University of Central Florida STARS
Electronic Theses and Dissertations, 2004-2019
2010
The Pdf Of Irradiance For A Free-space Optical Communications Channel: A Physics Based Model
David Wayne University of Central Florida
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STARS Citation Wayne, David, "The Pdf Of Irradiance For A Free-space Optical Communications Channel: A Physics Based Model" (2010). Electronic Theses and Dissertations, 2004-2019. 4300. https://stars.library.ucf.edu/etd/4300 THE PDF OF IRRADIANCE FOR A FREE-SPACE OPTICAL COMMUNICATIONS CHANNEL: A PHYSICS BASED MODEL
by
DAVID T. WAYNE B.S.E.E. Grand Valley State University, 2004 M.S.E.E. University Central Florida, 2006 M.S. University Central Florida, 2009
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Electrical Engineering and Computer Science in the College of Engineering and Computer Science at the University of Central Florida Orlando, FL
Summer Term 2010
Major Professor: Ronald L. Phillips
ABSTRACT
An accurate PDF of irradiance for a FSO channel is important when designing a laser radar,
active laser imaging, or a communications system to operate over the channel. Parameters such
as detector threshold level, probability of detection, mean fade time, number of fades, BER, and
SNR are derived from the PDF and determine the design constraints of the receiver, transmitter,
and corresponding electronics. Current PDF models of irradiance, such as the Gamma-Gamma,
do not fully capture the effect of aperture averaging; a reduction in scintillation as the diameter
of the collecting optic is increased. The Gamma-Gamma PDF of irradiance is an attractive
solution because the parameters of the distribution are derived strictly from atmospheric
2 turbulence parameters; propagation path length, Cn , l0, and L0. This dissertation describes a
heuristic physics-based modeling technique to develop a new PDF of irradiance based upon the
optical field. The goal of the new PDF is three-fold: capture the physics of the turbulent atmosphere, better describe aperture averaging effects, and relate parameters of the new model to measurable atmospheric parameters.
The modeling decomposes the propagating electromagnetic field into a sum of independent random-amplitude spatial plane waves using an approximation to the Karhunen-Loeve expansion. The scattering effects of the turbulence along the propagation path define the random-amplitude of each component of the expansion. The resulting PDF of irradiance is a double finite sum containing a K0 Bessel function. The newly developed PDF is a
generalization of the Gamma-Gamma PDF, and reduces to such in the limit.
ii An experiment was setup and performed to measure the PDF of irradiance for several receiver aperture sizes under moderate to strong turbulence conditions. The propagation path was instrumented with scintillometers and anemometers to characterize the turbulence conditions.
The newly developed PDF model and the GG model were compared to histograms of the experimental data. The new PDF model was typically able to match the data as well or better than the GG model under conditions of moderate aperture averaging. The GG model fit the data better than the new PDF under conditions of significant aperture averaging. Due to a limiting scintillation index value of 3, the new PDF was not compared to the GG for point apertures under strong turbulence; a regime where the GG is known to fit data well.
iii ACKNOWLEDGMENTS
I would like to thank Dr. Ronald Phillips and Dr. Larry Andrews for their time, commitment, and dedication to my education. I would like to thank Dr. John Stryjewski for his time, conversation, and direction regarding the practical aspects of experimentation and project management. I would like to thank Mr. Brad Griffis for his help, instruction, and guidance with the experimental setup. I would like to thank the researchers at Harris Corporation: Dr. Geoff Burdge and Mr.
Michael Borbath for their support and use of equipment and Mr. Robert Peach for his assistance in the early phases of data collection. Finally, I am indebted to Jamie for her endless patience, encouragement, and support.
iv TABLE OF CONTENTS
LIST OF FIGURES ...... ix
LIST OF TABLES ...... xiii
LIST OF SYMBOLS ...... xiv
1 INTRODUCTION ...... 1
2 BACKGROUND ...... 5
2.1 Random Processes ...... 5
2.1.1 Stationarity, homogeneity, isotropy, and ergodicity ...... 5
2.1.1.1 Testing for stationarity ...... 7
2.1.2 Probability density function ...... 9
2.1.2.1 Histogram ...... 13
2.1.3 Structure function...... 16
2.1.4 Averaging time ...... 18
3 ATMOSPHERIC TURBULENCE ...... 19
3.1 Solution Techniques for the Stochastic Helmholtz Equation ...... 19
2 3.2 Refractive Index Structure Parameter, Cn ...... 21
3.3 Power Spectrum Models ...... 24
3.4 Turbulence Generation ...... 28
3.4.1 Birth of an eddy ...... 32
2 3.5 Thermal Plumes and Near Ground Effects (Near-Ground Cn Fluctuations) ...... 33
3.6 Taylor’s Frozen Turbulence Hypothesis ...... 35
3.7 Scale Sizes of Optical Turbulence ...... 37
v 3.8 Stochastic Field Properties ...... 39
3.8.1 Second-order statistics ...... 40
3.8.2 Fourth-order statistics ...... 42
3.8.3 Coherence function of a stochastic field ...... 44
3.8.4 Karhunen-Loeve expansion ...... 45
3.9 Effects of Turbulence on Optical Waves ...... 49
3.9.1 Scintillation ...... 49
3.9.2 Aperture averaging ...... 56
4 PREVIOUS & CURRENT PDF MODELS ...... 60
4.1 Nakagami m-Distribution ...... 61
4.2 Lognormal Distribution ...... 64
4.3 K Distribution ...... 66
4.4 Universal Distribution ...... 69
4.5 I-K Distribution ...... 71
4.6 Generalized K-Distribution ...... 74
4.7 Lognormally Modulated Exponential ...... 76
4.8 Lognormally Modulated Rician ...... 78
4.9 Gamma-Gamma Distribution ...... 80
5 PHYSICAL MODELING ...... 86
5.1 Overview of Model Development ...... 87
5.2 Decomposing the Complex Envelope of a Stochastic Field ...... 89
5.3 Angular Spectrum ...... 92
5.4 2-D Fourier Transform of the Mutual Coherence Function ...... 95
vi 5.4.1 Relating angular spectrum and spatial spectrum ...... 98
5.5 Wavenumber PDF ...... 100
5.6 Probability Density Function of Received Irradiance ...... 102
5.7 Statistical Moments of Irradiance and Scintillation Index ...... 109
5.7.1 First moment of irradiance ...... 109
5.7.2 Second moment of irradiance ...... 112
5.7.3 Scintillation index ...... 114
5.8 Relating Model Parameters to Physical Quantities ...... 116
5.8.1 Sampling the spatial spectrum ...... 117
5.8.2 Sampling the wavenumber PDF ...... 120
6 EXPERIMENTATION ...... 122
6.1 Overview ...... 122
6.2 Equipment ...... 125
2 6.2.1 Scintillometer – path averaged Cn ...... 125
2 6.2.2 Scintillometer – path averaged Cn and l0 ...... 130
6.2.3 Transmitter system ...... 132
6.2.4 Receiver system ...... 133
6.3 Data ...... 135
7 DISCUSSION ...... 152
8 CONCLUSION ...... 159
APPENDIX A SCINTILLATON INDEX OF A SPHERICAL WAVE INCLUDING
APERTURE AVERAGING ...... 163
APPENDIX B HANKEL TRANSFORM OF MCF: b = 5/3 ...... 166
vii APPENDIX C CHARACTERISTIC FUNCTION ...... 172
APPENDIX D INTERPRETING POWER METER OUTPUT ...... 178
APPENDIX E NUMERICAL INACCURACIES OF NEW PDF AT LOW IRRADIANCE .. 182
LIST OF REFERENCES ...... 187
viii LIST OF FIGURES
2 Figure 1: 24-hour Cn profile of data taken over a 1km terrestrial path during the summer in
Florida...... 24
Figure 2: Eddy formation and breakdown...... 30
Figure 3: Power spectrum of turbulence fluctuations indicating important regions...... 31
Figure 4: Images of thermal plumes rising from a heated surface in a fluid [24]. Mechanical and inertial forces break the plumes apart as they rise...... 34
Figure 5: Relevant turbulent eddy scale sizes for an unbounded plane wave withλ = 532nm and
2 −− 3213 Cn ×= 105 m . The shaded region denotes scale sizes not contributing to scintillation under strong fluctuations...... 39
Figure 6: Effects of inner scale on the scintillation index [7]...... 52
Figure 7: Effects of inner and outer scale on the scintillation index [7]...... 52
Figure 8: Aperture averaging effects on the scintillation index of a spherical wave as a function of aperture size: moderate turbulence...... 57
Figure 9: Aperture averaging effects on the scintillation index of a spherical wave as a function of aperture size: strong turbulence...... 58
Figure 10: Flowchart of optical field analysis...... 88
Figure 11: Sketch of turbulent eddies refracting and scattering the optical field...... 89
Figure 12: Describing the wave vector with direction cosines [73]...... 94
Figure 13: Method of obtaining eigenvalues from sampling spatial spectrum...... 119
Figure 14: Method of calculating gammas from integrating the wavenumber PDF; N=20...... 121
ix Figure 15: Layout of equipment on 1km laser range. Three scintillometers were used to provide
2 path-averaged Cn and l0. Three 3-axis anemometers provided the crosswind at points along the path...... 124
Figure 16: Optical layout of transmitter system...... 133
Figure 17: Optical layout of receiver system...... 134
Figure 18: Receiving telescope with aperture in place...... 135
2 -14 Figure 19: New PDF, GG PDF, and histogram of data; 1.5mm aperture, Cn = 5.00 x10 , ρsp=
7.16mm...... 139
2 -14 Figure 20: New PDF, GG PDF, and histogram of data; 4.0mm aperture, Cn = 8.50 x10 , ρsp=
5.21mm...... 140
2 -14 Figure 21: New PDF, GG PDF, and histogram of data; 4.0mm aperture, Cn = 9.00 x10 , ρsp=
5.03mm...... 140
2 -14 Figure 22: New PDF, GG PDF, and histogram of data; 7.0mm aperture, Cn = 7.50 x10 , ρsp=
5.61mm...... 141
2 -13 Figure 23: New PDF, GG PDF, and histogram of data; 10.0mm aperture, Cn = 3.56 x10 , ρsp=
2.21mm...... 141
2 -14 Figure 24: New PDF, GG PDF, and histogram of data; 10mm aperture, Cn = 5.50 x10 , ρsp=
6.76mm...... 142
2 -14 Figure 25: New PDF, GG PDF, and histogram of data; 10mm aperture, Cn = 7.27 x10 , ρsp=
5.72mm...... 142
2 -13 Figure 26: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 3.70 x10 , ρsp=
2.15mm...... 143
x 2 -14 Figure 27: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 8.89 x10 , ρsp=
5.07mm...... 143
2 -14 Figure 28: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 7.00 x10 , ρsp=
5.85mm...... 144
2 -13 Figure 29: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 2.65 x10 , ρsp=
2.63mm...... 144
2 -13 Figure 30: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 1.30 x10 , ρsp=
4.04mm...... 145
2 -13 Figure 31: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 1.05 x10 , ρsp=
4.59mm...... 145
2 -13 Figure 32: New PDF, GG PDF, and histogram of data; 101.6mm aperture, Cn = 1.40 x10 , ρsp=
3.86mm...... 146
2 -13 Figure 33: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 3.47 x10 , ρsp=
2.24mm...... 146
2 -13 Figure 34: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 2.36 x10 , ρsp=
2.83mm...... 147
2 -13 Figure 35: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 1.65 x10 , ρsp=
3.50mm...... 147
2 -14 Figure 36: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 6.50 x10 , ρsp=
6.12mm...... 148
2 -13 Figure 37: Outer scale effects on the shape and scintillation index of the new PDF; Cn = 1 x10 l0 = 5mm...... 151
Figure 38: Low irradiance oscillation in PDF: M=30, N=60 ...... 185
xi Figure 39: Low irradiance oscillation in PDF: M=10, N=60 ...... 185
Figure 40: Low irradiance oscillation in PDF: M=5, N=60 ...... 186
xii LIST OF TABLES
Table 1: Atmospheric conditions observed during data collection...... 138
Table 2: Scintillation index as calculated from the new PDF, GG fit parameters, and experimental data...... 149
Table 3: Parameters for new PDF and GG model used to represent experimental data...... 150
xiii LIST OF SYMBOLS
⋅ ...... Ensemble Average
β0 ...... Rytov Variance for Spherical Wave
2 Cn ...... Refractive index structure parameter
DARPA ...... Defense Advanced Research Project Agency
DOC ...... Degree of Coherence
DPSK ...... Differential Phase Shift Keying
FSO ...... Free Space Optical
GG ...... Gamma Gamma
IQR ...... Interquartile Range
IR...... Infrared
ISTEF ...... Innovative Science and Technology Experimentation Facility
KL ...... Karhunen-Loeve
κm ...... Upper Wavenumber cutoff
κo ...... Lower Wavenumber cutoff
LED ...... Light Emitting Diode l0 ...... Inner scale of turbulence
L0 ...... Outer scale of turbulence
LNME ...... Log Normal Modulated Exponential
LNMR ...... Log Normal Modulated Rician
MCF ...... Mutual Coherence Function
xiv MLCD ...... Mars Laser Communication Development
MTF ...... Modulation Transfer Function
OOK ...... On-Off Keying
PDF ...... Probability Density Function
ρc ...... Correlation Width
ρ0 ...... Spatial Coherence Radius
RMS ...... Root Mean Square
SNR ...... Signal to Noise Ratio
WSF ...... Wave Structure Function
xv 1 INTRODUCTION
July 7, 1960 Maiman stimulated the emission of coherent light from a ruby rod. Shortly after,
his assistant joked he had created “a solution looking for a problem.” Since then many uses for
the laser have come about in physics, chemistry, medicine, military and the communications industry. The laser’s unique ability to generate coherent light made it extraordinary.
Preceding optical fibers, confocal optical waveguides were used to mitigate long distance propagation of light. The waveguides incorporated regularly spaced heaters causing the air to act as a lens and focus the light. This concept was adopted from millimeter wave transmission, but proved to be unsuccessful. In 1970 two key demonstrations took place: continuous laser emission from a semiconductor at room temperature and a low-loss glass optical fiber.
Development of these technologies led to the telecommunications boom in the 1980’s. Data rates of 2.5Gbps were achieved over a fiber, equating to 32,000 simultaneous phone conversations. In 1988 the first fiber optic cable was laid across the Atlantic Ocean. The development of erbium-doped amplifiers created a means of direct optical amplification of the optical signal within the fiber, thereby eliminating the optical-to-electrical conversion process previously required for signal amplification. The erbium-doped amplifier provided amplification across a broad band of wavelengths, and was a natural fit for wavelength-division multiplexing; the standard in today’s telecommunications [1].
1 In the mid-1960’s NASA tested laser transmission from the earth to the Gemini space capsule with mixed results. As technology matured with the development of pointing and tracking systems, space based laser communications became more realizable. In 2010, NASA had planned to test the first deep space laser communication link under the Mars Laser
Communication Development (MLCD) program [2]. The system promised to communicate nearly ten times faster than any existing radio link. However, budgetary issues in 2005 postponed the program. Outer space is an ideal medium to propagate light through; unlike earth, it is nearly a vacuum and there are no atmospheric obstacles.
Free space optical communications (FSOC) is becoming more of a commonplace. Technologies are advancing the use of LEDs indoor as combined lighting and short-range optical communication. Atmospheric models for propagation through turbulence have matured over the past fifty years since Tatarskii’s pioneering work [3, 4]. The electronics required to transmit and receive optical signals are shrinking and becoming more efficient due to advances in semiconductors. Point-to-point long-range FSOC has been advancing thanks to government
(DARPA) funded programs such as THOR, ORCLE, IRONT-2, and ORCA [5]. Ultraviolet radios take advantage of the scattering at short wavelengths to create a secure short-range channel with high signal to noise ratio (SNR). All major FSOC programs to this date have been based upon intensity modulation schemes; on-off keying (OOK), or more recently binary pulse- position modulation (BPPM). Future FSOC systems will take advantage of the signal gain and
SNR improvements from a coherent detection system, i.e. amplitude and phase detection.
2 Optical frequencies allow for higher bandwidth, faster data rates, smaller antennas, smaller size
and weight of components, increased directivity of EM radiation, and increased security. An
FSO communication system has the advantage to be deployed in situations where a fiber optic
cable is not practical. In the situation of a disaster site, FSO communication systems could be
quickly deployed and provide high bandwidth communications. However, at the short
wavelengths of optical waves, problems arise that are not a concern with RF communication.
The earth’s atmosphere has three main hurdles to overcome when using it as a communication
channel; absorption, scattering, and turbulence. Absorption of optical waves results in
attenuation, it occurs throughout the visible and IR spectrum. Absorption is a selective process
and results from specific molecules in the atmosphere having an absorption band at an optical
wavelength. Scattering occurs when a particle in the atmosphere is on the same order of
magnitude of the optical wavelength [6, 7]. The interaction of the particle and light wave causes
an angular redistribution of a portion of the radiated wave. Optical turbulence is a result of
fluctuations in the index of refraction along a propagation path. These fluctuations distort the
phase front and vary the temporal intensity of an optical wave. The combination of these
atmospheric effects on an optical system can cause phenomena such as beam spreading, image
dancing, beam wander, and scintillation [7].
The dominant noise source in an FSO communication system is atmosphere-induced intensity and phase fluctuations; scintillation. A larger aperture (collecting lens) can help reduce scintillation effects and improve SNR. Design criteria such as detector threshold level, probability of detection, mean fade time, number of fades, and SNR require knowledge of the
3 probability density function (PDF) of the received irradiance of the optical field. The PDF of the received irradiance is nonstationary by nature and is dependent upon the atmospheric turbulence parameters, transmitted beam characteristics, and receiver design parameters; such as aperture size and bandwidth. Nonetheless, an accurate PDF of the received irradiance is necessary to build a robust and reliable FSOC.
4 2 BACKGROUND
2.1 Random Processes
Any set of data collected from a physical phenomenon can be classified as either deterministic or
nondeterministic. A deterministic data set can be explicitly described by a mathematical formula. In a deterministic system, a repeated set of given initial conditions will always result in the same output. However, there is no way to predict the exact value of a nondeterministic phenomenon at an instant in time. Thereby these data are random by nature and can only be described by means of probability statements and statistical averages [8]. Any detection system in real life experiences noise and therefore requires a probabilistic treatment.
2.1.1 Stationarity, homogeneity, isotropy, and ergodicity
A stationary process has moments invariant with translations in time [9]. In nature, truly
stationary processes do not exist because there typically exists some stop time for the process [7,
10]. In some applications the random process does not change significantly over a finite sample
time and thus can be considered a stationary process. The condition of strict stationarity implies
all joint density functions of all orders describing the process to be independent of time. A more
relaxed and commonly used condition is stationarity in a wide-sense. A process is considered
5 wide-sense stationary if the first and second moments do not vary with time [7, 9]. In practice, a finite sample of a random phenomenon is typically used to represent the process [8, 11]. The entire sample may not be wide-sense stationary, but parts of it might be. In this case, stationary increments of the sample can be used to calculate statistics of the process.
Random functions of a vector spatial variable, R( ,, zyx ), and possibly time are called random fields [7]. The concept of homogeneity is introduced when considering spatial statistics. A random field is said to be homogeneous if the moments are invariant under a spatial translations.
A stationary random process in time is analogous to a homogeneous random field in space.
Isotropy is defined as uniform in all directions and therefore an isotropic field has no preferred direction for statistics, all average functions describing the statistics of the field remain unchanged regardless of a rotation in the coordinate system [10].
A process is called ergodic if the time or space average is equivalent to the ensemble average
[10]. Note that only stationary processes can be ergodic [8]. For an ergodic process, the statistics of the random process, such as the density functions and moments, can be determined from a single time sample.
Most experimental data of interest is fundamentally nonstationary; examples include ocean waves, atmospheric turbulence, and economic time-series data [8]. The assumptions of stationarity, homogeneity, isotropy, and ergodicity are necessary to apply any of the available turbulence theory. Unfortunately, atmospheric turbulence is a nonstationary process and caution must be taken when analyzing data records.
6 2.1.1.1 Testing for stationarity
Many procedures exist to test a time series data set for stationarity. The test procedures fall into
two main categories, parametric and nonparametric. Parametric tests assume a known
distribution of the data, they are more powerful, and generally require less data than
nonparametric tests to reach a conclusion. Nonparametric tests are commonly used and more
widely applicable because they require no assumption regarding the underlying nature of the
data. Consequently, they are less powerful than parametric tests [12]. Examples of parametric
tests are the Chi-squared and Kolmogorov-Smirnoff goodness-of-fit tests. Examples of
nonparametric tests are the sign test, Wilcoxon signed-rank test, runs test, and reverse
arrangements test [8, 12].
In order to test a random data set for stationarity, it must be assumed the data record will properly reflect the nonstationary character of the random process in question. Also, the data record needs to be significantly longer compared to the lowest frequency component in the data.
Under the previous two assumptions a single data record over time can be investigated. First, the sample record is divided into N equal time intervals where the data in each interval may be considered independent. Second, the mean square value is computed for each of the N intervals.
Those N mean square values are then subjected to any of the nonparametric tests. Bendat and
Piersol outline the application of the reverse arrangements test for stationarity [8].
7 For the reverse arragements test, we can write the N mean square values as i ix = ,...,3,2,1, N .
The number of times > xx ji is counted for i > j , each inequality is called a reverse arrangement.
The general definition for the number of reverse arrangements in a data record, A, is [8]
N −1 = ∑ AA i , (1) i=1
where
N = ∑ hA jii (2) ij += 1
and
⎧1 > xxif ji h ji = ⎨ . (3) ⎩0 otherwise
Assuming the N observations are independent, then the number of reverse arrangements is a
2 random variable A with mean and variance defined by μ A and σ A , respectively [8]
(NN −1) μ = (4) A 4
8 23 −+ 532 NNN ( )(NNN −+ 152 ) σ 2 = = . (5) A 72 72
If the number of reverse arrangements is close to the mean value, then the sample of random data is considered stationary.
To formally define upper and lower bounds based upon a specific level of significance, the z- score is used. If we always assume a level of significance of 5% then the upper and lower bounds are approximately defined by two standard deviations of the mean number of reverse arrangements, [13]
boundupper = μ + 2σ AA . (6) boundlower −= 2σμ AA
If the mean number of arrangements A lies within the bounds defined above, we can say with a
95% confidence level the data is stationary over the prescribed time interval.
2.1.2 Probability density function
The quantity observed in a trial of an experiment is a random variable. A random variable is a rule that assigns every outcome of an experiment a number. The random variable is a function mapping the set of outcomes to a set of numbers. Similarly, a random process is a function that
9 maps each outcome of an experiment to a function. A random process observed at a discrete point in time is a random variable. A random process is usually treated as a function of time, but can also be a function of space. [7-9]
A probability distribution for a random variable organizes the probabilities observed in an experiment. The distribution defines the range and probability of values a random variable can attain. The probability density function of a random variable describes the density of probability at each point within the sample space. The PDF is often displayed as a graph with the sample value on the abscissa and the corresponding probability on the ordinate. An estimate of the PDF of a random variable or process can be created from a sample data record through the use of a histogram with sufficiently small bins. [7-9]
If a random process is sampled a finite number of times, then a collection of random variables is obtained. Since each time sample of a random process is a random variable, then a random process has a collection of PDFs described by the joint PDF of order n [7],
x ( 2211 K txtxtxp nn ;,;;,;, ). (7)
In general, the complete joint PDF is required to describe a random process. However, under the assumption of a stationary and ergodic random process, the PDF of a single time sample is sufficient.
One of the fundamental properties of all PDFs is [14]
10 ∞ ∫ x ()dxxf = 1, (8) ∞−
i.e. the total area under a PDF is unity. The cumulative distribution function (CDF) can be derived from the PDF through integration [14]
x = duufxF , (9) x ()∫ x () ∞−
where u is a dummy variable of integration. To calculate the probability of an event being within an interval (a,b) from a continuous random variable, the PDF is integrated over an interval [14]
b Pr x =<< dxxfba . (10) ()()∫ x a
Often the conditional probability density function of a random variable is known and the unconditional density is of interest. The relationship between the unconditional density
yx (), yxf and the conditional densities x ( | y = yxf ) and y ( |x = xyf ) is [14]
yx ()= x ( |, y = ) ( ) = yy ( | x = ) x (xfxyfyfyxfyxf ), (11)
11 where Bayes’ theorem has been used relate the conditional densities and the marginal densities of each random variable, y ()yf and x (xf ). The total probability for x can be written by removing the condition y = y , this is done by taking the average of the conditional PDF over marginal PDF of y [14]
∞ ()= (| y = ) ()dyyfyxfxf , (12) x ∫ ∞− x y
and similarly for y
∞ ()= (|x = ) ()dxxfxyfyf . (13) y ∫ ∞− y x
As stated earlier, the PDF of a nonstationary process changes with time. Time-averaging of a nonstationary process will generally produce a severely distorted PDF [8]. Computing the PDF from a data sample with a nonstationary second moment will tend to exaggerate the probability density in the tails (high and low values) at the expense of intermediate values [8]. However, these distortions can be reduced by the selection of stationary increments within the finite sample.
12 2.1.2.1 Histogram
A histogram is a statistical tool for displaying frequency of events. The x-axis represents the data of interest and the y-axis represents how frequent the data occur over a specified non- overlapping interval, called a bin. A histogram represents numbers by area and is often displayed with a bar chart [15]. The bin widths of a histogram are typically of equal width, however, some data sets benefit from unequal bin widths. A PDF of the data can be estimated from a histogram with equal bin widths by normalizing the total area of the histogram to unity
[8]
N ˆ()xp = x , (14) NW
where N is the total number of data points, W is the bin width centered at x, and N x is the number of data values (bin height) within the range ±Wx 2 . A PDF of the data can also be estimated for unequal bin widths. For a histogram, the sum of the heights of each bin is equal to the total number of data points,
∑ x = NN . (15) binsall
13 First, consider creating a histogram in two parts, A and B, where A is the lower part of the histogram up to some arbitrary value and B is the remainder of the histogram. If the bins for
parts A and B were equal, = WW BA their histograms would be
N , Ax ˆ()xp , equalA = , (16) A NW
N , Bx N , Bx ˆ()xp ,equalB == . (17) B NW A NW
If two different bin widths were considered, WA and WB , where < WW BA then the heights of the bins in part B would need to be normalized to the area of the bins in part A [15],
WA N ,, unequalBx → N ,, equalBx . (18) WB
Their histograms for unequal bin widths would then be
N , Ax ˆ()xp ,unequalA = (19) A NW
N , Bx ˆ()xp ,unequalB = (20) B NW
14 When plotted on the same graph with their corresponding bin widths the partial histograms of A and B combine to display the PDF of the entire data set.
Selecting the proper bin size is essential when generating a histogram. If the bin size is too large important details will be smoothed out, if the bin size is too small the histogram will be jagged and the trend indiscernible. Several techniques exist for determining bin sizes based upon a given set of data. Two methods in particular are based upon minimizing the mean square error and are robust enough to support data sets of any size and range. Scott’s rule [16] defines the optimal bin size width, h, from the standard deviation of the data σ and the number of samples n,
5.3 σ h = . (21) n 31
Freedman and Diaconis [17] define the optimal bin size from the interquartile range (IQR), a measure of statistical dispersion, and the number of samples n,
*2 IQR h = . (22) n 31
15 2.1.3 Structure function
The structure function is introduced when a random process can no longer be considered stationary. Some examples of this are wind velocity fluctuations and temperature fluctuations; they are not stationary because their means are constant only over a short period of time [7]. To overcome this problem, the random process is considered to have stationary increments. Instead
of working directly with the random process, the function ( + 1 )− (txttx 1 ) is introduced. Such functions are considered to have a slow varying mean and can be easily described using structure functions [7].
In the study of turbulence, a random process, x(t), is written as the sum of the mean, m(t), and a
fluctuating part, x1(t), where 1 ()tx = 0 [7],
)()()( = + 1 txtmtx , (23)
where ⋅ denotes the ensemble average. The structure function associated with the random process x(t) is then [7],
2 x ( , 21 ) [ ( 1 )−= (txtxttD 2 )] (24)
2 2 []()1 ( 2 ) [ ( )−+−= (txtxtmtm 2111 )] . (25)
16 It is important to note that if the mean is slowly varying then ( 1 ) ≅ (tmtm 2 ) for t1 “close” to t2 the first term reduces to zero and the structure function is approximated by [7],
2 x ( , 21 ) [ ( )−≅ (txtxttD 2111 )] . (26)
This concept can be easily extended into the spatial domain. As discussed in a previous section, the spatial equivalent to stationary increments is locally homogeneous. This allows a random field to be written in the spatial domain as
( ) = ( )+ xmx 1 (RRR ), (27)
where R a position vector in space. Applying the same concept used for a random process with stationary increments, the structure function for a local homogeneity random field can be written as
D (), ( ) [ ( ) xxD ( +−== RRRRRR )]2 x 21 x 1 1 . (28)
17 2.1.4 Averaging time
There are three types of averaging: space, time, and ensemble. Space averaging is sampling a group of processes over a prescribed area at a particular instant [10]. Time averaging is sampling a single process for a sufficiently long realization. Ensemble averaging is sampling a group of identical processes at a particular instant. In practice, the ensemble average is rarely available and space and time averages are typically used [10].
A random process observed in real life is never completely homogenous or stationary. The best one can do is require that the variance falls to an acceptably small value over sufficiently large integrating times. It is difficult to measure the ensemble average of a random process. Instead, the assumption of ergodicity is made and thus the time or space average is equivalent.
When calculating higher order moments from experimental data, care must be taken to avoid detector saturation as well as assure a sufficient data sample. Higher order moments require a longer sample record than lower order moments to retain the same accuracy of lower order moments. For instance, the fourth moment weighs the extreme tails of the distribution much more heavily than lower order moments because it is looking at values occurring less frequently.
As an example take a zero mean Gaussian process, the sample time required to determine the fourth moment with the same accuracy as the second moment is more than five times as long
[10]. The necessary averaging, or sample, time must be considered when recording data, especially non-stationary data.
18 3 ATMOSPHERIC TURBULENCE
Classically, turbulence is derived from fluid dynamics. Dynamic mixing distinguishes turbulent flow from laminar, or smooth, flow. The Reynolds number defines a dimensionless ratio between inertial and viscous forces within a flow. Low Reynolds numbers characterize laminar flow where viscous forces dominate and the motion is smooth. High Reynolds numbers characterize turbulent flow where inertial forces dominate and the formation of random eddies and vortices occur [7, 10, 18].
Atmospheric turbulence is analogous to turbulence in a fluid; in the atmosphere the air is considered the fluid. The term atmospheric turbulence encompasses variations in wind, pressure, and temperature. A subset of atmospheric turbulence is optical turbulence, i.e. turbulence having a strong effect at optical wavelengths. Optical turbulence is defined as small random fluctuations in the refractive index of air. Solar heating creates a temperature gradient between the ground and the air. The temperature gradients cause the pressure of the air to change which causes the refractive index differences, this is optical turbulence.
3.1 Solution Techniques for the Stochastic Helmholtz Equation
Fundamentally, a propagating wave is governed by the wave equation. Assuming the time and space components of the wave function are separable, the Helmholtz equation results. An optical
19 wave propagating through an unbounded continuous medium with a smoothly varying refractive index obeys the stochastic Helomholtz equation [7]
2 +∇ 22 (R)UnkU = 0 (29)
where U is the complex amplitude of the field, ∇ 2 is the Laplacian operator, k = 2π λ is the wavenumber, and n2 ()R is the index of refraction in three dimensions. It is assumed the variations in the refractive index are slow compared to the frequency of the optical wave. The random refractive index creates a nonlinear stochastic partial differential equation, which to this day has no closed form solution [19]. There are several classic methods for approximately solving (29). Under weak or strong fluctuation conditions, as defined by the Rytov variance in section 3.9.1, both the parabolic equation method and the extended Huygens-Fresnel principle yield approximations to the first and second order moments of U, the latter method lending itself to spherical waves [7]. Under weak fluctuations two perturbation methods are commonly used, the Born approximation and the Rytov approximation [7]. The Born approximation writes the complex amplitude of the field as a sum of perturbations, while the Rytov approximation assumes a multiplication of exponential perturbation functions [7]. The Born approximation is limited to very short propagation distances and is not useful for most applications of optical wave propagation [7]. The Rytov approximation has fewer limitations and is the most widespread method used to solve the stochastic Helmholtz equation. Early work focused only on the first-order perturbation [3, 4], while later work demonstrated a need for the second-order term to calculate statistical moments of the optical field [7, 20].
20 2 3.2 Refractive Index Structure Parameter, Cn
2 Physically, the refractive-index structure parameter, Cn , is a measure of the strength of the fluctuations in the refractive index. The index of refraction of a medium is important when propagating light through it. The atmosphere exhibits random fluctuations in refractive index as the temperature and wind speed change. At a point R in space and time t, the index of refraction can be mathematically expressed as [7]
R, = + R,tnntn ( ) 10 ( ), (30)
where 0 ()R tnn ≅= 1, is the mean value of the index of refraction and 1 ()R,tn represents the
random deviation of ()R,tn from its mean value; thus we take, 1 (R tn ) = 0, . Typically time variations of the refractive index are slow compared to the frequency of the optical wave; therefore the wave is assumed to be monochromatic. The expression in (30) can then be written as [7]
(R) = 1+ nn 1 (R), (31)
where n()R has been normalized by its mean value n0.
The index of refraction for the atmosphere can be written for visible and IR wavelengths as [7]
21 P(R) n()R −6 ( ×+×+= 1052.71106.771 λ−− 23 ) T ()R , (32)
where λ, the optical wavelength, is expressed in μm, P(R) is the pressure in millibars at a point in space, and T(R) is the temperature in Kelvin at a point in space. Since the wavelength dependence for optical frequencies is very small; (32) can then be rewritten as [7]
P(R) n()R ×+≅ 10791 −6 T ()R . (33)
Since pressure fluctuations are usually negligible, the index of refraction exhibits an inverse relation with the random temperature fluctuations. This simple approximation only holds in the visible and near-IR regime. Extending the wavelength into the far-IR introduces other issues such as humidity.
Since 1 ()R tn = 0, is assumed, the spatial covariance Bn ( ,RR 21 )of n(R) can be expressed as
[7]
Bn (), 21 = Bn ( , 11 ) =+ ( )nn ( 1111 + RRRRRRRR ) . (34)
If the random field is both statistically homogeneous and isotropic, the spatial covariance
function can be expressed in terms of a scalar distance R −= RR 12 . Assuming statistically
22 homogeneous and isotropic turbulence, the related structure function exhibits asymptotic behavior [7]
3/22 ⎪⎧ n , 0 <<<< LRlRC 0 n ()RD = ⎨ ⎪ 2 − 23/4 , << lRRlC ⎩ n 0 0 , (35)
2 where Cn is the index-of-refraction structure parameter, l0 is the inner scale of turbulence, and L0 is the outer scale of turbulence. The inner and outer scales of turbulence act as a lower and upper
2 bound, respectively, for the fluctuations of the refractive index. Behavior of Cn at a point along the propagation path can be deduced from the temperature structure function obtained from point measurements of the mean-square temperature differences in two fine wire thermometers. With the use of (33), [7]
2 2 ⎛ −6 P ⎞ 2 Cn ⎜ ×= 1079 2 ⎟ CT ⎝ T ⎠ . (36)
2 -16 − 32 -12 − 32 Typical values for Cn are between 10 m for weak fluctuations and 10 m for strong
2 fluctuations. Figure 1 illustrates the behavior of Cn throughout a typical day. The data was taken using a commercial scintillometer at the Innovative Science and Technology
2 Experimentation Facility (ISTEF) laser range. When there is no sunshine, Cn is low. As the sun
2 begins to rise, Cn increases until it reaches a maximum in the middle of the day. As the sun
2 2 begins to set, Cn decreases. An interesting trend to note on the plot is the dip in Cn before and
23 2 after sunrise. These two dips are called the quiescent periods. This drop in Cn occurs due to the temperature gradient between the ground and atmosphere being minimal.
1.0E-12
8.1E-13
)
-2/3 6.1E-13 m (
2 n C Cn^2 (m^-2/3) 4.1E-13
2.1E-13
1.0E-14 0:00 2:00 4:00 6:00 8:00 10:52 13:24 15:26 17:26 19:26 21:26 23:26 Time
2 Figure 1: 24-hour Cn profile of data taken over a 1km terrestrial path during the summer in Florida.
3.3 Power Spectrum Models
The Riemann-Stieltjes integral allows the covariance (or correlation) function of a random process to be associated with a corresponding power spectrum. Through this relationship, the covariance function is the Fourier transform of the power spectrum and thus they are transform pairs [7],
24 ∞ i τω x ()τ = ∫ x ( )dSeB ωω (37) ∞−
1 ∞ S ()ω = −i τω ()dBe ττ (38) x 2π ∫ x ∞−
where ω represents angular frequency, τ represents time, Bx (τ ) is the covariance function, and
S x ()ω is the power spectrum. These general transform relations are know as the Wiener-
Khintchine theorem [7].
For theoretical work, the three-dimensional power spectrum is required. Using the Riemann-
Stieltjes integral a relationship between the three-dimensional power spectrum and the covariance function for a statistically homogeneous complex random field can be written as [7]
3 ⎛ 1 ⎞ ∞ K =Φ ⎜ ⎟ i •− RK R 3 RdBe (39) u ()⎜ ⎟ ∫∫∫ u () ⎝ 2π ⎠ ∞−
where K = (κ κ ,, κ zyx ) is the vector wave number in [rad/m] and R = ( ,, zyx ) is the spatial position vector. The equation simplifies when the random field is both homogeneous and isotropic and thus spherical symmetry can be assumed [7],
1 ∞ ()κ ==Φ ()sin (κ ) dRRRRB . (40) u 2 ∫ u 2 κπ 0
25 where κ = K is the magnitude of the wave number vector and R = R .
For work in atmospheric turbulence, the three-dimensional spatial power spectrum of the refractive index is of interest. The inverse Fourier transform of (40) yields the covariance function for the refractive index fluctuations [7],
4π ∞ RB Φ= sin dR κκκκ (41) n () ∫ n () ( ) R 0
where the subscript u has been replaced with n to denote the refractive index. Using the relationship between the refractive index structure function and covariance, the relation between the spectrum and the structure function is [7]
n ()= [ n (02 )− n (RBBRD )] ∞ ⎛ sin()κ R ⎞ . (42) 8 2 κκπ ⎜1−Φ= ⎟ dκ ∫ n ()⎜ ⎟ 0 ⎝ κ R ⎠
The Kolmogorov power-law spectrum can be deduced from the structure function in (35) [7],
− 3112 n ()κ =Φ 033.0 Cn κ 1, L0 κ <<<< 1 l0 , (43)
it represents the power spectral density for the refractive index fluctuations over the inertial
subrange, 1 L0 κ <<<< 1 l0 . The Kolmogorov spectrum is the most commonly used power
26 spectrum in optical turbulence calculations predominantly due to its simplicity. It is only valid within the inertial subrange, however it can be extended to represent all wavenumbers by
assuming an infinite outer scale ( L0 = ∞ ) and a zero inner scale (l0 = 0 ) [7].
Several other power spectrum models have been introduced based upon the 2/3 structure function. These other models include the effects of a finite inner and/or outer scale on the power spectrum. Tatarskii included inner scale effects by truncating the power spectrum at high wavenumbers (the dissipation range) with a Gaussian shaped function [7],
⎛ κ 2 ⎞ κ =Φ 033.0 C κ − 3112 ⎜− ⎟ κ >> L κ = 92.5;1,exp l . (44) n () n ⎜ 2 ⎟ 0 m 0 ⎝ κ m ⎠
Von Kármán included outer scale effects by truncating the power spectrum at low wavenumbers
[7]. The von Kármán spectrum was later combined with the Tatarskii spectrum to incorporate both inner and outer scale effects and thus be valid over all wavenumbers; the spectrum was named the modified von Kármán [7].
2 ⎧ 033.0 Cn , κ <<≤ 10 l0 ⎪ 2 2 611 ⎪()+ κκ 0 κ =Φ , (45) n () ⎨ 22 ⎪ 2 exp()− κκ m 033.0 Cn κκ m =∞<≤ 92.5;0, l0 ⎪ 2 2 611 ⎩ ()+ κκ 0
27 where κ 0 = Lc 0 and c is a constant typically assuming a value of 1 or 2π , the upper expression is the von Kármán and the lower expression is the modified version.
The above power spectrum models only have correct behavior within the inertial subrange and
do not incorporate the bump occurring at higher wavenumbers near 1 l0 [7]. Hill and Clifford first formulated a modified power spectrum for temperature fluctuations to incorporate the bump at higher wavenumbers, they also verified the bump in the spectrum through experimental temperature data [21]. Andrews developed a more tractable analytic expression to Hill’s modified spectrum for refractive index fluctuations, including an outer scale parameter, it is referred to as the modified atmospheric spectrum [7, 21],
⎡ 67 ⎤ 22 2 ⎛ κ ⎞ ⎛ κ ⎞ exp(− κκ ) κ =Φ C ⎢ + ⎜ ⎟ − 254.0802.11033.0 ⎜ ⎟ ⎥ l n () n ⎜ ⎟ ⎜ ⎟ 611 ⎢ κ κ ⎥ 2 2 , (46) ⎣ ⎝ l ⎠ ⎝ l ⎠ ⎦ ()+ κκ 0
κκ l =∞<≤ 3.3;0 l0
where κ 0 = Lc 0 and c is a constant typically assuming a value of 1 or 2π . The terms within the brackets account for the bump at high wavenumbers [7].
3.4 Turbulence Generation
Generally speaking, a turbulent eddy is a structured glob of fluid within the bulk fluid mass having a specific life history. Eddy size is defined by the outer scale L0 and inner scale l0 of
28 turbulence. The outer scale defines the largest eddy size, while the inner scale defines the smallest eddy size. All eddy sizes laying between the inner and outer scales make up the inertial subrange. When the power spectrum of eddies is considered, wavenumbers are typically used.
The wavenumber is defined as
2π k = , (47) λ
where λ is the wavelength or diameter of the eddy. For the Kolmogorov spectrum within the inertial subrange, the three-dimensional power spectral density has a slope of -11/3 (43).
For a high Reynolds number, i.e. strong mixing, the generation of the atmospheric turbulence spectrum can be described as follows [10]. Under stable conditions, energy is fed from the wind shear into the temperature gradient. At low wavenumbers, the spectrum is anisotropic.
Distortion of the large eddies by inertia forces, caused by fluctuating velocity gradients, transfer energy from the low wavenumber portion of the anisotropic spectrum to higher wavenumbers.
At increasing wavenumbers, feeding deteriorates and the spectrum becomes more isotropic due to pressure forces. The spectrum is considered isotropic in the ranges of inertial transfer and dissipation. The range of wavenumbers where no significant feeding or dissipation is occurring, only inertial transfer of energy, defines the inertial subrange. At large enough wavenumbers dissipation transpires, during which molecular action destroys eddy structure. Heat is neither gained or lost as a parcel of air is swirled into an eddy, this is because turbulence mixing of air is an adiabatic process [18]. Figure 2 illustrates the roles of solar heating (convective energy) and
29 wind shear (mechanical energy) in the formation of turbulent eddies. Figure 3 identifies the anisotropic feeding region, inertial subrange, and dissipation regions of a typical power spectrum of turbulence fluctuations.
Figure 2: Eddy formation and breakdown.
30
Figure 3: Power spectrum of turbulence fluctuations indicating important regions.
For atmospheric turbulence close to the ground an inertial subrange always exists above 1m under average conditions [10]. Typical values for L0 range from less than a meter near ground, up to hundreds of meter in the upper atmosphere [7, 18]. Near the ground, outer scale is
proportional to the height above ground according to 0 = 4.0 hL [3, 18]. Typical values for l0 range from millimeters near ground to tens of centimeters in the upper atmosphere [7].
A few statements can made be regarding the isotropy conditions of atmospheric turbulence: For wavenumbers small compared to the height above ground, isotropic turbulence can be assumed
[10]. In addition, if the scale of the motion is less than the height above the ground or less than the distance to the nearest inversion, the motions are essentially isotropic pg 161 [10].
31 Mechanical eddies of smaller wavelengths are more isotropic than eddies produced by heating and transfer little momentum. However, heat is transported more efficiently through larger eddies pg 109 [10].
There are four general regions in the atmosphere where the turbulence is expected to be high:
Near the ground, in convective clouds, in the surrounding region of jet streams and sharp troughs, and in the air flow on the lee of mountains. Close to the ground, the strength of turbulence is predominantly a function of the height above ground, ground roughness, and wind shear. The conditions of the surrounding environment contribute differently to each of the three components of motion in the atmosphere. Fluctuations in vertical motion near the ground are governed by wind speed and surface roughness. Fluctuations in lateral motion are governed by wind speed. Fluctuations in longitudinal motion near the ground are governed by wind speed especially over rough terrain. The temperature fluctuations, which drive the refractive index fluctuations, are produced by vertical velocity fluctuations. In general, wind blowing over a rough surface, such as the ground, creates mechanical turbulence.
3.4.1 Birth of an eddy
Temperature gradients between the hot ground and the cool air form local unstable air masses.
The sunlight passes through the air with out heating it too much. The sunlight heats the ground and the ground reradiates the heat due to its low heat capacity. The hot air then rises in the
32 presence of surrounding cold air in the form of plumes. The wind shears the rising blob of hot air from the ground and a turbulent eddy is formed. The temperature difference causes a density difference which drives the change in refractive index. As the eddy rises it cools from the surrounding air and expands adiabatically [18] while the wind continuously breaks the large turbulent cell into smaller ones. The expansion causes the density difference to decrease thus reducing the effectiveness of the eddy and eventually the eddy dissipates.
2 3.5 Thermal Plumes and Near Ground Effects (Near-Ground Cn Fluctuations)
During the day and within the first few kilometers of the atmosphere (surface layer) the temperature fluctuations, i.e., optical turbulence, are statistically non-stationary and inhomogeneous. Time records of temperature fluctuations, measured at a fixed location with a fine wire temperature sensor, show a pattern of “bursty” fluctuations with intervening periods of cooler air devoid of temperature fluctuation [22, 23]. These bursts are created by uneven heating of the ground, warming the air and its subsequent rising due to buoyancy forces. The spatial extent of the plumes may be tens of meters. This rising air creates plumes of optical turbulence, and, as the plumes rise, the higher cooler non-turbulent air sinks. Measurements show that these optical turbulence plumes can rise up to 200 meters [22]. Figure 4 shows such typical heat plumes forming in a fluid from a heated surface, rising due to buoyancy, and subsequently breaking into regions of strong turbulence. Optical turbulence has a similar structure and behavior.
33
Figure 4: Images of thermal plumes rising from a heated surface in a fluid [24]. Mechanical and inertial forces break the plumes apart as they rise.
In the atmosphere, the optical turbulence rises and the air cools with the decreasing temperature gradient above the earth’s surface, thereby reducing the optical turbulence. Additional reductions in optical turbulence occur indirectly from increased humidity. The convection and thermal pluming is reduced by the presence of ground moisture since part of the sun’s heating is used for evaporation instead of thermal pluming [18]. In the daytime, optical turbulence is
2 -14 -12 -2/3 strongest near the ground, characterized by Cn on the order of 10 to near 10 m . During this period the air temperature gradient is negative, and, with increasing altitude, it has been
2 −4/3 observed that Cn often decreases from the surface layer with a h altitude dependence [25], where h denotes altitude above ground.
34 At night, the Earth’s surface cools by radiation and becomes colder than the air, producing more stable conditions. This surface cooling produces a strong positive temperature gradient that can
2 reach tens or hundreds of meters or more. Within the temperature inversion, Cn will typically increase with increasing wind speed up to around 4 m/s, and then decrease with stronger wind
2 speeds exceeding 4 m/s. Consequently, the decrease in Cn with altitude at nighttime does not generally follow a h−4/3 altitude dependence; instead, similarity theory predicts the power-law relation h−2/3 representing more stable conditions [25].
3.6 Taylor’s Frozen Turbulence Hypothesis
Taylor’s hypothesis states that the evolving structure of optical turbulence is slow compared to the mean wind velocity [3]. Hence a propagating wave will pass through a spatially fixed realization of turbulence. The movement of this fixed random media allows for comparison of a wave propagating through a fixed media to measurements in the open atmosphere. If the turbulence evolves during a measurement there is no obvious way to relate the calculated spatial fluctuations to the temporal measurements.
Atmospheric parameters calculated from experimental measurements are typically determined from temporal statistics, as opposed to spatial statistics. Under the assumption of Taylor’s hypothesis, spatial statistics can be converted directly into temporal statistics with knowledge of the average crosswind speed. The hypothesis assumes the spatial distribution of the turbulence
35 along the path is frozen and is moved across the observation path by the mean crosswind.
Similar to clouds moving in the sky, the structure of turbulence is assumed to evolve slowly with respect to the crosswind speed. Therefore the speckle pattern produced by a laser beam propagating through turbulence would not change structure, but simply move across an observation plane in the direction of the mean crosswind. However, this assumption can not be made when the component of the mean wind speed parallel to the propagation path dominates.
Under certain conditions, Taylor’s hypothesis can become suspect. We can denote v⊥ and v|| as the perpendicular and parallel components of the average wind velocity, respectively. Since
fluctuations in the direction of the mean wind velocity are approximately 0.1, if ⊥ ≈ 1.0 vv || , then the fluctuating portion of perpendicular component is of the same order as the mean [23]. In this case, Taylor’s hypothesis does not hold. We can denote v and σ as the magnitude of the mean
wind speed and the standard deviation of the magnitude of the mean wind speed, σ ⊥ as the standard deviation of the perpendicular component of the wind speed, and α as the angle between the wind direction and the direction of propagation. From Tatarskii’s turbulence theory
[3],
2 2 = σσ 2 (48) ⊥ 3 we can then write [23]
σ 8.0 σ ⊥ = . (49) ⊥ vv sinα
36 ~ ~ Usually σ v 1.0~ ; therefore for α 90°= , σ v⊥⊥ ~ 10 percent. For α < 50° , σ v⊥⊥ > 1 and the validity of Taylor’s hypothesis becomes questionable [23].
3.7 Scale Sizes of Optical Turbulence
As a coherent optical wave propagates through optical turbulence, various eddies impress a spatial phase fluctuation on the wave front with an imprint of the eddy scale size. The accumulation of such phase fluctuations on the wavefront as the optical wave propagates leads to a reduction in “smoothness” of the wavefront. Hence, the turbulent eddies further away from the source experience a smoothness of the wavefront only on the order of the transverse spatial coherence radius ρ0 . After a wave propagates a sufficient distance L, only those turbulent eddies on the order of ρ0 or less are effective in producing further spreading and amplitude fluctuations on the wave [7].
Under strong irradiance fluctuations the spatial coherence radius also identifies a related large- scale eddy size near the source called the scattering disk kL ρ0 . Basically, the scattering disk is defined by the refractive cell size l at which the focusing angle θ f ~ Ll is equal to the average scattering angleθ D 1~ k ρ0 . That is, the field within a coherence area of size ρ0 at distance L from the source is assumed to originate from a scattering disk kL ρ0 near the source. Only
37 eddies equal to or larger than the scattering disk can contribute to the field within the coherence area [7].
Only under weak fluctuations do all scale sizes affect a propagating wave. Under strong fluctuations the propagating wave is affected by two distinct scale sizes; large-scale and small- scale. Large-scale contributions are refractive and come from turbulent cells larger than the
Fresnel zone kL or the scattering disk, whichever is larger. Small-scale contributions to scintillation are diffractive and come from turbulent cells smaller than the Fresnel zone or the spatial coherence radius ρ0 , whichever is smallest. In the regime of weak turbulence, all scale sizes contribute to scintillation and the Fresnel zone is the most important scale size [7]. Figure
5 shows the relationship of the cell sizes for an infinite plane wave.
38
Figure 5: Relevant turbulent eddy scale sizes for an unbounded plane wave withλ = 532nm and 2 −− 3213 Cn ×= 105 m . The shaded region denotes scale sizes not contributing to scintillation under strong fluctuations.
3.8 Stochastic Field Properties
A stochastic field is an electromagnetic field random in both space and time. A sample of a stochastic field is represented by a volume defined by some area of the field and some observation time interval. For an optical wave propagating through atmospheric turbulence, this space-time stochastic field can be considered a space-time narrow band fluctuation. That is, the temporal characteristics of the optical field are slow compared to the temporal oscillation
39 frequency of the wave. Therefore the stochastic field can be represented by its complex envelope. This section describes some techniques used when working with stochastic fields.
3.8.1 Second-order statistics
The mutual coherence function (MCF) is the second-order moment of the scalar complex envelope of the optical field ()r, LU . It can be written as [7]
∗ 2 Γ ()()()rr 212 = 1 rr 2 mWLULUL ][,,,,, (50)
where r1 and r2 are observation points in the receiver plane and L is the propagation distance along the positive z-axis from the transmitter. All other second-order statistics of the optical field stem from the MCF. The statistical quantities can be separated into two categories, those dealing with the amplitude of the optical field and those dealing with the phase of the optical field. The intensity, or irradiance of optical wave is given by the squared magnitude of the field
[7, 26]. Evaluating the MCF at identical observation points yields the mean irradiance [7]
( LI ) Γ= 2 ( rrr ,,, L). (51)
For a finite beam the amplitude related quantities are the short and long term beam spot size as well as beam wander. The short and long term beam spot sizes capture the additional beam 40 spread beyond diffraction due to turbulence. Beam wander describes the random steering of the instantaneous center of the beam away from the optical axis in the plane of the receiver.
Although not investigated in this paper, the additional spreading of the beam due to turbulence determines the loss of power at the receiver of an FSO system [7]. In addition, beam wander can lead to intermittency as well as additional fluctuations of the signal especially for a beam diameter on the order of the receiver.
The normalized MCF yields the degree of coherence (DOC), from which the wave structure function (WSF) is derived. The DOC is written as [7]
Γ ( rr 212 ,, L) DOC()rr 21 ,, L = ()()L ΓΓ rrrr ,,,, L 222112 , (52) ⎡ 1 ⎤ exp⎢−= ()rr 21 ,, LD ⎣ 2 ⎦⎥
where rr ,, LD is the WSF. The separation distance − rr at which the DOC falls to 1 ()21 21 e
defines the spatial coherence radius, ρ0 . The spatial coherence radius at the receiver describes the loss of spatial coherence of an initially coherent beam due to turbulence [7]. The WSF is written as
()rr 21 = χ ( 21 )+ S ( rrrr 21 ,,,,,, LDLDLD ) (53)
41 and is the sum of the log-amplitude structure function χ ( rr 21 ,, LD ) and the phase structure
function S ()rr 21 ,, LD . The rms angle-of-arrival and rms image jitter are derived from the phase structure function. Angle-of-arrival fluctuations in the optical wavefront in the plane of the receiver cause jitter (dancing) of the image in the focal plane [7]. Random movement of the image can cause blurred images as well as difficulty coupling light into a fiber.
3.8.2 Fourth-order statistics
Similar to second-order statistics, all fourth-order statistics emerge from the fourth-order cross- coherence function, or the fourth moment of the optical field. The fourth moment is written as
[7]
∗ ∗ Γ ()43214 = ( 1 ) ( 2 ) ( 3 ) (rrrrrrrr 4 ,,,,,,,, LULULULUL ) . (54)
It is seen from (50) that the mean irradiance is a second-order statistic. Therefore to examine fluctuations in irradiance, fourth-order statistics must be studied. Unlike second-order statistics, the phase is not a direct result of the fourth-order moment of the optical field [7]. This is because the fourth moment of the optical field is looking at irradiance variations. However, phase statistics such as phase variance, phase covariance function, phase structure function, and the temporal phase spectrum can be inferred from the fourth moment [7]. The phase variance and phase covariance function are nearly identical to their counterpart’s, scintillation index and
42 irradiance covariance and can therefore be derived from the fourth moment. It is important to have knowledge of the phase statistics for coherent heterodyne receivers and phase modulation techniques such as differential phase shift keying (DPSK) in an optical communications system
[7].
More importantly are the statistics of the irradiance fluctuations. The scintillation index, irradiance covariance function, correlation width, and temporal irradiance spectrum can be obtained directly from the fourth moment of the optical field [7]. The scintillation index describes the irradiance fluctuations at a single point, while the irradiance covariance function describes the correlation between irradiance at two points on the beam. Scintillation index is vital in predicting the performance of an FSO communication system or a laser radar system.
The irradiance covariance function is used to determine the correlation width ρc associated with irradiance fluctuations. The correlation width is commonly used to select an aperture size for a
receiver [7]. Aperture sizes on the order of ρc and smaller act like a point receiver, apertures
larger than ρc exhibit aperture averaging effects [7]. The correlation width is obtained by normalizing the irradiance covariance function and evaluating it at the zero or 1 e 2 crossing [7].
Invoking Taylor’s frozen turbulence hypothesis allows the calculation of temporal statistics from the spatial irradiance statistics and the mean crosswind speed [7]. The temporal irradiance function is one of the main characteristics of noise in an FSO communication system or a laser radar system [7].
43 3.8.3 Coherence function of a stochastic field
The complex envelope of a stochastic field must be considered random at each point in time, t, and space, r. Random fields are completely described by their probability density functions, i.e. all higher order moments of the envelope. In practice it is unrealistic to have knowledge of all higher order moments. Instead the random field is assumed to be wide-sense stationary and analysis can then be confined to second-order statistics associated with the field; particularly the coherence function. The time-space (mutual) coherence function of a stochastic field at points
(t ,r11 )and (t ,r22 ) is [7, 26]
∗ Γ (,;, rr 22112 )()()= tUtUtt ,, rr 2211 , (55)
where denotes the ensemble average. A stochastic field is said to be coherence-separable if its coherence function factors as [26]
Γ ()tt ,;, rr = Γt ( tt )Γs ( ,, rr 212122112 ), (56)
where the factor Γs (),rr 21 is called the spatial coherence function and Γt (),tt 21 is called the temporal coherence function of the field. The field is spatially homogeneous if the coherence function depends only on the spatial distance ρ = − rr 21 . The field is temporally stationary if the coherence function depends only on the time differenceτ = − tt 21 [26]. Finally, a stochastic
44 field is completely homogeneous if it is both spatially homogeneous and temporally stationary.
A completely homogeneous coherence separable field is said to be spectrally pure and can be written as [26]
Γ ()tt ,;, rr 22112 = Γ (τ )Γst (ρ). (57)
3.8.4 Karhunen-Loeve expansion
A deterministic function can be expanded using a Fourier series representation, the coefficients of the expansion are real numbers and the expansion basis consists of sinusoidal functions. The
Karhunen-Loeve (KL) expansion is for representing a stochastic process as a linear combination of orthogonal functions, the coefficients are uncorrelated random variables and the orthogonal set of the expansion basis depends on the covariance function of the stochastic process. Given a stochastic electromagnetic field ()r,tE over a specific observation space-time volume defined by [], + Ttt and []A , the expansion can be written as [26]
∞ ()r, = ∑ φ jj ()r,tatE (58) i=0
where [26]
45 +Tt * j = ∫∫ ()()φ j ,, ddtttEa rrr . (59) A t
The random coefficients a j are the modal coefficients describing the power in the envelope in each mode [26]. The functions φ j ()r,t form the complete set of orthogonal basis functions over the space-time volume of the electromagnetic field [26]. The convergence of the series is in the mean squared sense and requires bounded average energy in the field over the spatial area and time interval of the series expansion. For a KL expansion the correlation function of the field envelope and each member of the orthogonal function set satisfy the following integral relation for some positive constants γ j [26]
+Tt ∫∫Γ ()()tt 22112 φ j ,,;, rrrr 22 dtdt 22 = φγ jj ()r1 ,t 1 , (60) A t
2 this identifies the random coefficients a j as uncorrelated random variables andγ = a jj [26].
For a coherence separable field the left side of equation (60) reduces to [26],
+Tt ∫∫ t ()()()tt ΓΓ s 2121 φ j ,,, rrrr 22 dtdt 22 . (61) A t
The form of the equation shows that an eigenfunction solution always occurs as [26]
46 φ j (r1 , 1 ) = (r) jj (tgwt ). (62)
γ = γ γ jtjsj (63)
where the terms above satisfy [26]
+Tt ∫ Γt ()(), j 2221 = γ jjt ()tgdttgtt 1 (64) t
∫ Γs (),21 j ( ) rrrr 22 = γ wdw jjs (r1 ) (65) A
For a coherence separable field the temporal and spatial components are independent, consequently the eigenfunctions and eigenvalues factor into a product of spatial and temporal components. Therefore the eigenfunctions in (58) can be determined by from separate calculations of the eigenfunctions for both the time and space coherence kernels in (64) and (65)
[26]. The KL expansion of a coherence separable field is then [26]
()r,= ∑∑ , ji (r) ij (tgwEtE ) (66) ij
where [26]
+Tt ** , ji = ∫∫ ()()(), ij ddttgwtEE rrr (67) A t
47 The functions j ()tg describe the temporal modes of the field over the time interval[], + Ttt while the functions w j (r) describe the spatial modes of the field over the area A. The
2 corresponding γ γ itjs eigenvalues are the mean square energy in each of these modes, i.e. E , ji .
Collecting the energy in all spatial and temporal modes yields the total average energy over the space-time volume [26],
∞ ∞ +Tt 2 ∑∑ γγ itjs = ∫∫ (), ddttE rr . (68) ij=00= A t
The total energy over the area during a particular time mode i is [26]
∞ ∑γγ jsit (69) j=0
while the average energy over the time interval ( , + Ttt ) in a particular spatial mode j is [26]
∞ ∑γγ itjs (70) i=0
If the field is completely spatially coherent over the area A then the field will have only one spatial mode [26].
48 3.9 Effects of Turbulence on Optical Waves
Optical turbulence has many deleterious effects on a propagating optical wave. These effects include self-interference, diffractive and refractive effects, and wavefront steering and distortion.
The optical turbulence along the path introduces random phase shifts across the propagating wavefront. Interference between different parts of the wavefront cause random irradiance patterns in the plane of the receiver. The turbulent eddies on the order of the inner scale increase the diffraction of the beam therefore distorting the wavefront. Over longer distances i.e. stronger turbulence, eddies on the order of the outer scale contribute refractive effects, weakly focusing the beam and steering the wavefront. This chapter discusses some of the more common effects of turbulence as related to the PDF of received irradiance.
3.9.1 Scintillation
The scintillation index describes the fluctuations of the received irradiance after propagating through the atmosphere. The scintillation index includes both temporal variation (twinkling) and spatial variation (speckle). It is calculated through the normalized variance of the irradiance fluctuations,
2 − II 2 I 2 2 σ I = −= 1, (71) I 2 I 2
49 where I represents the irradiance of the optical wave and 〈I〉 denotes the ensemble average.
Scintillation index is used because irradiance is a positive quantity and as the fluctuations increase, the mean increases; therefore we normalize by the mean. Under weak turbulence
2 2 conditions, σ I < 1, while in moderate to strong turbulence, σ I ≥ 1 [7]. The scintillation index is typically plotted as a function of the Rytov variance, a parameter indicating the strength of turbulence.
The Rytov variance is used when studying the propagation of plane or spherical waves within the
Kolmogorov spectrum. The Kolmogorov spectrum represents the associated power spectral density for refractive-index fluctuations over the inertial sub-range (43). It is defined as
κ =Φ 033.0 C κ − 3/112 /1, L κ <<<< /1 l n () n 0 0 , (72)
where again κ represents the scalar spatial wave number. From this spectrum, the Rytov variances for a plane wave and spherical wave are
2 611672 σ R = 23.1 n LkC (73)
2 2 611672 0 σβ R == 5.04.0 n LkC , (74)
respectively. The Rytov variance physically represents the irradiance fluctuations of an unbounded plane or spherical wave in weak optical turbulence, but is otherwise considered a measure of optical turbulence strength when extended to strong fluctuations. Weak fluctuations
50 2 2 are associated with σ R < 1, moderate fluctuations are characterized by σ R 1~ , strong
2 2 fluctuations are associated with σ R > 1, and the saturation regime is defined by σ R ∞→ [7].
Scintillation increases until the focusing regime is reached. As the beam propagates further, it loses spatial coherence, weakening the focusing effect and thereby reducing scintillation. The large scales are not strong enough to focus the beam because the beam has lost coherence. The saturation effect was first observed by Gracherva and Gurvich [27].
The scintillation index is affected by both the inner and outer scales of optical turbulence. The inner scale effects are prominent in weak to moderate turbulence and cause a rise in scintillation index for increasing eddy size [7]. The outer scale effects occur in strong turbulence and reduce the scintillation index due the weak focusing effect of the larger sized eddies [7]. Figure 6 illustrates the effect of several different inner scale values on the scintillation index. Figure 7 shows both the inner and outer scale contributions to the scintillation index.
51
Figure 6: Effects of inner scale on the scintillation index [7].
Figure 7: Effects of inner and outer scale on the scintillation index [7].
52 Conventional Rytov theory is valid for describing scintillation in weak fluctuation conditions.
However, for situations involving strong turbulence a modification to the Rytov theory, called the extended Rytov theory, is required to incorporate the decreasing transverse spatial coherence of the optical wave as it propagates[7]. In addition to accounting for the additional loss in spatial coherence in strong turbulence, the extended Rytov theory treats the refractive index as an effective refractive index and it treats the received irradiance of an optical wave as a modulation
process. The random refractive index 1 (R,tn ) is replaced with an effective refractive index [7],
,1 e ( ) = X ( )+ nnn Y (RRR ), (75)
where nX ()R represents only the large-scale in homogeneities and nY (R) represents only the small-scale in homogeneities. The received irradiance is treated as a modulation process, where the small-scale (diffractive) fluctuations modulate the statistically independent large-scale
(refractive) fluctuations [7]. In general, the normalized irradiance can be written as [7]
I Iˆ == XY (76) I
where X arises from the large-scale turbulent eddy effects and Y arises from the statistically independent small-scale turbulent eddy effects [7]. Here the received irradiance is modeled as a multiply stochastic process, the same concept used to develop the PDF.
53 We are interested in the equation for scintillation index in strong turbulence. The second moment of the normalized irradiance is [7]
ˆ 2 22 2 2 = YXI ( X )(11 ++= σσ Y ), (77)
2 2 where the mean values of X and Y are assumed unity and that σ X and σ Y are normalized variances of X and Y [7]. The scintillation index as defined in (71) is
2 2 2 2222 I ( X )( Y ) 111 ++=−++= σσσσσσσ YXYX . (78)
Under the Rytov approximation, the log-amplitude variance and the scintillation index are related by [7]
2 2 2 2 σ I = ( σ χ ) =− (σ ln I )− σ χ <<1,1exp14exp , (79)
2 2 where 4 χ = σσ ln I is the variance of log-irradiance. The large-scale and small-scale scintillations can be defined by [7]
2 2 X = exp(σσ ln X )−1 , (80) 2 2 Y = exp()σσ lnY −1
54 2 2 where σ ln X and σ lnY are called the large-scale and small-scale log-irradiance variances, respectively [7]. Therefore, the general form for scintillation index in strong turbulence is [7]
2 2 2 2 I = (σσ ln I ) =− exp1exp ( ln X σσ lnY )−+ 1. (81)
2 The large-scale scintillation σ ln X incorporates effects of both the inner and outer scales of turbulence and can be expressed as [7]
2 2 2 σ ln ( oX , ) ln0 X ( )−= σσ ln0 X (LlLl 0 ). (82)
Due to spatial filtering effects brought on by the use of an effective refractive index in the
2 extended Rytov theory, the small-scale scintillation σ lnY incorporates effects of only the inner scale [7]
2 2 ln ( oY ) = σσ lnY (ll 0 ). (83)
APPENDIX A contains the equations for the scintillation index relevant to the experimentation performed.
55 3.9.2 Aperture averaging
As stated in section 3.8.2, the irradiance covariance function is used to determine the correlation
width ρc associated with irradiance fluctuations. Aperture sizes on the order of ρc and smaller
act like a point receiver and only “see” a single correlation patch. Apertures larger than ρc see several correlation patches and exhibit averaging effects, thus reducing scintillation at the receiver [7]. Aperture sizes beyond the irradiance correlation width increase the mean received signal level and decrease fluctuations in the received signal and therefore are intentionally used in direct detection systems to increase the SNR [7]. In strong fluctuations as aperture size is increased, an initial decrease in scintillation will occur then a flattening of the curve followed by another decrease; this is the two-scale phenomena discussed in the previous section. The aperture averaging first suppresses the small-scale fluctuations then suppresses the large-scale fluctuations for a sufficiently large aperture. Figure 8 and Figure 9 illustrate the reduction in scintillation index as aperture size increases for moderate and strong turbulence, respectively.
The plots were generated for a spherical wave with the equations in APPENDIX A, several curves are plotted to show the inner scale effects as well.
56
Figure 8: Aperture averaging effects on the scintillation index of a spherical wave as a function of aperture size: moderate turbulence.
57
Figure 9: Aperture averaging effects on the scintillation index of a spherical wave as a function of aperture size: strong turbulence.
The aperture averaging factor A is defined as the ratio of the reduction in scintillation index from a finite aperture compared to that of a point receiver [7]
2 σ (DGI ) A = 2 , (84) σ I ()0
2 where DG is the (hard) aperture diameter of the receiver and σ I (0) represents the scintillation index of a point aperture.
58 Measurements as early as the 1950’s revealed a shift in the relative spatial frequency content of the irradiance power spectrum towards lower frequencies when apertures larger than the correlation width were used [7]. The larger aperture sees more correlation patches and averages out the fast (small-scale) fluctuations; analogous to a low-pass filter.
59 4 PREVIOUS & CURRENT PDF MODELS
This section provides a summary, similar to others [28], of previous PDF models. Criteria such as existence of a closed form expression, theoretical or heuristic derivation, valid regimes of turbulence, number of PDF parameters, and relationship of parameters to physical quantities will be discussed for several common PDF models of irradiance.
Early studies in irradiance statistics focused on single family distributions, such as the modified
Rician and lognormal distribution. It was later shown irradiance statistics could not be accurately described with a single family distribution due to the nonstationary nature of the turbulence along the propagation path [29, 30]. The doubly stochastic behavior of the atmosphere, as well as land and sea clutter in radar applications, was described as a random process modulated by another random process [29, 31]. The technique used to develop PDF models treated the first-order PDF as a function of random parameters. Typically, the second- order PDF was of a single random parameter and described the fluctuations of the mean of the first-order PDF. Through the use of conditional statistics, the unconditional PDF of intensity was derived [31, 32].
Higher order moments were commonly used in the development and validation of a PDF; there were two principal uses. Parameters of the theoretical distribution could be inferred by comparing higher order moments of the theoretical distribution to calculated higher order moments of experimental data. Additionally, the accuracy of a PDF could be judged by
60 comparing higher order moments. One advantage of a heuristic model based on physical parameters is that it does not require parameters of the distribution to be inferred from observed moments [33].
Since the 1960’s, there has been considerable interest, both theoretically and experimentally, in an accurate PDF to describe the received irradiance of an optical wave after propagating through an arbitrarily strong turbulent path. In the limiting case of extremely weak turbulence, the general consensus is the lognormal PDF [4, 7, 11, 33-38]. While in the limiting case of extremely strong (infinite) turbulence, it is predicted the PDF will approach a negative exponential [33, 36, 39]. Difficulty arises when trying to derive a distribution valid in weak fluctuations all the way through saturation. Currently there is no PDF able to completely describe the irradiance statistics in all regimes of optical turbulence in the presence of aperture averaging. The ideal PDF of irradiance is valid in all regimes of turbulence, is valid for any receiving aperture size, has parameters related to physical atmospheric quantities, and has a closed mathematical form.
4.1 Nakagami m-Distribution
One of the earliest attempts at describing irradiance statistics for radio waves was made by
Nakagami [40]. Rice built upon Nakagami’s m-distribution by considering the sum of two
Gaussian random variables [41]. Both worked during the Second World War developing radar
61 technologies. The results were applied to optical wave propagation and named the modified
Rician distribution (also called the modified Rice-Nakagami distribution) [7]. The intensity distribution is derived from the first-order Born approximation. The field along the path is treated as the sum of the unperturbed and first-order perturbed circular complex Gaussian fields.
Taking the magnitude squared of the sum of the perturbed fields results in the intensity distribution. The modified Rician distribution is limited to conditions where the scintillation index is less than 2. Parry and Pusey show the modified Rician distribution is a lower bound when comparing theoretical higher order moments to that of data [42, 43].
The received field is modeled as a sum of a coherent portion and a scattered portion [44],
+= UUU S , (85)
where the average field U is taken as a constant A0 and US is the scattered field. The scattered field is written as a sum of contributions from many different particles [44],
N S = exp()φ ∑ += iYXiAU nn , (86) n=1
where = AX nn cos()φn and = AY nn sin(φn ). Both − AX 0 and Y are the sum of many independent contributors such that the central limit theorem is valid. The resulting PDF of the field is then [44]
62 1 ⎛ ( )2 +− YAX 2 ⎞ (),YXp = exp⎜− 0 ⎟ , (87) U 2 ⎜ 2 ⎟ 2 σπ s ⎝ 2σ s ⎠
2 where σ s is the variance of both − AX 0 and Y. The PDF of the amplitude of the field is [44]
2π = , dAYXpAp φ , (88) ()∫ U ( ) 0
thus
A ⎛ 2 + AA 2 ⎞ ⎛ AA ⎞ Ap exp⎜−= 0 ⎟ I ⎜ 0 ⎟ , (89) () 2 ⎜ 2 ⎟ 0 ⎜ 2 ⎟ σ s ⎝ 2σ s ⎠ ⎝ σ s ⎠
where I0( ) is a zero-order modified Bessel function of the first kind. Making the change of variable I =A2 yields the PDF of intensity
1 ⎛ + AI 2 ⎞ ⎛ IA ⎞ ()Ip = exp⎜− 0 ⎟ I ⎜ 0 ⎟ . (90) 2 ⎜ 2 ⎟ 0 ⎜ 2 ⎟ 2σ s ⎝ 2 s ⎠ ⎝ σσ s ⎠
63 4.2 Lognormal Distribution
The lognormal distribution is the most commonly used PDF under weak irradiance fluctuations.
The density has a single parameter, the scintillation index, and closed form expressions exist for the PDF, probability of fade, and mean number of fades [7]. By definition, a random process, X, is lognormally distributed if Y = ln(X) is normally distributed. The physical model behind the lognormal distribution assumes the optical wave is scattered forward once by many on-axis eddies along the path [45]. The result is a multiplication of many independent fields scattered from each eddy. Taking the logarithm and invoking the central limit results in a lognormal distribution of irradiance [19]. Tatarskii initially predicted the lognormal distribution for the intensity of a plane wave in weak fluctuation conditions [4]. The distribution is derived using the first-order Rytov approximation [4, 7, 19]. It has been shown that in order to properly describe the statistics of the optical wave, the second-order Rytov perturbation must be retained
[7]. When the non-gaussian second-order perturbation is retained, the irradiance distribution is no longer lognormal [11, 46, 47]. Theoretically and experimentally it has been verified the lognormal model is invalid outside of the weak fluctuation regime [19, 27, 42, 48], additionally, experimental data supports the lognormal distribution in weak turbulence [42, 43, 49, 50].
Nevertheless, the lognormal model can underestimate the probability of small intensity values of the PDF [36, 46, 51]. However, under conditions of strong turbulence and significant aperture averaging, lognormal statistics have been shown to be an accurate math model describing the experimental results [52, 53].
64 Modeling the turbulence as slabs perpendicular to the propagation path and larger than the outer scale of turbulence, the received optical field is a product of the original field multiplied by a large number of independent random modulations[19, 54],
N = ∏ i UMU 0 , (91) i=1
where Mi is the modulation of each slab along the path, U0 is the initial field. The received intensity, = UI 2 , according to Rytov theory is then [7, 11]
N I = ∏ ()2exp χ i (92) i=1
where χi is the first-order log-amplitude perturbation and obeys a Gaussian distribution. Taking the logarithm and applying the central limit theorem yields the logarithm of intensity obeying a
Gaussian distribution [7, 11, 19, 28],
2 1 ⎛ ( ()− lnln II ) ⎞ ()ln Ip = exp⎜− ⎟ , (93) 2 ⎜ 2σ 2 ⎟ 2 σπ ln I ⎝ ln I ⎠
2 where σ ln I is the variance of the log-irradiance, denotes the ensemble average, and I = 1.
Making a change of variable = ln II , the PDF of intensity is [7, 11, 28]
65 ⎛ 2 ⎞ ⎜ ⎛ 1 2 ⎞ ⎟ ⎜ln()I + σ ln I ⎟ 1 ⎜ ⎝ 2 ⎠ ⎟ ()Ip = exp⎜− ⎟, I > 0 , (94) 2 2σ 2 I 2 σπ ln I ⎜ ln I ⎟ ⎜ ⎟ ⎝ ⎠
where the change of variable has introduced a 1 I factor and from Rytov theory
2 2 ln I −= σ ln I 2 . Under weak fluctuations, σ ln I is often replaced with the scintillation index
2 σ I [7, 11].
4.3 K Distribution
As stated above, the lognormal model breaks down in the onset of moderate to strong turbulence.
The K distribution was adapted to the scattering of optical waves in attempt to find a model valid in the strong scattering regime. The K-distribution was originally derived for radar echo problems [29]. Nevertheless, it was shown to also provide good fit to experimental data of a laser beam propagating through extended turbulence [30, 42, 55, 56]. However, theoretically
Dashen [57] showed the third moment of the K-distribution was not derived exactly. It is a single parameter heuristic model generated from a two-dimensional random walk, using negative binomial number fluctuations to describe the birth-death process in a population [30, 55]. The birth-death process was thought to be analogous to the spontaneous creation of a large turbulent eddy which then birthed smaller daughter eddies through a cascade process, eventually terminating when the smallest eddy dissipated [55]. The K-distribution can also be derived as a 66 doubly stochastic modulation process through the use of conditional statistics. The first-order density is assumed to be governed by negative exponential statistics with a varying mean obeying a gamma distribution [7, 11, 58]. The parameter of the K distribution may be thought of as representing the effective number of scatterers along the path [29, 30, 55]. As the number of scatterers of the K distribution tends to infinity, the distribution approaches a negative exponential. The K distribution is not valid in weak turbulence where the scintillation index is below 1 [7, 11] and in some instances of strong turbulence into the saturation regime where the second moment begins to decrease [43, 51].
The heuristic derivation of the intensity PDF assumes a negative exponential distribution modulated by a gamma distribution. The conditional PDF of the intensity normalized by the second moment is
1 ()bIp ()IbI >−= 0,exp , (95) b
where b is the mean value of intensity. The distribution of the mean is assumed to obey a gamma distribution
()αα b α −1 ()bp = ()α bb α >>− 0,0,exp , (96) Γ()α
67 where α represents the effective number of discrete scatterers and Γ( ) is the gamma function.
The unconditional PDF of irradiance is then found using
∞ ()= ∫ ()()dbbpbIpIp , (97) 0
thus
2α ()α − 21 ()Ip = ()α α −1 ( α ) IIKI α >> 0,0,2 (98) Γ()α
where Kn( ) is an nth order modified Bessel function of the second kind.
68 4.4 Universal Distribution
In attempt to find a model valid in all regimes of turbulence, Phillips and Andrews derived the
Universal distribution. It is a heuristic model resulting from the sum of a specular component describing forward scattering and a diffuse component describing multiple scattering [59]. It has the functional form of the product of an exponential function and an infinite series of Laguerre polynomials [59]. The model agreed well when compared with higher-order moments of several experimental data sets ranging from weak to strong turbulence [59]. Unlike other irradiance distribution models, the Universal distribution uses three-parameters and therefore is able to better characterize the looping phenomena occurring in plots of higher-order moments versus the normalized second moment [59]. The Universal distribution was not a tractable PDF, the parameters were not directly related to physical atmospheric parameters and the PDF was difficult to work with due to the infinite series of Laguerre polynomials.
The scattered field U(t) is separated into a specular and diffuse component,
)( (AetU iθ += Re )e ωφ tii , (99)
where ω is the carrier frequency, A is the amplitude and θ is the phase of the specular component, and R is the amplitude and φ is the phase of the diffuse component. Both the specular and diffuse components are modeled with a Nakagami m-distributed amplitude and uniform phase,
69 2 AM MM −12 Ap )( = exp()− 2 AcMA > 0, , (100) Γ()cM M
2 Rm mm −12 Rp )( = exp()− 2 RbmR > 0, (101) Γ()bm m
where = Ac 2 , = Rb 2 , M is a parameter related to the reciprocal of the variance of A, m is a parameter related to the reciprocal of the variance of R, and Γ( ) is the gamma function. The
PDF of intensity, = tUtI )()( 2 , is written as [29]
1 ∞ )( = dzzRJzAJzIJzIp , (102) ∫ 0 ()0 ()0 () 2 0
where
∞ = dAzAJApzAJ (103) 0 ()∫ ()0 ( ) 0
and
∞ = dRzRJRpzRJ . (104) 0 ()∫ ()0 ( ) 0
70 The resulting PDF of intensity in (102) is
2 ∞ k k j Γ()mm − bmI ()−1 ⎛k ⎞ ()+Γ ()MmrjM ()Ip = e ∑ k ()bmIL ∑⎜ ⎟ , (105) Γ()Mb k=0 k! j=0 ⎝ j ⎠ ()+−Γ jkm
⎛k ⎞ where Ln( ) is the nth order Laguerre polynomial, ⎜ ⎟ is the binomial coefficient, and = bcr . ⎝ j ⎠
4.5 I-K Distribution
In attempt to extend the K-distribution to all regimes of turbulence, Phillips and Andrews introduced the I-K distribution [32, 41, 60]. The, appropriately named, I-K distribution is a generalization of the K-distribution, containing both I and K Bessel functions. It is a simplification of the infinite series of K-Bessel functions defining the Homodyned-K distribution derived by Link [61] and suggested by Jakeman [55]. Similar to the K-distribution, the I-K distribution is a doubly stochastic process, heuristically derived from two distributions. Two variations of the distribution exist, the first uses Nakagami’s m-distribution modulated by a negative exponential distribution [32]. In this case the optical field is represented by the sum of a deterministic quantity plus a random quantity [32]. The second uses an approximation of the
Rice-Nakagami distribution modulated by a gamma distribution [60]. In this case the optical field is represented by a deterministic component plus a random component, similar to the
Universal distribution [59, 60]. The I-K distribution has two parameters, one related to the 71 effective number of scatterers and the other a coherence parameter [28, 32]. Under strong scattering conditions where the coherence parameter goes to zero, the I-K distribution reduces to the K-distribution [32]. Closed form solutions exist for both the PDF and the CDF, allowing calculations of fade parameters [32]. Initially, the parameters of the I-K distribution had no physical relationship to atmospheric parameters and were obtained by solving simultaneous equations relating the experimental and theoretical second and third moments [32, 36, 60].
Later, expressions for the parameters of the distribution were developed in terms of the Rytov variance for a plane wave and the inner scale of turbulence [35]. Comparing statistical moments from phase screen experiments yielded positive results for single phase screens, but less promising results for multiple phase screens [41]. Experimentally, the I-K distribution was shown to overestimate in the peak of the PDF, but provide good agreement in the tails [36]. The
I-K distribution showed better fit with data collected in strong turbulence [36].
The received optical field is represented as a sum of scattered fields that have traveled different paths,
N ω ti ()= ∑ k ()exp θ k ()()+ k ()exp()φk ()titrtitaetU , (106) k=1
where ω is the carrier frequency. The first term inside the sum represents the coherent portion of the scattered field, both the amplitude a and the phase θ are deterministic quantities. The second term inside the sum represents the random component of the scattered field. It is assumed U(t) follows a Nakagami m-distribution leading to the conditional PDF of intensity, = tUtI )()( 2 ,
72 α −1 α ⎛ I ⎞ ⎛ 2α A ⎞ bIp = ⎜ ⎟ exp α 2 / ×+− IbIA II > 0, , (107) () ⎜ ⎟ []()α −1 ⎜ ⎟ b ⎝ A ⎠ ⎝ b ⎠
2 2 2 2 2 2 2 where I is the intensity, 1 2 ... +++= aaaA N , 1 2 ... +++= rrrb N , In( ) is a modified
Bessel function of the first kind, and the discrete parameter N representing the effective number of scatterers has been replaced with a continuous parameter α. The average random field is assumed to be circular complex Gaussian distributed and thus the average intensity b obeys negative exponential statistics,
1 ()bp exp()−= bb 0 , (108) b0
where b0 represents the mean value of b. The unconditional PDF of irradiance is then found using
∞ ()= ∫ ()()dbbpbIpIp , (109) 0
thus evaluating (109) yields
73 α −1 ⎧2α ⎛ I ⎞ ⎛ α ⎞ ⎛ α I ⎞ ⎪ ⎜ ⎟ ⎜2AK ⎟ I ⎜ ⎟,2 < AI 2 ⎜ ⎟ α −1 ⎜ ⎟ α −1 ⎜ ⎟ ⎪ b0 ⎝ A ⎠ ⎝ b0 ⎠ ⎝ b0 ⎠ Ip = . (110) () ⎨ α −1 ⎪2α ⎛ I ⎞ ⎛ α ⎞ ⎛ α I ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎪ α −1 2AI Kα −1 ,2 > AI b ⎜ A ⎟ ⎜ b ⎟ ⎜ b ⎟ ⎩⎪ 0 ⎝ ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
4.6 Generalized K-Distribution
Years after the K-distribution, Barakat introduced a generalization of the K-distribution extending it to the weak scattering regime [62]. The K-distribution assumed a uniform distribution of phases for the scattered field, Barakat’s generalization came about by assuming the phases were governed by the von Mises PDF [62]. In its most general form, the generalized
K-distribution is in an integral form. In the limiting cases of weak and strong scattering, closed form solutions exist [62]. The general distribution has three parameters, effective number of scatterers, a term describing the departure from uniform phases, and a free parameter with no physical meaning [62]. Jakeman simplified the derivation of the generalized K-distribution by treating the process as a biased random walk [63]. Experimental data [43] of the second and third normalized moments correspond well with those predicted by the generalized K- distribution [62].
The intensity PDF is derived from speckle theory [64]. The intensity at a point in a speckle pattern is the sum of N sinusoidal waves with nonuniform phases [64]. The complex amplitude of the scattered field U is represented by
74 N = ∑ n exp()iaU θ n , (111) n=1
where N is the discrete number of scatterers, an is the amplitude of each component, and θn is the phase. The assumption of von Mises phase statistics leads to a conditional PDF for intensity,
= UI 2 [65]
⎛ν I 21 ⎞ 1 ∞ = −N ν IINIp ⎜ ⎟ N 21 dIJaJ ωωωω , (112) ()0 ()0 ⎜ ⎟ ∫ 0 ()0 () ⎝ a ⎠ 2 0
where I is the intensity, υ describes the departure from uniform phase, a is the deterministic amplitude, J0( ) is a Bessel function of the first kind, and I0( ) is a modified Bessel function of the first kind. The number of scatterers, N, is modeled by negative binomial statistics,
N ⎛ N α −+ 1⎞ ( N /α ) Np = ⎜ ⎟ , N = ,...,1,0 (113) ()⎜ ⎟ N +α ⎝ N ⎠ ()+ N /1 α
where α is a real nonnegative free-parameter. The unconditional PDF of irradiance is then found using
∞ ()= ∫ ()()dNNpNIpIp , (114) 0
Thus with partial evaluation leads to
75 −α 1 ⎛ν I 21 ⎞∞ ⎧ N ⎡ ()aJ ω ⎤⎫ = IIp ⎜ ⎟ 11 −+ 0 21 dIJ ωωω . (115) () 0 ⎜ ⎟∫ ⎨ ⎢ ⎥⎬ 0 () 2 ⎝ a ⎠ 0 ⎩ α ⎣ I 0 ()ν ⎦⎭
4.7 Lognormally Modulated Exponential
An improvement to the K-distribution was proposed by Churnside and Hill [52]. The lognormally modulated exponential (LNME) PDF was a heuristic model derived from a negative exponential distribution modulated by a lognormal distribution [52]. The LNME distribution was similar to the K-distribution; a single parameter, doubly stochastic model only valid in strong turbulence. However, no closed form solution exists for the LNME distribution, only integral representation. The parameter represents the variance of the lognormal modulation factor, which is inferred from the scintillation index of experimental data [28, 52]. Higher order moment comparison showed disagreement between the LNME and experimental data, possibly due to detector saturation [52]. Simulations performed for both plane and spherical wave models implicated better agreement with the LNME for the spherical wave model and better agreement with the K-distribution for the plane wave model [56]. Experimentally, the LNME improved upon the K-distribution for a spherical wave [52].
The heuristic model assumes the small-scale scintillation spikes are modulated by the large irradiance patches [52]. The conditional PDF of the intensity normalized by the second moment is
76 1 ()zIp exp()−= zI , (116) z
where the mean value z represents irradiance modulation factor, z = (2exp χ ). The mean value is assumed lognormally distributed,
2 ⎡ ⎛ 1 ⎞ ⎤ ⎢ ⎜ln()z + σ 2 ⎟ ⎥ 1 2 z ()zp = exp⎢− ⎝ ⎠ ⎥ , (117) 2 ⎢ 2 ⎥ z 2 σπ 2σ z z ⎢ ⎥ ⎣⎢ ⎦⎥
2 where ln( ) is the natural logarithm and σz represents the variance of the logarithm of the irradiance modulation factor, z. The unconditional PDF of normalized irradiance is then found using
∞ ()= ∫ ()()dzzpzIpIp , (118) 0
thus the integral representation is
2 ⎡ ⎛ 1 ⎞ ⎤ ⎢ ⎜ln()z + σ 2 ⎟ ⎥ 1 ∞ dz I 2 z ()Ip = exp⎢ −− ⎝ ⎠ ⎥ . (119) 2 ∫ 2 ⎢ 2 ⎥ 2 σπ 0 z z 2σ z z ⎢ ⎥ ⎣⎢ ⎦⎥
77 4.8 Lognormally Modulated Rician
Shortly after the LNME was published, Churnside and Clifford reintroduced the lognormally modulated Rician (LNMR) PDF as a valid solution in all regimes of turbulence [33]. The PDF was previously proposed for irradiance fluctuations by Strohbehn in 1975 [66] and originally proposed by Beckmann in 1967 [67]. As the name suggests, the PDF is a Rician distribution modulated by a lognormal distribution [7, 11, 33]. The model is derived heuristically assuming the large eddies act as weak lenses producing the lognormal statistics and the small eddies introduce a random phase distortion reducing the coherence of the optical wave and produce
Rician statistics [33]. In the limit of weak turbulence the LNMR reduces to the lognormal distribution, in the limit of strong turbulence the LNMR reduces to the LNME distribution, and in the limit of saturated turbulence the LNMR approaches the negative exponential distribution
[33, 36, 51]. The LNMR is a two parameter doubly stochastic distribution claiming to be valid in all regimes of turbulence [33]. Similar to the LNME distribution, no closed form solution exists for the PDF, only integral representation; which for certain parameter values encounters convergence difficulties [7, 11]. The two parameters represent a coherence parameter and a lognormal modulation factor [7, 11, 33]. The parameters can be related to the measured second moment of intensity, however only in the limiting cases of weak or strong turbulence [33, 36].
Spherical wave data over short propagation paths and weak turbulence show good agreement with the LNMR density, except in the extreme tails of the PDF [36]. Spherical wave data over strong scintillation conditions show promising agreement with the LNMR [36]. Computer generated spherical wave data showed excellent comparison to the LNMR PDF in the weak to strong turbulence regime [51].
78 The scattered field U is written as
= ( +UUU GC )exp(χ + iS ) , (120)
where UC is a deterministic quantity, UG is a circular complex Gaussian random variable, and χ and S are real Gaussian random variables. The intensity, = UI 2 is given by
2 += UUI GC (2exp χ ), (121)
where +UU GC obeys a Rice-Nakagami distribution and (2exp χ ) obeys a lognormal distribution. The conditional PDF of the intensity is
1+ r ⎛ 1+ r ⎞ ⎛ (1+ )rr ⎞ zIp = exp r −− ⎜2 III ⎟ , (122) () ⎜ ⎟ 0 ⎜ ⎟ z ⎝ z ⎠ ⎝ z ⎠
2 2 where r is the coherence parameter C UU G , I0( ) is a modified Bessel function of the first kind, and the mean value z represents the irradiance modulation factor, z = ()2exp χ . The mean value is assumed lognormally distributed,
79 2 ⎡ ⎛ 1 ⎞ ⎤ ⎢ ⎜ln()z + σ 2 ⎟ ⎥ 1 2 z ()zp = exp⎢− ⎝ ⎠ ⎥ , (123) 2 ⎢ 2 ⎥ z 2 σπ 2σ z z ⎢ ⎥ ⎣⎢ ⎦⎥
2 where ln( ) is the natural logarithm and σz represents the variance of the logarithm of the irradiance modulation factor, z. The unconditional PDF of irradiance is then found using
∞ ()= ∫ ()()dzzpzIpIp , (124) 0
thus giving the integral representation of the PDF
⎡ 2 ⎤ ⎛ 1 2 ⎞ ∞ ⎢ ⎜ln()z + σ z ⎟ ⎥ 1+ r ⎛ 1+ r ⎞ ⎛ ()1+ rr ⎞ 1 ⎢ ⎝ 2 ⎠ ⎥ ()= dzIp exp⎜ r −− ⎟ 0 ⎜2 III ⎟ × exp − .(125) ∫ ⎜ ⎟ 2 ⎢ 2 ⎥ 0 z ⎝ z ⎠ z ⎠ z 2 σπ 2σ z ⎝ z ⎢ ⎥ ⎣⎢ ⎦⎥
4.9 Gamma-Gamma Distribution
The most recent model developed for the irradiance distribution was the Gamma-Gamma (GG)
PDF. The heuristic model was based upon developments in scintillation theory in the late 1990s.
The scintillation theory modeled received irradiance as a modulation process, I = xy, of large-
80 scale α and small-scale β turbulent eddies [68, 69]. The GG model was developed as a doubly stochastic process in a similar fashion to the K-distribution [7, 11, 58]. Similar to the K- distribution, the parameters of the GG distribution represent the effective number of scatterers.
The GG model assumed both the large and small-scale fluctuations obeyed a gamma distribution
[58]. Unlike the K-distribution, the GG distribution has two parameters one representing the effective number of large-scale scatterers and one representing the number of small-scale scatterers [58]. The two parameters are directly related to measured atmospheric parameters of
2 Cn and l0. The GG distribution reduces to the K-distribution when either α or β is unity. In the limit of saturated turbulence, the number of small-scale scatterers approaches unity and the number of large-scale scatterers becomes infinite thus reducing the GG distribution to the negative exponential distribution. The GG distribution has closed form solutions for both the
PDF and the CDF, although the latter containing generalized hypergeometric functions [7, 11,
28, 58, 70].
Another way to visualize the GG model is to model the large-scale and small-scale scatterers along the path as rare events, i.e. two independent Poisson processes. The distance between events in a Poisson process obeys an exponential distribution. Assuming an integer number of large-scale and small-scale scatterers, the received irradiance is a modulation of two discrete sums of exponential distributions. Assuming all the large-scale scatterers are independent and have equal mean values and all the small-scale scatterers are independent and have equal mean values, the discrete sum of exponentials reduces to a Gamma distribution. Recall that a gamma distribution can be thought of as the sum of an integer number of independent exponential
81 distributions with equal means. Therefore the received irradiance is a modulation of two Gamma distributions.
Computer generated data for both plane and spherical waves in weak to strong turbulence support the GG PDF [51, 56]. Comparison to plane wave data favors the GG distribution over the lognormal distribution in weak turbulence conditions [7, 11, 58]. Comparison to plane wave data for the GG distribution and the K-distribution in strong turbulence conditions yields nearly identical results, both matching the data [7, 11, 58]. The GG distribution also provided a good fit to the data when inner scale effects were considered [7, 11, 58]. Comparison to spherical wave data in weak and strong turbulence for the GG distribution, in the absence of inner scale, shows good agreement and good comparison with the LNMR distribution [7, 11, 58]. When inner scale effects are considered, the fit of the GG distribution to the data improves [7, 11, 58].
Comparison to spherical wave data in moderate turbulence yielded a worse fit than the LNMR distribution when the parameters of the GG distribution were chosen based upon atmospheric parameters [7, 11, 58]. A best fit of the GG distribution to the data, improved the results [7, 11,
58]. Discrepancies in the simulation data and the GG distribution could be accounted for by considering outer scale effects; which are inherent in simulation data due to numerical grid size
[7, 28]. Recent experimental measurements with both spherical waves and beam waves support the GG PDF [28, 34, 48]. However, data suggests the GG PDF is not able to properly account for the effects of aperture averaging [34]. Simulation data for a partially coherent beam has also shown good agreement with the GG PDF in all regimes of turbulence [38].
82 The heuristic model treats the irradiance of the received optical wave as a product of independent random variables, [7]
= xyI , (126)
where x arises from large-scale turbulent eddies and y arises from small-scale turbulent eddies.
The model assumes the large-scale and small-scale fluctuations obey gamma distributions. The conditional PDF of irradiance is [7]
()ββ xI β −1 ()xIp = exp()− β xIxI β >>> 0,0,0, , (127) x Γ()β
where the mean x has been fixed such that = xIy , β represents the effective number of small- scale scatterers, and Γ( ) is the gamma function. The mean value is assumed gamma distributed
[7]
()αα x α −1 ()xp = ()α xx α >>− 0,0,exp , (128) Γ()α
where α represents the effective number of large-scale scatterers. The unconditional PDF of irradiance is then found using
83 ∞ ()= ∫ ()()dxxpxIpIp , (129) 0
thus evaluating the integral yields [7]
( +βα ) 2 2()αβ ()βα −+ 12 ()Ip = KI −βα [ βα ] II > 0,2 , (130) ()()ΓΓ βα
where Kn( ) is an nth order modified Bessel function of the second kind.
The GG PDF is normalized such that the mean irradiance I = 1 is unity. Calculating the second moment yields [7]
2 ⎛ 1 ⎞⎛ 1 ⎞ I ⎜1+= ⎟⎜1+ ⎟ , (131) ⎝ ⎠⎝ βα ⎠
and therefore the scintillation index is
2 − II 2 2 ⎛ 1 ⎞⎛ 1 ⎞ σ I = ⎜1+= ⎟⎜1+ ⎟ −1. (132) I 2 ⎝ ⎠⎝ βα ⎠
The parameters of the model can be related to large-scale and small-scale scintillation according to [7]
84 1 1 α == 2 exp()σσ 2 −1 X ln X . (133) 11 β == 2 2 Y exp()σσ lnY −1
85 5 PHYSICAL MODELING
There are two core methods used to derive a PDF for irradiance fluctuations; mathematical and heuristic. There are several approaches commonly used to mathematically obtain a PDF for irradiance fluctuations: the central limit theorem, calculating moments, finding the characteristic functional, and trial and error [19]. When using the central limit theorem, the statistical theorems governing the use of it can make it difficult to apply to optical wave propagation problems. The moments of a distribution can be calculated and thus a characteristic function can be formed, the distribution can then be determined from a Fourier transform relationship. However, not all distributions are uniquely determined by their moments, particularly the lognormal [71]. The characteristic functional approach was used by Furutsu [72] for a Gaussian beam, but was not applicable to plane wave cases. Trial and error can be used by starting with a distribution based on an educated guess and fitting it to experimental data. Purely mathematical derivations of a
PDF for irradiance are impractical due the inaccuracies of the techniques used to solve statistically nonlinear differential equations. A more promising approach to determining a PDF is a heuristic method based on a physical situation. The propagation path is typically modeled as many scattering elements and as the optical wave passes through each element, part of the energy is scattered. This chapter will focus on the heuristic method for deriving a PDF for irradiance.
86 5.1 Overview of Model Development
We start with a field and propagate it through turbulence. The received field envelope ( ,rta r) is due to a random volume of time and space. We are interested in the probability of an event occurring within a particular volume of the received field envelope defined by the area of the detector and a time interval. The key to applying the statistical effect of the turbulence onto the field depends on the ability to express the random field envelope into an orthonormal Fourier series over the volume of interest. Using an approximation to the Karhunen-Loeve (KL) expansion, the field is decomposed spatially into a series of two-dimensional spatial plane waves with dimensions X and Y and temporally into a series of sinusoids. At the receiver, we assume the field is random and coherence separable. The assumption of a coherence separable field results in the eigenfunctions and eigenvalues of the KL expansion factoring into a product of temporal and spatial components.
We assume the temporal component of the field expansion is deterministic and therefore can represent the field expansion as a sum of spatial components. The number of spatial components is a function of the aperture size of the receiving telescope. Using Goodman [73], the spatial modes can be interpreted as angular modes, i.e. plane waves.
The amplitude of each spatial mode is a result of the transmitted field being scattered while propagating through the turbulence. The amplitude of each spatial mode is comprised as a sum of n scattered fields, where the field is scattered each time it encounters one of the N effective eddies along the path. We represent each scattered plane wave field contribution as a
87 conditionally zero-mean circular complex Gaussian random variable. Such that when the irradiance is calculated by the magnitude squared of each amplitude component, a conditionally exponential distribution arises. The irradiance at the receiver is found by summing the irradiance of each spatial mode over the total number of spatial modes, M.
Figure 10 outlines the procedure for deriving the irradiance of the field. Figure 11 illustrates the effect of the turbulent eddies acting as scatterers of the optical field.
Figure 10: Flowchart of optical field analysis. 88
Figure 11: Sketch of turbulent eddies refracting and scattering the optical field.
5.2 Decomposing the Complex Envelope of a Stochastic Field
Decomposition of the received optical field through the Karhunen-Loeve expansion requires the solution to the temporal and spatial equations (64) and (65), respectively. Several assumptions are made to simplify the problem such that the underlying physics can be understood. First, the received optical field is assumed homogeneous and coherence-separable. Second, the temporal component of the received optical field is assumed to be deterministic through Taylor’s frozen turbulence hypothesis. Third, the eigenfunctions of the field are approximated by two- dimensional spatial plane waves. Finally, the eigenvalues are approximated by sufficiently sampling the two-dimensional Fourier Transform of the mutual coherence function. After the preceding approximations have been made, the expansion of the received optical field is a general orthogonal expansion and no longer a KL expansion.
89 We start by assuming the temporal component of the received optical field is deterministic. It is therefore treated as a Fourier series expansion using sinusoids as the eigenfunctions, from section
3.8.4
exp( ω tis ) s ()tg = , (134) T
where i −= 1 and s is an integer. The exact solution the KL expansion of the spatial modes in
(65) with the Gaussian-like MCF results in eigenfunctions in the form of Hermite polynomials
[74]. To simplify the problem, Gagliardi and Saleh suggest an engineering approximation to the spatial eigenfunctions in the form of complex exponentials [26, 75]. For a rectangular receiver, the spatial eigenfunctions can be represented by two-dimensional plane waves [26]
[ π + π /2/2exp YqyiXpxi ] w j ()r = (135) A
where x and y are the spatial components of r, A is the area of the receiver with dimensions X by
Y, and p, q are integers. The spatial eigenfunctions can be thought of as spatial harmonics with spatial frequencies that are multiples of the periods X and Y [26]. In essence, the optical field within rectangular receiver area, A, is decomposed into a series of spatial plane waves. Solution of (65) for a circular receiver leads to eigenfunctions proportional to generalized prolate spheroidal functions; an unnecessary route beyond the scope of this dissertation [26]. The total random field envelope is given by
90 ∞ ∞ ()r, = jj ()r ∑∑ ss ()tgbwetE , (136) j=0 s=0
substituting yields
1 ∞ ∞ ∞ ()r,tE = ∑ p ()()()π /2exp ∑ q π /2exp ∑ s exp ω tisbYqyieXpxie . (137) TA p=0 q=0 s=0
Since the time variation is assumed deterministic, the randomness of the received optical field is characterized by only the spatial modes. Looking at the spatial component of the field for some finite number of spatial modes, M, within the receiver area, the field becomes
M ()r = ∑ weE jj ()r . (138) j=0
91 5.3 Angular Spectrum
We choose to work with the angular spectrum because of the receiver architecture. In practice, a lens of other optical concentrating device is always used to collect the received light and image it onto a detector. The lens creates the Fourier Transform of the correlation function onto the focal plane; it also creates the Fourier Transform of the circular aperture in the focal plane. It is then easier to understand the concept of an angle rather than a spatial distance in the transform plane because it is essentially the field of view of the receiving telescope. The angular spectrum physically represents the power in each angular mode captured by the detector.
The complex field at any plane can be decomposed into two-dimensional plane waves using two- dimensional Fourier transforming. The Fourier components of the field can be identified as individual plane waves propagating within the analysis plane. The field amplitude at any point or other parallel plane can be realized by summing the contributions of the complex amplitude of these plane waves [73].
To define the angular spectrum, consider a wave incident on a transverse ( , yx ) plane propagating in the positive z direction, with complex amplitude ( yxU 0,, ) at z = 0 . The objective is to calculate the resulting field ( ,, zyxU )after propagating a distance z from the first plane.
At the z = 0 plane, the field has the two-dimensional Fourier transform given by [73]
92 ∞ , ffA 0; = − 2exp0,, π + dydxyfxfjyxU . (139) ()()(YX ∫∫ []YX ) ∞−
By writing the field as the inverse two-dimensional Fourier transform, its representation as a decomposition of complex exponentials is apparent [73],
∞ ,0, = 2exp0;, π + dfdfyfxfjffAyxU . (140) ()∫∫ (YX )[] (YX ) YX ∞−
The complex exponential functions of the Fourier decomposed signal physically represent plane waves. A propagating plane wave has the general form [73]
r ()exp;,, [ ( r −⋅= 2 νπ trkjtzyxp )], (141)
where ν is the velocity of the propagating wave, r = + ˆˆ + zzyyxxr ˆ is a position vector, and
r k = 2 λπ is the wave vector with direction cosines (α β,, γ ) as shown in Figure 12 such that
[73]
r 2π ()ˆˆ ++= γβα zyxk ˆ . (142) λ
93
Figure 12: Describing the wave vector with direction cosines [73].
Ignoring time dependence, the complex phasor amplitude of the plane wave across a constant z- plane is [73]
r r ⎡ 2π ⎤ ⎡ 2π ⎤ ()exp,, ()=⋅= exp⎢ ()+ βα ⎥ exp⎢ ()γ zjyxjrkjzyxP ⎥ . (143) ⎣ λ ⎦ ⎣ λ ⎦
Note the direction cosines are related by [73]
1 −−= βαγ 22 . (144)
Using this physical definition of the Fourier components, a complex exponential function at the
z = 0 plane can be thought of as a plane wave propagating with direction cosines [73]
94 2 2 = f X = fY 1 ()()X −−= λλγλβλα ff Y . (145)
In the Fourier decomposition of U with spatial frequencies ( , ff YX ) the complex amplitude of
the plane wave is ()YX 0;, dfdfffA YX , where f X = α λ and fY = β λ . Therefore the angular spectrum of ()yxU 0,, is [73]
⎛α β ⎞ ∞ ⎡ ⎛α β ⎞⎤ A⎜ ,0; ⎟ = ∫∫ ()⎢− 2exp0,, π ⎜ + ⎟⎥ dydxyxjyxU . (146) ⎝ λ λ ⎠ ∞− ⎣ ⎝ λ λ ⎠⎦
5.4 2-D Fourier Transform of the Mutual Coherence Function
The MCF is defined as the second-order moment of the optical field after propagating through turbulence. The Fourier Transform of the correlation function of the received optical field yields the power spectral density of the field. The PSD can be interpreted as the power distribution in the random amplitudes of the plane waves, or using the angular spectrum, the angular power.
The angular spread of a spherical wave propagating through turbulence can be calculated from the two-dimensional Fourier transform of the MCF. Assuming a radially symmetric MCF, the
observation points are related by ρ −= rr 21 [7].
The magnitude of the MCF for a spherical wave is given by [7]
95 ⎡ 1 ⎤ 2 ()ρ IL 0 exp, −=Γ sp ()ρ, LD ⎥ (147) ⎣⎢ 2 ⎦
P0 where I0 = is the received irradiance without turbulence effects, P0 is the transmitted ()4π L 2 optical power and ()4π L 2 represents the power decrease as the propagating wave spreads.
sp ()ρ, LD is the wave structure function for a spherical wave, assuming Kolmogorov turbulence
[7],
22 − 231 ⎪⎧ 09.1 n LlkC 0 ρρ << l0 ρ, LD = (148) sp ()⎨ 3522 ⎩⎪ 09.1 n LkC ρ l0 ρ <<<< L0
The angular spectrum arises from the two-dimensional Fourier transform of the MCF. Due to the radial symmetry of the MCF, the two-dimensional Fourier transform becomes the Fourier-
Bessel transform, also known as the Hankel transform of zero order. The Hankel transform of zero order is defined by [14]
∞ = 2 2 drJfrg ρρρπρπ (149) H0 {}() ∫ ()(0 ) 0
The Hankel transform of the MCF is then defined as
96 ∞ ρ, rSL Γ==Γ 2,2 drJL ρρρπρπ (150) H {}()()()20 () ∫ 2 0 0
∞ b ()= π 0 ∫ exp2 ()− 0 ()2 drJaIrS ρρρπρ (151) 0
where b = 35,2 as shown in (148) and
⎧ 22 − 31 −2 ⎪ 55.0 n 0 [mLlkC ] ρ << l0 a = ⎨ (152) 22 − 35 ⎩⎪ 55.0 n []lmLkC 0 ρ <<<< L0
For the regime ρ << l0 ,b = 2 , see (148), the transform becomes
∞ 2 ()= π 0 ∫ exp2 ()− 0 ()2 drJaIrS ρρρπρ (153) 0
Using the integral relation [7]
∞ 2 22 1 ⎛ b ⎞ − xa ()dxbxJxe = exp⎜− ⎟, ba >> 0,0 (154) ∫ 0 2 ⎜ 2 ⎟ 0 2a ⎝ 4a ⎠
we have
97 π ⎛ ()π r 2 ⎞ = IrS exp⎜− ⎟. (155) () 0 ⎜ ⎟ a ⎝ a ⎠
For the regimel0 ρ <<<< L0 , b = 35 , see (148), the transform becomes
∞ 35 ()= π 0 ∫ exp2 ()− 0 ()2 drJaIrS ρρρπρ , (156) 0
where the integration is carried out in APPENDIX B. The spectrum in Watts is approximated by
⎧ π ⎛ ()π r 2 ⎞ 2π ⎪ I exp⎜− ⎟ r >> 0 ⎜ ⎟ ⎪ a a l0 ⎪ ⎝ ⎠ rS ≈ W (157) () ⎨ ⎛ ⎞ [] ⎪ 47.3 ⎜ ()π r 2 ⎟ 2π 2π I exp − r >>>> ⎪ 0 6 ⎜ 6 ⎟ 5 ⎜ 5 ⎟ l0 L0 ⎩⎪ a ⎝ 9.0 a ⎠
5.4.1 Relating angular spectrum and spatial spectrum
The spatial spectrum, S(r) can be expressed in terms of direction cosines. The direction cosines are given by
98 cosθ x = α = λ f x
cos y == λβθ f y . (158)
cos z = γθ
Since the direction cosines are related by
2 2 2 x y coscoscos θθθ z =++ 1, (159)
2 2 cos z 1 ()()X −−== λλγθ ff Y . (160)
It is assumed the spatial frequencies are symmetrical thus cylindrical coordinates can be used such that
222 += ffr yx . (161)
Substituting yields
22 z 1cos −= λθ r . (162)
Using the trigonometric identity
2 22 2 z 1cos λθ r −=−= sin1 θ z , (163)
99 22 2 λ r = sinθ z . (164)
Performing a Taylor expansion about θ z and retaining the first term for small angles yields
θ r ≅ z . (165) λ
This is the relationship between the radial parameter of the two-dimensional Fourier Transform of the MCF and the angular spread from the z-axis.
5.5 Wavenumber PDF
Fully developed atmospheric turbulence is a cascade process of large eddies on the order of the outer scale continually breaking into smaller eddies on the order of the inner scale and then dissipating. This continuum of eddies occurs approximately within the volume of the large eddy.
The three-dimensional power spectrum of the eddies represents average refractive power among
11 the continuum of eddies and obeys a − 3 power law within the inertial subrange,
− 3112 1 1 ()κ =Φ 033.0 Cn κ κ >>>> , (166) l0 L0
100 where κ = 2π R is interpreted as the spatial wavenumber for an eddy with size R. The power spectrum normalized by the total refractive power i.e. the integral of the power spectrum over all spatial wavenumbers, of the eddies can be interpreted as a PDF of the spatial wavenumbers present within the turbulence [76]. Here we are interested in the PDF of spatial wavenumbers lying within the inertial subrange and along a single dimension.
The three-dimensional power spectrum can be reduced to the one-dimensional power spectrum in the case of a statistically homogeneous and isotropic field using the relation [7, 76]
1 Vd (κ ) ()κ −=Φ , (167) 2 κπ d κ
where V ()κ is the one-dimensional spectrum. Solving for V (κ )yields
∞ V ()2 ∫ Φ−= ()d κκκπκ . (168) ∞−
Completing the integration gives
6π − 352 1 1 V ()κ = 033.0 Cn κ κ >>>> . (169) 5 l0 L0
101 To form the PDF of spatial wavenumbers lying within the inertial subrange, we first find the total refractive power within the inertial subrange
1 l0 − 32 − 32 9π 2 ⎡ 11 ⎤ ∫ ()dV κκ = 033.0 Cn ⎢ − ⎥ . (170) 5 ⎢ 0 lL 0 ⎥ 1 L0 ⎣ ⎦
The PDF is then the one-dimensional spectrum normalized by the total refractive power
2 κ − 35 1 1 p ()κ = κ >>>> (171) − 32 − 32 3 ⎡ 11 ⎤ l0 L0 ⎢ − ⎥ ⎣⎢ 0 lL 0 ⎦⎥
5.6 Probability Density Function of Received Irradiance
We begin with the random portion of the received spatial optical field as derived in 5.2, equation
(138)
M ()r = ∑ weE jj ()r , (172) j=0
102 where M is the number of spatial modes, w j (r) defines the spatial modes (plane waves), and e j
is the random amplitude of the spatial modes. The random amplitude coefficients e j are themselves a sum
n iθ iθk e j Re == ∑ k er , (173) k=1
where n is a random number representing the number of effective eddies, or scatterers, along the propagation path. The amplitude of each spatial mode is a sum of zero mean, circular complex,
Gaussian random variables with uniform phase; this is true assuming all scattering is weak [67].
The phase delay of the propagating field, introduced by the eddies, will undergo many cycles of
2π, leading to a uniform phase distribution [67]. The random variables represent the scattered components of the optical field after passing through a scatterer.
Due to the orthogonality of the spatial modes in the set w j (r)}{ ,
∫ ( ) jj ( )dww rrr =1, (174) A
the irradiance of the received field can be expressed as the integral of the magnitude squared of the optical field within the field of view of the receiver,
103 M 2 = ∑ eI j . (175) j=0
Assuming the random number n represents a large number of scatterers along the propagation
path the central limit theorem can be used [67]. Here we take e j to be approximately statistically independent. The random sum in (173) then tends to a joint distribution of Rayleigh distributed amplitude (R) and uniform distributed phase (θ) [67].
⎛ ⎞ 2r ⎜ r 2 ⎟ R ()rp = ⎜ ⎟ r ≥ 0,exp (176) R2 ⎜ R2 ⎟ ⎝ ⎠
We first derive the conditional PDF of irradiance for a fixed number of scatterers along the propagation path, n = n . The conditional variance of the amplitude is then [67]
2 2 k == nrnR λ j (177)
and the conditional PDF of the amplitude for a single spatial mode is
2r 2 j − nr λ jj ()j n nrP == e 0 rj ∞≤≤ . (178) nλ j
Making the change of variable to irradiance,
104 2 , == IrrI jjjj (179)
and forming the Jacobian
1 drj = dI j , (180) 2 I j
then substituting into the amplitude PDF
2 I j − nI λ jj 1 ()j n nIP == e (181) nλ j 2 I j
we get the irradiance PDF for a single spatial mode,
1 − nI λ jj ()j n nIP == e 0 I j ∞≤≤ . (182) nλ j
The PDF of irradiance is an exponential distribution with mean parameter nλ j . Therefore the variance of the received field amplitude is the mean of the received field irradiance. We now
2 have the PDF of the irradiance of a single spatial mode e j and need to derive the PDF for M spatial modes. To do this we use the property that the convolution of M PDFs is equivalent to
105 the product of M characteristic functions [9]. The derivation is carried out in APPENDIX C and yields
M A j − nI λ ()n nIP == ∑ e j 0 I ∞≤≤ , (183) j=1 nλ j
where
1 A j = . (184) M λ ∏1− m m=1 λ j ≠ jm
We must now calculate the unconditional PDF for irradiance,
∞ () ∫ ()n == ()dnnPnIPIP . (185) 0
We must first derive the PDF of n. We assume the effective scatterers are Poisson distributed and the distance between the effective scatterers is independent and exponentially distributed with a unique mean valueγ k . We assume there are a deterministic number of scatterers N along the propagation path. We can write the PDF for a single scatterer as
106 1 −n γ kk ()k = enP 0 nk ∞≤≤ . (186) γ k
A derivation similar to the conditional PDF of irradiance is used to derive the PDF of scatterers along the path. The resulting PDF is
N B ()nP = ∑ k e−n γ k 0 n ∞≤≤ , (187) k=1 γ k
where
1 Bk = . (188) N γ ∏1− l l=1 γ k ≠kl
Substituting into (185), the integral becomes
M N ∞ Aj B 1 − nI λ ()IP = k j −n γ k dnee (189) ∑ ∑ γλ ∫ n j=1 j k=1 k 0
We will use the integral representation [70]
107 p 1 ⎛ x ⎞ ∞ ⎛ x2 ⎞ xK = exp⎜ t −− ⎟ ()p+− 1 xdtt > 0 , (190) p () ⎜ ⎟ ∫ ⎜ ⎟ ⎝ 22 ⎠ 0 ⎝ 4t ⎠
where p ()xK is a modified Bessel function of the second kind of order p. For a zero-order modified Bessel function of the second kind we can write
1 ∞ ⎛ x 2 ⎞ xK exp⎜ t −−= ⎟ −1 xdtt > 0 . (191) 0 () ∫ ⎜ ⎟ 2 0 ⎝ 4t ⎠
We rewrite the integral in (189) as
∞ ⎛ − I n ⎞ n−1 exp⎜ − ⎟ dn (192) ∫ ⎜ n γλ ⎟ 0 ⎝ kj ⎠
and make a change of variable, γ k = tn to get
∞ ⎛ − I ⎞ t −1 exp⎜ − ⎟ dtt . (193) ∫ ⎜ γλ t ⎟ 0 ⎝ kj ⎠
Equating the terms in the exponential we get
I x2 I x =→= 2 . (194) γλ kj t 4 γλ kj
108 We can now write the PDF for irradiance as
M N A j B ⎛ I ⎞ ()IP = 2 k K ⎜2 ⎟ . (195) ∑∑ 0 ⎜ γλγλ ⎟ j 1 j k== 1 k ⎝ kj ⎠
5.7 Statistical Moments of Irradiance and Scintillation Index
As stated before, the scintillation index is vital in predicting the performance of an FSO communication system. The first and second moment of irradiance are required to calculate the scintillation index.
5.7.1 First moment of irradiance
The mean irradiance, also known as the first moment is calculated by
∞ = ∫ ()dIIpII . (196) 0
Substituting (195) we get
109 ∞ M N A j B ⎛ I ⎞ I = 2 k K ⎜2 ⎟ dII (197) ∫ ∑∑ 0 ⎜ γλγλ ⎟ 0 j 1 j k== 1 k ⎝ kj ⎠
and pulling out the constants
M N ∞ A j B ⎛ I ⎞ I = 2 k K ⎜2 ⎟ dII . (198) ∑∑ γλ ∫ 0 ⎜ γλ ⎟ j j k== 11 k 0 ⎝ kj ⎠
The integral can be solved using the relation [7]
∞ μ−1 ⎛12 μ ++ p ⎞ ⎛1 μ −+ p ⎞ μ ()dxaxKx Γ= ⎜ ⎟Γ⎜ ⎟ ∫ p μ+1 2 2 , (199) 0 a ⎝ ⎠ ⎝ ⎠ p ,0 μ ap >−>≥ 0,1
where p ()xK is a modified Bessel function of the second kind of order p and Γ()x is the gamma function. Making the change of variable
1 dvIv =→= dI 2 I , (200) = vI 2 = 2 dvvdI
and noting the limits stay the same we can concentrate on the integral
110 ∞ ⎛ ⎞ ∞ ⎛ ⎞ ⎜ I ⎟ ⎜ v ⎟ 2 K0 2 = KdII 0 2 2 dvvv . (201) ∫ ⎜ γλ ⎟ ∫ ⎜ γλ ⎟ 0 ⎝ kj ⎠ 0 ⎝ kj ⎠
2 Identifying μ ,0,3 ap === the integral becomes γλ kj
⎡ ⎤ ⎢ ⎥ ⎢ 22 ⎥ = 2⎢ ΓΓ 22 ⎥ 4 () () ⎢⎛ ⎞ ⎥ ⎢⎜ 2 ⎟ ⎥ . (202) ⎢⎜ γλ ⎟ ⎥ ⎣⎝ kj ⎠ ⎦ 2 ()γλ kj = 2
Substituting back into (198) yields
M N 2 M N A j Bk ( γλ kj ) I = 2 ∑∑ = jj ∑∑ BA γλ kk . (203) j j k== 11 γλ k 2 j=1 k=1
The first moment can be further simplified. Carrying out the sum for the first four terms reveals a pattern and the first moment can be rewritten as
M N I = j ∑∑ γλ k . (204) j k== 11
111 5.7.2 Second moment of irradiance
The second moment is calculated by
∞ = ∫ 22 ()dIIpII . (205) 0
Substituting (195) we get
∞ M N A j B ⎛ I ⎞ I 2 = 2 k K ⎜2 ⎟ 2 dII (206) ∫ ∑∑ 0 ⎜ γλγλ ⎟ 0 j 1 j k== 1 k ⎝ kj ⎠
and pulling out the constants
M N ∞ A j B ⎛ I ⎞ I 2 = 2 k K ⎜2 ⎟ 2 dII . (207) ∑∑ γλ ∫ 0 ⎜ γλ ⎟ j j k== 11 k 0 ⎝ kj ⎠
Using the integral relation in (199) we make the change of variable
1 dvIv =→= dI 2 I , (208) = vI 2 = 2 dvvdI
112 and noting the limits stay the same we can concentrate on the integral
∞ ⎛ ⎞ ∞ ⎛ ⎞ ⎜ I ⎟ 2 ⎜ v ⎟ 2 2 K0 2 = KdII 0 2 ()2 dvvv . (209) ∫ ⎜ γλ ⎟ ∫ ⎜ γλ ⎟ 0 ⎝ kj ⎠ 0 ⎝ kj ⎠
2 Identifying μ ,0,5 ap === the integral becomes γλ kj
⎡ ⎤ ⎢ ⎥ ⎢ 24 ⎥ = 2⎢ ΓΓ 33 ⎥ 6 () () ⎢⎛ ⎞ ⎥ . (210) ⎢⎜ 2 ⎟ ⎥ ⎢⎜ γλ ⎟ ⎥ ⎣⎝ kj ⎠ ⎦ 3 = 2()γλ kj
Substituting back into (207)
M N M N 2 A j Bk 3 2 2 I = 2 ∑∑ 2()γλ kj = 4 jj ∑∑ BA γλ kk (211) j j k== 11 γλ k j=1 k=1
The second moment can be further simplified. Carrying out the sum for the first four terms reveals a pattern and the second moment can be rewritten as
113 ⎛ M M −1 M ⎞⎛ N N−1 N ⎞ I 2 ⎜ λ2 += 22 ⎟⎜ γλλ 2 + 22 γγ ⎟ ⎜ ∑ j ∑∑r s ⎟⎜ ∑ k ∑∑ r s ⎟ ⎝ j=1 s =+11=sr ⎠⎝ k=1 s =+11=sr ⎠ ⎛ M M M −1 M ⎞⎛ N N N −1 N ⎞ ⎜ 2 λλ 2 ++= 2 ⎟⎜ 2 γγλλ 2 ++ 2 γγ ⎟ . (212) ⎜ j ∑∑ j ∑∑r s ⎟⎜ k ∑∑ k ∑∑ r s ⎟ ⎝ j 1 j== 1 s =+11=sr ⎠⎝ k 1 k== 1 s =+11=sr ⎠ ⎡ 2 ⎤ 2 M ⎛ M ⎞ ⎡ N ⎛ N ⎞ ⎤ ⎢ 2 += ⎜ ⎟ ⎥ ⎢ 2 + ⎜ γγλλ ⎟ ⎥ ⎢ j ⎜ j ⎟ ⎥ k ⎜∑∑∑∑ k ⎟ j=1 j=1 ⎢k=1 ⎝ k=1 ⎠ ⎥ ⎣⎢ ⎝ ⎠ ⎦⎥ ⎣ ⎦
5.7.3 Scintillation index
The scintillation is defined as the normalized second moment,
2 − II 2 I 2 2 σ I = −= 1. (213) I 2 I 2
Substituting (204) and (212) yields
⎡ 2 ⎤ 2 M ⎛ M ⎞ ⎡ N ⎛ N ⎞ ⎤ ⎢ 2 + ⎜ ⎟ ⎥ ⎢ 2 + ⎜ γγλλ ⎟ ⎥ ⎢ j ⎜ j ⎟ ⎥ ⎢ k ⎜∑∑∑∑ k ⎟ ⎥ ⎢ j=1 ⎝ j=1 ⎠ ⎥ ⎣k=1 ⎝ k=1 ⎠ ⎦ σ 2 = ⎣ ⎦ −1. (214) I 2 ⎡ M N ⎤ ⎢ j ∑∑ γλ k ⎥ ⎣⎢ j k== 11 ⎦⎥
114 The scintillation index can be rewritten as follows
⎡ 2 ⎤ 2 M ⎛ M ⎞ ⎡ N ⎛ N ⎞ ⎤ ⎢ 2 ⎜ ⎟ ⎥ ⎢ 2 ⎥ j + ∑∑ λλ j k + ⎜ γγ k ⎟ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜∑∑ ⎟ ⎥ ⎢ j=1 ⎝ j=1 ⎠ ⎥ k=1 ⎝ k=1 ⎠ σ 2 = ⎣ ⎦ ⎣ ⎦ −1 I 2 2 ⎡ M ⎤ ⎡ N ⎤ γ ⎢∑λ j ⎥ ⎢∑ k ⎥ ⎢ j=1 ⎥ ⎣⎢k=1 ⎦⎥ ⎣ ⎦ . (215) ⎛ ⎞ ⎜ M ⎟⎛ N ⎞ λ2 ⎜ 2 ⎟ ⎜ ∑ j ⎟⎜ ∑γ k ⎟ ⎜ j=1 ⎟ 1+= ⎜1+ k=1 ⎟ −1 ⎜ 2 ⎟⎜ 2 ⎟ ⎜ ⎡ M ⎤ ⎟ ⎡ N ⎤ ⎜ γ ⎟ ⎜ ⎢ λ j ⎥ ⎟ ⎢∑ k ⎥ ⎜ ∑ ⎟⎜ ⎢ ⎥ ⎟ ⎝ ⎣⎢ j=1 ⎦⎥ ⎠⎝ ⎣k=1 ⎦ ⎠
Carrying the multiplication of the two terms
⎛ ⎞ M N ⎜ M ⎟⎛ N ⎞ λ2 2 λ2 ⎜ 2 ⎟ ∑ j ∑γ k ⎜ ∑ j ⎟⎜ ∑γ k ⎟ j=1 ⎜ j=1 ⎟ σ 2 = + k=1 + 2 ⎜ k=1 ⎟ −1. (216) I 2 2 ⎜ 2 ⎟⎜ 2 ⎟ ⎡ M ⎤ ⎡ N ⎤ ⎜ ⎡ M ⎤ ⎟ ⎡ N ⎤ γ ⎜ γ ⎟ ⎢ λ j ⎥ ⎢∑ k ⎥ ⎜ ⎢ λ j ⎥ ⎟ ⎢∑ k ⎥ ∑ ⎢ ⎥ ⎜ ∑ ⎟⎜ ⎢ ⎥ ⎟ ⎣⎢ j=1 ⎦⎥ ⎣k=1 ⎦ ⎝ ⎣⎢ j=1 ⎦⎥ ⎠⎝ ⎣k=1 ⎦ ⎠
Since λ j and γ k are positive terms then the following inequality is valid for a single term in each of the summations
2 2 ⎡ M ⎤ M N N 2 ⎡ ⎤ 2 ⎢ j ⎥ ≥ ∑∑ λλ j and ⎢ k ⎥ ≥ ∑∑ γγ k . (217) ⎣⎢ j=1 ⎦⎥ j=1 ⎣⎢k=1 ⎦⎥ k=1
115 2 Therefore, the theoretical maximum allowable scintillation index is σ I = 3
5.8 Relating Model Parameters to Physical Quantities
In this section, the model parameters of the PDF, namely the λ's andγ 's , are related to physical measurable quantities of the optical wave. The solution of (65) would have yielded the exact eigenfunctions and eigenvalues for the spatial modes. However, we chose to simplify the problem via an engineering approximation which leads to spatial eigenfunctions of the form of two-dimensional plane waves. Saleh [75] points out that when the eigenfunctions of the KL expansion are approximated with harmonic eigenfunctions (sinusoids), the eigenvalues are then approximated by samples from the Fourier Transform of the correlation function. Recall the definition of the angular spectrum derived in section 5.3. The physical behavior of the received irradiance can be analyzed in two planes, aperture and focal. The two planes are related through the Fourier transform and have a reciprocal relationship. The physical interpretation of the model parameters are described in both the aperture plane and focal plane for completeness.
116 5.8.1 Sampling the spatial spectrum
Recall an approximation to the KL expansion was performed on the received optical field. The spatial spectrum of the optical wave was derived by taking the two-dimensional Fourier
Transform of the MCF. Assuming the received optical field is stationary, then in the limit of the radial distance ρ much larger than the width of the mutual coherence function Γs ()ρ at the 1/e point, we have [75]
2π λ ≅ ()rS jj , j = jjr = K,,2,1, M (218) ρcut
where j is an integer, ρcut is the radial distance much larger than the 1/e point of the MCF, each
λ j is the corresponding eigenvalue of the eigenfunctions defined by the approximation to the KL expansion, and (rS j ) is a sample of two-dimensional Fourier Transform of the MCF. This assumption can be made if the sampling is so fine such that Δr is infinitesimally small, however the units are not consistent. As stated in section 5.6, the variance of the received field amplitude is the mean of the received field irradiance. This implies each eigenvalue, λ should have units of irradiance [W/m2], while sampling S(r) yields units of [W]. This discrepancy is noted, however due to the nature of the approximations being made it is dismissed and the units are taken to be that of irradiance.
117 There is no formal definition of how much larger ρ should be compared to the 1/e point of the
MCF. After experimenting with many values a value of 250 times the 1/e point of the MCF was chosen, this allowed the model to avoid numerical issues over a broad range of conditions. The numerical issues encountered are discussed in APPENDIX E in more detail.
From (157), the spatial spectrum is only valid within the inertial subrange defined by the inner
1 2π and outer scales of turbulence. Upper and lower cutoff values were chosen as κ m = and 2 l0
2π κ0 = 2 respectively. Again, experimenting with several other cutoff values, the values L0 chosen allowed the model to avoid numerical issues over a broad range of conditions.
The total number M of λ's is related to aperture size. As the aperture size increases, M increases, this accounts for aperture averaging. Increasing M is physically equivalent to more spatial plane waves being collected in the aperture plane. On the other hand, increasing M is also physically equivalent to less angular modes being collected in the focal plane, i.e. the field of view is reduced with a larger aperture. Parseval’s theorem requires the integral of the spatial
spectrum yield the total irradiance, I 0 ; i.e. energy conservation. Since the spatial spectrum is
only sampled within the upper and lower cutoffs, the sum of the lambdas will never equal I 0 .
However, as defined in (204) the first moment, or mean irradiance, is the product of the sum of the lambdas and gammas. The PDF of irradiance is typically plotted with data normalized by the mean irradiance. In log space, this roughly centers the PDF about 0 dB. In order to normalize the theoretical PDF to a mean irradiance of unity, the following constraint is defined
118 M N I 1 j === ∑∑ γλ k . (219) j= k=11
This is implemented by always normalizing each lambda by the sum of all lambdas. There are many ways to scale the λ's andγ 's such that the condition of unit mean irradiance is satisfied.
However, the method presented is easy to implement and maintains scaling of the PDF such that the peak probability is always less than or equal to unity. This method also makes comparing the theoretical PDF to normalized measured data easier. Figure 13 illustrates how the spatial spectrum is sampled within the limits of the inertial subrange.
Figure 13: Method of obtaining eigenvalues from sampling spatial spectrum.
119 5.8.2 Sampling the wavenumber PDF
We are interested in the distribution of the number of effective scatterers along the propagation path. In section 5.5 a PDF was derived based upon the one-dimensional refractive index power spectrum of the atmosphere. Physically this represents the probability density of encountering eddies with a particular spatial wavenumber within the turbulence.
We assume the propagating wave encounters a finite deterministic number of scatterers along the path. The irradiance of each spatial mode is a product of the number of scattered components N contributing to the irradiance. The number of scattered components captured by an aperture is defined by N. For a point aperture N = 1, i.e. one scattered component of the wave is detected.
As the aperture size increases, it captures more scattered components of the received optical field in the aperture plane and N increases. On the other hand, increasing N is equivalent to receiving less angular components in the focal plane, i.e. the field of view is reduced with a larger aperture.
The γ 's will be calculated by integrating a portion of the wavenumber PDF, such that each
γ represents a probability and is unitless.
From (171), the wavenumber PDF is only valid within the inertial subrange defined by the inner and outer scales of turbulence. The wavenumber PDF can be generalized to any upper and lower cutoff values,
2 κ − 35 p ()κ = >>>> κκκ (220) 3 − − 3232 2 1 []0 − κκ m
120 where κ m and κ0 are the upper and lower cutoffs respectively. Upper and lower cutoff values
1 2π 2π were chosen as κ m = and κ0 = 2 respectively. The γ 's are then calculated by 2 l0 L0 dividing the valid region of the wavenumber PDF into N equally spaced segments,
N 1 N2 κm 21 K,,, j = () ()κκκκγγγ K ,,, ∫∫∫ ()dpdpdp κκ . (221) κ0 N1 Nk
Figure 14 shows an illustration of calculating the γ 's by integrating the wavenumber PDF, this graph corresponds to N = 20 .
Figure 14: Method of calculating gammas from integrating the wavenumber PDF; N=20.
121 6 EXPERIMENTATION
Irradiance data were collected over a 1km horizontal path for several different sized receiving apertures. The data were collected under moderate-to-strong turbulence conditions. The path was instrumented with scintillometers and anemometers to provide real time atmospheric data.
Histograms were formed with the data and compared to the existing Gamma-Gamma PDF model, as well as the new PDF developed.
6.1 Overview
The experiment was performed at the Innovative Science and Technology Experimental Facility
(ISTEF) laser range in Cape Canaveral, Kennedy Space Center, Florida. The range is 1km long and approximately 14m wide. There was 1-2m high vegetation surrounding the length of the range. The propagation path was trimmed grass kept at approximately 5cm tall. A mobile laser propagation laboratory was placed at the north end of the range to provide a clean and dry environment to operate equipment. The mobile laboratory was equipped with an isolated 122cm x 244cm optical bench supported from the ground separate from the mobile lab, 5kVA uninterruptible power supply, safety glasses, laser curtain, air conditioner, and a removable window to propagate through. The ISTEF laser lab was located at the north end of the range and provided an air conditioned environment for the transmitting equipment. Figure 15 illustrates the layout of the experimental setup on the ISTEF laser range.
122 The transmitter system consisted of a CW 532nm DPSS laser and optics to diverge the beam such that it was approximately a spherical wave at the receiver. The receiver system consisted of a 154mm (6”) refracting telescope outfitted with several removable apertures. The telescope was coupled with a high-speed, high-dynamic range power meter. The output of the power meter was sampled using a digitizer controlled with LabVIEW SignalExpress. In addition to the experimental setup, a suite of equipment was setup to characterize the atmosphere along the
2 optical propagation path. Three scintillometers were used to measure the Cn along the path, two of which were also capable of measuring the inner scale. Three sonic anemometers were positioned along the path to measure the three-dimensional wind vector.
123
2 Figure 15: Layout of equipment on 1km laser range. Three scintillometers were used to provide path-averaged Cn and l0. Three 3-axis anemometers provided the crosswind at points along the path.
Parallel to the experimental propagation path were three commercial scintillometers, one model
Scintec BLS900 and two model Scintec SLS-20. The BLS900 scintillometer provided a path-
2 averaged value of Cn and an estimate of the path-averaged crosswind speed over the 1km
2 propagation path. The SLS 20 scintillometers provided a path-averaged value of Cn and a path- average value of the inner scale of turbulence, l0 for a 100m path. The scintillometers were positioned on the laser range to verify the assumption of homogenous turbulence along the path.
In addition to the scintillometers, three 3-axis sonic anemometers were positioned along the path.
124 The reported crosswind from the anemometers was used to verify homogeneity as well as verify the assumption of frozen turbulence.
6.2 Equipment
This section goes into detail about the key system elements of the experimentation; scintillometers, transmitter, and receiver. Design details as well as theory of operation are discussed.
2 6.2.1 Scintillometer – path averaged Cn
2 The Scintec BLS 900 is a commercially available scintillometer capable of calculating Cn and cross wind speed directly from optical measurements. The instrument boasts an operable range of 0.5km-5km as well as valid measurements in all regimes of turbulence. The BLS series of scintillometers is based upon the assumption of weak turbulence and utilizes aperture averaging theory developed in the late 1970’s [77]. The scintillometer hardware consists of two incoherent transmitting disks and one receiving aperture, all approximately the same diameter. Proper
2 selection of transmit and receive apertures allow the scintillometer to provide accurate Cn measurements without taking into account the inner and outer scale effects.
125 Under weak irradiance fluctuations, the scintillation index and the log-amplitude variance are
2 2 directly proportional to Cn . Weak fluctuations become strong fluctuations by increasing Cn or the propagation distance, or both, for a fixed wavelength. Under strong irradiance fluctuations
2 (saturation) the linear relation between the log-amplitude variance and Cn no longer holds. The
BLS 900 is claimed to be relatively insensitive to strong fluctuation effects due to its large incoherent emission and reception apertures. Therefore, its operation is described by the use of weak scattering theory.
The BLS 900 has two transmitters and one receiver. The two transmitting disks consist of nearly
1000 infrared LEDs and are 15cm in diameter. Each transmitter disk has a different modulation frequencies, this reduces the background and allows the two signals to be separated by a single detector in the receiver. The receiver consists of a 14.5cm lens which collimates the incoming radiation onto two silicon photodiodes. One photodiode is used to sense the turbulence-induced fluctuations, while the other is strictly for alignment. The fluctuations received by the photodiode are processed with a signal-processing unit made by Scintec, the calculation results are then sent to a PC running Scintec’s BLSRun software[78].
The scintillometer measures the average received intensity from disk 1 and 2,
126 ⎡ ⎤ 1 σ11 B11 ⎢1ln += ⎥ (222) 4 ⎢ 2 ⎥ ⎣ I1 ⎦
⎡ ⎤ 1 σ 22 B22 ⎢1ln += ⎥ (223) 4 ⎢ 2 ⎥ ⎣ I 2 ⎦
⎡ ⎡ 1 1 ⎤⎤ ⎛ ⎞ 2 ⎛ ⎞ 2 1 ⎢ σ12 ⎢⎜ σ11 ⎟ ⎜ σ 22 ⎟ ⎥⎥ B12 ⎢1ln += ⎢ +1 + −11 ⎥⎥ . (224) 4 σσ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎢ 2211 ⎢⎝ I1 ⎠ ⎝ I 2 ⎠ ⎥⎥ ⎣ ⎣ ⎦⎦
Weak scattering theory relates the log-amplitude variances B11, B22, and covariance B12 to the three-dimensional spectrum of refractive index fluctuations [77]
∞ R ⎛ 2 ()− xRxk ⎞ == 4 2 dkBB Φ ()κκπ dxsin 2 ⎜ ⎟ × 2211 ∫∫n ⎜ 2kR ⎟ 00⎝ ⎠ 2 2 ⎡ ⎛ κ r xD ⎞⎤ ⎡ ⎛ κ t ()− xRD ⎞⎤ , (225) J1⎜ ⎟ ⎢ J1⎜ ⎟⎥ ⎢ 2R ⎥ 2R ⎢2 ⎝ ⎠⎥ ⎢2 ⎝ ⎠⎥ ⎢ κ r xD ⎥ ⎢ κ t ()− xRD ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 2R ⎦ ⎣ 2R ⎦
127 ∞ R ⎛ 2 ()− xRxk ⎞ ⎛ ⎛ x ⎞⎞ = 4 2 dkB Φ ()κκπ dxsin 2 ⎜ ⎟ ⎜κ SJ ⎜1− ⎟⎟ × 12 ∫∫n ⎜ 2kR ⎟ 0 ⎜ t R ⎟ 00⎝ ⎠ ⎝ ⎝ ⎠⎠ 2 2 ⎡ ⎛ κ r xD ⎞⎤ ⎡ ⎛ κ t ()− xRD ⎞⎤ , (226) J1⎜ ⎟ ⎢ J1⎜ ⎟⎥ ⎢ 2R ⎥ 2R ⎢2 ⎝ ⎠⎥ ⎢2 ⎝ ⎠⎥ ⎢ κ r xD ⎥ ⎢ κ t ()− xRD ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 2R ⎦ ⎣ 2R ⎦
where k = 2π λ is the wavenumber, x is the length coordinate along the propagation path, R is the propagation path length, Dr, and Dt are the receiver and transmitter diameters, respectively.
The two J1 Bessel function terms account for finite circular incoherent transmitting and receiving apertures.
The three-dimensional spectrum used by the scintillometer is given by [78]
− 3/112 n ()κ =Φ 033.0 n ( 0 ) (κκλκ LFfC 0 ) (227)
where f ()κλ0 and ()κLF 0 represent the effects of l0 and L0, respectively. The inner scale effects
approach unity as κ << /1 l0 and the outer scale effects approach unity as κ >> /1 L0 . Since the inner scale is typically on the order of millimeters and the outer scale is typically on the order of meters, the diameters chosen for the size of the radiating disks are not affected by the inner or outer scale. Also, by having the disks much larger than the Fresnel zone λ R , the wavelength dependence is also eliminated. These assumptions are used to create a scintillometer independent of inner, outer scale, and wavelength.
128 Inserting (227) into (225), the integration leads to the following approximate expression [77]
2 37 −3 37 −3 = α 11 trn = α 22 tr RDBRDBC , (228)
where the scaling factor α r = 629.4 for the BLS 900 and is dependent upon the ratio of the transmitting and receiving apertures [78].
For the case of strong turbulence the above equations are modified with the addition of a short- term modulation transfer function (MTF) [79]. The BLS accounts for the saturation effects in strong turbulence by using a look-up table to scale the log-amplitude variance defined in (225)
[78].
In weak turbulence, absorption fluctuations caused by a reduction in transmission along the path, such as fog and dust, are taken into account by the scintillometer. If not accounted for, the absorption fluctuations could be falsely interpreted as scintillation effects from the turbulence.
2 From the calculated Cn the BLS 900 computes a slew of other atmospheric parameters such as
Fried’s parameter, scintillation at a point detector for spherical and plane waves, and the structure function of phase. The BLS 900 also computes the cross wind speed from the time- lagged cross-covariance function [78].
129 2 6.2.2 Scintillometer – path averaged Cn and l0
2 The Scintec SLS 20 is a commercially available scintillometer capable of calculating Cn and l0 directly from optical measurements. The instrument is restricted to weak fluctuation conditions and therefore operates over a maximum range of 150m. The scintillometer hardware consists of two coherent emitters, separated by a fixed distance perpendicular to the propagation path. The receiver has two detectors separated by the same distance as the transmitters to observe the radiation. The detectors are aligned to produce two parallel paths between the transmitter and receiver [80].
Under weak irradiance fluctuations, the scintillation index and the log-amplitude variance are
2 directly proportional to Cn . Therefore equations (222) through (225) are valid for this scintillometer. The covariance of the logarithm of the amplitude of the received radiation (226) is simplified for this scintillometer on account of the two parallel paths with equal diameter transmit and receive elements [80]
2 ⎡ ⎛ κ xD ⎞⎤ J ⎜ ⎟ ∞ R ⎛ 2 ()− xRxk ⎞ ⎢ 1 2R ⎥ = 4 2 dkB Φ κκπ dxsin 2 ⎜ ⎟ κ dJ ⎢2 ⎝ ⎠⎥ , (229) 12 ∫∫n () ⎜ ⎟ 0 () ⎝ 2kR ⎠ ⎢ κ xD ⎥ 00 ⎢ ⎥ ⎣ 2R ⎦
where k = 2π λ is the wavenumber, x is the length coordinate along the propagation path, R is the propagation path length, D is the receiver and transmitter diameters, and d is the separation
130 distance of the transmitters and receivers. This formula is referenced from the operations manual of the instrument, but the covariance equation mixes plane wave and spherical wave theory without any justification; see pg. 280 of [7] for proper equations. The operation manual states the transmitter uses spatial filter pinholes; therefore it is speculated that spherical wave theory is more appropriate than plane wave theory. The log-amplitude variances B11, B22 are derived from
(229) by letting d = 0 .
The three-dimensional spectrum used by the scintillometer is given by [80]
− 3/112 n ()κ =Φ 033.0 n fC ( λκκ 0 ) (230)
where f ()κλ0 takes the form of Hill’s model and represent the effects of l0.
2 Both the log-amplitude variances B11, B22, and the covariance B12 are dependent upon both Cn and l0. However the correlation coefficient is dependent upon only l0
B B B r 12 12 === 12 . (231) BB 21 B1 B2
Therefore the instrument first calculates l0 from the correlation coefficient then references a
2 2 look-up table to calculate the inner scale effects on Cn . From the calculated Cn and l0 the SLS
20 computes a slew of other atmospheric turbulence parameters such as the structure constant for
131 temperature, the dissipation rate of the kinetic energy of turbulence, turbulent kinematic heat fluxes and momentum, turbulent heat flux, and turbulent momentum flux [80].
6.2.3 Transmitter system
The 532nm DPSS laser had a maximum output power of approximately 800mW. As the output power approached the maximum, the spatial mode structure transformed from the fundamental mode to TEM20. A spatial filter on the order of 1mm was positioned in the center of the fundamental mode to remove the additional spatial modes. Following the spatial filter was a defocused beam expander. By inputting collimated light from the laser and defocusing the beam expander, a good approximation to a spherical wave was achieved at the receiver. The estimated spot diameter at the receiver was approximately 4m, including turbulence effects, indicating a full-angle divergence of approximately 4mrad. Calculating the refractive beam parameter using the equation [7]
2z 0 =Λ 2 (232) kW0
yields 75, implying the receiver was very much in the far-field and thus the wavefront was approximately spherical at the receiver. Figure 16 illustrates the optical setup at the transmitter.
132
Figure 16: Optical layout of transmitter system.
6.2.4 Receiver system
The receiving telescope and corresponding optics were recycled from a previous experiment.
The telescope was setup as a pupil plane imager. The primary lens was a single 6” diameter refracting lens. A relay lens was placed at the focus of the primary lens such that it imaged the pupil plane of the system. The relay lens was chosen such that a 2mm image of the pupil plane was created at focus. The relay lens acts as a field stop for the telescope, defining the field of view of the optical system. The advantage of this system is its minimal dependence on the focus of the receiving telescope. To the first order, the telescope can change focus and there will minimal effect on the light received by the detector. This is advantageous when tracking objects at varying range, also when using Cassegrain type telescopes in an outdoor environment where thermal variations of the telescope tube cause a shift in focus. Figure 17 illustrates the optical components of the receiving telescope.
133 A New Focus 2101 power meter was purchased for this experiment. It has a high dynamic range, ~70dB, and a 5mm diameter silicon photodiode. Due to the large active area of the power meter, alignment in the focal plane of the relay lens was very forgiving. The output voltage of the power meter was recorded with a National Instruments NI-9234 digitizer. The digitizer has
24 bits of resolution and was operated at 51.25kS/s. The digitizer was coupled with a National
Instruments NI ENET 9163 Ethernet carrier. The acquired data was controlled via LabVIEW
SignalExpress over Ethernet.
Figure 17: Optical layout of receiver system.
134
Figure 18: Receiving telescope with aperture in place.
6.3 Data
2 A set of aperture averaging curves similar to those in Figure 9 was printed out for a range of Cn values expected during the data collection. These curves were used in conjunction with real-time
2 measurements of Cn from the scintillometers to decide what aperture size to collect data with.
All data were collected on October 2 2009 at the ISTEF laser range. Each data set was recorded
135 over a two minute collection period. The background was measured for twenty seconds before the start of each data set
Data were collected with a wide range of apertures such that different degrees of aperture averaging occurred. Due to the limiting scintillation index of 3 for the new PDF, not all data is presented, only those data sets where the calculated scintillation index was less than 3.
Each data set was processed by first converting the output of the power meter from volts to a measured optical power as outlined in APPENDIX D. The average background was then subtracted from each data point. The histogram of the background subtracted data set was then created using unequal bin widths as described in section 2.1.2.1. Unequal bin widths were necessary due to the large dynamic range of the recorded irradiance signal.
Table 1 presents the average atmospheric conditions observed over each two-minute data collection. The crosswind speed changes on a much slower time scale than the turbulence parameters. Throughout the three hours of data collection, the average wind speed did not change much. At the transmitter the average wind speed was approximately 0.05 m/s, with gusts up to 1.5 m/s in magnitude. At the middle of the path the average wind speed was approximately
0.1 m/s, with gusts up to 1.5 m/s in magnitude. At the receiver the average wind speed was approximately 0.02 m/s, with gusts up to 2 m/s in magnitude.
The detail of the PDF is best seen on a log-log scale. The abscissa is the irradiance value normalized by the mean irradiance ( IILog ) and the ordinate is the logarithm of the
136 probability density. To the first order, in the log-log domain the PDF has the shape of an upside down parabola or horseshoe. Physically this makes sense; the tips of the parabola represent extremely low probability events. These events correspond to the extreme irradiance values, deep fades and peak surges, which are known to be rare events. In each of the PDF plots below, three curves are plotted against points from the histogram. The new PDF is fit to the data with a best fit. The criteria of best fit are two-fold: how close did the theoretical curve match the histogram and how close was the theoretical scintillation index to that calculated from the data.
The GG is also fit to the histogram, but a fitting algorithm was used. For comparison, the GG is plotted with the parameters calculated according to scintillation theory. The best fit GG model is presented to show if the model is even capable of achieving the shape of the measured PDF. The theoretical GG model is presented to show how well scintillation theory predicts the parameters
α and β .
All of the plots were created with the corresponding atmospheric conditions presented in Table
1. The path length was fixed at 980m and the height above ground of the receiver was 2m. As discussed in section 3.4, a constant outer scale value of L0 = × = 8.024.0 mm was used for all the data sets. The outer scale is a statistical quantity and was not directly measured during the
2 data collection. Small variations in the outer scale undoubtedly took place as Cn and l0 changed over the three hours of data collection.
137 Table 1: Atmospheric conditions observed during data collection. Aperture Time Average Average l Diameter 0 (EST) C 2 (m-2/3) (mm) Size (mm) n 16:01 1.5 5.00 x 10-14 5.69 15:32 4.0 8.50 x 10-14 6.00 16:19 4.0 9.00 x 10-14 6.02 16:15 7.0 7.50 x 10-14 5.93 13:15 10.0 3.56 x 10-13 4.98 15:26 10.0 5.50 x 10-14 4.91 16:11 10.0 7.27 x 10-14 5.98 13:11 20.6 3.70 x 10-13 4.86 15:21 20.6 8.89 x 10-14 5.54 15:56 20.6 7.00 x 10-14 5.90 13:07 55.0 2.65 x 10-13 5.43 15:17 55.0 1.30 x 10-13 5.71 15:51 55.0 1.05 x 10-13 6.23 15:47 101.6 1.40 x 10-13 5.74 13:04 154.0 3.47 x 10-13 5.46 15:13 154.0 2.36 x 10-13 5.42 15:44 154.0 1.65 x 10-13 6.17 16:06 154.0 6.50 x 10-14 6.25
The following figures compare the new PDF developed in this dissertation and the GG PDF against the histogram of the experimental data. The title identifies the time of day the data set
2 was recorded, the aperture size on the receiving telescope, and the Rytov variance β0 for a spherical wave (as an indication to the strength of turbulence). The caption restates the aperture
2 size as well as the Cn from Table 1, the spatial coherence radius ρsp is also presented. Printed on each plot are the parameters used to create the PDF curves. The parameters M and N of the new PDF were selected by eye for best fit to the histogram data, as well as closest match to the scintillation index calculated from the data. The parameters α fit and β fit of the best-fit GG
PDF were selected using a Levenberg-Marquardt least-squares fitting algorithm implemented by the lsqcurvefit function in Matlab. The GG PDF was fit to the histogram data using α fit and
138 β fit as the free-parameters. The parameters αthry and βthry of the theoretical GG PDF were calculated using (80) and the equations for the scintillation index of a spherical wave under general irradiance conditions as outlined in APPENDIX A. In some of the plots, the new PDF is truncated at low irradiance values. This is due to numerical inaccuracies of the new model; these numerical issues are discussed in more detail in APPENDIX E.
2 -14 Figure 19: New PDF, GG PDF, and histogram of data; 1.5mm aperture, Cn = 5.00 x10 , ρsp= 7.16mm.
139
2 -14 Figure 20: New PDF, GG PDF, and histogram of data; 4.0mm aperture, Cn = 8.50 x10 , ρsp= 5.21mm.
2 -14 Figure 21: New PDF, GG PDF, and histogram of data; 4.0mm aperture, Cn = 9.00 x10 , ρsp= 5.03mm.
140
2 -14 Figure 22: New PDF, GG PDF, and histogram of data; 7.0mm aperture, Cn = 7.50 x10 , ρsp= 5.61mm.
2 -13 Figure 23: New PDF, GG PDF, and histogram of data; 10.0mm aperture, Cn = 3.56 x10 , ρsp= 2.21mm.
141
2 -14 Figure 24: New PDF, GG PDF, and histogram of data; 10mm aperture, Cn = 5.50 x10 , ρsp= 6.76mm.
2 -14 Figure 25: New PDF, GG PDF, and histogram of data; 10mm aperture, Cn = 7.27 x10 , ρsp= 5.72mm.
142
2 -13 Figure 26: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 3.70 x10 , ρsp= 2.15mm.
2 -14 Figure 27: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 8.89 x10 , ρsp= 5.07mm.
143
2 -14 Figure 28: New PDF, GG PDF, and histogram of data; 20.6mm aperture, Cn = 7.00 x10 , ρsp= 5.85mm.
2 -13 Figure 29: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 2.65 x10 , ρsp= 2.63mm.
144
2 -13 Figure 30: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 1.30 x10 , ρsp= 4.04mm.
2 -13 Figure 31: New PDF, GG PDF, and histogram of data; 55.0mm aperture, Cn = 1.05 x10 , ρsp= 4.59mm.
145
2 -13 Figure 32: New PDF, GG PDF, and histogram of data; 101.6mm aperture, Cn = 1.40 x10 , ρsp= 3.86mm.
2 -13 Figure 33: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 3.47 x10 , ρsp= 2.24mm.
146
2 -13 Figure 34: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 2.36 x10 , ρsp= 2.83mm.
2 -13 Figure 35: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 1.65 x10 , ρsp= 3.50mm.
147
2 -14 Figure 36: New PDF, GG PDF, and histogram of data; 154.0mm aperture, Cn = 6.50 x10 , ρsp= 6.12mm.
Table 2 compares the scintillation index values of the new PDF, GG, and experimental data. The scintillation index of the data is calculated from the background removed experimental data according to (71), scintillation index of the new PDF is calculated theoretically according to
(216), and the scintillation index of the GG model is calculated from (132) using the best fit parameters.
148 Table 2: Scintillation index as calculated from the new PDF, GG fit parameters, and experimental data. Aperture SI from Theoretical SI from Time Diameter SI from GG SI from GG Fit (EST) Size Data Theory New PDF Parameters (mm) Parameters 16:01 1.5 1.28 1.34 1.09 1.79 15:32 4.0 2.25 1.83 1.68 2.74 16:19 4.0 1.93 1.52 1.48 2.74 16:15 7.0 1.82 1.41 1.39 2.18 13:15 10.0 2.40 1.85 1.45 2.19 15:26 10.0 1.30 1.21 1.09 1.57 16:11 10.0 1.62 1.41 1.25 1.82 13:11 20.6 1.50 1.35 1.18 1.80 15:21 20.6 1.05 1.14 0.97 1.28 15:56 20.6 0.84 0.85 0.78 1.13 13:07 55.0 0.57 0.63 0.62 0.93 15:17 55.0 0.32 0.34 0.31 0.63 15:51 55.0 0.22 0.25 0.23 0.57 15:47 101.6 0.12 0.16 0.13 0.24 13:04 154.0 0.14 0.18 0.14 0.21 15:13 154.0 0.14 0.19 0.14 0.15 15:44 154.0 0.10 0.15 0.11 0.11 16:06 154.0 0.05 0.15 0.06 0.05
Table 3 compares the parameters of the PDF models used to represent the experimental data.
149 Table 3: Parameters for new PDF and GG model used to represent experimental data. New PDF GG Fit GG Theory Aperture Time Diameter (EST) Size (mm) M N αfit βfit αthry βthry 16:01 1.5 2.0 8.0 2.2 2.2 1.4 1.6 15:32 4.0 2.0 2.0 1.6 1.6 0.9 1.3 16:19 4.0 3.0 2.0 1.7 1.7 0.9 1.3 16:15 7.0 3.0 3.0 1.8 1.8 1.0 1.7 13:15 10.0 2.0 2.0 1.8 1.8 0.6 5.1 15:26 10.0 4.0 4.0 2.3 2.3 1.3 2.2 16:11 10.0 3.0 3.0 2.0 2.0 1.1 2.1 13:11 20.6 3.0 4.0 2.1 2.1 0.6 19.3 15:21 20.6 3.0 7.0 2.5 2.5 1.1 5.2 15:56 20.6 3.0 15.0 3.0 3.0 1.3 4.9 13:07 55.0 3.0 33.0 3.7 3.7 1.1 89.3 15:17 55.0 4.0 110.0 3.3 170.7 1.7 41.7 15:51 55.0 6.0 100.0 4.8 56.9 1.9 34.9 15:47 101.6 14.0 110.0 9.5 55.1 4.3 150.4 13:04 154.0 11.0 110.0 7.4 142.1 4.8 950.4 15:13 154.0 10.0 109.0 7.6 95.5 6.7 605.4 15:44 154.0 18.0 110.0 14.1 29.1 9.0 409.9 16:06 154.0 25.0 110.0 34.5 34.5 21.5 207.4
The outer scale has an impact on the shape of the new PDF as well as the scintillation index.
The outer scale has a more profound impact on the lower tail than upper tail. The outer scale and the scintillation index are directly related, a smaller outer scale results in a smaller scintillation index. As stated earlier, a constant value of L0 was used to produce all the plots. However, tweaking L0 on some of the curves may provide a better visual fit to the experimental data and a better match to the measured scintillation index. Figure 37 shows the effects of the outer scale on the shape of the PDF and the resulting scintillation index.
150
2 -13 Figure 37: Outer scale effects on the shape and scintillation index of the new PDF; Cn = 1 x10 l0 = 5mm.
151 7 DISCUSSION
The new PDF presented in this dissertation along with the GG model were compared to histograms of aperture averaging data. The data were collected using several aperture sizes over a range of turbulence conditions ranging from moderate to strong. The width of the PDF is directly related the variance of the data. The peak, or maximum value of PDF, is roughly the mean of the data. The PDF has two tails, the upper tail which describes the irradiance surges above the mean irradiance and the lower tail which describes the fades below the mean irradiance.
Figure 19 is for a 1.5mm aperture and represents the most benign case of turbulence. There is no aperture averaging taking place and the aperture is acting like a point receiver. The PDF appears to flatten out at the low tail; this is due to the fading of the signal falling into the noise of the detector. The new PDF and the numerically fitted GG model approximate the data very well in both tails and throughout the peak. The new PDF fits slightly better in the upper tail than either of the GG curves. The theoretical GG model does not correctly model the data below the peak of the PDF. The scintillation index calculated from the new PDF is nearly identical to that measured from the data, while the SI from the GG fit is low and the SI from the GG theory is high.
Figure 20 and Figure 21 are for a 4mm aperture, the turbulence conditions were moderately strong for both data sets. The aperture exhibited some aperture averaging, but not much. The
152 new PDF and the numerically fitted GG model had a good fit to the data in the upper tail. In the lower tail, the numerically fitted GG was not able bring the low tail down while still having the curve pass through the data points in the peak of the PDF. However, the new PDF was able to fit more of the data points in the low tail and still have a good fit to the data points in the peak of the
PDF. This speaks to an advantage of the new PDF, it is able to fit data with a large variance and still maintain the horseshoe shape. The theoretical GG model fits well in the upper tail, but is unable to fit any data below the peak of the PDF. In fact, the theoretical GG curve extends above the range of the plot axes and therefore is not visible below and irradiance value of -10dB.
For both data sets, the new PDF and the numerically fitted GG estimated the SI lower than the measured value. The SI vales from the new PDF were closer to the measured values than those calculated using the numerically fitted GG parameters. The SI values from the theoretical GG parameters were high for both data sets.
Figure 22 is for a 7mm aperture with moderately strong turbulence. The aperture performed some aperture averaging, slightly more than the 4mm aperture. The results are similar to the
4mm aperture, all three curves fit well in the high tail and the differences start in the low tail.
The GG model is not mathematically capable of reproducing the horseshoe shaped curve when the variance of the data is large. The theoretical GG model fits well in the upper tail, but is unable to fit any data below the peak of the PDF, similar to the 4mm aperture, it is not visible below and irradiance value of -10dB. The new PDF does a substantially better job of fitting the data in the low tail. Tweaking of the outer scale would have provided a nearly exact fit in the low tail with the new PDF and possibly a better estimate to the SI. Again, the SI vales from the
153 new PDF were closer to the measured values than those calculated using the numerically fitted
GG parameters. The SI values from the theoretical GG parameters were high for both data sets.
Figure 23, Figure 24, and Figure 25 are for a 10mm aperture, the turbulence conditions were strong in the first data set and moderately strong in other two. The aperture performed some aperture averaging, slightly more than the 7mm aperture. As expected, the data collected under stronger turbulence has a larger variance. All three curves fit the data well in the upper tail, but fall short when fitting the lower tail. The new PDF has a slightly better fit throughout the peak of the PDF than the numerically fitted GG. The theoretical GG model fits well in the upper tail, but is unable to fit any data below the peak of the PDF, similar to the 4mm aperture, it is not visible below and irradiance value of -10dB. The predicted SI was low for all three sets of parameters; however the theoretical GG was closest. In the two data sets with moderately strong turbulence the new PDF and numerically fitted GG had a better fit to the data. In both data sets all three curves fit the data well in the upper tail; however, the new PDF had a better fit to the data in the low tail. Tweaking of the outer scale would have provided a nearly exact fit in the low tail with the new PDF and possibly a better estimate to the SI. The SI vales from the new
PDF were closer to the measured values than those calculated using the numerically fitted GG parameters, both underestimated the measured SI. The SI values from the theoretical GG parameters were high for both data sets.
Figure 26, Figure 27, and Figure 28 are for a 20.6mm aperture, the turbulence conditions were strong in the first data set and moderately strong in other two. According to Figure 9 and Figure
8 the aperture averaged out almost all of the small-scale fluctuations under strong turbulence and
154 some of the small-scale fluctuations under moderate turbulence. Again, the data recorded under stronger turbulence conditions had a larger variance. All three curves fit the data well in the upper tail, but the numerically fitted GG and theoretical GG fall short when fitting the lower tail.
The new PDF has an almost exact fit to the data. Tweaking of the outer scale would have provided a nearly exact fit in the low tail with the new PDF and possibly a better estimate to the
SI. The SI values from the new PDF and the numerically fitted GG underestimated the measured value, while the theoretical GG overestimated the SI. In the two data sets with moderately strong turbulence the new PDF still provided and excellent fit to the data and numerically fitted GG had a better fit to the data than in the strong turbulence case. All three curves fit the data well in the upper tail, the new PDF had an excellent fit in the low tail, the numerically fitted GG had a close fit in the low tail, and the theoretical GG had a poor fit in low tail as well as in the peak of the
PDF. The numerically fitted GG was able to follow the data to the left of the peak better than the
4mm, 7mm, and 10mm apertures; this is due to a reduction in the variance of the data on account of the aperture averaging. For the data set presented in Figure 27, the new PDF and the theoretical GG overestimated the measured SI; the numerically fitted GG underestimated the SI and was closest to the measured SI. For the data set presented Figure 28, the new PDF predicted the exact SI, the numerically fitted GG underestimated and the theoretical GG overestimated. It is important to notice the excellent fit the new PDF provided for the data as well as the accurate prediction of the measured SI.
Figure 29, Figure 30, and Figure 31 are for a 55mm aperture, the turbulence conditions were strong in the first data set and less strong in other two. The aperture averaged out all the small- scale fluctuations under strong turbulence and nearly all the fluctuations under the moderate
155 turbulence. The data collected under the strong turbulence had a slightly larger variance than the other two data sets. The new PDF and the numerically fitted GG had an excellent fit to the data in both tails and the peak of the PDF. The theoretical GG had a good fit to the data in the upper tail, but a poor fit in the peak and low tail. The new PDF and the numerically fitted GG predicted the same SI but, both slightly overestimated the measured SI. The SI predicted by the theoretical GG was an overestimate. In the two data sets with less strong turbulence the new
PDF provided a good fit to the data, the numerically fitted GG provided an excellent fit to the data, and the theoretical GG had a poor fit. For the data presented in Figure 30, the new PDF and the numerically fitted GG had and excellent fit in the upper tail and throughout the peak of the PDF. Below -7dB, the numerically fitted GG outperformed the new PDF. The new PDF was truncated at -8dB to avoid displaying the oscillations due to numerical inaccuracies. The new
PDF and the numerically fitted GG nearly predicted the exact measured SI. Tweaking of the outer scale may have improved the fit in both the lower and upper tail of the new PDF, but it also would have changed the predicted SI. For the data presented in Figure 31 the new PDF had a good fit in the upper tail and the numerically fitted GG had and excellent fit in the upper tail, both had an excellent fit throughout the peak of the PDF. Similar to the data in Figure 30, below
-7dB the numerically fitted GG outperformed the new PDF; the new PDF was also truncated at -
8dB. The new PDF and the numerically fitted GG nearly predicted the exact measured SI.
Again, tweaking of the outer scale may have improved the fit in both the lower and upper tail of the new PDF, but it also would have changed the predicted SI. For both data sets with less strong turbulence, the theoretical GG had a terrible fit in both tails and the peak.
156 Figure 32 is for a 101.6mm aperture with strong turbulence. The aperture averaged out all of the small-scale fluctuations and some of the large-scale fluctuations. The new PDF and the theoretical GG had a poor fit in the upper tail, while the numerically fitted GG had a good fit in the upper tail. Both the new PDF and the GG fit the data in the peak of the PDF well, while the theoretical GG is too low. The new PDF and the numerically fitted GG have an excellent fit in the lower tail with the numerically fitted GG having a slightly better fit; the theoretical GG had a poor fit. Again, the new PDF was truncated at -8dB. The new PDF and the numerically fitted
GG nearly predicted the exact measured SI, the new PDF was slightly high and the numerically fitted GG was closer to the measured value. The theoretical GG overestimated the measured SI.
Tweaking of the outer scale would have improved the fit in the lower tail, but would not have helped with the fit in the upper tail. The new PDF was unable to fit the data in the upper tail due to the numerical issues discussed earlier. A limitation of the new PDF is a maximum value of
N = 110 . Since N is the parameter that ultimately controls the position of the upper tail, a higher value would be needed to fit the upper tail.
Figure 33, Figure 34, Figure 35, and Figure 36 are for a 154mm aperture, the turbulence conditions were strong for the first three data sets and moderately strong for the last data set.
The aperture averaged out all of the small-scale fluctuations and nearly all of the large scale fluctuations for each of the data sets. The new PDF was truncated at -8dB in all of the plots.
Figure 33 displays data taken over the strongest turbulence conditions out of all the data sets presented. The new PDF has a poor fit to the data in the upper tail, but fits well throughout the peak and low tail. The poor fit in the tail is attributed to the upper limit on N. The numerically fitted GG provides the best overall fit to the data; it slightly overestimates in the upper tail, but
157 has an excellent fit in the low tail and the peak. The theoretical GG has a poor fit in both tails as well as the peak of the PDF, but the overall fit to the data is an improvement compared to other apertures. The SI predicted by the numerically fitted GG was exactly the same as the calculated
SI from the data; the new PDF slightly overestimated the SI and the theoretical GG even more so. For the data presented in Figure 34 and Figure 35 the new PDF had a poor fit to the data in the upper tail, but fit well throughout the peak and low tail. The numerically fitted GG and theoretical GG had an excellent fit to the data in the upper tail and the peak of the PDF. In the low tail, the numerically fitted GG provided an excellent fit and the theoretical GG slightly overestimated the data. Overall the numerically fitted GG had the best fit for both sets of data.
The SI predicted by the numerically fitted GG was almost exactly the calculated SI, the new PDF and the theoretical GG overestimated the SI in both data sets. For the data presented in Figure
36 the new PDF had a poor fit in both the tails and the peak of the PDF. The new PDF overestimated the data in both the tails and underestimated the peak of the PDF. This was due in part to the upper limits on both M and N. An increase in M and N would have brought the lower and upper tails closer to the data. The numerically fitted GG and theoretical GG had an excellent fit to the data in the upper tail and the peak of the PDF. Both tracked each other and started to underestimate the data around -4dB. The new PDF overestimated the SI calculated from the data, while both the theoretical and numerically fitted GG almost exactly predicted the SI. In all except the last figure, tweaking of the outer scale would have improved the fit in the lower tail, but would not have helped with the fit in the upper tail.
158 8 CONCLUSION
A new model for the PDF of received irradiance has been developed based upon the optical field.
The PDF captures the physics of the propagation medium as well as the effects of a finite receiving aperture. The parameters of the PDF are selected from statistical properties of the received optical field and the random media, they depend upon the atmospheric turbulence
2 parameters; Cn , l0, and L0 as well. The new PDF along with the GG model were compared with experimental data. The new PDF outperformed the GG under conditions of minimal aperture averaging, while falling short to the GG under conditions of severe aperture averaging.
The new PDF is an extension of the derivation for the GG PDF. The GG PDF is a modulation of two gamma distributions. A gamma distribution is composed of the sum of a finite number of independent exponential distributions, each having the same mean and variance. The gamma distribution can be generalized by assuming each exponential distribution has a different mean.
Therefore the resulting generalization of the GG PDF is a double finite sum of exponentially distributed random variables, each with a unique mean and variance.
The GG has the advantage of using noninteger values for the parameters, although the concept of a fractional number of scatterers does not make physical sense. Using noninteger values for the parameters allows the GG to have a better fit to data from point apertures. The prediction of the alpha and beta parameters of the GG model using scintillation theory is not applicable for all cases of aperture averaging. In fact, the theoretical prediction of the parameters works best
159 under two cases, zero aperture averaging or complete aperture averaging. Any intermediate cases involving contributions of both scale sizes result in a poor fit to data when the theoretical parameters are used. For the 1.5mm – 10mm apertures, similar difficulties were experience by
Vetelino [34] when fitting the GG model to experimental data, results varied due to differences in path length and beam parameters.
In the presence of strong turbulence, the numerically fitted GG distribution agrees with some of the experimental data. The GG distribution works well for large apertures because all the small- scale and most of the large-scale fluctuations are averaged out and the distribution is approximately Gamma. The GG works well for small apertures because all of the large scale fluctuations are not seen by the detector and the distribution is approximately exponential.
Mathematically, the GG is capable of representing aperture averaging data. However in cases of small aperture averaging in strong turbulence, the PDF of irradiance has a wide horseshoe shape due to the large variance of the data and the GG is unable to fit the data. On the other hand, in cases of severe aperture averaging, the PDF of irradiance has a narrow horseshoe shape due to the small variance of the data and the new PDF is unable to fit the data due to restrictions brought on by numerical inaccuracies. The numerical inaccuracies of the new PDF were a result of the approximations made in the eigenfunctions and eigenvalues of the KL expansion. Small errors in the eigenvalues were exacerbated by the finite product of differences in the denominator of the new PDF and the K0 Bessel function for small values of irradiance. The technique used to select the lambdas and gammas has a profound impact on the shape and behavior of the PDF.
160 Many alternate methods were investigated to generate the lambdas and gammas, none of which proved to be fruitful.
In most cases, fitting data to the upper tail was not a problem for either the new PDF or the GG model. However, the low tail is where the fit needs to be most accurate; this part of the PDF is used to determine the fade parameters. Both the new PDF and the GG model have issues representing data with a large variance exhibiting small amounts of aperture averaging. The aperture averaging removes the small scale fluctuations and creates a horseshoe shaped PDF with a wide peak. Since both models are fundamentally a linear combination of negative distributions modulating another linear combination of exponential distributions, they fall short in nearly the same areas. The data suggests introducing a third modulation term would offer more flexibility in the shaping of the PDF by capturing more of the physics of the phenomena.
The additional modulation term could describe the temporal fluctuations which were considered deterministic while deriving the PDF in this dissertation.
Overall, the new PDF developed in this dissertation is a viable alternative to the GG model. This model serves as an excellent starting point for developing a heuristic PDF for irradiance based upon the optical field. Some suggested future research: Develop a method to select model parameters M and N of the new PDF. Perform a rigorous mathematical treatment of the limitations and restrictions of the PDF at low irradiance values such that the PDF converges properly without oscillations. Derive the exact solution of the KL expansion and examine the effects of the assumptions made in this dissertation. Derive the PDF including a third modulation term to add more versatility to the model. Collect experimental data with a
161 modulated source such that the benefits of AC coupling the receiver can be realized. Compare new PDF with experimental data over a longer path length and a wider variation of turbulence conditions.
162 APPENDIX A SCINTILLATON INDEX OF A SPHERICAL WAVE INCLUDING APERTURE AVERAGING
163 Elaborating the equations in section 3.9.1Under general irradiance fluctuations, weak or strong, the scintillation index including aperture averaging effects for a spherical wave is [7]
2 2 2 σ , ( GspI lD 0 ) = exp, [σ ln ( GX ,lD )−σ ln0 ( GX , LD 0 ) 2 512 − 65 ⎤ , (233) σ SP ()+ 69.0151.0 σ + SP ⎥ −1 + 90.01 d 2 ()σβ 512 + 62.0 d 2 β 512 ⎥ 0 SP 0 ⎦
2 611672 2 where β0 = 55.0 n LkC and σ SP is the spherical wave scintillation index given by [7]
2 2 ⎧ 2 1211 ⎡ ⎛11 −1 Ql ⎞ σ SP ≅ β0 ⎨ ( + 9140.065.9 Ql ) ⎢sin⎜ tan ⎟ ⎩ ⎣ ⎝ 6 3 ⎠ 61.2 ⎛ 4 Q ⎞ + sin⎜ tan −1 l ⎟ , (234) 2 41 ⎝ 3 3 ⎠ ()9 + Ql ⎤ ⎫ 52.0 ⎛ 5 −1 Ql ⎞ 50.3 ⎪ 2 − sin⎜ tan ⎟⎥ − ⎬ β0 < 1, 2 247 ⎝ 4 3 ⎠⎥ Q 65 ⎪ ()9 + Ql ⎦ l ⎭
2 where Ql = 89.10 lkL 0 . Also, [7]
67 2 2 ⎛ η QlXd ⎞ σ ln ()GX lD 0 = 04.0, β0 ⎜ ⎟ ⎝η + QlXd ⎠ , (235) 21 127 ⎡ ⎛ η ⎞ ⎛ η ⎞ ⎤ ⎢ +× 75.11 ⎜ Xd ⎟ − 25.0 ⎜ Xd ⎟ ⎥ ⎢ ⎜η + Q ⎟ ⎜η + Q ⎟ ⎥ ⎣ ⎝ lXd ⎠ ⎝ lXd ⎠ ⎦
164 67 2 2 ⎛ η 0 QlXd ⎞ σ ln ()GX LD 0 = 04.0, β0 ⎜ ⎟ ⎝η 0 + QlXd ⎠ , (236) 21 127 ⎡ ⎛ η ⎞ ⎛ η ⎞ ⎤ ⎢ +× 75.11 ⎜ Xd0 ⎟ − 25.0 ⎜ Xd0 ⎟ ⎥ ⎢ ⎜η + Q ⎟ ⎜η + Q ⎟ ⎥ ⎣ ⎝ 0 lXd ⎠ ⎝ 0 lXd ⎠ ⎦
56.8 η = , (237) X 2 61 + 20.01 β0 Ql
η 56.8 η = X = , (238) Xd 2 2 2 61 + 02.01 d η X d ++ 20.018.01 β0 Ql
η Q 56.8 Q η = Xd 0 = 0 , (239) Xd 0 η + Q 2 2 61 Xd 0 0 ++ 18.056.8 QdQ 0 + 20.0 β0 0 QQ l
kD2 64π L d = G , Q = . (240) 0 2 4L Lk 0
165 APPENDIX B HANKEL TRANSFORM OF MCF: b = 5/3
166 The integral to be solved is
∞ 35 ()= π 0 ∫ exp2 ()− 0 ()2 drJaIrS ρρρπρ (241) 0
The Bessel function can be represented by an infinite series [70]
∞ (− )k (bρ 21 )2k 0 ()bJ ρ = ∑ (242) k=0 ()kk +Γ 1!
where = 2π rb . Substituting this into (241)
∞ ()−1 k b2k ∞ ()= 2π IrS exp − a 35 2k dρρρρ (243) 0 ∑ 2k ∫ () k=0 ()kk +Γ 21! 0
Working on only the integral
∞ ∫ exp()− a 35 k+12 dρρρ (244) 0
5 ⎛ 6 3 ⎞ The exponent can be rewritten as k 12 ⎜ k +=+ ⎟ and the integral is then 3 ⎝ 5 5 ⎠
167 ∞ 6 3 k+ ∫ exp()− a 35 ()ρρ 35 5 5 dρ (245) 0
Making a substitution
= aulet ρ 35 = 3235 dadu ρρ 53 ⎛ u ⎞ 1 (246) ρ =→ ⎜ ⎟ dρ = du a 5 32 ⎝ ⎠ 3 a ρ
The integral becomes
6 3 6 3 ∞ k+ ∞ k+ ⎛ u ⎞ 5 5 1 ⎛ u ⎞ 5 5 1 ⎜ ⎟ e−u du = ⎜ ⎟ e−u du (247) ∫ a 5 32 ∫ a 5 156 0 ⎝ ⎠ 3 a ρ 0 ⎝ ⎠ 3 ()aua
The like terms are collected
∞ 6 3 6 6 3 6 3 ()k +− −1 k −+ −u 5 5 15 5 5 15 dueuuaaa (248) 5 ∫ 0
The terms are then arranged in the form of a gamma function integral
∞ 3 ()6 k 3 1−++− 6 6 k 18 −+ 1 a 5 5 15 u 5 15 −u due (249) 5 ∫ 0
168 Using the definition of the gamma function [70]
∞ ()=Γ ∫ 1 −− tz dtetz (250) 0
(248) becomes
3 (6 k +− 18 ) ⎛ 6 ⎞ a 5 15 ⎜ ()k +Γ 1 ⎟ (251) 5 ⎝ 5 ⎠
Substituting back into (243) yields
∞ k 2k (−1) b 3 (6 k+− 18 ) ⎛ 6 ⎞ = 2π IrS a 5 15 k +Γ 1 (252) () 0 ∑ 2k ⎜ ()⎟ k=0 ()kk +Γ 21! 5 ⎝ 5 ⎠
Rearranging terms yields
⎛ 6 ⎞ ⎜ ()k +Γ 1 ⎟ 6π ∞ ()−1 k b2k 5 rS = I ⎝ ⎠ (253) () 6 0 ∑ k 5 k=0 ⎛ 6 ⎞ ()k +Γ 1 5a ⎜4! ak 5 ⎟ ⎝ ⎠
In its current state, the infinite sum has no closed form. A closer look reveals the sum has the functional form of an exponential, if the ratio of gamma functions were unity [70]
169 ∞ xk ∑ = e x . (254) k=0 k!
Assuming the functional form of an exponential, a numerical approximation was made
⎛ 6 ⎞ ⎜ ()k +Γ 1 ⎟ ⎛ ⎞ ∞ ()−1 k b2k 5 ⎜ − b2 ⎟ ⎝ ⎠ ≈ exp92.0 . (255) ∑ k ⎜ 6 ⎟ k=0 ⎛ 6 ⎞ ()k +Γ 1 ⎜ 5 ⎟ ⎜4! ak 5 ⎟ ⎝ 4*9.0 a ⎠ ⎝ ⎠
The approximation has less than 5% error in area under the curve and a mean square error less than ×105 −4 .
(253) reduces to
⎛ ⎞ 6π ⎜ b2 ⎟ ()rS = 92.0 I0 exp⎜− ⎟ (256) 6 ⎜ 6 ⎟ 5a 5 ⎝ 4*9.0 a 5 ⎠
Substituting for b
⎛ ⎞ 6π ⎜ ()2π r 2 ⎟ ()rS = 92.0 I0 exp⎜− ⎟ (257) 6 ⎜ 6 ⎟ 5a 5 ⎝ 4*9.0 a 5 ⎠
170 Resulting in
⎛ 2 ⎞ I0 ⎜ ()π r ⎟ ()rS = exp47.3 ⎜− ⎟ (258) 6 ⎜ 6 ⎟ a 5 ⎝ 9.0 a 5 ⎠
171 APPENDIX C CHARACTERISTIC FUNCTION
172 First we calculate the corresponding characteristic function for the PDF of irradiance,
∞ ωIi ()ω =Φ j ()n = dInIPe (259) I j ∫ j ∞−
∞ 1 ω − nIIi λ jjj I ()ω =Φ dIee . (260) j nλ ∫ j 0
We make the change of variable = −Ix j and the function is now in the common form of a
Fourier transform
0 1 − ω xi nx λ j I ()ω =Φ dxee . (261) j nλ ∫ j ∞−
Using a Fourier transform table we get
1 ()ω =Φ . (262) I j 1− ni λω j
For the total irradiance, the convolution of M PDFs equals the product of M characteristic functions. We therefore multiply M characteristic functions
173 M 1 I ()ω =Φ ∏ . (263) j=1 1− ni λω j
A partial fraction expansion is done to change the product to a sum. The function can be rewritten as follows:
M 1 A j F ω)( = = , (264) M ∑ j=11− ni λω j ∏1− ni λω j j=1
where 1 ×−= FniA ωλω 1 . j ( j ) ()ω= inλ j
We further derive Aj
⎛ ⎞ ⎜ ⎟ ⎜ 1 ⎟ 1−= niA λω (265) j ()j ⎜ M ⎟ ⎜ 1− ni λω ⎟ ⎜ ∏ m ⎟ ⎝ m=1 ⎠
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ 1−= niA λω (266) j ()j ⎜ M ⎟ ⎜ 1− ni λω 1− ni λω ⎟ ⎜ []= jm ∏ m ⎟ ⎜ m=1 ⎟ ⎝ ≠ jm ⎠
174 1 A = (267) j M ∏1− ni λω m m=1 1 ≠ jm ω= inλ j
1 A j = . (268) M λ ∏1− m m=1 λ j ≠ jm
Substituting into (264)
M 1 1 I ()F ()ωω ==Φ ∑ (269) M λ 1− ni λω j=1 ∏1− m j m=1 λ j ≠ jm
To get the resulting PDF of irradiance, we must invert the characteristic function,
1 ∞ ()n == − ωIi Φ()denIP ωω (270) 2π ∫ ∞−
and substituting we get
175 ∞ M 1 A j ()n == enIP − ωIi dω . (271) 2π ∫ ∑1− ni λω ∞− j=1 j
We make the change of variable x = −ω and the function is now in the common form of a
Fourier transform
∞ M 1 A j ()n == enIP ixI dx . (272) 2π ∫ ∑1+ nix λ ∞− j=1 j
Rearranging and interchanging the order of integration and summation,
M ∞ A j 1 1 ()n nIP == eixI dx . (273) ∑ n 2πλ ∫ 1 λ + ixn j=1 j ∞− j
Using a Fourier transform table we get
M A j − nI λ ()n nIP == ∑ e j 0 I ∞≤≤ . (274) j=1 nλ j
The ratio A λ jj can be further simplified, starting with Aj
176 −1 M M M ⎛ λ ⎞ 1 λ j A ⎜1−= m ⎟ = = . (275) j ∏ ⎜ λ ⎟ ∏ 1 ∏ − λλ m=1 ⎝ j ⎠ m=1 ()− λλ m=1 ()mj ≠ jm ≠ jm mj ≠ jm λ j
Excluding the m = j term in the product results in M-1 products being formed, therefore
M M −1 1 A =λ jj ∏ (276) m=1 ()− λλ mj ≠ jm
and
M −2 A j λ j = . (277) M λ j ∏ ()− λλ mj m=1 ≠ jm
177 APPENDIX D INTERPRETING POWER METER OUTPUT
178 A decibel (dB) is a unit used to express relative differences in signal strength. A decibel is expressed as the base 10 logarithm of the ratio of the power of two signals
⎛ P1 ⎞ dB = log10 10 ⎜ ⎟ . (278) ⎝ P2 ⎠
A decibel referenced to one milliwatt (dBm) is the ratio of the input power (in Watts) to 1mW
⎛ P1 ⎞ dBm = log10 10 ⎜ ⎟ . (279) ⎝1mW ⎠
The output of the New Focus power meter can be written as either [81, 82]
mV ⎛ POPT ⎞ VLOG = log10*50 10 ⎜ ⎟ (280) dB ⎝ PZ ⎠
mV V = 50 ()− DD (281) LOG dB OPT Z
where “D” denotes the power in decibels as opposed to “P” which expresses power in Watts.
DOPT is the optical power in decibels above a reference level and DZ is the equivalent intercept power relative to the same level.
179 In order to convert the output of the New Focus 2101 power meter to true measured power, DZ must be calculated. Typically, one would take a measurement of a known source at the wavelength of interest and use that to calculate the intercept power. However, for this case we will use the approximate value of 2dBm of input power at 920nm corresponding to a 3.5V output, as stated in the users manual [82]. The user’s manual also estimates a 2dB drop in response at a wavelength of approximately 532nm.
mV V = ()22505.3 −− DdBdBm dB Z (282) DZ −= 70dBm
DOPT is the optical power as referenced to 1mW; therefore it is related to the input power by
⎛ Pin ⎞ DOPT = log10 10 ⎜ ⎟ (283) ⎝1mW ⎠
Substituting for DOPT in (280) above, the input optical is related to the output voltage by
⎛VLOG − 5.3 ⎞ Pin = 10^⎜ ⎟ []W (284) ⎝ 5.0 ⎠
The minimum detectable power of the detection system can be calculated. The digitizer has 24 bits of resolution and was operated over a range of -5V to +5V. The smallest detectable voltage is given by
180 10V ×= 1096.5 −7 V . (285) 224 bits
Substituting the minimum resolvable voltage for VLOG yields the minimum detectable power of the optical system,
⎛ −7 −× 5.31096.5 ⎞ P = ^10 ⎜ ⎟ ×≅ 1000.1 −7 W . (286) min ⎜ ⎟ ⎝ 5.0 ⎠
Therefore, any measured power value below ~100nV was considered noise and discarded. In reality the last one or two bits of the digitizer are lost due to noise, thus the digitizer has 22 usable bits to resolve the signal.
181 APPENDIX E NUMERICAL INACCURACIES OF NEW PDF AT LOW IRRADIANCE
182 Recall the PDF of irradiance derived herein,
M N A j B ⎛ I ⎞ ()IP = 2 k K ⎜2 ⎟ , (287) ∑∑ 0 ⎜ γλγλ ⎟ j 1 j k== 1 k ⎝ kj ⎠ where
M −1 N −1 λ j γ A = , B = k . (288) j M k N ∏()− λλ mj ∏()− γγ nk m=1 n=1 ≠ jm ≠kn
At low values of irradiance the new PDF exhibits oscillations, implying convergence issues. It is understood that a finite sum will always converge, however the finite sum representing the PDF of irradiance should be monotonically decreasing in the tails. Initially, the oscillations in the
PDF were thought to be from numerical inaccuracies in the K0 Bessel function. However, implementation of the PDF in both Mathematica and Matlab yielded identical results. Another possible cause of the convergence issues lies is the method of selecting of the λ’s and γ’s. It has been observed that different methods of selecting the λ’s and γ’s lead to different convergence results in the PDF. The oscillations in the low tail become more prevalent as the number of λ’s used is increased. As the number of λ’s is increased the spacing between subsequent λ’s is decreased, see Figure 13. The PDF has a finite product of the differences of the λ’s in the denominator and is therefore severely dependent on the differences of λ’s. It is believed the approximations made in the eigenfunctions and eigenvalues of the KL expansion introduce a small error. This error starts to manifest when the difference of many eigenvalues are multiplied
183 together. Small errors in the λ’s and γ’s are also exacerbated by the K0 Bessel function for small values of irradiance. For small arguments of the Bessel function it has an asymptotic behavior proportional to the negative logarithm [70]
+ 0 ( ) − ( ),ln~ zzzK → 0 (289)
whereas for large arguments, the Bessel function approaches zero exponentially. The oscillations can also be brought about by changing the sampling interval Δr . As the sampling interval is decreased, the λ’s become more similar and the oscillations begin at lower values of
M. The oscillations appear to be less dependent upon the γ’s.
The figures below are a typical example of the oscillations as the number of λ’s is changed and all other parameters are held constant.
184 0 10
-1 10
-2 10
-3 10
Probability Density -4 10
-5 10
-6 10 -25 -20 -15 -10 -5 0 5 10 15 Intensity (dB)
Figure 38: Low irradiance oscillation in PDF: M=30, N=60
0 10
-1 10
-2 10
-3 10
Probability Density -4 10
-5 10
-6 10 -25 -20 -15 -10 -5 0 5 10 15 Intensity (dB)
Figure 39: Low irradiance oscillation in PDF: M=10, N=60 185 0 10
-1 10
-2 10
-3 10
Probability Density -4 10
-5 10
-6 10 -25 -20 -15 -10 -5 0 5 10 15 Intensity (dB)
Figure 40: Low irradiance oscillation in PDF: M=5, N=60
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