18.矢量波函数(8-0).doc

18. Vector Wave FunctionsA. Vector wave equationl As discussed previously, in source-free region, the homogeneous Helmholtz equations (wave equations) for EM fields and potentials are given bywhere k is the intrinsic wave number,l The Laplacian operator is defined as andl Let represent , or , or in source-free region, then it will obey the following vector wave equation:B. Scalar wave equationl In rectangular coordinates, is denoted byand the homogenous vector Helmholtz equation for becomesorl Therefore the three rectangular components, , should satisfy the following homogeneous scalar Helmholtz equations:.l However, in a curvilinear coordinate system, only the rectangular coordinate components (if any) satisfy the homogeneous scalar Helmholtz equation and the others do not. For instance, in a cylindrical coordinate system,and its coordinate components satisfy, respectively, the scalar equations:l It is found that in this coordinate system, only the rectangular-coordinate component satisfies the homogeneous scalar Helmholtz equation and the other two components, and , do not.l Let denote or the rectangular-coordinate components of , , and , then it must satisfy the following homogeneous scalar wave equationC. Scalar wave functionsl The solution to the scalar wave equation is called the scalar wave function.l Although the scalar wave equation is set up only for the rectangular field components, the equation may be solved in various coordinate systems.l The solution to the scalar wave equation in rectangular coordinates is called the plane wave function, which is discussed in Chapter 4 of Harringtons book.l The solution to the scalar wave equation in cylindrical coordinates is called the cylindrical wave function, which appears in Chapter 5 of Harringtons book.l The solution to the scalar wave equation in spherical coordinates is called the spherical wave function, which is studied in Chapter 6 of Harringtons book.l The homogeneous scalar wave equation is usually solved for the scalar wave functions by means of the method of separation of variables in a certain coordinate system. l When the method of separation of variables is employed to approach the homogeneous scalar wave equation, a coordinate system should be properly chosen in order to make the coordinate surfaces coincided with the EM boundary so that the wave equation under consideration is separable. D. Vector wave functionsl In a curvilinear coordinate system, determination of solution to the vector wave equation means the determination of solution to the three scalar equations, which is usually a very difficult task to be fulfilled.l For example, in cylindrical coordinates, three scalar equations arein which the first two equations for and are mutually coupled together making the problem hard to be solved.l To facilitate the direct determination of solution to the homogeneous vector wave equation in a curvilinear coordinate system,following three vector functions, , , , are defined:where is a constant vector, and is scalar wave function to the following homogeneous scalar wave equation in curvilinear coordinates,l It is found from and thatl It is inferred from thatl It follows from andthatl In summary, the three vector functions, , , , are solutions to the vector wave equation, , namely all of them are the vector wave functions.l It is seen thatwhich means that and are mutually perpendicular. Namelyl It is also seen that which means that and are rotations of each other. l The above two conclusions indicate that the vector wave functions, , , , are not collinear in spatial direction and hence are mutually independent.E. Fields and potentials l In source-free region, the magnetic vector potential is the solution of the homogeneous vector wave equation,which may be expressed to be the following summation of vector wavefunctions:wherein which is the n-th scalar wave function to the following homogeneous scalar wave equation in a certain coordinate system,l It follows from the earlier-described propertiesthat the EM field will beandl Since both and are mutually related by rotations,andand and are also mutually related by rotations,it is rightful to expand both and in terms of and .l Finally and may be expanded in terms of scalar wave functions aswhere is the scalar wave function, satisfyingl The electric scalar potential is not employed yet in above derivation, nevertheless it is expressible in terms of the scalar wave functions. In factl Furthermore, the gradient of the electric scalar potential, , is denoted in terms of asl The same result can also be reached by using the Lorentz Gauge. In fact the electric scalar potential isl In summary, as long as the scalar w