Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides

The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwell’s equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode. The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers. In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwell’s equations.

The paper considers a class of smoothly irregular integrated optical multilayer waveguides, whose properties determine the characteristic features of guided propagation of monochromatic polarized light. An asymptotic approach to the description of such electromagnetic radiation is proposed, in which the solutions of Maxwell's equations are expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation. The gradient of the phase front of the adiabatic guided mode satisfies the eikonal equation with respect to the effective refractive index of the waveguide for the given mode.
The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the radiation phase front and the gradients of interfaces between thin homogeneous layers.
In the final part of the work, eigenvalue and eigenvector problems (differential and algebraic), describing adiabatic guided modes are formulated. The formulation of the problem of describing the single-mode propagation of adiabatic guided modes is also given, emphasizing the adiabatic nature of the described approximate solution of Maxwell's equations.

Introduction
Fundamental results in the theory of regular waveguides were obtained for closed (metallic) waveguides by A. N. Tikhonov and A. A. Samarskii [1], and for open (dielectric) waveguides by A. G. Sveshnikov [2] and V. V. Shevchenko [3]. Among the irregular waveguides, one can distinguish transversely irregular and longitudinally irregular waveguides. For transversely irregular waveguides, the equations and the corresponding solutions allow the separation of variables [4].
For closed longitudinally irregular waveguides, B. Z. Katsenelenbaum developed the method of cross sections [7], which was generalized for open longitudinally irregular waveguides by V. V. Shevchenko [8]. These models do not describe depolarization and hybridization of guided modes in irregular sections of waveguides. A. A. Egorov, L. A. Sevastyanov and A. L. Sevastyanov developed the foundations of the theory of smoothly irregular 3D dielectric and, in particular, integrated optical waveguides [9], [10], which was successfully applied to a number of three-dimensional integrated optical waveguides and smoothly irregular 3D waveguide devices based on them [11]- [13]. The mathematical basis of the model of adiabatic guided modes (AGMs) is the asymptotic method and the method of coupled modes. The asymptotic method for solving a boundary value problem for a system of differential equations with respect to a small parameter allows it to be reduced to a system of ordinary differential equations with special boundary conditions, the method of solving which is known. The coupling of two second-order equations for modes of two different polarizations when solving the original system of equations by the asymptotic method, manifests itself in the first approximation as a weak (of the order of ) coupling of two linear oscillators. It reflects the violation of the structure regularity caused by a change in the phase constant of smoothly irregular dielectric waveguides.
In this paper, we consider an approach to the construction of a model of propagation of electromagnetic radiation in integrated optical smoothly irregular waveguide structures. Traditionally, such models are described using Maxwell's equations. The paper considers only monochromatic radiation, depending on time as exp( ). Such time dependence of the solution allows considering a model of steady-state guided propagation of electromagnetic radiation.

Basic concepts and notations
Guided propagation of monochromatic polarized electromagnetic radiation in integrated optical waveguides is described by Maxwell's equations. The electromagnetic field is described using complex amplitudes. A material medium is considered, consisting of dielectric subdomains that fill the entire three-dimensional space. The permittivities of the subdomains are different and real, and the permeability is everywhere equal to the permeability of vacuum. It follows that in the absence of foreign currents and charges, the induced currents and charges are equal to zero.
In the absence of foreign charges and currents, the scalar Maxwell's equations follow from the vector ones, and the boundary conditions for the normal components follow from the boundary conditions for the tangential components [14]. The constitutive equations are assumed to be linear. Thus, the electromagnetic field in a space filled with dielectrics in the Gaussian system of units is described by equations where E, H are the electric and magnetic field strength vectors, D is the electric displacement vector, B is the magnetic flux density vector, is the velocity of electromagnetic waves in vacuum. At the interface between dielectric media 1 and 2, the tangential components of electric and magnetic field strengths satisfy the following boundary (matching) conditions: The asymptotic boundary conditions for guided modes at infinity ensure uniqueness of the solution of the problem (1)-(3).
In equations (1), is the medium permittivity, is the medium permeability. Let us denote by = √ the refractive index of the medium (hereinafterof a dielectric layer of the considered multilayer dielectric structure).

The considered class of objects
The object of our consideration is the guided propagation of monochromatic electromagnetic radiation of the optical range in thin-film integrated optical structures. Such structures are complex waveguide structures formed by the deposition of additional waveguide layers of various (smoothly irregular) geometric configurations on the base waveguide. As a base waveguide, we consider a regular planar three-layer waveguide filling the entire three-dimensional space (open waveguide): a substrate layer (substrate) is located in the lower half-space, then a guiding layer of constant thickness is located, and a cladding layer is located in the upper half-space. In this case, the guiding layer (core) is optically denser compared to the substrate and the cladding. By a thin-film waveguide, we mean a waveguide whose core thickness is comparable to the wavelength of propagating radiation.
Integrated optical waveguide structures are formed by introducing into the three-layer planar dielectric waveguide additional layers of variable thickness ℎ( , ). The additional waveguide layers are specified by the interface between the additional and waveguide layers = ℎ( , ) and directly affect phase distribution ( , ) and effective refractive index eff ( , ) of the composite waveguide system. The phase velocity is smaller and the effective refractive index eff ( , ) is greater in the locations where the total thickness of the waveguide layers is greater.
In particular, it can be a few-mode integrated optical waveguide implemented in the form of a three-layer dielectric "plate". On this structure, a local but smoothly irregular optical inhomogeneity is deposited, i.e., an additional dielectric layer of variable thickness. A thin-film waveguide generalized lens (TFWGL) (see Figure 1) is an example [11], [12]. In such integrated optical waveguide, a normal mode of the planar waveguide (allowed by the structure 1 , see Appendix) travelling from infinity (or from a point source in the plane of the planar waveguide) is deformed when meeting the localized irregularity (optical inhomogeneity). However, it preserves the structure of a 'transverse resonance', i.e., the structure of a standing wave in the 'transverse' (vertical) direction. Gorelyshev, Neishtadt et al. [20], [21] formulate this conditions as conservation of an adiabatic invariant.
After passing the irregularity region, the deformed "adiabatically invariant" guided (quasi)mode restores the initial (vertical) parameters of a normal mode of a regular waveguide (or transforms into a superposition of normal modes).
For convenience, let us define the Cartesian coordinate system so that all spatial subdomains corresponding to infinite dielectric layers be bounded by planes parallel to the -plane and surfaces asymptotically parallel to the -plane, so that hereafter = ( ), = 1. We will call smoothly irregular the optical structures satisfying the inequalities specified by the geometry of the additional waveguide layer: In weakly inhomogeneous 3D media the electromagnetic radiation propagation is described by locally planar waves or adiabatic approximations to the solutions of the Maxwell's equations, obtained using the asymptotic method [22]. By analogy with locally plane and locally spherical 3D waves [22], [23], we seek the guided propagation of electromagnetic monochromatic polarized radiation in a smoothly irregular integrated optical waveguide in the form of modified locally normal guided modes of reference waveguides 1 .
The adiabatic approximation of the solution of Maxwell's equation obtained in this way will preserve the adiabatic invariants that reflect the guided character of light propagation (the so called transverse resonance condition [15], [26]. This work is devoted to the search for a model of adiabatic guided propagation of monochromatic electromagnetic radiation in smoothly irregular integrated optical structures. The technique for finding it is based on an asymptotic approach in the form of locally normal guided modes of a locally planar reference waveguide. The adiabatic approximation of the solution of Maxwell's equations obtained in this way will preserve adiabatic invariants reflecting the waveguide nature (the so-called transverse resonance condition) of light propagation.

Basic equations of the adiabatic guided mode model
Let us recall the earlier assumptions made for the considered integrated optical waveguides and the electromagnetic radiation propagating along them.
1. Electromagnetic radiation is optical and monochromatic with a fixed wavelength ∈ [380; 780], nm. 2. The thickness of the guiding layer (core) of the base thin-film waveguide is comparable to the length of the propagating monochromatic electromagnetic radiation ∼ . 3. The surface of additional waveguide layer ( = ℎ( , )) satisfies the limiting conditions | ℎ | , | ℎ | ≪ 1.
4. The integrated optical waveguide is a material medium consisting of dielectric subdomains that fill the entire space. 5. The permittivities of the subdomains are different and real, and the permeability is everywhere equal to the magnetic permeability of vacuum. 6. There are no external currents and charges. It follows from this that in the absence of external currents and charges, the induced currents and charges are equal to zero. 7. A Cartesian coordinate system is introduced as follows: the interfaces between the dielectric media of the basic three-layer waveguide are parallel to the -plane. In this case, the subdomains of space corresponding to the cladding and substrate layers are semi-infinite, the additional waveguide layers are asymptotically parallel to the -plane, so that = ( ).
In Cartesian coordinates associated with the geometry of the substrate (or a three-layer planar dielectric waveguide), Maxwell's equations are written in the form To construct the model of adiabatic guided modes (AGMs) we represent the solutions of (4) in the form of locally normal guided modes of a locally planar reference waveguide (see [7], [8], [27]), which in the method of asymptotic expansion take the form In the notation ⃗ ( , , , ), ⃗ ( , , , ) the separation of with a semicolon means the following assumption and where ‖ ‖ is the Euclidean norm and is the circular frequency of the propagating monochromatic electromagnetic radiation.
Using the approach of the asymptotic expansion method with respect to the dimensional small parameter −1 [23], [28], [29], we substitute expressions (5), (6) into the system of equations (4) and equate the coefficients at equal powers of the small parameter −1 . As a result, with the relations (7) and (8) taken into account, in the zero approximation of the method of asymptotic expansion with respect to small parameter we arrive at a system of homogeneous equations: By means of simple transformations, we reduce it to the form of interest to us. Namely, from the relation (12) we get the expression which we substitute into equations (13)-(14) that take the form From relation (9) we get the expression and substitute it into equations (10)- (11), which take the form Ultimately, Maxwell's equation reduced in the zero approximation to two algebraic equations and four differential equations of the first order.
At any fixed values of ( , ), equations (15)-(16), (17)-(18) take the form of a system of ordinary differential equations of the first order. Hereinafter, we deal with the zero approximation of the asymptotic expansion with respect to the small parameter; therefore, the index of the order of smallness is omitted: In addition, for any fixed values of ( , ) the algebraic equations are valid

Remark 1.
In the proposed form (5) From the analysis carried out, we conclude that with the accepted assumptions the zero approximation to the guided solution of Maxwell's equations is given by the following relations: and as well as For thin-film multilayer waveguide, consisting of optically homogeneous layers, from (2) the conditions of the electromagnetic field matching at the interfaces between the media follow: From (3) the asymptotic conditions follow The system of equations (20), (24)

Algebraic model of adiabatic guided modes
In the case of a multilayer integrated optical waveguide consisting of homogeneous dielectric layers (possible, with complex permittivities) the relations (20), (22)-(24) are valid with generally non-horizontal interfaces between the layers (see Figure 2). In this case, in each inner layer the four-dimensional system of ordinary differential equations with constant coefficients has a four-parameter system of fundamental solutions. Then the general solutions of equations (20) in the layers take the form and ± = ± 0 √− + 2 + 2 .
The rest components of the electromagnetic field are calculated using formulas (21), and all the six components enter the explicit form of relations (22), (23).
In the substrate and cladding layers due to asymptotic conditions the two-dimensional systems of ordinary differential equations with constant coefficients have two-parametric systems of fundamental solutions.
The solution unique for all layers satisfies the matching conditions at the interfaces between the layers, i.e., determines a system of linear algebraic equations for indefinite coefficients, which specify the expression of particular solutions in the layers in terms of the systems of fundamental solutions.
At an arbitrary smooth interface between two dielectric media, described by the equation ( , , ) = − ℎ( , ) = 0 (see Figure 3), conditions (22) and (23) take the form  (25), (26). It is worth noting that only two of three components of the obtained vectors (25) and (26) are linearly independent. Therefore, for writing the boundary conditions we will use the following expressions: , where ⃗ = { ⃗ , ⃗ }. Moreover, for a planar boundary ( , , ) = -const the above expressions simplify to the following form: The above relations should be completed with the asymptotic condition (24). Thus, in each inner layer of a -layer waveguide the four-dimensional general solution is parametrized by four indefinite coefficients, in the substrate and cladding layers at the expense of asymptotic conditions the number of coefficients is reduced by two in each layer. Therefore, there is a total of 4( − 2) + 2 + 2 = 4( − 1) coefficients. To each of ( − 1) interfaces four equations correspond, making the total of 4( − 1). Thus, we have 4( − 1) linear homogeneous algebraic equations for 4( − 1) unknown coefficients.
Returning to the calculation of the electromagnetic field in the multilayer waveguide, we seek a nontrivial solution to the homogeneous system of linear algebraic equations with respect to the coefficient of expansion over the fundamental system of solutions in each dielectric layer: (ℎ( , ), ∇ℎ( , ), ( , ), ∇ ( , )) ⃗ = ⃗ 0.
Thus the system of homogeneous linear algebraic equations (27), nontrivially solvable under the condition (28), is an algebraic model of adiabatic guided modes in a smoothly irregular multilayer integrated optical waveguide. The roots of equation (28) are a set of eigenvalues and the solution of the system of equations (27) after substitution of each particular root is the corresponding eigenvector normalized by the condition ∥ ⃗ ∥ = 1.

Results
Thus, in the course of several stages of sequential reduction, we have formulated a number of problems of modeling the adiabatic guided modes in a smoothly irregular integrated optical multilayer waveguide.
is formulated as follows: The accompanying components of the electromagnetic field are calculated by the formulas: is formulated as follows: The accompanying components of the electromagnetic field are calculated by the formulas

Remark 2.
Similar to the method of cross sections [24], [25], [30], the solution of the last equation has the form

Conclusion
The development of methods for the rigorous and approximate analysis of smoothly irregular integrated optical waveguides requires the development of new mathematical models of the corresponding objects, as well as the use of new methods for studying the problems arising in this case. The fundamental problem of electrodynamics of smoothly irregular waveguide three-dimensional (3D) structures is the development of stable methods and algorithms for solving the corresponding Maxwell's equations.
The paper considers an approach to the formulation of the problem of propagation of electromagnetic radiation in integrated optical smoothly irregular waveguide structures. Traditionally, similar problems are formulated based on Maxwell's equations. In this paper, only monochromatic radiation is considered, which is reflected in the dependence of solution on the frequency of propagating radiation. This type of time dependence of the solution allows considering a steady-state electrodynamic problem for electromagnetic radiation.
The problem of finding the eigenvectors (guided modes) and eigenvalues is considered using the model of adiabatic guided modes in the framework of the zero approximation of the asymptotic expansion for a planar regular threelayer optical waveguide. Considering that the permittivity and permeability are piecewise constant functions, the problem is solved in each subdomain with constant values of , with subsequent matching of solutions at the interfaces between the dielectric media. In each layer , have constant values, and the construction of the entire fields ⃗ , ⃗ requires setting and solving the problem of finding ( , ).
The paper considers a class of smoothly irregular integrated-optical multilayer waveguides, the properties of which determine the characteristic features of waveguide propagation of monochromatic polarized light in them. An asymptotic approach to the description of this type of electromagnetic radiation is proposed, reducing the solutions of the system of Maxwell's equations to a form, which is expressed in terms of the solutions of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zero approximation.
The multilayer structure of waveguides allows one more stage of reducing the model to a homogeneous system of linear algebraic equations, the nontrivial solvability condition of which specifies the relationship between the gradient of the phase front of radiation and the gradients of interfaces between thin homogeneous layers.
In the final part of the work, eigenvalues and eigenvector problems (differential and algebraic) describing adiabatic guided modes are formulated. The problem of describing the single-mode propagation of adiabatic guided modes is also formulated with emphasis on the adiabatic nature of the described approximate solution of Maxwell's equations.

Appendix. Normal modes of a regular planar optical waveguide "Plane" guided modes
The monographic literature [15]- [19] widely describes normal modes of a regular planar dielectric waveguide propagating along the axis from −∞ to ∞ (and/or back). In the vertical direction (along the axis) they have the structure of standing waves, while in the horizontal direction along the waveguide they are travelling waves and are considered (not quite correctly) steady-state ('invariant') in the direction transverse with respect to the propagation direction (along the axis ). The vertical distribution of the electromagnetic field of a mode expressed in terms of the 'leading' transverse component of the electric field is given by the equation Two other components of the electromagnetic field of the mode are expressed in terms of the leading one by the formulas These three relations can be derived from Maxwell's equations in the form (4).
For modes analogous relations that follow from Maxwell's equations have the form Equations (35) and (37) in multilayer waveguides composed of uniform regular dielectric layers, in each subdomain of the real axis (at the intersection of each layer with the vertical axis) take the form of second-order linear ordinary differential equations with constant coefficients. Therefore, the most common form of solutions to these equations is obtained using a fundamental system of solutions. Consequently, the distributions of the corresponding leading components of the electromagnetic field are written in the form of expansions of general solutions in terms of (sin( ), cos( )) or (exp {± ( )}) and the expansion coefficients of the particular solution are determined from the boundary conditions (22)- (23).
These solutions, in particular, can be obtained according to the following algorithm.
For guided modes we get a homogeneous system of linear algebraic equations (SLAE) with matrix M ⊥4 ( ) with respect to variables + , + 1 , The homogeneous SLAE (̂) ⃗ = ⃗ 0 is nontrivially solvable if and only if its determinant is zero, det (̂) = 0.
For guided modes a system of homogeneous linear algebraic equations is obtained with the matrix M ⊥4 ( ) for unknowns + , + 1 , + 1 , − , the solutions of which yield the values of the unknown amplitude coefficients for guided modes.
If the expressions for the electromagnetic field strength are presented in complex form with the described dependence on the rest coordinates and time taken into account, i.e., in the form with real-valued amplitude ( ) and phase ( ), then the phase remains constant in time along , defines a travelling wave along , and is constant along . In other words, the phase front of the described solution to Maxwell's equations is "planar" (i.e., linear in the -plane) and defines a "plane" ( i.e., linear in the -plane) wave. There exist "forward" and "backward" waves travelling in opposite directions along the -axis. The can be identified with the "plane" (in the -plane) waves emitted by infinitely remote sources.

"Cylindrical" guided modes
Now let us proceed to the guided modes of a regular planar dielectric waveguide, excited by a source linear along the -axis and point-like in the -plane, localized at the point (0, 0 ). The structure of the modes along the -axis completely coincides with that of "plane" guided and leaky and modes. Let us analyze the structure of cylindrical guided modes propagating in the -plane where: -( , ) = for the modes from an infinitely remote source and -( , ) = √ 2 + ( − 0 ) 2 for the modes from a localized source.