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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 12292–12302
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Creating well-defined orbital angular momentum states with a random turbulent medium

Denis W. Oesch and Darryl J. Sanchez  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 12292-12302 (2012)
http://dx.doi.org/10.1364/OE.20.012292


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Abstract

Previous work by Allen, demonstrated that optical beams possess orbital angular momentum. Other work has shown that a random, phase-only disturbance can impart ±1 orbital angular momentum states to propagating waves. However, the field preceding the formation of these ±1 states was unknown. In this paper, we identify the unique field that leads to the formation of a pair of branch points, indicators of orbital angular momentum. This field is then verified in a bench-top optical experiment.

© 2012 OSA

1. Introduction

The discovery that light possesses orbital angular momentum (OAM) [1

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]

] has spurred wide ranging research topics. The light-matter interaction of OAM [2

2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

4

4. A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010). [CrossRef]

], the use of OAM as a basis for quantum key distribution [5

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

7

7. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). [CrossRef] [PubMed]

] in laser communication, addressing congestion in the radio-band for mobile communications [8

8. T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012). [CrossRef]

], the entanglement of OAM photons [9

9. S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003). [CrossRef]

, 10

10. B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011). [CrossRef] [PubMed]

], the structure and evolution of optical vortices [11

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

, 12

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

], and the application of OAM for astronomy [13

13. W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999). [CrossRef]

15

15. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008). [CrossRef]

], for instance.

Though OAM is a quantum property of light, it is manifested by the macroscopic field. Optical waves carrying OAM have a component of the field parallel to the direction of propagation. In Laguerre-Gaussian beams, for example, this is characterized by a Poynting vector that spirals about the beam axis [16

16. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995). [CrossRef]

, 17

17. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]

], while the amplitude of the field on axis is zero. Wave front sensors (WFS) measure a phase circulation about the zero in the amplitude.

In the research mentioned above, various methods have been developed to create light that carries OAM; spiral phase plates [18

18. S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004). [CrossRef] [PubMed]

], spatial light modulators [5

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

], and holograms [19

19. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008). [CrossRef] [PubMed]

], for example. Atmospheric turbulence has recently been added to this list of mechanisms for the formation of OAM in traveling optical waves. [20

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

] showed that a propagating wave acquires angular momentum from the fluctuations of the refractive index as a normal part of interacting with the atmosphere. Following propagation, this angular momentum can be found to be composed of both spin and orbital angular momentum, where the formation of branch points signals the appearance of OAM in the optical field [21

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

]. An explicit mechanism for the formation of branch points from random index fluctuations was not given. This paper demonstrates a mechanism through which the atmospherically imparted angular momentum can be transferred into orbital angular momentum. This differs from the previously stated methods for generating OAM states, which convert the field at the plane of the device.

Our work to this point has focused on the behavior of atmospheric branch point pairs in the pupil plane of a telescope. A branch point pair consists of two connected, counter rotating phase circulations, one positive and one negative. Sets of branch point pairs within the pupil plane can be grouped by velocities [22

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

]. These groups then can be used to infer the number and velocities of branch point producing turbulence layers. The density of branch points combined with the mean separation between paired points can be used to estimate the strength and distance of the turbulence [23

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

]. In this way, pupil plane measurements can be used to estimate the three dimensional structure of the branch point producing atmosphere based on a layered model [24

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

]. Here we investigate the phase conditions in the turbulence that leads to the aforementioned branch point pair and unlike the methods typically used to impart OAM to optical beams [5

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

,18

18. S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004). [CrossRef] [PubMed]

, 19

19. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008). [CrossRef] [PubMed]

], the identified phase is continuous.

In this work, we study the atmospheric conditions necessary for the creation of branch points. Specifically, we identify the continuous phase imparted to the field that leads to the formation of OAM. We begin by stating the geometry of our problem in Section 2. Section 3 presents the construction of the unique phase required for the formation of branch point pairs. This identified phase is validated in Section 4 with an experimental demonstration. Finally we summarize our results in Section 5.

2. Background

Passing through a turbulence layer, a traveling optical wave acquires phase variations. These phase variations are initially smooth, but with additional propagation and interference, branch points may form. Branch points indicate the presence of OAM photons [21

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

]. In this paper, we will use branch points when discussing measurement and OAM for the behavior of the field.

Our earlier work has identified a threshold distance, z0 [23

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

], following a turbulence layer before branch points will form.

It is known [25

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

], that following z0 the phase gradients can be separated into two orthogonal components. Now consider the possibility that there exists a component of the phase prior to z0 that goes on to form a branch point pair, and hence OAM. Here, we place that supposition on firm grounds and identify the part of the phase that goes on to create a branch point pair.

2.1. Geometry

Consider the propagation of an optical wave of radius R following a continuous, phase-only disturbance. Let the disturbance be sufficiently strong such that, after some distance, this field will develop branch points. The number of branch points that form depends on the strength of the disturbance and the distance propagated. To simplify the problem we assume that the disturbance is such that a single branch point pair is formed on the optical axis some distance, z > z0, after the disturbance.

Next, define two planes, Plane 1 at z = 1 and Plane 2 at z = 2, orthogonal to the direction of propagation, such that 1 < z0 < 2 and 21 = L. The planes are defined by the transverse coordinates, x and y, see Fig. 1. The positions of the positive and negative branch points in the pair are identified by (xp, yp) and (xn, yn) respectively. The branch point pair separation, δ, is given by (xp-xn)2+(yp-yn)2.

Fig. 1 Cartoon representation of the space of the Fresnel transform between Plane 1 and Plane 2 with indicated parameters and coordinates. The intended branch point pair, positive and negative are indicated by the red and green dots respectively in Plane 2.

The propagation between the planes is typically discussed in terms of the Fresnel number, NF=R2(λL). The Fresnel number is used in diffraction theory for scaling of an optical system [26

26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

] and is used here to relate the propagation between the planes to the functional behavior of the field.

2.2. Unique mapping

The Fresnel transform [26

26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

], uniquely relates the complex fields of Plane 1 and Plane 2 by
A2(x,y)eiϕ2(x,y)=--A1(ξ,η)eiϕ1(ξ,η)h(x-ξ,y-η)dξdη.
(1)
Where the propagation kernel is
h(x,y)=eikziλzexp[ik2z(x2+y2)],
(2)
with wavelength, λ, and wave number, k = 2π/λ. For clarity, we describe the field at Plane 1 as the precursor field. Then the precursor field, A1e1, and the field associated with a pair of branch points, A2e2, form a unique Fresnel transform pair [27

27. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).

].

3. The Field at Plane 1

Under our assumptions, the field at Plane 2 is A2ehid where the hidden phase [28

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

] is described by the equation,
ϕhid=tan1(yypxxp)tan1(yynxxn),
(3)
with a small separation, δ. We assume a uniform amplitude at Plane 2, specifically A2 = 1 for x2 + y2R2 and zero elsewhere. Equation (1) can then be solved for the field at Plane 1. The inverse Fresnel transform produces a unique field at Plane 1.

As an example, this was solved numerically according to angular spectrum propagation [29

29. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A 21, 2135–2145 (2004). [CrossRef]

] for a pair with a separation of 3 and a Fresnel number of 31, see Appendix A. The results of this calculation are shown in Fig. 2; the estimate of the unique precursor phase in Fig. 2(a) alongside the hidden phase of the branch point pair in Fig. 2(b). It is worth noting, using this construction, that the amplitude associated with the field at Plane 1 is weakly varying. However, as the focus here is to identify the phase-only disturbance relating to the formation of branch points, the amplitude can be treated as uniform for x2 + y2R2. This will be shown to be a good approximation by the demonstration in Section 4.

Fig. 2 Phases of the Fresnel transform pair for NF = 31 with a branch point separation, δ = 3. (a) The precursor phase. (b) The created hidden phase.

3.1. Functional behavior

The general closed-form algebraic function for the precursor phase, ϕpre, (ϕ1 from Eq. (1)) is not known, but it can be solved numerically for a range of different propagation distances and branch point pair separations. Doing so generates a set of precursor phases for examining the functional behavior of ϕpre over a large parameter space. This set was created using inverse transforms with NF ∈ [15,47] applied to hidden phases from many different pairs of points, δ ∈ [1

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]

,30

30. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).

], formed on the x-axis, see Appendix B. Close examination of the generated precursor phase for these solutions shows that it is a combination of symmetric and asymmetric components which can be fit particularly well using Bessel functions, Jν [30

30. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).

]. Specifically, functions of the form
Ps=c1J1[ωs(x2+y2)]-c0J0[ωs(x2+y2)],Pa=caJ122[ωa(x2+y2)]y,
(4)
give a good approximation to the symmetric, Ps, and asymmetric, Pa, components of the precursor phase.

Optimizing the fits of the combined function, ϕpre = Ps + Pa, across the set of generated precursor phases leads to
c1=4δNF5R,c0=3δNF5R,ωs=πNFR2,ca=2πδNFR2,ωa=πNF2R2.
(5)

3.2. The Precursor phase

The above results only apply to pairs created along the x-axis. In order to allow for translation and rotation of the branch point pair, we introduce two additional parameters; the midpoint between the pair as (x0, y0) and an angular dependence, θ about the midpoint for orienting the branch point pair. Substituting Eq. (5) into ϕpre = Ps + Pa and making the addition of these two new parameters gives the general form,
ϕpre=δNFR[45J1(πNF(rR)2)-35J0(πNF(rR)2)]+2πδNFR2J122(12πNF(rR)2)[(y-y0)cosθ-(x-x0)sinθ].
(6)
with r=(x-x0)2+(y-y0)2, x0=12(xp+xn) and y0=12(yp+yn).

3.3. Characteristics of the precursor phase

Changing the separation of the branch points in Plane 2 only results in a change of ’magnitude’ of the phase. To first order, the underlying functional behavior remains unchanged. As identified in the functional dependencies of c1, c0 and ca of Eq. (5), there is a linear relationship between the precursor phase and the branch point pair separation.

The symmetric and asymmetric components exhibit different behaviors. The symmetric term, Ps, forms the rings of the precursor phase, seen in Fig. 2(a). These are a form of Fresnel diffraction [26

26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

] resulting from the zero amplitude of a branch point acting as an obscuration to the back-propagated field. As Fresnel rings this underlying distribution of the precursor phase is dependent on the propagation distance between the planes. This is reflected in the dependence of the arguments of the Bessel functions, ωs and ωa, on the Fresnel number, NF, in Eq. (5).

In the asymmetric term, Pa, the branch point separation affects the steepness of the precursor phase between it’s extremum. The location of these extremum in Plane 1, seen in Fig. 2(a), lie on a line perpendicular to the one connecting the branch points in Plane 2 on Fig. 2(b). The slope through the origin along the line connecting the extremum of the precursor phase becomes more severe with increasing separation.

Interestingly, at some separation the slope reaches a point where a discontinuity remains in the precursor phase. This happens because there is a minimum preceding distance required to allow the pair to reach that separation. A residual discontinuity in the precursor phase indicates that the branch point δ is too large for the distance between the planes used in the transform. In other words, a residual discontinuity in Plane 1, indicates that 1 > z0 and the system does not follow our assumed geometry, leading to an incorrect solution.

4. Experimental demonstration

The numerical approach is limited because finite sampling of the field and finite array sizes limit proper matching of optical behavior. Propagating beams aren’t so restricted. Therefore, to verify this approach, we performed a table-top experiment with our laboratory system.

For this demonstration the goal is to create a branch point pair using phase applied to a continuous face-sheet deformable mirror (DM). The optical system consists of two planes separated by a Fresnel number of NF = 31. At Plane 1 sits the DM with a resolution of 31x31 actuators, and a temporal, self-referencing interferometer (SRI) [31

31. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126 (2004). [CrossRef]

] WFS is located at Plane 2. The SRI camera has a pixel density of 256x256. The DM is used to modify a plane wave at Plane 1. The beam is then propagated to Plane 2 and sampled by the WFS. This arrangement allows us to imprint the desired precursor phase on the beam and then sample the field following sufficient propagation for the formation of branch points.

Fig. 3 Images of the applied DM commands and hidden phase pair from our experimental set-up. (a) Calculated precursor phase from Eq. (6) for desired branch point pair arrangement. (b) Applied DM commands. (c) Measured hidden phase at the WFS from the applied DM commands. (d) Desired branch point phase calculated from Eq. (3).

After applying the DM commands in Plane 1, the phase is measured by the WFS in Plane 2, see Fig. 3(c). A pair of branch points are located at (xp, yp) ≃ (32,15) and (xn, yn) ≃ (23,2) for the positive and negative circulations respectively. This gives the created pair a measured separation of δ = 15.8 pixels. The position of the midpoint of the branch point pair rests at (x0, y0) = (27.5,8.5) with a rotation θπ/3.25.

For comparison, Fig. 3(d) shows the hidden phase for the intended pair calculated using Eq. (3). The differences in (x0, y0) and θ between the created and intended branch point pairs, Fig. 3(c) and (d), are due to a small translation and slight rotation between the DM and WFS coordinate planes.

The increased separation in the created pair results from either a slightly larger than expected gain factor in the DM control loop or an increased physical distance between the DM and WFS planes than expected. An increased gain factor in the conversion of the commands to the DM acts as a multiplier of the calculated precursor phase. The “magnitude” of the phase, ϕpre described by Eq. (6), is linear with respect to δ, so an increase to the scale of the phase is reflected as an increase in the separation of the points at a fixed distance. Branch point pairs, once formed, separate with additional propagation [23

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

]. Therefore, if the Fresnel number for our experimental planes is less than NF = 31, the pair will also appear with a larger separation.

5. Discussion and summary

We have identified that a unique precursor field associated with the formation of a pair of branch points exists. We experimentally demonstrated this field’s role in the formation of a pair of branch points using a continuous face-sheet DM.

This method provides a means of OAM state preparation through a continuous phase change to the beam. At this point we have only shown the case for a branch point pair with a first-order helical structure. That is to say, the transformed beam only carries ±1 OAM states. We have investigated the precursor phases associated with higher order pairs in simulation. There are issues however with measuring higher orders using our WFS, making an experimental demonstration difficult. However, it should be possible to create such states using this technique.

The precursor phase can be used to form OAM states at a distance, which may offer a novel approach for encoding information in a beam for laser communication. Further, atmospheric branch point pairs have been shown to be stable over long distances [32

32. D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express 2, 1046–1059 (2012). [CrossRef]

]. Through the introduction of a continuous phase and propagation, it may be possible to create photons with desired OAM states with reduced susceptibility to the degradation that has been seen in other approaches [6

6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]

, 7

7. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). [CrossRef] [PubMed]

].

This work verifies the mechanism a random turbulent medium, which imparts only continuous phase distortions to a traveling optical wave, uses in the creation of well defined orbital angular momentum states [21

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

].

A. Numerical solution for the precursor phase

Angular spectrum propagation [29

29. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A 21, 2135–2145 (2004). [CrossRef]

] uses Fourier transforms to approximate the Fresnel transform integral, Eq. (1), according to
A1(x,y)eiϕ1(x,y)=𝔉-1[𝔉{A2(x,y)eiϕ2(x,y)}H(z)],
(7)
where 𝔉{·} represents the two-dimensional Fourier transform. The propagator, H(z) is the Fourier transform of the propagation kernel, h(x,y) (Eq. (2)).

This approach was used in the calculation of the precursor phase shown in the example of Sec. 3, Fig. 2. However, direct application of Eq. (7) results in a combined phase pattern, shown in Fig. 4(a), composed of the precursor phase and a set of Fresnel rings due to the aperture function A2(x,y). The Fresnel rings phase can be reproduced by applying the inverse transform to only A2(x,y), a top-hat function with uniform, non-zero amplitude and constant phase, Fig. 4(b). The precursor phase then is the difference between the phases in Figs. 4(a) and (b), shown in Fig. 4(c).

Fig. 4 Back propagated phase. (a) Phase calculated from back-propagating the generated hidden phase for a pair of branch points. (b) Phase calculated from back-propagating a top-hat function; standard Fresnel rings. (c) The Precursor Phase; the difference of the back-propagated hidden phase (a) and the Fresnel rings (b).

B. Fitting precursor phases

The method in Appendix A was used to generate a wide range of precursor phases for analyzing the functional interdependencies of the propagation distance and branch point pair separation in the identified general form for ϕpre (getting Eq. (6) from Eq. (4)). This was done by generating a range of propagators, H(z) using Fresnel numbers between 15 and 47 and combining those with a set of hidden phases associated with branch point pairs of separations ranging from 1 to 30 pixels. For each combination of propagator and hidden phase, a precursor phase was generated using Eq. (7). Fits were then done to the symmetric and asymmetric cross sections of the resulting phase to identify the coefficients in Eq. (5).

Figure 5 shows some of these fits against a range of branch point separations for a constant Fresnel number, NF = 26, in the left column. In the middle and right columns the symmetric and asymmetric cross sections of the precursor phases are shown. Red and blue curves represent the precursor phase estimate from angular spectrum propagation, Eq. (7), and the identified general form, Eq. (6), respectively. As can be seen from comparing the images of the 2-D precursor phase in Fig. 5 column(b), the only change to the symmetric portion of the phase is an increase in the magnitude of the pattern.

Fig. 5 Empirical fit to the precursor phase for a range of branch point separations, δ for Fresnel transform based on a Fresnel number of 26. (a) Precursor phase associated with the indicated separations. The cross sections for the (b) symmetric and (c) asymmetric axis are shown. The red curve is from the phase shown in (a) and the blue curve is generated by Eq. (4) using the identified variables.

The asymmetric component of the phase, Fig. 5 column(c), shows similar increases with the separation but also demonstrates a slight deviation from the empirical fit with increasing δ. This deviation is almost entirely limited to the left side of the asymmetric curves, those values where the phase is greater than zero. This deviation is small however compared to the magnitude of the current term, Pa.

Figure 6 on the other hand, shows fits with respect to the Fresnel number for a constant separation, δ = 3. In this case, the general form matches well across all Fresnel numbers for both the symmetric and asymmetric fits to some radius. At larger Fresnel numbers the obscuration rings become suppressed in the numerical solution beyond some radius. This suppression can be matched by adding a weak Gaussian envelope function to the symmetric term. However, doing so increases the complexity of the functional dependencies in Eq. (5) without significant improvement in the resulting fits and so wasn’t included in this paper.

Fig. 6 Empirical fit to the precursor phase for a range of Fresnel numbers for a hidden phase based on a pair separation of 3. (a) Precursor phase associated with the indicated Fresnel numbers. The cross sections for the (b) symmetric and (c) asymmetric axis are shown. The red curve is from the phase shown in (a) and the blue curve is generated by Eq. (4) using the identified variables.

Acknowledgments

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

References and links

1.

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]

2.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Bull. Am. Phys. Soc. 75, 826–829 (1995).

3.

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009). [CrossRef]

4.

A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12, 083053 (2010). [CrossRef]

5.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]

6.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]

7.

G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). [CrossRef] [PubMed]

8.

T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012). [CrossRef]

9.

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003). [CrossRef]

10.

B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011). [CrossRef] [PubMed]

11.

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

12.

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

13.

W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. 521, L63–L66 (1999). [CrossRef]

14.

J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. 576, L73–L76 (2002). [CrossRef]

15.

N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008). [CrossRef]

16.

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36–40 (1995). [CrossRef]

17.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]

18.

S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004). [CrossRef] [PubMed]

19.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008). [CrossRef] [PubMed]

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

26.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

27.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).

28.

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

29.

Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A 21, 2135–2145 (2004). [CrossRef]

30.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).

31.

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126 (2004). [CrossRef]

32.

D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express 2, 1046–1059 (2012). [CrossRef]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(010.1285) Atmospheric and oceanic optics : Atmospheric correction

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: March 30, 2012
Revised Manuscript: April 19, 2012
Manuscript Accepted: April 19, 2012
Published: May 16, 2012

Citation
Denis W. Oesch and Darryl J. Sanchez, "Creating well-defined orbital angular momentum states with a random turbulent medium," Opt. Express 20, 12292-12302 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12292


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References

  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. A45, 8185–8189 (1992). [CrossRef] [PubMed]
  2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Bull. Am. Phys. Soc.75, 826–829 (1995).
  3. S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A26, 625–638 (2009). [CrossRef]
  4. A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12, 083053 (2010). [CrossRef]
  5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12, 5448–5456 (2004). [CrossRef] [PubMed]
  6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett.94, 153901 (2005). [CrossRef] [PubMed]
  7. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett.34, 142–144 (2009). [CrossRef] [PubMed]
  8. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14, 033001 (2012). [CrossRef]
  9. S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A67, 022303 (2003). [CrossRef]
  10. B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express19, 6671–6683 (2011). [CrossRef] [PubMed]
  11. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt.40, 73–87 (1993). [CrossRef]
  12. I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B20, 1169–1176 (2003). [CrossRef]
  13. W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett.521, L63–L66 (1999). [CrossRef]
  14. J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett.576, L73–L76 (2002). [CrossRef]
  15. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys.492, 883–922 (2008). [CrossRef]
  16. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun.121, 36–40 (1995). [CrossRef]
  17. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun.184, 67–71 (2000). [CrossRef]
  18. S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt.43, 688–694 (2004). [CrossRef] [PubMed]
  19. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt.47, A32–A42 (2008). [CrossRef] [PubMed]
  20. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express19, 25388–25396 (2011). [CrossRef]
  21. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express19, 24596–24068 (2011). [CrossRef] [PubMed]
  22. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express18, 22377–22392 (2010). [CrossRef] [PubMed]
  23. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE7816, 0501–0513 (2010).
  24. D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng.18, 467–483 (1992). [CrossRef]
  25. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt.31, 2865–2882 (1992). [CrossRef] [PubMed]
  26. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  27. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).
  28. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A15, 2759–2768 (1998). [CrossRef]
  29. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A21, 2135–2145 (2004). [CrossRef]
  30. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).
  31. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE5553, 112–126 (2004). [CrossRef]
  32. D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express2, 1046–1059 (2012). [CrossRef]

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