Study on power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere

Huiyun Wu; Shen Sheng; Zhisong Huang; Siqing Zhao; Hua Wang; Zhenhai Sun; Xiegu Xu

doi:10.1364/OE.21.004005

Optics Express
Vol. 21,
Issue 4,
pp. 4005-4016
(2013)
•https://doi.org/10.1364/OE.21.004005

Study on power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere

Huiyun Wu, Shen Sheng, Zhisong Huang, Siqing Zhao, Hua Wang, Zhenhai Sun, and Xiegu Xu

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Huiyun Wu, Shen Sheng, Zhisong Huang, Siqing Zhao, Hua Wang, Zhenhai Sun, and Xiegu Xu, "Study on power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere," Opt. Express 21, 4005-4016 (2013)

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- Atmospheric and Oceanic Optics
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About this Article
History
- Original Manuscript: November 7, 2012
- Revised Manuscript: December 15, 2012
- Manuscript Accepted: January 23, 2013
- Published: February 11, 2013

Abstract

As a new attractive application of the vortex beams, power coupling of annular vortex beam propagating through a two- Cassegrain-telescope optical system in turbulent atmosphere has been investigated. A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system is established, the general analytical expression of vortex beams with limited apertures and the analytical formulas for the average intensity distribution at the receiver plane are derived. Under the H-V 5/7 turbulence model, the average intensity distribution at the receiver plane and power coupling efficiency of the optical system are numerically calculated, and the influences of the optical topological charge, the laser wavelength, the propagation path and the receiver apertures on the power coupling efficiency are analyzed. These studies reveal that the average intensity distribution at the receiver plane presents a central dark hollow profile, which is suitable for power coupling by the Cassegrain telescope receiver. In the optical system with optimized parameters, power coupling efficiency can keep in high values with the increase of the propagation distance. Under the atmospheric turbulent conditions, great advantages of vortex beam in power coupling of the two-Cassegrain-telescope optical system are shown in comparison with beam without vortex.

1. Introduction

In recent years, optical beams carrying phase singularities which are known as the vortex beams, have become a live area of scientific research due to their importance in basic science and some attractive applications including microlithography, optical trapping, atom detection and optical communications [1–5]. The basic characteristics and the turbulent propagation properties of the vortex beams have been widely investigated [5–9]. Meanwhile, power coupling of optical system with the Cassegrain-telescope receiver has attracted much attention [10–14]. Previous researches showed that in the optical system, optical intensity gathers on the center and spreads out on the way up due to the influences of diffraction and the turbulence, when using a Cassegrain-telescope receiver, a large number of optical power losses due to the secondary mirror obstruction and the main mirror truncation [10,11]. The power losses can seriously degrade performance of the optical system, for example, the aerospace relay mirror system (ARMS) used in the outfield experiment at Kirtland Air Force Base N.M in 2006 employed a two-Cassegrain-telescope configuration, and the experimental results showed that power coupling efficiency of the ARMS was merely 50% [12,13]. How to reduce the serious power losses in optical system with the Cassegrain-telescope receiver has become an urgent work in the scientific research with great importance. Our recent research suggested that under the ideal adaptive optics conditions or the vacuum condition, vortex beams have great advantages in improving power coupling efficiency of optical system with the Cassegrain-telescope receiver [11,14]. Namely, power coupling in optical system with Cassegrain-telescope receiver may become a new attractive potential application of the vortex beams [11,14]. As a new area of scientific research, power coupling of vortex beam propagating through optical system with the Cassegrain-telescope receiver requires much practical investigation. In actuality, the adaptive optics installations in the optical systems can’t approach the ideal conditions, beams with dark hollow intensity spatial profiles are commonly used as the source and the propagation of laser beam is in a slant path through the turbulent atmosphere [10,15]. To the best of our knowledge, study on the characteristics of power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere has never been reported.

This paper is organized as follows. First, a typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere is established, and the analytical formulas for the average intensity distribution at the receiver plane are derived in Section 2. After that, numerical calculations and results analysis are provided in Section 3. In the end, the conclusions are outlined in section 4.

2. Models and theoretical analysis

A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere is shown in Fig. 1 . An annular vortex beam with phase singularity is used as the source, a Cassegrain telescope with central obscuration at the source plane is used as the beam transmitter, and another Cassegrain telescope is used as the beam receiver. The vortex beam propagates from the transmitter to the receiver through the atmospheric turbulence, and the uplink beam is received by the Cassegrain receiver.

Fig. 1 A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere.

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The optical field at the source plane can be expressed as

E_{0} (r, θ, 0) = E (r, θ, 0) t (r, a, b),

E (r, θ, 0) = A (r) \exp (i l θ),

t (r, a, b) = {\begin{matrix} 1 & a \leq r \leq b \\ 0 & e l s e \end{matrix},

where denote the polar coordinates at the source plane, denotes the optical field at the source plane, denotes the optical field of the vortex source, denotes the amplitude, denotes the optical topological charge, denotes the truncation function of the transmitter, denotes the inner radius and denotes the outer radius. In order to reduce power losses induced by the transmitter telescope, dark hollow vortex beams are used. The optical field of a dark hollow vortex beam can be expressed as

E (r, θ, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\begin{matrix} N \\ n \end{matrix}) [\exp (- \frac{n r^{2}}{ω_{0}^{2}}) - \exp (- \frac{n r^{2}}{σ ω_{0}^{2}})] \exp (i l θ),

where

N

is the expansion order,

n

is the offset of the corresponding Gaussian component,

(\begin{matrix} N \\ n \end{matrix})

represents the binomial coefficient,

ω_{0}^{}

is the Gaussian waist width,

σ

(

0 < σ < 1

) is a parameter concerning the circular dark hollow beam. When

N = 1

and

σ \to 0

, Eq. (4) represents a Gaussian vortex beam, and with

N

increases, Eq. (4) converts to a flattened vortex beam. The area of the dark region across a dark hollow vortex beam increases as

σ

N

increases [10,16]. By using the Gaussian functions, the truncation function of the transmitter can be expressed as [16,17]

t (r, a, b) = \sum_{w = 1}^{M} B_{w} [\exp (- \frac{C_{w}}{b^{2}} r^{2}) - \exp (- \frac{C_{w}}{a^{2}} r^{2})],

where

M

is the expansion order,

B_{w}

and

C_{w}

are the complex expansion coefficients. Thus, optical field at the source plane can be expressed as

\begin{array}{l} E_{0} (r, θ, 0) = \sum_{n = 1}^{N} \sum_{w = 1}^{M} \frac{{(- 1)}^{n - 1} B_{w}}{N} (\begin{matrix} N \\ n \end{matrix}) [\exp (- \frac{C_{w}}{b^{2}} r^{2}) - \exp (- \frac{C_{w}}{a^{2}} r^{2})] \\ \begin{matrix}  \end{matrix} \times [\exp (- \frac{n r^{2}}{ω_{0}^{2}}) - \exp (- \frac{n r^{2}}{σ ω_{0}^{2}})] \exp (i l θ) . \end{array}

Equation (6) represents the general analytical expression of vortex beams with limited apertures. By choosing different

l

, we can get different optical topological charges; by choosing different

ω_{0}^{}

σ

and

N

, we can get different intensity spatial profiles; by choosing different

a

and

b

, we can get different aperture functions.

By using the extended Huygens-Fresnel principle, the average intensity distribution at the receiver plane can be expressed as

\begin{array}{l} < I (R, φ, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{2 π} E_{0} (r_{1}, θ_{1}, 0) \exp {\frac{i k}{2 L} [R^{2} + r_{1}^{2} - 2 R r_{1} \cos (φ - θ_{1})]} \\ \begin{matrix}  \end{matrix} \times {E_{0} (r_{2}, θ_{2}, 0) \exp {\frac{i k}{2 L} [R^{2} + r_{2}^{2} - 2 R r_{2} \cos (φ - θ_{2})]}}^{*} \\ \begin{matrix}  \end{matrix} \times < \exp [ψ (R, φ, r_{1}, θ_{1}) + ψ^{*} (R, φ, r_{2}, θ_{2})] > r_{1} r_{2} d r_{1} d r_{2} d θ_{1} d θ_{2}, \end{array}

where

L

denotes the propagation distance,

(R, φ, L)

denote the polar coordinates at the receiver plane, k is the wave number, the asterisk denotes the complex conjugation, the < > indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere,

ψ (R, φ, r, θ)

represents the random part of the complex phase of a spherical wave that propagates from point

(r, θ, 0)

at the source plane to point

(R, φ, L)

at the receiver plane. The ensemble average term in Eq. (7) can be expressed as [18,19]

\begin{array}{l} < \exp [ψ (R, φ, r_{1}, θ_{1}) + ψ^{*} (R, φ, r_{2}, θ_{2})] > = \exp [- 0.5 D_{ψ} (r_{1}, r_{2}, θ_{1}, θ_{2})] \\ \begin{matrix}  \end{matrix} = \exp {- \frac{1}{ρ_{0}^{2}} [r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})]}, \end{array}

where

D_{ψ}

is the wave structure function,

ρ_{0} = {(0.545 \bar{C_{n}^{2}} k^{2} L)}^{- 3 / 5}

is the coherent length of spherical wave propagating in the turbulence,

{\bar{C}}_{n}^{2} = \frac{1}{H} \int_{0}^{H} C_{n}^{2} (h) d h

is the average value of the atmospheric refractive index structure constant along the propagation path,

H = L \cos ξ

is the vertical distance of the propagation path,

ξ

is the zenith angle,

C_{n}^{2} (h)

is the atmospheric refractive index structure constant along the vertical path,

h

is the altitude from the ground.

Inserting Eq. (6) and Eq. (8) into Eq. (7), the average intensity distribution at the receiver plane can be expressed as

\begin{array}{l} < I (R, φ, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \sum_{w = 1}^{M} \sum_{v = 1}^{M} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) B_{w} B_{v}^{*} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{2 π} \exp (- \frac{n r_{1}^{2}}{ω_{0}^{2}}) \exp (- \frac{m r_{2}^{2}}{ω_{0}^{2}}) \\ \begin{matrix}  \end{matrix} \times \exp (- \frac{C_{w}}{a^{2}} r_{1}^{2}) \exp (- \frac{C_{v}^{*}}{a^{2}} r_{2}^{2}) \exp {\frac{i k}{2 L} [r_{1}^{2} - 2 R r_{1} \cos (φ - θ_{1}) - r_{2}^{2} + 2 R r_{2} \cos (φ - θ_{2})]} \\ \begin{matrix}  \end{matrix} \times \exp [i l (θ_{1} - θ_{2})] \exp {- \frac{1}{ρ_{0}^{2}} [r_{1}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2}) + r_{2}^{2}]} r_{1} r_{2} d r_{1} d r_{2} d θ_{1} d θ_{2} . \end{array}

By using the following integral formulas [18–21]

\exp [\frac{i k R r}{α} \cos (φ - θ)] = \sum_{l = - \infty}^{\infty} i^{l} J_{l} (\frac{k R r}{α}) \exp [i l (φ - θ)],

\int_{0}^{2 π} \exp [- i l θ + 2 α R r \cos (θ - φ)] d θ = 2 π \exp (- i l φ) I_{l} (2 α R r),

I_{l} (x) = \sum_{m = 0}^{\infty} \frac{{(x / 2)}^{2 m + l}}{m! Γ (m + l)},

J_{l} (x) = \frac{{(- i)}^{l}}{2 π} \int_{0}^{2 π} \exp (i x \cos θ + i l θ) d θ,

\int_{0}^{\infty} t^{m - 1} \exp (- α t^{2}) J_{l} (β t) d t = \frac{α^{- m / 2}}{2 l!} {(\frac{β^{2}}{4 α})}^{l / 2} \exp (- \frac{β^{2}}{4 α}) Γ (\frac{l + m}{2}) F_{1} (\frac{- m + l + 2}{2}; l + 1; \frac{β^{2}}{4 α}),

and perform the related integrals in Eq. (9). We get that the average intensity distribution at the receiver plane is independent of

φ

and can be expressed as

< I (R, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \sum_{w = 1}^{M} \sum_{v = 1}^{M} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) B_{w} B_{v}^{*} [(T_{1} - T_{2} + T_{3} - T_{4}) - ({T^{'}}_{1} - {T^{'}}_{2} + {T^{'}}_{3} - {T^{'}}_{4})],

\begin{array}{l} T_{1} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) {(H_{1, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{2} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) {(H_{1, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{3} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) {(H_{1, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{4} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) {(H_{1, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{1} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) {(H_{2, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{2} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) {(H_{2, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{3} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) {(H_{2, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}_{1}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{4} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) {(H_{2, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

where

P_{1} = 0.5 l + t + s + 1,

P_{2} = - 0.5 l - t,

γ = \frac{1}{ρ_{0}^{2}},

H_{1, 1} = \frac{n}{ω_{0}^{2}} + \frac{C_{w}}{b^{2}} + \frac{1}{ρ_{0}^{2}} - \frac{i k}{2 L},

H_{1, 2} = \frac{n}{ω_{0}^{2}} + \frac{C_{w}}{a^{2}} + \frac{1}{ρ_{0}^{2}} - \frac{i k}{2 L},

H_{2, 1} = \frac{n}{σ ω_{0}^{2}} + \frac{C_{w}}{b^{2}} + \frac{1}{ρ_{0}^{2}} - \frac{i k}{2 L},

H_{2, 2} = \frac{n}{σ ω_{0}^{2}} + \frac{C_{w}}{a^{2}} + \frac{1}{ρ_{0}^{2}} - \frac{i k}{2 L},

{H^{'}}_{1, 1} = \frac{m}{ω_{0}^{2}} + \frac{C_{v}^{*}}{b^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{i k}{2 L},

{H^{'}}_{1, 2} = \frac{m}{ω_{0}^{2}} + \frac{C_{v}^{*}}{a^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{i k}{2 L},

{H^{'}}_{2, 1} = \frac{m}{σ ω_{0}^{2}} + \frac{C_{v}^{*}}{b^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{i k}{2 L},

{H^{'}}_{2, 2} = \frac{m}{σ ω_{0}^{2}} + \frac{C_{v}^{*}}{a^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{i k}{2 L},

J_{l} (.)

is the l-th order Bessel function of the first kind,

I_{l} (.)

is the l-th order modified Bessel function of the first kind,

Γ (.)

is the Gamma function and

F_{1} (.; .; .)

is the Kummer function. Equations from Eq. (15) to Eq. (23) represent the analytical expression of the average intensity distribution at the receiver plane. Power coupling efficiency of the two-Cassegrain-telescope optical system can be expressed as

η = \frac{2 π \int_{a^{'}}^{b^{'}} < I (R, L) > R d R}{\int_{a}^{b} \int_{0}^{2 π} E_{0} (r, θ, 0) E_{0}^{*} (r, θ, 0) r d r d θ},

where

a^{'}

and

b^{'}

denote the inner radius and the outer radius of the receiver, respectively.

3. Numerical calculations and results analysis

3.1 Average intensity distribution at the receiver plane

Parameters of the optical system are set as: the source is an ideal annular vortex beam with unit amplitude and 3.8um wavelength, the optical topological charge $l = 4$ , the dark hollow parameter $σ = 0.5$ , the source expansion order $N = 10$ , the Gaussian waist width $ω_{0} = 0.2 m$ , the inner radius of the transmitter $a = 0.15 m$ , the outer radius of the transmitter $b = 0.40 m$ , the expansion order of the transmitter truncation function $M = 16$ , the propagation distance $L = 10 k m$ and the zenith angle $ξ = 30^{\circ}$ . Distribution of the atmospheric structure constant is described as the H-V 5/7 turbulence model

C_{n}^{2} (h) = 8.2 \times 10^{- 56} V {(h)}^{2} h^{10} \exp (- h / 1000) + 2.7 \times 10^{- 16} \exp (- h / 1500) + C_{0} \exp (- h / 100),

V (h) = 5 + 30 \exp {{[- (h - 9400) / 4800]}^{2}},

where

h

is the altitude from the ground,

V (h)

is the wind speed along the vertical path,

C_{0}

is the nominal value of at ground level (the typical value is 4.0 × 10⁻¹⁴m^-2/3).

Figure 2 shows the optical field of the annular vortex source, and the dashed lines in Fig. 2(a) represent the truncation apertures of the transmitter. After calculation, we get that 97.5% power of the annular vortex beam can pass through the transmitter telescope.

Fig. 2 The optical field of the annular vortex source: (a) intensity distribution, (b) phase distribution.

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Figure 3 shows the average intensity distribution at the receiver plane, we can get that the average intensity distribution presents a central dark hollow profile. In order to show the influences of the optical topological charge, the laser wavelength, the propagation distance and the zenith angle on the average intensity distribution at the receiver plane, some numerical simulations are performed.

Fig. 3 The average intensity distribution at the receiver plane with the parameters ( $l = 4,$ $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $L = 10 k m,$ $ξ = 30^{\circ}$ ).

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Figure 4 shows the evolution of the average intensity distribution with the increase of the optical topological charge. When $l = 0$ , the source reduces to an annular beam without vortex, optical intensity gathers on the center and the average intensity distribution presents a complex profile. When with vortex ( $l \geq 1$ ), the average intensity on the center decreases and the central dark hollow generates. Figure 4(a) shows that when $l \leq 3$ , the width of the central dark hollow region increases with the increase of the optical topological charge, meanwhile radius of the beam spot keeps in the same. Figure 4(b) shows that with the further increase of the optical topological charge, both the width of the central dark hollow region and radius of the beam spot increases, the increment speed of the central dark hollow radius is much faster than that of the beam spot, and the peak value of the average intensity approximately keeps in equivalent.

Fig. 4 Evolution of the average intensity distribution with the increase of the optical topological charge ( $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $L = 10 k m,$ $ξ = 30^{\circ}$ ): (a) $l \leq 3,$ (b) $l \geq 4.$

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Figure 5 shows the average intensity distribution with different wavelength annular vortex sources. We can get that with the increase of the laser wavelength, optical intensity on the center decreases and the width of the central dark hollow region increases, meanwhile the beam spot spreads and the peak value of the average intensity decreases.

Fig. 5 Distribution of the average intensity with different laser wavelengths ( $l = 4,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $L = 10 k m,$ $ξ = 30^{\circ}$ ).

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Figure 6 shows the evolution of the average intensity distribution with the increase of the propagation distance. We can get that with the increase of the propagation distance, the width of the central dark region decreases, meanwhile the beam spot spreads and the peak value of the average intensity decreases.

Fig. 6 Evolution of the average intensity distribution with the increase of the propagation distance ( $l = 4,$ $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $ξ = 30^{\circ}$ ).

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Figure 7 shows the evolution of the average intensity distribution as the zenith angle of the propagation path increases. We can get that with the increase of the zenith angle, power on the center increases and the central dark hollow gradually disappears, meanwhile the beam spot spreads and the peak value of the average intensity decreases. The higher the zenith angle is, the faster the increment speed of the central intensity is. When $ξ = 90^{\circ}$ , the central intensity approaches to the maximum.

Fig. 7 Evolution of the average intensity distribution with the increase of the zenith angle ( $l = 4,$ $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $L = 10 k m$ ).

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3.2 Power coupling of the optical system

From the study above, we can get that the average intensity distribution at the receiver plane presents a central dark hollow profile, which is suitable for power coupling by the Cassegrain telescope with central obstruction and outer truncation; and parameters of the vortex source and the propagation path have great influences on the average intensity distribution. In order to show the basic characteristics of the power coupling and the influences of the parameters, we take the inner radius of the receiver $a^{'} \to 0$ (without the central obstruction), and perform some numerical calculations.

Figure 8 illuminates the evolution of the relations between power coupling efficiency and the receiver aperture as different parameter increases. We can get that power coupling efficiency keeps in zero at the beginning when the receiver aperture is low. Namely, when use a receiver telescope with central obscuration, power losses induced by secondary mirror obstruction can be quite low. With the increase of the receiver aperture, power coupling efficiency increases and the increment speed becomes faster. With further increases in the receiver aperture, the increment speed becomes slower and power coupling efficiency approaches to 100%. Results in Fig. 8 show that receiver aperture with zero power coupling efficiency is higher when the optical topological charge is higher, the laser wavelength is higher, the propagation distance is shorter or the zenith angle is smaller; the increment speed of the power coupling efficiency is faster when the optical topological charge is higher, the propagation distance is shorter, the zenith angle is smaller or the beam wavelength is lower.

Fig. 8 Evolution of the relations between power coupling efficiency and the receiver aperture as different parameter increases: (a) the optical topological charge, (b) the laser wavelength, (c) the propagation distance, (d) the zenith angle.

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In order to show the influence of the receiver apertures on the power coupling efficiency, we take the receiver apertures “a'=0.1m, b'=0.4m”, “a'=0.12m, b'=0.5m” and “a'=0.15m, b'=0.6m” as three examples, and calculate the power coupling efficiency under determinate conditions. The coming results are listed in Table 1 .

Table 1. Power Coupling Efficiency of the Optical System with Different Receiver Apertures ( $l = 4,$ $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $ξ = 30^{\circ},$ $a = 0.15 m,$ $b = 0.4 m$ ).

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Results show that power coupling efficiency of the optical system can be quite high when the propagation distance is short, and with the increase of the propagation distance, power coupling efficiency of the optical system degrades. Combing results in Fig. 8 and Table 1, we can get that parameters of the vortex source, the receiver apertures and the propagation path have great influences on the power coupling efficiency. Thus, in determining the optical system parameters, the optical topological charge, the laser wavelength, the propagation path and the receiver apertures need to be optimized to improve the power coupling efficiency. Results in Table 1 show that under the present conditions, telescope receiver with “a'=0.15m, b'=0.6m” is reasonable, and power coupling efficiency of the optical system can keep in high values with the increase of the propagation distance.

For comparison, we calculate power coupling efficiency of the optical system when using the laser source without vortex (l = 0), and the coming results are listed in Table 2 .

Table 2. ower Coupling Efficiency of the Optical System with Different Sources ( $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $ξ = 30^{\circ},$ $a = 0.15 m,$ $b = 0.4 m,$ $a^{'} = 0.15 m,$ $b^{'} = 0.6 m$ ).

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Form the results in Table 2, we can get that power coupling efficiency of the optical system with annular vortex source is significantly higher than that of the laser source without vortex. Under the atmospheric turbulent conditions, great advantages of vortex beams are shown in power coupling of the two-Cassegrain-telescope optical system.

4. Conclusions

In this paper, we have studied power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere. A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere in a slant path was established, the general analytical expression of vortex beams with limited apertures and the analytical formulas for the average intensity distribution at the receiver plane were derived. Under the H-V 5/7 turbulence model, the average intensity distribution at the receiver plane and the power coupling efficiency of the optical system were numerically calculated. From the analysis we can get that the average intensity distribution at the receiver plane presents a central dark hollow profile, which is suitable for power coupling by the Cassegrain telescope receiver. Parameters of the vortex source, the receiver apertures and the propagation path have great influences on the power coupling. In the optical system with optimized parameters, power coupling efficiency can keep in high values with the increase of the propagation distance. Under the atmospheric turbulent conditions, great advantages of vortex beams in power coupling of the two-Cassegrain-telescope optical system were shown in comparison with laser source without vortex.

Acknowledgments

The authors are indebted to the National High Technology Research and Development Program of China (2012AA02007).

References and links

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Fig. 1 A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere.

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Fig. 2 The optical field of the annular vortex source: (a) intensity distribution, (b) phase distribution.

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Fig. 3 The average intensity distribution at the receiver plane with the parameters (

l = 4,

λ = 3.8 u m,

σ = 0.5,

N = 10,

ω_{0} = 0.2 m,

L = 10 k m,

ξ = 30^{\circ}

View in Article | Download Full Size | PDF

Fig. 4 Evolution of the average intensity distribution with the increase of the optical topological charge (

λ = 3.8 u m,

σ = 0.5,

N = 10,

ω_{0} = 0.2 m,

L = 10 k m,

ξ = 30^{\circ}

): (a)

l \leq 3,

(b)

l \geq 4.

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Fig. 5 Distribution of the average intensity with different laser wavelengths (

l = 4,

σ = 0.5,

N = 10,

ω_{0} = 0.2 m,

L = 10 k m,

ξ = 30^{\circ}

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Fig. 6 Evolution of the average intensity distribution with the increase of the propagation distance (

l = 4,

λ = 3.8 u m,

σ = 0.5,

N = 10,

ω_{0} = 0.2 m,

ξ = 30^{\circ}

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Fig. 7 Evolution of the average intensity distribution with the increase of the zenith angle (

l = 4,

λ = 3.8 u m,

σ = 0.5,

N = 10,

ω_{0} = 0.2 m,

L = 10 k m

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Tables (2)

Table 1 Power Coupling Efficiency of the Optical System with Different Receiver Apertures ( $l = 4,$ $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $ξ = 30^{\circ},$ $a = 0.15 m,$ $b = 0.4 m$ ).

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Table 2 ower Coupling Efficiency of the Optical System with Different Sources ( $λ = 3.8 u m,$ $σ = 0.5,$ $N = 10,$ $ω_{0} = 0.2 m,$ $ξ = 30^{\circ},$ $a = 0.15 m,$ $b = 0.4 m,$ $a^{'} = 0.15 m,$ $b^{'} = 0.6 m$ ).

View Table | View all tables in this article

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

E_{0} (r, θ, 0) = E (r, θ, 0) t (r, a, b),

E (r, θ, 0) = A (r) \exp (i l θ),

t (r, a, b) = {\begin{matrix} 1 & a \leq r \leq b \\ 0 & e l s e \end{matrix},

E (r, θ, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\begin{matrix} N \\ n \end{matrix}) [\exp (- \frac{n r^{2}}{ω_{0}^{2}}) - \exp (- \frac{n r^{2}}{σ ω_{0}^{2}})] \exp (i l θ),

t (r, a, b) = \sum_{w = 1}^{M} B_{w} [\exp (- \frac{C_{w}}{b^{2}} r^{2}) - \exp (- \frac{C_{w}}{a^{2}} r^{2})],

\begin{array}{l} E_{0} (r, θ, 0) = \sum_{n = 1}^{N} \sum_{w = 1}^{M} \frac{{(- 1)}^{n - 1} B_{w}}{N} (\begin{matrix} N \\ n \end{matrix}) [\exp (- \frac{C_{w}}{b^{2}} r^{2}) - \exp (- \frac{C_{w}}{a^{2}} r^{2})] \\ \begin{matrix}  \end{matrix} \times [\exp (- \frac{n r^{2}}{ω_{0}^{2}}) - \exp (- \frac{n r^{2}}{σ ω_{0}^{2}})] \exp (i l θ) . \end{array}

\begin{array}{l} < I (R, φ, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{2 π} E_{0} (r_{1}, θ_{1}, 0) \exp {\frac{i k}{2 L} [R^{2} + r_{1}^{2} - 2 R r_{1} \cos (φ - θ_{1})]} \\ \begin{matrix}  \end{matrix} \times {E_{0} (r_{2}, θ_{2}, 0) \exp {\frac{i k}{2 L} [R^{2} + r_{2}^{2} - 2 R r_{2} \cos (φ - θ_{2})]}}^{*} \\ \begin{matrix}  \end{matrix} \times < \exp [ψ (R, φ, r_{1}, θ_{1}) + ψ^{*} (R, φ, r_{2}, θ_{2})] > r_{1} r_{2} d r_{1} d r_{2} d θ_{1} d θ_{2}, \end{array}

\begin{array}{l} < \exp [ψ (R, φ, r_{1}, θ_{1}) + ψ^{*} (R, φ, r_{2}, θ_{2})] > = \exp [- 0.5 D_{ψ} (r_{1}, r_{2}, θ_{1}, θ_{2})] \\ \begin{matrix}  \end{matrix} = \exp {- \frac{1}{ρ_{0}^{2}} [r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})]}, \end{array}

\begin{array}{l} < I (R, φ, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \sum_{w = 1}^{M} \sum_{v = 1}^{M} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) B_{w} B_{v}^{*} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{2 π} \exp (- \frac{n r_{1}^{2}}{ω_{0}^{2}}) \exp (- \frac{m r_{2}^{2}}{ω_{0}^{2}}) \\ \begin{matrix}  \end{matrix} \times \exp (- \frac{C_{w}}{a^{2}} r_{1}^{2}) \exp (- \frac{C_{v}^{*}}{a^{2}} r_{2}^{2}) \exp {\frac{i k}{2 L} [r_{1}^{2} - 2 R r_{1} \cos (φ - θ_{1}) - r_{2}^{2} + 2 R r_{2} \cos (φ - θ_{2})]} \\ \begin{matrix}  \end{matrix} \times \exp [i l (θ_{1} - θ_{2})] \exp {- \frac{1}{ρ_{0}^{2}} [r_{1}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2}) + r_{2}^{2}]} r_{1} r_{2} d r_{1} d r_{2} d θ_{1} d θ_{2} . \end{array}

\exp [\frac{i k R r}{α} \cos (φ - θ)] = \sum_{l = - \infty}^{\infty} i^{l} J_{l} (\frac{k R r}{α}) \exp [i l (φ - θ)],

\int_{0}^{2 π} \exp [- i l θ + 2 α R r \cos (θ - φ)] d θ = 2 π \exp (- i l φ) I_{l} (2 α R r),

I_{l} (x) = \sum_{m = 0}^{\infty} \frac{{(x / 2)}^{2 m + l}}{m! Γ (m + l)},

J_{l} (x) = \frac{{(- i)}^{l}}{2 π} \int_{0}^{2 π} \exp (i x \cos θ + i l θ) d θ,

\int_{0}^{\infty} t^{m - 1} \exp (- α t^{2}) J_{l} (β t) d t = \frac{α^{- m / 2}}{2 l!} {(\frac{β^{2}}{4 α})}^{l / 2} \exp (- \frac{β^{2}}{4 α}) Γ (\frac{l + m}{2}) F_{1} (\frac{- m + l + 2}{2}; l + 1; \frac{β^{2}}{4 α}),

< I (R, L) > = \frac{k^{2}}{{(2 π L)}^{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \sum_{w = 1}^{M} \sum_{v = 1}^{M} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) B_{w} B_{v}^{*} [(T_{1} - T_{2} + T_{3} - T_{4}) - ({T^{'}}_{1} - {T^{'}}_{2} + {T^{'}}_{3} - {T^{'}}_{4})],

\begin{array}{l} T_{1} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) {(H_{1, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{2} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 1} L^{2}}) {(H_{1, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{3} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) {(H_{1, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} T_{4} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{1, 2} L^{2}}) {(H_{1, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{1} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) {(H_{2, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{2} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 1} L^{2}}) {(H_{2, 1})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{3} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) {(H_{2, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) F_{1} (P_{2}_{1}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 2} L^{2}}) {({H^{'}}_{1, 2})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 1} L^{2}}) {({H^{'}}_{2, 1})}^{- P_{1}}], \end{array}

\begin{array}{l} {T^{'}}_{4} = \sum_{s = - \infty}^{\infty} \sum_{t = 0}^{\infty} \frac{{[Γ (P_{1})]}^{2} γ^{2 t + s + l}}{{(s!)}^{2} t! Γ (s + t + l)} {(\frac{k^{2} R^{2}}{4 L^{2}})}^{s} [\exp (- \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 H_{2, 2} L^{2}}) {(H_{2, 2})}^{- P_{1}}] \times \\ [\exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{1, 1} L^{2}}) {({H^{'}}_{1, 1})}^{- P_{1}} + \exp (- \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) F_{1} (P_{2}; s + 1; \frac{k^{2} R^{2}}{4 {H^{'}}_{2, 2} L^{2}}) {({H^{'}}_{2, 2})}^{- P_{1}}], \end{array}

η = \frac{2 π \int_{a^{'}}^{b^{'}} < I (R, L) > R d R}{\int_{a}^{b} \int_{0}^{2 π} E_{0} (r, θ, 0) E_{0}^{*} (r, θ, 0) r d r d θ},

C_{n}^{2} (h) = 8.2 \times 10^{- 56} V {(h)}^{2} h^{10} \exp (- h / 1000) + 2.7 \times 10^{- 16} \exp (- h / 1500) + C_{0} \exp (- h / 100),

V (h) = 5 + 30 \exp {{[- (h - 9400) / 4800]}^{2}},

Receiver apertures	a' = 0.1m, b' = 0.4m				a' = 0.12m, b' = 0.5m				a' = 0.15m, b' = 0.6m
Propagation distance (km)	5	10	20	30	5	10	20	30	5	10	20	30
Power efficiency (%)	97	88	64	45	99	98	84	68	99	98	93	82

Abstract

1. Introduction

2. Models and theoretical analysis

3. Numerical calculations and results analysis

3.1 Average intensity distribution at the receiver plane

3.2 Power coupling of the optical system

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (26)

Optics Express