This is the second of two papers demonstrating that photons with orbital angular momentum can be created in optical waves propagating through distributed turbulence. In the companion paper, it is shown that propagation through atmospheric turbulence can create non-trivial angular momentum. Here, we extend the result and demonstrate that this momentum is, at least in part, orbital angular momentum. Specifically, we demonstrate that branch points (in the language of the adaptive optic community) indicate the presence of photons with non-zero OAM. Furthermore, the conditions required to create photons with non-zero orbital angular momentum are ubiquitous. The repercussions of this statement are wide ranging and these are cursorily enumerated.
© 2011 Optical Society of America
Orbital angular momentum is a quantum state of a photon. On the other hand, a classical description better describes the macroscopic phenomenon of propagation through atmospheric turbulence.
Atmospheric turbulence is a distributed, but locally mild, disturbance. For optical beams propagating through it, its defining feature is randomly varying index fluctuations. Being locally mild, these fluctuations imprint a phase-only disturbance on the beam. Propagating further, these random phase fluctuations, due to diffraction, couple into the amplitude creating a beam with randomly varying phase and amplitude. Experimentally and in simulation, it has been shown that with further propagation, the scintillation can grow to such an extent that points of zero amplitude appear in the beam. Associated with these zeros are 2π circulations in phase--the zero with its associated circulation in phase are called branch points by the adaptive optic community . The zeros are experimentally unmeasurable due to averaging over finite pixel sizes. However, the phase can be measured by standard adaptive optic wavefront sensors and these measurements are used in adaptive optics to identify branch points. Not surprisingly, since branch points are created by a random phenomenon, in all measurements they appear and disappear randomly.
In contrast, orbital angular momentum (OAM) states are well defined and, in fact, their azimuthal functions, eimϕ, form a basis of the azimuthal coordinate, ϕ, in ℝ2. For instance, OAM has been shown to exist  in Laguerre-Gaussian beams. In general, the set of orbital angular momentum wave functions vary in functional form with m, but the two lowest states are given by a uniform variation in azimuthal angle in phase, specifically e±iϕ. In what follows, we are only concerned with these two states. It’s well known that all the non-zero OAM states cause the Poynting vector to helically spiral about the direction of propagation  and conversely a Poynting vector spiraling about the direction of propagation necessarily implies the existence of OAM (see Appendix A). Measurement of the helical structure with a wavefront sensor returns a 2π circulation in phase  identical to that found in branch points [4, 5]. Restricted, then, to the two lowest states and considering only the phase associated with branch points, there is a superficial similarity between branch points and the lowest OAM states.
However on physical grounds (see Appendix A), there are three fundamental problems with this identification. (a) The term in the wave equation that creates angular momentum is λ−2 smaller in the mean--10−14 at optical wavelengths--than the non angular momentum components, and this is so trivial as to preclude OAM in quantities of any consequence. On the other hand, in the right turbulence conditions, branch points are ubiquitous, and if branch points are to be shown to be markers for OAM states, this disparity of order 1014 that must be addressed. (b) Secondarily, momentum is conserved and since branch points randomly appear and disappear, they cannot be OAM states. (c) Tertiarily, atmospheric phase is random and this would seem to preclude--with vanishingly small probability--creation at the turbulence layer of the well defined OAM states.
These problems are addressed in-turn with the solution to the first appearing in a companion paper ; these results are synopsized in Section 2. The second problem is resolved in Section 3. Then, Section 4 addresses the third problem. At this point, having established that branch points indicate the presence of OAM, Section 6 discusses the repercussions of this connection mostly in the area of adaptive optics but also extending into astronomy, as well. The discussion is succinct, but necessary to enumerate the wide ranging repercussions of this work, as well as, elucidate our upcoming papers in this area. Finally, in Section 7 the work is summarized.
2. Overview/Summary of results and definition of terms
The basis for resolving the remaining two problems is twofold, first demonstrate that the seemingly random nature of branch points is a measurement/analysis problem. Secondly, demonstrate that a local smooth phase, after propagation, can lead to well-defined, but discontinuous, OAM phase.
Fortunately, these issues have been addressed implicitly by our previous results [5, 7–9] but not with respect to their relation to OAM. Here, we make explicit their connection, and the fundamental basis for solution of these problems reduces, with minor extension, to a review of our earlier work. Since propagation lies at the basis of this research, we begin with the wave equation.
2.1. Wave equations and angular momentum
Predictions of the field are made using the wave equation, which has many forms depending on the physics. For the physics of atmospheric propagation, the permeability equals one and the permittivity is real, greater than one and spatially varying. When we consider timescales that are long compared to the frequency of the electric field’s oscillation but short compared to atmospheric turbulence, the electric field can be written as E(r)ei2πνt, with ν the frequency of the light, r the spatial coordinate, and E not a function of time. Under these conditions, the wave equation is typically written12]. However, in our companion paper , it was shown that the addition of the third term in the right conditions causes the displacement current and the electric field to be in slightly different directions and this causes the electric field to have a non-trivial component in the direction of propagation.
2.2. Angular momentum
A necessary and sufficient condition for the existence of non-zero angular momentum is that the electric field have a component in the direction of propagation . Since ∇(E · ∇)logn creates non-trivial angular momentum as a normal and customary effect of propagation through turbulence , beams derived from Eq. 2 can have angular momentum. On the other hand, beams derived from Eq. 1 do not have one, and hence, they cannot have angular momentum.
Angular momentum is comprised of spin angular momentum plus orbital angular momentum. It was shown early in the quantum mechanics era  that spin angular momentum is mechanically measurable, specifically that it exerts a mechanical torque on a suspended half wave plate. Later, orbital angular momentum was shown to appear in Laguerre-Gaussian beams . This beam can be created from a Hermite-Gaussian beam using two cylindrical lenses separated by a propagation. Doing so adds ±1h̄ of orbital angular momentum to the beam. For OAM states, the amount of momentum is given by the eigenvalue, specifically, given a state eimϕ, the momentum is mh̄.
2.3. Branch points in the beam
Atmospheric fluctuations are smooth and hence, the index of refraction is also smooth. When the atmosphere imposes this random, but smooth, phase on a beam, parts of the beam get different tilts and this causes local regions within the beam to propagate in slightly different directions. As the beam propagates, sometimes these regions overlap and interfere to create a zero in amplitude. The phase associated with this zero is given by 
2.4. Branch points and OAM
If the superficial similarity (the spiral in phase) between branch points and the lowest OAM state is to be born out, since momentum is conserved, branch point helicity must be conserved. So, let mk be the helicity of the k = 1,..., K branch points. Conservation of momentum implies, Σjmj is constant which, in turn, implies that each branch point is persistent.
If we consider plane waves incident on atmospheric turbulence, prior to interacting with the atmosphere, Σjmj = 0. This implies that in order for branch points to be markers for OAM states, they must be created in pairs of opposite helicity. Our first work on this subject , demonstrated that branch points are created in pairs of opposite helicity; thus answering this second requirement.
It remains, then, to show that branch points are persistent. Unfortunately, in all experimental data, they apparently aren’t. To know why this is so, one must know how they are measured. This is done in Section 3.1.
To demonstrate that branch points really are persistent, one must enhance the standard techniques and use these enhancements to reevaluate experimental data. Our effort in this area is reiterated in Sections 3.3 and 3.4. Doing so conclusively demonstrates that branch points really are enduring features of the traveling wave.
3.1. Measurement of phase and branch points in that phase
To find branch points theoretically, phase is summed in a plane perpendicular to the direction of propagation in a closed path about a point, i.e.
To find branch points experimentally, phase is first measured using a Shack-Hartmann wavefront sensor  which measures the gradients of the phase, ∇ϕ(r). The gradients have been shown in the vein of Stokes Theorem to consist of two orthogonal components a pure divergence part--here called the least mean square  or irrotational phase--and a pure curl part--here called the rotational phase. (Note the rotational phase is shown in Eq. 3.) To find the branch points, the gradients are summed in a closed four pixel path, i.e.Eq. 4, the least mean square (LMS) phase sums to zero. However, unlike Eq. 4, the closed path is not arbitrarily small and this causes two problems. First, the accuracy in estimating ρ0 is half a pixel, not arbitrarily small as in Eq. 4. Secondly, since the path is dictated by pixel pitch, multiple branch points can be enclosed. If multiple branch points of the same helicity, say m = +1, are enclosed, then Eq. 5 returns 2π. If equal numbers of branch points of opposite helicity are enclosed, then Eq. 5 returns zero. In either case, unlike Eq. 4, an error results.
3.1.3. Adaptive optic systems and measurement of branch points
Given this paradigm for measuring phase, from a practical point of view wavefront sensors designed for adaptive optic systems are not well suited to measuring branch points since adaptive optic systems are designed to match the disturbance they are correcting. Specifically, at good observing sites and near zenith, their wavefront sensor subaperture sizes are tens of centimeters. This makes the accuracy of Eq. 5 tens of centimeters. Creation pairs with separation on the order of a subaperture size or smaller will only be measured when they straddle a wavefront sensor subaperture boundary. Hence, near their creation point when their separations are small, a wavefront sensor used for astronomical imaging will miss the presence of a good many branch points. However, as the creation pair moves and straddles subaperture boundaries, the branch points in the creation pair are measured causing a momentary flickering in the data.
In addition to practical considerations, how the measurements are used is also of importance. Classic adaptive optics uses proportional integral controllers primarily because they closely match the disturbance, but also for their simplicity and robustness. For these controllers, the LMS phase is fed on a frame-by-frame basis to the proportional integral controller. It is unnecessary to keep the raw frames since disturbance “memory” is retained in the integrator. So, standard practice is to not keep wavefront sensor frames and branch point phase is discarded along with the irrotational phase on a frame-by-frame basis.
Furthermore and more fundamentally, the summation of Eq. 5 can result in ±2π even when no branch point is present. The most common is camera noise. Read noise causes fluctuations in camera counts; these fluctuations randomly cause a circulation to be recorded by the integral. Furthermore, in regions of high local tilt, the circulation algorithm records the presence of two closely spaced circulations  one of positive helicity and one of negative. This, again, is strictly a figment of the mathematics. Additionally, tilt induced mod2π discontinuities can mask branch points . Finally, because branch points are created by a random phenomenon, each branch point is assumed to be an independent event, so discarding the rotational phase frame-by-frame is a viable means to reduce noise.
3.2. The measurement problem
All this adds up to why all circulations appear to be random. Specifically, (a) camera noise creates circulations which, because they are treated as branch points, give false readings. (b) Regions of high local tilt cause Eqs. 5 to report pairs of branch points when none exist. (c) 2π discontinuities mask branch points. (d) Frame-by-frame processing precludes detecting persistence. Finally, large subaperture sizes of standard wavefront sensors lead to (e) missing of a great many closely spaced creation pairs, and (f) imprecise measurement when they are measured.
3.3. New methods
To account for these issues, we have developed new hardware and algorithms [5, 9, 18–20]. Specifically, phase piston shifting algorithms move the 2π discontinuity which unmasks hidden branch points. Local tilt deconvolution removes regions of high local tilt and this removes measurement of false pairs. Keeping frame-to-frame data instead of just frame-by-frame (intra-frame) data allows for branch point tracking which was found to yield distinct velocity classes useful for noise mitigation. Tracking branch points allows us to test for persistence, and doing so, we have shown how branch points move in time. Construction of hardware to increase separation of branch point pairs allows for measurement of all pairs and also gives the ability to study branch points as they move through space. And finally, construction of a high density wavefront sensor allows for precise measurement of branch point positions.
We have found that the new hardware is essential. We have also found local tilt deconvolution to be redundant with piston shifting, and after initial testing, only piston shifting along with filtering by velocity class is used in our processing.
3.4. Branch points as enduring features of the wave
3.4.1. Reevaluation of experimental data
Once these changes were implemented and the data reevaluated, [5, 19, 21, 22] branch points’ true nature can be seen. Shown in Fig. 1 is a sample of our research  where we used the new hardware and algorithms on phase generated using a two turbulence layer atmosphere, with each screen having a different altitude, strength and velocity. A beam was propagated through this atmosphere and then propagated further to a telescope where a wavefront sensor was used to make time-resolved measurements of the beam’s phase. For each wavefront sensor frame, the beam’s phase was reconstructed and the frames stored in a three dimensional array. From this 3-D array of phase, circulations were found using Eq. 5 along with piston shifting, and then each position was stored in another 3-D array. A sample of such an array is shown in Fig. 1(a). Each data point is the location of a measured circulation with the red dots denoting +1 helicity and the green dots −1. In the plot, the two horizontal axes are spatial and the vertical is time. The data after filtering by velocity class is plotted in Figs. 1(b) and 1(c).
Most notable are the lines cutting through the plots. Each line created by dots of the same color is a track in time of one branch point. As each dot represents a zero in the electric field amplitude with the color of the zero indicating its helicity, the permanence of branch points represents the evolving zeros of the electric field. More interestingly, when a second layer is added, branch points persist through the second layer. Plotted in this way--grouped by velocity and tracked in time--it is immediately apparent that creation pairs are enduring features of the traveling wave.
We conclude that branch points are persistent features of the traveling wave. Illumination of this behavior was possible only after standard adaptive optic hardware/analysis was modified for this purpose.
4. OAM from turbulence
Having established that branch points are persistent, it remains to demonstrate how a well defined OAM state can be created from a random turbulence field. The key to this lies in understanding that discontinuous OAM phase can be created through propagation from continuous phase and also that this creation happens locally.
4.1. Creation of discontinuous phase from smooth phase
So then, consider a Hermite-Gaussian beam. This beam has a smooth phase and no OAM. It was shown  that an OAM state can be created from this beam using two cylindrical lenses separated by their focal length. Note, the phase applied by both lenses is smooth, yet after the propagation, the phase is discontinuous. Specifically, after the second lens the beam is a Laguerre-Gaussian beam containing ±1h̄ of angular momentum with its characteristic 2π circulation in phase.
In propagation through the atmosphere, there is typically not a Hermite-Gaussian beam incident to the turbulence layer and the random phase certainly does not take on globally cylindrical phase, but the smooth random phase can locally take on many shapes, and this random phase is indeed followed by propagation.
4.2. Atmospheric phase and branch point creation – the local atmosphere
To understand how random atmospheric phase can create branch points, we construct a first-order model. Since branch points are equivalently described by either the zero in amplitude or circulation in phase, here we only study creation of the zero in amplitude.
Consider, then, a plane wave incident on a thin atmospheric turbulence layer. The beam, immediately after propagating through it, contains a phase only disturbance. The coherence length of that disturbance is characterized by the Fried parameter, r0. To first order, within this coherence length the phase is flat, that is, within each r0 size patch, the beam has a constant tilt. This is shown pictorially in Fig. 2. On the left is the turbulence’s iso-OPD contours and on the right in black is the first order approximation of the phase as r0 size tilt patches. If we further assume a high Fresnel number so that the geometric approximation can be made, then the wave in each patch propagates in the direction of the arrows. In this simplified model, a zero in amplitude occurs when waves from different patches destructively interfere at a point. Such a point is shown as the green dot with the waves from the green arrows interfering at that point.
Considering the branch point at the green dot, note that only a localized region in Plane 1 is required to create it. Specifically, the extent of the parts of the beam in Plane 1 that can interfere in Plane 2 is limited by the maximum tilt in Plane 1. More generally, since atmospheric tilt is bounded, local regions in the turbulence plane create branch points in the measurement plane .
4.2.1. Decomposition of the turbulence layer field
This simplified model highlights the crucial nugget, this being that there are specific tilts in Plane 1 that are responsible for creation of the null at point p in Plane 2. That is, the tilts are comprised of two orthogonal sets, one responsible for creation of the branch point and the balance. Then interestingly recall that for a single branch point, the phase in Plane 2 is comprised of two orthogonal parts, a rotational phase entirely due to that branch point and the balance being irrotational phase. It seems natural then to presume that the field in the turbulence layer, Plane 1, is comprised of two parts, ELMS and ELMS–rot. The phase of both ELMS and ELMS–rot are irrotational. However, after propagation ELMS–rot creates the branch points and ELMS does not. In the cartoon in Fig. 2, these are depicted by the blue and green arrows, respectively. Branch points are created then by highly localized parts of the turbulence layer acting in highly constrained ways.
This simplistic model is fraught with theoretical perils, notably that linearity in phase does not equate to linearity in the field. Leaving these concerns to be fully addressed later , it can be shown that there is a field in Plane 1 that is sufficient to form a creation pair in Plane 2. Properly establishing this result rests on understanding how the Rytov approximation--the fundamental approximation used in all theoretical work in adaptive optics--fails. This is the subject of the aforementioned paper.
For our purpose here, it suffices to conclude that a mechanism exists for creating the well defined OAM states from a random turbulence layer, this being a proper local decomposition in the turbulence layer and this is well captured by the simplified model.
5. Branch points and orbital angular momentum
In the introduction, we pointed out that there are superficial similarities between branch points and the lowest OAM states, these being that both have an azimuthal dependance of eiϕ and a null in amplitude. There are, however, three fundamental problems with this identification. (a) The term in the wave equation that creates angular momentum is 10−14 times smaller at optical wavelengths than the non angular momentum components, and this appears to be so trivial as to preclude OAM. (b) Furthermore, since momentum is conserved and branch points seemingly randomly appear and disappear in all experimental data, they cannot be markers for momentum states. (c) Finally, the atmospheric disturbance is random and this seemingly precludes the creation of well defined OAM states.
Each issue has been answered, in turn. The first problem, (a), is addressed in a companion paper . The second problem, (b), is addressed in Section 3 where it is shown that branch points are enduring features of the traveling wave. The third problem, (c), is addressed in Section 4. Thus, we have shown that atmospheric turbulence followed by a propagation can create photons with non-zero orbital angular momentum.
6. Discussion, implications and further work
That branch points indicate the presence of photons carrying OAM raises many interesting ancillary implications. These mostly arise from the concordance with adaptive optics but extend into astronomy as well.
6.1. Sufficient conditions for the creation of OAM
Empirically branch points begin to appear when the Rytov parameter exceeds 0.1. Since branch points are markers for photons carrying OAM, this same condition applies to OAM photons. Specifically, the Rytov parameter for a plane wave is given by
6.2. The density of OAM
Since branch points are markers for OAM photons and since the density of branch points is known from simulation and experiment, an estimate can be made for OAM density within a propagating beam. Branch point density versus normalized propagation distance  is shown in Fig. 3. The curve plots normalized distance along the horizontal axis and branch point density times a function of turbulence strength along the vertical axis. This can now be reinterpreted as the number of OAM photons versus distance with an unknown constant multiplier to the axis.
We see that the density of OAM increases with increasing propagation distance, then saturates at larger distances. The dependency on turbulence strength is also seen in the saturation of density. While the functional form is easy to enumerate based on our previous work, calculation of the number of OAM photons is not because the probability density function for branch point formation is not known. Addressing this issue is beyond the scope of this paper.
6.3. The fixed ratio of oAM to angular momentum
Standard literature states that for well defined beams, the ratio of OAM to AM is fixed . This is not the case here since angular momentum is created in random local regions within the larger beam. Perhaps a means could be found to calculate the ratio in the local area, but this is beyond the scope of this paper.
6.4. The rytov approximation
There is a long standing open question in adaptive optics as to why the Rytov parameter saturates. Experimentally, it is well known that the curves roll over, but theory predicts an ever increasing value. A plot of this is shown in Fig. 4. Theoretical Rytov is plotted along the horizontal axis and the experimentally estimated Rytov along the vertical axis. The straight 45° line is the theoretical prediction. The black and blue curves below are measured data.
The answer to this is a natural consequence of the results presented here. To wit, the Rytov approximation begins with use of the standard wave equation, ∇2E + k2n2E = 0, and this precludes a ẑ component in the field with ẑ a unit vector in the direction of propagation. However, ∇2E + k2n2E + 2∇(E · ∇)logn = 0 is the proper form to use and the addition of ∇(E · ∇)logn creates an increasing ẑ term. That is, as the field propagates, it accumulates an ever growing term that standard theory discounts. Including this term impacts the derivation of the Rytov approximation. This is treated in more detail in .
6.5. Creation pair separation
Research has shown that the intra-creation pair separation δ, i.e. the distance between the branch points with positive and negative helicity comprising the pair, is well defined . This is shown in Fig. 5. The separation is shown on the vertical axis and the normalized propagation distance on the horizontal axis. We see that δ grows approximately as the root of the propagation distance.
A most interesting question is whether the creation pairs are entangled. If so, another interesting avenue of research would open; further investigation here is well beyond the scope of this paper.
6.6. Relation to the inner scale of turbulence
It’s long been known that to properly account for experimentally measured scintillation an increase at the highest spatial frequencies in the power of the spectrum over a strict Kolmogorov spectrum is needed. This increase is known as the Hill bump. In the companion paper, creation of angular momentum was shown to occur predominantly at the highest spatial frequencies that is, at the very frequencies of the Hill bump, and, hence, is moderated by the inner scale of turbulence . We can extend the companion paper’s result and further conclude that the number of OAM photons created by atmospheric turbulence is also governed by the inner scale.
The inner scale of turbulence is the scale length at which atmospheric fluctuations begin to be dissipated as heat. An interesting avenue of research would be “why does the regime of heat dissipation moderate OAM creation?”. This this is beyond the scope of this paper.
6.7. Implications to astrophysics
The physical mechanism for the creation of orbital angular momentum is a spatially varying permittivity of space near but always above one. This causes the displacement current to be in a different direction than the electric field, and this causes the electric field to have a component in the direction of propagation.
Consider then, galactic clouds and dust. These phenomena have spatially varying permittivity near but above one. Hence, light propagating through them will have a component of the electric field in the direction of propagation, and hence, a fraction of the photons will carry OAM. Even if the fraction of photons carrying OAM is small, the number of galactic non-zero OAM photons can be quite large. Studying this is the subject of a later paper . There the OAM density as a function of galactic radius will be investigated and the implication of galactic photons with non-zero OAM will be discussed.
We have demonstrated that propagation through atmospheric turbulence can create photons with angular momentum, and that this momentum is, in part, orbital angular momentum. In doing so, remarkably, we have demonstrated that a random macroscopic media can create well defined quantum states in photons and further that the conditions required to create non-zero OAM photons are ubiquitous.
A.1. OAM, the Poynting Vector, and Wavefront Sensor Measurements
It’s been shown that for a beam propagating in the ẑ direction, Ez ≠ 0 is a necessary and suffi-cient condition for the existence of orbital angular momentum .
A.1.1. Equating a Spiraling Poynting Vector with OAM
It’s then trivial to show that for atmospheric propagation, the Poynting vector spiraling about the direction of propagation is also a necessary and sufficient condition for the existence of orbital angular momentum. To wit, for a plane wave incident on atmospheric turbulence, μ = 1 and ɛ = ɛ(r). And since μ = 1 ⇒ Bz = 0,
Hence, Sx,y ≠ 0 is a necessary and sufficient condition for the existence of orbital angular momentum. In words, the Poynting vector spiraling about the direction of propagation indicates OAM.
A.1.2. 2π Circulations in Phase and OAM
Sx,y = EzBx,y implies that the Poynting vector spirals about the optic axis. This helical spiral, when sliced in a plane perpendicular to the direction of propagation, leads to the characteristic 2π circulation in the beam’s phase. For a Laguerre-Gaussian beam, it has been shown that a wavefront sensor measurement of the 2π circulation unambiguously indicates the presence of OAM .
A.1.3. The Problem with Identifying “Branch Points” as OAM
Since branch points are defined by their 2π circulation in phase, measurement of a branch point one would think that this implies the beam has a helical structure, and hence, OAM. However, unlike measurement of pristine Laguerre-Gaussian beams, wavefront sensor measurements of beams having traversed atmospheric turbulence return 2π circulations for many reasons other than branch points . These are briefly enumerated in Section 3.2. Hence, when discussing atmospheric measurements, 2π circulations in phase do not necessarily imply the existence of OAM.
We would like to express our gratitude to the Air Force Office of Scientific Research for their support of this research.
References and links
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6. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Optics Express (2011). Accepted for publication. [PubMed]
7. 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,” SPIE 7466, 0501–0512 (2009).
8. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - persistent pairs,” Optics Express (2011). Submitted for publication.
9. D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.
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12. See for instance Ref.  page 394.
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15. This analysis is true for any type of wavefront sensor. The Shack-Hartmann is presented here because it is well known. In our lab, we use a self-referencing interferometer.
16. This is in keeping with the notation in Fried’s seminal paper, Ref. .
17. First pointed out by Terry Brennan, the Optical Science Corporation, in a private conversation.
18. D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.
19. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).
20. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).
21. 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,” SPIE 7816, 0501–0513 (2010).
22. 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,” SPIE 7466, 0601–0610 (2009).
23. This result holds in general to include diffraction, not merely for the these simplifying assumptions Specifically, if the assumption were relaxed to allow diffraction, each small region after propagation would have non-zero components that extend to infinity. But other than in the local region, these components are trivial and would at most cause the location of the null to move slightly.
24. D. W. Oesch and D. J. Sanchez, “Studying the optical field in and through the failure of the Rytov approximation,” Optics Express (2011). In preparation.
25. D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.
26. D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.
27. We have taken great care in our research to make a distinction between the persistant topological features of the propagating wave from transient phenonena. The persistent topological features--pairs of zeros in amplitude with opposite winding number--we call branch points; all other ciculations, we label as noise. Prior to our work, this distinction was vague. As a point in fact, in the earlest days of adaptive optics all 2π circulations were lumped into what was then called the “slope discrepancy” or “null” space, and even today in the phase reconstruction process, standard wavefront sensors lump all 2π circulations into a single group so that they may be summarily discarded, hence Fried’s  terminology “hidden phase”.