Investigation on the behavior of a laser propagating through a random environment induced by wind

Xianwei Huang; Xiaohui Shi; Suqin Nan; Yanfeng Bai; Xiquan Fu

doi:10.1364/OE.27.009420

1. Introduction

It is known that the near field airflow induced by wind can be constructed when an object moves in free space. When a laser source is placed on a moving platform with high speed, the beam propagation can be largely influenced by a random airflow, and the optical field gets distorted [1–3]. In fact, beams propagation through a random media has attracted much attention due to the potential applications in optical imaging and optical communication, such as the beam evolution in atmosphere turbulence [4, 5] and ocean water [6, 7], and the deviation of the beam center and the decline of the beam quality are aggravated during propagation [8–11], in addition, the performance of the free space optical communication system is severely affected [12–14]. Among the factors influencing the turbulence, the wind velocity is an important one, and the wind leads to the random distribution of the airflow. As a result, investigating the beam propagation through a random environment induced by wind is significant for optical propagation and communication between two moving platforms. Up to now, much attention has been paid to the discussions about the airflow properties and the statistics in optics [15–17]. Later, a theoretical formula to describe the optical effects under the influence of wind was proposed, and the theory was determined by wind tunnel experiment [18, 19]. The optical distortion is governed by the square of the incoming wind velocity multiplied by the freestream density as well as the displacement thickness and the temperature mismatch between the freestream and underlying wall [20, 21]. With a high-speed Shack-Hartmann sensor, the optical aberration can be measured, and the effects of different aperture sizes and subsonic speeds are detailly discussed [22]. Two control methods have been derived to adaptively correct the aero-optical wavefronts, and the improvement of beam quality has been achieved [23]. However, these results mainly focus on the wave-front distortion when a laser beam transmits through an airflow environment. To the best of our knowledge, the behavior of beams propagation under the effects of the wind has not yet been reported, such as the beam propagation model and the beam deflection. And questions have been promoted: How is the beam propagation model established properly? How are the dynamics of beams propagation in an airflow environment? What are the effects of the wind velocity on beams propagation? This paper will clarify the questions.

In this paper, the aim is to investigate the behavior of a laser propagating through a random environment induced by wind, and the experiment is performed by the wind tunnel. The beam gradually deviates from the center, and the beam deflection occurs. The effect of the wind velocity on the beam deflection is detailly discussed. The simulation model of beams propagation through this kind of environment is established based on split-step beam propagation and multiple random phase screen method, and the numerical simulation agrees with the experimental results. In the end, we extend this theoretical model into the high-speed region, and the beam deflection evolution is analyzed, in addition, the difference of the beam deflection between turbulence and airflow is presented. It is the first time to investigate the beam deflection in a random environment induced by wind based on the theoretical beam propagation model. In optical communication, the optical signal can’t be detected if the beam deflection value is larger than the detector size, and then the bit error rate increases [24, 25]. Therefore, investigating the beam deflection is helpful to determine the suitable size of the detector and can find potential applications in optical imaging and optical communication.

2. Experimental results and analysis

The wind is an important factor that influences the turbulence, and the inhomogeneous distribution of the wind velocity leads to the random airflow which severely affects the beam propagation. It is noted that this kind of environment can be realized with the wind tunnel. The multiple phase screen construction is a common method in turbulence simulation, and the wind tunnel environment can be regarded as the single one among the multiple phase screen of turbulence. As dipicted in Fig. 1, the experiments have been performed at the test section of the wind tunnel to examine the influences of the wind velocity on beams propagation, and it is noted that the wall of the wind tunnel is adiabatic. The width of the wind tunnel is 45cm. The laser source (λ = 532nm) is employed to generate the input Gaussian beam with the width being w₀ = 1mm. The beam propagation through the airflow is reflected for four times by mirrors (M2–M5, the radius is 25.4mm), and we choose three locations (M1, M2 and M3) to detect the intensity distribution, then the length of the airflow interval at the three locations above is 2.31m (M1), 1.38m (M2) and 0.45m (M3), respectively. Finally, the intensity profile in x–y plane is recorded by a CCD (Stingray F-504B/C, interval between two pixels is 3.45 × 10⁻⁶m) at three locations, and the parameters of CCD are set as: the radius is 1.77mm, and the exposure time is 1ms.

Fig. 1 The experimental schematic of a laser beam propagating through the test section of the wind tunnel, and M2–M5 are mirrors; Light intensity is detected by CCD at the locations of M1, M2 and M3.

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At each detection location, the center position of beams is calculated as:

x_{n} = \frac{\sum_{i} x_{i} I_{n} (x_{i}, y_{i})}{\sum_{i} I_{n} (x_{i}, y_{i})}, y_{n} = \frac{\sum_{i} y_{i} I_{n} (x_{i}, y_{i})}{\sum_{i} I_{n} (x_{i}, y_{i})}

The intensity distribution is sampled for 100 times, and the average coordinates of the centroid is given by

\bar{x} = \frac{1}{N} \sum_{n = 1}^{N} x_{n}, \bar{y} = \frac{1}{N} \sum_{n = 1}^{N} y_{n}

The data points of the center of gravity are shown in Fig. 2, and it is obvious that the offset of beam center gradually increases for the larger airflow interval and wind velocity. The beam deviation under no wind condition mainly attributes to the instability of the experimental platform. The average beam deflection is easy to be get: $\bar{r} = \sqrt{{\bar{x}}^{2} + {\bar{y}}^{2}}$ . The experimental results are presented in Fig. 3. Figures 3(a)–3(c) depict the beam deflection versus measurement times at three detection locations. The wind velocity and propagation distance present a significant influence, and the larger wind velocity leads to the larger value of the beam deflection and the stronger fluctuation, in addition, the identical tendency is presented when the airflow interval increases comparing Fig. 3(a) with Fig. 3(c). However, the results above are the absolute beam deflection. In the experiment, the intensity distribution at three locations with no wind is also recorded for 100 times, and the beam deflection figure is not given. The average beam deflection at each location under different wind velocity values is shown in Fig. 3(d), where the average beam deflection under no wind condition is removed. Generally, the average beam deflection presents an increasing trend for the larger airflow interval and wind velocity. From Fig. 3(d), the average beam deflection is less than 0.1mm, which implies that the average beam deflection is lower than 2.83% comparing with the size of the detector.

Fig. 2 The data points of the center of gravity at three locations (M1, M2, M3) under different wind velocity cases.

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Fig. 3 The experimental results for beam deflection in airflow environment, (a)–(c) are the absolute beam deflection versus different velocity values at three locations, (d) is the average beam deflection that removes the no wind case.

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3. Comparison between simulation and experimental results

Here, we introduce a theoretical model to simulate the beam propagation in a random environment induced by wind, and the optical propagation can be modeled by:

i \frac{\partial A (x, y, z)}{\partial z} + \frac{1}{2 k} \nabla_{⊥}^{2} A (x, y, z) + k n_{1} A (x, y, z) = 0

where

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

, A(x, y, z) is the light field during propagation, k = 2π/λ is the wave number with λ being the wavelength of the source, n₁ is the refractive index caused by the airflow. Similar to the effect of atmosphere turbulence, the variation of the airflow leads to the change of the refractive index, and the perturbation can be considered as the phase modulation of the light field. From the analysis above, the beam propagation process, which can be regarded as the combination of the diffraction and phase modulation, is described as:

\begin{array}{l} A (x, y, z) & = & A_{diff} (x, y, z^{'}) exp [i k \int_{z^{'}}^{z} n_{1} (x, y, ξ) d ξ] \\ = & A_{diff} (x, y, z^{'}) exp (i S) \end{array}

where S is the random phase modulation due to the airflow disturbance, and A_diff (x, y, z′) is the light field after diffraction. The propagation of a wave field from the transmitter to the receiver is equally split over the sub-propagation distance Δz = L/M₀, where L is the total propagation distance and M₀ is the number of steps, and the wave field is multiplied with the random phase and propagates freely to the next screen at each step. The propagation method demonstrated above is called as split-step beam propagation and multiple random phase screen method [26, 27]. For one step during propagation, the light field after diffraction can be expressed as:

A_{diff} (x, y, z_{n}) = exp (- \frac{i}{2 k} Δ z \nabla_{⊥}^{2}) A (x, y, z_{n - 1})

According to Eqs. (4) and (5), the light field during propagation can be expressed as:

\begin{array}{l} A (x, y, z) & = & \prod_{n = 1}^{M_{0}} exp (- \frac{i}{2 k} Δ z \nabla_{⊥}^{2}) exp [i S (x, y, z_{n})] \\ \times A (x, y, z_{n - 1}) \end{array}

where Δz = z_n − z_n−1. The temperature disturbance of the wind tunnel can be ignored due to the adiabatic wall of the wind tunnel. From [28], the average phase difference of a laser source propagating through an airflow environment without considering the temperature can be approximately described as:

S_{airflow} = 1.7 \times 10^{- 5} \times \frac{2 π δ^{*} ρ_{0}}{λ sin (β) ρ_{SL}} M^{2}

where δ^* is the displacement boundary-layer thickness (δ/δ^* = q, δ is the boundary-layer thickness, q is a constant and is approximately equal to 8 as M < 0.1Ma [19]), ρ_SL = 1.229kg/m³ is the sea-level density, ρ₀ is the freestream density, β is the elevation angle between the laser and the wind tunnel wall, λ is the laser wavelength and M is the wind velocity (Mach number). From Eq. (6), the average phase difference is proportional to the displacement boundary-layer thickness and the square of the wind velocity. The random phase S can be obtained by constructing the random phase screen: the average phase difference is randomized, and the mean value of the random airflow phase is zero while the variance is S_airflow. According to the generation of the random phase S, the wind velocity can affect the variance of the random airflow phase, and then the randomness of the airflow phase is influenced. The larger wind velocity leads to the increase of the variance S_airflow, resulting in the enhanced randomness and phase disturbance of the airflow. From the construction process of the model introduced above, we have found several similarities between airflow and turbulence environment, on the one hand, the effects of airflow and turbulence on beams propagation are considered as the phase modulation, on the other hand, there exists the turbulence in the two kinds of environment, and then the phase screen construction method is reasonable because the turbulence noise follows Gaussian distribution. However, the structure of an airflow includes the laminar flow and the mixed flow apart from turbulence, thus it is not proper to describe the beam wander with well-defined formulas in turbulent environment, and the random modulation to the average phase difference S can be adopted.

With Eq. (5) and Eq. (6), a numerical simulation is performed. The input source is Gaussian type, and the parameters in the simulation are set as follows: the wavelength of the input source is 532nm, the width of the input Gaussian beam is 1mm, the freestream density is regarded as the air density ρ₀ = 1.225kg/m³, and β is set to be 90°. The comparison between the simulation results (the blue curve) and experimental results (the black diamond symbol with error bar) are presented in Fig. 4, where the larger wind velocity and airflow interval result in the increment and stronger oscillation of the average beam deflection. As depicted in Fig. 4, the evolution trend of the average beam deflection from numerical simulation is identical to the experimental case. In Figs. 4(a) and 4(b), the experimental results are generally larger than the simulation results for a small airflow interval, and only one experimental result is close to the simulation result. However, when the airflow interval increases, the experimental results match well with the simulation results as Fig. 4(c) shows. From Eq. (6), the results under longer airflow interval are equivalent to that under larger wind velocity, and one can also infer that the experimental and simulation results match well with each other for the relatively larger wind velocity. It can be demonstrated from Fig. 4 that the ratio between the experimental results and the simulation results is always lower than 10. From the analysis above, the simulations can agree with the experiments although existing the difference within an order of magnitude, which indicates the validity of the theoretical model, in addition, the ratio means that other factors such as the vibration of the experiment platform can affect the result because the experimental result is generally larger than the simulation one. In the end, we extend the theoretical model to simulate the effects of the large wind velocity on the average beam deflection, and the beam propagation distance within the airflow environment is set to be 0.45m, 1.38m and 2.31m, as shown in Fig. 5(a). One can find that the average beam deflection gradually increases with the wind velocity, and that the larger wind velocity leads to the stronger oscillation of the average beam deflection curve. From Eq. (6), the larger the wind velocity is, the stronger the disturbance of the airflow phase becomes, and then the stronger oscillation of the average beam deflection is presented, resulting in the transient decreasing trend. When the wind velocity approaches to 340m/s with the airflow interval being 2.31m, the average beam deflection can reach 2mm that is equal to 60% of the detector size. As the propagation distance within the airflow environment decreases to 1.38m or 0.45m, the average beam deflection decreases to 33.1% or 2.3% respectively comparing with the detector size.

Fig. 4 The comparison of the average beam deflection versus wind velocity between the numerical results (the blue curve) and the experimental results (the black diamond symbol with error bar) under different airflow interval values for (a) the length of airflow interval is 0.45m (detection location: M3), (b) the length of airflow interval is 1.38m (detection location: M2) and (c) the length of airflow interval is 2.31m (detection location: M1).

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Fig. 5 Numerical results for the average beam deflection under different propagation length for (a) versus wind velocity and for (b) versus turbulent strength.

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In addition, we present a comparison of the average beam deflection between airflow and turbulence environment in Fig. 5, where $C_{n}^{2}$ reperesents the turbulent strength. The Von Karman spectrum is used to generate a random phase screen of turbulence in the simulation [26, 27, 29], and the parameters are set as: the inner scale l₀ is 1mm, the outer scale L₀ is 10m, the spectral index of the Von Karman spectrum is 11/3 [30, 31], and other parameters are identical to that under the airflow case. With the increase of turbulence strength, the randomness and phase disturbance of the turbulence gets enhanced, thus the beam gradually deviates from the center, and the results can also be indicated in [24, 27]. Comparing with the average beam deflection results in Fig. 5(b), one can easily find that the influence of the airflow is much stronger under the identical propagation length. It is known that the beam deflection can cause the increase of the packet loss probability in communication due to the limitation of the aperture size, as a result, the ratio between the average beam deflection and the detector size is helpful to estimate the aperture size of the detector for communication and imaging system at the receiver plane.

4. Conclusion

In conclusion, we have investigated the behavior of a laser tramsmitting through a random environment induced by wind, and the experiment is performed with the help of the wind tunnel. The effects of the wind velocity and the propagation length on the average beam deflection are discussed in detail, and the larger wind velocity and airflow interval lead to the increasing average beam deflection. The theoretical model governing beams propagation is proposed based on the split-step beam propagation and multiple phase-screen method, and the simulation results agree with the experimental ones. Finally, we have extended the beam propagation into the high-speed condition, and it is found that the average beam deflection in airflow is much larger comparing with the turbulent case. The wind is an important factor causing the environment disturbance, therefore, our results are of importance for the future discussions in the long-distance beam propagation and communication under high wind velocity.

Funding

National Natural Science Foundation of China (61571183); Hunan Provincial Natural Science Foundation of China (2017JJ1014).

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Investigation on the behavior of a laser propagating through a random environment induced by wind

Abstract

1. Introduction

2. Experimental results and analysis

3. Comparison between simulation and experimental results

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (7)

Optics Express