The Pennsylvania State University. The Graduate School. Electrical Engineering Department OPTICAL PROPAGATION THROUGH THE OCEAN SURFACE.

Transcription

1 The Pennsylvania State University The Graduate School Electrical Engineering Department OPTICAL PROPAGATION THROUGH THE OCEAN SURFACE A Thesis in Electrical Engineering by Serdar Kizilkaya 1 Serdar Kizilkaya Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 1

2 The thesis of Serdar Kizilkaya was reviewed and approved* by the following: Timothy Kane Professor of Electrical Engineering Thesis Adviser Julio Urbina Assistant Professor of Electrical Engineering Kultegin Aydin Professor of Electrical Engineering Interim Department Head of Electrical *Signatures are on file in the Graduate School.

3 iii Abstract Usage of the laser in the studies through the ocean is very widespread and the each steps of the laser propagation through the ocean is a major topic to be analyzed independently. Objective of this study is specifically focusing on the interaction of the laser with the air-ocean interface and the analysis of the effects of the ocean surface with the gravity waves on the laser propagation. This thesis is based on a theoretical study which is supported with the gravity wave ocean surface model written in MATLAB programming language. The model is based on beam tracing approach rather than using ray tracing like in many similar studies. This thesis highlights brief information about the fields which use the laser propagation through the ocean surface and their history, the details about the airocean interface focused on the gravity waves, gravity wave ocean surface model based on beam tracing and the analysis about the effects of the gravity waves on the laser propagation.

4 iv Table of Contents List of Figures List of Tables Acknowledgments vi xii xiii CHAPTER 1 Introduction History of the Studies about the Ocean Surface Waves and Airborne Lidar Steps of the Ocean Surface Model Description of the Approach Used in the Thesis...6 CHAPTER Air-Ocean Interface Ocean Waves...1. Airborne Lidar Basics of the Single Mod Gaussian Laser Optical Propagation in the Atmosphere....5 Ocean Surface Capillary Wave Scale Gravity Wave Scale Beam Tracing through the Ocean Surface Refraction of the Laser Beam in the Air-Ocean Interface Reflection of the Laser Beam from the Air-Ocean Interface (Ocean to Air)...49 CHAPTER 3 Gravity Wave Ocean Surface Model Flow Chart of the Model Pseudocode of Model Written in MATLAB Programming Language Part 1M, Defining the Specifications about the Model Part M, Calculating Elevation Distributions of the Facets Part 3M, Calculation of the Daughter Beams Parameters...54

5 v 3..4 Part 4M, Calculation of the Mean Values Part 5M, Checking for Multiple Scattering Part 6M, Realization of Hemisphere Distribution of the Reflected and Refracted Beams The Analysis to Determine the Resolution Values, a and b The Analysis to Determine the Realization Number, M...73 CHAPTER 4 Results and Discussion Singular Wind Speed Results Singular Wind Speed Result Set with U 1 m / s Singular Wind Speed Result Set with U 5 m / s Singular Wind Speed Result Set with U 1 m / s Singular Wind Speed Result Set with U 15 m / s Singular Wind Speed Result Set with U m / s U / 4.7 The Mean Result Sets of 1 Realizations in Wind Speed Range 1 m s...18 CHAPTER 5 Conclusions and Future Works Conclusion Future Works Appendix A Additional Results Appendix B Notation Table...61 References...67

6 vi List of Figures Figure 1-1 Flowchart of the ocean surface model....5 Figure 1- Airborne lidar application....6 Figure -1 Schematic representation of the energy contained in the ocean surface waves Figure - Propagation of the Gaussian beam Figure -3 Model of the ocean surface with the triangular facets....4 Figure -4 Representation of the gravity wave surface model in Cartesian coordinate system Figure -5 The spectral domain ( uv, ) coordinate system representation...34 Figure -6 Paraxial approximation on the air-ocean interface Figure -7 Decomposition of 3-D beam representation into two -D beam representations Figure 3-1 The formation of the unit normal vector of the facet A` Figure 3- Separation of the electric field vector to its components through the ocean surface with gravity waves Figure 3-3 Representation of the facet A` with its neighbor facets....6 Figure 3-4 Representation of the upper hemisphere distribution of a reflected beam Figure 3-5 Resolution analysis of the gravity wave ocean surface model with realization number, M 1 7 Figure 3-6 Resolution analysis of the gravity wave ocean surface model with realization number, M 5 74 Figure 3-7 Resolution analysis of the gravity wave ocean surface model with realization number, M 5 74 Figure 3-8 Resolution analysis of the gravity wave ocean surface model with realization number, M Figure 3-9 Realization number analysis with resolution number values, ab Figure 4-1 Gravity wave ocean surface realization with wind speed, 5 U / m s and resolution values ab

7 vii Figure 4- Gravity wave ocean surface realization with wind speed, 15 U / m s and resolution values ab Figure D Upper Hemisphere Model...8 Figure D Lower Hemisphere Model...83 Figure 4-5 Focused part of the -D hemisphere model...83 Figure 4-6 Beam number distribution of the reflected beams on the hemisphere. U 1 m / s...84 Figure 4-7 Power percentage distribution of the reflected beams on the hemisphere U 1 m / s...84 Figure 4-8 Beam number distribution of the reflected beams on the graphical representation. U 1 m / s.85 Figure 4-9 Beam number distribution of the reflected beams in the 18 degrees azimuth angle ranges. U 1 m / s...86 Figure 4-1 Beam number distribution of the reflected beams in the 9 degrees elevation angle ranges. U 1 m / s...87 Figure 4-11 Beam number distribution of the underwater beams on the hemisphere. U 1 m / s...87 Figure 4-1 Power percentage of the underwater beams on the hemisphere. U 1 m / s...88 Figure 4-13 Beam number distribution of the underwater beams on the graphical representation. U 1 m / s...89 Figure 4-14 Beam number distribution of the underwater beams in the 18 degrees azimuth angle ranges. U 1 m / s...9 Figure 4-15 Beam number distribution of the underwater beams in the 9 degrees elevation angle ranges. U 1 m / s...91 Figure 4-16 Beam number distribution of the reflected beams on the hemisphere. U 5 m / s...9 Figure 4-17 Power percentage distribution of the reflected beams on the hemisphere. U 5 m / s...93

8 viii Figure 4-18 Beam number distribution of the reflected beams on the graphical representation. U 5 m / s...94 Figure 4-19 Beam number distribution of the reflected beams in the 18 degrees azimuth angle ranges. U 5 m / s...95 Figure 4- Beam number distribution of the reflected beams in the 9 degrees elevation angle ranges. U 5 m / s...96 Figure 4-1 Beam number distribution of the underwater beams on the hemisphere. U 5 m / s...96 Figure 4- Power percentage of the underwater beams on the hemisphere. U 5 m / s...97 Figure 4-3 Beam number distribution of the underwater beams on the graphical representation. U 5 m / s...98 Figure 4-4 Beam number distribution of the underwater beams in the 18 degrees azimuth angle ranges. U 5 m / s...99 Figure 4-5 Beam number distribution of the underwater beams in the 9 degrees elevation angle ranges. a b 96, M 1, U 5 m / s...1 Figure 4-6 Beam number distribution of the reflected beams on the hemisphere. U 5 m / s...11 Figure 4-7 Power percentage distribution of the reflected beams on the hemisphere. U 1 m / s...1 Figure 4-8 Beam number distribution of the reflected beams on the graphical representation. U 1 m / s...13 Figure 4-9 Beam number distribution of the reflected beams in the 18 degrees azimuth angle ranges. U 1 m / s...14 Figure 4-3 Beam number distribution of the reflected beams in the 9 degrees elevation angle ranges. U 1 m / s...15

9 ix Figure 4-31 Beam number distribution of the underwater beams on the hemisphere. U 1 m / s...15 Figure 4-3 Power percentage of the underwater beams on the hemisphere. U 1 m / s...16 Figure 4-33 Beam number distribution of the underwater beams on the graphical representation. U 1 m / s...17 Figure 4-34 Beam number distribution of the underwater beams in the 18 degrees azimuth angle ranges. U 1 m / s...18 Figure 4-35 Beam number distribution of the underwater beams in the 9 degrees elevation angle ranges. U 1 m / s...19 Figure 4-36 Beam number distribution of the reflected beams on the hemisphere. U 15 m / s...11 Figure 4-37 Power percentage distribution of the reflected beams on the hemisphere. U 15 m / s Figure 4-38 Beam number distribution of the reflected beams on the graphical representation. U 15 m / s...11 Figure 4-39 Beam number distribution of the reflected beams in the 18 degrees azimuth angle ranges. U 15 m / s Figure 4-4 Beam number distribution of the reflected beams in the 9 degrees elevation angle ranges. U 15 m / s Figure 4-41 Beam number distribution of the underwater beams on the hemisphere. U 15 m / s Figure 4-4 Power percentage of the underwater beams on the hemisphere. U 15 m / s Figure 4-43 Beam number distribution of the underwater beams on the graphical representation. U 15 m / s Figure 4-44 Beam number distribution of the underwater beams in the 18 degrees azimuth angle ranges. U 15 m / s...117

10 x Figure 4-45 Beam number distribution of the underwater beams in the 9 degrees elevation angle ranges. U 15 m / s Figure 4-46 Beam number distribution of the reflected beams on the hemisphere. U m / s Figure 4-47 Power percentage distribution of the reflected beams on the hemisphere. U 15 m / s...1 Figure 4-48 Beam number distribution of the reflected beams on the graphical representation. U m / s...11 Figure 4-49 Beam number distribution of the reflected beams in the 18 degrees azimuth angle ranges. U m / s...1 Figure 4-5 Beam number distribution of the reflected beams in the 9 degrees elevation angle ranges. U m / s...13 Figure 4-51 Beam number distribution of the underwater beams on the hemisphere. U m / s...13 Figure 4-5 Power percentage of the underwater beams on the hemisphere. U m / s...14 Figure 4-53 Beam number distribution of the underwater beams on the graphical representation. U m / s...15 Figure 4-54 Beam number distribution of the underwater beams in the 18 degrees azimuth angle ranges. U m / s...16 Figure 4-55 Beam number distribution of the underwater beams in the 9 degrees elevation angle ranges. U m / s...17 Figure 4-56 Mean of the elevation angles of the incident and transmitted beams in 1 realizations Figure 4-57 Mean of the elevation angles of the reflected beams according to the reference x y plane in 1 realizations Figure 4-58 Mean of the elevation angles of the underwater beams according to the reference x y plane in 1 realizations....13

11 xi Figure 4-59 Mean of the azimuth angles of the reflected beams in 1 realizations Figure 4-6 Mean of the azimuth angles of the transmitted beams in 1 realizations Figure 4-61 Mean of the gravity wave ocean surface s reflectance in 1 realizations Figure 4-6 Mean of the gravity wave ocean surface s transmittance in 1 realizations Figure 4-63 Mean of the reflected beams power values in 1 realizations Figure 4-64 Mean of the transmitted beams power values in 1 realizations Figure 4-65 Reflectance analysis Figure 4-66 Mean of the variances of the crosswind and upwind slopes in 1 realizations Figure 4-67 Mean of the maximum and minimum vertex elevation values in 1 realizations Figure 4-68 Wavelengths of the gravity waves with the maximum energy spectrum Figure 4-69 Width of the triads

12 xii List of Tables Table -1 Period and frequency bands of the ocean waves....1 Table 3-1 Computational analysis of the resolution values ab,...71 Table 4-1 Mean values of the radiuses of the curvatures of the spherical waves for the incident and the transmitted beams at the point A` when the wind speed, U is 1, 1 and m/s Table 4- The maximum and the minimum radius of the curvature of the spherical wave observed in the model Table 4-3 Comparison of the maximum wave height in the Beaufort scale and gravity wave ocean surface model....14

13 xiii Acknowledgements First of all, I would like to acknowledge my advisor, Dr. Tim Kane for his support to me and advices at every step of the research process. I would also like to thank my committee member, Dr. Julio Urbina who accepted to be a part of my master s thesis committee. I would like to specially thank my dear wife, Bahar Kizilkaya for her endless support to me and understanding that she showed during my study period. Last but not least, I would like to thank my precious son, Ruzgar Kizilkaya who changed my life totally in a good way and supported my studies with his existence.

14 1 CHAPTER 1 Introduction Equation Chapter 1 Section 1 The importance of the oceans is always in the top level for the human being s life due to the benefits of the oceans. From the beginning of the civilization, people have been looking for ways to improve their profits that they got from the oceans and for this reason they have done many researches to analyze the ocean and understand the features of it better. The first and the simplest method that people used in their observations was the visual survey which is still a valid method. Then in beginning of th century, the sonar was invented and people started to do underwater surveys. The invention of the radar followed the sonar and people learned many things about the ocean surface in long ranges. There are many other scientific methods used to analyze the ocean yet these two methods have major effects on the studies and eventually, the third major method, which has a broad application area, is laser. After the invention of the laser in 196s, the idea of using laser in the ocean studies was not late to come up. In beginning of the 197s the studies of the airborne laser hydrography or in another words, airborne lidar was started to give first outputs [1]. The applications of the airborne lidar systems are very widespread and the major topics are given below with some example studies. a. Oceanographic observations: i. Detecting of the ocean layers [], ii. Observations on the ocean waves, iii. Observation of the ocean temperature profile [3]. b. Biological monitoring: i. Studies about the ocean color [4], ii. Studies about the dissolved organic matter,

15 iii. iv. Studies about the phytoplankton, Studies about the organic detritus. c. Observations about the underwater animals. d. Bathymetry [1] e. Communication f. Detecting the underwater objects [5]: i. Mine detection [6], ii. Submarine detection, iii. Scuba diver detection [7]. In each system indicated above, there are some steps that the laser beam follows but when the system use an air platform, the interference of the laser beam with the ocean surface is inevitable. In this thesis, the propagation of the laser beam through the ocean surface is analyzed and the effects of the ocean surface to the laser beam are studied but before focusing on the study done in the thesis, a brief chronological process of the studies about the ocean waves and airborne lidar is given in the next part. 1.1 History of the Studies about the Ocean Surface Waves and Airborne Lidar With the developing technology, use of the oceans started to become beyond the fishing and the transportation. The energy produced from the different sources of the ocean is just one of the reasons that prompted people to do researches. But on the other side, still the forces of the oceans cannot be controlled by the people which can give serious damages to them. Thus, necessity to understand the forces on the ocean increased studies about forecasting the sea conditions. One of the best ways to understand the ocean forces is modeling the ocean in theoretically. The ocean waves are also results of the forces which affect the ocean. To model ocean waves, the characteristics of the ocean waves' properties, boundaries and the forces affecting the ocean have to be formulated.

16 3 It has been working on the ocean waves since the early 19 th century yet the first modern work in this topic was done in the beginning of the 194s. According to the Mitsuyasu [8], the first remarkable study was done by Sverdrup and Munk (1947) and also Mitsuyasu gives the following three points as the most important outcomes from their study: The first quantitative description of ocean surface waves, energy balance concept in the wave systems and empirical formulation. And then following years, many different spectrums which fit with capillary waves, gravity waves or capillary-gravity cascade waves were generated and some empirical formulas outputted based on statistical data gathered from limited area. In 1949, Duntley performed his first experimental study about the statistical properties of the slopes of the air-water interface. He tried to prove the hypothesis which was based on the Gaussian distribution valid for the slopes of the wind derived surface. He performed his research near a lake and used long wires along the water to measure the parameters of the water. Another famous and valid work on this topic was done by Cox and Munk [9] in 1954 with using sun glitter on the ocean surface. Even Duntley, Cox and Munk followed a similar concept at their results, the content of their results were different for wave-slope wind-speed law, which is explained in detail in part.5.1, like some of the other people concluded with different results. But then, Preisendorfer [1] chose the parameters of the Cox and Munk because, the research of the Duntley was performed near a lake but on the other side Cox and Munk's work was done in the Hawaiian area which is more suitable for a general case and also the wires used by Duntley caused some extra capillary waves which might affect the results. In 1955, the Neumann and Pierson made the first definite wave spectrum model that could be used to forecast the waveforms. And then Cote et al. revised the Neumann spectrum with the observations that they got from the Stereo Wave Observation Project (SWOP) and the results of this spectrum were clearly stated by Mobley and Preisendorfer [1], [11], [1]. The studies on the ocean surface waves have been continued and chronological order of them was told by the Mitsuyasu in detail. In this thesis, the directional energy spectrum of the Neumann is used which is in the 1 st generation of the numerical models about the ocean wave forecasting where there are three generations. The first reason about choosing the Neumann s spectrum is its feasibility. Neumann [13] stated this feature in the comparison paper of him, the calculated results of the Neumann s spectrum fitted with real

17 4 observations. The second reason about this choice, the Neumann s spectrum is working for only one specific wind force in each calculations, it is not for the cascade situation. Even this feature seems as a disadvantage, it enables to analyze the effects of the wind to the ocean surface and hence to the laser propagation. Additionally the Neumann s spectrum is valid for fully developed sea which is one the requested initial specifications of the model designed in the paper because, when it is reached to the fully developed sea condition, the steady state situation is provided and thus the analysis becomes valid for the general situations and easy to calculate. Another reason for choosing the Neumann s spectrum, the derivation of the spectrum is clearly told in many references ([1],[11],[1],[14], [15]). The last reason is about the wave frequency region that the Neumann s spectrum is valid. Neumann spectrum does not work for high frequencies; it is valid for only gravity waves which is the wave type in scope of the thesis. According to the Guenther [1], history of the airborne lidar started in 196s but real developments were achieved after the Naval Air Development Center supported the studies for the anti-submarine program in the late 196s. The first experimental studies were done in the 197s and the first reported success was achieved by receiving a return signal from 7 m depth in the 1971 with using Pulsed Light Airborne Depth Sounder. Since 1971, the studies to improve the airborne lidar technology have been continuing and today there are many active used airborne lidars designed by different countries especially, United States, Canada, Australia and Sweden. Some of the airborne lidar examples are given in part.. Beyond the applications in practice, many theoretical studies have been done about the airborne lidar. The cost of the experimental studies about the airborne lidar is a serious amount and this fact also directs the people to study in theoretical and design the theoretical models. In the theoretical studies about airborne lidar, generally one of the following two different methods is used: Formulation or ray tracing with Monte Carlo Simulation. But even the airborne lidar is studied theoretically in this thesis; beam tracing approach with Monte Carlo Simulation is used rather than formulation or ray tracing. In part 1.3, the reason to choose this method is explained with the other specifications of the ocean surface model designed in the thesis and in part 1., the steps of the ocean surface model, which is used in the thesis to analyze the effects of the ocean surface to the laser beam propagation, are explained.

18 5 1. Steps of the Ocean Surface Model According to aim of using airborne lidar, the system specifications can be very different and also very complex yet the results gotten at the end of the analysis in this thesis are wanted to be independent with any specific system and for this reason the pattern followed by the laser beam is tried to be kept general and simple. The flowchart given Figure 1-1 indicates the steps that laser beam followed and the visual representation of the typical airborne lidar application is given in Figure -1. Defining the Laser Transmitter Specifications Laser Propagation through the Atmosphere Interference with the Air-Ocean Interface Refraction through the Underwater Reflection Back to Upward Upward and Downward Beam Distribution Figure 1-1 Flowchart of the ocean surface model.

19 6 Air Platform with Lidar Air Incident Laser Beam Reflected Beam Air-Ocean Interface Underwater Transmitted Beam Figure 1- Airborne lidar application. 1.3 Description of the Approach Used in the Thesis As it is indicated above, propagation of the laser through ocean surface and the effects of the ocean surface to the laser beam are analyzed in the thesis and these are done by a theoretical model. While the model is being designed, some of the assumptions and approximations are used and also the wide contents of the ocean waves and airborne lidar are narrowed. Within this scope, description of the approach used in the thesis is explained below. a. The study is focused on the interaction of the laser propagation with the air-ocean interface. b. An airborne lidar which is located to an air platform is used as the laser source. c. The atmosphere is assumed that it is without the turbulence and any attenuation, in another word, the laser beam propagates in the atmosphere as it does in the vacuum. d. In modeling the ocean surface waves, the directional energy spectrum of the Neumann is used.

20 7 e. The content of the ocean surface waves is narrowed with the gravity waves. The reason of focusing on only gravity waves is incompatibility between the spatial resolution of the waves and beam tracing approach. This limitation is explained in part.5.1. f. In the realization of the ocean surface with the gravity waves, Fourier series are used. g. Linearly p-polarized laser beam, which has the parallel electric field oscillation to the incident plane, is used in the thesis. h. The cross-section surface area of the laser beam is kept as narrow as possible to use the paraxial approximation and do some of the assumptions which are explained in the next parts. While this kind of laser beam choice enables the simplicity in the calculations like as it is in the calculations with the rays, there are some disadvantages of using ultra-narrow laser beam. The first and most important one is about the human health; the intense power of the ultra-narrow laser beam can violate the eye safe situation when it is focused on a very small area. The other one is about a possible technical problem, the reflection/refraction of the narrow laser beam from a small surface of the ocean might create the biggest directional distribution mistakes because this situation is totally under the effect of the ocean surface randomness without any ensemble averaging effect of using a laser beam with a big cross-section surface area. This situation may cause some wrong outputs like the depth errors in bathymetry that Steinvall [16] stated. These problems can be solved by using an appropriate power for the human health or taking enough precautions and also ensemble average approach might be applied by using the Monte Carlo simulation with enough number of laser beams. The aim of using beam tracing rather than ray tracing is avoiding from the assumption the rays form a coherent beam after the superposition of the reflected/refracted rays in underwater. In the ray tracing approach, the laser beam is divided into numerous rays, these rays reflect/refract from different surfaces on the ocean surface and then reflected/refracted rays form the coherent reflected/refracted beam with the superposition principle. In this process, there is a big assumption that the rays form a coherent beam after the refraction but this is not the case in reality. To avoid from this assumption and form a more realistic approach, ultra-narrow beams are used and the propagation of these beams are followed with beam tracing approach.

21 8 i. In analyzing the propagation of the laser beam and the formation of the daughter beams, ABCD matrix approach is used. j. The model is valid for the ocean type which is offshore, deep (the depth of the water is much bigger than the wavelengths of the waves), clear and without whitecaps. This ocean type specification is preferred for the computational simplicity of the model. The first three conditions can be relatively met yet the fourth one is met with an assumption. Monahan s analysis [17] stated that the percentage of the ocean surface covered with the white caps is dependent on the wind speed with the following equation: W = where U is the wind speed at 1 m altitude over the ocean surface. For the speed U range from 1 m/s to m/s, the whitecap percentage, W varies from to.11. At this point, the ocean surface without whitecaps is a fair assumption for the indicated wind speed range. k. The ocean surface with the gravity waves is modeled with the facets which have specular surfaces. By degrading the complex ocean surface into small specular surfaces, level surface approach is wanted to be used on the facet s surfaces. By this way, it is possible to use geometrical optics phenomena in small scales. The specifications about the dimensions of the facets are indicated in part.5.. l. In the gravity wave ocean surface model, after the ocean surface is realized, it is assumed that it stays same during multiple scattering check. At all, this assumption is reasonable when the speed of light, 8 31 m/s and the largest and fastest recorded wave s speed, 3 m/s indicated by Mayo [18] are compared. m. The laser beam is always emitted with the normal incidence according to the reference ocean surface which is the imaginary ocean surface level with zero elevation. These are the major specifications and assumptions used in the thesis, the details about the model are indicated in the related parts of the thesis.

22 9 CHAPTER Air-Ocean Interface Equation Chapter Section 1 In this part, the theory of the optical propagation of laser light from an airborne lidar platform to the ocean surface is explained. The process, which is the more detailed version of the Figure 1-1, is broken up into 6 steps as it is given below. a. Specifying the lidar parameters and lidar platform specifications, b. Defining the atmospheric and oceanographic properties of the medium, c. Realizing an ocean surface with gravity waves, d. Beam tracing of the laser beam from source to air-ocean interface and determining the daughter beams, e. Transferring the incident beam s energy to the daughter beams, f. Getting the directional and power distribution of the transmitted and reflected beams respectively in underwater and above the water surface. The steps indicated above involve the downward propagation through the ocean and the upward reflection of the light through the air. In this thesis, the upward propagation of the reflected beams from an underwater object is not included. To discuss the effects of ocean waves on the reflected beams from an underwater object, the optical propagation of light in underwater has to be explained and the whole effects of water have to be applied on the light which are not going to be discussed in this thesis. Before focusing on the laser source, brief information is given below about the ocean waves.

23 1.1 Ocean Waves Ocean wave is a broad topic that the oceanographers have been working on it for years and there are many subtitles to go on deeply yet the whole details of the ocean waves are not indicated in here. Main point about this part is covering the reasons of choosing a specific type of ocean waves and giving brief information about its specifications. The details about the formation of the ocean waves are told by Kinsman [14] and Preisendorfer [1]. If there were not any effects to the air-ocean interface, it would be a totally specular surface which would have two different layers with different refractive indexes and specular interface. But in real world, there are some effects to the air-ocean interface which form the ocean waves. Kinsman classifies the ocean waves and some of the classifications are indicated below. The first classification is done according to the waves period and frequency bands as it is given Table -1. Wave Name Table -1 Period and frequency bands of the ocean waves. Source: [14]. Period Band (Second) Frequency Band (Cycles per second) Lower boundary Upper Boundary Lower boundary Upper Boundary 1 Capillary Ultragravity Gravity 1 1 Infragravity Long period 3 1 Transtidal (5min) (5min) Another classification done by Kinsman about the ocean waves depends on the energy that they contain. According to this classification, the gravity waves cover the majority of the power spectrum as it is shown in Figure -1.

24 11 Figure -1 Schematic representation of the energy contained in the ocean surface waves. Source: [14]. The third classification is about the disturbing force type. There are two types in this category which are free and forced waves. The free waves are generated by a force and then left free to keep going without any applied force like ship waves. The other one, forced wave type is always under a disturbing force to proceed like wind or tide waves. Wave oscillations need three things to continue: Equilibrium or stable, undisturbed state; a disturbing force to put out the equilibrium; and a restoring force to re-form the equilibrium. Beside the classification types told above, the most distinctive classification type is done according to restoring forces of the waves which are: Surface tension, gravity and Coriolis force. The reason to mention these classification types is not only for giving information about the waves but also to indicate the reasons of choosing gravity waves in analyzing the effects of the air-ocean interface. First of all, the frequency band of the gravity waves is easy to observe which might be helpful in the future experiments and secondly the gravity waves cover the majority of the energy spectrum of whole wave types, the model including the gravity waves might give a general idea about the impacts of the ocean surface on the laser propagation. Additionally, the gravity waves are wind waves which are in the forced wave type. Forced wave type has continuity while free waves have not, this property makes it easy to observe and produce a model about the forced wave types

25 1 There are some additional reasons to choose the gravity waves. One of them is the lack of a model individually worked on gravity waves while there are some detailed analysis done about the capillary waves effects, one of them is done by Preisendorfer and Mobley [11]. The reasons not to extend the content of the research with capillary waves are the restrictions about the adaptation of beam tracing approach on capillary waves which are told in detail in part.5.1 Even the capillary waves are kept out of the scope of the model which is used to analyze the effects of ocean surface to the laser propagation, the capillary waves are mentioned in the thesis to draw the boundaries of the gravity waves with dispersion theory, show the restrictions about the beam tracing approach in capillary waves and mentioning the importance of the capillary waves due to their existence in all circumstances. At this point, the dispersion theory is used to define the scale boundary between the gravity waves and the capillary waves. Dispersion Theory: Preisendorfer [1] used the Kevin-Helmholtz dynamic air-water surface model to describe the dispersion theory between capillary and gravity waves in deep water where the depth of water is bigger than the wavelength of the wave. The derived celerity (phase speed, c ) formula for the ocean waves is given below while the speed of the particles above and under the air-ocean interface is zero ( U, U ) which means that the ocean surface is only under the effects of surface tension (source of the capillary waves) and gravity (source of the gravity waves). a w c, U a, U w c k g Tk 1 ( w a) k ( ) w a (.1)

26 13 is the frequency of the wave, k is the wave number of the wave, c is the celerity of the wave when ( Ua, Uw ) condition is valid, T 1 is surface tension, w is density of water, a is the density of air and g is the gravitational acceleration. If the surface tension and the gravitational acceleration are separated into two parts and these two parts are squared, the following equation is derived. c 1 T g w a w a w a 1 (.) To simplify the equation (.), it is assumed that the w is one and a is zero, and then the limit of equation (.) is calculated according to zero or infinity and the critical wavelength is found, 1.7cm. It is the boundary wavelength between the capillary and gravity waves. The waves with the wavelength less than c c are called capillary waves and on the other side the waves with the wavelength bigger than c are called gravity waves. The minimum celerity, c m for both capillary and gravity waves is 3 cm/sec. To analyze the capillary and the gravity waves individually, it is assumed that T1 for the gravity waves and g for the capillary waves. g (.3) T 1 c As it is seen from equation (.3), celerity of gravity wave is directly proportional to the square of the wavelength of the gravity wave. 1 (.4) g c T1

27 14 On the other side, as it is seen from equation (.4), celerity of capillary wave is inversely proportional to the square of the wavelength of the capillary wave. The relation between the celerity and the wavelength of the capillary-gravity wave with constants c and c m is given in equation (.5). c 1 c m c c (.5) It is clear that, the wind waves are always formed from both capillary and gravity waves because there is not a case without any of the restoring forces; surface tension and gravity in real world, only there are situations which one of these two restoring forces dominates.. Airborne Lidar The airborne lidar is a laser source which is located in an air platform. It illuminates the ocean surface with different wavelengths used for the different aims; some of them are indicated in chapter 1. In this thesis, the airborne lidar which is specifically used for detecting underwater objects is used. An airborne is simply consisted of four main parts: Transmitter, receiver, power supply and software unit. In the thesis, details about the airborne lidar are not told except the specifications that used in the thesis. The first specification used in the thesis about the airborne lidar is the wavelength of the laser. It is chosen according to the attenuation spectrum of the ocean versus the wavelength and efficiency of the laser. One of the major works on the attenuation property of the ocean was done by Smith and Baker [19]. They got their data about the ocean from one of the clearest and purest waters, Sargasso Sea, because any possible particle intensity in the ocean affects the scattering and the spectrum of the attenuation in the ocean. The wavelength of the laser has to be in the minimum attenuation part of the attenuation spectrum and also the

28 15 laser working in the chosen wavelength has to be efficient because it is used in an air platform which has limited sources. One of the most widely being used laser in airborne lidar is the frequency doubled Nd:YAG (neodymium-doped yttrium aluminum garnet; Nd : Y3 Al5O 1 ) with the wavelength, 53 nm (green light). The wavelength with 53 nm does not have the least attenuation coefficient at the ocean according to the Smith and Baker yet it is efficient and it is in the acceptable region of the attenuation spectrum. For these reasons, the laser wavelength in the thesis is also chosen 53 nm. The second specification is the altitude of the platform that airborne lidar is located. The altitude of the airborne lidar platforms varies according to aim of the scanning and the platform s type. Some of the airborne lidar systems are: Scanning Hydrographic Operational Airborne Lidar (SHOAL) [], [1]; FOA Laser Airborne Sounder for Hydrography (FLASH/Hawk Eye) [16], Airborne Oceanographic Lidar (AOL) [3], []; Laser Airborne Depth Sounder (LADS) [3]; WRELADS [4]. In the model used in the thesis, the distance between the laser source and the air-ocean interface is wanted to be kept as close as possible because the laser beam is spreading with the propagation distance. This situation becomes a problem for the resolution values of the model especially when the beam tracing approach is applied to the capillary waves. The limitations about the cross-sectional surface area of the beam and the resolution values of the model are explained in part.5. The chosen altitude value for the laser transmitter in the model is 5 m. This value is reasonable when the air platforms (especially helicopters) minimum flight altitude or hover altitude of the helicopters is considered and also this distance is in the limit of many laser transmitters minimum distance for scanning. Choosing the specified values for the laser source is important for forming a quantitative model and also the chosen values can easily be changed without affecting the general results..3 Basics of the Single Mod Gaussian Laser In the model, the specifications of the linearly p-polarized single mod Gaussian beam are used. The fundamental properties of the single mod Gaussian beam are given below [5].

29 16 1-D Gaussian Laser Beam Form x x z x Phase Fronts z x z x w x z x x w wz () x z x Rz () x Figure - Propagation of the Gaussian beam. Electric Field (V/m) at the polar coordinate system rz, : w o 1 ik E( r, z) aˆ E exp i kz ( z) r w( z) w z R( z) (.6) â is the unit vector in the x y plane. Wave Number (1/m): k n (.7) n is the refractive index and is the wavelength of the beam.

30 17 Radius of the laser beam (m): r x y (.8) x and y are the Cartesian coordinates and r is the radius in polar coordinate system. Phase factor: 1 z ( z) tan z (.9) Radius of the curvature of the spherical wave (m): R( z) z 1 z z (.1) Beam radius ( m )(it is the value where the electric field amplitude is down by the factor 1/e compared to its value on the axis (r=)): w( z) w 1 z z (.11)

31 18 Inverse of the Gaussian beam parameter: i q z z iz R z w z n ( ) ( ) ( ) (.1) Amplitude of the electric field vector (V/m): r w E( x, y, z) E exp w ( z) w( z) (.13) Rayleigh range (m): z wn (.14) Complex constant: kw q i iz (.15) Initial (minimum) beam radius (m): w z n (.16)

32 19 Initial beam radius for the minimum spreading at distance z (m): w z (.17) n Irradiance W / m : I c E (.18) Power (W): c P E ds (.19) s Initial electric field amplitude (V/m): E P w c (.) is the permittivity of the medium.

33 .4 Optical Propagation in the Atmosphere In the thesis assumed that the atmosphere is homogeneous which means that it has no attenuation effect on the laser propagation. In this part, even the homogeneous atmosphere assumption is used, the atmospheric effects on the laser propagation are briefly mentioned. In reality, the atmosphere is not a homogenous medium; there are many fluctuations in it. The laser propagation comes across with the turbulence originated from the temperature fluctuation and this phenomena causes absorption and scattering. In another word, the turbulence originated from the temperature fluctuation is the refractive index fluctuation in the atmosphere which causes weak or strong turbulence. Generally the fluctuations in vertical axis is more than as it is in the horizontal axis because the horizontal correlation length, which is the length that the medium has the same effect on the propagation, is longer than it is in the vertical direction. This situation makes designed model in the thesis get into more serious turbulence effect. The main effects of the turbulence on the laser propagation, which are stated by Hadora [6], are given below. a. Random Beam Scanning: Along the propagation path, the fluctuations in the refractive index form different layers and the propagating beam refracts through these layers according to the Snell s Law. The forming of the layers is up to parallel and transverse correlation lengths. Due to these different layer formations, the beam does not propagate straight and it might miss the receiver according to the turbulence, beam diameter and receiver s cross-section surface area. b. Phase Change: The phase change is the result of the weak turbulence in the atmosphere which has two different types; phase variations parallel and perpendicular to the propagation direction. The phase variation parallel to the propagation direction is lack of time coherence which causes random frequency modulation. The refractive index variations along the propagation path modulates the phase of the beam and the beam arrives the receiver modulated in a randomly frequency. On the other hand, when the perpendicular phase variation exists, spatial coherence is degraded. The refractive index variation perpendicular to the

34 1 propagation distance affects each ray on the cross-section of the beam differently which results the beam with a random phase distribution at the receiver. c. Random Cross Section Change: This is result of the strong turbulence which has effects on the beam intensity beyond the phase change. Random cross section variation changes the amplitude (intensity) of the beam which is also called scintillation. By this effect, the intensity distribution of the laser gets into a random form. d. Polarization Change: The laser is a polarized beam which has a definite angular direction of polarization that is affected by the interface between the layers in the atmosphere. After each refractions through the layers, the angular direction of the polarization changes and at the receiver it results with a different angular direction of polarization. The polarization of the beam is affected by both parallel and perpendicular refractive index difference like the phase variation. In addition to these, the location of the majority of the turbulence is very important in the system design of the lidar. If there is turbulence close to the transmitter, the effects of it might be compensated to illuminate the correct location with requested form of laser yet if it exists close to the target, the shifting in the beam s properties just before the target cannot be compensated easily. But when the reflected beam propagation from the target to the upward is considered, the effect of the turbulence s location is not such important because the beam propagation is valid in both ways. The other point in atmospheric turbulence is the hardness of its measurement and the validation of this measurement in the broad area. There are many numerical models to measure the turbulence yet majority of them focused on the weak turbulence rather than the strong turbulence. The propagation of the laser through atmosphere is much broad topic to focus on yet the existence and importance of the atmospheric effect on the laser propagation is wanted to be emphasized in this brief part even it is not taken into account in the model. Another encouraging reason to ignore the atmospheric turbulence is the short propagation distance of the beam (5 m) which doesn t have that severe atmospheric turbulence effect on the beam.

35 .5 Ocean Surface To see the effects of the air-ocean interface on laser beam propagation by using beam tracing method, an ocean surface model has to be realized which also has to be common for the general purposes. Before getting into ocean wave realization theory, the assumptions and approaches used in the realization are indicated below. a. The upwind and crosswind slopes are calculated independently. b. The ergodic equivalence stationary random process independent of time is used. c. The ocean surface realization is realized in the steady wind. d. It is assumed that the ocean surface does not absorb any photons, it has only reflection and transmittance properties, it represents only the discontinuity of the refractive index of the medium; it is not a finite layer of material [1]. e. The wave spectra of the wind derived ocean wave is based on the rules which are derived from wind speed, fetch and wind duration information of the medium [1]. As it is explained in part.1, the ocean surface is formed by the wind waves which involve two wave types; capillary and gravity waves. The model used in the thesis is based on the gravity waves while the capillary waves are left at the out of the thesis s scope. The reason of this exception is explained in part.5.1 and the theory behind surface realization of the gravity wave ocean surface model is described in part Capillary Wave Scale Beyond the frequency of the waves, the most important features of them to categorize are their heights and slopes which mainly affect the laser propagation. To classify these features, the triangular wave facet model is used which is also based on wave-slope wind-speed law of Cox and Munk. To explain the capillary wave scale, the approaches of the Mobley [1] and Preisendorfer [11] are followed.

36 3 First of all, the wave-slope wind-speed law is briefly mentioned below which the capillary wave scale theory is based on. Wave-slope wind-speed law: When the wind blows across the ocean surface, it forms the capillary waves with the random elevations which are distributed with the normal distribution. This wave form has two different and independent slopes; upwind and crosswind slopes. Upwind slope is formed along the wind direction c and crosswind slope is formed perpendicular to the wind direction. They are calculated according to the elevation difference versus horizontal coordinates x (along the wind direction) and y (perpendicular to the wind direction). u u (.1) x c (.) y The experimental results of Cox and Munk [7] showed that even the distribution of the wave slopes have some upwind and downwind skewness which has a linear relation with wind speed and some peakedness without such significal relation with wind speed, the distribution of wave slopes is very close to the normal distribution. So the wave slopes have zero mean and variance which is linearly dependent to the wind speed. a U (.3) u u

37 4 c ac U (.4) a s m a s m u /, c / In equations (.3) and (.4), U is the wind speed in meters per second which is measured at an anemometer height 1.5 m above the mean sea level. a u and a c are constants found experimentally by Cox and Munk [9]. The variance of the normal distribution of the wave elevations is given below. a U (.5) q In the equation (.5), a, and q are unspecified for the capillary waves but there is no need to know them which is clarified below. A triangular facet model is realized on the mean horizontal sea level which is also formed by the triads. The triads are projections of the facets which have three different elevations in their vertices as drawn in Figure -3. Point A` Facet z 3 y Wind x Facet A` 1 Triad 1 Triad A` Centroid of the Triad Mean Level Ocean Surface Figure -3 Model of the ocean surface with the triangular facets.

38 5 To create the ocean surface with capillary waves in a finite area, mean level is formed by the triads with the fixed areas and secondly the elevations of the first facet s vertices, 1,, 3 are calculated with the normal distribution (mean is zero and variance is e ). Then, three points, which are located above or below the vertices of the related triad in z direction, are fixed in the finite area with the lengths defined in the previous step. When these three points are connected to each other, the first facet of the capillary wave surface is formed. For the following facets, only one more elevation value is needed because two vertices of the new facet are the vertices which are also belongs to the previous facet. The rest of the facets are realized by the same process. As it is shown in the Figure -3, γ is height and ε is width of the triad. The relations among, ε, crosswind slope, upwind slope and elevation values of the facet A` (it is the facet in Figure -3 with the thick lines) are given below according to the numbers used for the points 1,, 3 in the Figure u (.6) c (.7) The elevation of the facet is calculated from its centroid. The centroid elevation is the arithmetic mean of the vertices elevations (.8)

39 6 As indicated before, the mean of the facets vertex elevations is zero, j yet the correlation of it has a proper value for i, j=1,, 3 and where i j indicates Kronecker delta symbol and i j e i j x is used for ensemble average of x. To find the variances of the upwind and crosswind slopes, ensemble average is applied to the equations (.6) and (.7). 1 u (.9) c (.3) And then, by using j and i j e i j, the results become: u e (.31) c 3 e (.3) Again by using by using j and i j e i j, the variance of the facet elevation becomes:

40 7 e (.33) 3 If a ratio relation among a u, a c, triad dimensions,, wind speed and variance of the vertex elevation is established, the capillary wave surface realization depends on one unknown which is the wind speed of e the medium. This relation is set up by using the equations: (.3), (.4), (.5), (.31) and (.33). 3a 4a 1 e au U u c (.34) Equation (.34) shows that the dimension of the triad is independent to the wind speed which may be determined according to the resolution necessity of the model. Secondly, the vertex elevation variance is dependent to the wind speed, dimension of the triad and upwind/crosswind constant (if ε is used in defining the vertex elevation variance, a c has to be used instead of a u ). As seen in equation (.34), the dimensions of the triad have a definite ratio but there is not a limitation about how big or small they have to be. Mobley realizes a non-numerical model in his book [1] so he chooses 1 for the. But on the other side, in this thesis it is planned to use a numerical approach. For that reason, the dimensions of the triad used to form the capillary wave surface model have to be determined and at this point there is a definite limitation which is defined in part.1 with the dispersion theory. The maximum wavelength for the capillary waves is 1.7 cm which corresponds to the celerity 3 cm/s and for the faster capillary waves, the wavelength of the wave is decreasing. There are two questions to be answered here: What is the relation between wind speed and celerity? And what is the minimum wavelength that can be modeled and analyzed with beam tracing approach?

41 8 The answer of the second question is related about the part.1. First of all, to analyze the each incident beam and its daughter beams, one beam has to have interaction with only one facet, otherwise the incident beam has to be divided into multiple facets which would make it too hard to track the daughter beams. Also, determining the point where the beam hits the ocean surface or in another word while the beam hits a fixed point on the ocean surface, defining the distribution of the triads according to that fixed point is another issue that has be dealt with. To avoid those computational difficulties, determining the vertical projection of the triad s centroid is determined as the fixed point which is illuminated by the center of the incident beam (this point is named point A` in the rest of the thesis, Figure -3) and one beam-one facet interaction is used. To enable the one beam-one facet interaction assumption, the dimensions of the beam have to fit into the facet. It is already mentioned in part.1 that the maximum wavelength for the capillary wave is 1.7 cm and the minimum facet area in the model can be as small as the area of a triad if the all the vertex elevations of the facet are equal. So to make the model valid for all cases, the cross-sectional area of the incident beams at the centroid of the triad A` has to be smaller than the area of the triad which can maximally have the dimensions equal or smaller than 1.7 cm. To calculate the corresponding beam size fitting into the biggest triad of the capillary wave, it is assumed that the width of the triad is 1.7 cm and with using equation (.34) the height of the triad is calculated, 1.53 cm. On the other side, to calculate the cross-sectional area of the incident beam, the equations in part.3 are needed in addition to the specifications indicated in part.. Even the distance information between the laser transmitter and the point A` is variable with related to the elevation information of the facet, 5 meter which is the distance between the laser transmitter and horizontal mean level surface is used. This fixed distance is chosen because it is known that the elevations of the vertices have normal distribution which means that the mean of the vertices elevations is zero. The minimum initial beam radius for the minimum beam spreading at 5 meter away is calculated,.9 cm with using the equation (.17) and the broadened beam radius at 5 m away from the transmitter is calculated,.41 cm with using the equation (.11). To get a triad which is minimum equal and also bigger than the crosssection surface area of the incident beam, cross-section surface area of the incident beam has to be incircle of the triad. The radius of an incircle has the following ratio.

42 9 A r (.35) P P is the perimeter and A is the area of the triad. With using the ratio information between the height and width of the triad defined in equation (.34), the minimum beam radius value (.41 cm) and the equation (.35), the minimum width of the triad, which covers incident beam s cross-section surface area, is found 1.39 cm. This value can be accepted as the minimum wavelength of the capillary wave and the celerity corresponding to this minimum wavelength is calculated, 3.3 cm/sec with using equation (.5). The result of this analysis makes it clear that the range of the wave speed to be used in one beam-one facet approach is from 3 cm/sec to 3.3 cm/sec. At this point to understand the meaning of this narrow range, the related wind speed corresponding to this wave speed range has to be found which is also the answer of the first question indicated above. The relation between the wind speed and the wave speed is told with the wave-age term which has the equation as below. U cos( ) 1 (.36) c p U 1 (m/sec) is the wind speed at 1 m above the ocean surface, c (m/sec) is the peak phase speed of the p waves and is the angle between wave and wind directions. The wave age ratio varies in time due to change in the peak phase speed of the waves. As it is indicated in part.1, the wind-driven wave form, which gains the momentum form wind, are used in the thesis and according to the studies done about the wave age show that there is a wave-age range for the wind-driven waves. As it is indicated by Hanley et al. [8] and Mayo [18], the wave age ratio for the wind-driven waves is between.8 and 5 or bigger. Hanley et al. showed that the lower boundary of the wave-age range might be smaller than.8 yet in that case the wave is not a winddriven only, it also starts to contain the properties of the swells. If the wave age is under.15, the wave is

43 3 totally swell and the regime is totally inversed from wind-driven to wave-driven form. But the specific waveage for the wind-driven waves used in the thesis is.8 because it is the value that the fully developed sea conditions exist which is indicated by Hanley et al. and Mayo. The other parameter in equation (.36), is always zero in the thesis because it is assumed that waves are always formed along the wind direction. With using equation (.36), the wind speed range corresponding to the celerity range of the capillary waves (from 3 cm/sec to 3.3 cm/sec) with one facet-one beam approach is found from cm/sec to 19.4 cm/sec. This wind speed range is very low and also very narrow to get an idea of the capillary waves effects on the laser propagation because the analysis done in the thesis contains wind speed range which has 1m/sec lower boundary. Thus, the analysis of the capillary waves effects on the laser propagation is kept out of the scope of the thesis due to the approaches used in the model and the limitations mentioned above..5. Gravity Wave Scale The second wave type is the gravity wave whose effects on the laser propagation are analyzed in the thesis but before getting into its effects, the gravity wave scale has to be analyzed. In the thesis, the directional energy spectrum driven from the measurements done in the Stereo Wave Observation Project (SWOP) [9] is used in the Fourier series representation. The results in the SWOP fit with the energy spectrum of the Neumann and SWOP formed the wholeness of the Neumann energy spectrum [1]. To analyze the gravity wave scale, four references are used which are written by: Mobley [1], Preisendorfer [1], Kinsman [14], Chase et al.[9] and to form the gravity wave ocean surface model, the same steps used by Mobley are followed. According to the classical wave model that Preisendorfer explained, the celerity (wave speed) is g h c tanh (.37)

44 31 But in the deep water (when the depth is much bigger than the wavelength of the waves, h assumed that the depth is infinite and thus, the celerity becomes ), it is g c (.38) k According to the temporal-frequency wind-speed displacement law driven from the Neumann spectrum, there is a frequency called maximum frequency at which the highest density of the wave energy occurs. 3 g max (.39) U By using the equation (.38), the maximum wavelength where maximum energy occurs can be found. 3U max (.4) g The gravity wave ocean surface model is realized in a finite area which is enough for observing the single and multiple scattered light beam and its directional distribution. To make a feasible ocean surface model, it has to have a finite size which is based on wind speed and the some of the other parameters which are explained in chapter 3. Mobley forms a finite area in hexagonal shape but in the thesis, a finite area in rectangular form is used to avoid computational complexity. The numerical specifications of the scale are explained in chapter 3. The same geometrical model of the capillary wave surface model is used for gravity wave surface model. The main reason to use the triad shape form is about the form preserving property of the triad. While

45 3 the facet, which is the vertical projection of the facet, has the same form with triad (both facet and triad are in triangular forms), generally the other geometrical figures can t preserve their initial forms when the elevations of their vertices are changed. The dimensions of the gravity wave model will be X by Y meters, and in the Cartesian coordinate system formation, x axis is in downwind and y axis is in the crosswind direction at the meal level surface. In the thesis, the length of the finite gravity wave model in y axis is limited with four times the max and the length of the finite gravity wave model in x axis is limited with two times the max, and thus the dimensions of the finite gravity wave surface becomes: X by Y 4max meters which also satisfy max Nyquist rate. The length of the finite gravity wave model in y axis is doubled the length of the length of the finite gravity wave model in x axis because the triads are formed with two spacing in x axis and one spacing in y axis which forms a triad with equal width and length. The main effect on determining the size of triad is the wind speed because the dimensions of the finite area depend on the max which also depends on the wind speed. To determine the triads in the coordinate system, X and Y are divided respectively into a and b. By this way the vertices of the triads can be located with integers in the following ranges and spacing; X a i a, b j b, x and a Y y. In here a and b specify the resolution of the sampling and b they are called with resolution values in the rest of the thesis. Specification about the resolution values, a and b is indicated in chapter 3. The all definitions to specify formation of the ocean surface with gravity waves in the Cartesian coordinate system are showed in the Figure -4.

46 33 ( i, j) ( a, b) X ( i, j) ( a, b) j Facet A` Y j y ( i, j) ( a, b) i i x ( i, j) ( a, b) Figure -4 Representation of the gravity wave surface model in Cartesian coordinate system.

47 34 The all process of surface realization in is specified in Cartesian coordinate system yet the directional energy spectrum of the gravity waves is in cylindrical coordinate system so the formulation to define the relations between two coordinate systems is given with equation (.41) and also showed in the Figure -5. ( u, v) ( a, b) uv ( u, v) ( a, b) ( uv, ) k uv uv v, o uv ( u, v) ( a, b) j i uv ( u, v) ( a, b) Figure -5 The spectral domain ( uv, ) coordinate system representation of the Cartesian coordinate system in Figure -4. u v a1 x b1 y k uu, k vv u v (.41)

48 35 Actually the ( uv, ) represents the same coordinates with ( i, j ) yet due to the symmetry in the energy spectrum, E( k, k ) E( k, k ), a new set of coordinates is defined with ( uv, ) spectral domain coordinate system. u v u v ( u, v) W : u v 1, b or u 1, a & v b, b (.4) Set W involves only the coordinate system which is enough to define the whole system by using the symmetry property of the energy spectrum. After realizing the horizontal base of the triads, the only necessity to realize the ocean surface with gravity waves is determining the elevations of the vertices of the facets. At this point, the Fourier series expansion is used to define the elevation function for the nodal points ( i, j ) of the vertices [1], [11], [1]. ui vj ui vj ( i, j, m) buv ( m)cos cuv ( m)sin ( u, v) W a 1 b 1 a 1 b 1 (.43) In the equation (.43), m (1,,3 M) and M is the total realization number of the ocean surface used for the ensemble average approach and it is specified in chapter 3. The quantities b uv and c uv are independent, normally distributed random coefficients with zero mean and variances: uv uv u v b c E k, k u v (.44) E( k, k ) is the directional energy spectrum of the gravity waves derived from the observations during u v SWOP [1], [11], [1].

49 36 The derivation of E( k, k ) is given below: 4 ( u, v) ( uv, uv ),( ) 3 uv u v g E k k F m Cylindrical coordinates Cartesian coordinates uv 1 1, gkuv s Temporal frequency of the wave uv k 1 v tan ku Angle of k uv from x axis 1, uv u v 1 k k k m Wavenumber of the deep water in the gravity waves g F C f m s 6, uv exp ( uv, uv),( ) uvu Temporal frequency form of the energy spectrum of the wave f (, ) cos.3 cos 4 Empirical constant (.45) uv uv uv uv 1 exp C g uv U g 4 Empirical constant 5 Empirical constant.763, m s Gravity acceleration 9.8, ms To realize the gravity waves, wind speed information, realization number, and resolution values are needed which are specified in chapter 3.

50 37.6 Beam Tracing through the Ocean Surface As it is mentioned before, it is aimed to degrade the ocean surface into the specular surfaces and apply Fresnel reflectance to the linearly p-polarized incident laser beam. The general concept for the Gaussian laser beam is explained in part.3. In this part, the link between the Gaussian laser beam and the gravity ocean surface model is set up. In the model the output power of the laser, initial beam radius, wavelength and the initial angle of the laser beam are specified as design criteria in chapter 3. One of the most important reasons that the laser is chosen in these kind applications is almost collimated form of the laser beam even the it is not perfectly collimated. For that reason some specifications of the laser is defined for the truncated form of it. Measurement of the laser width is calculated according to the area, consisting of the 1/e of the whole electric field of the laser. On the other side, the given power of the laser is not truncated, so to make some comparison among the incident and daughter laser beams, the truncated power has to be calculated after propagation of the beam the distance, z between the laser transmitter and the surface of the facet A` by using the truncated beam width. To calculate the truncated power, first the electric field E has to be calculated from the initial beam power with using equation (.) and then the beam radius at facet A`, ws () z is also calculated by using ABCD law. ABCD matrix for propagation of Gaussian beam in the homogeneous and lossless medium is: 1 z M p 1 (.46) and by using ABCD law [5] approach for the complex beam parameter,

51 38 q1 () A B q() z q () C D 1 (.47) complex beam parameter of the propagated beam is found. The unknown variables in equation (.47) are transferred from equation (.46), A 1, B z, C, D 1. The complex beam parameter before the propagation is equal to the initial complex beam parameter, q1() q which is calculated by using equation (.15) and with using equation (.1) the incident beam specifications at the surface of facet A` can be found. R ( z) s w ( z) s z z z z a z n z (.48) Lastly putting the values together, the truncated power of the incident beam at the surface of facet A` is found. ws ( z) c r w S S Ps E exp rdr w ( z) w ( z) (.49) c Ps.8647 E w (.5) Refraction of the Laser Beam in the Air-Ocean Interface This part is about the calculations on the incident beam to analyze the effects of the ocean surface with gravity waves on the laser propagation.

52 39 The relation among the incident beam and the daughter beams is explained with the ABCD law and Snell's law which is given in equation (.51). n sin n sin a i w t i r (.51) In thesis, the following notation is used to explain the relation among the incident beam and the daughter beams: n a, the refractive index for air which is accepted 1 for the thesis and n w is the refractive index for ocean. i is the angle between the central axis (the ray in the center) of the incident beam and unit normal vector of the facet A`. t is the angle between the transmitted beam s central axis and the negative unit normal vector of the facet A`. r is the angle between the central axis (the ray in the center) of the reflected beam and unit normal vector of the facet A`. In the model, 3-D approach is used but the formulas in part.3 are only enough to define a -D Gaussian beam and reflection/transmission of it. At all, the incident beam is a perfectly symmetric Gaussian wave and for that reason, the results in two different -D realizations can be generalized for the 3-D realization with the using symmetry property yet this is not valid for both daughter beams. When the daughter beams are analyzed one by one, while the reflected beam preserves the circle wave form of the Gaussian wave, the transmitted beam is no longer in circle form, it transforms into the elliptical form unless the incident beam interacts with the ocean surface in normal angle. So whole calculations are going to be done in two different planes e.g. x z and y z planes and thus, 3-D realization is reduced into two different -D realizations. The reasons of using beam tracing rather than ray tracing is explained in chapter 1 yet there are some issues have to focused on while using beam tracing and one of them is dealing with incident angles of beam s different parts on the surface of facet A`. As indicated before, the Gaussian beam is not a collimated beam

53 4 which is expanding as it is propagating. If the beam is thought as the union of infinite rays, all the rays hit the surface of facet A` with different angles because also the rays forming the non-collimated beam are not collimated amongst. At this point, without using an assumption/approximation, ABCD matrix approach cannot be applied to the incident beam. When the values of z and ( S w S are compared, paraxial approximation z w ) can be used on the reflection and refraction calculations and the minor angle ripples can be neglected. In briefly, it is assumed that all the rays in the beam have the same incident angle with center ray of the incident beam. Display of this assumption is shown in the Figure -6. The another assumption in the beam form is about the measuring propagation distance, z. To measure z, the propagation distance of the center ray in the beam is referenced. There are two reasons about it. In the thesis, the elevation angels of the incident beams ( i ) to the air-ocean interface are very small, which are shown in chapter 4, and by this way the distances among the center ray and outer rays in the perpendicular direction to the propagation direction are very small. This situation allows the ABCD matrix approach to be used in the interacting of the laser beam with the dielectric surfaces which has different axis from the incident beam. By using these approximations, while the simplicity of the ray approach is being used in the calculations, the similarity of the model to the real situations is kept in a high level. This approach is illustrated in the Figure -6.

54 41 Center Ray of the Incident Beam Focused Part, The Collimated Beam Region z z focused Air-Ocean Interface wz () z refocused z refocused Refocused Part of the Incident Beam z z z focused z w() z refocused Figure -6 Paraxial approximation on the air-ocean interface. In this step, as same as it is used in determining the truncated power at surface of facet A`, ABCD law is used to find the specifications of daughter beams. To get the transmitted properties of the elliptical transmitted beam, all transformations are performed in two different sets of the ABCD matrices and the incident beam is analyzed in two planes ( x z and y z). By this way, there is going to be two sets of wz () and Rz () for transmitted beam.

55 4 ABCD matrix for the interface of Gaussian beam (all the ABCD matrices below can also be used in ray tracing approach) between two dielectric mediums in -D [3] is M t cost cosi n n a w cosi cos t (.5) To get the physical properties Rz () and wz () of the transmitted beam, another matrix includes both propagation and transmission matrices has to be calculated. M M M M pt t p pt cost cosi cost z cos i na cosi nw cost (.53) On the other side, the Gaussian wave has its own coordinate system. While Gaussian x, y are for Gaussian Gaussian defining the intensity distribution, z is for the propagation direction yet in the thesis, the intensity distribution is not analyzed, it is assumed that beam intensity has a uniform distribution. A, B, C and D values are picked up from equation (.53) for defining the Gaussian beam parameter of the transmitted beam and then equation (.47) becomes cost cost q1 () z cos i cos i q( z) (.54) na cosi n cos w t

56 43 n cos n cos q z q z (.55) w t w t ( ) 1() na cos i na cos i Using equation (.15) in (.55) and taking its inverse leads to 1 zn cos z n cos q z n z z n z z a i a i i () w cos t w cos t (.56) If we equalize equation (.1) with (.56) we obtain 1 zn cos z n cos 1 q ( z) R ( z) w ( z) n a i a i i i nw cos t z z nw t z z 1 zn cos R z n z z a i ( ) wcos t zn cos w z n n z z a i ( ) w wcos t cos w (.57) R n cos z z w t ( z) znacos i w ( z) cos t zn acos z z i (.58)

57 44 z facet y facet x facet z facet z facet x facet y facet x z plane facet facet facet facet y z plane Figure -7 Decomposition of 3-D beam representation into two -D beam representations. At this point, the orientation of the facet A` s coordinate system according to the mean level coordinate system is very important because the direction of the elliptical formation of the beam s cross-section is going to be realized according to it. For this reason, the unit normal vector of the facet A` has to be calculated for each realization and then according to the azimuth and elevation angles of unit normal vector of the facet A`, the orientation of the daughter beams direction versus the mean level surface coordinate system can be set up. The details about this issue are told in chapter 3. As it is told above, there are two sets of R () z and w () z while one set, which are represented by R () z and w () z, are determined according to local xfacet zfacet plane of facet A`, the other set, which are represented by R () z and w () z t t t, t

58 45 are determined according to local yfacet zfacet plane of facet A` (Figure -7). If assumed that the azimuth orientation of facet A` is in local x facet axis, the incident beam has an incident elevation angle and transmitted elevation angle different form zero ( i, t ) in local xfacet zfacet plane. On the other side, with the same assumption, incident elevation angel and also transmitted elevation angle are zero according to local y facet zfacet plane. For a more general explanation, the local facet facet x z plane of facet A` is renamed as the reflection plane and local yfacet zfacet plane of facet A` as the perpendicular plane to the reflection plane in the rest of the thesis. If the equations in (.58) are specified according to the local planes of the facet A`, the following two sets are derived. For local reflection plane, nwcos z z Rt ( z) zn cos w ( z) t t a cos zn t a z cos i z i (.59) For local perpendicular plane to the reflection plane, R t t nw z z ( z) zn w ( z) z a z zn a (.6) Rz () and wz () define the physical properties of the transmitted beam yet to calculate the transmitted power, transmittance information is necessary. By calculating the reflectance, and using energy conservation theory, the transmittance can be found in an easier way than the calculating it directly. On the reflected beam side, the

59 46 shape of the reflected beam is preserved as the continuation of the incident beam but on the other side the power it carries is attenuated by the reflectance. At this point the incident beam is parallel polarized along the x axis of mean level surface but on the other side, facet A` has its own local coordinate system which is not same with mean level surface s coordinate system in general. For each realization, a new local coordinate system and reflection plane of facet A` are defined. According to each reflection plane orientation versus mean level surface, the electric field vector makes an angle with reflection plane and if the parallel and perpendicular projections of the electric field vector of incident beam according to the reflection plane is calculated; for the parallel component, equation (.61), Fresnel reflection coefficient for the linear p polarization can be used and for the perpendicular component, equation (.6), Fresnel reflection coefficient for the linear s polarization can be used. r n cos n cos w i a t r _, i t na cost nw cosi Ei _ E (.61) r n E, i t n n E a cosi nw cost r _ a cosi w cost i _ (.6) For the derivation of the reflectance, equation (.19) is recalled for the incident beam s power. c Pi Ei ds (.63) s At this step, the electric field is separated into its components with using equations in (.64).

60 47 Er Er Er Ei c ( ) Ei Ei c ( ) Ei Ei Ei Ei Ei Ei Ei Ei Ei (.64) c ( ) and c ( ) are the facet A` s local coefficients to calculate the parallel and perpendicular components of the incident electric field according to the orientation of the facet A`. They are calculated according to the azimuth angle, of the unit normal vector of the facet A`. Lastly by using equation (.13), the power of the incident beam is defined with the components of the electric field. E E E E E E E i i i i i i i r w Ei Ei Ei _ Ei _ exp w ( z) w( z) c r w Pi Ei _ Ei _ exp ds w ( z) w( z) s (.65) If the reflected beam s electric field is broken into its components the following equation is derived. c Pr Er ds s c r w Pr Er _ Er _ exp ds w ( z) w( z) s (.66)

61 48 To represent the power of the reflected beam with using the components of the electric field amplitude term of the incident beam, the Fresnel reflected coefficients calculated in the equations (.61) and (.6) are used. c r w r i _ i _ exp w ( z) w( z) (.67) s P r E r E ds If the facet s local coefficients is applied to the equation (.67), power of the reflected beam is defined with electric field amplitude term of the incident beam. c r w r ( ) i _ ( ) i _ exp w ( z) w( z) (.68) s P r c E r c E ds Reflectance is found by calculating the ratio of the reflected beam s total power to the total the incident beam s total power. R P P r (.69) i And lastly, by using energy conservation theory, the transmittance is calculated. T 1 R (.7) Thus, the transmitted beam s power just beneath the surface is: P T P (.71) t i

62 49.6. Reflection of the Laser Beam from the Air-Ocean Interface (Ocean to Air) The reflected beam s radius and radius of the curvature of the spherical wave are found in similar way like properties of the transmitted beam yet instead of using transmission matrix in equation (.5), the reflection matrix for specular surface, which is defined below, is used. 1 M r 1 (.7) Final state of reflection matrix is M pr Mr M which is equal to p M and for the rest of the calculations to p find the wz () and Rz, () same steps are followed as it is done in the transmitted beam. Only one set is realized in reflected beam because as it told above, it preserves its circular symmetry. z R ( z) r w ( z) r z z z z n z (.73) These two parameters are exactly same with the beam properties at the surface because the main assumptions let us to get these results. The first one of these assumptions is the paraxial approximation used in the spreading of the beam and the other one is the specular property of the gravity ocean surface model. But on the other side, the power of the reflected beam is reduced by the reflectance which is calculated with the equation below. P R P (.74) r i

63 5 3 CHAPTER 3 Gravity Wave Ocean Surface Model Equation Chapter 3 Section 1 To test the validity of the theory indicated in the chapter and get the results of the effects of gravity wave ocean surface to the laser beam, a model written in MATLAB is designed in line with chapter. In this part the main m-file, which realizes ocean surface with the gravity waves and performs the whole calculations for the beam tracing, is told in detail. In the model, the transmitter is located at the zenith which causes the maximum beam transmission through water and also avoids high reflection percentage. The transmitter locations with various azimuth and elevation angles are left for the future works. While explaining the details about the code, the variables and the functions in the code are written with the same names as they are in the M-file to make the code easily followed and also double-quote is used to avoid the confusion with the variables used in the previous chapters. 3.1 Flow Chart of the Model a. Determine the specifications of: i. The transmitted laser beam, aa. bb. cc. dd. ee. ff. Output power, Wavelength, Emitting beam radius, Emitting azimuth angle, Emitting zenith angle, Altitude of the transmitter,

64 51 ii. iii. iv. Wind speed direction and range, Scale of the gravity waves and their resolution, Realization number, v. The constants about the medium. b. Calculate the elevation distribution of the facets according to the realization number and wind speed, c. Calculate the daughter beams (reflected and refracted beams) properties, d. Calculate the mean of the daughter beams properties versus realization number, e. Check the daughter beams for multiple scattering, f. Realize the hemisphere distribution of the daughter beams. 3. Pseudocode of Model Written in MATLAB Programming Language The main M-file of the gravity wave ocean surface model follows the steps indicated in the flow chart above. Before performing the steps, two important variables have to be decided which are the wind speed range and realization number. The wind speed range is between 1 and m/sec. 1- m/sec wind speed range almost covers the whole Beaufort scale except some extreme cases at the top range. Second major variable, the realization number, M has to be big enough to perform ensemble average approach in realistic way, but on the other side it also has to be small enough to make the model be calculated in a reasonable time. The realization number, M is chosen 1 in the model. In part 3.4, the analysis to determine the realization number is told in detail. The parts of the MATLAB code are specified with M capital at the end of the number of related part to avoid any possible confusion with the parts used in the thesis. The main part contains 6 parts.

65 Part 1M, Defining the Specifications about the Model The general specifications about the gravity wave ocean surface model are given in the part 1.3 and the details about the model with transmitter at the zenith are given below: a. The transmitter is located at the zenith and the transmitter surface is looking perpendicular to the reference x y plane. b. The transmitter altitude is 5 m high above the reference x y plane. c. The wavelength of the laser is 53 nm. d. The radius of the transmitter is calculated.9 cm with using equation (.17). It is the minimum radius for the minimum beam spreading at 5 m distance away from the transmitter. The other calculated laser parameters and their equations are consecutively: Rayleigh range, equation (.14), initial electromagnetic field supplied by 1 watt laser output, equation(.) and the power of the laser at the facet A` surface, equation (.19). e. The downwind direction is along the x axis of the reference coordinate system and the wind range is limited with 1- m/s. f. The realization number M is 1. g. For the refractive index of the ocean, 1.34 is chosen. As it is known, the refractive index of the ocean is dependent on many inputs like; pressure, salinity, temperature, the wavelength of the light and location of the ocean. According to the Austin and Halikas [31], when the wavelength of the light is 53 nm, for the typical sea water (salinity of the water %35 ) at the atmospheric pressure, the refractive index of the o o seawater varies between 1.34 and 1.34 versus the temperature range 3 Celsius. So using 1.34 as the refractive index of the ocean is a reasonable choice. h. Refractive index of air is assumed 1. i. As indicated in part.5., the size of the finite area is twice of the max. As the max is depend on the wind speed, the size of the finite area changes with the wind.

66 53 j. The resolution values a and b are 96. The detail about the resolution values of the triads is explained in parts 3.3and 3.4 with the realization number analysis. On the other side, the ratio between height and width of the triad is kept one and to obtain this ratio; two variables, horizontalcoefficent and verticalcoefficient are determined with 1/ ratio due to reason told in part.5. about the spacing formation of the triads in x and y coordinates. k. The resolution variables and sphere radius of the spherical distribution are told in part 3..6 in detail. 3.. Part M, Calculating Elevation Distributions of the Facets There are two main for loops at the beginning of Part M which are for the wind speed range and the realization number of the gravity wave ocean surface. These loops involve Part 3M, Part 5M and Part 6M. Actually Part 6M works only for one specific wind speed condition but it has to be inside the realization loop because it has to be in the wind speed range loop. To get the output of Part 6M, the wind speed range is fixed specifically to one wind speed value and while getting the outputs of the other parts in the wind speed range, Part 6M is deactivated. a. The Part M starts with the first main for loop for wind speed range. As it is mentioned above, wind speed range is between 1 and m/s with the 1 m/s steps. For each wind speed value, max is calculated with using equation (.4). max also is used to define the dimension variables of the finite area, X and Y with using the horizontalcoefficent and verticalcoefficient which are defined in part 1M. Then as it is explained in the Part.5.; x, y, u and v are calculated according to the wind speed. These parameters are calculated once for each specific wind speed value. As indicated in Figure -4, x and are respectively forming the triad s width and height. y

67 54 b. The rest of the Part M is in the second main for loop which is defined for the realization number. The random coefficients b and c in the elevation function of the facets vertices are generated with the multiplication of the variance with randn(n) function which is a built in function in MATLAB. randn(n) generates the random numbers in standard normal distribution with zero mean and variance one. The desired distribution of the coefficients is normal distribution with variance found from energy spectrum of Neumann which is given in the equations (.44) and (.45). So the calculated variance is multiplied with the randn(n) function to get the normal distribution with the desired variance. c. After generating the b and c coefficients corresponding to each nodes in the finite area, the elevations of the facets vertices are calculated with using equation (.43). At the end of Part M, one elevation distribution is derived corresponding to wind speed, 1 m/s. The second for loop generates elevation distributions as much as the M value, which is 1 in this model, with the wind speed, 1 m/s. When the loop of the realization number is completed, the loop of wind speed range jumps to second wind speed which is m/s and this goes on until the last realization is performed for elevation distribution in m/s wind speed Part 3M, Calculation of the Daughter Beams Parameters This part is for the analysis of the effects of the ocean surface with gravity waves to the laser beam propagation through it. As mentioned before, the coordinates of the point which is illuminated by the laser beam are fixed for the whole realizations and wind speed values (point A` on the facet A`, Figure -3). Locating the coordinates of the illuminated point is a predictable issue yet having the knowledge of the triad/facet/point corresponding to this coordinate is done under ensemble average approach. The gravity wave ocean surface model has a finite area which is defined in the thesis so the coordinates of the facet A can be chosen by the knowledge of laser transmission platform s coordinate and the emitting elevation angle of the laser beam. The point A` is chosen as a result of an assumption. This is a really fair assumption when the

68 55 realization number and randomness of the ocean surface is considered. The analysis in Part 3M is done with using point A`, facet A` and triad A`. a. The elevations of the vertices belong to facet A` are picked from the elevation distribution realized in Part M and assigned to three variables called n_1, n_ and n_3 which are shown in Figure -3. b. To verify the realness of the model, one of the arguments that can be checked is the maximum/minimum elevation value in the elevation distribution corresponding to the specific wind speed. Maximum and minimum elevations are calculated and recorded versus the realization number and the wind speed. c. Upwind and crosswind slopes are calculated by using equation (.6). d. Elevation of the facet A` s centroid, h_a is calculated with calculating the arithmetic mean of the elevations n_1, n_ and n_3. With using h_a and the altitude of the laser transmitter from the level surface, the propagation distance of the laser beam through the air-ocean interface is calculated. e. With the propagation distance information, the beam radius is calculated by using equations (.11) and (.16). f. Radius of the curvature of sphere beam wave is calculated by using equation (.1). g. To specify the daughter beam s properties, angular orientation of the facet A` is needed and the best way is determining the normal unit vector of the facet A` versus the reference coordinate system. Two vectors are defined which are: n_1, the vector from point n_1 to point n_ and n_13, the vector from point n_1 to point n_3. While defining these two vectors, the elevation of the point n_1 is accepted as the reference height. The unit normal vector of facet A`, n_normal is found with using the mutual relation between the vectors n_1 and n_13. normal (3.1)

69 56 The normal unit vector is calculated at point n_1 because the point n_1 is accepted as the reference point but the unit normal vector is same for the whole points on the facet A` because facet A` is a specular surface which has the same normal unit vector on its whole surface. h. With using equation (3.1), the unit normal vector of facet A` is found in Cartesian coordinate system and it is converted into spherical coordinate system (,, ) with cartsph built in function in o MATLAB. Azimuth angle THETA, ( ) is in the range 18,18 o in degrees or, in radian starting from positive x axis on the reference x y plane, it is positive in counter-clockwise and negative in clockwise. Elevation angle PHI, ( ) is in the range 9,9 in degrees or /, / in radian starting from reference x y plane, it is positive in positive z axis, negative in negative z axis. The radius is always one because it is a unit vector. o o i. To define the rest of the parameters easier, THETA is converted into ( 36 ) range. THETA_uw is azimuth angle of the transmitted beam which is in the reverse direction of the azimuth angle of the unit normal vector, THETA. THETA_reflected is the azimuth angle of the reflected beam which is same with THETA. j. PHI_incident, ( i ) is the elevation angle between the unit normal vector and the zenith ( 9 ). As mentioned before, the transmitter is located 5 meter above the reference x y plane at the zenith. In the angle calculations of the incident and daughter beams, the angle of the ray in the center of the beam is used as the angle of the whole beam as it is explained in part.6.1. k. PHI_transmitted, ( t ) is the elevation angle between the negative unit normal vector and transmitted beam s center ray which is calculated with equation (.51). PHI_uw, ( uw) is the elevation angle from the reference x y plane to the transmitted beam s center ray in the negative direction. PHI_reflected is the elevation angle from the reference x y plane to the reflected beam s center ray in

70 57 positive direction. The whole angular setup up between the unit normal vector and the incident beam is explained in Figure 3-1. z Incident Laser Beam normal 1 i c Raised Reference x y Plane to the Height of the Point A` Point A` Reference x y Plane Facet A` u 3 y Centroid of the Triad A` x Triad A` Figure 3-1 The formation of the unit normal vector of the facet A`. l. While R_transmitted is the transmitted beam s radius of the curvature of the spherical wave, w_transmitted is the radius of the transmitted beam at the point A` in the reflection plane and they are calculated with equation (.59). The radius of the transmitted beam at the point A` in the plane perpendicular to the reflection plane is equal to the beam radius of the incident beam, w_s at point A`. R_transmitted is the transmitted beam s radius of the curvature of the spherical wave at point A` in the plane perpendicular to the reflection plane and it is calculated with equation (.6). m. At this point, as it is told in part.6.1, according to the perpendicular and parallel electric field polarization components of the incident beam versus reflection plane, reflected beam s power,

71 58 reflectance, transmittance and the power of the transmitted beam are calculated with using equations (.61)-(.71). There are two coefficients, parallelcoefficient, c ( ) and perpendicularcoefficient c ( ) which are calculated according to the directional orientation of the unit normal vector of the facet A` versus the reference coordinate system. The coefficients divide the incident beam s electric field vector into two vectors which are parallel and perpendicular to the reflection plane. This phenomena is also so explained with Figure 3-. Reflection Plane z Incident Laser Beam E i Point A` i y i E i c ( ) cos i / / x E i Raised Reference x y Plane to the Height of the Point A` Projection of normal the Reference x y Plane Figure 3- Separation of the electric field vector to its components through the ocean surface with gravity waves.

72 Part 4M, Calculation of the Mean Values The analysis in Part 3M is done for each realization yet due to using random coefficients, ensemble averaging approach has to be performed to get a meaningful result. Ensemble averaging process is done by the following process: Realizing the gravity wave ocean surface (Part M), illuminating the facet A`, calculating the daughter beams properties (Part 3M), averaging the results and getting the mean values for the wind speed range which is done in Part 4M. PHI_incident_mean_facet is the mean of the incident beams elevation angles between the unit normal vector of the facet A` and zenith. PHI_reflected_mean_reference is the mean of the reflected beams elevation angles between the reflected beams center rays and reference x y plane. PHI_transmitted_mean_facet is the mean of the transmitted beams elevation angles between the negative unit normal vector of the facet A` and transmitted beams center rays. PHI_uw_mean_reference is the mean of the transmitted beams elevation angles between the reference x y plane and transmitted beams center rays. Up to now, while the averages of the angles are being calculated, the arithmetic mean is used yet the average of the azimuth angles cannot be done with arithmetic mean directly because they have a circle loop. To calculate the mean of the azimuth angles, the operations below are performed.

73 6 a a a a sin r cos r n1 n1 sin r r o o o mean mod mod arctan( ) 36,36 18,36 cos ar sin t cos t t mean M M M n1 M n1 r sin( ) n r cos( ) n t sin( ) n t cos( ) n sin a o r mod arctan( ) 36,36 cos ar a (3.) By using the sensitivity of the sin and cosine to modulo 36 o degree, the sum of the azimuth angles can be calculated among the whole realization for each wind speed value. Then with calculating the inverse tangent of these sums, the mean azimuth angle is automatically calculated with no need to averaging. As it is seen in equation (3.), there are two mod operations for mean of the reflected azimuth angle and one mod operation for transmitted azimuth angle. The first mod operation in the calculation of the mean reflected azimuth angle is to avoid the negative angles. And also before the second mod operation, there is 18 o degree addition to the result of the first mod operation, because the result of inverse tangent operation is limited in range o 9,9 o o degree which does not contain the range 9,7 degree and when the orientation of the unit normal vector of the facet A` s azimuth angles is analyzed, it is seen that general o o tendency is in the 9,7 degree sector. The reason for that, the upwind slope in the positive x axis is bigger than the crosswind slope in the positive y axis. Second mod operation is performed to prevent a result in excess of the 36 o degree. On the other side, in the calculation of the mean transmitted azimuth angle, there is no need to add 18 o degree because the transmitted beam is in the opposite direction of the azimuth angle of the facet A` s unit normal vector.

74 61 THETA_reflected_mean_reference is the mean of the reflected beams azimuth angles. THETA_uw_mean_reference is the mean of the transmitted beams azimuth angles. transmittedpower_mean is the mean of the transmitted beams power values. reflectedpower_mean is the mean of the reflected beams power values. reflectance_mean is the mean of the reflectance values in the different realizations. transmittance_mean is the mean of the transmittance values in the different realizations. maximumelevation_mean and minimumelevation_mean are respectively the mean of the maximum and minimum elevations in the different realizations. In briefly, the Part 4M is the mean of the second main for loop which is for the realization number Part 5M, Checking for Multiple Scattering The gravity wave ocean surface model is designed to analyze the effects of the gravity wave ocean surface to the laser propagation through air-ocean interface. In Part 3M, by calculating the properties of the daughter beams and averaging them over realization number in Part 4M, the directional and power distribution of the daughter beams are derived in the defined wind range. But it has to be checked that each beam interacts with one facet or if it interacts more than one, the effects of any multiple scattering have to be analyzed. The analysis of the Mobley [1] shows that the multiple scattering exist in different wind speeds with different emitting elevation angles for the capillary waves. In this part, the existence of multiple scattering is checked for the transmitter located 5 meter above the reference x y plane at zenith.

75 6 y Facet Which the Multiple Scattering Might be in the Extreme Case Facet A` Northwest Neighbor Facet Northeast Neighbor Facet x x Bisector South Neighbor Facet Facets Which the Multiple Scattering Might be in the Extreme Case Figure 3-3 Representation of the facet A` with its neighbor facets. As it is shown in the Figure 3-3, the facet A` has three neighbors and three points that multiple scattering might occur (yellow points in Figure 3-3). First of all, the neighbors are checked for multiple scattering and then three points are checked under Extreme Cases subtitle. When it is looked through reference x y plane from the zenith; the neighbor facet at the negative y and positive x region is named South Neighbor, the neighbor facet which intersects with y axis and exists at the positive y region is named Northwest Neighbor and the facet at positive x and y region is named Northeast Neighbor. a. In the first step, the neighbors which are going to be checked for multiple scattering have to be determined. In each realization, two neighbors are looked for possible multiple scattering, while one of them is related about the reflected beam, the other one is related with refracted beam. To find the related neighbors, angular sectors of the neighbors are defined. This calculation is done once for each wind speed

76 63 value so it is calculated just after the first main for loop which is for wind speed range and just before for loop of the realization number to avoid the unnecessary calculation. As it is shown in the Figure 3-3, first the bisector of the vertex at point n_1 is calculated and then the angular sectors of the neighbors are found. b. In the next step, the specifications of the neighbors are defined. To identify the daughter beams possible interaction with the neighbor facets, the normal unit vectors of the neighbor facets are calculated. The same method mentioned in part 3..3.g is used to find the unit normal vectors of the neighbor facets. n_normalsouth is the normal unit vector of the South neighbor facet, n_normalnorthwest is the normal unit vector of the Northwest neighbor facet and n_normalnortheast is the normal unit vector of the Northeast neighbor facet. c. The main idea of multiple scattering check depends on two steps which are elevation and slope check. For instance: Let s assume that the reflected beam s azimuth angle is in the South neighbor s sector and the refracted beam s azimuth angle is in the Northwest sector. While looking for multiple scattering in the reflection, if the vertex elevation of the South neighbor facet, n_south which is the uncommon point with facet A` is bigger than the elevation of the point A`, the next requirement is checked. But if the first condition cannot be met, thus multiple reflection is not the case for that specific realization because, if the elevation of the uncommon vertex s elevation is same or less than the elevation of point A`, even for the same elevation, the elevation angle of the reflected beam has to be zero or negative which means at least 45 degrees elevation angle or in another word 45 degrees wave slope is required. This is not a practical situation even for m/s wind speed in gravity waves. If the first requirement is met, the slope of the neighbor facet is checked. If the slope of the neighbor facet is bigger that the reflected beam s elevation angle according to the reference x y plane, there is a multiple reflections probability for this realization. The same process is checked for multiple refractions with reverse relations like; looking for smaller elevation of the uncommon vertex than the point A` and smaller slope of the neighbor facet than the elevation angle of the refracted beam (the angles and the slopes are negative in refraction case). In the whole process of the multiple scattering checks, hit or miss probability of the center rays in daughter beams to the neighbor facets is checked, the surface areas of the beams are ignored. This assumption is really fair when the surface area of

77 64 the beams and the facets are compared. The example given above about the multiple reflections check for the South neighbor is performed to all three neighbors for both multiple reflections and refractions according to the azimuth angle sector of the daughter beam in the specific realization. d. Beyond the three angular sectors in azimuth, three more possibilities exist. If the azimuth angle of the daughter beam is 9 degrees or in the negative value of the bisector or in the value of 18 o plus bisector (in degrees), the interaction possibility exists for the facets which are indicated in the Figure 3-3. The loop for angular sector search also looks for these three cases Part 6M, Realization of Hemisphere Distribution of the Reflected and Refracted Beams There are different hemisphere distribution types which perform different shapes of quads and polar caps with different resolutions. These different types can be seen in the topography. The chosen type for this thesis is very straight forward. The hemisphere is divided into equal angular diversity in both azimuth and elevation angles. According to how in detail the directional distribution is wanted to be analyzed, the resolution also can be adjusted. Mobley [1] used a similar type yet he preferred to use a polar cap. Using a polar cap makes you to see the top of the hemisphere in more general vision rather than in detail. Mobley chose this way and maybe he thought that there is no need to divide the top of the hemisphere anymore in horizontal but with this choice he decided the size of the polar cap and this becomes a little bit arbitrary choice. In the thesis, the imaginary sphere formed around the centroid of the triad A` is divided into 4 quads which have equal azimuth and elevation angle ranges. This means that, the both upper and lower hemispheres are divided into horizontal sectors with 18 degrees ranges and 1 vertical sectors with 9 degrees ranges. At the hemisphere distribution representation used in this thesis, it is assumed that each beam s cross-section surface hits only one quad. By this assumption, only the paths of the center rays of the daughter beams are analyzed as it is done in the previous parts.

78 65 The gravity wave ocean surface model designed in the thesis is not a non numerical model so the size of the hemisphere has to be also defined rather than using a unit hemisphere distribution. Even the propagation of the beam under the water is not included in the model, to see the directional and also power distribution of the daughter beams, the radiuses of the upper and lower hemisphere which are respectively for reflected beam and refracted beam distribution are chosen 5 meter. It is the same value with the altitude of the laser transmitter which might help to analyze the reflected beams effects on the transmitter in the next researches. Part 6M works for only one wind speed, not for a wind speed range. The aim for doing in this way is to get a directional distribution in both azimuth and elevation angles rather than a getting just one mean value. As it is indicated above, there are two hemispheres for reflection and refraction distributions. The both hemispheres origins are centered at centroid of triad A`. Fixing the origin of the hemisphere is good for dealing with daughter beams which reflect/refract from surfaces with different elevations yet on the other side, the beams do not come out from the origin of the hemisphere (Figure 3-4) which cannot be prevented in such a representation formed by the result of different surface realizations. In the hemisphere representation, the shifted origins of the daughter beams have to be calculated which cause some extra calculation for each realization. Part 6M is divided into 4 parts which are: Upper hemisphere (Distribution of the reflected beams) and elevation of point A is negative, lower hemisphere (Distribution of the transmitted beams) and elevation of point A is negative, lower hemisphere and elevation of point A is positive and upper hemisphere and elevation of point A is positive. Only the part which is including the conditions; upper hemisphere and point A with positive elevation is told in detail below, the rest three conditions have same principle with some small calculation differences.

79 66 Hit Point of the Reflected Beam to the Upper Hemisphere Incident Beam H ReflectedHsphereDistance Reflected Beam R Unit Normal Vector of the Facet A` PHI_incident Facet A` PHI_reflected Raised Reference x Plane y PHI_upper PHIUpperHsphere HA`O A` Point A` h_a Centroid of the Triad A` Reference x y Plane O Figure 3-4 Representation of the upper hemisphere distribution of a reflected beam. In the hemisphere distribution, two specific information of the daughter beams are needed to be distributed to the related quads; the elevation angle named PHIUpperHsphere (Figure 3-4) and azimuth angle of the daughter beam. The azimuth angle information is same as with THETA_reflected/THETA_uw, so this information is used for determining the quad column horizontally yet on the other side, the elevation angle information of the daughter beam cannot be used directly because, PHI_reflected/PHI_uw is the angle measured at the point A` and to determine vertically which quad is corresponding to the hit point of the daughter beam s center ray at the hemisphere s inner surface, the elevation angle PHIUpperHsphere has to be calculated. The details with the variable names used in the MATLAB code are indicated in the Figure 3-4. a. First HA`O, which is shown in Figure 3-4, is calculated (9 o - PHI_reflected ). Now in the triangular HA`O, the length of two sides, which are R and h_a, and one angle value, HA`O are known

80 67 and with using Cosine Law, which is shown at equation (3.3), ReflectedHsphereDistance is calculated and then with using Cosine Law again; first, the elevation angle PHI_upper and lastly, PHIUpperHsphere are computed. a b c bc cos (3.3) b. At this point, the quad information which is corresponding to the elevation angle, PHIUpperHsphere and azimuth angle, THETA_reflected is calculated with using the row and column resolution information of the hemisphere which are determined in Part 1M. And then the power of the reflected beam ( P_reflected ) corresponding to the related quad is added to that quad s power information and the number of the beams that hit on the related quad is also increased. This process is repeated for each realization in set M to find the reflected beams distribution in the case that the elevation of the point A` is positive. The last point about this part is about the dimension check between the quad surface area and the surface cross-section area of the reflected beam on the quad. The surface areas of the quads have to be big enough to fully cover the spread cross-section surface areas of the daughter beams. To check this condition, the smallest quad and the most spread beam are used. The minimum vertex elevation found in the model is m when the wind speed range is m/s and this elevation value is also used for the elevation of point A` to check the most extreme situation. The maximum elevation angle of the reflected beam from the reference x y plane is o degree which corresponds to the closest reflected beam to the narrowest part of the smallest quad and the longest propagation distance for the reflected beam. With using these two extreme conditions, the propagation distance of the reflected beam s center ray from point A` is calculated 6.44 m by the method explained above ( ReflectedHsphereDistance ). To calculate the beam radius of the reflected beam when it hits the hemisphere inner surface, one further step of part.6. has to be done which is

81 68 adding the propagation of the reflected beam from point A` to the point where the reflected beam s center ray hits the inner surface of the hemisphere. M 1 p1 p M p1 w ( z) r 1 z 1 z, M z 1 z1 1 z1 z z z z n z z 1 After doing the calculations above, the radius of the reflected beam at the inner surface of the hemisphere is calculated.77 cm with the specifications told above and the surface cross-section area of this beam is m 4. On the other side, the area of the smallest quad is found for the partial surface area of the sphere. m by equation (3.4) which is sphere 171 A r sindd (3.4) The ratio between the surface cross-section area of the reflected beam and the area of the smallest quad is which is approved the validity of the chosen hemisphere resolution value. Briefly, Part 6M computes the directional beam number and power distributions of the daughter beams for realization set M in a specific wind speed value. Up to here, the model is explained like one whole structure which is executing M times realizations for the wind speed range U 1 m / s except Part 6M yet to analyze the effects of the each wind speed on the laser beam and get the results of the part 6M, model is run individually for each wind speed value as it is said above. In the rest of the thesis, this type of analysis is called with the singular wind speed set.

82 The Analysis to Determine the Resolution Values, a and b The scale of gravity wave ocean surface model is based on the max, which is also based on wind speed, and resolution values of the triad dimensions. The first parameter is explained in the previous parts and this part focuses on the chosen resolution values for the thesis. As it is indicated in part.5., the dimensions of the defined finite area are X Y which can include at least two gravity waves with the wavelength max along x axis and at least four gravity waves along y axis. According to the approach told in chapter 1 about the specular surface, the defined finite area has to be divided into small enough surfaces, which are modeled in triad form, to be assumed as specular surfaces. The indicative parameters to define the size of the triads are resolution values a and b which are used to divide the defined finite area into the specified lengths and these lengths are used to form the triads. Mobley, whose approach is followed in the thesis, chooses 4 for a and b values to divide a finite area which is bigger than the twice of the finite area used in the thesis, so the resolution value that he used corresponds to 1 for the model used in this thesis. If the aim is forming triads with as small as possible surface areas, the resolution values have to be as big as possible but there are some limits about the resolution values. The first limit is the area of the facet which has to be big enough that the incident beam s surface area has to be smaller than it. To check this limit, the smallest finite area, which formed by the smallest wind speed 1 m/s, is picked. The max for 1 m/s is calculated.9611 m by using equation (.4). in the rest of the resolution value analysis, the same approach used in part.5.1 to define the minimum wavelength that can be modeled with beam tracing in capillary waves is followed. The area of the triad is referenced to define the resolution value. Thus, rather than calculating the beam radius at the facet A`, the beam radius at the triad is used which found.41 cm in part.5.1. To get a triad s area which is minimum equal and also bigger than the cross-section surface area of the incident beam, the calculations which are done in part.5.1 are repeated with using equation (.35). The only difference from the calculations done for the capillary waves, the ratio

83 7 between the height and width of the triads used for the gravity waves is 1 which is.9 in capillary waves. Ultimately, the minimum triad, which meets the first requirement, is found 1.3 cm and with using the relations defined in part.5., the maximum resolution values, ab, are calculated 144. But the triads formed with resolution value 144 don t meet the gravity waves lower wavelength limit (1.7 cm) when the wind speed is 1 m/s. When the minimum wavelength of the gravity waves is used to calculate the resolution values for 1 m/s with the same process explained above, the maximum resolution values, ab, are found 113. The limitation about the resolution value 144 is not valid when the wind speed is increase to m/s, because the max increases with increasing wind speed and the width of the triad becomes.67 cm for the resolution value 144 at m/s wind speed and it is obvious that this time the resolution value 144 meets the lower limit of the gravity wave s wavelength. In the rest of the resolution value analysis, the resolution value is increased by doubling the 1 up to 144. The resolution value 144 is chosen as the top resolution value in the resolution value analysis rather than 113 because it is aimed to use the top resolution value as high as possible to see the effects of the increasing resolution value on the model. The second limitation is computational. The model is written as a serial type MATLAB code whose speed is affected with the size of the for loops. The resolution values are the most used boundaries of the for loops which defines the computational time of the for loops. The code is performed with various resolution values and the analysis about the computational weight of this resolution values is explained in the Table 3-1. The analysis in the Table 3-1 is done with MATLAB profiler function. The model is run for one realization and one speed value. Repeat number column represents the repeat number of the main block for calculating the elevation distribution of the vertices of the triads, the reason to choose this block is the majority of its partial time to be executed. The values in the brackets at the same column are the ratios according to the resolution value 6. Total run time column represents the total run time of the whole model for one realization, the values are unitless because the total runtime is dependent on many parameters like the speed of the computer where the model is run and type of the programming language.

84 71 Resolution value a b Table 3-1 Computational analysis of the resolution values ab, Repeat Number Total Run Time Main Block Percentage (1) (14.) 1.9 % (14.19) 1.35 % (333.34) % ( ) % ( ) %99.3 Additionally, the result of the MATLAB profiler function is not constant and it cannot be accepted as a certain analysis result, it only gives an idea about the code s total and partial efficiencies. For that reason, the total run time of the model is accepted 1 when ab, is equal to 6 and the rest of the realizations with different resolution values are scaled according to the first one. The last column shows the percentage of the main block, which is mentioned above, in the total run time. It is clear that, the increase in the repeat number and total run time are not linear with resolution values. From 6 to 96, each step causes almost 15 times calculations in the main block and total run time has almost same increase in the resolution values: 4, 48 and 96. Increasing the resolution values is a serious computational issue and when it is thought that this model is run M times for each singular wind speed, keeping the resolution value in an acceptable level is very reasonable choice. But on the other side the effects of the resolution values on the results have to be analyzed. In the figure below, this issue is analyzed with calculating the mean of the elevation angles of the incident beams with various resolution value when the M is equal to 1. In the analyses presented at this and the following part, the gravity wave ocean surface model is run individually for each wind speed value and at each run the same bc, coefficients set is used to see the effects of the variations in the resolution values and

85 7 realization number to the results without any additional change in the model. But this situation is not case in chapter 4, the presented results are derived with totally random bc, coefficients sets. 8 Angle (Degree) a=b=6 a=b=1 a=b=4 a=b=48 a=b=96 a=b= Wind Speed (m/s) Figure 3-5 Resolution analysis of the gravity wave ocean surface model with realization number, M 1. In the figure above, the plot of the mean of the elevation angles increases until the resolution value reaches 144 and there is a significant ripple in the results except the results with resolution values 96 and 144. As it is told in part.5., the gravity wave ocean surface model is formed with Fourier series and the effect of the resolution value on the Fourier series summation is in a high level. As the resolution value increases, the gravity wave ocean surface model gets close to the real ocean surface features and the elevation angle of the incident beam gets into continuous pattern as it is seen in the result with resolution values 96 and 144. Additionally, there is an increase at the incident beam s elevation angle with increasing resolution values because, in the small resolution values, the slope of the waves are smoothed by big triad surfaces but on the other side, when high resolution values are used, the small triads can preserve the details of the slopes in the

86 73 waves. But there is one more fact in the graph that has to be paid attention, the realization number is 1 which is a very low number for ensemble averaging approach because, the randomness of the b and c coefficients cannot be removed totally. In conclusion, even it seems that the results with resolution value 144 gives out a continuous pattern, the computational hardness and the loss of flexibility in the design are still problems in using it and for these reasons, it is better to decide resolution value at the end of the realization number analysis which is given in the next part. 3.4 The Analysis to Determine the Realization Number, M The second analysis done before getting the results with using the model is for determining the realization number. The main point about the realization number is providing the ensemble averaging approach with the enough realization number value. Even there is not any similar analytic limit like the first limit indicated in part 3.3, there is a computational limit in determining the realization number. The realization number is proportional with the model run time because it is the second outer for loop after the wind speed range and thus, the increase in the realization number affects the run time in the same ratio while the doubling the resolution values causes almost 15 times increase in the total run time. Even the realization number doesn t affect the model so seriously in computational, keeping the total run time in the reasonable levels is intended. In the Figure 3-6, Figure 3-7and Figure 3-8, the same analysis of Figure 3-5 is done with different resolution values and realization numbers.

87 74 Mean of Elevation Angle of the Incident Beams Measured from the Normal of Facet (5 Realization) 8 Angle (Degree) a=b=6 a=b=1 a=b=4 a=b=48 a=b=96 a=b= Wind Speed (m/s) Figure 3-6 Resolution analysis of the gravity wave ocean surface model with realization number, M 5. Mean of Elevation Angle of the Incident Beams Measured from the Normal of Facet (5 Realization) 8 Angle (Degree) a=b=6 a=b=1 a=b=4 a=b=48 a=b=96 a=b= Wind Speed (m/s) Figure 3-7 Resolution analysis of the gravity wave ocean surface model with realization number, M 5.

88 75 Angle (Degree) a=b=6 a=b=1 a=b=4 a=b=48 a=b=96 a=b= Wind Speed (m/s) Figure 3-8 Resolution analysis of the gravity wave ocean surface model with realization number, M 1 As it is seen from the figures Figure 3-6,Figure 3-7 and Figure 3-8, the gravity wave ocean surface model gives the results in the continuous pattern only when ab, is equal to 96 or 144 and this means that, the resolution values ab, have to equal or bigger than 96. Even the resolution value 144 gives the best results in the analysis above, resolution value 144 doesn t meet the lower wavelength limit for the gravity waves and on the other side the results of the resolution value 96 is very close to the resolution value 144. Ultimately, resolution value 96 is chosen for the model in this thesis. After deciding the resolution values ab,, realization number M has to be chosen. In the Figure 3-9, the performance of the model is checked with various M values when ab, is equal to 96.

89 76 7 Mean of Elevation Angle of the Incident Beams Measured from the Normal of Facet (1 Realization) 6 5 Angle (Degree) 4 3 M=1 M=5 M=5 M= Wind Speed (m/s) ab. Figure 3-9 Realization number analysis with resolution number values, 96 Until the realization M is increased to 5, the plot of the mean of the elevation angles keeps on increasing yet when M reaches to 1, the plot decreases a little bit. This decrease is evaluated that, the mean values of the elevation angles of the incident beams reach their real limit values because the averaging effect of additional 5 realizations smoothens the extreme angle values. M is chosen 1 for the model. In conclusion, to analyze the effect of the gravity wave ocean surface to the laser propagation, the gravity wave ocean surface model is run with following parameters: a b 96, M 1, U 1 m / s.

90 77 Equation Chapter 4 Section 1 4 CHAPTER 4 Results and Discussion At this chapter, first the results of the model described in chapter 3 are presented and the below of them, there are discussion parts about the given results. The results are divided into three groups which are the numerical part given below, graphical representation of the singular wind speed results and lastly the mean results of the realization sets. While some of the results show the effects of the gravity wave ocean surface to the laser propagation and some of them are checking the validity of the model. The initial parameters of the laser at 5 m right above the centroid of the triad A` are: Wavelength is 53 nm, power is 1 Watt, initial beam radius is.9 cm, initial Rayleigh range is 5m, the incident elevation angle is 9 o degree (from the reference x y plane) and polarization is p-polarized along the downwind direction. The specifications of the model is: The resolution values ab 96, the realization number M 1 and wind range is U 1 m/s. Laser preserve its initial parameters along the atmospheric propagation except the expanding in its form but when the laser beam reaches to the facet A`, also the other parameters of the laser beam starts to be affected. The first change is on the power of laser which is independent from the surface effect. This effect is told in part.6 which is about the truncation of the laser beam. The power of the beam kept on beam tracing is.8467 Watt. The spreading effect of the propagation on the laser beam makes the beam radius and beam s radius of the curvature of the spherical wave increase. The beam radius at the ocean surface varies from.38 cm to.46 cm according to the wind speed and the realization because; the formation of the wave defines the propagation distance of the laser and it is obvious that both the maximum and minimum beam radiuses at the ocean surface are observed when the wind speed is maximum, m/s. These values are not the projected beam radius values on the facet A`, they are the radiuses of the beams at the end of the propagation that the center ray of the incident beam hits the point A`. The average beam radius for the whole speeds is

91 78.41 cm which is corresponding to the beam radius at the reference x y plane, in another word; it is the spread beam radius after 5 meter propagation in the lossless atmosphere. The radius of the curvature of the spherical wave, Rz () varies versus realization and wind speed like the radius of the beam. A brief analysis about the mean radius values of the curvatures of the spherical waves is given Table 4-1 and Table 4-. The analysis is done in 1 realizations for the maximum, the minimum and the medium wind speeds and also it involves the incident beam s radius of the curvature of the spherical wave just before interacting with airocean interface and transmitted beam s two different radiuses of the curvature of the spherical wave which are calculated by using equations (.59) and (.6). Table 4-1 Mean values of the radiuses of the curvatures of the spherical waves for the incident and the transmitted beams at the point A` when the wind speed, U is 1, 1 and m/s. a b 96, M 1 Wind Speed RS () z (m) Rt () z (m) Rt () z (m) The maximum and the minimum radius of the curvatures of the spherical waves are observed when the wind speed is m/s. Table 4- The maximum and the minimum radius of the curvature of the spherical wave observed in the model. a b 96, M 1, U m / s Wind Speed= m/s RS () z (m) Rt () z (m) Rt () z (m) Maximum Minimum As it is explained 3..5, the model is checking for any possible multiple scattering yet multiple scattering is not case for the gravity wave ocean surface model with given specifications. This is an expected result because the incident elevation angles are very small due to the location of the transmitter. If the emitting angle of the laser beam is increased, possibly multiple scattering will be observed in the gravity wave ocean surface model.

92 79 Up to now, the theory, calculations and some output values about the laser beam and ocean surface are given and in the rest of the thesis, there are some visual results which give more idea about model s performance than the theoretical expressions or quantitative results. The Figure 4-1and Figure 4- are just given to satisfy the visual requirement about model s reality. The Figure 4-1 is realized for 5 m/s wind speed which might be accepted as a gentle/moderate breeze and the Figure 4- is realized for 15 m/s which might be accepted as a high wind.

93 8 Figure 4-1 Gravity wave ocean surface realization with wind speed, 5 U / m s and resolution values ab 96.