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DESIGN AND CHARACTERIZATION OF AH ATMOSPHERIC TURBULENCE CHAMBER FOR LASER PROPAGATION by John Andrew DiUbaldo B.S. Colc:>rado State University, 1983 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of of Science Department of Electrical Engineering 1991 r...-.. ..
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This thesis for the Master of Science degree has been approved for the School of Electrical Engineering Arun K. Majumdar Edward T. Wall v Jan Bialasiewicz Date
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ABSTRACT DiUbaldo, John Andrew (M.S. Electrical Engineering) Design And Characterization Of An Atmospheric Turbulence Chamber For Laser Propagation. Thesisdirected by Professor Arun K. Majumdar. An atmospheric tur_bulence chamber has been designed and characterized for conducting experiments related to optical propagation in a turbulent atmosphere. Statistics of irradiance fluctuations were also determined for several propagation experiments. Laboratory generated turbulence on the scale of 1000 times that of the atmosphere were generated and statistics of irradiance measured tor turbulent conditions. The form and content of this abstract are approved. I recommend its publication. Signed Arun K. Majumdar iii
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DEDICATION I would like to thank Professor Arun K. Majumdar for pushing me forward on this-work until its completion. If it were not for him, I would not have finished this work. Dr. James of NOAA whose loan of equipment made this work possible. I would also like to thank my brother Pat for his assistance throughout and to give \ thanks to the rest of my fam1Ly for their continued support. iv
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CONTENTS Figures ....................................... viii Tables .......................................... xi CHAPTER 1. 2. INTRODUCTION .. : ........................ 1 STATISTICAL THEORY RELATED TO LASER 4 Optical Irradiance ............ 4 Atmospheric Turbulence ............. ? Probability Density Functions ...... s Observational Challenges .......... 10 3. CHAMBER DESIGN AND CONSTRUCTION ........... 14 4. THERMAL CHAMBER CHARACTERIZATION ........ 18 Measurements Of Temperature Structure Functions ............................. 2 0 Power Spectrum Of Temperature Fluctuations .................... 33 s. RECEIVER DESIGN ...... ............... 37 Component Selection ................. 37 Receiver Operational Circuits ....... 42 6. STATISTICAL MEASUREMENTS OF IRRADIANCE .... 44 Experimental Arrangement ......... 44 Experimental Results ............ 49 7. DISCUSSION OF RESULTS .................. 103 v
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rrequency Spectrum Measurements l03 Statistical Analysis 106 8. CONCLUSION ................................ 109 RBPBRDCBS . . . . . . . . . . . . . . . . . . . . . 111 vi
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FIGURES Figure 1. Transmitter And Receiver Scenario ............... s 2. Variance Of Log Irradiance .................. 6 3. Turbulence Eddies .................................. 8 4. Kolmogorov Spectrum . ............................ 8 5. Turbulence Chamber Schematic .......... 15 6. Temperature Probe Placement, View From Center Of Chamber ............ ............................ 17 7. Wfnd Velocity Along Length Of Chamber. ........ 19 8. Temperature Profile Along Length Of Chamber .... 21 9. Typical Signal v(t) From Output Of Log-Amp ...... 24 10. Typical Temperature Structure Measurement.For One Location Along Chamber .................. 26 11. Effects Of Pressure On Temperature Structure Function .......................................... 28 12. Effects Of Temperature On Temperature Structure Function .................. .......................... 29 13. Refractive-index cln measured along length of Chamber .......................................... 3 2 14. Power Spectrum Of Fluctuation W(f) .... 34 15. Power Spectrum Of Temperature Fluctuation 2 fW(f) . 35 16. Output Current v.s. Incident Light On Detector ... 38 vii
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17. Dual Amplifier Circuit Used To Capture Large Scale Spikes Of Irradiance ........................ 41 18. Photodiode Operational Circuits Used In Detection Schemes ................. 43 19. Test Configuration Used To Measure Laser Scintillation For One Path Measurements .......... 45 20. Test Configuration Used To Measure Laser Scintillation for Two Path Measurements .... 46 21. Normalized Power Spectra fW(f) .For One Path With Detector Only Receiver ............. so 22. Normalized Histogram For One Path With Detector Only .............................................. 51 23. Normalized Power Spectra fW(f) For One Path With Filter Only Receiver ...................... 54 24. Normalized Histogram For One Path With Filter Only Receiver .......................................... 55 25. Normalized Power Spectra fW(f) For One Path With Focusing Lens Receiver, Input Intensity Proportional To 0 .1 Volt .................... ................... 59 26. Normalized Histogram For One Path With Focusing Lens Receiver, Input Intensity Proportional To 0.1 Volt ............................................... 60 27. Normalized Power Spectra fW(f) For One Path With Focusing Lens Receiver, Input Intensity Proportional To 1 0 Volts ...................................... 6 3 viii
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28. Normalized Histogram For One Path With Focusing Lens Receiver, Input Intensity Proportional To 1.0 Volts ............................................. 64 29. Normalized Power Spectra fW(f) For One Path With Focusing Lens Receiver, 90 Degree Polarization Shift Input Intensity Proportional To 0.1 Volts ...... 68 30. Normalized Histogram For One Path With Focusing Lens Receiver, Input Intensity Proportional To 0.1 Volt, 90 Degree Polarization Shift ........... 69 31. Normalized Power Spectra fW(f) For One Path With Focusing Lens Receiver, 90 Degree Polarization Shift Input Intens1ty Proportional To 1.0 Volts ..... 72 32. Normalized Histogram For One Path With Focusing Lens Receiver, Input Intensity Proportional To 1.0 Volt, 90 Degree Polarization Shift .............. 73 33. Normalized Power Spectra fW(f) For One Path With Filter Only Receiver, 90 Degree Polarization Shift, And Amplified Channel Of Dual-Amp configuration . 77 34. Histogram For One Path With Filter Only Receiver, 90 Degree polarization Shift, And Amplified Channel Of Dual-Amp Configuration . .... 78 35. Normalized Power Spectra fW(f) For One Path With Filter Only Receiver, 90 Degree Polarization Shift, And Direct Channel Of Dual-Amp Configuration .... 81 ix
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36. Normalized Histogram For One Path With Filter Only Receiver, 90 Degree Polarization Shift, And Direct Channel Of Dual-Amp Configuration ............ 82 37. Normalized Power Spectra fW(f) For Two Path With Filter Only Receiver and Input Pressure= 45 psi .. 86 38. Normalized Histogram For Two Path With Filter Only Receiver And Input Pressure= 45 psi ........... 87 39. Normalized Power Spectra fW(f) For Two Path With Filter Only Receiver And Input Pressure= 60 psi .. 90 40. Normalized Histogram For Two Path With Filter Only Receiver And Input Pressure= 60 psi ....... 91 41. Normalized Power Spectra fW(f) For Two Path With Pinhole Only Receiver And Input Pressure = 45 psi ................................................ 9 5 42. Normalized Histograin For Two Path With Pinhole Only Receiver And Input Pressure= 45 psi ............. 96 43. Normalized Power Spectra fW(f) For Two Path With Pinhole Only Receiver And Input Pressure = 60 psi ................................................ 9 9 44. Normalized Histogram For Two Path With Pinhole Only Receiver And Input Pressure= 60 psi ......... lOO X
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TABLES Tables I. Statistical Data For One Path With Detector Only Receiver ......................................... 52 II. Statistical Data For One Path With Filter Only Receiver ... ." ... ................................. 56. III. Statistical Data For One Path With Focusing Lens Receiver, Input Intensity Proportional To 0.1 Volt .......................................... .. 61 IV. Statistical Data For One Path With Focusing Lens Receiver, Input Intensity Proportional To 1.0 Volts ............................................ 65 V. Statistical Data For One Path With Focusing Lens Receiver, 90 Degree Polarization Shift Input Intensity To 0.1 Volts ............. 70 VI. Statistical Data For One Path With Focusing Lens Receiver, 90 Degree Polarization Shift Input Intensity Proportional To 1.0 Volts ........... 74 VII. Statistical Data For One Path With Filter Only Receiver, 90 Degree Polarization Shift, And Amplified Channel Of Dual-Amp configuration ...... 79 VIII.Statistical Data F9r One Path With Filter Only Receiver, 90 Degree Polarization Shift, And Direct Channel Of Configuration ........ 83 xi I I
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IX. Statistical Data For Two Path With Filter Only Receiver and Input Pressure= 45 psi ............. 88 X. Statistical Data For Two Path With Filter Only Receiver And Input Pressure= 60 psi ............ 92 XI. Statistical Data For Two Path With Pinhole Only Receiver And Input Pressure= 45psi ............. 97 XII. Statistical Data For Two Path With Pinhole Only Receiver And Input Pressure= 60 psi .......... 101 xii
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CHAPTER 1 INTRODUCTION There has been a great deal of interest in laser propagation through the atmosphere over the years and much has been reported on it [1,2-5]. The interest comes from enormous gains that will be made in understanding the performance and design of electro-optical systems in the areas of imaging, optical communication and remote sensing, to name a few. The effects of laser propagation through the atmosphere are best described by the statistics of the laser scintillation in terms of its probability density function (PDF) [20]. Even though there has been much research in the area, very little has been accomplished in determining a general PDF that best describes the laser scintillation in the full range of weak to strong turbulence. However, it is generally accepted that the PDF for laser scintillation in weak turbulence will be log normal and that for strong turbulence it will be a negative exponential [14]. If the general PDF can be obtained, a plethora of new ideas and concepts will emerge. Experimentally, there are essentially two methods of 1
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attack in obtaining a PDF. The first method is to construct a histogram from laser scintillation data and compare it to known PDFs. The PDF chosen is the one that best matches the histogram curve. The second method is to compare moments of laser scintillation data with momentsof predicted PDFs. If all the moments are known, a unique PDF will emerge. Neither of these methods has yielded a general PDF to date. The reason is due to the difficulty in collecting reliable laser scintillation data [10]. The obstacles to overcome in collecting reliable scintillation data are many. The first and foremost i"s the unpredictability of atmospheric conditions. When collecting data on laser scintillation, the characteristics of the atmosphere most be completely known and constant during the data collection period. Other difficulties arise from the receiver configuration, data collection and signal processing techniques. To overcome the difficulties of laser scintillation data collection, an atmospheric turbulence chamber has been proposed and tested in this paper. The chamber has produced turbulence 1000 times that of the real atmosphere and is isotopic and homogenous. In addition, a technique is proposed and tested for collection and processing of the laser scintillation data from which 2
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reliable PDFs can be constructed. A discussion of the background of laser scintillation characterization and observational challenges is given in chapter 2. Followed by the chamber design requirements and construction in chapter 3 and the chamber thermal characterization in chapter. Chapter 5 covers the detection scheme used in collecting the laser scintillation data followed by statistical measurements of irradiance fluctuations in chapter 6. Finally, a discussion of the results and suggestions for future work are given in chapters 7 and 8, respectively. 3
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CHAPTER 2 STATISTICAL THEORY RELATED TO LASER PROPAGATION Electromagnetic waves when propagating through the atmosphere experience fluctuations in amplitude and phase. These fluctuations arise from variations in the index of refraction in the propagation path and are most notable for optical wavelengths [22]. For optical wavelengths the area of most interest are fluctuations in amplitude' (intensity) that are otherwise known as 11Scintillations11 To help in understanding what an irradiance.probability density function (PDF) is, a general distribution model of irradiance will be discussed and a model for the atmosphere will be described. Optical Irradiance A typical line-of-sight propagation scenario is shown in Figure 1. A laser (transmitter) beam propagates through the atmosphere and irradiates the detector (receiver). The irradiance, I, at the receiver is equal to the square of the amplitude, A, at the detector. The irradiance is observed at one point on the receiver and is proportional to the power received by a small receiver 4
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Electromagnetic waves incident upon turbulent atmosphere have amplitude and phase fluctuations 0 Transmitter Receiver Figure 1. Transmitter and receiver scenario. aperture, an aperture over which I is constant. The logarithm of the intensity of the wave in general has a Gaussian probability density function. so the strength of the scintillations is described in terms of variance" [6]. A plot of the of log irradiance is shown in Figure 2. Log irradiance variance increases with the strength of turbulence as the 11/6 power of the path length until it reaches a peak, 5
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OPTICAL SCINTILLATION Variance of Intensity Linear Path Length L Saturation .__,.,,_.,_, ,_, __ __ __ ,.,,.._,. .,,u--.. Figure 2. Variance of log irradiance. then it "saturates" and falls off to a lower level. Foroptical wavelengths this saturation occurs within a few kilometers and for microwave frequencies the saturation occurs in the tens of kilometers. It is this area of saturation that is most difficult to characterize and the area of interest for this paper. Atmospheric Turbulence Atmospheric and ocean turbulence can be described by 6
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the Kolmogorov model [22]. According to Kolmogorov, the turbulence eddies may be characterized by two sizes: the outer scale of turbulence, and the inner scale of turbulence, 10 reference Figure 3. We can then divide the characteristics of turbulence according to the size of the eddy into three regions: a) Input range (eddy size> L0), where energy is introduced into the turbulence and is anisotropic. b) Inertial range (L0 > eddy < 10 ) where kinetic energy dominates over the dissipation due to viscosity. c) Dissipation range (10 > eddy size), where dissipation dominates over kinetic energy. Considering these three ranges lead to the Kolmogorov spectrum shown in Figure 4. An example of how the Kolmogorov spectrum would affect optical propagation would be to describe the "large scale variation" (those in the input range) as causing the position of a light beam to vary over a period of a few seconds and "small scale variations" (those in the inertial range) as causing the rapid twinkling of a star. 7
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lo ----.c--I I Lo---- Figure 3. Turbulence eddies. r-----------------------------Kolmogorov Spectrum Input Inertial Dissipation ---L---------------------.. -----------Figure 4. Kolmogorov spectrum. 8
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Probability Densi t:r Functions In general, it is accepted that the PDF of optical scintillations will be a log-normal distribution for propagation through a weak turbulent atmosphere [14]. This distribution is derived from MaxWell's equations for a fluctuating medium which leads to ( .,a +IC2 (1 + ) B(r) -o where E is the electric field, k is the wave number and n is the index of refraction. By applying either the Born approximation or the Rytov approximation to (1) above gives a log-normal distribution [22]. Experimental data supports the log-normal prediction. However, the PDF for propagation through stronger turbulence over greater distances, where the irradiance, I, is well into the saturation region, is not understood well. The log riormal distribution has a predicted variance of I that will eventually go negative, a physical impossibility! It has been predicted that the density function in strong turbulence should approach a negative exponential in the limit of infinite turbulence but the extreme turbulence levels that would produce that are uncommon. Due to the difficulties in describing the PDF deep 9
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into the saturation region, a number PDFs have been proposed. Among them are Rice-Nakagami, Rayleigh, K and Log-normally Modulated Exponential [11]. The Rice' Nakagami and Rayleigh distributions are both based on circular gaussian statistics. Again, the Rice-Nakagami and Rayleigh distributions due not hold up well because they require turbulence levels approaching infinity. The K distribution was selected intuitively because it arises in the case of a strong scattering process (random phase fluctuations of many wavelengths) with two distinct scale sizes. This appeals intuitively because the Kolmogorov spectrum is based on turbulence eddies of two distinct sizes. Although the K distribution seems to fit better then the others mentioned, it has several problems. It can be applied only in cases of strong turbulence, and even where it can be applied it tends to underestimate the probability of high irradiances and thus to underestimate the higher-order moments. The Log-Normally Modulated Exponential PDF is obtained by assuming that measured small scale spikes of irradiance in the saturation region are exponentially distributed with a mean that varies depending on the log-normally distributed large-Hxale patches of irradiance. These assumptions are based on large-scale patches of irradiance having a log-normal distribution (This has 10
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been shown to be true when data has been filtered with apertures as small as the Fresnel zone size). The small scale patches of irradiance have the size of small-scale width of the irradiance covariance proportional to the wave coherence length. A heuristic picture then arises of the small scale spikes of irradiance being modulated by the large irradiance patches [11]. Observational Challenges Accurately recording strong scintillations of irradiance fluctuations has many difficulties. The difficulties involve receiver dynamic range, aperture size, sample size, surface reflections (mirages) and source divergence [10]. The problems of sample size, surface reflections and source divergence all develop from the randomness of the weather. If somehow we could control the weather we could then overcome these difficulties. The remaining problems are a technological difficulty and can be overcome by careful selection of components. We will examine the observational challenges posed by all these difficulties. Many of the PDFs proposed in the literature for strong scintillation have long tails [14]. If these PDFs are accurate, then the tails are from predicted scintillations of 500 tQ 1000 times the mean. If data is to be captured accurately then the detection scheme used 11
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should have a large dynamic range. If the detection scheme used does not have the dynamic range required then possibly the amplifier may saturate or if operating at the low end of the scale, valuable data may be lost in the noise. Effects of the size of the receiving aperture on the PDF must be investigated experimentally. It has been demonstrated that aperture size effects the resulting scintillation statistics. It has also been suggested that spikes of greater and greater irradiance are progressively smaller and smaller in spatial extent. If that is true, then the observed PDF will suffer from aperture averaging. Clearly more information is needed in this area. The randomness of the weather affects the sample size, source divergence and surface reflections when recording laser scintillation data. The sample size is affected by the time the weather, can be expected to remain unchanged during the collection period. If the sample size is made large then most likely the data collected will have been affected by various atmospheric conditions that occurred during the duration of the test. Source divergence and surface reflection are two more problems that must be overcome when recording reliable data. 12
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Information about the characteristics of the atmospheric turbulence at the time of recording the scintillation data is important. If information about the scale size of turbulence and refractive-index structure constant are known and these elements are homogenous along the propagation path then the effects of these parameters on the scintillation data can be determined [26]. 13
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CHAPTER 3 CHAMBER DESIGN AND CONSTRUCTION The turbulence chamber design requirements are to create turbulence in the inertial subrange of homogeneous isotropic turbulence, control and measure the strength of that turbulence and be able to propagate an optical signal through it. To maintain homogeneousness and control of the strength of the turbulence requires control over the wind velocities, temperatures and mixing of the air medium. The configuration shown in Fig. 5 allows for control of each of those parameters. The wind velocities are controlled by varying the input pressure to the chamber and or by varying the size of the hot and cold air jet nozzles. The thermostatically controlled oven allows for control of the heated air The turbulence chamber is a porcupine cylinder constructed of pyrex glass with dimensions of 1.5 meters in length and a diameter of 7.5 centimeters. Each of the porcupine nozzles of the chamber have a diameter of 1 centimeter. There are two columns of twenty (20) nozzles running the length of the chamber set perpendicular to two tangent parallel planes of the chamber. Connected to each column of nozzles are 1/4" o.d. teflon tubing 14
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REGULATOR 60 PSI OVEN Figure 5. Turbulence chamber schematic. ..... TURBULENCE CHAMBER HEATER CORE
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manifolds to direct to each nozzle an even air flow. One manifold for cool air flow and the other for hot air flow to the chamber. To minimize heat loss the warm air manifold is insulated with fiberglass. The warm air is generated by passing the pressurized air through a heater core located within a thermostatically controlled oven. The cool air flow is ambient air. Control of the air pressure to both the hot and cold manifolds is performed by one regulated air valve. The turbulent flow is generated when the hot and cold air flows mix in the chamber. Homogeneity is maintained by controlling the mixing of the air flows by adjusting the nozzle flow. The strength of the turbulence can be varied by either changing the difference in temperatures between the hot and cold air flows or by increasing the air flow velocities. The capability to measure the strength of the turbulence via thermocouple probes is realized by removing two air jets that face each other and replacing them with the thermocouples. The probes are each mounted on slides marked with 1 mm increments to mark probe location and movement. The configuration is shown in Fig. 6. 16
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TURBULENCE CHAMBER I SlJ LDGARITHIC L__ DIFFERENCE AMPLIFIER Sl-S2 A to D CONVERTER "<12 bits) THERMOCOUPLE PROBES S2 c DATA ACQUISITION SYSTEM Figure 6. Temperature probe placement view from center of chamber.
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CHAPTER 4 THERMAL CHAMBER CHARACTERIZATION To accurately define artificially generated atmospheric turbulence, information about the average wind velocity, temperature, temperature structure function, power spectra and statistical behavior all need to be known [26]. The following sections describe each of those required characterizations. Averaae Wind Velocity and Temperature Distribution The exit wind velocity of each nozzle were predicted for both the hot and cold flows from the input pressure and tubing diameter assuming an adiabatic, one dimensional, steady state flow of an incompressible fluid through a nozzle and the results with an input pressure of 45 psi are shown in Figure 7. Also shown in Figure 7 are the wind velocities measured on each end of the chamber with a straight line connecting the two. The slope is similar to the slope of the curve for the individual wind velocities of the nozzles plotted along the length of the chamber. The wind flow is fairly uniform along the length of the chamber with an average wind velocity of 0.18 m/s. A more uniform wind velocity can be obtained by further fine tuning of the air nozzles 18
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0.9 en 0.7 -0 u UJ en ........ en 0.6 a: UJ ::E 0.5 ) > .... H u 0.4 0 WIND VELOCITY AT ENDS WITH CONNECTING LINE .J UJ > c 0.3 z H PREDICTED EXIT VELOCITY FOR NOZZLE 0.2 0.1 0 0 2 4 6 8 10 12 LOCATION ALONG CHAMBER Figure 7. Wind velocity along length of chamber. 14 16 18 20
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sizes. The temperature was measured along the length of the chamber by inserting a thermometer through ports located along the top. For an oven setting of 300 degrees fahrenheit the temperatures are plotted in Fig. 8. Measurements of Temperature Structure Functions Turbulence is a nonstationary random fluctuation I process [20]. In order to measure this process, correlation functions normally used to describe the characteristics of the random variable are replaced with structure functions [26]. The underlying idea is to treat the process f(t) as random with stationary first increments. This approach was first introduced by Kolmogorov. From this approach the temperature structure function D (r) is defined as the mean square of the. temperature difference between two points, rl and r2: When the temperature field is homogeneous and isotropic, the structure function depends only on the distance between two points r=l rl-r21 : 20
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70 55 50 __ ._ __ ._ __ o 2 4 s a ro M re rn CHAMBER LOCATION Figure 8. Temperature profile along length of chamber. 21
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Experimentally, the temperature structure function can be measured with two thermocouples with their separation controlled [26]. This is accomplished by using micro-thermocouple of platinum with a frequency response equivalent to a low pass filter with a 3 db point of approximately 1 KHz. This frequency response is sufficient to record accurately the temperature fluctuations of the turbulence expected to have a predominant frequency component near 200 Hz. The \ thermocoupl.es are differentiated and then log-amplified for a greater dynamic recording range. The relationship between the output voltage of the difference amplifier and the temperature is given by The temperature structure function in relation to the measured voltage is then determined to.be The output is recorded on an analog-to-digital converter and stored on disk for signal processing. The temperature structure constant was measured along the entire length of the chamber using the 22
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arrangement described earlier, reference fig. 6, by removing opposite air jets and then placing the thermal couple probe in their place. Initially, the probes are placed as near to each other as possible without the delicate probes contacting each other and causing damage. This is a very approximate method and the initial separation is in the range of 0 to 1 mm. The oven and air pressure were turned on.up two hours before any data was recorded to reach thermal saturation of all the components related to the turbulence chamber. Tests were performed for mean operating temperatures of 50 and 55 degrees celsius and input pressures of 45 and 50 psi. Recordings were then made for the initial separation and separations made in 1 mm increments up to the width of the chamber. These measurements were made for the entire length of the chamber. The resulting measurements give a description of the chamber for various controlled conditions of wind velocity and temperature. A typical signal measured with the probes is shown in Fig. 9. In the inertial subrange of the homogeneous isotropic turbulence, the structure function will depend on the 2/3 power of distance between two probes [26] and is: 23
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-Q .48 r----r-,-T,-----r,----T,---,.r----r--,-r-1-Ta-----ra-! n -G.482 r u u -0.484 -o.486 r r \ \ I -0.49 r l, n -0.492 r \ -0.494 V(T)=2lLOG(TEMP 1 -TEMP2) i 1 i I I L I i -o.496 OL..____.J.5_-L10--1L5 _..l.20--2..L.5 -...-J301-..---L.35--4 ... 0 _.45:--:50" V(T), T = 0.0005 SECONDS Figure 9. Typical signal v(t) from output of log-amp. 24
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where the distance between two probes should be much larger than the inner scale of turbulence 10 but much smaller than the outer scale of turbulence L0 : 10 c r c L0 and Ct is the temperature structure constant and is a measurement of the turbulence strength. This 2/3 power is often called the similarity law [26]. A typical temperature structure function measurement is shown in Fig. 10 for one location along the length of the chamber for a fixed temperature of 56 degrees c and input pressure of 50 psi. The curve demonstrates that the measurement follows the 2/3 law for a probe separation of less than lmm to approximately 2 em. The results begin to deviate from the 2/3 line at 2 em due to exceeding the outer scale size of turbulence where the correlation between temperature fluctuations no longer holds. For all of the structure function measurements there were slight deviations from the 2/3 line when the correlation is still good. These slight deviations can be accounted for in the error of determining the exact separation between probes. The initial error plus an estimated error of .25 mm for each probe separation increment werenot accounted for. Using the error squared 25
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1u I--I I I I I I I I I I I I I I I I I I c 0 .... of.) u c -:J IL GJ t.. :J of.) u :J t.. too ..... rn GJ t.. :J of.) Ill t.. GJ a. E GJ .... 10_,, I I I I I I I I I I I I I I I I I I I I I I I I I I I to-t to0 to' 102 Probe Separtion, meters Figure 10. Typical temperature structure measurement for one locqtion along chamber.
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approximation formula almost all the deviations from the 2/3 line are within the approximate expected error. Temperature structure functions were also measured for variations in temperature and input pressure. The effects for variations in pressure and temperature are shown in Figures 11 and 12, respectively. Figure 11 contains two curves ofthe temperature structure function, both curves are for a common temperature but different input pressures. We see that with the input pressure reduced the strength of the turbulence is also reduced. The same results were also obtained by holding the pressure constant and varying the reference Figure 12. The uniformity of the strength of the turbulence within the chamber was evaluated from the temperature structure functions. The strength of the turbulence was determined from each structure function for a probe separation of 2 mm. This probe separation was chosen because it is within the small scale size and outer scale size of turbulence and the correlation between probe separation still holds. A measure of the strength of the turbulence is best described in terms of the refractive 27
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Probe Location @ K, Temp 59 c. Pressure c 50 & 45 psi 10' c: 0 ... ..... u c ::J u. CD L ::J ..... u ::J L 100 ..... UJ CD L ::J ..... 10 L Cll a. E Cll 1-10_1 , , , , , , , , , , , , , 10-1 10 101 102 Probe Separation, meters Figure 11. Effects of pressure on temperature structure function.
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Probe Location @ K, Temp = 59 and 54 degrees C 101 c 0 ..... ..., u c :::1 lL CD t. :::1 ..., u :::1 t. 100 ..., (I) CD t. :::1 ..., ca t. CD a. E Q) ..... 10_1 , , , , , , , , , , 10-4 10-3 10-2 10-1 Probe Separation, meters Figure 12. Effects of temperature on temperature function.
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index structure function and can be expressed in the same 2/3 law as the temperature structure function DIJ (r) c!r:a/3. The refractive index structure constant is determined -from the temperature structure function by using the equation of refractive index as a function of temperature and pressure with wavelength as a parameter. The refractive index of the air at the temperature (T) K, the pressure (p) mb, and the wavelength 1 um can be approximated by the following equation [26] Since the refractive-index under the constant pressure depends only on the temperature, we obtain c 77 .6P 11 + 0 I 00753 10-'C-. D Ta r 12 .. 30
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For the normal atmospheric pressure p = 1,013.25 mb and the wavelength 1 = 0.6328 um (typical ReNe wavelength) and the average temperature of the turbulent atmosphere of T = 326 K, the CnA2 were obtained and plotted in Figure 13. The values obtained all fluctuate about a mean of approximately 1.5 X 10A-11 mA-2/3. This is a turbulence strength approximately 1000 times that of typical atmospheric turbulence. The fluctuations about the mean are attributed to the uneven air jet velocities along. the. length of. the chamber. Adj.ustments were never made to correct for the uneven air flow but the results obtained are nearly homogenous and will suffice for the remainder of this paper. It is predicted that with the fine tuning of the air jet vel9cities the strength of turbulence can become very nearly perfectly homogenous. 31
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'C QJ t. co ::J 0" (I) ...., c I1J ...., ID c 0 u QJ t. ::J ...., u ::J t. ...., (I) X QJ 'C c H I QJ > rl ...., u co t. .... QJ a: Figure 13. x1o-u 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 Location In Chamber Along Z-Axis Refractive-index C02 measured along length of chamber. 18 20
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Power Spectrum Of Temperature Fluctuations The power spectra of the temperature fluctuations were determined for one location along the chamber for a probe separation of 2 mm and is shown in Figure 14. This was obtained by the Welch method of power spectrum estimation. As can be seen from the figure, the estimated power spectrum is proportional to the f-5/3 law over the frequency region of 3-35 Hz. At frequencies greater than 35 Hz the power spectrum follows the f-7 law, which was obtained by K in the viscous dissipation region. At frequencies below 1 Hz no discernable information can be obtained due to the limitations of the FFT used. However, it is predicted by Strobehn that the power spectra at that end will approach a constant value. In order to determine the transition frequency from the inertial subrange to the viscous dissipation range, we also calculated f.2P(f) and is also shown in Figure 15. The upper crossing point of the f-5/3 line for inertial subrange with the f-7 line is fh=65 Hz. Now, the inner scale of turbulence equal to lo of the inertial subrange is given by lo=u/fh [26], by inserting the average wind velocity u = 0.18 m/sec gives lo = 2.71 mm. If the lower crossing point is estimated to be 3 Hz this gives a Lo=6.2 em which is in agreement with the estimate from the temperature structure function curves. The 33
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>..... ID c Ql 0 rl IU u Ql c. Ul 'QI X 0 Q. "0 ID N r-4 rl tU E c... 0 z 100 1 i I I I F1-rl I \ I I I I I I I I I I I I I I I 13 POWER SPECTRA OF TEMPERATURE FLUCTUATIONS SLOPE -7 -s' I I I I I I I I I I I I I I I I I I \ I I I I I I I I I 10 1oo 10l 102 103 Frequency Figure 14. Power spectrum of temperature fluctuation W(f) as a function of frequency (Hz) where slopes of respective regions are indicated.
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lU .. .... SLOPE 113 SLOPE _5 Jl: .... .... r > 101 L \ t-1-1 m z I1J c ...J a: tu I1J D. m a: I1J 3: 100 c UJ N .... :E a: 0 z -1 I I I I I I I I I I I I I I I I I I I I I I \ ,, !ft. !!. I I I 101oo 101 102 103 FREQUENCY, HZ Figure 15. Power spectra of temperature fluctuation f2W(f) as a function of frequency (Hz) where slopes of respective regions are indicated.
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Reynolds number for these scales of turbulence are Rel=ulo/v.and ReL=uLo/v where vis the kinematic viscosity of air, v=l.S X 10-5 [26] are given by Rel=33 and ReL=768, respectively. 36
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CHAPTER 5 RECEIVER DESIGN To accurately capture irradiance scintillation for producing genuine PDFs of the scintillation, the receiver design must have a wide dynamic range and the characteristics of the detection. must be thoroughly understood [10]. In addition, the for capturing and storing-the data for later signal processing must not corrupt or filter the data. The choices involved in designing a receiver are in the detector, aperture size, lens configuration and amplifier. For irradiance scintillation, the selection of the aperture and lens configuration are best achieved experimentally and will not be investigated here. The remaining choice of selecting the detector and the amplifier will be examined and made. The storage and signal processing techniques will also be discussed. Component Selection The detector chosen must have an achievable range of linearity of 3 to 4 orders of magnitude. The photodiode has this characteristic. It is extremely linear respect to the amount of incident light and has a fast response time making it ideally suited for this application. Figure 16 shows the result of plotting the 37
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VB. Ll5HI 3000 en 2500 !:i 0 > H ::f 2000 I .... I V OUT AMPLIFIED :E ..: :::3 0 1500 .l&J CD ..: 1..J 0 > 1000 I I 5001-I V OUT PASSIVE 50 100 150 200 250 300 ILLUMINANCE. MICRO-LUX Figure 16. output current vs incident light on detector.
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photodiode output current as a function of incident light. Ideally, the operating point chosen for the detector would be in the center of the curve, but for our applications we are most interested in capturing large scale spikes of irradiance that are 1000 times the mean without saturating the detector. Therefore, the operating point chosen will be at lower point of the curve, the large scale spikes to be faithfully recorded. Operating near the bottom of the curve, however, causes the small scale spikes of -irradiance-to be located within or very near the noise. This problem will be mitigated using noise subtraction techniques. If the noise is signal-independent, additive noise such as that expected from detector dark-current-shot noise and preamplifier thermal noise, one can reliably correct the measured signal statistics for the effects of low signalto-noise ratio as long as the noise statistics are known precisely. If the noise subtraction techniques are to be reliable, the analog to digital conversion process must not filterthe small changes. The data acquisition system we are using in association with detection scheme has a maximum input of 5 volts. If amplifier saturation is to be avoided for fluctuations 1000 times the mean, the operating mean selected will be in the order of 39 .. ... "'-
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millivolts. The small scale spikes of irradiance will then be in the order of microvolts. The analog to digital conversion of 12 bits provided by the acquisition system is only capable of a resolutions in voltage of 0.25 millivolts (5 volts/212). The smaller scale spikes of scintillation and the noise may not be faithfully recorded at the operating point we have chosen. This is avoided by recording the signal directly and simultaneously recording the signal amplified as shown in Figure 17. The amplified signal,V1(t), will have a much greater mean and therefore small variations will be amplified greater than 0.25 millivol.ts and the 12 bit digital to analog conversion will not filter any vital information. The amplified signal will saturate for the large scale scintillations but the saturated points can be thrown away and replaced by good points from VJ(t). The process is as follows: 1.) Start with v1Ck), k is the sample number. 2.) Locate the points where v1Ck) is greater than 5 Vdc. 3.) At these locations, pick up 1000*Va(k) and replace v1(k) at these locations only. 4.) The result is V' (k) that is lOOO*V(k) and is free from detector and amplifier saturation. 40
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V -LIGHT VR(_L) L AMPLIFIER Vt (t)=AV Figure 17. Dual amplifier circuit used to capture lagre scale spikes of irradiance.
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Using this method, we can reconstruct accurately the laser scintillation and be that we have captured all fluctuations. Receiver Operational Circuits Two operational circuits were selected and are shown in Figure 18. Figure 18(a) offers a faster response time but has some limitations in linearity. Figure 18(b) which uses an operational amplifier gives nearly ideal linearity and offers the flexibility of changing the value of Rf to measure intensities over a large range. The input intensities were varied to determine the range of linearity of the two circuits. Linearity over 3 orders of magnitude was achieved by both circuits. 42
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(a) Reverse Bias Circuit LIGHT"<..... v. ( Vout=A X Ish X RL L--...... -----o (b) Op-Amp Circuit Rt Vout = -(Ish X Rl) LIGHT Figure 18. Photodiode operational circuits.
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CBAPTBR 6 STATISTICAL MBASURBMBNTS OF IRRADIANCB EXPERIMENTAL ARRANGEMENT A number of experimental arrangements were used to measure the irradiance scintillations. A collimated laser beam with a 1 mm diameter cross section from a single-mode He-Ne gas laser (Spectra Physics model 119, wavelength = 6328 angstroms, and output power of 0.1 mW) was propagated down the turbulence chamber as shown in Figure 19. The intensity was varied by adjusting the neutral density filters placed in the path of the beam and measurements were taken for different intensities. Three receivers alone were used in the detection of the propagated beam. The three receivers consisted of one with a 0.5 mm pinhole in front of the detector. The second receiver used a focusing lens and the third was a detector with only a filter in front. The three receivers were used in combination with one and two paths down the chamber and different polarizations of the laser beam. The second path through the chamber was done by placing a corner cube reflector at one end of the chamber and setting up the detection scheme at the other end as shown in Figure 20. The polarization was changed by 44
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MIRROR Figure 19. LENS(f=40MM)/APERTURE D DETECTOR TURBULENCE CHAMBER I FILTER ATTENUATDRS AMPLIFIEN D: LASER 0 DATA ACQUISITION SYSTEM Test configuration used to measure laser scintillation for one path measurements.
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I Figure 20. MIRRORS --------------' t\. \. TURBULENCE CHAMBER DETECTOR \ CORNER CUBE REFLECTOR ATTENUATDRS D: LASER 0 DATA ACQUISITION SYSTEM Test configuration used to measure laser scintillation for two path case.
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reflecting the light off two mirrors before propagating the beam down the length of the chamber. For each of the arrangements just described, two sample sets each of 8000 samples at a sample rate of 2000 hz was taken for various configurations. From the sampled data, normalized histograms p(I/*) and power spectra plots fW(f) were generated and the higher order moments, cumulants, and central moments up to the eight order were determined. For comparison, the moments of the data were compared to three PDFs commonly proposed in the literature for describing irradiance fluctuations. They are the Gamma ,Log-Normally Modulated Exponential and the Log-Normal distributions [1] The Gamma function has the form: p (I) _!_ Ill-1 e(-..!) p r(m) p with moments r(k+m) m,.JJ: m .It mk r(m) The log-normally modulated function has the form: 47 *
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with the moments And the log-normal function has the form: p (I) 1 exp [lna (Ia) ] IJ2sln(.m..z) with the moments The moments used for the data were normalized for the comparison by where is the average intensity. The estimates for the normalized moments of the data ar& then given by 1 N Hk--E v(N) i: Nl 48
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EXPERIMENTAL RESULTS The first measurement taken was for one path propagation down the turbulence chamber with the filter only receiver and vertical polarization of the laser beam. Prior to any measurements the chamber was prepared by turning on the heat and the air pressure at 45 psi and waiting until the temperature in the chamber saturated. The chamber temperature saturated after approximately one hour at a temperature of 57 degrees. centigrade. The laser beam was centered down the length of the chamber and aligned onto the detector for maximum output (The alignment was done with the chamber at ambient temperature and no air pressure prior to the start of the test), The intensity of the beam was adjusted for a voltage from the output of the receiver of V0 = 0.1 volts by selection of a filter. With the room darkened two sample sets each of 8000 samples at a sample rate of 2000 Hz were taken. A filter (fc = 0.6238 micrometers) was placed in front of the laser and again two sets of data were taken and stored for processing. The power spectra plots, normalized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 19 & 20 and Table I and Figures 21 & 22 and Table II for the data with a filter and 49
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.... 3: .... > 1-"H en z UJ 0 ...J c( a: 1u UJ a. en a: l1J 3: 0 a. 0 UJ N H ...J c( ::E a: 0 z too I I I I 1 I I I I I I I I I I I I I I I I I I I IOJ fW(f) FOR ONE WAY PASS, DETECTOR ONLY o-s I I I I I I I I I I I I I I I I I I I I I I I Qll I I I 1 10 101 102 103 FREQUENCY (HZ) Figure 21. Normalized power spectra fW(f) for one 'path with detector only receiver.
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A .... v ...... .... Q. ..... l&. .... 5 CD .... rn .... 0 > .... .... -' 1-t 21PDF FOR ONE WAY, DETECTOR ONLY ... 1.5111'1-m 0.5 ... 0 J I I 0 0.5 1 1.5 2 2.5 3 NORMALIZED IRRADIANCE I/* Figure 22. Normalized histogram for one path with detector only receiver. *
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TABLE I. Statistical Data For One Way Pass, Detector Only, Without Filter, Input PSI = 45. ORDER 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 n MOMENTS m 1.1606e-001 1.4229e-002 1. 8487e-Oo3 2.5527e-004 3.7564e-005 5.9080e-006 9.9588e-007 1.8034e-007 CENTRAL CUMULANT MOMENTS u k 1.1606e-001 7.5877e-004 2.1213e-005 9.3246e-007 8.2680e-008 7.0203e-009 1.3968e-005 0 7.5877e-004 2.1213e-005 2.6596e-006 2.4364e-007 2.8686e-008 1.3972e-005 6.1282e-010 -3.6271e-004 NORMALIZED MOMENTS m' 1 1.0563e+OOO 1.4069e+OOO 2.7839e+OOO 2.4174e+OOO 3.5110e+OOO 5.4781e+OOO 52 CENTRAL MOMENTS u' CUMULANTS k' l.OOOOe+OOO l.OOOOe+OOO 1.0149e+OOO 1.0149e+OOO 4.6196e+OOO 1.6196e+OOO 1.5363e+001 5.2135e+OOO 6.5666e+001 1.6070e+001 1.1611e+006 1.1608e+006 1.848e+003 -1.0943e+009
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TABLE I. (contd.) EXPERIMENTAL VS PDFs ORDER m' m' m' n GAMMA LOG-NRM LOG-M\NRM MODULATED 2 1 1 1 3 1.0061e+OOO 9.5344e-001 1.0033e+OOO 4 1.0240e+OOO 4.8375e-001 1.0127e+OOO 5 1.0596e+OOO 4.6651e-002 1.0312e+OOO 6 1.1203e+OOO 1.9556e-004 1.0625e+OOO 7 1.2162e+OOO 5.1348e-009 1.1108e+OOO 8 1.3609e+OOO 7.5895e-017 1.1810e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 5.1270e-002 volts 3.0518e-001 volts 53
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10o I 1 I I I I I I I I I I I I I I I I I I I I I I I I I I .... I .... > .... .... en ffi 0 ;i u UJ CL en 0 UJ N .... ;i 0 z 10-l 10-2 10-s 100 Figure 23. fW(f) FOR ONE WAY PASS,DETECTOR ONLY 101 102 103 FREQUENCY (HZ) Normalized power spectra fW(f) for one path with filter only receiver.
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1\ .... v ...... .... -Q. .... .... lL .... 1-ffi .... en .... c > .... .... _. .... $! m 0 a: Q. 1.61-PDF FOR ONE WAY PATH,DETECTOR ONLY 1.41. r-1.21. 110.81-,... 0.61-1-,.... 0. 4 1. 0.2 10 I _rt] 111111111.111111111. I tinn-rn, I I I 0 0.5 1 1.5 2 2.5 3 3.5 NORMALIZED IRRADIANCE I/* Figure 24. Normalized histogram for one path with filter only receiver. *
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TABLE II. Statistical Data For One Way Pass,Detector With Filter, Input PSI = 45. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 MOMENTS m 1.6323e-001 2.8921e-002 5.5733e-003 1.1698e-003 2. 6771e-00.4 6.6836e-005 1.8189e-005 5.3826e-006 CENTRAL MOMENTS u 0 2.2772e-003 1.0917e-004 2.4575e-005 3.6618e-006 7.7100e-007 1.7823e-004 4.4691e-008 NORMALIZED MOMENTS m 1 1.0855e+OOO 1.2815e+OOO 1.6478e+OOO 2.3104e+OOO 3.5337e+OOO '5.8914e+OOO 1.0681e+001 CENTRAL MOMENTS u 1.0000e+OOO 1.0046e+OOO 4.7389e+OOO 1.4797e+001 6.5289e+001 3.1628e+005 1.6619e+003 56 CUMULANT k 1.6323e-001 2.2772e-003 1.0917e-004 9.0174e-006 1.1758e-006 1.6667e-007 1.7808e-004 -3.2670e-003 CUMULANTS 1.0000e+OOO 1.0046e+OOO 1.7389e+OOO 4.7514e+OOO 1.4114e+001 3.1601e+005 -1.2149e+008 k' 0
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TABLE II.(contd.) ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL-VS PDFs m GAMMA 1 1.0083e+OOO 1.0319e+OOO 1.0782e+OOO 1.1553e+OOO 1.2732e+OOO 1.4443e+OOO m' LOG-NRM MODULATED 1 8.1188e-001 2.1948e-001 4.5491e-003 9.6133e-007 1.6398e-013-1.1237e-024 m' LOG-M\NRM 1 1.0020e+OOO 1.0074e+OOO 1.0174e+OOO 1.0327e+OOO 1.0526e+OOO 1.0748e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 5.3711e-002 volts 5.0049e-001 volts 57
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without a filter, respectively. The next set of data was taken with the same polarization but with the receiver with the focusing lens,focal point = 2 cm,radius = 4mm. The receiver was aligned for maximum incident light with the output voltage of the receiver adjusted to 0.1 volts with the chamber inactive. The chamber was then activated and the input pressure set to 45 psi. After temperature saturation two sets of data were taken as before. The air pressure was then turned off and the intensity of the beam down the chamber increased so that the incident light on the detector produced an output voltage of V0 = 1.0 volts. The input pressure was turned back on to 45 psi and measurements were repeated after the chamber temperature saturated. The incident light on the detector was then increased to 1.0 volts and two more sets of data taken. The power spectra plots, normalized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 25 & 26 and Table III and Figures 27 & 28 and Table IV for the input intensities proportional to 0.1 and 1.0 volts, respectively. 58
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... I ... > .... ..... ffi Cl ...J a: 1u UJ a. en a: UJ 3 0 a. c ..... ...J z so so-s so Figure 25. DETECTOR WITH FOCUSING LENS 10' 102 FREQUENCY (HZ) Normalized power spectra fW(f) for one path with focusing lens receiver, input intensity proportional to 0.1 volt.
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A .... v ....... .... a. z 0 .... .... ;:::) u. z 0 .... ... :::l CD .... a: ... U) .... c > ... .... _J .... CD CD 0 a: a. 7 6 5 4 3 2 1 -0 0 Figure 26. --PDF FOR ONE WAY PATH FOCUSING LENS WITH DETECTOR r ---I J 0.2 0.4 0.6 0.8 1 1.2 1.4 NORMALIZED IRRADIANCE I/* Normalized histogram for one path focusing lens receiver, input intensity proportional to 0.1 volt. *
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TABLE III. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 Hx 7 8 Statistical Data For One Way Pass, Detector With Focusing Lens, Input PSI =45. MOMENTS m CENTRAL MOMENTS u 8.8953e-002 7.9394e-003 7.1143e-004 6.3999e-005 5.7798e-006 5.2400e-007 4.7691e-008 4.3572e-009 0 2.6857e-005 4.1726e-007 -3.3016e-008 3.2137e-009 -2.8495e-010 1.5766e-006 -2.2586e-012 5 NORMALIZED MOMENTS CENTRAL MOMENTS CUMULANT k 8.8953e-002 2.6857e-005 4.1726e-007 -3.5180e-008 3.1016e-009 -2.7281e-010 1.5766e-006 -4.544Je-007 CUMULANTS m u k' 1 1Hx034e+OOO 1.0108e+OOO 1.0222e+OOO 1.0378e+OOO 1.0577e+OOO 1.0822e+OOO 1.1116e+OOO 1.0000e+OOO 2.9979e+OOO -4.5773e+001 8.5971e+002 -1.4709e+004 1.5704e+010 -4.3411e+006 61 2.9979e+OOO -4.8773e+001 8.2973e+002 ... 1.4082e+004 1.5704e+010 -8.7342e+011
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TABLE III. ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL VS PDFs m' GAMMA 1 NaN NaN NaN NaN NaN NaN m' m' LOG-NRM LOG-M\NRM MODULATEDHx 1 1.3051e+OOO 2.3893e+OOO 6.1102e+OOO 2.0738e+001 8.5652e+001 3.8289e+002 1 1.0006e+OOO 1.0016e+OOO 1.0032e+OOO 1.0053e+OOO 1.0079e+OOO l.OllOe+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 6.8359e-002 volts 1.0986e-001 volts 62
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100 ..... 3: ..... >= IH Cl) 1o-t z UJ CJ "_J < a: Iu UJ 11. Cl) a: UJ 3: 0 11. fW(f) FOR TWO WAY PATH CJ UJ N DETECTOR WITH FOCUSING LENS H _J < ::: a: 0 z -3 II I I I 101oo 101 102 103 FREQUENCY (HZ) Figure 27. Normalized power spectra fW(f) for one path with focusing lens receiver, input intensity proportional to 1.0 volt.
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1\ H v ....... H a. z 0 H b" z z 0 H .... = m H a: trn H c > tH _. H m c( m 0 a: a. 81-71PDF FOR ONE WAY PATH DETECTOR WITH FOCUSING LENS 61-51-41-31-2111-0 I I I I I ,..oJ'}f] 1111111!111111 0 0.2 0.4 0.6 NORMALIZED IRRADIANCE Figure 28. Normalized histogram 'for :one path intensity proportional to o. ----
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TABLE IV. Statistical Data For One Way Pass, Detector With Focusing Lens, Increased Intensity, Input PSI = 45. l 2 3 4 5 6 7 8 ORDER n ORDER n l 2 3 4 5 6 7 8 MOMENTS m l.l758e+OOO 1.3854e+OOO 1.6366e+OOO 1.9382e+OOO 2.3013e+OOO 2.7393e+OOO 3.2689e+OOO 3.9l05e+OOO CENTRAL MOMENTS u 0 2.8200e-003 9.2282e-004 -l.0606e-003 1.2944e-003 -1.522le-003 l.1028e+002 -2.1055e-003 NORMALIZED MOMENTS CENTRAL. MOMENTS m' u' l 1.0020e+OOO l.OOOOe+OOO 1.0067e+OOO 6.1623e+OOO l.Ol40e+OOO -1.3337e+002 l.0239e+OOO 3.065le+003 l.0365e+OOO -6.7870e+004 l.0519e+OOO 9.2603e+010 1.0702e+OOO -3.3292e+007 65 cuMULANT k l.1758e+OOO 2.8200e-003 9.2282e-004 -1.0845e-003 1.2684e-003 -l.485le-003 1.1028e+002 -7.1039e-003 CUMULANTS k' l.OOOOe+OOO 6.l623e+OOO -1.3637e+002 3.0035e+003 -6.6220e+004 9.2603e+010 -l.l233e+008
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TABLE IV. (contd.) ORDER n 2 J 4 5 6 7 8 EXPERIMENTAL VS PDFs m' GAMMA 1 NaN NaN NaN NaN NaN NaN m' LOG-NRM MODULATED 1 1.3157e+OOO 2.4882e+OOO 6.9008e+OOO 2.7550e+001 1.5120e+002 1.0656e+OOJ m' LOG-M\NRM 1 1.0006e+OOO 1.0016e+OOO l.OOJ2e+OOO 1.005Je+OOO 1.0079e+OOO 1.0109e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 9.2041e-001 volts 1.3843e+OOO volts 66
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Using the same receiver with the focusing lens as before the test was repeated but with a new horizontal polarization. Two sets of data were taken for an incident light producing 0.1 volts output. The intensity was increased as before to 1.0 volts output and again two sets of data were taken. The power spectra plots, norm_alized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 29 & 30 and Table Vand Figures 31 & 32 and Table VI for the input intensities proportional to 0.1 and 1.0 volts, respectively. 67
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.... 3 .... > .... H en ffi Q ..J oC a: .... (J UJ a. en a: UJ 3 0 a. 0 UJ N ..... .J oC 0 z to-t to-s 100 Figure 29. fW(f) FOR ONE WAY PATH DETECTOR WITH FOCUSING LENS 90 DEGREE POLARIZATION SHIFT tot 102 103 FREQUENCY (HZ) Normalized power spectra fW(f) for one path with focusing lens receiver,90 degree polarization shift input intensity proportional to 0.1 volt. .,
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1\ 2.51 111n 90 DEGREE t-4 PDF FOR ONE WAY PATH v ...... .... DETECTOR WITH n111111 POLARIZATION SHIFT -a. z FOCUSING LENS 0 .... 2 1-u z :::l lL z 0 t-4 1.5 1-:::l m t-4 a: 1-rn t-4 Q 1 > 1-t-4 _. t-4 m m 0.5 0 a: Q. 0 I I 1 -" I I I I I I I !1 I I I I I I I I !1 I I I I I I I I !I I I I I I I I I !I I I I CO, ..._..,__ I 0 Figure 30. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 NORMALIZED IRRADIANCE I/* Normalized histogram for one path with focusing lens receiver, input intensity proportional to 0.1 volt, 90 degree polarization shift. *
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TABLE V. Statistical Data For One Way Pass, Detector With Focusing Lens, 90 Degree Polarization Shift. 1 2 3 4 5 6 7 8 ORDER ORDER n 1 2 3 4 5 6 7 8 n MOMENTS CENTRAL CUMULANT MOMENTS m u 8.8504e-001 0 8.0224e-001 1.8947e-002 7.4542e-001 1.8764e-003 7.1015e-001 9.1712e-004 6.9371e-001 6.0305e-004 6.9483e-001 -1.1311e-004 7.1346e-001 1.7231e+001 7.5075e-001 -1.8472e-004 NORMALIZED k 8.8504e-001 1.8947e-002 1.8764e-003 -1.5985e-004 2.4753e-004 -2.0492e-004 1.7231e+001 -2.2620e-001 MOMENTS CENTRAL MOMENTS CUMULANTS m' u' 1 1.0242e+OOO 1.0753e+OOO 1.1575e+OOO 1.2775e+OOO 1.4458e+OOO 1.6774e+OOO 1.9944e+OOO 1.0000e+OOO 1.0000e+OOO 7.1947e-001Hx7.1947e-001 2.5547e+OOO -4.4526e-001 1.2204e+001 5.0093e+OOO -1.6630e+001 -3.0128e+001 1.8404e+007 1.8404e+007 -1.4334e+003 -1.7553e+006 70 k'
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TABLE -v. (contd.) ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL_ VS PDFs m' GAMMA 1 -1. 0014e+OOO 1.0050e+OOO 1.0114e+OOO 1.0212e+OOO 1. 0_346e+OOO 1.0520e+OOO m' LOG-NRM MODULATED 1 1.1532e+OOO 1.2837e+OOO 9.4133e-001 2.6093e-001 1.3378e-002 5.3516e-005 m' LOG-M\NRM 1 1.0009e+OOO 1.0028e+OOO 1.0059e+OOO 1.0102e+OOO 1.0155e+OOO 1.0213e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 5.1758e-001 volts 1.5039e+ooo volts 71
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& ... I ... > .... H .en ffi c ..J 4( a: .... u en I .... a. to-2 c UJ N H 0 to-a too Figure 31. fW(f) FOR ONE WAY PASS DETECTOR WITH FOCUSING LENS 90 DEGREE POLARIZATION SHIFT to FREQUENCY (HZ) t02 103 Normalized power spectra fW(f) for one path with focusing lens receiver, input intensity proportional to 1.0 volt, 90 degree polarization shift.
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" PDF FOR ONE WAY PATH 90 DEGREE ..... v ...... ..... DETECTOR WITH POLARIZATION SHIFT a. FOCUSING LENS z 0 H +u z z 0 H .... ::l m H a: .... Ul .... c i .... _J H m ""' m 0 0.5 a: a. 0 I 1 1 + d I I I I I I hi IIIII Ill hi Ill IIIII hill I Ill II hi f"n""Ln 1 I 0 Figure 32. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 i.B NORMALIZED IRRADIANCE I/* Normalized histogram for one path with focusing lens receiver, input intensity proportional to 1.0 volt, 90 degree polarization shift. *
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TABLE VI. Statistical Data For One Way Pass,Detector With Focusing Lens, Increased Intensity,90 Degree Polarization Shift. 1 2 3 4 5 6 7 8 ORDER ORDER n 1 2 3 4 5 6 7 8 n MOMENTS CENTRAL CUMULANT MOMENTS m u k 1.0730e-001 0 1.0730e-001 1.1731e-002 2.1690e-004 2.1690e-004 1.3072e-003 1.8335e-006 1.8335e-006 1.4843e-004 8.1199e-008 -5.9941e-008 1.7172e-005 1.0979e-008 7.0026e-009 2.0237e-006 -6.6745e-010 -6.5912e-010 2.4288e-007 6.4189e-006 2.9685e-008 -9.6859e-012 -2.9639e-005 NORMALIZED MOMENTS CENTRAL CUMULANTS MOMENTS m' u' k' 1 1.0188e+OOO 1.0000e+OOO l.OOOOe+OOO 1.0580e+OOO 5.7396e-001 5.7396e-001 1.1196e+OOO 1.7259e+OOO -1.2741e+OOO 1.2070e+OOO 1.5846e+001 1.0106e+001 1.3256e+OOO -6.5407e+001 -6.459le+001 1.4827e+OOO 4.2711e+007 4.2710e+007 1.6888e+OOO -4.3760e+003 -1.3391e+010 74
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TABLE VI. (contd.) ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL VS PDFs m' GAMMA 1 1.0007e+OOO 1.0023e+OOO 1.0049e+OOO 1.0136e+OOO 1.0200e+OOO m' LOG-NRM MODULATED 1 1.1912e+OOO 1.5160e+OOO 1.5689e+OOO 8.7950e-001 1.5774e-001 4.7712e-003 m' LOG-M\NRM 1 1.0004e+OOO 1.0010e+OOO 1.0016e+OOO 1.0019e+OOO 1.0020e+OOO 1.0015e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 6.3477e-002 volts 1.7090e-001 volts 75
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To expand on the dynamic range of the detection scheme the dual amplifier design discussed in Chapter 2 was used for the one path propagation configuration with a filter only in front of the detector. The horizontal polarization and alignment done for the previous test were unchanged. Only the new receiver was in place and aligned to the laser beam. To account for the increased .sensitivity of the new receiver the output voltage of channel 2 was adjusted to 2.5 mV. Four sample sets of 1600 samples each at a sample rate of 1000 Hz were then taken at an input pressure of 45 psi and an average saturated chamber temperature of 57 degrees centigrade. The sample size and rate had to be changed due to the limitations of the data acquisition system used for all of the tests. The incident light was increased for an output mean voltage of 25 mV and the tests repeated. The power spectra plots, normalized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 33 & 34 and Table VII and Figures 35 & 36 and Table VIII for channel zero and one ,respectively. 76
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-""" """ > .... 1-1 Ul 10-1 z UJ a ...J <( a: .... u UJ a. Ul a: UJ 0 a. 10-2 a UJ N 1-1 ...J <( X a: 0 z fW(f) FOR ONE WAY PATH AMPLIFIED CHNL OF DUAL AMP CONFIGURATION 10-3 100 10' 102 103 Figure 33. FREQUENCY (HZ) Normalized power spectra fW(f) for one path with filter only receiver. 90 degree polarization shift, and amplified channel of dual-amp configuration.
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1\ t-1 v ....... t-1 -a. z 0 .... 1-u z z 0 t-1 1-:::J m H lr 1-CD H c H -' .... m m 0 lr a. Figure 34. 10 -8 6 -4 2 0 0 PDF FOR ONE WAY,DETECTOR WITH FILTER AMPLIFIED CHANNEL OF DUAL AMP CIRCUIT I I I 0.2 0.4 0.6 0.8 NORMALIZED IRRADIANCE I/* l I 1 I j I j --' r-r-II 1 1.2 Normalized histogram for one path with filter only receiver. 90 degree polarization shift, and amplified channel of dual-amp configuration. *
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TABLE VII. ORDER n 1 2 3 4 5 6 7 8 ORDER n 2 3 4 5 6 7 8 1 Statistical Data For Two Way Pass, Detector With Filter, Dual Amp Configuration, Ampli.fied Channel. MOMENTS m 1.2131e-002 7.3659e-004 4.4799e-005 2.7291e-006 1.6653e-007 1.0178e-0'08 6.2314e-010 3.8213e-011 CENTRAL MOMENTS u 0 5.8943e-004 2.1563e-005 1.1407e-006 5.4823e-008 2.7055e-009 3.0359e-010 6.5710e-012 NORMALIZED CUMULANT k 1.2131e-002 5.8943e-004 2.1563e-005 9.8435e-008 -7.2273e-008 -5.8860e-009 3.3729e-010 -2.1888e-004 MOMENTS CENTRAL MOMENTS CUMULANTS m' 1 5.0053e+OOO 2.5094e+001 1.2602e+002 6.3388e+002 3.1937e+003 1.6118e+004 8.1477e+004 u' l.OOOOe+OOO l.OOOOe+OOO 1.5068e+OOO 1.5068e+OOO 3.2833e+OOO 2.8333e-001 6.4996e+OOO -8.5683e+OOO 1.3211e+001 -2.8743e+001 6.1062e+001 6.7842e+001 5.4439e+001 -1.8133e+009 79 k'
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TABLE VII.(contd.) ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL VS PDFs m' GAMMA 1 5.5640e-001 2.1467e-001 6.3439e-002 1.5201e-002 3.0647e-003 5.3353e-004 m' LOG-NRM MODULATED 1 2.1173e-003 8.3907e-011 5.1648e-025 3.9623e-048 2.9722e-082 1.6804e-129 m' LOG-M\NRM 1 2.0012e-001 8.0139e-003 6.4223e-005 1.0300e-007 3.3056e-011 2.1230e-015 Mimimum And Maximum Intensity Fluctuations minimum maximum 5.3711e-002 6.8359e-002 80
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.... 3: .... .... en z UJ c ...J c( a: t-u LLI a. en a: LLI Jl: 0 a. c UJ N H ...J c( :::E a: 0 z Figure 35. 10-l 10-2 10-a 100 DIRECT CHANNEL OF DUAL AMP 90 DEGREE POLARIZATION SHIFT 10' FREQUENCY (HZ) 102 103 Normalized power spectra fW(f) for one path with filter only receiver. 90 degree polarization shift, and direct channel of dual-amp configuration.
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1\ .... v ....... H a. z 0 H .... u I&. z 0 H :J m H a: .... en H c >-.... .... _. .... m c( m 0 a: Q. 8 7 -6 -5 4 3 -2 1 0 0 Figure 36. DIRECT CHANNEL OF DUAL AMP CONFIGURATION. DETECTOR WITH -FILTER ONLY. 90 DEGREE POLARIZATION SHIFT -I I I I 0.2 0.4 0.6 0.8 1 1.2 1.4 NORMALIZED IRRADIANCE I/* Normalized histogram for one path with filter only receiver. 90 degree polarization shift, and direct channel of dual-amp configuration. *
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TABLE VIII. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 Statistical Data For One Way Pass, Detector With Filter, Dual Amp Configuration Direct Channel. MOMENTS m 4.5470e-003 1.0358e-004 2.3656e-006 5.4171e-008 1.2439e-009 2.8643e-011 6.6140e-013 1.5316e-014 CENTRAL MOMENTS u 8.2902e-005 1.1407e-006 .2713e-008 4.1181e-010 7.6787e-012 3.2121e-013 2.6774e-015 NORMALIZED MOMENTS m' 1 5.0098e+OOO 2.5163e+001 1.2673e+002 6.3998e+002 3.2409e+OOJ 1.6459e+004 8.3819e+004 CENTRAL MOMENTS u' l.OOOOe+OOO 1.5112e+OOO 3.3047e+OOO 6.5809e+OOO 1.3477e+001 6.1917e+001 5.6683e+001 83 CUMULANT k 4.5470e-003 8.2902e-005 1.1407e-006 2.0945e-009 -5.3385e-010 -1.6484e-011 3.4382e-013 -4.3298e-006 k' 1.0000e+OOO 1.5112e+OOO 3.0475e-001 -8.5310e+OOO -2.8931e+001 6.6274e+OOl -9.1666e+010
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TABLE VIII.(contd.) EXPERIMENTAL VS PDFs ORDER m m m n GAMMA LOG-NRM LOG-M\NRM MODULATED 2 1 1 1 3 5.5688e-001 2.1058e-003 2.0013e-001 4 2.1526e-001 8.1585e-011 8.0163e-003 5 6.3798e-002 4.7382e-025 6.4270e-005 6 .5349e-002 3.2266e-048 1.0314e-007 7 3.1106e-003 1.9559e:..082 3.3132e-011 8 5.4498e-004 7.8168e-130 2.1305e-015 Mimimum And Maximum Intensity Fluctuations minimum maximum 1.9531e-002 2.6855e-002 84
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The path length of the propagating beam-was increased to two paths down the length of the chamber by placing a corner cube reflector at one end of the chamber and then reflecting the to the detector at the other end after the beam had traveled up and down the chamber as shown in Figure 20. The filter only receiver was used first. The input pressure was set to 45 psi and the measurements were taken after the chamber temperature saturated. samples of 8000 samples each were taken at a sample rate of 2000 Hz. The input pressure was then increased to 60 psi and again chamber temperature saturation was achieved and two data sets of 8000 samples each were taken at a sample rate of 2000 Hz. The power spectra plots, normalized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 37 & 38 and Table IX and Figures 39 & 26 and Table X for the input intensities proportional to 0.1 and 1.0 volts, respectively. 85
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... 3: ... .&v.-.... Ul to- z UJ c ..J .... u UJ D.. Ul a: UJ 3: 0 D.. 10-2 c UJ N .... ;l :E a: 0 z Figure 37. 10-3 too fW(f) FOR TWO WAY PATH,DETECTOR ONLY to t02 103 FREQUENCY (HZ) Normalized power spectra fW(f) for two path with filter only receiver and input pressure = 45 psi.
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II. 3.5 L PDF FOR TWO WAY, 1-t v ....... 1-t I DETECTOR ONLY Q 3 z 0 1-t .... -U 2.5 z :::l. IJ.. z 0 1-t .... ::I m .... a: .... en 1-t c >-.... H ...J 1 1-t m m 0 a: Q. 0 0 0.2 0.4 0.6 O.B 1 1.2 1.4 1.6 1.8 NORMALIZED IRRAD.IANCE 1/ * Figure 38. Normalized histogram for two path with filter only receiver and input pressure = 45 psi. *
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TABLE IX. Statistical Data For Two Way Pass, Detector With Filter, Input PSI = 45. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 MOMENTS m 8.88!6e-002 8.0375e-003 7.4131e-004 6.9662e-005 6.6679e-006 6.4995e-007 6.4497e-008 6.5144e-009 CENTRAL MOMENTS u 0 1.4917e-004 9.4637e-007 3.9202e-008 4.0671e-009 -2.1679e-010 1.7082e-006 -2.1435e-012 NORMALIZED MOMENTS m' 1 1.0189e+OOO 1.0581e+OOO 1.1195e+OOO 1.2065e+OOO 1.3241e+OOO 1.4794e+OOO 1.6824e+OOO CENTRAL MOMENTS u' l.OOOOe+OOO 5.1944e-001 1.7618e+OOO 1.4965e+001 ..!..6.5311e+001 4.2137e+007 -4.3291e+003 88 CUMULANT k 8.8816e-002 1.4917e-004 9.4637e-007 -2.7553e-008 2.6554e-009 -2.1388e-010 1.7082e-006 -1.4019e-005 CUMULANTS l.OOOOe+OOO 5.1944e-001 -1.2382e+OOO 9.7707e+OOO -6.4436e+001 4.2137e+007 -2.8312e+010 k'
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TABLE IX. (contd.) EXPERIMENTAL. VS PDFs ORDER m m m .n GAMMA LOG-NRM LOG-M\NRM MODULATED 2 1 1 1 3 1.0006e+OOO 1.1910e+OOO 1.0003e+OOO e 1.0018e+OOO 1.5166e+OOO 1.0005e+OOO 5 1.0038e+OOO 1.5754e+OOO 1.0004e+OOO 6 1.0065e+OOO 8.9385e-001 9.9972e-001 7 1.0099e+OOO 1.6497e-001 9.9824e-001 8 1.0142e+OOO 5.2872e-OOJ 9.9569e-001 Mimimum And Maximum Intensity Fluctuations minimum maximum 4.6387e-002 volts 1.4404e-001 volts 89
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... 3: ... > en z UJ c _. 4 a: u UJ a. en a: UJ 3: 0 a. c UJ N _. 4 ::E a: Q z Figure 39. to- to-s 100 fW(f) FOR TWO WAY PATH,DETECTOR ONLY INPUT PSI "" 60 10' 102 103 FREQUENCY (HZ) Normalized power spectra fW(f) for two path with filter only receiver and input pressure = 60 psi.
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1\ .... 2.51-H Q. z 0 .... ... 21z 0 .... ... i .... en H c > ... H ...J H m 0 If 1.51-110.5r o 0 Figure 40. PDF FOR TWO WAY DETECTOR ONLY INPUT PSI .. 60 ---. Jf 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 NORMALIZED IRRADIANCE I/* Normalized histogram for two path with filter only receiver and input pressure = 60 psi. *
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TABLE X. Statistical Data For Two Way Pass,Detector With Filter, Input PSI = 60. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 MOMENTS m CENTRAL MOMENTS u 8.7248e-002 o 7.7826e-003 1.7045e-004 7.0971e-004 9.5182e-007 6.6124e-005 6.1698e-008 6.2911e-006 4.0599e-009 6.1o9oe-oo7 -1.5428-o1o 6.0525e-008 1.5371e-006 6.1163e-009 -1.7460e-012 NORMALIZED MOMENTS m' 1 1.0224e+OOO 1.0686e+OOO 1.1411e+OOO 1.2444e+OOO 1.3850e+OOO 1.5727e+OOO 1.8216e+OOO CENTRAL MOMENTS u' 1.0000e+OOO 4.2771e-001 2.1235e+OOO 1.0703e+001 -3.1153e+001 2.3773e+007 -2.0683e+003 92 CUMULANT 8.7348e-002 1.7045e-004 9.5182e-007 -2.5465e-008 2.4375e-009 -1.7252e-010 1.5371e-006. -1.8304e-005 CUMULANTS 1.0000e+OOO 4.2771e-001 -8.7646e-001 6.4260e+OOO -3.4835e+001 2.3773e+007 -2.1684e+010 k' k
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TABLE X. (contd.) EXPERIMENTAL VS PDFs ORDER m' m' m' n GAMMA LOG-NRM LOG-M\NRM MODULATED 2 1 1 1 3 1.0004e+OOO 1.1676e+OOO 9.9992e-001 4 1.0011e+OOO 1.3780e+OOO 9.9917e-001 5 1.0019e+OOO 1.1927e+OOO 9.9719e-001 6 1.0028e+OOO 4.7631e-001 9.9353e-001 7 1.0039e+OOO .8546e-002 9.8784e-001 8 1.0052e+OOO 6.1810e-004 9.7986e-001 Mimimum And Maximum Intensity Fluctuations minimum maximum 4.3945e-002 volts 1.5625e-001 volts .93
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A pinhole was added to the filter only receiver setup of the two path configuration and two data sets were taken for input pressures of 45 and 60 psi. The power spectra plots, normalized histograms and the higher-order moments, cumulants and central moments up to the eight order along with comparisons to other predicted PDFs are shown in Figures 41 & 42 and Table XII and Figures 43 & 44 and Table XIII for the input pressures of 45 and 60 psi, respectively. 94
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... 3: ... > t-.... rn z bJ c _. -c a: tu bJ a. rn a: bJ 3: 0 a. c bJ N t-1 _. -c :::E a: 0 z 1u to-' to-2 to-a 100 Figure 41. I I I I I I I I I I I I 1 , -r"""l fW(f) FOR TWO WAY PATH,PINHOLE DETECTOR 10' 102 10!1 FREQUENCY (HZ) Normalized power spectra fW(f) for two path with pinhole only receiver and input pressure = 45 psi.
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A .... v ...... .... a. z 0 .... ... u z z 0 .... t-=-CD .... t-en .... 0 > ... .... ...1 .... m ..: CD 0 a: Q. Figure 42. 1.8 r-1.6 r-1.4 r-1.2 r-1 r-O.B r-0.6 r-0.4 r-0.2 r-0 0 1-PDF FOR TWO WAY PATH .........-I PINHOLE DETECTOR -I ., I -.........-I l --1--r:-7"1-I I 0.5 1 1.5 2 2.5 NORMALIZED IRRADIANCE I/* Normalized histogram for two path with pinhole only receiver and input pressure = 45 psi. *
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TABLE XI. Statistical Data For Two Way Pass,Detector With Pinhole, Input PSI = 45, Trial 1. ORDER n 1 2 3 4 5 6 7 8 ORDER n 2 3 4 5 6 7 8 MOMENTS m 2.4702e-002 6.2978e-004 1.6587e-005 4.5163e-007 1.2724e-008 3.7140e-010 1.1249e-011 3.5422e-013 CENTRAL MOMENTS u 0 1.9570e-005 6.3332e-008 1.3662e-009 2.0843e;...011 1.9880e-013 2.3827e-010 5. 2548.e-017 NORMALIZED MOMENTS CENTRAL MOMENTS m' u' 1 1.0321e+OOO 1.0000e+OOO 1.1004e+OOO 7.3151e-001 1.2129e+OOO 3.5670e+OOO 1.3833e+OOO 1.2301e+001 1.6346e+OOO 2.6522e+001 2.0043e+OOO 7.1859e+006 2.5549e+OOO 3.5823e+002 97 CUMULANT k 2. 4e-002 1.9570e-005 6.3332e-008 2.1716e-010 8.4483e-012 -1.7493e-014 2.3827e-010 CUMULANTS k' 1.0000e+OOO 7.3151e-001 5.6701e-001 4.9862e+OOO -2.3337e+OOO 7.1857e+006 -1.6449e+012
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TABLE XI. (cont.) EXPERIMENTAL VS PDFs ORDER m' m' m' n GAMMA LOG-NRM LOG-M\NRM MODULATED 2 1 1 1 3 1.0020e+OOO 1.1011e+OOO 1.0010e+OOO 4 1.0074e+OOO 1.0159e+OOO 1.0036e+OOO 5 1.0184e+OOO 4.5999e-001 1.0089e+OOO 6 1.0370e+OOO 4.6806e-002 1.0180e+OOO 7 1.0664e+OOO 3.8010e-004 1.0329e+OOO 8 1.1101e+OOO 6.6731e-008 1.0556e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 1.2207e-002 volts 5.3711e-002 volts 98
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.... 3:: ..... > 1-.... en z UJ c ..J 4( a: .... u UJ a. en a: UJ 3:: 0 a. c UJ N 1-1 ...J 4( :::E a: 0 z 10u to-s to-a 10-3 10D Figure 43. fW(f) FOR TWO WAY PATH, PINHOLE DETECTOR 101 102 103 FREQUENCY (HZ) Normalized power spectra fW(f) for two path with pinhole only receiver and input pressure = 60 psi.
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1\ t-1 v ......... H -a. z 0 H 1u z ::::: u. z 0 H 1-::::: Ill t-1 a: .... en t-1 Cl > .... H ...J t-1 Ill oC Ill 0 a: a. Figure 44. 3.5 3 r2.5 2 1.5 11 r-0.5 1-0 0 I --PDF FOR TWO WAY PATH r---PINHOLE DETECTOR INPUT PSI 60 f--r-I -' I -I r-r---.. I r I 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 NORMALIZED IRRADIANCE I/* Normalized histogram for two path with pinhole only receiver and input pressure = 60 psi. *
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TABLE XII. ORDER n 1 2 3 4 5 6 7 8 ORDER n 1 2 3 4 5 6 7 8 Statistical Data For Two Way Pass, Detector With Pinhole, Input PSI-= 60. MOMENTS. m 4.0071e-002 1.6305e-003 6.7439e-005 2.8360e-006 1.2129e-007 5.2764e-009 2.3352e-010 1.0515e-011 CENTRAL MOMENTS u 0 2.4861e-005 1.0915e-007 7.4038e-010 7.8204e-011 -1.9967e-012 6.3866e-009 -3.7219e""':'015 NORMALIZED e MOMENTS CENTRAL MOMENTS CUMULANT k 4.0071e-002 2.4861e-005 1.0915e-007 -1.1139e-009 5.1067e-Oll -1.9309e-012 6.3865e-009 -3.8940e-007 CUMULANTS m' u' k' 1 1.0155e+OOO 1.0481e+OOO 1.1000e+OOO 1.1740e+OOO 1.2745e+OOO 1.4077e+ooo 1.5818e+OOO l.OOOOe+OOO. 8.8053e-001 1.1978e+OOO 2.5375e+001 -1.2993e+002 8.3353e+007 -9.7421e+003 101 l.OOOOe+OOO 8.8053e-001 -1.8022e+OOO 1.6570e+001 -1.2566e+002 8.3353e+007 -1.0193e+012
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TABLE XII.(contd.) ORDER n 2 3 4 5 6 7 8 EXPERIMENTAL VS PDFs m GAMMA 1 1.0012e+OOO 1.0040e+OOO 1.0091e+OOO 1.0168e+OOO 1.0276e+OOO 1.0418e+OOO m LOG":'"'NRM MODULATED 1 1.2137e+OOO 1.6560e+OOO 2.0143e+OOO 1.5245e+OQO 4.4560e-001 2.7922e-002 m LOG-M\NRM 1 1. 0009e+OOO 1.0031e+OOO 1.0068e+OOO 1.0122e+OOO 1.0195e+OOO 1.0288e+OOO Mimimum And Maximum Intensity Fluctuations minimum maximum 2.6855e-002 volts 6.5918e-002 volts 102
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CHAPTER 7 DISCUSSION OF RESULTS Frequency Spectrum Measurements The temporal-frequency spectra of irradiance fluctuations were obtained by applying he Welch method of power spectrum analysis [27]. The irradiance data was divided into 8 sections of 1024 samples each and Hanning windowed and then the power spectral density was calculated. Before the power spectra was processed, the signal was filter through a Butterworth second order filter [28] wwith a 3dB cutoff of 200 Hz. A filter with a cutoff of 100 Hz was also done for comparison and the results were the same for the spectra of interest here and is not shown. The spectra W(f) were multiplied by frequency f and normalized to obtain Tatarski's power spectra of the form: U(f) :fW(E) fwcf> d:t 0 103
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Looking at the one path propagation case with a filter only detection scheme the normalized power spectrum U (f) depends on the frequency f as f-S/J in the range of frequencies from fL = 45 to fa = 150 Hz. This is in agreement with the theory predicted for Kolmogorov-Obukhov's atmospheric turbulence in the inertial subrange, i.e. for frequencies in the region where Here v! is the transverse component of the average wind velocity and is equal to 0.06 meters/sec. The most effective scale size, lp, of turbulence in this case is For frequencies above 150 Hz the slope U(f) appears to be equal to f-4 The scale size as determined from this break point is given by 104
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The theoretical peak frequency fc is given by f Q and for the one path case is approximately 25 Hz. The measured value of very nearly 50 Hz is larger than the predicted break point. This seems to suggest that the shift in frequency spectra to the higher side is a result of approaching the saturation region. The one path case with a detector with lens receiver design showed a higher fL then the one path cases with similar polarization vector. The shift in frequency would tend to indicate a stronger strength of turbulence but this can not be true since the strength is controlled. This suggests that the detection scheme used affects the laser scintillation statistics, this has been demonstrated in other research and there is continuing interest in that topic [10]. The one path case with lens and 90 degree polarization shift has similar cutoff frequencies to that of the one path case with filter only. The polarization shifted the cutoff frequencies to the left, representative of weaker turbulence for that polarization. This is not supported with the results obtained for the one path case with filter only and 90 105
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polarization shift used in the dual amplifier configuration. In that case the peak frequency was greater than that measured for the one pass case with filter only (These two cases of filter only and different polarizations are not identical in that different detectors were used which will affect the scintillation statistics and therefore should not be used comparatively) . The slope below the peak frequency is approximately one third for all the one path cases except for the dual amp configuration has a slope on one. For the two path case with filter only, the observed peak frequency occurs at even higher frequencies which is to be expected from the longer path traveled [1]. The two path case with a pinhole showed very nearly identical results. This is somewhat surprising since research indicates the aperture size will affect the scintillation statistics [10]. The predicted peak frequency for the two path cases are much less than that recorded but the differences again are attributed to approaching the saturation region. Below that the power spectra appears to have a slope of one. Statistical Analysis Normalized histograms of irradiance fluctuations for the two path lengths and various receivers were plotted The moments, central moments and the cumulants [24] were 106
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also computed for each case and so were the normalized valU}!S of each. Moments were generated from the normalized moments of the data for the gamma, lognormally modulated and log-normal distributions and then ratios of the normalized data moments vs the distributions were made for comparison. The higher or.der moments can give information about the shape of the probability density function of the irradiance fluctuations [19]. The higher the order of the moments, the greater the contribution by the tail of the probability density function. Even-and odd-order moments give information about the width and the lack of symmetry of the distribution, respectively. The comparison of moments indicates that the lognormal distribution best fits thecurve of the data for one path cases. Up two the eighth order the ratios were very near one for all the one path cases when compared to the log-normal distribution. The gamma distribution was close but was not as good a fit. The log-normally modulated curve departed quickly from the data. When the .path length was increased to two paths the moments began to depart more from the log-normal distribution and began to converge on the gamma distribution. There was some improvement for_the log-normally modulated curve but the moments still departed drastically from the data. The 107
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log-normal distribution is generally accepted to describe the laser scintillation for the weak turbulence regime and that as the strength of turbulence is increased it will depart from the data as occurred here. The gamma distribution is best used to describe stronger strength of turbulence regimes. Comparison of the histograms for the one and two path case show that the histogram is taller and thinner then the one path case. This is as expected [1]. Reconstruction Of Signal From Dual Amp The data recorded for the dualamplifier circuit never recorded any large scale fluctuations in irradiance and therefore no special reconstruction techniques were used. This method is still determined to be advantageous because it was not properly implemented in this experiment. To capture the large scale fluctuations the laser scintillation data must be recorded over a very long period over constant conditions which the chamber provides. This was not carried out to its fullest due to the limitations of the data storage facilities used in this experiment. 108
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CHAPTER 8 CONCLUSION Turbulent flow with an average refractive-index fluctuations cnl approximately one thousand times larger than that found in the typical atmosphere have been 'generated in the laboratory. The turbulence was characterized to be homogenous and isotropic down the length of the chamber. It was also demonstrated that the strength can be controlled via temperature and exit velocity of the nozzles. This leads to the possibility of generating reproducible turbulence in varying strength and locations within the chamber. The turbulence chamber was used to perform optical propagation experiments under well controlled and reproducible conditions. Statistics of irradiance fluctuations were measured for fluctuations in the diffraction region and strong turbulence regimes. The moments of the irradiance data were compared to gamma, log-normally modulated and log-normal distributions and found to compare with the log-normal distribution for the one path case and to converge to the gamma distribution for the two path case. Experiments were conducted to demonstrate the effects of various receivers on the statistics of irradiance fluctuations. Experiments were conducted on 109
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receivers with apertures, focusing lens and with filters i all to have demonstrated various effects on the data. The polarization of the laser down the propagation path was also determined to have an affect on the irradiance fluctuations as well. The characteristic feature of the -8/3 power law behavior of temporal-frequency spectra for irradiance fluctuations was clearly demonstrated for numerous conditions of path length and receiver design. This clearly shows the existence of the inertial subrange of.-: .... ....... -laboratory simulated turbulence. The shift to the of the in the case of strong turbulence has also been demonstrated. Reliable irradiance scintillation data are needed and can now be obtained to compare with new and pending approximate solutions of thte field-moment equations. By applying the chamber to the numerous experiments yet to be performed ,reliable and reproducible scintillation data can be achieved. 110
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