# An Empirical Model of Angle-of-Arrival Variance for a Gaussian Wave Propagation through Non-Kolmogorov Turbulence

Authors: Chao Gao; Libin Jiao; Xiaofeng Li
Journal: International Journal of Optics
 (2020)

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## Abstract

This paper proposes an empirical model of the angle-of-arrival (AOA) variance for a Gaussian wave propagating through the weak non-Kolmogorov turbulence. The proposed model is approximately expressed as the linear weighted average between the AOA variances of the plane and spherical waves. The Monte Carlo method is applied to validate the proposed model. The numerical simulations indicate that, under the geometrical optics approximation, the AOA variance for a Gaussian wave is insensitive to the change of the diffraction parameter and can be closely approximated by a simple linear relationship in the refraction parameter. These two properties ensure the validity of the empirical model.


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## Body

## 1. Introduction

Optical wireless communication technology has attracted widespread attention during the past few decades. It adopts unguided laser beam propagating in random media to carry the digital signal for data transmission with high data rates. The random media, however, may contain numerous invisible turbulent eddies, which fluctuate the refractive index inside the random media and distort the equiphase wavefront of the laser beam. Considering that the equiphase wavefront in the vacuum is undistorted and uniform, AOA can be defined as the angle between the normal vectors of the distorted and undistorted equiphase wavefront. It is generally accepted that AOA could hardly degenerate the quality of the imaging spot, but will lead the focus position to deviate from the optimal point. This phenomenon is equivalent to the geometric misalignment between the receiving and transmitting antennas [1–3].AOA stochastically fluctuates with the changes in both temporal and spatial dimensions, and the degree of the AOA fluctuation is conventionally described by its variance. In recent years, theoretical models of the AOA variances for plane and spherical waves have been fully analyzed based on different turbulent power spectrums [4–15]. However, the research on the AOA variance for a Gaussian wave still needs further discussion. Gao et al. [16] proposed a heuristic model of the AOA variance for a Gaussian wave propagation through non-Kolmogorov turbulence, but lacked adequate evidence to ensure the validity of the expressions. Regarded as an increment to the previous literature studies, this paper not only investigates the similar scenario but also conducts simulations for experimental verification. The rest of the paper is organized as follows. Section 2 presents the theoretical models of the AOA variances for different types of laser beams. In Section 3, numerical simulations are performed, followed by conclusions in Section 4.


## 2. Theoretical Models

### 2.1. AOA Variances for Plane and Spherical Waves

The AOA variance for either plane or spherical wave can be calculated by(1)σAOA2=DSdkd2,where d is the aperture diameter of the collecting lens, k=2π/λ is the angular wavenumber with the wavelength λ, and(2)DSρ=4π2k2L∫01dξ∫0+∞dκ×κΦnκfκ,ξ,is the phase structure function [1]. In equation (2), ρ is the scalar separation between two observation points, L is the propagation optical path length, ξ is the normalized path coordinate, and κ is the scalar spatial wavenumber which regularized the scale of the turbulence eddy. Φnκ and fκ,ξ in equation (2) represent the turbulent power spectrum and the weight function, respectively. This paper applies the benchmark non-Kolmogorov model:(3)Φnκ=AαC^n2κ−α,as the turbulent power spectrum, where α∈3,4 is the general spectral power law value, Aα=Γα−1/4π2cosαπ/2 is a function related to α, Γz=∫0+∞xz−1e−xdx is the gamma function, and C^n2 is the scalar generalized atmospheric structure parameter [14]. The weight function for the plane wave fpκ,ξ and for the spherical wave fsκ,ξ are defined as(4)fpκ,ξ=1−J0κρ×1+cosLκ2kξ,fsκ,ξ=1−J0κξρ×1+cosLκ2kξ1−ξ,where J0x=∑n=0+∞−1n/n!2x/22n is the Bessel function of the first kind with zero order.To achieve the closed-form expression of equation (1), this paper invokes the geometrical optics approximation (GOA) to reduce equation (2). For terrestrial horizontal wireless optical communication link, the Fresnel length lF=λL is usually in the order of magnitude of centimeter, whereas d is designed in the order of magnitude of decimeter, i.e., d>lF is almost satisfied. For ρ>lF, the lens effect of the turbulent eddy can be dealt with GOA, which yields Lκ2/k≪1. Consequently, equation (4) leads to(5)fpκ,ξ≈21−J0κρ,fsκ,ξ≈21−J0κξρ.Based on the equation for−3<p<−1 and a>0 [17]:(6)∫0+∞xp1−J0axdx=−a−p−12pΓ1+p2Γ1−p2−1,the double integrals in equation (2) can be reduced to(7)∫01dξ∫0+∞dκ×κΦnκfpκ,ξ≈−AαC^n2ρα−222−α×Γ2−α2Γα2−1,∫01dξ∫0+∞dκ×κΦnκfsκ,ξ≈−1α−1AαC^n2ρα−222−α×Γ2−α2Γα2−1.Thus, based on GOA, the AOA variances for plane and spherical wave can be rewritten as(8)σAOA,p2≈−π2LAαC^n2dα−424−α×Γ2−α2Γα2−1,σAOA,s2≈σAOA,p2α−1.It is found that under GOA, the AOA variances are independent toλ. This property has been confirmed under various situations [15, 16, 18].


### 2.2. AOA Variances for a Gaussian Wave

The Gaussian beam is a representative type of the electromagnetic wave, with normally distributed transverse electric field and intensity [19]. The mathematical description of a Gaussian wave usually depends on the wavelength λ and the waist radius w0. For the optical wireless communication, more parameters are necessary to determine the location of both the transmitter and the receiver. The curvature parameter Θ0=1−L/R0 and the Fresnel ratio Λ0=2L/kW02, two nondimensional scalars, are related to the transmitter, where R0 is the radius of curvature of the phase wavefront at the transmitter and W0 is the radius where the intensity reduces to 1/e2 of the axial values at the transmitter. Similarly, there are also three nondimensional scalars, the refraction parameter Θ=Θ0/Θ02+Λ02, the complementary parameter Θ¯=1−Θ, and the diffraction parameter Λ=λL/πW2, related to the receiver, where W=W0Θ02+Λ02 is the beam radius in a vacuum at the receiver.It is known that both plane and spherical waves are particular types of the Gaussian waves; therefore, their AOA variances should take the form as follows:(9)σAOA,p2=σAOA,G2Λ=0,Θ=1,σAOA,s2=σAOA,G2Λ=0,Θ=0.For the given Gaussian wave with arbitrary values ofΛ and Θ, it is difficult to achieve the analytic formulae of σAOA,G2. As an alternative, σAOA,G2 may be approximatively expressed by σAOA,p2 and σAOA,s2. On the one hand, according to the definition of Λ, it follows that(10)limλ⟶0Λ=0.On another hand, equation (8) implies that, under GOA, both σAOA,p2 and σAOA,s2 are independent to λ. Thus, a heuristic inference can be made that σAOA,G2 may be insensitive to the change of Λ when d>lF, i.e.,(11)σAOA,G2≈σAOA,G2Λ=0.Furthermore,σAOA,G2Λ=0 may be closely approximated by a simple linear relationship in Θ, i.e.,(12)σAOA,G2Λ=0≈ΘσAOA,p2+1−ΘσAOA,s2.Equation (12) is inspired by [20], which investigated the on-axis scintillation index for a Gaussian wave in the saturation regime.


## 2.1. AOA Variances for Plane and Spherical Waves

The AOA variance for either plane or spherical wave can be calculated by(1)σAOA2=DSdkd2,where d is the aperture diameter of the collecting lens, k=2π/λ is the angular wavenumber with the wavelength λ, and(2)DSρ=4π2k2L∫01dξ∫0+∞dκ×κΦnκfκ,ξ,is the phase structure function [1]. In equation (2), ρ is the scalar separation between two observation points, L is the propagation optical path length, ξ is the normalized path coordinate, and κ is the scalar spatial wavenumber which regularized the scale of the turbulence eddy. Φnκ and fκ,ξ in equation (2) represent the turbulent power spectrum and the weight function, respectively. This paper applies the benchmark non-Kolmogorov model:(3)Φnκ=AαC^n2κ−α,as the turbulent power spectrum, where α∈3,4 is the general spectral power law value, Aα=Γα−1/4π2cosαπ/2 is a function related to α, Γz=∫0+∞xz−1e−xdx is the gamma function, and C^n2 is the scalar generalized atmospheric structure parameter [14]. The weight function for the plane wave fpκ,ξ and for the spherical wave fsκ,ξ are defined as(4)fpκ,ξ=1−J0κρ×1+cosLκ2kξ,fsκ,ξ=1−J0κξρ×1+cosLκ2kξ1−ξ,where J0x=∑n=0+∞−1n/n!2x/22n is the Bessel function of the first kind with zero order.To achieve the closed-form expression of equation (1), this paper invokes the geometrical optics approximation (GOA) to reduce equation (2). For terrestrial horizontal wireless optical communication link, the Fresnel length lF=λL is usually in the order of magnitude of centimeter, whereas d is designed in the order of magnitude of decimeter, i.e., d>lF is almost satisfied. For ρ>lF, the lens effect of the turbulent eddy can be dealt with GOA, which yields Lκ2/k≪1. Consequently, equation (4) leads to(5)fpκ,ξ≈21−J0κρ,fsκ,ξ≈21−J0κξρ.Based on the equation for−3<p<−1 and a>0 [17]:(6)∫0+∞xp1−J0axdx=−a−p−12pΓ1+p2Γ1−p2−1,the double integrals in equation (2) can be reduced to(7)∫01dξ∫0+∞dκ×κΦnκfpκ,ξ≈−AαC^n2ρα−222−α×Γ2−α2Γα2−1,∫01dξ∫0+∞dκ×κΦnκfsκ,ξ≈−1α−1AαC^n2ρα−222−α×Γ2−α2Γα2−1.Thus, based on GOA, the AOA variances for plane and spherical wave can be rewritten as(8)σAOA,p2≈−π2LAαC^n2dα−424−α×Γ2−α2Γα2−1,σAOA,s2≈σAOA,p2α−1.It is found that under GOA, the AOA variances are independent toλ. This property has been confirmed under various situations [15, 16, 18].


## 2.2. AOA Variances for a Gaussian Wave

The Gaussian beam is a representative type of the electromagnetic wave, with normally distributed transverse electric field and intensity [19]. The mathematical description of a Gaussian wave usually depends on the wavelength λ and the waist radius w0. For the optical wireless communication, more parameters are necessary to determine the location of both the transmitter and the receiver. The curvature parameter Θ0=1−L/R0 and the Fresnel ratio Λ0=2L/kW02, two nondimensional scalars, are related to the transmitter, where R0 is the radius of curvature of the phase wavefront at the transmitter and W0 is the radius where the intensity reduces to 1/e2 of the axial values at the transmitter. Similarly, there are also three nondimensional scalars, the refraction parameter Θ=Θ0/Θ02+Λ02, the complementary parameter Θ¯=1−Θ, and the diffraction parameter Λ=λL/πW2, related to the receiver, where W=W0Θ02+Λ02 is the beam radius in a vacuum at the receiver.It is known that both plane and spherical waves are particular types of the Gaussian waves; therefore, their AOA variances should take the form as follows:(9)σAOA,p2=σAOA,G2Λ=0,Θ=1,σAOA,s2=σAOA,G2Λ=0,Θ=0.For the given Gaussian wave with arbitrary values ofΛ and Θ, it is difficult to achieve the analytic formulae of σAOA,G2. As an alternative, σAOA,G2 may be approximatively expressed by σAOA,p2 and σAOA,s2. On the one hand, according to the definition of Λ, it follows that(10)limλ⟶0Λ=0.On another hand, equation (8) implies that, under GOA, both σAOA,p2 and σAOA,s2 are independent to λ. Thus, a heuristic inference can be made that σAOA,G2 may be insensitive to the change of Λ when d>lF, i.e.,(11)σAOA,G2≈σAOA,G2Λ=0.Furthermore,σAOA,G2Λ=0 may be closely approximated by a simple linear relationship in Θ, i.e.,(12)σAOA,G2Λ=0≈ΘσAOA,p2+1−ΘσAOA,s2.Equation (12) is inspired by [20], which investigated the on-axis scintillation index for a Gaussian wave in the saturation regime.


## 3. Numerical Simulations

This section adopts the Monte Carlo method to validate equations (11) and (12), respectively. In our simulations, the distorted equiphase wavefront is generated by the stochastic phase screen with the first 496 terms of the Zernike polynomials, whose theoretical formulae are presented in Section 4.1. Afterwards, over 50 points are stochastically sampled on the surface of the distorted equiphase wavefront, and the corresponding AOA can be computed by their normal vectors. For each combination of optical parameters, the simulations are repeated 100 times, and the AOA variance can be statistically estimated by the empirical data. Other default settings are listed as follows: α∈3.2,3.5,3.8, C^n2=1.0×10−15m3−α, λ=1.55×10−6m, k≈4.05×106rad/m, L=4000m, and d=0.1m>lF=0.078m.To validate equation (12), Λ is fixed at 0, whereas Θ is assigned to 11 points linearly spaced between and including 0 and 1. Particularly, Θ=1 refers to the plane wave, and Θ=0 refers to the spherical wave. The simulation results are depicted in Figure 1, where the dots stand for the experimental values, and the lines stand for the theoretical values. It can be identified that there is linear relationship between σAOA,G2Λ=0 and Θ. Thus, equation (12) is valid within the range of the allowable error.Figure 1
Validation of equation (12).To validate equation (11), Θ is fixed at 0.5. Considering that the optical wireless communication systems usually adopt the collimators to parallel the outgoing beam at the transmitting antenna, Λ is assigned to 11 points linearly spaced between and including 0 and 0.5. Figure 2 illustrates the simulation results. It is evident that due to the observational errors, the experimental values of σAOA,G2 fluctuate around the theoretical values of σAOA,G2Λ=0, and the margin of the relative error is less than 15%. Thus, equation (11) is valid for d>lF. This phenomenon can be explained physically. The validity of equation (11) is associated with the validity of equation (10). The condition “d>lF” ensures the validity of GOA, which implies that diffraction effects are negligible and can be generally characterized by the limiting case with λ⟶0. Hence, equation (10) is valid under GOA.Figure 2
Validation of equation (11).


## 4. Conclusions

This paper investigates the AOA variance for a Gaussian wave propagating through the weak non-Kolmogorov turbulence along a horizontal link. An empirical model is deduced from the results of plane and spherical waves under the GOA. The stochastic phase screens based on the Zernike terms are generated to validate the proposed model. The numerical simulations indicate that, under the GOA:(1)
The AOA variance for a Gaussian wave is insensitive to the change of the diffraction parameter(2)
The AOA variance for a Gaussian wave can be closely approximated by a simple linear relationship in the refraction parameterThese two points ensure the validity of our proposed model.It should be pointed out that the phase screen based on the Zernike terms is capable of characterizing the low-frequency component, but is insufficient to describe the high-frequency component. However, the high-frequency component, associated with the small-scale turbulent eddies, makes little contribution to the phase fluctuation of laser beam. Considering that AOA is directly connected with the phase fluctuation, the forementioned weakness could hardly invalidate the conclusions.


### 4.1. Stochastic Phase Screen with Zernike Polynomials

The Zernike polynomials are a set of binary functions orthogonal on the unit disk, whose expressions in the polar coordinate system take the form [21]:(13)Zjρ,θ=Zn,mρ,θ=n+1Rn,mρAmθ,where j=n,m is the index ordered by Noll’s sequence, ρ is the radius, and θ is the azimuth. Rn,mρ is the radial component:(14)Rn,mρ=∑k=0n−m/2−1kn−k!k!n+m/2−k!n−m/2−k!ρn−2k,and Amθ is the angular component:(15)Amθ=2cosmθ,m>0,2sinmθ,m<0,1,m=0.LetΨ be the distorted equiphase wavefront. It can be decomposed as [22, 23](16)Ψ=∑j=1+∞cjZj,where cj are undetermined coefficients. To generate the stochastic phase screen, the covariance between cj1 and cj2 should be(17)covcj1,cj2=Bαdr0α−2Nn1,n2Mm1,m2×−1n1+n2−m1−m22,where j1=n1,m1, j2=n2,m2, r0 is the Fried coherent length, and(18)Bα=23−αα8α−2Γ2α−2α−2/2×Γα2Γ−1−α2Γα+1.Nn1,n2 is the factor related to the radial degrees:(19)Nn1,n2=Γn1+n2−α+22×Γ−1n1+n2+α+42×Γ−1n1−n2+α+22×Γ−1n2−n1+α+22×n1+1n2+1,and Mm1,m2 is the logical Kronecker symbol related to the azimuthal frequencies:(20)Mm1,m2=modj1−j2,2=0∨m1m2=0∧m1=m2.The covariance matrix C about cj is real and symmetrical, so there must be a unitary matrix U such that S=UCUT is diagonal. U can be obtained by the singular value decomposition. R is a Gaussian random variable, with zero mean and variance given by S. The components of the vector UTR are the desired Zernike coefficients cj.The influences ofΘ and Λ on the stochastic phase screen can be delivered by r0. In the weak fluctuation region [24],(21)r0=2.1−4α−1Γα/2π2AαC^n2k2LΓ1−α/21/α−2×25−αb+α−2Γα2Λα21/2−α,where(22)b=1−Θα−11−Θ,Θ≥0,1+Θα−11−Θ,Θ<0.


### 4.2. Tangent Plane and Normal Vector

The equiphase wavefront generated by the stochastic phase screen can be regarded as a surfaceS in the three-dimensional space. Mathematically, S can be described by the implicit function(23)Fx,y,z=0.If these partial derivatives ∂F/∂x, ∂F/∂y and ∂F/∂z are continuous over S, and(24)∂F∂x2+∂F∂y2+∂F∂z2≠1,S is called smooth.For arbitrary pointP0=x0,y0,z0∈S, its tangent plane is(25)∂F∂xP0×x−x0+∂F∂yP0×y−y0+∂F∂zP0×z−z0=0,and the corresponding normal line takes the form as follows:(26)x−x0∂F/∂xP0=y−y0∂F/∂yP0=z−z0∂F/∂zP0.The normal vector n⇀=∂F/∂xP0,∂F/∂yP0,∂F/∂zP0 is orthogonal to the tangent plane.For the stochastic phase screen with the Zernike polynomials, the equiphase wavefront takes the form as(27)Ψx,y−z=0with the normal vector n⇀=∂Ψ/∂x,∂Ψ/∂y,−1. Particularly, n0⇀=0,0,−1 is the normal vector of the undistorted equiphase wavefront.Since AOA can be defined as the angle between the normal vector of the distorted and undistorted equiphase wavefront, it follows that(28)∠AOA=arccosn⇀⋅n0⇀n⇀⋅n0⇀=arccos1∂Ψ/∂x2+∂Ψ/∂y2+1.


## 4.1. Stochastic Phase Screen with Zernike Polynomials

The Zernike polynomials are a set of binary functions orthogonal on the unit disk, whose expressions in the polar coordinate system take the form [21]:(13)Zjρ,θ=Zn,mρ,θ=n+1Rn,mρAmθ,where j=n,m is the index ordered by Noll’s sequence, ρ is the radius, and θ is the azimuth. Rn,mρ is the radial component:(14)Rn,mρ=∑k=0n−m/2−1kn−k!k!n+m/2−k!n−m/2−k!ρn−2k,and Amθ is the angular component:(15)Amθ=2cosmθ,m>0,2sinmθ,m<0,1,m=0.LetΨ be the distorted equiphase wavefront. It can be decomposed as [22, 23](16)Ψ=∑j=1+∞cjZj,where cj are undetermined coefficients. To generate the stochastic phase screen, the covariance between cj1 and cj2 should be(17)covcj1,cj2=Bαdr0α−2Nn1,n2Mm1,m2×−1n1+n2−m1−m22,where j1=n1,m1, j2=n2,m2, r0 is the Fried coherent length, and(18)Bα=23−αα8α−2Γ2α−2α−2/2×Γα2Γ−1−α2Γα+1.Nn1,n2 is the factor related to the radial degrees:(19)Nn1,n2=Γn1+n2−α+22×Γ−1n1+n2+α+42×Γ−1n1−n2+α+22×Γ−1n2−n1+α+22×n1+1n2+1,and Mm1,m2 is the logical Kronecker symbol related to the azimuthal frequencies:(20)Mm1,m2=modj1−j2,2=0∨m1m2=0∧m1=m2.The covariance matrix C about cj is real and symmetrical, so there must be a unitary matrix U such that S=UCUT is diagonal. U can be obtained by the singular value decomposition. R is a Gaussian random variable, with zero mean and variance given by S. The components of the vector UTR are the desired Zernike coefficients cj.The influences ofΘ and Λ on the stochastic phase screen can be delivered by r0. In the weak fluctuation region [24],(21)r0=2.1−4α−1Γα/2π2AαC^n2k2LΓ1−α/21/α−2×25−αb+α−2Γα2Λα21/2−α,where(22)b=1−Θα−11−Θ,Θ≥0,1+Θα−11−Θ,Θ<0.


## 4.2. Tangent Plane and Normal Vector

The equiphase wavefront generated by the stochastic phase screen can be regarded as a surfaceS in the three-dimensional space. Mathematically, S can be described by the implicit function(23)Fx,y,z=0.If these partial derivatives ∂F/∂x, ∂F/∂y and ∂F/∂z are continuous over S, and(24)∂F∂x2+∂F∂y2+∂F∂z2≠1,S is called smooth.For arbitrary pointP0=x0,y0,z0∈S, its tangent plane is(25)∂F∂xP0×x−x0+∂F∂yP0×y−y0+∂F∂zP0×z−z0=0,and the corresponding normal line takes the form as follows:(26)x−x0∂F/∂xP0=y−y0∂F/∂yP0=z−z0∂F/∂zP0.The normal vector n⇀=∂F/∂xP0,∂F/∂yP0,∂F/∂zP0 is orthogonal to the tangent plane.For the stochastic phase screen with the Zernike polynomials, the equiphase wavefront takes the form as(27)Ψx,y−z=0with the normal vector n⇀=∂Ψ/∂x,∂Ψ/∂y,−1. Particularly, n0⇀=0,0,−1 is the normal vector of the undistorted equiphase wavefront.Since AOA can be defined as the angle between the normal vector of the distorted and undistorted equiphase wavefront, it follows that(28)∠AOA=arccosn⇀⋅n0⇀n⇀⋅n0⇀=arccos1∂Ψ/∂x2+∂Ψ/∂y2+1.



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