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 Optical Society of America
The discovery that light possesses orbital angular momentum (OAM)  has spurred wide ranging research topics. The light-matter interaction of OAM [2–4], the use of OAM as a basis for quantum key distribution [5–7] in laser communication, addressing congestion in the radio-band for mobile communications , the entanglement of OAM photons [9, 10], the structure and evolution of optical vortices [11, 12], and the application of OAM for astronomy [13–15], 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, 17], 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 , spatial light modulators , and holograms , for example. Atmospheric turbulence has recently been added to this list of mechanisms for the formation of OAM in traveling optical waves.  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 . 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 . 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 . 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 . 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 [5,18, 19], 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.
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 . 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 , following a turbulence layer before branch points will form.
It is known , 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.
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 = ℓ1 and Plane 2 at z = ℓ2, orthogonal to the direction of propagation, such that ℓ1 < z0 < ℓ2 and ℓ2 – ℓ1 = 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 .
The propagation between the planes is typically discussed in terms of the Fresnel number, . The Fresnel number is used in diffraction theory for scaling of an optical system  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 , uniquely relates the complex fields of Plane 1 and Plane 2 by27].
3. The Field at Plane 1
Under our assumptions, the field at Plane 2 is A2eiϕhid where the hidden phase  is described by the equation,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  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 + y2 ≤ R2. This will be shown to be a good approximation by the demonstration in Section 4.
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 ∈ [1,30], 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 . Specifically, functions of the form
Optimizing the fits of the combined function, ϕpre = Ps + Pa, across the set of generated precursor phases leads to
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,
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  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 ℓ1 > 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)  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.
We define the pair separation to be 12 pixels and set the pair center to (20,10) with a θ = π/3 rotation in our WFS plane. This places the positive and negative branch points at approximately (23.00,15.19) and (17.00,4.80) respectively. Next we calculated the precursor phase in the DM plane as was done in Sec. 3. A plot of the result is shown in Fig. 3(a). This precursor phase is then binned down to the 31x31 actuator density of our DM, Fig. 3(b). The binned phase is shown in numbers of waves before being scaled into DM commands. Interestingly, the range of the binned phase is on the order of a half of a wave, peak-to-valley.
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 θ ≃ π/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 . 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 . 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, 7].
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 .
A. Numerical solution for the precursor phaseEq. (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).
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.
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.
We would like to express our gratitude to the Air Force Office of Scientific Research for their support in funding this research.
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