Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence

Italo Toselli; Szymon Gladysz

doi:10.1364/OE.394468

1. Introduction

For many years sound has been used as a primary method for underwater communications. However, the data transmission rate of acoustic systems is low because typical frequencies associated with underwater acoustics are between tens of hertz and hundreds of kilohertz. A higher bandwidth can be achieved using visible light to transfer data underwater. However, the underwater environment presents challenges for propagation of laser beams. In clean ocean water, the extinction due to absorption and scattering probably limits the useful range of underwater communications to ∼100 m in the blue-green spectral window. Additionally, underwater optical turbulence stemming from minute differences in water’s refractive index and salinity generates beam broadening and scintillation as additional challenges. Experiments show that the magnitude of the distortions that are caused by temperature and salinity fluctuations in water can significantly exceed those encountered in atmospheric turbulence [1–2]. Levels of turbulence underwater can be much higher than those typically found in the atmosphere because the density of water is orders of magnitude higher than that of air and, consequently, the dependence of water’s refractive index on temperature fluctuations is much greater. As an example, in the blue-green spectral region, dn/dT ∼ −8×10⁻⁵ (K⁻¹) in water compared with ∼ −1×10⁻⁶ (K⁻¹) in air [3]. While a great number of scientific publications have been devoted to the topic of modelling and measurement of water’s absorption and scattering properties, the study of underwater optical turbulence is relatively new compared to the study of turbidity contributions. This stems from the fact that absorption and scattering are the limiting factors for underwater imaging and laser communications. In clear ocean water, only ∼10⁻⁸ of the input flux emitted by a transmitter would arrive at a receiver positioned 100 m from the source, and this number would get orders of magnitude lower for harbor-type water [4].

Similarly to Tyson’s analysis in Kolmogorov atmospheric turbulence [5], later partially extended to the non-Kolmogorov case [6], in this paper we theoretically investigated the effectiveness of adaptive optics (AO) correction for Gaussian beams in oceanic turbulence. In this context, the “effectiveness of AO” refers to the capability of AO to reduce scintillation. We make the distinction between oceanic and underwater turbulence because the former semi-explicitly contains the salinity effect, which is very important for scintillation modelling as will be shown later. Our analysis is confined to direct-detection systems, i.e. those where the intensity carries information and the phase is ignored. We used the oceanic turbulence power spectrum to describe the fluctuations of the index of refraction [7]. We also made use of the Zernike spatial filter functions to model the action of idealized AO as the removal of a certain number of Zernike modes from the turbulence-affected wavefront of a Gaussian beam [8]. The use of this spatial-frequency-domain approach to fully describe both the oceanic turbulence and the AO correction process, allows us to theoretically investigate the reduction of scintillation for each removed Zernike mode (up to a certain number of modes N) for different values of the temperature and salinity components of oceanic turbulence. In fact, in contrast to atmospheric turbulence, where essentially only temperature fluctuations are important (pressure plays a marginal role), for applications related to laser beam propagation through the ocean, salinity impact is also important.

The key component of any analysis involving light propagation through a random medium, the spectrum of refractive-index fluctuations, is a linear combination of the temperature spectrum, the salinity spectrum and the temperature-salinity co-spectrum [9]. The resulting spectrum exhibits two “bumps” at high spatial frequencies: one around the Kolmogorov microscale (inner-scale) and another one for frequencies around ten times higher than the Kolmogorov microscale (cf. Fig. 1). Such a double bump can heavily affect the metrics of interest such as scintillation [10]. For this reason, we focus our attention on the impact of both temperature and salinity on scintillation, probability of fade, mean signal-to-noise ratio (SNR) and mean bit error rate (BER), under the action of AO.

Fig. 1. Scaled oceanic power spectrum

Download Full Size | PDF

We will show that scintillation can be reduced by a factor of ∼7 when removing 15 modes and when the aperture size of the transceiver is large enough to capture 4-5 speckles of the oceanic turbulence-affected beam. Consequently, the optical system performances in terms of probability of fade, signal-to-noise ratio and bit error rate can be remarkably increased using AO pre-compensation, if the optimum diameter is used for both apertures (transmitter and receiver).

2. Oceanic turbulence power spectrum

In our analysis the fluctuations of the index of refraction in the ocean are modeled by the following three-dimensional power spectrum by Nikishov & Nikishov [7]

(1)$${\Phi _n}(\kappa )= 0.388 \cdot {10^{ - 8}} \cdot {\varepsilon ^{ - 1/3}} \cdot {\kappa ^{ - 11/3}} \cdot [{1 + 2.35{{({\kappa \eta } )}^{2/3}}} ]\cdot \frac{{{X_T}}}{{{w^2}}} \cdot ({{w^2} \cdot {e^{ - {A_T} \cdot \delta }} + {e^{ - {A_S} \cdot \delta }} - 2w \cdot {e^{ - {A_{TS}} \cdot \delta }}} )$$

Here, κ=(κ_x²+κ_y²+κ_z²)^1/2 is the spatial wavenumber, ɛ is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 10⁻¹⁰ to 10⁻¹ [m² /s³], X_T is the rate of dissipation of mean-squared temperature ranging from 10⁻¹⁰ in deep ocean to 10⁻² [K² /s] on the surface, η is the Kolmogorov microscale, w is a unit-less parameter giving the ratio of temperature and salinity contributions to the refractive index spectrum, where $w ={-} 5$ means temperature-dominated turbulence and $w \to 0$ salinity dominated. Note that this oceanic power spectrum is not valid for the case w = 0. In addition, the remaining parameters are:

(2)$${A_T} = 1.863 \cdot {10^{ - 2}};\textrm{ }{A_S} = 1.9 \cdot {10^{ - 4}};\textrm{ }{A_{TS}} = 9.41 \cdot {10^{ - 3}};\textrm{ }\delta = 8.284 \cdot {({\kappa \eta } )^{\frac{4}{3}}} + 12.978 \cdot {({\kappa \eta } )^2}$$

and the constant $0.388 \cdot {10^{ - 8}}$ is obtained from $\frac{1}{{4\pi }} \cdot \alpha _T^2 \cdot {C_0}$ with ${\alpha _T} = 2.6 \cdot {10^{ - 4}}[\textrm{L/deg}]$ and ${C_0} = 0.72$ (known as Obukhov-Corrsin constant) [11]. We plot in Fig. 1 the oceanic power spectrum scaled by $\frac{1}{{0.388 \cdot {{10}^{ - 8}}}} \cdot \frac{{{\varepsilon ^{1/3}}}}{{{X_T}}} \cdot {\kappa ^{11/3}} \cdot \frac{{{w^2}}}{{{w^2} + 1 - 2w}} \cdot {C_0}$ to better show the bumps. Note that the scaled power spectrum approaches the value ${C_0} = 0.72$ when $\kappa \eta \to 0$. Spectrum given by Eq. (1) is the basis of the calculations of scintillation before and after AO correction, which is the main parameter used to calculate the performance metrics such as SNR or BER. In this paper, we will show the strong effect of the bump due to higher salinity (lower $|w |$), clearly visible in Fig. 1, on scintillation and, in turn, on optical system performance before and after AO correction.

3. On-axis scintillation index without adaptive optics

Oceanic optical turbulence is generated by eddies with sizes ranging from the outer scale (on the order of meters) to the inner scale (on the order of millimeters). These eddies act as many lenses that locally introduce aberrations across the wavefront of the beam. Propagation of the distorted wavefront will generate constructive and destructive interference responsible for intensity fluctuations across the beam. This effect is known as scintillation. The scintillation pattern is characterized by several dark and bright regions called speckles moving randomly across the beam cross-section. The parameter used to quantify the intensity fluctuations is the scintillation index which is defined as:

(3)$$\sigma _I^2 = \frac{{\left\langle {{I^2}} \right\rangle - {{\left\langle I \right\rangle }^2}}}{{{{\left\langle I \right\rangle }^2}}}\; \; \;$$

where I stands for intensity measured in the receiver’s pupil plane, and in the following analysis the on-axis point is taken as the measurement place.

The refraction and diffraction properties of a propagating Gaussian beam can be described using a couple of geometric parameters at the transmitter and the receiver side, see Fig. 2.

Fig. 2. Commonly used parameters to describe propagation characteristics of a Gaussian laser beam in the transmitter plane (${{\Theta }_0}$ and ${{\Lambda }_0}$) and in the receiver plane (${{\Theta }_1}$ and ${{\Lambda }_1}$). Additionally, the following definitions are used: ${F_0}$ and ${W_0}$ – transmitter-plane wavefront radius of curvature and Gaussian intensity beam radius (1/e² value), respectively, and $k \equiv 2\pi /\lambda $ where $\lambda $ is the wavelength of the laser.

Download Full Size | PDF

These parameters enter the scintillation index equation, obtained by using the Rytov method. The analysis is quite complex and here we give only the final result [12]:

(4)$$\begin{array}{l} \sigma _I^2 = 8{\pi ^2} \cdot {k^2} \cdot L \cdot {\mathop{\rm Re}\nolimits} \left\{ {\int\limits_0^1 {\int\limits_0^\infty {\kappa \cdot {\Phi _n}(\kappa )\cdot {e^{ - \frac{{\Lambda L}}{k} \cdot {\xi^2} \cdot {\kappa^2}}} \cdot \left[ {1 - {e^{ - j\frac{{L \cdot \xi \cdot ({1 - \bar{\Theta }\xi } )}}{k} \cdot {\kappa^2}}}} \right]d\kappa d\xi } } } \right\}\\ = 8{\pi ^2} \cdot {k^2} \cdot L \cdot {\chi _T} \cdot {\varepsilon ^{ - \frac{1}{3}}} \cdot 0.388 \cdot {10^{ - 8}} \cdot {w^{ - 2}} \cdot \\ \times \left[ {{w^2}\left( {{I_{T, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{T,3}}} \right) + \left( {{I_{S, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{S,3}}} \right) - 2w\left( {{I_{TS, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{TS,3}}} \right)} \right] \end{array}$$

where $\xi \equiv 1 - \frac{z}{L},\; {\bar{\Theta }} \equiv 1 - {\Theta }$ and

(5)$$\begin{array}{l} {I_{i = [{T,S,TS} ], - 11/3}} = \sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{2}} \frac{{{{({{A_i} \cdot {a_3}} )}^n}}}{{n!}}\Gamma \left( {\frac{2}{3}n - \frac{5}{6}} \right).\\ \times \left\{ {{{({{A_i} \cdot {a_4}} )}^{\left( {\frac{5}{6} - \frac{2}{3}n} \right)}}{ \cdot_2}{F_1}\left\{ {\frac{2}{3}n - \frac{5}{6},\frac{1}{2};\frac{3}{2}; - \frac{{\Lambda L}}{{k \cdot {a_4}}} \cdot \frac{1}{{{A_i}}}} \right\} - {\mathop{\rm Re}\nolimits} \int\limits_0^1 {{{\left[ {{A_i} \cdot {a_4} + \frac{{\Lambda L}}{k} \cdot {{\xi }^2} + j\frac{L}{k} \cdot {\xi }({1 - \bar{\Theta } {\xi }} )} \right]}^{\left( {\frac{5}{6} - \frac{2}{3}n} \right)}}} } \right\} \end{array}$$

(6)$$\begin{array}{l} {I_{i = [{T,S,TS} ],3}} = \sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{2}} \frac{{{{({{A_i} \cdot {a_3}} )}^n}}}{{n!}}\Gamma \left( {\frac{2}{3}n - \frac{1}{2}} \right).\\ \times \left\{ {{{({{A_i} \cdot {a_4}} )}^{\left( {\frac{1}{2} - \frac{2}{3}n} \right)}}{ \cdot_2}{F_1}\left\{ {\frac{2}{3}n - \frac{1}{2},\frac{1}{2};\frac{3}{2}; - \frac{{\Lambda L}}{{k \cdot {a_4}}} \cdot \frac{1}{{{A_i}}}} \right\} - {\mathop{\rm Re}\nolimits} \int\limits_0^1 {{{\left[ {{A_i} \cdot {a_4} + \frac{{\Lambda L}}{k} \cdot { {\xi }^2} + j\frac{L}{k} \cdot {\xi }({1 - \bar{\Theta } {\xi }} )} \right]}^{\left( {\frac{1}{2} - \frac{2}{3}n} \right)}}} } \right\} \end{array}$$

Here ${a_3} = 8.284 \cdot {\eta ^{\frac{4}{3}}},\textrm{ }{a_4} = 12.978 \cdot {\eta ^2}$, ₂F₁(.) is the hypergeometric-type function, L is the path length and j is the imaginary unit. The values of constants ${A_T}$, ${A_S}$ and ${A_{TS}}$ are given in Eq. (2).

We remark that the Rytov method to describe the optical effects of turbulence is only valid when the Rytov variance is lower than unity (weak turbulence regime). Rytov variance is defined as the scintillation index of a plane wave, and can be obtained from the previous equation by setting $ {\Theta } = 1$ and $ {\Lambda } = 0$.

4. On-axis scintillation index with adaptive optics

In the following analysis it is assumed that AO is implemented in the pre-compensation mode, i.e. the outgoing beam on the transmitter side is pre-aberrated in such a way that the turbulent aberrations on its path to the receiver will get cancelled. In the ideal case of perfect phase conjugation the beam arrives at the receiver with a corrected phasefront and thus scintillation at the receiver will also be reduced. Correction of both phase and scintillation is also possible on the receiver side in the post-compensation mode with two deformable mirrors [13], but such schemes are iterative and too slow for communications applications. Figure 3 shows schematically the evolution of a pre-compensated wavefront along the propagation path.

Fig. 3. Schematic representation of pre-compensation AO.

Download Full Size | PDF

The mechanics of AO implementation such as availability of the beacon, the modality of wavefront sensing, characteristics of the deformable mirror used, etc. are not considered here. Adaptive optics is mathematically included in the scintillation equations using Zernike filter functions which are given by [14]

(7)$$\begin{array}{l} {F_{\textrm{even}\; m,n}}({\kappa ,\gamma D,\phi } )= ({n + 1} ){\left[ {\frac{{2{J_{n + 1}}({\kappa \gamma D/2} )}}{{\kappa \gamma D/2}}} \right]^2}\textrm{2co}{\textrm{s}^2}m\phi \\ {F_{\textrm{odd}\; m,n}}({\kappa ,\gamma D,\phi } )= ({n + 1} ){\left[ {\frac{{2{J_{n + 1}}({\kappa \gamma D/2} )}}{{\kappa \gamma D/2}}} \right]^2}\textrm{2si}{\textrm{n}^2}m\phi \\ {F_{m = 0,n}}({\kappa ,\gamma D,\phi } )= ({n + 1} ){\left[ {\frac{{2{J_{n + 1}}({\kappa \gamma D/2} )}}{{\kappa \gamma D/2}}} \right]^2} \end{array}$$

Note that the expressions contain the products of Bessel and trigonometric functions as well as D, the transmitter’s diameter, multiplied by the propagation parameter $\gamma $ which depends on the geometrical beam parameters defined in Fig. 3, and which physically scales up or down the aperture diameter D along the path, depending on whether the beam is divergent or convergent.

Because the filter functions act layer by layer along the propagation path, the AO-corrected power spectrum can be defined as [12]

(8)$${\Phi _n}{(\kappa )_{\textrm{AO}}} = {\Phi _n}(\kappa )\cdot \left[ {1 - \sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} } \right]$$

Inserting the AO-corrected oceanic spectrum into the scintillation equation allows us to express the scintillation index with AO as the difference between the scintillation index without AO (Eq. (4)) and the scintillation index reduction term due to AO correction (c), as

(9)$$\sigma _{I,\textrm{ AO}}^2(L )= \sigma _I^2(L )- \sigma _{I,\textrm{ c}}^2\left( {\sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} } \right)$$

where the latter is given by

(10)$$\begin{array}{l} \sigma _{I,\textrm{ c}}^2 = 8{\pi ^2} \cdot {k^2} \cdot L \cdot \\ \times {\mathop{\rm Re}\nolimits} \left\{ {\int\limits_0^1 {\int\limits_0^\infty {\kappa \cdot {\Phi _n}(\kappa )\cdot \sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} \cdot {e^{ - \frac{{\Lambda L}}{k} \cdot {\xi^2} \cdot {\kappa^2}}} \cdot \left[ {1 - {e^{ - j\frac{{L \cdot \xi \cdot ({1 - \bar{\Theta }\xi } )}}{k} \cdot {\kappa^2}}}} \right]d\kappa d\xi } } } \right\} \end{array}$$

This final expression serves as input to calculations of the probability of fade, mean signal-to-noise ratio, and mean bit error rate, under the assumption of perfect removal of the first N Zernike modes. The results of these calculations without aperture averaging are shown in [12]. However, in Section 6, we show the more complete results including aperture averaging of the collecting lens on the receiver side.

5. Scintillation with aperture averaging without adaptive optics

A real free-space communication system operating underwater should be symmetric and bidirectional. In such systems, the beam, after propagation, is focused on the detector by the collecting lens at the receiver side, and therefore scintillation is reduced by aperture averaging. In this section the theoretical analysis of aperture-averaged scintillation is carried out.

The normalized variance of the power fluctuations in the receiver plane is the same as the flux variance of irradiance fluctuations defined by [15]

(11)$$\sigma _I^2({{D_G}} )= \frac{{\left\langle {{P^2}} \right\rangle - {{\left\langle P \right\rangle }^2}}}{{{{\left\langle P \right\rangle }^2}}}$$

where ${D_G}$ is the diameter of the receiver and P is the received optical power. In the following analysis the aperture-averaged value of the intensity fluctuations is calculated in the focal plane at distance ${L_f}$ from the collecting lens of the receiver aperture (see Fig. 4).

Fig. 4. The same system configuration as shown in Fig. 3 with scintillation averaged by the aperture, and measured in the focal plane using a sensor (photo-detector).

Download Full Size | PDF

The flux variance of irradiance fluctuations can be expressed as

(12)$$\begin{array}{l} \sigma _I^2({{D_G}} )= 8{\pi ^2} \cdot {k^2} \cdot L \cdot \\ \times \int\limits_0^1 {\int\limits_0^\infty {\kappa \cdot {\Phi _n}(\kappa )\cdot {e^{ - {\kappa ^2} \cdot \frac{{\gamma _{ave}^2 \cdot D_G^2}}{{16}}}} \cdot \left[ {1 - \cos \left( { - \frac{{L \cdot {\kappa^2}}}{k}\frac{{({{\Omega _G} - {\Lambda _1}} )}}{{({{\Omega _G} + {\Lambda _1}} )}}\xi \cdot ({1 - {{\bar{\Theta }}_1}\xi } )} \right)} \right]d\kappa d\xi } } ,\textrm{ }{\Omega _G} \ge {\Lambda _1}\\ = 8{\pi ^2} \cdot {k^2} \cdot L \cdot {\chi _T} \cdot {\varepsilon ^{ - \frac{1}{3}}} \cdot 0.388 \cdot {10^{ - 8}} \cdot {w^{ - 2}} \cdot \\ \times \left[ {{w^2}\left( {{I_{T, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{T, - 3}}} \right) + \left( {{I_{S, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{S, - 3}}} \right) - 2w\left( {{I_{TS, - 11/3}} + 2.35{\eta^{\frac{2}{3}}}{I_{TS, - 3}}} \right)} \right] \end{array}$$

where $\gamma _{ave}^2 = \frac{{{\Omega _G}}}{{{\Lambda _1} + {\Omega _G}}} \cdot [{{{({1 - {{\bar{\Theta }}_1}\xi } )}^2} + {\Lambda _1} \cdot {\Omega _G} \cdot {\xi^2}} ]; {\bar{\Theta }_1} = 1 - {\Theta _1};$ and the couples of beam parameters $({{\Theta _1},{\Lambda _1}} )$, $({{\Theta _0},{\Lambda _0}} )$ are defined in Fig. 4. The terms corresponding to temperature, salinity and temperature-salinity contributions are given by

(13)$$\begin{array}{l} {I_{i = [{T,S,TS} ], - 11/3}} = \sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{2}} \frac{{{{({{A_i} \cdot {a_3}} )}^n}}}{{n!}}\Gamma \left( {\frac{2}{3}n - \frac{5}{6}} \right) \cdot \\ \times \left\{ {{\mathop{\rm Re}\nolimits} \int\limits_0^1 {{{\left( {{A_i} \cdot {a_4} + \gamma_{ave}^2 \cdot \frac{{D_G^2}}{{16}}} \right)}^{\left( {\frac{5}{6} - \frac{2}{3}n} \right)}}d\xi } } \right.\left. { - {\mathop{\rm Re}\nolimits} {{\int\limits_0^1 {\left[ {{A_i} \cdot {a_4} + \gamma_{ave}^2 \cdot \frac{{D_G^2}}{{16}} + j\frac{L}{k}\frac{{({{\Omega _G} - {\Lambda _1}} )}}{{({{\Omega _G} + {\Lambda _1}} )}}\xi \cdot ({1 - {{\bar{\Theta }}_1}\xi } )} \right]} }^{\left( {\frac{5}{6} - \frac{2}{3}n} \right)}}d\xi } \right\} \end{array}$$

(14)$$\begin{array}{l} {I_{i = [{T,S,TS} ],\textrm{ - }3}} = \sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{2}} \frac{{{{({{A_i} \cdot {a_3}} )}^n}}}{{n!}}\Gamma \left( {\frac{2}{3}n - \frac{1}{2}} \right) \cdot \\ \times \left\{ {{\mathop{\rm Re}\nolimits} \int\limits_0^1 {{{\left( {{A_i} \cdot {a_4} + \gamma_{ave}^2 \cdot \frac{{D_G^2}}{{16}}} \right)}^{\left( {\frac{1}{2} - \frac{2}{3}n} \right)}}d\xi } } \right.\left. { - {\mathop{\rm Re}\nolimits} {{\int\limits_0^1 {\left[ {{A_i} \cdot {a_4} + \gamma_{ave}^2 \cdot \frac{{D_G^2}}{{16}} + j\frac{L}{k}\frac{{({{\Omega _G} - {\Lambda _1}} )}}{{({{\Omega _G} + {\Lambda _1}} )}}\xi \cdot ({1 - {{\bar{\Theta }}_1}\xi } )} \right]} }^{\left( {\frac{1}{2} - \frac{2}{3}n} \right)}}d\xi } \right\} \end{array}$$

Here ${\Omega _G} = \frac{{2L}}{{kW_G^2}}$, and $W_G^{}$ is the effective radius related to the diameter of the “soft aperture” at the receiver side by ${D_G} = 8W_G^2$.

We again remark that the Rytov method to describe the effects of optical turbulence is only valid when the Rytov variance is lower than unity (weak turbulence regime). Also note that scintillation vanishes when ${\Omega _G} = {\Lambda _1}$, i.e. when the radius of the collecting lens equals the radius of the diffraction limited incident beam in free-space, W.

6. Scintillation with aperture averaging and adaptive optics

Similarly to Section 4, it is assumed that AO is implemented in the pre-compensation mode, however, in this section we include also the aperture averaging effect of the collecting lens at the receiver side.

For this analysis we still use the AO-corrected power spectrum, Eq. (8), but now the scaling parameter inside the Zernike filter functions, Eq. (7), becomes [15]

(15)$$\gamma ={-} \frac{L}{{{L_f}({{\Lambda _1} + {\Omega _G}} )}}[{{\Lambda _1}\xi + j({1 - {{\bar{\Theta }}_1}\xi } )} ]$$

As mentioned earlier, the parameter $\gamma$ physically scales the size of the beam along its path. The first part of the path is the distance L from the transmitter to the receiver aperture and the second part is the distance ${L_f}$ after the collecting lens. Note also that the condition ${\Omega _G} \ge {\Lambda _1}$ in Eq. (12) means that the diffraction-limited beam size at the lens should be equal to or greater than the size of the lens.

Inserting the AO-corrected oceanic spectrum into the scintillation equation allows us to express the scintillation corrected by AO as the difference between the aperture-averaged scintillation without AO (Eq. (12)), and the aperture-averaged scintillation reduction term due to AO correction (c), i.e.:

(16)$$\begin{array}{l} {\sigma _I}^2{({{D_G}} )_{\textrm{AO}}} = 8{\pi ^2} \cdot {k^2} \cdot L \cdot \\ \times {\mathop{\rm Re}\nolimits} \left\{ {\int\limits_0^1 {\int\limits_0^\infty {\kappa \cdot {\Phi _n}(\kappa )\cdot \left[ {1 - \sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} } \right] \cdot {e^{ - {\kappa^2} \cdot \frac{{\gamma_{ave}^2 \cdot D_G^2}}{{16}}}} \cdot \left[ {1 - {e^{ - j\frac{{L \cdot {\kappa^2}}}{k}\frac{{({{\Omega _G} - {\Lambda _1}} )}}{{({{\Omega _G} + {\Lambda _1}} )}}\xi \cdot ({1 - {{\bar{\Theta }}_1}\xi } )}}} \right]d\kappa d\xi } } } \right\}\\ = {\sigma _I}^2({{D_G}} )- 8{\pi ^2} \cdot {k^2} \cdot L \cdot \\ \times {\mathop{\rm Re}\nolimits} \left\{ {\int\limits_0^1 {\int\limits_0^\infty {\kappa \cdot {\Phi _n}(\kappa )\cdot \sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} \cdot {e^{ - {\kappa^2} \cdot \frac{{\gamma_{ave}^2 \cdot D_G^2}}{{16}}}} \cdot \left[ {1 - {e^{ - j\frac{{L \cdot {\kappa^2}}}{k}\frac{{({{\Omega _G} - {\Lambda _1}} )}}{{({{\Omega _G} + {\Lambda _1}} )}}\xi \cdot ({1 - {{\bar{\Theta }}_1}\xi } )}}} \right]d\kappa d\xi } } } \right\}\\ = {\sigma _I}^2({{D_G}} )- \sigma _{I,\textrm{ c}}^2\left( {{D_G},\sum\limits_{i = 1}^N {{F_i}({\kappa ,\gamma D,\Phi } )} } \right) \end{array}$$

We consider the case of a symmetric laser communication system, where the diameters of the transmitter, D, and receiver, D_G, are identical.

We plot in Fig. 5 the uncorrected scintillation divided by the scintillation with N modes removed as a function of the aperture diameter in units of the first Fresnel zone as defined in [15], equal to $\sqrt {L/k} $. We set $\lambda $ = 417 nm, ε = 10⁻⁵ m²/s³, η=10⁻³ m, X_T = 10⁻⁷ K²/s, w = -3, and consider a collimated beam at the transmitter for two cases of propagation distances: (left) L = 100 m and (right) L = 150 m).

Fig. 5. Scintillation of a Gaussian beam scaled by the scintillation of an AO-corrected Gaussian beam as a function of the transmitter diameter scaled by the first Fresnel zone for different numbers of Zernike modes removed, L=100 meters case (top) and L=150 meters case (bottom). The peaks represent the best aperture size to make AO most effective. We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m.

Download Full Size | PDF

From these plots we can visually extract the optimum aperture diameter, defined as the aperture diameter that maximizes the effectiveness of AO. We deduce from Fig. 5 (left) that for L = 100 m, if we remove 20 modes the optimum diameter is around 5.3 times the first Fresnel zone. Similarly, we infer from Fig. 5 (right) that for L = 150 m and removal of N = 20 modes, the optimum diameter is about 4.3 times the first Fresnel zone. Scintillation is reduced by a factor of ∼7 for both cases when removing 15 modes and when the optimum aperture diameter is used. Note that the optimum diameter size depends on the propagation distance, the wavelength, the aperture diameter and the number of Zernike modes removed.

The theory has produced the expected result, similar to that known from astronomical AO, that the optimal aperture size should be several times larger than the dominant turbulence scale (speckle size). In astronomical AO (see Ref. [16]) this scale is given by the Fried parameter r₀ but here, because of much shorter distances, it is the first Fresnel zone $\sqrt {L/k} $.

It is also interesting to observe that the correlation width ${\rho _c}$ associated with irradiance fluctuations of a collimated beam propagated a distance L in weak turbulence is ${\rho _c} \cong \sqrt {L/k}$ whereas in strong turbulence it is ${\rho _c} \cong {\rho _0}$, where ${\rho _0}$ is the transverse coherence length of the beam [15]. From this fact we posit, that validity of the inferences based on Fig. 5, with regards to the optimum aperture size, can be extended to the strong turbulence regime if aperture sizes are represented in units of ${\rho _0}$, instead of $\sqrt {L/k} $.

The correlation width ${\rho _c}$ is defined as the 1/e² point of the normalized covariance function of irradiance fluctuations, which describes how the irradiance fluctuations at one point in the beam are correlated with those at the symmetric point with respect to the center of the beam. It is well known that aperture sizes larger than ${\rho _c}$ can lead to significant reduction in scintillation by the action of aperture averaging [15]. This result is confirmed also for oceanic turbulence, and here it is extended to the case where a combined action of AO and aperture averaging is present. Physically, this result means that for an AO system to be effective, the aperture size should be able to capture 4-5 speckles of the turbulence-affected beam. Aperture sizes on the order of the correlation width ${\rho _c}$ or smaller will act like “point apertures” and, as a consequence, neither aperture averaging nor an effective AO action will take place. These considerations are very important for building direct-detection free-space optical communications terminals with AO.

In Fig. 6 we show the decrease of scintillation index as a function of the number of Zernike modes removed if the optimum identical diameter for both the transmitter and the receiver is used. We can also clearly see from Fig. 6 that high salinity (w = -1) results in a higher scintillation due to the increase of the bump in the oceanic power spectrum (see Fig. 1). It can also be inferred, for the specific cases analyzed here, that the effect of AO correction starts to wane at around 10 modes.

Fig. 6. Scintillation index for three different salinity values as a function of the number of Zernike modes removed if the optimum diameter for the transmitter and the receiver is used. Adaptive optics is clearly effective for scintillation reduction. We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m

Download Full Size | PDF

Additionally, in Fig. 7 we plot the scintillation index as a function of the propagation distance for different numbers of Zernike modes removed. Again, the benefit of AO for reducing scintillation is evident. In particular, at a distance of 80 meters, scintillation is reduced from 0.32 to 0.05 by removing 10 modes and diminishing gains are observed when removing higher number of modes (black and red lines almost overlap).

Fig. 7. Scintillation index as a function of propagation distance for different numbers of Zernike modes removed. Here we used a divergent beam and a higher X_T than Fig. 6. We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m.

Download Full Size | PDF

7. Performance of free-space optical communications in oceanic turbulence

We investigate the benefits of AO to improve optical system performance for laser beam propagation in oceanic turbulence. Performance of communication systems is usually quantified by probability of fade, signal-to-noise ratio (SNR), and bit error rate (BER). Our analysis considers only shot-noise-limited systems utilizing direct detection. In this scheme, schematically shown in Fig. 8, the corrected wavefront arrives at the optical sensor, which generates an electric current signal, which is then filtered by a front-end filter to remove out-of-band noise. In our analysis the beam is supposed to be perfectly focused on the photo-detector (which is positioned exactly at the focal plane).

Fig. 8. Schematic illustration of a receiver side of a direct-detection system. The transmitted wave shows a pre-distorted phase because of AO pre-compensation.

Download Full Size | PDF

Note that the transmitted wave shows a pre-distorted phase because of AO pre-compensation (cf. Fig. 3). As explained in Section 4, it is assumed that the outgoing beam on the transmitter side is pre-aberrated in such a way that the turbulent aberrations on its path to the receiver will get cancelled. This method requires the knowledge of the phase aberrations the beam is going to acquire along the path, and such information can be obtained from the phase of an incoming beacon positioned at the receiver side. Once the phase profile of the incoming beacon is known, phase-conjugation is applied to the outgoing beam.

7.1 Probability of fade analysis

Given a model for the probability density function (PDF) of the pupil-plane irradiance fluctuations, the probability of fade describes the percentage of time the irradiance of the received signal is below some prescribed threshold value I_T. It is defined by

(17)$$\Pr ({I < {I_T}} )= \int_0^{{I_T}} {{p_I}(I )dI}$$

The PDF most often used in atmospheric turbulence under the assumption of weak irradiance fluctuations is the lognormal model and the resulting probability of fade is then

(18)$$\Pr ({I < {I_T}} )= \frac{1}{2} \cdot \left\{ {1 + \textrm{erf}\left[ {\frac{{\frac{1}{2} \cdot \sigma_I^2{{({{D_G}} )}_{\textrm{AO}}} - 0.23 \cdot {F_T}}}{{\sqrt 2 \cdot \sigma_I^{}{{({{D_G}} )}_{\textrm{AO}}}}}} \right]} \right\}$$

where $\sigma _I^2{({{D_G}} )_{\textrm{AO}}}$ is given by Eq. (16), and F_T is the ratio of mean intensity at the receiver, after aperture averaging, and I_T is the decision threshold in dB:

(19)$${F_T} = 10 \cdot {\log _{10}}\left( {\frac{{\left\langle I \right\rangle }}{{{I_T}}}} \right)[dB]$$

First, we suppose that for weak oceanic turbulence the lognormal PDF holds as well. Next, using the same optimum diameter for both apertures, we plot in Fig. 9 the probability of fade as a function of the number of Zernike modes removed for several values of salinity.

Fig. 9. Probability of fade as a function of the Zernike modes removed for different values of salinity. We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m.

Download Full Size | PDF

We deduce from Fig. 9 that for this specific scenario of propagation, a probability of fade of 10⁻⁶ can be reached by removing at least 14 modes in channels with low to medium salinity.

7.2 Signal-to-noise ratio

SNR is often used to measure the system performance at a receiver. In absence of turbulence, SNR is defined by the ratio of mean signal current to root-mean-square shot noise power:

(20)$$SN{R_0} = \frac{{{i_s}}}{{{\sigma _N}}}$$

In turbulence, SNR is described by a mean SNR given by [15]

(21)$$\left\langle {SNR} \right\rangle = \frac{{\left\langle {{i_s}} \right\rangle }}{{{\sigma _N}}} = \frac{{SN{R_0}}}{{\sqrt {\frac{{{P_{S0}}}}{{\left\langle {{P_S}} \right\rangle }} + \sigma _I^2{{({{D_G}} )}_{\textrm{AO}}} \cdot SN{R_0}^2} }} \cong \frac{{SN{R_0}}}{{\sqrt {1 + \sigma _I^2{{({{D_G}} )}_{\textrm{AO}}} \cdot SN{R_0}^2} }}$$

Here, without losing generality, we suppose that the ratio of the on-axis received power without turbulence Ps₀ and the mean received power including turbulence < Ps > is close to unity, or in other words, the spread due to turbulence and beam wander can be neglected [15]. However, even if we suppose a more realistic value of Ps₀, let’s say double < Ps>, results are essentially not affected considering practical SNR₀ values (usually higher than 15-20 dB). In addition, it should be noted, that even a very high SNR₀ does not lead to a reasonable < SNR > because of scintillation, which is the dominant factor in this case:

(22)$${\left\langle {SNR} \right\rangle _{SN{R_0} \to \infty }} \cong \frac{1}{{\sigma _I^{}{{({{D_G}} )}_{\textrm{AO}}}}}$$

We plot in Fig. 10, the mean SNR with turbulence, <SNR>, as a function of the SNR without turbulence, SNR₀, for several numbers of Zernike modes removed from the wavefront. Both figures show the same scenario of propagation for two different salinity values. The high-salinity case is shown in Fig. 10 (left) and the zero-salinity case is shown in Fig. 10 (right). The positive impact of AO is clearly visible for both cases: increasing the number of removed modes brings the < SNR > closer to the ideal case of no turbulence, which is plotted as the black dashed line.

Fig. 10. Mean signal to noise ratio < SNR > as a function of the free-space signal-to-noise ratio SNR₀ for different numbers of removed Zernike modes: strong salinity case, w = -1 (top) and no salinity case, w = -5 (bottom). We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m.

Download Full Size | PDF

7.3 Bit error rate

In presence of turbulence, BER is considered a conditional probability that must be averaged over the PDF of the random signal to determine the mean BER

(23)$$\Pr (E) = \left\langle {BER} \right\rangle = \frac{1}{2}\int\limits_0^\infty {{p_I}(u )} \cdot \textrm{erfc}\left( {\frac{{\left\langle {SNR} \right\rangle u}}{{2\sqrt 2 }}} \right)du$$

where ${p_I}(u )$ is the lognormal PDF associated with the irradiance power fluctuations, erfc(.) is the complementary error function, and u is the normalized signal current

(24)$$u = \frac{{{i_s}}}{{\left\langle {{i_s}} \right\rangle }};\left\langle u \right\rangle = 1$$

We plot in Fig. 11 the mean BER as a function of the mean signal-to-noise ratio for several numbers of removed Zernike modes for the case of medium salinity, w = -3. It can be deduced from Fig. 11 that < BER > can be improved by more than three orders of magnitude for < SNR> = 25 dB if 15 Zernike modes are removed from the wavefront by AO. The improvement of < BER > through AO is reduced for lower < SNR > values.

Fig. 11. Mean bit error rate as a function of the mean signal-to-noise ratio for several values of N – the number of Zernike modes removed from the wavefront (medium salinity case, w = -3). We set ε = 10⁻⁵ m²/s³ and η=10⁻³ m.

Download Full Size | PDF

8. Summary

In this paper the theoretical analysis of the effectiveness of AO for free-space communications systems operating in oceanic turbulence has been presented. The analysis includes aperture averaging effects, is derived for the weak fluctuations regime, and is valid only for direct detection systems. We found that scintillation can be reduced by a factor of ∼7 when removing 15 modes and when the aperture size of the transceiver was large enough to capture 4-5 speckles of the oceanic turbulence-affected beam. Specifically, for the two considered scenarios of propagation, L = 100 m and L = 150 m, if 20 modes are removed, we found that the optimal transmitter/receiver size is 5.3 and 4.3 times the correlation width ${\rho _c}$, respectively. This translates to an aperture size of 1.365 and 1.357 cm, respectively. In weak turbulence regime ${\rho _c}$ is approximately equal to the first Fresnel zone $\sqrt {L/k} $, whereas in strong turbulence regime ${\rho _c}$ approaches the spatial coherence radius ${\rho _0}$. Consequently, the optical system performances in terms of probability of fade, signal-to-noise ratio and bit error rate can be remarkably increased using AO pre-compensation, if the optimum diameter is used for both apertures (transmitter and receiver). These considerations are very important for building direct-detection free-space optical communications terminals equipped with AO. Finally, recently several papers discussed also the propagation of laser beams in anisotropic oceanic turbulence [17–18]. The presented analysis could easily accommodate more complicated, anisotropic spectra, but the general approach would remain the same and the trends of the results can be expected to persist.

Funding

Office of Naval Research Global (N62909-17-1-2037).

Disclosures

The authors declare no conflicts of interest.

References

1. G. Nootz, E. Jarosz, F. R. Dalgleish, and W. Hou, “Quantification of optical turbulence in the ocean and its effects on beam propagation,” Appl. Opt. 55(31), 8813–8820 (2016). [CrossRef]

2. J. P. Montoya, M. Segel, S. Gladysz, M. Kremer, and K. Stein, “Measurements of Temperature and Image Motion Structure Functions in a Rayleigh-Bénard Water Tank,” in Imaging and Applied Optics 2019 (COSI, IS, MATH, pcAOP), OSA Technical Digest (Optical Society of America, 2019), paper PTh4C.3.

3. F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49(16), 3224–3230 (2010). [CrossRef]

4. W. Cox and J. Muth, “Simulating channel losses in an underwater optical communication system,” J. Opt. Soc. Am. A 31(5), 920–934 (2014). [CrossRef]

5. Robert K. Tyson, “Bit-error rate for free-space adaptive optics laser communications,” J. Opt. Soc. Am. A 19(4), 753–758 (2002). [CrossRef]

6. Italo Toselli and Szymon Gladysz, “Efficiency of adaptive optics correction for Gaussian beams propagating through non-Kolmogorov turbulence,” in Imaging and Applied Optics 2014, paper PM3E.6, (2014).

7. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000). [CrossRef]

8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

9. X. Yi and I. B. Djordjevic, “Power spectrum of refractive-index fluctuations in turbulent ocean and its effect on optical scintillation,” Opt. Express 26(8), 10188–10202 (2018). [CrossRef]

10. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68(8), 1067–1072 (1978). [CrossRef]

11. O. Korotkova, “Light Propagation in a Turbulent Ocean,” Prog. Opt. 64, 1–43 (2019). [CrossRef]

12. I. Toselli and S. Gladysz, “Adaptive optics correction of scintillation for oceanic turbulence-affected laser beams,” Proc. SPIE 10787, 1078709 (2018). [CrossRef]

13. M. C. Roggemann and D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37(21), 4577–4585 (1998). [CrossRef]

14. Richard J. Sasiela, Electromagnetic wave propagation in turbulence. Evaluation and application of Mellin transforms, 2nd ed. (SPIE Publications, 2007).

15. Larry Andrews and Ronald Phillips, Laser beam propagation through random media, 2nd ed. (SPIE Publications, 2005).

16. R. W. Wilson and C. R. Jenkins, “Adaptive optics for astronomy: theoretical performance and limitations,” Mon. Not. R. Astron. Soc. 278(1), 39–61 (1996). [CrossRef]

17. M. C. Gökçe, “Average capacity analysis of underwater optical wireless communication links over anisotropic strong oceanic turbulence channels,” J. Opt. Soc. Am. A 36(12), 2040–2047 (2019). [CrossRef]

18. G. Xu and J. Lai, “Scintillation index and BER performance for optical wave propagation in anisotropic underwater turbulence under the effect of eddy diffusivity ratio,” Appl. Opt. 59(8), 2551–2558 (2020). [CrossRef]

Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence

Abstract

1. Introduction

2. Oceanic turbulence power spectrum

3. On-axis scintillation index without adaptive optics

4. On-axis scintillation index with adaptive optics

5. Scintillation with aperture averaging without adaptive optics

6. Scintillation with aperture averaging and adaptive optics

7. Performance of free-space optical communications in oceanic turbulence

7.1 Probability of fade analysis

7.2 Signal-to-noise ratio

7.3 Bit error rate

8. Summary

Funding

Disclosures

References

Cited By

Figures (11)

Equations (24)

Optics Express