Propagation of an Airy beam through the atmosphere

Xiaoling Ji; Halil T. Eyyuboğlu; Guangming Ji; Xinhong Jia

doi:10.1364/OE.21.002154

1. Introduction

In 1979, Berry and Balazs reported that Airy wave packets were a nondiffracting solution to the free-space Schrödinger equation with the context of quantum mechanics [1]. In 2007, Airy beams were studied theoretically and experimentally with the context of optics [2, 3]. In recent years, Airy beams have attracted more attentions because the exotic features of Airy beams make them of interest in some applications. Airy beams have the ability to transversely accelerate (or self-bend) on propagation, i.e., to propagate along parabolic trajectories in vacuum [2–5]. Airy beams have self-healing features, i.e., exhibit remarkable resilience against perturbations and tend to reform on propagation [6]. The scintillation of an Airy beam array is significantly reduced and close to the theoretical minimum [7] and it is possible to obtain substantial scintillation reductions even in the case of a single Airy beam [8]. For an Airy beam, the centroid position is unchanged on propagation in vacuum [5, 9, 10], and the centroid position and skewness are independent of turbulence [10].

When a beam propagates through a medium, a fraction of the beam energy is absorbed by the medium. This absorbed power heats the medium and alters the index of refraction of the path, and leads to a distortion of the beam itself, which is called the thermal blooming [11]. The thermal blooming is the result of the nonlinear interaction of the radiation with the propagation path which is heated by absorption of a fraction of the radiation itself. The thermal blooming was studied by using the four-dimensional (4D) computer code of the time-dependent propagation of high power laser beams through the atmosphere [12, 13]. The estimation of thermal blooming degradation in adaptive-optics was shown in Ref [14]. Reference [15] presented an overview of thermal blooming.

When an Airy beam propagates through the atmosphere, a fraction of the Airy beam energy will be absorbed by the atmosphere, and the thermal blooming may occur. Then, the interesting questions arise: Can an Airy beam retain its shape and the structure when the Airy beam propagates through the atmosphere just like in vacuum? Is the centroid position of Airy beams still unchanged on propagation in the atmosphere just like in vacuum? What is the effect of thermal blooming on the propagation of Airy beams? It is noted that the analysis in the present paper covers only thermal blooming, excluding the effects of atmospheric turbulence.

2. Theoretical model

Maxwell’s wave equation in Fresnel approximation is written as [12, 13]

2 i k \frac{\partial U}{\partial z} = \nabla_{⊥}^{2} U + k^{2} δ ε U,

where

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

,

δ ε = n^{2} / n_{0}^{2} - 1

is the hydrodynamically induced change in permittivity, n is the refractive index, n₀ is the refractive index before perturbation, k is the wave number related to the wave length λ by k = 2π/ λ, U is a slowly varying field amplitude. It is noted that the retarded time is dropped in Eq. (1) for convenience.

The intensity I is given in terms of U, i.e.,

I = {| U |}^{2} \exp (- α z),

where α is the absorption coefficient.

Letting $U^{n}$ be the complete solution to Eq. (1) at $z = z^{n}$ , the solution at $z = z^{n} + Δ z$ may be written as [12, 13]

U^{n + 1} = \exp (- \frac{i}{4 k} Δ z \nabla_{⊥}^{2}) \exp (- \frac{i k}{2} \int_{z^{n}}^{z^{n} + Δ z} δ ε d z) \exp (- \frac{i}{4 k} Δ z \nabla_{⊥}^{2}) U^{n} .

Equation (3) shows that, propagating the field over a distance $Δ z$ consists of a vacuum propagation of the field over a distance $Δ z / 2$ , an incrementing of the phase in accordance with nonlinear medium changes, and the followed by a vacuum propagation of the resulting field over a distance $Δ z / 2$ . In fact, after the first upgrading of the phase, the half steps of propagation can be combined into single propagation step.

On the other hand, the hydrodynamic equation in the isobaric condition is expressed as [12]

\frac{\partial ρ}{\partial t} + v \cdot \nabla ρ = - \frac{(γ - 1) α}{c_{s}^{2}} I,

where ρ and v are perturbations in density and velocity, c_s is the sound speed, and γ is the specific heat ratio.

The field of Airy beams at the initial plane (z = 0) can be expressed as [2, 7, 10]

U (x, y, z = 0) = Ai (\frac{x}{w_{0}}) \exp (a \frac{x}{w_{0}}) Ai (\frac{y}{w_{0}}) \exp (a \frac{y}{w_{0}}),

where w₀ and a are the arbitrary transverse scale and the exponential truncation factor, respectively.

The centroid position and the mean-squared beam width of optical beams are defined as [16]

\bar{x} = \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x I (x, y, z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) d x d y}, \bar{y} = \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} y I (x, y, z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) d x d y},

and

w_{x}^{2} = \frac{4 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(x - \bar{x})}^{2} I (x, y, z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) d x d y}, w_{y}^{2} = \frac{4 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(y - \bar{y})}^{2} I (x, y, z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) d x d y},

respectively.

Based on Eqs. (3) and (4) together with Eqs. (2) and (5), the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere is obtained by means of a discrete Fourier transform method, which is described in [12, 13] in detail and omitted here. In addition, changes of centroid position and mean-squared beam width of Airy beams propagating through the atmosphere also can be studied when Eqs. (6) and (7) are considered by using a discrete method in the 4D computer code.

3. Numerical calculation results and analysis

In the numerical examples, calculation parameters are taken as γ = 1.4, n₀ = 1.00035, c_s = 340m/s, α = 1.252 × 10⁻⁵/m, $λ = 10.6 μm$ , w₀ = 0.05m and a = 0.2. The standard atmosphere density ρ₀ = 1.302461kg/m³ is taken as the initial value to solve Eq. (4).

The changes of three-dimensional (3D) intensity distribution I(x, y, z) and its counter lines versus the propagation distance z, the time t, the power P and the cross wind velocity v_x in x direction are examined by using the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere and are depicted in Figs. 1 -8, respectively. It is mentioned that in Figs. 1-8 the “12.5 mm” is referring to multiplication for the numerals on the transverse axes, and the middle value of the transverse axes is corresponding to the position of the propagation axis z. From Figs. 1 and 2 , it can be seen that an Airy beam can’t retain its shape and the structure when the Airy beam propagates through the atmosphere except for the short propagation distance (e.g., z = 0.1km in Fig. 1(b) and Fig. 2(b)). The physical reason is the effect of thermal blooming of atmosphere. When the propagation distance z increases, the thermal blooming causes the center lobe to dip centrally, spread and decrease. In contrast with the center lobe, the side lobes are less affected by thermal blooming because the energy within the side lobe is lower than that within the center lobe, such that the intensity maximum of the side lobe may be larger than that of the center lobe (see Fig. 1(e) and 1(f)). When the z is long enough, the side lobe may dip centrally due to thermal blooming (e.g., z = 1.2km in Fig. 1(f)). In this paper, we adopt the Strehl ratio S_R to describe the effect of thermal blooming on the maximum intensity, which is defined as S_R = I_max/I_0max, where I_max and I_0max are the maximum intensity in the atmosphere and in vacuum respectively. The smaller S_R means the maximum intensity is more affected by thermal blooming. When z = 0, 0.1km, 0.4km, 0.6km, 0.8km and 1.2km (see Figs. 1(a)-1(f)), we have S_R = 1, 0.899, 0.411, 0.307, 0.273 and 0.204, respectively.

Fig. 1 3D intensity distribution I(x, y, z) for different values of the propagation distance z when P = 5 × 10⁵W, t = 0.6s and v = 0. (a) z = 0, (b) z = 0.1km, (c) z = 0.4km, (d) z = 0.6km, (e) z = 0.8km, (f) z = 1.2km.

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Fig. 2 Counter lines of I(x, y, z) for different values of the propagation distance z when P = 5 × 10⁵W, t = 0.6s and v = 0. (a) z = 0, (b) z = 0.1km, (c) z = 1.2km.

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Figures 3 -6 show that the effect thermal blooming increases as the time t and the power P increase. However, when the time t is short enough (e.g., t = 0.024s in Fig. 3(b) and Fig. 4(b) ) and the power P is low enough (e.g., P = 10⁴W in Fig. 5(b) and Fig. 6(b) ), the effect of thermal blooming can be ignored, and the Airy beam can retain its shape and the structure, which is similar to the behavior in vacuum. On the other hand, for Figs. 3(a)-3(f) we have S_R = 1, 0.762, 0.411, 0.285, 0.247 and 0.226, respectively. For Figs. 5(a)-5(f) we have S_R = 1, 0.877, 0.397, 0.280, 0.247 and 0.201, respectively.

Fig. 3 3D intensity distribution I(x, y, z) for different values of the time t when P = 6 × 10⁵W, z = 1km and v = 0. (a) in vacuum, (b) t = 0.024s, (c) t = 0.12s, (d) t = 0.256s, (e) t = 0.4s, (f) t = 0.616s.

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Fig. 4 Counter lines of I(x, y, z) for different values of the time t when P = 6 × 10⁵W, z = 1km and v = 0. (a) in vacuum, (b) t = 0.024s, (c) t = 0.616s

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Fig. 5 3D intensity distribution I(x, y, z) for different values of the power P when t = 0.8s, z = 1km and v = 0. (a) in vacuum, (b) P = 10⁴W, (c) P = 10⁵W, (d) P = 2 × 10⁵W, (e) P = 3 × 10⁵W, (f) P = 6 × 10⁵W.

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Fig. 6 Counter lines of I(x, y, z) for different values of the power P when t = 0.8s, z = 1km and v = 0. (a) in vacuum, (b) P = 10⁴W, (c) P = 6 × 10⁵W.

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Figures 7 and 8 give the 3D intensity distribution I(x, y, z) and its counter lines versus the cross wind velocity v_x in x direction. It can be seen that the center lobe takes as a crescent-like pattern (see Fig. 7(b) and Fig. 8(b) where v_x = 0.15m/s), which is associated with thermal distortion. In addition, it is clear that the cross wind results in a decrease of the effect of thermal blooming. The physical reason is that the cross wind constantly brings cooler air onto the beam path, allowing more resistance to the effect of thermal blooming. For example, for the center lobe, the central dip disappears, the intensity maximum increases, and the spreading decreases due to the cross wind. When the cross wind velocity is large enough (e.g., v_x = 1m/s in Fig. 7(f) and Fig. 8(c)), the intensity distribution is some similar to that of the Airy beam propagating in vacuum. On the other hand, for Figs. 7(a)-7(f) we have S_R = 0.223, 0.384, 0.563, 0.652, 0.698 and 0.744, respectively.

Fig. 7 3D intensity distribution I(x, y, z) for different values of the cross wind velocity v_x when t = 0.64s, z = 1km and P = 6 × 10⁵W. (a) v_x = 0, (b) v_x = 0.15m/s, (c) v_x = 0.4m/s, (d) v_x = 0.6m/s, (e) v_x = 0.8m/s, (f) v_x = 1m/s.

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Fig. 8 Counter lines of I(x, y, z) for different values of the cross wind velocity v_x when t = 0.64s, z = 1km and P = 6 × 10⁵W. (a) v_x = 0, (b) v_x = 0.15m/s, (c) v_x = 1m/s.

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The changes of mean-squared beam width w (w_x = w_y = w) versus the propagation distance z, the time t and the power P are plotted in Figs. 9 , 10 , and 11 , respectively. It can be seen that, w in the atmosphere is larger than that in vacuum due to the effect of thermal blooming. In the atmosphere w increases with increasing z, t and P, but in vacuum w is unchanged versus t and P. The changes of centroid position versus the propagation distance z, the time t and the power P are also examined by using the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere. It can be shown that the changes of centroid position versus z, t and P are very small, e.g., only within 1mm for the cases in Figs. 9-11, and the numerical results are all omitted here. It is mentioned that the centroid doesn’t change in vacuum or turbulence [10], but changes in the atmosphere. When an optical beam starts with the decentred intensity at the source plane, the thermal blooming will also be decentred. Thus, the decentred field phase distortion will appear due to thermal blooming when the optical beam propagates through the atmosphere, which results in a change of the centroid.

Fig. 9 Mean-squared beam width w (w_x = w_y = w) versus the propagation distance z when P = 3 × 10⁵W, t = 0.6s and v = 0.

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Fig. 10 Mean-squared beam width w (w_x = w_y = w) versus the time t when P = 6 × 10⁵W, z = 1km and v = 0.

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Fig. 11 Mean-squared beam width w (w_x = w_y = w) versus the power P when t = 0.8s, z = 1km and v = 0.

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The changes of centroid position and the mean-squared beam width versus the cross wind velocity v_x in x direction are given in Figs. 12 and 13 , respectively. Figure 12 indicates that the change of centroid position $\bar{x}$ is not monotonic versus the cross wind velocity v_x, and there exists a minimum, where the centroid position is the furthest away from the propagation axis z. On the other hand, the centroid position $\bar{y}$ is nearly unchanged versus v_x. Figure 13 shows that w_x and w_y are all decrease with increasing v_x, but w_x is smaller than w_y. It implies that, in the atmosphere the effect of the cross wind velocity v_x on the intensity distribution of an Airy beam in x direction is larger than that in y direction.

Fig. 12 Centroid position $\bar{x}$ , $\bar{y}$ versus the cross wind velocity v_x when t = 0.64s, z = 1km and P = 6 × 10⁵W.

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Fig. 13 Mean-squared beam width w_x, w_y versus the cross wind velocity v_x when t = 0.64s, z = 1km and P = 6 × 10⁵W.

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4. Conclusions

In this paper, the changes of the intensity distribution, the centroid position and the mean-squared beam width of an Airy beam propagating through the atmosphere have been examined by using the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere. It has been shown that an Airy beam cannot retain its shape and the structure when the Airy beam propagates through the atmosphere due to thermal blooming. When the propagation distance z or the time t is short enough, or the beam power P is low enough, the effect of thermal blooming can be ignored, and the Airy beam can behave as that in vacuum. The thermal blooming causes the center lobe to dip centrally, spread and decrease. In contrast with the center lobe, the side lobes are less affected by thermal blooming because the energy within the side lobe is lower than that within the center lobe, such that the intensity maximum of the side lobe may be larger than that of the center lobe. However, the cross wind can reduce the effect of thermal blooming.

The dependence of centroid position $\bar{x}$ on the cross wind velocity v_x is not monotonic, and there exists a minimum, but the centroid position $\bar{y}$ is nearly independent of v_x. The mean-squared beam width w_x and w_y are all decrease with increasing v_x, but w_x is smaller than w_y. In addition, changes of centroid position versus z, t and P are very small. In the atmosphere w increases with increasing z, t and P, but in vacuum w is independent of t and P. The results obtained in this paper are very useful for applications of Airy beams.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under grant 61178070, and by the financial support from Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province under grant 12TD008.

References and links

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Propagation of an Airy beam through the atmosphere

Abstract

1. Introduction

2. Theoretical model

3. Numerical calculation results and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (13)

Equations (7)

Optics Express