Abstract

The estimation accuracy of a point-diffraction interferometer is examined with two phase-shifting schemes: spatial and temporal. Under the assumption of plane- or spherical-wave propagation through isotropic turbulence that can be accurately represented as a series of thin phase screens, results that are valid for any scintillation regime are obtained by use of the invariance with a propagation of the mutual coherence function. It is established that the estimation accuracy of the spatial phase-shifting point-diffraction interferometer is invariant with scintillation. Upper and lower bounds on the performance of the temporal phase-shifting point-diffraction interferometer are developed. Wave optical simulation results are presented that validate the analytic predictions for the two phase-shifting schemes. The results and techniques presented can be used to assess the appropriate phase-shifting scheme given finite resources, such as a limited number of pixels in a detector array or a restricted detector frame rate.

© 2002 Optical Society of America

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References

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  1. D. L. Fried, “Reconstructor formulation error,” in Optical Pulse and Beam Propagation III, Y. B. Band, ed., Proc. SPIE4271, 1–21 (2001).
  2. V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Tech. Rep. TT-68-50464 (National Science Foundation, Washington, D.C., 1968); available from the National Technical Information Service, Springfield, Va.
  3. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  4. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  5. J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
    [CrossRef] [PubMed]
  6. J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation,” Appl. Opt. 41, 3674–3684 (2002).
    [CrossRef] [PubMed]
  7. D. L. Fried, “Wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  8. D. L. Fried, “Diffusion analysis for propagation of mutual coherence,” J. Opt. Soc. Am. 58, 961–969 (1968).
    [CrossRef]
  9. D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
  10. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
  11. M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
    [CrossRef]
  12. E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
    [CrossRef]

2002 (2)

2001 (3)

1998 (2)

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

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

1992 (1)

1968 (1)

Barchers, J. D.

Beresnev, L. A.

Carhart, G. W.

Fried, D. L.

Justh, E. W.

Krishnaprasad, P. S.

Link, D. J.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).

Tatarskii, V. I.

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Tech. Rep. TT-68-50464 (National Science Foundation, Washington, D.C., 1968); available from the National Technical Information Service, Springfield, Va.

Vaughn, J. L.

Vorontsov, M. A.

Appl. Opt. (3)

Atmos. Oceanic Opt. (1)

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

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

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

Other (3)

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

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Tech. Rep. TT-68-50464 (National Science Foundation, Washington, D.C., 1968); available from the National Technical Information Service, Springfield, Va.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).

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Figures (5)

Fig. 1
Fig. 1

Geometry for the (a) SPSPDI and (b) TPSPDI. In (a) U() is the complex field in the subaperture whereas Û(′) is the estimate of the complex field at the coordinate ′. The prime coordinates ′ are denoted by the open circles, whereas is the continuous spatial coordinate. In (b) the four measurements obtained from temporal phase shifting and the estimate of the complex field at time t 0 are illustrated.

Fig. 2
Fig. 2

Performance results obtained with wave optical simulation of the SPSPDI as a function of the Rytov number for several values of l/ r 0. The performance is largely invariant with scintillation.

Fig. 3
Fig. 3

Comparison of analytic and simulation results for the performance of the SPSPDI. Agreement is consistent with expectation.

Fig. 4
Fig. 4

Comparison of the general and asymptotic (geometrical-optics) result for the formulation error of the SPSPDI. Agreement is good for l/ r 0 < 0.1.

Fig. 5
Fig. 5

Validation of the upper and lower bounds on estimation accuracy of the TPSPDI. The quantity S[ U(′, t 0), Û T (′, t 0)], representing the performance of the TPSPDI, is shown in (a), (c), (e), and (g) for l/ r 0 equal to 1/4, 1/2, 1, and 2, respectively. The quantity S[ Û(′, t 0), Û T (′, t 0)], representing the specific Strehl loss that is due to temporal phase shifting, is shown in (b), (d), (f), and (h) for l/ r 0 equal to 1/4, 1/2, 1, and 2, respectively. The simulation points that fall outside the upper bounds are attributed to the outer-scale-like effect imposed by use of periodic phase screens.

Equations (48)

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Iθ=1αSdr¯|Ur¯+Uref|2,
=1αSdr¯A2r¯+Aref2+2Ar¯Aref cosϕr¯-θ,
Uˆ=I0-Iπ+iI3π/2-Iπ/2,
=1αSdr¯ArefAr¯cosϕr¯+i Sdr¯ArefAr¯sinϕr¯,
=ArefαSdr¯Ur¯.
Uˆr¯=1l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryUr¯.
SUr¯, Uˆr¯=Ur¯Uˆ*r¯2Ur¯U*r¯Uˆr¯Uˆ*r¯,
Ur¯Uˆ*r¯=1l2-l/2l/2drx-l/2l/2dryU0Ur¯.
Ur¯Uˆ*r¯=4 01/2drx01/2dryUlr¯,
Uˆr¯Uˆ*r¯=1l4-l/2l/2drx1-l/2l/2dry1-l/2l/2drx2×-l/2l/2dry2Ur¯1-r¯2.
Uˆr¯Uˆ*r¯=4l40ldrx-0ldry-Ur¯-l-rx-×l-ry-.
Uˆr¯Uˆ*r¯=4 01drx01dryUlr¯1-rx1-ry.
Iθ, t=1αSdr¯|Ur¯, t+Uref|2,
=1αSdr¯A2r¯, t+Aref2+2Ar¯, tAref cosϕr¯, t-θ.
ReUˆTr¯, t0=I0, t-3Ts/2-Iπ, t+Ts/2,
=14l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryArefUr¯, t-3Ts/2+U*r¯, t-3Ts/2+Ur¯, t+Ts/2+U*r¯, t+Ts/2+A2r¯, t-3Ts/2-A2r¯, t+Ts/2,
i ImUˆTr¯, t0=iI3π/2, t+3Ts/2-Iπ/2, t-Ts/2,
=14l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryArefUr¯, t+3Ts/2-U*r¯, t+3Ts/2+Ur¯, t-Ts/2-U*r¯, t-Ts/2+iA2r¯, t+3Ts/2-iA2r¯, t-Ts/2.
Dr¯, τ=2.91k020LdzCn2z|r¯+v¯zτ|5/3,
UΔ¯, τ=Ur¯0, t0U*r¯0+Δ¯, t0+τ,
U˜Δ¯, τ=Ur¯0, t0Ur¯0+Δ¯, t0+τ,
U3Δ¯, τ=Ur¯0, t0Ur¯0+Δ¯, t0+τU*×r¯0+Δ¯, t0+τ,
Δ¯, τ=Ur¯, tU*r¯, tUr¯+Δ¯, t+τU*×r¯+Δ¯, t+τ.
Qdr¯1Qdr¯2Ar¯1-r¯2, τ=Qdr¯1Qdr¯2Ar¯1-r¯2, -τ,
Qdr¯1Qdr¯2Ar¯1-r¯2, τ=Qdr¯1Qdr¯2Ar¯2-r¯1, τ,
Qdr¯Ar¯, τ=Qdr¯Ar¯, -τ,
Qdr¯Ar¯, τ=Qdr¯A-r¯, τ.
Ur¯, t0UˆT*r¯, t0=12l2-l/2l/2drx1-l/2l/2×dry1Ur¯, 3Ts/2+U r¯, Ts/2.
Ur¯, t0UˆT*r¯, t0=12-1/21/2drx1-1/21/2dry1×Ulr¯, 3Ts/2+Ulr¯, Ts/2.
Uˆr¯, t0UˆT*r¯, t0=12l4-l/2l/2drx1-l/2l/2dry1×-l/2l/2drx2-l/2l/2dry2×Ur¯1-r¯2, 3Ts/2+Ur¯1-r¯2, Ts/2.
Uˆr¯, t0UˆT*r¯, t0=12l4-lldrx--lldry-×Ur¯-, 3Ts/2+Ur¯-, Ts/2×|l-rx-||l-ry-|.
Uˆr¯, t0UˆT*r¯, t0=12-11drx-11dry×Ulr¯, 3Ts/2+Ulr¯, Ts/2×|1-rx||1-ry|.
UˆTr¯, t0UˆT*r¯, t0est=12-11drx-11dryUlr¯, 0+Ulr¯, 2Ts×|1-rx||1-ry|.
14-1/21/2drx1-1/21/2dry1-1/21/2drx2-1/21/2dry2lr¯1-r¯2, 0)-lr¯1-r¯2, 2Ts.
Q/ldr1Q/ldr2lr¯1-r¯2, 0)Q/ldr1Q/ldr2lr¯1-r¯2, 2Ts.
SUBUr¯, t0, UˆTr¯, t0=|Ur¯, t0UˆT*r¯, t0|2Ur¯, t0U*r¯, t0UˆTr¯, t0UˆT*r¯, t0est,
SUBUˆr¯, t0, UˆTr¯, t0=|Uˆr¯, t0UˆT*r¯, t0|2Uˆr¯, t0Uˆ*r¯, t0UˆTr¯, t0UˆT*r¯, t0est.
SLBUr¯, t0, UˆTr¯, t0=|Ur¯, t0UˆT*r¯, t0|2Ur¯, t0U*r¯, t0Uˆr¯, t0Uˆ*r¯, t0,
SLBUˆr¯, t0, UˆTr¯, t0=|Uˆr¯, t0UˆT*r¯, t0|2Uˆr¯, t0Uˆ*r¯, t02.
ϕˆr¯=1l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryϕr¯.
σSPSPDI2=ϕr¯-1l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryϕr¯2.
ϕr¯=1l2rx-l/2rx+l/2ry-l/2ry+l/2drxdryϕr¯
σSPSPDI2=1l4rx-l/2rx+l/2drx1ry-l/2ry+l/2dry1rx-l/2rx+l/2drx2×ry-l/2ry+l/2dry2ϕr¯-ϕr¯1ϕr¯-ϕr¯2.
σSPSPDI2=12l4-l/2l/2drx1-l/2l/2dry1-l/2l/2drx2-l/2l/2dry2×Dr¯1+Dr¯2-Dr¯1-r¯2.
σSPSPDI2=4 0l/2drx0l/2dryDlr¯+2 01drx01dryDlr¯1-rx1-ry.
U3r¯, r¯, r¯=expiϕr¯+ϕr¯-ϕr¯,
=exp-12ϕr¯+ϕr¯-ϕr¯2.
U3r¯, r¯, r¯=exp-12Dr¯-r¯+Dr¯-r¯-Dr¯-r¯+ϕ2r¯.

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