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Average capacity of an underwater wireless communication link with the quasi-Airy hypergeometric-Gaussian vortex beam based on a modified channel model

Hang Chen, Peng Zhang, Shuang He, Hui Dai, Yunlong Fan, Yuanxin Wang, and Shoufeng Tong

Open Access Open Access

Abstract

Prompted by alleviating the random perturbation of underwater channel and enhancing the performance for the orbital angular momentum (OAM) -based underwater wireless optical communication (UWOC), the quasi-Airy Hypergeometric-Gaussian (QAHyGG) vortex beam is first proposed and demonstrated. Moreover, an underwater channel model is first modified for more accurate simulated results of the propagation property of various beams. Based on the modified model, the transmission and communication performance of three different OAM-carrying beams (the Gauss vortex (GV) beam, the Hypergeometric-Gaussian (HyGG) vortex beam, and the QAHyGG vortex beam) are comparatively studied. In addition, the parameters optimization of the QAHyGG vortex beam is made for further enhancing the average capacity. The results show that the QAHyGG vortex beam exhibits higher received power and lower crosstalk probability under different channel conditions. The average capacity of the QAHyGG vortex beam has enhanced by ∼8% and ∼27% compared with the HyGG vortex beam and the GV beam at 100m, respectively. The QAHyGG vortex beam is more suitable in an OAM-based UWOC system with a limited-size receiving aperture or lower transmit power. Besides, the average capacity will improve effectively at longer distances with the optimized beam parameters. These research results can provide advances in designing the practical OAM-based UWOC system.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, the UWOC system that uses OAM information carriers has attracted extensive attention. This is because OAM modes can form an infinite N qubit basis and provide a new type of communication link coding scheme with high security, wide bandwidth, and lower-cost [1–8]. However, since the OAM mode is related to the wave-front spatial structure of the carrier, ocean turbulence, absorption, and scattering can inevitably cause the distortion of the OAM mode and crosstalk between the energy states of the OAM mode [9–15]. To reduce the perturbation caused by seawater, scholars have mainly focused on the Gaussian Schell model vortex beam [9], perfect Laguerre-Gauss beam [10], Hermite-Gauss vortex beam [11], and some quasi-nondiffractive beams, such as the Bessel–Gauss (BG) vortex beam [12], Airy vortex beam [13], Lommel–Gauss beam [14], and HyGG vortex beam [15]. The application of novel beams to enhance channel capacity has become a new trend in the OAM-based UWOC field.

Herein, some novel beams have attracted an upsurge of interest in OAM-based optical communication due to their abrupt autofocusing property [14,15,20,22,33]. This property can effectively depress the beam spreading in the turbulent medium, thus reducing the optical power loss and channel crosstalk at the receiver and capturing higher-order OAM beams [22]. As a member of the autofocusing beams family, the HyGG vortex beam also possesses higher OAM transferring efficiency than that of the BG vortex beam and self-reconstruction after an obstacle [15–19,21]. Furthermore, it can be reduced to various beam modes for some special values of beam parameters [16], which exhibits great potential applications in wireless optical communication. However, the autofocusing ability of the HyGG vortex beam is weaker than the ring Airy vortex beam (RAVB) due to the limit of its own phase structure. Particularly, a quasi-ring Airy vortex beam (QRAVB) [22] was successfully generated in experiments via controlling the radial phase distribution of the GV beam. The experiment proved that QRAVB has stronger anti-turbulence and anti-attenuation abilities than the GV or BG vortex beam due to its stronger autofocusing ability. Inspired by this simple but efficient strategy (controlling the radial phase distribution), we intend to add an Airy-type phase onto the HyGG vortex beam to create a novel autofocusing beam (the QAHyGG vortex beam). The QAHyGG vortex beam may possess more robust anti-turbulence and anti-attenuation abilities than the HyGG vortex beam or the QRAVB in an oceanic channel.

To verify the QAHyGG vortex beam possesses robust propagation through an oceanic channel, an effective OAM-based UWOC model is necessary to be established. Recently, considerable efforts have been put into establishing the UWOC channel model [23–31]. However, these developed models mainly focused attention on the propagation of collimated beams. Concurrently, [32] indicated that the coordinate transformation is necessary for propagating diverging or converging beams. Inspired by [32,33] simulated the propagation of autofocusing beams in atmospheric turbulence via combining the method of linear non-adaptive coordinate transformation (LNCT) and subharmonic compensation technique [38]. Whereas the transmission trajectory of autofocusing beams is always nonlinearly varied, and the phase structure function (PSF) of phase screens will change with the variation of screen size [23]. This model may take some extra errors in the simulation of beams’ propagation in a turbulence channel. Hence, we propose a method of non-linear adaptive coordinate transformation (NACT) and employ the PSF-based extrusion technique [39] to establish a model for the OAM-based UWOC channel. The modified model can automatically adjust the grid spacing of transmission planes to accommodate the changing beam spreading during split-step propagation. In addition, the PSF of phase screens is independent of the grid spacing. To the best of our knowledge, the QAHyGG vortex beam and the modified UWOC model are proposed for the first time. Meanwhile, the QAHyGG vortex beam established methods, its transmission and communication performance in the underwater channel, and the best selection of its parameters are explored for the first time by us.

In this paper, by utilizing the modified UWOC model, we demonstrate that the transmission and communication performance of the QAHyGG vortex beam is better than that of the GV and HyGG vortex beams. The proper selection of the QAHyGG vortex beam parameters is also discussed in detail. In section 2, we proposed the established principle and the structure of the QAHyGG vortex beam and the modified model for the OAM-based UWOC channel. In section 3, we first verify the effectiveness of the modified model for the OAM-based UWOC channel. Then, the evolution of average intensity distribution and spiral spectrum for three different OAM-carrying beams with various transmission distances is demonstrated and compared. Moreover, the influence of transmission distance, channel conditions, UWOC system parameters, and beam parameters on the average capacity of the QAHyGG vortex beam is investigated in detail. Finally, a conclusion is given in section 4.

2. Beam-established principle and the modified channel model

2.1 Established principle and the structure of the QAHyGG vortex beam

By adding an Airy-type radial phase (proportional to r3/2 [22]) distribution onto the HyGG vortex beam at the original plane [15–19], we can get the electric field distribution of the QAHyGG beam in the source plane (z = 0), which can be expressed in cylindrical coordinates as below:

$$E({\boldsymbol r} )= {E_0}{C_{pl}}{\left( {\frac{r}{{{\omega_0}}}} \right)^{p + |l |}}\textrm{exp} \left( {\frac{{ - {r^2}}}{{{\omega_0}^2}} + il\theta } \right)\textrm{exp} ({i{p_0}k{r^{3/2}}} ), $$
where E0 denotes the on-axis amplitude of the light source, p is the hollowness parameter of the QAHyGG beam, and l corresponds to the number of OAM. |r|=r, r = (x,y) = (r,θ) is the two-dimensional position vector in the source plane. θ denotes the azimuthal angle and ω0 is the beam waist. The normalized constant Cpl is given by ${C_{pl}} = \sqrt {{2^{p + |l |+ 1}}/[{\pi \Gamma ({p + |l |+ 1} )} ]} $, which ensures that the QAHyGG vortex beam carries a finite power as long as p ≥ -|l|. Γ(x) is the Gamma function. p0 is the autofocusing factor, one can control the divergence of the QAHyGG beam, and k is the wave number of light related to the wavelength λ in vacuum by k = 2π/λ. ${\left( {\frac{r}{{{\omega_0}}}} \right)^{p + |l |}}\textrm{exp} \left( {\frac{{ - {r^2}}}{{{\omega_0}^2}}} \right)$ is a Gaussian-parabolic function of the order of p+|l| [16], which modulates the intensity distribution of the QAHyGG beam in the source plane. exp(-ilθ) is a singular phase factor, which shapes the phase front to be helical. exp(-ip0kr3/2) is an Airy-type phase factor used to modulate the radial phase distribution. When p0 = 0, the QAHyGG vortex beam degrades into the HyGG vortex beam; when p0≠0, p = -|l|, the QAHyGG vortex beam degrades into the QRAVB.

To theoretically demonstrate the transmission of the QAHyGG vortex beam in free space, the intensity and phase distribution at an arbitrary distance z can be determined through the Fresnel diffraction integral [22,32].

Figure 1 exhibits the propagation property of the QAHyGG vortex beam in free space. The related intensity patterns and phase distributions at the locations of white dash lines are shown in Fig. 1(b1)-(b4) and Fig. 1(c1)-(c4), respectively. It is quite clear that the beam sharply auto-focuses after a certain distance of propagation. Before the intensity increases sharply, the radius of the QAHyGG vortex beam decreases as the propagation distance increases. After autofocusing occurs, we can see that the intensity of the profile still keeps the dark hollow ring, and a dark hollow channel forms because of the vortex on the axis along the propagation. The focal distance is about z = 24m in the simulation.

 figure: Fig. 1.

Fig. 1. Numerical demonstrations of the QAHyGG vortex beam (λ=450nm, p = 2, l = 1, w0 = 2mm, p0 = 8 × 10−4) propagating in free space. (a) numerically simulated side-view propagation of the QAHyGG vortex beam; (b1)-(b4) snapshots of the transverse intensity patterns taken at the planes marked by the dashed lines in (a); (c1)-(c4) the corresponding phase distributions at different planes marked in (a).

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The white dash lines in Figs. 2 and 3 denote the positions of the focal point. Figure 2 displays the propagation dynamics of the QAHyGG vortex beam with different autofocusing factors p0 and beam waists ω0 in free space. It can be observed that the propagation trajectory of the QAHyGG vortex beam can be controlled by ω0 and p0. Furthermore, due to the modulation of the Airy-type phase factor, the propagation dynamics of the QAHyGG vortex beam seem more similar to the performance of ring Airy vortex beam [33] as increasing ω0 and p0. In addition, the autofocusing position of the QAHyGG vortex beam moves away from the source plane as ω0 increases or p0 decreases. Thus, a higher value of ω0 and a smaller value of p0 can effectively suppress the beam spreading for a longer distance.

 figure: Fig. 2.

Fig. 2. Propagation dynamics of the QAHyGG vortex beam (λ=450nm, p = 2, l = 1) with different autofocusing factors p0 and beam waists ω0 in xz plane.

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 figure: Fig. 3.

Fig. 3. Propagation dynamics of the QAHyGG vortex beam (λ=450nm, ω0 = 2mm, p0 = 8 × 10−4) with different hollowness parameters p and OAM numbers l in xz plane.

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In Fig. 3, we examine the propagation dynamics of the QAHyGG vortex beam with different hollowness parameters p and OAM numbers l. It can be observed that the dark hollowness of the QAHyGG vortex beam, at the source plane, enlarges with the increasing value of p or l due to the modulation of the Gaussian-parabolic function in Eq. (1). Moreover, the autofocusing position will drift away from the source plane as p increases. Meanwhile, the intensity distribution is more centralized with an increasing value of p at 100m. Compared with p, the value of l has less influence on the autofocusing positions. In addition, the radius of the dark hollow channel will increase as increasing l. Hence, a higher value of p can effective suppress the beam spreading for a longer distance.

2.2 Modified model for the OAM-based UWOC channel

As shown in Fig. 4, the transmission process can be separated into refraction and diffraction parts [35]. The turbulence effect is treated as thin phase screens that only change the light propagation directions [36]. Then, all the lights keep their directions until reaching the next phase screen, with the light intensity attenuating due to absorption and scattering effects along the path [37]. In the end, the spatial light is captured by a probe. In comparison with other UWOC models [23–31], our model’s grid spacing of every transmission plane can adjust size automatically, according to the variation of the beams’ effective radius in free space.

 figure: Fig. 4.

Fig. 4. Schematic description of the modified model for OAM-based UWOC channel.

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Figure 5 illustrates the simulation scheme of beams’ propagation in our modified model. The whole simulation procedure can be separated into three different but interrelated parts. First, we use the NACT to obtain the grid spacing of transmission planes. Then, by utilizing the grid spacing parameters, sample points, and other initial parameters, the random phase screens are produced via the PSF-based extrusion technique [39]. After that, every phase screen is fixed with the same spacing Δz in the transmission path of beams. Finally, we introduce the attenuation operator [23] and use the split-step propagation technique [32] to simulate the transmission of beams in an oceanic channel.

 figure: Fig. 5.

Fig. 5. Propagation procedure flow chart used in the modified model. (Left) Coordinate Transformation Procedure (Middle) Phase Screens Generation Procedure (Right) Beam Propagation Procedure.

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Based on the above arguments, the change of field induced by the modified model is described as:

$$\scalebox{0.9}{$\displaystyle E({{{\boldsymbol r}_n}} )= \prod\limits_{j = 1}^{n - 1} {\left\{ {A[{{{\boldsymbol r}_{j + 1}}} ]{F^{ - 1}}\left[ {{{\boldsymbol f}_i},\frac{{{{\boldsymbol r}_{j + 1}}}}{{{m_j}}}} \right]{Q_2}\left[ { - \frac{{\varDelta z}}{{{m_j}}},{{\boldsymbol f}_j}} \right]F[{{{\boldsymbol r}_j},{{\boldsymbol f}_j}} ]\frac{1}{{{m_j}}}Q\left[ {\frac{{1 - {m_j}}}{{\varDelta z}},{{\boldsymbol r}_j}} \right]\mathop D\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]\mathop T\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]} \right\}} \{{E({{\boldsymbol r}_1})} \},$}$$
where $E({{{\boldsymbol r}_1}} )$ is the input optical field; $\varDelta z$ is the spacing distance between the layers, n is the number of layers; $\mathop T\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]= \textrm{exp}[{\textrm{ - }i\varPhi ({\varDelta z,{{\boldsymbol r}_j}} )} ]$ is the turbulence operator. The phase $\varPhi ({\varDelta z,{{\boldsymbol r}_j}} )$ can be obtained from the PSF-based extrusion technique [39]. $\mathop D\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]= \textrm{exp}\left[ { - \sqrt {a + b} \varDelta z} \right]$ is the attenuation operator, a and b are the absorption coefficient and the scattering coefficient of seawater [37], respectively. $Q[{\bullet} ]$, $F[{\bullet} ]$, ${Q_2}[{\bullet} ]$ and ${F^{ - 1}}[{\bullet} ]$ are operator notations served as describing the Fresnel diffraction integral equation, which have the same definitions in [32]. ${m_j} = \frac{{{\delta _{j + 1}}}}{{{\delta _j}}} = \frac{{{R_{j + 1}}}}{{{R_j}}}$ is the scaling factor from plane j to plane j + 1, δj and δj + 1 are grid spacings in the j-th plane and j + 1-th plane, ${R_j} = \sqrt {\frac{{2\int {{r_j}^2I({{{\boldsymbol r}_j}} )dS} }}{{\int {I({{{\boldsymbol r}_j}} )dS} }}} $ and ${R_{j + 1}} = \sqrt {\frac{{2\int {{r_{j + 1}}^2I({{{\boldsymbol r}_{j + 1}}} )dS} }}{{\int {I({{{\boldsymbol r}_{j + 1}}} )dS} }}} $ are effective radius [34] in the j-th plane and j + 1-th plane when the vortex beam propagates in free space. S is the receiver aperture area, I(rj) and I(rj + 1) are intensity distribution in free space of the vortex beam in the j-th plane and j + 1-th plane, respectively, which can be estimated roughly via the conventional LNCT approach [33]. Consequently, we only need to preset the source plane grid spacing δ1 properly, the grid spacing of subsequent planes can be calculated by the scaling factor mj. A[rj + 1] = exp(-rj + 1/σj + 1)16 is the super-Gaussian function (SGF), and σj + 1 = 0.47Nsδj + 1 [32] is considered as the “width” of SGF, which make the optical field not attenuate in the center of the j + 1-th plane but close to zero at the edge of it. Ns is the number of sampling points on one dimension of planes.

The power spectrum of oceanic turbulence characterizes turbulence features versus the spatial frequency. In [47], a new spectrum of the refractive-index fluctuations, including the outer scale was developed for the unstable stratification ocean, and is claimed that this oceanic spectrum agrees better with the experimental data. However, it should also be noted that the PSF of the new spectrum is still under research [5]. Hence, in order to improve the simulation accuracy and compare the result with other OAM-based UWOC studies, a more widely used Nikishov spectral model is adopted as [40]:

$$\begin{aligned} \varphi ({\boldsymbol \kappa } )&= 0.388C_n^2{w^{ - 2}}{(\kappa )^{\frac{{ - 11}}{3}}}\left[ {1 + 2.35{{({\kappa \eta } )}^{\frac{2}{3}}}} \right]\textrm{ }\\ \textrm{ } &\times [{{w^2}\textrm{exp} ({ - {A_T}\delta } )+ \textrm{exp} ({ - {A_S}\delta } )- 2w\textrm{exp} ({ - {A_{TS}}\delta } )} ]\end{aligned}, $$
where |κ|=κ, κ=(κx,κy) is the two-dimensional spectrum position vector. $C_n^2 = {10^{ - 8}}{\varepsilon ^{{{ - 1} / 3}}}{\chi _T}$ is the equivalent temperature structure parameter [41] that denotes the temperature fluctuation strength of oceanic turbulence for given w and η. ɛ is the rate of dissipation of kinetic energy per unit mass of fluid changing from 10−10m2/s3 to 10−1m2/s3. χT is the rate of dissipation of mean-squared temperature changing from 10−10K2/s to 10−4K2/s. η is the Kolmogorov micro scale. w is the ratio of temperature and salinity contributions to the refractive index spectrum varying from −5 to 0 for underwater turbulence, where −5 represents the dominating temperature-induced, 0 denotes the salinity-induced oceanic turbulence. AT = 1.863 × 10−2, AS = 1.9 × 10−4, ATS = 9.41 × 10−3, and δ=8.284(ηκ)4/3 + 12.978(ηκ)2.

The theoretical expression of the underwater turbulence PSF is used as [42]:

$$\begin{aligned} D_\varPhi ^{theory} &= {\varepsilon ^{ - 1/3}}({{{\chi _{\boldsymbol T}}} / {{w^2}}}){r^{5/3}}[3.063 \times {10^{ - 7}}{k^2}\varDelta z \times (1.116{w^2} - 2.235w + 1.119)\\ \textrm{ } &- \varDelta {z^3}(1.841{w^2} - 40.341w + 2077)] \end{aligned}. $$
Furthermore, due to crosstalk between different OAM modes of the vortex beam at the receiver, the spiral spectrum can be given as [15]
$${P_l} = \frac{{{{\left|{\int {\int {M_l^\ast ({r,\phi ,z} ){E_{{l_0}}}({r,\phi ,z} )rdrd\phi } } } \right|}^2}}}{{\sum\nolimits_{m ={-} \infty }^{ + \infty } {{{\left|{\int {\int {M_m^\ast ({r,\phi ,z} ){E_{{l_0}}}({r,\phi ,z} )rdrd\phi } } } \right|}^2}} }}, $$
where Ml is the analyzing field with OAM state l, the * indicates the complex conjugate of the corresponding variable. ${E_{{l_0}}}$ is the received field with transmitted OAM state l0. For l = l0, Pl = l0 is defined as the detection probability of the signal OAM mode l0, which shows the purity of the transmitted OAM mode; for l≠l0, Pl≠l0 is defined as the crosstalk probability, which denotes the probability of a vortex beam to change its signal OAM mode.

In our simulations, we use the model from [43]. To elaborate, the OAM crosstalk distribution follows the Gaussian distribution illustrated in [44], which motives the use of signal-to-noise-and-crosstalk ratio (SNCR) instead of signal-to-noise ratio (SNR) as a figure of merit, which is defined as

$$SNCR = \frac{{{E_b}}}{{{E_c} + {N_0}}} = \frac{{{P_{Rx}} \times {P_{l = {l_0}}}}}{{{P_{Rx}} \times ({1 - {P_{l = {l_0}}}} )+ {N_0}}}, $$
where Eb is the power of signal OAM mode, N0 is the additive white Gaussian noise (AWGN) power introduced by the communication system, and Ec is the power of OAM crosstalk. PRx is the power of the received field. The OAM crosstalk is therefore included in our simulations via considering that a portion of the total noise power originates from OAM crosstalk.

For signal modulation of on-off keying (OOK), the bit-error rate (BER) of OAM channels is derived as [44]

$$BER = \frac{1}{2}erfc\sqrt {\frac{{SNCR}}{2}} , $$
where erfc(x) denotes the complementary error function.

With the assumption of an OAM-based UWOC link [10], we can approximate the OAM channel as a symmetric channel N = 2H + 1. Using the concept of information capacity of the multi-level symmetric channel [45], the average capacity of optical communication link with N symmetric OAM channels is defined as [10]

$$C = {\log _2}N + (1 - BER){\log _2}(1 - BER) + BER{\log _2}\frac{{BER}}{{N - 1}}. $$
Besides, the relation between the received OAM mode l and H is l = −H, −H + 1, ···, 0, ···, H − 1, H.

3. Numerical experiment and discussion

3.1 Modeling verification

To verify the above modified UWOC model, we respectively confirm the accuracy of the NACT method and PSF-based extrusion technique via comparing the simulation results with the theoretical predictions under the same values of parameters. The key values of simulation parameters are given in Tables 1 and 2 unless other variable parameters are specified in our calculation, and the final simulation results (average intensity, OAM spiral spectrum, BER, and average capacity) are averaged by 500 independent realizations. In addition, to avoid the intensity distribution of beams exceeding the edge of the source plane, the grid spacing δ1 is proportional to the effective radius R1.

Tables Icon

Table 1. Underwater channels parameters values used in the simulations

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Table 2. OAM-based UWOC system parameters values used in the simulations

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To confirm the NACT method is effective in our simulations, the simulated results by the NACT-based split-step propagation technique should be compared with the theoretical predictions. Thus, the theoretical electric field distribution of the HyGG vortex beam at an arbitrary distance z in free space is defined as [16]

$$\begin{aligned} {E_H}({\textrm{r},z} )&= {C_{pl}}\frac{{\varGamma \left( {|l |+ \frac{p}{2} + 1} \right)}}{{\varGamma ({|l |+ 1} )}}{\left( {\frac{z}{{{z_R}}} + i} \right)^{ - |l |- \frac{p}{2} - 1}}{i^{|l |+ 1}}{\left( {\frac{r}{{{\omega_0}}}} \right)^{|l |}}{\left( {\frac{z}{{{z_R}}}} \right)^{\frac{p}{2}}}{\left( {\frac{{kr}}{z}} \right)^{|l |}} \times \\ &\textrm{ }{\textrm{ }_1}{F_1}\left( { - \frac{p}{2}, |l |+ 1;\frac{{z_R^2{r^2}}}{{\omega_0^2({{z^2} + iz{z_R}} )}}} \right)\textrm{exp} \left( { - \frac{{i{z_R}{r^2}}}{{\omega_0^2({{z^2} + i{z_R}} )}}} \right)\textrm{exp} [il\theta ] \end{aligned}, $$
where zR = (kω02)/2 is the Rayleigh range, 1F1 (a,b;x) is the confluent hypergeometric function. Other parameters in Eq. (9) have the same definitions as Eq. (1). When p = -|l|, Eq. (9) denotes the electric field distribution of the GV beam at an arbitrary distance z in free space.

Moreover, the simulated electric field distribution of the HyGG vortex beam in free space can be calculated by Eq. (2) without turbulence and attenuation operators. Thus, the simulated result for propagating the HyGG vortex beam in free space is described as

$${E_H}({{{\boldsymbol r}_n}} )= \prod\limits_{j = 1}^{n - 1} {\left\{ {A[{{{\boldsymbol r}_{j + 1}}} ]{F^{ - 1}}\left[ {{{\boldsymbol f}_i},\frac{{{{\boldsymbol r}_{j + 1}}}}{{{m_j}}}} \right]{Q_2}\left[ { - \frac{{\varDelta z}}{{{m_j}}},{{\boldsymbol f}_j}} \right]F[{{{\boldsymbol r}_j},{{\boldsymbol f}_j}} ]\frac{1}{{{m_j}}}Q\left[ {\frac{{1 - {m_j}}}{{\varDelta z}},{{\boldsymbol r}_j}} \right]} \right\}} \{{{E_H}({{\boldsymbol r}_1},z = 0)} \}, $$
where EH(r1,z = 0) denotes the electric field of the HyGG vortex beam in the source plane, which can be obtained by setting p0 = 0 in Eq. (1). Moreover, the electric field of the GV beam can be obtained by setting p0 = 0 and p = -|l| in Eq. (1). By utilizing Eqs. (1), (9), and (10), we can compare the theoretical and simulated results of the GV beam and the HyGG vortex beam.

Based on Eqs. (1), (9), and (10), the theoretical predictions and the simulated results by the NACT-based split-step propagation technique of the GV and HyGG vortex beams are shown in Fig. 6, respectively. It shows that simulated results almost coincide with the theoretical intensity and phase distributions. Therefore, the proposed NACT method is effective in propagating beams.

 figure: Fig. 6.

Fig. 6. Theory and simulation comparison with intensity and phase of the GV beam (l = 1) and the HyGG vortex beam (p = 2, l = 1) in free space.

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In Fig. 7, every technique’s PSF is generated from the produced phase screens which are defined as ${D_\varPhi }(r )= \left\langle {{{[{\varPhi ({{\boldsymbol r}^{\prime} + r} )- \varPhi ({{\boldsymbol r}^{\prime}} )} ]}^2}} \right\rangle $, where r’ represents the coordinate on a phase screen. A statistical average of 1000 random phase screens is carried out and then compared with the theoretical PSF $D_\varPhi ^{theory}$. Where $D_\varPhi ^{DFT}$, $D_\varPhi ^{SUB}$, and $D_\varPhi ^{EXT}$ are the PSF of the traditional DFT technique, subharmonic compensation technique, and PSF-based extrusion technique respectively. Figure 7 displays the PSF of the PSF-based extrusion technique that can best match the theoretical PSF. Consequently, Figs. 6,7 indicate that our modified model can accurately simulate the propagation of beams in UWOC channels.

 figure: Fig. 7.

Fig. 7. The PSF of the simulation model with three different techniques.

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3.2 Numeric analysis

(1) Effect of distances on transmission and communication performance

Figure 8 shows the relationship between average intensity distribution and propagation distances of three different OAM-carrying beams in seawater. It can be seen in the source plane, the average intensity distribution of the GV beam is the most centralized. However, owing to the autofocusing property, the beam spreading of the QAHyGG vortex beam and the HyGG vortex beam is gradually less than that of the GV beam with the increase of the transmission distance. The peak average intensity of the QAHyGG vortex beam is ∼3 times higher than that of the GV beam and ∼1.5 times higher than that of the HyGG vortex beam at 50m. When transmission distance increases to 100m, the average intensity distribution of the QAHyGG vortex beam is similar to that of the HyGG vortex beam. Compared with the HyGG vortex beam, the QAHyGG vortex beam spreading is obviously suppressed via adding an Airy-type radial phase during the propagation process, it thus has less beam spreading and higher light intensity than that of the HyGG vortex beam.

 figure: Fig. 8.

Fig. 8. Average intensity distribution comparisons with respect to transmission distance z in seawater.

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Figure 9 displays the spiral spectrum of three vortex beams at different transmission distances. The energy of the transmission OAM mode is gradually leaking into adjacent OAM modes as the transmission distance increases. Crosstalk to adjacent channels increases by increasing transmission distance. Moreover, the detection probability of the signal OAM mode of the QAHyGG vortex beam is ∼10% higher than that of the HyGG vortex beam and the GV beam at 50m. When the propagation distance increases to 100m, the signal OAM mode detection probability of the QAHyGG vortex beam is still ∼7% higher than that of the HyGG vortex beam, and ∼18% higher than that of the GV beam. The QAHyGG vortex beam can more effectively depress the spread of the signal OAM mode compared to the HyGG and GV beams. These results can be explained by Fig. 8. The QAHyGG beam undergoes fewer turbulence eddies due to its less beam spreading in the UWOC channel. Thus, the turbulence-induced distortion of the signal OAM mode wavefront also decreases. Of course, the level of OAM crosstalk generated by turbulent channels also decreases, thereby depressing the spread of the signal OAM mode.

 figure: Fig. 9.

Fig. 9. Crosstalk from the transmitted OAM channels to neighboring channels: (a) 0m; (b) 50m; (c) 100m.

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Figure 10 indicates that the BER of the OAM channel decreases with the increase of transmission distance, but the variation trend of average capacity is opposite. The result is consistent with that of the conclusion in [10]. Moreover, at 50m, the BER of the QAHyGG vortex beam reduces by ∼75%, and the average capacity of it enhances by ∼7% compared with the HyGG and GV beams. At 100m, the BER of the QAHyGG vortex beam reduces by ∼21% and the average capacity of it enhances by ∼8% compared with the HyGG vortex beam; the BER of the QAHyGG vortex beam reduces by ∼43% and the average capacity of it enhances by ∼27% compared with the GV beam. The communication performance (BER and average capacity) of the QAHyGG vortex beam is better than that of the HyGG and GV beam in the UWOC channel. This phenomenon can be elucidated by Figs. 8,9. The QAHyGG vortex beam will travel through less turbulent eddies with different refractive indexes due to its less beam spreading in the ocean channel, thereby mitigating the wavefront aberration and intermodal crosstalk. In addition, the average signal intensity per unit area of the QAHyGG vortex beam is stronger as increasing transmission distance, more emission information thus can be detected by the probe. Consequently, the utilization of the QAHyGG vortex beam will have more contributions to enhance communication performance in a medium to long (50m−100m) UWOC link.

(2) Effects of seawater conditions on average capacity

 figure: Fig. 10.

Fig. 10. BER (a) and average capacity (b) comparisons with respect to transmission distance z

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We investigate the effects of the equivalent temperature structure parameter $C_n^2$ and the ratio of temperature and salinity contributions w on the average capacity of OAM channels with three different vortex beams as shown in Fig. 11. It is clear from the Fig. 11 that the average capacity decreases with increasing $C_n^2$ or w. Moreover, it is worth noting that the average capacity drops sharply as w→0 in Fig. 11(b). The reason is that salinity-driven turbulence is the main influencing factor as w→0, and temperature-driven turbulence dominates as w→−5 [20]. Since salinity fluctuations are much more effective than temperature fluctuations in determining the effect of oceanic turbulence, thus increasing the values of w as w→0 significantly leads to the degradation of the average capacity of the OAM-based UWOC system. These results are in accordance with [10–15,20]. In addition, the average capacity of the QAHyGG vortex beam is always higher than that of the GV beam and the HyGG vortex beam. The difference between the GV and HyGG vortex beam is almost negligible with increasing $C_n^2$ or w due to their analogous spiral spectrum at 50m. As the transmission distance increases to 100m, the capacity of the HyGG vortex beam is higher than that of the GV beam. The reason for this result is that the beam spreading of the HyGG vortex beam is depressed by its autofocusing property with increasing transmission distance. The HyGG vortex beam thus undergoes fewer turbulence eddies and reduces the wavefront distortion. Meanwhile, since the QAHyGG beam has a stronger autofocusing ability, its average intensity distribution is more centralized, and the beam spreading of it is less than the HyGG and GV beams. The QAHyGG vortex beam thus depresses the spread of the signal OAM mode and enhances the average capacity in oceanic channels with different turbulence strengths.

 figure: Fig. 11.

Fig. 11. Average capacity comparisons with respect to (a) the equivalent temperature structure parameter $C_n^2$ and (b) the ratio of temperature to salinity contributions w of 50m and 100m UWOC links, respectively.

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Figure 12 shows the average capacity will decrease as the increase of absorption and scattering coefficients. The capacity of the QAHyGG vortex beam is greater than that of the HyGG vortex beam and the GV beam as the increase of attenuation coefficient (a + b). When the attenuation coefficient equals 0.16m−1, the average capacity of the QAHyGG vortex beam enhances by ∼34% compared with the HyGG vortex beam, and ∼117% compared with the GV beam at 50m. The reason for these results is that the power penalty of the signal source will increase more severely with the increasing attenuation coefficient, which will directly reduce the SNCR and average capacity. Moreover, when the attenuation coefficient is large, the extra power loss will be induced by the threshold of the probe. The extra power loss will lead to lower received light intensity and less information of the signal OAM mode. However, the QAHyGG beam possesses more centralized light intensity distribution and less intermodal crosstalk at 50m. Therefore, it can effectively mitigate the influence of oceanic absorption and scattering. Figure 8 also indicates that the channel capacity at 100m decreases more severely with increasing attenuation coefficient due to the longer communication link. Meanwhile, the average capacity of the QAHyGG and HyGG vortex beam almost cannot be distinguished with increasing attenuation coefficient at 100m. This can be explained by Figs. 8,9. Although the intermodal crosstalk of the QAHyGG vortex beam is smaller than that of the HyGG vortex beam, the average intensity distribution between the QAHyGG and HyGG vortex beams is similar at 100m. However, the SNCR is mainly affected by the received light intensity with the increasing attenuation coefficient at 100m. Hence, the average capacity of the QAHyGG and HyGG vortex beams is similar as the attenuation coefficient increases at 100m.

(3) Effects of UWOC system parameters on average capacity

 figure: Fig. 12.

Fig. 12. Average capacity comparisons with respect to attenuation coefficients (a + b) of 50m and 100m UWOC links, respectively.

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As shown in Fig. 13, the average capacity of beams decreases first and then maintains a steady state with the increasing receiver aperture radius Ra at 50m. The reason for this phenomenon is that the channel capacity is mainly affected by the turbulence-induced intermodal crosstalk at 50m. The crosstalk probability of three OAM-carrying beams increases as Ra increases. When the increasing receiver aperture can entirely capture the light spot, the crosstalk probability no longer increases [46]. The average capacity of three beams thus is independent of Ra. These results are in consistent of [10]. Moreover, the QAHyGG vortex beam, due to its stronger autofocusing property, thereby having less wavefront aberration and intermodal crosstalk at 50m. Consequently, its channel capacity curve with respect to Ra is flatter and higher than that of the HyGG and GV beams at 50m. Different from the variation trend at 50m, the average capacity first increases rapidly and then decreases gradually, after that, it remains almost unchanged with the increasing Ra at 100m. The channel capacity of QAHyGG and HyGG beams is explicitly higher than that of the GV beam when Ra equals 0.3cm. The reason for these results can be explained in two respects as follows: long distance and small receiver aperture will inevitably lead to more power penalty. When the received light power approaches the AWGN power, the average capacity is mainly influenced by the power of the received light. Consequently, we can observe the average capacity of three vortex beams increases conspicuously when Ra increases from 0.3cm to 0.5cm at 100m. With the increase of Ra, the received light power is sufficiently larger than the AWGN power. Hence, the variation trend of the average capacity at 100m is analogous to that at 50m when Ra increases from 0.7cm to 2.5cm. On the other hand, the average intensity distribution of the QAHyGG vortex beam and the HyGG vortex beam is similar but more centralized than the GV beam. Thus, the power loss of the QAHyGG and HyGG vortex beam is less than the GV beam in a finite receiver aperture. We can conclude that the channel capacity of the QAHyGG beam is always higher and it varies more gradually with increasing aperture radius regardless of the length of UWOC links. The QAHyGG vortex beam thus is more suitable when more signal OAM modes are required to be captured by the larger receiving aperture in the OAM-based UWOC system.

 figure: Fig. 13.

Fig. 13. Average capacity comparisons with respect to the receiver aperture radius Ra of 50m and 100m UWOC links, respectively.

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The turbulence strength of the ocean can be quantitatively defined regarding the average capacity data in Fig. 14. Here, we consider weak turbulence as $C_n^2 = {10^{ - 15}}{K^2}/{m^{ - 2/3}}$; medium turbulence as $C_n^2 = {10^{ - 14}}{K^2}/{m^{ - 2/3}}$; strong turbulence as $C_n^2 = {10^{ - 13}}{K^2}/{m^{ - 2/3}}$ for given w and η.

 figure: Fig. 14.

Fig. 14. Average capacity comparisons with respect to the transmit power PTx under different turbulence strength of (a) 50m and (b) 100m UWOC links, respectively.

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As illustrated in Fig. 14, it is clear that the average capacity will decline with the intensification of turbulence strength. In addition, the average capacity remains almost unchanged with decreasing transmit power when the transmit power is relatively high. As the transmit power further decreases, the average capacity drops steeply. This is because when the transmit power is relatively high, the turbulence-induced intermodal crosstalk is independent of transmit power, the value of average capacity thus maintains a almost stable state. As the transmit power further declines, apart from the turbulence-induced intermodal crosstalk, there are two significant factors that reduce the channel capacity: (1) the low received power directly leads to the reduction of SNCR; (2) the presence of a detection threshold of the probe can lead to an incomplete light speckle pattern received, which will further aggravate intermodal crosstalk and power penalty. In Fig. 14(a), the QAHyGG vortex beam possesses an improvement of the average capacity compared with the HyGG and GV beams, and the improvement increases with the decrease of transmit power. It is reasonable, the QAHyGG beam has less power penalty and intermodal crosstalk due to its stronger autofocusing property at 50m under diverse turbulence conditions. In Fig. 14(b), when the transmit power is higher, the QAHyGG beam has higher channel capacity due to its less intermodal crosstalk under medium or strong turbulence conditions. As the transmit power further decreases, the QAHyGG and the HyGG vortex beam almost synchronously decrease owing to the similar light intensity distribution between them at 100m.

(4) The beam parameters optimization for average capacity

Figure 15 displays that the average capacity of the QAHyGG vortex beam up to an autofocusing parameter p0 or a beam waist ω0 increases and then after a marked peak point decreases with increasing the p0 or ω0. Hence, the average capacity peaks at p0 = 8 × 10−4, ω0 = 2mm of a 50m UWOC link, at p0 = 4 × 10−4, ω0 = 3mm of a 100m UWOC link. The reason for this result is that the beam spreading is smaller for a larger ω0, whereas the power density of the transmitted OAM state decreases with the increase of the ω0 for the given emitted energy. Meanwhile, according to the researched results in Fig. 2, a larger p0 can strongly depress the beam spreading only at a short distance, and a smaller p0 can weakly depress the beam spreading at a long distance. Consequently, the smaller p0 and the larger ω0 of the QAHyGG vortex beam will through fewer turbulence eddies, thereby mitigating wavefront distortion and improving the purity of signal OAM mode, finally, enhancing the average capacity at a longer distance. Hence, a smaller ω0 and a larger p0 should be chosen for a shorter distance, and a larger ω0 and a smaller p0 should be selected for a longer distance when the QAHyGG vortex beam is utilized in designing the OAM-based UWOC system.

 figure: Fig. 15.

Fig. 15. Average capacity of the QAHyGG vortex beam with respect to the autofocusing factor p0 and initial beam waist ω0 of (a) 50m and (b) 100m UWOC links, respectively.

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The average capacity of the QAHyGG vortex beam bring by the hollowness parameter p and the OAM number l0 is discussed in detail and shown in Fig. 16. p ≥ -|l0| must be satisfied in Eq. (1) to ensure the finite power of the QAHyGG vortex beam. Thus, in Fig. 16, the bar of p = −1, l0 = 0 is meaningless. The average capacity decreases along the positive l0 axis. In the case of OAM number l0 = 1, the average capacity of the QRAVB (p = −1, l0 = 1) is smaller than that of the QAHyGG beam associated with p ≥ 0. And the capacity of the QAHyGG beam with p ≥ 0 is generally larger than p = −1. Meanwhile, when l0 is small, a smaller p is more beneficial to enhance the average capacity. As l0 increases, a larger p can more effectively enhance the average capacity of the QAHyGG vortex beam. Moreover, comparing Fig. 16(a) with Fig. 16(b), the QAHyGG vortex beam with a larger p seems more suitable for longer transmission distances. These results can be elucidated by Fig. 3. Figure 3 indicates that the autofocusing position will drift away from the source plane with a higher value of p. Meanwhile, the intensity distribution is more centralized with an increasing p at 100m. Therefore, a higher p can be more effective in suppressing the beam spreading for a longer distance. As a result, the QAHyGG vortex beam with a larger p will undergo fewer turbulence eddies for a longer UWOC link, thereby mitigating the wavefront aberration and intermodal crosstalk. Furthermore, the received power of the QAHyGG beam will increase with increasing p, which can effectively enhance the SNCR of a longer UWOC link.

 figure: Fig. 16.

Fig. 16. Average capacity of the QAHyGG vortex beam with respect to the OAM number l0 and the hollowness parameter p of (a) 50m and (b) 100m UWOC links, respectively.

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4. Conclusion

In summary, we have proposed the QAHyGG vortex beam and established a modified model for the OAM-based UWOC channel. The modified model is verified to effectively simulate various beams’ propagation features in the UWOC channel. Based on the modified model, the evolution of the average intensity and the spiral spectrum, as well as the influence of channel and UWOC system parameters on the average capacity for three OAM-carrying beams are discussed in detail. We also analyze the effect of the QAHyGG vortex beam parameters on the average capacity. It is found that a smaller receiving aperture will reduce the average capacity for a long underwater channel. The presence of a probe threshold will further aggravate the intermodel crosstalk and power loss when the received power is lower. Compared with the HyGG vortex and GV beams, the QAHyGG beam has less intermodal crosstalk due to its less beam spreading in the UWOC channel. The QAHyGG vortex beam also has stronger anti-turbulence and anti-attenuation abilities owing to the modulation of an Airy-type radial phase. The average capacity of it has enhanced by ∼8% and ∼27% compared with the HyGG vortex beam and the GV beam at 100m, respectively. The QAHyGG vortex beam is more suitable in an OAM-based UWOC system with a finite receiving aperture or lower transmit power. In addition, the larger values of the QAHyGG vortex beam waist ω0 or hollowness parameter p, and the smaller values of autofocusing parameter p0 or OAM number l0 can effectively improve the average capacity for a longer distance. We can enhance the average capacity of the QAHyGG vortex beam through the oceanic channel via selecting initial parameters properly. Finally, as a superior information carrier, the QAHyGG vortex beam can be employed with some innovative techniques, such as channel coding [48], adaptive optics [27], multiple-input multiple-output [49], and convolutional neural network [3] to establish a more robust OAM-based UWOC system.

Funding

National Natural Science Foundation of China (62231005); Department of Science and Technology of Jilin Province (20200801053GH).

Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments, which significantly improved the presentation of this paper.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (16)
Fig. 1.
Fig. 1. Numerical demonstrations of the QAHyGG vortex beam (λ=450nm, p = 2, l = 1, w0 = 2mm, p0 = 8 × 10−4) propagating in free space. (a) numerically simulated side-view propagation of the QAHyGG vortex beam; (b1)-(b4) snapshots of the transverse intensity patterns taken at the planes marked by the dashed lines in (a); (c1)-(c4) the corresponding phase distributions at different planes marked in (a).

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Fig. 2.
Fig. 2. Propagation dynamics of the QAHyGG vortex beam (λ=450nm, p = 2, l = 1) with different autofocusing factors p0 and beam waists ω0 in xz plane.

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Fig. 3.
Fig. 3. Propagation dynamics of the QAHyGG vortex beam (λ=450nm, ω0 = 2mm, p0 = 8 × 10−4) with different hollowness parameters p and OAM numbers l in xz plane.

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Fig. 4.
Fig. 4. Schematic description of the modified model for OAM-based UWOC channel.

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Fig. 5.
Fig. 5. Propagation procedure flow chart used in the modified model. (Left) Coordinate Transformation Procedure (Middle) Phase Screens Generation Procedure (Right) Beam Propagation Procedure.

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Fig. 6.
Fig. 6. Theory and simulation comparison with intensity and phase of the GV beam (l = 1) and the HyGG vortex beam (p = 2, l = 1) in free space.

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Fig. 7.
Fig. 7. The PSF of the simulation model with three different techniques.

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Fig. 8.
Fig. 8. Average intensity distribution comparisons with respect to transmission distance z in seawater.

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Fig. 9.
Fig. 9. Crosstalk from the transmitted OAM channels to neighboring channels: (a) 0m; (b) 50m; (c) 100m.

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Fig. 10.
Fig. 10. BER (a) and average capacity (b) comparisons with respect to transmission distance z

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Fig. 11.
Fig. 11. Average capacity comparisons with respect to (a) the equivalent temperature structure parameter $C_n^2$ and (b) the ratio of temperature to salinity contributions w of 50m and 100m UWOC links, respectively.

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Fig. 12.
Fig. 12. Average capacity comparisons with respect to attenuation coefficients (a + b) of 50m and 100m UWOC links, respectively.

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Fig. 13.
Fig. 13. Average capacity comparisons with respect to the receiver aperture radius Ra of 50m and 100m UWOC links, respectively.

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Fig. 14.
Fig. 14. Average capacity comparisons with respect to the transmit power PTx under different turbulence strength of (a) 50m and (b) 100m UWOC links, respectively.

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Fig. 15.
Fig. 15. Average capacity of the QAHyGG vortex beam with respect to the autofocusing factor p0 and initial beam waist ω0 of (a) 50m and (b) 100m UWOC links, respectively.

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Fig. 16.
Fig. 16. Average capacity of the QAHyGG vortex beam with respect to the OAM number l0 and the hollowness parameter p of (a) 50m and (b) 100m UWOC links, respectively.

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Tables (2)

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Table 1. Underwater channels parameters values used in the simulations

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Table 2. OAM-based UWOC system parameters values used in the simulations

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Equations (10)

Equations on this page are rendered with MathJax. Learn more.

$$E({\boldsymbol r} )= {E_0}{C_{pl}}{\left( {\frac{r}{{{\omega_0}}}} \right)^{p + |l |}}\textrm{exp} \left( {\frac{{ - {r^2}}}{{{\omega_0}^2}} + il\theta } \right)\textrm{exp} ({i{p_0}k{r^{3/2}}} ), $$
$$\scalebox{0.9}{$\displaystyle E({{{\boldsymbol r}_n}} )= \prod\limits_{j = 1}^{n - 1} {\left\{ {A[{{{\boldsymbol r}_{j + 1}}} ]{F^{ - 1}}\left[ {{{\boldsymbol f}_i},\frac{{{{\boldsymbol r}_{j + 1}}}}{{{m_j}}}} \right]{Q_2}\left[ { - \frac{{\varDelta z}}{{{m_j}}},{{\boldsymbol f}_j}} \right]F[{{{\boldsymbol r}_j},{{\boldsymbol f}_j}} ]\frac{1}{{{m_j}}}Q\left[ {\frac{{1 - {m_j}}}{{\varDelta z}},{{\boldsymbol r}_j}} \right]\mathop D\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]\mathop T\limits^ \wedge [{\varDelta z,{{\boldsymbol r}_j}} ]} \right\}} \{{E({{\boldsymbol r}_1})} \},$}$$
$$\begin{aligned} \varphi ({\boldsymbol \kappa } )&= 0.388C_n^2{w^{ - 2}}{(\kappa )^{\frac{{ - 11}}{3}}}\left[ {1 + 2.35{{({\kappa \eta } )}^{\frac{2}{3}}}} \right]\textrm{ }\\ \textrm{ } &\times [{{w^2}\textrm{exp} ({ - {A_T}\delta } )+ \textrm{exp} ({ - {A_S}\delta } )- 2w\textrm{exp} ({ - {A_{TS}}\delta } )} ]\end{aligned}, $$
$$\begin{aligned} D_\varPhi ^{theory} &= {\varepsilon ^{ - 1/3}}({{{\chi _{\boldsymbol T}}} / {{w^2}}}){r^{5/3}}[3.063 \times {10^{ - 7}}{k^2}\varDelta z \times (1.116{w^2} - 2.235w + 1.119)\\ \textrm{ } &- \varDelta {z^3}(1.841{w^2} - 40.341w + 2077)] \end{aligned}. $$
$${P_l} = \frac{{{{\left|{\int {\int {M_l^\ast ({r,\phi ,z} ){E_{{l_0}}}({r,\phi ,z} )rdrd\phi } } } \right|}^2}}}{{\sum\nolimits_{m ={-} \infty }^{ + \infty } {{{\left|{\int {\int {M_m^\ast ({r,\phi ,z} ){E_{{l_0}}}({r,\phi ,z} )rdrd\phi } } } \right|}^2}} }}, $$
$$SNCR = \frac{{{E_b}}}{{{E_c} + {N_0}}} = \frac{{{P_{Rx}} \times {P_{l = {l_0}}}}}{{{P_{Rx}} \times ({1 - {P_{l = {l_0}}}} )+ {N_0}}}, $$
$$BER = \frac{1}{2}erfc\sqrt {\frac{{SNCR}}{2}} , $$
$$C = {\log _2}N + (1 - BER){\log _2}(1 - BER) + BER{\log _2}\frac{{BER}}{{N - 1}}. $$
$$\begin{aligned} {E_H}({\textrm{r},z} )&= {C_{pl}}\frac{{\varGamma \left( {|l |+ \frac{p}{2} + 1} \right)}}{{\varGamma ({|l |+ 1} )}}{\left( {\frac{z}{{{z_R}}} + i} \right)^{ - |l |- \frac{p}{2} - 1}}{i^{|l |+ 1}}{\left( {\frac{r}{{{\omega_0}}}} \right)^{|l |}}{\left( {\frac{z}{{{z_R}}}} \right)^{\frac{p}{2}}}{\left( {\frac{{kr}}{z}} \right)^{|l |}} \times \\ &\textrm{ }{\textrm{ }_1}{F_1}\left( { - \frac{p}{2}, |l |+ 1;\frac{{z_R^2{r^2}}}{{\omega_0^2({{z^2} + iz{z_R}} )}}} \right)\textrm{exp} \left( { - \frac{{i{z_R}{r^2}}}{{\omega_0^2({{z^2} + i{z_R}} )}}} \right)\textrm{exp} [il\theta ] \end{aligned}, $$
$${E_H}({{{\boldsymbol r}_n}} )= \prod\limits_{j = 1}^{n - 1} {\left\{ {A[{{{\boldsymbol r}_{j + 1}}} ]{F^{ - 1}}\left[ {{{\boldsymbol f}_i},\frac{{{{\boldsymbol r}_{j + 1}}}}{{{m_j}}}} \right]{Q_2}\left[ { - \frac{{\varDelta z}}{{{m_j}}},{{\boldsymbol f}_j}} \right]F[{{{\boldsymbol r}_j},{{\boldsymbol f}_j}} ]\frac{1}{{{m_j}}}Q\left[ {\frac{{1 - {m_j}}}{{\varDelta z}},{{\boldsymbol r}_j}} \right]} \right\}} \{{{E_H}({{\boldsymbol r}_1},z = 0)} \}, $$
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