Guided Electromagnetic Waves Propagating in a Two-Layer Cylindrical Dielectric Waveguide with Inhomogeneous Nonlinear Permittivity

The paper focuses on the problem of monochromatic electromagnetic TM wave propagation in a two-layer circular cylindrical dielectric waveguide. The space outside the waveguide is filled with isotropic medium having constant permittivity. The inner core of the waveguide is filled with isotropic medium having constant permittivity; the cladding of the core is filled with isotropic inhomogeneous nonlinear permittivity (the nonlinear term is expressed by Kerr law). Existence of guided modes which depend harmonically on z (the waveguide axis coincides with z-axis) is proved and their localization is found. Numerical results including different type of nonlinearities are presented. A comparison with the linear case is given.The existence of a new propagation regime is predicted.


Introduction
The paper studies the problem of monochromatic electromagnetic TM (transverse-magnetic) wave propagation in a two-layered circle cylindrical dielectric waveguide with nonlinear permittivity inside one of its layers.Here we talk only about intensity-dependent permittivity.We do not consider multiple harmonic generation or other nonlinear effects that in rigorous statement involve time-dependent Maxwell's equations.The nonlinear permittivity is described by the Kerr law.Kerr law is one of the most important dependencies in nonlinear optics; see, for example, [1][2][3], and for newest experimental observation see [4][5][6].
The physical problem is reduced to a nonlinear transmission eigenvalue problem for a system of nonlinear ordinary differential equations.Eigenvalues of the problem correspond to propagation constants (PCs) of the waveguide.The full set of PCs of a waveguide is one of the most important characteristics of the waveguide; this characteristic is used for waveguide's designing.One of the main methods to study the problem is the small parameter method.Since the Kerr law is characterised by a small constant factor in front of the nonlinear term (coefficient of the nonlinearity), then this approach is justified.Numerical results are based on a numerical method that does not depend on the smallness of the parameter [7].As is known the Kerr law is described by an unbounded function; in order to demonstrate difference between unbounded and bounded nonlinear permittivities we presented numerical results for both cases; we also gave a comparison between linear and nonlinear cases.Numerical results given here demonstrate not only those eigenvalues that are predicted by the main theorem of this work but new eigenvalues that correspond to a new nonlinear propagation regime.
Most of these papers are devoted to studying of polarized waves in waveguides filled with a homogeneous nonlinear 2 Advances in Mathematical Physics medium.From aforementioned studies only [7,13,15,16] focus on inhomogeneous nonlinear permittivity.
Multilayered cylindrical linear homogeneous waveguide was studied in [17,18], and one of the practical applications for nonlinear two-layered waveguides is shown in [19].
Results obtained in this paper together with the results given in [16] give an opportunity to consider a very intriguing phenomenon of coupled electromagnetic TE-TM wave propagation in a two-layered cylindrical waveguide.This problem will be treated in a separate paper.For different types of phenomena of coupled wave propagation see [20][21][22].

Governing Equations
Consider three-dimensional space R 3 with cylindrical coordinate system .In this space a two-layer circular cylindrical waveguide is placed; the waveguide axis coincides with .(The waveguide is unlimited in  direction.)The waveguide is filled with isotropic inhomogeneous nonlinear medium.The space outside Σ is filled with isotropic medium characterised by constant permittivity.Throughout the paper we assume that everywhere  =  0 , where  0 > 0 is the permeability of free space.Geometry of the problem is shown in Figure 1.
Consider monochromatic electromagnetic field propagating at a frequency  along the surface of Σ.The quantities E, H are called the complex amplitudes [1]; E = (  ,   ,   ) T and H = (  ,   ,   ) T , where (⋅) T denotes the transpose operation.
Complex amplitudes E, H of a monochromatic electromagnetic field must satisfy time-harmonic Maxwell's equations the continuity conditions for the tangential components of the field at the interfaces (on the boundary of the waveguide)  =  1 ,  =  2 , and the radiation condition at infinity: the electromagnetic field decays as (|| −1 ) when  → ∞.The solutions to Maxwell's equations are sought in the entire space.
In the whole space the dielectric permittivity has the form  = ε 0 , where 1 ,  3 are positive real constants,  0 > 0 is the permittivity of free space, and

Statement of the Problem
Consider TM-polarized waves in the monochromatic mode It is assumed that waves propagating along the boundary of Σ depend harmonically on  [1,8,16,23].Substituting the TM waves into (3) one is convinced that the field does not depend on .Thus the components of the field have the form where  is a real spectral parameter of the problem which defines unknown PCs (without loss of generality we suppose  > 0).In what follows arguments of functions will often be omitted.

Main Nonlinear Equations
Taking into account the boundedness condition of the field in any finite region and the radiation condition at infinity solutions to (9) outside the cladding have the form where and  1 are the modified Bessel functions [27].The constant  1 is assumed to be known (see the definition of  M ()); the constant  2 is determined using conditions (11).Solutions (12) are real as where Thus system (9) in the waveguide cladding can be written in the form In the Kerr law the coefficient  is supposed to be small [2,3,28]; for this reason the small parameter method can be used to study the problem  M ().
For proving existence of solutions to  M () it is necessary at first to prove existence of solutions to the linear problem  M (0), which is  M () for  = 0.

Existence of Solutions to 𝑃 M (0)
The problem  M (0) derived from  M () is difficult to investigate.For this reason we revert to (8) and consider the original problem for  = 0.The problem obtained in such a way is also denoted as  M (0).
Let  :=  2 .Introduce the notation  3 := H  ; from system (8) for  = 0 one obtains where As the components   and   are continuous at the interfaces one obtains the following conditions for  3 : and in view of ( 12) Taking into account conditions (17) and solutions (18) one obtains boundary conditions of the third kind (which depend nonlinearly on the spectral parameter): where the functions  1 ,  2 are easily found.Thus the problem  M (0) is a (linear) problem of Sturm-Liouville type for (16) with conditions (19).At first one needs to consider yet another Sturm-Liouville: Let {  , V  ()} ∞ =1 be the complete system of eigenvalues and eigenfunctions orthonormal in the space  2 ( 1 ,  2 ; ()) with weight () := 1/ 2  of boundary value problem (20).It is known [29] that all the eigenvalues are real and simple (of multiplicity 1).To be more precise, there are not more than a finite number of positive eigenvalues and an infinite number of negative ones (  → −∞ for  → ∞).We arrange eigenvalues in the ascending order: Solvability of  M (0) is established by the following statement (see proof in [29]).This statement asserts that required eigenvalues of  M (0) lie between eigenvalues of problem (20).
Note 2. Thickness  2 −  1 of the waveguide cladding can be chosen in such a way that the problem  M (0) will have solutions.

Nonlinear Integral Equations and Problem 𝑃 M (𝛼)
Let  :=  2 .Consider the boundary value problem Since all the coefficients of  M V = 0 are continuous and nonzero on the segment [ 1 ,  2 ], then the equation has two linearly independent continuous solutions defined on the same segment.Furthermore, it is clear that the eigenvalues of problem (22) do not coincide with the eigenvalues of the problem  M (0).Thus one obtains that a unique Green's function  M (, ) of the boundary value problem exists in a neighbourhood of each eigenvalue of  M (0).
For brevity we use the following notation: for the value of  at the point  ± 0 we often write () because it is clear from the context whether it is the left or right limit; the symbol  M always means  M (, ) (if necessary the dependence on  is explicitly indicated).
Using the second Green's formula one obtains Let V =  M .Using ( 23), ( 25) one finds From the previous formula one obtains the integral representation of a solution  2 () to system (15): Using solutions (12), the second column of transmission conditions (11), and integrating by parts one gets from the previous formula where The following calculation is required below: Using the previous formula one obtains from ( 28) Using this expression one finds an integral representation for a solution  1 to system (15).Combining these results one finally obtains where  1 ⩽  ⩽  2 ; ℎ 1 () := (/ 2 2 ())ℎ  2 ().Rewrite system (32) in an operator form.Introduce a linear operator where the linear operator J is defined by the formula the matrix linear integral operator K is defined by the formula where and g = ( 1 ,  2 )  .System (32) can be rewritten as where h = (ℎ 1 , ℎ 2 )  is defined by formulae ( 29), (32).Equation ( 37) is studied in It should be noted that (37) contains the unknown constant  2 .This constant can be expressed through  1 , which is assumed to be known (see Section 8).Expressing  2 through  1 and substituting it into (37) one obtains an operator Ñ and a vector h instead of N and h, respectively, where Ñ and h are bounded.An example of how this can be done is seen, for example, in [21].Note 3.An iterative procedure to determine approximate eigenvalues of  M () can be formulated and grounded.On this way one can prove convergence of approximate eigenvalues obtained at each step of the iterative procedure to exact ones.However, here we do not formulate these results, since the main purpose of the mathematics used here is to give a rigorous proof of solvability of the problem  M ().For numerical calculations it is more convenient to use an approach based on solving an auxiliary Cauchy problem [7,16].

Operator Equation Studying
It follows from general properties of Green's function that the functions   are piecewise continuous in the (closed It is easy to prove the following. ] be a matrix integral operator with bounded and piecewise continuous kernels   (, ) in the square Π.Then the operator K is bounded and Consider the algebraic cubic equation with respect to  0 [8,12]: If the inequality holds, then (38) has two nonnegative roots  * and  * [8,12].

Advances in Mathematical Physics
The following two statements are proved in [8,12].

Dispersion Equation
Using the first column of transmission conditions (11) and solutions ( 12) one finds Using the first equation from (32) the previous formulae can be rewritten in the form where Excluding the unknown  2 from (42) one obtains the DE in the form where As is noted in Section 7 the constant  2 can be expressed through the constant  1 from the system (42).Taking this into account (37) can be rewritten in a form which does not contain  2 ; thus all estimations indicated in the previous section are substantiated.
For  = 0 from (44) one obtains () = 0. Clearly, this equation determines eigenvalues (and only them) of the problem  M (0).In Section 5 it is proved that  M (0) has not more than a finite number of simple eigenvalues.Hence one obtains that the equation () = 0 has not more than a finite number of simple roots (which coincide with eigenvalues of the problem  M (0)).
Let the problem  M (0) have  eigenvalues  *  such that Since each  *  is a simple root of the equation () = 0, then for each  *  there exists a segment Λ  := [ *  −    ,  *  +     ] such that the function () has different signs at the endpoints of the segment (where    > 0,    > 0 are determined in the way that Green's function  M (, ; ) is continuous on Λ  ).Under Λ  we consider a segment of maximum possible length.
It is clear that the maximum value of () is bounded on each Λ  .Moreover, by choosing appropriate  the product () can be made as small as necessary.
Consider the dispersion equation Φ() = 0.It is clear that () is continuous and changes its sign when  varies from  *  −    to  *  +    .Since () is bounded on Λ  , then it is always possible to chose sufficiently small  in order that the equation Φ() = 0 will have at least  roots λ ∈ ( Since the function Φ() is continuous, then the equation Φ() = 0 has a root λ inside Λ  ; to be more precise    < λ <    .We can choose  0 = min{ 2 1 , M/ 3 0 }.It follows from this theorem that there exist axisymmetric guided TM-polarized waves without attenuation in circular cylindrical dielectric waveguides filled with nonmagnetic isotropic medium with Kerr nonlinearity.This result generalizes the well-known corresponding assertion for homogeneous dielectric waveguides [17] and for inhomogeneous dielectric waveguides [30] filled with a linear medium (when  = 0).Note 5.An estimation for  0 is given in the proof of Theorem 1.
Red lines in Figures 2-4, 11-13 correspond to the linear inhomogeneous case, and blue curves correspond to the nonlinear inhomogeneous case (in the same figure the inhomogeneity is the same for both types of curves).
There is a vertical dashed line in Figures 2-4, 11-13.Eigenvalues (or PCs) of a particular problem are points of intersections of this dashed line with the dispersion curves; first three points in Figures 2-4 are marked (the smallest value, red dot, corresponds to the LP and the other two (green and black dots) correspond to the Kerr case); first four points

Figure 1 :
Figure 1: Geometry of the problem.
[8,12].The operators K, J do not coincide with operators of the same name from[8,12].For this reason Statements 2-4 are not repetitions of similar statements from[8,12]but they are proved in the same way.