Propagation of Gaussian Schell-model beams through a jet engine exhaust

Chaoliang Ding; Chaoliang Ding; Chaoliang Ding; Olga Korotkova; Olga Korotkova; Daliang Li; Daomu Zhao; Liuzhan Pan; Liuzhan Pan

doi:10.1364/OE.381242

1. Introduction:

With the rapid development of the optical systems operating through the turbulent boundary layer of Earth, including imaging systems, LIDARs, and free-space optical communications, the statistical changes in light beams propagating through free space and linear random media have been extensively investigated, both theoretically and experimentally [1]. In these studies, the source partial coherence became one of the key mitigation tools of optical turbulence (e.g. [2,3]). After that, the propagation of various classes of partially coherent beams through atmospheric turbulence has received a great deal of attention because of their superiority in self-reshaping of spectra, spectral density and polarization state [4–23].

In the classic turbulent regime the statistics of the refractive index are predominantly isotropic and homogeneous and can be described by the Kolmogorov’s power spectrum [1]. However, in a number of situations, such as the presence of stratification or the proximity of boundaries, atmospheric turbulence may exhibit different statistical behavior, including anisotropy [24], the presence of coherent structures [25], inverse cascade [26], etc. Since publication of results in [24] the presence of turbulent anisotropy has been measured in a number of campaigns [27–32] and studied theoretically, including [33,34] where a convenient theoretical approach was developed (see also [35]) and various statistics of a laser beam interacting with it were calculated, including the long-term beam spread and the scintillation index [36]. Thereafter, the evolution of laser beams and also various partially coherent beams through atmospheric turbulence with anisotropic non-Kolmogorov power spectrum were investigated in a large number of studies, mostly theoretically and by in-lab simulations involving spatial light modulators and specialized air chambers [37–52].

On the other hand, recently airborne laser systems arouse considerable interest in view of a number of potential applications relating to security of aircrafts and communications between them [53–58]. Usually, airborne platforms meet some challenges for laser systems because of their substantial external perturbations, mainly caused by turbulence in the neighborhood of the jet engine plume. Moreover, it was shown that the power spectrum of the jet engine plume carries anisotropic statistics and might produce similar effects on optical beams, as compared to those caused by natural anisotropic turbulence, e.g., unequal beam spread along two mutually orthogonal directions, transverse to the optical axis.

Therefore, it is quite important not only to investigate the propagation of laser beams through jet engine exhaust (as was already done [53]) but also suggest the ways to mitigate the turbulence-induced beam degradation. The goal of this paper is to suggest the use of partial coherence for mitigation of jet engine plume effects on optical beams, primarily on the anisotropy acquired by the beam’s spectral density (average intensity) and its degree of coherence. As the model for this study we use the beams radiated by scalar Gaussian Schell-Model (GSM) sources, which provide the access to arbitrary choice of size, radius of curvature and degree of coherence [59].

2. Propagation of GSM beams through a jet engine exhaust

Consider a GSM source located in the z = 0 plane and radiating a beam-like optical field propagating along the z-direction, in a jet engine exhaust (see Fig. 1). The two-dimensional position vector in the source plane is denoted as r = (x, y). Let the jet stream plume be ejected from the positive z-direction, along the optical axis. After propagation at distance z from the source, the position vector in the plane transverse to beam axis is denoted as ρ=(ξ,η). The second-order statistical properties of the GSM source are characterized by the cross-spectral density (CSD) function of the form [59]

(1)$${W^{(0)}}({{\boldsymbol r}_1},{{\boldsymbol r}_2},\omega ) = \exp \left( { -\frac{{{\boldsymbol r}_1^2 + {\boldsymbol r}_2^2}}{{4\sigma_0^2}}}\right)\exp \left[ { - \frac{{{{({{\boldsymbol r}_2} -{{\boldsymbol r}_1})}^2}}}{{2\delta_0^2}}} \right],$$

where r₁=(x₁, y₁) and r₂=(x₂, y₂) are two-dimensional position vectors of two source points; σ₀ and δ₀ denote the incident root-mean-squared (rms) beam width and rms coherence width, respectively, ω stands for angular frequency. For brevity, we will ignore the dependence of the CSD function on ω throughout the paper.

Fig. 1. Schematic diagram for propagation of a GSM beam through a jet engine exhaust.

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According to the extended Huygens-Fresnel principle and paraxial approximation, the CSD function of a partially coherent beam propagating through a linear turbulent medium at distance z from its source can be expressed as the following formula [59]

(2)$$\begin{aligned} W({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} ) &= \frac{1}{{{\lambda ^2}{z^2}}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{W^{(0)}}({{{\boldsymbol r}_1},{{\boldsymbol r}_2}} )} } \exp \left[ { - \frac{{ik}}{{2z}}{{({{\boldsymbol \rho }_1} - {{\boldsymbol r}_1})}^2} + \frac{{ik}}{{2z}}{{({{\boldsymbol \rho }_2} - {{\boldsymbol r}_2})}^2}} \right]\\ &\quad \times \left\langle {\exp [{\psi \ast ({{\boldsymbol r}_1},{{\boldsymbol \rho }_1},z) + \psi ({{\boldsymbol r}_2},{{\boldsymbol \rho }_2},z)} ]} \right\rangle {d^2}{{\boldsymbol r}_1}{d^2}{{\boldsymbol r}_2}, \end{aligned}$$

where ρ₁=(ξ₁, η₁) and ρ₂=(ξ₂, η₂) denote transverse two points in the plane z > 0. Here, k = 2π/λ is the wave number with λ being the wave length of light, while ψ(r, ρ, z) denotes the complex phase perturbation induced by the refractive-index fluctuations of the random medium from point (r, 0) to point (ρ, z). The star denotes complex conjugate and the angular brackets stand for ensemble average over the medium fluctuations.

According to Ref. [60] (see also [39]), the term in the angular brackets <·> in Eq. (2) can be expressed, for general anisotropic power spectra of turbulence, as

(3)$$\begin{array}{l} \left\langle {\exp [{\psi \ast ({{\boldsymbol r}_1},{{\boldsymbol \rho }_1},z) + \psi ({{\boldsymbol r}_2},{{\boldsymbol \rho }_2},z)} ]} \right\rangle \\ \qquad = \exp \left\{ { - 2\pi {k^2}z\int\limits_0^1 {dt\int {{d^2}{\boldsymbol \kappa }{\Phi _n}({\boldsymbol \kappa })[1 - \exp (t{{\boldsymbol \rho }_d} \cdot {\boldsymbol \kappa } + (1 - t){{\boldsymbol r}_d} \cdot {\boldsymbol \kappa })]} } } \right\}, \end{array}$$

where ${{\boldsymbol \rho }_d} = {{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}$ and ${{\boldsymbol r}_d} = {{\boldsymbol r}_1} - {{\boldsymbol r}_2}$, and ${\boldsymbol \kappa } = ({\kappa _x},{\kappa _y})$ is the two-dimensional spatial frequency vector. In this paper, we will consider propagation of optical beams within an exhaust region from a jet engine with anisotropic power spectrum of Ref. [53]

(4)$${\Phi _n}({\boldsymbol \kappa }) = 0.033C_n^2 \cdot \left\{ \begin{array}{l} \frac{{{{({{L_{0x}}{L_{0y}}} )}^{11/6}}}}{{{{[{1 + {{({\kappa_x}{L_{0x}})}^2} + {{({\kappa_y}{L_{0y}})}^2}} ]}^{^{11/6}}}}}\exp \left( { - \frac{{\kappa_x^2}}{{\kappa_{mx}^2}} - \frac{{\kappa_y^2}}{{\kappa_{my}^2}}} \right)\\ \quad + Q{\left[ {{{\left( {\frac{{2\pi }}{{{L_s}}}} \right)}^2} + {\kappa^2}} \right]^{ - \frac{{11}}{6}}}\exp ({ - {{{\kappa^2}} \mathord{\left/ {\vphantom {{{\kappa^2}} {\kappa_m^2}}} \right.} {\kappa_m^2}}} )\end{array} \right\}, $$

where L_0x and L_0y are outer scales in the x and y directions, respectively, κ_m=c(α)/l₀ with ${l_0} = \sqrt {l_{0x}^2 + l_{0y}^2} $ being the inner scale, κ_mx=c(α)/l_0x, κ_my=c(α)/l_0y with l_0x and l_0y being the inner scales in the x and y directions, respectively. Comparing with Eq. (2) of Ref. [53], we implement in Eq. (4) the finite inner scale, for the sake of the integral convergence and for obtaining more realistic results (see also Ref. [39]). Provided l₀→0, the exponential functions in Eq. (4) will tend to 1, implying that the power spectrum will reduce to Eq. (2) of Ref. [53]. Further, we set for the general power slope α the coefficient:

(5)$$c(\alpha )= {\left[ {\frac{{2\pi \Gamma (5 - \alpha /2)A(\alpha )}}{3}} \right]^{\frac{1}{{\alpha - 5}}}}, $$

where A(α) is defined by expression

(6)$$A(\alpha )= \frac{1}{{4{\pi ^2}}}\Gamma ({\alpha - 1} )\cos \left( {\frac{{\alpha \pi }}{2}} \right),\textrm{ }3\;<\;\alpha\;<\;4. $$

For our calculations α = 11/6. We rewrite Eq. (4) as a sum of two terms

(7)$${\Phi _n}({\boldsymbol \kappa }) = {\Phi _{n1}}({\boldsymbol \kappa }) + {\Phi _{n2}}({\boldsymbol \kappa }), $$

where

(8)$${\Phi _{n1}}({\boldsymbol \kappa }) = 0.033C_n^2 \cdot \frac{{{{({{L_{0x}}{L_{0y}}} )}^{11/6}}}}{{{{[{1 + {{({\kappa_x}{L_{0x}})}^2} + {{({\kappa_y}{L_{0y}})}^2}} ]}^{^{11/6}}}}}\exp \left( { - \frac{{\kappa_x^2}}{{\kappa_{mx}^2}} - \frac{{\kappa_y^2}}{{\kappa_{my}^2}}} \right), $$

(9)$${\Phi _{n2}}({\boldsymbol \kappa }) = 0.033C_n^2 \cdot Q{\left[ {{{\left( {\frac{{2\pi }}{{{L_s}}}} \right)}^2} + {\kappa^2}} \right]^{ - \frac{{11}}{6}}}\exp \left( { - \frac{{{\kappa^2}}}{{\kappa_m^2}}} \right). $$

According to Eqs. (8) and (9), we also rewrite Eq. (3) as a sum of two terms:

(10)$$\left\langle {\exp [{\psi \ast ({{\boldsymbol r}_1},{{\boldsymbol \rho }_1},z) + \psi ({{\boldsymbol r}_2},{{\boldsymbol \rho }_2},z)} ]} \right\rangle = A + B, $$

where

(11)$$A = \exp \left\{ { - 2\pi {k^2}z\int\limits_0^1 {dt\int {{d^2}{\boldsymbol \kappa }{\Phi _{n1}}({\boldsymbol \kappa })[1 - \exp (t{{\boldsymbol \rho }_d} \cdot {\boldsymbol \kappa } + (1 - t){{\boldsymbol r}_d} \cdot {\boldsymbol \kappa })]} } } \right\}, $$

(12)$$B = \exp \left\{ { - 2\pi {k^2}z\int\limits_0^1 {dt\int {{d^2}{\boldsymbol \kappa }{\Phi _{n2}}({\boldsymbol \kappa })[1 - \exp (t{{\boldsymbol \rho }_d} \cdot {\boldsymbol \kappa } + (1 - t){{\boldsymbol r}_d} \cdot {\boldsymbol \kappa })]} } } \right\}. $$

For term A in Eq. (11), we use coordinates changes: ${\kappa _x} = {\kappa ^{\prime}_x}/{\mu _x},{\kappa _y} = {\kappa ^{\prime}_y}/{\mu _y}$, where µ_x and µ_y denote the anisotropic factors in two mutually orthogonal directions. In the inertial subrange, when the turbulent eddies transport the energy from the outer scale to the inner scale, we can assume that the ratio of the outer and the inner scale in the x direction to the y direction are constants. According to this assumption, the outer and the inner scales in the x and y direction can be expressed as L_0x = µ_xL₀, L_0y =µ_yL₀, l_0x=µ_xl₀ and l_0y=µ_yl₀.

Under these assumptions, Eq. (11) can be written as

(13)$$A = \exp \left\{ { - \frac{{2\pi {k^2}z}}{{{\mu_x}{\mu_y}}}\int\limits_0^1 {dt\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {d{{\kappa^{\prime}}_x}d{{\kappa^{\prime}}_y}{{\Phi^{\prime}}_{n1}}({\boldsymbol \kappa^{\prime}})[1 - \exp (t{{{\boldsymbol \rho^{\prime}}}_d} \cdot {\boldsymbol \kappa^{\prime}} + (1 - t){{{\boldsymbol r^{\prime}}}_d} \cdot {\boldsymbol \kappa^{\prime}})]} } } } \right\}, $$

where ${{\boldsymbol \rho ^{\prime}}_d} = ({{{\xi _d}} \mathord{\left/ {\vphantom {{{\xi_d}} {{\mu_x},}}} \right.} {{\mu _x},}}{{{\eta _d}} \mathord{\left/ {\vphantom {{{\eta_d}} {{\mu_y}}}} \right.} {{\mu _y}}})$ and ${{\boldsymbol r^{\prime}}_d} = ({{{x_d}} \mathord{\left/ {\vphantom {{{x_d}} {{\mu_x},}}} \right.} {{\mu _x},}}{{{y_d}} \mathord{\left/ {\vphantom {{{y_d}} {{\mu_y}}}} \right.} {{\mu _y}}})$ resulting in the modified power spectrum

(14)$${\Phi ^{\prime}_{n1}}(\kappa ^{\prime}) = 0.033C_n^2 \cdot {\left( {\frac{{{\mu_x}{\mu_y}L_0^2}}{{1 + {{\kappa^{\prime}}^2}L_0^2}}} \right)^{\frac{{11}}{6}}}\exp \left( { - \frac{{l_0^2}}{{c{{(\alpha )}^2}}}{{\kappa^{\prime}}^2}} \right). $$

We now change the Cartesian coordinates to polar, i.e., $d{\kappa ^{\prime}_x}d{\kappa ^{\prime}_y} = \kappa ^{\prime}d\kappa ^{\prime}d\theta $ and integrate over θ. Then, Eq. (13) takes form

(15)$$A = \exp \left\{ { - \frac{{4{\pi^2}{k^2}z}}{{{\mu_x}{\mu_y}}}\int\limits_0^1 {dt\int\limits_0^\infty {\kappa^{\prime}d\kappa^{\prime}{{\Phi^{\prime}}_{n1}}(\kappa^{\prime})[1 - {J_0}({\kappa^{\prime}|{t{{{\boldsymbol \rho^{\prime}}}_d} + (1 - t){{{\boldsymbol r^{\prime}}}_d}} |} )]} } } \right\}. $$

If the coordinate points are sufficiently close to the optical axis, i.e., under the paraxial approximation, the Bessel function in Eq. (15) can be expanded to be J₀(x)≈1-x²/4 by Taylor series. After integrating over t, κ’, we get

(16)$$A = \exp \left\{ { - \frac{{{\pi^2}{k^2}z{T_A}({\xi_d^2 + {\xi_d}{x_d} + x_d^2} )}}{{3\mu_x^2}}} \right\}\exp \left\{ { - \frac{{{\pi^2}{k^2}z{T_A}({\eta_d^2 + {\eta_d}{y_d} + y_d^2} )}}{{3\mu_y^2}}} \right\}, $$

where

(17)$$\begin{aligned} {T_A} &= \frac{1}{{{\mu _x}{\mu _y}}}\int_0^\infty {{{\kappa ^{\prime}}^3}{{\Phi ^{\prime}}_{n1}}(\kappa ^{\prime})d\kappa ^{\prime}} \\ & = 0\textrm{.033}C_n^2{({\mu _x}{\mu _y})^{\frac{5}{6}}}\left[ {\frac{{\left( {5 + \frac{6}{{\kappa_m^2L_0^2}}} \right)\exp \left( {\frac{1}{{\kappa_m^2L_0^2}}} \right)\Gamma \left( {\frac{1}{6},\frac{1}{{\kappa_m^2L_0^2}}} \right)}}{{10{{\left( {\frac{1}{{\kappa_m^2}}} \right)}^{\frac{1}{6}}}}} - \frac{3}{5}{{\left( {\frac{1}{{L_0^2}}} \right)}^{\frac{1}{6}}}} \right] \end{aligned}. $$

On substituting from Eqs. (1) and (16) into Eq. (2) and after integrating, we find that the CSD function of the GSM beam from term A in the plane z > 0 factorizes as:

(18)$${W_A}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )= {W_{xA}}({{\xi_1},{\xi_2}} ){W_{yA}}({{\eta_1},{\eta_2}} ), $$

where

(19)$$\begin{aligned} {W_{xA}}({{\xi_1},{\xi_2}} ) &= \frac{1}{{\sqrt {{\Delta _{xA}}(z)} }}\exp \left( { - \frac{{\xi_1^2 + \xi_2^2}}{{4\sigma_0^2{\Delta _{xA}}(z)}}} \right)\exp \left[ { - \frac{{ik}}{{2{R_{xA}}(z)}}({\xi_1^2 - \xi_2^2} )} \right]\\ &\quad \times \textrm{exp}\left\{ { - \left[ {\frac{1}{{2\delta_0^2{\Delta _{xA}}(z)}} + \frac{{{\pi^2}{k^2}{T_A}z}}{{3\mu_x^2}}\left( {1 + \frac{2}{{{\Delta _{xA}}(z)}}} \right) - \frac{{{\pi^4}{k^2}T_A^2{z^4}}}{{18\mu_x^4{\Delta _{xA}}(z)\sigma_0^2}}} \right]{{({{\xi_1} - {\xi_2}} )}^2}} \right\}, \end{aligned}$$

and parameters Δ_xA(z) and R_xA(z) are given by expressions

(20)$${\Delta _{xA}}(z) = 1 + \left[ {\frac{1}{{4{k^2}\sigma_0^4}} + \frac{1}{{{k^2}\sigma_0^2}}\left( {\frac{1}{{\delta_0^2}} + \frac{{2{\pi^2}{k^2}{T_A}z}}{{3\mu_x^2}}} \right)} \right]{z^2}, $$

(21)$${R_{xA}}(z) = z + \frac{{\sigma _0^2z - {\pi ^2}{T_A}{z^4}/3\mu _x^2}}{{({\Delta _{xA}}(z) - 1)\sigma _0^2 + {\pi ^2}{T_A}{z^3}/3\mu _x^2}}. $$

Factor ${W_{yA}}({{\eta_1},{\eta_2}} )$ has the same form as ${W_{xA}}({{\xi_1},{\xi_2}} )$ by using y, η₁and η₂ in place of x, ξ₁and ξ₂ in Eq. (19), respectively.

For term B in Eq. (12), we change the integral variables d²κ from the Cartesian coordinates to polar, i.e., d²κ=κdκdθ, and integrate over θ. Then Eq. (12) turns out to be

(22)$$B = \exp \left\{ { - 4{\pi^2}{k^2}z\int\limits_0^1 {dt\int {\kappa d\kappa {\Phi _{n2}}(\kappa )[1 - {J_0}(\kappa |{t{{\boldsymbol \rho }_d} + (1 - t){{\boldsymbol r}_d}} |)]} } } \right\}. $$

By applying approximation J₀(x)≈1-x²/4 and integrating over t, κ, we deduce that

(23)$$B = \exp \left[ { - \frac{1}{3}{\pi^2}{k^2}z{T_B}({\xi_d^2 + {\xi_d}{x_d} + x_d^2} )} \right]\exp \left[ { - \frac{1}{3}{\pi^2}{k^2}z{T_B}({\eta_d^2 + {\eta_d}{y_d} + y_d^2} )} \right], $$

where

(24)$$\begin{aligned} {T_B} &= \int_0^\infty {{\kappa ^3}{\Phi _{n2}}(\kappa )d\kappa } \\ & = 0\textrm{.033}C_n^2Q\left[ {\frac{{\left( {5 + \frac{{24{\pi^2}}}{{\kappa_m^2L_s^2}}} \right)\exp \left( {\frac{{4{\pi^2}}}{{\kappa_m^2L_s^2}}} \right)\Gamma \left( {\frac{1}{6},\frac{{4{\pi^2}}}{{\kappa_m^2L_s^2}}} \right)}}{{10{{\left( {\frac{1}{{\kappa_m^2}}} \right)}^{\frac{1}{6}}}}} - \frac{3}{5}{{\left( {\frac{{4{\pi^2}}}{{L_s^2}}} \right)}^{\frac{1}{6}}}} \right] \end{aligned}. $$

On substituting from Eqs. (1) and (23) into Eq. (2) and after integrating, the CSD function of the GSM beam from term B can be derived in the plane z > 0 as:

(25)$$\begin{aligned} {W_B}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} ) &= \frac{1}{{{\Delta _B}(z)}}\exp \left( { - \frac{{{\boldsymbol \rho }_1^2 + {\boldsymbol \rho }_2^2}}{{4\sigma_0^2{\Delta _B}(z)}}} \right)\exp \left[ { - \frac{{ik}}{{2{R_B}(z)}}({{\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2} )} \right]\\ &\quad \times \textrm{exp}\left\{ { - \left[ {\frac{1}{{2\delta_0^2{\Delta _B}(z)}} + \frac{{{\pi^2}{k^2}{T_B}z}}{3}\left( {1 + \frac{2}{{{\Delta _B}(z)}}} \right) - \frac{{{\pi^4}{k^2}T_B^2{z^4}}}{{18{\Delta _B}(z)\sigma_0^2}}} \right]{{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2}} \right\}, \end{aligned}$$

where parameters Δ_B(z) and R_B(z) are given by expressions

(26)$${\Delta _B}(z) = 1 + \left[ {\frac{1}{{4{k^2}\sigma_0^4}} + \frac{1}{{{k^2}\sigma_0^2}}\left( {\frac{1}{{\delta_0^2}} + \frac{{2{\pi^2}{k^2}{T_B}z}}{3}} \right)} \right]{z^2}$$

(27)$${R_B}(z) = z + \frac{{\sigma _0^2z - {\pi ^2}{T_B}{z^4}/3}}{{({\Delta _B}(z) - 1)\sigma _0^2 + {\pi ^2}{T_B}{z^3}/3}}. $$

For scalar random beams the spectral density (SD) and the spectral degree of coherence (DOC) are defined by expressions [59]

(28)$$S({\boldsymbol \rho },\omega ) = {W_{}}({{\boldsymbol \rho },{\boldsymbol \rho },z} )= {W_A}({{\boldsymbol \rho },{\boldsymbol \rho },z} )+ {W_B}({{\boldsymbol \rho },{\boldsymbol \rho },z} ), $$

(29)$$\begin{aligned} \mu ({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z) &= \frac{{{W_{}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )}}{{\sqrt {S({{\boldsymbol \rho }_1},z)S({{\boldsymbol \rho }_2},z)} }}\\ & = \frac{{{W_A}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )+ {W_B}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )}}{{\sqrt {[{{W_A}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_1},z} )+ {W_B}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_1},z} )} ][{{W_A}({{{\boldsymbol \rho }_2},{{\boldsymbol \rho }_2},z} )+ {W_B}({{{\boldsymbol \rho }_2},{{\boldsymbol \rho }_2},z} )} ]} }}. \end{aligned}$$

3. Numerical calculation and analysis

In this section, we will illustrate the evolution of the SD and the DOC of the GSM beam propagating through a jet engine exhaust according to the analytical expressions derived in section 2 by a number of numerical examples. The parameters used in numerical calculations are chosen to be σ₀=5mm, δ₀=5mm, λ=632.8nm, $C_n^2 = 1.6 \times {10^{ - 9}}{\textrm{m}^{ - 2/3}}$, L₀=0.5m, L_S=1mm, µ_x=0.7, µ_y=1.4 and Q = 6 [53] unless different values are specified. In addition, we let the inner scale length be l₀=2/3mm ≈ 0.67mm, which originates from the experimental results of Ref. [61] (see also Ref. [62]). In Fig. 2, we plot the anisotropic power spectrum Φ_n of the refractive index fluctuations along with x and y directions, according to Eq. (4), for different values of inner scale l₀. It is shown that the inner scale length has a noticeable impact on the power spectrum only in the high spatial frequency region (above 10³ m⁻¹) if l₀=1mm, or larger, and the influence of the inner scale on the power spectrum is almost negligible when the l₀=0.01mm or smaller.

Fig. 2. Anisotropic power spectrum of the refractive-index fluctuations, from the jet engine exhaust along the x and y directions, according to Eq. (4).

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Figure 3 presents the density plots of the SD of the GSM beams at some propagation distances for three different values of source coherence width:δ₀=5mm, δ₀=0.3mm and δ₀=5µm. We note right away that our analysis only pertains to very small propagating distances, on the order of meters, in striking difference to the natural anisotropic turbulence where the beam can be analyzed on propagation at kilometer ranges and farther, depending on the local strength of the refractive-index fluctuations (see Ref. [39]). We plot the results at 10 m and 20 m to illustrate that anisotropy effects may increase further with increasing distances, even though such ranges might be out of the realistic scenario. However, if a beam is affected by a jet stream in the beginning of a path and then travels much further in free space or in a natural atmospheric turbulence, it can be considered as an effective phase screen [1] equivalent to somewhat milder extended turbulence with the same anisotropy ratio in transverse direction.

Fig. 3. The spectral density of a GSM beam at some propagation distances from the source for three different values of the source coherence width (First row:δ₀=5mm, second row:δ₀=0.3mm, third row: δ₀=5µm).

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It is seen from Figs. 3(a)–3(d) that, for the nearly coherent source, the beam profile transitions from circular to elliptical upon propagation. However, with decreasing coherence width, the transition from circular shape to elliptical becomes less noticeable [see Figs. 3(e)–3(h)]. When the source coherence width become much smaller, e.g. δ₀=5µm, the SD profile remains circular up to the propagation distance of about z = 20 m. Hence, the GSM beam with very low coherence may weaken the anisotropic effect of turbulence generated from a jet engine as compared to that with highly coherent source.

Next, we will perform some explanations and illustrations. The SD of the GSM beam is a sum of W_A in Eq. (18) and W_B in Eq. (25). In Eq. (25), there is no anisotropic term. Therefore, W_A plays a more important role on determining the anisotropic features of the SD. Hence, in the following, we will emphasize the effect of W_A on the whole SD.

The root-mean-square beam width of the GSM beam from term A along the x or y direction in the plane z > 0 can be found by comparing Eq. (18) with Eq. (1), i.e.,

(30)$${\sigma _i}(z) = {\sigma _0}\sqrt {{\Delta _{iA}}(z)} = {\left[ {\sigma_0^2 + \frac{{{z^2}}}{{4{k^2}\sigma_0^2}} + \frac{{{z^2}}}{{{k^2}\delta_0^2}} + \frac{{2{\pi^2}{T_A}{z^3}}}{{3\mu_i^2}}} \right]^{\frac{1}{2}}},\textrm{ }(i = x,y). $$

Equation (30) implies that the propagating beam width has four contributions: the first term is the initial SD width; the second term and third term indicate the free-space diffraction rate of the GSM beam upon propagation, induced by σ₀ and δ₀, respectively. Diffraction of the second term and third term is isotropic, depending only on the propagation distance and the wavelength. The fourth term indicates turbulence-induced diffraction, which is anisotropic and dependent on the anisotropy parameter µ_i. Figure 4 shows the second-fourth terms of Eq. (30) versus propagation distance, for four different values of source coherence width. It is clearly seen that the influence of the fourth term is very small when the propagation distance is short. Hence, the SD profile remains circular at small ranges from the source. For the high source coherence δ₀=5 mm, with increasing propagation distance z, turbulence-induced diffraction begins to play a more important role because the value of the fourth term changes with z³, which is much greater than those of the second and third term, i.e., z². As a result, the SD obtains an elliptical shape when the propagation distance is sufficiently large, e.g. z = 20 m. However, with the decrease in the source coherence, the coherence-induced diffraction gradually plays a major role comparing with the turbulence-induced diffraction [see third term in Fig. 4(d)]. In addition, the third term is isotropic. That is why the SD profile retains circular profile when the source coherence is very low. What’s more, the contribution from the fourth term in the x direction is larger than that of the fourth term in the y direction. Hence, the major axis of the SD’s elliptical distribution is the x axis.

Fig. 4. Second-fourth terms of Eq. (30) versus propagation distance, for four different values of source coherence width.

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The density plots for the modulus of the DOC of the GSM beams at two points ρ₁=(ξ, η) and ρ₂=(0, 0) are given in Fig. 5. As can be seen from Figs. 5(a)–5(d) the DOC distribution exhibits elliptical shape upon propagation when the source coherence width is large, e.g., δ₀=5 mm. And the major axis of the DOC profile is along y direction (in contrast with that for the SD profile). However, with the decreasing source coherence width, the evolution of the DOC profile from circular to elliptical occurs at smaller ranges [see Figs. 5(f)–5(h) and 5(j)–5(l)]. For example, when δ₀=5µm, the DOC profile is almost circular at the propagation distance z = 5 m. For larger propagation distances, the DOC profile starts acquiring elliptical shape. In order to give the physical explanation, we examine the root-mean-square value of the DOC of the GSM beam, obtained by comparison of Eq. (18) with Eq. (1):

(31)$${\delta _i}(z) = {\left[ {\frac{1}{{\delta_0^2{\Delta _{iA}}(z)}} + \frac{{2{\pi^2}{k^2}{T_A}z}}{{3\mu_i^2}} + \frac{{2{\pi^2}{k^2}{T_A}z}}{{3\mu_i^2{\Delta _{iA}}(z)}}\left( {2 - \frac{{{\pi^2}{T_A}{z^3}}}{{6\mu_i^2\sigma_0^2}}} \right)} \right]^{ - \frac{1}{2}}},(i = x,y). $$

Equation (31) shows that all of three terms in the square brackets affect the coherence width of the GSM beam. The first term $\textrm{1/}\delta _0^2{\Delta _{iA}}(z)$ indicates the coherence-induced diffraction, which results in the increase of coherence width upon propagation. The second term is the turbulence-induced decoherence effects, attenuating the beam coherence with increasing propagation distance. In the third term, there are complex interactions between the coherence-induced diffraction and the turbulence-induced decoherence effect, which result in a somewhat complicated evolution of the DOC.

Fig. 5. Density plots for the modulus of the DOC of a GSM beam at some propagation distances for three different values of the source coherence width (First row:δ₀=5mm, second row:δ₀=0.3mm, third row: δ₀=5µm).

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In Fig. 6, we demonstrate the changes in the first-third terms of Eq. (31) and with propagation distance z, for three different values of the source coherence width. As can be seen from Figs. 6(a) and 6(b) the influence of the first term on the coherence width is practically negligible, while the third term displays the non-monotonic changes near the value of zero, with increasing propagation distance z. However, the contribution from the second term becomes dominant already after a short propagation range. Hence, the DOC distribution exhibits elliptical shape upon propagation at relatively short distances when the source coherence widthδ₀=5 mm. Moreover, when δ₀→∞, i.e., in the coherent limit case, the effect of the first term can be ignored. With decreasing source coherence width, the effect of the first term is dominant for short propagation distance [see Figs. 6(e) and 6(f)], playing a more important role with the decrease of source coherence [see Figs. 6(g) and 6(h)]. Thus, the overall effect of anisotropy of turbulence on the DOC distribution is very weak for short propagation distance. That is why the DOC profile remains nearly circular for short propagation distance [see Fig. 5(j)]. For large propagation distances, the second term becomes dominant, relating to the turbulence-induced decoherence effect. Hence, at sufficiently large distances from the source plane, the DOC always keeps elliptical shape.

Fig. 6. First-third terms of Eq. (31) versus propagation distance, for four different values of the source coherence width.

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In Fig. 7 we show the evolution of the coherence width with propagation distance for different values of the source coherence widths. It is shown that the coherence widths in the x/y direction decreases with increasing propagation distance when the source coherence width is large [see Fig. 7(a) and 7(b)]. However, for low source coherence, the coherence width of the propagating beam displays the non-monotonic changes with increasing propagation distance [see Figs. 7(c) and 7(d)].

Fig. 7. Coherence widthin x and y direction versus propagation distance for four different values of source coherence width.

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4. Conclusion

We have investigated in detail the interaction of the scalar GSM beam with a region in space containing a jet engine exhaust. Such medium can be described by means of an anisotropic power spectrum of refractive index fluctuations with Kolmogorov power law, having very high values of the refractive-index structure parameter and very low values of the inner scale. Hence, as compared with natural turbulence, an optical beam propagating in such medium is mostly affected by small scales, which produce the scattering-like effects and limit the paraxial regime to very small ranges. But even over these small distances the effect of anisotropy can be substantial.

Our main contribution to this research area are the derived analytical expressions for the SD and the DOC of the GSM beams and their analysis upon propagation, as a whole quantity and segregated into several terms, indicating the effects of the source coherence, of the turbulent anisotropy and their interplay. Our numerical examples illustrate that the profile of the SD and the DOC of the GSM beams gradually transfer from a circular shape to an elliptical shape upon propagation for highly coherence sources. However, when the source coherence width is short, the SD’s shape remains circular for long propagation distance, while the DOC’s shape remains circular shape for short propagation distance, implying that the GSM beams with sufficiently low source coherence are insusceptible to the anisotropy of turbulence form jet engine exhaust, and thus, can be used as the mitigation tool. This is the major practical outcome of our study. Another possibility, not explored in this paper, but readily seen to be feasible, is modulation of the source DOC in anisotropic manner [63]. If the major semi-axis of the source DOC is orthogonal to that of the jet stream, then at some propagation distances their effects must cancel. This idea can be exploited when the directionality of the beam is also of importance. The obtained results might have some potential applications in optical imaging, ranging, tracking and communication systems that are to operate through the immediate jet engine exhaust environments.

Funding

National Natural Science Foundation of China (61675094, 61575091, 11874321).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Propagation of Gaussian Schell-model beams through a jet engine exhaust

Abstract

1. Introduction:

2. Propagation of GSM beams through a jet engine exhaust

3. Numerical calculation and analysis

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Equations (31)

Optics Express