Optics and Photonics Journal, 2013, 3, 143147 doi:10.4236/opj.2013.32B035 Published Online June 2013 (http://www.scirp.org/journal/opj) Study on Turbulence Effects for Beam Propagation in Turbulent Atmosphere Pingping Pan College of physics and communication electronics, Jiangxi normal university, Nanchang, 330022 Received 2013 ABSTRACT Based on the theory for the turbulence and the Rytov method, the propagation formulas of the scintillation index and the mean square angleofarrival fluctuation for the beam propagation in atmospheric turbulence have been derived respec tively. The propagation properties of the two turbulence effects have been investigated, and the effects of the character istic 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 char acteristics parameters of the turbulence by using the measurement of the scintillation index and the angleofarrival fluctuation has been proposed. Keywords: Atmospheric Turbulence; Rytov Method; Scintillation Index, Mean Square AngleofArrival Fluctuations; Constant of the Refractive Index Fluctuations 1. Introduction The study on the propagation of laser beams through turbulence is of great importance for many practical ap plications, such as the remote sensing, atmospheric opti cal communications, and track system [15]. There are a few affecting factors on the beam propagation in atmos phere, such as atmosphere attenuation, atmospheric ab sorption, scattering, and atmospheric turbulence effect etc. The atmospheric turbulence effects include the beam spreading, the angleofarrival fluctuations, the scintilla tion index and the bit error rate, and they have more ob vious influence on the performance of a communication system. [610]. The scintillations and the angleofarrival fluctuations are more significant effects of atmospheric turbulence on the beam propagation. The effect of the angle of arrival may increase the fieldofview requirements in a di rectdetection optical receiver and degrade the perform ance a heterodyne system [1113]. 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 at mospheric optic links [1415]. In this paper, the propaga tion properties of the scintillations and the angleofar rival fluctuations have been investigated, the effects of the turbulent parameters and the beam parameters have been discussed, and a method for estimating the charac teristics parameters of the turbulence has been proposed. 2. Turbulence Effects for Beam Propagation in Atmosphere 2.1. Mean Square AngleofArrival Fluctuations When light travels through the atmosphere, it experiences phase fluctuations due to turbulence. The mean square angleofarrival fluctuation is related to the phase struc ture function through the expression below [14], 12 2 2 (0, , () s a DR kR pp )L (1) Here the mean square angleofarrival 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 22 11 221 (,), , (,)(0,0) xy xy ppR ppppp (2) The phase structure function described as 2 121 2 ( ,,)[( ,)(,)] s DLSLSLPPPP (3) where <> denotes the ensemble average over the medium statistics, 1 (, )[ (, )*(, )] 2 SLL L j ppp (4) the asterisk represents the complex conjugate. In Rytov method, ψ denotes the fluctuations of the complex am Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL. 144 plitude in turbulence [4], 1 2 3FS FS ' (,)'(,') (,') 2(,) exp(' ) ' V k LdrnzE EL ik pp' p rr rr zp' z (5) EFS (P, z) is the field in free space and '(',')(',','), '''' xyx y zppzddpdpd rPr (6) 11 (,')(',',') xy nznppzp' (7) expresses the random part of the refractive index, which can be written in the form of a Fourier–Stieltjes repre sentation [5] for a statistically homogeneous and iso tropic medium. The field distribution of Gaussian beam can be written as, '2 '2 '' 11 11 2 0 ( ,,0)exp() y Ex yw (8) where w0 is the waist width and (x1 ′, y1 ′) is the transversal coordinates at the z=0 plane. Then, by using the paraxial form of the extended HuygensFresnel principle, the field distribution of Gaussian beam through free space can be obtained, 2 22 0 111 1 2 0 22 22 01 1 22 0 (,, )exp[() 2 2 () ] 42 FS kw ik Exyzikz xy z kw iz kw xy zikzw (9) Assume that the turbulent atmosphere is statistically homogenous and isotropic, the source and the medium statistics are independent. The mean square angleof arrival fluctuation of Gaussian beam in turbulence can be derived, 221/3 0 0 11/62 22 2 0 1 1/62 2 0.1628( 14.45.5663() 1 Re{[] 1(1) []}) (1 ) m an m m CL dt Lt k kj L t 1/3 (10) Re means the real part, Cn 2 is the structure constant of the refractive index fluctuations of the turbulence. κm =5.92/l0, l0 is the inner scale of turbulence; κ0=1/L0, L0 is the outer scale of turbulence. And 2 0 1222 2 00 0 1, (/ ) kw jLt kwL wkwjL (11) 2.2. Scintillation Index The scintillation index in atmospheric turbulence at the L receiver plane is given by [810] 24(,,B 12 ) Iz pp (12) 121 2 (, ,)(,)(BzL L ,)L pppp (13) is the correlation function of the logamplitude L fluctua tions at the receiver plane z =L, (,) 0.5[(,LL) *(,)] ppp (14) and Bχ can be written as y * 0 [Re()Re()]( ) nx Bd dd z (15) Φn (κ) denotes the spatial power spectra of the refractive index fluctuations of the turbulent atmosphere, Φn(κ) =0 in free space, and κ= (κx 2+κy 2)1/2. 2 0 2 0 22 0 2 0 exp( ) (0, )2 ()(2 ) exp[] 2(2 ) FS kw ikikz Ez izkw zLt Ltikw kizkw (16) Letting x=0 and y=0 in EFS(x, y, z), the onaxis field distribution E FS (0, z) can be obtained. Tatarskii power spectra is represented as 2 2 11/3 2 ( )0.033exp() nn m C (17) After some tedious integral calculation, the scintilla tion index of Gaussian beam in turbulence can be de rived, 22 1 2222 0 22 2 00 5/6 0 2 0 2(1) 1 0.8816Re {[] (1 4) (1)(12)1 []} (1 2) In m m wL t CkLkLwk L tiLtwdt ik iLwk (18) 3. Numerical Examples and Analysis farrival fluc wave propagation through at RMSAOA is zero in the initial plane, and it increases 3.1. AngleofArrival Fluctuations In this section, the variation of the angleo tuations have been studied, and the effects of the turbu lent parameters and the beam parameters have been dis cussed. Eq. (10) is numerically evaluated and used in plotting the root mean square of the angleofarrival (RMSAOA) fluctuations. Figure 1 gives the variation of the angleofarrival fluctuations, and the calculation pa rameters are Cn 2 = 1014 m2/3, w0 = 30 mm, λ = 1.55 μm, L = 3 km, l0 = 10 mm, L0 = 25 m. The wave front tilt of optical mospheric turbulence gives rise to an angleofarrival fluctuation. Figure 1(a) shows the effects of constant of the refractive index fluctuations, C = Cn 2/1015 m2/3. The Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL. 145 versus the propagation distance L. The RMSAOA in creases when beams propagate through stronger turbu lence, 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 l0 and the outer scale of the turbulence Lare displayed in Fig ur 0 e 1(b). The change range of the inner scale l0 in turbu lence has been adopted between 1mm and 20mm usually, so the variation of RMSAOA under the change range of l0 is represented. Examining Figure 1(b), the RMSAOA decreases for the larger l0 and smaller L0. Figure 1. Variation of the angleofarrival fluctuations in turbulence. R the intensity fluctuations at the re Figure 1(c) provides the effects of the beam parame ters including the waist width w0 and wavelength λ. The MSAOA decreases first and increases then, and there exits a minimum during the variation of w0. In our graphs, two infrared wavelengths are employed, representing the most commonly utilized wavelengths in current space links. From Figure 1(c), the RMSAOA 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 RMSAOA happens is smaller. If the waist width is very big, the RMSAOA of the beam will increase. On the other hand, if w0 is very small, it is not easy to obtain larger power in the initial plane, so the propagation dis tance is limited in practice. Therefore, the appropriate waist width and wavelength should be selected. 3.2. Scintillations Scintillations reflect ceiver plane, and the variation of the scintillation index in turbulence is provided in Figure 2. The calculation parameters Cn 2 = 1014 m2/3, w0 = 30 mm, λ = 1.55 μm，L = 3 km, l0 = 10 mm are taken. In Figure 2(a), the scintilla tion index rises against propagation distance and through stronger turbulence. Furthermore the effect of the turbu lence on the scintillation index is more obvious when the propagation distance increases. The variation of scintilla tion index given in this paper is consistent with the re sults of Figure 2. Variation of the scintillations in turbulence. Copyright © 2013 SciRes. OPJ
P. P. PAN ET AL. 146 related literature [13]. As observed from Figure 2(b), there exits a minimum for the scintillation index during the variation of w0, and both the scintillation index and the corresponding minimum are larger for the smaller l0. In a word, the effect of the turbulence on beam propa gating 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. 3.3. Estimation of the Structure Constant of the Turbulence so stant of the turbulence can be estimated roughly. Secondly, if the scintillation index is scintnd turbulent parameters (see Figure The relations of the turbulence effects and turbulent pa rameters are illustrated in Figure 3 and Figure 4 respec tively, 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 l0 is adopted between 1 mm and 20 mm, whereas the range of the structure constant spans several magnitudes. Firstly, if the RMSAOA is measured by experiments, by using the relation of the RMSAOA 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. 4. Conclusions In this paper, the propagation formulas of the angleof arrival 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 RMSAOA and the scintillation Figure 3. Relation of the RMSAOA and turbulent parame ters. Figure 4. Relation of the scintillation index and turbulent parameters. index. The effects of the turbulent parameters and the beam parameters have been analyzed quantitatively. The angleofarrival 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 RMSAOA and the scintillation index measured by experiments. REFERENCES s of anomalous hollow beams in a turbulent at mosphere,” Optics Communications, Vol. 281, No. 21, [1] Y. Cai, H. T. Eyyuboğlu and Y. Baykal, “Propagation propertie 2008, pp. 52915297 doi:10.1016/j.op tcom.2008.07.080 [2] Y. Dan, B. Zhang and P. pan, “Propagation of Partially Coherent FlatTopped Beams Through A Turbulent At mosphere,” Journal of Optical Society of America A, Vol. 25, No.9, 2008, pp. 22232231. doi:10.1364 /JOSAA.25.0022 23 [3] P. Pan, Y. Dan, B. Zhang, N. Qiao and Y. Lei, “Propaga tion Properties of FlatTopped Beams in A Turbulent Atmosphere,” Journal of Modern Optics, Vol. 56, No. 12, 2009,pp.13751382.doi:10.1080/09500340903 145049 [4] A. Ishimaru, “Fluctuations meters of Spherical Waves Propagtmosphere,” Radio Partially Coherent in the Para ating in A Turbulent A Science, Vol. 4, 1969, pp. 209305. [5] A. Ishimaru, Wave Propagation and Scattering in Ran dom Media, New York San Francisco, London, 1978, pp. 93101. [6] G. Gbur and E. Wolf, “Spreading of Beams in Random Media,” Journal of the Optical Society of America A, Vol. 19, No.8, 2002, pp. 15921598. doi:10.1364 /JOSAA.19.0015 92 [7] X. Ji, X. Li, S. Chen, E. Zhang and B. Lü, “Spatial Cor Copyright © 2013 SciRes. OPJ
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