Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere

Guoquan Zhou

doi:10.1364/OE.19.024699

1. Introduction

With the identical spatial extension, the angular spreading of a Lorentzian distribution is higher than that of a Gaussian distribution [1]. Therefore, Lorentz beams provide more appropriate theoretical models than Gaussian beams to describe the radiation emitted by a single mode diode laser [1–3]. The paraxial propagation of a Lorentz beam has been investigated in free space [4]. The vectorial structure of a Lorentz beam has been examined in the far-field regime [5]. Based on the vectorial Rayleigh-Sommerfeld integral formulae, the analytical propagation expression of a Lorentz beam beyond the paraxial approximation has been derived [6]. Based on the second-order moments, the beam propagation factors and the kurtosis parameters of a Lorentz beam have been calculated [7]. Based on the method of the truncated second-order moments, the analytical expression of the generalized beam propagation factor for a truncated partially coherent Lorentz beam has been presented [8]. The propagation properties of a Lorentz beam in uniaxial crystals orthogonal to the optical axis have been also evaluated by using two different methods [9, 10]. The coherent combination of some certain Lorentz beams results in a super-Lorentzian beam [11]. The link between relativistic Hermite polynomials and Lorentz beams has been demonstrated [12]. Due to the crucial applications in optical communications and remote sensing, the average intensity and the spreading properties of various kinds of laser beams including the Lorentz beam in a turbulent atmosphere have been extensively investigated [13–18]. The propagation of the partially coherent Lorentz beams through a paraxial ABCD optical system has been also examined in a turbulent atmosphere [19].

However, the output of low beam power of a single diode laser hinders its applications. Accordingly, coherent diode laser arrays have been proposed to provide the efficient high-power output [20–24]. Naturally, the Lorentz beam array represents an appropriate theoretical model to describe a coherent diode laser array. Formulae for the intensity distributions of coherent and incoherent combined one-dimensional Lorentz beam array have been derived by using the representation of the cross-spectral density of the far-field [25]. When extended to the two-dimensional case, the Lorentz beam array in Ref [25]. is a rectangular arrangement. The propagation properties of a P × Q rectangular Lorentz beam array have been investigated in free space [26]. The reported researches on the Lorentz beam array are all confined to a rectangular arrangement. As the important Gaussian beam array is a radial arrangement [27, 28] and the Gaussian beam is replaced by the Lorentz beam in the theoretical model of diode lasers, one maybe more interested in a radial Lorentz beam array. The researches on the propagation of beam arrays in a turbulent atmosphere are useful to optical communications and remote sensing [29–32], therefore, the propagation of a radial Lorentz beam array in turbulent atmosphere is examined in the remainder of this paper. The average intensity and the spreading properties of a radial Lorentz beam array in turbulent atmosphere are mainly concerned.

2. Propagation of a radial phase-locked Lorentz beam array in turbulent atmosphere

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The scheme of a radial Lorentz beam array in the source plane is shown in Fig. 1 . The radial Lorentz beam array considered here is phase-locked (PL).

Fig. 1 Schematic diagram of a radial PL Lorentz beam array in the source plane.

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The optical field of the n-th Lorentz beam element in the source plane z = 0 takes the form as [4]:

E_{n} (r_{0}, 0) = \frac{1}{w_{0 x} w_{0 y} [1 + {(x_{0} - a_{n x})}^{2} / w_{0 x}^{2}] [1 + {(y_{0} - a_{n y})}^{2} / w_{0 y}^{2}]} \exp (i φ_{n}),

where r₀ = x₀e_x + y₀e_y. e_x and e_y are the two transverse unit vectors in the Cartesian coordinate system, respectively. w₀_x and w₀_y are the parameters related to the beam widths in the x- and y-directions, respectively. (a_nx, a_ny) is the center of the n-th beamlet located at the source plane and is given by

a_{n x} = R \cos φ_{n}, a_{n y} = R \sin φ_{n}, φ_{n} = n φ_{0} = 2 n π / N, \begin{matrix} n = 1, 2, 3, \dots, N, \end{matrix}

where R is the radius and φ_n is the initial phase of the n-th beamlet. The optical field of the PL Lorentz beam array composed of N beamlets in the source plane is expressed as

E (r_{0}, 0) = \sum_{n = 1}^{N} E_{n} (r_{0}, 0) .

By directly using Eqs. (1) and (3), one cannot obtain the analytical expression of a beam profile for a radial PL Lorentz beam array propagating in turbulent atmosphere. Therefore, we have to expand the Lorentz distribution, which is the best choice. As the Lorentz distribution is expanded into the linear superposition of Hermite-Gaussian functions [33], the optical field of the radial PL Lorentz beam array in the source plane can be rewritten as follows:

E (r_{0}, 0) = \frac{π}{2 w_{0 x} w_{0 y}} \sum_{n = 1}^{N} \sum_{m_{1} = 0}^{M} \sum_{m_{2} = 0}^{M} σ_{2 m_{1}}^{} σ_{2 m_{2}}^{} H_{2 m_{1}} (\frac{{x^{'}}_{0}}{w_{0 x}}) H_{2 m_{2}} (\frac{{y^{'}}_{0}}{w_{0 y}}) \exp (- \frac{{x^{'}}_{0}^{2}}{2 w_{0 x}^{2}} - \frac{{y^{'}}_{0}^{2}}{2 w_{0 y}^{2}} + i φ_{n}),

where

{x^{'}}_{0} = x_{0} - a_{n x}, {y^{'}}_{0} = y_{0} - a_{n y} .

M is the term number of the expansion. $H_{2 m_{1}} (\cdot)$ and $H_{2 m_{2}} (\cdot)$ are the 2m₁-th and 2m₂-th order Hermite polynomials, respectively. $σ_{2 m_{1}}^{}$ and $σ_{2 m_{2}}^{}$ are the weight coefficients and are listed in Table 1 [33]. With increasing the integer m, however, the value of the weight coefficient σ₂_m dramatically decays. In the practical calculations, therefore, M takes a small integer. For an example, the original expression of the Lorentz beam in the source plane well coincides with the expansion in the Hermite-Gaussian functions with M = 5.

Table 1. Value of the weight coefficient σ₂_m.

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The propagation of the radial PL Lorentz beam array in a turbulent atmosphere can be investigated by using the following extended Huygens-Fresnel integral [34–36]:

E (r, z) = - \frac{i k}{2 π z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{0}, 0) \exp [- \frac{i k}{2 z} {(r_{0} - r)}^{2} + ψ (r_{0}, r)] d x_{0} d y_{0},

where r = xe_x + ye_y. (r, z) is the observation plane. ψ(r₀, r) is the solution to the Rytov method that represents the random part of the complex phase. k = 2π/λ is the wave number. λ is the optical wavelength. The average intensity of the radial PL Lorentz beam array in the observation plane is given by

\begin{array}{l} < I (r, z) > = \frac{k^{2}}{4 π^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{01}, 0) E^{*} (r_{02}, 0) \exp [- \frac{i k}{2 z} {(r_{01} - r)}^{2} + \frac{i k}{2 z} {(r_{02} - r)}^{2}] \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \times < \exp [ψ (r_{01}, r) + ψ^{*} (r_{02}, r)] > d r_{01} d r_{02}, \end{array}

where the angle brackets indicate the ensemble average over the medium statistics, and the asterisk denotes the complex conjugation. The last ensemble average term in the above equation can be expressed as follows [34]:

< \exp [ψ (r_{01}, r) + ψ^{*} (r_{02}, r)] > = \exp [- \frac{{(r_{01} - r_{02})}^{2}}{ρ_{0}^{2}}],

where

ρ_{0}^{} = {(0.545 C_{n}^{2} k^{2} z)}^{- 3 / 5}

is the spherical wave lateral coherence length. C_n² is the structure constant of the atmospheric turbulence. In Eq. (8), the quadratic structure functions are employed. Substituting Eqs. (4) and (8) into Eq. (7) and using the following mathematical formulae [37]:

\int_{- \infty}^{\infty} H_{2 m_{1}} (x) \exp [- {(x - y)}^{2} / Ω] d x = \sqrt{π Ω} {(1 - Ω)}^{m_{1}} H_{2 m_{1}} [y {(1 - Ω)}^{- 1 / 2}],

H_{2 {m^{'}}_{1}} (x) = \sum_{l = 0}^{{m^{'}}_{1}} \frac{{(- 1)}^{l} (2 {m^{'}}_{1})!}{l! (2 {m^{'}}_{1} - 2 l)!} {(2 x)}^{2 {m^{'}}_{1} - 2 l},

{(x + y)}^{2 m_{1}} = \sum_{u = 0}^{2 m_{1}} (\begin{matrix} 2 m_{1} \\ u \end{matrix}) x^{2 m_{1} - u} y^{u},

\int_{- \infty}^{\infty} x^{2 t} \exp (- b x^{2} + 2 c x) d x = (2 t)! \sqrt{\frac{π}{b}} {(\frac{c}{b})}^{2 t} \exp (\frac{c^{2}}{b}) \sum_{s = 0}^{t} \frac{1}{s! (2 t - 2 s)!} {(\frac{b}{4 c^{2}})}^{s},

we can obtain the analytical average intensity of the radial PL Lorentz beam array in the observation plane:

< I (r, z) > = \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} I_{n_{1}, n_{2}} (x, z) I_{n_{1}, n_{2}} (y, z) \exp [i (φ_{n_{1}} - φ_{n_{2}})],

with

I_{n_{1}, n_{2}} (x, z)

and

I_{n_{1}, n_{2}} (y, z)

being given by

\begin{array}{l} I_{n_{1}, n_{2}} (j, z) = \frac{k π}{4 z} \sqrt{\frac{1}{α_{1 j} α_{2 j}}} \sum_{m_{1} = 0}^{M} \sum_{{m^{'}}_{1} = 0}^{M} \exp (\frac{β_{1 j}^{2}}{4 α_{1 j}} + \frac{β_{2 j}^{2}}{4 α_{2 j}} - \frac{a_{n_{2 j}}^{2}}{2 w_{0 j}^{2}} - \frac{a_{n_{1 j}}^{2}}{ρ_{0}^{2}} + \frac{i k a_{n_{1 j}}^{2}}{2 z} - \frac{i k a_{n_{1 j}} j}{z}) σ_{2 m_{1}}^{} σ_{2 {m^{'}}_{1}}^{} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \times {(1 - \frac{1}{α_{1 j}})}^{m_{1}} \sum_{l_{1} = 0}^{m_{1}} \frac{{(- 1)}^{l_{1}} (2 m_{1})!}{l_{1}! (2 m_{1} - 2 l_{1})!} \sum_{l_{2} = 0}^{2 m_{1} - 2 l_{1}} (\begin{matrix} 2 m_{1} - 2 l_{1} \\ l_{2} \end{matrix}) γ_{j}^{2 m_{1} - 2 l_{1} - l_{2}} δ_{j}^{l_{2}} \sum_{l_{3} = 0}^{{m^{'}}_{1}} \frac{{(- 1)}^{l_{3}} (2 {m^{'}}_{1})!}{l_{3}! (2 {m^{'}}_{1} - 2 l_{3})!} \\ \begin{matrix}  \end{matrix} \times \sum_{l_{4} = 0}^{2 {m^{'}}_{1} - 2 l_{3}} (\begin{matrix} 2 {m^{'}}_{1} - 2 l_{3} \\ l_{4} \end{matrix}) {(\frac{- 2 a_{n_{2 j}}}{w_{0 j}})}^{2 {m^{'}}_{1} - 2 l_{3} - l_{4}} (l_{2} + l_{4})! \sum_{s = 0}^{[(l_{2} + l_{4}) / 2]} \frac{α_{2 j}^{s - l_{2} - l_{4}} β_{2 j}^{l_{2} + l_{4} - 2 s}}{s! (l_{2} + l_{4} - 2 s)!}, \end{array}

where j = x or y in the all locations including the subscript. [(l₂ + l₄)/2] gives the greatest integer less than or equal to (l₂ + l₄)/2, and the auxiliary parameters are defined as follows:

α_{1 j} = \frac{1}{2} + \frac{w_{0 j}^{2}}{ρ_{0}^{2}} - \frac{i k w_{0 j}^{2}}{2 z}, α_{2 j} = \frac{1}{2} + \frac{w_{0 j}^{2}}{ρ_{0}^{2}} + \frac{i k w_{0 j}^{2}}{2 z} - \frac{w_{0 j}^{4}}{α_{1 j} ρ_{0}^{4}},

β_{1 j} = \frac{i k w_{0 j} a_{n_{1 j}}}{z} - \frac{2 w_{0 j} a_{n_{1 j}}}{ρ_{0}^{2}} - \frac{i k w_{0 j} j}{z}, β_{2 j} = \frac{a_{n_{2 j}}}{w_{0 j}} + \frac{β {}_{1 j}w_{0 j}^{2}}{α_{1 j} ρ_{0}^{2}} + \frac{2 w_{0 j} a_{n_{1 j}}}{ρ_{0}^{2}} + \frac{i k w_{0 j} j}{z},

γ_{j} = \frac{β {}_{1 j}}{{(α_{1 j}^{2} - α_{1 j})}^{1 / 2}}, δ_{j} = \frac{w_{0 j}^{2}}{{(α_{1 j}^{2} - α_{1 j})}^{1 / 2} ρ_{0}^{2}} .

The effective beam size of the radial PL Lorentz beam array in the x- and y-directions of the observation plane is defined as [38]:

W_{j z} = \sqrt{2 \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{2} < I (r, z) > d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} < I (r, z) > d x d y}} .

Substituting Eq. (13) into Eq. (18), the analytical effective beam sizes of the radial PL Lorentz beam array yield

W_{x z} = \sqrt{\frac{2 \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} A_{n_{1}, n_{2}} (x, z; 2) A_{n_{1}, n_{2}} (y, z; 0) \exp [i (φ_{n_{1}} - φ_{n_{2}})]}{\sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} A_{n_{1}, n_{2}} (x, z; 0) A_{n_{1}, n_{2}} (y, z; 0) \exp [i (φ_{n_{1}} - φ_{n_{2}})]}},

W_{y z} = \sqrt{\frac{2 \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} A_{n_{1}, n_{2}} (x, z; 0) A_{n_{1}, n_{2}} (y, z; 2) \exp [i (φ_{n_{1}} - φ_{n_{2}})]}{\sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} A_{n_{1}, n_{2}} (x, z; 0) A_{n_{1}, n_{2}} (y, z; 0) \exp [i (φ_{n_{1}} - φ_{n_{2}})]}},

with

A_{n_{1}, n_{2}} (j, z; 0)

and

A_{n_{1}, n_{2}} (j, z; 2)

being given by

\begin{array}{l} A_{n_{1}, n_{2}} (j, z; v) = \sum_{m_{1} = 0}^{M} \sum_{{m^{'}}_{1} = 0}^{M} \exp (\frac{ξ_{j}^{2}}{4 α_{1 j}} + \frac{η_{j}^{2}}{4 α_{2 j}} - \frac{a_{n_{2 j}}^{2}}{2 w_{0 j}^{2}} - \frac{a_{n_{1 j}}^{2}}{ρ_{0}^{2}} + \frac{i k a_{n_{1 j}}^{2}}{2 z}) σ_{2 m_{1}}^{} σ_{2 {m^{'}}_{1}}^{} {(1 - \frac{1}{α_{1 j}})}^{m_{1}} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \times \sum_{l_{1} = 0}^{m_{1}} \frac{{(- 1)}^{l_{1}} (2 m_{1})!}{l_{1}! (2 m_{1} - 2 l_{1})!} \sum_{l_{2} = 0}^{2 m_{1} - 2 l_{1}} (\begin{matrix} 2 m_{1} - 2 l_{1} \\ l_{2} \end{matrix}) \sum_{t_{1} = 0}^{2 m_{1} - 2 l_{1} - l_{2}} (\begin{matrix} 2 m_{1} - 2 l_{1} - l_{2} \\ t_{1} \end{matrix}) p_{j}^{2 m_{1} - 2 l_{1} - l_{2} - t_{1}} \\ \begin{matrix}  \end{matrix} \times q_{j}^{t_{1}} δ_{j}^{l_{2}} \sum_{l_{3} = 0}^{{m^{'}}_{1}} \frac{{(- 1)}^{l_{3}} (2 {m^{'}}_{1})!}{l_{3}! (2 {m^{'}}_{1} - 2 l_{3})!} \sum_{l_{4} = 0}^{2 {m^{'}}_{1} - 2 l_{3}} (\begin{matrix} 2 {m^{'}}_{1} - 2 l_{3} \\ l_{4} \end{matrix}) {(\frac{- 2 a_{n_{2 j}}}{w_{0 j}})}^{2 {m^{'}}_{1} - 2 l_{3} - l_{4}} (l_{2} + l_{4})! \\ \begin{matrix}  \end{matrix} \times \sum_{s = 0}^{[(l_{2} + l_{4}) / 2]} \frac{α_{2 j}^{s - l_{2} - l_{4}}}{s! (l_{2} + l_{4} - 2 s)!} \sum_{t_{2} = 0}^{l_{2} + l_{4} - 2 s} (\begin{matrix} l_{2} + l_{4} - 2 s \\ t_{2} \end{matrix}) η_{j}^{l_{2} + l_{4} - 2 s - t_{2}} μ_{j}^{t_{2}} 2^{- t_{1} - t_{2} - 1 - v} \\ \begin{matrix}  \end{matrix} \times \exp (\frac{τ_{2 j}^{2}}{4 τ_{1 j}}) τ_{1 j}^{- 1 - (t_{1} + t_{2} + v) / 2} 2^{t_{1} + t_{2} + v} {\sqrt{τ_{1 j}} [1 + {(- 1)}^{t_{1} + t_{2}}] Γ (\frac{t_{1} + t_{2} + 1 + v}{2}) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \times {}_{1}F_{1} (- \frac{t_{1} + t_{2} + v}{2}; \frac{1}{2}; - \frac{τ_{2 j}^{2}}{4 τ_{1 j}}) - τ_{2 j} [(^{- 1) t_{1} + t_{2}} - 1] Γ (\frac{t_{1} + t_{2} + 2 + v}{2}) \\ \begin{matrix}  \end{matrix} \times {}_{1}F_{1} (- \frac{t_{1} + t_{2} + v - 1}{2}; \frac{3}{2}; - \frac{τ_{2 j}^{2}}{4 τ_{1 j}})}, \begin{matrix}  \end{matrix} v = 0 \begin{matrix} o r \end{matrix} 2, \end{array}

where ₁F₁(⋅)is a Kummer function and Γ(⋅) is a Gamma function. The auxiliary parameters are defined as

ξ_{j} = \frac{i k w_{0 j} a_{n_{1 j}}}{z} - \frac{2 w_{0 j} a_{n_{1 j}}}{ρ_{0}^{2}}, η_{j} = \frac{a_{n_{2 j}}}{w_{0 j}} + \frac{ξ_{j} w_{0 j}^{2}}{α_{1 j} ρ_{0}^{2}} + \frac{2 w_{0 j} a_{n_{1 j}}}{ρ_{0}^{2}}, p_{j} = \frac{ξ_{j}}{{(α_{1 j}^{2} - α_{1 j})}^{1 / 2}},

q_{j} = \frac{- i k w_{0 j}}{z {(α_{1 j}^{2} - α_{1 j})}^{1 / 2}}, μ_{j} = \frac{i k w_{0 j}}{z} (1 - \frac{w_{0 j}^{2}}{α_{1 j} ρ_{0}^{2}}),

τ_{1 j} = \frac{k^{2} w_{0 j}^{2}}{4 z^{2} α_{1 j}} + \frac{k^{2} w_{0 j}^{2}}{4 z^{2} α_{2 j}} {(1 - \frac{w_{0 j}^{2}}{α_{1 j} ρ_{0}^{2}})}^{2}, τ_{2 j} = \frac{i k w_{0 j} η_{j}}{2 z α_{2 j}} (1 - \frac{w_{0 j}^{2}}{α_{1 j} ρ_{0}^{2}}) - \frac{i k w_{0 j} ξ_{j}}{2 z α_{1 j}} - \frac{i k a_{n_{1 j}}}{z} .

Though the analytical expressions of the average intensity and the effective beam size of the radial PL Lorentz beam array derived here seem very complicated, the term number in the expressions of the average intensity and the effective beam size will not be too large. Moreover, the Kummer function and the Gamma function can be directly used in software Mathmatica. Compared with the direct integrals that the Hermite-Gaussian expansion is not adopted, therefore, the calculations of the average intensity and the effective beam size are convenient and fast by using the analytical formulae derived here.

3. Numerical calculations and analyses

Now, the average intensity and the spreading of a radial PL Lorentz beam array in turbulent atmosphere are calculated by using the formulae derived above. In the following numerical calculations, the beam propagation we consider is with horizontal path. Figures 2 -6 represent the contour graph of normalized intensity distribution of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. The optical wavelength is set to be λ = 0.8μm. w₀_x = w₀_y results in the symmetry of the beam spot, which is the familiar case. Therefore, we consider w₀_x being equal to w₀_y. C_n² = 10⁻¹⁴m^-2/3, w₀_x = w₀_y = 2mm, R = 3mm, and N = 6 in Fig. 2. In the near-field such as $z = z_{r} = k w_{0 x}^{2}$ , there is a central dark region in the contour graph of normalized intensity distribution. With the increase of the propagation distance z, the central dark region disappears, and the on-axis intensity increases from zero to the maximum value. Moreover, the normalized intensity distribution in the central region tends to be uniform. In Fig. 3 , C_n² = 10⁻¹⁵m^-2/3 and the rest of parameters are same as those in Fig. 2. When the structure constant of the atmospheric turbulence decreases, the propagation distance where the central dark region disappears apparently increases, which can be interpreted as follow. The coherence of the radial PL Lorentz beam array will be damaged by the atmospheric turbulence. If the structure constant of the atmospheric turbulence decreases, the extent of damage of the coherence of the radial PL Lorentz beam array also decreases. In the source plane, the beam spot of the radial PL Lorentz beam array takes on a central dark distribution. The better the coherence is, the longer the axial propagation distance within which the beam spot doesn’t distort is. With decreasing the structure constant of the atmospheric turbulence, therefore, the propagation distance where the central dark region disappears increases. w₀_x = w₀_y = 1mm in Fig. 4 , and the other parameters are same as those in Fig. 2. In the near-field such as z = z_r, we can distinctly distinguish six petals from the whole pattern. However, the six petals are conjoint. When z = 5z_r, the outer distribution in the contour graph of normalized intensity takes on a gear shape. When z = 100z_r, the on-axis intensity is still zero and the two isolated side lobes are located at the y-axis. When z = 1500z_r, the on-axis intensity is no longer zero and two side lobes are connected to the central dominating spot. R = 5mm in Fig. 5 , and the rest of parameters are same as those in Fig. 2. Comparing Fig. 4 with Fig. 5, the effect of the decrease of w₀_x on the beam pattern is equivalent to that of the increase of R on the beam pattern, which leads to the similar beam pattern in the different observation planes. Moreover, When z = 100z_r, the on-axis intensity is nonzero. When z = 200z_r, the on-axis intensity is nearly the maximum. N = 8 in Fig. 6, and the other parameters are same as those in Fig. 2. Comparing Fig. 6 with Fig. 2, we nearly cannot distinguish one from the other. When the number of the beamlet is large enough, the effect of the further increase of the beamlet is saturated. To sum up, the fill-factor in the near-field pattern is low in the case of w₀_x being small or R being large. If one wants to get a dark hollow uniform beam in the near-field, the parameters should be appropriate, e.g., w₀_x is close to R and N is large enough. As optical communications and remote sensing is involved in the far distance and a radial PL Lorentz beam array propagating in turbulent atmosphere finally turns out to be a solid beam, a radial PL Lorentz beam array is applicable to optical communications and remote sensing.

Fig. 2 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 350zr.

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Fig. 6 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 8, w₀_x = w₀_y = 2mm, R = 3mm, and C_n² = 10⁻¹⁴m^-2/3. (a) z = z_r. (b) z = 5z_r. (c) z = 100z_r. (d) z = 350z_r.

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Fig. 3 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w₀_x = w₀_y = 2mm, R = 3mm, and C_n² = 10⁻¹⁵m^-2/3. (a) z = z_r. (b) z = 5z_r. (c) z = 100z_r. (d) z = 1500z_r.

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Fig. 4 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w₀_x = w₀_y = 1mm, R = 3mm, and C_n² = 10⁻¹⁴m^-2/3. (a) z = z_r. (b) z = 5z_r. (c) z = 100z_r. (d) z = 1500z_r.

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Fig. 5 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w₀_x = w₀_y = 2mm, R = 5mm, and C_n² = 10⁻¹⁴m^-2/3. (a) z = z_r. (b) z = 5z_r. (c) z = 100z_r. (d) z = 200z_r.

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To further reveal the spreading properties of a radial PL Lorentz beam array in turbulent atmosphere, the effective beam sizes of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere are depicted in Figs. 7 and 8 . As the effective beam sizes in the x- and y-directions have the similar variational rule, only W_xz is shown in Figs. 7 and 8. The radial Lorentz beam array with the larger w₀_x and w₀_y spreads more rapidly, which seems to contradict the spreading of a single laser source. The combination of the different initial phase and the dark hollow distribution in the source plane maybe result in the difference. When reducing to R = 0 and N = 1, the result obtained here is consistent with the behavior of a single laser source. The radial Lorentz beam array spreads more rapidly in turbulent atmosphere for a larger structure constant of the atmospheric turbulence. The structure constant of the refractive index fluctuations of the turbulence increasing denotes that the turbulence strengthens, which results in the large effective beam size. The radial PL Lorentz beam array with the larger radial radius spreads more rapidly. As to the effective beam size, the effect of the increase of w₀_x is similar to that of the increase of R. When the radius of the beam array is not far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet under the condition of the initial number of beamlet being large. When the radius of the beam array is far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is first sensitive to the increase of the number of beamlet under the condition of the initial number of beamlet being not large enough. If the initial number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. The variational law revealed in Figs. 7 and 8 is consistent with that denoted by Figs. 2-6.

Fig. 7 The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. N = 6 and R = 3mm (a) C_n² = 10⁻¹⁴m^-2/3. w₀_x and w₀_y take different values. (b) w₀_x = w₀_y = 2mm, and C_n² takes different value.

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Fig. 8 The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. w₀_x = w₀_y = 2mm and C_n² = 10⁻¹⁴m^-2/3. (a) N = 6, and R takes different values. (b) R = 3mm, and N takes different value.

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4. Conclusions

Based on the extended Huygens-Fresnel integral and some mathematical techniques, the analytical average intensity and the analytical effective beam size of a radial PL Lorentz beam array are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a radial PL Lorentz beam array are numerically examined in turbulent atmosphere. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a radial PL Lorentz beam array in turbulent atmosphere are analyzed. Upon the propagation in turbulent atmosphere, the radial PL Lorentz beam array first takes on a dark hollow distribution and finally evolves to be a solid brilliant spot. Anyone of w₀_x, w₀_y, R, N, and C_n² will affect the propagation distance within which the radial PL Lorentz beam array propagating in turbulent atmosphere keeps a dark hollow distribution. The radial PL Lorentz beam array in turbulent atmosphere with the larger w₀_x and w₀_y or with the larger radial radius or for a larger structure constant of the atmospheric turbulence spreads more rapidly. When the number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. This research is useful to the practical application of a coherent diode laser array in turbulent atmosphere, which is used for optical communications, remote sensing, and optical imaging. When the parameters are not appropriate, the near-field beam pattern of the radial PL Loretnz beam array is not ideal due to the low fill-factor. If one wants to achieve a higher fill-factor, the P × Q rectangular PL Lorentz beam array maybe alternative, which can be handled in the similar analytical approach as here.

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grant No. 10974179 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y1090073. The author is indebted to the reviewers for valuable comments.

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m	σ₂_m	m	σ₂_m	m	σ₂_m
0	0.7399	7	0.3773 × 10⁻⁹	14	0.8608 × 10⁻²¹
1	0.9298 × 10⁻²	8	0.1415 × 10⁻¹⁰	15	0.1255 × 10⁻²²
2	0.5382 × 10⁻²	9	0.2782 × 10⁻¹²	16	0.1948 × 10⁻²⁴
3	0.1112 × 10⁻³	10	0.7633 × 10⁻¹⁴	17	0.2578 × 10⁻²⁶
4	0.1356 × 10⁻⁴	11	0.1361 × 10⁻¹⁵	18	0.3522 × 10⁻²⁸
5	0.3008 × 10⁻⁶	12	0.2958 × 10⁻¹⁷	19	0.4264 × 10⁻³⁰
6	0.1769 × 10⁻⁷	13	0.4764 × 10⁻¹⁹	20	0.5270 × 10⁻³²

Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere

Abstract

1. Introduction

2. Propagation of a radial phase-locked Lorentz beam array in turbulent atmosphere

3. Numerical calculations and analyses

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (24)

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