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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 10 — May. 12, 2008
  • pp: 6963–6973
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Measurement of optical intensity fluctuation over an 11.8 km turbulent path

Yijun Jiang, Jing Ma, Liying Tan, Siyuan Yu, and Wenhe Du  »View Author Affiliations


Optics Express, Vol. 16, Issue 10, pp. 6963-6973 (2008)
http://dx.doi.org/10.1364/OE.16.006963


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Abstract

An 11.8km optical link is established to examine the intensity fluctuation of the laser beam transmission through atmosphere turbulence. Probability density function, fade statistic, and high-frequency spectrum are researched based on the analysis of the experimental data collected in each season of a year, including both weak and strong fluctuation cases. Finally, the daily variation curve of scintillation index is given, compared with the variation of refractive-index structure parameter C2 n , which is calculated from the experimental data of angle of arrival. This work provides the experimental results that are helpful to the atmospheric propagation research and the free-space optical communication system design.

© 2008 Optical Society of America

1. Introduction

Atmospheric turbulence has an important effect on the light waves. The refractive index fluctuations can affect the speed of wave-fronts propagating and produce fluctuations in amplitude and phase of a laser beam propagating through it. Atmospheric refraction of the light energy can cause random temporal and spatial distribution of the light intensity [1–2

1. D. L. Fried, G. E. Mevers, and M. P. Keister, “Measurements of laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 787–797 (1967). [CrossRef]

]. These effects result in optical intensity fluctuations at the receiving plane and severely influence the performance of free-space optical communication systems.

According to Kolmogorov theory [3

3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

], if the atmosphere turbulence is homogeneous and isotropic, for weak fluctuation, the probability distribution of received intensity should be lognormal. For strong fluctuation far into the saturation regime, probability distribution is assumed to be governed by the negative exponential distribution. Besides, a model was established by Andrews to describe the behavior of light intensity from weak to strong fluctuation [4

4. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash “Theory of optical scintillation,” J. Opt. Soc. Am. 16, 1417–1429 (1974). [CrossRef]

, 5

5. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, Bellingham, 2001). [CrossRef]

].

Experimental observations have been conducted to evaluate turbulence-introduced intensity scintillation and its effects on free-space optical communications [6–15

6. Y. Han Oh, J. C. Ricklin, E. S. Oh, and F. D. Eaton, “Evaluating optical turbulence effects on free-space laser communication: modeling and measurements at ARL’s A_LOT facility,” Proc. SPIE 5550, 247–255 (2004). [CrossRef]

]. The Kolmogorov theory has been certified by many of them, but there are still some experiments that show disagreements with it, especially on high-frequency power spectrum [8

8. A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

, 16–18

16. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20, 1359–1361 (1995). [CrossRef] [PubMed]

]. It is not clear what is lying behind the differences. Turbulence intensity, wavelength, temperature gradients, moisture, all these factors can make the results different.

Fig. 1. An aerial photo of the optical path (data from maps. google.com)

Landscape along the path is complex, mainly including a river and several roads and building arrays. Non-uniform landscape makes the microclimate along the optical path complicated and then affects the measured data. Besides, vibration of the buildings and temperature gradients can also influence the experimental results.

Noting the similarity of our work to the experiment mentioned in Refs. [7

7. A. Tunick, “Statistical analysis of optical turbulence intensity over a 2.33 km propagation path,” Opt. Express. 15, 3619–3628 (2007). [CrossRef] [PubMed]

, 8

8. A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

], many statements and data analysis methods in this paper are in similar format to them for easy comparison of the results of two experiments, especially in the analysis processes of probability density function and high-frequency spectrum (section 3.1 and 3.3).

2. Experimental setup

The experimental setup is shown in Fig. 2. The transmitter is a frequency doubled Nd: YAG laser with the wavelength of 532nm and the maximum output power of 100mW. Full power is transmitted only in some winter days because the smoking caused by city heating system attenuates the light power severely. The original divergence angle of the laser transmitter is 1.3mrad and a beam expander (4mm input diameter and 24mm output diameter) is used to compress the beam width to 250µrad. The laser source and the beam expander are mounted on a 3-dimensional (700mm×450mm) adjustable optics table.

Beam diameter at the receiving plane is about 3–4m, but is hard to determine exactly because of the effects of beam wander and beam spread. A Cassegrain telescope (diameter, D=127mm; focal length, f=1900mm) is placed on the top of the Electronic Information Centre to collect the input light power.

Fig. 2. Configuration of the experimental setup

The output of the Cassegrain telescope is divided into two beams by a light splitter. One of the two beams is focused on a Pin photo-detector by a lens (f=78mm) to measure the intensity scintillation, and the other is focused on a CCD camera by its own lens (f=50mm) to measure the fluctuation of angle of arrival (AOA). Either the Pin or the CCD has a 532nm wave filter in front of it to mitigate the influence of the background light. In addition, a light attenuator is used to prevent power saturation of the CCD camera, depending on the received laser power. These devices are mounted on the other 3-dimensional adjustable optics table (Fig. 3). Output signal of the Pin is sampled at the rates of 3000Hz or 5000Hz every 30 seconds by an analog/data (AD) card inserted in a computer.

A temperature and humidity recorder is used to record the temperature and humidity every 3 minutes and a wind velocity recorder is also used to measure wind speed at the receiving side of the link.

Fig. 3. Photograph of the receiving optics and the 3-dimensional adjustable table

3. Data analysis

The experiment was performed from October 2006 to the end of the year of 2007, lasting more than one year. Table 1 summarizes the date and climate conditions for an initial data set consisting of four weak fluctuation (σ 2 I<1) and two strong fluctuation (σ 2 I>1) cases. Considering the dynamic range (60dB) of the AD card, data selection is mainly based on whether both high and low irradiance part of the intensity fluctuation can be included.

Table 1. Climate characteristics for the experimental data set

table-icon
View This Table

3.1 Probability density function

Usually, scintillation index is used to indicate the strength of the intensity fluctuation. It is defined as the normalized variance of the intensity fluctuations, which can be expressed as

σI2=I2I21,
(1)

where I is light intensity and <·> means ensemble average. For weak fluctuations, scintillation index is equal to the Rytov variance which can be calculated as

σ12=1.23Cn2k76L116,
(2)

pw(I)=12πσI21Iexp[(lnI+σI22)22σI2],
(3)

For strong fluctuation far into the saturation regime, PDF is assumed to be governed by the negative exponential distribution [19

19. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001). [CrossRef]

],

ps(I)=1Iexp[II].
(4)

Fig. 4. Histograms of normalized light intensity and lognormal fitting curves (red lines) for trials T1(a), T2(b), T3(c) and T4(d).

R=Y·yY·yDY·Dy,
(5)

where DY and Dy are variances of Y and y, respectively.

Fig. 5. R 2 versus σ 2 I for weak fluctuation trials

Fig. 6. Histograms of normalized light intensity and negative exponential fitting curves (red lines) for trials T5(a) and T6(b).

Other data trials of strong fluctuation are analyzed too. We found that R 2 slightly falls down when σ 2 I rises (Fig. 7). It seems that in strong fluctuation region, the negative exponential approximation is less effective to larger σ 2 I trials than to smaller σ 2 I trials. However, R 2 is still above the level of 0.98 even though σ 2 I is greater than 3, so the negative exponential distribution is still a nice approximation for strong fluctuation trials.

Fig. 7. R 2 versus σ 2 I for strong fluctuation trials

3.2 Probability of fade

The probability of fade is defined by the probability that the received light intensity is below some given threshold IT. It can be calculated as [3

3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

]

P(IIT)=0ITp(I)dI,
(6)

where p(I) is PDF of the intensity. The bit-error-rate (BER) of an on-off-keyed communication system can be given by multiplying the probability of fade with a factor of 0.5. It is customary to express IT in decibels, which can be described as the fade margin

FT=10lg(IIT).
(7)
Fig. 8. Fade statistics for trials T2, T4 and T5.

The probability of fade for T2, T4 and T5 are presented in Fig. 8. It can be seen in Fig. 8 that the probability of fade falls down rapidly with the increasing of FT. Besides, the required FT for a given BER is larger for the trial that has a greater σ2I and the BER also rises with the increasing of σ 2 I for a given FT. To achieve a BER of 10-4, about 13dB fade margin is required for T2 (σ 2 I=0.4991), 20dB is required for T4 (σ 2 I=0.9074), and more than 35dB is required for T5 (σ 2 I=1.2249). To achieve a BER of 10-5, the required fade margin will be 15dB, 23dB and more than 40dB for T2, T4 and T5, respectively.

3.3 High-frequency spectrum

Usually, it is accepted that the high-frequency spectrum has a -8/3 power-law dependence [3

3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

], but there are also some studies that indicate the high-frequency spectrum to be a -11/3, -14/3, or -17/3 power-law dependence [8

8. A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

, 20

20. R. Rao, S. Wang, X. Liu, and Z. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A 16, 2755–2762 (1999). [CrossRef]

, 21

21. R. Rao and Z. Gong, “High-frequency behavior of the temporal spectrum of laser beam propagating through turbulence,” Proc. SPIE 4926, 175–180 (2002). [CrossRef]

].

Fig. 9. Power spectra for trials T1(a), T2(b), T3(c) and T5(d). Power exponents of high-frequency spectra are annotated on each graph

Power spectra for trials T1, T2, T3 and T5 are plotted in Fig. 9. High-frequency spectra of all these trials have the negative exponential power law dependence, but the power exponents of them are different. The power exponents of high-frequency spectra for T1, T2, T3 and T5 are -11/3, -16/3, -17/3 and -16/3, respectively. It was mentioned that the measured power exponent was -17/3 in reference [8

8. A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

], so it seems that our result is more complicated than the result in reference [8

8. A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

] and any other reports mentioned above.

Figure 9 shows only four power spectra. In fact, more data trials are analyzed for further research of the high-frequency spectrum, totally containing 24 weak fluctuation cases and 11 strong fluctuation cases. Power spectrum is made and power exponent of high-frequency spectrum is calculated for each of these data trials. These power exponents (α) are plotted in Fig. 10. As it is shown in Fig. 10, power exponent varies in a broad range from -8/3 to -21/3. The values of -8/3 and -11/3 can only be observed when σ 2 I is smaller than 0.4. For weak fluctuation trials, α is in the interval of -21/3 to -13/3 mostly. For strong fluctuation trials, α is in the interval of -21/3 to -15/3. Mean values are -5.48 and -5.78 for weak fluctuation and strong fluctuation trials, respectively, close to the value of -17/3. It is hard to tell what makes the high-frequency power-law complicated. Beam parameter [21

21. R. Rao and Z. Gong, “High-frequency behavior of the temporal spectrum of laser beam propagating through turbulence,” Proc. SPIE 4926, 175–180 (2002). [CrossRef]

], validity of the Kolmogrov model [22

22. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]

], non-uniform wind along the path [23

23. Y. Wang, C. Fan, X. Wu, J. Zhan, and Z. Gong, “Effects of non-uniform wind on the arrival angle temporal spectrum of spherical wave,” Proc. SPIE 4125, 98–101 (2000). [CrossRef]

], Turbulence scale and boundary [20

20. R. Rao, S. Wang, X. Liu, and Z. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A 16, 2755–2762 (1999). [CrossRef]

, 24

24. E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43, 6151–6156 (2004). [CrossRef] [PubMed]

], all these factors can affect the experimental results.

Fig. 10. Power exponents of high-frequency spectrum versus σ 2 I

3.4 Daily variation of σ2I and C2n

For a Gaussian-beam wave, the variance of AOA can be expressed as [3

3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

]

σβ2=1.093Cn2LD13[a+0.618Λ116(kD2L)13],
(8)

a=1Θ831Θ,
Θ=1LF,
Λ=2LkW2,
(9)

F (FL, if L is long enough) and W denote the receiver plane phase front radius of curvature and beam radius, respectively. In our experiment, k and L are on the order of 107 and 104, respectively, W is about 3–4m, and D=0.127cm. Based on these parameters, the approximation of FL, a≈1 and Λ≈0 can be made, and Eq. (8) can be simplified to

σβ21.093Cn2LD13,
(10)

So C 2 n values can be calculated from the experimental data of AOA by

Cn2σβ21.093LD13.
(11)

A CCD was used to measure the variance of AOA. A computer was used to calculate the centroid coordinate of the CCD image and record it at a rate of 1000Hz. The variance of AOA can be calculated from the variance of the centroid coordinate, and then C 2 n can be calculated by Eq. (11).

The 10 minutes average values of σ 2 I and C 2 n are presented in Fig. 11 and Fig. 12, respectively, which were measured in a complete diurnal period of 27 March 2007. In Fig. 11 and Fig. 12, both σ 2 I and C 2 n have greater values in the daytime than at night. In the daytime, two curves have similar profiles. Both of them increase before midday, maximize at noon and then decline in the afternoon. But at night, C 2 n appears more stable than σ 2 I. The C 2 n curve of Fig. 12 is similar to the measurement results in reference [7

7. A. Tunick, “Statistical analysis of optical turbulence intensity over a 2.33 km propagation path,” Opt. Express. 15, 3619–3628 (2007). [CrossRef] [PubMed]

].

Fig. 11. Daily variation of scintillation index
Fig. 12. Daily variation of C 2 n

We consider that the night deviation of two curves is mainly caused by misalignment of the laser beam. In fact, the scintillation index of a Gaussian beam can be expressed by

σI2=σ02+4.42σ12Λ56r2W2
(12)

where σ 2 0 is the on-axis scintillation index and r is the distance between the receiving point and the beam center. Equation (12) indicates that σ 2 I increases with square of distance transverse to the optical axis [25

25. 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, 2719–2726 (1994). [CrossRef]

], so if the light is not central received, the measured σ 2 I will be larger, but the measured C 2 n will not be different because it is independent of r. In our experiment, the beam direction will slowly drift with time, especially at night, which means r increases with time. This can explain why two figures are inconsistent with each other in some period of time.

There are multiple potential causes of the beam drift including thermal deformation of the building and bending of the optical line-of-sight induced by large scale vertical dependencies in the atmospheric refractive index.

4. Summary and conclusions

Acknowledgments

This work is supported by the program of excellent team in Harbin institute of technology.

References and links

1.

D. L. Fried, G. E. Mevers, and M. P. Keister, “Measurements of laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 787–797 (1967). [CrossRef]

2.

T. Chiba, “Spot dancing of the laser beam propagated through the turbulent atmosphere,” Appl. Opt. 10, 2456–2461 (1971). [CrossRef] [PubMed]

3.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

4.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash “Theory of optical scintillation,” J. Opt. Soc. Am. 16, 1417–1429 (1974). [CrossRef]

5.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, Bellingham, 2001). [CrossRef]

6.

Y. Han Oh, J. C. Ricklin, E. S. Oh, and F. D. Eaton, “Evaluating optical turbulence effects on free-space laser communication: modeling and measurements at ARL’s A_LOT facility,” Proc. SPIE 5550, 247–255 (2004). [CrossRef]

7.

A. Tunick, “Statistical analysis of optical turbulence intensity over a 2.33 km propagation path,” Opt. Express. 15, 3619–3628 (2007). [CrossRef] [PubMed]

8.

A. Tunick, “Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path,” Opt. Express. 15, 14115–14122 (2007). [CrossRef] [PubMed]

9.

I. I. Kim, B. McArthur, and E. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001). [CrossRef]

10.

Y. E. Yenicea, R. Li, M. Takabeb, and T. Arugab, “Atmospheric turbulence measurements through stellar observations,” Proc. SPIE 3615, 316–324 (1999). [CrossRef]

11.

D. Romain, M. Larkin, G. Ghayal, B. Paulson, and G. Nykolak, “Optical wireless propagation theory vs. experiment,” Proc. SPIE 4214, 38–45 (2001). [CrossRef]

12.

A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, “Comparison between experimental and theoretical probability of fade for free space optical communications,” Proc. SPIE 4873, 79–89 (2002). [CrossRef]

13.

W. Brown, B. Wallin, D. Lesniewski, D. Gooding, and J. Martin, “The experimental determination of on-off keying laser communications probability models and a comparison with theory,” Proc. SPIE 6105, 61050U (2006). [CrossRef]

14.

M. J. Curley, B. H. Peterson, J. C. Wang, S. S. Sarkisov, S. S. Sarkisov II, G. R. Edlin, R. A. Snow, and J. F. Rushing, “Statistical analysis of cloud-cover mitigation of optical turbulence in the boundary layer,” Opt. Express. 14, 8929–8946 (2006). [CrossRef] [PubMed]

15.

H. Yuksel and C. C. Davis, “Aperture averaging for studies of atmospheric turbulence and optimization of free space optical communication links” Proc. SPIE 5892, 58920P (2005). [CrossRef]

16.

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20, 1359–1361 (1995). [CrossRef] [PubMed]

17.

C. Rao, W. Jiang, and N. Ling, “Atmospheric parameters measurements for non-Kolmogorov turbulence with Shack-Hartmann wavefront sensor,” Proc. SPIE 3763, 84–91 (2006). [CrossRef]

18.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995). [CrossRef]

19.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001). [CrossRef]

20.

R. Rao, S. Wang, X. Liu, and Z. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A 16, 2755–2762 (1999). [CrossRef]

21.

R. Rao and Z. Gong, “High-frequency behavior of the temporal spectrum of laser beam propagating through turbulence,” Proc. SPIE 4926, 175–180 (2002). [CrossRef]

22.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]

23.

Y. Wang, C. Fan, X. Wu, J. Zhan, and Z. Gong, “Effects of non-uniform wind on the arrival angle temporal spectrum of spherical wave,” Proc. SPIE 4125, 98–101 (2000). [CrossRef]

24.

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43, 6151–6156 (2004). [CrossRef] [PubMed]

25.

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, 2719–2726 (1994). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.3310) Atmospheric and oceanic optics : Laser beam transmission

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: January 30, 2008
Revised Manuscript: February 26, 2008
Manuscript Accepted: April 11, 2008
Published: May 1, 2008

Citation
Yijun Jiang, Jing Ma, Liying Tan, Siyuan Yu, and Wenhe Du, "Measurement of optical intensity fluctuation over an 11.8 km turbulent path," Opt. Express 16, 6963-6973 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6963


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References

  1. D. L. Fried, G. E. Mevers, and M. P. Keister, "Measurements of laser beam scintillation in the atmosphere," J. Opt. Soc. Am. 57, 787-797 (1967). [CrossRef]
  2. T. Chiba, "Spot dancing of the laser beam propagated through the turbulent atmosphere," Appl. Opt. 10, 2456-2461 (1971). [CrossRef] [PubMed]
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).
  4. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, "Theory of optical scintillation," J. Opt. Soc. Am. 16, 1417-1429 (1974). [CrossRef]
  5. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Optical Engineering Press, Bellingham, 2001). [CrossRef]
  6. Y. Han Oh, J. C. Ricklin, E. S. Oh, and F. D. Eaton, "Evaluating optical turbulence effects on free-space laser communication: modeling and measurements at ARL�??s A_LOT facility," Proc. SPIE 5550, 247-255 (2004). [CrossRef]
  7. A. Tunick, "Statistical analysis of optical turbulence intensity over a 2.33 km propagation path," Opt. Express 15, 3619-3628 (2007). [CrossRef] [PubMed]
  8. A. Tunick, "Statistical analysis of measured free-space laser signal intensity over a 2.33 km propagation path," Opt. Express 15, 14115-14122 (2007). [CrossRef] [PubMed]
  9. I. I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications," Proc. SPIE 4214, 26-37 (2001). [CrossRef]
  10. Y. E. Yenicea, R. Li, M. Takabeb, and T. Arugab, "Atmospheric turbulence measurements through stellar observations," Proc. SPIE 3615, 316-324 (1999). [CrossRef]
  11. D. Romain, M. Larkin, G. Ghayal, B. Paulson, and G. Nykolak, "Optical wireless propagation theory vs. experiment," Proc. SPIE 4214, 38-45 (2001). [CrossRef]
  12. A. Al-habash, K. W. Fischer, C. S. Cornish, K. N. Desmet, and J. Nash, "Comparison between experimental and theoretical probability of fade for free space optical communications," Proc. SPIE 4873, 79-89 (2002). [CrossRef]
  13. W. Brown, B. Wallin, D. Lesniewski, D. Gooding, and J. Martin, "The experimental determination of on-off keying laser communications probability models and a comparison with theory," Proc. SPIE 6105, 61050U (2006). [CrossRef]
  14. M. J. Curley, B. H. Peterson, J. C. Wang, S. S. Sarkisov, S. S. SarkisovII, G. R. Edlin, R. A. Snow, and J. F. Rushing, "Statistical analysis of cloud-cover mitigation of optical turbulence in the boundary layer," Opt. Express. 14, 8929-8946 (2006). [CrossRef] [PubMed]
  15. H. Yuksel and C. C. Davis, "Aperture averaging for studies of atmospheric turbulence and optimization of free space optical communication links" Proc. SPIE 5892, 58920P (2005). [CrossRef]
  16. M. S. Belen'kii, "Effect of the stratosphere on star image motion," Opt. Lett. 20, 1359-1361 (1995). [CrossRef] [PubMed]
  17. C. Rao, W. Jiang, and N. Ling, "Atmospheric parameters measurements for non-Kolmogorov turbulence with Shack-Hartmann wavefront sensor," Proc. SPIE 3763, 84-91 (2006). [CrossRef]
  18. R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 6-16 (1995). [CrossRef]
  19. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, "Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media," Opt. Eng. 40, 1554-1562 (2001). [CrossRef]
  20. R. Rao, S. Wang, X. Liu, and Z. Gong, "Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere," J. Opt. Soc. Am. A 16, 2755-2762 (1999). [CrossRef]
  21. R. Rao and Z. Gong, "High-frequency behavior of the temporal spectrum of laser beam propagating through turbulence," Proc. SPIE 4926, 175-180 (2002). [CrossRef]
  22. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995). [CrossRef]
  23. Y. Wang, C. Fan, X. Wu, J. Zhan, and Z. Gong, "Effects of non-uniform wind on the arrival angle temporal spectrum of spherical wave," Proc. SPIE 4125, 98-101 (2000). [CrossRef]
  24. E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different turbulent fields," Appl. Opt. 43, 6151-6156 (2004). [CrossRef] [PubMed]
  25. 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, 2719-2726 (1994). [CrossRef]

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