## Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence |

Optics Express, Vol. 20, Issue 4, pp. 4232-4247 (2012)

http://dx.doi.org/10.1364/OE.20.004232

Acrobat PDF (1006 KB)

### Abstract

By use of the generalized von Kármán spectrum model that features both inner scale and outer scale parameters for non-Kolmogorov turbulence and the extended Rytov method that incorporates a modified amplitude spatial-frequency filter function under strong-fluctuation conditions, theoretical expressions are developed for the scintillation index of a horizontally propagating plane wave and spherical wave that are valid under moderate-to-strong irradiance fluctuations. Numerical results show that the obtained expressions also compare well with previous results in weak-fluctuation regimes. Based on these general models, the impacts of finite inner and outer scales on the scintillation index of an optical wave are examined under various non-Kolmogorov fluctuation conditions.

© 2012 OSA

## 1. Introduction

1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE **4489**, 23–34 (2002). [CrossRef]

3. H. G. Sandalidis, “Performance of a laser Earth-to-satellite link over turbulence and beam wander using the modulated gamma-gamma irradiance distribution,” Appl. Opt. **50**(6), 952–961 (2011). [CrossRef] [PubMed]

4. L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

9. D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE **2120**, 43–55 (1994). [CrossRef]

11. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. **88**(1), 66–77 (2008). [CrossRef]

## 2. Generalized von Kármán spectrum

*κ*denotes the magnitude of the spatial-frequency with units of

*rad/m*,

*α*is the generalized power law that takes all values ranging from 3 to 4;

*A*(

*α*) is the generalized amplitude factor that has the form

*A*(

*α*) = Γ (

*α*– 1) · cos(

*απ*/2)/(4

*π*

^{2}) with Γ(·) being the Gamma function [24]. The factor

*m*

^{3−}

*. The remaining parameters*

^{α}*κ*and

_{m}*κ*

_{0}are basically reciprocals, respectively, of the inner scale

*l*

_{0}and outer scale

*L*

_{0}of refractive index fluctuations, that is,

*κ*=

_{m}*c*(

*α*)/

*l*

_{0}with

*c*(

*α*) = {Γ[(5 –

*α*)/2] ·

*A*(

*α*) · 2

*π*/3}

^{1/(}

^{α}^{−5)}and

*κ*

_{0}= 2

*π*/

*L*

_{0}. The inner and outer scale parameters

*l*

_{0}and

*L*

_{0}are characteristic length scales associated with the smallest and largest scale sizes, respectively, in the turbulent atmosphere that causes random fluctuations in refractive index. The inner scale is generally on the order of millimeters while the outer scale is on the order of meters. It is noted that the spectrum model Eq. (1) reduces to the generalized Kolmogorov spectrum [12, Eq. (2)] when

*l*

_{0}tends to zero and

*L*

_{0}tends to infinite, and reduces to the conventional von Kármán spectrum [6

6. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

*α*= 11/3,

*A*(

*α*) = 0.033, and

*m*

^{−2/3}.

25. R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell, M. R. Suite, and L. M. Thomas, “Analysis of long-term measurements of laser propagation over the Chesapeake Bay,” Appl. Opt. **48**(12), 2388–2400 (2009). [CrossRef] [PubMed]

4. L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

6. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

6. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

4. L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

20. B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A **28**(5), 912–916 (2011). [CrossRef]

21. L. Y. Cui, B. D. Xue, L. Cao, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express **19**(18), 16872–16884 (2011). [CrossRef] [PubMed]

## 3. The plane-wave spatial coherence radius and phase structure function

### 3.1. Spatial coherence radius

15. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE **6747**, 67470B (2007). [CrossRef]

15. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE **6747**, 67470B (2007). [CrossRef]

*k*= 2

*π*/

*λ*is the wave number at the wavelength

*λ*, and

*L*is the path length. However, for an arbitrary spectrum model, we should rely on the more general parameter

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

**11**(10), 2719–2726 (1994). [CrossRef]

*ρ*(

_{pl}*α*) is the spatial coherence radius of a plane wave [15

15. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE **6747**, 67470B (2007). [CrossRef]

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

19. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express **18**(20), 21269–21283 (2010). [CrossRef] [PubMed]

*ρ*is the separation distance between two observation points at the receiver plane;

*J*

_{0}(·) is a Bessel function of the first kind [24]. By substituting Eq. (1) into Eq. (4) and expanding the Bessel function in the integrand in a Maclaurin series, we find Then, by making an appropriate change of variable, we deduce that where we have used [24, Eq. (10.28)] to evaluate the integral in terms of the confluent hyper-geometric function of the second kind

*U*(

*a;c;x*) [24]. In the present study, it suffices to know the form of the WSF only in the asymptotic regime

*ρ*≪

*l*

_{0}. For small

*ρ*, we can approximate Eq. (6) by the first term of the series to obtain Under most conditions of atmospheric turbulence we find that

*U*(

*a;c;x*) ≅ Γ(

*c*– 1)

*x*

^{1−c}/Γ(

*a*),0 <

*x*≪ 1 [24, Eq. (10.30a)], we obtain When

*α*= 11/3, Eq. (8) reduces to

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

*ρ*, we deduce the plane-wave spatial coherence radius as It is noted that the spatial coherence radius given by Eq. (9) is consistent with conventional Kolmogorov result [4

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

*α*is set to 11/3.

### 3.2. Phase structure function

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

7. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**(6), 1417–1429 (1999). [CrossRef]

**6747**, 67470B (2007). [CrossRef]

*ρ*≪

*l*

_{0}.

*ξ*is the normalized path length parameter. For small separation distances in which

*ρ*≪

*l*

_{0}, we can use the small-argument approximation for the Bessel function (i.e.,

*J*

_{0}(

*κρ*) ≅ 1 – (

*κρ*)

^{2}/4) to obtain If we write the cosine function in Eq. (11) as cos

*x*= Re(

*e*

^{−ix}) through use of Euler’s formula, the substitution of Eq. (1) into Eq. (11) leads to where

*α*= 11/3, Eq.(15) reduces to

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

## 4. Scintillation index in the strong fluctuation regime

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

7. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**(6), 1417–1429 (1999). [CrossRef]

*et al*. [21

21. L. Y. Cui, B. D. Xue, L. Cao, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express **19**(18), 16872–16884 (2011). [CrossRef] [PubMed]

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

**6747**, 67470B (2007). [CrossRef]

*τ*is a normalized distance variable and the exponential function acts like a low-pass spatial filter defined by the plane-wave PSF

*D*(

_{S,pl}*ρ*,

*α*). The function

*w*(

*τ*,

*ξ*) is defined by [4

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

*ρ*(

_{pl}*α*) is much less than the inner scale of turbulence

*l*

_{0}, the PSF for a plane wave based on the generalized von Kármán spectrum can be calculated by Eq. (15). Hence, for the plane wave case, we find that Also, the sine function in Eq. (17) may be approximated by its leading term, which yields By substituting Eq. (1), Eq. (19), and Eq. (20) into Eq. (17), we deduce that

*U*(

*a;c;x*) [24, Eq. (10.30a)] and then obtain

*α*Γ(2 −

*α*/2)

*ρ*(

_{pl}*α*) ≪

*l*

_{0}[based on Eq. (9)], and defined

## 5. Scintillation index for plane wave

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

7. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A **16**(6), 1417–1429 (1999). [CrossRef]

**6747**, 67470B (2007). [CrossRef]

*G*(

*κ*,

*l*

_{0},

*L*

_{0},

*α*) is an amplitude (irradiance) spatial filter that consists of a large-scale spatial filter

*G*(

_{X}*κ*,

*l*

_{0},

*L*

_{0},

*α*) and a small-scale spatial filter

*G*(

_{Y}*κ*,

*α*). To include the effects from the inner scale and outer scale, we represent the large-scale and small-scale spatial filters by where

*f*(

*κ*,

*l*

_{0},

*α*) is a factor that describes inner scale modifications of the basic non-Kolmogorov power law, and

*g*(

*κL*

_{0}) describes outer scale effects. The parameter

*κ*in the large-scale filter function is a cutoff spatial frequency for the large-scale turbulent cell effects, and

_{X}*κ*in the small-scale filter function is a cutoff spatial frequency for the small-scale turbulent cell effects. In this fashion,

_{Y}*G*(

*κ*,

*l*

_{0},

*L*

_{0},

*α*) acts like a linear spatial filter that only permits low-pass spatial frequencies

*κ*<

*κ*and high-pass spatial frequencies

_{X}*κ*>

*κ*at a given propagation distance

_{Y}*L*. The effects of mid-range scale sizes which contribute little to scintillation under strong fluctuations can therefore be filtered out by using

*G*(

*κ*,

*l*

_{0},

*L*

_{0},

*α*). Note that outer-scale effects and the inner-scale factor

*f*(

*κ*,

*l*

_{0},

*α*) is assumed to act only on the large-scale fluctuations, i.e., the small-scale filter has exactly the same form as assumed for zero inner scale [15

**6747**, 67470B (2007). [CrossRef]

*κ*depends on inner scale. In our modeling of inner scale and outer scale effects, we choose

_{Y}*κ*

_{0}= 8

*π*/

*L*

_{0}for the purpose of capturing the underlying physics, but also for mathematical tractability. In this context, the large scale filter function takes the form where

*η*=

*Lκ*

^{2}/

*k*in the second step, and defined

*l*

_{0}→ 0 (i.e.,

*Q*(

_{m}*α*) → ∞), Eq. (38) reduces to which is the large-scale log-irradiance variance result for the zero scale case [15

**6747**, 67470B (2007). [CrossRef]

*η*, we use the asymptotic result [7

_{X}**16**(6), 1417–1429 (1999). [CrossRef]

21. L. Y. Cui, B. D. Xue, L. Cao, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express **19**(18), 16872–16884 (2011). [CrossRef] [PubMed]

*c*

_{1}(

*α*) can be deduced by imposing

*Laser Beam Propagation through Random Media*, 2nd ed. (SPIE Optical Engineering Press, 2005). [CrossRef]

**6747**, 67470B (2007). [CrossRef]

*η*, we impose

_{Y}**16**(6), 1417–1429 (1999). [CrossRef]

*κ*for the small-scale filter does depend on the inner scale [see (51)] and, hence, small-scale scintillation described by Eq. (52) also depends on the inner scale, particularly, in the weak fluctuation regime. Outer-scale effects are negligible here. Finally, by combining Eq. (46), Eq. (47), and Eq. (52), the SI for an infinite plane wave in the presence of a nonzero inner scale and a finite outer scale is given by

_{Y}## 6. Scintillation index for spherical wave

**19**(18), 16872–16884 (2011). [CrossRef] [PubMed]

*l*

_{0}→ 0 (i.e.,

*Q*(

_{m}*α*) → ∞), Eq. (55) reduces to Similar to the plane-wave case, we rely on Eq. (56) rather than Eq. (55) to determine

*η*. Thus, we impose the conditions

_{X}14. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express **18**(2), 451–462 (2010). [CrossRef] [PubMed]

*η*[Eq. (40)], we derive where and By substituting Eq. (58) into Eq. (55), we obtain Similarly, the large-scale scintillation due to outer scale effects is derived by The small-scale log-irradiance variance is once again given by Eq. (49), but in this case

_{X}*η*is described by Hence, the small-scale log-irradiance variance becomes If we combine Eq. (61), Eq. (62), and Eq. (64), we obtain the SI for a spherical wave in the presence of a finite inner scale and outer scale described by

_{Y}## 7. Numerical results

*λ*= 1.55

*μm*, the generalized structure parameter

*L*to vary. In this fashion, the increment of the path length corresponds to increases in turbulence strength.

*α*(solid line). We take the atmospheric conditions described by the generalized von Kármán spectrum with inner scale size of

*l*

_{0}= 3mm and outer scale

*L*

_{0}= ∞. For comparison purposes, we also plot the SI based on the asymptotic behavior Eq. (41) and Eq. (54) under weak irradiance fluctuations (dotted line) and the asymptotic theory approximation Eq. (24) and Eq. (28) for strong irradiance fluctuations (dashed line). As expected, our general scintillation models Eq. (53) and Eq. (65) track the corresponding weak and strong turbulence curves fairly closely, providing visual indications that calculations and assumptions used to develop them are correct. In addition, we note that when power law is lower than 11/3 (≈ 3.67), the SI increases initially within the regime of weak irradiance fluctuations with increasing values of the path length. It then increases beyond unity and reaches its maximum value. And after that, the SI gradually decreases, saturating at a level on the order of unity as the path length increases without bound. This general behavior is basically the same as that for conventional von Kármán spectrum (also see [8. Fig. 4 and 5]). However, as the value of the power law decreases, peak scintillation occurs at shorter and shorter path lengths. We also note that for power law values higher than 11/3, the peak scintillation phenomenon vanishes and the SI increases monotonically toward a limiting value above unity with the increment of the path length. This behavior is similar to what is observed for the zero inner scale case [15

**6747**, 67470B (2007). [CrossRef]

*U*(

*a;c;x*) that appears in Eq. (22). Such approximation has been successfully applied to develop expression for the SI in the saturation regime based on the conventional von Kármán spectrum with

*α*= 11/3 [8, Appendix H]. To justify the validity of such approximation for other values of

*α*, we evaluate the integral in Eq. (22) numerically. All of the parameters selected in numerical integration of Eq. (22) are the same as that used in the evaluation of Eq. (24) except that

*L*

_{0}= 1000m (a finite outer scale parameter is required in calculating Eq. (22), so, we use a large outer scale value to approximate an infinite outer scale.) The results of the numerical integration of Eq. (22) are also plotted in Fig. 1(a) with several values of

*α*(open circles). It can be seen that the approximation results based on Eq. (24) agree well with the numerical results of Eq. (22) for all

*α*values, indicating that the approximation that we have made is valid. From Fig. 1(b) it is clear that the approximation results based on Eq. (28) are also valid in the spherical-wave case. In addition, note in Fig. 1(a) and 1(b) that the total SI Eq. (53) and Eq. (65) match with the SI in the saturation regime Eq. (24) and Eq. (28) at different path lengths for different

*α*values. This is because the validity of Eq. (24) and Eq. (28) are determined by the conditional statement in Eq. (24) that is a function of

*α*. For example, the conditional statement in Eq. (24) is satisfied at

*L*= 2000m for

*α*= 3.07 (in this case, the conditional statement is 113≫1), but for

*α*= 3.87,

*L*need to increase to 6000m to make the conditional statement established (in this case, the conditional statement is 125≫1). The studies based on the generalized Kolmogorov spectrum [15

**6747**, 67470B (2007). [CrossRef]

*α*= 3.07 at

*L*= 2000m is equal to 9.8 which is not satisfied

*α*= 3.07 does indeed enter into the saturation regime at

*L*= 2000m.

*L*

_{0}= ∞ and inner scale values

*l*

_{0}= 1mm and

*l*

_{0}= 10mm, illustrating the effect of the inner scale alone on scintillation. The second group of curves are for the same inner-scale values but with finite outer scale value

*L*

_{0}= 1m. In Fig. 2(a) we set

*L*= 200m which corresponds to weak fluctuation conditions. In this figure, we note that with a given inner scale value, the SI curves of the finite outer scale case and the infinite outer scale case coincide regardless of the value of

*α*. This implies that the outer scale has negligible effect on scintillation under weak fluctuations, consistent with previous results in this regime [21

**19**(18), 16872–16884 (2011). [CrossRef] [PubMed]

*α*<3.2, the SI predicted for the smaller inner scale value (

*l*

_{0}= 1mm) is less than that predicted for the larger inner scale value (

*l*

_{0}= 10mm). However, the situation is opposite for power law values in the range of 3.2<

*α*<4. This is a consequence of the behavior in the generalized von Kármán spectrum as a function of inner scale and power law. In Fig. 2(b),

*L*is set to 1000m, corresponding to stronger conditions of non-Kolmogorov turbulence. Here we note that for power law values lower than 3.7 but not close to 3, the SI predicted for

*l*

_{0}= 10mm is significantly larger than that predicted for

*l*

_{0}= 1mm. We also note that the presence of a finite outer scale causes a slight drop in the SI within the range 3<

*α*<3.7 except for

*α*close to 3. In Fig. 2(c),

*L*increases to 2000m. Here we note that for either the finite outer scale or the infinite outer scale cases, the gap between two inner scale SI curves becomes narrow when compared with that depicted in Fig. 2(b). Such narrowing gap implies that the effect of inner scale on scintillation begins to diminish as turbulence strength increases. We also note that the presence of a finite outer scale leads to increased scintillation reduction in this figure than in Fig. 2(c), particularly for larger values of inner scale and power law. Such increased reduction of scintillation implies that the outer-scale effect on the SI becomes stronger as the strength of turbulence increases. In Fig. 2(d), we set

*L*= 6000m, representing an extremely strong fluctuation condition. From this figure it is clear that the inner scale has no appreciable effect on scintillation when outer scale is finite. Although there exists a gap between two inner scale SI curves in the infinite outer scale case, it becomes narrower in comparison with Fig. 2(c), also implying that the effects of inner scale on scintillation weaken with increasing strength of turbulence. On the other hand, for a given inner scale, the gap between the finite outer-scale curve and the infinite outer-scale curve increases apparently, especially for larger values of

*α*. This indicates that the outer-scale effects cannot be ignored for larger power law values under extremely strong fluctuation conditions.

*L*, with inner scale values of

*l*

_{0}= 3 and 8mm and outer scale

*L*

_{0}= ∞ (solid lines). The dashed lines are comparable scintillation values that arise with all parameters the same except

*L*

_{0}= 1m. We set

*α*= 3.07, 3.37, 3.67, and 3.87 in the Fig. 3(a), 3(b), 3(c), and 3(d), respectively. It is observed that for an arbitrary value of

*α*, the outer scale has a negligible effect on the SI of a spherical wave under weak fluctuations associated with short path lengths. However, with an increase in path lengths, the presence of a finite outer scale in the scintillation model is quite clear. Namely, the outer-scale effect initially reduces scintillation at a steeper rate toward its limiting value of unity than would occur with an infinite outer scale. Also, for an arbitrary value of

*α*, the influence of inner scale on scintillation is not significant under weak fluctuation conditions. Under moderate-to-strong fluctuations, the SI predicted for the smaller inner scale value (

*l*

_{0}= 3mm) is always less than that predicted for the larger inner scale value (

*l*

_{0}= 8mm). Specifically, for power law values lower than 11/3 (viz.,

*α*= 3.07, 3.37), the increment of inner scale significantly increases the SI near its peak value. As the path length increases, the gap between the larger and smaller inner scale SI curves becomes narrower and narrower, suggesting that inner-scale effect on scintillation tends to diminish with the increment of turbulence strength. For power law values higher than 11/3 (viz.,

*α*= 3.87), the predicted SI over the whole moderate-to-strong fluctuation regimes is only slightly higher for a large inner-scale value than for a small inner-scale value.

## 8. Conclusion

## Acknowledgments

## References and links

1. | L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE |

2. | A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. |

3. | H. G. Sandalidis, “Performance of a laser Earth-to-satellite link over turbulence and beam wander using the modulated gamma-gamma irradiance distribution,” Appl. Opt. |

4. | L. C. Andrews and R. L. Phillips, |

5. | W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A |

6. | W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A |

7. | L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A |

8. | K. J. Mayer, Effect of Inner Scale Atmospheric Spectrum Models on Scintillation in All Optical Turbulence Regimes, Ph.D. dissertation, University of Central Florida, 2007. |

9. | D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE |

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11. | A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. |

12. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE |

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14. | L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express |

15. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE |

16. | P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication. |

17. | J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. |

18. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE |

19. | L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express |

20. | B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A |

21. | L. Y. Cui, B. D. Xue, L. Cao, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express |

22. | J. Cang and X. Liu, “Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model,” Opt. Express |

23. | B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE |

24. | L. C. Andrews, |

25. | R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell, M. R. Suite, and L. M. Thomas, “Analysis of long-term measurements of laser propagation over the Chesapeake Bay,” Appl. Opt. |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: December 20, 2011

Revised Manuscript: January 29, 2012

Manuscript Accepted: January 29, 2012

Published: February 6, 2012

**Citation**

Xiang Yi, Zengji Liu, and Peng Yue, "Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence," Opt. Express **20**, 4232-4247 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4232

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### References

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