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

Authors: E. Yu. Smol’kin; D. V. Valovik
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

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
onz (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.


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## Body

## 1. 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, [1–3], and for newest experimental observation see [4–6].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.Currently there has been significant progress in studying of polarised electromagnetic TE and TM wave propagation in waveguide structures (for a planar waveguide, see [8–11]; for a circular cylindrical waveguide, see [1, 8, 12–15]; for a circular two-layered cylindrical waveguide, see [7, 16]) filled with nonlinear dielectric permittivities. In particular, theorems on existence and localization of eigenvalues in some of these problems have been proved.Most of these papers are devoted to studying of polarized waves in waveguides filled with a homogeneous nonlinear 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 [20–22].


## 2. Governing Equations

Consider three-dimensional spaceR3 with cylindrical coordinate system Oρφz. In this space a two-layer circular cylindrical waveguide
(1)Σ≔ρ,φ,z:0⩽ρ<R1,0⩽φ<2π∪(ρ,φ,z):R1⩽ρ⩽R2,0⩽φ<2π
is placed; the waveguide axis coincides with Oz. (The waveguide is unlimited in z 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 μ=μ0, where μ0>0 is the permeability of free space. Geometry of the problem is shown in Figure 1.Figure 1
Geometry of the problem.Consider monochromatic electromagnetic field(2)E~=ReEe-iωt,H~=ReHe-iωt
propagating at a frequency ω along the surface of Σ. The quantities E, H are called the complex amplitudes [1]; E=(Eρ,Eφ,Ez)T and H=(Hρ,Hφ,Hz)T, where (·)T denotes the transpose operation.Complex amplitudesE, H of a monochromatic electromagnetic field must satisfy time-harmonic Maxwell’s equations
(3)rot⁡H=-iωεE,rot⁡E=iωμH,
the continuity conditions for the tangential components of the field at the interfaces (on the boundary of the waveguide) ρ=R1, ρ=R2, and the radiation condition at infinity: the electromagnetic field decays as O(|ρ|-1) when ρ→∞. The solutions to Maxwell’s equations are sought in the entire space.In the whole space the dielectric permittivity has the formε=ε~ε0, where
(4)ε~=ε1,0⩽ρ<R1,ε2(ρ)+αE2,R1⩽ρ⩽R2,ε3,ρ>R2,ε1, ε3 are positive real constants, ε0>0 is the permittivity of free space, and
(5)E2=Ee-iωt,eρ2+Ee-iωt,eφ2+Ee-iωt,ez2;
here ek is an orthonormal vector in k direction, (·,·) is the Euclidean scalar product, ε2(ρ)∈C1[R1,R2], ε2(ρ)>0, and α>0 is a real constant.Note 1.
InsideΣ the quantities ε1 and ε2+α|E|2 are scalars. It is possible to consider an anisotropic case in which one of these quantities (or both) is a diagonal tensor.


## 3. Statement of the Problem

Consider TM-polarized waves in the monochromatic mode(6)Ee-iωt=e-iωtEρ,0,EzT,He-iωt=e-iωt0,Hφ,0T.It is assumed that waves propagating along the boundary ofΣ depend harmonically on z [1, 8, 16, 23]. Substituting the TM waves into (3) one is convinced that the field does not depend on φ. Thus the components of the field have the form
(7)Eρ=Eρ(ρ)eiγz,Ez=Ez(ρ)eiγz,Hφ=Hφ(ρ)eiγz,
where γ is a real spectral parameter of the problem which defines unknown PCs (without loss of generality we suppose γ>0). In what follows arguments of functions will often be omitted.Letk02:=ω2μ0ε0. Substituting the complex amplitudes E, H with components (7) into Maxwell’s equations (3) one obtains
(8)iγEρ-Ez′=iωμHφ,iγHφ=iωεEρ,ρ-1ρHφ′=-iωεEz.
Introducing the notations u1(ρ;γ):=Eρ(ρ;γ) and u2(ρ;γ):=iEz(ρ;γ) one obtains from (8) that
(9)γ2-k02ε~u1+γu2′=0,γρu1′+ρu2′′+k02ε~ρu2=0,
where u1, u2 are real functions and γ2>k02max⁡{ε1,ε3}.It is also assumed that functionsu1, u2 are sufficiently smooth:
(10)u1∈C0,R1∩CR1,R2∩CR2,+∞∩C1(0,R1)∩C1(R1,R2)∩C1(R2,+∞),u2∈C0,+∞∩C10,R1∩C1R1,R2∩C1R2,+∞∩C2(0,R1)∩C2(R1,R2)∩C2(R2,+∞).Tangential components of electromagnetic field are known to be continuous at interfaces [23, 24]. Taking into account that the tangential components are Ez and Hφ and using formulae (8) one obtains
(11)[ε~u1]ρ=R1=0,[u2]ρ=R1=0,[ε~u1]ρ=R2=0,[u2]ρ=R2=0,
where [v]ρ=s=lim⁡ρ→s-0⁡v(ρ)-lim⁡ρ→s+0⁡v(ρ).ProblemPM(α) is to prove existence of quantities γ^ such that for prescribed value of the field at a boundary of Σ there exist nontrivial functions u1(ρ;γ^), u2(ρ;γ^) defined for ρ∈[0,+∞), which must satisfy system (9), conditions (10), (11), and the radiation condition at infinity.The quantitiesγ^ solving problem PM(α) are called eigenvalues or PCs; corresponding functions u1(ρ;γ^), u2(ρ;γ^) are called eigenfunctions (see, e.g., [25, 26]).


## 4. 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
(12)u1(ρ)=-C1γk1-1I1(k1ρ),0⩽ρ<R1,C2γk3-1K1(k3ρ),ρ>R2;u2(ρ)=C1I0(k1ρ),0⩽ρ<R1,C2K0(k3ρ),ρ>R2,
where k12:=γ2-k02ε1, k32:=γ2-k02ε3; I0, I1, K0, and K1 are the modified Bessel functions [27]. The constant C1 is assumed to be known (see the definition of PM(α)); the constant C2 is determined using conditions (11). Solutions (12) are real as γ2>k02max⁡{ε1,ε3}.Let(13)LM:=ddρpddρ+ε2ρ,W:=γqu2u1′-ρu2u2,
where
(14)k22(ρ):=k02ε2(ρ)-γ2,p(ρ):=ε2(ρ)ρk22(ρ),q(ρ):=ρk22(ρ),u2=u12+u22.
Thus system (9) in the waveguide cladding can be written in the form
(15)u1=k2-2(ρ)(γu2′-αk02u2u1),LMu2=αW.In the Kerr law the coefficientα is supposed to be small [2, 3, 28]; for this reason the small parameter method can be used to study the problem PM(α).For proving existence of solutions toPM(α) it is necessary at first to prove existence of solutions to the linear problem PM(0), which is PM(α) for α=0.


## 5. Existence of Solutions toPM(0)

The problemPM(0) derived from PM(α) is difficult to investigate. For this reason we revert to (8) and consider the original problem for α=0. The problem obtained in such a way is also denoted as PM(0).Letλ:=γ2. Introduce the notation u3:=ρHφ; from system (8) for α=0 one obtains
(16)Lu3=0,ρ∈[R1,R2],
where L:=(d/dρ)((1/ε2ρρ)·(d/dρ))+k02/ρ-λ/ε2ρρ.As the componentsHφ and Ez are continuous at the interfaces one obtains the following conditions for u3:
(17)[u3]ρ=R1=0,[u3]ρ=R2=0,[u3′]ρ=R1=0,[u3′]ρ=R2=0,
and in view of (12)(18)u3(ρ)=-C1ωε0ε1k1-1ρI1(k1ρ),0⩽ρ<R1,C2ωε0ε3k3-1ρK1(k3ρ),ρ>R2.Taking into account conditions (17) and solutions (18) one obtains boundary conditions of the third kind (which depend nonlinearly on the spectral parameter):
(19)u3′(R1)-g1(λ)u3(R1)=0,u3′(R2)+g2(λ)u3(R2)=0,
where the functions g1, g2 are easily found. Thus the problem PM(0) is a (linear) problem of Sturm-Liouville type for (16) with conditions (19).At first one needs to consider yet another Sturm-Liouville:(20)Lv=0,v′(R1)=v′(R2)=0.Letλn,vn(ρ)n=1∞ be the complete system of eigenvalues and eigenfunctions orthonormal in the space L2R1,R2;r(ρ) with weight r(ρ):=1/ε2ρ 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 (λn→-∞ for n→∞). We arrange eigenvalues in the ascending order: ⋯<λi+1<λi<⋯<λ2<λ1.Solvability ofPM(0) is established by the following statement (see proof in [29]). This statement asserts that required eigenvalues of PM(0) lie between eigenvalues of problem (20).Statement 1.
Letε2(ρ)∈C1[R1,R2], ε2(ρ)>0 for ρ∈[R1,R2], and problem (20) have m+1 positive eigenvalues λj, where m⩾1(j=1,m+1¯), such that
(21)0<k02max⁡{ε1,ε3}<λm+1<λm<⋯<λ2<λ1.
Then there exist at least m simple eigenvalues λi*>0(i=1,m¯) of the problem PM(0).In this case PCsγ~i of guided TM waves in Σ are defined as γ~i=λi*∈(λi+1,λi), i=1,m¯.Note 2.
ThicknessR2-R1 of the waveguide cladding can be chosen in such a way that the problem PM(0) will have solutions.


## 6. Nonlinear Integral Equations and ProblemPM(α)

Letλ:=γ2. Consider the boundary value problem
(22)LMv=0,v(R1)=v(R2)=0,ρ∈[R1,R2].Since all the coefficients ofLMv=0 are continuous and nonzero on the segment [R1,R2], 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 PM(0). Thus one obtains that a unique Green’s function GM(ρ,s) of the boundary value problem
(23)LMGM=-δ(ρ-s),GM(R1,s)=GM(R2,s)=0mmmmmmmmmmmmmmmmmmnR1⩽ρ,s⩽R2
exists in a neighbourhood of each eigenvalue of PM(0).For brevity we use the following notation:(24)GM′(A,s):=∂ρGM(ρ,s)ρ=A,GM′′(A,B):=∂sρ2GM(ρ,s)ρ=As=B,GM′(ρ,A):=∂sGM(ρ,s)s=A,GM′′(A,s):=∂sρ2GM(ρ,s)ρ=A,GM′′(ρ,A):=∂sρ2GM(ρ,s)s=A;
for the value of f at the point A±0 we often write f(A) because it is clear from the context whether it is the left or right limit; the symbol GM always means GM(ρ,s) (if necessary the dependence on γ is explicitly indicated).Using the second Green’s formula one obtains(25)∫R1R2(vLMu-uLMv)dρ=p(ρ)(u′v-uv′)R1R2.Letv=GM. Using (23), (25) one finds
(26)∫R1R2GMLMu-uLMGMdρ=p(R1)u(R1)GM′(R1,s)-pR2uR2GM′(R2,s).From the previous formula one obtains the integral representation of a solutionu2(s) to system (15):
(27)u2s=-α∫R1R2GMWdρ+pR1u2R1GM′R1,s-p(R2)u2(R2)GM′(R2,s).Using solutions (12), the second column of transmission conditions (11), and integrating by parts one gets from the previous formula
(28)u2s=αγ∫R1R2∂ρGM·qu2u1dρ+α∫R1R2GM·ρu2u2dρ+h2(s),
where
(29)h2s≔C1pR1I0k1R1GM′R1,s-C2p(R2)K0(k3R2)GM′(R2,s).The following calculation is required below:(30)∂∂s∫R1R2∂ρGM·qu2u1dρ=∂∂s∫R1s-0∂ρGM·qu2u1dρ+∫s+0R2∂ρGM·qu2u1dρ=∫R1R2∂sρ2GM·qu2u1dρ+q(s)us2u1(s)·∂ρGMρ=s-0-∂ρGMρ=s+0=∫R1R2∂sρ2GM·qu2u1dρ-us2u1(s)ε2(s).Using the previous formula one obtains from (28)(31)u2′s=αγ∫R1R2∂sρ2GM·qu2u1dρ+α∫R1R2∂sGM·ρu2u2dρ-αγus2u1(s)ε2s+h2′(s).
Using this expression one finds an integral representation for a solution u1 to system (15). Combining these results one finally obtains
(32)u1s=αγ2k22s∫R1R2∂sρ2GM·qu2u1dρ+αγk22s∫R1R2∂sGM·ρu2u2dρ-αk22(s)γ2ε2(s)+k02us2u1(s)+h1(s),u2s=αγ∫R1R2∂ρGM·qu2u1dρ+α∫R1R2GM·ρu2u2dρ+h2(s),
where R1⩽s⩽R2; h1(s):=γ/k22sh2′(s).Rewrite system (32) in an operator form. Introduce a linear operator
(33)N:=K+J,
where the linear operator J is defined by the formula
(34)J=-1k22(s)γ2ε2(s)+k021000,
the matrix linear integral operator K is defined by the formula
(35)Kg=∫R1R2K(ρ,s)g(ρ)dρ,
where
(36)Kρ,s=Knmρ,sn,m=12=γ2q(ρ)∂sρGMk22(s)γρ∂sGMk22(s)γq(ρ)∂ρGMρGM,
and g=(g1,g2)T.System (32) can be rewritten as
(37)u=αNu2u+h,
where h=(h1,h2)T is defined by formulae (29), (32).Equation (37) is studied in C[R1,R2]=C[R1,R2]×C[R1,R2] with norm uC2=u1C2+u2C2, where uC=max⁡x∈[R1,R2]⁡|u(x)|.It should be noted that (37) contains the unknown constant C2. This constant can be expressed through C1, which is assumed to be known (see Section 8). Expressing C2 through C1 and substituting it into (37) one obtains an operator N~ and a vector h~ instead of N and h, respectively, where N~ 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 ofPM(α) 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 PM(α). For numerical calculations it is more convenient to use an approach based on solving an auxiliary Cauchy problem [7, 16].


## 7. Operator Equation Studying

It follows from general properties of Green’s function that the functionsKij are piecewise continuous in the (closed) square Π¯=[R1,R2]×[R1,R2]. Thus the operator K:C[R1,R2]→C[R1,R2] is bounded. Obviously, the operator J:C[R1,R2]→C[R1,R2] is also bounded.It is easy to prove the following.Statement 2.
LetK:C[R1,R2]→C[R1,R2] be a matrix integral operator with bounded and piecewise continuous kernels Knm(x,y) in the square Π¯. Then the operator K is bounded and KC→C⩽M, where M2=2(max⁡j=1,2⁡K1jC→C2+max⁡j=1,2⁡K2jC→C2).Consider the algebraic cubic equation with respect tor0 [8, 12]:
(38)αNr03+h=r0.If the inequality(39)0⩽h<233αN-1
holds, then (38) has two nonnegative roots r* and r* [8, 12].The following two statements are proved in [8, 12].Statement 3.
Ifα⩽A2, where A=2(3h3N)-2, then (37) has a unique continuous solution u∈C[R1,R2] in the ball Br*≡u:u⩽r*.Note thatA is independent of α and A>0.Statement 4.
Let kernels ofN and h in (37) depend continuously on the parameter γ∈Γ0 on a real interval Γ0 and
(40)h(γ)<233αNγ-1.
Then (37) has a unique continuous solution u(ρ;γ) which depends continuously on γ∈Γ0; that is, u(ρ;γ)⊂C[R1,R2]×Γ0.Note 4.
The operatorsK, 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.


## 8. Dispersion Equation

Using the first column of transmission conditions (11) and solutions (12) one finds
(41)-C1ε1γk1-1I1k1R1=ε2(R1)u1(R1+0)+αuR1+02u1(R1+0),C2ε3γk3-1K1k3R2=ε2R2u1R2-0+αuR2-02u1R2-0.Using the first equation from (32) the previous formulae can be rewritten in the form
(42)-C1ε1γk1-1I1(k1R1)=αγk22(R1)F(R1)+ε2(R1)h1(R1),C2ε3γk3-1K1k3R2=αγk22R2FR2+ε2R2h1R2,
where
(43)Fs≔γε2s∫R1R2∂sρGM(ρ,s)·qu2u1dρ+ε2(s)∫R1R2∂sGMρ,s·ρu2u2dρ-2γus2u1(s).Excluding the unknownC2 from (42) one obtains the DE in the form
(44)C1g(λ)=αQ(λ),
where
(45)gλ:=R1R2k1k3ε22(R1)ε22(R2)I0(k1R1)K0·k3R2GM′′(R1,R2)GM′′(R2,R1)-ε1k24(R1)I1(k1R1)+R1k1ε22(R1)·I0k1R1GM′′R1,R1×ε3k24R2K1k3R2+R2k3ε22(R2)K0(k3R2)GM′′(R2,R2),Qλ≔k1k22R1ε3k24R2K1k3R2+R2k3ε22mmmmmmmk24R2·R2K0k3R2GM′′R2,R2·FR1-R2k1k22R1k3ε2R1ε2R2K0·(k3R2)GM′′(R2,R1)F(R2).As is noted in Section7 the constant C2 can be expressed through the constant C1 from the system (42). Taking this into account (37) can be rewritten in a form which does not contain C2; thus all estimations indicated in the previous section are substantiated.


## 9. Existence of Solutions toPM(α)

Consider the functionΦ(λ)≡C1g(λ)-αQ(λ). If λ=λ^ is such that Φ(λ^)=0, then eigenvalues γ^ of the problem PM(α) are determined from the equation λ^=γ^2.Forα=0 from (44) one obtains g(λ)=0. Clearly, this equation determines eigenvalues (and only them) of the problem PM(0). In Section 5 it is proved that PM(0) has not more than a finite number of simple eigenvalues. Hence one obtains that the equation g(λ)=0 has not more than a finite number of simple roots (which coincide with eigenvalues of the problem PM(0)).Let the problemPM(0) have m eigenvalues λi* such that
(46)k02max⁡{ε1,ε3}<λm*<λm-1*<⋯<λ2*<λ1*.Since eachλi* is a simple root of the equation g(λ)=0, then for each λi* there exists a segment Λi:=λi*-δi′,λi*+δi′′ such that the function g(λ) has different signs at the endpoints of the segment (where δi′>0, δi′′>0 are determined in the way that Green’s function GM(ρ,s;λ) is continuous on Λi). Under Λi we consider a segment of maximum possible length.It is clear that the maximum value ofQ(λ) is bounded on each Λi. Moreover, by choosing appropriate α the product αQ(λ) can be made as small as necessary.Consider the dispersion equationΦ(λ)=0. It is clear that g(λ) is continuous and changes its sign when λ varies from λi*-δi′ to λi*+δi′′. Since Q(λ) is bounded on Λi, then it is always possible to chose sufficiently small α in order that the equation Φ(λ)=0 will have at least m roots λ^i∈λi*-δi′,λi*+δi′′, i=1,m¯.The following theorem is the main result of this paper.Theorem 1.
Let0<ε0<min⁡(ε1,ε3) and let there be m⩾1 solutions λi* of the problem PM(0) such that
(47)max⁡ε1,ε3<λm*<λm-1*<⋯<λ2*<λ1*.
Then there is α0>0 such that for any α⩽α0 at least m eigenvalues γ^i of the problem PM(α) exist and γ^i∈λi*-δi′,λi*+δi′′, i=1,m¯.Proof.
LetΛ:=⋃i=1m‍Λi. Green’s function GM(ρ,s;λ) exists for all λ∈Λ. Clearly, the function A(λ)=2/3h(λ)3N(λ) is continuous with respect to λ∈Λ. Let A1=min⁡λ∈Λ⁡A(λ) and α<A12. According to Statement 3 there is a unique solution u=u(λ) to (37) for any λ∈Λ. This solution is continuous and u⩽r*=r*(λ). Let r0=max⁡λ∈Λ⁡r*(λ). Estimating Q(λ) one obtains Qλ⩽Cr03, where C is a constant.
The functiong(λ) is continuous and equation g(λ)=0 has at least one root λi*∈Λi. Let M1=min⁡λ∈Λ⁡g(λ), M2=min⁡λ∈Λ⁡g(λ). Then the value M~=min⁡{M1,M2} is positive and does not depend on α.
Ifα⩽M~/Cr03, then for each λi* there exist λi′ and λi′′ such that λi*-δi′<λi′<λi*<λi′′<λi*+δi′′ and it is true that
(48)C1g(λi′)-αQ(λi′)·C1g(λi′′)-αQ(λi′′)<0.
Since the functionΦ(λ) is continuous, then the equation Φ(λ)=0 has a root λ^i inside Λi; to be more precise λi′<λ^i<λi′′. We can choose α0=min⁡A12,M~/Cr03.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 α = 0).Note 5.
An estimation forα0 is given in the proof of Theorem 1.


## 10. Numerical Results

We use abbreviations “LP” for “linear problem” and “NLP” for “nonlinear problem.”For calculations a numerical method suggested in [7, 16] was used. The following nonlinearities and inhmogeneities were considered.(1)
Kerr law:ε=ε2(ρ)+αu2:
(a)
ε2(ρ)=εc+1/ρ,(b)
ε2ρ=εc,(c)
ε2(ρ)=εc+ρ.(2)
Law with saturation:ε=ε2(ρ)+αu2/(1+βu2):
(a)
ε2(ρ)=εc+1/ρ,(b)
ε2(ρ)=εc,(c)
ε2ρ=εc+ρ.The termεc is a positive real constant; we use εc>max⁡{ε1,ε3}. (In fact, cases 1(b) and 2(b) are homogeneous.)In the figures the dependenceγ≡γ(ΔR) (Figures 2–4, 11–13) and eigenfunctions u1(ρ), u2(ρ) (Figures 5–10, 14–19) are plotted. We call the dependence γ(ΔR) the dispersion curve (DC).Figure 2
Case 1(a). Marked eigenvalues:γ~≈2.482 (red dot, LP), γ^≈2.546 (green dot, NLP), and γ^≈3.940 (black dot, NLP).Figure 3
Case 1(b). Marked eigenvalues:γ~≈2.422 (red dot, LP), γ^≈2.464 (green dot, NLP), and γ^≈3.980 (black dot, NLP).Figure 4
Case 1(c). Marked eigenvalues:γ~≈2.554 (red dot, LP), γ^≈2.624 (green dot, NLP), and γ^≈3.988 (black dot, NLP).Figure 5
Case 1(a): eigenfunctionsu1(ρ) (R1=2, R2=3.5). LP: γ~≈2.482 (red); NLP: γ^≈2.546 (green), γ^≈2.940 (black).Figure 6
Case 1(a): eigenfunctionsu2ρ (R1=2, R2=3.5). LP: γ~≈2.482 (red); NLP: γ^≈2.546 (green), γ^≈2.940 (black).Figure 7
Case 1(b): eigenfunctionsu1(ρ) (R1=2, R2=3.5). LP: γ~≈2.422 (red); NLP: γ^≈2.464 (green), γ^≈2.980 (black).Figure 8
Case 1(b): eigenfunctionsu2(ρ) (R1=2, R2=3.5). LP: γ~≈2.422 (red); NLP: γ^≈2.464 (green), γ^≈2.980 (black).Figure 9
Case 1(c): eigenfunctionsu1(ρ) (R1=2, R2=3.2). LP: γ~≈2.554 (red); NLP: γ^≈2.624 (green), γ^≈2.988 (black).Figure 10
Case 1(c): eigenfunctionsu2(ρ) (R1=2, R2=3.2). LP: γ~≈2.554 (red); NLP: γ^≈2.624 (green), γ^≈2.988 (black).Figure 11
Case 2(a). Marked eigenvalues:γ~≈2.482 (red dot, LP), γ^≈2.5425, γ^≈3.12, and γ^≈3.855 (green, blue, and black dots, resp., NLP).Figure 12
Case 2(b). Marked eigenvalues:γ~≈2.422 (red dot, LP), γ^≈2.4625, γ^≈3.1925, and γ^≈3.7925 (green, blue, and black dots, resp., NLP).Figure 13
Case 2(c). Marked eigenvalues:γ~≈2.554 (red dot, LP), γ^≈2.62, γ^≈3.1925, and γ^≈3.8425 (green, blue, and black dots, resp., NLP).Figure 14
Case 2(a): eigenfunctionsu1(ρ) (R1=2, R2=3.5). LP: γ~≈2.482 (red); NLP: γ^≈2.5425 (green), γ^≈3.12 (blue), and γ^≈3.855 (black).Figure 15
Case 2(a): eigenfunctionsu2(ρ) (R1=2, R2=3.5). LP: γ~≈2.482 (red); NLP: γ^≈2.5425 (green), γ^≈3.12 (blue), and γ^≈3.855 (black).Figure 16
Case 2(b): eigenfunctionsu1(ρ) (R1=2, R2=3.5). LP: γ~≈2.422 (red); NLP: γ^≈2.4625 (green), γ^≈3.1925 (blue), and γ^≈3.7925 (black).Figure 17
Case 2(b): eigenfunctionsu2(ρ) (R1=2, R2=3.5). LP: γ~≈2.422 (red); NLP: γ^≈2.4625 (green), γ^≈3.1925 (blue), and γ^≈3.7925 (black).Figure 18
Case 2(c): eigenfunctionsu1(ρ) (R1=2, R2=3.2). LP: γ~≈2.554 (red); NLP: γ^≈2.62 (green), γ^≈3.1925 (blue), and γ^≈3.8425 (black).Figure 19
Case 2(c): eigenfunctionsu2(ρ) (R1=2, R2=3.2). LP: γ~≈2.554 (red); NLP: γ^≈2.62 (green), γ^≈3.1925 (blue), and γ^≈3.8425 (black).The following parameters were used for the calculations:ε1=4, εc=9, ε3=1, R1=2, 2<R2<8, k0=1, and C1=1; for the Kerr nonlinearity α=0.01 (Figures 2–10).Dispersion curves for the cases 1(a,b,c), 2(a,b,c) are shown in Figures2–4, 11–13, respectively.In Figures2–4, 11–13 the vertical axis corresponds to γ2>4, and the horizontal axis corresponds to the thickness 0<ΔR=R2-R1<6 of the cylindrical cladding (see Figure 1).Red lines in Figures2–4, 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 Figures2–4, 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 in Figures 11–13 are marked (the smallest value, red dot, corresponds to the LP and the other three (green, blue, and black dots) correspond to the case with a saturated nonlinearity).Vertical dashed line in Figures2, 3, and 4 corresponds to ΔR=1.5, ΔR=1.5, and ΔR=1.2, respectively.In Figures5–10, 14–19 the vertical axis corresponds to the value of the plotted functions u1(ρ),u2(ρ) and the horizontal axis corresponds to the value ρ.In Figures5–10, 14–19 eigenfunctions u1(ρ),u2(ρ) for the cases 1, 2, and 3, respectively, are plotted; the color of a curve corresponds to the color of marked eigenvalue in Figures 2–4, 11–13.Vertical dashed line in Figures11, 12, and 13 corresponds to ΔR=1.5, ΔR=1.5, and ΔR=1.2, respectively.


## 11. Discussion

Applying the analytical method, which is used in this paper, it is possible to consider an arbitrary continuous nonlinearity with a small multiplier. If one chooses sufficiently small value of the multiplier, then the analytical method allows proving existence of eigenvalues in the nonlinear problem which are close to eigenvalues of the corresponding linear problem. For example, nonlinearities with saturation likeε=ε2(ρ)+αu2/(1+βu2) or ε=ε2(ρ)+α(1-e-βu2) [28] can also be treated with this method. However, as it is seen from the plots above for different type of nonlinearities, in the nonlinear problems there exist more eigenvalues than Theorem 1 predicts. Up to now the authors are not aware of any rigorous mathematical method which allows solving nonlinear eigenvalue problems of the type considered in this paper (for cylindrical geometry). In other words, it is an open question how to study theoretically this and similar problems [1, 7, 12–16] completely.If we compare red (LP) and blue (NLP) dispersion curves in Figures2–4 we can notice that in LPs there are not more than a finite number of eigenvalues (this fact is well known). In each nonlinear case there are eigenvalues which are close to corresponding linear case (these eigenvalues can be determined with a perturbation theory; existence of these very eigenvalues is proved analytically in this paper). At the same time, it is easy to see, from Figures 2–4, 11–13, that in each nonlinear case there are new eigenvalues which may correspond to a new propagation regime (we call them “purely nonlinear” eigenvalues).In Figures2–4 in groups of three eigenvalues are marked. The red dot (smallest value) is an eigenvalue in the linear problem (with α=0). The green dot is an eigenvalue of the nonlinear problem. When one passes to the limit α→0 then the green dot tends to the red dot. Choosing several values of α, 0<α1<α2<α3, we did not see from calculations that the black dot tends to an eigenvalue of the linear problem when α decreases; this eigenvalue likely can not be determined with a perturbation theory.In Figures11–13 groups of four eigenvalues are marked. The green dot is close to a solution of the linear problem (red dot). Here there are new eigenvalues as well: blue and black dots which also may correspond to a new propagation regime. However, in this case if α decreases, then the new eigenvalues disappear.Whether these mathematically predicted “purely” nonlinear waves really exist is a hypothesis that can be proved or disproved in an experiment.It is also important to note that the higher an eigenvalue is, the higher the maximum of the corresponding eigenfunction is (see Figures5–10, 14–19). Probably, eigenfunctions with sufficiently high maxima correspond to the case in which chosen nonlinear law is no longer valid.



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