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Propagation of Bessel and Airy beams through atmospheric turbulence

We investigate, through simulation, the modifications to Bessel and Airy beams during propagation through atmospheric turbulence. We find that atmospheric turbulence disrupts the quasi-non-diffracting nature of Bessel and Airy beams when the transverse coherence length (Fried parameter) nears the initial aperture diameter or diagonal respectively. The turbulence induced transverse phase distortion limits the effectiveness of Bessel and Airy beams for applications requiring propagation over long distances in the turbulent atmosphere.


I. Introduction
Phase distortions acquired from turbulent fluctuations in the refractive index modify the propagation of laser beams through atmosphere [1][2][3][4][5][6][7]. In weak atmospheric turbulence, the modifications can be described as three separate effects: wander, spreading, and scintillation each of which can be detrimental to applications requiring long range propagation of the beam [4,7]. For instance, beam wander, or deflections in the beam's centroid due primarily to the large scale fluctuations, can result in the beam missing a target entirely. Beam spreading, expansion of the beam beyond vacuum diffraction due primarily to small scale fluctuations, distributes the beam power over a larger area, reducing the intensity, delivered power, and efficiency. And scintillation, fluctuations in the beam's intensity, can result in image distortion or higher bit-error-rates in optical communication systems [8].
For many applications, the beams' initial transverse intensity profile has a single, on-axis peak such as a Gaussian, a 'flat-top' generated by a circular aperture, or some approximation thereof. Any additional peaks or non-monotonic decreases in the profile are often categorized as higher-order mode aberrations, and because they result in enhanced diffraction of the beam are usually undesirable [9]. Bessel and Airy beams are an exception. Bessel beams have cylindrically symmetric profiles with a central peak surrounded by concentric rings each possessing nearly the same amount of energy.
During propagation the outer rings diffract inward fueling the center of the beam with energy and maintaining constant on-axis intensity. The inward diffraction also results in self-healing of the beam: when the beam's propagation path is partially obstructed the profile can nearly reform itself down-stream [10][11][12][13][14].
Airy beams, on the other hand, do not have cylindrical symmetry. The profile appears waffle-like with the intensity peaking in one corner and dropping with distance from the peak. While the Airy beam's centroid follows a straight trajectory, the peak intensity propagates along a curved path. These curved paths approximately follow the ballistic trajectory of projectiles and exhibit the same 'stalling' in motion that gravity produces in launched projectiles [14][15][16][17]. As with the Bessel beam, the interference of diffracting beamlets in the Airy beam's waffle-like pattern results in self-healing [14].
True Bessel and Airy beams are, however, only theoretical. Like plane waves, their field profiles would need to extend to infinity, requiring an unphysical, infinite amount of energy. In practice, the transverse profile of Bessel and Airy beams must be truncated at a finite radius or 'super-apertured,' for instance with a Gaussian profile, changing how the beam propagates [11]. Nevertheless, even when apertured, the desirable features of Bessel and Airy beams survive over extended distances in vacuum.
As a result, it has been suggested that Bessel beams [18] and Airy beams [14,19] could be useful for remote sensing or directed energy applications. If the desirable properties of Bessel and Airy beams are to be harnessed for these applications, they must be robust to phase distortions resulting from atmospheric turbulence [20][21][22][23].
Here we investigate, through the use of propagation simulations, how turbulence modifies the propagation of Bessel and Airy beams. For both beams, we find the extent that each beam is modified by atmospheric turbulence depends on the transverse beam size. The transverse coherence length (Fried parameter), describing the transverse distance over which the phase fronts of the laser beam remain correlated, decreases with propagation distance. As a result, large diameter beams acquire transverse phase distortions that modify their propagation before they undergo standard diffraction. Small diameter beams, on the other hand, diffract before their transverse coherence is disrupted.
Furthermore we find that the nature of turbulence-induced beam spreading differs between Bessel and Airy beams and Gaussian beams: the simultaneous diffraction of many rings or beamlets make the Bessel and Airy beams resistant to spreading in turbulence. Despite the Bessel beams resistance to spreading, the power delivered to the initial aperture area decreases as the number of rings increases. This suggests that the most effective Bessel beam for delivering power is a Bessel beam with zero rings, essentially a Gaussian beam.
In section II, we describe a model for propagation through turbulence. Section III discusses the simulation technique, namely a pseudo-spectral propagation algorithm for the paraxial equation with turbulence-induced refractive index fluctuations included through phase screens. Section IV presents results for propagation of Gaussian, Bessel, and Airy beams through turbulence and simple scalings to describe the observed phenomena. Section V ends the manuscript with a summary of our results and conclusions.

II. Propagation through turbulence
We write the transverse electric field of the laser beam as a plane wave carrier modulating an envelope as follows: δ is to be determined. We limit our investigation to beams with duration long enough and power low enough such that the envelope is independent of time. The evolution of the beam envelope is then determined by the steady-state, enveloped paraxial equation The refractive index fluctuations, ( ) where 2 n C is the refractive index structure constant. Using Eq. (4), the index structure function can be shown to be To assess the modifications to a laser beam in atmospheric turbulence, it is common to consider the solution to Eq. (3) in the Frauhofer diffraction limit. In particular, the ensemble averaged, on-axis intensity, The exponential in Eq. (5a) includes the phase structure function . Up to a numerical factor this is the same distance at which the Fried parameter Fried parameter, also referred to as the transverse coherence length, describes the transverse distance over which the phase fronts of the laser beam remain correlated. In other words, transverse distortions on the size of 0 r and modifications to beam propagation can be expected when the Fried parameter is smaller than the beam diameter.

A. Simulation algorithm
For intervals of propagation, If the propagation interval is also much larger than the outer scale, 0 s z L ∆ >> , the accumulated phase can be approximated as [2,5] where r a and i a are independent Gaussian random variables. Equation (7) provides the basis for the phase screen approximation in which the laser beam acquires the accumulated transverse phase at discrete axial points along the propagation path, and is propagated in vacuum between these points.
To simulate the propagation, we use a pseudo-spectral, split step algorithm. A typical axial advance, propagating the beam forward by z ∆ , involves three steps. In the first step, the transverse Fourier transform of the envelope is advanced a half step, / 2 z ∆ by applying the diffraction propagator: the overbar denotes the transform with respect to the transverse plane. For the second step, the phase screen is applied in coordinate space Finally the beam is advanced the second half step in the transverse Fourier domain, note that a phase screen does not need to be applied at every advance but the interval between applications should satisfy the constraint that 0

B. Simulation details
We simulate the propagation of initially collimated laser beams with wavelengths

A. Gaussian Beam
We begin by reviewing the propagation of Gaussian beams through vacuum and turbulence. The Rayleigh range, where 0 w is the initial 1 / e field width, defines the length scale over which a Gaussian beam diffracts in vacuum. In particular, the on-axis intensity of a Gaussian beam drops, A more quantitative examination and scaling with 0 w is presented in Fig. (2). , suggesting that the beam has spread more from the edges than the center. Higher moments of the intensity distribution could be used to examine this effect, but we save this for future investigations.

B. Bessel Beam
Our goal now is to explore modifications to Bessel beam propagation in atmospheric turbulence. During vacuum propagation, each ring of the Bessel beam can be considered a separate beam undergoing its own diffraction. Each ring possesses near equal energy which it transports, through diffraction, outward from its initial radius and inward towards the center of the beam. The inward diffraction of the rings supplies the center of the beam with energy. This process maintains the on-axis intensity until the inward diffraction of the outer ring reaches the beam center. We can estimate the diffraction length of the Bessel beam by finding the distance at which the spot size associated with outer-ring expands to the total beam size. Noting that the Bessel function zeroes approach even spacing, we write the outer ring's spot size as ~/ ( 1)  . Figure 1(b) shows that the concentric, closed ring structure observed in Fig. 1(a)   Gaussian beams, small and large beams are susceptible to standard diffraction and turbulence induced spreading and wander respectively.

C. Fixed Aperture Comparison
In the previous section the effect of beam size on atmospheric propagation of fixed, 14 ring Bessel beams was examined. We now fix the beam size and power, and vary the number of rings. We chose an aperture size of 30 cm , approximating the size of a beam director for directed energy applications [7]. As before, the beams are hard- indicates that a beam's sensitivity to on-axis intensity loss in turbulence increases with its number of rings. As illustrated by the red and green curves, the decay in on-axis intensity relative to the vacuum values increases with r N . The result of these two effects is that the ideal beam profile for applications requiring high peak intensity depends on the distance to the target.

D. Airy Beam
We now explore modifications to Airy beam propagation in atmospheric turbulence. Like the rings of the Bessel beam, each beamlet of the Airy beam can be considered a separate beam undergoing its own diffraction. During propagation through vacuum, the beamlets interfere such that the Airy beam's center of mass follows a straight line while its intensity maximum follows a parabolic trajectory, drifting in the transverse plane. In particular, the transverse position of the intensity peak, , p ⊥ r , evolves according to    respectively. Figure 7(d) shows that the maximum intensity of the Airy beams is affected by turbulence in the same way the on-axis intensity of Gaussian and Bessel beams are affected.

V. Summary and Conclusions
We have investigated the propagation of Gaussian, Bessel, and Airy beams through atmospheric turbulence. We have also examined the scaling of power and intensity delivered to a target with the number of rings in a fixed aperture area Bessel beam. Despite the Bessel beams resistance to spreading, the power delivered to an area equal to the beam director decreases as the number of rings increases. As a result, the most effective Bessel beam for delivering power is a Bessel beam with zero rings, which approximates a Gaussian beam. Finally, the optimal number of rings for delivering on-axis intensity to a target depends on the distance to the target.       Normalized on axis intensity at one diffraction as a function of aperture diagonal. In (a-d) the ratio of the Fried parameter to beam diagonal, green, is plotted for reference.