Solution of Maxwell’s Equations for Cylindrical Symmetry Waveguides

The solution of Maxwell’s equations for a piecewise homogeneous medium of cylindrical symmetry has been obtained. The parameters of the cylindrical waveguide modes have been calculated on its basis. The conclusions are con-firmed by numerical calculation of the first four modes of a hollow metal waveguide operating as a mode convector.


Introduction
Fiber-optic waveguides serve as a transfer medium for optical communication and information transmission systems. Over the past few decades, there has been a steady improvement in both the theory of waveguides [1] [2] and their manufacturing technologies with significant improvements in technical characteristics. The scope of their application is also expanding [1] [2] [3] [4] [5].
In addition to transmitting an optical signal over a distance, waveguides perform the functions of elements of integrated optoelectronic circuits [1] [2]. The physical and mathematical model of such a waveguide is defined by the term "planar". In it, the light wave propagates within the plane, that is, in two-dimensional space. The mathematical model of the propagation of an electromagnetic wave in such a structure is determined by an accurate calculation. A stronger localization of directed optical energy occurs in channeling waveguides, where the electromagnetic wave propagates in a single isolated direction. Such optical structures hollow waveguides with a metal shell and dielectric waveguides, where the core is made of a material that is more optically dense than the shell material. In the latter structure, the direction of light in an optical fiber is carried out by the effect of full internal reflection at its borders. The hollow waveguide transmits the optical signal with minimal distortion. The shell can be metal or dielectric multilayer. Therefore, hollow structures can be used as a mod Converter [3] [4] [5].
The cylindrical waveguide supports modes that differ in radial and azimuthal mode numbers. An axisymmetric waveguide of any radius can excite an infinite number of modes with different azimuthal mode numbers. A larger waveguide will support many modes that differ in the number of zeros of the field in the cross section.
However, even in such a channel, it is possible to excite one mode if the conditions are such that the modes of high orders are strongly attenuated. In articles [3] [4] [5], the parameters of the modes of a cylindrical hollow waveguide with a metal shell are calculated. In the waveguide [4], all modes except the symmetrical transverse magnetic wave are suppressed. This waveguide is used as a mode Converter for high-power microwave sources. Its parameters are calculated in such a way as to highlight a single mode.
To solve such a problem, the Helmholtz equation is usually solved in a cylin- Using this method, the parameters of all waveguide modes are calculated. The essence of the method is a rule that allows you to write the integration constants of Maxwell's equations on all the interface boundaries medium of a layered structure. There is no need to "stitch" solutions of equations, since the continuity of tangential components of the field is a property of the General solution. The result of calculating the problem with the parameters defined in the article [4]. Result corresponds to the theoretical conclusions and data obtained in [4].

Theoretical Solution of the Problem
In order to obtain a general solution of Maxwell's equations, let's write down a wave equation derived from the system of Maxwell's equations excluding unknown functions-vectors of dielectric and magnetic induction, as well as the electric field strength: where ω is the frequency of the electromagnetic wave and ε is the permittivity of a two-component piecewise homogeneous medium: (ε 1 , ε 2 is the permittivity of regions 1 and 2 respectively).
Since the interface between media with different permittivity has a cylindrical symmetry, let's calculate in a cylindrical coordinate system. Let's write down the equations that define the rotor and the vector product of vectors in cylindrical coordinates: here ijk ∈ is a Levi-Civita symbol, We also use the following equation for cylindrical coordinates: First of all, let's define the vector product of the permittivity gradient on the magnetic field strength rotor.

H H
Let's consider two types of waves propagating in a cylindrical waveguide: transverse electric ( 0 z E = ) and transverse magnetic ( 0 z H = ) waves.
Thus, in a transverse magnetic wave, only the φand r-components of the magnetic field strength differ from zero. For the φ-component, Equation (1) has the form: First of all, let's find the distribution of fields in a transverse magnetic wave.
For the purpose of applying the method of separating variables, let's present the magnetic field strength function as a product of functions that depend on only one variable: R(r) and Φ(φ).
Substituting this representation of the magnetic field strength in the wave equation, we transform it to the form: The equation that defines the angular dependence of the field, let's define it as follows: here m is a wonderful mod number, 0 Φ is a constant.
To apply the method of separating variables to Equation (2), let's define the relationship between the radial and azimuthal components of the magnetic field.
To this end, let's write the dependence of the basic orts in the cylindrical coordinate system (e r , e φ ) and the cyclic orts (e 1 , e −1 ), the field expansion of which determines the complex form of the electromagnetic field recording. The intermediate result contains the Cartesian unit vectors (e x , e y ).
Since Equation (1)  In this case, the polarization vector rotates relative to the Z axis in the direction of right rotation, which corresponds to the minus sign in the exponent (3). Then the last term in the left part of Equation (2) can be defined by the following relation: For the wave component defined by the e −1 vector, the following equation is true: Here there is a plus sign in the exponent (3), and the last term on the left side of Equation (2) is defined by the relation: If we substitute expressions (4)-(5) in (2) and divide the equation by the azimuthal function Φ − (φ) (or Φ + (φ)), we get the same equations for the radial function. That is, the field of the General transverse magnetic wave is determined by the General equation: The dependence on the z coordinate is determined by the multiplier exp(k z z), where k z is the wave propagation constant. Assuming For a piecewise homogeneous medium, the last term on the left side of this equation is the Dirac Delta function.
Note that the equality defined by the differentiation formula of the product of functions, known from the course of mathematical analysis, is valid: Let's enter a notation: The solution of Equation (7)  Assume that the function a(r) is odd, and therefore the equality is true: The integral in the exponent is already defined above and is equal to: Analyzing the structure of Equations (7) and (10) Let's proceed to the calculation of the phase, the direct integral equation for which, obtained by integration (6), has the form: As you can see, for m > 0, the phase on the cylinder axis is zero, but for a mode with a zero azimuthal mode number, the phase at r = 0 is not zero, and the field on the axis is infinite, which does not correspond to the physical meaning of the problem. In this case, you can perform integration using formulas (22) (23).
The final result for a function a(r) equal to zero on the cylinder axis has the form:

Numerical Calculation of Modes Parameters of a Cylindrical Waveguide Structure
Equations (24) (25) can be used to calculate the parameters of the modes of a cylindrical waveguide of any material composition. If the waveguide is a pure