Abstract

A novel wave-front sensor that estimates phase from Fourier intensity measurements is described, and an explicit expression is found and numerically evaluated for the Cramér–Rao lower bound on integrated rms wave-front phase estimation error. For comparison, turbulence-aberrated wave-front phases and corresponding noisy Fourier intensity measurements were computer simulated. An iterative phase-retrieval algorithm was then used to estimate the phase from the Fourier intensity measurements and knowledge of the shape of an aperture through which the wave front passed. The simulation error approaches the lower bound asymptotically as the noise is reduced.

© 1989 Optical Society of America

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References

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  1. J. T. Foley, R. R. Butts, “Uniqueness of phase retrieval from intensity measurements,”J. Opt. Soc. Am. 71, 1008–1014 (1981).
    [CrossRef]
  2. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  3. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  4. R. A. Gonsalves, “Phase retrieval from modulus data,”J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  5. H. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968), pp. 66–73, 79–85, 96–97, 178–182, 437–441.
  6. J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
    [CrossRef]
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,”J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  9. W. Wolfe, G. Zissis, eds., The Infrared Handbook (Office of Naval Research, Washington, D.C., 1978), Chap. 6.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 526.
  11. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).
    [CrossRef]
  12. J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  13. J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

1987

1986

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

1983

1982

1981

1976

1965

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 526.

Butts, R. R.

Cederquist, J. N.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

Fienup, J. R.

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

Foley, J. T.

Fried, D. L.

Gonsalves, R. A.

Kryskowski, D.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).
[CrossRef]

Robinson, S. R.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

Van Trees, H.

H. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968), pp. 66–73, 79–85, 96–97, 178–182, 437–441.

Wackerman, C. C.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 526.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Other

W. Wolfe, G. Zissis, eds., The Infrared Handbook (Office of Naval Research, Washington, D.C., 1978), Chap. 6.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 526.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt. Instrum. Eng.74, 225–233 (1976).
[CrossRef]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound for Fourier modulus wavefront sensor,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 156–159.

H. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968), pp. 66–73, 79–85, 96–97, 178–182, 437–441.

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Figures (6)

Fig. 1
Fig. 1

Model of the Fourier intensity wave-front sensor system.

Fig. 2
Fig. 2

Fourier intensity wave-front sensor normalized lower bound e0/σϕ versus light level Pc for σϕ = π/2 and (curve A) a point object and (curves B–E) extended objects with ratios of the field coherence area πL2 to the sensor aperture area A of 10−1, 10−2, 10−3, and 10−4, respectively.

Fig. 3
Fig. 3

Fourier intensity wave-front sensor lower bound versus Pc for a point object and σϕ of (curve A) π/2, (curve B) π, (curve C) 2π, and (curve D) 4π. (a) Normalized lower bound e0/σϕ; (b) absolute lower bound e0.

Fig. 4
Fig. 4

Comparison of simulated atmospheric phase covariance (solid curve) with Gaussian covariance (dashed curve) used in analysis.

Fig. 5
Fig. 5

Fourier intensity wave-front sensor simulation for a point object, σϕ = π/2, and D/r0 ≃ 12.7. Intensity in the measurement plane is shown (A) without noise and (B and C) for light levels Pc of 1.1 × 105 and 1.1 × 103 photons per coherence cell, respectively. Wave-front phases shown are (D) the actual phase and (E and F) phase estimates obtained from B and C, respectively, with a random initial phase estimate. The phases are wrapped with −π phase (black) and +π phase (white). Normalized errors e/σϕ are (E) 0.059 and (F) 0.55.

Fig. 6
Fig. 6

Comparison of Fourier intensity wave-front sensor lower bound with simulation phase-estimate errors versus light level Pc for a point object, σ = π/2 and D/r0 ≃ 12.7. Curve A represents the normalized lower bound e0/σϕ. Normalized simulation errors e/σϕ are given for initial phase estimates of (curve B) the actual phase and (curve C) a random phase.

Equations (30)

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e 2 = 1 A A E { [ ϕ ˜ ( x ) - ϕ ( x ) ] 2 } d x ,
e 0 2 = 1 A A J - 1 ( x , x ) d x .
J - 1 ( x , y ) + A A J - 1 ( x , p ) R ( p , q ) K ( q - y ) d p d q = K ( x - y ) ,
R ( x , y ) = ( 4 η 2 T 2 / N ) exp [ - 2 K ( 0 ) ] Re { P A A h ( u , x ) × h * ( u , p ) M ( x , p ) exp [ K ( x - p ) + K ( y - q ) ] × { h * ( u , y ) h ( u , q ) M * ( y , q ) exp [ K ( x - y ) - K ( x - q ) - K ( y - p ) + K ( p - q ) ] - h ( u , y ) × h * ( u , q ) M ( y , q ) exp [ - K ( x - y ) + K ( x - q ) + K ( y - p ) - K ( p - q ) ] } d u d p d q } ;
h ( u , x ) = [ W ( x ) / i λ F ] exp ( - i 2 π u · x / λ F ) ,
( λ F ) - 2 P exp ( - i 2 π u · x / λ F ) d u = δ ( x ) .
R ( x , y ) = [ 4 η 2 T 2 / ( λ F ) 2 N ] W ( x ) W ( y ) exp [ - 2 K ( 0 ) + 2 K ( x - y ) ] × Re { A W ( p ) M ( x , p ) exp [ 2 K ( x - p ) + 2 K ( y - p ) ] × { W ( y - x + p ) M * ( y , y - x + p ) × exp [ - 3 K ( y - p ) - K ( 2 x - y - p ) ] - W ( y + x - p ) M ( y , y + x - p ) × exp [ - 3 K ( x - y ) - K ( x + y - 2 p ) ] } d p } .
K ( x - y ) = σ ϕ 2 exp ( - x - y 2 / r ϕ 2 ) ,
M ( x , y ) = I exp ( - x - y 2 / L 2 ) ,
R ( x , y ) [ 4 η 2 T 2 I 2 A 0 / ( λ F ) 2 N ] W ( x ) W ( y ) × exp { - 2 σ ϕ 2 [ 1 - exp ( - x - y 2 / r ϕ 2 ) ] } ,
R ( x , y ) [ 4 η 2 T 2 I 2 A 0 / ( λ F ) 2 N ] exp ( - 2 σ ϕ 2 x - y 2 / r ϕ 2 ) .
e 0 2 = 2 σ ϕ 2 0 f exp ( - f 2 ) 1 + P exp { - f 2 [ 1 + ( 2 σ ϕ 2 ) - 1 ] } d f ,
P = 2 η 2 T 2 I 2 A 0 ( π r ϕ 2 ) 2 / [ ( λ F ) 2 N ] .
e 0 2 σ ϕ 2 ln ( 1 + P ) P ,
P = P / [ 1 + ( 2 σ ϕ 2 ) - 1 ] .
0 exp [ - f 2 ( 2 σ ϕ 2 ) - 1 ] exp ( - f 2 ) f d f 0 exp ( - f 2 ) f d f = 1 1 + ( 2 σ ϕ 2 ) - 1 ,
I A = I FT π ( λ F / r 0 ) 2 .
P = 2 π 4 η 2 T 2 I FT 2 A 0 σ ϕ 4 ( λ F ) 2 ( 1.86 ) 4 A 2 N .
s = η T I FT / ν ,
N s = η T I FT / ν .
s 2 / N s = η T I FT / ν ,
P = π 4 η T I FT A 0 σ ϕ 4 ( λ F ) 2 ( 1.86 ) 4 A 2 ν .
P = π 2 η T I A 0 σ ϕ 2 ( π r ϕ 2 ) ( 1.86 ) 2 A ν .
P c = π r ϕ 2 η T I / ν .
P = 2.87 σ ϕ 2 P c A 0 / A .
ϕ ( x ) = j = 1 a j g j ( x ) ,
μ j g j ( x ) = A K ( x - y ) g j ( y ) d y
I ( a , u ) = E ( A A h ( u , x ) h * ( u , y ) M ( x , y ) × exp { i [ ϕ ( x ) - ϕ ( y ) ] } d x d y ) ,
A A R ( x , y ) g j ( x ) g k ( y ) d x d y = E [ 2 η 2 T 2 N P I ( a , u ) a j I ( a , u ) a k d u ] .
E ( exp { i j = 1 a j [ g j ( x ) - g j ( y ) ] } ) = exp [ - K ( 0 ) + K ( x - y ) ]

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