Spatially partially coherent beam parameter optimization for free space optical communications

Deva K. Borah; David G. Voelz

doi:10.1364/OE.18.020746

1. Introduction

It is known that the use of a spatially partially coherent beam (PCB) can improve the performance of free space laser communications through atmospheric turbulence [1], [2]. One of the reasons for this improvement is a reduction in coherent interference, which lowers the intensity fluctuation (scintillation) at the receiver. The reduction in intensity fluctuations of Gaussian beams with decreasing source spatial coherence was observed and reported as early as 1980s [3]. However, the divergence of a PCB is greater than an analogous coherent beam, causing a reduction of the signal level that degrades the performance of an optical communication system. Therefore, for designing a laser communication system, an important issue is the trade-off between reduction in scintillation and reduction in the mean signal intensity at the receiver and how this trade-off can be achieved through optimization of the PCB parameters.

In previous work [4], Schulz showed that a coherent beam maximizes the expected intensity whereas a PCB minimizes the scintillation index (SI). He provided an optimizing criterion for the modes of the PCB for minimizing the SI. Voelz and Xiao described a heuristic metric for determining near-optimal link performance that incorporates both the reduction in scintillation and the increase in beam spread [5]. Their work considered a collimated Gaussian Schell-model (GSM) beam propagating through turbulence, and investigated optimization of the transverse coherence length (l_c) as a function of beam size, wavelength, turbulence strength, and propagation distance under the heuristic metric criterion. For a more detailed discussion on PCBs, please see the references listed in [1], [5].

There is also a large body of literature that focuses on the SI of different beams propagating through the atmospheric turbulence channels. More recently, the works in [6] and [7] consider Bessel-Gaussian and Laguerre Gaussian beams and demonstrate that lower scintillations at on-axis and off-axis positions can be obtained using certain beam orders. Other recent works include multibeam investigations of scintillation [8], and propagation properties of a stochastic GSM beam [9]. However, since the performance of a communication system, in terms of bit error rates or outage probabilities, typically depends nonlinearly on beam parameters, including the SI, the exact trade-offs for these beams for communication purposes are not obvious.

Our work in this paper focuses on scalar PCBs for optical communications. Although PCBs generally lower the SI, this alone does not guarantee performance improvement in communication systems. In fact, in certain communications scenarios, the use of a PCB can degrade performance, and a coherent beam is rather the optimal beam configuration. The choice of a PCB over a coherent beam is influenced by several parameters including wavelength, phase front radius of curvature, and beamwidth. To our best knowledge, studies to date have not provided an analytical framework that leads to simple intuitive conclusions regarding the effects of these parameters on coherence length optimization of PCBs. In general, previous work involves extensive evaluations of a cost function or a non-linear optimization to merely confirm the potential benefits of using a PCB for a given parameter set. Detailed interpretation of PCB performance results is generally difficult using the available mathematical expressions.

In our work, the problem of optimizing the transverse coherence length of a PCB for minimal outage probability in a communications link is considered. This optimization approach and metric are more specific to the optical communications problem than have been considered previously. The study concerns the GSM beam and performance optimization as a function of various link parameters. We are not aware of other work in the literature that considers optimization of beam parameters for PCBs except the studies reported in [4], [5]. A significant extension of previous work is the inclusion of beam focus (phase front radius of curvature) and beamwidth as study parameters. A numerical test is proposed for confirming possible improvement with a PCB. We demonstrate that large initial beam size, small propagation distance and short wavelength are favorable conditions for performance enhancement using a PCB.

The paper is organized as follows. Section 2 presents the GSM beam model. In Section 3, the communication performance metric and the simplified cost function are described. Section 4 presents the derivation of the SI for PCBs in a series form. The coherence length optimization condition is discussed in detail, and a numerical test to confirm performance gains due to PCBs is given. This section also discusses the effects of various parameters, such as the phase front radius of curvature and the beamwidth, on coherence length optimization. Finally, Section 5 concludes our study and outlines directions for future research.

2. Beam model

We consider a Gaussian Schell model (GSM) beam [10], [11]. After propagation through the atmospheric turbulence channel over a distance z, the mean intensity profile, Ī(ρ), on the receiver plane is modeled as [12],

\bar{I} (ρ) = K_{n} \frac{W_{0}^{2}}{W^{2}} exp (- \frac{2 ρ^{2}}{W^{2}})

where ρ is the radial distance from the mean beam center on the transverse plane, $K_{n} = W_{r}^{2} / W_{0}^{2}$ is a normalization constant so that the total power is $(1 / 2) π W_{r}^{2}$ , W_r is the reference beam radius or beamwidth, W ₀ is the effective beam radius at the transmitter, W is the receiving beam size given by

W = W_{0} {[r_{0}^{2} + (ξ_{s} + \frac{2 W_{0}^{2}}{ρ_{0}^{2}}) z_{0}^{2}]}^{1 / 2},

r ₀ = 1 − z/F ₀ is a focusing parameter, F ₀ is the phase-front radius of curvature at the transmitter, $z_{0} = 2 z / (k W_{0}^{2})$ is normalized distance, k = 2π/λ is the wave number, λ is the wavelength, $ξ_{s} = 1 + 2 W_{0}^{2} / l_{c}^{2}$ is the source coherence parameter, l_c is the spatial coherence length, $ρ_{0} = {(0.55 C_{n}^{2} k^{2} z)}^{- 3 / 5}$ , and $C_{n}^{2}$ is the index of refraction structure parameter of the atmosphere expressed in units of m^−2/3. Noting that the Rytov variance is $σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} z^{11 / 6}$ , the SI $σ_{I}^{2}$ on or close to the beam center, including beam wander effects for an untracked beam, is given by (p. 274 in [12])

σ_{I}^{2} = 4.42 σ_{R}^{2} z_{e}^{5 / 6} \frac{σ_{pe}^{2}}{W_{0}^{2} (r_{0}^{2} + ξ_{s} z_{0}^{2})} + 3.86 σ_{R}^{2} {0.4 {[{(1 + 2 r_{e})}^{2} + 4 z_{e}^{2}]}^{5 / 12} cos [\frac{5}{6} {tan}^{- 1} (\frac{1 + 2 r_{e}}{2 z_{e}})] - \frac{11}{16} z_{e}^{5 / 6}}

where

\begin{array}{l} r_{e} = \frac{r_{0}}{r_{0}^{2} + ξ_{s} z_{0}^{2}}, & z_{e} = \frac{ξ_{s} z_{0}}{r_{0}^{2} + ξ_{s} z_{0}^{2}} \end{array}

and

σ_{pe}^{2} = ζ_{1} {(\frac{λ z}{2 W_{0}})}^{2} {(\frac{2 W_{0}}{F_{p}})}^{5 / 3} [1 - ζ_{2} {(\frac{F_{p}^{2}}{C_{r}^{2} W_{0}^{2}} + ζ_{3})}^{- 1 / 6}]

with ζ ₁ = 0.54, ζ ₂ = 8/9, and ζ ₃ = 0.5 for a focused beam, while ζ ₁ = 0.48, ζ ₂ = 1 and ζ ₃ = 1 for a collimated beam, and $F_{p} = {(0.16 C_{n}^{2} k^{2} z)}^{- 3 / 5}$ . Let us write $σ_{I}^{2} = σ_{I, bw}^{2} + σ_{I, nw}^{2}$ , where $σ_{I, bw}^{2} = 4.42 σ_{R}^{2} z_{e}^{5 / 6} \frac{σ_{pe}^{2}}{W_{0}^{2} (r_{0}^{2} + ξ_{s} z_{0}^{2})}$ is due to beam wander effects and essentially corresponds to the radial component of the SI evaluated at ρ = σ_pe, and $σ_{I, nw}^{2}$ represents the remaining term in (2). Note that we consider the received signal at the beam center and, therefore, $σ_{I, bw}^{2}$ does not contain ρ.

3. Performance metric

We consider the outage probability, P_out, which is the probability that the received signal-to-noise ratio (SNR) falls below a threshold SNR resulting in unacceptable error rates. For a given noise level, this occurs when the intensity I becomes less than a threshold intensity I_th. We will use this simplified approach in this work. Thus, we write

P_{out} = \int_{0}^{I_{th}} p (I) dI

where p(I) is the probability density function of I. Using the log-normal (LN) model for I [12], we can write

P_{out} = \int_{0}^{I_{th}} \frac{1}{I \sqrt{2 π σ_{lnI}^{2}}} exp {- \frac{{[ln (I / \bar{I}) + 0.5 σ_{lnI}^{2}]}^{2}}{2 σ_{lnI}^{2}}} dI

where $σ_{lnI}^{2} = ln (1 + σ_{I}^{2})$ . It can be easily shown that

P_{out} = Q (- y_{th})

where

y_{th} = \frac{ln I_{th} - ln \bar{I} + 0.5 σ_{lnI}^{2}}{σ_{lnI}},

and $Q (x) = 1 / \sqrt{2 π} \int_{x}^{\infty} exp (- y^{2} / 2) dy$ is the Gaussian tail probability. For weak turbulence and small P_out, y_th is a large negative number. Under these conditions, we need to minimize

y_{th} \approx \frac{ln I_{th} - ln \bar{I} + 0.5 σ_{lnI}^{2}}{σ_{lnI}} \approx \frac{1}{σ_{lnI}} ln (I_{th} / \bar{I})

For weak turbulence, $σ_{I}^{2} \approx σ_{lnI}^{2}$ , and so we consider the following cost function for minimization,

φ (l_{c}) = \frac{1}{σ_{I}} ln (\frac{I_{th}}{\bar{I}})

Note that for I_th < Ī, the value of the above cost function is a negative number, and hence zero is not the minimum value of the above cost function in general.

4. PCB Parameter selection

For certain beam configurations and turbulence scenarios, the optimal beam is simply the coherent beam. In other words, l_c = ∞ will minimize (4), and PCB is not even necessary. However, this can be ascertained only after performing a non-linear optimization on (4) with respect to l_c, or evaluating the cost function (4) or outage probability (3) for various values of l_c. This becomes more difficult when one studies the effects of combinations of other parameters, such as F ₀ and W ₀, over l_c. We show in this section that this can be done without conducting extensive optimization. Toward that end, we first provide an alternative series expansion for the SI and develop interpretations.

4.1. Expressions for the Scintillation Index

Let us first consider the SI given by (2). This index depends on l_c through trigonometric expressions. As l_c increases or decreases, it is difficult to interpret from (2) whether $σ_{I}^{2}$ will also increase or decrease. We define $x = (1 + 2 r_{e}) / (2 z_{e}) = (r_{0}^{2} + ξ_{s} z_{0}^{2} + 2 r_{0}) / (2 ξ_{s} z_{0})$ , and then show in the Appendix how $σ_{I, nw}^{2}$ can be expressed in a series form. We obtain

σ_{I}^{2} = 4.42 σ_{R}^{2} z_{e}^{5 / 6} [\frac{σ_{pe}^{2}}{W_{0}^{2} (r_{0}^{2} + ξ_{s} z_{0}^{2})} + 0.8733 g (x)]

where

g (x) = 0.7127 (1 + \frac{5}{72} x^{2} - \frac{455}{4! 6^{4}} x^{4} + \frac{43225}{144 \times 6^{6}} x^{6} - \dots) - 11 / 16

for |x| < 1, while

g (x) = 0.7127 {(x^{2})}^{\frac{5}{12}} {cos (\frac{5 π}{12}) [1 + \frac{5}{72} {(\frac{1}{x})}^{2} - \frac{455}{24 \times 6^{4}} {(\frac{1}{x})}^{4} - \dots] \pm sin (\frac{5 π}{12}) [\frac{5}{6} (\frac{1}{x}) - \frac{35}{6^{4}} {(\frac{1}{x})}^{3} + \dots]} - \frac{11}{16}

for the case |x| > 1, with the positive or negative sign selection before the sine function corresponding to the sign of x. Note that, in (5), the non-wander part is $σ_{I, nw}^{2} = 3.86 σ_{R}^{2} z_{e}^{5 / 6} g (x)$ . When |x| is close to unity, the series converges slowly, and more terms in the series expansion are required for accurate representation.

4.2. Coherence length optimization

In order to obtain the optimum l_c, we differentiate the cost function φ(l_c) with respect to l_c and set it to zero to obtain

\frac{d σ_{I}^{2}}{{dl}_{c}} = (\frac{1}{W_{r}^{2}}) [\frac{\bar{I} σ_{I}^{2}}{ln (I / I_{th})}] \frac{32 z^{2}}{k^{2} l_{c}^{3}}

where we have used

\frac{d \bar{I}}{{dl}_{c}} = {(\frac{W_{0}}{W_{r}})}^{2} {\bar{I}}^{2} z_{0}^{2} \frac{4 W_{0}^{2}}{l_{c}^{3}}

Equation (8) shows that when l_c is optimal, the derivative of the SI equals a target value, $T (l_{c}) = (1 / W_{r}^{2}) 32 \bar{I} σ_{I}^{2} z^{2} / [ln (\bar{I} / I_{th}) k^{2} l_{c}^{3}]$ , represented by the right-hand-side (RHS) of (8). The SI derivative, $d σ_{I}^{2} / {dl}_{c}$ , is given by

\frac{d}{{dl}_{c}} σ_{I}^{2} = 3.6833 σ_{R}^{2} z_{e}^{- 1 / 6} (\frac{d z_{e}}{d l_{c}}) [\frac{σ_{p e}^{2}}{W_{0}^{2} (r_{0}^{2} + ξ_{s} z_{0}^{2})} + 0.8733 g (x)] + 4.42 σ_{R}^{2} z_{e}^{5 / 6} [\frac{4 σ_{pe}^{2} z_{0}^{2}}{{(r_{0}^{2} + ξ_{s} z_{0}^{2})}^{2} l_{c}^{3}} + 0.8733 g' (x)]

where

g' (x) = 0.7127 (\frac{10}{72} x - \frac{4 \times 455}{4! 6^{4}} x^{3} + \frac{6 \times 43225}{144 \times 6^{6}} x^{5} + \dots) (\frac{dx}{{dl}_{c}})

for |x| < 1, and

\begin{array}{l} g' (x) = & 0.7127 [(\frac{5}{6}) {(x^{2})}^{- \frac{7}{12}} x {cos (\frac{5 π}{12}) [1 + \frac{5}{72} {(\frac{1}{x})}^{2} - \frac{455}{24 \times 6^{4}} {(\frac{1}{x})}^{4} - \dots] \pm sin (\frac{5 π}{12}) \\ [\frac{5}{6} (\frac{1}{x}) - \frac{35}{6^{4}} {(\frac{1}{x})}^{3} + \dots]} + {(x^{2})}^{\frac{5}{12}} {cos (\frac{5 π}{12}) [- \frac{10}{72} {(\frac{1}{x})}^{3} + \frac{4 \times 455}{24 \times 6^{4}} {(\frac{1}{x})}^{5} - \dots] \\ \pm sin (\frac{5 π}{12}) [- \frac{5}{6} {(\frac{1}{x})}^{2} + \frac{3 \times 35}{6^{4}} {(\frac{1}{x})}^{4} - \dots]}] (\frac{dx}{{dl}_{c}}) \end{array}

for |x| > 1. In (9), we have to use

\frac{{dz}_{e}}{{dl}_{c}} = - \frac{8 {zr}_{0}^{2}}{{kl}_{c}^{3} {(r_{0}^{2} + ξ_{s} z_{0}^{2})}^{2}},

and

\frac{dx}{{dl}_{c}} = \frac{π (3 - \frac{4 z}{F_{0}} + \frac{z^{2}}{F_{0}^{2}})}{2 λ z} \frac{l_{c}}{{(1 + \frac{l_{c}^{2}}{2 W_{0}^{2}})}^{2}}

is required in (10) and (11). We now present the following proposition.

Proposition 1

For a sufficiently small value of I_th so that I_th < Ī, if the derivative $\frac{d}{{dl}_{c}} σ_{I}^{2}$ is larger than or equal to the target value T(l_c) at a high l_c value, then an optimum finite l_c exists.

Proof

First observe that the target value T(l_c) increases rapidly as l_c → 0, while T(l_c) → 0 as l_c → ∞. The SI derivative $d σ_{I}^{2} / {dl}_{c}$ , on the other hand, approaches zero when l_c → 0. This can be seen by observing that (12) and (13) both approach zero, x → 0.5z ₀, and z_e → 1/z ₀ as l_c → 0. If the SI derivative becomes larger than the target value for a large l_c, then it implies that the SI derivative has crossed the target curve at least at one l_c value, where the condition (8) is satisfied. This gives an optimum l_c value.

The above propostion provides a useful test to guarantee optimization benefit from l_c. All we need to do is to calculate the SI derivative and the target value at a large l_c value, say at l_c = 100W ₀. If the scintillation derivative is larger than the target value, then finite optimized l_c exists.

Since Ī decreases with a decrease in l_c, it may not be possible to guarantee I_th < Ī for smaller values of l_c when the outage probability is larger. In that case, T(l_c) becomes negative when I_th > Ī. This occurs when l_c becomes smaller than a certain minimum value l_c ₀, which can be obtained from (1) in the form

l_{c 0} = \sqrt{{[\frac{k^{2} W_{0}^{2}}{8 z^{2}} {\frac{W_{r}^{2}}{W_{0}^{2} I_{th}} - [r_{0}^{2} + (1 + \frac{2 W_{0}^{2}}{ρ_{0}^{2}}) z_{0}^{2}]}]}^{- 1}}

In that case, one also needs to evaluate the SI derivative and T(l_c) at an l_c value close to but greater than l_c ₀, making sure that T(l_c) is higher than the SI derivative at that point. Then Proposition 1 is used to verify if the SI derivative is larger than or equal to T(l_c) at a large l_c value, in which case an optimum finite l_c is guaranteed.

Figure 1 shows outage probability versus coherence length in meters, obtained numerically from (3) for $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , W ₀ = 0.05 m, I_th = 0.01, F ₀ = ∞, and λ = 1.55μm at distances of 1000, 1500 and 2000 meters. The Rytov variance for these distances corresponds to 0.1991, 0.4187 and 0.7095 respectively. The reference beamwidth W_r is 0.025 m, which will be used in all our numerical examples. Observe that the optimal l_c increases with z, and the benefit from optimizing l_c decreases with increasing z. We show the corresponding SI derivative and target values given by (8) in Fig. 2. Note that the meeting points of the SI derivative and target values correspond to the optimal l_c values. As z increases, both the SI derivative and target values increase but target values increase more than the SI derivative for any given increase in z. Observe that for low outage probability or sufficiently small I_th, the RHS of (8) is a positive quantity. Therefore, for optimal l_c to exist, the derivative $d σ_{I}^{2} / {dl}_{c}$ must be positive. Note that since (12) is negative, the first term in (9) is negative. Therefore, the second term in (9) must be positive and large for the SI derivative to cross the target values. Although it is difficult to make a general statement without considering all terms involved, roughly speaking, a large dx/dl_c helps to create a situation where an optimal finite l_c value exists. This implies a small z, a small λ, and a large W ₀.

Fig. 1 Outage probability for $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , W ₀ = 0.05 m, F ₀ = ∞, and λ = 1.55μm at distances of 1000, 1500 and 2000 meters.

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Fig. 2 The derivative of the scintillation index and target values for the same parameters given in Fig. 1

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4.3. Effects of phase front radius of curvature

Differentiating the cost function with respect to F ₀, and setting it to zero, we obtain the following condition to find optimal F ₀

\frac{d σ_{I}^{2}}{d F_{0}} = {(\frac{W_{0}}{W_{r}})}^{2} \frac{4 z σ_{I}^{2} \bar{I}}{ln (I_{th} / \bar{I})} (1 - \frac{z}{F_{0}}) \frac{1}{F_{0}^{2}}

Note that from the physical behavior of focusing, for F ₀ less than but close to z, the SI increases with an increase in F ₀ as the beam wander effects begin to dominate significantly when F ₀ approaches z. So, the left-hand-side (LHS) of (14) is a positive quantity in this region. The RHS is also a positive quantity in this region for low threshold values (I_th < Ī), and the optimality condition is satisfied in this region.

Figure 3 shows outage probabilities versus F ₀ for λ = 1μm, $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , W ₀ = 0.05 m, l_c = ∞, and I_th = 0.1. A larger I_th than the previous figures is selected so that P_out does not become too low. The optimal F ₀ values for the z values of 1000 m, 1250 m, and 1500 m are 860 m, 1030 m and 1180 m respectively. These values are less than z. A natural question with respect to this set of results would be: will there be further performance improvement by optimizing over l_c? To answer this question, we can run optimization over l_c for each possible value of F ₀, requiring significant computations. Fortunately, we can simply use Proposition 1, and produce the SI derivative and the target values given by (8) for different values of F ₀. The results so obtained are shown in Fig. 4. We observe that the SI derivative values are larger than the target values for all values of F ₀. This implies that further improvement in P_out can be achieved using l_c optimization for each of these F ₀ values.

Fig. 3 Outage probability versus F ₀ for $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , W ₀ = 0.05 m and λ = 1μm.

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Fig. 4 The SI derivative with respect to l_c and target values for the parameters given in Fig. 3

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4.4. Effects of beamwidth

The conditions for optimum beamwidth can be obtained by differentiating the cost function with respect to W ₀ and setting it to zero. This produces

\frac{d σ_{I}^{2}}{{dW}_{0}} = \frac{4 σ_{I}^{2}}{W_{0} ln (I_{th} / \bar{I})} [1 - \frac{8 z^{2} \bar{I}}{k^{2} W_{r}^{2}} (\frac{1}{l_{c}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{1}{W_{0}^{2}})]

We show beamwidth effects in Fig 5 by computing (3) for four cases of (z, λ) pairs under $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , F ₀ = ∞, l_c = ∞, and I_th = 0.025. These cases correspond to (z, λ) values of (1500 m, 1 μm), (1500 m, 1.55 μm), (2000 m, 1 μm), and (2000 m, 1.55 μm). The optimized W ₀ values for the cases are found to be 0.016 m, 0.02 m, 0.018 m, and 0.024 respectively. In Fig. 6, we plot the SI derivative with respect to W ₀, and the target values given by the RHS of (15) for l_c = 0.02 m, z = 1500 m, λ = 1μm, F ₀ = ∞, and I_th = 0.025. The optimal value for W ₀, obtained from calculating outage probability (3), is found to be 0.019 m. This agrees with Fig. 6, where the SI derivative and the target curve are found to meet around the optimal value. To gain insight on the effects of l_c for a given value of W ₀, consider two cases:

Fig. 5 Outage probability versus beamwidths for $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , collimated beam, and l_c = ∞. Cases 1, 2, 3 and 4 refer to (z, λ) pairs of (1500 m, 1 μm), (1500 m, 1.55 μm), (2000 m, 1 μm), and (2000 m, 1.55 μm) respectively.

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Fig. 6 SI derivatives and target values with respect to beamwidths for z = 1500 m, λ = 1μm, F ₀ = ∞, I_th = 0.025. The optimal value for W ₀ is found to be 0.019 m.

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Case 1: Suppose

W_{0}^{4} ≫ \frac{4 z^{2} ξ_{s}}{k^{2}}

That is,

W_{0}^{2} ≫ \frac{4 z^{2}}{k^{2} l_{c}^{2}} + \sqrt{\frac{16 z^{4}}{k^{4} l_{c}^{4}} + \frac{4 z^{2}}{k^{2}}}

In this case, using $z_{0} = 2 z / ({kW}_{0}^{2})$ , we can write

z_{e} = \frac{2 z / k}{\frac{W_{0}^{2}}{ξ_{s}} (1 + \frac{4 z^{2} ξ_{s}}{k^{2} W_{0}^{4}})} \approx \frac{2 z ξ_{s}}{{kW}_{0}^{2}} (1 - \frac{4 z^{2} ξ_{s}}{k^{2} W_{0}^{4}})

and also

x = \frac{3 {kW}_{0}^{2}}{4 z ξ_{s}} (1 + \frac{4 z^{2} ξ_{s}}{3 k^{2} W_{0}^{4}})

so that

z_{e} \approx \frac{3}{2 x}

To keep the discussion simple, consider a collimated beam (F ₀ = ∞), and ignore beam wander effects. The SI can then be expressed as

σ_{I}^{2} \approx 3.86 σ_{R}^{2} z_{e}^{5 / 6} g (x)

Therefore, the SI becomes a function of x. Now, from (18), we can write

x = \frac{3 k}{4 z (1 / W_{0}^{2} + 2 / l_{c}^{2})}

so W ₀ and l_c affect SI via x in nearly similar ways. As l_c increases, x also increases. From (6) and (19), we observe that when |x| < 1, the SI will also increase if |x| is close to unity, otherwise the SI will decrease. If |x| > 1, then we can see from (7) and (19) that the SI will increase with increasing |x|.

Case 2: We next consider the case

W_{0}^{4} ≪ \frac{4 z^{2} ξ_{s}}{k^{2}}

That is,

W_{0}^{2} ≪ \frac{4 z^{2}}{k^{2} l_{c}^{2}} + \sqrt{\frac{16 z^{4}}{k^{4} l_{c}^{4}} + \frac{4 z^{2}}{k^{2}}}

For this condition, we can write

z_{e} \approx \frac{{kW}_{0}^{2}}{2 z} (1 - \frac{k^{2} W_{0}^{4}}{4 ξ_{s} z^{2}})

and

x = \frac{z}{{kW}_{0}^{2}} (1 + \frac{3 k^{2} W_{0}^{4}}{4 z^{2} ξ_{s}})

We get

z_{e} \approx \frac{1}{2 x} \approx \frac{{kW}_{0}^{2}}{2 z}

It appears that in this region, x is not affected much by l_c. Therefore, SI is relatively unaffected by l_c. However, the mean intensity increases with l_c. Therefore, not much benefit can be obtained by optimizing l_c for W ₀ selected in this region, and coherent beam tends to perform better.

To understand the benefits of optimizing l_c for different values of W ₀, we consider the SI derivative with respect to l_c and the target values given by (8). Figure 7 shows the derivative and the target values for different values of W ₀ under $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , λ = 1μm, F ₀ = ∞ evaluated at a large value of l_c = 0.5 m. For the z = 1500 m case, we see from Fig. 5 that the optimal W ₀ is 0.016. For the same parameters, the target values just meet the derivative curve in Fig. 7 at about W ₀ = 0.016. Since the target curve does not exceed the SI derivative curve for this W ₀, performance improvement by optimizing l_c is not guaranteed. In fact, for all the optimal W ₀ values observed in Fig. 5, we could not obtain further improvement in performance by optimizing l_c, i.e., l_c = ∞ gives best performance. Observe from Fig. 7 that for very small values of W ₀, benefits from optimizing l_c is not guaranteed. For larger values of W ₀, there is guaranteed benefit from optimizing l_c. This agrees with our theory observations that for small W ₀, the SI becomes nearly independent of l_c with no potential benefits from small l_c values. Also, for longer distance, the minimum W ₀ required for benefit from l_c optimization increases.

Fig. 7 SI derivatives with respect to l_c and target values to determine benefits of PCBs at various beamwidths for the parameters λ = 1μm and F ₀ = ∞.

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5. Conclusion

Partially coherent beams can provide significant performance improvement in free space optical communications, but such improvements may not be realized under all beam configurations and channel conditions. In order to analyze the effects of a PCB, a more intuitive series expression for the scintillation index of atmospheric turbulence is derived. Using this expression, a numerical test for confirming the performance improvement due to coherence length optimization is developed. The effects of different parameters, including the phase front radius of curvature and beamwidth are studied. The results show improvements in outage probability by several orders of magnitude from the use of PCBs for smaller distances, lower wavelengths and larger beamwidths. An important direction for future research is to study the characteristics of coherence length optimization when the receiver employs aperture averaging. Since aperture averaging also reduces scintillation, we expect further overall performance improvement with the relative improvement due to the PCB requiring additional investigation.

Appendix

Consider the second part of the SI (2). Define x = (1 + 2r_e)/2z_e and let |x| < 1. Using

{tan}^{- 1} x = \frac{1}{2 j} ln (\frac{1 + jx}{1 - jx})

we can write

\begin{array}{l} cos {\frac{5}{6} {tan}^{- 1} (\frac{1 + 2 r_{e}}{2 z_{e}})} & = \frac{1}{2} [exp (j \frac{5}{6} {tan}^{- 1} x) + exp (- j \frac{5}{6} {tan}^{- 1} x)] \\ = \frac{1}{2} {exp [\frac{5}{12} ln (\frac{1 + jx}{1 - jx})] + exp [- \frac{5}{12} ln (\frac{1 + jx}{1 - jx})]} \\ = \frac{{(1 + jx)}^{5 / 6} + {(1 - jx)}^{5 / 6}}{2 {(1 + x^{2})}^{5 / 12}} \end{array}

Using the expansion (1 + y)^q = 1 + qy + (1/2!)q(q − 1)y² + ⋯ in (24), and recalling (2), we get the following expression after a few simplification steps,

σ_{I, nw}^{2} = 3.86 σ_{R}^{2} z_{e}^{5 / 6} [0.7127 (1 + \frac{5}{72} x^{2} - \frac{455}{4! 6^{4}} x^{4} + \frac{43225}{144 \times 6^{6}} x^{6} - \dots) - 11 / 16]

When |x| > 1, we substitute y = 1/x and proceed as follows. Observe that

{tan}^{- 1} x = \pm \frac{π}{2} + \frac{1}{2 j} ln (\frac{1 - jy}{1 + jy})

where the positive or the negative sign is taken according to the sign of x. Next, using

cos [\frac{5}{6} {tan}^{- 1} (\frac{1 + 2 r_{e}}{2 z_{e}})] = \frac{1}{2} [e^{\pm j \frac{5 π}{12}} {(\frac{1 - jy}{1 + jy})}^{5 / 12} + e^{\mp j \frac{5 π}{12}} {(\frac{1 + jy}{1 - jy})}^{5 / 12}]

and performing several steps for simplification, we obtain

σ_{I, nw}^{2} = 3.86 σ_{R}^{2} z_{e}^{5 / 6} [0.7127 {(x^{2})}^{5 / 12} {cos (\frac{5 π}{12}) [1 + \frac{5}{72} {(\frac{1}{x})}^{2} - \frac{455}{24 \times 6^{4}} {(\frac{1}{x})}^{4} - \dots] \pm sin (\frac{5 π}{12}) [\frac{5}{6} (\frac{1}{x}) - \frac{35}{6^{4}} {(\frac{1}{x})}^{3} + \dots]} - \frac{11}{16}]

Acknowledgments

The authors would like to thank the Air Force Office of Scientific Research (AFOSR) for providing funding support to conduct this research under the Transformational Communications Advanced Technology Study (TCATS) program.

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Spatially partially coherent beam parameter optimization for free space optical communications

Abstract

1. Introduction

2. Beam model

3. Performance metric

4. PCB Parameter selection

4.1. Expressions for the Scintillation Index

4.2. Coherence length optimization

Proposition 1

Proof

4.3. Effects of phase front radius of curvature

4.4. Effects of beamwidth

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (7)

Equations (42)

Optics Express