Study on Turbulence Effects for Beam Propagation in Turbulent Atmosphere

Based on the theory for the turbulence and the Rytov method, the propagation formulas of the scintillation index and the mean square angle-of-arrival fluctuation for the beam propagation in atmospheric turbulence have been derived respectively. The propagation properties of the two turbulence effects have been investigated, and the effects of the characteristic parameters of turbulence and the beam parameters have been discussed. The results show that the variation of the two turbulence effects depends on the structure constant of the refractive index fluctuations Cn2, the inner scale of the turbulence l0, the waist width of source beam w0 and the wave length λ. Moreover, there are two parameters including Cn2 and l0 which show more significant effects in atmosphere. Consequently, a new method for determining the characteristics parameters of the turbulence by using the measurement of the scintillation index and the angle-of-arrival fluctuation has been proposed.


Introduction
The study on the propagation of laser beams through turbulence is of great importance for many practical applications, such as the remote sensing, atmospheric optical communications, and track system [1][2][3][4][5].There are a few affecting factors on the beam propagation in atmosphere, such as atmosphere attenuation, atmospheric absorption, scattering, and atmospheric turbulence effect etc.The atmospheric turbulence effects include the beam spreading, the angle-of-arrival fluctuations, the scintillation index and the bit error rate, and they have more obvious influence on the performance of a communication system.[6][7][8][9][10].
The scintillations and the angle-of-arrival fluctuations are more significant effects of atmospheric turbulence on the beam propagation.The effect of the angle of arrival may increase the field-of-view requirements in a direct-detection optical receiver and degrade the performance a heterodyne system [11][12][13].It is well known that the atmospheric turbulence affects the received intensity, resulting in the intensity fluctuations at the receiver plane.Consequently, the optical scintillation take place in atmospheric optic links [14][15].In this paper, the propagation properties of the scintillations and the angle-of-arrival fluctuations have been investigated, the effects of the turbulent parameters and the beam parameters have been discussed, and a method for estimating the characteristics parameters of the turbulence has been proposed.

Mean Square Angle-of-Arrival Fluctuations
When light travels through the atmosphere, it experiences phase fluctuations due to turbulence.The mean square angle-of-arrival fluctuation is related to the phase structure function through the expression below [14], Here the mean square angle-of-arrival fluctuations are evaluated at a radial distance R, L is the path length, i.e., the length of the atmospheric link, and k = 2π/λ is the wave number, λ is wavelength.The second phase point lies at ( , ), , ( , ) (0,0) The phase structure function described as ( , ') ( ', ', ') x y n z n p p z  p' (7) expresses the random part of the refractive index, which can be written in the form of a Fourier-Stieltjes representation [5] for a statistically homogeneous and isotropic medium.
The field distribution of Gaussian beam can be written as, where w 0 is the waist width and (x 1 ) is the transversal coordinates at the z=0 plane.Then, by using the paraxial form of the extended Huygens-Fresnel principle, the field distribution of Gaussian beam through free space can be obtained, Assume that the turbulent atmosphere is statistically homogenous and isotropic, the source and the medium statistics are independent.The mean square angle-ofarrival fluctuation of Gaussian beam in turbulence can be derived, Re means the real part, C n 2 is the structure constant of the refractive index fluctuations of the turbulence.κ m =5.92/l 0 , l 0 is the inner scale of turbulence; κ 0 =1/L 0 , L 0 is the outer scale of turbulence.And

Scintillation Index
The scintillation index in atmospheric turbulence at the ) is the correlation function of the log-amplitude and Bχ can be written as Φ n (κ) denotes the spatial power spectra of the refractive index fluctuations of the turbulent atmosphere, Φ n (κ) =0 in free space, and κ= (κ x 2 +κ y 2 ) 1/2 .2) Letting x=0 and y=0 in E FS (x, y, z), the on-axis field distribution E FS (0, z) can be obtained.Tatarskii power spectra is represented as After some tedious integral calculation, the scintillation index of Gaussian beam in turbulence can be derived, ) )

Numerical Examples and Analysis
f-arrival flucwave propagation through at RMS-AOA is zero in the initial plane, and it increases

Angle-of-Arrival Fluctuations
In this section, the variation of the angle-o tuations have been studied, and the effects of the turbulent parameters and the beam parameters have been discussed.Eq. ( 10) is numerically evaluated and used in plotting the root mean square of the angle-of-arrival (RMS-AOA) fluctuations.Figure 1 gives the variation of the angle-of-arrival fluctuations, and the calculation parameters are C n 2 = 10 -14 m -2/3 , w 0 = 30 mm, λ = 1.55 μm, L = 3 km, l 0 = 10 mm, L 0 = 25 m.
The wave front tilt of optical mospheric turbulence gives rise to an angle-of-arrival fluctuation.

R
the intensity fluctuations at the re-Figure 1(c) provides the effects of the beam parameters including the waist width w 0 and wavelength λ.The MS-AOA decreases first and increases then, and there exits a minimum during the variation of w 0 .In our graphs, two infrared wavelengths are employed, representing the most commonly utilized wavelengths in current space links.From Figure 1(c), the RMS-AOA of the beam with λ = 0.85 μm is larger than λ = 1.55 μm.Moreover, the waist width where the value of minimum for the RMS-AOA happens is smaller.If the waist width is very big, the RMS-AOA of the beam will increase.On the other hand, if w 0 is very small, it is not easy to obtain larger power in the initial plane, so the propagation distance is limited in practice.Therefore, the appropriate waist width and wavelength should be selected.

Scintillations
Scintillations reflect ceiver plane, and the variation of the scintillation index in turbulence is provided in Figure 2. The calculation parameters C n 2 = 10 -14 m -2/3 , w 0 = 30 mm, λ = 1.55 μm，L = 3 km, l 0 = 10 mm are taken.In Figure 2(a), the scintillation index rises against propagation distance and through stronger turbulence.Furthermore the effect of the turbulence on the scintillation index is more obvious when the propagation distance increases.The variation of scintillation index given in this paper is consistent with the results of related literature [13].As observed from Figure 2(b), there exits a minimum for the scintillation index during the variation of w 0 , and both the scintillation index and the corresponding minimum are larger for the smaller l 0 .
In a word, the effect of the turbulence on beam propagating in turbulence is very obvious, so the beam quality may be affected, and especially the effect of the constant of the refractive index fluctuations is not neglected.Consequently, it is significative to estimate the structure constant of the turbulence.

Estimation of the Structure Constant of the Turbulence
so stant of the turbulence can be estimated roughly.Secondly, if the scintillation index is scint nd turbulent parameters (see Figure The relations of the turbulence effects and turbulent parameters are illustrated in Figure 3 and Figure 4 respectively, which implies a method for the estimation of the structure constant of the turbulence.We noted that in our lution, the change range of the inner scale l 0 is adopted between 1 mm and 20 mm, whereas the range of the structure constant spans several magnitudes.Firstly, if the RMS-AOA is measured by experiments, by using the relation of the RMS-AOA and turbulent parameters (see Figure 3), the structure con measured by experiments, by using the relation of the illation index a 2 4), C n can be estimated once again.Applying the two change ranges, we can estimate the range of the structure constant of the turbulence more accurately.

Conclusions
In this paper, the propagation formulas of the angle-ofarrival fluctuation and the scintillation index for the beam propagation in atmospheric turbulence have been obtained respectively, indicating that there are similarities in the derivation of the RMS-AOA and the scintillation  index.The effects of the turbulent parameters and the beam parameters have been analyzed quantitatively.The angle-of-arrival fluctuation and the scintillation index rise with the increasing propagation distance and the structure constant of the turbulence, and the decreasing inner scale, moreover there exits a minimum for them during the variation of waist width.Finally, the range of the structure constant of the turbulence can be estimated from the RMS-AOA and the scintillation index measured by experiments.

Figure 1 (
a) shows the effects of constant of the refractive index fluctuations, C = C n 2 /10 -15 m -2/3 .The Copyright © 2013 SciRes.OPJ versus the propagation distance L. The RMS-AOA increases when beams propagate through stronger turbulence, which can be revealed from Eq. (10) straightway.In addition, the effect of the turbulence is more obvious for larger L. The effects of the inner scale of the turbulence l 0 and the outer scale of the turbulence L are displayed in Figur 0 e 1(b).The change range of the inner scale l 0 in turbu- lence has been adopted between 1mm and 20mm usually, so the variation of RMS-AOA under the change range of l 0 is represented.Examining Figure 1(b), the RMS-AOA decreases for the larger l 0 and smaller L 0 .

Figure 1 .
Figure 1.Variation of the angle-of-arrival fluctuations in turbulence.

Figure 2 .
Figure 2. Variation of the scintillations in turbulence.

Figure 3 .
Figure 3. Relation of the RMS-AOA and turbulent parame-

Figure 4 .
Figure 4. Relation of the scintillation index and turbulent parameters.