Abstract

A scintillation resistant sensor that allows retrieval of an input optical wave phase using a multi-aperture phase reconstruction (MAPR) technique is introduced and analyzed. The MAPR sensor is based on a low-resolution lenslet array in the classical Shack–Hartmann arrangement and two high-resolution photo-arrays for simultaneous measurements of pupil- and focal-plane intensity distributions, which are used for retrieval of the wavefront phase in a two stage process: (a) phase reconstruction inside the sensor pupil subregions corresponding to lenslet subapertures and (b) recovery of subaperture averaged phase components (piston phases). Numerical simulations demonstrate the efficiency of the MAPR technique in conditions of strong intensity scintillations and the presence of wavefront branch points.

© 2012 Optical Society of America

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2002 (1)

2001 (1)

1999 (1)

1998 (2)

1994 (1)

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

1992 (2)

V. Y. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. 9, 1515–1524 (1992).
[CrossRef]

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef]

1988 (2)

1986 (1)

1982 (2)

1979 (1)

V. U. Zavorotnyi, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

1977 (1)

1976 (2)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Al-Habash, M. A.

Andrews, L. C.

Arnold, R.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Barchers, J. D.

Barrett, T.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Beresnev, L. A.

Carhart, G. W.

Cuellar, L.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fienup, J. R.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gonsalves, R. A.

Hardy, J. W.

Hopen, C. Y.

Ivanov, V. Y.

V. Y. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. 9, 1515–1524 (1992).
[CrossRef]

Johnson, P.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Justh, E. W.

Koliopoulos, C. L.

Kravtsov, Yu A.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).

Krishnaprasad, P. S.

Lefebvre, J. E.

Lefebvre, M.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Link, D. J.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Paxman, R. G.

Phillips, R. L.

Rego, A.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Roddier, F.

Roggemann, M. C.

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

Rousset, G.

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999), pp. 91–130.

Rytov, M. C.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).

Sandler, D. G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sivokon, V. P.

M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

V. Y. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. 9, 1515–1524 (1992).
[CrossRef]

Smith, G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Spivey, B.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Tatarskii, V. I.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).

Taylor, G.

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

Vaughn, J. L.

Vorontsov, M. A.

Wackerman, C. C.

Welsh, B.

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

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Opt. Soc. Am. (5)

R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
[CrossRef]

V. Y. Ivanov, V. P. Sivokon, and M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. 9, 1515–1524 (1992).
[CrossRef]

J. W. Hardy, J. E. Lefebvre, and C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

D. G. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, and B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/r0,” J. Opt. Soc. Am. 11, 858–873 (1994).
[CrossRef]

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

J. Opt. Soc. Am. A (6)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Radiophys. Quantum Electron. (1)

V. U. Zavorotnyi, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

Other (4)

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999), pp. 91–130.

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

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Figures (13)

Fig. 1.
Fig. 1.

Notional schematics of (a) MAPR, (b) Shack–Hartmann, and (c) lens analyzer wavefront sensors.

Fig. 2.
Fig. 2.

Block diagram of data processing in the MAPR wavefront sensor using parallel computation of local phases {φ˜l(r)} with the Gerchberg–Saxton algorithm (computational blocks {GSl}) and piston phase reconstruction based on stochastic parallel gradient descent (SPGD) optimization techniques.

Fig. 3.
Fig. 3.

Examples of densely packed lenslet array configurations for different lenslet shapes: (a) circular, (b) hexagonal, and (c) rectangular. The dot in (a) and dashed rectangles in (b) and (c) indicate the region (Λlk) used for piston phase reconstruction.

Fig. 4.
Fig. 4.

Performance factor η characterizing the ratio of computational costs for phase reconstruction using the MAPR sensor with the array of nl×nl lenslets and the lens analyzer (GS technique): results of computer simulations (solid curve) and approximation using Eq. (6) with κ=17 (dashed curve). The sensor input fields were generated using the technique described in Subsection 3.A.

Fig. 5.
Fig. 5.

Gray-scale images corresponding to computer-generated optical field intensity [(a), (b)] and phase [(c), (d)] distributions at the MAPR sensor pupil plane for D/r0=8 [(a), (c)] and D/r0=12 [(b), (d)]. Phase distributions [(c), (d)] are shown inside a 2π range [between π (black) and +π (white)]. (e), (f) The corresponding phase interference patterns. The interference pattern “forks” inside the white circles indicate branch points.

Fig. 6.
Fig. 6.

Aperture averaged scintillation index for the MAPR sensor input fields used in the numerical simulations versus D/r0 for L=0.1Ldif.

Fig. 7.
Fig. 7.

Intensity distributions in the MAPR sensor lenslet focal plane for input fields shown in Fig. 5: (a) D/r0=8 and (b) D/r0=12.

Fig. 8.
Fig. 8.

(a)–(d) Phase distributions and (e)–(h) the corresponding interference patterns obtained after n=200 phase reconstruction iterations using the pupil- and focal-plane intensity distributions from Figs. 5(a) and 5(b) and 7(a) and 7(b). The phase distributions are shown before [(a), (b)] and after [(c), (d)] piston phase recovery. The corresponding interferograms of the residual phase φ(r)φ˜(r) are shown before [(e), (f)] and after [(g), (h)] piston phase computation.

Fig. 9.
Fig. 9.

Phase reconstruction convergence process in the MAPR wavefront sensor with a 3×3 lenslet array (solid curves) and in the lens analyzer/GS sensor (dashed curves) for (a) D/r0=8 and (b) D/r0=12.

Fig. 10.
Fig. 10.

Averaged Strehl ratio evolution curves for the MAPR (3×3 lenslet array configuration) and lens analyzer/GS sensors with different initial conditions: Shack–Hartmann-based (solid curves), random (dashed curves), and spatially uniform (dotted curves) initial phase functions. The sensor input field realizations correspond to D/r0=8.

Fig. 11.
Fig. 11.

Performance curves (Strehl ratios versus D/r0) for the MAPR 3×3 lenslet array (solid curve) and lens analyzer/GS (dashed curve) sensors. The length of the vertical lines indicates the standard deviation obtained for the corresponding Strehl ratio value.

Fig. 12.
Fig. 12.

Averaged Strehl ratio for the residual phase achieved after 200 MAPR phase reconstruction iterations versus number of lenslets nl in rows and columns of a rectangular lenslet array for D/r0=4, 8, and 12.

Fig. 13.
Fig. 13.

Averaged Strehl ratios versus photo-array noise parameter σξ for the MAPR sensor with 3×3 and 2×2 lenslet arrays (solid curves) and lens analyzer/GS sensor (dashed curve) for input fields with D/r0=8.

Equations (8)

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J(Δ1,,ΔNl)=l=1Nlϵl(0)(Δ1,,ΔNl),
ϵl(0)(Δ1,,ΔNl)=klMl{[Ωlρlk(rrlkb){φ˜l(r)+Δj}d2rΩkρkl(rrklb){φ˜k(r)+Δk}d2r]2}
Δl(n+1)=Δl(n)+γ(n)δJ(n)δΔl(n),l=1,,Nl,
φ˜(r)=l=1Nl[φ˜l(r)+Δ˜l].
η=CLACMAPR=NitLANplogNpNitMAPRNslogNs=NitLANllogNpNitMAPR(logNplogNl)NitLANitMAPRNl.
ηNitLANitMAPRnl2=κnl2,
σI2=1S{[IP(r)]2IP(r)21}d2r,
St(n)=||Ain(r)|exp[iδ(r,n)]d2r|2/[|Ain(r)|d2r]2,

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