Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path

Xiuxiang Chu; Guoquan Zhou

doi:10.1364/OE.15.007697

1. Introduction

In recent years the propagation of laser beams in turbulent atmosphere has become a very lively area of scientific research and application. The propagation properties of many types of laser beams have been studied widely-especially the average intensity distribution and scintillation at the receiver plane [1–14]. Meanwhile, some software has been developed to simulate the propagation of laser beams in turbulent atmosphere [9]. However, many of these studies have been restricted to the unapertured cases in the horizontal path. In the propagation of laser beams in turbulent atmosphere, both diffraction and atmospheric turbulence reduce the number of laser beams that are received by the receiver due to aperture. The effects of aperture on propagation require much practical investigation. In actuality, study of laser beam propagation through turbulent atmosphere in a slant path is more important due to the fact that in many applications, such as both free-space optical communications and deep-space optical communications, the propagation of laser beams is in a slant path. For application of the laser beam in the atmosphere, Cassegrain telescopes, for beam expansion and low-speed focus control, are often used as transmitters. To receive more of the laser beam’s power, and then shrink it with a laser beam control system, Cassegrain telescopes are often used as receivers.

2. Analysis of theory

The two-Cassegrain-telescopes system configuration is shown in Fig. 1. A Cassegrain telescope with central obscuration is used to project a laser beam to a target. Another Cassegrain telescope with central obscuration is used as a receiver. H is the altitude between the transmitter and the receiver, ζ is the zenith angle, L is the propagation distance of the laser beam along z. In the Cartesian coordinate system, the source plane is located at the transmitter aperture (z=0), the receiver plane is located at the receiver aperture (z=L). The (x,y) and (p,q) denote the transverse coordinates of the source plane and receiver plane, respectively. When laser beams enter into the transmitter telescope, the power will be lost due to the limitation of the telescope. Laser beams at the source plane can be regarded as a collimated laser beam limited by aperture of the primary mirror and the obscuration of the second mirror of the Cassegrain telescope. If E ₀(x,y,0) denotes the optical field of the reflected collimated laser beam from the primary mirror without any loss of energy, the optical field at the source plane can been expressed as

Fig. 1. Transmitter and receiver configuration.

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E (x, y, 0) = t (x, y, 0) E_{0} (x, y, 0),

with

t (x, y, 0) = {\begin{matrix} 1 & a^{2} \leq x^{2} + y^{2} \leq b^{2} \\ 0 & others \end{matrix},

where a and b are the radius of the secondary and primary mirror, respectively. The loss due to the aperture on the primary mirror is neglected. In order to reduce the loss due to the transmitter telescope, dark hollow beams are adapted in this paper. The optical field of a circular dark hollow beam (with circular symmetry) at z=0 can be expressed as the following finite sum of Gaussian beams [15]:

E_{0} (x, y, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\begin{matrix} \begin{matrix} N \\ n \end{matrix} \end{matrix}) [\exp (- \frac{{nx}^{2} + {ny}^{2}}{w_{0}^{2}}) - \exp (- \frac{{nx}^{2} + {ny}^{2}}{{σw}_{0}^{2}})],

where N is the order of a circular dark hollow beam, $(\begin{matrix} N \\ n \end{matrix})$ represents a binomial coefficient, w0 is the beam waist width, σ (0<σ<1) is a parameter concerning the circular dark hollow beam. If we set σ → 0 , Eq. (3) reduces to a flattened Gaussian beam [16]. When N=1 and σ → 0 , Eq. (3) reduces to a Gaussian beam. Contour plots (see Fig. 1 in Ref [11]) show that the area of the dark region across a dark hollow beam increases as N or σ increases. By using the extended Huygens-Fresnel principle, the average intensity distribution at the receiver plane can been expressed as [2]

< I (p, q, L) > = \frac{k^{2}}{{(2 πL)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} E (x, y, 0) E^{*} (ξ, η, 0) \times \exp (\frac{ik}{2 L} [{(p - x)}^{2} + {(q - y)}^{2} - {(p - ξ)}^{2} - {(q - η)}^{2}]) \times < \exp [ψ (x, y, p, q) + ψ^{*} (ξ, η, p, q)] > dxdyd ξdη

where k is the wave number, the asterisk denotes the complex conjugation, and the <> indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. Φ(x,y,p,q) represents the random part of the complex phase of a spherical wave that propagates from point (x, y, 0) at the source plane to the point (p, q, L) at the receiver plane and can be expressed as [17]

< \exp [ψ (x, y, p, q) + ψ^{*} (ξ, η, p, q)] > = \exp [- 0.5 D_{ψ} (x - ξ, y - η)]

In Eq. (5), D_Φ (x-ξ,y-η) is the wave structure function, ρ ₀ is the coherence length. In longdistance transmission systems, the coherence length may be reduced by propagation factors such as turbulence, scattering, and diffraction. In turbulent atmosphere the coherence length of spherical wave is

ρ_{0} = {(0.545 {\bar{C}}_{n}^{2} k^{2} L)}^{\frac{- 3}{5}}

where

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

Here, structure constant C ² _n (h) is altitude dependent. In this paper the ITU-R model presented in 2001 [18] is selected to describe the model of the structure constant and can be expressed as

C_{n}^{2} (h) = 8.148 \times 10^{- 56} V^{2} h^{10} \exp (\frac{- h}{1000}) + 2.7 \times 10^{- 16} \exp (\frac{- h}{1500}) + C_{0} \exp (\frac{- h}{1000})

where h is the altitude from the ground, V = [v_g ² + 30.69v_g + 348.91)^1/2 is the wind speed along the vertical path, v_g is the ground wind speed (in this paper we set v_g=0), C ₀ is the nominal value of at ground level (typically value is 1.7×10^-14/m ^-2/3). Generally, the hard-edge aperture function can be expanded as the sum of complex Gaussian functions with finite numbers [19, 20]. Therefore Eq. (2) can be expressed as

t (x, y) = \sum_{j = 1}^{M} B_{j} {\exp [- \frac{C_{j}}{b^{2}} (x^{2} + y^{2})] - \exp [- \frac{C_{j}}{a^{2}} (x^{2} + y^{2})]},

where the complex constants B_j and C_j, are the expansion coefficients. M is the number of the expansion coefficients. In this paper we adopt the expansion coefficients in Ref [19] (M=16).

From Eqs. (9), (5), and (4) and setting R = [p ² + q ²)^1/2 , the average intensity at receiver plane of the laser beams limited by the Cassegrain telescope can be written as

< I (R, L) > = \frac{ρ_{0}^{4} k^{2}}{{4 L}^{2} N^{2}} \sum_{j = 1}^{M} \sum_{s = 1}^{M} \sum_{m = 1}^{N} \sum_{n = 1}^{N} B_{j} B_{s}^{*} (\begin{matrix} N \\ m \end{matrix}) (\begin{matrix} N \\ n \end{matrix}) {\frac{1}{β_{1} β_{2} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1} β_{2} ρ_{0}^{4} - 1)}] + \frac{1}{β_{1}^{'} β_{2}^{'} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (β_{1}^{'} ρ_{0}^{2} + β_{2}^{'} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1}^{'} β_{2}^{'} ρ_{0}^{4} - 1)}] - \frac{1}{β_{1} β_{2}^{'} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2}^{'} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1} β_{2}^{'} ρ_{0}^{4} - 1)}] - \frac{1}{β_{1}^{'} β_{2} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (β_{1}^{'} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1}^{'} β_{2} ρ_{0}^{4} - 1)}] - \frac{1}{α_{1} {α_{2} ρ}_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (α_{1} α_{2} ρ_{0}^{4} - 1)}] - \frac{1}{α_{1}^{'} α_{2}^{'} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2}^{'} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (α_{1}^{'} α_{2}^{'} ρ_{0}^{4} - 1)}] + \frac{1}{α_{1} α_{2}^{'} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{2}^{'} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (α_{1} α_{2}^{'} ρ_{0}^{4} - 1)}] + \frac{1}{α_{1}^{'} α_{2} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (α_{1}^{'} α_{2} ρ_{0}^{4} - 1)}]}

where

β_{1} = \frac{n}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{s}^{*}}{b^{2}} - \frac{ik}{2 L},

β_{2} = \frac{m}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{j}}{b^{2}} + \frac{ik}{2 L},

β_{1}^{'} = \frac{n}{{σw}_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{s}^{*}}{b^{2}} - \frac{ik}{2 L},

β_{2}^{'} = \frac{m}{{σw}_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{j}}{b^{2}} + \frac{ik}{2 L},

α_{1} = \frac{n}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{s}^{*}}{a^{2}} - \frac{ik}{2 L},

α_{2} = \frac{m}{w_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{j}}{a^{2}} + \frac{ik}{2 L},

α_{1}^{'} = \frac{n}{{σw}_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{s}^{*}}{a^{2}} - \frac{ik}{2 L},

α_{2}^{'} = \frac{m}{{σw}_{0}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{C_{j}}{a^{2}} + \frac{ik}{2 L} .

From Eqs. (10) to (18), some limiting cases can be obtained. For example, if we assume σ → 0 and a=0, the average intensity of a circular flattened Gaussian beam with aperture in turbulent atmosphere can be obtained as

< I (p, q, L) > = \frac{ρ_{0}^{4} k^{2}}{{4 L}^{2} N^{2}} \sum_{j = 1}^{M} \sum_{s = 1}^{M} \sum_{m = 1}^{N} \sum_{n = 1}^{N} (\begin{matrix} N \\ m \end{matrix}) (\begin{matrix} N \\ n \end{matrix}) \frac{B_{j} B_{s}^{*}}{β_{1} β_{2} ρ_{0}^{4} - 1} \exp [- \frac{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1} β_{2} ρ_{0}^{4} - 1)}] .

It agrees with the Eq. (10) in Ref [14]. If we set b → ∞ and a=0, the average intensity of a circular dark hollow beam without aperture in turbulent atmosphere can be obtained. It agrees with the results in Ref. [11]. Diffraction and turbulence cause power in the projected beam to move out beyond the edge of the receiving aperture and into the secondary mirror of the receiving Cassegrain telescope. If a Cassegrain telescope with central obscuration is used on the receiver side, light beyond the edge of the primary mirror and into the central obscuration will be lost. The power of the dark hollow beam before entering the transmitter telescope is

P_{0} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{0}^{2} (x, y, 0) dxdy = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) \frac{{πmnw}_{0}^{2} {(1 - σ)}^{2} (1 + σ)}{(m + n) (mσ + n) (m + nσ)} .

The power distribution of the laser beam at the receiver plane just before the receiver telescope is

P (R, L) = 2 π \int_{0}^{R} < I (R, L) > RdR

where

T_{1} = \frac{1}{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1} β_{2} ρ_{0}^{4} - 1)}] - 1}

T_{2} = \frac{1}{k^{2} ρ_{0}^{2} (β_{1}^{'} ρ_{0}^{2} {+ β_{2}^{'} ρ}_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (β_{1}^{'} ρ_{0}^{2} + {β_{2}^{'} ρ}_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1}^{'} β_{2}^{'} ρ_{0}^{4} - 1)}] - 1}

T_{3} = \frac{1}{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2}^{'} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (β_{1} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1} β_{2}' ρ_{0}^{4} - 1)}] - 1}

T_{4} = \frac{1}{k^{2} ρ_{0}^{2} (β_{1}^{'} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (β_{2}^{'} ρ_{0}^{2} + β_{2} ρ_{0}^{2} - 2) R^{2}}{{4 L}^{2} (β_{1}^{'} β_{2}^{'} ρ_{0}^{4} - 1)}] - 1}

{T'}_{1} = \frac{1}{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{2} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{0} ρ_{0}^{2} - 2) R^{2}}{4 L^{2} (α_{1} α_{2} ρ_{0}^{4} - 1)}] - 1}

T_{2}^{'} = \frac{1}{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2}^{'} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2}^{'} ρ_{1}^{2} - 2) R^{2}}{4 L^{2} (α_{1}^{'} α_{2}^{'} ρ_{0}^{4} - 1)}] - 1}

T_{3}^{'} = \frac{1}{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{2}^{'} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (α_{1} ρ_{0}^{2} + α_{2}^{'} ρ_{1}^{2} - 2) R^{2}}{4 L^{2} (α_{1} α_{2}^{'} ρ_{0}^{4} - 1)}] - 1}

T_{4}^{'} = \frac{1}{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2} ρ_{0}^{2} - 2)} {\exp [- \frac{k^{2} ρ_{0}^{2} (α_{1}^{'} ρ_{0}^{2} + α_{2} ρ_{1}^{2} - 2) R^{2}}{4 L^{2} (α_{1}^{'} α_{2} ρ_{0}^{4} - 1)}] - 1} .

From Eqs. (20) to (29), the efficiency of the power coupling can be obtained. It should be pointed out that the expression of the hard-edge aperture function expanded by using a sum of finite-term complex Gaussian function is approximate. Therefore, the analytical formulas in this paper involving the hard-edge aperture function are approximate expressions. Studies w2 show that in the far field, for example Fresnel number $F_{w} \leq 1 (F_{w} = \frac{w_{0}^{2}}{λz}),$ the method can provide satisfactory results [19, 21].

3. Numerical calculation and analysis

3.1. Power distribution of dark hollow beam

It is important to know the power distribution of the laser beam, especially the peak value position of the power density when we use Cassegrain telescope as a transmitter. There are two types of power density to describe the power distribution when the study involves in the truncation of the aperture and the obscuration of the transmitter telescope. One is the intensity I(r,0) = E ² (r,0) (power on unit area). Another one is the power at annular aperture with unit radius and can be expressed as

S (r, 0) = 2 πrI (r, 0)

where $r = \sqrt{x^{2} + y^{2}}$ . Figure 2 shows the variation of the peak value position of I(r, 0) and S(r, 0) with different parameters. The vertical axis denotes the coordinates of the position where the value of I(r, 0) or S(r, 0) is the maximum.

Fig. 2. Variations of the peak value position with different parameters: (a) with parameter σ, (b) with waist width w ₀, (c)with the order N.

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From Fig. 2 we can see that the coordinates of the peak value positions of I (r, 0) and S (r, 0) are different when the parameters of the dark hollow beams are the same. The value of coordinates of the peak value position of S(r, 0) is bigger than that of I (r, 0). Meanwhile, the coordinates of peak value positions are changed with the change of the parameters. Figure 2 shows that the value of the coordinates of the peak value position of S (r, 0) is increasing with σ, N, and w ₀. The principles for I (r, 0) are the same, but the speed of the change of the coordinates versus w ₀ is faster than those with σ and N. In this paper the parameters of the dark hollow beams are selected as σ = 0.7, w ₀ = 0.2m, and N = 3. Numerical calculation shows that the coordinates of the peak value position of I (r, 0) and σ (r, 0) are 0.24m and 0.26m, respectively. In order to optimize the parameters of the receiver Cassegrain telescope, the profile of the power in bucket (P(r, 0)/P ₀) of the dark hollow beam without the limitation of the transmitter telescope is plotted in Fig. 3, where P(r,0) = ∫^r ₀ S(r,0)dr .

Fig. 3. Variations of the normalized power of the dark hollow beam before entering the transmitter telescope versus r.

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Figure 3 shows that the power of the dark hollow beams is concentrated on an annular aperture. From Figs. 3 and 4 we can optimize the parameters of the dark hollow beams to satisfy the need of the transmitter or to optimize the parameters of aperture to satisfy the need of the dark hollow beams. For example, if the transmitter aperture is selected as a=0.15 m and b=0.4 m, 98% power of the dark hollow beams can pass through the transmitter telescope.

3.2. Average intensity distribution at the receiver plane

To show the characteristic of the average intensity distribution at the receiver plane, some numerical simulations are performed. The parameters of the dark hollow beams are selected as σ=0.7, w ₀ 0=0.2m, and N =3. The normalized intensity of the dark hollow beams before entering the transmitter is defined as

I_{N} (x, y, 0) = \frac{E^{2} (x, y, 0)}{I_{Max} (x, y, 0)},

where I_Max (x, y,0) is the peak value of I(x,y,0) (I(x,y,0) = E ² (x, y,0)). The normalized average intensity at the receiver plane is defined as

〈 I_{N} (x, y, 0) 〉 = \frac{〈 I (p, q, L) 〉}{I_{Max} (x, y, 0)} .

To show the evolution of the normalized average intensity profile, the normalized intensity profile of the dark hollow beams before entering the transmitter is depicted in Fig. 4(a). Meanwhile, the profiles of the normalized average intensity with different propagation distance are plotted in Figs. 4(b), 4(c), and 4(d).

Fig. 4. Evolution of the normalized intensity distribution of the dark hollow beam in a slant path with λ=3.8μm and ζ=π/6: (a) before reaching the source plane, (b) with L=10km (F_w =1), (c) with L=15km (F_w = 0.7), and (d) with L=30km (F_w = 0.34).

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From Fig. 4 we can see that the peak value of the average normalized intensity profile decreases and the dark hollow gradually disappears with the increase of the propagation distance. When the width of the annular aperture (b-a) is larger, namely the truncation by the annular aperture is small, the profiles of the normalized average intensity with annular aperture is similar to those without the limitation of the annular aperture [see the solid curves and the dashed curves in Figs. 4(b) and 4(c)]. But with more truncation, the profile of the normalized average intensity is complex [see the dotted curve in Fig. 4(b)]. With further increases in the propagation distance, the peak of the profile of the normalized average intensity around the central point gradually disappears and concentrates on the centre. The speed of the concentration with more truncation is faster than that with less truncation or without truncation. If we chose the same parameters as those in Ref. 11, numerical calculation shows that the effects of turbulence on the average intensity distribution are smaller in a slant path than those in a horizontal path.

3.3. Efficiency of the power coupling

From the study above we can see that the power of the dark hollow beams gradually concentrates on the center of the receiver plane with the increase of the propagation distance. Meanwhile, the beam spots spread. If we use a Cassegrain telescope as a receiver, the loss of the power will increase due to the size of the receiver aperture and the central obscuration. From Eqs. (21) to (30), the efficiency of the power coupling can be expressed as η = P(R,L)/P ₀, if the radius of the receiver telescope (without obscuration on the center) is R. Figure 5 illuminates the efficiency of the power coupling with different parameters.

Fig. 5. Variation of the efficiency of the power coupling versus R with λ=3.8μm and ζ=π/6: (a) with L=10km (F_w = 1), (b) with L=15km (F_w = 0.7), (c) with L=30km (F_w = 0.34).

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Figure 5(a) shows that the increase of the efficiency is much slower at the beginning when the propagation distance is shorter. Namely, when we use a telescope with central obscuration as a receiver the loss due to the obscuration is smaller. With the increase of the radius R, the speed of the increase of the efficiency becomes faster. However, with further increases in R, the speed of the increase of the efficiency becomes slower and reaches to zero. Namely 100% energy transmitted from the transmitter enters into the receiver aperture. From Figs. 5(b) and 5(c) we can see that the speed of the increase of the efficiency gradually reaches to a constant within a magnitude of R with the increase of the propagation distance. Namely the power is more diffused with longer propagation distance. The principle for a truncated beam is the same way, but the speed of the variations is faster than this without truncation. From the discussion above we can see that the size of the central obscuration and receiver aperture at the receiver plane is an important factor to influence the efficiency of the power coupling. To show the effect of the receiver telescope and propagation distance on the efficiency of the power coupling, some values of the efficiency with different parameters are listed in Table.1.

Table 1. Efficiency of the power coupling with different parameters.

View Table

In order to show the relations between the efficiency of the power coupling and the altitude from the ground, Fig. 6 is plotted. Figure 6 illuminates that the power of the dark hollow beams spreads with the increase of the altitude. Namely, the distribution of the energy trends to uniform with respect to the radius of the receiver plane. The results agree with those in Fig. 5. Because the effects of diffraction and turbulence on the efficiency of the power coupling have close relations with the wavelength of a laser beam, the variations of the coupling efficiency with the wavelength are plotted in Fig. 7.

Fig. 6. Relations between the efficiency of the power coupling and the altitude from the ground with the parameters λ=3.8μm, σ=0.7, w ₀=0.2m, N =3, ζ=0, a=0.15m and b=0.4m.

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Fig. 7. Relations between efficiency of power coupling and wavelength with the parameters H=30 km, σ=0.7, w ₀=0.2m, N =3, ζ=0, a=0.15m and b=0.4m.

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From Fig. 7 we can see that the effects of turbulence and diffraction on the efficiency of the power coupling are smaller with λ=1.06 μm than those with λ=3.8μm and λ=10.6μm. The density of the power on the center of receiver plane is larger with λ=3.8μm than that with λ=1.06μm and λ=10.6μm; namely, the loss due to obscuration of the receiver telescope is bigger with λ=3.8μm than that with λ=1.06μm and λ=10.6μm. The transmitted power moving out beyond the edge of the receiving aperture is bigger with λ=10.6μm than that with λ=1.06μm and λ=3.8μm.

4. Conclusion

In this paper we have studied the characteristics of dark hollow beams with the truncation of Cassegrain telescope when they pass through turbulent atmosphere in a slant path. Analytical formulas of the average intensity at the receiver plane and the efficiency of the power coupling are derived. In order to increase the efficiency of power coupling of the two-Cassegrain-telescopes system, the power distributions at the source and receiver plane are numerically calculated and discussed. From the analysis we can see that the transmitter telescope easily satisfies the requirements for dark hollow beams to pass through. For short distances where the effects of diffraction and turbulence are not strong, the power coupling is quite high. As the propagation distances get longer, the dark hollow beams converge to the central and the spot spreads. The central obscuration blocks more of the transmitted power, while more of the transmitted power moves out beyond the edge of the receiving aperture. The energy received by the receiver telescope will decrease due to the aperture size and the obscuration of the central part. In some applications, it is impractical to enlarge the size of the receiver aperture and throw off the central obscuration to receive more of the energy. In order to acquire more of the energy, the parameters of the laser beams and the telescopes need optimizing to compensate for some of the loss.

Acknowledgments

The authors are indebted to the reviewers for their valuable advice.

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Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path

Abstract

1. Introduction

2. Analysis of theory

3. Numerical calculation and analysis

3.1. Power distribution of dark hollow beam

3.2. Average intensity distribution at the receiver plane

3.3. Efficiency of the power coupling

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (34)

Optics Express

Parameters	λ=3.8μm, σ=0.7, w ₀=0.2m, N =3, ζ=π/6
Outer radius of the transmitter (m)	0.35	0.35	0.3	0.35	0.35	0.3
Inner radius of the transmitter (m)	0.15	0.15	0.2	0.15	0.15	0.2
Outer radius of the receiver (m)	0.35	0.35	0.3	0.35	0.35	0.3
Inner radius of the receiver (m)	0	0.15	0.2	0	0.15	0.2
Propagation distance (km)	10	10	10	30	30	30
Efficiency of the power coupling	80%	71%	25%	61%	33%	11%