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Table Of Contents TABLE OF CONTENTS 1 CHAPTER ONE - INTRODUCTION 4 1.1Introduction 4 1.2Types of Fibers for the 1.2.1Solid core fibers IR 1.2.2Liquid core fibers 1.2.3Hollow waveguides 1.2.4Why 1.3The hollow waveguides Research Subject and Main Goal 1.4References for chapter one 6 6 16 16 24 25 27 CHAPTER TWO - ATTENUATION MECHANISMS IN HOLLOW WAVEGUIDES 32 List of Symbols For Chapter 2 32 2.1Reflection 34 from a thin film 2.2Scattering from rough surfaces 2.2.1The Rayleigh criterion 2.2.2The general solution for scattering from rough surface surface 2.2.3Scattering from a randomly rough 2.2.4The normally distributed surface 2.3Measurements of surface roughness 43 43 45 50 55 58 2.4Summary of chapter 2 62 2.5Refrences for chapter 2 63 CHAPTER THREE - RADIATION PROPAGATION THROUGH HOLLOW WAVEGUIDES 65 List of Symbols For Chapter 3 65 3.1Theory 3.1.1The 3.1.2The 67 67 71 74 mode approach ray model 3.1.3Ray propagation through straight waveguides 2 3.1.4Ray propagation through bent waveguide 3.2Experimental results and 3.2.1Experimental setups theoretical calculations 3.2.2The 3.2.3The attenuation of straight hollow waveguides attenuation of bent hollow waveguides 3.2.4The dependence of the attenuation on the waveguide’s radius 3.2.5The dependence of the attenuation on the coupling conditions 3.3Discussion 3.3.1Straight hollow waveguides 3.3.2Bent hollow waveguides 3.3.3The 3.3.4The dependence on the waveguide’s inner diameter influence of coupling conditions 82 87 87 90 95 102 104 110 110 111 112 113 3.4Conclusion 114 3.5Summary 114 of chapter 3 3.6References for chapter 3 116 CHAPTER FOUR - PULSE DISPERSION IN HOLLOW WAVEGUIDES 119 List of Symbols For Chapter 4 119 4.1Theory 4.1.1Pulse 4.1.2Pulse 120 120 121 dispersion in straight and smooth multi-mode fibers dispersion in real hollow waveguides 4.2Experimental results and discussion 4.2.1Experimental setup 4.2.2Experimental results and discussion 122 122 123 4.3Summary 127 4.4References for chapter four 129 CHAPTER FIVE - IMPROVING THE HOLLOW WAVEGUIDES 130 List of Symbols For Chapter 5 130 5.1Theory 5.1.1Bloch theory 5.1.2Maxwell equations a multilayer mirror for the infrared 135 135 143 147 results and discussion– multilayer mirror 149 5.1.3Designing 5.2Experimental 3 5.2.1Dielectric mirror 5.2.2Metal dielectric mirror 5.3Hollow waveguides made of multilayer films 150 152 160 5.4Conclusion 163 5.5Summary 163 5.6Refrences for chapter five 165 CHAPTER SIX – SUMMARY 167 6.1Summary 167 6.2Future 168 work 4 Chapter One - Introduction 1.1 Introduction In 1966 Kao and Hockham [1.1] described a new concept for a transmission medium. They suggested the possibility of information transmission by optical fibers. In 1970 scientists at Corning Inc. [1.2], fabricated silica optical fibers with a loss of 20 dB/km. This relatively low attenuation (at the time) encouraged scientists from around the world that perhaps optical communication could become a reality. A concentrated effort followed, and by the mid 1980s there were reports of low loss silica fibers that were close to the theoretical limit. Today, silica based fiber optics is a mature technology with major impacts in telecommunications, laser power transmission, sensors for medicine, industry, military, as well as other optical and electro optical systems. While silica based fibers exhibit excellent optical properties out to about 2m, other materials are required for transmission of longer wavelengths in the infrared (IR). These materials can be glassy, single crystalline and polycrystalline. Examples of such materials include fluoride and chalcogenide glasses, single crystalline sapphire, and polycrystalline silver halide. Depending on their composition these materials can transmit to beyond 20m. Consequently, optical fibers made from these materials enables numerous practical applications in the IR. For example, IR transmitting fibers can be used in medical applications such as for laser surgery and in industrial application such as metal cutting and machining using high power IR laser sources (e.g. Er:YAG, CO, CO2 lasers). 5 More recently, there has been considerable interest in using IR transmitting fibers in fiber optic chemical sensors systems for environmental pollution monitoring using absorption, evanescent, or diffused reflectance spectroscopy since practically all molecules posses characteristic vibration bands in the IR. Aside from chemical sensors, IR fibers can be used for magnetic field, current and acoustic sensing, thermal pyrometry, medical applications, IR imaging, IR countermeasures and laser threat warning systems. While low loss silica fibers are highly developed for transmission lines in telecommunications applications, the IR transmitting materials still have large attenuation and need to be improved. The materials described above are used to fabricate solid core fibers; however there is another class of fibers based on hollow waveguides, which has been investigated, primary for CO2 laser power transmission. These waveguides posses hollow core and are based on hollow tubes with internal metallic coating with or without a dielectric coating. These waveguides may be good candidates for transmitting infrared radiation. This chapter has two main sections. The first section of this chapter is a survey of the current research of IR fibers and waveguides. It describes each type of fiber shortly and at the end of the section compares the different types. The second section outlines the objective of this thesis. It describes the major contribution of my work to the field of hollow waveguides to the IR region. 6 1.2 Types of Fibers for the IR IR fibers can be used for many applications. In the last two decades IR materials and fibers were investigated intensively in order to produce commercial IR fibers for different applications. These fibers are divided into three categories; solid core fibers, liquid core fibers and hollow waveguides. Each category can be divided to subcategories see table 1.1. Subcategory Solid Core Glass Crystalline Liquid Core Hollow Waveguides Type Silica based Fluoride based Chalcogenide Single crystal Polycrystalline Fused silica Examples Na2O-CaO-SiO2 ZABLAN Sapphire AgBrCl, KRS-5 C2Cl4 Metal/ Dielectric Refractive index <1 Table 1.1 – Categories of IR Fibers 1.2.1 Solid core fibers Solid core fibers guide the laser radiation through total internal reflection. These fibers are made of different kinds of glasses, single crystalline materials and polycrystalline materials. 1.2.1.1 Glass fibers Silica and silica based fibers Silica based glass fibers [1.3] can be optically transparent from the near ultraviolet (NUV) to the mid-infrared (MIR) range of the electromagnetic spectrum. Optical fibers made from these glasses are widely used in the near infrared (NIR) at wavelengths close to the zero material dispersion (1310nm) 7 and minimum loss (1550nm) wavelengths of silica. Such fibers provide the backbone of modern optical telecommunication networks. Since the late 1970s these fibers has been manufactured routinely. It is possible to manufacture very long fibers with very low attenuation (0.2dB/km). Multicomponent glasses, specifically soda-lime silicate (Na2O-CaO-SiO2) and sodium borosilicate (Na2O-B2O3-SiO2) and related compositions, in which silica comprises less then 75% mol of the glass, were early candidates for optical communication fibers. Core cladding index differences were typically achieved by varying the concentration or type of the alkali in the respective glasses or by adding GeO2 to the core glass. Graded index profiles in the fiber could be tailored by using crucible designs, which permitted more, or less, interfacial contact and interdiffusion between the core and cladding glasses during the fiber draw. Up to the mid 1970’s significant efforts were made to fabricate low loss multicomponents telecommunication fibers. It was recognized that Rayleigh scattering in many multicomponent silicate glasses could be lower than in high silica content glasses and that the achievable losses were largely determined by extrinsic impurities. Many innovative approaches were tried to minimize the concentrations of these impurities. The efforts yielded fibers with losses as low as 3.4 dB/km at 840nm. Further loss reduction was dependent on reducing –OH contamination to sub parts per million and transition metals to low ppm levels. The intrinsically lower strength, reliability, and radiation hardness of these fibers also present significant obstacles for their practical utilization. 8 High silica content fibers are compositionally simple. In most instances, the cladding glass is 100% SiO2 while the core glass is 90 to 95% SiO2 with a few percent of dopants to increase the refractive index in order to achieve a guiding structure. The cladding glass in the vicinity of the core may also be doped to achieve specific refractive profiles. These fibers are sensitive to moisture. Given the opportunity, moisture can diffuse from the surface to the core of the fiber with an attendant increase in attenuation at communication wavelengths due to overtone and combination absorptions of OH vibration. Exposure to ionizing radiation can produce defect centers in fibers, which also contributes to optical loss. Natural radiation, which is approximately 0.1 to 1 rad/year can be sufficient to produce significant degradation over system lifetime. For practical purposes the strength of a fiber should be sufficient to withstand initial handling stresses including those generated in cabling and deployment. Furthermore this strength should not degrade during the system lifetime. Fluoride glass based fibers Fluoride glasses based on ZrF4 [1.3] are predicted to have minimum optical loss of less than 0.01 dB/km, which is more than an order of magnitude lower than the 0.12 dB/km predicted and practically realized for silica fibers. This phenomenon is related to fact that these are low phonon frequency glasses and, hence, the multiphonon energy is shifted to longer wavelengths. In addition, fluoride glasses possess low non-linear refractive indices and, in some cases, a 9 negative thermo optic coefficient (dn/dT). Furthermore, these glasses are excellent hosts for rare earth elements. As a result, there are many applications for optical fibers, such as low loss repeater-less links for long distance telecommunications, fiber lasers and amplifiers, as well as infrared laser power delivery. More recently, there has been interest in using fluoride fibers in remote fiber optic chemical sensor systems for environmental monitoring, using diffuse reflectance and absorption spectroscopy. Historically, the first fluoride glasses were based on beryllium and have been known since the 1920’s. These glasses are very stable and have many unique properties including good UV transparency, low index of refraction, low optical dispersion and low non-linear refractive index. However, the combination of high toxicity, volatility, and hygroscopic nature of BeF2 poses serious problems in melting, forming, handling, and disposal of these glasses. These glasses were investigated primarily because of their high resistance to damage. A second kind of HMF glasses is fluorozirconate glasses. They are so named since the major component is ZrF4. The glass formed is ZrF4-BaF2 (ZB). The main disadvantage of these glasses is their crystallization instability. It is possible to add more metal fluoride to facilitate stable glasses. For example the addition of ThF4 and LaF3 form ZBT and ZBL. Unfortunately these glasses are not stable enough for practical uses. In addition Th is radioactive. It has been demonstrated that the addition of a few percent of AlF3 to fluorzirconate glasses greatly enhances the glass stability. The glasses ZrF4- 10 BaF2-LaF3-AlF3 (ZBLA) and ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAN) are very stable and may be used to fabricate fibers. All fluoride glasses have excellent optical properties and transmit more than 50% between 0.25m and 7m. While fluoroaluminate glasses do not transmit as far as fluorozirconate glasses, non-ZrF4 based on heavy metal can transmit to even longer wavelengths. As a result, fluoride glasses are candidate materials for both bulk windows and fibers for the IR. The ultra low loss application for fluoride glass fibers have not been realized because of problems associated with microcrystallization, crucible contamination and bubble formation during fiber fabrication. Nevertheless, fibers with losses greater then than 10dB/km are routinely obtained and can be used for chemical sensor application as well as high power UV, visible and IR laser transmission. Chalcogenide glass based fibers Chalcogenide compounds [1.3] of some elements belonging to groups 4B and 5B in the periodic table exhibit excellent glass forming ability. Based on the wide infrared transmission range of the As2S3 glass, various glass compositions have been developed as optical component materials for the 3 to 5m and 8 to 12m bands. Chalcogenide glasses can be classified into three groups: sulfide, selenide and telluride. Glass formation can be achieved when chalcogen elements are melted and quenched in evacuated silica glass ampoules with one or more elements, such as As, Ge, P, Sb, Ga, Al, Si, etc. The properties of the glasses change 11 drastically with glass composition. For example while some sulfide glasses are transparent in the visible wavelength region, the transmission range of selenide and telluride glasses shift to the IR region with increasing the contents of selenium and tellurium. The mechanical strength and thermal and chemical stabilities of chalcogenide glasses, which are typically lower than oxide glasses, are sufficient for practical fiber applications. The attenuation of chalcogenide fibers depends on the glass compound and the fiber’s drawing technique. Typical attenuation of sulfide glass fibers is 0.3 dB/m at 2.4m. The attenuation of selenide fibers is 10dB/m at 10.6m and 0.5dB/m at 5m. The attenuation of telluride fibers is 1.5dB/m at 10.6m and 4dB/m at 7m. Chalcogenide glass fibers may be used for temperature monitoring, thermal imaging, chemical sensing and laser power delivery. The main disadvantage of these fibers is the poisonous nature of some of the materials they are made of, a thing that makes them hard to handle and unsuitable for medical application. 1.2.1.2 Crystalline Fibers About 80 crystals are listed as IR optical materials [1.3]. Many of these materials have similar IR transmission characteristics, but may possess very different physical properties. Table 1.2 lists the optical properties of several crystals. While low transmission loss is usually a prime consideration, mechanical strength cannot be overlooked. Fibers that tend to be brittle will be difficult to bend and therefore lose much of their attractiveness as optical fibers. 12 Crystal Si Ge Al2O3 BaF2 CaF2 CsBr CsI CuCl MgF2 KCl AgBr KRS-5 TlCl ZrO2-Y2O3 Transmission Range (m) 1.2-7 1.8-23 0.15-6.5 0.14-15 0.13-10 0.21-50 0.25-60 0.19-30 0.11-9 0.2-24 0.45-35 0.6-40 0.4-30 0.35-7 Refractive Index 3.426 4 1.7 1.45 1.4 1.662 1.739 1.88 1.337 1.457 2 2.36 2.193 2.009 Table 1.2 -IR Materials Single crystal fibers Unlike glass fibers, which are pulled at high speed from a heated preform, single crystal fibers [1.3] have to be grown at much slower rate from a melt. Long distance transmission using crystalline fibers is therefore not practical. Instead, the early development of crystalline IR transmitting fibers was driven primarily by the interest in fibers with good transmission at the 10.6m wavelength of the CO2 laser. Such fibers could deliver laser power to targets for surgery, machining, welding and heat treatment. Excellent IR optical materials, such as the halides of alkali metals, silver and thallium, were considered as promising candidates for fiber development. More recently, solid-state lasers with output near 3m have emerged as excellent medical lasers because of the strong water absorption at that wavelength in human tissues. Currently silica based fibers do not transmit at that wavelength. Fluoride glass fibers, on the other hand, have excellent 13 transmission in the 2-3m region, but their chemical stability in wet environment is a problem. Therefore, single crystal fibers that are free from the above constraints and that can handle high laser power are sought for new medical lasers, and sapphire fiber is a prime candidate for fiber delivery of Er:YAG laser. Besides applications in optical fiber beam delivery, single crystal fibers also find potential use in fiber-based sensors. In applications where sensor must operate in harsh environment, the optical property of fiber materials is not the only consideration. High melting temperature, chemical inertness, and mechanical strength often dictate the choice for fiber materials. Sapphire is one example of a single crystal that possesses an unusual combination of these properties. The main advantage of single crystal fiber is the purity of the material. Fibers made from very pure crystals have low transmission loss due to low absorption and scattering losses. These fibers have some disadvantages. They are hard to manufacture, it is hard to fabricate long fibers, and some of them are brittle or made of toxic materials. Sapphire fibers Sapphire fibers have good physical properties. Sapphire is a strong material, it is chemically stable, it is not soluble in water, it has high melting temperature and it is biocompatible. Sapphire fibers are transparent up to 3.5m [1.4,1.5]. 14 Sapphire fibers are grown from a melt at rates up to 1mm/min [1.6]. They are made as core only fibers (without cladding) with diameters in the range of 0.18mm to 0.55mm. Their bending radius depends on their diameter and ranges between 2.8cm to 5.2cm. Sapphire fibers have attenuation of 2dB/m at 2.93m and maximum power delivery of 600mJ. AgBr Fibers Bridges et al. [1.7] fabricated AgBr fibers using a pulling method. In this method a reservoir of single crystal material is heated and the material is then pulled through a nozzle to form the fiber. Using this method they fabricated fibers up to 2m with diameters between 0.35mm to 0.75mm at a rate 2cm/min. These fibers are able to transmit up to 4W of CO2 laser at 10.6m without any damage and attenuation of about 1dB/m. 1.2.1.3 Polycrystalline fibers The first to propose the fabrication of infrared fibers from crystalline materials (AgCl) was Kapany [1.8] in the mid 1960s. However it took another decade until the fabrication of polycrystalline infrared fibers was reported by Pinnow et al.[1.9]. These authors fabricated fibers made of thallium halide crystals, TlBr-TlI (KRS-5). The motivation for fabricating fibers from heavy metal halides was to realize predictions of ultra low attenuation of these materials in the IR. 15 The first attempts to fabricate IR fibers from KRS-5 , KCl, NaCl, CsI, KBr, and AgCl [1.3] resulted in optical losses of two orders of magnitude higher than those of the original crystals. Much of the work over the years on these fibers was concentrated on finding the origin of the fibers loss and improving the fabrication process. Besides the low theoretical optical loss of polycrystalline materials, there are some practical requirements from these materials. First, the crystal must be deformed plastically in a typical temperature range with speeds higher than 1cm/min. This requirement is needed in order to manufacture a long fiber in a reasonable amount of time. Second, the crystal must be optically isotropic, due crystallographic reorientation. Therefore it must posses a cubic crystal structure. Third, the composition of the crystal must be of solid solution. Finally, the recrystallization process of the materials must be a one that does not cause the degradation of the optical material. From this point of view the suitable materials are thallium halides, silver halides and alkali halides. Thallium Halides Thallium halides fibers made of TlBr-TlI (KRS-5) are fabricated using the extrusion method [1.10]. These fibers have a low theoretical attenuation about 6.5dB/km. However due to material impurities and scattering the achieved attenuation is much lower, in the range of 120-350dB/km. The scattering effects [1.11] in these fibers are caused by three factors. The first is scattering by surface imperfections, the 16 second is due to residual strains and the last is due to the grain boundaries and dislocation lines. Furthermore these fiber have additional drawbacks [1.12] such as aging effects, sensitivity to UV light and solubility in water. Silver Halides Polycrystalline silver halides fibers are made of AgClxBr1-x by Kaztir et al. in Tel-Aviv University [1.13, 1.14]. These waveguides are manufactured using the extrusion method. The optical losses of these waveguides are very low, 0.15dB/m at 10.6m. The attenuation is caused by bulk scattering, absorptions in material defects and extrinsic absorption. 1.2.2 Liquid core fibers Liquid core fibers are hollow silica tubes filled with liquid which is transparent in the infrared [1.17]. The measured attenuation of a liquid core fiber filled with C2Cl4 is very high (about 100dB/m) for 3.39m. fibers with liquid core made of CCl4 have attenuation of about 4dB/m at 2.94m. The only potential advantage of these fibers over solid core fibers is that it is easier to manufacture a clear liquid. Hence there are less losses due to scattering. 1.2.3 Hollow waveguides The concept of using hollow pipes to guide electromagnetic waves was first described by Rayleigh in 1897 [1.3]. Further understanding of hollow waveguides was delayed until the 1930s when microwaves generating 17 equipment was first developed and hollow waveguides for these wavelengths were constructed. The success of these waveguides inspired researchers to develop hollow waveguides to the IR region. Initially these waveguides were developed for medical uses especially high power laser delivery. But more recently they have been used to transmit incoherent light for broad band spectroscopic and radiometric applications [1.15, 1.16]. Hollow waveguides present an attractive alternative to other types of IR fibers. They can transmit wavelength at a large interval (well beyond 20m); their inherent advantage of having an air core enable them to transmit high laser power (2.7kW [1.16]) without any damage to the waveguide. Moreover they have a relatively simple structure and low cost. However, these waveguides also have some disadvantages. They have a large attenuation when bent; their NA (numerical aperture) is small and they are very sensitive to the coupling conditions of the laser beam. Hollow waveguides may be divided into two categories. The first include waveguides whose inner wall materials have refractive indices greater than one. These waveguides are also known as leaky waveguides. Thus the guidance is done via total reflection. The second includes waveguides whose inner wall has a refractive index smaller than one. These are known as attenuated total reflection (ATR) waveguides and the guidance mechanism is similar to solid core fibers. 18 1.2.3.1 Leaky Waveguides Leaky waveguides are made of hollow tubes that are internally coated by thin metal and dielectric films. The choice of the hollow tube and the thin layers materials depends on the application and coating technique. Table 1.3 summarizes the different kinds of materials used to manufacture hollow leaky waveguides. These waveguides may have a circular or rectangular cross section. Tube Teflon, Poly-Imide, fused silica, glass Copper, Stainless steel Nickel, Plastic, Glass Plastic, Glass Metal Layer Ag Dielectric Layer AgI Cu CuO ,ZnS, ZnSe Ag, Ni, Al Si, ZnS, Ge Ag AgI Silver Ag AgBr Stainless steel Plastic Ag Ag, Al, Au PbF2 ZnSe, PbF4 Table 1.3 – Types of hollow waveguide Group Croitoru et al. [1.18] Croitoru et al. [1.20] Miyagi et al. [1.25] Harrington et al. [1.28] Morrow et al. [1.29] Luxar et al. [1.31] Laakman et al. [1.31] 19 Croitoru et al. This group from Tel-Aviv University [1.18-1.22] develops hollow waveguides for a large wavelengths. waveguides range The are of Figure 1.1 Cross Section of a Hollow Waveguide hollow made of Black – tube Gray – metal layer Light gray – dielectric layer different kind of tubes, mainly fused silica, Teflon and polyimide. The deposited thin using films a are patented electro less method in which solutions containing Ag are passed through the tubes and coat it. The dielectric layer is built by iodination of some of the Ag layer to create a AgI layer. Figure 1.1 shows a cross section of the waveguide. The choice of the tube depends on the application and is an optimization among the desirable characteristics such as flexibility, roughness and chemical inertness. Fused silica tubes are very smooth [1.23] and flexible (at small bore radius). However their thermal conductivity is very poor. Plastic tubes are very flexible. But they also have a large surface roughness and are damaged easily at high temperatures. In the past few years there has been an attempt to develop hollow waveguides using metal tubes. Croitoru et al. have made hollow waveguides for different wavelengths. However mainly waveguides for CO2 lasers at 10.6m and Ee:YAG lasers at 2.94m were manufactured, since these wavelengths are used in many medical applications. Table 1.4 summarizes the characteristics of these waveguides. 20 Inner Diameter [mm] Attenuation [dB/m] Maximum input power Mode of operation NA =10.6m Fused Silica Teflon 0.7 1 0.2-0.5 0.2-0.75 50 W 80 W CW, Pulse CW, Pulse 0.0338 0.1563 =2.94m Fused Silica Teflon 0.7 1 0.8 1.1 900mJ 1650mJ Pulse Pulse 0.0475 - Table 1.4 – Characteristics of Hollow Waveguides Made by Croitoru et al. The group also investigated hollow waveguides for Ho:YAG laser at 2.1m that transmit up to 1200mJ and have attenuation of 1.5dB/m, waveguides for CO2 laser at 9.6m that have attenuation of 2.8dB/m and waveguides for free electron lasers at 6-7m that have attenuation of 1.8-4 dB/m [1.24]. Miyagi et al. Miyagi and his co-workers [1.25-1.26] have pioneered the development and theoretical characterization of hollow waveguides. They used a three steps method to fabricate the waveguides. In the first step a metal rod, usually made of aluminum, was placed in a sputtering chamber where it was coated by dielectric and metal thin films. Next, the coated pipe was placed in an electroplating tank where a thick metal layer was deposited on top of the sputtered layers. Finally, the metal rod was etched leaving a hollow waveguide structure similar to the one in figure 1.1. Miyagi et al. fabricated hollow waveguides for different wavelengths. They measured about 0.3dB/m for hollow waveguides optimized for CO 2 lasers at 10.6m and about 0.5dB/m for Er:YAG lasers at 2.94m. Recently this group has made hollow waveguides with dielectric thin films made of plastic materials [1.27]. 21 Harrington et al. Harrington’s group [1.28] has made hollow waveguides similar to those made by Croitoru et al. They deposit thin Ag and AgI layers using wet chemistry method similar to that of Croitoru and co-workers. Using this method waveguides of bore size between 0.25mm to 0.75mm and up to 13m long have been made. The measured attenuation of a 0.5mm bore size waveguides is about 0.5dB/m for CO2 lasers at 10.6m and about 1dB/m for Er:YAG lasers at 2.94m. Morrow et al. The waveguides developed by Morrow and coworkers [1.29] are constructed from silver tubes. The tube is made of silver and there is no need to deposit a metal layer inside the tube. The first step is to etch the bore of the tube in order to smoothen it. Then a dielectric layer (AgBr) is deposited using a wet chemistry method. The attenuation of a 1mm bore hollow waveguide for CO2 lasers at 10.6m is less than 0.2dB. However the beam shape quality of this waveguide is very poor due to mode mixing which is caused by the surface roughness of the silver tube. Laakman et al. and Luxar et al. Laakman et al. [1.30] have developed first a hollow waveguide with a rectangular cross section. They used Ag, Au or Al as a metal layer and ZnSe or ThF4 as a dielectric layer. Such waveguides had attenuation of about 1.7dB/m. Later, the group begun to fabricate hollow waveguides with circular cross section. Their fabrication technique involved depositing a Ag film on a metal 22 sheet and then over coating it with a dielectric layer, PbF2 [1.31]. The same method was used by Luxar et. al as well. The attenuation of such waveguides with bore size of 0.75mm was about 0.5dB/m for CO2 lasers at 10.6m. Garmire et al. and Kubo et al. Garmire et al. [1.32] developed hollow waveguides with rectangular cross sections. They use polished metal strips made of Au or Al separated by brass spacers 0.25mm to 0.55mm thick. The distance between the spacers was several millimeters. The measured attenuation of such a hollow waveguide with cross section of 0.5mm by 7mm was about 0.2dB/m. The maximum power that was transmitted through it was 940W. Kubo et al. [1.33] fabricated similar hollow waveguides. They used Al strips and Teflon spacers. Such a waveguide with a cross section of 0.5mm by 8mm transmitted laser power of 30W and the measured attenuation was about 1dB/m. These waveguides had several advantages. They had small attenuation, they were made of cheap and common materials and they were able to transmit high laser power for a long time without damaging the waveguides. However they also had some drawbacks. Their size made them inapplicable for medical applications and for several industrial ones. There was a need for two focusing lenses in order to couple the laser beam into the waveguide, and they had a large attenuation when bent. 23 1.2.3.2 Attenuated Total Reflection (ATR) Hollow Waveguides Hollow waveguides with n<1 were suggested by Hidaka et al. [1.34] in 1981. In these waveguides the air core (n=1) has a refractive index greater than the inner walls, therefore the guiding mechanism is the same as in regular core clad fibers. To be useful the ATR must have an anomalous dispersion in the region of the laser wavelength. Haikida et al. made ATR hollow waveguides for CO2 lasers. These waveguides were made of glass tubes with bore size of 0.6mm to 1mm, which are made of lead and germanium doped silicates. By adding heavy ions to the silica glass, it was possible to create the anomalous needed for CO2 laser guiding. Such waveguides were able to transmit laser power of up to 20W and attenuation of 5.5dB/m [1.35]. Another kind of ATR hollow waveguides was made of chalchogenide glass [1.36]. These waveguides had a bore size of 0.1mm to 0.3mm and were several meters in length. The attenuation of these waveguides was very high (5dB/m). Gregory et al. [1.37] have developed ATR waveguides made of sapphire. The attenuation of these waveguides was much lower than that of the previous one, 1.5dB/m and they were able to transmit about 2KW of laser power [1.38]. 24 1.2.4 Why hollow waveguides The following table summarizes the advantages and drawbacks of each type of fiber. Fiber type Glass Advantages Known and established manufacturi ng process Can be made with very small diameter Single crystal High melting point (sapphire). Low theoretical attenuation. Polycrystalline Low theoretical attenuation. Drawbacks Brittle Some of the materials are toxic. Impurities cause scattering High attenuation Moisture sensitive Ionization radiation sensitive Some of the materials are sensitive to UV radiation. Some of the materials are toxic. Hard to manufacture. Brittle Soluble in water. Impurities cause scattering Some of the materials are sensitive to UV radiation. Some of the materials are poisonous. Hard to manufacture. Brittle Soluble in water. Impurities 25 Liquid core Non toxic Hollow waveguides Able to transmit high power Low attenuation Non toxic materials Easy to manufacture Support a wide range of wavelengths cause scattering The fibers are brittle Large attenuation Sensitive to temperature beyond a certain value. Some of the tube have large surface roughness. Hard to manufacture long fibers Surface roughness causes scattering Table 1.5 – Advantages and Drawbacks of Infrared Fibers As can be seen from the table, hollow waveguides are good candidates for transmitting infrared radiation. They support a wide range of wavelengths from the x ray region [1.39] through the visible and to the IR. This characteristic makes them able to transmit a wide selection of lasers for many applications. Hollow waveguides may deliver very high power. This characteristic makes them usable for industrial applications such as cutting metals. In addition they are non-toxic which makes them very suitable for medical applications. Moreover they may also be used for military applications such counter measure detection, and for civilian applications such as imaging, spectrometry, thermometry and signal delivering. 1.3 The Research Subject and Main Goal 26 The purpose of this research is to develop a new type of hollow waveguides and to improve the hollow waveguides that are made by Croitoru et al. These goals will be achieved by a theoretical and experimental study of laser propagation through hollow waveguides. The development of a new type of hollow waveguides will be based on the notion of photonic crystals and relating it to hollow waveguides. The theoretical study will be based on the development of an improved ray model. Although many ray models have been suggested over the years, none of them could accurately predict the transmission of laser radiation through hollow waveguides. I have been successful in introducing the effect of surface roughness into the ray model. This has enabled me to determine the influence of the waveguide’s physical parameters (length, inner diameter, roughness), the coupling conditions (focal length of the coupling lens, off center coupling) and the shape of the laser beam at the entrance to the waveguide on its transmission and energy distribution at the distal end. This ray model can also determine the pulse dispersion caused by the waveguide. In the future, the introduction of different coupling devices and tips can be included as well. The experimental study is based on the measurement of the waveguide’s transmission and laser beam shape as a function of the above parameters. The experimental setup consists of a light source (laser, monochromator), a detector (power/ energy meter, beam analyzer) and several types of hollow waveguides. The experimental setup for the pulse dispersion measurement consists of a Qswitched Er:YAG laser, fast detector, oscilloscope and a hollow waveguide. 27 One of the ways to improve the current hollow waveguides and to develop new types of hollow waveguides is by using photonic crystals structures. Photonic crystals are periodic structure of dielectric materials that introduce energy bandgaps for photons. These bandgaps are similar to those introduced to electrons in crystals with periodic potentials. Such bandgaps cause radiation with specific wavelengths and at any incidence angle to fully reflect (R=1) from the photonic crystal structure. Introducing such a structure to hollow waveguides will enable us to improve their performance and enlarge their usage possibility. Although periodic dielectric structures have been investigated for many years, I am suggesting to use a metal and dielectric structure that may consist of fewer number of layers. 1.4 References for chapter one 1.1 C. K. Kao and G. A. Hockham, Proc. IEE, 133, 1158, (1966). 1.2 F. P. Karpon, D. B. Keck and R. D. Maurer, Appl. Phys. Lett., 17, 423, (1970). 1.3 J. S. Sanghera and I.D. Aggarwal “Infrared Fibers Optics”, CRC Press 1998. 1.4 G. N. Merberg, J. A. Harrington, “Optical and Mechanical Properties of Single-Crystal Sapphire Optical Fibers”, Applied Optics 1993, v. 32, p. 3201.] 1.5 G. N. Merberg, “Current Status of Infrared Fiber Optics for Medical Laser Power Delivery”, Lasers in Surgery and Medicine 1993, p. 572. 28 1.6 R. W. Waynant, S. Oshry, M. Fink, “Infrared Measurements of Sapphire Fibers for Medical Applications”, Applied Optics 1993, v. 32, p. 390. 1.7 T. J. Bridges, J.S. Hasiak and A. R. Strand, “Single Crystal AgBr Infrared Optical Fibers”, Optics Letters 1980, v. 5, p. 85-86. 1.8 N. S. Kapany, “Fiber Optics: Principles and Applications”, Academic Press, 1962. 1.9 D. A. Pinnow, A. L. Gentile, A. G. Standlee and A. J. Timper, “Polycrystalline Fiber Optical Waveguides for Infrared”, Applied Physics Letters 1978, v. 33, p. 28-29. 1.10 M. Ikedo, M. Watari, F. Tateshi and H. Ishiwatwri, “Preparation and Characterization of the TLBr-TlI Fiber for a High Power CO2 Laser Beam”, J. of Applied Physics 1986, v. 60, p. 3035-3036. 1.11 V. G. Artjushenko, L. N. Butvina, V. V. Vojteskhosky, E. M. Dianov and J. G. Kolesnikov, “Mechanism of Optical Losses in Polycrystalline KRS-% Fibers”, J. of Lightwave Technology 1986, v. 4, p. 461-464. 1.12 M. Saito, M. Takizawa and M. Miyagi, “Optical and Mechanical Properties of Infrared Fibers”, J. of Lightwave Technology 1988, v. 6, p. 233239. 1.13 A. Saar, N. Barkay, F. Moser, I. Schnitzer, A. Levite and A. Katzir, “Optical and Mechanical Properties of Silver Halide Fibers”, Proc. SPIE 843 1987, p. 98-104. 1.14 A. Saar and A. Katzir, “Intristic Losses in Mixed Silver Halide Fibers”, Proc. SPIE 1048 1989, p. 24-32. 29 1.15 M. Saito, Y. Matsuura, M. Kawamura and M. Miyagi, “Bending Losses of Incoherent Light in Circular Hollow Waveguides”, J. Opt. Soc. Am. A 1990, v. 7. p. 2063-2068. 1.16 A. Hongo, K. Morosawa, K. Matsumoto, T. Shiota and T. Hashimoto, “Transmission of Kilowatt-Class CO2 Laser Light Through Dielectric Coated Metallic Hollow Waveguides for Material Processing”, Applied Optics 1992, v. 31, p. 6441-6445. 1.17 H. Takahashi, I. Sugimoto, T. Takabayashi, S. Yoshida, “Optical Transmission Loss of Liquid-Core Silica Fibers in the Infrared Region”, Optics Communications 1985, v.53, p.164. 1.18 I. Gannot, J. Dror, A. Inberg, N. Croitoru, “Flexible Plastic Waveguides Suitable For Large Interval of The Infrared Radiatio Spectrum”, SPIE 1994. 1.19 J. Dror, A. Inberg, R. Dahan, A. Elboim, N. Croitoru, “Influence of Heating on Prefornence of Flexible Hollow Waveguides For the Mid-Infrared” 1.20 I. Gannot, S. Schruner, J. Dror, A. Inberg, T. Ertl, J. Tschepe, G. J. Muller, N. Croitoru, “Flexible Waveguides For Er:YAG Laser Radiation Delivery”, IEEE Transaction On Biomedical Engineering, 1995, v. 42, p.967. 1.21 A. Inberg, M. Oksman, M. Ben-David and N. Croitoru, “Hollow Waveguide for Mid and Thermal Infrared Radiation”, J. of Clinical Laser Medicine & Surgery 1998, v. 16, p.125-131. 1.22 A. Inberg, M. Oksman, M. Ben-David, A. Shefer and N. Croitoru, “Hollow Silica, Metal and Plastic Waveguides for Hard Tissue Medical Applications”, Proc. SPIE 2977 1997, p.30-35. 30 1.23 M. Ben-David, A. Inberg, I. Gannot and N. Croitoru, “The Effect of Scattering on the Transmission of IR Radiation Through Hollow Waveguides”, J. of Optoelectronics and Advanced Materials, 1999, No. 3, p-23-30. 1.24 I. Gannot, A. Inberg, N. Croitoru and R. W. Waynant, "Flexible Waveguides for Free Electron Laser Radiation Transmission", Applied Optics, Vol. 36, No. 25, pp 6289-6293, September 1997. 1.25 M. Miagi, K. Harada, Y. Aizawa, S. Kawakami, “Transmission Properties of Dielectric-Coated Metallic Waveguides For Infrared Transmission”, SPIE 484 Infrared Optical Materials and Fibers III 1984. 1.26 Y. Matsuura, M. Miagi, A. Hongo, “Dielectric-Coated Metallic Holllow Waveguide For 3m Er:YAG, 5m CO, and 10.6m CO2 Laser Light Tranamission”, Applied Optics 1990, v. 29, p. 2213. 1.27 Y. Wang, A. Hongo, Y. Kato, T. Shimomura, D. Miura and M. Myagi, “Thickness and Uniformity of Fluorocabon Polymer Film Dynamically Coated Inside Silver Hollow Glass Waveguide" 1.28 J. Harrington, “A Review of IR Transmitting Hollow Waveguides”, Fibers and Integrated Optics 2000, v. 19, p. 211-217. 1.29 P. Bhardwaj, O. J. Gregory, C. Morrow, G. Gu and K. Burbank, “Preformance of a Dielectric Coated Monolithic Hollow Metallic Waveguide”, Material Letters 1993, v. 16, p. 150-156. 1.30 K. D. Laakman, Hollow Waveguides, 1985, U.S. Patent no. 4652083. 1.31 K. D. Laakman and M. B. Levy 1991, U.S. Patent no. 5,500,944. 31 1.32 E. Garmire “Hollow Metall Waveguides With Rectangular Cross Section For High Power Transmission”, SPIE 484 Infrared Optical Materials and Fibers III 1984. 1.33 U. Kubo, Y. Hashishin, “Flexible Holloe Metal Light Guide For Medical CO2 Laser”, SPIE 494 Novel Optical Fiber Techniques For Medical Application 1984. 1.34 T. Haidaka, T. Morikawa and J. Shimada, “Hollow Core Oxide Glass Cladding Optical Fibers For Middle Infrared Region”, J. Applied Physics 1981, V. 52, p. 4467-4471. 1.35 R. Falciai, G. Gireni, A. M. Scheggi, “Oxide Glass Hollow Fibers For CO2 Laser Radiation Transmission”, SPIE 494 Novel Optical Fiber Techniques For Medical Application 1984 1.36 A. Bornstein, N. Croitoru, “Chalcognide Hollow Fibers”, Journal Of NonCrystaline Materials 1985, p. 1277. 1.37 C. C. Gregory and J. A. Harrington, “Attenuation, Modal, Polarization Properties of n<1 Hollow Dielectric Waveguides”, Applied Optics 1993, v. 32, p. 5302-5309. 1.38 R. Nubling and J. A. Harrington, “Hollow Waveguide Delivery Systems for High Power Industrial CO2 Lasers”, Applied Optics 1996, v. 34, p. 372380. 1.39 F. Pfeiffer, “X-Ray Waveguides”, Diploma Thesis, July 1999. 32 Chapter Two - Attenuation Mechanisms In Hollow Waveguides List of Symbols For Chapter 2 i, ..... Angle of incidence t ............. Transmittance angle r ............. Reflectance angle rp ............. Reflection coefficient for the amplitude of p state polarization rs ............. Reflection coefficient for the amplitude of s state polarization tp ............. Transmission coefficient for the amplitude of p state polarization ts.............. Transmission coefficient for the amplitude of s state polarization Rp ............ Reflection coefficient for the power of p state polarization Rs ............ Reflection coefficient for the power of s state polarization n .............. Index of refraction Wavelength ........ Phase shift d .............. Dielectric layer thickness ............. Surface roughness r ............ Path difference k.............. Wave vector R,r .......... Radius vector ............. Angular frequency E ............. Electric field vector H............. Magnetic field vector .............. Scattering coefficient .............. Separation parameter T ............. Correlation distance 33 As was seen in the previous chapter, hollow waveguides are composed of two thin films (metal and dielectric), which are deposited on the inner wall of a hollow tube. In order to understand the transmission of the laser beam through such a waveguide we should examine how the characteristics of the dielectric layer (thickness, index of refraction and roughness) influence the transmission of the waveguide. The thickness of the dielectric layer and its index of refraction influence the reflection coefficient of the thin layer. The roughness of the surface influences the scattering of the incident laser beam. This chapter deals with the two main attenuation mechanisms, which influence the transmission of the laser through the hollow waveguide. The first is the reflection of the laser beam from a thin layer, and the second is the scattering of the laser beam from a rough surface. Understanding these mechanisms will enable us to analyze in the next chapter the dependence of the waveguide’s attenuation on the waveguide’s characteristics, its radius of bending and the coupling conditions. Although these two mechanisms had been discussed in many papers, I chose to include them for two reasons. The first is that they are the most important building blocks of the ray model that I will develop in the next chapter. The second is the scattering theory that will be described here has not yet been implemented to hollow waveguides. Furthermore these two mechanisms may be used as a tool for designing hollow waveguides. The chapter ends with the description of roughness measurements made by an atomic force microscope. These measurements show the roughness of the inner 34 layers and the height distribution of the scattering centers. These measurements will be used as parameters for the theoretical calculations that will be presented in the next chapter. 2.1 Reflection from a thin film First we will examine the properties of the reflection coefficient of a thin film, especially how the characteristics of the dielectric film (thickness and index of refraction) influence the Figure 2.1 Reflection and refraction at a boundary reflection coefficient. When electromagnetic impinges on radiation a boundary i r n1 n2 t between two materials with different index of refraction it undergoes two processes, reflection and refraction (figure 2.1). According to the law of reflection and Snell’s law the relations between the angle of incidence, i, the angle of reflection, r, and the angle of refraction, t, are [2.1] i=r (2.1.1) n1sin(i)=n2sin(t) (2.1.2) The reflection (r) and transmission (t) coefficients are given by Fresnel’s coefficients for TE (s) and TM (p) polarization [2.2] rp n1 cos t n2 cos i n1 cos t n2 cos i rs n1 cos i n2 cos t n1 cos i n2 cos t (2.1.3) (2.1.4) 35 tp 2n1 cos i n1 cos t n2 cos i (2.1.5) ts 2n1 cos i n1 cos i n2 cos t (2.1.6) and the reflection coefficient for the power are given by1 R p R s rp2 rs2 (2.1.7) When the material, which the radiation (at a certain wavelength) is advancing through, is completely transparent, its index of refraction is real. If the material is only partially transparent such as in the case of metals and semiconductors, its index of refraction is complex and can be expressed as n n ik where k is the extinction coefficient, and n (2.1.8) Figure 2.2 Reflection from a thin layer is the I r2 index of refraction. 2 Since we are dealing with 1 t2t’2r2 n2 t2r1 t2r1r’2 n1 reflection from a thin layer 0 t2t1 t2r1r’2t1 n0 the reflection coefficient is influenced not only by the change in the index of reflection but also by the multiple reflections inside the thin layer. Figure 2.2 shows the path of a ray through a thin layer with a thickness d and index of refraction n1. 1 The reflected electric, Er, field is related to the incidence electric field, Ei, by Er=rEi.The electric power is the power of the electric field (Er)2=(rEi)2=r2Ei2=REi2 36 The ray that is reflected from the lower boundary undergoes a phase shift of e -i where 2 n1d cos1 (2.1.9) As we can see from figure 2.2, the reflectance from the thin layer is the sum of all the internal reflections plus the first reflection. The reflection is then given by rtot r2 r1e 2i 1 r1r2e 2i (2.1.10) and the reflection coefficient for the power is R r12 r22 2r1r2 cos( 2 ) 1 r12 r22 2r1r2 cos( 2 ) (2.1.11) This expression is correct for any polarization (TE and TM). For an unpolarized radiation we have to take the average of the two polarizations. In the case of unpolarized radiation the reflection coefficient is given by Rtot Rs Rp 2 (2.1.12) As was described in the first chapter, hollow waveguides are made of two thin layers (Ag and AgI), that are deposited on the inner wall of a hollow tube, and an air core. The air and the AgI are dielectric materials, which absorb little of the radiation. The Ag layer is a metal layer with a complex index of refraction, which absorbs some of the radiation. In order to get maximum reflectance we need to find which state of polarization is reflected better, and how the dielectric layer properties (thickness, index of refraction) influence the radiation’s reflection. 37 In order to calculate the reflection coefficients of the different layers one has to know the index of refraction of Ag and AgI. The index of refraction of Ag depends on the radiation wavelength [2.3-2.5]. For infrared radiation Beatie [2.4] found the following relations for the index of refraction and absorption of Ag n a2 b4 (2.1.13) k c d3 (2.1.14) where is measured in m and a=0.12, b=1.3e10-4, c=7.2, d=3.6e10-3. The index of refraction of AgI is 2.2 for all wavelengths [2.5]. 38 The first hollow waveguides, which Figure 2.3 Reflection Coefficient Vs. Angle of Incidence were developed, had only a metal film. Using equations 2.1.32.1.6 one can calculate the reflection coefficients (Rs, Rp, Rtot) of the metal layer for the different types of polarization. Figure 2.3 shows the reflection coefficients silver from layer =10.6m (i.e. a for Figure 2.4 Reflection Coefficient Vs. Angle of Incidence CO2 laser). We can see from the figure that the layer does not reflect well TM polarization at large angles of incidence. To overcome this drawback Miyagi [2.6] suggested adding a dielectric layer over the metal layer. The dielectric layer could be made of several materials Si, Ge or as in our case AgI. As can be seen from figure 2.4, the dielectric layer improves the reflection of TM polarization at large angles. The dielectric layer decreases slightly the reflection coefficients of the TE 39 polarization but the total reflection coefficient increases. Therefore the introduction of a dielectric layer improves the hollow waveguide. If we examine equations 2.1.9 and 2.1.10 closely we can see that the reflectance of the waveguide depends on the laser’s wavelength, the materials deposited on the inner wall, the thickness of the dielectric layer and the angle of incidence. Since in most applications the laser’s wavelength is constant, we need to adapt the materials and the thickness of the dielectric layer to it. If we need to transmit several wavelengths, the parameters of the materials need to be adapted to all of them. We can see from equation 2.1.11 that the index of refraction of the inner layers plays a major role in determining the waveguide’s reflectance. In order to achieve maximum transmission we need to select materials with indices of refraction, which are suitable to the wavelengths we want to transmit. The angle of incidence and the thickness of the dielectric layer determine the nature of interference (constructive or destructive) of the incident radiation. We need to build the hollow waveguide with the appropriate thickness of the dielectric layer [2.7]. In order to build waveguides suitable for transmitting a certain laser wavelength we wrote a computer program which calculates the reflection coefficient as a function of the thin layer parameters (thickness, index of refraction), the laser wavelength and angle of incidence. Since we know the wavelength of the laser, which we want to transmit, we first have to choose the material for the dielectric layer. The dielectric layer needs to 40 have two characteristics: it has to be transparent at the laser wavelength, and it has to have the appropriate thickness and index of refraction, which enable maximum reflectance for various angle of incidence. Figures 2.5 to 2.7 show the reflection coefficient as a function of the index of refraction, the layer thickness and the angle of incidence. For the calculation we took the laser wavelength to be =10.6m, which is the wavelength of CO2 laser and assumed that the metal layer is silver. [m] Figure 2.5 Reflection coefficient vs. thickness and index of refraction of the dielectric layer =850, =10.6m, metal layer – Ag 41 [m] [degrees] Figure 2.6 Reflection coefficient vs. thickness of the dielectric layer and the laser’s angle of incidence n=2.2, =10.6m, metal layer – Ag [degrees] Figure 2.7 Reflection coefficient vs. index of refraction of the dielectric layer and the laser’s angle of incidence d=0.8m, =10.6m, metal layer – Ag 42 Figure 2.8 shows the reflection coefficient as a function of the laser wavelength and thickness of the dielectric layer. For the calculation the laser’s angle of incidence is 85o, the index of refraction of the dielectric layer (n) is 2.2, which is the index of refraction of AgI, and the thickness of the dielectric layer is 0.8m. The metal layer is silver. [m] [m] Figure 2.8 Reflection coefficient vs. thickness of the dielectric layer and the laser’s wavelength nd=2.2, metal layer – Ag We can use the above graphs in order to choose the thickness and index of refraction of the dielectric layer suitable for the laser wavelength and its angle of incidence. Notice that for d=0 the index of refraction has no influence and the reflection coefficient is the one for a metal layer. As an example let us design a waveguide for CO2 laser (=10.6m). According to the above graphs we will get maximum reflectance (0.992R1) for d1m and n2. 43 2.2 Scattering from rough surfaces The second attenuation mechanism that affects the transmission of the hollow waveguide is the scattering of the laser beam from a rough surface. The tube in which the thin layers are deposited is rough and as we shall see later the deposition of the thin layers increases the roughness of the waveguide’s wall. 2.2.1 The Rayleigh criterion Before attempting to determine the scattering coefficient quantitatively, we shall first consider a more Figure 2.9 Specular reflection and difused scattering elementary question. When does a smooth surface become a rough one, or for what values of wavelength, Specular Reflection Diffuse Scattering surface roughness and angle of incidence does specular reflection change to diffused scattering (figure 2.9)? Rayleigh suggested [2.8] Figure 2.10 Derivation of the Rayleigh criterion a way of formulating the 2 1 relation involving these parameters: Consider two rays 1 and 2 (figure 2.10) incident on a surface with irregularities of heights at a grazing angle . The path difference between the two rays is 44 r 2 sin (2.2.1) and hence the phase difference is 2 r 4 sin (2.2.2) If the phase difference is small, the two rays will be almost in phase as they are in the case of perfectly smooth surface. If the phase difference increases, the two rays will interfere until for where they will be in phase opposition and cancel each other. If there is no energy flow in this direction, then it must have been redistributed in other directions, for it can not have been lost. Thus for the surface scatters and hence is rough, while for 0 it reflects specularly and is smooth. We may now establish a value of phase difference between theses two extremes to distinguish between “rough” from “smooth. We may arbitrarily choose the value of half way between the two cases, i.e. . By by substituting this value in equation 2.2.2 we obtain the relation 2 known as the “Rayleigh criterion”, namely that a surface is considered smooth for 8 sin (2.2.3) However there seems to be a little point in assessing an exact dividing line, which must in essence be more or less conventional. A safer way of expressing the basic idea of the Rayleigh criterion is to use the right hand side of equation 2.2.2 as a measure of effective surface roughness. It may then be stated that a surface will be effectively smooth only under two conditions: 45 0 or 0 (2.2.4) i.e. the surface roughness is small in comparison to the radiation wavelength or when the grazing angle is small. If either one tends to zero then according to equation 2.2.2 the phase difference will tend to zero, thus implying that the surface is smooth. 2.2.2 The general solution for scattering from rough surface In this section we will derive the general solution for scattering from a rough surface [2.8]. The rough surface will be given by the function x, y (2.2.5) The mean level of the surface is the plane z=0 (figure 2.11). All quantities associated with the incident wave will be denoted by the subscript 1 and those associated with the scattered field - by the subscript 2. Thus the incident wave is E1 and the scattered wave is E2. The medium in the space z is assumed to be free space. We shall assume E1 to be a harmonic plane wave of unit amplitude: E1 exp ik1 r i t (2.2.6) where k1 is the propagation vector, (2.2.7) Figure 2.11 Basic notion which will always lie in the xz plane (this plane will 2 k 1 k1 z r k1 r k2 1 2 x 46 denote the plane of propagation of the rays in the ray model in the next chapter, thus I will discuss only the influence of scattering on the propagation vector in that plane) (figure 2.11) and r is the radius vector r xx0 yy 0 zz 0 (2.2.8) In particular, for points on the surface S, we have r xx0 yy 0 x, y z 0 (2.2.9) The angle of incidence, included between the direction of propagation of E1 and the z axis, will be denoted by 1; the scattering angle, included between z0 and k2, where k 2 k1 2 (2.2.10) will be denoted by 2 (figure 2.11). Let P be an observation point and let R’ be the distance from P to a point x, y, x, y on the surface S. Then the scattered field is given by the Helmholtz integral E1 P 1 4 E E n n dS (2.2.11) where exp ik 2 R ' R' (2.2.12) In order to deal with plane waves, rather then spherical ones, we let R ' , i.e. we move the point P to the Fraunhofer zone of diffraction and then as seen in figure 2.12 47 to P R0=OP (x,y ) k2 r OC k R’=BP k2 C r B O X Figure 2.12 k 2 R ' k 2 R0 k 2 r (2.2.13) where R0 is the distance of P from the origin, so that exp k 2 R0 k 2 r R' (2.2.14) Within this approximation the field on surface S will be E S 1 R E1 (2.2.15) and E 1 R E1k1 n n S (2.2.16) 48 where n is the normal to the surface at the considered point and R is the reflection coefficient of a smooth plane given in section 2.1. The second relation follows from the first by differentiating the incident and reflected waves or by taking the magnetic field as H S 1 RH1 (2.2.17) and using Maxwell equation E H t (2.2.18) To simplify the calculations we will limit ourselves to the case of one dimensionally rough surface i.e. x, y x (2.2.19) which is constant along the y coordinate, so that n always lies in the plane of incidence xz in figure (2.13). Then k1 1 n$n x x Figure 2.13 1 1 arctan ' x Substituting 2.2.14, 2.2.15 and 2.2.16 in 2.2.13 we find (2.2.20) 49 E2 i exp ikR0 Rv p n exp iv r dS 4 R0 S (2.2.21) where v k sin 1 sin 2 x 0 k cos 1 cos 2 z 0 (2.2.22) p k sin 1 sin 2 x 0 k cos 2 cos 2 z 0 (2.2.23) n x0 sin z 0 cos (2.2.24) r xx0 x z 0 (2.2.25) dS sec dx (2.2.25) tan ' x (2.2.26) For a surface extending from x=-L to x=L we may thus write 2.2.20 in the scalar form ik exp ikR0 ' L a b exp iv x x iv z dx 4 R0 L E2 (2.2.27) where a 1 r sin 1 1 R sin 2 (2.2.28) b 1 r cos 2 1 R cos 1 (2.2.29) To get rid of the factor in front of integral 2.2.27, we normalize the expression 2.2.27 by introducing a scattering coefficient E2 E20 (2.2.30) 50 E20 is the field reflected in the direction of specular reflection (1=2) by a smooth, perfectly conducting plane of the same dimensions under the same angle of incidence at the same distance. E20 is given by E20 k exp ikR0 L cos 1 R0 (2.2.31) and the scattering coefficient is given by 1 , 2 L F2 exp iv r 2 L L (2.2.32) with F2 1 , 2 sec 1 1 cos 1 2 cos 1 cos 2 (2.2.33) 2.2.3 Scattering from a randomly rough surface The rough surfaces met in nature are best described by the statistical distribution of their deviation from a certain mean level. This does not yet, however, describe the surface completely, since this distribution does not tell us whether the hills and valleys of the surface are crowded, close together, or whether they are far apart. A second function, the correlation function or the autocorrelation coefficient, describes this aspect of the surface. It turns out that the statistical distribution suffices for determining the mean value of the field, but to calculate its variance (or power) the general two dimensional distribution is needed. In this section we shall apply the general solution derived in the previous section to the case where x, y is a random function. Again we will limit ourselves to the case of one dimensional roughness, x . 51 Let x be a random variable assuming values z with a probability density w(z): Let the mean value be 0 (2.2.34) and consider the mean value of the integral L exp iv r dx L L exp iv x x exp iv z z dx exp iv z L L exp iv x dx (2.2.35) x L We notice that exp iv z wz exp iv z dz v 2 z (2.2.36) z is the definition of the characteristic function vz , associated with the distribution w(z). We therfore have L exp iv r dx L L v z exp iv x x dx (2.2.37) L Substituting 2.2.36 and 2.2.37 in 2.2.32, we find the important relation vz 0 (2.2.38) where 0 sin cvx L (2.2.39) is the scattering coefficient of a smooth surface. Note that as L>>, 0 will be equal to unity for the specular direction (1=2, vx=0), but it will rapidly tend to zero as 2 leaves the specular direction. Hence will equal zero for any direction of scattering except for a narrow wedge 2 The characteristic function v of a distribution p(z) is v pz exp ivzdz 52 about the specular direction; if vz equals zero in the specular direction will equal to zero in all directions. However, it should be remembered that is a complex quantity and that we may not infer 0 from 0 (unless is real and non negative). Since is a complex quantity, its mean value is of little use except as a stepping stone to determine the mean value of * (2.2.40) Note that the mean square of 2.2.40 is * 2 E22 2 E20 (2.2.41) whitch is proportional to the mean scattered power: It is also related to the variance of , denoted by D, and the variance of the scattered field DE2 by * * D * 1 DE2 3 (2.2.42) 2 E20 The root mean square value of is RMS * (2.2.43) and the mean scattered power is given by 2 P2 12 Y0 E2 E2* 12 Y E20 * where Y0=1/120 is the admittance of free space. 3 D * * (2.2.44) 53 Thus we can find all the quantities of interest by determining the value * . To find * , we shall assume that the surface S is large enough, to get rid of edge effect terms. From 2.2.32 we have * F2 22 4L L L exp iv x x 1 x2 exp iv z 1 2 dx1dx2 (2.2.45) L L where 1 x1 2 x2 (2.2.46) From the theory of statistics 4 exp iv z 1 2 W z , z exp iv 1 2 z 1 2 dz1dz2 2 v z v z (2.2.47) is the two dimensional characteristic function of the distribution W(z1,z2), which will be equal to the product of the distribution w(z1) and w(z2) only if 1 and 2 are independent. Now we can assume that x is a “purely random” process, that does not contain any non random periodic components. 1 and 2 will obviously be independent if they are far apart, i.e. for large values of the “separation parameter”, defined by x1 x2 (2.2.48) But when is small, i.e. when x1 and x2 are taken near each other, 1 and 2 will be correlated, and when =0, they will be identical. Thus the distribution of , w(z), is insufficient to determine the two dimensional characteristic function 4 2 v1 , v2 p z, y exp iv z iv y dzdy 2 1 2 54 2.2.47. In addition to the statistical distribution of we must know the correlation function L B lim 1 x x dx L 2 L L (2.2.49) or its autocorrelation coefficient, which is related to the correlation function by C B D 2 1 2 1 2 12 1 2 (2.2.50) Since 0 we get C 1 2 12 (2.2.51) It follows directly from equation 2.2.49 that lim C 1 (2.2.52) 0 i.e. full correlation (linear dipendence) and lim C 0 (2.2.53) i.e. independence. For a purely random surface C will decrease monotonously from its maximum value C 0 1 to C 0 . Let the distance in which C drops to the value e-1 be T. This distance, which will be called the “correlation distance”, is smaller then L. From 2.2.42, 2.2.45, 2.2.38 and 2.2.36 we find that D * * F22 2 2L L L exp iv x x 1 x 2 2 v z ,v z 2 v z *2 v z dx1 dx 2 L L (2.2.54) when 1 and 2 are independent 55 2 v z ,v z 2 v z *2 v z (2.2.55) but 1 and 2 are independent for all but small ; hence the square bracket in 2.2.54 will vanish for all but small . Substituting 2.2.48 in 2.2.54 we get F22 D 2 2L exp iv v ,v v v d L x 2 z z 2 * 2 z z (2.2.56) L with the only significant contribution to the integral coming from the region near =0. Equation 2.2.54 shows that to determine the variance we must know the dimensional distribution W(z1, z2; ) of the surface, i.e. know the dimensional distribution of at two points x1 and x2 separated by any distance . 2.2.4 The normally distributed surface We take the normal distribution as the most important and typical of a rough surface for substitution in the formulae of the preceding section. Let be normally distributed with the mean value 0 (2.2.57) and standard deviation . The distribution of is given by z2 1 wz exp 2 2 2 (2.2.58) The standard deviation , which in this case is due to 2.2.57 and is also the root mean square of , describes the roughness of the surface. The characteristic distribution of 2.2.36 is v exp 12 2vz2 (2.2.59) 56 The rough surface is not uniquely described by the statistical distribution of , as it tells us nothing about the distances between the hills and valleys of the surface; i.e. about the density of irregularities. This is described by the autocorrelation coefficient given by 2.2.50. The autocorrelation coefficient is an even function of . We take as a sufficiently general autocorrelation coefficient the function 2 C exp 2 T (2.2.60) where T is the “correlation distance”. Substituting 2.2.59 in 2.2.38 and expressing vz explicitly, we find that 2 2 0 exp 2 cos 1 cos 2 2 (2.2.61) since 0 vanishes everywhere except near the direction of specular reflection (vx=0) spec 1 4 cos 1 2 exp 2 (2.2.62) This result was also obtained experimentally by several others such as Bennett et al. [2.9] The two dimensional normal distribution of two random variables 1, 2, with mean values zero and variances 2, and whitch is correlated by a correlation coefficient C, is wz1 , z2 z 2 2Cz z z 2 exp 1 2 1 2 2 2 2 1 C 2 1 C 2 1 The characteristic function of this distribution is given by (2.2.63) 57 2 vz ,vz exp vz2 21 C (2.2.64) Substituting 2.2.60 into this expression and expanding it in an exponential series we have 2 vz ,vz exp vz2 2 vz2 m m 2exp m m 0 2 (2.2.65) T 2 We shall work with the quantity vz2 2 . For briefness we therefore introduce a new symbol g, where g v z 2 cos 1 cos 2 (2.2.65) Using 2.2.59 we then find v ,v v v e gm! exp m 2 z z z g * z m m 0 2 T 2 (2.2.66) Substituting 2.2.66 into 2.2.56 we obtain D F2 gm m 2 ivx g e exp d T2 2 L m 0 m! (2.2.67) where we have replaced the integration limits by ; this is permitted since the integral receives significant contributions only from the region near =0. Using the integral e at cos btdt a e b2 4a a 0 (2.2.68) we then find D F 2T 2L gm v 2T 2 exp x 4 m m0 m! m eg Using 2.2.38 and 2.2.41 we can also write (2.2.69) 58 * sin v L 2 F 2T x e vx L 2L g In order to find the scattering coefficient we should compute gm v 2T 2 exp x 4m m m! m 0 (2.2.70) Figure 2.14 Normalized scattering coefficient vs. scattering angle (/=0.0001,T/=1000, L=100) the series in equation 2.2.70. The series is calculated numerically dotted line - /=0.0001 solid line - /=0.01 dashed line - /=1 until the last term is less then 0.01%. The scattering coefficient was calculated using a MATLAB program. Figures 2.14 and 2.15 show the scattering coefficient for various angles of incidence and Figure 2.15 Normalized scattering coefficient vs. scattering angle (=80,T/=1000, L=100) surface roughness respectively. We can see that as the angle of incidence decreases, or as the surface roughness increases, the normalized scattering coefficient increases. This is also predicted from equation 2.2.62. 2.3 Measurements of surface roughness In order to apply the above theory to real hollow waveguides (as will be explained in the next chapter) we first have to measure the surface roughness and its distribution [2.10]. 59 The surface roughness was measured using an atomic force microscope (AFM). We measured the surface roughness of fused silica, Teflon and glass waveguides coated with Ag and AgI thin filmes. We cut several the waveguides open and measured the roughness of little pieces. The distribution of the surface height was calculated using the AFM software. Table 2.1 summarizes the surface heights measurements. As can be seen from the table, the smoothest surface is that of fused silica waveguides and the roughest surface is that of the Teflon waveguides. It is clear from the table that except for the fused silica tube, the addition of the AgI layer smoothen the surface roughness hence decreasing the surface scattering. Waveguide type Teflon Fused silica Glass Ag layer roughness (nm) 298 10 42 AgI layer roughness (nm) 100 24 20 Table 2.1 Surface roughness of different waveguides Figures 2.16 to 2.17 a, b and c show the surface roughness of Ag layer, AgI layer and the height distribution of the AgI layer for glass and Teflon waveguides respectively. Similar results were obtained for fused silica waveguides. We can see from the figures that the AgI layers looks smoother than Ag layers. 60 Figure 2.16 AFM measurements for glass waveguide’s Ag and AgI layer (a) Ag layer (b) AgI layer 61 (c) AgI height distribution Figure 2.17 AFM measurements for Teflon waveguide’s Ag and AgI layer (a) Ag layer 62 (b) AgI layer (c) AgI height distribution 2.4 Summary of chapter 2 This chapter laid the foundation for the ray model that I will discuss in the next chapters. It described the main attenuation mechanisms in hollow waveguides. These mechanisms determine the performance of hollow waveguides. The reflection from the thin film determines the wavelength that will be transmitted by the waveguide and its minimal attenuation, while the surface roughness 63 determines the amount of scattering the laser beam encounters while propagating through the waveguide. Surface scattering has not been applied to hollow waveguides yet and I will use the above theoretical analysis as one of building blocks of my ray model. I also measured the roughness of the different layers within different types of tubes. As was shown in the above table, different tubes have different roughness. Hence they impose different roughness on the deposited layers. The differences in the roughness among the tubes are the cause for the variation in attenuation among them. This will be shown in the next chapter, where I will use the above measurements results as parameters for the ray model and the theoretical calculations. 2.5 Refrences for chapter 2 2.1 D. Marcuse, “Theory of Dielectric Optical Waveguides”, ch. 1, Accademic Press, 1991. 2.2 Chopra, “Thin Film Phenomena”, ch. 2, McGraw Hill 1969. 2.3 E. D. Palik: “Handbook of Optical Constants of Solids”, D.W. Lynch and W.R. Hunter, “Comments on optical constants of metals and an introduction to the data of several metals - Silver”, p. 275, Accademic press. 2.4 J. R. Beatie, “The annomalous skin effect and the IR properties of silver and aluminum”, Physica 1957, v. 23, p. 898. 2.5 W. G. Drisco;l and W. Vaughan, “Handbook of Optics”, McGraw Hill 1978. 64 2.6 M. Miyagi, A. Hongo and S. Kawakami, “Transmission characteristics of dielectric coated metalic waveguides for infrared transmission: slab waveguide model”, IEEE J. Quantun Electronics, 1983, v. 9, p. 136. 2.7 M. Alaluf, J. Dror, R. Dahan, N. Croitoru, “Plastic hollow fibers as a selective mid-IR radiation transmission medium”, J. Appl. Phys., 1992, v. 72, p. 3878. 2.8 P. Beckman and A. Spizzichino “The Scattering of Electromagnetic Waves from Rough Surfaces”, Pergamon Press, 1963. 2.9 H. E. Bennet “Specular reflectance of aluminized ground glass and height distribution of surface irregularities”, J. Opt. Soc. Am. 1963, v. 53, p. 1389. 2.10 M. Ben-David, A. Inberg, I. Gannot and N. Croitoru, “The Effect of Scattering on the Transmission of IR Radiation Through Hollow Waveguides”, J. of Optoelectronics and Advanced Materials, 1999, No. 3, p-23-30. 65 Chapter Three - Radiation Propagation Through Hollow Waveguides List of Symbols For Chapter 3 .............. Propagation constant ............. Attenuation constant a, T, r ...... Waveguide’s inner radius .............. Polar distribution of the ray model r .............. Radial distribution of the ray model R ............. Reflection coefficient n .............. Index of refraction Wavelength ............. Angle of entrance Angle of propagation d, .......... Dielectric layer thickness ............. Surface roughness k.............. Wave vector ............. Angular frequency E ............. Electric field vector S.............. Scattering coefficient I .............. Laser beam energy z .............. The distance between two refractions p .............. Number of times a ray impinges on the waveguide’s wall l............... waveguide’s length A ............. Attenuation T ............. Transmission 0 ............ Laser spot size f .............. focal length D ............. lens diameter 66 y .............. Deviation from the waveguide’s center P.............. laser power R ............. Bending radius .............. The ratio of the experimental M2 factor to the theoretical one In the previous chapter I introduced the two main attenuation mechanisms in hollow waveguides, Fresnel reflection related properties (thin film thickness, index of refraction etc.) and the surface roughness. These mechanisms are structural mechanisms. In this chapter I will relate the attenuation mechanisms to the transmission/ attenuation of the hollow waveguide. There are two methods to analyze the propagation of infrared radiation through hollow waveguides. The first is known as the mode model. The mode model solves Maxwell equations in cylindrical coordinates and derives from the solution the attenuation of the hollow waveguide as a function of the waveguide parameters. The second method is the ray model. This model is applicable as long as the laser wavelength is small compared to the waveguide’s inner diameter. This condition is satisfied in our case, where the inner diameter is larger than 0.2mm and the wavelength is smaller than 12m. This chapter is composed of two main parts. The first part is the theoretical part, in which I will describe briefly the mode approach and develop the improved ray model. The improved ray model will be the base of the theoretical study of hollow waveguides. The main improvements, that I have introduced in this model, are: introduction of surface roughness to the 67 attenuation calculations and geometrical calculations for bent hollow waveguides and off center coupling. Although the ray model is not new. The approach I have developed enables to take into account many of the hollow waveguide’s parameters as well as the laser beam and coupling conditions parameters. This ray model, as will be demonstrated, gives very accurate results when compared to the experimental results. The second part of this chapter describes the experimental results, which were obtained using different hollow waveguides. The experimental results are compared to the theoretical calculations and are found to be in good agreement with them. This study enables me to identify the drawbacks of the current generation of hollow waveguides and to suggest a way to improve and develop the next generation of hollow waveguides. This will be done in chapter five. Note: This chapter deals only with the spatial aspect of laser propagation through hollow waveguides. The temporal aspect will be dealt in the next chapter. 3.1 Theory 3.1.1 The mode approach In the mode approach, we solve Maxwell equations in cylindrical coordinates with the appropriate boundary conditions. The solutions to this problem give the propagation conditions of each mode along the hollow waveguide. Many researchers used this method to derive the attenuation of hollow waveguides. Among the first who analyzed the propagation of each mode in hollow waveguides were Marcatili and Schmeltzer [3.1]. They analyzed the 68 attenuation coefficients of each mode in hollow metallic and dielectric waveguides, assuming the waveguide’s inner diameter is much larger then the laser wavelength. According to Marcatili and Schmeltzer the propagation constants of the different modes is given by nm 2 2 1 unm n 1 1 Im 2 2a a (3.1.1) and the attenuation coefficients is given by 3 u nm 3 Re n 2a a 2 nm (3.1.2) where a is the waveguide’s inner diameter, unm is the mth root of the equation J n1 unm 0 (3.1.3) and n is a function of the propagation mode and the index of refraction. For the HE11 mode, which is the main mode of a CO2 laser, 1 2 1 2 n 2 1 (3.1.4) Equation 3.1.2 indicates that the waveguide’s attenuation is proportional to a-3. Hence as the waveguide diameter decreases the attenuation increases. This theoretical result is not accurate. Experiments done by Harrington [3.2] have shown that the waveguide’s attenuation is proportional to a-2 and nor a-3. While Marcatili and Schmeltzer analyzed the mode propagation in hollow metallic waveguides, Miyagi et al. analyzed the mode propagation in hollow metallic-dielectric waveguides [3.3, 3.4]. 69 Miyagi et al. used the following argument. Let us look at a hollow cylindrical waveguide with inner radius T and index of refraction n0 1 . The waveguide is coated by a metallic layer with index of refraction n-ik, and a number of dielectric layers made of two materials of width iT and index of refraction ain0 (i=1,2) and a layer of width near the air core. Figure 3.1 shows the index of refraction profile for such a waveguide. Miyagi assumes that there are no energy losses in the dielectric layers in the dielectric layers and that a1<a2. n(x ) n0 1 2 a1 n ik a2 Dielectric layers metal 1 1 Figure 3.1 The index of refraction profile of Miyagi’s waveguide The layers widths satisfy the condition i ai2 1 2 n0 k0T 1 2 (3.1.5) The attenuation constant is given by n0 k0 u02 F n0 k0T 3 (3.1.6) where F is a constant that is a function of and the polarization of the laser radiation. 70 When there is an odd number of dielectric layers m 2m p 1 (3.1.7) the minimum of the attenuation coefficients is achieved when a 1 2 1 1 2 a 1 n0 k0T tan 1 2 a1 1 4 1 mp m p a1 C 2 a2 s (3.1.8) where s is an integer and C is given by C a12 1 1 a22 1 (3.1.9) For the hybrid modes Fmin 2 1 mp a1 Fmetal C 1 1 2 a12 1 2 a1 a2 2m p 2 m p C 2 Fmetal Fdiel (3.1.10) where Fmetal n n k2 2 (3.1.11) For a waveguide without a dielectric layer the attenuation is given by n0 k0 u02 F 0 3 n0 k0T (3.1.12) where F 0 for the hybrid mode is equal to n 2 . The above equations show that the bigger the metal index of refraction the smaller the waveguides attenuation. This is true for the infra red region where k>>n. The addition of the dielectric layers causes the attenuation to decrease since Fdiel 1 . As an example let us take a waveguide made of Al (n-ik=20.558.6i) and Ge (n=4). The calculated attenuation is 0.08dB/m while the calculated attenuation without a dielectric layer is 12dB/m. 71 3.1.2 The ray model According to the ray model [3.5, 3.6] one can decompose the laser beam into separate rays, and use the laws of geometrical optics to calculate the ray propagation through the waveguide. This model may be used since in our case <<ID, where is the wavelength of the coupled laser beam and ID is the inner diameter of the waveguide’s cross section. We assume that multiple incidences on the metal and dielectric layers guide the rays, by refraction and reflection. The dielectric layer has a normal distribution of heights (equation 2.2.58) as was measured experimentally. The “traditional” ray model used the following conditions to describe the laser beam propagation through hollow waveguides: 1. Fresnel’s equations (equations 2.1.3-2.16) give the reflection coefficient each time the ray impinges the waveguide’s wall. 2. The rays propagate only frontally and not rotationally (there are no skewed rays). According to Miyagi [3.7] the contribution of skew rays is of the second order and can be neglected. 3. Two coordinates represent the laser beam cross-section and the point the ray enters into the waveguide; r (with Gaussian distribution) and (with uniform distribution) (figure 3.2a), where 0 r R (R is the laser spot size) and 0 2 . The angle also determines the plane in which the ray propagates. 72 4. The angle of entrance, , (figure 3.2b) has a Gaussian distribution. This angle determines the angle of propagation, , by =90-. r Figure 3.2 The ray parameters 5. The rays have random polarization, TE or TM. 6. The total energy of the laser beam, I, is the sum of the energy of all rays, and is given by I I i r , , ri , i (3.2.1) i where r, and are the standard deviations of the Gaussian beam size and the angle respectively. The traditional ray model and mode approach did not give accurate results when trying to compare their results with the experimental ones. One of the reasons is that these models did not take into account the surface roughness. The surface roughness causes scattering ,which (as shall be seen in the next 73 chapter) increases the attenuation and changes the energy distribution outside the waveguide. Miyagi et al. [3.8] tried to use equation 3.3.14 in order to take into account the influence of the surface roughness. They assumed that the ray continues at the same angle after it impinges on the waveguide’s wall and is attenuated according to equation 3.3.14. This is of course not accurate since the surface scatters the ray’s energy in a range of directions and changes the angle of incidence. To improve the ray model we introduced the influence of surface roughness as described in chapter 2. The new ray model adds the following assumptions: 1. The surface of the dielectric layer is rough. 2. The roughness centers are distributed randomly. 3. The scattering is produced only on the surface of the incident layer and not inside the AgI layer, since the AgI layer is more granular than the Ag layer, hence roughness is greater. 4. The scattered energy is taken only in the positive direction; scattered energy in the negative direction is assumed lost. The scattering coefficient (S) is given by 90 S S 0 90 S (3.2.2) 90 5. The scattering of the ray changes the ray’s angle of propagation. The new angle is the average angle of the scattered energy. 6. The laser beam is decomposed to a minimum of 105 rays. 74 Using these assumptions one can calculate the attenuation and the beam shape outside the waveguide as a function of the waveguide’s parameters (length, inner diameter), and the coupling conditions (focal length of the coupling lens, off center movement of the ray). When the laser beam hits the waveguide’s inner wall it undergoes two processes: reflection from a thin layer and scattering. According to the assumptions mentioned above, the energy of each ray after one incident with the inner wall is I i I i 0 R S (3.2.3) where R() is the reflection coefficient, given by Fresnel law, and S() is the scattering coefficient (equation 3.2.2). Using the assumptions mentioned above one can write a computer program, which can simulate the propagation of a laser beam through hollow waveguides. The simulation uses ray tracing to find how different parameters (length, inner diameter, bending etc.) affect the attenuation of different waveguides and energy distribution outside them. 3.1.3 Ray propagation through straight waveguides First let us look at a straight waveguide. In this section I will analyze the influence of the waveguide’s geometrical parameters (length, inner diameter) and the coupling conditions (focal length of the coupling lens and off center coupling) on the waveguide’s attenuation and energy distribution of the laser beam outside the waveguide. 3.1.3.1 The influence of the waveguide’s geometrical parameters 75 In order to analyze the influence of the waveguide’s geometrical parameters let us first assume that the waveguide is perfectly smooth i.e. S()=1 [3.10]. Using this assumption, the distance a ray passes between two refractions from the waveguide’s wall, zi, can be calculated using figure 3.3 and is given by zi 2 2r tan i (3.3.1) z i Figure 3.3 Ray propagation in a smooth waveguide where r is the waveguide’s radius. The number of times a ray impinges on the waveguide’s wall of length l, pi, is given by l pi int zi l int 2r tan i (3.3.2) The total reflection coefficient of the ray is the multiplication of all the reflections it passed on the way. It is given by Rtotal i R i i p (3.3.3) The transmission, T, and the attenuation, A, are given by Ti I i ,out I i ,in (3.3.4) Ai 10 log( T ) Using equations 3.3.2 and 3.3.3 we get 1 log R i Ai 10int 2r tan i (3.3.5) 76 We can conclude from equation 3.3.5 that the attenuation is proportional to the waveguide’s length. Equation 3.3.5 also helps us to find the dependence of the attenuation on the waveguide’s radius. The waveguide’s radius appears in the equation explicitly and indirectly through the propagation angle i . The angle of propagation is determined by the coupling lens focal length and by the laser beam spot size. The laser spot size at the entrance to the waveguide is given by 0 1. 9 f D (3.3.6) and the maximum angle at the entrance is tan D f (3.3.7) The angle of incidence and the angle at the entrance are related by 90 . The maximum spot size is limited by the waveguide’s radius r r max 2 0.95 f D (3.3.8) Using equations 3.3.5 – 3.3.8 it is possible to find the dependence of the attenuation on the waveguide’s radius l l int int r 2r tan 2r 0.95 int 0.95 l 1 2 2 4r r (3.3.9) The reflection coefficient does not depend on the waveguide’s radius. Hence the attenuation is inversely proportional to r2. This result is in accordance with Harrington’s [3.2] experimental results. 77 More recently Harrington et al. [3.21] showed experimentally that the attenuation varies as 1/r3. It is possible to show that for small angles of incidence the ray model gives a 1/r3 dependence for the attenuation. Let us look at equations 3.3.5 through 3.3.9. In the derivation of the attenuation dependence I did not take into account the influence of the waveguide’s radius on the reflection coefficient. According to Saito et al. [3.22] for small angles of incidence R( ) 1 C (3.3.10) and 1 (3.3.11) r using equations 3.3.10 and 3.3.11, expanding the log(R) into a power series and taking the second term in the series (the first one equals to zero) gives us log R log 1 C C 1 r (3.3.12) equation 3 along with equation 3.3.9 in the dissertation gives us A 1 r3 (3.3.13) which agrees with the calculations of Marcatilli et al. and Miyagi et al. theoretical calculations and with Harrington et al. experimental results. The result mentioned above may become more complicated when we introduce the surface roughness into the equations. Miyagi [3.11] tried to introduce the effect of surface roughness by multiplying the reflection coefficient by 1 4n cos 2 S exp 2 (3.3.14) 78 This equation assumes that the ray continues to propagate at the same angle after it impinges the wall. The results obtained by Miyagi’s model were not in agreement with the experimental results. Using the theory developed in chapter 2 we can calculate the effect of the surface roughness and insert it into the attenuation. In using this theory one has to remember that each time the ray impinges at the waveguide’s wall it changes its angle of propagation. Thus the next time it impinges at the wall, the reflection coefficient and the scattering coefficient will be different. This calculation could be done numerically by a ray-tracing program, which was developed especially for this purpose. The theoretical results of the ray model for different waveguide’s parameters are shown in the experimental section along with the experimental results. 3.1.3.2 The influence of the coupling conditions on the waveguide’s attenuation 3.1.3.2.1 The coupling lens In the previous section we saw that the waveguide’s attenuation depends on the angle of propagation of the laser beam. The angles of propagation of the rays, which constitute the laser beam, are determined by the coupling lens (equations 3.3.7-3.3.8). The smaller the focal length the smaller the angles of incidence, thus the increase in the reflection coefficient and scattering, which increase the attenuation. The focal length also determines the size of the laser’s spot at the waveguide’s entrance (equation 3.3.7). The smaller the focal length the smaller the laser’s 79 spot size. Clearly we want a spot size smaller then the waveguide’s radius. This would help us to couple the laser beam to the waveguide and it does not damage the waveguide’s wall. As one can see there is a conflict between the need for large and small focal lengths. On one hand we need a large focal length, in order to get large angles of incidence. On the other hand a small focal length causes a small laser’s spot that helps us to couple the laser beam to the waveguide. We need to find an optimum focal length. Harrington [3.12] found the desired ratio between the laser’s spot size and the waveguide’s inner diameter should be 0.6. 3.1.3.2.2 Off center coupling Till now we assumed that the laser beam enters the waveguide at its geometrical center. Practically it is very difficult to couple the laser beam right at the geometrical center of the waveguide. Let us now examine what happens when the laser beam does not enter at the geometrical center of the waveguide. This is known as off center coupling. Let us look at a laser beam that enters the waveguide with inner diameter r, at distance y from the center of the waveguide (see figure 3.4). r y O R 80 Figure 3.4 One of the assumptions of the ray model is that there are no skewed rays. Hence when the laser beam does not enter at the center of the waveguide’s cross section the rays will propagate in a waveguide with an effective radius which is smaller or equal to the waveguide’s true radius. In order to find the effective radius each ray sees, let us look at figure 3.5. O’ 90+ reff y O Figure 3.5 The geometry of an off center coupling Using the sine theorem sin y cos r (3.3.15) and the effective radius is given by reff r cos (3.3.16) Using equation 3.3.15 and 3.3.16 one can see that the effective radius depends on the initial deviation of the ray from the center of the waveguide, the waveguide’s radius and the angle . Figure 3.6 shows the effective radius as a function of for different deviations and r=0.5mm. 81 Effective Radius Vs. Teta Waveguide's IR= . mm . . . . . (mm ) Effective Radius . . Teta (degrees) Y= . m m Y= . m m Y= . m m Y= . m m Figure 3.6 Effective radius vs. teta As can be seen from figure 3.6 the longer the deviation the smaller the effective radius (except for =90o,270o ). As was explained earlier, as the radius of the waveguide decreases the attenuation increases. In the case of off center coupling, the effective radius is smaller than the waveguide’s radius and hence the attenuation increases. Figure 3.7 shows the effective radius vs. for different waveguides radii. As can be seen from the figure as the waveguide’s radius increases the effect of off center coupling decreases since the effective radius is longer. 82 Effective Radius Vs. Teta 0.6000 0.4000 0.3000 0.1000 ] 0.2000 [mm Effective Radius 0.5000 0.0000 0 100 200 300 Teta [deg] Reff (ID=0.5mm) Reff (ID=0.7mm) Reff (ID=1mm) Figure 3.7 Effective radius vs. teta Combining the above arguments with the assumptions of the ray model enable us to determine the effect of off center coupling on the waveguide’s attenuation and the energy distribution of the laser beam outside it. The theoretical results are shown in the experimental section with the experimental ones. 3.1.4 Ray propagation through bent waveguide The main purpose of optical fibers is to deliver the laser radiation through bent trajectories. However bending the fiber causes the attenuation to increase due to the change in the angle of propagation. When dealing with bending one has to remember that there are two ways a ray can propagate. The first is the normal way in which the ray impinges on the inner and the outer walls of the waveguide (figure 3.8a). The second is known as the whispering gallery mode (WGM), in which the ray impinges only on the outer wall (figure 3.8b). 83 (a) (b) Figure 3.8 Ray propagation in a bent waveguide, (a) the normal way, (b) whispering gallery mode (WGM) When a ray impinges on a bent waveguide’s wall it changes its angle of propagation. Since this change usually decreases the angle, the attenuation increases. Figure 3.9 shows the geometry of a bent waveguide. From the geometry one can calculate the change in the angle of incidence. R Rr 2sin R sin ' out (3.4.1) 0 where R is the bending radius, R0 – is the waveguide radius and rout is the place the ray enters the curvature. 84 ’ R+2R0 R+rout O Figure 3.9 The geometry for a bent waveguide Figure 3.10 shows ' as a function of for various radii of curvature. As can be seen from the figure the change in the angle of incidence increases as the radius of the curvature decreases. That change causes the attenuation to increase. 85 100 90 70 60 50 40 20 ] 30 [deg angle of incidence after bending 80 10 0 0 10 20 30 40 50 60 70 80 90 angle of incidence before bending [deg] R=40cm R=10cm R=5cm R=1cm Figure 3.10 ’ as a function of As we mentioned earlier there are two modes of propagation in a bent waveguide: the normal mode and whispering gallery mode (WGM). Not each ray can propagate in WGM. The condition for such mode of propagation is derived using the following geometry. 100 86 x R R+d d Figure 3.11 The geometry of WGM propagation The condition for WGM propagation is then given by [3.13] 0 x 2d (3.4.2) x R d 1 sin Combining the above arguments with the assumptions of the ray model enable us to determine the effect of bending on the waveguide’s attenuation and the energy distribution of the laser beam outside it. 87 3.2 Experimental results and theoretical calculations 3.2.1 Experimental setups In order to characterize hollow waveguides under different conditions, I had to use different experimental setups. This section describes the experimental setups that were used. 3.2.1.1 Experimental method for measuring the attenuation of straight waveguides One of the most important parameters that characterize a hollow waveguide is its attenuation. There are two methods to measure the waveguide’s attenuation: a destructive one and a non-destructive one. The first method, which is also the most common one, is known as the cutback method [3.14]. In this method we measure the output power from a hollow waveguide with a known length. Afterwards we cut a piece of the waveguide and measure the output power and the waveguide’s length again. The waveguide’s attenuation is calculated as follows: for a given waveguide the attenuation per unit length, α, is measured in units of dB/m. The relation between the output power of a waveguide with different lengths, l, is Pl1 Pl 2 10 A 10 (3.5.1) where A is the total attenuation of the waveguide. Using equation 3.5.1 we get A 10 log P l1 P l2 and the attenuation per unit length is given by (3.5.2) 88 A l1 l2 (3.5.3) The following experimental setup was used to measure the waveguide’s attenuation. The laser beam was coupled to the hollow waveguide using a focusing lens, and its power at the waveguide’s output was measured using a detector. During the experiment the laser output power was kept constant. Surgical knife was to cut the waveguides. laser focusing lens hollow waveguide detector Figure 3.12 – Experimental Setup For Measuring The Waveguide’s Attenuation Although the cutback method is the most common one to measure the hollow waveguides attenuation, it has one major drawback. During the measurement process the waveguide is being destroyed. An alternative method that overcomes this drawback is the non-destructive method [3.15]. In this method the laser beam enters the waveguide at different points using another hollow waveguide. Usually a fused silica waveguide with inner diameter of 0.7mm is used to couple the laser beam into a waveguide with inner diameter of 1mm, a 0.5mm one is used to couple the beam into a 0.7mm waveguide etc. The non destructive method is a repetitive process and the attenuation is calculated in the same manner as in the first method. The two methods produce similar results. 89 3.2.1.2 Experimental method for measuring the attenuation of bent waveguides Hollow waveguide are used to deliver infrared radiation under straight and bent trajectories. Therefore it is very important to measure the waveguide’s attenuation as a function of the bending radius. Knowing this dependence will enable us to learn more about the hollow waveguides limitations and to design more suitable waveguides for medical and industrial applications. The following experimental setup (figure 3.13) was used to measure the waveguide’s attenuation as a function of the radius of curvature. The laser beam was coupled to the hollow waveguide using a focusing lens, the waveguide was bent using a bending rail with different radii of curvature, and the output power was measured using a detector. During the experiment the laser output power was again kept constant. The result of the experiment is a graph of the waveguide’s attenuation as a function of 1/R (R is the radius of curvature). waveguide bent rail laser focusing lens detector Figure 3.13 – Experimental Setup For Measuring A Bent Waveguide’s Attenuation 90 It is possible to measure the energy distribution outside the waveguide if we use a beam profiler instead of a regular detector. In this case we need to work with a pulsed laser or to use a chopper before the beam profiler. 3.2.2 The attenuation of straight hollow waveguides 3.2.2.1 Experimental results and theoretical calculations The attenuation of the different types of straight waveguides (fused silica, glass, Teflon and polyimide) was measured using one of the two methods that were described in section 3.5.1. I used a Synrad CO2 laser, a 50mm lens a an Ophir power meter or a Spiricon beam profiler. Figures 3.14 to 3.20 show the attenuation as a function of the waveguide’s length for different type of waveguides. Table 3.1 shows the measured attenuation, the calculated one, and the correlation between the experimental results and the theoretical model calculations for different type of hollow waveguides. 1.6 1.4 1.2 1 0.8 0.6 [dB ] Attenuation [dB] 0.4 0.2 0 60 70 80 90 100 110 Waveguide's Length [cm] experimental theoretical Figure 3.14 – Theoretical And Experimental Attenuation of a Straight Waveguide: Fused Silica Waveguide, ID=0.5mm, L=1m, Laser Wavelength=10.6m 91 [dB ] Attenuation 2 1.8 1.6 [dB] 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Waveguide's Length [m] experimental theoretical Figure 3.15 – – Theoretical And Experimental Attenuation of a Straight Waveguide: Fused Silica Waveguide, ID=0.7mm, L=1.9m, Laser Wavelength=10.6m 0.4 0.35 0.25 0.2 0.15 [dB ] Attenuation [dB] 0.3 0.1 0.05 0 50 55 60 65 70 75 80 85 90 Waveguide's length [cm] experimental theoretical Figure 3.16 – – Theoretical And Experimental Attenuation of a Straight Waveguide: Fused Silica Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m 92 [dB ] Attenuation 2 1.8 1.6 [dB] 1.4 1.2 1 0.8 0.6 0.4 0.2 0 40 50 60 70 80 90 100 110 Waveguide's Length [cm] experimental theoretical Figure 3.17 – Theoretical And Experimental Attenuation of a Straight Waveguide: Teflon Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m 1.4 1.2 0.8 0.6 0.4 [dB ] Attenuation [dB] 1 0.2 0 50 60 70 80 90 100 110 Waveguides's Length [cm] experimental theoretical Figure 3.18 Theoretical And Experimental Attenuation of a Straight Waveguide: Teflon Waveguide, ID=2mm, L=1.1m, Laser Wavelength=10.6m 93 [dB ] Attenuation 0.90 0.80 0.70 [dB] 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0 10 20 30 40 50 Waveguide's Length [cm] experimental theoretical Figure 3.19 – Theoretical And Experimental Attenuation of a Straight Waveguide: Glass Waveguide, ID=1.6mm, L=0.9m, Laser Wavelength=10.6m 1.2 1 0.6 0.4 [dB ] Attenuation [dB] 0.8 0.2 0 50 60 70 80 90 Waveguide's Length [cm] experimental theoretical Figure 3.20 – Theoretical And Experimental Attenuation of a Straight Waveguide: Polyimide Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m 94 Waveguide’s Inner Experimental Theoretical Correlation Experimental Type Diameter Attenuation Attenuation Method [mm] [dB/m] [dB/m] Fused Silica 0.5 0.78 0.81 0.7208 Cut back 0.7 0.54 0.67 0.9628 Cut back 1.0 0.43 0.35 0.9767 Nondestructive Teflon 1.0 1.86 1.64 0.9312 Cut back 2.0 1.59 1.56 0.9348 Glass 1.6 1.43 1.46 0.9699 Nondestructive Polyimide 1.0 2.30 2.46 0.9834 Nondestructive Table 3.1 – Theoretical And Experimental Attenuation For Different Types Of Hollow Waveguides 3.2.2.1 Experimental and theoretical beam shapes Knowing the attenuation of each type of hollow waveguide is not enough. For many applications, especially medical ones, it is also important to know what is the energy distribution at the output of the waveguide. Figures 3.21 and 3.22 show the experimental beam shape (a) and theoretical one (b) of a fused silica hollow waveguide (ID=1mm) and Teflon hollow waveguide (ID=1mm). One can see the similarity between the experimental beam shapes and the theoretical ones. 95 (a) (b) Figure 3.21 – Experimental (a) And Theoretical (b) Beam Shapes: Straight Fused Silica Waveguide, ID=1mm, L=1m (a) (b) Figure 3.22 – Experimental (a) And Theoretical (b) Beam Shapes: Straight Teflon Waveguide, ID=1mm, L=1m 3.2.3 The attenuation of bent hollow waveguides 3.2.3.1 Experimental results and theoretical calculations The attenuation of the different types of waveguides (fused silica, Teflon and polyimide) as a function of the bending radius was measured using the experimental method that was described in section 3.2.1. I used the same equipment as in the measurement of straight waveguides. Figures 3.23 to 3.28 show the attenuation as a function of the radius of curvature for different type of waveguides. Table 3.2 shows the dependence of the attenuation on the bending radius for the experimental results, the theoretical calculations ones, 96 and the correlation between the experimental results and the theoretical method. The dependence of the attenuation on the bending radius is given by A Rx (3.3.1) where x is given in table 3.2. 3.00 2.50 2.00 1.50 1.00 [dB ] Attenuation [dB] 0.50 0.00 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1/R [1/cm] Experimental Theoretical Figure 3.23 – Theoretical And Experimental Attenuation of Bent Waveguides: Fused Silica Waveguide, ID=0.5mm L=1m, Laser Wavelength 10.6m 3.5 3 2 1.5 1 [dB ] Attenuation [dB] 2.5 0.5 0 0 0.02 0.04 0.06 0.08 0.1 1/R [1/cm] experimental theoretical Figure 3.24 – Theoretical And Experimental Attenuation of Bent Waveguides: Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m 97 2.5 2 1.5 1 [dB ] Attenuation [dB] 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 1/R [1/cm] Experimental Theoretical Figure 3.25 Theoretical And Experimental Attenuation of Bent Waveguides: Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m 3.5 3 2 1.5 1 [dB ] Attenuation [dB] 2.5 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1/R [1/cm] Experimental Theoretical Figure 3.26 – Theoretical And Experimental Attenuation of Bent Waveguides: Teflon Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m 98 3 2.5 2 1.5 1 [dB ] Attenuation [dB] 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1/R [1/cm] Experimental Theoretical Figure 3.27 Theoretical And Experimental Attenuation of Bent Waveguides: Teflon Waveguide, ID=2mm L=1m, Laser Wavelength 10.6m 2.5 2 1.5 1 [dB ] Attenuation [dB] 0.5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1/R [1/cm] Experimental Theoretical Figure 3.28 – Theoretical And Experimental Attenuation of Bent Waveguides: Polyimide Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m 99 Waveguide’s Type Fused Silica Teflon Polyimide Inner Diameter [mm] 0.5 0.7 1.0 1.0 2.0 1.0 Experimental Theoretical Correlation Dependence Dependence -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0.9475 0.9891 0.7441 0.9549 0.9561 0.9742 Table 3.2 – Theoretical And Experimental Dependence Of The Attenuation On The Bending Radius For Different Types Of Hollow Waveguides 3.2.3.2 Energy distribution at the end of the waveguide [3.16] Knowing the attenuation as a function of length and radius of curvature is not sufficient. For medical purposes, knowing the beam shape of the laser beam outside the hollow waveguide is very important. Simple Gaussian beam shape will enable us to make clean cuts, while more complex beam shapes, will cause damage to the surrounding area. The new ray model also enables us to calculate the energy distribution outside the hollow waveguide. In order to compare between the two beam shapes let us define the parameter which is the ratio of the M2 factor of the experimental beam shape to the M2 [3.17] of the calculated beam shape. The beam quality M2 is the ratio of the laser beam's multimode diameter, Dm, to the diffractionlimited beam diameter, d0 [3.25]. D M m d0 M x2,exp x 2 M x ,cal 2 2 y M y2,exp M y2,cal (3.3.2) 100 A good correlation is achieved when is close to one. Figures 3.29 to 3.33 show the experimental beam shapes (a) and the theoretical ones (b) for a bent fused silica hollow waveguide (ID=1mm), for radii of curvatures of 20cm, 28cm, 36cm, 40cm, and 44cm respectively. The beam shapes were obtained using a pyroelectric camera (made by Spiricon). As can be seen from the images, the experimental beam shapes and theoretical ones are quite similar. . (a) (b) Figure 3.29 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides: Fused Silica Waveguide, ID=1mm, R=20cm (a) (b) Figure 3.30 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides: Fused Silica Waveguide, ID=1mm, R=28cm 101 (a) (b) Figure 3.31 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides: Fused Silica Waveguide, ID=1mm, R=36cm (a) (b) Figure 3.32 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides: Fused Silica Waveguide, ID=1mm, R=40cm (a) (b) Figure 3.33 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides: Fused Silica Waveguide, ID=1mm, R=44cm In order to get a quantitative result that describes the similarity of the beam shapes we calculated x and y. Table 3.3 summarize x and y for various radii of curvatures. 102 Radius of Curvature (cm) 20 28 36 40 44 x y 0.81 1.15 1.25 1.08 0.87 0.9 1.2 1.02 1.05 1.03 Table 3.3 3.2.4 The dependence of the attenuation on the waveguide’s radius 3.2.4.1 Experimental results and theoretical calculations I measured the attenuation of fused silica hollow waveguides as a function of the waveguide’s radius. Figure 3.34 shows the attenuation as a function of the waveguide’s radius. The correlation between the experimental results and the theoretical ones is practically one (0.9999). It is possible to find the dependence of the attenuation on the waveguide’s radius. The experimental results show that the attenuation is proportional to r-n where n=2.12±0.07 and the theoretical results show the attenuation is proportional to r-n where n=2.15±0.05. 1.6 1.4 0.8 0.6 [dB ] Attenuation [dB] 1.2 1 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Waveguide's radius [mm] experimental theoretical Figure 3.34 – Experimental And Theoretical Attenuation, Straight Fused Silica Waveguides 103 3.2.4.2 Energy distribution at the end of the waveguide [3.20] As was explained in the previous section it is important to measure the beam shape of the laser beam outside the hollow waveguide. Figures 3.34 to 3.36 show the experimental beam shapes (a) and the theoretical ones (b) for fused silica hollow waveguides with different radius. Again I used the test in order to get a quantitative result (table 3.4). (a) (b) Figure 3.24 – Experimental (a) And Theoretical (b) Beam Shapes: Fused Silica Waveguide, ID=0.5mm, L=1m (a) (b) Figure 3.25 – Experimental (a) And Theoretical (b) Beam Shapes: Fused Silica Waveguide, ID=0.7mm, L=1m 104 (a) (b) Figure 3.26 – Experimental (a) And Theoretical (b) Beam Shapes: Fused Silica Waveguide, ID=1mm, L=1m Waveguide’s Radius (mm) 0.25 0.35 0.5 x y 1.02 1.09 0.85 0.97 0.99 0.77 Table 3.4 3.2.5 The dependence of the attenuation on the coupling conditions 3.2.5.1 The influence of the focal length of the coupling lens The laser beam has to be coupled to the waveguide. The most common way to couple it is by using a simple lens. As I showed in section 3.1 the focal length of the coupling lens determines the laser spot size at the entrance to the waveguide and the laser beam divergence. When trying to couple a laser beam, one has to consider two elements: the laser spot size and the laser beam divergence. Smaller spot sizes enable us to couple the laser beam easily and prevent wall damage. However it also mean a large divergence angle and thus a larger attenuation. Figures 3.37 through 3.40 show the attenuation of a fused silica hollow waveguides (figures 3.37-3.39) and a glass hollow waveguide (figure 3.40). 105 Table 3.5 shows the correlation between the theoretical results and the experimental ones. 6.00 5.00 4.00 3.00 2.00 [dB ] Attenuation [dB] 1.00 0.00 0 20 40 60 80 100 Focal Length [mm] experimental theoretical Figure 3.37 – Theoretical And Experimental Attenuation for Different Focal Lengths: Fused Silica Waveguide, ID=0.5mm L=1m, Laser Wavelength 10.6m 6.00 5.00 3.00 2.00 [dB ] Attenuation [dB] 4.00 1.00 0.00 0 50 100 150 200 250 Focal Length [mm] experimental theoretical Figure 3.38 – Theoretical And Experimental Attenuation for Different Focal Lengths: Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m 106 [dB ] Attenuation 5.00 4.50 4.00 [dB] 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0 50 100 150 200 250 Focal Length [mm] experimental theoretical Figure 3.39 – Theoretical And Experimental Attenuation for Different Focal Lengths: Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m 6.00 5.00 3.00 2.00 [dB ] Attenuation [dB]4.00 1.00 0.00 0 50 100 150 200 250 Focal Length [mm] experimental theoretical Figure 3.40 – Theoretical And Experimental Attenuation for Different Focal Lengths: Glass Waveguide, ID=1.6mm L=1m, Laser Wavelength 10.6m Waveguide’s Type Fused Silica Glass Inner Diameter [mm] 0.5 0.7 1.0 1.6 Table 3.5 Correlation 0.9999 0.9891 0.9715 0.9918 107 3.2.5.2 Off center coupling Coupling the laser beam into the hollow waveguide is not easy. If the laser beam does not enter the waveguide at the center of its cross section it traverses the waveguide near the waveguide’s wall. This increases the laser beam attenuation due to increased absorption in the waveguide’s wall. 3.2.5.2.1 Experimental results and theoretical calculations The coupling position of the laser beam into the waveguide was changed by changing the position of the hollow waveguide by using a micrometer XYZ positioner. Figures 3.41 to 3.43 show the attenuation as a function of the distance from the waveguide’s cross section center, for different waveguides. As can be seen from the graph and from table 3.6, the experimental results and theoretical ones are very close. 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 (dB ) Attenuation [dB] -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Distance From The Waveguide's Center (mm) experimental theoretical Figure 3.41 – Theoretical And Experimental Attenuation for Off Center Coupling: Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m 108 8.00 7.00 6.00 [dB] Attenuation 5.00 4.00 (dB ) 3.00 2.00 1.00 0.00 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Distance From The Waveguide's Center (mm) experimental theoretical Figure 3.42 – Theoretical And Experimental Attenuation for Off Center Coupling: Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m 6.00 5.00 4.00 Attenuation [dB] 3.00 (dB ) 2.00 1.00 0.00 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Distance From The Waveguide's Center (mm) experimental theoretical Figure 3.43 – Theoretical And Experimental Attenuation for Off Center Coupling: Glass Waveguide, ID=1.6mm L=1m, Laser Wavelength 10.6m 109 Waveguide’s Type Fused Silica Glass Inner Diameter [mm] 0.7 1.0 1.6 Correlation 0.9897 0.9693 0.9834 Table 3.6 3.2.5.2.2 Energy distribution at the end of the waveguide [3.20] It is important to measure the beam shape of the laser beam outside the hollow waveguide. Figure 3.44 to 3.46 show the experimental beam shapes (a) and the theoretical ones (b) for different deviations from the center of a fused silica hollow waveguide. (a) (b) Figure 3.44 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.15mm from the center (a) (b) Figure 3.45 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.3mm from the center 110 (a) (b) Figure 3.46 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.4mm from the center It can be seen that the theoretical beam shapes are similar to the experimental ones. 3.3 Discussion 3.3.1 Straight hollow waveguides As can be seen from the graphs (figures 3.14 to 3.20) and from table 3.1, the theoretical calculations are similar to the experimental results. Moreover the correlation between the experimental results and the theoretical calculations are more than 0.9 in most cases. It is noticed that there is a difference between the waveguides attenuation. The difference between waveguides within one type is due to the differences in the waveguide’s inner diameter and the surface roughness (the surface roughness varies for different tubes). The larger the waveguide’s inner diameter the smaller its attenuation. This was shown theoretically in section 3.1. The difference in attenuation among waveguides with the same inner diameter that are made of different types of tubes is due to the differences in the surface scattering. In chapter 2 I showed the measurements of the surface roughness for 111 different types of waveguides. The larger the surface roughness the larger the attenuation. Since Teflon waveguides have larger surface roughness their attenuation is larger as well. The surface roughness influences the beam shape of the transmitted laser beam as well. We can see from Figures 3.21 and 3.22 that the beam shape of the Teflon waveguide is more complex and it contains higher modes than the beam shape of the fused silica waveguide. The difference in the beam shapes is due to the scattering from the surface. The larger the surface roughness the larger the scattering and as a consequence there is more mode coupling between lower modes and higher modes. This coupling increases the attenuation and gives rise to a more complex beam shapes. The scattering from the rough surface changes the angle of incidence of the rays and thus mode coupling occur. Although our experimental results fits the theoretical calculations of the ray model they are different from the ones obtained by Harrington et al. [3.21]. Our experimental results differs from the ones achieved by Harrington et al. since we did not used optimal coupling conditions and took into account the surface roughness. 3.3.2 Bent hollow waveguides As can be seen from figures 3.23 to 3.28 and table 3.2 the results of the theoretical calculations are similar to the experimental results. Moreover the correlation between the experimental results and the theoretical calculations is more than 0.95 for most cases. 112 As expected the attenuation increases with the decrease in the bending radius. This is due to the change in the angle of incidence caused by bending the waveguide. Bending the waveguide causes the angle of incidence to decrease, thus increasing the attenuation. The change in the angle of incidence also causes mode coupling. Usually lower modes are coupled to higher ones. It is worthwhile to note that for most cases the attenuation increases when the bending radius decreases. However for special bending radius the attenuation decreases due to the presence of whispering gallery modes. As can be seen from table 3.2 the attenuation changes as 1/R. This result is with accordance to other works that has been done in the past [3.18, 3.19]. These works also show that the attenuation dependence on the bending radius is -1. The correlation between the experimental beam shapes and the theoretical beam shapes is very good. The shapes differ by no more than 20%. The similarity between the beam shapes can also be seen through figures 3.29 to 3.33. Although the original shape of the laser beam at the waveguide’s entrance is Gaussian, the beam shape the beam shape at the distal end is more complex. This is due to mode coupling caused by bending. 3.3.3 The dependence on the waveguide’s inner diameter In the theoretical section (section 3.1) I showed that the waveguide’s attenuation is proportional to r-n where n=2, where r is the waveguide’s radius. In this section I showed that n=2.12 and 2.15 for the experimental and theoretical attenuation respectively. This result is very close to the theoretical 113 prediction. The deviation is due to the influence of scattering, which is taken into account in the simulations, but not in the analytical derivation of the attenuation dependence (equation 3.3.9). The experimental beam shapes and the theoretical ones are similar, as can be seen from figures 3.34 to 3.36. The deviation of the factor is less than 25%, which shows a good similarity between the beam shapes. As can be seen from figures 3.34 trough 3.36 the beam shapes of different waveguides differ from one another. The beam shapes of the waveguides that have larger inner diameter have more secondary modes. This difference is due to the fact that higher modes have smaller angles of incidence with the waveguides wall and thus are highly attenuated. Since the attenuation is a function of the waveguides radius, higher modes will be more attenuated in smaller core waveguides than in larger core ones. This is clearly shown in the beam shapes figures. 3.3.4 The influence of coupling conditions The coupling conditions have a great influence on the attenuation and beam shapes of hollow waveguides. The coupling lens determines the spot size of the laser beam and its numerical aperture (NA). The smaller the focal length of the coupling lens the larger the NA and thus the attenuation increases. This is clearly shown in figures 3.37 to 3.40. One has to use the appropriate lens when coupling the laser beam to the hollow waveguides. It is well known that the desired ratio between the spot size and the waveguide’s inner diameter is 0.6. 114 Coupling the laser beam off center with regards to the waveguide’s crosssection causes the attenuation to increase and the beam shape to be spoiled. It is obviously seen from figures 3.41 to 3.46 that when the beam is further out of the center, the beam is “spoiled”, symmetry is gone, the main peak is reduced and higher modes appear. We can also see that our model predicts well the experimental results. The loss of symmetry is due to the large attenuation of the lower modes, which, propagate now near the waveguide’s wall. When the lower modes are highly attenuated, the higher modes are more emphasized. 3.4 Conclusion The comparison between the experimental result and the theoretical calculation that was made in this chapter shows that the theoretical model gives the correct results. The model predicts accurately the waveguide’s attenuation and the laser beam shape outside the hollow waveguide for complicated situations such as bending and off center coupling. This is useful for designing new kinds of hollow waveguides and improving the current ones. The model is also useful in finding new applications for hollow waveguides. As was mentioned earlier, infrared radiation may be used for medical applications. The model may predict for example how much the laser beam will be attenuated when it is transmitted through endoscopes and how the beam shape would look like at its end. 3.5 Summary of chapter 3 In this chapter I described the improved ray model. Although ray models have been used in the past, they have not taken into account the surface roughness in 115 such a manner that they could predict correctly the performance of hollow waveguides. Moreover, the improved ray model has more features that enables analyze complex situations correctly. Whispering gallery modes enable to analyze bent hollow waveguides correctly and off center coupling may predict the complexity in the beam shape. The second section of the chapter is a characterization of hollow waveguides, their performance and the parameters that influence their attenuation and the shape of the output beam. The experimental results are also used to examine the validity of the improved ray model. As can be seen from the above graphs and beam shapes, the improved ray model predicts the performance of hollow waveguides very well. The theoretical calculations and the experimental results enable us to better understand the influence of different parameters on the performance of hollow waveguides and to design hollow waveguides that suit a desired application. 116 3.6 References for chapter 3 3.1 E. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers”, The Bell System Technical Journal, July 1964, p. 1783. 3.2 J. A. Harrington, “Laser Poewr Delivery in Infrared Fiber Optics”, SPIE 1649, Optical Fibers in Medicne 1992. 3.3 M. Miyagi and S. Kawakami, “Design Theory of Dielectric-Coated Circular Metalic Wavwguides for Infrared Transmission”, Journal of Lightwave Technology, 1984, p.116. 3.4 miyagi 3.5 D. Mendelovic, E. Goldenberg, S. Rushin, J. Dror and N. Croitoru, “Ray Model for Transmission of Metalic-Dielectric Hollow Bent Cylindrical Waveguides”, Applied Optics, 1989, v. 28, p. 708. 3.6 O. Morhhaim, D. Mendelovic, I. Gannot, J. Dror and N. Croitoru, “Ray Model for Transmission of Infrared Radiation Through Multibent Cylindrical Waveguides”, Optical Engineering, 1991, v. 30, p. 1886. 3.7 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Spectral attenuation of incoherent IR light in circular hollow waveguides”, Applied Optics v. 27, n. 20, p. 4169-4170. 3.8 K. Takatani, Y. Matsuura and M. Miyagi, “Theoretical and experimental investigations of loss behavior in the infrared quartz hollow waveguides with rough inner surfaces”, Applied Optics v. 34, n. 21, p. 4352-4357. 117 3.9 A. Inberg , M. Ben-David, , M. Oksman, A. Katzir and N. Croitoru, “Theoretical and Experimental Studies of Infrared Radiation Propagation in Hollow Waveguides”, Optical Engineering, 2000, 39, 5, 1384-1390. 3.10 M. Ben-David, A. Inberg, A. Katzir and N. Croitoru, “Hollow Waveguides for Infrared Radiation”, Chemistry and Chemical Engineering, Journal of the Chemical Engineers and Chemist, The Association of Engineers and Architects in Israel, No. 34-35, 12-18, 1999. 3.11 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light”, J. Opt. Soc. Am. A, v. 6, n. 3, p. 423-427 3.12 R. L. Kozodoy, A. T. Pagkalinawan, J. A. Harrington, “Small bore hollow waveguides for delivery of 3m laser radiation”, Applied Optics v. 35, n. 7, p. 1077-1082. 3.13 M. Ben-David, I. Gannot, A. Inberg and ,N. Croitoru, “Bending Effect on IR Hollow Waveguides Transmission”, EBIOS 2000, 4-8 July, Amsterdam Netherland. 3.14 M.Bass “Handbook of Optics”, second edition, McGraw Hill 1995. 3.15 R. Dahan, J. Dror, A. Inberg and N. Croitotu, “Nondestructive Method for Attenuation Measurement in Optical Hollow Waveguides”, Optics Letters, 1995, v. 20. 3.16 M. Ben-David, I. Gannot, A. Inberg and ,N. Croitoru, “Bending Effect on IR Hollow Waveguides Transmission”, EBIOS 2000, 4-8 July, Amsterdam Netherland 118 3.17 O. Svelto, “Principles of Lasers”,1998, 4th ed.:481-482, Plenum Press 3.18 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light”, J. Opt. Soc. Am. A, v. 6, n. 3, p. 423-427 3.19 R. L. Kozodoy, A. T. Pagkalinawan, J. A. Harrington, “Small bore hollow waveguides for delivery of 3m laser radiation”, Applied Optics v. 35, n. 7, p. 1077-1082. 3.20 I. Gannot, M. Ben-David, A. Inberg, N. Croitoru, and R. Waynant, “Beam Shape analysis of waveguide delivered IR lasers”, Optical Engineering, Accepted for publication 2001. 3.21 Y. Matsuura, T. Abel, and J. A. Harrington, "Optical properties of smallbore hollow glass waveguides", Applied Optics, 34, 6842-6847, (1995). 3.22 Saito, Sato, and Miyagi, JOSA A vol. 10, pg. 277-282 (1993). 3.23 Abe et al., IEEE Transactions on Microwave Theory and Techniques, v. 39 n. 2, February 1991. 3.24 Matsuura et al., Applied Optics, v. 31, n. 30 October (1992). 3.25 O. Svelto, “Principles of Lasers”. 119 Chapter Four - Pulse Dispersion In Hollow Waveguides List of Symbols For Chapter 4 c .............. Speed of light fast .......... The travel time for the fastest mode slow ......... The travel time for the slowest mode ............ Pulse dispersion Angle of propagation L ............. waveguide’s length The previous chapter dealt with the influence of hollow waveguides on the spatial distribution of the laser beam. However the laser beam is influenced by the hollow waveguide in the time domain as well. For medical purposes such as laser tissue interaction and diagnostics there is a need to understand the delivery of short laser pulses through hollow optical waveguides. In this chapter, I describe a theoretical model for calculating the pulse dispersion within the hollow optical waveguide and the experimental results of pulse dispersion, which correspond to these calculations [4.5]. Although pulse dispersion has been discussed in many papers, such as in [4.6], it is always with regard to a single mode of propagation and the calculations are done using mode theory. However in many cases the laser beam is a multi-mode which makes the calculation very difficult, and previous calculations did not take into account difficult situations such as surface roughness, bending, off center coupling etc. 120 The ray model that I have developed enables to calculate the pulse dispersion for many complicated situations. Moreover it is very straightforward, simple to code into a computer and it does not require a long running time. These advantages along with good agreement between the theoretical calculations and the experimental results (as will be shown), make the ray model a very useful tool in evaluating the pulse dispersion. 4.1 Theory Different light components (modes or wavelengths) travel through the waveguide with different velocities or through different optical paths. Therefore, laser pulses that travel along the fiber may become longer by a certain amount of time that we will call . This phenomenon is well known and is called pulse dispersion. In a multi-mode fiber, the dominant dispersion mechanism is the modal or inter-modal dispersion (i.e., different modes travel in different optical paths because of the different incidence and reflectance angles at and from the reflecting/refracting layers). In a single-mode fiber, the dominant dispersion mechanism is the chromatic dispersion (i.e., different wavelengths travel with different velocities). 4.1.1 Pulse dispersion in straight and smooth multi-mode fibers Different modes travel at different optical paths through the fiber and the result is a modal dispersion that can be evaluated using figure 4.1 [4.1]. 121 2 0 Figure 4.1 The shortest path traveling mode in figure 1 corresponds to ray 0, which travels directly to the destination along the fiber axis. The travel time for that mode is fast Ln1 c (4.1.1) where L is the fiber length, and n is the refractive index of the core. 1 The longest path of a traveling mode corresponds to ray 2, which travels with the maximum angle with respect to the fiber axis. (This is limited by NA of a fiber). In the hollow waveguide case there is no real NA but a limiting input angle, which is 25o full angle or “NA” of 0.22). The travel time for this mode is slow Ln1 c cos (4.1.2) The modal dispersion is obtained by subtracting the two equations [4.8] Ln1 1 1 c cos (4.1.3) As can be seen from equation 4.1.3 the pulse dispersion depends on the fiber material properties, its length and the incidence angle of the input ray. 4.1.2 Pulse dispersion in real hollow waveguides The calculation mentioned above is useful only for straight and smooth waveguides. However in a practical real life use, hollow waveguides are being bent and they have a certain amount of surface roughness. Bending and roughness cause a change of the incidence angle that leads to increasing pulse 122 dispersion. In order to calculate the real pulse dispersion under these conditions, we can use the ray model, which we described in chapter 3 [4.2]. Our ray model calculates the new angle of incidence after each time a ray impinges on the waveguide’s wall taking into account the roughness of the surface, which causes scattering of the impinging ray. Knowing the correct angle of incidence enables us to calculate the ray trajectory along the waveguide and from it, the time it takes the ray to exit at the distal tip. The pulse dispersion is therefore given by: slow fast (4.1.4) 4.2 Experimental results and discussion 4.2.1 Experimental setup There are several methods to measure the pulse width of a laser beam. One of them is to use an autocorelator setup [4.4]. We measured the pulse dispersion directly, i.e. we measured the pulse width at the waveguide entrance and the pulse width at the output of the waveguide. The pulse dispersion is the difference between them. We chose this method since the theoretical calculations have shown that the pulse dispersion would be in the order of nano-seconds, i.e. the magnitude of the pulse itself. Figure 4.2 shows the experimental setup that was used to measure the pulse width of the laser beam after propagating through the waveguide. We used a Qswitched Er-YAG laser (MSQ, Israel); the laser wavelength was 2.m and pulse width 70nm. The waveguide was manufactured by our group. We used a 3m long fused silica waveguide with inner diameter of 0.7mm. The waveguide 123 was bent at a constant radius (8cm) at different bending angles i.e. several turns for angles greater than 360o. For each bending we measured the pulse width using a fast IR detector (VIGO by Boston Electronics, MA). The received signal was captured by HP 500 MHz digital oscilloscope and transferred to a Microsoft EXCEL program on the PC through HP BenchLink XL Software. There are several ways to couple the laser beam to the waveguide, using conventional lens or other coupling devices. We coupled the laser beam to the waveguide using a glass funnel [4.3]. The funnel has two functions. The first is to smoothen the laser beam energy. The second is to decrease the laser’s N.A., because the funnel has a low N.A.. The funnel we used has NA of 0.033, which corresponds, to a very small input angle of 3.8o full angle. Oscilloscope Computer Q switched Funnel waveguide Fast IR detector Er-YAG Figure 4.2 – Experimental Setup 4.2.2 Experimental results and discussion We measured the pulse dispersion at various bending angles. Figure 4.3 (a and b) shows the experimental results (points), their best fit curve (dotted line) and the theoretical calculations for a waveguide with (solid line) and without roughness (squares). As can be seen from figure 4.3 there is a good correlation 124 between the experimental results and the theoretical ones (the calculated correlation between the curve fit and the theoretical calculations is 0.92). Moreover it can be seen that the major contribution to the pulse dispersion is due to the surface roughness. The figure also shows that the pulse dispersion increases as the angle of bending increases. This is due to mode coupling between lower modes of propagation and higher ones. The mode coupling is caused by two mechanisms. The first is scattering from a rough surface. Each time a ray impinges on the waveguide’s wall it scatters, which means that its angle of reflection is different from the angle of incidence. Since each angle corresponds to a different mode of propagation, mode coupling occurs. The second mechanism is bending. Here changing the radius of the curvature changes the angle of incidence, and therefore the mode of propagation. The two mechanisms cause lower modes to couple to higher ones. The coupling cause a longer optical path and thus to a significant pulse dispersion even in a straight waveguide. Pulse Dispersion [nsec] 125 50 40 30 20 10 0 0 200 400 600 800 Bending [degrees] experimental best fit theoretical - with roughness theoretical - without roughness (a) Pulse dispersion [nsec] 0.25 0.2 0.15 0.1 0.05 0 0 200 400 600 800 Bending [degrees] theoretical - without roughness (b) Figure 4.3 – Experimental and Theoretical Results (a – all results, b – without roughness) The measurements and calculations correspond to the optical funnel used. However, calculation can be made for each input angle, which is generated by a focusing lens. Usually, we use lenses with focal lengths between 50 and 150 mm. We repeated those calculations to even a broader range of 25 to 200 mm. 126 We obtained pulse broadening as a function of focal length. This can be seen in the graph in Fig 4.4. As can be seen from the figure a smaller focal length leads to a larger pulse dispersion. The theoretical pulse dispersion is about 10% of the original for a 25mm focal length, and almost negligible compared to the original pulse at focal lengths higher than 100mm. These theoretical calculations are in agreement with the experimental results, which were obtained by Prastisto et. Al [4.4]. They measured the pulse dispersion of micro pulses from FEL laser. They found an increase of about 50% in the pulse width for a small focal length and a negligible at longer focal lengths. Figure 4.4 Although the relative pulse dispersion of Paratisto et al., Matsuura et al. [4.7] and ours are similar (about 50%). The absolute value of the pulse dispersion in each experiment is different. The difference among the different pulse dispersion measurements (Partisto et al. Matsuura et. al and ours) is due to different experimental conditions, laser type, pulse width, mode distribution, 127 fiber length, bending radius and coupling conditions. While Matsuura et al. use almost a single mode laser beam and grazing angle coupling conditions (f=300mm) our beam structure is more complex and is composed of a lot of modes. This leads to larger pulse dispersion. I used the ray model to calculate the pulse dispersion of Matsuura et al. laser pulse using the same setup they used (f=300mm, l=1m, ID=1mm, pulse width 196fs and =775nm) for a waveguide without roughness and with roughness. The results are summarized in table 4.1. Pulse dispersion (fs) Ben-David et al. ray model MatSuura et al. without roughness with roughness Straight waveguide 21 25 17 Bending waveguide 46 127 Max: 178 (R=50cm) Average: 133 Table 4.1 As can be seen from the table the results are similar. 4.3 Summary I studied the temporal behavior of a laser pulse after it was transmitted through an IR hollow core optical waveguide. Using the ray model enables to analyze the pulse dispersion in hollow waveguides. The model takes into account the waveguide’s rough surface and enables to estimate the pulse dispersion in complicated situations such as bending and multi-mode laser beams. We measured experimentally pulse broadening and compared it to our theoretical calculations. The experimental results were in agreement with the calculations. 128 These results help us to understand the laser pulse behavior in hollow waveguides. It will help us to characterize laser pulses to transmitted tissue and may assist in designing surgical or diagnostic procedures. 129 4.4 References for chapter four 4.1 L. Kazovsky, S. Benedetto, A. Willner, “Optical Fiber Communication Systems”, Artech House, 1996. 4.2 I. Gannot, M. Ben-David, A. Inberg, N. Croitoru, and R. Waynant, “Beam Shape analysis of waveguide delivered IR lasers”, Optical Engineering, January 2002. 4.3 I. Ilev and R. Waynant, "Grazing-incidence-based hollow taper for infrared laser-to-fiber coupling", Applied Physics Letters 74, 2921 (1999). 4.4 H.S. Pratisto, S. R. Uhlhorn and E. Duco Jansen, “Beam Delivery of the Vanderbilt Free Electron Laaser with Hollow Wave Guides: Effect on Temporal and Spatial Pulse Propagation”, Fiber and Integrated Optics; 20, p. 83-94, 2001. 4.5 I. Gannot, M. Ben-David and Ilko K. Ilev, “Pulse Dispersion in Hollow Optical Waveguides”, submitted for publication. 4.6 Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljacic, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers”, Optics Express; v. 9, n. 13, p. 748-778, December 2001. 4.7 Y. Matsuura and M. Miyagi, “Delivery of Fentosecond Pulses by Flexible Hollow Fibers”, J. of Applied Physics, v.91 n.2, 2002. 4.8 D. Sadot, Notes from the course “Optical Communication” Ben-Gurion University”. 130 Chapter Five - Improving The Hollow Waveguides List of Symbols For Chapter 5 c .............. Speed of light B ............. Magnetic field E ............. Electric field .............. Charge density J .............. current density D ............. Displacement field .............. Dielectric constant r .............. radius vector t............... time ............. angular frequency ............. Operator a .............. Lattice constant u .............. Wave function iAngle of incidence tAngle of transmittance rAngle of reflection k.............. Wave vector d .............. Layer thickness r .............. Reflection coefficient t............... Transmission coefficient Wavelength 131 As we saw in chapter 3, hollow waveguides have two drawbacks: high attenuation when bend and high sensitivity to the laser beam’s angle of incidence. Moreover, each waveguide is “tuned” for delivering only a specific wavelength [5.13]. One of the ways to overcome these drawbacks is to introduce the notion of photonic crystals. Photonic crystals are materials patterned with a periodicity in dielectric constant, which can generate (as will be shown later) a range of “forbidden” frequencies called photonic bandgaps. Photons with energies lying in the bandgap cannot propagate through the medium. This provides a way to construct materials with high reflectance (R=1), such as a “perfect mirror”, laser cavities etc. The theoretical study of photonic crystal began in the early sixties. Yariv et al. [5.1] showed that a periodic multilayer structure of dielectric materials might reflect photons completely, thus generating a “perfect mirror”. Later on Joannopoulos et al. [5.2] showed that the phenomena could be expanded to two and three dimensions. Perfect mirrors made of dielectric materials have been made by many companies and research groups [5.6, 5.8]. Most recently Fink et al. [5.7] reported of a “perfect” mirror made of polystyrene and tellurium. Although the idea of photonic crystals has been known for many years, only one dimensional structures i.e. multi-layer mirrors have been built. Only recently there have been successful attempts to manufacture photonic crystal fibers. These fibers could be categorized into two groups. The first one is multi layer hollow waveguides [5.14] and the second one is “holey fibers” [5.15]. 132 Building long multi-layer hollow waveguide is not an easy task and was not accomplished yet due to manufacturing problems. One of the first attempts at achieving a PBG structure was the “holey” fiber; an ordinary, pure silica fiber with a triangular array of axial air holes running the length of it; see Figure 5.0. A single, missing air hole in the center, the “defect”, was predicted to exhibit wave guidance by the PBG effect [5.16]. It did exhibit wave guidance, but the structure was not precise enough to produce the required band gaps. In 1999, Crystal Fibre A/S was founded in Denmark to mass-produce and commercially sell these fibers. Although they do not operate on the photonic bandgap principle, these fibers are commonly called Photonic Crystal Fibers (PCFs). pitch cladding a Air holes Figure 5.0 General structure of a holey waveguide Triangular lattice fibers have some advantages over the more exotic designs that have been proposed for true PCFs. They can be produced with ordinary silica fiber drawing technologies [5.17]. Hollow silica tubes are bundled together in the desired configuration. Solid glass rods are added to the center to 133 form the defect, and the whole structure is heated and drawn out on an ordinary fiber-drawing tower [5.18]. While the holes do not retain their relative sizes and locations perfectly, the general triangular lattice structure survives. The resulting fibers are not regular enough to exhibit photonic bandgaps, but they are regular enough to display wave guidance based on the “effective index difference” between the defect and the lattice. Crystal fibers have some very desirable optical properties. They can be made single mode over a large bandwidth, while maintaining large core diameter, and still have reasonable performance at low wavelengths. They are also comparatively simple to manufacture, and the critical parameter a/(this parameter determines the bandwidth) can be adjusted with relative ease. While crystal fibers have certain desirable optical properties, they also have some undesirable mechanical properties. The bridges” between air holes are thin and prone to breakage, especially given the brittle nature of silica. Crystal fibers tend to crack around the holes and lose chunks of glass as the fiber pieces are separated. Finally, it is difficult to use fiber-bonding epoxies on a crystal fiber, as the epoxy could plug the air holes, increasing their refractive index and killing wave guidance at the entrance of the fiber. In this chapter I will propose a way to improve the current hollow waveguides by using a multi-layer structure also known as a photonic crystal. The first part of the chapter describes the theory behind photonic crystals. It includes a 134 description of photonic crystals, using Bloch theory and the more conventional Maxwell equation approach. The second part describes our efforts to design and manufacture a multi-layer mirror as a step in fabricating a multi-layer hollow waveguide. Many researches around the world have been designing and manufacturing multilayer mirrors. However none of the methods used are applicable to the manufacturing of long hollow waveguides. We propose a new approach, using metal and dielectric layers instead of two dielectric layers. This approach has two advantages: the first is that the method of deposition may be used for long tubes and the second one is that using metal and dielectric layers enables us to use a smaller number of layers (in fact two pairs is enough) as opposed to the large number of layers needed for dielectric structures. The last section shows a theoretical calculation that compares the performance of multi-layer hollow waveguide to a conventional one. 135 5.1 Theory There are two ways for analyzing photonic crystals. The first is to look at the material as a lattice and to solve Maxwell equations using the symmetry principles, which lead to Bloch theory [5.1, 5.2]. The second is to solve Maxwell equations with the appropriate boundary conditions [5.3]. 5.1.1 Bloch theory 5.1.1.1 The macroscopic Maxwell equations All macroscopic electromagnetic phenomena, including the propagation of light in a photonic crystal, is governed by the four macroscopic Maxwell equations. In cgs units, they are B 0 1 B 0 c t D 4 E H (5.1.1) 1 D 4 J c t c where E and H are the macroscopic electric and dielectric and magnetic fields, D and B are the displacement and magnetic induction fields, and and J are the free charges and currents. We will restrict ourselves to propagation within mixed dielectric medium with no free charges or currents J 0 . Next we relate D to E and B to H with the constitutive relations appropriate to our problem. Quite generally, the components Di of the displacement field D are related to the components Ei via the power series, Di ij E j k ij Ei E j OE 3 j j (5.1.2) 136 However, for many dielectric materials, it is good approximation to employ the following assumptions. First, we assume that the fields’ strengths are small, so that we are in the linear region, and we can use only the first term in 5.1.2. Second, we assume that the material is macroscopic and isotropic, so that Er, , Dr, are related by a scalar dielectric constant r, . Third, we ignore any explicit frequency dependence of the dielectric constant. We simply choose the value of the dielectric constant appropriate to the frequency range of the physical system we are considering. Fourth, we focus only on low loss dielectrics, which means we can treat r as purely real. When all is said and done, we have Dr r Er . An equation similar to (5.1.2) relates B and H. However, for most dielectric materials the magnetic permeability is very close to unity and we may assume B=H. With all these assumptions in place, the Maxwell equations (5.1.1) become H r, t 0 1 H r, t 0 c t r Er 0 Er, t H r, t (5.1.3) 1 Er, t 0 c t In general both E and H are complicated functions of time and space. But since the Maxwell equations are linear, we can separate out the time dependence by expanding the fields into a set of harmonic modes. Maxwell equations impose restrictions on a field pattern that happens to vary sinusoidally (harmonically) with time. The solutions for the Maxwell equations are Hr, t Hr ei t Er, t Er ei t (5.1.4) 137 To find the equations for the mode profiles of a given frequency, we insert the above equations into (5.1.3). The two divergence equations give the simple conditions: Hr Er 0 (5.1.5) These equations have a simple physical interpretation. There are no point sources or sinks of displacements and magnetic fields in the medium. The two curls relate E(r) and H(r): i H r 0 c i H r r Er 0 c Er (5.1.6) We can decouple these equations in the following way. Divide the bottom, equation of (5.1.6) by r , and then take the curl. Then use the first equation to eliminate E(r). The result is an equation entirely in H(r): 1 Hr Hr r c 2 (5.1.7) This is the master equation. In addition to the divergence equation (5.1.5), it determines H(r). Now we can find the magnetic field for a given photonic crystal, and using the first equation (5.1.6) we can derive the electric field ic Hr Er r (5.1.8) From the master equation we can derive the allowable electromagnetic modes. If H(r) is an allowable mode, the other results will just be constants times the original function H(r). This situation arises often in mathematical physics, and is called an eigenvalue problem. If the result of an operation on a function is 138 just the function itself, multiplied by some constant, then the function is called an eigenfunction or eigenvector of the operator, and the multiplicative constant is called the eigenvalue. In this case we identify the left side of the master equation as an operator acting on H(r) to make it look like an eigenvalue problem: H r H r c (5.1.9) 1 Hr Hr r (5.1.10) 2 where The eigenvectors H(r) are the field patterns of the harmonic modes and the eigenvalues / c 2 are proportional to the squared frequencies of those modes. Notice that the operator is a linear operator. Our eigenvalue problem resembles the eigenvalue problems in quantum mechanics, where a Hermitian operator (the Hamiltonian) is operated on the wave function and the eigenvalues can be derived by the variation principle and symmetry properties. These useful similarities can be useful in the electromagnetic case as well. 5.1.1.2 Translational symmetry and solid-state electromagnetism In both classical mechanics and quantum mechanics, we learned the lesson that symmetries of a system allow one to make general statements about that system’s behavior. 139 There are several types of symmetries: inversion, continuous translation and discrete translation. Since a photonic crystal material has a periodic structure of the dielectric constant we will consider translation symmetry. Photonic crystals, like the familiar crystals of atoms, have discrete translation symmetry. That is, they are not invariant under translations of any distance – only under distances that are a multiple sum of a fixed step length. The simplest example of such a system is a structure that is repetitive in one direction (figure 5.1). z a y x Figure 5.1 For this system we have continuous translational symmetry in the x direction, but we have also discrete translational symmetry in the y direction. The basic step length is the lattice constant a, and the basic step vector is called the primitive lattice vector, which in this case is a ayˆ . Because of the symmetry, r r a . By repeating this translation, we see that r r R for any R that is an integral multiple of a. Because of the translational symmetries, must commute with all the translation operators in the x direction, as well as the translation operators for lattice vectors R layˆ (l is an integer) in the y direction. With this knowledge 140 we can identify the modes of as simultaneous eigenfunctions of both translational operators. These eigenfunctions are plane waves: Tdxˆ eikx x eikx x d eikx d eikx x TR e ik y y e ik y y la e e ik y la (5.1.11) ik y y Where Tdx and TR are the translation operators in x and y directions respectively. We can begin to classify the modes by specifying k x and k y . However not all values of k y yield different eigenvalues. Consider two modes, one with wave vector k y and the other k y 2 . A quick insertion into the above equations a shows that they have the same TR eigenvalues. In fact, all of the modes with wave vectors of the form k y m 2 , where m is an integer, form a degenerate a set: they all have the same TR eigenvalues of e integral multiple of b ik y la . Augmenting k y by an 2 leaves the state unchanged. We call b the primitive a reciprocal lattice vector. Since any linear combination of these degenerate eigenfunctions is itself an eigenfunctions with the same eigenvalues, we can take linear combination of our original modes to put them in the form H k x ,k y r eikx x ck y ,m z e m i k y mb y eikx x e ik y y c z e k y ,m imby eikx x e ik y y uk y y, z (5.1.12) m where the c’s are expansion coefficients to be determined by the explicit solution, and uk y, z is a periodic function of y. y 141 The discrete periodicity in the y direction leads to a y dependence for H that is simply the product of a plane wave with a y periodic function. We can think of it as a plane wave, as it would be in free space, but modulated by a periodic function because of the periodic lattice: H ... y....e ik y y uk y y... (5.1.13) This result is commonly known as Bloch’s theorem. In solid-state physics the above equation is known as Bloch state. One key fact about Bloch state is that the Bloch state with wave vector k y and the Bloch state with the wave vector k y mb are identical. The k y ’s that differ by integral multiple of b are identical in the physical point of view. Thus the mode frequencies must also be periodic in k y : k y k y mb . In fact we only need to consider k y in the region a k y a which is called the Brillouin zone. This result can be expanded to three dimensional photonic crystal [5.2]. 5.1.1.3 Photonic band structure We saw that it is possible to derive the solutions for the wave equations from symmetry principles. All the information about the modes is given by the wave vector k and the periodic function uk r . To solve for uk r we insert the Bloch state into the master equation: 142 k H k Hk c 2 1 k ikr e ikr u k r e u k r r c 2 ik 1 ik uk r k uk r r c 2 (5.1.14) k k u k r u k r c 2 Here we have defined k as a new Hermitian differential operator that depends on k: 1 ik k ik r (5.1.15) The function u, and therefore the mode profiles, are determined by the eigenvalue problem in the forth equation in 5.1.14, subjected to the condition uk r uk r R (5.1.16) Because of the periodic boundary condition, we can regard the eigenvalue problem as being restricted to a unit cell of the photonic crystal. This leads to a discrete spectrum of eigenvalues. We can find for each value of k an infinite set of modes with discretely spaced frequencies, which we can label with a band index n. For a multilayer film, which is composed of two materials, Yablonovich [5.4] showed that there could exist a band gap in the frequency band structure. In this band gap light with frequency that lies within the band gap is not allowed to propagate and is reflected completely. 143 5.1.2 Maxwell equations Consider the linearly polarized wave shown in figure 5.2 [5.3], impinging on a thin dielectric film between two semi-infinite transparent media. Each wave ErI, E’rII, EtII, and so forth, represents the resultant of all possible waves traveling in that direction, at that point in the medium. The summation process is therefore built in. The boundary conditions require that the tangential components of both the electric field, E, and the magnetic field, H, be continuous across the boundaries. HiI EiI ErI kiI iI I n0 EtI iII d ErII EiII n1 II ns tII HtII EtII KtII Figure 5.2 Fields at the boundaries 144 At boundary I ' EI EiI ErI EtI ErII (5.2.1) and HI 0 EiI ErI n0 cos iI 0 EtI ErII' n1 cos iII 0 0 (5.2.2) where use is made at the fact that E and H in non-magnetic media are related through the index of refraction and the unit propagation vector: H 0 ˆ nk E 0 (5.2.3) At boundary II EII EiII ErII EtII (5.2.4) and H II 0 EiII ErII n1 cos iII 0 EtII ns cos tII (5.2.5) 0 0 the substrate having an index ns. A wave that transverses the film once undergoes a phase shift of k0 h k0 2n1d cos iII 2 , so that EiII EtI exp k0 h (5.2.6) and ErII ErI exp k0 h (5.2.7) so that equations 5.2.4 and 5.2.5 can now be written as EII EtI exp k0 h ErI exp k0 h (5.2.8) and 145 H II EtI exp k0 h ErI exp k0 h 0 n1 cos iII (5.2.9) 0 These last two equations can be solved for EtI and E ' rII , which when substituted into 5.2.1 and 5.2.2, yield EI EII cosk0 h H II i sin k0 h Y1 (5.2.10) and H I EII Y1i sin k0 hcosk0 h H II cosk0 h (5.2.11) where Y1 0 n1 cos iII 0 (5.2.12) When E is in the plane of incidence, the above calculations result in similar equations, provided that now Y1 0 n1 0 cos iII (5.2.13) In matrix notation, the above linear relations take the form EI cosk0 h H I Y1i sin k0 h i sin k0 h E Y1 II cosk0 h H II (5.2.14) or EI EII H M I H I II (5.2.15) The characteristic matrix M I relates the fields at the two adjacent boundaries. It follows, therefore, that if two overlaying films are deposited on the substrate, there will be three boundaries or interfaces, and now 146 EII EIII H M II H II III (5.2.16) Multiplying both sides of this expression by M I we obtain EI EIII H M I M II H (5.2.17) I III In general, if p is the number of layers, each with a particular value of n and h, then the first and the last boundaries are related by E p 1 EI H M I M II .....M p H (5.2.18) I p 1 The characteristic matrix of the entire system is the resultant of the product (in the proper sequence) of the individual 2x2 matrices, that is m12 m M M I M II .....M p 11 m21 m22 (5.2.19) From the above equations we can derive the reflection and transmission coefficients. By reformulating equation 5.2.14 and setting Y0 0 n0 cos iI 0 0 Ys ns cos tII 0 (5.2.20) we obtain EiI ErI EtII E E Y M I E Y (5.2.21) rI 0 iI tII s When the matrices are expanded the last relation becomes 1 r m11t m12Ys t and (5.2.22) 147 1 r Y0 m21t m22Yst (5.2.23) where r E ErI , t tII EiI EiI (5.2.24) Consequently, r Y0 m11 Y0Ys m12 m21 Ys m22 Y0 m11 Y0Ys m12 m21 Ys m22 (5.2.25) t 2Y0 Y0 m11 Y0Ys m12 m21 Ys m22 (5.2.26) and To find r or t for any configuration of films, we need only to compute the characteristic matrices for each film, multiply them, and then substitute the resulting matrix elements into the above equations. 5.1.3 Designing a multilayer mirror for the infrared As stated before, one of the solutions to the attenuation problems in hollow waveguides may be the use of a multilayer film inside the hollow tube. In order to get “perfect” reflection we will design a multilayer film, which totally reflects radiation around a given wavelength 0 . The simplest form of a photonic crystal is the quarter wave stack, which is a multilayer film where each layer has a width of a quarter wavelength. We will use equation 5.2.25 for the design. As an example let us design a multilayer mirror for a laser with a wavelength 6m. We will use two dielectric materials that have indices of refraction, which are far apart. Such materials could be Germanium (n=4) and Zinc Selenide 148 (n=2.4) [5.5]. A mirror made of Germanium and Zinc Selenide has layers that are 0.375m and 0.625m thick respectively. Figure 5.3 shows the calculated reflectance of a multilayer film made of different number of pairs of Ge and ZnSe, as a function of wavelength, for 0 o angle of incidence. As can be seen from the figure the more pairs there are the sharper is the region where the reflectance is “perfect”. Figure 5.3 Reflectance Vs. Wavelength (dotted line – 2 pairs, dashed line – 4 pairs, solid line – 6 pairs) Figure 5.4 shows the reflectance of a multilayer mirror with 6 pairs of layers as a function of wavelength for different angles of incidence. As can be seen from the figure the reflectance pattern shifts to shorter wavelengths but its shape stays the same. 149 Figure 5.4 Reflectance Vs. Wavelength (dotted line – 0o, dashed line – 30o, solid line – 60o) These theoretical results may help us understand the dependence of a multilayer film on the laser wavelength and the laser beam angle of incidence, thus helping us in designing a multilayer film that reflects a certain wavelength range. 5.2 Experimental results and discussion– multilayer mirror Commercial companies and research teams have been building onedimensional photonic crystals for many years. They reported the use of films made of GaAS - AlAs [5.6], polystyrene – tellurium (PS – Te) [5.7] and Na2AlF – ZnSe [5.8]. These mirrors usually have a large number of layers and exhibits a 100% reflection for a large interval of wavelengths. 150 Before applying the above-mentioned theory to hollow waveguides we tried to build two kinds of multilayer mirror. The first was a dielectric mirror made of Si and SiO2. The second was a metal dielectric mirror made of silver and silver iodine. 5.2.1 Dielectric mirror Since most of the teams and companies have made dielectric multilayer mirrors we tried to build a mirror made of Si and SiO2. The refractive index of Si is 3.415 and the one of SiO2 is 1.54 [5.5]. Since their refractive indices are far apart they may be a good candidates for a “perfect” mirror. 5.2.1.1 Deposition of the layers The layers were deposited using the sputtering technique on a glass plate. We used two targets, one made of Si and one made of SiO2.The mirror was build for the reflection of near IR radiation around 0=700nm. The thickness of each layer was determined by the time material is sputtered on the glass. We built mirrors made of one pair of Si-SiO2, 2 pairs, 4 pairs and 6 pairs. 5.2.1.2 Reflection measurements The reflection measurements were made using a FTIR (Brucker Germany), which emits radiation in the visible and near IR (400nm< < 1200nm). Figures 5.5 and 5.6 show the experimental results (fig. 5.5) and the theoretical calculations (fig. 5.6) of the reflectance as a function of the wavelength for the different number of pairs respectively. 151 Reflectivity Vs. Wavelength 1 0.9 0.8 Reflectivity 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 400 500 600 700 800 900 1000 1100 Wavelength [nm] 1 pair 2 pairs 4 pairs 6 pairs Figure 5.5 Experimental reflectance as a function of for a multilayer mirror made of Si and SiO2 Figure 5.6 Experimental reflectance as a function of for a multilayer mirror made of 4 layers of Si and SiO2 Blue – Ag (Si-SiO2), Green – Ag 2(Si-SiO2) , Red - Ag 4(Si-SiO2), Light green - Ag 6(SiSiO2) As may be seen from the figures there is a good agreement between the experimental results and theoretical ones only for the mirrors made of one pair and two pairs. 152 The deviation between the experimental results and the theoretical ones is due to the inability to exactly control the thickness of the layers and a poor adherence between the pairs. Since the structure is very sensitive to the changes in the layers thickness, any deviation will cause the reflectance to drop from one. 5.2.2 Metal dielectric mirror Manufacturing multilayer mirrors is not an easy task. It is necessary to control the thickness of the deposited layers very accurately otherwise the reflection would decrease. Moreover it is very difficult to use the current coating methods in order to deposit multilayer thin films inside a hollow tube. Another possibility is to make multilayer mirrors made of metal and dielectric materials [5.12]. Metals reflect radiation very well and if made thin enough (of the order of nm) they also transmit some of the radiation and almost do not absorb it. We tried to manufacture a multilayer mirror made of silver (Ag) and silver iodine (AgI). We are very familiar with these materials and know how to deposit them inside a hollow tube. 5.2.2.1 Deposition of the layers The deposition of the layers was done using a chemical electro less method. The main advantage of this method over other methods, such as CVD, is the ability to deposit thin layers inside hollow tubes and not just coat simple plates. This will enable us to develop in the future “lossless” hollow waveguides. We used Ag for the metal layer and AgI for the dielectric layer. The Ag and AgI layers were deposited on a sapphire slide. The sapphire slides were 153 cleaned, and their surface was activated in PdCl2 solution before silver electro less plating. An Alpha-step 500 was used to measure the thickness of the deposited Ag and AgI layers. The deposition of Ag was made from AgNO3 solution (0.05 M), buffered by ammonia and citric acid to pH = 10-10.6, at room temperature. Ammonia and citric acid were also used to complex metal-ions. The minute quantities of additives were introduced for brightness and softness of the deposit and for stabilization of the electrolyte. Hydrazine hydrate was used as a reducer. To obtain the pair Ag/AgI layer, an Ag layer of thickness dAg >70 nm was deposited and by reaction with iodine a part dAg was transformed into AgI layer. Since the density of AgI is smaller than that of Ag, it was easy to obtain values of thickness of AgI of dAgI 50 nm, without reducing drastically dAg. 5.2.2.2 Properties of thin Ag layers In order to use thin silver layers in a multilayer structure we must first determine their optical properties, especially for which thickness they are transparent and what is their index of refraction. In order to determine the transmission of a thin silver film we deposited thin silver films of different dAg on a sapphire plate with known transmission properties and measured their transmission. The measurement was done using a FTIR (Brucker, Germany). Figure 5.7 shows the transmission of the sapphire plate, a sapphire plate coated with a 30nm thick Ag film and another with a 50nm thick AgI film, as a function of wavelength. As can be seen from figure 5.7, the values of transmission of the Ag+sapphire and AgI+sapphire layers 154 were identical to the transmission of the sapphire itself, in the region where the transmission of sapphire is higher than 40%. This proves that the thin films are transparent in the region where the substrate in not absorbing. 100 Sapphire 1 mm Ag 30 nm 90 80 70 Transmission, % 60 50 40 30 20 10 0 100 4 5 6 7 6 7 90 8 9 8 9 Sapphire 1 mm AgI 50 nm 80 70 60 50 40 30 20 10 0 4 5 Wavelength, m Figure 5.7 – Transmission of Ag+Sapphire and AgI+Sapphire as a function of wavelength Figure 5.8 shows the transmission of different silver layers as a function of wavelength, for various values if dAg. As can be seen from the figure, thin Ag films are transparent for dAg<75nm. Using thin Ag films of thickness less than 75nm will enable us to build a multilayer mirror. 155 T ransmission 1 0 .8 0 .6 0 .4 0 .2 0 3 4 5 6 Wa v e le n g th [m ic ro n ] 36 nm 50 nm 75nm 120 nm Figure 5.8 – Transmission of Ag as a function of wavelength for various dAg The advantage of using metallic materials such as silver is that for a certain interval of wavelengths range the ratio of the index of refraction of the metal to that of the dielectric material (nm/nd) is large enough to obtain the behavior of a photonic crystal. This enables us to use a small number of pairs in order to manufacture an omnidirectional mirror .The index of refraction of Ag is a function of wavelength and changes between 3.74 at 6m to 14 at 14m. Such index of refraction yields the large ratio needed to obtain a photonic crystal 1.7 n Ag n AgI 6.36 6m 14m (5.4.1) The following graph (figure 5.9) shows the calculated reflectance of our structure (Ag/AgI multilayer 4 pairs) as a function of wavelength for three angles of incidence (blue – 80o, green – 45o and red – 25o). As can be seen from the figure perfect reflectance and omni-directionality are achieved for 156 the two wavelengths range. The boarder one is between 10.5m to 14m and the narrower one is between 6.5m to 7.5m Figure 5.9 – The reflectance of a Ag/AgI multilayer mirror (4 pairs) as a function of wavelength for various angles of incidence blue – 80o, green – 45o and red – 25o 5.2.2.3 Reflectance measurements Using the above electro less method we manufactured multilayer mirrors made of pairs of Ag-AgI. We made mirrors with 1 to 4 pairs of Ag-AgI layers on a thick silver substrate. Such a substrate reflects back the part of the radiation, which might be transmitted by the multilayer structure. This makes it possible to avoid losses due to partial photonic mirror effects. The main difference between a good mirror made of a perfect thick Ag and the multilayer maybe demonstrated by (omnidirectional). the angle dependence of the reflected radiation 157 The reflectance of the mirrors as a function of wavelength was measured using a FTIR (Brucker – Equinox 55). Figure 5.9 shows the reflectance of a thick Ag layer as a function of wavelength for various angles of incidence. As can be seen, the reflectance (R) decreases as the angle of incidence () increases. The Reflectance change is from R=95% at =100 to R=70% at =600. 1 0.5 0 3 4 5 6 7 8 9 10 Wavelength (m) 10 degrees 20 degrees 40 degrees 50 degrees 30 degrees Figure 5.9 – The reflectance of a thick Ag layer for various angles of incidence Figure 5.10 a and b show the reflectance of a 4 pairs mirror as a function of wavelengths for various angles of incidence: (a) experimental results and (b) theoretical calculations. As can be seen from the figures there is a good agreement between the theoretical calculations and the experimental results. There is a negligible dependence of the reflectance on the angle of incidence. The difference in the reflectance is less than 2%. This result shows that the reflectance is insensitive to the angle of incidence (omnidirectional) as opposed to the strong dependence of the reflectance of the thick Ag layer. R eflectan ce 158 1 0 .9 0 .8 0 .7 0 .6 0 .5 6 8 10 12 14 W a v e le n g th [ m ] 1 0 d e g re e s 2 0 d e g re e s 4 0 d e g re e s 5 0 d e g re e s 3 0 d e g re e s (a) 1 0 .9 Reflectance 0 .8 0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 6 7 8 9 10 11 12 13 14 Wav eleng th [m] 1 0de gre es 2 0d egre es 30d eg rees 40 deg ree s 50 de gree s (b) Figure 5.10 – The reflectance of 4 pairs multilayer mirror for various angles of incidence. a – experimental results, b – theoretical calculation Figures 5.11 a and b show the reflectance as a function of wavelength for mirrors made of different numbers of pairs of Ag and AgI: (a) experimental results and (b) theoretical calculations for the angle of incidence of 600. The theoretical calculations were made using the method described in [5.10]. As 159 can be seen from these figures, there is a good agreement between the theoretical calculation and the experimental results. As the number of layers increases the reflectance increases. A mirror that is made of 4 pairs has a reflectance in the range of 98% to 100%. However for practical applications it can be seen that the increase in the reflectance is getting smaller with the Refle ctance number of pairs and for certain wavelength range two pairs might be sufficient. 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 6 8 10 12 14 Wavelength [ m] 2 pairs 3 pairs 4 pairs Reflectance (a) 1.005 1 0.995 0.99 0.985 0.98 0.975 0.97 0.965 0.96 0.955 6 7 8 9 10 11 12 13 Wavelength [micron] 1pair 2pairs 3pairs 4pairs (b) Figure 5.11 – The reflectance of multilayers mirror for various number of Ag-AgI pairs, angle of incidence 60o; a – experimental results, b – theoretical calculation 14 160 5.3 Hollow waveguides made of multilayer films In order to examine the influence of multilayer structure on the performance of hollow waveguides the above theoretical calculation was inserted into our ray model (presented in chapter 3) [5.11], which predicts the attenuation and beam shape of a hollow waveguide under different conditions. Figure 5.12 shows the calculated transmission of a bent hollow waveguide (ID=1mm, l=1m) made of the traditional structure (tube, metal layer, dielectric layer) as a function of the bending radius. Figure 5.13 shows the calculated transmission of a bent hollow waveguides (ID=1mm, l=1m) with a multilayer structure as a function of the bending radius. Transmission Vs. Radius of Curvature 1 0.95 0.9 Transmission 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 50 100 150 200 250 300 350 Radius of Curvature [mm] 400 Figure 5.12 – Transmission vs. Radius Of Curvature Traditional Hollow Waveguide (theoretical calculation) 450 500 161 Transmission Vs. Radius of Curvature 0.999 Transmission 0.9985 0.998 0.9975 0 50 100 150 200 250 300 350 Radius of Curvature [mm] 400 450 500 Figure 5.13 – Transmission vs. Radius Of Curvature Multilayer Hollow Waveguide (theoretical calculation) Figure 5.14 and 5.15 show the calculated transmission of hollow waveguide as a function of the focal length of the coupling lens for a traditional and multilayer waveguides respectively. 162 1 0.98 transmission 0.96 0.94 0.92 0.9 0.88 0.86 20 40 60 80 100 focal length [mm] 120 140 160 140 160 Figure 5.14 – Transmission vs. Focal Length Traditional Waveguide (theoretical calculation) 0.9994 0.9992 transmission 0.999 0.9988 0.9986 0.9984 0.9982 20 40 60 80 100 focal length [mm] 120 Figure 5.15– Transmission vs. Focal Length Multilayer Hollow Waveguide (theoretical calculation) 163 As can be seen from the figures the transmission of the multilayer waveguide is much better then the transmission of the traditional waveguide. This due to the fact that the omnidirectional mirror’s reflectance is less influenced by the angle of incidence then a thin film deposited on a metal layer. 5.4 Conclusion Hollow waveguides are suitable for delivering infrared radiation even though they have some drawbacks. As we showed the performance of the hollow waveguides can be improved as we introduce the notion of multilayer mirror to the calculation. 5.5 Summary In this chapter I proposed a way to improve the performance of hollow waveguides by using the concept of photonic crystals. I surveyed two methods for dealing with multilayer mirrors: Bloch theory and Maxwell equations. I used these theories in order to design a multilayer mirror for the IR. We tried to make experimentally two types of multilayer mirrors. The first one was made of Si and SiO2. This experiment was not successful due to insufficient control over the layers thickness. The second one was made from silver and silver iodine. I showed that thin silver layers are transparent in the IR and that a multilayer mirror made from these materials has the desired properties of a photonic crystal. Using these materials we will be able to manufacture multilayers inside a hollow tube to fabricate a multilayer hollow waveguide. 164 I introduced the concept of photonic crystals to the improved ray model and showed that hollow waveguides made of multilayers have better properties than the conventional hollow waveguides. To date we have not been successful in manufacturing a multilayer hollow waveguides due to technical problems relating to the control of the layer’s thickness. 165 5.6 Refrences for chapter five 5.1 P. Yeh, A. Yariv and C. S. Hong, “Electromagnetic propagation in periodic media I. General Theory”, J. Opt. Soc. Am., V. 67, No. 4, p. 423-438, April 1977. 5.2 J. D. Joannopoulos, R. D. Meade, J. N. Winn, “Photonic Crystals – Molding the Flow of Light”, p. 3-53, Princeton University Press 1995. 5.3 E. Hecht, “Optics”, 2nd edition, p. 363-368, Addison – Wesley Publishing 1987. 5.4 E. Yablonovich, “Inhibited spontaneous emission in solid-state physics and electronics”, Physical Review Letters, V.58, No. 20, p. 2059-2062, May 1987. 5.5 M. Bass, “Handbook of Optics”, V. 2, McGraw-Hill, 1995. 5.6 M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari and W. H. Steier, “Wide bandwidth distributed bragg reflectors using oxide/GaAs multilayers”, Electronic Letters, V.30, No. 14, p. 1147-1148, July 1994. 5.7 Y. Fink, J. N. Winn,, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos and E. L. Thomas, “A dielectric omnidirectional reflector”, Science, V. 282, p. 16791682, November 1998. 5.8 S. Bains, “Dielectric stack can be used as omnidirectional mirror”, OE Reports, p. 3, June 1999. 5.9 J. R. Beattie, “The anomalous skin effect and infrared properties of silver and aluminum”, Physica XXIII, p.898-902, 1957. 166 5.10 M. Ben-David, I. Gannot, A. Inberg and N. N. Croitoru, “Optimizing Hollow Waveguides Performance Through Omnidirectional Mirrors”, BIOS 2001. 5.11 A. Inberg , M. Ben-David, , M. Oksman, A. Katzir and N. Croitoru, “Theoretical and Experimental Studies of Infrared Radiation Propagation in Hollow Waveguides”, Optical Engineering, 2000, 39, 5, 1384-1390. 5.12 M. Ben-David, I. Gannot, A. Inberg, G. Rezevin and N. Croitoru, “Electroless deposited broadband omnidirectional multilayer reflectors for midinfrared (MIR) lasers”, BIOS 2002. 5.13 J. Harrington, “A Review of IR Transmitting Hollow Waveguides”, Fibers and Integrated Optics 2000, v. 19, p. 211-217. 5.14 www.omni-guide.com 5.15 J.C. Knight, T.A. Birks, P.St.J. Russell and D.M. Atkin, “All-silica singlemode fiber with photonic crystal cladding,” Opt. Lett. 21 (1547-1549) 1996; Errata, Opt.Lett. 22 (484-485) 1997 5.16 Jes Broeng et al. Waveguidance by the photonic bandgap effect in optical fibres. Journal of Optics A, pages 477–482, July 1999 5.17 Jes Broeng. Photonic crystal fibers: A new class of optical waveguides. Optical Fiber Technology, pages 305–330, July 1999 5.18 Jes Broeng et al. Review paper: Crystal fibre technology. DOPS, 2000. 167 Chapter Six – Summary 6.1 Summary Hollow waveguides are very good candidates for the delivery of infrared radiation. They have relatively low attenuation; they are non-toxic and may be suitable for many applications such as industrial and medical ones. In order to develop better hollow waveguides it is important to understand the attenuation mechanism, and the physical parameters that influence the delivery of the radiation through hollow waveguides. I developed a tool (a computer simulation program) that uses an improved ray model. The improvements made in this model include surface roughness, whispering gallery modes and different coupling conditions. Using the ray model enabled me to determine the attenuation/ transmission of different kinds of hollow waveguides and the energy distribution of the laser radiation after propagating through them. The improved ray model calculations were found to be in good agreement with the experimental measurements. The ray model was also used to predict the pulse dispersion caused by a hollow waveguide. It was shown that bending the hollow waveguide significantly increases the pulse dispersion. This is because of coupling between lower modes of propagation and higher ones. The experimental measurements and the theoretical calculations revealed the hollow waveguide’s drawbacks; i.e. high attenuation when bent, high sensitivity to surface roughness and to the coupling conditions. Introducing the notion of photonic crystals and designing a multilayer waveguide might 168 overcome these drawbacks. Theoretical calculations for such a waveguide show that it is the right approach for solving the hollow waveguides problems. Proving theoretically that photonic crystals may improve the performance of hollow waveguides is very exciting. However, building a multilayer hollow waveguide is very difficult. Since we have lots of experience with flow chemistry we have chosen to use silver and silver iodine as the layers. This method although unorthodox in the field of photonic crystals decreases the number of pairs needed and thus make the manufacturing less difficult. We have shown that thin silver layers, of the order of tens of nano-meters, behave as a dielectric layer and may be used as a building block for a multilayer mirror or hollow waveguide. We have built multilayer mirrors made of silver and silver iodine. The mirrors show the properties of photonic crystals and the two materials may serve as the building blocks for a multilayer hollow waveguide. In conclusion an improved ray model helps us understand the attenuation mechanisms in hollow waveguides and is used in order to improve them. One of the ways to improve the current hollow waveguides is by introducing photonic crystals and manufacturing a multilayer hollow waveguide, 6.2 Future work Further research of the issues that were investigated in this thesis includes further development of the ray model. The model can be expanded to include heat dissipation along the waveguide. This will help us understand the factors that may cause damage to the waveguides and to find constraints under which 169 these waveguides could be utilized.. Furthermore it can be expand to applications such as thermal imaging and tissue welding. There is a lot of work to be done in the field of multilayer hollow waveguides in particular designing, manufacturing and investigating multi-layer hollow waveguides and adopting them as a mean for radiation delivery in many applications.