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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1046–1059
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Aggregate behavior of branch points - persistent pairs

Denis W. Oesch, Darryl J. Sanchez, and Carolyn M. Tewksbury-Christle  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1046-1059 (2012)
http://dx.doi.org/10.1364/OE.20.001046


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Abstract

Light propagating through atmospheric turbulence acquires spatial and temporal phase variations. For strong enough turbulence conditions, interference from these phase variations within the optical wave can produce branch points; positions of zero amplitude. Under the assumption of a layered turbulence model, our previous work has shown that these branch points can be used to estimate the number and velocities of atmospheric layers. Key to this previous demonstration was the property of branch point persistence. Branch points from a single turbulence layer persist in time and through additional layers. In this paper we extend persistence to include branch point pairs. We develop an algorithm for isolating persistent pairs and show that through experimental data that they exist through time and through additional turbulence.

© 2011 OSA

1. Introduction

A branch point in adaptive optics is defined as a point in the optical field where the amplitude goes to zero and the phase becomes undefined [1

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

,2

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

]. About this point of undefined phase, there is an associated helicity, a 2π circulation, described by the complex function e±, where θ describes the azimuthal angle.

These circulations contain a 2π discontinuity extending outward from each branch point. Two branch points of opposite helicity can be partnered to share their discontinuity, then referred to as a branch cut. Pairing branch points is typically an exercise in placing branch cuts for phase unwrapping [3

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

]. In this case, branch cuts are a tool used to find a solution to the two dimensional phase problem.

In the singular optics community the term for these phenomena is “optical vortex”. There has been significant research into the formation and development of these vortices [4

4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974). [CrossRef]

8

8. M. Chen and L. Roux, “Evolution of the scintillation index and the optical vortex density in speckle fields after removal of the least-squares phase,” J. Opt. Soc. Am. A 27, 2138–2143 (2008). [CrossRef]

] in propagating fields. However, the term “vortex” has also been attributed to beams carrying orbital angular momentum (OAM) [9

9. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

11

11. I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vorticies,” J. Opt. Soc. Am. B 20, 1169–1176 (2003). [CrossRef]

]. While this work contributes to the ongoing investigation of these two fields we wish to emphasize the localized nature of the phenomena and its role in wave front sensor (WFS) phase data, and so use the term from the adaptive optics community [1

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

, 2

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

, 12

12. 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]

, 13

13. T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008). [CrossRef] [PubMed]

], “branch point”.

Branch points form from interference effects within a traveling optical wave. When a plane wave interacts with a layer of the atmosphere it adopts variations in its phase according to the random structure of the turbulence. Through propagation, the phase variations couple into the amplitude. For sufficient propagation distances, interference between different regions of the wave leads to the formation of branch point pairs [14

14. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

]. These effects create a time dependent pupil plane branch point distribution encoded with atmospheric turbulence information.

The distribution of branch points in experimental WFS data appear random when viewed frame by frame. That is, branch points often seem to appear and disappear in the measurement. The segmented nature of the detector, pixel averaging, noise, and phase wrapping discontinuities cause difficulties in identifying all of the branch points. Along with the apparently random branch points, branch cuts also appear to shift from one arrangement to another between frames as unwrapping algorithms attempt to find the best least squares fit to the current phase measurement.

However, we have shown that pupil plane branch point measurements can be used to estimate the number and velocity of atmospheric layers [15

15. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010). [CrossRef] [PubMed]

]. In order for this information to be obtained branch points must persist, both in time as well as through additional turbulence. In this paper we show persistence is also a property of branch point pairs.

The existence of persistent pairs supports ongoing research at the Atmospheric Simulation and Adaptive-optic Laboratory Testbed (ASALT) at Starfire Optics Range. In our work characterizing the three dimensional atmosphere, persistent pairs provide an additional parameter in the form of the separation of the paired points. Like branch point density [16

16. 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,” Proc. SPIE 7466, 0601–0610 (2009).

], the separation of paired branch points from a single layer has been empirically shown to depend on the strength and distance of the turbulence layer [17

17. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

]. However for the separation to be a useful measurement, we must demonstrate that branch point pairs from one layer are identifiable through additional turbulence. Further we must be able to associate individual branch points with specific turbulence layers. These can both be accomplished through the identification of persistent pairs.

Additionally, ASALT has been researching the formation of angular momentum from atmospheric turbulence, specifically investigating the similarity between the helicity in the phase associated with branch points and that found in beams with orbital angular momentum [9

9. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

,18

18. J. Leach, S. Keen, M. J. Padget, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]

20

20. K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vorticies with a shack-hartmann wavefront sensor,” Opt. Express 18, 15448–15460 (2010). [CrossRef] [PubMed]

]. To this end, persistence of branch point pairs is fundamental to meeting the requirement of conservation of momentum given the usual apparently random nature of branch points in WFS data. Through this effort [21

21. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express 19, 25388–25396 (2011). [CrossRef]

, 22

22. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011). [CrossRef] [PubMed]

] we demonstrate the connection between branch points and photons carrying orbital angular momentum, bringing together the work of these different research fields.

We begin, in Section 2, with a brief review of our experimental setup and its underlying assumptions. Next, in Section 3, we present the process for finding persistent branch points in WFS data. In Section 4, we demonstrate the pairing algorithm with experimental data. Then we discuss the results of these experiments in Section 5.

2. Problem formulation and assumptions

We use a self referencing interferometer (SRI) [23

23. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004). [CrossRef]

] WFS, that returns modulo-2π phase on a 256x256 spatial grid. Our WFS gathers short exposure frames at a constant frequency. These measurements include both photon and read noise.

The gradients of the phase about each 2x2 pixel region of the camera, are used to identify the presence of a branch point [1

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

]. For each of these regions, the sum of the gradients is calculated. If the magnitude of the sum is greater than 2π a branch point is assigned to the intersection of the four pixels. An array of these locations of size 255x255, is called the polarity array. It is composed of zeros or ±1 for those positions of branch points with positive or negative helicity respectively.

The atmospheric turbulence model consists of separated horizontal layers with no turbulence in between. Each layer is isotropic, imparts only phase changes to the propagating wave and is uncorrelated with respect to other turbulence layers. Propagation through a turbulence layer is modeled using geometric optics and wave optics models propagation between layers [24

24. D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992). [CrossRef]

]. This model, where the WFS exposure time is short enough that the turbulence is essentially frozen, has been experimentally demonstrated [25

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

27

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

]. The light is modeled as a monochromatic point source sufficiently far to have plane waves at the top of the atmosphere.

The Rytov number, σχ2, is used in characterizing atmospheric turbulence, written as
σχ2=0.5631k07/60LCn2(z)z5/6dz,
for Kolmogorov turbulence [28

28. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).

], where k0 is the wave number, Cn2(z) profiles the turbulence strength, 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 σχ20.1. Roggemann notes their appearance at σχ2>0.05 [29

29. M. C. Roggeman 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]

]. Rytov is a description of the degree of scintillation in the optical beam. The Rytov number is known to saturate at σχ20.4. In our earlier work [16

16. 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,” Proc. SPIE 7466, 0601–0610 (2009).

] we found that the branch point density similarly saturates at about this point. For the work done here, the generated turbulence is in the non-saturated regime.

Fig. 1 The phase associated with branch points. (a) phase of a single branch point. (b) the phase of a branch point pair. Note, the phase for a single branch point extends to infinity, i.e. it occupies the extent of the propagating beam. The phase for a pair of branch points also extends to infinity, but goes to zero as the distance from the pair increases.

3. Pairing algorithm

Starting with temporal SRI WFS data comprised of multiple frames of mod2π phase our pairing algorithm has four main stages; (1) branch point identification, (2) determining branch point velocities, (3) 2-D pairing, and (4) velocity filtering. The process is shown at a cartoon level in Fig. 2.

Fig. 2 Cartoon of the steps in the identification of persistent pupil plane branch point pairs. Red and green dots indicate the positive and negative circulations respectively. (a) The empty polarity array. (b) The polarity array with ±1’s at the locations of positive and negative circulations respectively. (c)Diagram showing the estimated velocities from the positive branch point locations. (d)Blue lines indicate the initial pairings following frame by frame use of a walking algorithm. (e)Pairings following the use of piston shifting technique. (f)Overlay of measured velocity to incorporate 3rd dimension of WFS data.

In what follows the experimental data consists of 100 consecutive frames of 256x256 pixels per frame.

3.1. Branch point identification

Given the 256x256x100 array of WFS data, we create a polarity array of 255x255x100 zeros. A few frames of this array are represented in Fig. 2(a). Branch points are identified, as described in Section 2. Then at the locations of the branch points the polarity array filled with ±1 appropriately. The locations of positive and negative circulations are shown as red and green dots in Fig. 2(b). The polarity array forms the basis of subsequent processing.

3.2. Estimation of branch point velocities

3.3. 2-D pairing

Two dimensional pairing relies on a “walking” algorithm that, in a given frame of WFS data, follows a branch cut from one branch point to its partner. Isolating the branch cuts from phase data simplifies the work by creating a 2-D map of the paths between branch points. We’ve developed a simple modification to the elementary circulation technique, called the difference of gradients, that identifies the 2π discontinuities in the phase data

3.3.1. The difference of gradients

To locate 2π discontinuities in a frame of WFS data, we compare the signs of the x and y gradients computed as
δxi,j=ϕi,jϕi+1,jδyi,j=ϕi,jϕi,j+1,
(1)
with those computed in mod2π space,
δexi,j=mod2π(ϕi,jϕi+1,j)δeyi,j=mod2π(ϕi,jϕi,j+1).
(2)
We find that the signs of Eqs. (1) and (2) are reversed in the presence of a discontinuity. This leads to a simple process for reducing a frame of WFS data to a map of 2π discontinuities. Combining the signs of Eqs. (1) and (2), creates two arrays composed only of delta functions at the positions of discontinuities,
Δxi,j=(sign(δxi,j)sign(δexi,j))/2Δyi,j=(sign(δyi.j)sign(δeyi,j))/2.
(3)
The arrays of Eq. (3), by the nature of the differences in Eqs. (1) and (2), are 256x255 and 255x256 for x and y respectively. Shifting and adding the absolute values of these arrays creates a 255x255 map of the 2π discontinuities,
BCi,j=|Δxi,j|+|Δxi+1,j|+|Δyi,j|+|Δyi,j+1|.
(4)
This is shown pictorially in Fig. 3.

Fig. 3 (a) Sample WFS phase and (b) a map of the discontinuities from difference of gradients method, as produced by Eq. (4).

This map is the input to the walking algorithm. Starting from one branch point the algorithm simply follows the discontinuity until it locates a partner of opposite polarity. These two points are added to a list of possible pairs, see Fig. 2(d).

3.3.2. Scanning piston

Unfortunately, due to wrapping discontinuities, not all of the branch points are connected in a single frame using the walking algorithm. As there is no difference between the branch cut 2π discontinuities and those due to wrapping effects from the phase being represented in mod2π space, this approach can lead to incorrect or missed pairs. A sample of this is shown in Fig. 4. Here several branch point pairs have been highlighted in the insets to show how the discontinuities fell in the difference of gradient method. Sometimes the pair is clearly connected by a branch cut. However, wrapping discontinuities can interact with branch cuts in such a way as to obscure the correct branch cut placement.

Fig. 4 A sample map of discontinuities with enlarged sections showing some of the branch point pairs and how the discontinuities are distributed around them.

While the location of the wrapping discontinuities depends on the global piston of the measured phase, the branch cuts do not. Therefore, by adding piston to the phase, the 2π discontinuities not associated with the branch points move, providing a different view of the branch cuts, see Fig. 5. Through varying the amount of added piston between 0 and 2π while applying the walking algorithm at each step, provides a large set of possible branch point pairs for a given frame of WFS data. Sorting these pairings according to frequency greatly reduces the number of errors due to phase wrapping.

Fig. 5 Sample WFS phase (left), (a) original and (b) with a piston shift of π added. Maps of discontinuities from the difference of gradients method (right), (c) from the original phase and (d) from the phase with the added piston.

3.4. Velocity filtering

Up to this point, the pairing has been applied on a frame by frame basis. Now we look at the polarity array as a three-dimensional object, see Fig. 6(a). Again, the red and green points indicate the positive and negative circulations respectively. The blue lines indicate where 2-D pairings have been identified. Also note, the common velocity component stands out strongly in this view of a sample of single layer turbulence.

Fig. 6 Single layer example of pairings using only single frame techniques. (a) The polarity array shown in three dimensions and (b) a close-up of the array showing numerous 2-D only pairings.

A close-up, Fig. 6(b), reveals some of the issues discussed in the 2-D pairing sections. Paired points moving in time through the polarity array should appear as a collection of ladders. However due to their close spacing the 2-D pairing approach occasionally makes errors which are seen in the close-up as blue line segments crossing between the ladders.

Comparing branch point pairs from frame to frame now, we can filter the pairings according to their frequency of occurrences in time. Identifying those pairs that persist through time gives further confidence that these are correct branch point pairings. In Fig. 7(a) the polarity array is shown after filtering the pairs by velocity. The close up in Fig. 7(b) shows a more orderly series of pairings than what was seen in Fig. 6(b). All of the blue line segments now show neatly arranged ladder like structures in the polarity array.

Fig. 7 Single layer example of pairing now incorporating velocity filtering. (a) The polarity array shown in three dimensions and (b) a close-up of the array showing numerous 2-D only pairings.

4. Two layer demonstration

Having demonstrated the technique in the idealized, single layer case, we now demonstrate the technique in a two layer experiment. Table 1 shows five simulated turbulence conditions. For each configuration, three tests were conducted. The first test, used only the low altitude layer for generating the turbulence. The second test, similarly only used the high altitude layer. For the final test in each configuration both layers were included in generating the turbulence. This approach provides a means of comparison of how well the technique worked at separating the branch points between the two layers.

Table 1. Turbulence conditions for one and two layer simulations. Dashes indicate when a phase wheel wasn’t included in the optical path.

table-icon
View This Table

4.1. The experimental apparatus

The atmospheric turbulence simulator (ATS) [30

30. S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004). [CrossRef]

] that provides the turbulence for the ASALT adaptive optical systems creates a one or two layer atmosphere where the standard parameters [28

28. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).

]; Fried’s parameter (r0), Rytov number ( σχ2) and Greenwood Frequency (fG), are controllable and repeatable. All of the data presented here was collected with a 256x256 pixel resolution SRI WFS, scaled to a 1.5 meter aperture. The placement of the phase wheels selects the r0 and σχ2 parameters for the atmospheric turbulence. Rotating the phase wheels provides simulated wind speed for each layer.

4.2. Turbulence parameters

Table 1 displays the ATS configuration parameters alongside the simulated atmospheric conditions by layer. The cumulative pupil plane parameters, r0 and σχ2 are shown for quick comparison. Dashes indicate where a phase wheel is not contributing to the generated turbulence.

4.3. Close-up of Case 1

To begin the discussion, we look closely at Case 1. Figure 8 shows the results of the persistent pairing technique, with red and green points as before for the circulations. Fig. 8(a) shows all of the identified circulations. The two layer velocities are clearly evident in the two sets of sloping branch point trails. Each velocity appears to be represented by several strong trails of branch point pairs moving steadily through the polarity array. Additionally there are many weaker trails that parallel both velocities.

Fig. 8 3D examination of Case 1. (a)Identified circulations for two layer configuration. (b) Persistent pairs identified for the low altitude layer. (c) Persistent pairs identified for the high altitude layer. (d) Circulations not found to follow one of the measured velocities, noise circulations.

Figures 8(b) and 8(c) show the identified persistent pairs for the low and high altitude layers respectively. The two sets of branch point trails visible in Fig. 8(a) are now split between the two layers shown in Figs. 8(b) and 8(c). Figure 8(d) shows the circulations that were not found to pair consistently along one of the velocity components.

4.3.1. Examination of the unpaired points

It is expected that points which don’t repeat consistently along the velocity are noise, but examination of the unpaired points shows some trail like behavior, points that appear to match the velocity of the high altitude layer. Typically these apparent trails nearly overlap existing pairs and are caused by additional circulations appearing more frequently around persistent pairs than elsewhere. These additional circulations shift their location and orientation relative to the branch points they appear to follow frame by frame, which is why they aren’t selected by the pairing algorithm as an additional persistent pair. Aside from their proximity to the identified pairs these points behave in all other ways as noise circulations. We acknowledge, that some branch points may be discarded, but this does not interfere with demonstrating the existence of persistence pairs in our layered atmospheric model.

What is also interesting is that the two layer case results in more persistent pairs associated to both layers. Noise circulations increase as well. We are looking into modeling these effects. We have ongoing work to examine the interaction of multiple layers and the formation of these additional circulations.

4.3.2. 1 to 2 layer comparison

Fig. 9 One and Two layer comparison. Top row: Persistent pairs identified for (a) the lower layer and (b) the upper layer from the combined two layer atmosphere. Bottom row: Persistent pairs identified for (c) the lower layer alone and (d) the upper layer alone (right). Pairs in the isolated layers appear in the combined layer results - indicating that the persistent pair method finds pairs through additional turbulence.

The top row shows the persistent pairs identified from the two layer atmosphere case, the low altitude layer in Fig. 9(a) and the high altitude layer in Fig. 9(b). The bottom row shows the same two layers but taken individually, Case 1a and 1b. Where the low altitude layer was alone in generating the turbulence, Fig. 9(c) shows the identified persistent pairs. The persistent pairs of the isolated high altitude layer is shown in Fig. 9(d).

The low altitude layer, Fig. 9(c), shows one long lasting branch point pair’s trail very clearly, as well as three other much less pronounced trails. Looking at the pairs identified in the two layer case for the low altitude turbulence, Fig. 9(a), the strongest trail of the single low altitude layer case, Fig. 9(c), is still identifiable in Fig. 9(a). Further the less pronounced trails may show more frequent detections in the two layer case.

The high altitude layer, Fig. 9(d), shows two very strong trails and two minor ones. The strongest two are again displayed in Fig. 9(b), while the weaker trails are definitely showing more detections that in the single layer case.

It is clear that the branch point pairs produced by the isolated layers, bottom row, are still present in the two layer case, top row.

4.4. Other configuration results

Case 1 supports our conjecture of the persistence of branch point pairs. Now we examine the rest of the configurations, Fig. 10, following what was shown in Fig. 9. In this graphic, the Case numbers are identified along the left side. The column marked (a) represents the configurations for the isolated low altitude layer, while the column labeled (b) identifies those cases where only the high altitude layer was used. The (c) column represents the full polarity array collected for the combined two layer experiment in each case. While the columns marked (c,1) and (c,2) identify the results from the pairing technique applied to the two layer cases for the low and high layers respectively.

Fig. 10 One and Two layer comparisons for all five configurations, on Table 1. Each row is proceeded by the Case number for the presented results. Identified persistent pairs for (a) the isolated low altitude layer, (b) the isolated high altitude layer, (c) the full polarity array for the combined two layer experiment and then (c,1) and (c,2) the low and high altitude layer pairs identified from our technique respectively.

In all cases the persistent pairs identified in the single layer cases, Fig. 10 columns (a) and (b), are captured again in the two layer cases, Fig. 10 column (c).

5. Discussion

We have demonstrated, within the constraints and assumptions of our model, that branch point pairs are persistent even when encountering additional turbulence. That they remain an enduring feature of the traveling wave is significant in our ongoing work to understand atmospheric turbulence.

The establishment of persistence in branch points pairs supports the ASALT work developing the connection between branch points and the formation of orbital angular momentum in atmospheric turbulence [21

21. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express 19, 25388–25396 (2011). [CrossRef]

, 22

22. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011). [CrossRef] [PubMed]

]. Further, given that branch point pairs can endure through additional turbulence, this identifies persistence as a property of creation pairs [14

14. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

], pairs formed together that are still identifiable as pairs following propagation. Then the separation of those pairs may serve as a probe of the atmospheric turbulence conditions.

However, much research remains to be done. For instance, with changing turbulence strength, our ability to isolate persistent pairs becomes more difficult. In the weakest example, Case 5, on Fig. 10(c,2), only a few detections of the strong trail in Fig. 10(b) are seen. However, looking at the full polarity array for that Case, Fig. 10(c), it is clear that the high altitude layer is strongly represented. The pairing algorithm however had difficulty separating out those persistent pairs for Fig. 10(c,2). While the pairs may persist, the frequency with which they are detected may decrease significantly over the single layer case depending on the strength of the additional turbulence increases. Interestingly, Case 5 represents the weakest strength turbulence layers, which therefore required greater propagation distances to achieve comparable Rytov parameters to the other configurations.

Furthermore, multiple layers result in more branch points than each layer alone would produce. The complex interactions of turbulence strength and propagation distances, multiple layers, distributed turbulence, as well as the limits of branch point persistence are reserved for future papers.

Acknowledgments

We would like to express our gratitude to the Air Force Office of Scientific Research for their support in funding this research.

References and links

1.

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

2.

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

3.

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

4.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974). [CrossRef]

5.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]

6.

I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. 11, 1644–1652 (1994). [CrossRef]

7.

F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. 21, 664–670 (2004). [CrossRef]

8.

M. Chen and L. Roux, “Evolution of the scintillation index and the optical vortex density in speckle fields after removal of the least-squares phase,” J. Opt. Soc. Am. A 27, 2138–2143 (2008). [CrossRef]

9.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

10.

J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).

11.

I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vorticies,” J. Opt. Soc. Am. B 20, 1169–1176 (2003). [CrossRef]

12.

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]

13.

T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008). [CrossRef] [PubMed]

14.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

15.

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010). [CrossRef] [PubMed]

16.

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,” Proc. SPIE 7466, 0601–0610 (2009).

17.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

18.

J. Leach, S. Keen, M. J. Padget, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]

19.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

20.

K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vorticies with a shack-hartmann wavefront sensor,” Opt. Express 18, 15448–15460 (2010). [CrossRef] [PubMed]

21.

D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express 19, 25388–25396 (2011). [CrossRef]

22.

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011). [CrossRef] [PubMed]

23.

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004). [CrossRef]

24.

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

25.

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

26.

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

27.

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

28.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).

29.

M. C. Roggeman 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]

30.

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004). [CrossRef]

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: August 12, 2011
Revised Manuscript: November 10, 2011
Manuscript Accepted: December 1, 2011
Published: January 4, 2012

Citation
Denis W. Oesch, Darryl J. Sanchez, and Carolyn M. Tewksbury-Christle, "Aggregate behavior of branch points - persistent pairs," Opt. Express 20, 1046-1059 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1046


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References

  1. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt.31, 2865–2882 (1992). [CrossRef] [PubMed]
  2. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A15, 2759–2768 (1998). [CrossRef]
  3. D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).
  4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A336, 165–190 (1974). [CrossRef]
  5. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt.40, 73–87 (1993). [CrossRef]
  6. I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am.11, 1644–1652 (1994). [CrossRef]
  7. F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am.21, 664–670 (2004). [CrossRef]
  8. M. Chen and L. Roux, “Evolution of the scintillation index and the optical vortex density in speckle fields after removal of the least-squares phase,” J. Opt. Soc. Am. A27, 2138–2143 (2008). [CrossRef]
  9. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992). [CrossRef] [PubMed]
  10. J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev.90, 133901 (2003).
  11. I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B20, 1169–1176 (2003). [CrossRef]
  12. 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]
  13. T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express16, 6985–6998 (2008). [CrossRef] [PubMed]
  14. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE7466, 0501–0512 (2009).
  15. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express18, 22377–22392 (2010). [CrossRef] [PubMed]
  16. 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,” Proc. SPIE7466, 0601–0610 (2009).
  17. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE7816, 0501–0513 (2010).
  18. J. Leach, S. Keen, M. J. Padget, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express14, 11919–11924 (2006). [CrossRef] [PubMed]
  19. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett.22 (2010).
  20. K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vorticies with a shack-hartmann wavefront sensor,” Opt. Express18, 15448–15460 (2010). [CrossRef] [PubMed]
  21. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express19, 25388–25396 (2011). [CrossRef]
  22. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express19, 24596–24068 (2011). [CrossRef] [PubMed]
  23. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE5553 (2004). [CrossRef]
  24. D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng.18, 467–483 (1992). [CrossRef]
  25. M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE3353, 1092–1099 (1998). [CrossRef]
  26. M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A17, 1650–1658 (2000). [CrossRef]
  27. L. Poyneer, M. van Dam, and J. P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemtry,” J. Opt. Soc. Am. A26, 833–846 (2009). [CrossRef]
  28. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).
  29. M. C. Roggeman 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]
  30. S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE5553, 290–300 (2004). [CrossRef]

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