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 [1,2]. About this point of undefined phase, there is an associated helicity, a 2π circulation, described by the complex function e^±iθ, 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]. 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 [4–8] in propagating fields. However, the term “vortex” has also been attributed to beams carrying orbital angular momentum (OAM) [9–11]. 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, 2, 12, 13], “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]. 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]. 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], 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]. 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 [9,18–20]. 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, 22] 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

In this work, we are using the same measurements and atmospheric model presented with our previous research [15].

We use a self referencing interferometer (SRI) [23] 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]. 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]. This model, where the WFS exposure time is short enough that the turbulence is essentially frozen, has been experimentally demonstrated [25–27]. The light is modeled as a monochromatic point source sufficiently far to have plane waves at the top of the atmosphere.

The Rytov number, ${\sigma }_{\chi }^2$, is used in characterizing atmospheric turbulence, written as

The helical phase surrounding each branch point, shown in Fig. 1(a), contains a 2π discontinuity extending outward from the branch point. Figure 1(b) shows two branch points of opposite helicity sharing a common branch cut. In typical branch cut placing problems as those presented in [3], the work is of pairing branch points is being done on a single frame of phase data. Although the locations of branch points in the wrapped phase data are fixed by the turbulence, the arrangement of the branch cuts is not and in order to obtain a correct placement additional information is needed [3]. The suggestion of additional information usually takes the form of guide maps or other metrics to weigh how the branch cuts are placed. In previous work, no consideration is given to using a time series of phase data as is collected with a WFS in adaptive optics. Our additional constraint relies on identifying the arrangement of branch cuts through time. This additional constraint allows for proper pairing of branch points, as is shown in the next section.

 

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.

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3. Pairing algorithm

Starting with temporal SRI WFS data comprised of multiple frames of mod_2π 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.

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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

For every identified branch point in the polarity array an instantaneous velocity vector, Δx/Δtî+ Δy/Δtĵ, is calculated with respect to every other point of like polarity throughout the 255x255x100 array, with î and ĵ representing unit vectors in the x and y directions. Our previous research has shown that a histogram of all these velocity components contains strong correlation peaks that signify the branch point group velocities [15]. The estimated instantaneous velocity vectors for the positive circulations are shown in Fig. 2(c) in purple. Further processing is then done only using the well-defined, peak 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

 

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

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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 mod_2π 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.

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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.

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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.

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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.

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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.

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4.1. The experimental apparatus

The atmospheric turbulence simulator (ATS) [30] that provides the turbulence for the ASALT adaptive optical systems creates a one or two layer atmosphere where the standard parameters [28]; Fried’s parameter (r₀), Rytov number ( ${\sigma }_{\chi }^2$) and Greenwood Frequency (f_G), 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 ${\sigma }_{\chi }^2$ 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 ${\sigma }_{\chi }^2$ 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.

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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

Next we look closer at the persistent pairs identified from the two layers, Case 1c, with the test configurations for the individual layers, Case 1a and 1b, in Fig. 9 to determine how well the pairing algorithm works with additional turbulence.

 

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.

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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.

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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).

Also overall density for the two layer case is higher than the sum of the individual layers. As was shown with Case 1 earlier, all cases show additional pairs related to both layers.

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