Propagation of vector vortex beams through a turbulent atmosphere

Wen Cheng; Joseph W. Haus; Qiwen Zhan

doi:10.1364/OE.17.017829

1. Introduction

Propagation of laser beams through a turbulent atmosphere has important applications in many areas including free space optical communications (FSOC), Laser Radar (LADAR), Light Detection and Ranging (LIDAR), remote sensing and imaging. The turbulence strongly affects the properties of the laser beams propagating through it and consequently limits the performance of these systems. Understanding the propagation properties of laser beams through a turbulent atmosphere is critical in the optimization of these systems. Previously, spherical wave, plane wave or fundamental Gaussian beams were treated in most of the studies [1]. Higher order Gaussian or various other modified Gaussian beams have been studied recently [2–9]. It has been found that the initial beam properties (beam shape, phase, coherence etc) strongly affect the propagation results. Thus it is desirable to be able to handle more complicated initial beam conditions. Due to the complicated stochastic nature of the turbulence, analytical expressions for laser beam propagation only exist for limited cases and typically results are restricted to lower order statistics. Consequently numerical modeling becomes a necessary tool to achieve insights into the propagation of complex beam shapes. With modern computing power, numerical study of the propagation of complicated laser beams through turbulence becomes feasible and affordable, enabling the investigation of laser beams with nearly arbitrary initial conditions.

Recently, there are growing interests in applying optical vortex beams for remote sensing and free space optical communications [10–13]. The optical vortex is one type of optical singularity with undefined properties [14]. Most of the previous studies in this area dealt with scalar vortex beams with a phase singularity. Vector vortex with a singularity in the state of polarization has attracted lots of interests in other areas, particularly for those so-called cylindrical vector beams [15]. However, not much attention has been paid to these types of beams as they propagate through turbulence. Recently the second order properties such as the average irradiance and degree of polarization of a radially polarized beam propagating through turbulence have been studied analytically [16]. It has been found that the beam will be depolarized and lose its characteristic polarization and intensity pattern. However, higher order statistics such as scintillations for these beams propagating through the turbulence has not been investigated due to the lack of analytical expressions.

In this paper, we present numerical investigations of higher order statistical properties of a vector vortex beam propagating through a turbulent atmosphere and compare with a fundamental Gaussian beam and a scalar vortex beam with topological charge of + 1 propagating through the same turbulent atmosphere. We show that the existence of the vector vortex can be identified by a polarization signature at longer propagation distances than the scalar vortex even though characteristic vortex structure disappears in the irradiance images. This indicates the potential advantages of using vector vortex to mitigate atmospheric effects and enables a more robust free space communication channel with longer link distance.

2. Numerical modeling method

In order to numerically investigate the propagation properties of vector vortex beams through an extended turbulent atmosphere, we adopted the multiple phase screen method by Martin and Flatte [17,18]. In this work, von Kármán type index power spectrum is used to describe the atmosphere turbulence:

Φ_{n} (K) = 0.033 C_{n}^{2} \times \exp [- {(\frac{K l_{0}}{2 π})}^{2}] {(K^{2} + {(\frac{2 π}{L_{0}})}^{2})}^{- 11 / 6},

where

l_{0}

and

L_{0}

are the inner and outer scales of the turbulence, C_n ² is refractive index structure constant that represents the atmospheric turbulence strength. In this study, the inner and outer scales are chosen to be 1 cm and 3 m respectively. The necessary number of screens strongly depends on the strength of the turbulence measured by C_n ² and the propagation distance Z. In our study, for each propagation distance and turbulence, 20 equally spaced random screens are generated and used.

The numerical modeling procedure is summarized in Fig. 1 . A given initial laser beam is propagated through screen separation δz = Z/20 to the next plane using angular plane wave spectrum method [19]. Then the random phase screen represents the accumulated turbulence effect through this propagation distance; it is multiplied with the field at that position and the resultant field propagates under diffraction to the next screen. This algorithm is repeated until the final distance is reached. For each propagation distance, 500 independent realizations were calculated to provide the sufficient statistics for the calculations of average irradiance, polarization properties and scintillation index (SI).

Fig. 1 Propagation procedure flow chart used in the numerical model. (Left) Beam Propagation Procedure (Middle) Sample Random Phase Screen (Right) Random Phase Generation Procedure.

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The random phase screen is generated through filtering white Gaussian noise to obtain the random field with the desired second-order statistics [20]. A 512x512 array of pseudorandom complex numbers A + iB are generated with Matlab. This array of complex numbers is then multiplied by $Δ_{K}^{- 1} \sqrt{Φ_{θ} (K)}$ , where K is the transverse wave number, $Δ_{K}^{- 1} = 2 π / N Δ$ is the wave number increment, N = 512 is the number of sampling points and Δ is spatial sampling interval; $Φ_{θ} (K)$ is the random phase power spectrum given by $Φ_{θ} (K) = 2 π K^{2} δ_{z} Φ_{n} (K)$ , with $Φ_{n} (K)$ being the refractive-index power spectrum density. This result is then inverse Fourier transformed to yield the real-space random phase field θ₁ + iθ₂. In our simulation, we choose the real part θ₁ as the random screen. The numerical model has been validated by repeating published results for Gaussian beam propagation through turbulence [21,22].

In this paper, we numerically study the propagation properties of vector vortex beams by comparing them with the fundamental Gaussian and scalar vortex beams under the same turbulence conditions. For the scalar vortex, we used the well known Laguerre-Gauss LG_pl modes:

u (r, ϕ, z) = E_{0} {(\sqrt{2} \frac{r}{w (z)})}^{l} L_{p}^{l} (2 \frac{r^{2}}{w {(z)}^{2}}) \frac{w_{0}}{w (z)} \exp [- i φ_{p l} (z)] \exp [i \frac{k}{2 q (z)} r^{2}] \exp (i l ϕ),

where

L_{p}^{l} (x)

is the associated Laguerre polynomials, q(z) is the beam parameter, w(z) is the beam size, w₀ is the beam size at beam waist, and

φ_{p l} (z) = (2 p + l + 1) \tan^{- 1} (z / z_{0})

is the Gouy phase shift with

z_{0} = π w_{0}^{2} / λ

being the Rayleigh range. For l = p = 0, the solution reduces to the fundamental Gaussian beam solution that is used in this paper for comparison study. For l≠0, the LG mode has a vortex phase term

e^{i l ϕ}

. In this paper, we choose a scalar vortex with topological charge l = + 1. The initial beam is taken at the beam waist z = 0.

Radial polarization is used as a special example of vector vortex in this paper. The initial electric field distribution is taken to be:

\vec{E} (r, ϕ, z = 0) = E_{0} {(\sqrt{2} \frac{r}{w_{0}})}^{l} L_{p}^{l} (2 \frac{r^{2}}{w_{0}^{2}}) \exp [- r^{2} / w_{0}^{2}] {\vec{e}}_{r},

where

{\vec{e}}_{r}

is the unit vector along radial direction. Mathematically, a radial polarization can be decomposed into a linear superposition of two orthogonally polarized Hermite-Gauss (HG) modes [15](Fig. 2 ). Under the assumption we use for the model, there is no birefringence introduced by the turbulence, thus the two components can be treated independently. Each of the components propagates through the turbulent atmosphere separately using the procedure illustrated in Fig. 1. Then the final results for both components are used to synthesize the final result. For all of these beams, the wavelength is chosen to be λ = 2μm and the Gaussian beam waist is chosen to be w₀ = 7cm. The spatial window size is fixed at 3 m for all the calculations, which corresponding to a spatial sampling grid size of 5.859 mm.

Fig. 2 Radial x-polarized beam as a superimposition of two orthogonally, linearly y-polarized HG modes.

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3. Numerical modeling results with weak and strong turbulence

With the validated numerical model, we studied the propagation properties of scalar vortex, vector vortex beams and compared them with the fundamental Gaussian beam through two turbulence strengths with C_n ² equals to 10⁻¹⁴ m^-2/3 and 10⁻¹² m^-2/3. The results are summarized in the movies shown in Figs. 3 -6 . Beam intensity images are captured every 100 meters of the total 2km propagation distance. Corresponding linescans across the beam center for each image are also shown.

Fig. 3 Beam irradiance movies for the propagation of three kinds of beams through a weak turbulence (C_n ² = 10⁻¹⁴ m^-2/3) (a) Fundamental Gaussian (Media 1), (b) Scalar vortex beam (Media 2) and (c) Vector vortex beam (Media 3).

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Fig. 6 Irradiance linescan movies for the propagation of three kinds of beams through a strong turbulence (C_n ² = 10⁻¹² m^-2/3) (a) Fundamental Gaussian (Media 10), (b) Scalar vortex beam (Media 11) and (c) Vector vortex beam (Media 12).

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Fig. 4 Irradiance linescan movies for the propagation of three kinds of beams through a weak turbulence (C_n ² = 10⁻¹⁴ m^-2/3) (a) Fundamental Gaussian (Media 4), (b) Scalar vortex beam (Media 5) and (c) Vector vortex beam (Media 6).

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From the simulations, all three beam types eventually acquire Gaussian average irradiance patterns, as expected. The fundamental Gaussian beam maintains the overall Gaussian profile while the beam expands in size. For the donut-shaped scalar and vector vortex beams, the beam intensity fills in the center gradually as they propagate. The characteristic of the existence of vortex has been lost in the irradiance distribution after a certain propagation distance. For stronger turbulence, such annihilation occurs earlier at around 700m, as opposed to the weaker turbulence case which is still not obvious after 2km of propagation.

Also noticed from the irradiance distribution, the average intensity fluctuation is higher for stronger turbulence. To quantitatively characterize this fluctuation, we calculated the scintillation index (SI) for each of the propagation distance of all of three types of beams. The SI is defined as [1]:

σ_{I}^{2} = \frac{〈 I^{2} 〉 - {〈 I 〉}^{2}}{{〈 I 〉}^{2}}

where I is the irradiance and the angle brackets <•> represent an ensemble average. The value of SI represents degree of coherence of optical wave through atmosphere turbulence. Lower SI means that the beam can keep relative higher beam coherence which would be more desirable in free space optical communications.

The axial SI is calculated for Gaussian beam. Unlike the fundamental Gaussian beam that has a central peak, both the charge + 1 scalar vortex and the radially polarized vector vortex have donut shape irradiance distribution in free space propagation. Thus we adopted a modified SI calculation to give a fair comparison. For each propagation distance, the corresponding beam size w(z) of the vortex beam is calculated using the free space propagation. Then the irradiance within an annular area between radii of (1 + 5%) w(z) and (1-5%) w(z) of the numerically calculated vortex beams propagated through turbulence are used for the SI calculation given by Eq. (4). The +/−5% annular width is necessary to ensure there are sufficient number of data of the beam irradiance for the SI calculation. The results are summarized in Fig. 7 . The SI for vectorial vortex beam shows consistent lower value than scalar vortex case for both C_n ² = 10⁻¹⁴ m^-2/3 and C_n ² = 10⁻¹² m^-2/3 situations. Especially for C_n ² = 10⁻¹² m^-2/3, the two curves are separated by a considerable difference. This demonstrates the potential advantage of using vectorial vortex with respect to the scalar vortex beams.

Fig. 7 Scintillation index vs. propagation distance with (a) C_n ² = 10⁻¹⁴ m^-2/3, and (b) C_n ² = 10⁻¹² m^-2/3.

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4. Polarization properties of vector vortex propagating through turbulence

For vector vortex, an important property is its degree of polarization (DOP). This is summarized in the movies shown in Figs. 8 and 9 . As the vector vortex beam propagates, it loses its DOP radially from beam center due to decoherence effect caused by the turbulence. This result confirms the analytical results reported in a previous research [16].

Fig. 8 Movies for (a) Stokes parameter S₁ (Media 13), (b) Stokes parameter S₃ (Media 14) and (c) Linescan of Degree of Polarization (Media 15) for the propagation of vector vortex beam through a weak turbulence.

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Fig. 9 Movies for (a) Stokes parameter S₁ (Media 16), (b) Stokes parameter S₃ (Media 17) and (c) Linescan of Degree of Polarization (Media 18) for the propagation of vector vortex beam through a strong turbulence.

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5. Discussions and conclusions

The scalar vortex beam has been proposed for optical communications as information channel. One of the challenges is to detect the existence of the singularity points. If we only consider the irradiance pattern under the strong turbulence case (Fig. 5 , Movies 8 and 9), after 700 m propagation, the beam profiles of both scalar and vectorial vortex beams appear to break down and no characteristic of vortex can be observed. Thus the “information” is lost. However, the image of the Stokes parameter S₁ (Fig. 9) still clearly shows the vector vortex beam polarization singularity. This polarization characteristic can still be observed until 1500 m, more than twice the 700 m propagation distance at which the irradiance characteristic disappears. Eventually, the polarization characteristic also diminishes after further propagation. By examining these Stokes images, it is possible to identify the existence of the vectorial vortex, even though the characteristic vortex structure has disappeared in the irradiance images. This indicates that the vector vortex beam may enable a more robust communication channel than the scalar vortex or fundamental Gaussian channels with improved link distance and signal-to-noise-ratio (SNR). The advantage of vector vortex beams is due to the fact that these beams can be decomposed into two orthogonally polarized LG modes (Fig. 2). These two LG modes are somewhat spatially separated. As they propagate through the turbulence, they are spatially uncorrelated and will experience partially different randomness in the atmosphere, which leads to better information preservation. We point out that, although radial polarization is used as one example to illustrate the unique characteristics of vector vortex propagation through turbulence, the same effect is expected for azimuthally polarized laser beams. Explorations into even more complicated spatially variant polarization states may enable further mitigation of the turbulence effect on their propagation and allow the development of better FSOC systems.

Fig. 5 Beam irradiance movies for the propagation of three kinds of beams through a strong turbulence (C_n ² = 10⁻¹² m^-2/3) (a) Fundamental Gaussian (Media 7), (b) Scalar vortex beam (Media 8) and (c) Vector vortex beam (Media 9).

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Acknowledgement

Wen Cheng is supported by the Dayton Area Graduate Studies Institute (DAGSI) Scholarship. The authors are thankful for this support.

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Propagation of vector vortex beams through a turbulent atmosphere

Abstract

1. Introduction

2. Numerical modeling method

3. Numerical modeling results with weak and strong turbulence

4. Polarization properties of vector vortex propagating through turbulence

5. Discussions and conclusions

Acknowledgement

References and links

Supplementary Material (18)

Cited By

Figures (9)

Equations (4)

Optics Express