Wave propagation in randomly perturbed weakly coupled waveguides

# Wave propagation in randomly perturbed weakly coupled waveguides

Liliana Borcea111Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. borcea@umich.edu    Josselin Garnier222Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France. josselin.garnier@polytechnique.edu
###### Abstract

We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation and evanescent modes with amplitudes that are random fields, due to scattering. We obtain a detailed characterization of these amplitudes and thus quantify the transfer of power between the two waveguides in terms of their separation distance. The results show that, no matter how small the fluctuations of the interfaces are, they have significant effect at sufficiently large distance of propagation, which manifests in two ways: The first effect is well known and consists of power leakage from the guided modes to the radiation ones. The second effect consists of blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power. Its quantification is the main practical result of the paper.

Key words. Random waveguides, directional coupler, power leakage, equipartition of power.

AMS subject classifications. 35Q60, 35R60.

## 1 Introduction

Guided waves have applications in electromagnetics [9], optics and communications [22, 27], imaging underwater [19, 5, 14, 15], imaging of and in tunnels [3] and so on. The classical theory of guided waves is for ideal waveguides with perfectly reflecting straight walls and filled with homogeneous media, where the wave equation can be solved using separation of variables. The wave field is represented as a superposition of finitely many guided modes, which are waves that propagate along the axis of the waveguide, and infinitely many evanescent modes which decay away from the source. These modes do not interact with each other and thus have constant amplitude determined by the wave source [9, 27].

Motivated by applications in imaging and communications, the classical theory has been extended to waveguides filled with random media [22, 12, 13, 1], with randomly perturbed boundaries [2, 1] and with slowly changing cross-section [6, 7, 22]. Weakly guiding waveguides with confining graded-index profile affected by small random perturbations have also been analyzed in [11, 25]. The resulting mode coupling theory quantifies the interaction between the modes induced by scattering, and the consequent randomization of the wave field.

We consider waveguides with penetrable boundaries, where the guiding effect is due to a medium of high index of refraction embedded in a homogeneous background. Such waveguides are analyzed in [19, 15, 17] in the context of underwater acoustics [28] and are of great importance in optics and communications [22, 26, 27]. Motivated by the latter applications, we consider a waveguide system made of two parallel step-index waveguides, which is known as a directional coupler in integrated optics [4], [27, Chapter 10]. The classical analysis of this system, described in [18], [27, Chapter 10] and [22, Chapter 10], is based on the observation that the transverse profiles of the guided waves are essentially supported in the step-index waveguides and decay outside. When the step-index waveguides are nearby, the decaying tails penetrate the neighboring waveguide and transfer of power can occur. This is the sole coupling mechanism in ideal directional couplers and for synchronous waveguides (with identical guided mode phase velocity) there is a complete, periodic transfer of power. That is to say, if the source emits power in one step-index waveguide, this is transferred to the other waveguide and back in a periodic manner, at regular distance intervals. These intervals depend on the separation between the step-index waveguides. The larger this distance is, the weaker the coupling and the farther the waves must travel for the transfer of power to occur.

We introduce a mathematical analysis of a randomly perturbed directional coupler, where the interfaces that separate the medium with high index of refraction from the background have small amplitude random fluctuations on a scale similar to the wavelength. The classic approach in [27, Chapter 10], which is based solely on the guided modes, is inadequate in this case, because scattering at the random interfaces induces mode coupling. We take into account all the modes, the guided, radiation and evanescent ones, and quantify how their interaction affects the performance of the directional coupler. The analysis is focused on the case of well separated waveguides, where the deterministic coupling is weak. It applies to an arbitrary number of guided modes, but we describe in depth the results for the case of single guided mode step-index waveguides. We show that mode coupling induced by the random fluctuations is present no matter how far apart the waveguides are and it has two effects: The first effect is well known [22, Chapter 10] and consists of power leakage from the guided modes to the radiation modes. Our analysis captures it and shows that the leaked power is self-averaging i.e., it is independent of the realization of the random processes that model the fluctuations of the interfaces. The other effect consists of the blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power between the guided waves. Its quantification in terms of the waveguide separation and amplitude of the random fluctuations is the main practical result of the paper.

The paper is organized as follows: We begin in section LABEL:sect:form with the mathematical formulation of the problem and then state the results in section LABEL:sect:results. Their derivation is in sections LABEL:sect:homogLABEL:sect:randomTwo. We end with a summary in section LABEL:sect:sum.

## 2 Formulation of the problem

We study the propagation of a time harmonic wave in a medium with index of refraction defined below, that models two step-index waveguides of width , separated in the transverse direction by the distance , as illustrated in Figure LABEL:fig:setup. The wave field is denoted by and it solves the two-dimensional Helmholtz equation

 (∂2x+∂2z)p(z,x)+[kn(ε)(z,x)]2p(z,x)=f(x)δ′(z), \hb@xt@.01(2.1)

with radiation condition at infinity, where is the wavenumber and models a source supported at the origin of the range coordinate .

In the case of ideal (unperturbed) waveguides, the index of refraction is range independent and equal to

 n(0)(x)={n if x∈(−d/2−D,−d/2)∪(d/2,d/2+D),1 otherwise, \hb@xt@.01(2.2)

with . We are interested in perturbed waveguides, where the index of refraction

 n(ε)(z,x)={n if x∈(\@fontswitchD(ε)1(z),\@fontswitchD(ε)2(z))∪(\@fontswitchDε3(z),\@fontswitchD(ε)4(z)),1 otherwise, \hb@xt@.01(2.3)

jumps across four randomly fluctuating interfaces

 \@fontswitchD(ε)1(z) =−d/2−D+εDν1(z)1(0,L(ε))(z), \@fontswitchD(ε)2(z) =−d/2+εDν2(z)1(0,L(ε))(z), \@fontswitchD(ε)3(z) =d/2+εDν3(z)1(0,L(ε))(z), \@fontswitchD(ε)4(z) =d/2+D+εDν4(z)1(0,L(ε))(z). \hb@xt@.01(2.4)

The fluctuations are modeled by the zero-mean, bounded, independent and identically distributed stationary random processes with smooth covariance function

 \@fontswitchR(z)=E[νq(0)νq(z)],q=1,…,4. \hb@xt@.01(2.5)

These satisfy strong mixing conditions as defined for example in [23, section 2]. The typical amplitude of the fluctuations is much smaller than and it is modeled in (LABEL:eq:Interfaces) by the small and positive dimensionless parameter .

We study the wavefield at , satisfying

 p(z,x)∈\@fontswitchC0((0,+∞),H10(R)∩H2(R))∩\@fontswitchC2((0,+∞),L2(R)),z>0, \hb@xt@.01(2.6)

and to set radiation conditions, we suppose that the random fluctuations are supported in the range interval . We will see that net scattering effect of these fluctuations becomes of order one at range scales of order , so we let . We will also see that for the assumed smooth covariance , the guided waves propagate mostly in the forward direction and do not interact with any fluctuations at , which is why we neglect them.

The goal of the paper is to quantify how scattering at the random interfaces (LABEL:eq:Interfaces) affects the coupling of the two step-index waveguides centered at . We consider in particular the case of a sufficiently large separation distance between the waveguides, where the deterministic coupling is very weak but the coupling induced by the random fluctuations is still present.

## 3 Statement of results

We state here the main results of the paper, derived in sections LABEL:sect:homogLABEL:sect:randomTwo by decomposing the wavefield into guided, radiation and evanescent modes of the waveguide system made of two parallel step-index waveguides illustrated in Figure LABEL:fig:setup. While our analysis applies to an arbitrary number of guided modes, in this section we consider only the case where an isolated step-index waveguide has only one guided mode. This case captures all the essential aspects of the problem and arises when

 kD√n2−1<π. \hb@xt@.01(3.1)

The two ideal step-index waveguides centered at do not interact in the waveguide system when . Thus, we can write the wavefield for large in terms of the unique guided mode of the step-index waveguide centered at , modeled by the index of refraction

 ns(x)={n if x∈(d/2,d/2+D),1 otherwise, \hb@xt@.01(3.2)

and in terms of the unique guided mode of the step-index waveguide centered at , modeled by the index of refraction . Here is the eigenfunction of the Helmholtz operator for the eigenvalue . This is defined in Lemma LABEL:lem.singlebeta as the unique solution in the interval of

 √n2k2−β2√β2−k2tan(D2√k2n2−β2)=1,β∈(k,nk). \hb@xt@.01(3.3)

The expression of the eigenfunction is [20, Section 2], [22, Chapter 1]

 ϕ(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(2η+D)−1/2cos(ξD2)exp[η(x−d2)],x≤d2,(2η+D)−1/2cos[ξ(x−d2−D2)],d2≤x≤d2+D,(2η+D)−1/2cos(ξD2)exp[−η(x−d2−D)],x≥d2+D. \hb@xt@.01(3.4)

It has a peak centered at of width , and an exponentially decaying tail, at the rate , where

 ξ=√n2k2−β2,η=√β2−k2. \hb@xt@.01(3.5)

### 3.1 Coupling of ideal waveguides

We show in section LABEL:sect:homog that under the assumption (LABEL:eq:assumeD) and for sufficiently large separation distance , the solution of

 (∂2x+∂2z)p(0)(z,x)+[kn(0)(x)]2p(0)(z,x)=f(x)δ′(z), \hb@xt@.01(3.6)

with radiation condition at infinity, takes the form***We use consistently the index for the mode amplitudes and wave field in the ideal (unperturbed) two step-index wave guide system.

 p(0)(z,x)=∑t∈{e,o}a(0)t√βteiβtzϕt(x)+O(z−2),z>0. \hb@xt@.01(3.7)

The terms in the sum model the guided modes of the waveguide system, whereas the remainder accounts for the radiation and evanescent modes [20, Section 3]. The guided modes are waves that propagate along the range coordinate and have transverse profiles essentially supported in the two step-index waveguides. They are defined by the even and odd eigenfunctions and of the operator

 \@fontswitchH(x)=∂2x+n(0)(x), \hb@xt@.01(3.8)

with given in (LABEL:eq:no), for the eigenvalues and satisfying

 βt∈(k,nk),t∈{e,o}. \hb@xt@.01(3.9)

These eigenfunctions are given explicitly in section LABEL:sect:discrSpect. They have peaks close to the center axes of the step-index waveguides and decay outside their transverse support, as illustrated in Figure LABEL:fig:eigenfunctions. Each guided mode is multiplied in (LABEL:eq:phom) by the constants

 a(0)t=√βt2∫Rdx¯¯¯¯¯¯¯¯¯¯¯¯ϕt(x)f(x),t∈{e,o}, \hb@xt@.01(3.10)

called the mode amplitudes, that are determined by the source. Throughout the paper the bar denotes complex conjugate.

We assume that the waveguides separation distance is sufficiently large, so that

 exp(−ηd)≪1, \hb@xt@.01(3.11)

with defined in (LABEL:eq:defxieta), and obtain that the eigenfunctions can be approximated by

 ϕe(x) =ϕ(|x|)+O(e−ηd),ϕo(x)=sgn(x)ϕ(|x|)+O(e−ηd),x∈R, \hb@xt@.01(3.12)

where “sgn” is the sign function. The accuracy of this approximation is illustrated in the right plot of Figure LABEL:fig:eigenfunctions. Consequently, the transverse profile of the even guided mode presents two positive peaks centered at , with exponentially decaying tails, while the transverse profile of the odd guided mode presents one positive peak centered at , one negative peak centered at , and exponentially decaying tails. The form of these peaks is proportional to the unique eigenfunction (LABEL:eq:phi) of the single-mode step-index waveguide.

We also obtain that the wave numbers are

 \hb@xt@.01(3.13)

with

 β′ =ηβ(1+η2ξ2)(1η+D2), \hb@xt@.01(3.14)

so the wavefield (LABEL:eq:phom) in the ideal waveguide system takes the form

 p(0)(z,x)=ϕ(|x|)√βeiβz[1(0,∞)(x)u(0)+(z)+1(−∞,0)(x)u(0)−(z)]+O(e−ηd)+O(z−2). \hb@xt@.01(3.15)

Here we introduced the range-dependent amplitudes of the waves propagating in the two step-index waveguides

 u(0)+(z) =(a(0)e+a(0)o)cos(β′ze−ηd)+i(a(0)e−a(0)o)sin(β′ze−ηd), \hb@xt@.01(3.16) u(0)−(z) =(a(0)e−a(0)o)cos(β′ze−ηd)+i(a(0)e+a(0)o)sin(β′ze−ηd), \hb@xt@.01(3.17)

with indexes “” corresponding to the waveguide centered at .

Equation (LABEL:eq:Po1) shows that the wavefield consists of two components, the first one is centered at with the form and the second one is centered at with the form . This is similar to the case of two independent, single-mode step-index waveguides, except that in (LABEL:eq:Po1) the amplitudes vary in , due to coupling. We obtain from (LABEL:eq:dlarge) and (LABEL:eq:up0LABEL:eq:um0) that for of the order of the wavelength,

 u(0)±(z) ≈u(0)±(0)=a(0)e±a(0)o. \hb@xt@.01(3.18)

However, at large , satisfying

 z=exp(ηd)Z,Z>0, \hb@xt@.01(3.19)

oscillate periodically in . For example, if the source gives the amplitudes

 a(0)e=a(0)o=a(0)2, \hb@xt@.01(3.20)

according to (LABEL:eq:Amplo), so that and , at range (LABEL:eq:largez) we have

 u(0)+(eηdZ)=a(0)cos(β′Z),u(0)−(eηdZ)=ia(0)sin(β′Z). \hb@xt@.01(3.21)

In conclusion, the total wave power in the ideal waveguide system at large range is essentially supported in the two step-index waveguides. The wave power in the waveguide centered at is proportional to , and equation (LABEL:eq:uoscZhom) shows that it oscillates slowly and periodically. At scaled distance , , the wave power is concentrated in the waveguide centered at , whereas at , , the wave power is concentrated in the waveguide centered at . These periodic oscillations have been reported in the literature [27, Chapter 10]. The standard method to analyze them is not to start from the analysis of the modes of the waveguide system, as we do in section LABEL:sect:homog, but to simplify by assuming that the modes can be represented as a weighted sum of the guided modes of the two waveguides. This simplified approach does not allow to take into account the role of evanescent and radiation modes, which are critical to the study of random waveguides in sections LABEL:sect:randomLABEL:sect:randomTwo, with results described next.

### 3.2 Coupling of random waveguides

The analysis of the solution of the Helmholtz equation (LABEL:eq:fouriertransform) with index of refraction (LABEL:eq:modelpert1) is carried out in sections LABEL:sect:randomLABEL:sect:randomTwo. It shows that under the assumptions (LABEL:eq:assumeD), (LABEL:eq:dlarge) and at large range , where scattering at the random interfaces (LABEL:eq:Interfaces) becomes significant, there are three distinguished regimes that determine the coupling between the waveguides:

1. The “moderate coupling” regime, where the separation distance is moderately large, satisfying

 1≫exp(−ηd)≫ε2. \hb@xt@.01(3.22)
2. The “weak coupling” regime, where is large enough so that

 exp(−ηd)=O(ε2). \hb@xt@.01(3.23)
3. The “very weak coupling” regime, where is so large that

 exp(−ηd)≪ε2≪1. \hb@xt@.01(3.24)

We now describe the results in each of these three regimes.

#### 3.2.1 Moderate coupling

At large range and in the regime defined by (LABEL:eq:modReg), the solution of (LABEL:eq:fouriertransform) with radiation condition at infinity and with index of refraction (LABEL:eq:modelpert1) is

 p(zε2,x)=ϕ(|x|)√βeiβzε2[1(0,∞)(x)u+(z)+1(−∞,0)(x)u−(z)]+o(1), \hb@xt@.01(3.25)

where are random processes and denotes a residual that tends to zero as . This residual accounts for the radiation and evanescent components of .

Similar to (LABEL:eq:Po1), we have a propagating wave centered at and another wave centered at . The coupling between the two waveguides is described by the random variations of the complex amplitudes , the analogues of (LABEL:eq:up0LABEL:eq:um0). To write their expressions, we introduce the notation

 Δβt=βt−β,t∈{e,o}, \hb@xt@.01(3.26)

which takes into account that the residual in (LABEL:eq:betaeoapr) is not negligible under the assumption (LABEL:eq:modReg) at range . We have

 u±(z) =ae(z)eiΔβezε2±ao(z)eiΔβozε2, \hb@xt@.01(3.27)

where is a random, Markovian process defined at , with initial condition given in (LABEL:eq:Amplo) for , and with the infinitesimal generator in Theorem LABEL:prop:1. To describe the results, it suffices to state from there that

 \hb@xt@.01(3.28)

with probability one, and that

 E[ao(z)¯¯¯¯¯¯¯¯¯¯¯¯ae(z)]=a(0)o¯¯¯¯¯¯¯¯a(0)eexp(−(Γ+Λ)z), \hb@xt@.01(3.29)

with positive and defined by

 Λ \hb@xt@.01(3.30) Γ =k4(n2−1)2D2β2(2η+D)2cos4(ξD2)ˆ\@fontswitchR(0), \hb@xt@.01(3.31)

in terms of

 ξγ=√k2n2−γ,ηγ=√k2−γ, \hb@xt@.01(3.32)

and the power spectral density , the Fourier transform of the covariance (LABEL:eq:covar).

We have therefore from (LABEL:eq:uepsplus) and (LABEL:eq:detpower) that the total power of the guided waves decays exponentially, at the rate ,

 |u+(z)|2+|u−(z)|2=2(|ae(z)|2+|ao(z)|2)2=2(|a(0)e|2+|a(0)o|2)exp(−Λz). \hb@xt@.01(3.33)

This decay models the transfer of power from the guided modes to the radiation modes, induced by scattering at the random interfaces (LABEL:eq:Interfaces).

The imbalance of power between the two waveguides is quantified by

 \@fontswitchP(z) =|u+(z)|2−|u−(z)|2|u+(z)|2+|u−(z)|2=2Re{ae(z)¯¯¯¯¯¯¯¯¯¯¯¯ao(z)exp[i(βe−βo)zε2]}|ae(z)|2+|ao(z)|2, \hb@xt@.01(3.34)

and its expectation is

 \hb@xt@.01(3.35)

To explain what this gives, consider the source excitation (LABEL:eq:assumea0) with , so that the initial wavefield is supported in the waveguide centered at . Then, equation (LABEL:eq:powertransf) becomes

 \hb@xt@.01(3.36)

and it describes the competition between the deterministic and random coupling of the waveguides. The cosine in (LABEL:eq:imb1) models the deterministic coupling which induces periodic oscillations of the power, as in section LABEL:sect:resHomog. The random coupling is modeled by the exponential decay in , at the rate . It shows that as the range increases, the power tends to become equally distributed among the two waveguides. This decay is present in (LABEL:eq:powertransf) as well, so the equipartition of power at large is independent of the initial condition generated by the source.

#### 3.2.2 Weak coupling

When the separation distance between the waveguides satisfies (LABEL:eq:larged), the wavefield has the same expression as in (LABEL:eq:pmoder), (LABEL:eq:uepsplus), but the random processes have different statistics.

The total power of the guided waves is still given by (LABEL:eq:powerdecay), and decays at the same rate defined in (LABEL:eq:defLambda). However, the expectation of the imbalance of power between the two waveguides satisfies the damped harmonic oscillator equation

 [∂2z+2Γ∂z+(2θβ′)2]E[P(z)]=0, \hb@xt@.01(3.37)

with defined in (LABEL:eq:Betaprime) and

 θ=ε−2exp(−ηd). \hb@xt@.01(3.38)

Based on the value of , which is independent of by assumption (LABEL:eq:larged), we distinguish three regimes, which we describe for the source excitation (LABEL:eq:assumea0), with and therefore :

1. When , the solution of (LABEL:eq:harmosc) is

 E[P(z)]= e−Γz[cosh(√Γ2−(2θβ′)2z)+Γ√Γ2−(2θβ′)2sinh(√Γ2−(2θβ′)2z)].

This tends to as , meaning that when the two waveguides are very far apart, there is no transfer of power between them. This is just as in the ideal (deterministic) waveguide system. However, unlike in ideal waveguides, the power is transferred from the guided modes to the radiation ones, as described by the exponential decay in (LABEL:eq:powerdecay).

For a finite , we have as , so the random coupling distributes the power evenly among the two waveguides.

2. In the critical case we have

 E[P(z)]=(1+Γz)exp(−Γz). \hb@xt@.01(3.39)

As in the previous case, the random coupling equidistributes the power among the two waveguides, in the limit .

3. When , the solution of (LABEL:eq:harmosc) is

 E[P(z)]=e−Γz[cos(√(2θβ′)2−Γ2z)+Γ√(2θβ′)2−Γ2sin(√(2θβ′)2−Γ2z)].

It displays periodic oscillations induced by the deterministic coupling of the waveguides, but these oscillations are damped due to the random coupling. In particular, if , we get

 E[P(z)]≈e−Γzcos(2θβ′z), \hb@xt@.01(3.40)

in agreement with (LABEL:eq:imb1).

We plot in Figure LABEL:fig:imbalance the imbalance ratio as a function of normalized by the deterministic coupling distanceAt the distance the power is fully transferred from one step-index waveguide to the other one, when there are no random perturbations. . From this plot and from equation (LABEL:eq:powerdecay) we conclude that wave scattering at the random interfaces (LABEL:eq:Interfaces) has two net effects:

1. It induces a self-averaging loss (or leakage) of total power, due to the coupling of the guided modes with the radiation modes.

2. It causes a blurring of the periodic (deterministic) power transfer from one waveguide to the other.

The blurring effect at item 2 is the main practical result of the paper. It depends on the effective parameter and it is significant as soon as becomes of order one. Therefore, the deterministic transfer of power is very sensitive to the random fluctuations of the interfaces (LABEL:eq:Interfaces).

#### 3.2.3 Very weak coupling

When the distance between the waveguides is so large that (LABEL:eq:huged) holds, the wavefield has the same expression as (LABEL:eq:pmoder), but the wave amplitudes have different statistics. They model the only coupling in this regime, between the guided and radiation modes, which generates effective wave power leakage. Explicitly, we show in section LABEL:sect:randomTwo that the wave amplitudes converge in probability, as , to the deterministic function satisfying

 |u+(z)|2=|u(0)+(0)|2exp(−Λz),|u−(z)|2=|u(0)−(0)|2exp(−Λz). \hb@xt@.01(3.41)

Although deterministic, this function is not as in the ideal waveguide system, where is constant in (because in (LABEL:eq:up0-LABEL:eq:um0) in this regime). Instead, it decays exponentially at rate , due to the power leakage. The imbalance of power between the waveguides is constant

 \@fontswitchP(z)=|u+(z)|2−|u−(z)|2|u+(z)|2+|u−(z)|2=|u(0)+(0)|2−|u(0)−(0)|2|u(0)+(0)|2+|u(0)−(0)|2, \hb@xt@.01(3.42)

so in the case of the source excitation (LABEL:eq:assumea0),

## 4 Analysis in ideal waveguides

The analysis in this section is classical and follows the lines of [20]. It derives the results stated in section LABEL:sect:resHomog by expanding the wave field on a complete set of eigenmodes. The proof of the completeness of this set is the most delicate part and it is carried out in [20] by the Levitan-Levinson method [8, Chapter 9].

Recall the Helmholtz operator (LABEL:eq:HelmIdeal) in the transverse coordinate, with index of refraction given in (LABEL:eq:no), and note that it is self-adjoint with respect to the scalar product associated to the -norm,

 (ϕ1,ϕ2) :=∫R¯¯¯¯¯¯¯¯¯¯¯¯¯ϕ1(x)ϕ2(x)dx. \hb@xt@.01(4.1)

Its spectrum is where are called the guided mode wavenumbers. They are positive and satisfy the order relation

 k2<β2t,Nt<⋯<β2t,1

where the index stands for the even and odd eigenfunctions in the transverse coordinate .

We describe next the eigenfunctions for the discrete spectrum and the improper eigenfunctions for the continuum spectrum, and explain that they form a complete set. We use them to decompose the wavefield into guided, radiation and evanescent modes with amplitudes determined by the source.

### 4.1 Discrete spectrum

There are two sets of discrete eigenvalues and eigenfunctions: the first one associated with the even modes in and the second one associated with the odd modes.

The -th even eigenfunction for the eigenvalue is defined by

 ϕe,j(x)Ae,j=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩exp(−ηe,jd2)sin(ξe,jD)(1+ξ2e,jη2e,j)cosh(ηe,jx),x∈[0,d2],ξe,jηe,jcos[ξe,j(x−d2−D)]−sin[ξe,j(x−d2−D)],x∈[d2,d2+D],ξe,jηe,jexp[−ηe,j(x−d2−D)],x∈[d2+D,∞), \hb@xt@.01(4.3)

and for . Here

 ξe,j=√k2n2−β2e,j,ηe,j=√β2e,j−k2, \hb@xt@.01(4.4)

and is the normalization constant

 Ae,j= [12exp(−ηe,jd)sin2(ξe,jD)(1+ξ2e,jη2e,j)2(sinh(ηe,jd)ηe,j+d) \hb@xt@.01(4.5)

calculated so that has unit -norm. Moreover, satisfies the dispersion relation

 (1+ξ2e,jη2e,j)e−ηe,jd=[1−ξe,jηe,jtan(ξe,jD2)][1+ξe,jηe,jcotan(ξe,jD2)], \hb@xt@.01(4.6)

with given by (LABEL:def:xiej), which ensures the continuity of (LABEL:eq:phie) and its derivative. The number of solutions of (LABEL:eq:dispersione) is denoted by and when is large, it depends on the value of .

The -th odd eigenfunction for the eigenvalue is defined similarly,

 ϕo,j(x)Ao,j=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩exp(−ηo,jd2)sin(ξo,jD)(1+ξ2o,jη2o,j)sinh(ηo,jx),x∈[0,d2],ξo,jηo,jcos[ξo,j(x−d2−D)]−sin[ξo,j(x−d2−D)],x∈[d2,d2+D],ξo,jηo,jexp[−ηo,j(x−d2−D)],x∈[d2+D,∞), \hb@xt@.01(4.7)

with for and

 ξo,j=√k2n2−β2o,j,ηo,j=√β2o,j−k2. \hb@xt@.01(4.8)

The normalization constant is given by

 Ao,j \hb@xt@.01(4.9)

so that has unit norm and satisfies the dispersion relation

 −(1+ξ2o,jη2o,j)e−ηo,jd=[1−ξo,jηo,jtan(ξo,jD2)][1+ξo,jηo,jcotan(ξo,jD2)], \hb@xt@.01(4.10)

with given by (LABEL:eq:xio), for , which ensures that (LABEL:eq:phio) and its derivative are continuous.

### 4.2 Continuous spectrum

For there are two improper eigenfunctions, even and odd, denoted by , for . We write their expression below in terms of the parameters

 ξγ=√k2n2−γ,ηγ=√k2−γ. \hb@xt@.01(4.11)

The even eigenfunctions satisfy for , and are defined by

 ϕe,γ(x)Ae,γ=ξγηγcos(ηγx),for x∈[0,d2], \hb@xt@.01(4.12)

and by

 ϕe,γ(x)Ae,γ=ξγηγcos[ξγ(x−d2)]cos(ηγd2)−sin[ξγ(x−d2)]sin(ηγd2), for x∈[d2,d2+D], \hb@xt@.01(4.13)

and by

 −sin(ξγD)sin(ηγd2)] +cos(ξγD)sin(ηγd2)] for x≥d2+D, \hb@xt@.01(4.14)

with normalization constant given by

 Ae,γ= (2πηγ)−1/2{[ξγηγcos(ξγD)cos(ηγd2)−sin(ξγD)sin(ηγd2)]2 +ξ2γη2γ[ξγηγsin(ξγD)cos(ηγd2)+cos(ξ