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Akustik Fiberlerde Dalga Yayılımı

Akustik Fiberlerde Dalga Yayılımı

##### Dosyalar

##### Tarih

1997

##### Yazarlar

Üçer, Mürvet (Kırcı)

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu çalışmada, haberleşme ve radar sistemlerinde yaygın olarak kullanılan ultrasonik elemanlardan heksagonal anizotropiye sahip akustik kılıflı fiber ayrıntılarıyla incelenmiştir. Akustik kılıf kalınlığı sonsuz geniş, boyunun da sonsuz uzun olduğu kabul edilmiştir. Heksagonal anizotropiye sahip kılıflı fiberin akustik dalga yayılımı için şimdiye kadar kullanılan geleneksel yöntem karmaşık sayılarla analiz yapılacak şekilde yemden ele alınmış ve dağılma bağıntıları verilmiştir. Elde edilen dağılma bağıntılarından faydalanılarak torsiyonel, radyal-eksenel ve fleksural modlara ilişkin dağılma eğrileri çizdirilmiştir. Ayrıca izotropik durum için ve kılıfsız fiber yapısı için dalga yayılımı gözden geçirilmiştir. Bu tezde anlatılan yeni yöntem, sınır koşullarında görünen gerilme, mekanik yer değiştirme gibi büyüklüklerin durum değişkeni olarak tanımlanmasına dayanmaktadır. Bu yöntem bir bakıma lineer sistemlerdeki durum değişkenleri yöntemine benzemektedir. Dağılma bağıntıları durum denkelmlerine benzer yapıdaki denklemlerden faydalanılarak elde edilmektedir. Bu tezde verilen yeni yöntemden faydalanılarak yine fleksural, torsiyonel ve radyal-eksenel modlar için dağılma eğrileri elde Bu eğrilerin elde edilmesinde ZnS, CdS ve CdSe gibi heksagonal simetriye sahip kristallerin malzeme değerleri kullanılmıştır. Kılıflı fiber için piezoelektrik etki de gözönüne alınarak yeni yöntemle dalga yayılımı incelenmiştir. Fleksural, radyal-eksenel ve torsiyonel modlar için dağılma bağıntıları elde edilmiştir. Bu elde edilen dağılma bağıntılarından faydalanılarak sayısal hesaplamalar yapılmıştır. Bu bölümde ZnS ve CdS'e ilişkin malzeme değerleri kullanılarak dağılma eğrileri elde edilmiştir.

Recently ultrasonic devices which we can see in various application areas have been improved very much. Although the existence of the acoustic wave was known since 19th century, the improvement on devices has been done recently. This improvement is due to developement in fabrication technique and researches on the analysis methods. These devices are various filters (as band pass filter, matched filter), delay lines, convolvers imaging probes etc. The developement in design and theoretical analysis methods has provided wide usage of devices in the telecommunication and radar systems. One can observe the difference in the geometrical structure which can be either rectangular prism or cylindric due to the application area concerned. The first of these different geometrical structures is used in making various filters, correlators and convolvers. There are many papers related with the rectangular prism structure on which interdigital transducer is placed. The methods which have been improved for this structure, is used later also for the cylindrical geometry. In this thesis, acoustic fibers, which is the one of the second group mentioned above has been investigated. Acoustic fibers are versatile compenents as imaging probe and delay lines because of their advantages. These advantages are low loss in high frequency, advanced imaging resolution, smaller wave length than fiber diameter and mechanical flexibility. One of the application areas of acoustic fibers is delay line since electromagnetic velocity is faster than acoustic velocity. Besides the long delay line, many ultrasonic applications using cladded fibers are also of interest for instance, ultrasonic imaging probes were developed using cladded fibers and these probes could be arranged in to an imaging array. Because of the low acoustic loss the probes using cladded fibers can operated at a higher frequency than probes with the silica fibers(lGHz.). Another important and interesting characteristic of the single cristal cladded fibers is that several crystals as LiNbOs are both transparent for optical and acoustical waves. One could utilize one Vlll cladded fibers or an array of cladded fibers to provide some of optical and ultrasonic functions. These above applications require a comprehensive understanding of the acoustic wave propagation characteristics in cladded fibers. Most single crystals are anisotropic in acoustic properties. So one of the main questions concerning whether a cladded fiber is suitable for practical applications is how the wave dispersion is affected by the material anisotropy. The acoustic wave propagation characteristics are well understood, but the knowledge of acoustic wave propagation in anisotropic fiber waveguides is in its initial stages. For acoustics, even without the consideration of piezoelectricity, the material stiffness constants which need to be considered can be up to 21 in number in a triclinic crystal. In this thesis only the cladded fibers of hegzagonal crystal symmetries are investigated. Although stiffness constant matrix is fourth order tensor, it is represented as a 6x6 matrix because of the crystal symmetries. Stiffness constant matrix has many different coefficients other than zero, which changes from 2 till 21 due to the class of crystal. The most complex stiffness matrix belongs to triclinic class crystals which have 21 different stiffness constants. Monoclinic class crystals follow this in complexity, which have 13 stiffness constants. The structures belonging to these class crystals are too difficult to invetigate. In the literature isotropic, hexagonal, cubic and trigonal class crystals are investigated because their analysis is easier than the above mentioned ones. In the investigation of the dispersion characteristics of the acoustic fibers, the first works are related with the cylindrical acoustic waveguides. The analysis which investigates torsional mode for circular cylindrical fiber was improved by Thurston. In the work which has been done by Waldron the tickness of the clad is chosen infinity large. This work has been further studied in details by Thurston. Choosing the thickness of the clad large enough makes it possible to ignore the acouctic field in far field. This approximation is a need in investigating the guided mode analysis. Therefore this analysis has given only for the torsinal mode investigation. Mirsky has investigated other modes (radial- axial modes) besides torsinal for the hollow circular cylindrical fibers. For the hegzagonal anisotropic and nonpiezoelectric crystal fibers has been further carried by Dai. He has completed this analysis for various modes in the weakly guided acoustic waveguides. In all these works, the IX theory is a complicated one where the equations that has to be solved are partial differential equations. There are two main difficulties in the analysis of the anisotropic fiber waveguides. The first one is the material anisotropy which differs due to class, crystal belongs. These classes are mentioned above. The second one is the geometrical structure of the waveguide. The classical methods used in analyzing the problem of the wave propagation are the finite difference method or the finite element method in the circular cylindrical acoustic anisotropic fibers. Considering the regular geometry of the cylindrical fiber,finite difference method seems to be preferable. However, because of the existence of the material anisotropy and due to the cylindrical structure, even the formulation of the numerical solution is very complex. Therefore numerical methods are assisted by analytical perturbation methods. Because many material constants are involved, acoustic wave propagation is very complicated. Nevertheless several researchers have used numerical methods to analyze isotropic acoustic fiber waveguides and the vibration of anisotropic acoustic rods. Analytical solutions usually provide a better physical insight to a problem than numerical methods. They often serve as guidelines for numerical approaches to investigate more complicated problems. Thus it is desirable to obtain such an analytical result if it exists. But here, it is only possible to obtain for unisotropic cladded fibers in the implicit form. The first step in searching for the solutions of the anisotropic cladded fibers is to formulate the problem using the physical magnitudes. In all these works, these physical magnitudes such as stress, strain and displacements, which show up in the acoustic wave equations, are expressed by scalar potential functions. As a result helmholtz differential equations are obtained, where the variables are displacements which are represented by scalar potentials. After expressing the solution in implicit form, the boundary conditions of the stresses and displacements have been used. This analysis is based on expressing displacements and stresses as scalar potential functions. In this thesis the displacements and stresses which are used in the boundary conditions, are taken as state variables. A new method based on this fundamental concept is presented. This method converts the second order partial differrential equations (with respect to space) to the first order ordinary differrential equations for the case of acoustic wave. Because the physical magnitudes have been expressed directly as state variables, there is no need for the scalar potantial functions which was used before. The dispersion equation which gives the dispersion characteristic is obtained from the solutions of the equations by using the boundary conditions. This method is an improved version of the one which has been given for the multilayer structures. In this case first order vector matrix differential equations related to acoustic wave propagation is similar to the ones governing the wave equations in time varying linear systems. The cladded fibers are assumed to be infinity long while the claddings are also of infinite thickness. The axes due to the material of the core and cladding is same. The axes due to the material is named crystallines, the fiber axes is the crystalline Z(0,0,1) axis. Figure 1.1 shows the coordinate system used in this thesis. Then cartesian coordinate system (x,y,z) is chosen such that the coordinates x,y and z coincide with the crystalline X,Y and Z axis, respectivly.The cylindrical coordinate system (r,0,z) is designated with the z axis being the same as the cartesian z-axis and the angle 9 is in x-y plane and measured counterclockwise from the x-axis. In order to follow the customs in the literature, in the cylindrical coordinate system, the angular coordinate 9 is used for acoustic propagation investigations. The core and cladding regions will be designated by the subscript ( i ) i=l and 2 Respectively The time dependence of the acoustic waves is assumed to be exp(j©t), where ©is the angular frequency and t is time. P will be used to represent the propagation constant of acoustic waves and phase velocity is V=co/J3. Other symbols are defined in each chapter. In order to investigate the acoustic wave propagation of the cladded fiber, Newton's second law, quasistatic field equations, constitutive equations and strain displacement relations will be used. Constitutive equations shows the relation between the stress, strain and electric field. In this thesis these equations are for the hexagonal situation. In the second and third chapters, the stress- strain equation will be used as the constitutive equations, this is the property of nonpiezoelectric situation, where Ty denotes stress, cyu denotes stiffness, Ski denotes strain tensors. Stress Tjj and strain Ski are represented with the matrix notation insted of tensor because of the crystal symmetries. Similarly the forth order stiffness tensor is represented by a stiffness matrix. These crystal symmetries gives the following equalities Cijkr=CkHj-Cjiik=Cikji. Therefore in this thesis, the matrix and the vector notation will be used instead of tensor notation. Thus Sy, Ty and c^ki becomes Si, Tj and en respectively. In the second chapter an analysis method is given which is based on expressing the displacement conpenents with the scalar potential functions. As the acoustic cladded fiber is investigated in the cylindrical coordinates (r,0,z), Newton's second law will be written in the X! cylindrical coordinate system. The stresses expressed in this equation are due to displacements and stiffness constants by using constitutive relations and strain displacement relations. Displacements are expressed with the scalar potential functions as convenient with the fundamental concept of the method. If this method is followed, equations will be third order differential equations with respect to space (radyal compenent) The solutions of these equations end in Bessel differential equations. Therefore the general solution of the stresses and the displacements will be the combination of the Bessel functions. In this case boundary conditions will be as follows: a) In the boundary for both regions, the radyal compenents of stress are continious. b) In the boundary for both regions, displacement compenets are continious. Applying to the boundary conditions to the general solution of the stresses and displacements, equations are arranged and the dispersion equation is obtained. Since the determinant of the a" matrix vanishes which has been written from these equations, the dispersion equation has been obtained as a result. The compenents of a,j matrix are the Bessel functions. Therefore for situations n=0 and n>lmod analysis has been done. For n=0 a*j has been seperated into two different matrices which are 2x2 and 4x4. Torsional modes have been obtained from the 2x2 matrix and radial-axial modes have been obtained from the 4x4 matrix. For n>l, aij has been used completly,and in this case general flexural modes are analysed. Here mode analysis has been revised under the various approaches. In the third chapter, a new method has been given where the stresses and displacements are the state variables. In the boundary conditions only these state variables are seen. Since this method directly expresses the mechanical magnitudes related with space as the state variables in the equations, the scalar potential functions are no use. As the structure is acoustic cladded fiber in cylindrical geometry. The general solutions related with these equations are combination of Bessel functions. If in the new obtained equations, these general solutions are subsituted, the coefficients of the general solution will be found carrying out linear algebric equations. Thus the stresses and displacements which were seen as the state variables, have been obtained as the Bessel functions. Applying boundary conditions in these solutions and arranging the equations one can obtain a;j. Determinant of ay gives the dispersion relation. Then for n=0 and n>l mode analysis have been done. The aim is to show torsinal, radial-axial and general flexsural modes. These result Xll are shown with the dispersion curves(V-fa) where V is phase velocity, fa is the product of frequency and radius of the cladded fiber. These curves are same as the ones in the second chapter. In the fourth chapter acoustic wave propagation of the cladded fiber has been investigated where the piesoelectric effect is also concerned. In this chapter the same geometric structure and properties have been considered. The dispersion relation has beeen obtained by using the state variable method. The result are very convenient as shown in figures.

Recently ultrasonic devices which we can see in various application areas have been improved very much. Although the existence of the acoustic wave was known since 19th century, the improvement on devices has been done recently. This improvement is due to developement in fabrication technique and researches on the analysis methods. These devices are various filters (as band pass filter, matched filter), delay lines, convolvers imaging probes etc. The developement in design and theoretical analysis methods has provided wide usage of devices in the telecommunication and radar systems. One can observe the difference in the geometrical structure which can be either rectangular prism or cylindric due to the application area concerned. The first of these different geometrical structures is used in making various filters, correlators and convolvers. There are many papers related with the rectangular prism structure on which interdigital transducer is placed. The methods which have been improved for this structure, is used later also for the cylindrical geometry. In this thesis, acoustic fibers, which is the one of the second group mentioned above has been investigated. Acoustic fibers are versatile compenents as imaging probe and delay lines because of their advantages. These advantages are low loss in high frequency, advanced imaging resolution, smaller wave length than fiber diameter and mechanical flexibility. One of the application areas of acoustic fibers is delay line since electromagnetic velocity is faster than acoustic velocity. Besides the long delay line, many ultrasonic applications using cladded fibers are also of interest for instance, ultrasonic imaging probes were developed using cladded fibers and these probes could be arranged in to an imaging array. Because of the low acoustic loss the probes using cladded fibers can operated at a higher frequency than probes with the silica fibers(lGHz.). Another important and interesting characteristic of the single cristal cladded fibers is that several crystals as LiNbOs are both transparent for optical and acoustical waves. One could utilize one Vlll cladded fibers or an array of cladded fibers to provide some of optical and ultrasonic functions. These above applications require a comprehensive understanding of the acoustic wave propagation characteristics in cladded fibers. Most single crystals are anisotropic in acoustic properties. So one of the main questions concerning whether a cladded fiber is suitable for practical applications is how the wave dispersion is affected by the material anisotropy. The acoustic wave propagation characteristics are well understood, but the knowledge of acoustic wave propagation in anisotropic fiber waveguides is in its initial stages. For acoustics, even without the consideration of piezoelectricity, the material stiffness constants which need to be considered can be up to 21 in number in a triclinic crystal. In this thesis only the cladded fibers of hegzagonal crystal symmetries are investigated. Although stiffness constant matrix is fourth order tensor, it is represented as a 6x6 matrix because of the crystal symmetries. Stiffness constant matrix has many different coefficients other than zero, which changes from 2 till 21 due to the class of crystal. The most complex stiffness matrix belongs to triclinic class crystals which have 21 different stiffness constants. Monoclinic class crystals follow this in complexity, which have 13 stiffness constants. The structures belonging to these class crystals are too difficult to invetigate. In the literature isotropic, hexagonal, cubic and trigonal class crystals are investigated because their analysis is easier than the above mentioned ones. In the investigation of the dispersion characteristics of the acoustic fibers, the first works are related with the cylindrical acoustic waveguides. The analysis which investigates torsional mode for circular cylindrical fiber was improved by Thurston. In the work which has been done by Waldron the tickness of the clad is chosen infinity large. This work has been further studied in details by Thurston. Choosing the thickness of the clad large enough makes it possible to ignore the acouctic field in far field. This approximation is a need in investigating the guided mode analysis. Therefore this analysis has given only for the torsinal mode investigation. Mirsky has investigated other modes (radial- axial modes) besides torsinal for the hollow circular cylindrical fibers. For the hegzagonal anisotropic and nonpiezoelectric crystal fibers has been further carried by Dai. He has completed this analysis for various modes in the weakly guided acoustic waveguides. In all these works, the IX theory is a complicated one where the equations that has to be solved are partial differential equations. There are two main difficulties in the analysis of the anisotropic fiber waveguides. The first one is the material anisotropy which differs due to class, crystal belongs. These classes are mentioned above. The second one is the geometrical structure of the waveguide. The classical methods used in analyzing the problem of the wave propagation are the finite difference method or the finite element method in the circular cylindrical acoustic anisotropic fibers. Considering the regular geometry of the cylindrical fiber,finite difference method seems to be preferable. However, because of the existence of the material anisotropy and due to the cylindrical structure, even the formulation of the numerical solution is very complex. Therefore numerical methods are assisted by analytical perturbation methods. Because many material constants are involved, acoustic wave propagation is very complicated. Nevertheless several researchers have used numerical methods to analyze isotropic acoustic fiber waveguides and the vibration of anisotropic acoustic rods. Analytical solutions usually provide a better physical insight to a problem than numerical methods. They often serve as guidelines for numerical approaches to investigate more complicated problems. Thus it is desirable to obtain such an analytical result if it exists. But here, it is only possible to obtain for unisotropic cladded fibers in the implicit form. The first step in searching for the solutions of the anisotropic cladded fibers is to formulate the problem using the physical magnitudes. In all these works, these physical magnitudes such as stress, strain and displacements, which show up in the acoustic wave equations, are expressed by scalar potential functions. As a result helmholtz differential equations are obtained, where the variables are displacements which are represented by scalar potentials. After expressing the solution in implicit form, the boundary conditions of the stresses and displacements have been used. This analysis is based on expressing displacements and stresses as scalar potential functions. In this thesis the displacements and stresses which are used in the boundary conditions, are taken as state variables. A new method based on this fundamental concept is presented. This method converts the second order partial differrential equations (with respect to space) to the first order ordinary differrential equations for the case of acoustic wave. Because the physical magnitudes have been expressed directly as state variables, there is no need for the scalar potantial functions which was used before. The dispersion equation which gives the dispersion characteristic is obtained from the solutions of the equations by using the boundary conditions. This method is an improved version of the one which has been given for the multilayer structures. In this case first order vector matrix differential equations related to acoustic wave propagation is similar to the ones governing the wave equations in time varying linear systems. The cladded fibers are assumed to be infinity long while the claddings are also of infinite thickness. The axes due to the material of the core and cladding is same. The axes due to the material is named crystallines, the fiber axes is the crystalline Z(0,0,1) axis. Figure 1.1 shows the coordinate system used in this thesis. Then cartesian coordinate system (x,y,z) is chosen such that the coordinates x,y and z coincide with the crystalline X,Y and Z axis, respectivly.The cylindrical coordinate system (r,0,z) is designated with the z axis being the same as the cartesian z-axis and the angle 9 is in x-y plane and measured counterclockwise from the x-axis. In order to follow the customs in the literature, in the cylindrical coordinate system, the angular coordinate 9 is used for acoustic propagation investigations. The core and cladding regions will be designated by the subscript ( i ) i=l and 2 Respectively The time dependence of the acoustic waves is assumed to be exp(j©t), where ©is the angular frequency and t is time. P will be used to represent the propagation constant of acoustic waves and phase velocity is V=co/J3. Other symbols are defined in each chapter. In order to investigate the acoustic wave propagation of the cladded fiber, Newton's second law, quasistatic field equations, constitutive equations and strain displacement relations will be used. Constitutive equations shows the relation between the stress, strain and electric field. In this thesis these equations are for the hexagonal situation. In the second and third chapters, the stress- strain equation will be used as the constitutive equations, this is the property of nonpiezoelectric situation, where Ty denotes stress, cyu denotes stiffness, Ski denotes strain tensors. Stress Tjj and strain Ski are represented with the matrix notation insted of tensor because of the crystal symmetries. Similarly the forth order stiffness tensor is represented by a stiffness matrix. These crystal symmetries gives the following equalities Cijkr=CkHj-Cjiik=Cikji. Therefore in this thesis, the matrix and the vector notation will be used instead of tensor notation. Thus Sy, Ty and c^ki becomes Si, Tj and en respectively. In the second chapter an analysis method is given which is based on expressing the displacement conpenents with the scalar potential functions. As the acoustic cladded fiber is investigated in the cylindrical coordinates (r,0,z), Newton's second law will be written in the X! cylindrical coordinate system. The stresses expressed in this equation are due to displacements and stiffness constants by using constitutive relations and strain displacement relations. Displacements are expressed with the scalar potential functions as convenient with the fundamental concept of the method. If this method is followed, equations will be third order differential equations with respect to space (radyal compenent) The solutions of these equations end in Bessel differential equations. Therefore the general solution of the stresses and the displacements will be the combination of the Bessel functions. In this case boundary conditions will be as follows: a) In the boundary for both regions, the radyal compenents of stress are continious. b) In the boundary for both regions, displacement compenets are continious. Applying to the boundary conditions to the general solution of the stresses and displacements, equations are arranged and the dispersion equation is obtained. Since the determinant of the a" matrix vanishes which has been written from these equations, the dispersion equation has been obtained as a result. The compenents of a,j matrix are the Bessel functions. Therefore for situations n=0 and n>lmod analysis has been done. For n=0 a*j has been seperated into two different matrices which are 2x2 and 4x4. Torsional modes have been obtained from the 2x2 matrix and radial-axial modes have been obtained from the 4x4 matrix. For n>l, aij has been used completly,and in this case general flexural modes are analysed. Here mode analysis has been revised under the various approaches. In the third chapter, a new method has been given where the stresses and displacements are the state variables. In the boundary conditions only these state variables are seen. Since this method directly expresses the mechanical magnitudes related with space as the state variables in the equations, the scalar potential functions are no use. As the structure is acoustic cladded fiber in cylindrical geometry. The general solutions related with these equations are combination of Bessel functions. If in the new obtained equations, these general solutions are subsituted, the coefficients of the general solution will be found carrying out linear algebric equations. Thus the stresses and displacements which were seen as the state variables, have been obtained as the Bessel functions. Applying boundary conditions in these solutions and arranging the equations one can obtain a;j. Determinant of ay gives the dispersion relation. Then for n=0 and n>l mode analysis have been done. The aim is to show torsinal, radial-axial and general flexsural modes. These result Xll are shown with the dispersion curves(V-fa) where V is phase velocity, fa is the product of frequency and radius of the cladded fiber. These curves are same as the ones in the second chapter. In the fourth chapter acoustic wave propagation of the cladded fiber has been investigated where the piesoelectric effect is also concerned. In this chapter the same geometric structure and properties have been considered. The dispersion relation has beeen obtained by using the state variable method. The result are very convenient as shown in figures.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1997

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1997

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1997

##### Anahtar kelimeler

Akustik dalga,
Dalga yayılması,
Lifler,
Acoustic wave,
Wave propagation,
Fibers