Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model Arkadi Zilberman,1,* Ephim Golbraikh,2 and Norman S. Kopeika1 1
Electrical and Computer Engineering Department, Ben-Gurion University of the Negev, POB 653, Be’er-Sheva, Israel 2
Physics Department, Ben-Gurion University of the Negev, POB 653, Be’er-Sheva, Israel *Corresponding author:
[email protected] Received 3 June 2008; revised 25 September 2008; accepted 14 October 2008; posted 28 October 2008 (Doc. ID 96879); published 24 November 2008
Turbulence properties of communication links (optical and microwave) in terms of log-amplitude variance are studied on the basis of a three-layer model of refractive index fluctuation spectrum in the free atmosphere. We suggest a model of turbulence spectra (Kolmogorov and non-Kolmogorov) changing with altitude on the basis of obtained experimental and theoretical data for turbulence profile in the troposphere and lower stratosphere. © 2008 Optical Society of America OCIS codes: 010.1330, 010.1300.
1. Introduction
It is well-known (see, for example, [1,2]) that random spatial and temporal fluctuations in the refractive index of the atmosphere due to turbulent mixing of air masses of different temperatures and humidity give rise to variations in the amplitude and phase of an electromagnetic (EM) wave known as scintillations both across the wavefront (or receiver aperture) and in time. For systems at millimeter wave and optical bands, these effects give rise to power losses and add fading in communication channels that degrade performance. To establish highly reliable optical communications links, quantitative estimates of various statistical quantities that are associated with atmospheric turbulence-induced scintillations are necessary. It is also important for low-fade margin systems operating at frequencies above 10 GHz and at low elevation angles ( ðλLÞ1=2 > l0 :
Here the source is at z ¼ 0, and the observation point is at L. These expressions are valid only for small apertures (point detector) with a diameter of d < d0 ≈ ðλLÞ1=2 ¼ ðλh sec φÞ1=2 , where d0 corresponds to the correlation length of intensity fluctuations when the propagation path length L satisfies the condition l0 < ðλLÞ1=2 < L0 . The variance σ X 2 is dependent on the turbulence model assumed for the atmosphere. For α in vertical and slant-path propagation, a three-layer model in form of Eq. (7) can be used with α ≡ αðzÞ. For the case of constant β along the path, as might approximate a horizontal path, Eqs. (7) and (8) become 6388
3 < α < 5 ðplane waveÞ;
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σ 2X ¼ BðαÞ ·
ð9Þ
Γðα=2ÞΓðα=2Þ 6α α AðαÞk 2 βL2 ; ΓðαÞ
3 < α < 5 ðspherical waveÞ:
ð10Þ
For O–K turbulence (α ¼ 11=3), we have the logamplitude variance in the classical form, where β ≡ Cn 2 . For N-O–K turbulence, β ≠ Cn 2 and needs to be recalculated. Following [19], β can be chosen such that the power spectrum for N-O–K turbulence at a specific wavenumber K n is constant, i.e., corresponds to the O– K spectrum, and β is found using the relation β¼
Að11=3Þ 2 α11=3 Cn K n : AðαÞ
ð11Þ
As mentioned earlier, the scintillations are primarily caused by irregularities having the size of a Fresnel zone, i.e., ½λL1=2 for a wavelength λ and a path length L. For visible or NIR wavelengths and path length of a few kilometers, the scale sizes that cause scintillations are a few centimeters. On the other hand, for millimeter waves (K, Ka, or V band), the scale sizes of refractive index fluctuations that produce the most powerful scintillations (or the most effective eddies) are of the order of a few meters. Thus the different regions of the turbulence spectrum influence the log-amplitude scintillations at corresponding wave bands and are important in consideration and analyzing microwave and optical systems. To define K n , we introduce a concept of turbulence intensity, which can be characterized by βðλ; LÞ and determined in such a way that the refractive turbulence essentially affects the log-amplitude fluctuations at different wave bands (optical or millimeter) and distances as a result of influence of different turbulence scale sizes. This concept is similar to that suggested in [25] for the turbulence minimum intensity, Cn 2 min ðλ; LÞ, where the turbulence starts to affect the beam focusing at given wavelength λ and distance L.Following this concept, K n in Eq. (11) can be chosen as K n ≈ ðk=LÞ1=2 < 1=l0 . It relates the spatial wavenumber to the correlation length of intensity scintillations. Thus the equation for β is given by β¼
Að11=3Þ 2 pffiffiffiffiffiffiffiffiffi α11=3 Cn ð k=LÞ ; AðαÞ
ð12Þ
where β ≡ βðzÞ and α ≡ αðzÞ for vertical or slant-path propagation. As follows from experimental data [26–28], the value of refractive index structure parameter Cn 2 for
microwave bands can be larger than in the visible or NIR region of the EM spectrum by 1–2 orders of magnitude. We note that Φn ðKÞ enters into various convolutions for propagation parameter calculation. Therefore its dependence on the wavelength should be represented in the form of effective spectral index α [21,23]. For horizontal propagation, substituting Eq. (12) into Eqs. (9) and (10) yields 2 7 11 σ 2X ¼ BðαÞAð11=3Þk6 C2n L 6 ; α 3 < α < 5 ðplane waveÞ;
σ 2X ¼ BðαÞ ·
ð13Þ
Γðα=2ÞΓðα=2Þ 7 11 Að11=3Þk6 C2n L 6 ; ΓðαÞ
3 < α < 5 ðspherical waveÞ:
ð14Þ
For α ¼ 10=3 (Helical N-O–K turbulence), the constants in Eq. (13) and (14) are equal to 0.393 and 0.192, respectively. Ground-level scintillations can be accurately modeled by a plane wave in the case of space-to-ground propagation (downlink). For an uplink, where the atmospheric turbulence begins just outside the transmitting aperture, a spherical wave can be assumed. Thus for plane wave downlink propagation (source at L), substituting Eq. (12) into Eq. (7) yields σ 2X
Z
α 2
L
7 6
¼ BðαÞðsec φÞ Að11=3Þk
0
3 < α < 5:
5
C2n ðzÞz6 dz; ð15Þ
For a spherical wave and uplink propagation (source at z ¼ 0) yields α
Z 7
σ 2X ¼ BðαÞðsec φÞ2 Að11=3Þk6 3 < α < 5:
0
L
α2 z 2 5 C2n ðzÞ 1 z6 dz; L
ture constant Cn 2 [6]. This model is generally regarded as representative of continental conditions. We will use the H–V Cn 2 profile with v ¼ 21 m= sec to represent the H–V 5=7 model. The values of 5 and 7 in the H–V 5=7 profile refer to the values of the atmospheric coherence length in centimeters and the isoplanatic angle in microradian, respectively, at λ ¼ 0:55 μm. At the wavelength of 1:55 μm, the corresponding values are approximately 14 cm and 20 μrad, respectively. All calculations correspond to zenith angle φ ¼ 0. Figure 2 shows changing structure constant β with altitude calculated for pure helical turbulence and different wave bands: Ka band 32 GHz and λ ¼ 1:55 μm. The values of β are different for optical and millimeter wave bands in calculations of the log-amplitude fluctuations. It also coincides with experimental data [26–28]. The same case is shown in Fig. 3 but for our threelayer turbulence model. There is a difference in behavior of the structure parameter β for the one-layer and three-layer models starting above the boundary layer (∼2 km) and reaching maximal values at the tropopause altitude region (∼10 km). The strong turbulence layer in the low stratospheric region has been introduced in the H–V model as an empirical parameter. However, in our threelayer model, it is a logical consequence of the transition between layers with different turbulence properties. In this region, AðαÞ in Eq. (12) has a maximum at α ≈ 4:4. On the other hand, as follows from Fig. 3, magnitude of the maximum depends on the wavelength for amplitude scintillations of the EM wave. The width of the transition zone (and width of maxima region in Fig. 3) is an unknown free parameter of the three-layer model. The log-amplitude variance as a function of α for Earth–space downlink and uplink propagation is shown in Fig. 4. The scintillation effect is slightly reduced for higher power laws, where α → 5. Here the
ð16Þ
In the case of arbitrary power law exponent, the expressions are different from the “classical” solution by the constants and path weighting for a spherical wave. 4. Discussion
The theoretical model presented here is applied for the calculation of EM wave propagation parameters. In order to explain the importance of different regions of the turbulence spectrum in analyzing microwave and optical systems, an example of calculation results for wavelengths 1:55 μm and Ka band (32 GHz) is presented. Numerical results are based on the Hufnagel– Valley (H–V) vertical atmospheric model for struc-
Fig. 3. βðzÞ as a function of altitude for three-level model with b1 ¼ 8 and b2 ¼ 10 calculated for λ ¼ 1:55 μm (dash) and 32 GHz (dash-dot). The solid curve corresponds to the H–V 5=7 model. 1 December 2008 / Vol. 47, No. 34 / APPLIED OPTICS
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Fig. 4. Log-amplitude variance as a function of α for Earth–space propagation: downlink (solid curve) and uplink (dashed curve). λ ¼ 1:55 μm and angle from zenith φ ¼ 0 for the H–V 5=7 turbulence model.
spectrum has a steeper slope (reduced energy of small eddies), and the phase effects become dominant. The refractive index fluctuation variance is generally influenced by large scale sizes. A comparison of models with different numbers of turbulent layers is shown in Fig. 5. As we can see, including additional turbulent layers into the model affects the log-amplitude fluctuations. For the threelayer model, the variability of fluctuations starts at some upper boundary layer. For propagation in helical turbulence with a flatter slope, the influence of small scales increases (more energy in small scales corresponding to Fresnel zone size) and, as a result, causes larger scintillations. The quantitative data of Fig. 5 suggest that the three-layer model, which includes two layers of N-O–K turbulence, indicate that turbulence effects in many situations are more severe than previously thought. The predictions based on the three-layer model can be tested experimentally by measuring the scintillation index (or log-intensity variance) at different elevations and Cn 2 . This would allow evaluation of log-intensity variance to be compared theoretically to the measurements.
This work was supported by the Israel Science Foundation (ISF) (grant 730/06). The authors express their deep gratitude to the reviewers of this paper.
5. Conclusions
References
We propose a three-layer model of the atmosphere, which considers dependence of the turbulence spectral exponent on altitude. It was shown that this dependence essentially influences parameters that describe the main characteristics of EM wave propagation through Earth’s atmosphere. Equations [7,8,19,20] are the general solutions for estimation of log-amplitude (or intensity) variance of EM wave propagation in turbulent media with arbitrary power spectrum of refractive index fluctuation. They can be used for applications in microwave and optical wave bands. The log-amplitude variance is basic to communication reliability effects. 6390
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Fig. 5. Log-amplitude variance as a function of distance for different models: (a) uplink propagation (spherical wave) and (b) downlink propagation (plane wave). Three-layer (dash), twolayer (dash-dot), and O–K one-layer (solid). λ ¼ 1:55 μm and angle from zenith φ ¼ 0 for the H–V 5=7 turbulence model.
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