With the rapid development of technologies for underwater communication, imaging and sensing, it has been become ever more important to deeply understand how oceanic turbulence affect the propagation of laser beams^[1]. In recent years, the effects of oceanic turbulence on the intensity^[2], polarization^[3], scintillation^[4], beam spreading^[5] and other characteristics of laser beams have been studied^[6-7]. However, most of researches on laser beams are assumed that the field source with so-called Gaussian Schell-model (GSM) beams correlations, where the degree of coherence (DOC) is uniformly distributed and independent of the position of the two points^[8]. Gori et al.^[9] investigated the sufficient conditions for the cross-spectral density (CSD) matrix to satisfy the non-negative definition, and the result was shown that laser beams could expression some special non-uniformly correlation. Subsequently, scholars have been studied on the scattering and propagating characteristics of the non-uniformly correlated beams, the results demonstrated that non-uniformly correlated beams have the self-focusing effect^[10], lateral shifted intensity maximum^[11] and lower scintillation^[12]. Lajunen and Saastamoinen^[8] found that the partially coherent beams with spatially varying correlations have the lateral shifted intensity maximum during propagation. Tong and Korotkova^[13]explored the phenomenon that the off-axis intensity maximum of the non-uniformly correlated scalar beams were inhibited in isotropic atmosphere turbulence. At the same time, some scholars had studied the generation and propagation characteristics of non-uniformly correlated laser beams by experiments^[14–16]. Mei^[17]proposed and studied uniformly Sinc-correlated beams, and the result was shown that it had flat profiles in the far field. In the next year, Mei and Korotkova et al.^[18] proposed the concept of non-uniformly Sinc-correlated beams and found that this beam had the self-focusing and lateral shifted intensity maximum effects in free space. Studies have shown that blue-green lasers with wavelengths between 450-580 nm have strong penetrability and less propagation loss in seawater, so it is more suitable for propagation in seawater^[19]. Therefore, the study of the propagation characteristics of the non-uniformly sinc-correlated blue-green laser beam in oceanic turbulence is of great significance.

In this paper, we provide the theoretical analysis in section 2 for the propagation model of non-uniformly sinc-correlated blue-green laser beams in oceanic turbulence. In Section 3, the influence of oceanic turbulence parameters and propagation distances on the intensity and lateral shifted intensity maximum of the non-uniformly sinc-correlated blue-green laser beam are numerically discussed. It gives the conclusions in Section 4.

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]

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]

$f\left( {\bf{\rho }} \right)$ is an arbitrary function. At this time, the CSD of Eq. (2) can be expressed as^[21]

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

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]

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

$\mu \left( {{{\bf{\rho }}_{\bf{1}}},{{\bf{\rho }}_{\bf{2}}}} \right)$ 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

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]

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]

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

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

where

Now let ${{\bf{\rho '}}_{\bf{1}}} = {{\bf{\rho '}}_{\bf{2}}} = r$, Eq. (13) satisfies the following form

where

Now the spectral intensity $I$ of the non-uniformly Sinc-correlated blue-green laser beams has the following form at the point $\left( {r,z} \right)$

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]

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

${\lambda _T}$ 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.

We plot the intensity variations and the lateral shifted intensity maximum of non-uniformly sinc-correlated laser beams propagating through free space and oceanic turbulence according to Eqs. (12)−(18) as $\sigma = 1\;{\rm{mm}}$, $\lambda = 532\;{\rm{nm}}$. ${r_{\max }}$ is the x-coordinate corresponding to the peak of the curve.

Figure 2(a) shows the intensity variation of the non-uniformly sinc-correlated laser beam propagating through free space. Figure 2(b) illustrates evolution of intensity on the $r - z$plane. It can be seen that the non-uniformly Sinc-correlated blue-green laser beams has the self-focusing effect and lateral shifted intensity maximum, and the maximum intensity is not taken at the center of the laser beams. The intensity obeys a non-uniformly distribution and no longer satisfies the symmetry.

Figure 3(a) demonstrates the intensity of a non-uniformly Sinc-correlated blue-green laser beams propagating through oceanic turbulence. The intensity evolution in the $r - z$ plane is given in Fig. 3(b). It is shown that the oceanic turbulence significantly suppresses the intensity of the non-uniformly Sinc-correlated blue-green laser beams.

In Fig.4, we plot the intensity variation of non-uniformly Sinc-correlated laser beams in free space and oceanic turbulence with $r$ when the propagation distances z = 0 m, 40 m, 80 m, 120 m. The corresponding parameters are as follows ${\lambda _T} = {10^{ - 8}}\;{{\rm{K}}^2} \cdot {{\rm{s}}^{ - 1}}$, $\varepsilon = {10^{ - 7}}\;{{\rm{m}}^2} \cdot {{\rm{s}}^{ - 3}}$, w = −2.5, η = 10⁻³ m. In Fig. 4(b), when z = 0 m, 40 m, 80 m, 120 m, the corresponding r_max = 0 mm, 2.08 mm, 2.01 mm, 1.96 mm. It can be seen that the oceanic turbulence has suppression on intensity and the lateral shifted intensity maximum. When z = 0 m, the intensity distribution is still uniformly in both free space and oceanic turbulence. As the propagation distance increases, the uniformly distribution gradually degenerates into non-uniformly distribution. Specially, the oceanic turbulence accelerates this degradation. This characterizes is similar to the propagation of non-uniform partially coherent laser beams in free space^[28].

Figures. 5 and 6 show the ${\lambda _{{T}}}$ and $w$ on the intensity of the non-uniformly sinc-correlated blue-green laser beams at a certain propagation distance. The other parameters are taken as $\varepsilon = {10^{ - 7}}\;{{\rm{m}}^2} \cdot {{\rm{s}}^{ - 3}}$, $w = - 2.5$, $\eta = {10^{ - 3}}\;{\rm{m}}$ in Fig. 5, and $\varepsilon = {10^{ - 7}}{{\rm{m}}^2} \cdot {{\rm{s}}^{ - 3}}$, ${\lambda _T} = {10^{ - 8}}{{\rm{K}}^2} \cdot {{\rm{s}}^{ - 1}}$, $\eta = {10^{ - 3}}{\rm{m}}$ in Fig. 6. In Fig. 5, when ${\lambda _T} = {10^{ - 7}},{10^{ - 8}},{10^{ - 9}}$, the corresponding (a) r_max = 0.26 mm, 0.57 mm, 0.69 mm, (b) r_max = 0.43 mm, 0.65 mm, 0.7 mm.

In Fig. 6, when $w = - 1, - 2.5, - 4$, the corresponding (a) r_max = 0.4 mm, 0.57 mm, 0.63 mm, (b) r_max = 0.57 mm, 0.65 mm, 0.68 mm.

It can be concluded from Figs. 5 and 6 that when the propagation distance $z = 35\;{\rm{m}}$, with the increase of ${\lambda _{{T}}}$ and $w$, the intensity is suppressed, and the lateral shifted intensity maximum is also reduced. This is similar to the propagation characteristics of a non-uniformly laser beams in oceanic turbulence^[29]. When the propagation distance $z = 85\;{\rm{m}}$, the intensity is still suppressed with the increase of ${\lambda _{\rm{T}}}$ and $w$. In the same time, it is found that the suppression is stronger. The lateral shifted intensity maximum is less affected because the oceanic turbulence accelerates the degradation of the non-uniformly Sinc-correlated blue-green laser beams. When the propagation distance is large enough, the intensity distribution degenerates from a non-uniformly distribution to uniformly distribution.

When propagation distances $z = 35\;{\rm{m }}$ and $z = 85\;{\rm{m }}$, we plot the curve that the effect of $\varepsilon $ on intensity of non-uniformly Sinc-correlated blue-green laser beams in Fig. 7. The other parameters are taken as ${\lambda _T} = {10^{ - 8}}\;{{\rm{K}}^2} \cdot {{\rm{s}}^{ - 1}}$, $w = - 2.5$, $\eta = {10^{ - 3}}\;{\rm{m}}$. When $\varepsilon = {10^{ - 5}},{10^{ - 7}},{10^{ - 9}}$, the corresponding (a) r_max = 0.67 mm, 0.57 mm, 0.39 mm, (b) r_max = 0.69 mm, 0.65 mm, 0.56 mm. When the propagation distance $z = 35\;{\rm{m}}$, it is indicated that the lateral shifted intensity maximum and intensity increases with the increase of $\varepsilon $. When the propagation distance $z = 85\;{\rm{m}}$, the intensity increases with the increase of $\varepsilon $. Due to the accelerated degradation of the non-uniformly sinc-correlated blue-green laser beams in the oceanic turbulence, the lateral shifted intensity maximum changes very little. It can be seen from Figs. 5, 6 and 7 that when the propagation distance increase, ${\lambda _{{T}}}$ has greater influence on intensity than $w$ and $\varepsilon $. As can be seen from Figs. 5, 6 and 7, when ocean turbulence parameters ${\lambda _{{T}}}$, $w$ and $\varepsilon $ are constant, ${r_{\max }}$ increases with the increase of propagation distance.

In summary, the propagation characteristics of non-uniformly Sinc-correlated blue-green laser beams are discussed in oceanic turbulence. When the root-mean-square width is different, we explore the cross-spectral density variations. The results are shown that the oceanic turbulence can suppress the lateral shifted intensity maximum and intensity of the non-uniformly Sinc-correlated blue-green laser beams, and accelerate the degradation of this beam, and affect the beam’s intensity self-focusing. Considering the propagation distance $z = 35\;{\rm{m}}$, the lateral shifted intensity maximum and intensity decrease with the increase of ${\lambda _T}$ and $w$, and increase with the increases of $\varepsilon $, the results of the propagation distance $z = 85\;{\rm{m}}$ are similar to $z = 35\;{\rm{m}}$.

As the degradation of the non-uniformly Sinc-correlated laser beams is accelerated by the effect of oceanic turbulence, the influences of oceanic turbulence parameters ${\lambda _T}$, $w$ and $\varepsilon $ on the lateral shifted intensity maximum is relatively small when z = 85 m. In addition, with the propagation distance increases, the influence of ${\lambda _T}$ on the intensity is greatest.

Our works provide a theoretical basis for the experimental research of the blue-green laser under the background of oceanic turbulence, and further study of the underwater communication and detection research.