## Orbital angular momentum in optical waves propagating through distributed turbulence |

Optics Express, Vol. 19, Issue 24, pp. 24596-24608 (2011)

http://dx.doi.org/10.1364/OE.19.024596

Acrobat PDF (859 KB)

### Abstract

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 OSA

## 1. Introduction

*π*circulations in phase--the zero with its associated circulation in phase are called branch points by the adaptive optic community [1

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

*all*measurements they appear and disappear randomly.

*e*, form a basis of the azimuthal coordinate,

^{im}^{ϕ}*ϕ*, in ℝ

^{2}. For instance, OAM has been shown to exist [2

2. 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,” Physical Review A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*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 [2

2. 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,” Physical Review A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*π*circulation in phase [3

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

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

5. 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,” Optics Express **18**, 22377–22392 (2010). [CrossRef] [PubMed]

*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 10

^{14}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.

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]

## 2. Overview/Summary of results and definition of terms

5. 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,” Optics Express **18**, 22377–22392 (2010). [CrossRef] [PubMed]

### 2.1. Wave equations and angular momentum

**E**(

**r**)

*e*

^{i2πνt}, 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 written with

*k*the wave number and

*n*(

**r**) the index of refraction. Using this formulation, the electric and magnetic field are orthogonal to each other and perpendicular to the direction of propagation.

### 2.2. Angular momentum

**E**· ∇)log

*n*creates non-trivial angular momentum as a normal and customary effect of propagation through turbulence [6

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]

14. R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review **50**, 115–125 (1936). [CrossRef]

2. 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,” Physical Review A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*h̄*of orbital angular momentum to the beam. For OAM states, the amount of momentum is given by the eigenvalue, specifically, given a state

*e*, the momentum is

^{im}^{ϕ}*mh̄*.

### 2.3. Branch points in the beam

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

*x,y*) are the coordinates in the measurement plane,

*ρ*= (

_{i}*x*) are the locations of the branch points, and

_{i}, y_{i}*m*= ±1 are the helicity of the branch points. The helicity is the direction the branch point’s 2

_{k}*π*spiral in phase. In all experimental data,

*ρ*and

_{k}*m*appear to be random variables.

_{k}### 2.4. Branch points and OAM

*m*be the helicity of the

_{k}*k*= 1,...,

*K*branch points. Conservation of momentum implies, Σ

*is constant which, in turn, implies that each branch point is persistent.*

_{j}m_{j}*= 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 [7], demonstrated that branch points are created in pairs of opposite helicity; thus answering this second requirement.*

_{j}m_{j}## 3. Persistence

*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.

### 3.1. Measurement of phase and branch points in that phase

#### 3.1.1. Theoretical

_{ρ0}is a closed path about point

*ρ*

_{0}, and

*ϕ*is the phase. This integral can assume three values, zero and ±2

*π*. If the result is ±2

*π*, a circulation is said to exist at point

*ρ*

_{0}. It has become common practice to call all circulations branch points. Furthermore, since Ω

_{ρ0}is arbitrary, it can be chosen both so that it encircles only one branch point and also so that

*ρ*

_{0}is known to arbitrary precision. Thus, there is not possibility of missing branch points or misestimation of their position.

#### 3.1.2. Experimental

*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 [16] 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. As with the continuous integral in 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

*ρ*

_{0}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 2

*π*. 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

*π*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 [17] one of positive helicity and one of negative. This, again, is strictly a figment of the mathematics. Additionally, tilt induced

*mod*

_{2}

*discontinuities can mask branch points [18]. 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.*

_{π}### 3.2. The measurement problem

*π*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

### 3.4. Branch points as enduring features of the wave

#### 3.4.1. Reevaluation of experimental data

5. 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,” Optics Express **18**, 22377–22392 (2010). [CrossRef] [PubMed]

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).

## 4. OAM from turbulence

### 4.1. Creation of discontinuous phase from smooth phase

**45**, 8185–8189 (1992). [CrossRef] [PubMed]

*h̄*of angular momentum with its characteristic 2

*π*circulation in phase.

### 4.2. Atmospheric phase and branch point creation – the local atmosphere

*r*

_{0}. To first order, within this coherence length the phase is flat, that is, within each

*r*

_{0}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

*r*

_{0}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.

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.

#### 4.2.1. Decomposition of the turbulence layer field

*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,

**E**

*and*

_{LMS}**E**

*. The phase of both*

_{LMS–rot}**E**

*and*

_{LMS}**E**

*are irrotational. However, after propagation*

_{LMS–rot}**E**

*creates the branch points and*

_{LMS–rot}**E**

*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.*

_{LMS}## 5. Branch points and orbital angular momentum

*e*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

^{iϕ}^{−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.

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]

## 6. Discussion, implications and further work

### 6.1. Sufficient conditions for the creation of OAM

### 6.2. The density of OAM

### 6.3. The fixed ratio of oAM to angular momentum

### 6.4. The rytov approximation

^{2}

**E**+

*k*

^{2}

*n*

^{2}

**E**= 0, and this precludes a

*ẑ*component in the field with

*ẑ*a unit vector in the direction of propagation. However, ∇

^{2}

**E**+

*k*

^{2}

*n*

^{2}

**E**+ 2∇(

**E**· ∇)log

*n*= 0 is the proper form to use and the addition of ∇(

**E**· ∇)log

*n*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 [25].

### 6.5. Creation pair separation

*δ*, i.e. the distance between the branch points with positive and negative helicity comprising the pair, is well defined [7]. 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.

### 6.6. Relation to the inner scale of turbulence

### 6.7. Implications to astrophysics

## 7. Conclusion

## A. Appendix

## A.1. OAM, the Poynting Vector, and Wavefront Sensor Measurements

*ẑ*direction,

*E*≠ 0 is a necessary and suffi-cient condition for the existence of orbital angular momentum [13].

_{z}### A.1.1. Equating a Spiraling Poynting Vector with OAM

*μ*= 1 and

*ɛ*=

*ɛ*(

**r**). And since

*μ*= 1 ⇒

*B*= 0, Then, note

_{z}*E*≠ 0 implies

_{z}*S*≠ 0 where the notation

_{x,y}*S*≠ 0 means

_{x,y}*S*≠ 0 or

_{x}*S*≠ 0. Therefore, OAM implies

_{y}*S*≠ 0. Conversely,

_{x,y}*S*≠ 0 implies

_{x,y}*E*≠ 0.

_{z}*S*≠ 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.

_{x,y}### A.1.2. 2*π* Circulations in Phase and OAM

*S*=

_{x,y}*E*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 2

_{z}B_{x,y}*π*circulation in the beam’s phase. For a Laguerre-Gaussian beam, it has been shown that a wavefront sensor measurement of the 2

*π*circulation unambiguously indicates the presence of OAM [3

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

### A.1.3. The Problem with Identifying “Branch Points” as OAM

*π*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 2

*π*circulations for many reasons other than branch points [27

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 2*π* 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 2*π* circulations into a single group so that they may be summarily discarded, hence Fried’s [4] terminology “hidden phase”.

*π*circulations in phase do not necessarily imply the existence of OAM.

## Acknowledgments

## References and links

1. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Applied Optics |

2. | 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,” Physical Review A |

3. | J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express |

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

5. | 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,” Optics Express |

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 |

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. |

10. | R. J. Sasiela, |

11. | J. W. Goodman, |

12. | See for instance Ref. [11] page 394. |

13. | J. D. Jackson, |

14. | R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review |

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. [4]. |

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 “ |

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 |

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 |

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 |

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 2 |

**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

**Citation**

Darryl J. Sanchez and Denis W. Oesch, "Orbital angular momentum in optical waves propagating through distributed turbulence," Opt. Express **19**, 24596-24608 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24596

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### References

- D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Applied Optics31, 2865–2882 (1992). [CrossRef] [PubMed]
- 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,” Physical Review A45, 8185–8189 (1992). [CrossRef] [PubMed]
- J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express14, 11919–11924 (2006). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A15, 2759–2768 (1998). [CrossRef]
- 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,” Optics Express18, 22377–22392 (2010). [CrossRef] [PubMed]
- 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]
- 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,” SPIE7466, 0501–0512 (2009).
- 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.
- 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.
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE Press, Bellingham, Wa, USA, 2007), 2nd ed.
- J. W. Goodman, Statistical Optics (John Wiley & Sons, New York, New York, 2000), Wiley Classics Library ed.
- See for instance Ref. [11] page 394.
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, USA, 1975), 2nd ed.
- R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review50, 115–125 (1936). [CrossRef]
- 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.
- This is in keeping with the notation in Fried’s seminal paper, Ref. [4].
- First pointed out by Terry Brennan, the Optical Science Corporation, in a private conversation.
- 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.
- 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).
- 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,” SPIE7816, 0601–0616 (2010).
- 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,” SPIE7816, 0501–0513 (2010).
- 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,” SPIE7466, 0601–0610 (2009).
- 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.
- 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.
- D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.
- D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.
- 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 [4] terminology “hidden phase”.

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