterodyne array phasing,” Proc. SPIE 1416, 266-277 (1991).
[CrossRef]

1988 (2)

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” Proc. SPIE 976, 37-47 (1988).

H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416-425 (1988).
[CrossRef]

1980 (1)

1977 (1)

1974 (1)

1966 (1)

1962 (1)

J. Opt. Soc. Am. (5)

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

Opt. Lett. (2)

Proc. SPIE (3)

J. N. Cederquist, J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Digital shearing laser interferometry for heterodyne array phasing,” Proc. SPIE 1416, 266-277 (1991).
[CrossRef]

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion,” Proc. SPIE 976, 37-47 (1988).

J. R. Fienup, A. M. Kowalczyk, and J. E. Van Buhler, “Phasing sparse arrays of heterodyne receivers,” Proc. SPIE 2241, 127-131 (1994).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmorgorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (4)

Provided courtesy of Jet Propulsion Laboratories (J. B. Breckinridge).

J. W. Goodman, Statistical Optics (Wiley, 2000), Sec. 5.6, pp. 207-211.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

J. R. Fienup, J. N. Cederquist, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Heterodyne array phasing by digital shearing laser interferometry,” in IRIS Specialty Group on Active Systems Meeting Digest, October 16-18, 1990.

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 (18)

Fig. 1
Fig. 1

Layout for recording digital holograms.

Fig. 2
Fig. 2

Flowchart for simulating digital holography data.

Fig. 3
Fig. 3

Incoherent intensity image I ( ξ , η ) used for simulations.

Fig. 4
Fig. 4

Phase error ϕ ( x , y ) used for simulations in units of waves. The phase error is a random-draw atmospheric phase screen [16] (with tip and tilt subtracted) with D r 0 = 8 , where D is the width of the detector and r 0 is Fried’s parameter.

Fig. 5
Fig. 5

Flowchart for reconstructing an object field from a digital hologram.

Fig. 6
Fig. 6

DSLI performance versus number of available speckle realizations for P F = 640 photoelectrons/speckle. Each data point is the average of results from five trials with independent speckle and noise realizations. The dotted curve is a theoretical prediction of DSLI performance based on Eqs. (29, 30) (ignoring photon and detector noise).

Fig. 7
Fig. 7

DSLI results for specific trials. Each column corresponds to a different number of speckle realizations N and shows in successive rows (a) an ideal speckle-averaged image (with no phase errors or noise), (b) a speckle-averaged image degraded by both phase errors and noise, and speckle-averaged reconstructed images corresponding to (c) the 10th-order polynomial and (d) the point-by-point phase-error estimates.

Fig. 8
Fig. 8

Sharpness metric M 1 with α = 2 performance versus number of available speckle realizations for P F = 640 photoelectrons/speckle.

Fig. 9
Fig. 9

Sharpness metric M 1 with α = 2 results for specific trials. Each column corresponds to a different number of speckle realizations N and shows in successive rows (a) an ideal speckle-averaged image (with no phase errors or noise), (b) a speckle-averaged image degraded by both phase errors and noise, and speckle-averaged reconstructed images corresponding to (c) the 10th-order polynomial and (d) the point-by-point phase-error estimates.

Fig. 10
Fig. 10

Sharpness metric M 1 and M 2 performance versus α and sharpness metric M 3 performance for P F = 640 photoelectrons/speckle and N = 20 speckle realizations.

Fig. 11
Fig. 11

Sharpness metric M 1 with α = 1.01 performance versus number of available speckle realizations for P F = 640 photoelectrons/speckle.

Fig. 12
Fig. 12

Sharpness metric M 1 and M 2 performance versus α for P F = 640 photoelectrons/speckle with different numbers of available speckle realizations.

Fig. 13
Fig. 13

Sharpness metric M 2 with α = 0.5 performance versus number of available speckle realizations for P F = 640 photoelectrons/speckle.

Fig. 14
Fig. 14

DSLI performance versus object beam intensity for N = 6 speckle realizations.

Fig. 15
Fig. 15

Sharpness metric M 1 with α = 1.01 performance versus object beam intensity for N = 6 speckle realizations.

Fig. 16
Fig. 16

Sharpness metric M 1 with α = 1.01 results for specific trials. Each column corresponds to a different object beam intensity P F with different speckle realizations and shows in successive rows (a) an ideal speckle-averaged image (with no phase errors or noise), (b) a speckle-averaged image degraded by both phase errors and noise, and speckle-averaged reconstructed images corresponding to (c) the 10th-order polynomial and (d) the point-by-point phase-error estimates.

Fig. 17
Fig. 17

Sharpness metric M 1 with α = 1.25 performance versus object beam intensity for N = 6 speckle realizations.

Fig. 18
Fig. 18

Sharpness metrics M 1 and M 2 performance versus α for N = 6 speckle realizations with different object beam intensities.

Tables (2)

Tables Icon

Table 1 Statistics-Based Sharpness Metrics Based on the Form of Eq. (23)

Tables Icon

Table 2 Correlation-Based Sharpness Metrics Based on the Form of Eq. (24)

Equations (31)

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

H ( x , y ) = R ( x , y ) + F ( x , y ) 2 ,
R ( x , y ) = R 0 exp [ i ( k x x + k y y ) ] ,
F ( x , y ) = 1 i λ z exp ( i k z ) exp [ i π λ z ( x 2 + y 2 ) ] f ( ξ , η ) exp [ i π λ z ( ξ 2 + η 2 ) ] exp [ i 2 π λ z ( x ξ + y η ) ] d ξ d η ,
H ̃ ( f x , f y ) = R 0 2 δ ( f x , f y ) + F ̃ ( f x , f y ) F ̃ ( f x , f y ) + R 0 * F ̃ ( k x 2 π + f x , k y 2 π + f y ) + R 0 F ̃ * ( k x 2 π f x , k y 2 π f y ) ,
R ε ( x , y ) = R ( x , y ) exp [ i ϕ R ( x , y ) ] ,
F ε ( x , y ) = F ( x , y ) exp [ i ϕ F ( x , y ) ] ,
H ε ( x , y ) = R ε ( x , y ) + F ε ( x , y ) 2 = R ( x , y ) + F ( x , y ) exp [ i ϕ ( x , y ) ] 2 ,
ϕ ( x , y ) = ϕ F ( x , y ) ϕ R ( x , y ) .
G ( x , y ) = F ( x , y ) exp [ i ϕ ( x , y ) ] .
G n ( x , y ) = F n ( x , y ) exp [ i ϕ ( x , y ) ] .
S x ( x , y ) = 1 N n = 1 N G n ( x , y ) G n * ( x Δ , y ) ,
S y ( x , y ) = 1 N n = 1 N G n ( x , y ) G n * ( x , y Δ ) ,
S x ( x , y ) = exp { i [ ϕ ( x , y ) ϕ ( x Δ , y ) ] } 1 N n = 1 N F n ( x , y ) F n * ( x Δ , y ) .
1 N n = 1 N F n ( x , y ) F n * ( x Δ , y ) = 1 λ 2 z 2 exp { i π λ z [ x 2 ( x Δ ) 2 ] } 1 N n = 1 N f n ( ξ , η ) f n * ( ξ , η ) × exp [ i π λ z ( ξ 2 ξ 2 + η 2 η 2 ) ] × exp { i 2 π λ z [ ξ Δ + ( ξ ξ ) x + ( η η ) y ] } d ξ d η d ξ d η ,
1 N n = 1 N f n ( ξ , η ) f n * ( ξ , η ) κ I ( ξ , η ) δ ( ξ ξ , η η ) ,
S x ( x , y ) = κ λ 2 z 2 exp { i [ ϕ ( x , y ) ϕ ( x Δ , y ) ] } exp [ i π λ z ( 2 x Δ Δ 2 ) ] × I ( ξ , η ) exp ( i 2 π λ z ξ Δ ) d ξ d η = κ λ 2 z 2 exp { i [ ϕ ( x , y ) ϕ ( x Δ , y ) ] } exp [ i π λ z ( 2 x Δ Δ 2 ) ] I ̃ ( Δ λ z , 0 ) ,
S y ( x , y ) κ λ 2 z 2 exp { i [ ϕ ( x , y ) ϕ ( x , y Δ ) ] } exp [ i π λ z ( 2 y Δ Δ 2 ) ] I ̃ ( 0 , Δ λ z ) .
ϕ ̂ ( x , y ) ϕ ( x , y ) + π λ z ( x 2 + y 2 ) + a x + b y + c ,
F ̂ n ( x , y ) = G n ( x , y ) exp [ i ϕ ̂ ( x , y ) ] .
f ̂ n ( ξ , η ) = 1 λ z exp [ i π λ z ( ξ 2 + η 2 ) ] F ̂ n ( x , y ) exp [ i 2 π λ z ( x ξ + y η ) ] d x d y ,
I ̂ ( ξ , η ) = 1 N n = 1 N f ̂ n ( ξ , η ) 2 .
M = ( ξ , η ) Γ [ I ̂ ( ξ , η ) ] ,
M = ( ξ , η ) ( ξ , η ) D Γ [ I ̂ ( ξ , η ) I ̂ ( ξ ξ , η η ) ] ,
f n ( ξ , η ) = I ( ξ , η ) [ N ( 0 , 0.5 ) + i N ( 0 , 0.5 ) ]
ϕ ̂ ( x , y ) = k C k ψ k ( x , y ) ,
E DSLI = ( x , y ) W ( x , y ) W ( x Δ , y ) { ϕ ̂ ( x , y ) ϕ ̂ ( x Δ , y ) arg [ S x ( x , y ) ] } 2 + ( x , y ) W ( x , y ) W ( x , y Δ ) { ϕ ̂ ( x , y ) ϕ ̂ ( x , y Δ ) arg [ S y ( x , y ) ] } 2 ,
E = ( x , y ) W ( x , y ) exp { i [ ϕ ̂ ( x , y ) + a + b x + c y ] } exp [ i ϕ ( x , y ) ] 2 .
σ ϕ 2 = [ ( x , y ) W ( x , y ) ] 1 ( x , y ) W ( x , y ) × { arg [ exp { i [ ϕ ̂ ( x , y ) + a + b x + c y ϕ ( x , y ) ] } ] } 2 ,
σ ϕ ( N ) = σ DSLI ( N ) 0.6558 [ 1 + ln ( 256 2 ) ] ,
σ DSLI ( N ) = 1 μ 2 2 μ 2 N
μ ( f x , f y ) = I ̃ ( f x , f y ) I ̃ ( 0 , 0 ) ,

Metrics