Abstract

A simulation study is presented that evaluates the ability of a unit-shear, shearing interferometer to estimate a complex field resulting from propagation through extended turbulence. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and weaknesses of the shearing interferometer in the two-dimensional space of the Fried parameter r 0 and the Rytov number. The performance of the shearing interferometer is compared with that of a Hartmann sensor in the Fried and Hutchin geometries. Although the effects of additive measurement noise (such as read noise, shot noise, amplifier noise) are neglected, the fundamental characteristics of the measurement process are shown to distinguish the performance of the various wave-front sensors. It is found that the performance of a shearing interferometer is superior to that of a Hartmann sensor when the Rytov number exceeds 0.2.

© 2002 Optical Society of America

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References

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  1. J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).
  2. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  3. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–379 (1977).
    [CrossRef]
  4. D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
    [CrossRef]
  5. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  6. D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
  7. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  8. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  9. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
    [CrossRef]
  10. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  11. G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Ind., 1996).

2002 (1)

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

2000 (1)

1998 (2)

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

1977 (1)

1974 (1)

1965 (1)

Barchers, J. D.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).

Clifford, S. F.

Fried, D. L.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[CrossRef]

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Ind., 1996).

Hudgin, R. H.

Lawrence, R. S.

Link, D. J.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).

Ochs, G. R.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).
[CrossRef]

Tyler, G. A.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Ind., 1996).

Appl. Opt. (1)

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1612–621 (2002).

Atmos. Oceanic Opt. (1)

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

J. Opt. Soc. Am. (3)

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

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Other (3)

G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Ind., 1996).

D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
[CrossRef]

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Measurements and points at which the field is to be reconstructed for a unit-shear, shearing interferometer in the Hutchin geometry. The open circles represent a section of a larger array of points on which the complex field is to be reconstructed (i.e., the points in K′). The solid square is the subaperture region in which the interference patterns of the dashed squares are measured. The interference patterns are combined to form an estimate of the phase difference between the pair of filled circles. A large array of phase difference measurements in the horizontal and vertical directions is obtained and then reconstructed to form an estimate of the complex field at the points in K′.

Fig. 2
Fig. 2

Measured log-amplitude variance of the fields used in this paper is shown as a function of the Rytov number. As expected, the agreement is excellent for small values of the Rytov number, a slight amplification is observed in the intermediate regime, and saturation occurs for large values of the Rytov number.

Fig. 3
Fig. 3

Estimation accuracy of the unit-shear, shearing interferometer by use of a least-squares (solid curves) or complex exponential reconstructor (dashed curves) for l/r 0 = 1/4 (circles), l/r 0 = 1/2 (squares), and l/r 0 = 1 (left triangles). The estimation Strehl falls off rapidly when the Rytov number exceeds 0.2 and a least-squares reconstructor is used. However, better performance is obtained when a complex exponential reconstructor is used, particularly when l/r 0 ≤ 1/2.

Fig. 4
Fig. 4

Estimation accuracy of the unit-shear, shearing interferometer in the absence of intensity fluctuctions is shown as a function of the Rytov number for l/r 0 = (a) 1/4, (b) 1/2, and (c) 1. Each curve corresponds to the ability of either the least-squares (solid curves) or complex exponential reconstructor (dashed curves) to form an estimate of the least-squares (no symbol) or the hidden (diamonds) phase at the prime coordinates. Also shown is the Strehl loss that is due to the hidden phase S when there is no wave-front correction (dashed-dotted). The least-squares reconstructor ignores the hidden phase when branch points are present (Rytov numbers greater than 0.2). The complex exponential reconstructor provides a better estimate of the hidden phase for similar conditions. For large values of l/r 0, the least-squares reconstructor cannot be used to estimate the least-squares phase because of the presence of many phase differences between subapertures outside the range (-π, π). [In (a) the complex exponential reconstructor curve for ϕ and that for ϕ lie on each other and cannot be distinguished.] The increase in the least-squares curve for ϕ (solid curves) with increasing Rytov number—seen strongly in (b) and weakly in (c)—is due to the conversion of phase perturbations to scintillation, so that with increasing Rytov number, there are fewer occurrences of phase difference measurements outside the range (-π, π) that corrupt the phase estimate obtained by the least squares reconstructor.

Fig. 5
Fig. 5

Strehl loss (or gain) that is due to scintillation is shown as a function of the Rytov number for l/r 0 = 1/4 (circles) and l/r 0 = 1 (triangles) for the unit-shear, shearing interferometer by use of a complex exponential reconstructor. For both the least-squares (solid curves) and hidden (dashed curves) phase, scintillation has little impact on estimation accuracy.

Fig. 6
Fig. 6

Comparison of the performance of a unit-shear, shearing interferometer with that of Hartmann sensors in the Hutchin and Fried geometries for l/r 0 equal to (a) 1/4, (b) 1/2, and (c) 1. The shearing interferometer exhibits superior performance in nearly all conditions.

Equations (10)

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U r¯=Ωwfsrec Ur¯.
SU1r¯, U2r¯=|kK U1r¯kU2*r¯k|2kK U1r¯kU1*r¯kkK U2r¯kU2*r¯k.
Δx,yϕ=tan-1dr¯Wr¯Ur¯-l¯x,y2Ur¯+l¯x,y2sinϕ r¯+l¯x,y2-ϕr¯-l¯x,y2dr¯Wr¯Ur¯-l¯x,y2Ur¯+l¯x,y2cosϕ r¯+l¯x,y2-ϕ r¯-l¯x,y2,
dr¯Wr¯U*r¯-l¯x,y2Ur¯+l¯x,y2.
Sϕ=GTG-1GTPVGϕ,
ϕ=PVargexpiϕexp-iϕ.
Sexpiϕr¯, expi arg Ωwfsrecexpiϕr¯,
Sexpiϕr¯, expi arg Ωwfsrecexpiϕr¯,
Qscint,=Sexpiϕr¯, expi argΩwfsrecAr¯expiϕr¯Sexpiϕr¯, expi argΩwfsrecexpiϕr¯,
Qscint,=Sexpiϕr¯, expi argΩwfsrecAr¯expiϕr¯Sexpiϕr¯, expi argΩwfsrecexpiϕr¯,

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