Abstract

Strong turbulence causes phase discontinuities known as branch points in an optical field. These discontinuities complicate the phase unwrapping necessary to apply phase corrections onto a deformable mirror in an adaptive optics (AO) system. This paper proposes a non-optimal but effective and implementable phase unwrapping method for optical fields containing branch points. This method first applies a least-squares (LS) unwrapper to the field which isolates and unwraps the LS component of the field. Four modulo-2π-equivalent non-LS components are created by subtracting the LS component from the original field and then restricting the result to differing ranges. 2π phase jumps known as branch cuts are isolated to the non-LS components and the different non-LS realizations have different branch cut placements. The best placement of branch cuts is determined by finding the non-LS realization with the lowest normalized cut length and adding it to the LS component. The result is an unwrapped field which is modulo-2π-equivalent to the original field while minimizing the effect of phase cuts on system performance. This variable-range ‘ϕLS+ϕnon-LS’ unwrapper, is found to outperform other unwrappers designed to work in the presence of branch points at a reasonable computational burden. The effect of improved unwrapping is demonstrated by comparing the performance of a system using a fixed-range ‘ϕLS+ϕnon-LS’ realization unwrapper against the variable-range ‘ϕLS+ϕnon-LS’ unwrapper in a closed-loop simulation. For the 0.5 log-amplitude variance turbulence tested, the system Strehl performance is improved by as much as 41.6 percent at points where fixed-range ‘ϕLS+ϕnon-LS’ unwrappers result in particularly poor branch cut placement. This significant improvement in previously poorly performing regions is particularly important for systems such as laser communications which require minimum Strehl ratios to operate successfully.

© 2008 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
  2. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (John Wiley & Sons, Inc., 1998).
  3. D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15, 2759-2768 (1998).
    [CrossRef]
  4. D. L. Fried, "Adaptive optics wave function reconstruction and phase unwrapping when branch points are present," Opt. Commun. 200, 43-72 (2001).
    [CrossRef]
  5. J. Barchers, "The performance of wavefront sensors in strong turbulence," Proc. SPIE 4839, 217 (2003).
    [CrossRef]
  6. D. L. Fried and J. L. Vaughn, "Branch cuts in the phase function," Appl. Opt. 31, 2865-2882 (1992).
    [CrossRef] [PubMed]
  7. D. L. Fried, "The nature of the branch point problem in Adaptive Optics," Proc. SPIE 3381, 38-46 (1998).
    [CrossRef]
  8. C. A. Primmerman, "Atmospheric-compensation experiments in strong-scintillation conditions," Appl. Opt. 34, 2081-2088 (1995).
    [CrossRef] [PubMed]
  9. M. C. Roggemann, "Branch-point reconstruction in laser beam projection through turbulence with finite-degreeof-freedom phase-only wavefront correction," J. Opt. Soc. Am. A 17, 53-62 (2000).
    [CrossRef]
  10. Wikipedia (2007). Website, URL http://en.wikipedia.org/wiki/Vector-potential.
  11. J. Barchers, "Personal Communication," (2007).
  12. T. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," Proc. SPIE 5553, 112-126 (2004).
    [CrossRef]
  13. L. C. Andrews and R. C. Philips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).
  14. T. J. Brennan and P. H. Roberts, AO Tools The Adaptive Optics Toolbox-Users Guide version 1.2 (Optical Sciences Company, 2006).

2004 (1)

T. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," Proc. SPIE 5553, 112-126 (2004).
[CrossRef]

2003 (1)

J. Barchers, "The performance of wavefront sensors in strong turbulence," Proc. SPIE 4839, 217 (2003).
[CrossRef]

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, "The nature of the branch point problem in Adaptive Optics," Proc. SPIE 3381, 38-46 (1998).
[CrossRef]

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

1995 (1)

1992 (1)

Barchers, J.

J. Barchers, "The performance of wavefront sensors in strong turbulence," Proc. SPIE 4839, 217 (2003).
[CrossRef]

Fried, D. L.

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, "The nature of the branch point problem in Adaptive Optics," Proc. SPIE 3381, 38-46 (1998).
[CrossRef]

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

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

Primmerman, C. A.

Rhoadarmer, T.

T. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," Proc. SPIE 5553, 112-126 (2004).
[CrossRef]

Roggemann, M. C.

Vaughn, J. L.

Appl. Opt. (2)

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]

Proc. SPIE (3)

J. Barchers, "The performance of wavefront sensors in strong turbulence," Proc. SPIE 4839, 217 (2003).
[CrossRef]

D. L. Fried, "The nature of the branch point problem in Adaptive Optics," Proc. SPIE 3381, 38-46 (1998).
[CrossRef]

T. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," Proc. SPIE 5553, 112-126 (2004).
[CrossRef]

Other (6)

L. C. Andrews and R. C. Philips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).

T. J. Brennan and P. H. Roberts, AO Tools The Adaptive Optics Toolbox-Users Guide version 1.2 (Optical Sciences Company, 2006).

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

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms, and Software (John Wiley & Sons, Inc., 1998).

Wikipedia (2007). Website, URL http://en.wikipedia.org/wiki/Vector-potential.

J. Barchers, "Personal Communication," (2007).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Wrapped phase with only wrapping cuts. (b) Unwrapped version of (a). Note that the unwrapped phase is smooth.

Fig. 2.
Fig. 2.

Wrapped Phase with both wrapping and branch cuts. If this phase were to be unwrapped, it would not be smooth.

Fig. 3.
Fig. 3.

Poor unwrap and minimum cut distance unwrap of phase field w/branch points

Fig. 4.
Fig. 4.

Intensity overlaid by branch cuts using LS unwrapper to eliminate wrapping cuts

Fig. 5.
Fig. 5.

Poor unwrapping, phase and intensity overlaid by branch cuts

Fig. 6.
Fig. 6.

Branch cuts of four different unwrap realizations

Fig. 7.
Fig. 7.

Field estimation Strehl versus integrated cut intensity

Fig. 8.
Fig. 8.

Comparison of closed-loop AO performance between variable and fixed ϕnon-LS range ‘ϕLS +φnon-LS ’ unwrappers.

Fig. 9.
Fig. 9.

CDF comparisons between variable range ϕnon-LS and fixed range ϕnon-LS unwrappers

Tables (5)

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Table 1. Normalized cut lengths for 0.4 log-amplitude variance field

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Table 2. Normalized cut lengths for 0.8 log-amplitude variance field

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Table 3. Normalized cut lengths from various unwrappers, 0.4 log-amplitude variance field

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Table 4. Normalized cut lengths from various unwrappers, 0.8 log-amplitude variance field

Tables Icon

Table 5. Average Strehl results for 1000 frame simulation

Equations (5)

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= s G T Gφ = G T s ( G T G) -1 G T Gφ = ( G T G) 1 G T s φ LS = ( G T G) 1 G T s
Gφ = s WGφ = Ws (WG) T WGφ = (WG) T Ws G T W T Wgφ = G T W T Ws G T W 2 Gφ = G T W 2 s ( G T W 2 G) -1 G T W 2 Gφ = ( G T W 2 G) 1 G T W 2 s φ LS = ( G T W 2 G) 1 G W 2 s,
φ LS = LS( φ Tot ) and φ nonLS = 𝒲( φ Tot LS( φ Tot ))
𝒰 ( ϕ Tot ) = ϕ LS + ϕ non LS ,
S = a 1 = 1 N b 1 = 1 N F a 1 b 1 E a 1 b 1 * 2 a 2 = 1 N b 2 = 1 N F a 2 b 2 F a 2 b 2 * a 3 = 1 N b 3 = 1 N E a 3 b 3 E a 3 b 3 * ,

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