## The Aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers |

Optics Express, Vol. 18, Issue 21, pp. 22377-22392 (2010)

http://dx.doi.org/10.1364/OE.18.022377

Acrobat PDF (1021 KB)

### 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

## 1. Introduction

*r*

_{0}, whose value is typically between 5 and 30 cm for visible wavelengths. In the absence of atmospheric turbulence, the spatial resolution of a telescope is proportional to 1/

*D*, where D is the diameter of the telescope, and is typically a few meters or larger; in the presence of atmospheric turbulence, the spatial resolution is proportional to 1/

*r*

_{0}. The reduction in resolution as a result of atmospheric turbulence is due to the random index of refraction fluctuations in both position and time. Significant effort has been expended over the past several decades to develop ways to remove atmospheric turbulence blurring in order to increase spatial resolution.

3. M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” 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]

## 2. Problem formulation and assumptions

*π*phase of the field on a discrete

*N*by

*N*spatial grid at discrete time intervals. It is necessary that our WFS provide

*F*measurements equally spaced in time by Δ

*t*and that the exposure time for each measurement is short enough so that the atmosphere is essentially frozen. Further we expect that these measurements shall include components due to photon and read noise reflecting real-world systems for measuring phase.

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]

*r*

_{0}, given by and the Rytov parameter,

*k*

_{0}is the wave number,

*z*is the altitude with the telescope at

*z*= 0, and

*L*is the maximum height of the turbulence. We have observed experimentally that branch points begin to appear in the data at

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

*N*turbulence layers, the

*h*. Here,

_{i}*δ*represents the standard Delta function. The layered-turbulence-model approximations to

*r*

_{0}and

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]

## 3. Branch points

*U*(

**r**) =

*A*(

**r**)

*expi*[

*ϕ*(

**r**)] be a two-dimensional complex field with position vector

**r**, amplitude

*A*(

**r**), and phase

*ϕ*(

**r**). Where bold text is used to identify vector quantities. When

*A*(

**r**) = 0,

*ϕ*(

**r**) is undefined and the field is said to have a branch point at

**r**. Because the presence of any uncertainty or spatial or temporal averaging of the values of

*U*(

**r**) will make it effectively impossible to identify locations where

*A*(

**r**) = 0 in the field, a more robust method of identifying branch points is needed. To this end, let

**g**

*(*

_{ϕ}**r**) be the spatial gradient of

*ϕ*(

**r**) at

**r**. Then the optical field is said to have a branch point at

**r**′ if, for any arbitrarily small contour

*C*enclosing

**r**′, we have where

*μ*is a parameter defining a location on contour

*C*,

**r**(

*μ*) is the corresponding location, and

**t**(

*μ*) is a vector tangent to the contour at

**r**(

*μ*) such that |

**t**(

*μ*)|

*dμ*is the length of the contour between

*μ*and

*μ*+

*dμ*[9

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

*x*2 block of pixels for every 2

*x*2 block possible in the

*N*by

*N*frame of phase data.

*π*; instead, a continuum of values clustered about zero, +2

*π*and −2

*π*will be computed. To account for this noise, we say that a branch point exists when with

*ɛ*a small constant. For the purposes of discussion, at any

**r**for which Eq. (4) holds, we will say that a “circulation” in phase exists at that point and simply call it a “circulation”.

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]

*ɛ*= 0 to be adequate for our purposes and is what is used in processing all our experimental data.

## 4. Application of branch points to identifying turbulence properties

*N*turbulence layers with measurements in the presence of random and fixed-pattern noise.

### 4.1. Illustrative example

**v**

_{1}, sufficiently distant from the telescope that branch points can form. Consider the situation where we have three frames of measured phase of the pupil field, collected at times

*t*

_{1},

*t*

_{2}, and

*t*

_{3}, where

*t*

_{i}_{+1}=

*t*+ Δ

_{i}*t*. Assume that, after estimating the locations of the circulations in each of the frames, each frame is found to contain two circulations. One of the circulations in each frame is due to the turbulence layer and the other is due to spatially-random measurement noise.

*t*

_{1}and draw vectors from those at time

*t*

_{1}to each circulation from times

*t*

_{2}and

*t*

_{3}. Similarly we connect those from time

*t*

_{2}to the ones from time

*t*

_{3}. These vectors appear as arrows in Fig. 1, with arrows originating from circulations at

*t*

_{1}in blue and those from

*t*

_{2}in black. Due to the simplified example, the circulations due to branch points are known, and vectors in Fig. 1 between branch points are shown with thick lines.

*θ*with respect to the horizontal axis, and speed, |Δ

**r**/Δ

*t*|. In this simple example, all of the vectors in the set have exactly the same velocity vector. Let

*M*be the demagnification factor mapping distance on the sky to distance in the measurement plane, and let

*θ*be the rotation of the on-sky coordinate system relative to the measurement plane coordinate system. Then our estimate of the velocity vector direction is equal to

_{r}*θ*+

*θ*, and our estimate of the velocity magnitude is equal to

_{r}*M*Δ|

**r**/Δ

*t*|.

### 4.2. Generalization of the method

*N*. In this case, we can identify the existence of

*N*turbulence layers by the presence of

*N*unique velocity sets. A velocity set is a collection of measurements of the same velocity vector within WFS phase data. However, if two turbulence layers are translating in the same direction and at the same angular velocity, they will be indistinguishable and identified as a single layer

## 5. Experimental verification of theory

### 5.1. The experimental apparatus

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.

*x*256. For all of the turbulence conditions used we collected 200 frames of data producing phase data arrays 256

*x*256

*x*200 in size. We also define a new array with size 255

*x*255

*x*200, called the polarity array,

*p*(

*x,y,t*). The positions, (

*x,y*), of the polarity array correspond to the locations of the intersections of the four wavefront sensor pixels for each frame

*t*. The locations of the circulations in each frame of the phase data are found as described in Section 3. At the location of each identified circulation, a ±1 is placed, depending on the sign of Equation 4, in the polarity array. This transforms the collected frames of phase data to a series of maps of the positions and polarities of the identified circulations indicated by values of 0, ±1.

*V⃗*=

*V*+

_{x}î*V*

_{y}*ĵ*. To determine the x-component of the velocity we begin with a cross section of the polarity array identified by

*p*(

*x*, 1,

*t*). This ‘x-t slice’ is of size, 255

*x*200. This slice will contain both positive and negative circulations and though we don’t know which circulations are branch points and which are noise, we do know that branch points don’t change polarity. So we only calculate velocities,

*V*= Δ

_{x}*x*/Δ

*t*, between like circulations. The determined velocities are arranged as a histogram in what we call the velocity distribution,

*D*(

*v*), for the x-component. More velocities are calculated from each of the remaining 254 ‘x-t’ slices and added to the velocity distribution to create a full image of all possible velocities for the x-component. The velocity distribution for the y-component is compiled in the same manner, using ‘y-t’ slices of the polarity array.

_{x}*D*(

*v*) and

_{x}*D*(

*v*) when plotted against the range of possible velocities. These spikes occur at the locations of the branch points respective velocity components as will be seen in the experimental demonstration.

_{y}### 5.2. Initial demonstration

*p*(

*x,y,t*), calculated for each atmospheric configuration. Since the direction of the turbulence motion is parallel to the x-axis of our WFS we will focus on the x-component of the results. For ease of viewing the polarity array as a whole, summations of the array along the y-axis are shown in Fig. 2 plots (a), (c), and (e) for each configuration. These are projections of array onto the

*x*–

*t*plane, identified as

*Proj*

_{x–t}*p*(

*x, y,t*) in the plots. Branch point trajectories, called trails, are clearly visible in these projections. The velocity distribution for the x-components,

*D*(

*v*) are shown in Figs. 2 plots (b), (d), and (f) for each configuration. The peaks, characteristic of turbulence layers, are readily apparent.

_{x}*pixel*/

*frame*. The contribution from noise terms is nearly non-existent as would be expected be for low density.

*pixel*/

*frame*. The contribution from noise is still very weak however in the projection of the x-t slices there is evidence of more noise circulations than in the previous case. This case demonstrates the validity of the assumption of branch point persistence for the frozen flow model. Even though both phase wheels are moving, the strong narrow peak indicates there isn’t a significant interaction between these branch points and the low altitude phase wheel.

*pixel/frame*. The zero term is part of the cross-term noise and the other two peaks match those found for the same velocities in the previous two configurations. Also, the broad distribution is indicative of noise and cross terms beginning to be non-negligible in the velocity distribution.

### 5.3. Experimental measurement of turbulence layer velocity

*x*–

*t*plane, and the right column the resulting

*D*(

*v*). The slowest turbulence layer velocities are plotted on the top row increasing to the bottom. The characteristic peaks in

_{x}*D*(

*v*) are clearly visible in all cases. The measured velocities corresponding to these peaks is presented in the next section on Table 3.

_{x}*v*is the turbulence layer velocity (in meters/second),

_{turb}*v*is the measured velocity assigned to the branch point motion (in pixels/frame). The constants for conversion to output space are obtained by noting the pupil occupies 240 pixels on the WFS camera and a sensor pixel in output space is 1.5 m / 240 and the system frame rate is 1 kHz.

_{bp}### 5.4. Measurement of the number of turbulence layers

#### An Artificially Large Number of Turbulence Layers

*p*(

*x,y,t*) was constructed from the compilation of the five data sets shown in Fig. 3 and this amalgamated data set is plotted in Fig. 5(a).

*D*(

*v*) was then calculated as before and is plotted in Fig. 5(b). As can be seen,

_{x}*p*(

*x, y,t*) is filled with large numbers of branch point trails and also noise. Even so, after processing, peaks in the velocity distribution are still seen in

*D*(

*v*).

_{x}## 6. Discussion and summary

## Acknowledgments

## References and links

1. | J. W. Hardy, |

2. | D. C. Ghiglia and M. D. Pritt, |

3. | M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” SPIE |

4. | M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. |

5. | D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Elect. Eng. |

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 |

7. | R. J. Sasiela, |

8. | M. C. Roggemann and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. |

9. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. |

10. | E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. |

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,”, |

12. | T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” in “Advanced Wavefront Control: Methods, Devices, and Applications II,” , |

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 “ |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: July 23, 2010

Revised Manuscript: September 14, 2010

Manuscript Accepted: September 20, 2010

Published: October 7, 2010

**Citation**

Denis W. Oesch, Darryl J. Sanchez, and Charles L. Matson, "The Aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers," Opt. Express **18**, 22377-22392 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-21-22377

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### References

- 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).
- M. Schöck and E. J. Spillar, "Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor," Proc. SPIE 3353, 1092-1099 (1998). [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]
- 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]
- 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]
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (Bellingham, Wa, USA, 2007), 1st ed.
- 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]
- 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).

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