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Assuming that
${{\bf{\rho }}_{\bf{1}}}$ ,${{\bf{\rho }}_{\bf{2}}}$ is two spatial points at the source plane, the CSD can be described by[20]$${W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = \left\langle {E\left( {{{\bf{\rho }}_{\bf{1}}}} \right){E^ * }\left( {{{\bf{\rho }}_{\bf{2}}}} \right)} \right\rangle $$ (1) where
$E\left( \rho \right)$ is electric field at the point${\bf{\rho }}$ , * indicate complex conjugate,$\left\langle {} \right\rangle $ represent the ensemble averaging. As we all know, the correlation function for optical fields cannot be chosen at wish owing to the non-negative definiteness restrictions, and the non-negative definiteness restrictions refers that, for any$f\left( {\bf{\rho }} \right)$ , the CSD must satisfy the inequality[9]$$\int {\int {{{\rm{d}}^2}{{\bf{\rho }}_{\bf{1}}}} } {{\rm{d}}^2}{{\bf{\rho }}_{\bf{2}}}{W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)f\left( {{{\bf{\rho }}_{\bf{1}}}} \right)f\left( {{{\bf{\rho }}_{\bf{2}}}} \right) \geqslant 0$$ (2) $f\left( {\bf{\rho }} \right)$ is an arbitrary function. At this time, the CSD of Eq. (2) can be expressed as[21]$${W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = \int {p\left( v \right)H_0^ * \left( {{{\bf{\rho }}_{\bf{1}}},v} \right){H_0}\left( {{{\bf{\rho }}_{\bf{2}}},v} \right){\rm{d}}v} $$ (3) where the
$p\left( v \right)$ is an arbitrary nonnegative weighting function, the${H_0}\left( {{\bf{\rho }},v} \right)$ is an arbitrary kernel function. From Ref. [22],${H_0}\left( {{\bf{\rho }},v} \right)$ has the form$${H_0}\left( {{\bf{\rho }},v} \right) = \tau \left( {\bf{\rho }} \right)\exp \left[ { - if\left( {\bf{\rho }} \right)v} \right]$$ (4) The
$\tau \left( {\bf{\rho }} \right)$ is a complex amplitude. When the kernel function${H_0}\left( {{\bf{\rho }},v} \right)$ of Eq. (3) is known, different CSD can be obtained by selecting different weighting function[23]. The$p\left( v \right)$ and${H_0}\left( {{\bf{\rho }},v} \right)$ is given by[18]$$p\left( v \right) = \frac{1}{a} \cdot {\rm{rect}}\left( {\frac{v}{a}} \right) = \left\{ {\begin{array}{*{20}{c}} {1/a,\;\;\; \left| v \right| \leqslant a/2} \\ {0,\;\;\; \left| v \right| > a/2} \end{array}} \right.$$ (5) $${H_0}\left( {{\bf{\rho }},v} \right) = \tau \left( {\bf{\rho }} \right)\exp \left[ { - 2{\text{π}} iv{{\left( {{\bf{\rho }} - {\rho _0}} \right)}^2}} \right]$$ (6) where
$\tau \left( {\bf{\rho }} \right) = \exp \left[ { - {{\bf{\rho }}^2}/\left( {2{\sigma ^2}} \right)} \right]$ ,$\sigma $ is the root-mean-square width.${\rm{rect}}\left( x \right)$ is a rectangular function with width$a$ , and$a$ is a positive constant. Substituting Eqs. (5) and (6) in Eq. (3), it is obtained$$ {W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = \exp \left[ { - \left( {{\bf{\rho }}_{\bf{1}}^{\bf{2}} + {\bf{\rho }}_{\bf{2}}^{\bf{2}}} \right)/\left( {2{\sigma ^2}} \right)} \right]\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) $$ (7) $\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)$ is the DOC of the light field, usually defined as$\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = W\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)/\sqrt {I\left( {{{\bf{\rho }}_{\bf{1}}}} \right)I\left( {{{\bf{\rho }}_{\bf{2}}}} \right)} $ , where$I\left( {\bf{\rho }} \right) = W\left( {{\bf{\rho }},{\bf{\rho }}} \right)$ is the spectral intensity. In this paper, the DOC of non-uniformly Sinc-correlated beams has the following form$$\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = {\rm{sinc}}\left\{ {c\left[ {{{\left( {{{\bf{\rho }}_{\bf{1}}} - {\rho _0}} \right)}^2} - {{\left( {{{\bf{\rho }}_{\bf{2}}} - {\rho _0}} \right)}^2}} \right]} \right\}$$ (8) From Eq. (8), we can see that the DOC include an extra shift by
${\rho _0}$ , where${\rho _0} = 0.7\sigma $ is a real constant[24]. The parameter$c$ is the amplification factor and can be used to control the beams profile of far field[17].It can be seen from Eq. (8) that as
$\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right) = $ $ {\rm{sinc}}\left[ {c\left( {{{\bf{\rho }}_{\bf{1}}} - {{\bf{\rho }}_{\bf{2}}}} \right)} \right]$ , we get the correlation function of uniformly sinc-correlated beam.In Fig.1, we plot the CSD of the non-uniformly Sinc-correlated blue-green laser beams according to Eqs. (7) and (8) when the wavelength
$\lambda = $ 532 nm and root-mean-square width$\sigma = 1\; {\rm{mm}},\;3\;{\rm{mm}}$ . From Ref. [16], we do not discuss the effect of amplification factor$c$ on the CSD, we make$c$ as a value of 8. It is indicated that the DOC of the non-uniformly Sinc-correlated laser beams are related to the lateral coordinate and takes the maximum value near the point${\rho _0}$ , which is different from the GSM beams[8].Let us consider that a non-uniformly Sinc-correlated laser beams propagating close to
$z$ axis from the source plane$z = 0$ to the half-space$z \geqslant 0$ in oceanic turbulence. With the help of the generalized Huygens-Fresnel principle, the CSD between two points$\left( {{{{\bf{\rho '}}}_1},z} \right)$ and$\left( {{{{\bf{\rho '}}}_{\bf{2}}},z} \right)$ in any propagation plane are satisfied the following equation[25]$$\begin{split} & W\left( {{{{\bf{\rho '}}}_{\bf{1}}},{{{\bf{\rho '}}}_{\bf{2}}},z} \right) =\\ & \quad\frac{{{k^2}}}{{4{{\text{π}} ^2}{z^2}}}\iint {{W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)}\times \\ & \quad\exp \left[ { - ik\frac{{{{\left( {{{{\bf{\rho '}}}_{\bf{1}}} - {{\bf{\rho }}_{\bf{1}}}} \right)}^2}{\rm{ - }}{{\left( {{{{\bf{\rho '}}}_{\bf{2}}} - {{\bf{\rho }}_{\bf{2}}}} \right)}^2}}}{{2z}}} \right]\times \\ & \quad \left\langle {\exp \left[ {\phi \left( {{{{\bf{\rho '}}}_{\bf{1}}},{{\bf{\rho }}_{\bf{1}}}} \right) + {\phi ^ * }\left( {{{{\bf{\rho '}}}_{\bf{2}}},{{\bf{\rho }}_{\bf{2}}}} \right)} \right]} \right\rangle {{\rm{d}}^2}{{\bf{\rho }}_{\bf{1}}}{{\rm{d}}^2}{{\bf{\rho }}_{\bf{2}}} \end{split} $$ (9) where
${W^{\left( 0 \right)}}\left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)$ is the CSD in the source plane.$k = 2{\text{π}} /\lambda $ is wave number.$\left\langle {\exp \left[ {\phi \left( {{{{\bf{\rho '}}}_{\bf{1}}},{{\bf{\rho }}_{\bf{1}}}} \right) + {\phi ^ * }\left( {{{{\bf{\rho '}}}_{\bf{2}}},{{\bf{\rho }}_{\bf{2}}}} \right)} \right]} \right\rangle $ can be expressed as[23]$$\begin{split} & \left\langle {\exp \left[ {\phi \left( {{{{\bf{\rho '}}}_{\bf{1}}},{{\bf{\rho }}_{\bf{1}}}} \right) + {\phi ^ * }\left( {{{{\bf{\rho '}}}_{\bf{2}}},{{\bf{\rho }}_{\bf{2}}}} \right)} \right]} \right\rangle=\\ & {\rm{exp}}\left\{ { - 4{{\text{π}} ^2}{k^2}z\int {\int {\kappa {{\varPhi _n}}\left( \kappa \right)} } \left\{ {1 - {J_0}\left[ {\left| {\left( {1 - \gamma } \right){{u}} + \gamma {{q}}} \right|\kappa } \right]} \right\}} \right\}{\rm{d}}\kappa {\rm{d}}\gamma \end{split} $$ (10) In Eq. (10),
${{u}} = {{\bf{\rho }}_{\bf{1}}}^\prime - {{\bf{\rho }}_{\bf{2}}}^\prime $ ,${{q}} = {{\bf{\rho }}_{\bf{1}}} - {{\bf{\rho }}_{\bf{2}}}$ and${J_0}$ is the Bessel function of zero order.$\phi \left( {{\bf{\rho '}},{\bf{\rho }}} \right)$ is the complex phase perturbation. Eq. (10) can be calculated as$$\left\langle {\exp \left[ {\phi \left( {{{{\bf{\rho '}}}_{\bf{1}}},{{\bf{\rho }}_{\bf{1}}}} \right) + {\phi ^ * }\left( {{{{\bf{\rho '}}}_{\bf{2}}},{{\bf{\rho }}_{\bf{2}}}} \right)} \right]} \right\rangle = \exp \left\{ { - P\left[ {{{{u}}^2} + {{uq}} + {{{q}}^2}} \right]} \right\}$$ (11) In Eq. (11),
$P = \dfrac{{{{\text{π}} ^2}{k^2}z}}{3}\int_0^\infty {{\kappa ^3}{\varPhi _n}\left( \kappa \right)} {\rm{d}}\kappa $ means the turbulence strength. The${\varPhi _n}\left( \kappa \right)$ is the spatial power spectrum of the refractive index fluctuation in oceanic turbulence, and$\kappa $ is the spatial angular frequency.Introducing new variables
${{U}} = \left( {{{{\bf{\rho '}}}_{\bf{1}}} + {{{\bf{\rho '}}}_{\bf{2}}}} \right)/2$ ,${{Q}} = {{\bf{\rho '}}_{\bf{1}}} - {{\bf{\rho '}}_{\bf{2}}}$ and putting Eqs. (7), (8), (10) and (11) into Eq. (9), and exchange integral order, Eq. (9) can be simplified as$$W\left( {{{{\bf{\rho '}}}_{\bf{1}}},{{{\bf{\rho '}}}_2},z} \right) = \frac{{{k^2}}}{{4{{\text{π}} ^2}{z^2}}}\int {p\left( v \right){H^ * }\left( {{{{\bf{\rho '}}}_{\bf{1}}},z,v} \right)H\left( {{{{\bf{\rho '}}}_{\bf{2}}},z,v} \right){\rm{d}}v} $$ (12) where
$$\begin{split} & {H^ * }\left( {{{{\bf{\rho '}}}_{\bf{1}}},z,v} \right)H\left( {{{{\bf{\rho '}}}_{\bf{2}}},z,v} \right) = \\ & \quad\frac{\sigma }{{\omega \left( {z,v} \right)}}\exp \left[ { - {{\left( {\frac{{k\sigma }}{{2z}}} \right)}^2}{{\left( {{{{\bf{\rho '}}}_1} - {{{\bf{\rho '}}}_{\bf{2}}}} \right)}^2} - \frac{{ik}}{{2z}}\left( {{{{\bf{\rho '}}}_{\bf{1}}}^2 - {{{\bf{\rho '}}}_{\bf{2}}}^2} \right)} \right] \times \\ & \quad\exp \left\{ { - \frac{1}{{{\omega ^2}\left( {z,v} \right)}}{{\left[ {\frac{{{{{\bf{\rho '}}}_{\bf{1}}} + {{{\bf{\rho '}}}_{\bf{2}}}}}{2} - \frac{{ik{\sigma ^2}}}{{2z}}\left( {{{{\bf{\rho '}}}_{\bf{1}}} - {{{\bf{\rho '}}}_{\bf{2}}}} \right)-\frac{{4{\text{π}} vz{\rho _0}}}{k}} \right]}^2}} \right\} \end{split} $$ (13) Now let
${{\bf{\rho '}}_{\bf{1}}} = {{\bf{\rho '}}_{\bf{2}}} = r$ , Eq. (13) satisfies the following form$${\left| {H\left( {r,v,z} \right)} \right|^2} = \frac{\sigma }{{\omega \left( {z,v} \right)}}\exp \left[ { - \frac{{{{\left( {r - 4{\text{π}} vz{\rho _0}/k} \right)}^2}}}{{{\omega ^2}\left( {z,v} \right)}}} \right]$$ (14) where
$$\begin{split} & \omega \left( {z,v} \right)= \\ & \quad\sqrt {{{\left( {\dfrac{z}{{k\sigma }}} \right)}^2} + {\sigma ^2}{{\left( {1 - \dfrac{{4{\text{π}} zv}}{k}} \right)}^2} + \dfrac{{{{\text{π}} ^2}{k^2}z}}{3}\int_0^\infty {{\kappa ^3}{\varPhi _n}\left( \kappa \right)} {\rm{d}}\kappa } \end{split} $$ (15) Now the spectral intensity
$I$ of the non-uniformly Sinc-correlated blue-green laser beams has the following form at the point$\left( {r,z} \right)$ $$I\left( {r,z} \right) = W\left( {r,r,z} \right) = \frac{{{k^2}}}{{4{{\text{π}} ^2}{z^2}}}\int {p\left( v \right){{\left| {H\left( {r,z,v} \right)} \right|}^2}{\rm{d}}v} $$ (16) Oceanic turbulence is different from atmosphere turbulence, and the refractive index fluctuation of seawater is caused by the both change of temperature and salinity[2]. Assuming that the oceanic turbulence is isotropic and uniform, the absorption and scattering effects of seawater on laser beams are ignored, at this time the oceanic turbulence spectrum is given by[26-27]
$$ {\varPhi _n}\left( \kappa \right)= 0.388 \times {10^{ - 8}}{\varepsilon ^{ - 1/3}}{\kappa ^{ - 11/3}}\left[ {1 + 2.35{{\left( {\kappa \eta } \right)}^{2/3}}} \right]f\left( {\kappa ,w,{\lambda _T}} \right) $$ (17) where
$\eta = {10^{ - 3}}\;{\rm{m}}$ is the Kolmogorov internal scale and$\varepsilon $ is the rate of dissipation of kinetic energy per unit mass of fluid ranging from${10^{ - {\rm{10}}}}\;{{\rm{m}}^2} \cdot {{\rm{s}}^{ - 3}}$ to${10^{ - {\rm{4}}}}\;{{\rm{m}}^2} \cdot {{\rm{s}}^{ - 3}}$ , and$f\left( {\kappa ,w,{\lambda _T}} \right)$ has the form$$f\left( {\kappa ,w,{\lambda _T}} \right) = \frac{{{\lambda _T}}}{{{w^2}}}\left( {{w^2}{e^{ - {A_T}\delta }} + {e^{ - {A_S}\delta }} - 2w{e^{ - {A_{TS}}\delta }}} \right)$$ (18) ${\lambda _T}$ is the rate of dissipation of mean-square temperature, which varies from oceanic surface to deep water layer is${10^{{\rm{ - 10}}}}\;{{\rm{K}}^2} \cdot {{\rm{s}}^{ - 1}}$ to${\rm{1}}{{\rm{0}}^{ - 2}}\;{{\rm{K}}^2} \cdot {{\rm{s}}^{ - 1}}$ .$w$ indicates the relative strength of temperature and salinity fluctuations, and it describes the contribution of both to the change in oceanic power spectrum. In the oceanic medium, the value of$w$ ranges from −5 to 0, when$w = - 5$ , the oceanic turbulence is caused by temperature-induced, and$w = 0$ gives the oceanic turbulence of salinity-induced. The other parameters are evaluated as${A_T} = 1.863 \times {10^{ - 2}}$ ,${A_S} = 1.9 \times {10^{ - 4}}$ ,${A_{TS}} = 9.41 \times {10^{ - 3}}$ , δ = 8.284(κη)4/3 + 12.978(κη)2.
Propagation characteristics of non-uniformly Sinc-correlated blue-green laser beam through oceanic turbulence
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摘要: 根据广义惠更斯-菲涅尔原理,建立了非均匀辛格相关蓝绿激光波束在海洋湍流中的传输模型。基于交叉谱密度函数,讨论了不同传播距离下波束光强变化。数值计算了波束光强和光强最大值横向偏移受海洋湍流参数的影响。结果表明,传播距离和海洋湍流参数对非均匀辛格相关蓝绿激光波束的光强自聚焦现象有一定影响。当传播距离一定时,温度均方耗散率对光强自聚焦的影响大于湍流动能耗散率和温度盐度波动相对强度。Abstract: The propagation model of non-uniformly Sinc-correlated blue-green laser beam in oceanic turbulence was developed according to generalized Huygens-Fresnel principles. Based on the cross-spectral density, intensity variations in different propagation distances were discussed. When the oceanic turbulence parameters were varied, intensity and lateral shifted intensity maximum were numerically simulated. The results show that the propagation distance and ocean turbulence parameters have a certain influence on the intensity self-focusing effect of the non-uniformly Sinc-correlated blue-green laser beam. When the propagation distance is certain, the effect of the rate of dissipation of mean-square temperature on the intensity self-focusing is greater than the rate of dissipation of kinetic energy and the relative strength of temperature and salinity fluctuations.
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