Spreading of a Lorentz-Gauss Vortex Beam Propagating through Oceanic Turbulence

Abstract

Based on the extended Huygens-Fresnel principle, the analytical equation for a Lorentz-Gauss vortex beam propagating through oceanic turbulence has been derived. The spreading properties of a Lorentz-Gauss vortex beam propagating through oceanic turbulence are analyzed in detail using numerical examples. The results show that a Lorentz-Gauss vortex beam propagating through stronger oceanic turbulence will spread more rapidly, and the Lorentz-Gauss vortex beam with higher topological charge M will lose its initial dark center more slowly.

I. INTRODUCTION

The propagation of laser beams through random media is a topic that has been of considerable interest for a long time, because of its connection to applications, including wireless optical communication and imaging systems [1-4]. Over the years, the evolution properties of various laser beams propagating through random media have been reported, such as a stochastic electromagnetic beam [5], radially polarized beams [6], astigmatic stochastic electromagnetic beams [7], electromagnetic vortex beams [8], Gaussian Schell-model vortex beams [9], partially coherent annular beams [10], partially coherent flat-topped vortex hollow beams [11], coherent Gaussian array beams [12-14], partially coherent Hermite-Gaussian linear array beams [15],fourth-order mutual coherence function of laser beams [16], flat-topped vortex hollow beams [17], partially coherent cylindrical vector beams [18], chirped Gaussian pulsed beams[19], partially coherent four-petal Gaussian beams [20], and partially coherent four-petal Gaussian vortex beams [21].

Recently a new beam called a Lorentz beam has been introduced, to describe the output of a laser diode [22]. Since then, the propagation properties of Lorentz and Lorentz-Gauss beams have been widely investigated. In recent years the propagation properties of Lorentz and Lorentz-Gauss beams in uniaxial crystals have been analyzed [23-26], and the propagation properties of Lorentz and Lorentz-Gauss beams in random media have been widely studied [27-32]. However, the evolution properties of a Lorentz-Gauss vortex beam in oceanic turbulence have not been reported. In this paper, the spreading properties of a Lorentz-Gauss vortex beam propagating through oceanic turbulence have been illustrated and analyzed using numerical examples.

II. PROPAGATION OF THE BEAM THROUGHOCEANIC TURBULENCE

In the Cartesian coordinate system, the optical field of a Lorentz-Gauss vortex beam propagating along the z-axis at the source plane z = 0 can be written as [33]:

\(E\left(\mathbf{r}_{0}, 0\right)=\frac{w_{0 x} w_{0 y}}{\left(w_{0 x}^{2}+x_{0}^{2}\right)\left(w_{0 y}^{2}+y_{0}^{2}\right)} \exp \left(-\frac{x_{0}^{2}+y_{0}^{2}}{w_{0}^{2}}\right)\left(x_{0}+i y_{0}\right)^{M}\)       (1)

where \(\mathbf{r}_{0}=\left(x_{0}, y_{0}\right)\) is the position vector at the source plane z = 0, w0x and w0 y are the parameters related to the beam widths of the Lorentz part of the Lorentz-Gaussvortex in the x and y directions respectively, w0 is the waist width of the Gaussian part of the Lorentz-Gaussvortex beam, and M is the topological charge of the Lorentz-Gauss vortex beam. Recalling the relationship of the Lorentz distribution and Hermite-Gaussian function [33]:

\(\begin{aligned} \frac{1}{\left(x^{2}+w_{0 x}^{2}\right)\left(y^{2}+w_{0 y}^{2}\right)}=& \frac{\pi}{2 w_{0 x}^{2} w_{0 y}^{2}} \sum_{m=0}^{N} \sum_{n=0}^{N} \sigma_{2 m} \sigma_{2 n} H_{2 m}\left(\frac{x}{w_{0 x}}\right) H_{2 n}\left(\frac{y}{w_{0 y}}\right) \\ & \times \exp \left(-\frac{x^{2}}{2 w_{0 x}^{2}}-\frac{y^{2}}{2 w_{0 y}^{2}}\right) \end{aligned}\)        (2)

where N is the number of the expansion and σ2m andσ2n are the expanded coefficients, which can be found in ref [34]. As the even number 2m increases, the values ofσ2m decrease dramatically, so N will not be large in the numerical calculations. The 2mth-order Hermite polynomial H x 2m ( ) can be expressed as [35]:

\(H_{2 m}(x)=\sum_{l=0}^{m} \frac{(-1)^{l}(2 m) !}{l !(2 m-2 l) !}(2 x)^{2 m-2 l}\)       (3)

recalling the following equation [35]

\((x+i y)^{M}=\sum_{l=0}^{M} \frac{M ! i^{l}}{l !(M-l) !} x^{M-l} y^{\prime}\)       (4)

Within the framework of the paraxial approximation, the average intensity of a Lorentz-Gauss vortex beam propagating through oceanic turbulence at the receiver plane z can be expressed as:

\(\begin{aligned} \langle I(\mathbf{r}, z)\rangle=& \frac{k^{2}}{4 \pi^{2} z^{2}} \iiint \int_{-\infty}^{+\infty} E\left(\mathbf{r}_{10}, 0\right) E^{*}\left(\mathbf{r}_{20}, 0\right) \exp \left[-\frac{i k}{2 z}\left(\mathbf{r}-\mathbf{r}_{10}\right)^{2}+\frac{i k}{2 z}\left(\mathbf{r}-\mathbf{r}_{20}\right)^{2}\right] \\ & \times\left\langle\exp \left[\psi\left(\mathbf{r}, \mathbf{r}_{10}\right)+\psi^{*}\left(\mathbf{r}, \mathbf{r}_{10}\right)\right]\right) d \mathbf{r}_{10} d \mathbf{r}_{20} \end{aligned}\)        (5)

where r =  ( x ,y) is the position vector at the output plane,\(k=2 \pi / \lambda\) is the wavenumber, and  \(\psi\left(\mathbf{r}_{0}, \mathbf{r}\right)\) is the complex phase perturbation due to the random medium. The last term in angle brackets of Eq. (5) for a Lorentz-Gaussvortex beam propagating through oceanic turbulence can expressed as [5-12]:

\(\left\langle\exp \left[\psi\left(\mathbf{r}_{10}, \mathbf{r}\right)+\psi^{*}\left(\mathbf{r}_{20}, \mathbf{r}\right)\right]\right\rangle=\exp \left[-\frac{\left(\mathbf{r}_{1}-\mathbf{r}_{2}\right)^{2}}{\rho_{0}^{2}}\right]\)       (6)

where ρ0 is the coherence length of a spherical wave propa-gating in oceanic turbulence and \(\rho_{0}^{2}=3 / \pi^{2} k^{2} z \int_{0}^{\infty} d \kappa \kappa \Phi(\kappa)\) ,k is the spatial frequency. Φ(κ ) is the one-dimensional spatial power spectrum of oceanic turbulence, which can be written as:

\(\Phi(\kappa)=0.388 \times 10^{-8} \varepsilon^{-11 / 3}\left[1+2.35(\kappa \eta)^{2 / 3}\right] f\left(\kappa, \zeta, \chi_{T}\right)\)       (7)

where ε is the rate of dissipation of turbulent kineticenergy per unit mass of fluid, which may vary from \(10^{-1} m^{2} s^{-3} \text {to } 10^{-10} m^{2} s^{-3}, \eta=10^{-3}\)  being the Kolmogorovmicro (inner) scale, and

\(f\left(\kappa, \zeta, \chi_{T}\right)=\frac{\chi_{T}}{\zeta^{2}}\left[\zeta^{2} \exp \left(-A_{T} \delta\right)+\exp \left(-A_{S} \delta\right)-2 \zeta \exp \left(-A_{T S} \delta\right)\right]\)       (8)

with χ_r being the rate of dissipation of mean square temperature, ranging from \(10^{-4} K^{2} s^{-1} \text {to } 10^{-10} K^{2} s^{-1}, A_{T}=1.863 \times 10^{-2}\) , \(A_{S}=1.9 \times 10^{-4}, A_{T S}=9.41 \times 10^{-3}, \delta=8.284(\mathrm{K} \eta)^{4 / 3}+12.978(\mathrm{K} \eta)^{2}\) . \(\zeta\)is the relative strength of temperature and salinity fluctuations, which in ocean waters can vary from -5 to0, with a value of 0 correspondings to the case when salinity-driven turbulence prevails, and -5 the case when temperature-driven turbulence dominates.

By substituting Eq. (1) into Eq. (5) and recalling the following integral equation [34]:

\(\int_{-\infty}^{+\infty} x^{n} \exp \left(-p x^{2}+2 q x\right) d x=n ! \exp \left(\frac{q^{2}}{p}\right)\left(\frac{q}{p}\right)^{n} \sqrt{\frac{\pi}{p}} \sum_{k=0}^{\frac{m}{2}} \frac{1}{k !(n-2 k) !}\left(\frac{p}{4 q^{2}}\right)^{k}\)        (9)

after some tedious calculation we can obtain

\(\begin{array}{c} I(x, y, z)=\frac{k^{2}}{4 \pi^{2} z^{2}}\left(\frac{\pi}{2 w_{0 x} w_{0 y}}\right)^{2} \sum_{m=0}^{N} \sum_{m=0}^{N} \sigma_{2 m 1} \sigma_{2 n 1} \sum_{l=0}^{M} \frac{M ! i^{n}}{l 1 !(M-l 1) !} \\ \times \sum_{m 2=0}^{N} \sum_{n=0}^{N} \sigma_{2 m 2} \sigma_{2 n 2} \sum_{l_{2=0}}^{M} \frac{M !(-i)^{12}}{12 !(M-12) !} I(x, z) I(y, z) \end{array}\)       (10)

where I  ( x, z) can be expressed a

 (11)

with

\(a_{x 1}=\frac{1}{2 w_{0 x}^{2}}+\frac{1}{w_{0}^{2}}+\frac{1}{\rho_{0}^{2}}+\frac{i k}{2 z}\)       (12a)

\(b_{x}=\frac{1}{2 w_{0 x}^{2}}+\frac{1}{w_{0}^{2}}+\frac{1}{\rho_{0}^{2}}-\frac{i k}{2 z}-\frac{1}{a_{x}}\left(\frac{1}{\rho_{0}^{2}}\right)^{2}\)       (12b)

\(c_{x}=\frac{1}{a_{x}} \frac{1}{\rho_{0}^{2}} \frac{i k}{2 z} x-\frac{i k}{2 z} x\)       (12c)

and I  (y , z) can be expressed as

(13)

with

\(a_{y}=\frac{1}{2 w_{0 y}^{2}}+\frac{1}{w_{0}^{2}}+\frac{1}{\rho_{0}^{2}}+\frac{i k}{2 z}\)       (14a)

\(b_{y}=\frac{1}{2 w_{0 y}^{2}}+\frac{1}{w_{0}^{2}}+\frac{1}{\rho_{0}^{2}}-\frac{i k}{2 z}-\frac{1}{a_{y}}\left(\frac{1}{\rho_{0}^{2}}\right)^{2}\)       (14b)

\(c_{y}=\frac{1}{a_{y}} \frac{1}{\rho_{0}^{2}} \frac{i k}{2 z} y-\frac{i k}{2 z} y\)       (14c)

Equations (10)~(14) are the analytical expression at the receiver plane z for a Lorentz-Gauss vortex beam propagating through oceanic turbulence; the average intensity of a Lorentz-Gauss vortex beam propagating through oceanic turbulence can be obtained using the derived equation.

III. NUMERICAL RESULTS AND DISCUSSION

In this section, the spreading properties of a LorentzGauss vortex beam propagating through oceanic turbulence are investigated at the receiver plane z using numerical examples. In these numerical examples, the parameters of the Lorentz-Gauss vortex beam and oceanic turbulence are set arbitrarily as 

\(\lambda=417 \mathrm{nm}, w_{0 x}=w_{0 y}=5 \mathrm{mm}, w_{0}=10 \mathrm{mm}\), \(\mathrm{M}=1, \quad \chi_{T}=10^{-8}, \quad \varepsilon=10^{-7}, \text {and } \zeta=-2.5\)

The normalized average intensity of a Lorentz-Gaussvortex beam propagating through oceanic turbulence is illustrated in Figs. 1~3. In Fig. 1, the parameters are set as w_ox ; in Fig. 2, 0 0 15 w w mm x y = = ; and Fig. 3shows the normalized intensity of a Lorentz-Gauss vortex beam for various w0x and w0 y . As can be seen, a Lorentz-Gauss vortex beam propagating through oceanic turbulence can retain its initial dark center similar to the source beam at short propagation distances; as the propagation distance increases, the beam loses its initial dark center and evolves into a flat-topped beam and gauss beam; and beams with different w w 0 0 x y = and w0will have different evolution properties at the same propagation distance. A Lorentz-Gauss vortex beam with smaller w w 0 0 x y = will have a smaller beam spot at the source plane and at the short propagation distances (Figs.1(a), 2(a), and 3(a)), while a beam with smaller w w 0 0 x y =will evolve into a flat-topped Gauss beam more rapidly as the propagation distance increases (Fig. 3(d)).

FIG. 1. Normalized average intensity of a Lorentz-Gaussvortex beam with M = 1 propagating through oceanic turbulence: (a) z = 20 m, (b) z = 60 m, (c) z = 120 m, (d) z = 180 m.

FIG. 2. Normalized average intensity of a Lorentz-Gaussvortex beam with M = 1 propagating through oceanic turbulence: (a) z = 20 m, (b) z = 60 m, (c) z = 120 m, (d) z = 180 m.

FIG. 3. Cross section of the normalized average intensity of a Lorentz-Gauss vortex beam propagating through oceanic turbulence,for the different w0x and w0 y : (a) z = 20 m, (b) z = 60 m, (c) z = 120 m, (d) z = 180 m.

The influence of topological charge M on the spreading properties of a Lorentz-Gauss vortex beam propagating in oceanic turbulence are given in Fig. 4. From Fig. 4(a), it can be found that a Lorentz-Gauss vortex beam with larger M will have a dark center, and a beam with larger M can keep its initial dark center better than one with smaller M,as the propagation distance z increases.

FIG. 4. Cross section of the normalized average intensity of a Lorentz-Gauss vortex beam propagating through oceanic turbulence,for different M: (a) z = 20 m, (b) z = 180 m.

To investigate the influence of the strength of oceanic turbulence on the spreading properties of a Lorentz-Gaussvortex beam, the normalized intensity of a beam propagating through oceanic turbulence for different Tx , ε , and ς is illustrated in Figs. 5~7 respectively. From Fig. 5, it can be found that a beam propagating through oceanic turbulence will lose its initial dark center and evolve into a Gaussbeam more rapidly with increasing oceanic parameter Tx ,as the propagation distance increases. The strength of oceanic turbulence increases as the oceanic parameter Txincreases. as the oceanic parameter ς increases, the strength of oceanic turbulence will become stronger. So, in Fig. 6it can be seen that a Lorentz-Gauss vortex beam will evolve into a Gauss beam more quickly as the oceanic parameter ς increases. From Fig. 7, it can be seen that a Lorentz-Gauss vortex beam propagating through oceanic turbulence with smaller ε will evolve into a Gauss beam more rapidly. Small ε indicates stronger oceanic turbulence. Thus, a Lorentz-Gauss vortex beam propagating through stronger oceanic turbulence (due to the increase of oceanic parameters Tx and ς , or the decrease of oceanic parameterε ) will evolve into a Gauss beam more rapidly.

 

FIG. 5. Cross section of the normalized average intensity of a Lorentz-Gauss vortex beam with M = 1 propagating through oceanic turbulence, for different χT : (a) z = 20 m, (b) z = 180 m.

FIG. 6. Cross section of the normalized average intensity of a Lorentz-Gauss vortex beam with M = 1 propagating through oceanic turbulence, for different ς : (a) z = 60 m, (b) z = 180 m.

FIG. 7. Cross section of the normalized average intensity of a Lorentz-Gauss vortex beam with M = 1 propagating through oceanic turbulence, for different ε : (a) z = 60 m, (b) z = 180 m

IV. CONCLUSIONS

Based on the Huygens-Fresnel integral of beam propagation through random media, the propagation equation of a Lorentz-Gauss vortex beam propagating through oceanic turbulence has been derived. The average intensity properties and spreading properties of a Lorentz-Gaussvortex beam propagating through oceanic turbulence are analyzed using numerical examples. The results show that a Lorentz-Gauss vortex beam propagating through oceanic turbulence can keep its initial dark center at short propagation distances, but the beam will lose its initial dark center and evolve into a Gaussian-like beam as the propagation distance z increases. It is also found that a Lorentz-Gauss vortex beam with higher M has a larger dark center, and will lose its initial dark center more slowly. Finally, it is found that a Lorentz-Gauss vortex beam propagating through stronger oceanic turbulence(corresponding to the increase of oceanic parameters Txand ς , or the decrease of oceanic parameter ε ) will spread faster as the propagation distance z increases. This results have potential applications in underwater laser communication using a Lorentz-Gauss vortex beam.