During the last decade, underwater unmanned vehicles(UUVs) have been widely developed and deployed^[1-2], facilitating a variety of offshore and maritime applications. However, the problem arises as how to efficiently exchange information between a UUVs and an unmanned aerial vehicle(UAV) controller. In this case, the UUV cannot be maneuvered remotely by a data cable, as the link is posed as an air-sea two-section wireless link. Traditionally, acoustic waves are used for underwater communications^[3-4], but the data rate is restricted to ~100 kbps due to both the low frequency of acoustic carriers and the harsh underwater environment. Moreover, the refraction at the ocean surface makes it highly inefficient for acoustic waves to transmit signals to an above sea platform. On the other hand, whereas electromagnetic waves are prevalent in aerial communications, they are attenuated so fast in sea water that those within 1 meter will be completely absorbed because sea water has much higher electrical conductivity than air.

Optical waves can be considered as a good compromise between their acoustic and electromagnetic counterparts, allowing decent penetrability through sea water as well as high transmission rate. Free-space optical communications(FSOC) has found its position among various modern network infrastructures, while its applications in communications between under water and above water platforms is seldom addressed in literature.

In any aerial optical link, the existence of atmospheric turbulence will inevitably impose wavefront distortion to the propagating laser beam^[5-6]. As a result, in the far-field, the beam will be deflected and form optical speckles, leading to intensity fluctuations at the receiver. This is called optical scintillation^[7]. The same thing happens for an underwater optical link, where the turbulence is mainly caused by temperature and salinity gradients. Another major adverse factor in underwater environments is absorption, but since there is no other effective means to compensate for that except to emit more optical power, we will not investigate that aspect here. On the other hand, regardless of the amount of power that is used of the source beam, it is of no help to alleviate the optical scintillation because it is essentially a multiplicative noise. Only by properly designing the spatial properties of the optical source beam can the scintillation be alleviated.

In this study, we will examine the optical link performance between UUVs and low-to-medium altitude UAVs in terms of optical scintillation and signal-to-noise ratio(SNR) at the receiver end. The optical beams considered here are the Gaussian beam and the annular beam. The Gaussian beam is the most commonly used beam type because most lasers' output conforms to Gaussian profiles, and numerous research papers concerning Gaussian beam turbulence propagation have been published^[8-10]. Recent studies also reveal that annular beams are more resilient to turbulence induced intensity fluctuations, indicating it to be potentially better than Gaussian beams in certain application scenarios^[11]. Although the study of annular beams propagation in atmospheric slant paths has already been carried out^[12-14], no results related to air-sea combined links can be found to date. The performance of the two beam types are compared using wave optics simulation(WOS) numerical method, which is based on fast Fourier transformation(FFT) to compute the Fresnel diffraction integral^[15-17]. The turbulence effects are emulated by random phase screens which are arranged to be uniformly distributed for the underwater section of the link. For the aerial part of the link, the phase screens are non-uniformly arranged so as to better reflect the altitude-dependent characteristic of the atmosphere.

The rest of this paper is structured as follows. In Section Ⅱ the channel model is introduced in a numerical manner and the basics of WOS is briefly explained. Section Ⅲ provides the simulation results of Gaussian and annular beams propagation through different ranges of air-sea channels, from the perspective of aperture-averaged scintillation and ensembled SNR. Some discussions concerning the obtained results are also included in this part. Section Ⅳ is a conclusion and summary of the work presented.

Generally, an optical link between a UUV and a UAV comprises two parts: an underwater section and an aerial section, which is illustrated in Fig. 1. Taking into account the fact that laser beams dissipate within 100m in the sea water, we only consider 50m depth of sea water in the vertical direction. Moreover, recent field-measured data in the Yellow Sea and the Bohai Sea reveal that in autumn and winter the vertical variations of oceanic parameters are almost negligible in the upper layer waters^[18], which is why we will model the oceanic turbulence as homogeneous in our case, independent of depth. Therefore, as indicated in Fig. 1, the oceanic turbulence is modeled by equally separated, statistically equivalent phase screens.

Figure 1.  Air-sea two-section optical link model

The power spectrum density(PSD) model of oceanic refractive index fluctuations adopted here is the one developed by Nikishov^[19], which can be expressed as:

in which the following denotations have been defined:

among which κ is the spatial wave number, η is the Kolmogorov microscale, and ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid ranging from 10^-1 to 10^-10 m²/s³. For constant gradients of mean temperature and salinity, ω defines the ratio of temperature to salinity contributions to the spectrum, varying from -5(corresponding to temperature dominated turbulence) to 0(corresponding to salinity dominated turbulence). C₀=0.72(the Obukhov-Corrsin constant), C₁ is a free parameter to be determined by comparison with the experiment and its value is taken as 2.35. K_T is eddy thermal diffusivity and K_S is the diffusivity of salt. A_T=1.683×10^-2, A_S=1.9×10^-4, A_TS=9.41×10^-3, α=2.6×10^-4 L/deg, and β=1.75×10^-4 L/g. χ_T and χ_S represent the rate of dissipation of mean squared temperature and that of salinity, respectively. χ_TS denotes the correlation of χ_T and χ_S.

The phase screen generation process is to randomize the spatial PSD and then FFT it^[20].

On the other hand, the altitude-dependent atmosphere density directly leads to a vertically varying atmospheric turbulence profile, for which the best acknowledged model is the H-V_5-7, also shown in Fig. 1, where the horizontal scale marks the turbulence strength. To better reflect the unevenness of the H-V_5-7 turbulence model and fully utilize the limited computational power, nonuniformly arranged phase screens are used to represent the effects of turbulent wavefront distortions. The partition is based on the path integral of the structure constant and the atmospheric turbulence PSD used here is the modified von Karman spectrum.

where C_n²(h) is the atmospheric refractive index structure constant at altitude h, κ_l= 5.92/l₀, κ_m=5.92/L₀, and where l₀ and L₀ are the inner scale and the outer scale.

The WOS numerical method provides a viable approach to studying complex optical propagation problems. In the scenario considered here, it is nearly impossible to derive analytically expressed results, comprising both underwater and slant-path atmospheric channels, so it is reasonable to use WOS instead. The basic principle of WOS is the split-step propagation between separate phase screens, which can be written as:

where r is the transverse position vector, U_i(r_i) is the optical field at the ith screen, ψ_i(r_i) is the wavefront distortion induced by the turbulence within the ith interval, L is the path length, and m is the number of phase screens. For the specific implementation of WOS, the readers can refer to related literature [20], which we will not reiterate here.

As indicated above, the optical channel that is to be discussed here is a two-section air-sea path, which is comprised of a 50 m underwater path that is submitted to oceanic turbulence, and a vertical atmospheric path whose length is variable between 1, 2 and 5 km. The optical beams considered here are the Gaussian beam and the annular beam. An annular beam can be represented by the difference of two concentric Gaussian beams^[21]. To make these two comparable, the launching power of the annular beam at the source plane is made to be equal to a Gaussian beam with width parameter w₀=50 mm. Their profiles and the cross-section of the annular beam are shown in Fig. 2.

Figure 2.  Comparison of Gaussian beam and annular beam with equal power

In the oceanic turbulence model of Eq.(1), the relevant parameters are set as:η=2 mm, ε=10^-5 m²/s³, ω=-2.5, χ_T=10^-8 K²/s. For the von Karman atmospheric spectrum, we have set the inner scale at l₀=5 mm and the outer scale at L₀=5 m. In the H-V_5-7 model^[21], the ground level turbulence strength is C_n²(0)=10^-5 m^-2/3 and the pseudo wind speed is 27 m/s. Moreover, the wavelength λ is assumed to be 1 μm and 1 000 computation loops are averaged for every single result below. Both the uplink and the downlink are examined.

Fig. 3 shows the beam wander of the Gaussian beam and the annular beam versus the propagation distance. The first thing that should be noted is that it can be found that the downlink beam wander is much less than that of the uplink. This is because in an air-sea optical path, the strongest part of the turbulence lies at the underwater portion and the near sea-level portion, which is close to the uplink transmitter. As a result, the beam is heavily distorted at the beginning of the propagation. During subsequent propagation the distorted wavefront will have enough time to fully diffract and refract, essentially magnifying the earlier imposed distortion. Therefore, in the far-field, the beam exhibits severe beam wander. By contrast, in the downlink the stronger turbulence distortion is imposed to the beam at a relatively later stage of the propagation, leaving less time for the distorted wavefront to evolve, so that the phase distortion does not convert into phase tip and tilt as it does in the case of the uplink.

Figure 3.  Beam wanders of the Gaussian beam and the annular beam in (a)uplink, (b)downlink

The other thing that is well worth noting is that in either case, whether it in the uplink or in the downlink, the wanders of annular beam less than that of the Gaussian beam.

For any FSO link, the aperture-averaged scintillation index, also known as irradiance flux variance, is a vital system performance metric^[21]. The aperture scintillation indices of the Gaussian beam and the annular beam are compared and the results for three different distances, namely, 1 km, 2 km and 5 km are all provided in Fig. 4. Fig. 4(a)-4(c) is the results for the uplink, where we can see that for the 1 km and the 2 km links, the annular beam always has smaller scintillation once the aperture diameter D is larger than 20 mm. For the 5 km uplink in Fig. 4(c), the annular beam scintillation is significantly reduced compared to that of the Gaussian beam. For the downlinks, the annular beam seems to be better than the Gaussian beam as well, although the advantage might be minor.

Figure 4.  Scintillation index versus receiver aperture diameter D for (a)1 km uplink, (b)2 km uplink, (c)5 km uplink, (d)1 km downlink, (e)2 km downlink, (f)5 km downlink

Optical scintillation degrades the reliability of the FSO link, which can be evaluated by the probability of fade. Fading is when the received signal level lies below a prescribed threshold. For the case in Fig. 4, the corresponding probability of fade is given in Fig. 5. According to the results, it is safe to say that in the uplink, the annular beam always allows for a smaller probability of fade. For the downlink, if the receiver aperture is not too small, almost no fade would happen.

Figure 5.  Probability of fade versus receiver aperture diameter D for (a)1 km uplink, (b)2 km uplink, (c)5 km uplink, (d)1 km downlink, (e)2 km downlink, (f)5 km downlink

Last but not least, the mean SNR is calculated and the results are shown in Fig. 6. Except for the 1 km downlink, the annular beam outperforms the Gaussian beam in terms of SNR.

Figure 6.  Mean SNR versus receiver aperture diameter D for (a)1 km uplink, (b)2 km uplink, (c)5 km uplink, (d)1 km downlink, (e)2 km downlink, (f)5 km downlink

Using numerical simulation approach, we have presented a study of the optical link evaluation for the air-sea two-section paths between UUVs and UA- Vs. To our knowledge this is the first time such a type of link configuration is examined in the context of free-space optical communications. The main characteristic of this type of turbulent link is that the turbulence is aggregated at either the transmitter end or the receiver end, forming into an asymmetric phase perturbation structure. In this case, the performance of the Gaussian beam and the annular beam are compared and we can conclude that, under most circumstances, annular beams are obviously a more preferable choice. Generally, in the uplink, beam jitter caused by large-scale turbulence eddies near the transmitter is significantly more severe, leading to worse performance in scintillation, yet the SNR can reach an acceptable level if the receiver aperture is large enough. In the downlink, the degradation caused by turbulence effects is somewhat mild in comparison to that found in the uplink. However, it should be noted that in any air-sea path, the attenuation of the sea water is another major source of the SNR ceiling.

It is also important to notice that the annular beam is quite easy to generate and use, meaning that the switch from Gaussian beams to annular beams will not add much to the overall system cost. Taken that into consideration, combined with the performance improvement that annular beams can provide, as demonstrated by this study, there seems to be no reason to not use annular beams instead of Gaussian beams in FSO systems. This study is supposed to benefit future FSO transceiver designs and implementations.