Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence

Yongping Huang; Bin Zhang; Zenghui Gao; Guangpu Zhao; Zhichun Duan

doi:10.1364/OE.22.017723

1. Introduction

Recently, much attention has been paid to the laser beams with phase singularities propagating through ideal atmospheric turbulence and non-Kolmogorov turbulence [1–7]. It was shown that the spreading of partially coherent vortex beam is less affected by atmospheric turbulence than partially coherent non-vortex, and the topological charge can be used as the information carrier in optical communication [1, 2]. The intensity distribution, coherent vortex evolution and orbital angular momentum entangled states of vortex beam in non-Kolmogorov turbulence were investigated in [5–7]. On the other hand, the oceanic turbulence is another important random medium, but the turbulent properties of the underwater turbulence are determined by temperature and salinity fluctuation of the oceanic water [8–10]. It was shown that the degree of polarization, coherence, spectral changes, scintillation index and intensity distributions of electromagnetic beams through oceanic turbulence are affected by oceanic turbulence [11–19]. This paper is devoted to study the evolution behavior of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence. The analytical expressions for the oceanic turbulence quantity $T (η, ε, χ_{T}, ω)$ and the cross-spectral density as well as average intensity of GSM vortex beams through oceanic turbulence are derived in Sec.2 and Sec.3. Then, Sec.3 and Sec.4 illustrate the evolution behavior of GSM vortex beams through oceanic turbulence numerically. Finally, the main conclusion is drawn in Sec.5.

2. The cross-spectral density of Gaussian Schell-model vortex beams through oceanic turbulence

In Cartesian coordinates the field distribution of a vortex beam at the z = 0 source plane is expressed as

U (ρ, z = 0) = u (ρ) {[ρ_{x} + i s g n (m) ρ_{y}]}^{| m |},

where u(ρ) represents the profile of the background beam envelope, ρ≡(ρ_x, ρ_y) is two-dimensional coordinate vectors at the plane z = 0, sgn(.) is the sign function, m is the topological charge.

The cross-spectral density of GSM vortex beams at the plane z = 0 is written as

\begin{array}{l} W_{}^{(0)} (ρ_{1}, ρ_{2}, 0) = [^{(ρ_{1 x} ρ_{2 x} + ρ_{1 y} ρ_{2 y}) + i sgn (m) (ρ_{1 x} ρ_{2 y} - ρ_{2 x} ρ_{1 y})] m} \\ \times \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w_{0}^{2}}) e x p (- \frac{{| ρ_{1} - ρ_{2} |}^{2}}{2 σ_{0}^{2}}) . \end{array}

In Eq. (2) ρ₁ = (ρ₁_x, ρ₁_y) and ρ₂ = (ρ₂_x, ρ₂_y) are positions of two points at the plane z = 0, respectively. In this paper the m is assumed to be $\pm$ 1, in the case for m = 0, Eq. (2) degenerates into the cross spectral density of GSM non-vortex beam, w₀ is the waist width for the Gaussian part, σ₀ is spatial correlation length.

According to the extended Huygens–Fresnel principle, the cross-spectral density function at the z plane (z>0) of GSM vortex beams through oceanic turbulence is written as

\begin{array}{l} W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = (^{\frac{k}{2 π z}) 2} \iint d^{2} ρ_{1} \iint d^{2} ρ_{2} W_{}^{(0)} (ρ_{1}, ρ_{2}, 0) \\ \times \exp {- \frac{i k}{2 z} [{({ρ^{'}}_{1} - ρ_{1})}^{2} - {({ρ^{'}}_{2} - ρ_{2})}^{2}]} \\ \times 〈 e x p [ψ^{*} ({ρ^{'}}_{1}, ρ_{1}) + ψ ({ρ^{'}}_{2}, ρ_{2})] 〉, \end{array}

where k denotes the wave number related to the wavelength by k = 2π/λ.

{ρ^{'}}_{1}

and

{ρ^{'}}_{2}

are the positions of two points at the z plane, respectively. The asterisk * specifies the complex conjugate,

〈 \cdot 〉

represents the ensemble average. ψ(ρ'_i, ρ_i) (i = 1, 2) represents the random part of the complex phase of a spherical wave due to the existence of the turbulence, whose ensemble average of the turbulent ocean water can be expressed as [20]

\begin{array}{l} 〈 \exp [ψ^{*} ({ρ^{'}}_{1}, ρ_{1}) + ψ ({ρ^{'}}_{2}, ρ_{2})] 〉 \\ = \exp {- 4 π^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} d κ d ξ κ Φ_{n} (κ) [1 - J_{0} (κ | (1 - ξ) ({ρ^{'}}_{1} - {ρ^{'}}_{2}) + ξ (ρ_{1} - ρ_{2}) |)]} \\ = \exp {- \frac{π^{2} k^{2} z}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ [{({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2} + ({ρ^{'}}_{1} - {ρ^{'}}_{2}) (ρ_{1} - ρ_{2}) + {(ρ_{1} - ρ_{2})}^{2}]} \\ = \exp {- k^{2} z T (η, ε, χ_{T}, ω) [{({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2} + ({ρ^{'}}_{1} - {ρ^{'}}_{2}) (ρ_{1} - ρ_{2}) + {(ρ_{1} - ρ_{2})}^{2}]}, \end{array}

where J₀ (.) is the Bessel function of the first kind and zero order, and

T (η, ε, χ_{T}, ω) = \frac{π^{2}}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ,

denoting the strength of the oceanic turbulence. Φ_n(κ) in Eq. (5) is the spatial power spectrum of the refractive-index fluctuations of the turbulent ocean water.

For the sake of simplicity, we consider the influence of oceanic turbulence due to temperature and salinity fluctuations on the beam, using the spatial power spectrum of oceanic turbulence model in [9, 11]

\begin{array}{l} Φ_{n} (κ, η, ε, χ_{T}, ω) = 0.388 \times 10^{- 8} ε^{- 1 / 3} κ^{- 11 / 3} \frac{χ_{T}}{ω^{2}} [1 + 2.35 (^{κ η) 2 / 3}] \\ \times (ω^{2} e^{- A_{T} δ} + e^{- A_{S} δ} - 2 ω e^{- A_{T S} δ}), \end{array}

where

δ = 8.284 {(κ η)}^{4 / 3} + 12.978 {(κ η)}^{2},

η is the Kolmogorov micro scale, ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in range from 10⁻⁴m²/s³ to10⁻¹⁰m²/s³, χ_T being the rate of dissipation of mean-square temperature, taking values in the range from 10⁻⁴K²/s to 10⁻¹⁰K²/s [10]. ω is the relative strength of temperature and salinity fluctuation, which in the ocean water the value can range from 0 to −5, the minus sign of the parameter ω denotes that there is a reduction in temperature and an increase in salinity with depth. 0 corresponding to the case when temperature-driven turbulence dominates, −5 corresponding to the situation when salinity-driven turbulence prevails [10].

Substituting Eqs. (6) and (7) and $A_{T} = 1.863 \times 10^{- 2}$ , $A_{S} = 1.9 \times 10^{- 4}$ as well as $A_{T S} = 9.41 \times 10^{- 3}$ [14] into Eq. (5), Eq. (5) can be simplified to

T (η, ε, χ_{T}, ω) = 1.2765 \times 10^{- 8} ω^{- 2} ε^{- 1 / 3} η^{- 1 / 3} χ_{T} (47.5708 - 17.6701 ω + 6.78335 ω^{2}) .

Obviously, the

T (η, ε, χ_{T}, ω)

increases with the increasing χ_T and decreasingω, ε and η.

Introducing two variables of integration s = (ρ₁ + ρ₂)/2, t = ρ₁-ρ₂ and with Eqs. (4)–(8) substituting into Eq. (3), we can obtain

\begin{array}{l} W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = (^{\frac{k}{2 π z}) 2} e x p [- \frac{i k}{2 z} ({ρ^{'}}_{1}^{2} - {ρ^{'}}_{2}^{2})] e x p [- T (η, ε, χ_{T}, ω) k^{2} z (^{{ρ^{'}}_{1} - {ρ^{'}}_{2}) 2}] \\ \times \iint d^{2} s \iint d^{2} t [(s^{2} - \frac{t^{2}}{4}) \mp i (s_{x} t_{y} - s_{y} t_{x})] \\ \times \exp (- a t^{2} - \frac{i k}{z} s t) \exp (- \frac{2}{w_{0}^{2}} s^{2}) \exp [\frac{i k}{z} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) s] \\ \times \exp {t [\frac{i k}{2 z} ({ρ^{'}}_{1} + {ρ^{'}}_{2}) - T (η, ε, χ_{T}, ω) k^{2} z ({ρ^{'}}_{1} - {ρ^{'}}_{2})]}, \end{array}

where

a = \frac{1}{2} (\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}) + k^{2} z T (η, ε, χ_{T}, ω),

s_x = (ρ₁_x + ρ₂_x)/2, t_x = ρ₁_x-ρ₂_x, s_y = (ρ₁_y + ρ₂_y)/2, t_y = ρ₁_y-ρ_2y, the symbol ‘

\mp

’ corresponds to m = ± 1 at the source plane z = 0.

Using the integral formulae [21]

\int \exp (- p x^{2} + 2 q x) d x = \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}},

\int x \exp (- p x^{2} + 2 q x) d x = \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}} (\frac{q}{p}),

\int x^{2} \exp (- p x^{2} + 2 q x) d x = \frac{1}{2 p} \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}} (1 + \frac{2 q^{2}}{p}),

after tedious but straightforward integral calculations, we deliver

\begin{array}{l} W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = (^{\frac{k}{2 π z}) 2} e x p [- \frac{i k}{2 z} ({ρ^{'}}_{1}^{2} - {ρ^{'}}_{2}^{2})] \\ \times \exp [- k^{2} z T (η, ε, χ_{T}, ω) {| {ρ^{'}}_{1} - {ρ^{'}}_{2} |}^{2}] \times [(N_{1} - N_{2}) \mp (N_{3} - N_{4})], \end{array}

where

N_{1} = \frac{π^{2}}{a C} E_{x} E_{y} \exp (\frac{F_{x}^{2} + F_{y}^{2}}{C}) (\frac{F_{x}^{2} + F_{y}^{2}}{C_{}^{2}} + \frac{1}{C}),

N_{2} = \frac{π^{2} w_{0}^{2}}{8 D} e x p (- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {| {ρ^{'}}_{1} - {ρ^{'}}_{2} |}^{2}) \times \exp (\frac{G_{x}^{2} + G_{y}^{2}}{D}) (\frac{G_{x}^{2} + G_{y}^{2}}{D_{}^{2}} + \frac{1}{D}),

N_{3} = \frac{i π^{2} w_{0}}{\sqrt{2 a C D}} E_{x} \frac{F_{x} G_{y}}{C D} \exp [- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {({ρ^{'}}_{1 y} - {ρ^{'}}_{2 y})}^{2}] \exp (\frac{F_{x}^{2}}{C} + \frac{G_{y}^{2}}{D}),

N_{4} = \frac{i π^{2} w_{0}}{\sqrt{2 a C D}} E_{y} \frac{F_{y} G_{x}}{C D} \exp [- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x})}^{2}] \exp (\frac{F_{y}^{2}}{C} + \frac{G_{x}^{2}}{D}),

b = \frac{i k}{z},

C = \frac{2}{w_{0}^{2}} - \frac{b^{2}}{4 a},

D = a - \frac{b^{2} w_{0}^{2}}{8},

E_{x} = e x p {\frac{1}{4 a} {[\frac{i k}{2 z} ({ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}) - T (η, ε, χ_{T}, ω) k^{2} z ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x})]}^{2}},

F_{x} = \frac{1}{2} [\frac{i k}{z} ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x}) - \frac{i k b}{4 a z} ({ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}) + \frac{b k^{2}}{2 a} T (η, ε, χ_{T}, ω) ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x})],

G_{x} = \frac{1}{2} [\frac{i k}{2 z} ({ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}) - T (η, ε, χ_{T}, ω) k^{2} z ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x}) - \frac{b^{2} w_{0}^{2}}{4} ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x})] .

Using of the symmetry, E_y, F_y and G_y are obtained from E_x, F_x and G_x by replacing ρ′₁_x and ρ′₂_x with ρ′₁_y and ρ′₂_y, respectively.

Equations (14)–(24) provide the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 propagating through oceanic turbulence. It follows from Eq. (14) that the cross–spectral density of GSM vortex beams depends on the oceanic turbulence and beam parameters such as the Kolmogorov micro scaleη, rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, rate of dissipation of mean-square temperatureχ_T, relative strength of temperature salinity fluctuation ω, spatial correlation length σ₀, waist width w₀, sign and values of topological charge m, as well as the propagation distance z, and the positions of two points at the z plane. In addition, for the case of T(η, ε, χ_T, ω) = 0 in Eqs. (14)–(24), which can simplify to the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 in free space.

3. The evolution behavior of average intensity of Gaussian Schell-model vortex beams through oceanic turbulence

Substituting ${ρ^{'}}_{1} = {ρ^{'}}_{2} = {ρ^{'}}_{}$ into Eq. (14), the average intensity at the z plane is expressed as

\begin{array}{l} I ({ρ^{'}}_{}, z) = W ({ρ^{'}}_{}, {ρ^{'}}_{}, z) \\ = (^{\frac{k}{2 π z}) 2} \iint d^{2} s \iint d^{2} t [(s^{2} - \frac{t^{2}}{4}) \mp i (s_{x} t_{y} - s_{y} t_{x})] \\ \times \exp (- a t^{2} - \frac{i k}{z} s t) \exp (- \frac{2}{w_{0}^{2}} s^{2}) \exp (\frac{i k}{z} {ρ^{'}}_{} t), \end{array}

after straightforward integral calculations deliver the expression

\begin{array}{l} I ({ρ^{'}}_{}, z) = (^{\frac{k}{2 π z}) 2} {\frac{π^{2}}{a C} (\frac{k^{2} {ρ^{'}}^{2}}{4 a z^{2} C^{2}} + \frac{1}{C}) \\ \times \exp [\frac{k^{2}}{4 a z^{2}} {ρ^{'}}^{2} (\frac{1}{C} - 1)] - \frac{π^{2} w_{0}^{2}}{8 D} (\frac{i k {ρ^{'}}^{2}}{2 z D^{2}} + \frac{1}{D}) \exp (\frac{i k {ρ^{'}}^{2}}{2 z D})} . \end{array}

Eq. (26) is the analytical expression of a GSM vortex in oceanic turbulence average intensity, Eq. (26) together with Eqs. (8), (10), (19)–(21) imply that the average intensity depends on correlation length σ₀, waist width w₀, topological charge |m|, and parameter T(η, ε, χ_T, ω) (including the Kolmogorov micro scaleη, the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, the rate of dissipation of mean-square temperatureχ_T and the relative strength of temperature and salinity fluctuation ω), but is independent of the sign of m. It’s worth noting that Eq. (26) reduces to the analytical expression of a GSM vortex in free space for the case of T(η, ε, χ_T, ω) = 0 in Eq. (10), namely,

a^{'} = \frac{1}{2} (\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}),

C^{'} = \frac{2}{w_{0}^{2}} - \frac{b^{2}}{4 a^{'}},

D^{'} = a^{'} - \frac{b^{2} w_{0}^{2}}{8},

\begin{array}{l} I^{'} ({ρ^{'}}_{}, z) = (^{\frac{k}{2 π z}) 2} {\frac{π^{2}}{a^{'} C^{'}} (\frac{k^{2} {ρ^{'}}^{2}}{4 a^{'} z^{2} {C^{'}}^{2}} + \frac{1}{C^{'}}) \\ \times \exp [\frac{k^{2}}{4 a^{'} z^{2}} {ρ^{'}}^{2} (\frac{1}{C^{'}} - 1)] - \frac{π^{2} w_{0}^{2}}{8 D^{'}} (\frac{i k {ρ^{'}}^{2}}{2 z {D^{'}}^{2}} + \frac{1}{D^{'}}) \exp (\frac{i k {ρ^{'}}^{2}}{2 z D^{'}})}, \end{array}

which is consistent with the Eq. (9) in [4] for the case of C_n² = 0.

Illustration numerical examples are given in Figs. 1(a)–1(e), which plot normalized intensity distributions I/I_max (I_max—maximum intensity) of a GSM vortex beam versus the slanted axis ρ′_r ( ${ρ^{'}}_{r} = \sqrt{{ρ^{'}}_{x}^{2} + {ρ^{'}}_{y}^{2}}$ ) for different values of (a) propagation distance z, (b) relative strength of temperature and salinity fluctuation ω, (c) rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (d) rate of dissipation of mean-square temperature χ_T, (e) spatial correlation length σ₀, where the calculation parameters w₀ = 3cm, λ = 1060nm and η = 10⁻³m are fixed in every figure, while the rest parameters including σ₀ = 2.5cm, ω = −2.5, ε = 10⁻⁷ m²/s³, χ_T = 10⁻⁹ K²/s and z = 300m are allowed to vary in different figures. As can be seen from Fig. 1(a), the average intensity of GSM vortex beams propagating through oceanic turbulence undergoes several stages of evolution like as GSM vortex beams propagating through ideal Kolmogorov atmospheric turbulence [4]. At the z = 0 plane there exists a zero intensity at the center ρ′_r = 0, where the phase becomes singular. With increasing propagation distance to z_flat = 560m, a flat-topped intensity profile occurs, and finally evolves into a Gaussian one at z_Gau = 700m. In Figs. 1(b)–1(e), we can see that at the given propagation distance (e.g., z = 300m), the evolution behavior of average intensity of GSM vortex beams in oceanic turbulence is significantly affected by the oceanic turbulence parameters includingω, ε and χ_T_, and beam parameter σ₀. From Figs. 1(a)–1(e), it is interesting to find that the beam profile will first take on a hollow shape, and then approach a Gaussian distribution, for example, when ω = −0.5 in Fig. 1(b), ε = 10⁻¹⁰ m²/s³in Fig. 1(c), χ_T = 10⁻⁷ K²/s in Fig. 1(d) and σ₀ = 1cm, the I/I_max of a GSM vortex beam in oceanic turbulence extend to a Gaussian profile. The larger χ_T, the smallerω, ε and spatial correlation length and the longer propagation distance z are, the faster the change of the beam profile is. We can see from Eq. (8) and Figs. 2(a) and 2(b) (the calculation parameters are the same as those of Fig. 1) that the larger χ_T and the smallerω and ε are, the stronger the oceanic turbulence is. Therefore, Figs. 1(a)–1(e) indicate that the evolution behavior of intensity depends on the oceanic turbulence strength, the beam coherent degree and the propagation distance. At the same time, it is shown that the larger χ_T and z or the smaller ω and ε are, the larger the spreading of the beam width is, but the effect on the beam width by the spatial correlation length is not significant for the GSM vortex beams through oceanic turbulence.

Fig. 1 Evolution of normalized intensity profiles of GSM vortex beams propagating in oceanic turbulence.

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Fig. 2 T(η,ω,ε,χ_T) versus the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (a) for different values of the rate of dissipation of mean-square temperatureχ_T and (b) for different values of the relative strength of temperature and salinity fluctuation ω.

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4. The evolution behavior of coherent vortices of Gaussian Schell-model vortex beams through oceanic turbulence

The spectral degree of coherence is defined as [22]

μ ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = \frac{W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z)}{{[I ({ρ^{'}}_{1}, z) I ({ρ^{'}}_{2}, z)]}^{1 / 2}},

and the position of coherence vortices is determined by [23]

Re [μ ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z)] = 0,

Im [μ ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z)] = 0.

where Re and Im stand for the real and imaginary parts of

μ ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z)

, respectively. The topological charge and its sign of coherence vortices are determined by the sign principle [24].

Figure 3 gives curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 through oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m, where ρ′₁ = (6cm, 9cm), m = + 1, σ₀ = 2.5cm are kept fixed, and the other calculation parameters are the same as in Fig. 1. Figure 4 plots contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m, where the calculation parameters are the same as in Fig. 3. Figures 3(a), 3(b), 4(a), and 4(b) indicate that there exists a coherent vortex A with m = + 1 at z = 50m whose position is (0.3783cm,-0.1178cm) and two coherent vortices A and B with m = + 1, −1 appear at z = 220m. The singularity A moves to the position (3.27cm, 0.2321cm), while a new singularity B with m = −1 appears at the position (−7.351cm, 11.88cm). Therefore, the position and number of coherent vortices change with increasing propagation distance through oceanic turbulence.

Fig. 3 Curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 in oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m.

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Fig. 4 Contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m.

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Figure 5 plots that the position and number of coherent vortices of a GSM vortex beam with m = + 1 through oceanic turbulence for different values ofω,ε, χ_T, where z = 300m is kept fixed, the other calculation parameters are same as those of Fig. 1, the white and black dots represent m = −1 and m = + 1, respectively. It can be seen that there appears only a coherent vortex with m = + 1 and the topological charge is conserved within a certain propagation distance. But as the propagation distance increases, two coherent vortices with m = + 1 and m = −1 appear, whose positions change and gradually move closer. From Figs. 5(a)–5(c), we can see that the larger ω is, the larger the distance z_c for the conservation of the topological charge. It can be seen from Figs. 5(b), 5(d) and 5(e) that the larger ε is, the larger z_c is, for the case of ε = 10^-10, 10⁻⁷ and 10⁻⁵, z_c corresponding to 70m, 150m and 250m, respectively. The position and number of coherent vortices of GSM vortex beams with m = + 1 propagation through oceanic turbulence for different values of χ_T are plotted in Figs. 5(b), 5(f) and 5(g). We can see that the position and number of coherent vortices change with propagation distance, and two coherent vortices with m = + 1 and m = −1 appear at z = 150m in Fig. 5(b), 70m in Fig. 5(f) and 40m in Fig. 5(g) respectively. Obviously, the smaller the χ_T is, the larger the distance z_c for the conservation of the topological charge, which indicates that the distance for the conservation of the topological charge increases with increasingω, ε and decreasingχ_T. The physical reason can be seen from Fig. 2 that the largerω, ε and smaller χ_T are, the smaller the value of T(η, ε, χ_T, ω) is, which indicates that the weaker the oceanic turbulence is, the less effect on the conservation of the topological charge by the weaker oceanic turbulence. And then the positions of two singularities constantly change and the separation distance between them gradually becomes smaller, and this condition is more significant for the larger the rate of dissipation of mean-square temperature χ_T in Fig. 5(g).

Fig. 5 Position and number of coherent vortices of a GSM with m = + 1 propagation through oceanic turbulence for different values of ω, ε, χ_T.

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5. Concluding remarks

In this paper, we have studied the evolution behavior of coherent vortices and average intensity of GSM vortex beams propagating through oceanic turbulence in detail. It has been shown analytically and numerically that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence and beam parameters, including the rate of dissipation of turbulent kinetic energy per unit mass of fluidε, rate of dissipation of mean-square temperatureχ_T, relative strength of temperature salinity fluctuationsω, spatial correlation length and topological charge of the beams, as well as the propagation distance. The larger χ_T, the smaller ω, ε and the shorter spatial correlation length and the longer propagation distance z are, the faster the evolution of the beam intensity profile is. The smaller χ_T and the larger ω and ε are, the longer the distance appearing two coherent vortices with m = + 1 and m = −1, namely, the larger the distance z_c for the conservation of the topological charge. The results obtained in this paper would be useful for potential applications of vortex beams in oceanic optical communications.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant No.61275203, National Natural Science Foundation of China and Civil Aviation Administration of China under grant No. 61079023. A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department under grant No.14ZA0268, the Program of Yibin Municipal Science and Technology Bureau under grant No.2013SF020, and the Scientific Research Project of Yibin University under grant No. 2013YY04.

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Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence

Abstract

1. Introduction

2. The cross-spectral density of Gaussian Schell-model vortex beams through oceanic turbulence

3. The evolution behavior of average intensity of Gaussian Schell-model vortex beams through oceanic turbulence

4. The evolution behavior of coherent vortices of Gaussian Schell-model vortex beams through oceanic turbulence

5. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (5)

Equations (33)

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