Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere

Xiuxiang Chu

doi:10.1364/OE.15.017613

1. Introduction

With the wide application of laser beams in the atmosphere, such as optical communications and the propagation of high power laser beams, the propagation of laser beams in turbulent atmosphere has been studied extensively. In recent years the effects of turbulent atmosphere on the propagation of laser beams with various forms of excitation are investigated, such as the effects of turbulence on the average intensity and the scintillation of laser beams [1–17]. In practice some optical elements are often used to improve system performance. It is necessary to determine various parameters of a laser beam passing through an optical system, for example, the spot size, radius of curvature and intensity distribution. In free space many models of laser beam in ABCD optical system with or without hard-edge aperture have been studied [18–21]. But to the best of my knowledge, many models of laser beams through an ABCD optical system in turbulent atmosphere have not been taken into account. In the applications of a laser beam in turbulent atmosphere, a series of optics is often used to direct or redirect a laser beam to a distant target plane. Therefore, the analysis of propagation of a laser beam through an ABCD optical system in turbulent atmosphere is important. A cosh-Gaussian beam can be regarded as the superposition of four decentered Gaussian beams with the same waist width and can be generated by using an ordinary Gaussian beam as an incident beam in an appropriate transmission or reflection aperture [22]. Considerable studies have been made to search for its application in free-space optical systems. In the presence of atmosphere turbulence, average intensity distribution and scintillation of cosh-Gaussian beams without optical element on its propagation path have been investigated [2, 3, 16 and 17].

2. The average intensity of a cosh-Gaussian beam through an optical system in turbulent atmosphere

In the Cartesian coordinate system, a cosh-Gaussian beam propagates along the z-axis, the source plane and the receiver plane are located at (x ₀, y ₀, z=0) and (x, y, z=L), respectively. The optical field of a cosh-Gaussian beam at the source plane is of the form [19]

E_{0} (x_{0}, y_{0}, 0) = \exp [- \frac{x_{0}^{2} + y_{0}^{2}}{w_{0}^{2}} - \frac{i k (x_{0}^{2} + y_{0}^{2})}{2 F}] \cosh (Ω_{0} x_{0}) \cosh (Ω_{0} y_{0}),

where (x ₀, y ₀) denotes the transverse coordinates of the source plane, (x, y) denotes the transverse coordinates of the receiver plane, L denotes the optical distance along the z-axis, w ₀ is the initial waist width of the beam associated with the Gaussian part, k=2π/λ is the wave number andλ is wavelength, Ω₀ is the parameter associated with the cosh part, and F is the phase front radius of curvature (F>0 for a convergent beam and F<0 for a divergent beam). Based on Huygens-Fresnel diffraction integral, the fields at receiver plane of a beam through an ABCD optical system in turbulent atmosphere can be expressed as [23]

E (x, y, L) = {\frac{1}{i λ B} \exp (i k L) \int}_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{0} (x_{0}, y_{0}, 0)

where A, B and D are the geometrical ray-matrix elements for the complete optical system between source and receiver planes, Ψ(x ₀, y ₀, x, y) is the solution to the Rytov method that represents the random part of the complex phase. The average intensity at the receiver plane is

〈 I (x, y, L) 〉 = \frac{1}{λ^{2} B^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{0} (x_{01}, y_{01}, 0) E_{0}^{*} (x_{02}, y_{02}, 0)

where angle brackets indicate the ensemble average over the medium statistics, asterisk denotes the complex conjugation. The last term of Eq. (3) can be denotes as [24]

〈 \exp [ψ (x_{01}, y_{01}, x, y) + ψ (x_{02}, y_{02}, x, y)] 〉 = \exp [- D_{w} (r_{01} - r_{02}) ⁄ 2],

where D_w(r ₀₁–r ₀₂) is the wave structure function. If a quadratic approximation for the wave structure function is employed, D_w(r ₀₁–r ₀₂) can be expressed as

D_{w} (r_{01} - r_{02}) ⁄ 2 = [{{(x_{01} - x_{02})}^{2} + (y_{01} - y_{02})}^{2}] ⁄ ρ_{0}^{2},

where ρ ₀ is the spherical-wave lateral coherence length. If a laser beam passes through an optical system ρ ₀ can be expressed as [23]

ρ_{0} = B σ_{0} = {B [1.46 k^{2} {C_{n}^{2} \int}_{0}^{L} d z b^{5 ⁄ 3} (z)]}^{- 3 ⁄ 5},

where C ² _n is the structure constant, b(z)is the corresponding matrix element for a ray whose origin is at the receiver plane and whose endpoint is at a distance z from receiver plane. Substituting Eqs. (1), (4) and (5) into Eq. (3), the average intensity can be obtained as

〈 I (x, y, L) 〉 = \frac{w_{0}^{2}}{4 w^{2}} \exp [- \frac{2 (x^{2} + y^{2})}{w^{2}} + \frac{4 B^{2} Ω_{0}^{2}}{k^{2} w^{2}}] {\cos (\frac{4 B Ω_{0}}{k w^{2}} x) + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}}) \cosh [\frac{2 (B - A F) Ω_{0} w_{0}^{2} x}{F w^{2}}]}

where w is the 1/e ² average intensity radius of a fundamental Gaussian beam after passing through an ABCD optical system in turbulent atmosphere and can be written as [23]

w^{2} = 4 B^{2} ⁄ (k^{2} w_{0}^{2}) + {w_{0}^{2} (A - B ⁄ F)}^{2} + 8 (k^{2} σ_{0}^{2}) .

The three terms on the right-hand side of Eq. (8) represent the contributions to the beam spread that are due to diffraction, geometrical magnification and turbulence, respectively. From Eqs. (7) and (8) we can see that the average intensity and the intensity radius depend on only ray-matrix elements A and B.

3. Special cases and numerical calculation

As an elementary example, the case of a single thin lens of focal length f located at the propagation path is considered, as shown in Fig. 1. The source plane and the receiver plane are conjugate image plane, namely, 1/f=1/z ₁+1/z ₂. If we set z ₁>f, it is easy to find that A=m=-z ₂/z ₁ and B=0, where m denotes the magnification.

Fig. 1. Propagation geometry of image system in turbulent atmosphere.

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Therefore, the average intensity at the image plane is

〈 I (x, y, L) 〉 = \frac{w_{0}^{2}}{4 w^{2}} \exp [- \frac{2 (x^{2} + y^{2})}{w^{2}}] [1 + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}}) \cosh (\frac{2 m Ω_{0} w_{0}^{2} x}{w^{2}})] [1 + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}}) \cosh (\frac{2 m Ω_{0} w_{0}^{2} y}{w^{2}})],

where

w^{2} = m^{2} w_{0}^{2} + 8 ⁄ (k^{2} σ_{0}^{2}),

b (z) = {\begin{matrix} z_{2} z ⁄ z_{1} & 0 < z \leq z_{1} \\ L - z & z_{1} < z \leq L \end{matrix} .

Substituting Eq. (11) into Eq. (6), it is easy to obtain

σ_{0} = {z_{2}^{- 1} (0.5475 k^{2} C_{n}^{2} L)}^{- 3 ⁄ 5} .

From Eqs. (8) and (10) we can see that when a beam passes through a lens, the contributions to the beam spread at the image plane that are due to diffraction disappear and the contributions to the beam spread that are due to turbulence become small. Because the cosh-Gaussian beam adopted in this paper is x-y symmetric, the properties of one-dimensional intensity distributions can reveal the properties of a two-dimensional intensity distribution. From Eq. (7) the one-dimensional average intensity of a collimated cosh-Gaussian beam (F→∞) for direct propagation (B=L, A=1) in turbulent atmosphere can be derived as

〈 I (x, L) 〉 = \frac{w_{0}}{2 w} \exp (- \frac{2 x^{2}}{w^{2}} + \frac{2 L^{2} Ω_{0}^{2}}{k^{2} w^{2}}) [\cos (\frac{4 L Ω_{0}}{k w^{2}} x) + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}}) \cosh (\frac{2 Ω_{0} w_{0}^{2} x}{w^{2}})],

where

w^{2} = w_{0}^{2} + 4 L^{2} ⁄ (k^{2} w_{0}^{2}) + 8 L^{2} ⁄ (k^{2} ρ_{0}^{2}) .

The one-dimensional average intensity at focal plane (L=F) of a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere can be represented as

〈 I (x, L) 〉 = \frac{w_{0}}{2 w} \exp (- \frac{2 x^{2}}{w^{2}} + \frac{2 L^{2} Ω_{0}^{2}}{k^{2} w^{2}}) [\cos (\frac{4 L Ω_{0}}{k w^{2}} x) + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}})],

where

w^{2} = 4 L^{2} ⁄ (k^{2} w_{0}^{2}) + 8 L^{2} ⁄ (k^{2} ρ_{0}^{2}) .

The one-dimensional average intensity at image plane of a cosh-Gaussian beam through a lens in turbulent atmosphere is

〈 I (x, L) 〉 = \frac{w_{0}}{2 w} \exp (- \frac{2 x^{2}}{w^{2}}) [1 + \exp (\frac{4 Ω_{0}^{2} w_{0}^{2}}{k^{2} σ_{0}^{2} w^{2}}) \cosh (\frac{2 m Ω_{0} w_{0}^{2} x}{w^{2}})] .

To see the effects of a lens on the intensity distributions, the one-dimensional normalized average intensity profiles of the beam through a lens and for direct propagation with different parameters are plotted in Fig. 2. Normalized average intensity I_N(x,L) at receiver plane is defined as

I_{N} (x, L) = 〈 I (x, L) 〉 ⁄ Max [〈 I (x_{0}, 0) 〉],

where Max[〈I(x ₀,0)〉] denotes the maximum value of the one-dimensional average intensity at the source plane.

Fig. 2. The normalized average intensity profiles of a cosh-Gaussian beam in turbulent atmosphere. (a) At image plane of a lens with w ₀=0.2m and Ω₀=30m ^-1; (b) At receiver plane for direct propagation with F→∞, w ₀=0.2m and Ω0=30m ^-1; (c) At focal plane (L=F) for direct propagation with w ₀=0.2m and Ω₀=30m ^-1; (d) At image plane of a lens with L=20km.

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The parameters are adopted as L=2z ₁=2z ₂=4f, C ² _n=10^-14 m ^-2/3, and λ=1.06×10^-6 m. From Fig. 2(a) we can see that the normalized average intensity profiles at the image plane of a cosh-Gaussian beam through a lens in turbulent atmosphere are the hollow shapes when L=10km, 16km and 20km. With increase in propagation distance the peak values of the normalized average intensity decrease, meanwhile, the central dips become smaller and the beam spot spreads. From Eqs. (9) and (10) we can see that the changes of the average intensity profiles is only due to the effects of turbulence. Figure 2(b) shows that the intensity profile of a collimated cosh-Gaussian is hollow shape at L=10km. But, with increase in propagation distance, the central dip of the hollow shape gradually disappears [see the dashed line in Fig. 2(b)]. With further increase in the propagation distance (L=16km) the central dip disappears completely and the hollow shape evolve into a bell shape [see the dotted line in Fig. 2(b)]. From Eqs. (13) and (14) we can see that the evolution of the average intensity profiles of a collimated cosh-Gaussian is due to the effects of diffraction and turbulence. The average intensity profiles of a focused cosh-Gaussian beam at the focal plane are bell shapes [see Fig. 2(c)]. The peak values decrease with the increase in the propagation distance. Eqs. (15) and (16) denote that the decreases of the peak value at the focal plane with the increase in the propagation distance are due to the diffraction, geometrical magnification and the turbulence. From Figs. 2(a), 2(b) and 2(c) we can see that the hollow shapes of a cosh- Gaussian beam after passing through an image system can be sustained for longer propagation distance. From Eqs. (9) and (10) we can also see that the intensity distributions has a close relation with w ₀ and Ω₀. Since the intensity distribution at source plane is different with different Ω₀, the different intensity distribution at source plane results in the different intensity distributions at the image plane [see the solid line and dotted line in Fig. 2(d)]. Because large initial waist width can reduce the effects of turbulence on intensity distribution, the hollow shapes can be sustained for longer propagation distance with large initial waist width [see the dashed line and dotted line in Fig. 2(d)].

4. Conclusion

The propagation of a cosh-Gaussian beam through an arbitrary ABCD optical system in turbulent atmosphere is studied. To show the effects of an ABCD system on the propagation of a laser beam in turbulent atmosphere, an elementary ABCD system of a single lens is studied. From the studies and the comparisons with the properties of a collimated cosh-Gaussian and a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere, we can see that a cosh-Gaussian beam through a lens can have a shape similar to that of the initial beam for a longer propagation distance in turbulent atmosphere than that of a collimated cosh-Gaussian beam for direct propagation in turbulent atmosphere. With the increment in the propagation distance, the average intensity radius of a cosh-Gaussian beam at the image plane after passing through a thin lens will be smaller than that of a focused cosh-Gaussian beam at the focal plane for direct propagation in turbulent atmosphere. Meanwhile, the intensity distributions at the image plane with different w ₀ and Ω₀ are also studied. The investigations show that the average intensity distribution and radius of a cosh-Gaussian beam can be improved by using optical elements on its propagation path in turbulent atmosphere.

Acknowledgments

This work is supported by Zhejiang Provincial Education Committee Foundation of China.

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Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere

Abstract

1. Introduction

2. The average intensity of a cosh-Gaussian beam through an optical system in turbulent atmosphere

3. Special cases and numerical calculation

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Equations (22)

Optics Express