Abstract

Optical waves propagating through atmospheric turbulence develop spatial and temporal variations in their phase. For sufficiently strong turbulence, these phase differences can lead to interference in the propagating wave and the formation of branch points; positions of zero amplitude. Under the assumption of a layered turbulence model, we show that these branch points can be used to estimate the number and velocities of atmospheric layers. We describe how to carry out this estimation process and demonstrate its robustness in the presence of sensor noise.

© 2010 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 Inc., New York, NY, USA, 1998), 1st ed.
  2. D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, Inc., New York, NY, 1998).
  3. M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998).
    [CrossRef]
  4. M. Schöck and E. J. Spillar, "Method for a quantitative investiagation of the frozen flow hypothesis," J. Opt. Soc. Am. 17, 1650-1658 (2000).
    [CrossRef]
  5. D. C. Johnston and B. M. Welsh, "Estimating the contribution of different parts of the atmosphere to optical wavefront aberration," Comput. Elect. Eng. 18, 467-483 (1992).
    [CrossRef]
  6. L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
    [CrossRef]
  7. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (Bellingham, Wa, USA, 2007), 1st ed.
  8. M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beamprojection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. 17, 53-62 (2000).
    [CrossRef]
  9. D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. 15, 2759-2768 (1998).
    [CrossRef]
  10. E. O. Le Bigot and W. J. Wild, "Theory of branch-point detection and its implementation," J. Opt. Soc. Am. 16, 1724-1729 (1999).
    [CrossRef]
  11. S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, "Simple laboratory system for generating well-controlled atmospheric-like turbulence," in "Advanced Wavefront Control: Methods, Devices, and Applications II," presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds., (2004), vol. 5553, pp. 290-300.
  12. T. A. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," in "Advanced Wavefront Control: Methods, Devices, and Applications II," presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds., (2004), vol. 5553.
  13. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, "The Aggregate Behavior of Branch Points - Branch Point Density as a Characteristic of an Atmospheric Turbulence Simulator," in "2009 SPIE Annual Conference," R. Carerras, T. Rhoadharmer, and D. Dayton, eds., (SPIE, 2009).

2009 (1)

L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
[CrossRef]

2000 (2)

M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beamprojection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. 17, 53-62 (2000).
[CrossRef]

M. Schöck and E. J. Spillar, "Method for a quantitative investiagation of the frozen flow hypothesis," J. Opt. Soc. Am. 17, 1650-1658 (2000).
[CrossRef]

1999 (1)

E. O. Le Bigot and W. J. Wild, "Theory of branch-point detection and its implementation," J. Opt. Soc. Am. 16, 1724-1729 (1999).
[CrossRef]

1998 (2)

M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998).
[CrossRef]

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

1992 (1)

D. C. Johnston and B. M. Welsh, "Estimating the contribution of different parts of the atmosphere to optical wavefront aberration," Comput. Elect. Eng. 18, 467-483 (1992).
[CrossRef]

Fried, D. L.

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

Johnston, D. C.

D. C. Johnston and B. M. Welsh, "Estimating the contribution of different parts of the atmosphere to optical wavefront aberration," Comput. Elect. Eng. 18, 467-483 (1992).
[CrossRef]

Koivunen, A. C.

M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beamprojection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. 17, 53-62 (2000).
[CrossRef]

Le Bigot, E. O.

E. O. Le Bigot and W. J. Wild, "Theory of branch-point detection and its implementation," J. Opt. Soc. Am. 16, 1724-1729 (1999).
[CrossRef]

Poyneer, L.

L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beamprojection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. 17, 53-62 (2000).
[CrossRef]

Schöck, M.

M. Schöck and E. J. Spillar, "Method for a quantitative investiagation of the frozen flow hypothesis," J. Opt. Soc. Am. 17, 1650-1658 (2000).
[CrossRef]

M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998).
[CrossRef]

Spillar, E. J.

M. Schöck and E. J. Spillar, "Method for a quantitative investiagation of the frozen flow hypothesis," J. Opt. Soc. Am. 17, 1650-1658 (2000).
[CrossRef]

M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998).
[CrossRef]

van Dam, M.

L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
[CrossRef]

Véran, J. P.

L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
[CrossRef]

Welsh, B. M.

D. C. Johnston and B. M. Welsh, "Estimating the contribution of different parts of the atmosphere to optical wavefront aberration," Comput. Elect. Eng. 18, 467-483 (1992).
[CrossRef]

Wild, W. J.

E. O. Le Bigot and W. J. Wild, "Theory of branch-point detection and its implementation," J. Opt. Soc. Am. 16, 1724-1729 (1999).
[CrossRef]

Comput. Elect. Eng. (1)

D. C. Johnston and B. M. Welsh, "Estimating the contribution of different parts of the atmosphere to optical wavefront aberration," Comput. Elect. Eng. 18, 467-483 (1992).
[CrossRef]

J. Opt. Soc. Am (1)

L. Poyneer, M. van Dam, and J. P. Véran, "Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry," J. Opt. Soc. Am 26, 833-846 (2009).
[CrossRef]

J. Opt. Soc. Am. (4)

M. Schöck and E. J. Spillar, "Method for a quantitative investiagation of the frozen flow hypothesis," J. Opt. Soc. Am. 17, 1650-1658 (2000).
[CrossRef]

M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beamprojection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. 17, 53-62 (2000).
[CrossRef]

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

E. O. Le Bigot and W. J. Wild, "Theory of branch-point detection and its implementation," J. Opt. Soc. Am. 16, 1724-1729 (1999).
[CrossRef]

Proc. SPIE (1)

M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998).
[CrossRef]

Other (6)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press Inc., New York, NY, USA, 1998), 1st ed.

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, Inc., New York, NY, 1998).

S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, "Simple laboratory system for generating well-controlled atmospheric-like turbulence," in "Advanced Wavefront Control: Methods, Devices, and Applications II," presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds., (2004), vol. 5553, pp. 290-300.

T. A. Rhoadarmer, "Development of a self-referencing interferometer wavefront sensor," in "Advanced Wavefront Control: Methods, Devices, and Applications II," presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds., (2004), vol. 5553.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, "The Aggregate Behavior of Branch Points - Branch Point Density as a Characteristic of an Atmospheric Turbulence Simulator," in "2009 SPIE Annual Conference," R. Carerras, T. Rhoadharmer, and D. Dayton, eds., (SPIE, 2009).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (Bellingham, Wa, USA, 2007), 1st ed.

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Figures (5)

Fig. 1
Fig. 1

Branch point and noise circulation locations from three temporal measurements and connecting arrows for the example in Section 4.1. Each subplot shows three measurement frames. For each subplot, the locations of the circulations associated with t1 are shown on the bottom frame of the three frame stack; while those for t2 and t3 are shown on the middle and top layer respectively. (a) Three measurement frames showing one branch point (red) and one noise circulation (yellow) in each frame - all assumed to be positive for this illustration. (b) Blue arrows originate from circulations at time t1 and extend to all circulations in t2 and t3. Solid arrows connect different branch point measurements, while dashed arrows include noise circulations. (c) Black arrows originate from circulations at t2 and point to circulations in t3. With solid and dashed arrows indicating connections as previously. (d) Superpositions of all connecting arrows to show how velocity measurements begin to stack with this method. Here three of the twelve arrows have the same direction (solid arrows), all of the others have different directions (dashed arrows).

Fig. 2
Fig. 2

Initial demonstration data for Table 1. Left column: x-t projections of p(x,y,t). Right column: matching velocity distributions, D(vx). Top row: Table Configuration 1: r0 = 6.91, σ χ 2 = 0.19. Middle row: Table Configuration 2: r0 = 8.12, σ χ 2 = 0.29. Bottom row: Table Configuration 3: r0 = 6.25, σ χ 2 = 0.47. In each case both phase wheels were spinning, vbench = (1, −3). In Config. 1 the branch points are associated with the low altitude phase wheel and the spike is at 0.5 pixels/frame. In Config. 2 they are associated with the high altitude phase wheel and the spike exists at −1.5 pixels/frame. In Configuration 3, branch points form from both layers and two non-zeros spikes are seen at −1.5 and 0.5 pixels/frame respectively.

Fig. 3
Fig. 3

Plotted data for the Table Configurations 4 – 8. The x-t projection of the polarity array is shown in the left column and the corresponding velocity distributions, D(vx), on the right column. Note, the vertical lines seen in some of the projections to the left are bad frames of wavefront sensor data; they have no significant effect on the peaks of D(vx), demonstrating the robustness of the technique.

Fig. 4
Fig. 4

Velocity magnitudes estimated versus theoretical. The correlation between the set velocity and measured velocity is remarkably high. Also note, there appears to be only four low altitude measurements because configurations #4 and #5 have the same low altitude turbulence layer velocity, and this causes the two measurements to overlay in the plots.

Fig. 5
Fig. 5

Results for our composite ten layer turbulence atmosphere. (a) The projection of the polarity array onto the x-t plane. (b) The velocity distribution, D(vx), for the x-component. (c) A closeup of D(vx) with red lines overlaid at the positions of the pre-set velocities. Velocity peaks corresponding to atmospheric layers are clearly evident.

Tables (3)

Tables Icon

Table 1 Turbulence parameters used to demonstrate both that our method is invariant with respect to turbulence strength and also that the assumption of branch point persistence has experimental basis. Bold print is used to identify the branch point producing layers.

Tables Icon

Table 2 Turbulence parameters for the tests in Section 5.3. The turbulence conditions are shown in columns 3–6, while the corresponding ATS parameters are displayed in columns 7–8.

Tables Icon

Table 3 Turbulence layer velocity information. Configuration number and predicted velocities based on the set phase wheels step sizes as was shown in Table 2 compared with the estimated velocities determined from the locations of the peaks as displayed in Fig. 4, along with a percent error between the estimated and theoretical velocities.

Equations (6)

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r 0 = [ 0.423 k 0 2 0 L C n 2 ( z ) d z ] 3 / 5
σ χ 2 = 0.5631 k 0 7 / 6 0 L C n 2 ( z ) z 5 / 6 d z ,
C n 2 ( z ) C n 2 ( h 1 ) δ ( z h 1 ) + C n 2 ( h 2 ) δ ( z h 2 ) + + C n 2 ( h N ) δ ( z h N )
C t ( μ ) g ϕ ( r ) d μ = ± 2 π
| C t ( μ ) g ϕ ( r ) d μ | 2 π ɛ
v t u r b = 1000 ( 1.5 240 ) v b p ,

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