Spatial distributions of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence

Jianlong Li; Baida Lü; Shifu Zhu

doi:10.1364/OE.17.011399

Optics Express
Vol. 17,
Issue 14,
pp. 11399-11408
(2009)
•https://doi.org/10.1364/OE.17.011399

Spatial distributions of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence

Jianlong Li, Baida Lü, and Shifu Zhu

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Jianlong Li, Baida Lü, and Shifu Zhu, "Spatial distributions of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence," Opt. Express 17, 11399-11408 (2009)

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Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: March 24, 2009
- Revised Manuscript: May 10, 2009
- Manuscript Accepted: June 3, 2009
- Published: June 23, 2009

Abstract

The formulas of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence are derived by using Maxwell’s equations. Expressions expressed by elements of electric cross spectral density matrixes of the magnetic and the mutual cross spectral density matrix are obtained for the partially coherent electromagnetic beams. Taken the partially coherent Cosh-Gaussian (ChG) electromagnetic beam as a typical example, the spatial distributions of the energy and energy flux density in atmospheric turbulence are numerically calculated. It is found that the turbulence shows a broadening effect on the spatial distributions of the energy and energy flux density. Some interesting results are obtained and explained with regard to their physical nature.

1. Introduction

The propagation of the laser beams in turbulent atmosphere has an important significance in such fields as the laser radar, laser satellite communications, laser-guided and laser warning, etc. [1]. In the early 1990’s, Wu etc. and Boardman [2, 3] found that the effect of atmospheric turbulence on the partially coherent beam was smaller than the completely coherent beam. The study is restricted to the scalar case [4–15]. The effect of turbulence on the beam quality, on the beam energy concentration and on maximum peak light intensity was researched in literature [16]. To our knowledge, as yet the effect of the turbulence on the energy and energy flux density of partially coherent electromagnetic beams have not been derived and studied. With the development of the micro-optics, the rigorous theory of the light intensity of partially coherent electromagnetic beam has become one of the important application problems. Based on the classic electromagnetic field [17], the analytical expressions for the energy and energy flux formulas of partially coherent electromagnetic beam are derived. Taking the partially coherent ChG electromagnetic beam as a typical example, we analyze the application of these formulas, and numerically calculate the spatial distributions of energy and energy flux density of partially coherent ChG electromagnetic beam. Also, we comparatively study the effect of turbulence on the spatial distributions of energy and energy flux density.

2. The energy and energy flux density of partially coherent electromagnetic beam

Suppose that a quasi-monochromatic partially coherent electromagnetic beam perpendicularly incidents on the plane z=0, and the electric field is expressed as E(ρ _s0, 0)=E_x (ρ _s0, 0)i+ E_y(ρ _s0, 0)j+0k(s=1,2), where i, j, k stand for the unit vectors along the x, y, z axes respectively, and ρ _s0=(x _s0, y _s0) is the point vector at the plane. Based on the Rayleigh—Sommerfeld diffraction integral [18], the electric field E(ρ _s, z_s) at the plane z_s of the electric field E (ρ _s0, 0) propagating in atmospheric turbulent given by

E_{α} (ρ_{s}, Z_{s}) = - \frac{1}{2 π} \iint_{(z = 0)} E_{α} (ρ_{s 0}, 0) \frac{\partial}{{\partial z}_{s}} (\frac{\exp (ik R_{s})}{R_{s}}) \exp [ψ_{α} (ρ_{s}, ρ_{s 0}, Z_{s})] d^{2} ρ_{s 0}, {(α = x, y)}^{'}

+ E_{y} (ρ_{s 0}, 0) \frac{\partial}{\partial y_{s}} (\frac{\exp (ik R_{s})}{R_{s}}) \exp [ψ_{y} (ρ_{s}, ρ_{s 0}, z_{s})] d^{2} ρ_{s 0}'

where

R_{s} = \sqrt{{{(x_{s} - x_{s 0})}^{2} + (y_{s} - y_{s 0})}^{2} + z_{z}^{2},}

k = 2 π ⁄ λ (wave number),

Ψ_α(ρ _s, ρ _s0,z_s) is the α axis random phase factor of atmospheric turbulence. In the space z>0, the average field of the two point’s electric fields can be expressed as

E = \frac{1}{2} [(E_{x}^{*} (ρ_{1}, t) + E_{x} (ρ_{2}, t + τ)) i + (E_{y}^{*} (ρ_{1}, t) + E_{y} (ρ_{2}, t + τ)) j + (E_{z}^{*} (ρ_{1_{1}}, t, ω) + E_{z} (ρ_{2}, t + τ)) k] .

The magnetic field corresponding to Eq. (3) can be obtained by using the Maxwell’s equations and omitted here.

The energy and energy flux density of the electromagnetic field are expressed as

w = \frac{E \cdot D}{2} + \frac{B \cdot H}{2} (energy density),

S = E \times H, (energy flux density),

where H, B, D denote the magnetic field intensity, the magnetic flux density and electric flux density, respectively. The first and second items of Eq. (4a) are the electric energy density (expressed by w_e) and the magnetic energy density (expressed by w_h), respectively. The physical nature of energy flux density vector is the energy of the electromagnetic waves pass through the cross-sectional plane in unit time.

Suppose that electromagnetic radiation field is ergodic, and then its ensemble average for time can be expressed as

< M^{*} (r_{1}, t + τ) N (r_{2}, t) > = lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} M^{*} (r_{1}, t + τ) N (r_{2}, t) dt,

where T, M(·) (or N(·)) and * denote the measurement time, the physical quantity E or H and the complex conjugate.

Substitute Eq. (3) into the first item of the Eq. (4a), and take time ensemble average on both sides, can be given by

< w_{e} (ρ_{1}, ρ_{2}, ω) > = \frac{ε_{0}}{4} \underset{α = x, y, z}{Σ} W_{vv}^{(ee)} (ρ_{1}, ρ_{2}, ω),

is the time-average electric energy density, and where ε₀ denotes the medium conductivity, “<>” denotes the time ensemble average and

W_{vv}^{(ab)} (ρ_{1}, ρ_{2}, ω) = < M_{v}^{*} (ρ_{1}, ω) N_{v} (ρ_{2}, ω), >, (a, b = e, h; M, N = E, H),

denotes the elements of the cross spectral density matrix of partially coherent electromagnetic beams.

So the electric energy density of the partially polychromatic coherent electromagnetic fields is

< w_{e} (ρ_{1}, ρ_{2}) > = \frac{ε_{0}}{4} \int_{0}^{\infty} \underset{v = x, y, z}{Σ} W_{vv}^{(ee)} (ρ_{1}, ρ_{2}, ω) d ω,

According to the time-harmonic Maxwell’s equations, the magnetic field energy density corresponding to the electric field energy density expressed by Eq. (5), that is,

< w_{h} (ρ_{1}, ρ_{2}, ω) > = \frac{1}{ω^{2} μ_{0}^{2}} [(\frac{\partial_{2}}{\partial z_{1} \partial y_{2}} - \frac{\partial^{2}}{\partial y_{1} \partial z_{2}}) W_{yz}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial y_{1} \partial y_{2}} W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial z_{1} \partial z_{2}} W_{yy}^{(ee)} (ρ_{1}, ρ_{2}, ω) +

is the time-average magnetic energy density, and the magnetic energy density of the partially polychromatic coherent electromagnetic fields is

< w_{h} (ρ_{1}, ρ_{2}) > = \frac{μ_{0}}{4} \int < w_{h} (ρ_{1}, ρ_{2}, ω) > d ω .

where µ ₀ and ω denote the medium permeability and the angular frequency of the electromagnetic field, respectively. In lossless medium, the electric field energy density equals to the magnetic field energy density. Adding up the energy densities of electric and magnetic field, respectively was expressed by Eqs. (5) and (6), can obtain the total energy density of partially coherent electromagnetic beam, and this result is one of the main outcomes obtained in this paper.

According to Maxwell’s equations, Eqs. (3) and (4b), the analytical expressions of the energy flux vector of partially coherent electromagnetic beam can be obtained, that is

< s (ρ_{1}, ρ_{2}, ω) > = \frac{1}{2} [(W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω) i + (W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω),

is the time- average power density. So the average electromagnetic energy flux density of the partially polychromatic coherent electromagnetic fields is

< S (ρ_{1}, ρ_{2}) > = \frac{1}{2} \int_{0}^{+ \infty} [(W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω)) i + (W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{zx}^{(he)} (ρ_{1}, ρ_{2}, ω)) j

As can be seen from Eq. (7), the energy flux vector of partially coherent electromagnetic beam is different between the crossed-elements of electric-magnetic and magnetic-electric mutual cross spectral densities matrixes. Based on Eq. (5a) and the Maxwell’s equations, the elements of mutual cross spectral density matrix in Eq. (7) can be expressed as

W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{yy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{2}} - \frac{\partial W_{yx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{2}}],

W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{xz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{1}} - \frac{\partial W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{1}}],

W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{2}} - \frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{2}}],

W_{zx}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{yx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{1}} - \frac{\partial W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{1}}],

W_{xy}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{2}} - \frac{\partial W_{xz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{2}}],

W_{xy}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{i}{ω μ_{0}} [\frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{1}} - \frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{1}}],

other elements of mutual cross spectral density matrix of the partially coherent electromagnetic beams can be obtained in similar process. When the incident electromagnetic field is known, the energy and energy flux density of partially coherent electromagnetic field can be calculated by Eqs. (5)–(8). These formulas derived in this paper would be benefit to the research on the partially coherent electromagnetic field.

When the random phase item caused by the atmospheric turbulence in Eqs. (5)–(7) are zero, these formulas can applicable to calculate the energy and energy flux density in free space.

3. An application example: the energy and energy flux density of the partially coherent electromagnetic ChG beams in atmospheric turbulence

Suppose the electric of a partially coherent ChG electromagnetic field linearly polarized in the x direction at the plane z=0 is E(ρ ₀,0)=E_x(ρ ₀,0)i+E_y(ρ ₀,0)j, and various components of it take the forms as follows[19] :

E_{x} (ρ_{0}) = \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \cosh (Ω_{0} x_{0}) \cosh (Ω_{0} y_{0})'

E_{y} (ρ_{0}) = 0,

where w ₀ denotes the waist width of the Gaussian part, Ω₀ is a parameter associated with the cosh part, and ρ ₀=(x ₀,y ₀) is the vector at the plane z=0.

By introducing a Gaussian term of the spectral coherent degree, the fully coherent beam can be extended to the partially coherent one. Thus, the cross-spectral density matrix elements of linearly polarized partially coherent ChG beams are expressed as [20]

{[w_{α b}^{(e e)} (ρ_{01}, ρ_{02})]}^{(0)} = {\begin{array}{c} \cosh (Ω_{0} x_{01}) \cosh (Ω_{0} y_{01}) \cosh (Ω_{0} x_{02}) \cosh (Ω_{0} y_{02}) \\ \exp (- \frac{ρ_{01}^{2} + ρ_{02}^{2}}{w_{0}^{2}}) \exp (- \frac{{|ρ}_{01} - ρ_{02} |^{2}}{σ_{0}^{2}}) & a = b = x, \\ 0 & otherwise \end{array}

where σ ₀ denotes the correlation length, ρ_0s=x_0s i+y_0s j.

From Eqs. (1) (5a) and Eq. (10), the cross spectral density matrix of the partially coherent ChG electromagnetic beams at the half space z≥0 can be written as

[{\hat{W}}^{(ee)} (ρ_{10}, ρ_{20})] = [\begin{array}{c} W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) & 0 & W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) \\ 0 & 0 & 0 \\ W_{xz}^{(ee)} (ρ_{10}, ρ_{20}) & 0 & W_{zz}^{(ee)} (ρ_{10}, ρ_{20}) \end{array}] \times,

where

W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) = \frac{z_{1} z_{2}}{4 π^{2}} \int \int \int \int_{(z = 0)} {[W_{xx}^{(ee)} (ρ_{01}, ρ_{02})]}^{(0)} \frac{\exp [ik (R_{2} - R_{1})]}{R_{1}^{3} R_{2}^{3}} (ik R_{2} - 1),

W_{zz}^{(ee)} (ρ_{10}, ρ_{20}) = \frac{1}{4 π^{2}} \int \int \int \int_{(z = 0)} (x_{1} - x_{01}) (x_{2} - x_{02}) {[W_{xx}^{(ee)} (ρ_{01}, ρ_{02}, 0)]}^{(0)} \frac{\exp [ik (R_{2} - R_{1}]}{R_{1}^{2} R_{2}^{2}} (ik R_{2} - 1),

W_{xz} (ρ_{10}, ρ_{20}) = - \frac{z}{4 π^{2}} \int \int \int \int_{(z = 0)} (x_{2} - x_{02}) W_{xx}^{(0) (ρ_{01}, ρ_{02}, 0) \frac{\exp [ik (R_{2} - R_{1})]}{R_{1}^{2} R_{2}^{2}} (ik R_{2} - 1)},

W_{zx} (ρ_{10}, ρ_{20}) = W_{xz}^{*} (ρ_{10}, ρ_{20}) .

For simplicity, suppose that the random phase in each coordinate axis is same when the partially coherent ChG electromagnetic beams propagating in atmospheric turbulence, and the statistical ensemble average over the random complex phase of a spherical wave in a homogeneous turbulent atmosphere could be approximately expressed as [21]

< \exp [ψ_{α} (ρ_{1}, ρ_{10}, z_{1}) + ψ_{b}^{*} (ρ_{2}, ρ_{20}, z_{2})] >

where ρ ² ₀(z ₁,z ₂) is revised to

ρ_{0}^{2} (z_{1}, z_{2}) = {(0.545 C_{n}^{2} k^{2})}^{- 3 ⁄ 5} {(z_{1} z_{2})}^{- 3 ⁄ 10},

is the coherence length of a spherical wave propagating in turbulent atmosphere. C_n² is a refractive index structure constant.

We use the far-field approximation to R ₁and R ₂, i.e.

R_{s} \approx ρ_{s} - \frac{x_{s} x_{s} 0 + y_{s} y_{s} 0}{ρ_{s}},

ρ_{s} = \sqrt{x_{s}^{2} + y_{s}^{2} + z_{s}^{2}},

and employing the following equations

\int_{- \infty}^{\infty} \exp (- α x^{2} + cx) dx = \sqrt{\frac{π}{α}} \exp (\frac{c^{2}}{4 α}),

the non-zone elements of the electric cross spectral density matrix in turbulent atmosphere diffraction space can be written as

W_{xx}^{(ee)} (ρ_{1}, ρ_{2}) = \frac{π^{2} z_{1} z_{2} f_{0}}{16 A A_{1}} {\exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} + Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} + Ω_{0})}^{2}}{4 A_{1}}]},

W_{zx}^{(ee)} (ρ_{1}, ρ_{2}) = x_{1} z_{2} W_{xx}^{(ee)} (ρ_{1}, ρ_{2}) + \frac{z_{2} π^{2} f_{0}}{32 A A_{1}^{2}} [(B_{5} - Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{5} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{6} - Ω_{0}) \exp [\frac{h_{6}^{2}}{4 A} + \frac{{(B_{6} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{5} - Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{5} - Ω_{0})}^{2}}{4 A_{1}}] +

W_{zz}^{(ee)} (ρ_{1}, ρ_{2}) = x_{1} x_{2} w_{xx}^{(ee)} (ρ_{1}, ρ_{2}) - \frac{π^{2} f_{0}}{32 A A_{1}^{2}} {x_{1} [(B_{1} - Ω_{0}) \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{2} - Ω_{0}) \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{1} - Ω_{0})

w_{zx}^{(ee)} (ρ_{1}, ρ_{2}, z) = {[w_{zx}^{(ee)} (ρ_{1}, ρ_{2}, z)]}^{*},

where

A = \frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}, h_{1} = c_{3} - Ω_{0}, h_{2} = c_{3} + Ω_{0}, h_{3} = c_{4} - Ω_{0}, h_{4} = c_{4} + Ω_{0}, h_{5} = c_{1} - Ω_{0}, h_{6} = c_{1} + Ω_{0}, c_{1} = - \frac{ik x_{2}}{ρ_{2}},

Substitute Eqs. (16) and (17) into Eqs. (5)–(8), then the energy and energy flux density of partially coherent ChG electromagnetic beams in turbulent atmosphere can be obtained.

Suppose that the wavelength of partially coherent ChG electromagnetic beam λ=1.06µm, the waist width w ₀=0.2mm, the spatial correlation parameter β=[1+(w ₀/σ ₀)²]^-1/2=0.8, Ω₀=0.05, and the structure constant C ² _n=10^-16, the energy and energy flux density of partially coherent electromagnetic beams at z=10.0m plane can be calculated. The energy flux density depicted in following Fig. 1b-5b is the module of Poynting vector.

Figure 1 gives the spatial distributions and contour map of energy and energy flux density of the partially coherent ChG electromagnetic beams with the spatial correlation parameter β=0.8 at the plane z=10.0m in free space when the maximum of energy and energy flux density is normalized. Compare the two contour maps, it is shown that the spatial distribution of energy flux density is more concentrated than that of energy density.

Figure 2 gives the spatial distributions and contour map of energy and energy flux density of the partial coherent ChG electromagnetic beams with the spatial correlation parameter β=0.8 at the plane z=10.0m in turbulent atmosphere with C²_n=10^-16 when the maximum of energy and energy flux density is normalized. Compare with Fig. 1, it is shown that atmospheric turbulence has a significant effect on the spatial distributions of energy and energy flux density of the partially coherent ChG electromagnetic beam. From Figs. 2 (a) and (b), the atmospheric turbulence has a broadening effect on the spatial distributions of energy and energy flux density of partially coherent ChG electromagnetic beam, however, the degree of broadening effect caused by the atmospheric turbulence on the energy flux density is less than that on the energy density.

Figure 3 shows the spatial distributions and contour of the energy and energy flux of the partially coherent ChG electromagnetic beam with the spatial correlation parameter β=0.8 at the plane z=10.0m in turbulent atmosphere with C²_n=10^-15. Compare with Fig. 2, it is shown that the effect of turbulent atmosphere on the energy spatial distribution of energy density of the partially coherent ChG electromagnetic beams is more obvious than that on the energy flux spatial distribution, in other words, the energy flux density of partially coherent electromagnetic beams has less sensitivity on the turbulent atmosphere than the energy density.

Figure 4 gives the spatial distribution and contour of the energy flux of the partially coherent ChG electromagnetic beams with the spatial correlation parameter β=0.8 and β=0.6 at the plane z=10.0m in turbulent atmosphere with C ² _n=10^-16. The figures show that the energy flux density of partially coherent electromagnetic field is sensitivity on the spatial correlation parameter, and the broadening effect caused by atmosphere turbulent of partially coherent ChG electromagnetic beam with a smaller spatial correlation parameter is more than that of partially coherent ChG electromagnetic beam with a bigger spatial correlation parameter.

Fig. 1. The spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in free space. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour

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Fig. 2. The spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C ² _n=10^-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Fig. 3. The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C ² _n=10^-15. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Fig. 4. The spatial distribution and contour of the energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.6 and β=0.8 at the plane z=10.0m in turbulent atmosphere with C ² _n=10^-16. (a) β=0.8 (b) β=0.6

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Fig. 5. The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=5.0 m in turbulent atmosphere with C ² _n=10-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Figure 5 gives the spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with the spatial correlation parameter β=0.8 at the plane z=5.0m in turbulent atmosphere with C ² _n=10-16. Compare with Fig. 3, it is shown that the broadening effect of energy and energy flux density increases with the increasing of the propagation distance. It may be due to the increasing of the transmission distance, the impact of turbulent atmosphere on the electromagnetic energy and energy flux is a cumulative process.

The main physical reasons caused above results are that there exists a spherical perturbation on each sub-center of transmission wave front and it is coupled with the turbulence-induced wave front random phase. This leads to the spatial widths of energy and energy density spread with the increasing of the propagation distance. The additional wave-front random phase generated by turbulence is superimposed on the original wave-front random phase, while the randomness of original wave-front would reduce the effects of atmospheric turbulence on the wave-front. Therefore, the turbulence has a smaller effect on the spatial distribution of the energy flux density of partially coherent electromagnetic beams which with a smaller spatial correlation parameter.

4. Conclusion

Based on the classical electromagnetic theory, the formulas of energy and energy flux density of partially coherent electromagnetic beams in atmosphere turbulent have been derived in this paper. The elements of magnetic and mutual cross spectral density matrix have been also successively expressed by the elements of electric cross spectral density. At the same time, we take the partially coherent ChG electromagnetic beam as a typical example to analyze the application of these formulas, and numerically calculate the spatial distributions of energy and energy flux density of partially coherent ChG electromagnetic beam with these formulas derived in this paper. It is shown that the spatial distributions of energy and energy flux density is determined by the structure constant C ² _n, spatial correlation parameter β and the location of observation points. In free space, the spatial distribution of energy flux density is more concentrated than that of energy density. Atmospheric turbulence has a broadening effect on the spatial distributions of energy and energy flux density of the partially coherent ChG electromagnetic beams. However, the degree of broadening effect caused by the atmosphere turbulent on the energy density is different from that on the energy flux density. The turbulence has a smaller effect on the spatial distribution of the energy flux density of partially coherent electromagnetic beams with smaller spatial correlation parameters. The results in this article are of great practical significance.

Acknowledgments

This work was supported by the Open Foundation of the Sate Key Laboratory of Optical Technologies for Micro-Fabrication & Micro-Engineering, Chinese Academy of Science. One of authors (J. Li) is grateful to the reviewer for his helpful comments.

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Fig. 2. The spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C ² _n =10^-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Fig. 3. The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C ² _n =10^-15. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Fig. 5. The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=5.0 m in turbulent atmosphere with C ² _n =10-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

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Equations (58)

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E_{α} (ρ_{s}, Z_{s}) = - \frac{1}{2 π} \iint_{(z = 0)} E_{α} (ρ_{s 0}, 0) \frac{\partial}{{\partial z}_{s}} (\frac{\exp (ik R_{s})}{R_{s}}) \exp [ψ_{α} (ρ_{s}, ρ_{s 0}, Z_{s})] d^{2} ρ_{s 0}, {(α = x, y)}^{'}

E_{z} (ρ_{s}, z_{s}) = \frac{1}{2 π} \int \int_{(z = 0)} [E_{x} (ρ_{s 0}, z_{s}) \frac{\partial}{\partial x_{s}} (\frac{\exp (ik R_{s})}{R_{s}}) \exp [ψ_{x} (ρ_{s}, ρ_{s 0}, z_{s}))]

+ E_{y} (ρ_{s 0}, 0) \frac{\partial}{\partial y_{s}} (\frac{\exp (ik R_{s})}{R_{s}}) \exp [ψ_{y} (ρ_{s}, ρ_{s 0}, z_{s})] d^{2} ρ_{s 0}'

R_{s} = \sqrt{{{(x_{s} - x_{s 0})}^{2} + (y_{s} - y_{s 0})}^{2} + z_{z}^{2},}

k = 2 π ⁄ λ (wave number),

E = \frac{1}{2} [(E_{x}^{*} (ρ_{1}, t) + E_{x} (ρ_{2}, t + τ)) i + (E_{y}^{*} (ρ_{1}, t) + E_{y} (ρ_{2}, t + τ)) j + (E_{z}^{*} (ρ_{1_{1}}, t, ω) + E_{z} (ρ_{2}, t + τ)) k] .

w = \frac{E \cdot D}{2} + \frac{B \cdot H}{2} (energy density),

S = E \times H, (energy flux density),

< M^{*} (r_{1}, t + τ) N (r_{2}, t) > = lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} M^{*} (r_{1}, t + τ) N (r_{2}, t) dt,

< w_{e} (ρ_{1}, ρ_{2}, ω) > = \frac{ε_{0}}{4} \underset{α = x, y, z}{Σ} W_{vv}^{(ee)} (ρ_{1}, ρ_{2}, ω),

W_{vv}^{(ab)} (ρ_{1}, ρ_{2}, ω) = < M_{v}^{*} (ρ_{1}, ω) N_{v} (ρ_{2}, ω), >, (a, b = e, h; M, N = E, H),

< w_{e} (ρ_{1}, ρ_{2}) > = \frac{ε_{0}}{4} \int_{0}^{\infty} \underset{v = x, y, z}{Σ} W_{vv}^{(ee)} (ρ_{1}, ρ_{2}, ω) d ω,

< w_{h} (ρ_{1}, ρ_{2}, ω) > = \frac{1}{ω^{2} μ_{0}^{2}} [(\frac{\partial_{2}}{\partial z_{1} \partial y_{2}} - \frac{\partial^{2}}{\partial y_{1} \partial z_{2}}) W_{yz}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial y_{1} \partial y_{2}} W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial z_{1} \partial z_{2}} W_{yy}^{(ee)} (ρ_{1}, ρ_{2}, ω) +

(\frac{\partial^{2}}{\partial z_{1} \partial x_{2}} \frac{\partial^{2}}{\partial x_{1} \partial z_{2}}) W_{xz}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial z_{1} \partial z_{2}} W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω) +

(\frac{\partial^{2}}{\partial y_{1} \partial x_{2}} - \frac{\partial^{2}}{\partial x_{1} \partial y_{2}}) W_{xy}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} W_{yy}^{(ee)} (ρ_{1}, ρ_{2}, ω) - \frac{\partial^{2}}{\partial y_{1} \partial y_{2}} W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω),

< w_{h} (ρ_{1}, ρ_{2}) > = \frac{μ_{0}}{4} \int < w_{h} (ρ_{1}, ρ_{2}, ω) > d ω .

< s (ρ_{1}, ρ_{2}, ω) > = \frac{1}{2} [(W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω) i + (W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω),

- W_{zx}^{(he)} (ρ_{1}, ρ_{2}, ω) j + (W_{xy}^{(eh)} (ρ_{1}, ρ_{2}, {ω) - W}_{xy}^{(he)} (ρ_{1}, ρ_{2}, ω)) k]

< S (ρ_{1}, ρ_{2}) > = \frac{1}{2} \int_{0}^{+ \infty} [(W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω)) i + (W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{zx}^{(he)} (ρ_{1}, ρ_{2}, ω)) j

+ (W_{xy}^{(eh)} (ρ_{1}, ρ_{2}, ω) - W_{xy}^{(he)} (ρ_{1}, ρ_{2}, ω)) k] d ω .

W_{yz}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{yy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{2}} - \frac{\partial W_{yx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{2}}],

W_{yz}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{xz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{1}} - \frac{\partial W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{1}}],

W_{zx}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{zz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{2}} - \frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{2}}],

W_{zx}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{yx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{1}} - \frac{\partial W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{1}}],

W_{xy}^{(eh)} (ρ_{1}, ρ_{2}, ω) = \frac{- i}{ω μ_{0}} [\frac{\partial W_{xx}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{2}} - \frac{\partial W_{xz}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial x_{2}}],

W_{xy}^{(he)} (ρ_{1}, ρ_{2}, ω) = \frac{i}{ω μ_{0}} [\frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial y_{1}} - \frac{\partial W_{zy}^{(ee)} (ρ_{1}, ρ_{2}, ω)}{\partial z_{1}}],

E_{x} (ρ_{0}) = \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \cosh (Ω_{0} x_{0}) \cosh (Ω_{0} y_{0})'

E_{y} (ρ_{0}) = 0,

{[w_{α b}^{(e e)} (ρ_{01}, ρ_{02})]}^{(0)} = {\begin{array}{c} \cosh (Ω_{0} x_{01}) \cosh (Ω_{0} y_{01}) \cosh (Ω_{0} x_{02}) \cosh (Ω_{0} y_{02}) \\ \exp (- \frac{ρ_{01}^{2} + ρ_{02}^{2}}{w_{0}^{2}}) \exp (- \frac{{|ρ}_{01} - ρ_{02} |^{2}}{σ_{0}^{2}}) & a = b = x, \\ 0 & otherwise \end{array}

[{\hat{W}}^{(ee)} (ρ_{10}, ρ_{20})] = [\begin{array}{c} W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) & 0 & W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) \\ 0 & 0 & 0 \\ W_{xz}^{(ee)} (ρ_{10}, ρ_{20}) & 0 & W_{zz}^{(ee)} (ρ_{10}, ρ_{20}) \end{array}] \times,

W_{xx}^{(ee)} (ρ_{10}, ρ_{20}) = \frac{z_{1} z_{2}}{4 π^{2}} \int \int \int \int_{(z = 0)} {[W_{xx}^{(ee)} (ρ_{01}, ρ_{02})]}^{(0)} \frac{\exp [ik (R_{2} - R_{1})]}{R_{1}^{3} R_{2}^{3}} (ik R_{2} - 1),

\times (- 1 - ik R_{1}) < \exp [ψ_{x} (ρ_{01}, ρ_{10}, z) + ψ_{x} (ρ_{02}, ρ_{20}, z)] > d^{2} ρ_{01} d^{2} ρ_{02}

W_{zz}^{(ee)} (ρ_{10}, ρ_{20}) = \frac{1}{4 π^{2}} \int \int \int \int_{(z = 0)} (x_{1} - x_{01}) (x_{2} - x_{02}) {[W_{xx}^{(ee)} (ρ_{01}, ρ_{02}, 0)]}^{(0)} \frac{\exp [ik (R_{2} - R_{1}]}{R_{1}^{2} R_{2}^{2}} (ik R_{2} - 1),

\times (- 1 - ik R_{1}) < \exp [ψ_{z} (ρ_{01}, ρ_{10}, z) + ψ_{z} (ρ_{02}, ρ_{20}, z)] > d^{2} ρ_{01} d^{2} ρ_{02}

W_{xz} (ρ_{10}, ρ_{20}) = - \frac{z}{4 π^{2}} \int \int \int \int_{(z = 0)} (x_{2} - x_{02}) W_{xx}^{(0) (ρ_{01}, ρ_{02}, 0) \frac{\exp [ik (R_{2} - R_{1})]}{R_{1}^{2} R_{2}^{2}} (ik R_{2} - 1)},

\times (- 1 - ik R_{1}) < \exp [ψ_{x} (ρ_{01}, ρ_{10}, z) + ψ_{z} (ρ_{02}, ρ_{20}, z)] > d^{2} ρ_{01} d^{2} ρ_{02}

W_{zx} (ρ_{10}, ρ_{20}) = W_{xz}^{*} (ρ_{10}, ρ_{20}) .

< \exp [ψ_{α} (ρ_{1}, ρ_{10}, z_{1}) + ψ_{b}^{*} (ρ_{2}, ρ_{20}, z_{2})] >

\approx \exp {- \frac{1}{ρ_{0}^{2} (z_{1}, z_{2})} [{(ρ_{20} - ρ_{10})}^{2} + {(ρ_{2} - ρ_{1})}^{2} + (ρ_{1} - ρ_{2}) (ρ_{10} - ρ_{20})]},

ρ_{0}^{2} (z_{1}, z_{2}) = {(0.545 C_{n}^{2} k^{2})}^{- 3 ⁄ 5} {(z_{1} z_{2})}^{- 3 ⁄ 10},

R_{s} \approx ρ_{s} - \frac{x_{s} x_{s} 0 + y_{s} y_{s} 0}{ρ_{s}},

ρ_{s} = \sqrt{x_{s}^{2} + y_{s}^{2} + z_{s}^{2}},

\int_{- \infty}^{\infty} \exp (- α x^{2} + cx) dx = \sqrt{\frac{π}{α}} \exp (\frac{c^{2}}{4 α}),

W_{xx}^{(ee)} (ρ_{1}, ρ_{2}) = \frac{π^{2} z_{1} z_{2} f_{0}}{16 A A_{1}} {\exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} + Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} + Ω_{0})}^{2}}{4 A_{1}}]},

\times {\exp [\frac{h_{3}^{2}}{4 A} + \frac{{(B_{3} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{3}^{2}}{4 A} + \frac{{(B_{3} + Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} + Ω_{0})}^{2}}{4 A_{1}}]}

W_{zx}^{(ee)} (ρ_{1}, ρ_{2}) = x_{1} z_{2} W_{xx}^{(ee)} (ρ_{1}, ρ_{2}) + \frac{z_{2} π^{2} f_{0}}{32 A A_{1}^{2}} [(B_{5} - Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{5} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{6} - Ω_{0}) \exp [\frac{h_{6}^{2}}{4 A} + \frac{{(B_{6} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{5} - Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{5} - Ω_{0})}^{2}}{4 A_{1}}] +

(B_{5} - Ω_{0}) \exp [[\frac{h_{6}^{2}}{4 A} + \frac{{(B_{6} + Ω_{0})}^{2}}{4 A_{1}}]] {\exp [\frac{h_{3}^{2}}{4 A} + \frac{{(B_{3} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{3}^{2}}{4 A} + \frac{{(B_{3} + Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} - Ω_{0})}^{2}}{4 A_{1}}]},

W_{zz}^{(ee)} (ρ_{1}, ρ_{2}) = x_{1} x_{2} w_{xx}^{(ee)} (ρ_{1}, ρ_{2}) - \frac{π^{2} f_{0}}{32 A A_{1}^{2}} {x_{1} [(B_{1} - Ω_{0}) \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{2} - Ω_{0}) \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{1} - Ω_{0})

\times \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} + Ω_{0})}^{2}}{4 A_{1}}] + (B_{2} + Ω_{0}) \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}]] + x_{2} [(B_{5} - Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{5} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{6} - Ω_{0}) \times

\exp [\frac{h_{6}^{2}}{4 A} + \frac{{(B_{6} + Ω_{0})}^{2}}{4 A_{1}}] + (B_{5} + Ω_{0}) \exp [\frac{h_{5}^{2}}{4 A} + \frac{{(B_{6} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{6} - Ω_{0}) \exp [\frac{h_{6}^{2}}{4 A} + \frac{{(B_{6} - Ω_{0})}^{2}}{4 A_{1}}]] - [\frac{1}{A} (B_{1} - Ω_{0}) \times

\exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{1} + Ω_{0}) \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}]) + \frac{h_{1}}{2 A} ((B_{2} - Ω_{0}) \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} - Ω_{0})}^{2}}{4 A_{1}}] + (B_{2} - Ω_{0}) \times

\exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} + Ω_{0})}^{2}}{4 A_{1}}])] - \frac{B}{4 A A_{1}} [2 A_{1} + {(B_{2} - Ω_{0})}^{2}) \exp [\frac{h_{2}^{2}}{4 A} + \frac{{(B_{2} + Ω_{0})}^{2}}{4 A_{1}}] + (2 A_{1} + {(B_{2} + Ω_{0})}^{2}) \exp [\frac{h_{2}^{2}}{4 A} +

\frac{{(B_{2} + Ω_{0})}^{2}}{4 A_{1}}] + (2 A_{1} + {(B_{1} - Ω_{0})}^{2}) \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} - Ω_{0})}^{2}}{4 A_{1}}] + (2 A_{1} + {(B_{1} + Ω_{1})}^{2}) \exp [\frac{h_{1}^{2}}{4 A} + \frac{{(B_{1} + Ω_{0})}^{2}}{4 A_{1}}]]} {\exp [\frac{h_{3}^{2}}{4 A} +

\frac{{(B_{3} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{3}^{2}}{4 A} + \frac{{(B_{3} + Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} - Ω_{0})}^{2}}{4 A_{1}}] + \exp [\frac{h_{4}^{2}}{4 A} + \frac{{(B_{4} + Ω_{0})}^{2}}{4 A_{1}}]}

w_{zx}^{(ee)} (ρ_{1}, ρ_{2}, z) = {[w_{zx}^{(ee)} (ρ_{1}, ρ_{2}, z)]}^{*},

A = \frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}, h_{1} = c_{3} - Ω_{0}, h_{2} = c_{3} + Ω_{0}, h_{3} = c_{4} - Ω_{0}, h_{4} = c_{4} + Ω_{0}, h_{5} = c_{1} - Ω_{0}, h_{6} = c_{1} + Ω_{0}, c_{1} = - \frac{ik x_{2}}{ρ_{2}},

c_{2} = - \frac{ik y_{2}}{ρ_{2}}, c_{3} = \frac{ik x_{1}}{ρ_{2}}, c_{4} = \frac{ik y_{1}}{ρ_{2}}, B = \frac{2}{σ}, A_{1} = A - \frac{B^{2}}{4 A}, B_{1} = c_{1} + \frac{B h_{1}}{2 A}, σ = p_{0} σ_{0} ⁄ \sqrt{p_{0}^{2} + σ_{0}^{2}}, B_{2} = c_{1} + \frac{B h_{2}}{2 A},

B_{3} = c_{2} + \frac{B h_{3}}{2 A}, B_{4} = c_{2} + \frac{B h_{4}}{2 A}, B_{5} = c_{3} + \frac{B h_{5}}{2 A}, B_{6} = c_{3} \frac{B h_{6}}{2 A}, f_{0} = \frac{\exp [ik (ρ_{2} - ρ_{1})]}{4 π^{2} ρ_{1}^{3} ρ_{2}^{3}} (ik ρ_{2} - 1) (- 1 - ik ρ_{1}),

Abstract

1. Introduction

2. The energy and energy flux density of partially coherent electromagnetic beam

3. An application example: the energy and energy flux density of the partially coherent electromagnetic ChG beams in atmospheric turbulence

4. Conclusion

Acknowledgments

References and links

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Figures (5)

Equations (58)

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