The spatial coherence radius in moderate-to-strong maritime turbulence is derived on the basis of the modified Rytov approximation. Models are developed to simulate the spiral spectrum of Airy beams propagating through moderate-to-strong maritime turbulence. In the moderate-to-strong irradiance fluctuation region, we analyze the effects of maritime turbulence on the spread of the spiral spectrum of Airy beams in a horizontal propagation path. Results indicate that the increment in the inner-scale significantly increases the received power. By contrast, the outer-scale elicits a negligible effect on the received power if the ratio of the inner-scale to the outer-scale is less than 0.01. The outer-scale affects the received power only if the ratio is greater than 0.01. The performance of a light source is essential for the received power of Airy beams carrying orbital angular momentum (OAM) through moderate-to-strong maritime turbulence. Airy beams with longer wavelengths, smaller OAM numbers, larger radii of the main ring, and smaller diameters of the circular aperture are less affected by maritime turbulence. Autofocusing of Airy beams is beneficial for the propagation of the spiral spectrum in a certain propagation distance. These results contribute to the design of optical communication systems with OAM encoding for moderate-to-strong maritime turbulence.
© 2016 Optical Society of America
Understanding the characteristics of optical propagation through atmospheric turbulence in the maritime environment is attractive and complementary for free space optical communications between marine vessels. Humidity and temperature fluctuations in maritime environments may complicate optical propagation to a greater extent than those in terrestrial environments; the correlation and cospectrum of temperature and humidity fluctuations must be considered . According to Hill’s experimental data , a new marine atmospheric spectrum that describes atmospheric turbulence in a general maritime environment is proposed . Considering the effect of the inner-scale of turbulence, researchers  developed the irradiance fluctuation expressions of plane and spherical waves on the basis of the new maritime spectrum in weak optical turbulence. In maritime environments, Vetelino et al.  proposed theoretical scintillation expressions under moderate-to-strong Kolmogorov turbulence. In weak non-Kolmogorov maritime atmospheric turbulence channels, scintillation and aperture averaging of Gaussian beams were investigated .
The propagation of a spiral spectrum with orbital angular momentum (OAM) through atmospheric turbulence has been extensively investigated in the field of optical wireless communications owing to its potential use in multivalued encoding of information and applications in communication systems [6–10]. A key motivation for this idea is that OAM can take any integer value and span an infinite dimensional orthonormal basis [11, 12]. This property may provide a potential solution in the context of wireless communications [12, 13]. However, OAM is susceptible to atmospheric turbulence because of its natural spatial structure. Atmospheric turbulence can induce the spread of the spiral spectrum of OAM, crosstalk among neighboring channels, and attenuation of the information capacity of communication channels. Several models and experiments have recently been established and implemented for the spread of the input mode power over neighboring OAM modes, or a spiral spectrum, resulting in crosstalk between the channels through weak fluctuation region [8, 14–16]. To mitigate the effect of turbulence on OAM, non-diffracting beams [8, 9, 15] are adopted to reduce scattering artefacts and to increase the quality of OAM encoding. Airy beam, a member of the non-diffracting family, is a potential candidate because of several properties, including non-diffraction , self-healing [18, 19], and autofocusing [20, 21].
In this study, the spiral spectrum of Airy beams is investigated in the moderate-to-strong turbulence of maritime atmosphere. We initially develop the theoretical expression of the spatial coherence radius through moderate-to-strong maritime turbulence via the modified Rytov approximation. We then obtain the expressions of the probability distribution function and normalized received power of OAM as Airy beams propagate in moderate-to-strong maritime turbulence.
2. Effective Kolmogorov spectrum for moderate-to-strong maritime turbulence
Kolmogorov spectrum is the traditional spatial power spectrum of refractive-index fluctuations when laser beams propagate in weak atmospheric turbulence. This spectrum is described by the following expression
Rytov approximation is extended to the moderate-to-strong fluctuation regime because of theoretically investigation of the inner-scale effects under weak fluctuations and outer-scale effects under strong fluctuations. To develop a model under moderate-to-strong turbulence, spatial frequency filter functions of the refractive index spectrum are introduced and the Kolmogorov spectrum is replaced by the effective Kolmogorov spectrum 
If the inner-scale effects are important, the expression of for a maritime environment is defined by [3, 4]
The outer-scale effects are adopted by denoting as 
The filter functions and are identified as 3, 4, 22]22]3]
At this point, the effective marine spectrum for a receiver with a finite aperture in moderate-to-strong maritime atmospheric turbulence becomes
3. Probability distribution and normalized received power of spiral spectrum mode
Consider an aperture truncated Airy beam that initially has a transverse spatial wave function corresponding to an eigenstate of OAM. We write the initial radial Airy profile at the source plane (z = 0) in cylindrical coordinates as [21, 23]
Under the paraxial appromation, the normalized Airy model in the z plane through free-space assumes the following form 
In the regions of moderate-to-strong maritime atmospheric turbulence and in the half-space , the complex amplitude of the Airy beam can be written as
As beam propagates through the maritime atmospheric turbulence, the effect of the refractive index fluctuations perturbs the complex amplitude of the wave so that it is no longer guaranteed to be in the original eigenstate of orbital angular momentum. The function can be written as a superposition of plane waves with phase  as follows
Using the form of the phase structure function given in [25,26], we obtain
Based on the integral expression Eqs. (12), (16), and (19), we obtain the radial distribution functions of the signal mode probability density as and crosstalk probability density as
The relative power of the spiral harmonics marked with m in the paraxial regime of light propagation is determined by 
The received power is defined as relative power of the original OAM mode in the receiver plane (i.e. ) and the crosstalk power is the relative power in the receiver plane that is found to be in OAM mode [6, 29].
4. Numerical results and discussion
In this section, we will carry out a study of the spiral spectrum of Airy beams in moderate-to-strong maritime atmospheric turbulence by using the formulae derived in the previous section. Numerical calculations are performed to illustrate the effects of the inner-scale, outer-scale, diameter of the receiver and light source on the transmission of OAM of Airy beams in moderate-to-strong turbulence of maritime atmosphere.
Figure 1 shows the relationship of the received power of the detected OAM states (the probability for achieving the original OAM state ) and the propagation distance z with several values of the wavelength. The parameters are set as and . Turbulence of maritime atmosphere is known to vary from moderate to strong fluctuation regions by increasing the index of refraction structure parameter and propagation distance z or decreasing wavelength . For simplicity, we let z vary when is given. In this case, longer propagation distance directly corresponds to a stronger fluctuation of maritime atmospheric turbulence. At a short fixed propagation distance, the received power of Airy beams slightly fluctuates with short wavelength. This effect is caused by the distribution and spread of signal mode probability density in the radial direction as the wavelength increases. At the same time, however, a shorter wavelength corresponds to a larger wave number, which induces relatively strong scintillations. Airy beams with short wavelengths are susceptible to strong fluctuations, resulting in one benefit of a larger wavelength for OAM propagation under maritime atmospheric turbulence.
The received power of the detected OAM states and crosstalk power of Airy beams through maritime environment are depicted in Figs. 2(a) and 2(b). For practical optical communication with the OAM mode, larger OAM quantum numbers correspond to the larger radius of spiral spectrum. This indicates that the received signal carried the maximum OAM modes is limited by the radius of the confined aperture in the optical system. For a given receiving aperture radius, the received power with a smaller OAM number is higher than that with a larger OAM number, as shown in Fig. 2(a). When OAM numbers further enlarge [Fig. 2(a)], the influence of OAM number on the received power can be ignored for a fixed optical system. The crosstalk power for different is represented by the histograms in Fig. 2(b), taking parameters . The bars labeled with describe the received powers that maintain the launched mode after propagation in maritime atmospheric turbulence. The other bars indicate that the energy spreads to other OAM states caused by turbulence, and the corresponding received power becomes noise (crosstalk power). For short propagation distances, the crosstalk power is negligible. As propagation distance increases ( in middle fluctuation regions), the crosstalk power only found in the neighboring channel () becomes evident. As the propagation distance further increases ( in strong fluctuation regions), the crosstalk power is conspicuous in all of the channels, and the received power is difficult to differentiate from noises.
Figure 3(a) illustrates the received power of the detected OAM states for various radii of the main ring of Airy beams. As increases, the maximum intensity of Airy beams elevates for propagation . As a consequence, in Fig. 3(a), the received power boosts as increases for a fixed propagation distance. For [ in Fig. 3(a)], the Airy beams decay slowly with oscillations of Airy tails. As becomes sufficiently large, the third term on the right hand of Eq. (12) becomes significant. In Fig. 3(a), the oscillations become increasingly notable as rises. As z increases, the amplitude of Airy beams slowly decay; concurrently, the power tends to concentrate in a small area because of the decrease of the radius of Airy beams, and leads to an increase in amplitude of the Airy beam . The balance of these two effects results in the abrupt autofocusing of Airy beams, which causes the received power to significantly rise for a large at about . For even larger values of z, the received powers are suppressed by the increasing fluctuation of the maritime atmosphere. Thus, the difference induced by in the received power diminishes gradually over long distances. To illustrate the effect of aperture diameter D on the received power through marine environment, we plot Fig. 3(b). As aperture diameter D increases [the numerator of Eq. (23)], the total of the signal carried initially launched OAM states actually rises. Meanwhile, the denominator of Eq. (23) is constructed with primitive transmitting OAM states and crosstalk among all OAM channels. As D increases, the ratio of the detected OAM signal to noise decreases because of the decays with oscillations of Airy beams along the radial direction. Thus, the denominator of Eq. (23) increases more quickly than the numerator. As a consequence, a larger aperture diameter D leads to a lower received power [Fig. 3(b)]. These results imply that to optimize the received signal in such OAM communication, adopting an auto-tracking system is highly necessary.
Figures 4(a) and 4(b) reveal the effects of inner- and outer-scale on the received power. The inner- and outer-scale define the lower and upper limits, respectively, of scale sizes of eddies in the inertial range. Smaller inner scales induce the more severe degradation of the received power, as indicated in Fig. 4(a). This result is achieved because of the existence of a greater number of turbulent cells in turbulence for smaller inner scales. By comparing Figs. 4(a) and 4(b), we note that the influence of the outer-scale of turbulence on the received power is less than that of the inner-scale. This finding can be explained directly by and in Eq. (10). For clarity, the influences of the inner- and outer-scale on the spatial coherence radius are explored in Figs. 4(c) and 4(d). At , the spatial coherence radius increases until it reaches a maximum value as the inner-scale increases; then decreases, saturating at a level as the inner-scale further increases. Instead, almost increases linearly as the inner-scale increases at . In Figs. 4(c) and 4(d), we observe that the spatial coherence radius is not sensitive to the change in outer-scale. For , the effect of the outer-scale becomes insignificant. The ratio of to is only larger than 0.01; the outer-scale gradually plays a role, then the larger outer-scale corresponds to the larger spatial coherence radius. These findings explain why the higher outer-scale, in Fig. 4(b), generates larger received power.
In this study, theoretical models for the spatial coherence radius were derived by utilizing the modified Rytov method when the spiral spectrum of the Airy beams through moderate-to-strong maritime atmospheric turbulence. On the basis of the developed models, we analyzed the received power and crosstalk power of the Airy beams that carried OAM through moderate-to-strong maritime atmospheric turbulence. Our results showed that the increment of the inner-scale significantly alleviates the spread of the spiral spectrum and increases the received power of OAM. By contrast, the outer-scale elicits almost no effect on the received power at . The outer-scale began to work on the spatial coherence radius and the received power at . The performance of the light source plays a key role in the received power of the OAM under moderate-to-strong maritime atmospheric turbulence. The spiral spectrum of the Airy beams is less affected by turbulence under longer wavelengths, smaller OAM numbers, larger radii of the main ring and smaller diameters of the circular aperture. Autofocusing of Airy beams is beneficial for the propagation of the spiral spectrum in a certain propagation distance. Our study is useful for the selection of light source and the design of an optical communication system with OAM encoding through moderate-to-strong maritime atmospheric turbulence.
This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128), the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51517).
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