Why should the Kolmogorov Spectrum ONLY apply to Locally Homogeneous and Isotropic (LHI) media? These conditions are equivalently equal to translationally and rotationally invariance, which requires the LINEARITY property. This is especially the case, not only due to the vast several decades of use of Shack-Hartmann Wavefront Sensors, but also because the Kolmogorov spectrum is the spatial frequency Fourier dual of the structural (spatial) coherence of the cross-section. For underwater use, the lateral dimension is y and the depth is z. Therefore, the wavefront propagates approximately in the x-direction. The linearity property of the Fourier transform requires that the component signals (in space) a * sin(ky y) + b sin(kz z) = a F_{est}{ky) + a F_{est}{kz), the spatial spectrum transforms like the spatial signal. A vast experience with testing supports this requirement that the Kolmogorov Spectrum ONLY apply to LHI conditions: __________________________________________________________________/ related references, atmospheric and underwater optics through UWA. __________________________________________________________________\ Anisoplanatic studies and Fried parameter estimation via multi-channel laser communication system Aleksandr Sergeyev; Michael Aerospace Conference, 2013 IEEE Year: 2013 Pages: 1 - 9, DOI: 10.1109/AERO.2013.6496844 IEEE Conference Publications \bibitem{ppr:Sergeyev} Aleksandr Sergeyev and Michael Roggemann, ``Anisoplanatic studies and Fried parameter estimation via multi-channel laser communication system,'' \texttt{Aerospace Conference, IEEE Conference Publications, DOI: 10.1109/AERO.2013.6496844}, 2013. Pages 1-9. More at @book{Roggemann, author="Michael C. Roggemann and Byron Welsh", title="Imaging through turbulence", publisher="CRC Press", address="Boca Raton, Florida", year=1996 } @book{Roggemann, author="Michael C. Roggemann and Byron Welsh", title="Imaging through turbulence", publisher="CRC Press", address="Boca Raton, Florida", year=1996} \bibitem{book:GoodmanSO85} Joseph W. Goodman, "Statistical Optics." \texttt{Wiley - Interscience Publications, John Wiley & Sons, Inc., New York}, 1985. Joseph W. Goodman, "Statistical Optics," Wiley - Interscience Publications, John Wiley & Sons, Inc., New York, 1985. Covariance is on page 17, jointly circularly complex [Gaussian] random on p. 42, finite power 68 & then Parseval's theorem, Wiener-Khinchin p. 73-75, 80, 121-124, Basic structure functions p. 79, Definitions on p. 183, van Cittert-Zernike theorem (p. 207), 241, 247, resolution as a function of the mutual intensity J and the complex coherence factor mu, 326 ff culminating in the sparrow limit on page 328, 345, 348-355, Fineup's method to estimate phase from spatial power spectra 344-347, averaged OTF's and PSF's on page 366, OTFs via the structure function on page 376, wavespeed structure function and the structure constant C_n^2 on page 391. PSF's on page 366, 431, the plot with the Fried parameter used for responsivity is also on page 441, extended to short-exposure. D. L. Fried, "Limiting Resolution Looking Down Through the Atmosphere," Journal of the Optical Society of America, Volume 56, page 1383, 1966 @article{ppr:FriedParameter66, title={Limiting resolution looking down through the atmosphere}, author={Fried, David L}, journal={JOSA}, volume={56}, number={10}, pages={1380--1384}, year={1966}, publisher={Optica Publishing Group} } The best complete reference (although it is atmospheric optics): @book{Andrews, author="Larry C. Andrews and Ronald L. Phillips", title="Laser Beam Propagation through Random Media", publisher="SPIE Optical Engineering Press", note="ISBN 0-8194-5948-8, QC976.L36A63", address="New York", year=2005} Andrews, Larry C. and Ronald L. Phillips. Laser Beam Propagation through Random Media. SPIE Optical Engineering Press, New York, 1998. Ergodicity and wide sense stationarity appears on page 25. See page 29 (Riemann-Stieltjes), page 32 (statistically isotropic) and pages 30 & 33 (Wiener-Khinchin). Pages 77-79 discuss output plane beam parameters such as the Fresnel ratio, focusing, and divergence. Rytov variance is on page 132 A summary of image degradation effects and resolution methods is on pages 141-142. Beam wander, Fresnel zone size and Fante’s relation are on page 147. Fractional fade time and expected number of fades is on page 244. Mean fade time is on page 246. The Strehl ratio is defined in terms of output beam parameters on page 281. Strong fluctuation theory scintillation is on page 366. _____________________________/ Underwater optics: Notification from Szymon Gladysz , Workshop on non-Kolmogorov Turbulence and Associated Phenomena, 1 - 3 July 2019, Buhlsche Mühle, Ettlingen, GERMANY, Fraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB, Remove the 4 spaces that were added to the domain name to activate the link: https :// newsletter .fraunhofer.de https :// www .onr .navy .mil/en/Science-Technology/ONR-Global Italo Toselli and Szymon Gladysz, "Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence," Opt. Express 28, 17347-17361 (2020) Toselli, Italo, and Szymon Gladysz. "Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence." Optics Express 28, no. 12 (2020): 17347-17361. @article{Toselli:20, author = {Italo Toselli and Szymon Gladysz}, journal = {Opt. Express}, keywords = {Atmospheric turbulence; Laser beam propagation; Laser communications; Optical turbulence; Spatial frequency; Visible light}, number = {12}, pages = {17347--17361}, publisher = {Optica Publishing Group}, title = {Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence}, volume = {28}, month = {Jun}, year = {2020}, url = {https://opg.optica.org/oe/abstract.cfm?URI=oe-28-12-17347}, doi = {10.1364/OE.394468}, abstract = {We theoretically investigate the effectiveness of adaptive optics correction for Gaussian beams affected by oceanic turbulence. Action of an idealized adaptive optics system is modeled as a perfect removal of a certain number of Zernike modes from the aberrated wavefront. We focused on direct detection systems and we used the aperture-averaged scintillation as the main metric to evaluate optical system performances. We found that, similar to laser beam propagation in atmospheric turbulence, adaptive optics is very effective in improving the performance of laser communication links if an optimum aperture size is used. For the specific cases we analyzed in this study, scintillation was reduced by a factor of \&\#x223C;7 when 15 modes were removed and when the aperture size of the transceiver was large enough to capture 4-5 speckles of the oceanic turbulence-affected beam.}, } R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). Robert J. Noll, "Zernike polynomials and atmospheric turbulence*," J. Opt. Soc. Am. 66, 207-211 (1976) https://opg.optica.org/josa/abstract.cfm?URI=josa-66-3-207 _________________________________________________/ Both atmospheric optics and underwater acoustics Akira Ishimaru, "Wave Propagation and Scattering in Random Media," Volume 2, "Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing," Academic Press, ISBN 0-12-374702-3, New York, 1978. In using covariance functions B_n(r_d) on page 333 the assumption of "If the medium is assumed to be statistically [and locally] homogeneous and isotropic." (The LHI assumption.) This leads to the definition and use of the structure constant C_n^2 on page 337, "Considering the three ranges [input, inertial, and dissipation], we write the Kolmogorov spectrum as follows." Which is based upon this LHI assumption written many pages earlier. On page 391 an analysis of the temporal spectrum compares information attainable above the Greenwood and Tyler frequencies represented by the spatial frequency related to cross-current speed omega/U_t. @book{book:Ishimaru78, author="Akira Ishimaru", title="Wave Propagation and Scattering in Random Media", publisher="Prentice-Hall Inc.", edition="First", address="Englewood Cliffs, NJ 07632", isbn="0-12-374702-3", note="In using covariance functions $B_n(r_d)$ on page 333 the assumption of ``If the medium is assumed to be statistically [and locally] homogeneous and isotropic.'' (The LHI assumption.) This leads to the definition and use of the structure constant $C_n^2$ on page 337, ``Considering the three ranges [input, inertial, and dissipation], we write the Kolmogorov spectrum as follows.'' Which is based upon this LHI assumption written many pages earlier. Page 359, filter functions. On page 391 an analysis of the temporal spectrum compares information attainable above the Greenwood and Tyler frequencies represented by the spatial frequency related to cross-current speed $\omega/U_t$", year=1984 } Purely underwater acoustics (UWA), Flatté (1974) and Colosi (2016) separately: Stanley M Flatté, "Sound transmission through a fluctuating ocean," Cambridge University Press, The Pitt Building, Trumpington Street, Cambridge CB2 1RP, England, ISBN 0-521-21940-X, 1979. Variables are different, such the propagation constant (spatial frequency) q in Equation 5.2.1 on page 76. The dimensionless sound speed change parameter U described on page 76 to be orders of magnitude lower for internal waves, is in the first row of a list of parameters with their numerical values in Table 5.1 on page 77. Structure functions are defined on page 95. The statistics of the oceans are described on page 100 including the statement as to their anisotropy and statistical inhomogeneity. His Figure 8.2 shows Doppler broadening is dominated by spread on page 125. A geometric analysis of the Rytov extension is found in pages 130-143. The unsaturated region is defined on page 140, saturated on 142. Doppler spread descriptions appear on pages 145-9 (spread and wander of pulses). The "frozen ocean assumption" is on page 144. Page 167-172 ff uses a Green's function G(x, y) to calculate the spatial power spectrum and the phase of the pressure X(x) which was obtained by the WKB approximation. @book{book:Flatte79, author = "Stanley M Flatt$\acute{e}$ ", title = "Sound transmission through a fluctuating ocean", publisher = "Cambridge University Press", address = "The Pitt Building, Trumpington Street, Cambridge CB2 1RP, England", isbn = "0-521-21940-X", note = " Variables are different, such the propagation constant (spatial frequency) q in Equation 5.2.1 on page 76. The dimensionless sound speed change parameter U described on page 76 to be orders of magnitude lower for internal waves, is in the first row of a list of parameters with their numerical values in Table 5.1 on page 77. Structure functions are defined on page 95. The statistics of the oceans are described on page 100 including the statement as to their anisotropy and statistical inhomogeneity. His Figure 8.2 shows Doppler broadening is dominated by spread on page 125. A geometric analysis of the Rytov extension is found in pages 130-143. The unsaturated region is defined on page 140, saturated on 142. Doppler spread descriptions appear on pages 145-9 (spread and wander of pulses). The "frozen ocean assumption" is on page 144. Page 167-172 ff uses a Green's function G(x, y) to calculate the spatial power spectrum and the phase of the pressure X(x) which was obtained by the WKB approximation." year = "1979" } Purely underwater acoustics (UWA), Colosi (2016): Colosi, John A., "Sound propagation through the stochastic ocean." Cambridge University Press, 2016. Page 10, equation 1.1 describes the effect on sound speed by vertical displacement of internal waves. Page 12 starts with a comparison of background sound channels and their effect as a normal mode structure. The next section below this discussion indicates that the weak versus strong fluctuations in wavespeed were addressed in acoustics with the Born and Rytov approximations versus the Feynman path integral method, respectively. Page 13 defines the scintillation index (SI) in Equation 1.2 while describing the unsaturated regime. It and the partially and fully saturated regimes are defined as and after the page turns to 14. Page 20 The "frozen flow" assumption is stated on page 41 just before equation (2.8). The Born approximation is compared to the Rytov approximation of the phase - which some say is superior to Born, yet "it is clear these claims are unfounded" - page 77 ff. Internal waves and the use of the WKB method are on page 113 and Appendix B of Chapter 2 on page 102-3. A section of the Garret-Munk (GM) spectrum model using WKB follows on page 113 and 118 ff. Rationale for assuming internal waves are everywhere and always affecting a-comms -- page 163. And that the upper turning points affect a-comms the most, p. 165. In developing the Feynman path integral of Chapter 7, and its quantum and statistical mechanics history, and using the term saturation to mean 'strong fluctuation theory,' and showing that this method of calculating the normal modes that statistical acoustics calls 'internal waves' (page 47), Colosi describes how the stationary phase selection from this variational method is a * deterministic * analysis result on page 274 just after his Equation 7.7. Assume the preliminary structure function V12 is LHI (stationary phase R.V.) in developing D(1,2) on page 279. Markov approximation for acoustics for a small turbulence condition applies more easily with large range using "the fact that the spectrum and correlation are cosine transform pairs," page 283. The coherence depth and width are on page 284 and 286. The next page has an estimate of the structure function for small ray separation with respect to the depth separation z_o being calculated in that section. The scintillation index is again defined, SI = var(I), on page 387 after equation 8.121. Colosi, John A., "Sound propagation through the stochastic ocean." Cambridge University Press, 2016. Page 10, equation 1.1 describes the effect on sound speed by vertical displacement of internal waves. Page 12 starts with a comparison of background sound channels and their effect as a normal mode structure. The next section below this discussion indicates that the weak versus strong fluctuations in wavespeed were addressed in acoustics with the Born and Rytov approximations versus the Feynman path integral method, respectively. Page 13 defines the scintillation index (SI) in Equation 1.2 while describing the unsaturated regime. It and the partially and fully saturated regimes are defined as and after the page turns to 14. Page 20 The "frozen flow" assumption is stated on page 41 just before equation (2.8). The Born approximation is compared to the Rytov approximation of the phase - which some say is superior to Born, yet "it is clear these claims are unfounded" - page 77 ff. Internal waves and the use of the WKB method are on page 113 and Appendix B of Chapter 2 on page 102-3. A section of the Garret-Munk (GM) spectrum model using WKB follows on page 113 and 118 ff. Rationale for assuming internal waves are everywhere and always affecting a-comms -- page 163. And that the upper turning points affect a-comms the most, p. 165. In developing the Feynman path integral of Chapter 7, and its quantum and statistical mechanics history, and using the term saturation to mean 'strong fluctuation theory,' and showing that this method of calculating the normal modes that statistical acoustics calls 'internal waves' (page 47), Colosi describes how the stationary phase selection from this variational method is a * deterministic * analysis result on page 274 just after his Equation 7.7. Assume the preliminary structure function V12 is LHI (stationary phase R.V.) in developing D(1,2) on page 279. Markov approximation for acoustics for a small turbulence condition applies more easily with large range using "the fact that the spectrum and correlation are cosine transform pairs," page 283. The coherence depth and width are on page 284 and 286. The next page has an estimate of the structure function for small ray separation with respect to the depth separation z_o being calculated in that section. The scintillation index is again defined, SI = var(I), on page 387 after equation 8.121.