Attenuation of Amplitude and Phase Fluctuations of Optical Wave in Coherent Turbulence
With the use of the model of 3D spectrum (3.32), we can obtain equations for the variance of fluctuations of the log amplitude aX = {X) (under conditions of applicability of the method of smooth perturbations [4]), the variance of displacements of the energy centroid of laser beam [176] a] = {pt), and the variance of displacements of the image of optical sources [177] at =(pt). Consider the parameters b_{x}, b,, and b ,, being ratios of these variances for coherent and incoherent Kolmogorov turbulence
Let a_{t} be the radius of optical receiver, a_{e}(x) be the effective radius of optical beam at the path with the length x (a(x) > a_{e}(0)); A be wavelength, R_{F} = (Ax)^{1/2 }be the radius of the first Fresnel zone. The calculations show that in the coherent turbulence (v = 5/6) the variances at, at, and a can be found (accurate to constant factors) from known equations for them in Kolmogorov turbulence (v = 1/3) through replacements a_{e}(x) ^ L_{0}; a_{t} ^ L_{0}; l_{0} ^ l_{0}^{4/7} L_{0}^{3/7} at l_{0} ^ R_{F} and x ^ x (R_{F}/L_{0})^{6/11} at l_{0} ^ R_{f}; respectively.
Actually, consider, for example, displacements of images of astronomical sources. For the case of incoherent Kolmogorov turbulence, the variance of angular displacements (jitter) of images at (a_{a} = aJF_{t}, F_{t} is the focal length of the receiving telescope) can be expressed through the integral value I of the structure characteristic of refractive index Ct (integral intensity of incoherent turbulence) in the known way [4]:
where a_{t} and h_{0} are, respectively, the radius of entrance aperture of the telescope and the height of the aperture center above the underlying surface, в is the zenith angle of an observed astronomical object (measured at the place of receiver location from the zenith direction), Ct(h) is the structure characteristic of fluctuations of the refractive index of air dependent on the height h above the underlying surface (vertical profile of C^{4}). For every value of the angle в, the value of I
determines the integral intensity of atmospheric Kolmogorov turbulence at optical paths of the given tilt angle. This equation allows reconstruction of the integral intensity of Kolmogorov turbulence I from measured values of the variance of jitter a..
As can be seen from Equation 3.32, in coherent turbulence (v = 5/6), the spectrum of atmospheric turbulence differs from the case of Kolmogorov turbulence (v = 1/3). Therefore, the equation for the variance aO, changes. After calculation of the corresponding integrals, we obtain the approximate estimative representation for the variance aO, in coherent turbulence [126,127,140]:
where L_{0} = L_{0}(h_{0}) is the outer (exponential) scale of turbulence at the height of the center of the receiving aperture h_{0}, I_{c} is the integral intensity of coherent turbulence at optical paths with the given tilt angle. It can be seen from this equation that in coherent turbulence the variance of jitter aO. is independent of the receiver radius a_{t} in contrast to the case of Kolmogorov turbulence. In place of the receiver radius a, the equation for the variance of jitter includes the outer scale of turbulence L_{0}. Consequently, theoretical representation of the variance of jitter in coherent turbulence (v = 5/6) follows (accurate to a constant factor) from the corresponding equation in Kolmogorov turbulence (v = 1/3), as should be expected, by the replacement a _{t} ^ L_{0}.
We believe, for comparison, that the coherent and incoherent turbulences have the same outer exponential scales L_{0}, inner scales l_{0}, and the turbulence intensity C. (or Cl). Then
Since the values of a_{e}(x), a,, l_{0}, and R_{F} are usually much smaller than the outer scale of turbulence L_{0} [a_{e}(x), a, l_{0}, R_{p}^ L_{0}], it is seen from here that for typical optical paths and typical values of the source and receiver parameters these ratios are small: b_{x}, b, b_{t} ^ 1. This means that, in comparison with incoherent Kolmogorov turbulence, in coherent turbulence, the significant attenuation of both the amplitude (under conditions of weak radiation intensity fluctuations [4]) and phase (refractive) fluctuations of optical radiation takes place.
In Reference 174, it was found experimentally that for large receiver size, the variance of displacements of astronomical images (see Figure 3.34; measurements at the top of a 2000m high mountain; displacements of the image of moon disk edge; the maximal receiver radius a = 22 cm; afternoon transient turbulent conditions; northern wind through Sayan mountain ridge and the deep river valley) is
Figure 3.34 Normalized rootmeansquare deviation of jitter of astronomical images. Comparison of traditional incoherent theory with coherent theory and experiment [174].
independent of the receiver radius a_{t} in contrast to the usual dependence at ~ a^{1/3 }for the Kolmogorov spectrum (dashed line in Figure 3.34).
This result can now be explained by the predominant action of one large coherent structure during the measurements carried out in Reference 174.
In 2010—2015, it was conducted the experimental observations of the effect of attenuation of phase (refractive) fluctuations of optical radiation at the coherent turbulence [114,116,126,140142]. For this purpose, optical measurements were carried out analogous to those reported in Reference 174. The measurements were conducted at the Sayan Sun Observatory of the Institute of SolarTerrestrial Physics SB RAS (Russia) with the automated horizontal sun telescope. Simultaneously with optical measurements, the state of the nearsurface atmosphere was monitored with the mobile ultrasonic meteorological system.
As an example, we present the data of the experiment of 2011 (Figure 3.35). The main parameters of the experiment were the following: sun zenith angle в и 55°; structure characteristic of refractive index fluctuations at a height of 4.5 m from the underlying surface Ct = 4.2 • 10^{15} cm^{2/3}; the angular radius of the astronomical source (solar disk edge) corresponded to the limiting angular resolution of the used receiver of 0.1 arc s.
The measured results have shown that when large coherent structures (spectrum of temperature fluctuations W_{T} in the inertial range is proportional to f^{8/3}) are present in the atmosphere over the Sayan Sun Observatory (Russia), the results of measurements coincide with predictions of the coherent theory (horizontal line in Figure 3.35). If in the atmosphere, there are no large coherent structures (incoherent turbulence, W_{T} ^ f^{5/3}), our results coincide with data of the traditional
Figure 3.35 Rootmeansquare deviation a_{a} of jitter of astronomical image of the solar disk edge as a function of diameter of the telescope entrance aperture 2a_{t}. Sayan Sun Observatory (Russia). Summer measurements of August 4, 2011. Experimental points correspond to spectra: either W_{T} ~ f^{8/3} (open squares, coherent turbulence) or W_{T} ~ f^{5/3} (closed circles, Kolmogorov turbulence).
incoherent theory (slant line in Figure 3.35). As can be seen from Figure 3.35, the standard deviation of image jitter of the solar disk edge in coherent turbulence for the same aperture is smaller (more than two times for small receivers) than for the Kolmogorov turbulence. Consequently, in the presence of large coherent structures (area of coherent turbulence) in the atmosphere, refractive fluctuations of optical radiation decrease significantly. This means that the effect of attenuation of phase (refractive) fluctuations of optical radiation in coherent turbulence is confirmed experimentally.
Thus, in References 114, 117119, 123126, 128, 129, 131, 133, 134, 139, 140, 142, the effect of attenuation of radiation fluctuations in coherent turbulence was found theoretically and experimentally. The effect appears in the presence of large coherent structures (areas of coherent turbulence) in the atmosphere, consists in attenuation of phase (refractive) fluctuations of optical radiation in comparison with Kolmogorov turbulence, and is caused by the faster decrease of the spectrum of coherent turbulence and the smaller contribution of smallscale components.
The effect of attenuation of refractive fluctuations decreases the jitter and improves the quality of astronomical images. Therefore, for installation of ground based telescopes, we can recommend regions, over which during measurements there are large coherent structures [123125,128,131,140142]. At the same time, large coherent structures themselves can be found from characteristic jitter of astronomical images. It should be noted that, for every groundbased astronomical observatory, time intervals of the best images are individual, and their position in the 24h interval and duration are determined by the regional meteorological situation.
The experiments of 2010—2015 carried out under different meteorological conditions extended our understanding of the influence of coherent turbulence on the propagation of optical radiation in a turbulent medium. The measurements show that over the territory of the Sayan Sun Observatory the coherent turbulence is observed for a long time (up to 20—120 min), usually, at the northern wind (from mountains of Sayan ridge). In summer, this wind is usually observed at night. Therefore, in this observatory, night astronomical observations are preferable. At the southern wind (from relatively smooth underlying surface, usually observed in daytime), the maximal average lifetime of areas of coherent turbulence ranges from 10 to 30 min. The lifetimes of turbulence of different types (time intervals of the presence of turbulence of one type, when it does not alternate to other type) observed in daytime optical measurements of different years are given in Table 3.3.
In general, we can say that long daytime astronomical observations in Sayan Sun Observatory are accompanied by often transitions from the Kolmogorov to the coherent turbulence (especially, in late fall at a strong wind). This transition corresponding to the alternation of turbulence type gives intermittence in the jitter of astronomical images, which shows itself in the frequent alternation of intervals of strong and weak image jitter. Since coherent turbulence leads to improvement of the quality of optical images [123—125,128,131,140—142], this effect can be considered as positive for shortexposure measurements (within tens of minutes), when the highestquality images can be selected from a series of recorded images based on the meteorological data.
Table 3.3 Average Lifetimes of Turbulence of Different Types Observed in Daytime Optical Measurements
Date of Measurements 
Wind Direction (deg) 
Horizontal Wind Velocity (m/s) 
Time of Continued Observation (min) 

of Coherent Turbulence 
of Kolmogorov Turbulence 

June 19, 2010 
190230 
4.4 
614 
833 
August 4, 2011 
2040 
1.9 
712 
815 
July 14, 2012 
4070 
1.7 
1839 
729 
September 25, 2013 
180200 
6.0 
1026 
67 