## Localization of angular momentum in optical waves propagating through turbulence |

Optics Express, Vol. 19, Issue 25, pp. 25388-25396 (2011)

http://dx.doi.org/10.1364/OE.19.025388

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### Abstract

This is the first in a series of papers demonstrating that photons with orbital angular momentum can be created in optical waves propagating through distributed turbulence. The scope of this first paper is much narrower. Here, we demonstrate that atmospheric turbulence can impart non-trivial angular momentum to beams and that this non-trivial angular momentum is highly localized. Furthermore, creation of this angular momentum is a normal part of propagation through atmospheric turbulence.

© 2011 OSA

## 1. Introduction

1. 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]

*e*

^{±iϕ}where

*ϕ*is the azimuthal coordinate. It has been shown that the states can be created in the laboratory, but to create even the lowest order states takes careful preparation--creation of higher order states with azimuthal dependence of

*e*where

^{im}^{ϕ}*m*is an arbitrary integer, is more difficult.

^{2}

**E**+

*k*

^{2}

*n*

^{2}

**E**= 0 with

**E**the electric field,

*k*the wave number, and

*n*the spatially varying index of refraction. The solutions to which are well known to have no angular momentum. So, it’s not clear what, if any, a quantum effect would have on adaptive optics performance. However, it has been shown [2], that traveling waves can contain angular momentum and that this momentum is comprised of spin and orbital components. And in fact, the standard theoretical adaptive optic analysis [3, see for instance] initially begins with terms in the wave equation which can lead to angular momentum, but then by rough calculation, the term is discounted due to its relative size, 10

^{−14}as small in the mean, vis-á-vis the other components in the wave equation. However, as is shown in Section 3 the rough calculation overlooks details which show the possibility that local effects can make the angular momentum term non-trivial.

## 2. Definition of terms and overview/summary of results

**E**(

**r**)

*e*, that is a spatially varying part times a quickly oscillating time part. With these restrictions, the apropos wave equation is given by [3, see for instance] with

^{i}^{ω}^{t}*k*the wavenumber,

*n*the atmospheric index of refraction. Traditionally in the field of atmospheric propagation, the last term in Eq. 1 is discounted due to it’s relative size. The argument goes that with the index given by

*n*= 1 +

*n*

_{1}> 1 with |

*n*

_{1}| ≤ 10

^{−3}, then at optical wavelengths

*k*

^{2}

*n*

^{2}≈ 10

^{12}. Then since log(

*n*) ≈

*n*

_{1}≪ 1, the third term is much less than the second term.

**×**

*ρ***E**×

**H**

*d*

**with**

*ρ***the transverse coordinates. Hence, a necessary and sufficient condition for the existence of non-zero angular momentum is with**

*ρ**ẑ*the direction of propagation. With Eq. 2 establishing the criterion, we will show in the next section that the last term in Eq. 1 acts as an angular momentum source term. (Note in passing that if we had restricted ourselves to the customary Helmholtz wave equation, ∇

^{2}

**E**+

*k*

^{2}

*n*

^{2}

**E**= 0, then terms of the form of Eq. 2 cannot occur since for any propagation given by the free space Helmholtz equation,

**E**and

**H**are orthogonal to each other and perpendicular to the direction of propagation.)

*h̄*of momentum. The Poynting vector precesses helically because of a component of the electric field in the

*ẑ*direction. It has been shown that the ratio of the

*ẑ*direction to the transverse direction is

*λ*. At optical wavelengths this is approximately equal to 10

^{−6}. Hence, we will look for effects of this size. We will show in the following section that because atmospheric turbulence is a random process, there exists localized regions in the atmosphere where this is false with non-trivial probability.

**E**· ∇)log

*n*is small. However, in the next section we show that in regions where high frequency fluctuations are present, ∇(

**E**· ∇)log

*n*can be non-trivially large. For any given atmosphere, the total energy in these high frequency patches is governed by the inner scale of turbulence. Calculation of the probability of finding such a patch of atmosphere is beyond the scope of this paper.

## 3. Algebraic development

### 3.1. The source of angular momentum

*ẑ*·

**E**=

*ẑ*·

**B**= 0 and the third term in Eq. 1 reduces in the

*ẑ*direction to We have used the fact that the components of

**E**are constant,

*n*(

**r**) describes a physical process with

*n*(

**r**) > 1, therefore log(

*n*(

**r**)) is smooth, so the derivatives can be interchanged. Since the index varies isotopically, the conditions for creation of angular momentum are ubiquitous. The physical cause of the appearance of this term is that as the wave interacts with the atmospheric constituents, the electric field and the displacement current are in different directions; that is, although ∇ ·

**B**= 0, here ∇ ·

**E**≠ 0, and this has the effect of scattering the components of

**E**into the

*ẑ*direction. Once this first interaction occurs, as shown in Eq. 3, the beam contains a component in the direction of propagation and thereafter much richer interactions occur. The size of angular momentum term will be calculated under this richer condition.

*3.2. Calculation of size, i.e. of* ||*z*̂· ∇(**E** · ∇)log*n*||

*n*(

**r**) = 1 +

*n*

_{0}+

*n*

_{1}(

**r**) with

*n*

_{1}(

**r**) a zero mean random variable and

*n*

_{0}(

**r**) a constant measured to be approximately 10

^{−3}. Then the spectral decomposition of the varying part is [6]

*n*

_{1}(

**r**) = ∫

*dν*(

**)**

*κ**e*

^{iκ·r}with ∫ · a Riemann-Stiejles integral. Then note,

*n*(

**r**) > 1, so

*|n*

_{1}(

**r**)| <

*n*

_{0}; hence to one part in 10

^{3}, log

*n*(

**r**) =

*n*

_{0}+

*n*

_{1}(

**r**). Furthermore, since

*n*(

**r**) is smooth, its Fourier transform is smooth which allows the derivative and integral to be interchanged, and ∇ log

*n*(

**r**) =

*i*∫

*dν*(

**,**

*κ**z*)

*κ**e*

^{iκ·r}. Then, the norm of the source term for angular momentum is We wish to study the wave as the angular momentum term builds. So, consider a small region (but big enough so that the ensemble averages are meaningful) near the turbulence boundary. In this case we use the solution to the free space wave equation as the seed beam while recognizing that it will soon contain a

*ẑ*term, i.e. let

**E**(

**r**) =

**A**(

**r**)

*e*

^{ik·r}. Then resubstituting and assuming the derivative of turbulence induced scintillation and phase is small, one obtains where

**is the three coordinates of the spatial spectrum,**

*κ**k*= 2

_{z}*π*/

*λ*since the atmosphere is isotropic, and

*f*(

**) the spectrum (see Appendix A.2). Note in passing that as the longitudinal term,**

*κ**ẑ*·

**E**≠ 0, grows and becomes appreciable with respect to the transverse term,

*ẑ*·

**E**= 0, the assumptions here would have to be revisited and

**E**be recalculated. However, this adds an additional level of complexity that does not change the main conclusion of this paper.

### 3.3. ||*k*^{2}*n*^{2}**E**||

### 3.4. Comparison of size – the kolmogorov spectrum

*ẑ*direction. In the central equality, the integral form is intentionally kept to explicitly enumerate the difference between the angular momentum and non-angular momentum terms. The most striking is the proportionality

*κ*

^{4}in the

*ẑ*direction. This makes it strikingly apparent that the third term in Eq. 1 is dominated by high spatial frequencies. In the right most equality, the aforementioned proportionality to

*λ*

^{2}--for which this ratio is typically discounted--appears. Since the inertial range of the Kolmogorov spectrum is unbounded, i.e. the spectrum is −11/3 to all scales, there exists a frequency in the spectrum above which the numerator dominates the denominator, i.e. for which the angular momentum term dominates the customary term. In particular, That is, the size of the term that produces angular momentum can become arbitrarily larger than the customary term that is kept.

### 3.5. Comparison of size – the von karman spectrum

*κ*[3]. Substituting into Eq. 7 Immediately obvious, unlike the Kolmogorov spectrum, this ratio is finite in all cases, but also note, there is an amplification at the frequencies closest to

_{i}*κ*. Since for the atmosphere,

_{i}*κ*≫ 1, the value of the numerator will be dominated by frequencies closest to

_{i}*κ*. Whether it is nontrivial is solely a function of

_{i}*κ*.

_{i}*κ*is the inverse of the inner scale of turbulence,

_{i}*L*. The inner scale is the characteristic length at which the atmosphere becomes viscous and turbulence (motion) gets converted to heat. Measurements have found

_{i}*L*to be 1 – 5

_{i}*mm*at sea level and centimeters at higher altitudes. Following standard practice, consider

*κ*constant; then the integrals can be performed. This is done numerically for three values of the innerscale,

_{i}*L*= {0.1

_{i}*mm*, 1

*mm*, 10

*mm*}; these values bound the range of typically measured inner scales. The values of Eq. 9 are {55,2.7,0.12} × 10

^{−6}respectively and recall that 10

^{−6}would be significant. Based on this rough calculation, the effect is on the cusp of detectability.

*m*, consider patches of this size. Then propagation over 1

*Km*will yield 100 independent source patches of angular momentum, each contributing equally as a source. In this example, the value of the ratio in Eq.9 is multiplied 100-fold. And furthermore, since propagation through turbulence is not restricted to 1

*Km*, the ratio could grow very much bigger.

## 4. Discussion

### 4.1. The probability of non-zero angular momentum

*κ*>

*κ*

^{thresh}given that

*κ*<

_{i}*κ*

^{thresh}where

*κ*

^{thresh}is the threshold for detection of angular momentum. This calculation is the complementary problem to the Fried “Lucky Imaging Problem” [7]. Whereas there, Fried calculated the probability that locally (over the telescope diameter) the atmosphere is unusually quiescent, i.e. is comprised mostly of low frequency components, here, we are interested in calculating the probability that locally (over the atmosphere interrogated by an optical beam) the atmosphere is unusually active. Enumerating this calculation is beyond the scope of this paper.

### 4.2. The innerscale and κ_{i}

### 4.3. Further work

*E*term locally, propagation through extended turbulence will cause this term to accumulate. In fact, it is well known that propagation through extended turbulence causes branch points to appear, and that propagation through very much extended turbulence causes the branch point density to grow to the extend that they are a nuisance to adaptive optic performance--a nice corroboration of results and insight into the physical process of formation of angular momentum.

_{z}*κ*>

*κ*

_{thresh}is beyond the scope of this paper. Also, the result here applies to a local patch; once the longitudinal component is non-trivial with respect to the transverse components, the expressions leading to Eq. 4 must be reevaluated; this is beyond the scope of this paper.

## 5. Summary

## A. Appendix

## A.1. The Hilbert space of index fluctuations

### A.1.1. Construction of the Hilbert space

*n*

_{1}}. Since

*n*

_{1}describes a physical process,

*n*

_{1}∈

*L*

_{2}with

*L*

_{2}the space of square integrable functions. Note, that since we are describing atmospheric fluctuations which by definition does not include the vacuum, 0 ∉ {

*n*

_{1}}. Also note, that {

*n*

_{1}} ⊂

*L*

_{2}since the atmosphere is finite,

*max*({

*n*

_{1}}) ≪ 1, and

*min*({

*n*

_{1}}) > −

*n*

_{0}.

*n*

_{1}}. Since

*L*

_{2}is complete and since {

*n*

_{1}} ⊂

*L*

_{2}, this completion necessarily lies in

*L*

_{2}. Choose the smallest such completion and call it ℋ

*. ℋ*

_{n}*is a Hilbert space.*

_{n}### A.1.2. The covariance as an inner product

*V*is a vector space,

*x,y,z*∈

*V*, and

*α*∈ ℂ

*R*(·,·) denote the covariance function. Then, the covariance function is defined as, following Gikhman [5, p. 9] with

*R*(

*x,y*) the covariance function,

*M*(·) the expectation operator,

^{*}the complex conjugate, and

*ξ*(·) a function that maps

*V*→

*V*. For our purposes, let

*ξ*(

*x*) =

*x*. Then, Gikhman shows

*R*(·,·) is an inner product on ℋ

*, we will show the four parts of the definition in Eq. 10. To begin, let*

_{n}*x,y,z*∈ ℋ

*. So,*

_{n}*M*(

*x*) =

*M*(

*y*) =

*M*(

*z*) = 0. Using the definition of the covariance function, it is trivial to show the first part of (i), and also (ii), (iii), and (iv) in the definition, namely The only point needing clarification is the second part of (i), i.e. Since we are describing atmospheric fluctuations, 0 ∉ {

*n*

_{1}}. But note, ℋ

*is the completion of {*

_{n}*n*

_{1}} and 0 ∈ ℋ

*. Then since*

_{n}*M*(

*x*

^{2}) > 0,

*R*(

*x, x*) = 0 if and only if

*x*= 0.

*is a Hilbert space of a random process (the atmospheric fluctuations). The Fourier transform of this space is also a Hilbert space and is isometric to the first. So, the norm (inner product) in ℋ*

_{n}*equals the norm (inner product) of the transformed variable in the other. In the text, the norm is always evaluated in the transformed space.*

_{n}## A.2. The spectrum of turbulence

*κ*is the spatial frequency. This is the Kolmogorov spectrum.

*L*, and lower bounds,

_{o}*L*, that the atmosphere can attain. This is captured by the von Karman spectrum with

_{i}*L*and

_{i}*L*are respectively the inner and outer scale. The inner scale is the length at which turbulence gets dissipated by heat. For our purposes, the inner scale is the smallest scale at which turbulence structures are supported.

_{o}## Acknowledgments

## References and links

1. | 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 |

2. | J. D. Jackson, |

3. | R. J. Sasiela, |

4. | D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Exp. (2011). Accepted for publication. |

5. | I. I. Gikhman and A. V. Skorokhod, |

6. | L. C. Andrews and R. L. Phillips, |

7. | D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,” J. Opt. Soc. Am. |

8. | M. Reed and B. Simon, |

**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, "Localization of angular momentum in optical waves propagating through turbulence," Opt. Express **19**, 25388-25396 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25388

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### References

- 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]
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, USA, 1975), 2nd ed.
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE Press, Wa, USA, 2007), 2nd ed.
- D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Exp. (2011). Accepted for publication.
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover Publications, Inc., New York, USA, 1969), 1st ed. An english translation of the original work.
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, WA, USA, 1998), 2nd ed.
- D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,” J. Opt. Soc. Am.40, 1651–1658 (1977).
- M. Reed and B. Simon, Methods of Modern Mathematical Analysis, I: Functional Analysis (Academic Press, New York, USA, 1980), revised and enlarged ed.

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