Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

Guoquan Zhou

doi:10.1364/OE.19.003945

1. Introduction

Hermite-sinusoidal-Gaussian (HSG) beams have been introduced as exact solutions of the paraxial wave equation [1,2]. Cosh-Gaussian beams are the special cases of HSG beams. The propagation of cosh-Gaussian beams in free space and through complex optical systems has been studied [1–4]. The propagation characteristics of cosh-Gaussian beams in turbulent atmosphere have been also extensively investigated [5–9]. On the basis of the superposition of laser beams, a group of virtual sources that generates a cosh-Gaussian beam has been presented [10]. By means of the theorem of the vectorial structure and the method of stationary phase, the vectorial structure of cosh-Gaussian beam has been examined in the far-field [11]. The coherent combination of certain cosh-Gaussian beams results in flattened beams with an axial shadow [12]. The incoherent combination of some cosh-Gaussian beams with different parameters has been shown to be a flattened Gaussian beam [13]. The higher-order cosh-Gaussian beam, which is defined as a higher-order cosh function multiplied by a Gaussian function, can be produced by coherent superposition of cosh-Gaussian beams or by superposition of decentered Gaussian beams with the same waist width. The higher-order cosh-Gaussian beam is a broader beam model than the cosh-Gaussian beam. By varying the beam parameters, the higher-order cosh-Gaussian beam in the source plane will take on a Gaussian-like distribution, a flattened distribution, and a dark hollow distribution. Therefore, the higher-order cosh-Gaussian beam is an interesting beam model. Based on the second-order moment, the beam propagation factor and the kurtosis parameter of a higher-order cosh-Gaussian beam have been derived [14]. The fractional Fourier transform has been applied to treat the propagation of higher-order cosh-Gaussian beams, and the properties of a higher-order cosh-Gaussian beam have been demonstrated in the fractional Fourier transform plane [15]. The vectorial structural characteristics of a higher-order cosh-Gaussian beam have been revealed in the far-field reference plane [16]. Propagation properties of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis have been examined [17]. The research in the propagation of laser beams in a turbulent atmosphere is vital to the applications in free-space optical communications and remote sensing. To the best of our knowledge, the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere has not been reported elsewhere. In the remainder of this paper, therefore, the propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere is investigated. Analytical propagation formulae of the average intensity, the effective beam size, and the kurtosis parameter are derived and illustrated by a numerical example.

2. Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The higher-order cosh-Gaussian beam in the source plane z = 0 takes the form as

E {}_{n}{(x_{0}, y_{0}, 0)} = E {}_{n}{(x_{0}, 0)} E {}_{n}{(y_{0}, 0)},

with E_n(x ₀, 0) and E_n(y ₀, 0) given by

E {}_{n}{(j_{0}, 0)} = \cosh^{n} (Ω j_{0}) \exp (- \frac{j_{0}^{2}}{w_{0}^{2}}), \begin{matrix} n = 1, 2, \end{matrix} 3, \dots,

where j ₀ = x ₀ or y ₀. n is the beam order. Ω is the cosh parameter, and w ₀ is the waist width of Gaussian part. If n = 0, Eq. (1) reduces to be the well-known Gaussian beam. If n = 1, Eq. (1) reduces to be a cosh-Gaussian beam. E_n(j ₀, 0) can also be written in the form as

E {}_{n}{(j_{0}, 0)} = \sum_{m = 0}^{n} C_{m} \exp [- {(j_{0} - b_{m})}^{2} / w_{0}^{2}],

with the coefficients C_m and b_m given by

C_{m} = a_{m} \exp [{(m - \frac{n}{2})}^{2} δ], a_{m} = \frac{n!}{2^{n} m! (n - m)!}, b_{m} = (m - \frac{n}{2}) \frac{δ}{Ω},

where δ = w ₀ ²Ω². Therefore, E_n(j ₀, 0) can be produced by superposition of n + 1 decentered Gaussian beams with the same waist width. Based on the extended Huygens-Fresnel diffraction integral, the higher-order cosh-Gaussian beam propagating through a paraxial and real ABCD optical system in turbulent atmosphere can be obtained by

\begin{array}{l} E (x, y, z) = \frac{1}{i λ B} \exp [i k z + \frac{i k D (x^{2} + y^{2})}{2 B}] \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{n} (x_{0}, y_{0}, 0) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \times \exp {\frac{i k}{2 B} [A (x_{0}^{2} + y_{0}^{2}) - 2 (x x_{0}^{} + 2 y y_{0}^{})] + ψ (x_{0}, y_{0}, x, y)} d x_{0} d y_{0}, \end{array}

where k = 2π/λ with λ being the optical wavelength. ψ(x ₀, y ₀, x, y) is the solution to the Rytov method that represents the random part of the complex phase. The optical system of concern is orthogonal and is described by the same ABCD matrix in each of the mutually orthogonal planes, x-z and y-z. A, B, C, and D are the transfer matrix elements of the paraxial optical system. Moreover, there is no inherent aperture between the source and the output planes, which denotes that A, B, C, and D are all realvalued. Therefore, the average intensity of the higher-order cosh-Gaussian beam passing through a paraxial and real ABCD optical system in turbulent atmosphere is found to be

\begin{array}{l} < I (x, y, z) > = \frac{1}{λ^{2} B^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{n} (x_{01}, y_{01}, 0) E_{n}^{*} (x_{02}, y_{02}, 0) < \exp [ψ (x_{01}, y_{01}, x, y) + ψ^{*} (x_{02}, y_{02}, x, y)] > \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \times \exp {\frac{i k}{2 B} [A (x_{01}^{2} - x_{02}^{2} + y_{01}^{2} - y_{02}^{2}) - 2 x (x_{01}^{} - x_{02}^{}) - 2 y (y_{01}^{} - y_{02}^{})]} d x_{01} d y_{01} d x_{02} d y_{02}, \end{array}

where the angle brackets denote the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The asterisk means the complex conjugation. The ensemble average term is given by

< \exp [ψ (x_{01}, y_{01}, x, y) + ψ^{*} (x_{02}, y_{02}, x, y)] > = \exp [- \frac{{(x_{01} - x_{02})}^{2} + {(y_{01} - y_{02})}^{2}}{ρ_{0}^{2}}],

where ρ ₀ is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and is defined as

ρ_{0} = B {[1.46 k^{2} C_{n}^{2} \int_{0}^{L} b^{5 / 3} (z) d z]}^{- 3 / 5},

where C_n ² is the constant of refraction index structure and describes the turbulence level. b(z) corresponds to the approximate matrix element for a ray propagating backwards through the system. L is the axial distance between the source and the output planes. Substituting Eqs. (1), (3), and (7) into Eq. (6), the average intensity of the higher-order cosh-Gaussian beam in the output plane reads as

< I (x, y, z) > = < I (x, z) > < I (y, z) >,

with <I(x, z) and <I(y, z) given by

< I (j, z) > = \frac{π}{λ B \sqrt{α_{1 j} α_{2 j}}} \sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} \exp (\frac{β_{1 j}^{2}}{4 α_{1 j}} + \frac{β_{2 j}^{2}}{4 α_{2 j}}),

where j = x or y (hereafter). The parameters α ₁ _j, α ₂ _j, β ₁ _j, and β ₂ _j are defined by

α_{1 j} = \frac{1}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{i k A}{2 B}, α_{2 j} = \frac{1}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} - \frac{1}{α_{1 j} ρ_{0}^{4}} - \frac{i k A}{2 B}, β_{1 j} = \frac{2 b_{l}}{w_{0}^{2}} + \frac{i k j}{B}, β_{2 j} = \frac{2 b_{m}}{w_{0}^{2}} - \frac{i k j}{B} + \frac{β_{1 j}}{α_{1 j} ρ_{0}^{2}} .

The effective beam size of the higher-order cosh-Gaussian beam in the j-direction of the output plane yields

W_{j} = {[\frac{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{2} < I (x, y, z) > d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} < I (x, y, z) > d x d y}]}^{1 / 2} = \frac{1}{\sqrt{2} ξ} {[\frac{\sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} (2 ξ_{j} + η_{j}^{2}) \exp (τ)}{\sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} \exp (τ)}]}^{1 / 2},

with τ given by

τ = \frac{η_{j}^{2}}{4 ξ_{j}} + \frac{b_{l}^{2}}{α_{1 j} w_{0}^{4}} + \frac{γ_{1 j}^{2}}{4 α_{2 j}},

where γ ₁ _j, γ ₂ _j, ξ_j, and η_j are defined by

γ_{1 j} = \frac{2 b_{m}}{w_{0}^{2}} + \frac{2 b_{l}}{α_{1 j} w_{0}^{2} ρ_{0}^{2}}, γ_{2 j} = \frac{i k}{α_{1 j} B ρ_{0}^{2}} - \frac{i k}{B}, ξ_{j} = \frac{k^{2}}{4 α_{1 j} B^{2}} - \frac{γ_{2 j}^{2}}{4 α_{2 j}}, η_{j} = \frac{i k b_{l}}{B α_{1 j} w_{0}^{2}} + \frac{γ_{1 j} γ_{2 j}}{2 α_{2 j}} .

The kurtosis parameter, which is defined by the fourth-order moment, is employed to describe the flatness degree of the beams and is an important parameter to valuate the beam propagation. The kurtosis parameter of the higher-order cosh-Gaussian beam in the j-direction turns out to be

\begin{array}{l} K_{j} = \frac{[\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{4} < I (x, y, z) > d x d y] [\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} < I (x, y, z) > d x d y]}{{[\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{2} < I (x, y, z) > d x d y]}^{2}} \\ \begin{matrix}  \end{matrix} = \frac{\sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} \exp (τ) \sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} (12 ξ_{j}^{2} + 12 ξ_{j} η_{j}^{2} + η_{j}^{4}) \exp (τ)}{{[\sum_{m = 0}^{n} \sum_{l = 0}^{n} a_{m} a_{l} (2 ξ_{j} + η_{j}^{2}) \exp (τ)]}^{2}} . \end{array}

3. The numerical results and analyses

Now, the average intensity and spreading of a higher-order cosh-Gaussian beam in turbulent atmosphere are calculated by using the formulae derived above. For simplicity, no other optical element is placed in turbulent atmosphere, which denotes that the matrix elements are A = 1, B = z, C = 0, and D = 1. As the x- and y-directions are separable in the formulae derived above, only the x-direction is considered hereafter. Moreover, λ is set to be 0.8μm. Figures 1 and 2 represent the normalized intensity distributions of higher-order cosh-Gaussian beams with different beam parameters in the reference plane z = 1km and z = 2km in turbulent atmosphere. In Figs. 1 and 2, C_n ² is set to be 10⁻¹⁴m^-2/3. Under different conditions of the beam parameters, the higher-order cosh-Gaussian beam in the source plane can be a Gaussian-like beam, a flattened beam, and a dark hollow beam. The intensity distributions of a flattened beam and a dark hollow beam will change upon propagation. When the propagation distance is large enough, the intensity distributions of a flattened beam and a dark hollow beam will tend to a Gaussian-like distribution. With varying one of the cosh parameter Ω, the beam order n, and the Gaussian waist w ₀, the normalized intensity distribution of a higher-order cosh-Gaussian beam propagating in turbulent atmosphere also takes on different distribution, which is shown in Fig. 1. From Figs. 1 and 2, we can conclude the following conclusion. When one of the cosh parameter Ω, the beam order n, and the Gaussian waist w ₀ is large, the normalized intensity distribution of a higher-order cosh-Gaussian beam will undergo first the dark hollow distribution, then the fattened distribution, and finally the Gaussian-like distribution with increasing the propagation distance in turbulent atmosphere. To further reveal the spreading properties of a higher-order cosh-Gaussian beam in turbulent atmosphere, the effective beam sizes of higher-order cosh-Gaussian beams versus the propagation distance z in turbulent atmosphere are depicted in Fig. 3 , where C_n ² = 10⁻¹⁴m^-2/3. When the propagation distance in turbulent atmosphere is small, the higher-order cosh-Gaussian beam with the large cosh parameter has the large effective beam size. When the propagation distance in turbulent atmosphere is large enough, the effective beam sizes of higher-order cosh-Gaussian beams with the different cosh parameters are approximately equivalent. The higher-order cosh-Gaussian beam with the large beam order has the large effective beam size. However, the difference between the effective beam sizes of higher-order cosh-Gaussian beams with different beam orders will shrink upon propagation in turbulent atmosphere. The higher-order cosh-Gaussian beam with the smaller Gaussian waist first has the smaller effective beam size and finally has the larger effective beam size with increasing the propagation distance in turbulent atmosphere. The gradients of the curves in Fig. 3 denote that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters. Figure 4 shows the kurtosis parameters of higher-order cosh-Gaussian beams versus the propagation distance z in turbulent atmosphere. When first propagating in turbulent atmosphere, the higher-order cosh-Gaussian beam with the small cosh parameter, or the small beam order, or the small Gaussian waist has the large kurtosis parameter. When the propagation distance in turbulent atmosphere is large enough, the kurtosis parameters of the higher-order cosh-Gaussian beams all tend to 3, which is the value of the kurtosis parameter of the Gaussian distribution. The influence of the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere is shown in Fig. 5 . We find that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for a larger structure constant.

Fig. 1 Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 1km in turbulent atmosphere. (a) n = 2 and w ₀ = 0.02m. (b) w ₀ = 0.02m and Ω = 60m⁻¹. (c) n = 2 and Ω = 60m⁻¹.

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Fig. 2 Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km in turbulent atmosphere. (a) n = 2 and w ₀ = 0.02m. (b) w ₀ = 0.02m and Ω = 60m⁻¹. (c) n = 2 and Ω = 60m⁻¹.

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Fig. 3 The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w ₀ = 0.02m. (b) w ₀ = 0.02m and Ω = 60m⁻¹. (c) n = 2 and Ω = 60m⁻¹.

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Fig. 4 The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w ₀ = 0.02m. (b) w ₀ = 0.02m and Ω = 60m⁻¹. (c) n = 2 and Ω = 60m⁻¹.

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Fig. 5 (a) Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km. (b) The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. (c) The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. n = 2, Ω = 80m⁻¹, and w ₀ = 0.02m.

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4. Conclusions

The propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral, the analytical expressions of the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam are derived in turbulent atmosphere. The average intensity distribution and spreading properties of a higher-order cosh-Gaussian are numerically examined. The higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters and a larger structure constant. This research is useful to the practical applications in free-space optical communications and remote sensing involving the higher-order cosh-Gaussian beam.

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 10974179) and Zhejiang Provincial Natural Science Foundation of China (No. Y1090073).

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Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

Abstract

1. Introduction

2. Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

3. The numerical results and analyses

4. Conclusions

Acknowledgements

References and links

Cited By

Figures (5)

Equations (15)

Optics Express