Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides

This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwell’s equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM & ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions – adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.


Introduction
In works [1]- [5] a cycle of studies of the propagation of polarized light in integrated-optical smoothly irregular thin-film waveguides was carried out within the framework of the model of adiabatic waveguide waves. They showed the advantages of the model and its advantages over other models in the description of open dielectric waveguides [6]- [8]. At the same time, until recently, the question of substantiating this model remained open. In work [9] the substantiation of the model was carried out, which is a reduction of a more complex in use general model based on Maxwell's equations. In the present work, within the framework of the model of adiabatic waveguide waves, the problem of stationary propagation of polarized light in a smoothly irregular integral-optical waveguide is posed, an auxiliary problem for eigenvalues and eigenfunctions (adiabatic waveguide modes) is formulated and solved. The solution of the stationary problem by the generalized Kantorovich method is proposed, its solution is obtained in the single-mode propagation mode.

Basic concepts and notation
Waveguide propagation of monochromatic polarized electromagnetic radiation in integrated optical waveguides is described by Maxwell's equations. The electromagnetic field is described using complex amplitudes. The material environment is considered, consisting of dielectric subdomains that fill the entire three-dimensional space. The latter means that the dielectric constants of the subdomains are different and real, and the magnetic permeability is everywhere equal to the magnetic permeability of the vacuum. It follows from the foregoing that in the absence of external currents and charges, the induced currents and charges are equal to zero.
In the absence of external charges and currents, the scalar Maxwell equations follow from the vector ones, and the boundary conditions for the normal components follow from the boundary conditions for the tangential components. The constitutive equations of connection in the case under consideration are assumed to be linear. Thus, the electromagnetic field in a space filled with dielectrics in the Gaussian system of units is described by the equations [10]: where E, H are the vectors of electric and magnetic field strength; D is the electric displacement vector, B is the magnetic flux density vector; is the velocity of electromagnetic wave propagation in vacuum. In this case, the boundary conditions and the asymptotic boundary conditions at infinity are assumed to hold for guided modes, which ensures the uniqueness of the solution to problem (1)-(2). In equations (1): is the permittivity of the medium; is the permeability of the medium, E, H are the electric and magnetic field strength vectors. We denote by = √ the refractive index of the medium (here and below -of a layer of the multilayer dielectric structure under consideration). All subdomains are infinite and bonded by planes parallel to the -plane and surfaces, asymptotically parallel to the -plane, so that below we have = ( ), = 1.

Model of adiabatic guided modes
In Ref. [9] the adiabatic approximation of the guided solution of Maxwell's equations is found in the form: where After additional differentiations from four first-order ODEs, four secondorder ODEs are obtained, two of which take the form: in the case when the layers (three or four) of a multilayer waveguide are homogeneous.
The rest four components are calculated from the system of linear algebraic equations (SLAE)

Waveguides regular in y and electromagnetic fields
First, let us consider the case when neither the integrated optical waveguide geometry, nor the solutions to Maxwell's equations for the adiabatic guided mode (AGM) depend on one of the horizontal coordinates, i.e., the case / ≡ 0. For fields harmonic in time in the Cartesian system of coordinates, the system of Maxwell's equations has the form: In the case / ≡ 0, system (8) takes the form A substitution of the first and the third equations (9) into the second ones leads to the equivalent systems for ТЕ polarization and for TM polarization.
In this case, the system of equations (6), (7) is split into two independent subsystems: for ТЕ polarization: and for ТМ polarization: For thin film multilayer waveguide consisting of optically homogeneous layers, the system of equations (12) and (13) should be completed with the conditions of the electromagnetic field matching at the interfaces between the media that follow from (2): and the asymptotic conditions that follow from (3):

Setting of the physical problem
The solution of the first equation of system (10) is found using the generalized Kantorovich method [11], [12], which is analogous to the method of separation of variables proposed in [13]. We find the solutions to an auxiliary problem analogous to a problem of finding eigenvalues and eigenfunctions that depend on a parameter.
Auxiliary eigenvalue problem For each fixed we consider the problem where ( ) = ( ), with the boundary (asymptotic) conditions Let us normalize the eigenfunctions with a condition It is known that for any fixed ∈ ℝ problem (17)- (19) is a problem of finding normal guided ТЕ modes of a regular planar reference waveguide [14], [15]. At any real-valued , and any finite thickness of the reference waveguide it allows a finite number TE of forward and TE of backward ТЕ modes [16]- [18].
We restrict ourselves to considering such smoothly irregular waveguides in which the number of guided modes is constant throughout the change and the degree of irregularity is so small that the transformation of the energy of guided modes is limited by the adiabatic approximation. In this case, we seek a solution to problem in the form of an expansion: Substituting expansion (20), (21) into Eqs. (10) with relations (19) taken into account, after cumbersome but not complicated transformations, we arrive at a system of ODEs for the expansion coefficient functions TE ( ).

Solution of a single-mode problem for zero contribution to the AGM
As an example, let us carry out the above calculations for the particular case (20), (21), which describes the single-mode propagation of a TE-polarized AGM, namely:̃( , ) = TE ( ) ( ; ), satisfying conditioñ( We differentiate (23) with respect to and substitute the result into the first equation of the system (10) taking into account (17) (following [13]). As a result, we arrive at a system consisting of a single ODE We substitute the solution of Eq. (24) into relation (22) and obtain the ultimate explicit form of the component̃of the electromagnetic field of the single-mode TE polarized AGM: By means of the second equation of the system (10) we obtaiñ and using the third equation of the system (10) we obtaiñ In analogy with the above calculations, we solve the system (11) using the auxiliary problem The eigenfunctions are normalized by the condition The single-mode solution for the TM-polarized AGM has the form ( , ) = TM (0)

Search for adiabatic guided mode phase using the Cauchy method
In the model of adiabatic guided modes, in addition to the coefficient functions TE ( ) and TM ( ), the explicit dependence on the horizontal coordinate is present in the exponential factor exp{ − 0 ( , )} in expression (4) [9]. The mode evolution in the horizontal direction, besides the dependences (24) for the coefficient functions in the case / ≡ 0, is formed by the dependence of / on the obtained eigenvalues of problems (17). It is convenient to formulate the description of phase evolution law ( ) in terms of the algebraic model of adiabatic guided modes [9] for thin-film waveguides. A constructive representation of this dependence for thin-film waveguides consisting of homogeneous layers is obtained in Refs. [19]- [21]. Namely, the general solutions in the homogeneous layers have the form where ( ) = √ 2 ( ) − 2 , = , , .

Basic equations of the adiabatic guided mode model
The problem of propagation of polarized electromagnetic radiation in regular waveguides was successfully solved both in closed [27]- [30] and in open waveguides [31]. In both cases the base method is the method of separation of variables, which reduces the initial problem to an auxiliary spectral problem with a discrete spectrum in the case of closed waveguides and a mixed spectrum in the case of open ones.
Problems with dielectric and magnetic filling of waveguides, intricate in the cross section but regular in the longitudinal direction (along the radiation propagation axis) were reduced to considerably more complicated spectral problems, which have been subsequently also solved (see, e.g., [32]- [39]). These physical models gave rise to new mathematical problems and description of new electromagnetic phenomena, which are beyond the present discussion.