Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean

1. Introduction

In recent years, optical beams with orbital angular momentum (OAM) propagating in random media have attracted much attention due to the applications in optical wireless communication. The main motivation is that beams carrying OAM can realize arbitrary base-N quantum digits in principle [1,2], which offers the additional degrees of freedom for information coding and increases information capacity of the communication link. Due to the presence of various constituents in the ocean and the fluctuations of the refractive index of water, the propagation of optical signal through an oceanic medium will suffer attenuation and wavefront distortion [3,4]. It is well known that oceanic turbulence will distort OAM modes of optical beams, which introduces intermodal crosstalk between channels. After Nikishov et al. [5] presented the power spectrum of oceanic turbulence, there are great attentions paid to understand the laws of OAM modes propagation in the oceanic turbulence [6–10]. Huang et al. [9] analyzed the influence of beam parameters, the rate of dissipation of turbulent and temperature salinity fluctuations on the average intensity of Gaussian Schell-model vortex beam propagating through oceanic turbulence, and showed that the partially coherent vortex beams have robust turbulent resistance. Liu et al. [10] studied the evolution properties of partially coherent flat-topped vortex hollow beams propagating through oceanic turbulence, and revealed that this beam evolves into the Gaussian-like beam for the long distance propagation and this beam with higher order M will lose its initial dark hollow center slower. The effects of oceanic turbulence and light source parameters on the received probability of partially coherent Laguerre–Gaussian (LG) vortex beam was investigated [6], and showed that the partially coherent vortex beam provides worse performance than fully coherent vortex beam. The channel capacity of Bessel–Gauss Beams propagating through turbulent ocean was investigated [7], which revealed that BG beams are better than LG beams to mitigate the effects of oceanic turbulence because of the non-diffraction and self-healing characteristics of BG beams.

The experimental and theoretical results demonstrated that turbulence can be anisotropic [11–17]. Due to the earth's rotation, the turbulence of ocean will also be anisotropic [18,19]. The average polarizability of Gaussian Schell-model quantized beams and the spreading and wandering of Gaussian–Schell model laser beams in anisotropic oceanic turbulence were studied respectively [18,19]. However, the energy distribution of OAM modes of LG beams decreases with the increasing quantum number of OAM modes, and the crosstalk between OAM modes increases with increasing the quantum number of OAM modes [2,7]. We cannot make use of the high order OAM modes of LG beams to improve the capacity of the free-space optical links. To improve capacity of the free-space optical links with the high order OAM modes, a more suitable vortex beam should be found. To the best of our knowledge, the transmission properties of Hermite-Gaussian (HG) vortex beams carrying OAM in anisotropic turbulent ocean have not been reported yet.

In this paper, we first develop a new angular spectrum of the vortex beam and derive a new HG vortex beam based on the angular spectrum theory. Then, we study the propagation of the new HG vortex beam in anisotropic turbulent ocean, and establish the model of the received probability distribution of OAM modes. In the end, we discuss the influences of the HG vortex beam parameters and the anisotropic turbulent ocean on probability density of signal OAM modes and crosstalk OAM modes in weak horizontal oceanic turbulent channels.

2. Anisotropic oceanic turbulence

In this paper, we study the HG vortex beam propagation in the turbulence of the pure seawater [20]. Due to the earth’s rotation, the turbulence of the ocean will be anisotropic [11]. For the sake of simplicity, here we employ the method that the anisotropic power spectrum has been proposed in atmospheric turbulence [15–17] and consider the turbulent anisotropy only existing along the propagation direction of the beam as the discussion of Ref [18]. We can present the anisotropic power spectrum of oceanic turbulence by an effective anisotropic factor $\zeta$. The anisotropic spectrum of oceanic turbulence can be expressed as

3. Propagation of HG vortex beams in the absence of turbulence

The propagation of a monochromatic scalar light field $u\left(x,y,z;\omega \right)$ in free space is represented by the angular spectrum for the half-space $z>0$ [21,22]:

For the lowest-order Gaussian beam field, the product of the divergence $\Delta \theta \left(z;\omega \right)$ and transverse spread $\Delta \omega \left(z;\omega \right)$ reaches a minimum and can be represented by [22]:

The expressions of the divergence and transverse spread for the $x-y$ symmetry can be defined as

The product of the divergence and transverse spread in $x$ or $y$ direction meets the minimum respectively. In $x$ direction, the product can be expressed by [22]:

Applying the Cauchy-Schwartz inequality

Equation (7) can reach its minimum in the condition that $k_x^2{\left|a\left(k_x,k_y;\omega \right)\right|}^2$ and ${\left|\frac{\partial a\left(k_x,k_y;\omega \right)}{\partial k_x}\right|}^2$are proportional to each other. The angular spectrum for the Gaussian beams can be represented by

It is known that the angular spectrum of the HG beam is given by [21,22]

Inserting the vortex factor ${\left(k_x-\text{i}{k}_y\right)}^m$ into the angular spectrum of the HG beam, we can obtain a new angular spectrum. We call this new angular spectrum as the angular spectrum of the HG vortex beam and express it as

Substituting Eq. (10) into Eq. (2), the propagation of the HG vortex beam in free space of absence turbulence can be investigated by modulating the angular spectrum

According to these relations

By setting the new variables${\gamma }_a=x+\text{i}y$ and ${\gamma }_b=x-\text{i}y$, the partial derivative becomes ${\partial }_x-\text{i}{\partial }_y=\partial {\gamma }_a$. Therefore, Eq. (16) can be rewritten by

With the help of the integral expression [21–23]

Neglecting the constant factor, we can obtain the HG vortex beams by using the following mathematical formulae ${\gamma }_a{\gamma }_b=x^2+y^2$ and ${\left(x+y\right)}^m={\displaystyle \sum _{u=0}^m\frac{m!}{(m-u)!u!}}{x}^{m-u}{y}^u$ as

 

Fig. 1 Intensity profiles of HG vortex beams in the absence of turbulence (a) $n=2,m=2$, (b) $n=2,m=4$,(c) $n=2,m=6$. When $z=0$ in the plane, the intensity of the HG vortex beam decreases with deviating from the optical axis center. In addition, the intensity forms annular shapes along the optical axis center with increasing quantum number $m$.

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In particular, when $m=0$, the new HG vortex beam can degrade into HG beam as

Moreover, let $n=0$ and $r^2=x^2+y^2$, the new HG vortex beam can be reduced as ${\text{LG}}_m^0$ beam [24] and can be given by

4. Probability density of OAM modes in oceanic turbulence

The HG vortex beam $u\left(r,\phi ,z\right)$ is a superposition of eigenstates and the conditional probability of obtaining a measurement of the OAM of a photon$l_z=l\hslash$ is obtained by summing the probabilities associated with that eigenvalue [2],

The aberrations are random, so by taking the ensemble average, the probability that a measurement of the OAM will yield a value $l_z=l\hslash$ is found as follows,

By the superposition theory of plane waves with phase $\exp \left(\text{i}l\phi \right)$, the $a_l\left(r,z\right)$ can be written as

The complex amplitude of the HG vortex beam is given by [25]

The ensemble average of the superposition coefficients $〈a_l(r,z)a_l^{\ast }(r,z)〉$ is represented as

Substituting Eq. (28) into Eq. (24), Eq. (24) can be rewritten as

Since the aberrations introduced by oceanic turbulence are random variables, the coherence function of the phase can be expressed as

Make use of the relations ${\left|\left(r_1-r_2\right)\right|}^{5/3}={\left\{{\left(r_1-r_2\right)}^2\right\}}^{5/6}$and ${\left(r_1-r_2\right)}^2=2r^2\left[1-\cos \left(\phi -{\phi }^{\prime }\right)\right]$, the coherence function of the phase ${〈\exp \left[\text{i}\psi \left(r,\phi \right)-\text{i}{\psi }^{*}\left(r,{\phi }^{\prime }\right)\right]〉}_{at}$ is given by

In Markov approximation, i.e. ${\kappa }^{\prime }=\zeta {\kappa }_{\rho }$, the spatial coherence radius ${\rho }_{oc\varsigma }$ can be written as

After a long but straightforward calculation [4], we can obtain the lateral coherence length of the spherical wave in anisotropic oceanic turbulence. The Eq. (34) is reduced to

Based on the Eq. (30) and Eq. (33), we can obtain the received probability [16]

For $l=m$, $P\left(l|m\right)=P\left(m\right)$ is the received probability of transmitting OAM modes $m$. In addition, when $l\ne m$, $P\left(l|m\right)=P\left(l\right)$ is the received probability of transmitting OAM modes with index $m$ followed by detection of OAM mode l.

5. Numerical analysis

We now investigate the statistical properties of OAM modes of the HG vortex beam propagating through oceanic turbulence by numerical simulation. In the following analysis, the calculation parameters of HG vortex beams are $\lambda =0.4173\mu m$, $\eta =1\mathrm{mm}$, $w_0=1\mathrm{cm}$, $\varpi =-0.5$, $\epsilon ={10}^{-6}$, $z=150m$, ${\chi }_T={10}^{-8}$, $\zeta =2$, $m=2$, $l=2$, $n=2$, unless other variable parameters are specified in calculation.

Figure 2 displays the received probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence for several values of parameters $n$ and $m$. It can be seen that the received probability of signal OAM modes decreases monotonously with increasing propagation distance. Figure 2(a) indicates that the received probability of signal OAM modes of the ${\mathrm{LG}}_2^0$ beam is smaller than the HG vortex beam, when the transmission distance is greater. It means that the HG vortex beam has a stronger resistance to the oceanic turbulence than the ${\mathrm{LG}}_2^0$ beam, and the resistance increases with the increase of transmission distance. In addition, from Fig. 2(a-b), the received probability of signal OAM modes decreases with increasing OAM mode number $m$ for a given propagation distance. Therefore, the HG vortex beam with smaller $m$ is less affected by oceanic turbulence than the ${\mathrm{LG}}_m^0$ beam from the aspect of the signal OAM probability in long distance transmission.

 

Fig. 2 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence for several values of parameter $n$ (a) $m=2$, (b) $m=4$ respectively.

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The discussion of Fig. 2 is related to the received probability of signal OAM modes. The intermodel crosstalk between channels can be caused by optical OAM beams propagation through oceanic turbulence, which represents the OAM energy level difference $\Delta l=\left|l-m\right|\ne 0$. Figure 3 displays the difference between the HG vortex beams and the conventional LG beam propagation through oceanic turbulence causing the received probability of crosstalk OAM modes. We find that the received probability of crosstalk OAM modes of the conventional LG beam is obviously larger than that of the HG vortex beam. That is to say, the effects of oceanic turbulence on the HG vortex beam compared to the conventional LG beam are significantly less. The results will be very useful for underwater optical communications.

 

Fig. 3 Probability of crosstalk OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance $z$ for different modes $\Delta l$.

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For the HG vortex beam propagation in oceanic turbulence, the received probability of signal OAM modes will be improved in anisotropic turbulence in Fig. 4. Figure 4 also shows that the received probability of signal OAM modes of the HG vortex beam as functions of propagation distance $z$ for different anisotropic factor $\zeta$. For a given value of propagation distance $z$, when the anisotropic factor $\zeta$ increases, the signal probability of HG vortex beams increases.

 

Fig. 4 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance $z$ for different anisotropic factor $\zeta$.

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To comprehend the regularity of the received probability of signal OAM modes of the HG vortex beam, we calculate the influence of wavelength $\lambda$ for different inner scale $\eta$ in Fig. 5. We deduce from Fig. 5 that, for any given inner scale $\eta$, there is an increase of the received probability of signal OAM modes with increasing $\lambda$. The result shows that the HG vortex beam with longer wavelength is less affected by oceanic turbulence. In addition, Fig. 5 also shows that the received probability of signal OAM modes increases with the increase of $\eta$ for any given $\lambda$. The phenomenon can be explained that the larger inner scale of turbulent eddy results in small light scattering, and then the small light scattering produces a small energy loss. Thus, the received probability of signal OAM modes of the HG vortex beam can be improved under the greater inner scale of oceanic turbulence.

 

Fig. 5 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus wavelength $\lambda$ for different inner scale $\eta$.

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We also analyze the influence of the dissipation rate of temperature variance ${\chi }_t$ and the source’s transverse size $w_0$ on the received probability of signal OAM modes for the HG vortex beam. From Fig. 6, we can see that, as the source’s transverse size $w_0$ increases, the received probability of signal OAM modes of the HG vortex beam increases. This is caused by the larger source’s transverse size to make a smaller beam spread and beam wander which are caused by the diffraction and refringence of the vacuum and oceanic turbulence, respectively. The smaller beam spread and beam wander lead to the smaller transformations of the space distribution of beams propagation in the link. In addition, as would be expected, the received probability of signal OAM modes of the HG vortex beam decreases with increasing dissipation rate of temperature variance ${\chi }_t$ from Fig. 6, especially in the larger source’s transverse size $w_0$.

 

Fig. 6 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the dissipation rate of temperature variance ${\chi }_t$_{for different the source’s transverse size $w_0$.}

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To further comprehend the influence of oceanic turbulence parameters, in particular the rate of dissipation of kinetic energy per unit mass of fluid $\epsilon$ and the ratio of temperature and salinity contributions to the refractive index spectrum $\varpi$ on the received probability of the signal OAM mode of the HG vortex beam. Figure 7 shows that the received probability of the signal OAM mode is a function of $\epsilon$ for different $\varpi$. It is evident from Fig. 7 that the larger value of $\varpi$ will obtain the larger received probability of signal OAM modes. This implies that the salinity fluctuations dominated turbulence affects the HG vortex beam propagation more than the temperature fluctuations dominated turbulence. Figure 7 also shows that the larger rate of dissipation of kinetic energy per unit mass of fluid enhances the detection probability of the signal OAM mode of the HG vortex beam. This conclusion comes from the larger rate of dissipation of kinetic energy per unit mass of fluid reducing oceanic turbulence.

 

Fig. 7 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the rate of dissipation of kinetic energy per unit mass of fluid $\epsilon$ for different the ratio of temperature and salinity contributions to the refractive index spectrum $\varpi$.

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6. Conclusion

We develop a new angular spectrum of the vortex beam based on the angular spectrum of the HG beam and the vortex factor ${\left(k_x-\text{i}{k}_y\right)}^m$, which evolves a new HG vortex beam. Then we establish the model of the received probabilities of OAM modes of the HG vortex beam propagation through anisotropic oceanic turbulence. The influence of the HG vortex beam on the received probabilities of signal and crosstalk OAM modes shows that the HG vortex beam is superior to the LG beam propagation in anisotropic oceanic turbulence for long-distance transmission. We also find that the HG vortex beam has robust turbulent resistance under the condition of anisotropic oceanic turbulence, longer $\lambda$, $\eta$, $\varpi$, $\epsilon$, $z$, $w_0$ or smaller ${\chi }_T$. The HG vortex beam can realize the goal for enhancing the link capacity of underwater optical communication.