Wave propagation in randomly perturbed weakly coupled waveguides
WWAVE PROPAGATION IN RANDOMLY PERTURBED WEAKLYCOUPLED WAVEGUIDES
LILIANA BORCEA ∗ AND
JOSSELIN GARNIER † Abstract.
We present an analysis of wave propagation in a two step-index, parallel waveguidesystem. The goal is to quantify the effect of scattering at randomly perturbed interfaces between theguiding layers of high index of refraction and the host medium. The analysis is based on the expansionof the solution of the wave equation in a complete set of guided, radiation and evanescent modes withamplitudes that are random fields, due to scattering. We obtain a detailed characterization of theseamplitudes and thus quantify the transfer of power between the two waveguides in terms of theirseparation distance. The results show that, no matter how small the fluctuations of the interfacesare, they have significant effect at sufficiently large distance of propagation, which manifests in twoways: The first effect is well known and consists of power leakage from the guided modes to theradiation ones. The second effect consists of blurring of the periodic transfer of power between thewaveguides and the eventual equipartition of power. Its quantification is the main practical result ofthe paper.
Key words.
Random waveguides, directional coupler, power leakage, equipartition of power.
AMS subject classifications.
1. Introduction.
Guided waves have applications in electromagnetics [9], opticsand communications [22, 27], imaging underwater [19, 5, 14, 15], imaging of and intunnels [3] and so on. The classical theory of guided waves is for ideal waveguides withperfectly reflecting straight walls and filled with homogeneous media, where the waveequation can be solved using separation of variables. The wave field is represented asa superposition of finitely many guided modes, which are waves that propagate alongthe axis of the waveguide, and infinitely many evanescent modes which decay awayfrom the source. These modes do not interact with each other and thus have constantamplitude determined by the wave source [9, 27].Motivated by applications in imaging and communications, the classical theoryhas been extended to waveguides filled with random media [22, 12, 13, 1], with ran-domly perturbed boundaries [2, 1] and with slowly changing cross-section [6, 7, 22].Weakly guiding waveguides with confining graded-index profile affected by small ran-dom perturbations have also been analyzed in [11, 25]. The resulting mode couplingtheory quantifies the interaction between the modes induced by scattering, and theconsequent randomization of the wave field.We consider waveguides with penetrable boundaries, where the guiding effect isdue 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]. Motivatedby 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, Chapter10] and [22, Chapter 10], is based on the observation that the transverse profiles ofthe guided waves are essentially supported in the step-index waveguides and decayoutside. When the step-index waveguides are nearby, the decaying tails penetratethe neighboring waveguide and transfer of power can occur. This is the sole coupling ∗ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. [email protected] † Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, 91128 Palaiseau Cedex, France. [email protected] a r X i v : . [ m a t h . A P ] D ec + D d − d − D − d x z n ( ε ) ( z, x ) = n n ( ε ) ( z, x ) = n n ( ε ) ( z, x ) = 1n ( ε ) ( z, x ) = 1n ( ε ) ( z, x ) = 1 Fig. 2.1 . Illustration of two waveguides with fluctuating interfaces, filled with a medium withindex of refraction n > . The waves propagate along the range axis z . The waveguides have width D and are separated by the distance d . mechanism in ideal directional couplers and for synchronous waveguides (with identi-cal guided mode phase velocity) there is a complete, periodic transfer of power. Thatis to say, if the source emits power in one step-index waveguide, this is transferredto the other waveguide and back in a periodic manner, at regular distance intervals.These intervals depend on the separation between the step-index waveguides. Thelarger this distance is, the weaker the coupling and the farther the waves must travelfor the transfer of power to occur.We introduce a mathematical analysis of a randomly perturbed directional cou-pler, where the interfaces that separate the medium with high index of refraction fromthe background have small amplitude random fluctuations on a scale similar to thewavelength. The classic approach in [27, Chapter 10], which is based solely on theguided modes, is inadequate in this case, because scattering at the random interfacesinduces mode coupling. We take into account all the modes, the guided, radiationand evanescent ones, and quantify how their interaction affects the performance ofthe directional coupler. The analysis is focused on the case of well separated waveg-uides, where the deterministic coupling is weak. It applies to an arbitrary number ofguided modes, but we describe in depth the results for the case of single guided modestep-index waveguides. We show that mode coupling induced by the random fluctu-ations 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 theguided modes to the radiation modes. Our analysis captures it and shows that theleaked power is self-averaging i.e., it is independent of the realization of the randomprocesses that model the fluctuations of the interfaces. The other effect consists of theblurring of the periodic transfer of power between the waveguides and the eventualequipartition of power between the guided waves. Its quantification in terms of thewaveguide separation and amplitude of the random fluctuations is the main practicalresult of the paper.The paper is organized as follows: We begin in section 2 with the mathematicalformulation of the problem and then state the results in section 3. Their derivationis in sections 4–6. We end with a summary in section 7.
2. Formulation of the problem.
We study the propagation of a time harmonicwave in a medium with index of refraction n ( ε ) ( z, x ) defined below, that models twostep-index waveguides of width D , separated in the transverse direction x by thedistance d , as illustrated in Figure 2.1. The wave field is denoted by p ( z, x ) and it olves the two-dimensional Helmholtz equation (cid:0) ∂ x + ∂ z (cid:1) p ( z, x ) + (cid:2) k n ( ε ) ( z, x ) (cid:3) p ( z, x ) = f ( x ) δ (cid:48) ( z ) , (2.1)with radiation condition at infinity, where k is the wavenumber and f ( x ) models asource supported at the origin of the range coordinate z .In the case of ideal (unperturbed) waveguides, the index of refraction is rangeindependent and equal ton (0) ( x ) = (cid:26) n if x ∈ ( − d/ − D, − d/ ∪ ( d/ , d/ D ) , n >
1. We are interested in perturbed waveguides, where the index of refractionn ( ε ) ( z, x ) = (cid:26) n if x ∈ ( D ( ε )1 ( z ) , D ( ε )2 ( z )) ∪ ( D ε ( z ) , D ( ε )4 ( z )) , D ( ε )1 ( z ) = − d/ − D + εDν ( z ) (0 ,L ( ε ) ) ( z ) , D ( ε )2 ( z ) = − d/ εDν ( z ) (0 ,L ( ε ) ) ( z ) , D ( ε )3 ( z ) = d/ εDν ( z ) (0 ,L ( ε ) ) ( z ) , D ( ε )4 ( z ) = d/ D + εDν ( z ) (0 ,L ( ε ) ) ( z ) . (2.4)The fluctuations are modeled by the zero-mean, bounded, independent and identi-cally distributed stationary random processes { ν q ( z ) } ≤ q ≤ with smooth covariancefunction R ( z ) = E [ ν q (0) ν q ( z )] , q = 1 , . . . , . (2.5)These satisfy strong mixing conditions as defined for example in [23, section 2]. Thetypical amplitude of the fluctuations is much smaller than D and it is modeled in(2.4) by the small and positive dimensionless parameter ε .We study the wavefield at z >
0, satisfying p ( z, x ) ∈ C (cid:0) (0 , + ∞ ) , H ( R ) ∩ H ( R ) (cid:1) ∩ C (cid:0) (0 , + ∞ ) , L ( R ) (cid:1) , z > , (2.6)and to set radiation conditions, we suppose that the random fluctuations are sup-ported in the range interval (0 , L ( ε ) ). We will see that net scattering effect of thesefluctuations becomes of order one at range scales of order ε − , so we let L ( ε ) = L/ε .We will also see that for the assumed smooth covariance R ( z ), the guided waves prop-agate mostly in the forward direction and do not interact with any fluctuations at z <
0, which is why we neglect them.The goal of the paper is to quantify how scattering at the random interfaces (2.4)affects the coupling of the two step-index waveguides centered at x = ± ( d + D ) /
2. Weconsider in particular the case of a sufficiently large separation distance d between thewaveguides, where the deterministic coupling is very weak but the coupling inducedby the random fluctuations is still present. . Statement of results. We state here the main results of the paper, derivedin sections 4–6 by decomposing the wavefield into guided, radiation and evanescentmodes of the waveguide system made of two parallel step-index waveguides illustratedin Figure 2.1. 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 hasonly one guided mode. This case captures all the essential aspects of the problem andarises when kD (cid:112) n − < π. (3.1)The two ideal step-index waveguides centered at ± ( d/ D/
2) do not interactin the waveguide system when d → ∞ . Thus, we can write the wavefield for large d in terms of the unique guided mode φ ( x ) e iβz of the step-index waveguide centered at x = d/ D/
2, modeled by the index of refractionn s ( x ) = (cid:26) n if x ∈ ( d/ , d/ D ) , φ ( − x ) e iβz of the step-index waveguide cen-tered at x = − d/ − D/
2, modeled by the index of refraction n s ( − x ). Here φ ( x ) isthe eigenfunction of the Helmholtz operator ∂ x + n s ( x ) for the eigenvalue β . This β is defined in Lemma 4.1 as the unique solution in the interval ( k, nk ) of (cid:112) n k − β (cid:112) β − k tan (cid:32) D (cid:112) k n − β (cid:33) = 1 , β ∈ ( k, nk ) . (3.3)The expression of the eigenfunction φ ( x ) is [20, Section 2], [22, Chapter 1] φ ( x ) = (cid:16) η + D (cid:17) − / cos( ξ D ) exp[ η ( x − d )] , x ≤ d , (cid:16) η + D (cid:17) − / cos (cid:2) ξ ( x − d − D ) (cid:3) , d ≤ x ≤ d + D, (cid:16) η + D (cid:17) − / cos( ξ D ) exp (cid:2) − η ( x − d − D ) (cid:3) , x ≥ d + D. (3.4)It has a peak centered at x = d/ D/ /ξ , and an exponentially decayingtail, at the rate η , where ξ = (cid:112) n k − β , η = (cid:112) β − k . (3.5) We show in section 4 that under theassumption (3.1) and for sufficiently large separation distance d , the solution of (cid:0) ∂ x + ∂ z ) p (0) ( z, x ) + (cid:2) k n (0) ( x ) (cid:3) p (0) ( z, x ) = f ( x ) δ (cid:48) ( z ) , (3.6)with radiation condition at infinity, takes the form ∗ p (0) ( z, x ) = (cid:88) t ∈{ e,o } a (0) t √ β t e iβ t z φ t ( x ) + O ( z − ) , z > . (3.7) ∗ We use consistently the index (0) for the mode amplitudes and wave field in the ideal (unper-turbed) two step-index wave guide system. 4 -5 0 5 φ ( x ) -0.8-0.6-0.4-0.200.20.40.6 φ e φ o x -5 0 5 φ ( x ) -0.8-0.6-0.4-0.200.20.40.6 φ e φ o Fig. 3.1 . The eigenfunctions φ e ( x ) (solid blue line) and φ o ( x ) (dashed red line) calculated atwavenumber k = 2 π , for the waveguide system with index of refraction n = 1 . . For reference wealso plot the eigenfunction φ ( x ) of the step-index waveguide centered at x = d/ D/ with the blackdotted line. The abscissa is in units of the waveguide width D , which is equal to the wavelength.We consider a separation d = D (left plot) and d = 4 D (right plot) between the waveguides. The terms in the sum model the guided modes of the waveguide system, whereasthe O ( z − ) remainder accounts for the radiation and evanescent modes [20, Section3]. The guided modes { φ t ( x ) e iβ t z } t = e,o are waves that propagate along the rangecoordinate z and have transverse profiles essentially supported in the two step-indexwaveguides. They are defined by the even and odd eigenfunctions φ e ( x ) and φ o ( x ) ofthe operator H ( x ) = ∂ x + n (0) ( x ) , (3.8)with n (0) ( x ) given in (2.2), for the eigenvalues β e and β o satisfying β t ∈ ( k, nk ) , t ∈ { e, o } . (3.9)These eigenfunctions are given explicitly in section 4.1. They have peaks close to thecenter axes x = ± ( d/ D/
2) of the step-index waveguides and decay outside theirtransverse support, as illustrated in Figure 3.1. Each guided mode is multiplied in(3.7) by the constants a (0) t = √ β t (cid:90) R dx φ t ( x ) f ( x ) , t ∈ { e, o } , (3.10)called the mode amplitudes, that are determined by the source. Throughout the paperthe bar denotes complex conjugate.We assume that the waveguides separation distance d is sufficiently large, so thatexp( − ηd ) (cid:28) , (3.11)with η defined in (3.5), and obtain that the eigenfunctions can be approximated by φ e ( x ) = φ ( | x | ) + O (cid:0) e − ηd (cid:1) , φ o ( x ) = sgn( x ) φ ( | x | ) + O (cid:0) e − ηd (cid:1) , x ∈ R , (3.12)where “sgn” is the sign function. The accuracy of this approximation is illustrated inthe right plot of Figure 3.1. Consequently, the transverse profile of the even guidedmode presents two positive peaks centered at x = ± ( d/ D/ x = d/ D/
2, one negative peak centered at x = − d/ − D/ nd exponentially decaying tails. The form of these peaks is proportional to theunique eigenfunction (3.4) of the single-mode step-index waveguide.We also obtain that the wave numbers are β e = β + β (cid:48) e − ηd + o (cid:0) e − ηd (cid:1) , β o = β − β (cid:48) e − ηd + o (cid:0) e − ηd (cid:1) , (3.13)with β (cid:48) = ηβ (cid:0) η ξ (cid:1)(cid:0) η + D (cid:1) , (3.14)so the wavefield (3.7) in the ideal waveguide system takes the form p (0) ( z, x ) = φ ( | x | ) √ β e iβz (cid:104) (0 , ∞ ) ( x ) u (0)+ ( z ) + 1 ( −∞ , ( x ) u (0) − ( z ) (cid:105) + O (cid:0) e − ηd (cid:1) + O ( z − ) . (3.15)Here we introduced the range-dependent amplitudes of the waves propagating in thetwo step-index waveguides u (0)+ ( z ) = ( a (0) e + a (0) o ) cos (cid:0) β (cid:48) ze − ηd (cid:1) + i ( a (0) e − a (0) o ) sin (cid:0) β (cid:48) ze − ηd (cid:1) , (3.16) u (0) − ( z ) = ( a (0) e − a (0) o ) cos (cid:0) β (cid:48) ze − ηd (cid:1) + i ( a (0) e + a (0) o ) sin (cid:0) β (cid:48) ze − ηd (cid:1) , (3.17)with indexes “ ± ” corresponding to the waveguide centered at x = ± ( d + D ) / x = d/ D/ φ ( | x | ) e iβz (0 , ∞ ) ( x ) and the secondone is centered at x = − d/ − D/ φ ( | x | ) e iβz ( −∞ , ( x ). This is similarto the case of two independent, single-mode step-index waveguides, except that in(3.15) the amplitudes u ± ( z ) vary in z , due to coupling. We obtain from (3.11) and(3.16–3.17) that for z of the order of the wavelength, u (0) ± ( z ) ≈ u (0) ± (0) = a (0) e ± a (0) o . (3.18)However, at large z , satisfying z = exp( ηd ) Z, Z > , (3.19) u (0) ± ( z ) oscillate periodically in z . For example, if the source gives the amplitudes a (0) e = a (0) o = a (0) , (3.20)according to (3.10), so that u (0)+ (0) = a (0) and u (0) − (0) = 0, at range (3.19) we have u (0)+ (cid:0) e ηd Z (cid:1) = a (0) cos( β (cid:48) Z ) , u (0) − (cid:0) e ηd Z (cid:1) = ia (0) sin( β (cid:48) Z ) . (3.21)In conclusion, the total wave power in the ideal waveguide system at large range z is essentially supported in the two step-index waveguides. The wave power in thewaveguide centered at x = ± ( d/ D/
2) is proportional to | u (0) ± ( z ) | , and equation(3.21) shows that it oscillates slowly and periodically. At scaled distance Z = mπ/β (cid:48) , m ∈ N , the wave power is concentrated in the waveguide centered at x = d/ D/ Z = (1 / m ) π/β (cid:48) , m ∈ N , the wave power is concentrated in thewaveguide centered at x = − d/ − D/
2. These periodic oscillations have been reported n the literature [27, Chapter 10]. The standard method to analyze them is not tostart from the analysis of the modes of the waveguide system, as we do in section 4,but to simplify by assuming that the modes can be represented as a weighted sumof the guided modes of the two waveguides. This simplified approach does not allowto take into account the role of evanescent and radiation modes, which are critical tothe study of random waveguides in sections 5–6, with results described next. The analysis of the solution p ( z, x ) ofthe Helmholtz equation (2.1) with index of refraction (2.3) is carried out in sections5–6. It shows that under the assumptions (3.1), (3.11) and at large range z/ε ,where scattering at the random interfaces (2.4) becomes significant, there are threedistinguished regimes that determine the coupling between the waveguides:1. The “moderate coupling” regime, where the separation distance d is moder-ately large, satisfying 1 (cid:29) exp( − ηd ) (cid:29) ε . (3.22)2. The “weak coupling” regime, where d is large enough so thatexp( − ηd ) = O ( ε ) . (3.23)3. The “very weak coupling” regime, where d is so large thatexp( − ηd ) (cid:28) ε (cid:28) . (3.24)We now describe the results in each of these three regimes. At large range z/ε and in the regime defined by(3.22), the solution of (2.1) with radiation condition at infinity and with index ofrefraction (2.3) is p (cid:16) zε , x (cid:17) = φ ( | x | ) √ β e iβ zε (cid:2) (0 , ∞ ) ( x ) u + ( z ) + 1 ( −∞ , ( x ) u − ( z ) (cid:3) + o (1) , (3.25)where u ± ( z ) are random processes and o (1) denotes a residual that tends to zero as ε →
0. This residual accounts for the radiation and evanescent components of p .Similar to (3.15), we have a propagating wave φ ( | x | ) e iβz/ε (0 , ∞ ) ( x ) centered at x = d/ D/ φ ( | x | ) e iβz/ε ( −∞ , ( x ) centered at x = − d/ − D/ u ± ( z ), the analogues of (3.16–3.17). To write their expressions,we introduce the notation ∆ β t = β t − β, t ∈ { e, o } , (3.26)which takes into account that the residual in (3.13) is not negligible under the as-sumption (3.22) at range O ( ε − ). We have u ± ( z ) = a e ( z ) e i ∆ β e zε ± a o ( z ) e i ∆ β o zε , (3.27)where (cid:0) a e ( z ) , a o ( z ) (cid:1) is a random, Markovian process defined at z ≥
0, with initial con-dition a t (0) = a (0) t given in (3.10) for t ∈ { e, o } , and with the infinitesimal generatorin Theorem 5.1. To describe the results, it suffices to state from there that | a e ( z ) | + | a o ( z ) | = (cid:0) | a (0) e | + | a (0) o | (cid:1) exp( − Λ z ) , (3.28) ith probability one, and that E (cid:2) a o ( z ) a e ( z ) (cid:3) = a (0) o a (0) e exp (cid:0) − (Γ + Λ) z (cid:1) , (3.29)with positive Λ and Γ defined byΛ = k ( n − D β (cid:0) η + D (cid:1) cos (cid:16) ξD (cid:17) (cid:90) k dγπη γ √ γ (cid:34) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) (cid:35) (cid:98) R ( β − √ γ ) , (3.30)Γ = k ( n − D β (cid:0) η + D (cid:1) cos (cid:16) ξD (cid:17) (cid:98) R (0) , (3.31)in terms of ξ γ = (cid:112) k n − γ, η γ = (cid:112) k − γ, (3.32)and the power spectral density (cid:98) R ≥
0, the Fourier transform of the covariance (2.5).We have therefore from (3.27) and (3.28) that the total power of the guided wavesdecays exponentially, at the rate Λ, | u + ( z ) | + | u − ( z ) | = 2 (cid:0) | a e ( z ) | + | a o ( z ) | (cid:1) = 2 (cid:0) | a (0) e | + | a (0) o | (cid:1) exp( − Λ z ) . (3.33)This decay models the transfer of power from the guided modes to the radiationmodes, induced by scattering at the random interfaces (2.4).The imbalance of power between the two waveguides is quantified by P ( z ) = | u + ( z ) | − | u − ( z ) | | u + ( z ) | + | u − ( z ) | = 2Re (cid:8) a e ( z ) a o ( z ) exp (cid:2) i ( β e − β o ) zε (cid:3)(cid:9) | a e ( z ) | + | a o ( z ) | , (3.34)and its expectation is E (cid:104) P ( z ) (cid:105) = 2Re (cid:8) a (0) e a (0) o exp (cid:0) i ( β e − β o ) zε (cid:1)(cid:9) | a (0) e | + | a (0) o | exp( − Γ z ) . (3.35)To explain what this gives, consider the source excitation (3.20) with a (0) e = a (0) o , sothat the initial wavefield is supported in the waveguide centered at x = d/ D/ E (cid:104) P ( z ) (cid:105) = cos (cid:104) ( β e − β o ) zε (cid:105) exp( − Γ z ) , (3.36)and it describes the competition between the deterministic and random coupling ofthe waveguides. The cosine in (3.36) models the deterministic coupling which inducesperiodic oscillations of the power, as in section 3.1. The random coupling is modeledby the exponential decay in z , at the rate Γ. It shows that as the range increases, thepower tends to become equally distributed among the two waveguides. This decay ispresent in (3.35) as well, so the equipartition of power at large z is independent ofthe initial condition generated by the source. .2.2. Weak coupling. When the separation distance d between the waveguidessatisfies (3.23), the wavefield p ( z/ε , x ) has the same expression as in (3.25), (3.27),but the random processes ( u + ( z ) , u − ( z )) have different statistics.The total power of the guided waves is still given by (3.33), and decays at thesame rate Λ defined in (3.30). However, the expectation of the imbalance of powerbetween the two waveguides satisfies the damped harmonic oscillator equation (cid:2) ∂ z + 2Γ ∂ z + (2 θβ (cid:48) ) (cid:3) E [ P ( z )] = 0 , (3.37)with β (cid:48) defined in (3.14) and θ = ε − exp( − ηd ) . (3.38)Based on the value of θ , which is independent of ε by assumption (3.23), we distinguishthree regimes, which we describe for the source excitation (3.20), with u ( ε ) − (0) = 0 andtherefore E [ P (0)] = 1:1. When 2 θβ (cid:48) < Γ, the solution of (3.37) is E [ P ( z )] = e − Γ z (cid:104) cosh (cid:0)(cid:112) Γ − (2 θβ (cid:48) ) z (cid:1) + Γ (cid:112) Γ − (2 θβ (cid:48) ) sinh (cid:0)(cid:112) Γ − (2 θβ (cid:48) ) z (cid:1)(cid:105) . This tends to 1 as θ →
0, 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 fromthe guided modes to the radiation ones, as described by the exponential decay in(3.33).For a finite θ , we have E [ P ( z )] → z → ∞ , so the random coupling distributesthe power evenly among the two waveguides.2. In the critical case 2 θβ (cid:48) = Γ we have E [ P ( z )] = (1 + Γ z ) exp( − Γ z ) . (3.39)As in the previous case, the random coupling equidistributes the power among thetwo waveguides, in the limit z → ∞ .3. When 2 θβ (cid:48) > Γ, the solution of (3.37) is E [ P ( z )] = e − Γ z (cid:104) cos (cid:0)(cid:112) (2 θβ (cid:48) ) − Γ z (cid:1) + Γ (cid:112) (2 θβ (cid:48) ) − Γ sin (cid:0)(cid:112) (2 θβ (cid:48) ) − Γ z (cid:1)(cid:105) . It displays periodic oscillations induced by the deterministic coupling of the waveg-uides, but these oscillations are damped due to the random coupling. In particular,if 2 θβ (cid:48) (cid:29) Γ, we get E [ P ( z )] ≈ e − Γ z cos(2 θβ (cid:48) z ) , (3.40)in agreement with (3.36).We plot in Figure 3.2 the imbalance ratio E [ P ( z )] as a function of z normalizedby the deterministic coupling distance † z θ = 1 / (2 θβ (cid:48) ). From this plot and fromequation (3.33) we conclude that wave scattering at the random interfaces (2.4) hastwo net effects: † At the distance πz θ the power is fully transferred from one step-index waveguide to the otherone, when there are no random perturbations. 9 /z θ h P ( z ) i -1-0.500.51 g = 0 g = 1 / g = 1 / g = 1 g = 2 Fig. 3.2 . Imbalance ratio (cid:104)P ( z ) (cid:105) := E [ P ( z )] as a function of z/z θ , where z θ = 1 / (2 θβ (cid:48) ) . Weillustrate the result for different values of g = Γ z θ , given in the legend. Note how the effective cou-pling coefficient Γ reduces the deterministic and periodic transfer of power and causes the imbalanceratio to tend to .
1. It induces a self-averaging loss (or leakage) of total power, due to the couplingof the guided modes with the radiation modes.2. It causes a blurring of the periodic (deterministic) power transfer from onewaveguide to the other.The blurring effect at item 2 is the main practical result of the paper. It dependson the effective parameter Γ and it is significant as soon as Γ z θ becomes of orderone. Therefore, the deterministic transfer of power is very sensitive to the randomfluctuations of the interfaces (2.4). When the distance d between the waveguides is solarge that (3.24) holds, the wavefield has the same expression as (3.25), but the waveamplitudes have different statistics. They model the only coupling in this regime, be-tween the guided and radiation modes, which generates effective wave power leakage.Explicitly, we show in section 6 that the wave amplitudes converge in probability, as ε →
0, to the deterministic function ( | u + ( z ) | , | u − ( z ) | ) satisfying | u + ( z ) | = | u (0)+ (0) | exp( − Λ z ) , | u − ( z ) | = | u (0) − (0) | exp( − Λ z ) . (3.41)Although deterministic, this function is not as in the ideal waveguide system, where( | u (0)+ ( z ) | , | u (0) − ( z ) | ) is constant in z (because β (cid:48) e − ηd z (cid:28) P ( z ) = | u + ( z ) | − | u − ( z ) | | u + ( z ) | + | u − ( z ) | = | u (0)+ (0) | − | u (0) − (0) | | u (0)+ (0) | + | u (0) − (0) | , (3.42)so in the case of the source excitation (3.20), P ( z ) (cid:39) P (0) = 1 .
4. Analysis in ideal waveguides.
The analysis in this section is classical andfollows the lines of [20]. It derives the results stated in section 3.1 by expanding thewave field on a complete set of eigenmodes. The proof of the completeness of this setis the most delicate part and it is carried out in [20] by the Levitan-Levinson method[8, Chapter 9]. ecall the Helmholtz operator (3.8) in the transverse coordinate, with index ofrefraction n (0) ( x ) given in (2.2), and note that it is self-adjoint with respect to thescalar product associated to the L -norm,( φ , φ ) := (cid:90) R φ ( x ) φ ( x ) dx. (4.1)Its spectrum is ( −∞ , k ) ∪ (cid:8)(cid:0) β t,j (cid:1) ≤ j ≤ N t , t ∈ { e, o } (cid:9) , where β t,j are called the guidedmode wavenumbers. They are positive and satisfy the order relation k < β t,N t < · · · < β t, < k n , (4.2)where the index t ∈ { e, o } stands for the even and odd eigenfunctions in the transversecoordinate x .We describe next the eigenfunctions for the discrete spectrum and the impropereigenfunctions for the continuum spectrum, and explain that they form a completeset. We use them to decompose the wavefield into guided, radiation and evanescentmodes with amplitudes determined by the source. There are two sets of discrete eigenvalues and eigen-functions: the first one associated with the even modes in x and the second oneassociated with the odd modes.The j -th even eigenfunction φ e,j ( x ) for the eigenvalue β e,j is defined by φ e,j ( x ) A e,j = exp( − η e,j d ) sin( ξ e,j D ) (cid:16) ξ e,j η e,j (cid:17) cosh( η e,j x ) , x ∈ (cid:2) , d (cid:3) , ξ e,j η e,j cos (cid:2) ξ e,j ( x − d − D ) (cid:3) − sin (cid:2) ξ e,j ( x − d − D ) (cid:3) , x ∈ (cid:2) d , d + D (cid:3) , ξ e,j η e,j exp (cid:2) − η e,j ( x − d − D ) (cid:3) , x ∈ (cid:2) d + D, ∞ ) , (4.3)and φ e,j ( − x ) = φ e,j ( x ) for x ≥
0. Here ξ e,j = (cid:113) k n − β e,j , η e,j = (cid:113) β e,j − k , (4.4)and A e,j > A e,j = (cid:34)
12 exp( − η e,j d ) sin ( ξ e,j D ) (cid:32) ξ e,j η e,j (cid:33) (cid:32) sinh( η e,j d ) η e,j + d (cid:33) + (cid:32) ξ e,j η e,j − (cid:33) sin(2 ξ e,j D )2 ξ e,j + (cid:32) ξ e,j η e,j + 1 (cid:33) D + 2 sin ( ξ e,j D ) η e,j + ξ e,j η e,j (cid:35) − / (4.5)calculated so that φ e,j has unit L -norm. Moreover, β e,j ∈ ( k, nk ) satisfies the dis-persion relation (cid:32) ξ e,j η e,j (cid:33) e − η e,j d = (cid:20) − ξ e,j η e,j tan (cid:16) ξ e,j D (cid:17)(cid:21) (cid:20) ξ e,j η e,j cotan (cid:16) ξ e,j D (cid:17)(cid:21) , (4.6)with ξ e,j , η e,j given by (4.4), which ensures the continuity of (4.3) and its derivative.The number of solutions β e,j ∈ ( k, nk ) of (4.6) is denoted by N e and when d is large,it depends on the value of kD √ n − he j -th odd eigenfunction for the eigenvalue β o,j is defined similarly, φ o,j ( x ) A o,j = exp( − η o,j d ) sin( ξ o,j D ) (cid:16) ξ o,j η o,j (cid:17) sinh( η o,j x ) , x ∈ (cid:2) , d (cid:3) , ξ o,j η o,j cos (cid:2) ξ o,j ( x − d − D ) (cid:3) − sin (cid:2) ξ o,j ( x − d − D ) (cid:3) , x ∈ (cid:2) d , d + D (cid:3) , ξ o,j η o,j exp (cid:2) − η o,j ( x − d − D ) (cid:3) , x ∈ (cid:2) d + D, ∞ (cid:1) , (4.7)with φ o,j ( − x ) = − φ o,j ( x ) for x ≥ ξ o,j = (cid:113) k n − β o,j , η o,j = (cid:113) β o,j − k . (4.8)The normalization constant A o,j > A o,j = (cid:34)
12 exp( − η o,j d ) sin ( ξ o,j D ) (cid:32) ξ o,j η o,j (cid:33) (cid:32) sinh( η o,j d ) η o,j − d (cid:33) + (cid:32) ξ o,j η o,j − (cid:33) sin(2 ξ o,j D )2 ξ o,j + (cid:32) ξ o,j η o,j + 1 (cid:33) D + 2 sin ( ξ o,j D ) η o,j + ξ o,j η o,j (cid:35) − / (4.9)so that φ o,j has unit L norm and β o,j ∈ ( k, nk ) satisfies the dispersion relation − (cid:32) ξ o,j η o,j (cid:33) e − η o,j d = (cid:34) − ξ o,j η o,j tan (cid:16) ξ o,j D (cid:17)(cid:35)(cid:34) ξ o,j η o,j cotan (cid:16) ξ o,j D (cid:17)(cid:35) , (4.10)with ξ o,j , η o,j given by (4.8), for j = 1 , . . . , N o , which ensures that (4.7) and itsderivative are continuous. For γ ∈ ( −∞ , k ) there are two improper eigen-functions, even and odd, denoted by φ t,γ ( x ), for t ∈ { e, o } . We write their expressionbelow in terms of the parameters ξ γ = (cid:112) k n − γ, η γ = (cid:112) k − γ. (4.11)The even eigenfunctions satisfy φ e,γ ( − x ) = φ e,γ ( x ) for x ≥
0, and are defined by φ e,γ ( x ) A e,γ = ξ γ η γ cos( η γ x ) , for x ∈ (cid:104) , d (cid:105) , (4.12)and by φ e,γ ( x ) A e,γ = ξ γ η γ cos (cid:104) ξ γ (cid:16) x − d (cid:17)(cid:105) cos (cid:16) η γ d (cid:17) − sin (cid:104) ξ γ (cid:16) x − d (cid:17)(cid:105) sin (cid:16) η γ d (cid:17) , for x ∈ (cid:104) d , d D (cid:105) , (4.13)and by φ e,γ ( x ) A e,γ = cos (cid:104) η γ (cid:16) x − d − D (cid:17)(cid:105)(cid:34) ξ γ η γ cos( ξ γ D ) cos (cid:16) η γ d (cid:17) − sin( ξ γ D ) sin (cid:16) η γ d (cid:17)(cid:35) − ξ γ η γ sin (cid:104) η γ (cid:16) x − d − D (cid:17)(cid:105)(cid:34) ξ γ η γ sin( ξ γ D ) cos (cid:16) η γ d (cid:17) + cos( ξ γ D ) sin (cid:16) η γ d (cid:17)(cid:35) for x ≥ d D, (4.14) ith normalization constant A e,γ > A e,γ =(2 πη γ ) − / (cid:40)(cid:34) ξ γ η γ cos( ξ γ D ) cos (cid:16) η γ d (cid:17) − sin( ξ γ D ) sin (cid:16) η γ d (cid:17)(cid:35) + ξ γ η γ (cid:34) ξ γ η γ sin( ξ γ D ) cos (cid:16) η γ d (cid:17) + cos( ξ γ D ) sin (cid:16) η γ d (cid:17)(cid:35) (cid:41) − / . (4.15)The odd eigenfunctions satisfy φ o,γ ( − x ) = − φ o,γ ( x ) for x ≥
0, and are definedby φ o,γ ( x ) A o,γ φ o,γ = ξ γ η γ sin( η γ x ) , for x ∈ (cid:104) , d (cid:105) , (4.16)and by φ o,γ ( x ) A o,γ = ξ γ η γ cos (cid:104) ξ γ (cid:16) x − d (cid:17)(cid:105) sin (cid:16) η γ d (cid:17) + sin (cid:104) ξ γ (cid:16) x − d (cid:17)(cid:105) cos (cid:16) η γ d (cid:17) for x ∈ (cid:104) d , d D (cid:105) , (4.17)and by φ o,γ ( x ) A o,γ = cos (cid:104) η γ (cid:16) x − d − D (cid:17)(cid:105)(cid:34) ξ γ η γ cos( ξ γ D ) sin (cid:16) η γ d (cid:17) + sin( ξ γ D ) cos (cid:16) η γ d (cid:17)(cid:35) − ξ γ η γ sin (cid:104) η γ (cid:16) x − d − D (cid:17)(cid:105)(cid:34) ξ γ η γ sin( ξ γ D ) sin (cid:16) η γ d (cid:17) − cos( ξ γ D ) cos (cid:16) η γ d (cid:17)(cid:35) for x ≥ d D, (4.18)with normalization constant A o,γ > A o,γ =(2 πη γ ) − / (cid:40)(cid:34) ξ γ η γ cos( ξ γ D ) sin (cid:16) η γ d (cid:17) + sin( ξ γ D ) cos (cid:16) η γ d (cid:17)(cid:35) + ξ γ η γ (cid:34) ξ γ η γ sin( ξ γ D ) sin (cid:16) η γ d (cid:17) − cos( ξ γ D ) cos (cid:16) η γ d (cid:17)(cid:35) (cid:41) − / . (4.19) Remark 4.1.
Note that the improper eigenfunctions φ t,γ ( x ) , for t ∈ { e, o } , arenot in L ( R ) . However, we can define for any ϕ ∈ L ( R )( φ t,γ , ϕ ) = lim M → + ∞ (cid:90) M φ t,γ ( x ) ϕ ( x ) dx + lim M → + ∞ (cid:90) − M φ t,γ ( x ) ϕ ( x ) dx, where the limit holds in L ( −∞ , k ) . The normalizing constants A t,γ given in (4.15) and (4.19) are such that, for any test function a ∈ L ( −∞ , k ) , (cid:90) M (cid:12)(cid:12)(cid:12) (cid:90) k −∞ φ t,γ ( x ) a ( γ ) dγ (cid:12)(cid:12)(cid:12) dx M → + ∞ −→ (cid:90) k −∞ | a ( γ ) | dγ. They depend continuously on γ and are bounded on ( −∞ , k ) with A t,γ (cid:39) (cid:0) π (cid:112) | γ | (cid:1) − / as γ → −∞ . .3. Completness. The proof of completness is based on the Levitan-Levinsonmethod [8, Chapter 9] and is the same as that in [20, Section 2.3]. We state directlythe result:For any ϕ ∈ L ( R ) we have the Parseval identity( ϕ, ϕ ) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 (cid:12)(cid:12) ( φ t,j , ϕ ) (cid:12)(cid:12) + (cid:88) t ∈{ e,o } (cid:90) k −∞ (cid:12)(cid:12) ( φ t,γ , ϕ ) (cid:12)(cid:12) dγ, and the map which assigns to ϕ the coefficients of its spectral decomposition ϕ (cid:55)→ (cid:16) ( φ t,j , ϕ ) , j = 1 , . . . , N t ; ( φ t,γ , ϕ ) , γ ∈ ( −∞ , k ); t ∈ { e, o } (cid:17) is an isometry from L ( R ) onto C N e + N o × L ( −∞ , k ) . Moreover, there exists aresolution of the identity Π of the operator (3.8) such that: for any ϕ ∈ L ( R ) andany −∞ ≤ r ≤ r (cid:48) ≤ + ∞ ,Π ( r,r (cid:48) ) ( ϕ )( x ) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 ( φ t,j , ϕ ) φ t,j ( x ) ( r,r (cid:48) ) ( β t,j )+ (cid:88) t ∈{ e,o } (cid:90) min( k ,r (cid:48) ) r ( φ t,γ , ϕ ) φ t,γ ( x ) dγ ( r, + ∞ ) ( k ) , and for any ϕ in the domain of H ,Π ( r,r (cid:48) ) ( H ϕ )( x ) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 β t,j ( φ t,j , ϕ ) φ t,j ( x ) ( r,r (cid:48) ) ( β t,j )+ (cid:88) t ∈{ e,o } (cid:90) min( k ,r (cid:48) ) r γ ( φ t,γ , ϕ ) φ t,γ ( x ) dγ ( r, + ∞ ) ( k ) . Using the results in the previous section, we canwrite the solution p (0) ( z, x ) of the Helmholtz equation (3.6) as a superposition ofmodes p (0) ( z, x ) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 p (0) t,j ( z ) φ t,j ( x ) + (cid:88) t ∈{ e,o } (cid:90) k p (0) t,γ ( z ) φ t,γ ( x ) dγ + (cid:88) t ∈{ e,o } (cid:90) −∞ p (0) t,γ ( z ) φ t,γ ( x ) dγ. (4.20)The first sum contains the guided modes, where p (0) t,j ( z ) are one-dimensional wavesthat propagate along z , with wavenumber β t,j , ∂ z p (0) t,j ( z ) + β t,j p (0) t,j ( z ) = 0 , z (cid:54) = 0 , for j = 1 , . . . , N t , t ∈ { e, o } . (4.21)The other two sums contain the radiation modes and evanescent modes, where ∂ z p (0) t,γ ( z ) + γp (0) t,γ ( z ) = 0 , z (cid:54) = 0 , for γ ∈ ( −∞ , k ) , t ∈ { e, o } . (4.22) he radiation modes correspond to γ ∈ (0 , k ), so that p (0) t,γ ( z ) are one-dimensionalwaves that propagate along z at wavenumber √ γ . The evanescent modes are decayingexponentially ‡ in z on the range scale 1 / (cid:112) | γ | , for γ < p (0) t,j ( z ) = a (0) t,j (cid:112) β t,j e iβ t,j z (0 , ∞ ) ( z ) + b (0) t,j (cid:112) β t,j e − iβ t,j z ( −∞ , ( z ) , j = 1 , . . . , N t ,p (0) t,γ ( z ) = a (0) t,γ γ / e i √ γz (0 , ∞ ) ( z ) + b (0) t,γ γ / e − i √ γz ( −∞ , ( z ) , γ ∈ (0 , k ) ,p (0) t,γ ( z ) = a (0) t,γ | γ | / e − √ | γ | z (0 , ∞ ) ( z ) + b (0) t,γ γ / e √ | γ | z ( −∞ , ( z ) , γ ∈ ( ∞ , , for t ∈ { e, o } , with constant mode amplitudes determined by the source in (2.1), a (0) t,j = − b (0) t,j = (cid:112) β t,j φ t,j , f ) , j = 1 , . . . , N t , (4.23) a (0) t,γ = − b (0) t,γ = γ / φ t,γ , f ) , γ ∈ (0 , k ) , (4.24) a (0) t,γ = − b (0) t,γ = | γ | / φ t,γ , f ) , γ ∈ ( −∞ , . (4.25)Substituting in (4.20), we obtain the modal expansion of the wavefield p (0) ( z, x ). We describe here the specialsituation considered in section 3.1, where the assumptions (3.1) and (3.11) hold andthere is a single guided mode per step-index waveguide, as we now explain.
Lemma 4.1.
Under the assumption kD √ n − < π , equation (cid:112) n k − β (cid:112) β − k tan (cid:0) (cid:112) n k − β D (cid:1) = 1 (4.26) has a unique solution β in ( k, nk ) , whereas equation (cid:112) n k − β (cid:112) β − k cotan (cid:0) (cid:112) n k − β D (cid:1) = − has no solution in ( k, nk ) . Furthermore, when the distance d between the waveguidesis large enough, so that exp( − ηd ) (cid:28) , with η = (cid:112) β − k , equations (4.6) and (4.10) have a single solution, meaning that N e = N o = 1 . The proof of the lemma is in appendix A. We let henceforth β be the uniquesolution of (4.26) in the interval ( k, nk ) and recall the definition (3.5) of ξ and η interms of β . Since N e = N o = 1, we simplify notation as β t, (cid:32) β t , for t ∈ { e, o } , andobtain from (4.6) and (4.10) that β e = β + β (cid:48) e − ηd + o (cid:16) e − ηd (cid:17) , β o = β − β (cid:48) e − ηd + o (cid:16) e − ηd (cid:17) , (4.28) ‡ We neglect the case γ = 0 because there is no coupling of the modes in ideal waveguides, andtherefore it has a negligible contribution in (4.20). Mode coupling occurs in randomly perturbedwaveguides, so the case γ = 0 must be dealt with carefully, as explained in section 5.6.15 ith β (cid:48) = − (cid:40) ∂∂β s (cid:34) (cid:112) k n − β s (cid:112) β s − k tan (cid:32) (cid:112) k n − β s D (cid:33)(cid:35)(cid:41) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β s = β = ηβ (cid:0) η ξ (cid:1)(cid:0) η + D (cid:1) , Similarly, we let φ t, ( x ) (cid:32) φ t ( x ), for t ∈ { e, o } , and obtain from (4.3)–(4.9) that φ e ( x ) = φ ( x ) + O (cid:0) exp( − ηd ) (cid:1) , φ o ( x ) = φ ( x ) + O (cid:0) exp( − ηd ) (cid:1) , x ≥ . (4.29)Substituting in (4.20) and obtaining from [20, Section 3.2] that the radiation andevanescent modes have an O ( z − ) contribution § , we obtain equations (3.7) and (3.15–3.17) used in section 3.1 to describe the deterministic coupling between the idealstep-index waveguides.
5. Analysis in randomly perturbed waveguides.
Let us introduce the no-tation ∆ k = k ( n −
1) (5.1)and rewrite the index of refraction defined in (2.3–2.4) as (cid:0) n ( ε ) ( z, x ) (cid:1) = (cid:0) n (0) ( x ) (cid:1) + ( n − V ( ε ) ( z, x ) , V ( ε ) ( z, x ) = (cid:88) q =1 V ( ε ) q ( z, x ) , (5.2)with V ( ε )1 ( z, x ) = − ( − d/ − D, − d/ − D + εDν ( z )) ( x ) (0 , + ∞ ) ( ν ( z ))+ ( − d/ − D + εDν ( z ) , − d/ − D ) ( x ) ( −∞ , ( ν ( z )) ,V ( ε )2 ( z, x ) = ( − d/ , − d/ εDν ( z )) ( x ) (0 , + ∞ ) ( ν ( z )) − ( − d/ εDν ( z ) , − d/ ( x ) ( −∞ , ( ν ( z )) ,V ( ε )3 ( z, x ) = − ( d/ ,d/ εDν ( z )) ( x ) (0 , + ∞ ) ( ν ( z ))+ ( d/ εDν ( z ) ,d/ ( x ) ( −∞ , ( ν ( z )) ,V ( ε )4 ( z, x ) = ( d/ D,d/ D + εDν ( z )) ( x ) (0 , + ∞ ) ( ν ( z )) − ( d/ D + εDν ( z ) ,d/ D ) ( x ) ( −∞ , ( ν ( z )) . To explain this reformulation, we illustrate in Figure 5.1 the fluctuations εDν ( z ) ofthe interface at x = − d/ − D . At range z we have ν ( z ) <
0, so ( z , x ) with x ∈ ( − d/ − D + εDν ( z ) , − d/ − D ) is in the waveguide and therefore, (cid:0) n ( ε ) ( z , x ) (cid:1) = (cid:0) n (0) ( x ) (cid:1) + ( n − V ( ε )1 ( z , x ) = 1 + ( n −
1) = n . At range z we have ν ( z ) >
0, so ( z , x ) with x ∈ ( − d/ − D, − d/ − D + εDν ( z ))is outside the waveguide and (cid:0) n ( ε ) ( z , x ) (cid:1) = (cid:0) n (0) ( x ) (cid:1) − ( n − V ( ε )1 ( z , x ) = n − ( n −
1) = 1 . The same reasoning applies to all the interfaces.Our goal in this section is to analyze the solution p ( z, x ) of (2.1), with index ofrefraction (5.2) and radiation condition at infinity, and in particular, to derive theresults stated in section 3.2. § Note that in [10] the decay is O ( z − ) because the source is f ( x ) δ ( z ). In our case the decay is O ( z − ), because we have the z derivative of the solution in [10], due to the source f ( x ) δ (cid:48) ( z ).16 = − d − Dx = − d z z Fig. 5.1 . Illustration of fluctuations of the bottom interface at x = − d/ − D . The idealinterfaces are drawn with the dotted lines, whereas the fluctuations are drawn with the full line. Weidentify two ranges z and z so that ν ( z ) < and ν ( z ) > . The completness result of section 4.3 allows theexpansion of p ( z, x ), z by z , in terms of the eigenfunctions defined in sections 4.1–4.2, p ( z, x ) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 p t,j ( z ) φ t,j ( x ) + (cid:88) t ∈{ e,o } (cid:90) k p t,γ ( z ) φ t,γ ( x ) dγ + (cid:88) t ∈{ e,o } (cid:90) −∞ p t,γ ( z ) φ t,γ ( x ) dγ. (5.3)Here p t,j ( z ) are complex-valued amplitudes of guided modes with wavenumber β t,j satisfying (4.6) or (4.10). They propagate along z and satisfy the following one-dimensional Helmholtz equations at z > ∂ z p t,j ( z ) + β t,j p t,j ( z ) = − ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C ( ε ) t,j,t (cid:48) ,l (cid:48) ( z ) p t (cid:48) ,l (cid:48) ( z ) − ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k −∞ C ( ε ) t,j,t (cid:48) ,γ (cid:48) ( z ) p t (cid:48) ,γ (cid:48) ( z ) dγ (cid:48) , j = 1 , . . . , N t . (5.4)Similarly, p t,γ ( z ) are complex-valued amplitudes of modes that are propagating for γ ∈ (0 , k ) (radiation modes) and decaying for γ < ∂ z p t,γ ( z ) + γp t,γ ( z ) = − ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C ( ε ) t,γ,t (cid:48) ,l (cid:48) ( z ) p t (cid:48) ,l (cid:48) ( z ) − ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k −∞ C ( ε ) t,γ,t (cid:48) ,γ (cid:48) ( z ) p t (cid:48) ,γ (cid:48) ( z ) dγ (cid:48) , γ ∈ ( −∞ , k ) . (5.5)The source terms in these equations are due to the random fluctuations, which aresupported at z ∈ (0 , L/ε ). These induce mode coupling, modeled by the randomcoefficients C ( ε ) t,j,t (cid:48) ,l (cid:48) ( z ) = (cid:16) φ t,j , φ t (cid:48) ,l (cid:48) V ( ε ) ( z, · ) (cid:17) , (5.6) C ( ε ) t,j,t (cid:48) ,γ (cid:48) ( z ) = (cid:16) φ t,j , φ t (cid:48) ,γ (cid:48) V ( ε ) ( z, · ) (cid:17) , (5.7) C ( ε ) t,γ,t (cid:48) ,l (cid:48) ( z ) = (cid:16) φ t,γ , φ t (cid:48) ,l (cid:48) V ( ε ) ( z, · ) (cid:17) , (5.8) C ( ε ) t,γ,t (cid:48) ,γ (cid:48) ( z ) = (cid:16) φ t,γ , φ t (cid:48) ,γ (cid:48) V ( ε ) ( z, · ) (cid:17) . (5.9) ecalling the definition (5.2) of V ( ε ) ( z, x ) and using Taylor expansions of the eigen-functions φ t,j ( x ) and φ t,γ ( x ) around x = ± d/ x = ± ( d/ D ), we obtain apower series (in ε ) expression of these coefficients C ( ε ) t,j,t (cid:48) ,l (cid:48) ( z ) = εC t,j,t (cid:48) ,l (cid:48) ( z ) + ε c t,j,t (cid:48) ,l (cid:48) ( z ) + o ( ε ) , (5.10) C t,j,t (cid:48) ,l (cid:48) ( z ) = − Dν ( z )[ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) − d − D (cid:17) + Dν ( z )[ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) − d (cid:17) − Dν ( z )[ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) d (cid:17) + Dν ( z )[ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) d D (cid:17) , (5.11) c t,j,t (cid:48) ,l (cid:48) ( z ) = − D ν ( z )2 ∂ x [ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) − d − D (cid:17) + D ν ( z )2 ∂ x [ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) − d (cid:17) − D ν ( z )2 ∂ x [ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) d (cid:17) + D ν ( z )2 ∂ x [ φ t,j φ t (cid:48) ,l (cid:48) ] (cid:16) d D (cid:17) , (5.12)and similarly for C ( ε ) t,j,t (cid:48) ,γ (cid:48) , C ( ε ) t,γ,t (cid:48) ,l (cid:48) , C ( ε ) t,γ,t (cid:48) ,γ (cid:48) .Since we consider propagation distances of the order of ε − , we neglect fromnow on the o ( ε ) terms in these expansions, because they give no contribution in thelimit ε →
0. To simplify the presentation, we will also not write the ε terms, eventhough they are not negligible. They contribute to the expression of the infinitesimalgenerators of the limit processes via terms that are proportional to ∆ k β t,j E [ c t,j,t,j ( z )].Because ∂ x ( φ t,j ) is odd, for t ∈ { e, o } , we obtain from (5.12) that E [ c t,j,t,j ( z )] = D E [ ν ( z ) + ν ( z )] ∂ x φ t,j (cid:16) d D (cid:17) − D E [ ν ( z ) + ν ( z )] ∂ x φ t,j (cid:16) d (cid:17) = D R (0) (cid:104) ∂ x φ t,j (cid:16) d D (cid:17) − ∂ x φ t,j (cid:16) d (cid:17)(cid:105) . Moreover, in the case of two weakly coupled single guided component waveguides, E [ c t, ,t, ( z )] = − D R (0) ξ sin( ξD ) (cid:16) η + D (cid:17) − . (5.13)We will incorporate these terms in the statements of the results without giving addi-tional details of their derivation. They correspond to effective deterministic phases ofthe mode amplitudes, in the limit ε →
0, as explained in Remark 5.2.
The guided modes can be decom-posed further in forward and backward guided modes. This decomposition is basicallythe method of variation of parameters for the perturbed Helmholtz equations (5.4–5.5), where we define the complex valued amplitudes { a t,j ( z ) , b t,j ( z ) , j = 1 , . . . , N t } and { a t,γ ( z ) , b t,γ ( z ) , γ ∈ (0 , k ) } , (5.14)for t ∈ { e, o } , such that p t,j ( z ) = 1 (cid:112) β t,j (cid:16) a t,j ( z ) e iβ t,j z + b t,j ( z ) e − iβ t,j z (cid:17) ,∂ z p t,j ( z ) = i (cid:112) β t,j (cid:16) a t,j ( z ) e iβ t,j z − b t,j ( z ) e − iβ t,j z (cid:17) , j = 1 , . . . , N t , (5.15) nd p t,γ ( z ) = 1 γ / (cid:16) a t,γ ( z ) e i √ γz + b t,γ ( z ) e − i √ γz (cid:17) ,∂ z p t,γ ( z ) = iγ / (cid:16) a t,γ ( z ) e i √ γz − b t,γ ( z ) e − i √ γz (cid:17) , γ ∈ (0 , k ) . (5.16)Substituting (5.15–5.16) in (5.4–5.5), we obtain that the amplitudes (5.14) satisfy thefollowing first-order system of stochastic differential equations ∂ z a t,j ( z ) = iε ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:112) β t (cid:48) ,l (cid:48) β t,j (cid:104) a t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) − β t,j ) z + b t (cid:48) ,l (cid:48) ( z ) e i ( − β t (cid:48) ,l (cid:48) − β t,j ) z (cid:105) + iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,j,t (cid:48) ,γ (cid:48) ( z ) √ γ (cid:48) (cid:112) β t,j (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) − β t,j ) z + b t (cid:48) ,γ (cid:48) ( z ) e i ( −√ γ (cid:48) − β t,j ) z (cid:105) dγ (cid:48) + iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,j,t (cid:48) ,γ (cid:48) ( z ) (cid:112) β t,j p t (cid:48) ,γ (cid:48) ( z ) e − iβ t,j z dγ (cid:48) , (5.17) ∂ z a t,γ ( z ) = iε ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,γ,t (cid:48) ,l (cid:48) ( z ) √ γ (cid:112) β t (cid:48) ,l (cid:48) (cid:104) a t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) −√ γ ) z + b t (cid:48) ,l (cid:48) ( z ) e i ( − β t (cid:48) ,l (cid:48) −√ γ ) z (cid:105) + iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( z ) √ γ (cid:48) γ (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) −√ γ ) z + b t (cid:48) ,γ (cid:48) ( z ) e i ( −√ γ (cid:48) −√ γ ) z (cid:105) dγ (cid:48) + iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,γ,t (cid:48) ,γ (cid:48) ( z ) √ γ p t (cid:48) ,γ (cid:48) ( z ) e − i √ γz dγ (cid:48) , (5.18) ∂ z b t,j ( z ) = − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:112) β t (cid:48) ,l (cid:48) β t,j (cid:104) a t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) + β t,j ) z + b t (cid:48) ,l (cid:48) ( z ) e i ( − β t (cid:48) ,l (cid:48) + β t,j ) z (cid:105) − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,j,t (cid:48) ,γ (cid:48) ( z ) √ γ (cid:48) (cid:112) β t,j (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) + β t,j ) z + b t (cid:48) ,γ (cid:48) ( z ) e i ( −√ γ (cid:48) + β t,j ) z (cid:105) dγ (cid:48) − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,j,t (cid:48) ,γ (cid:48) ( z ) (cid:112) β t,j p t (cid:48) ,γ (cid:48) ( z ) e iβ t,j z dγ (cid:48) , (5.19) ∂ z b t,γ ( z ) = − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,γ,t (cid:48) ,l (cid:48) ( z ) √ γ (cid:112) β t (cid:48) ,l (cid:48) } (cid:104) a t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) + √ γ ) z + b t (cid:48) ,l (cid:48) ( z ) e i ( − β t (cid:48) ,l (cid:48) + √ γ ) z (cid:105) − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( z ) √ γ (cid:48) γ (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) + √ γ ) z + b t (cid:48) ,γ (cid:48) ( z ) e i ( −√ γ (cid:48) + √ γ ) z (cid:105) dγ (cid:48) − iε ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,γ,t (cid:48) ,γ (cid:48) ( z ) √ γ p t (cid:48) ,γ (cid:48) ( z ) e i √ γz dγ (cid:48) . (5.20)The right-hand sides in these equations are supported at z ∈ (0 , L/ε ) and model themode coupling induced by scattering at the random interfaces (2.4). This couplingcauses the randomization of the amplitudes (5.14) and therefore of the wavefield. .3. Role of the evanescent modes. Equations (5.17–5.20) show that theforward and backward going amplitudes of both the guided and radiation modes arecoupled to each other and to the evanescent modes p t,γ ( z ), for γ ∈ ( −∞ ,
0) and t ∈ { e, o } . These mode amplitudes satisfy ∂ z p t,γ ( z ) + γp t,γ ( z ) = − ε (cid:2) g t,γ ( z ) + g ev t,γ ( z ) (cid:3) , z (cid:54) = 0 , (5.21)where g t,γ ( z ) and g ev t,γ ( z ) are supported at z ∈ (0 , L/ε ) and are defined by g t,γ ( z ) =∆ k (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,γ,t (cid:48) ,l (cid:48) ( z ) (cid:112) β t (cid:48) ,l (cid:48) (cid:104) a t (cid:48) ,l (cid:48) ( z ) e iβ t (cid:48) ,l (cid:48) z + b t (cid:48) ,l (cid:48) ( z ) e − iβ t (cid:48) ,l (cid:48) z (cid:105) + ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( z ) √ γ (cid:48) (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i √ γ (cid:48) z + b t (cid:48) ,γ (cid:48) ( z ) e − i √ γ (cid:48) z (cid:105) dγ (cid:48) , (5.22) g ev t,γ ( z ) =∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,γ,t (cid:48) ,γ (cid:48) ( z ) p t (cid:48) ,γ (cid:48) ( z ) dγ (cid:48) , z ∈ (cid:16) , L/ε (cid:17) . (5.23)We now explain that the solution of (5.21) can be expressed in terms of theamplitudes (5.14) when ε is small enough. This result has already been obtained insimilar frameworks [15] and it allows us to obtain a closed system of equations for theguided and radiation mode amplitudes.Let us invert the operator ∂ z + γ in (5.21) using the Green’s function that satisfiesthe radiation condition (i.e., it decays away from the source). This gives p t,γ ( z ) = ε (cid:112) | γ | (cid:90) L/ε (cid:2) g t,γ ( z (cid:48) ) + g ev t,γ ( z (cid:48) ) (cid:3) e − √ | γ || z − z (cid:48) | dz (cid:48) + a (0) t,γ | γ | / e − √ | γ | z , (5.24)at z >
0, for γ < t ∈ { e, o } . The first term in this expression comes fromthe random mode coupling induced by scattering at the random interfaces (2.4). Thesecond term comes from the source. Since we will consider propagation distances z ofthe order of ε − , we can neglect this second term.Using (5.23) and the notation p ev t = ( p t,γ ( z )) z> ,γ< for t ∈ { e, o } , we obtain that (cid:0) I − ε J t (cid:1) p ev t = ε G t , (5.25)where I is the identity operator, J t is the integral operator[ J t p ev t ] γ ( z ) = ∆ k (cid:112) | γ | (cid:88) t (cid:48) ∈{ e,o } (cid:90) L/ε dz (cid:48) (cid:90) −∞ dγ (cid:48) C t,γ,t (cid:48) ,γ (cid:48) ( z (cid:48) ) p t (cid:48) ,γ (cid:48) ( z (cid:48) ) e − √ | γ || z − z (cid:48) | (5.26)coming from the g ev t,γ term in (5.24), and the right-hand side is[ G t ] γ ( z ) = ∆ k (cid:112) | γ | (cid:90) L/ε g t,γ ( z (cid:48) ) e − √ | γ || z − z (cid:48) | dz (cid:48) , (5.27)for z > , γ <
0. The operator J t is bounded from C ((0 , ∞ ) , L ( −∞ , ε → P ( I − ε J t is invertible) = 1 , nd therefore p ev t = ε G t + O ( ε ) . (5.28)Furthermore, equations (5.17–5.20) give that a t (cid:48) ,l (cid:48) ( z (cid:48) ), b t (cid:48) ,l (cid:48) ( z (cid:48) ), a t (cid:48) ,γ (cid:48) ( z (cid:48) ), b t (cid:48) ,γ (cid:48) ( z (cid:48) )are equal to a t (cid:48) ,l (cid:48) ( z ), b t (cid:48) ,l (cid:48) ( z ), a t (cid:48) ,γ (cid:48) ( z ), b t (cid:48) ,γ (cid:48) ( z ) up to terms of order ε , as long as | z − z (cid:48) | = O (1), where the exponential in (5.27) contributes.Gathering the results we obtain p t,γ ( z ) = ε ∆ k (cid:112) | γ | (cid:88) t (cid:48) ∈{ e,o } (cid:90) L/ε N t (cid:48) (cid:88) l (cid:48) =1 (cid:40) C t,γ,t (cid:48) ,l (cid:48) ( z (cid:48) ) (cid:112) β t (cid:48) ,l (cid:48) (cid:104) a t (cid:48) ,l (cid:48) ( z ) e iβ t (cid:48) ,l (cid:48) z (cid:48) + b t (cid:48) ,l (cid:48) ( z ) e − iβ t (cid:48) ,l (cid:48) z (cid:48) (cid:105) + (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( z (cid:48) ) √ γ (cid:48) (cid:104) a t (cid:48) ,γ (cid:48) ( z ) e i √ γ (cid:48) z (cid:48) + b t (cid:48) ,γ (cid:48) ( z ) e − i √ γ (cid:48) z (cid:48) (cid:105) dγ (cid:48) (cid:41) e − √ | γ || z − z (cid:48) | dz (cid:48) + O ( ε ) , (5.29)for z > , γ < t ∈ { e, o } . Substituting this expression into (5.17-5.20) we get aclosed system for the guided and radiation mode amplitudes (5.14). The resulting stochastic system for the guided andradiation mode amplitudes has a right-hand side which is proportional to ε . Therefore,the amplitudes are not affected by scattering at the random interfaces (2.4) until thewaves travel at long enough range, dependent of ε . Note that the O ( ε ) coupling termsin (5.17–5.20) have zero expectation, so we need a central limit theorem type scaling,with range z (cid:32) z/ε , in order to observe net scattering effects.We rename the mode amplitudes in this long range scaling as a ( ε ) t,j ( z ) = a t,j (cid:16) zε (cid:17) , b ( ε ) t,j ( z ) = b t,j (cid:16) zε (cid:17) j = 1 , . . . , N t ,a ( ε ) t,γ ( z ) = a t,γ (cid:16) zε (cid:17) , b ( ε ) t,γ ( z ) = b t,γ (cid:16) zε (cid:17) , γ ∈ (0 , k ) , (5.30)for t ∈ { e, o } . Recalling that the random fluctuations are supported in the rangeinterval (0 , L/ε ) and using the radiation conditions (i.e., the waves are outgoing at z > L ), we obtain a ( ε ) t,j ( z = 0) = a (0) t,j , b ( ε ) t,j ( z = L ) = 0 , j = 1 , . . . , N t , (5.31) a ( ε ) t,γ ( z = 0) = a (0) t,γ , b ( ε ) t,γ ( z = L ) = 0 , γ ∈ (0 , k ) , (5.32)where a (0) t,j and a (0) t,γ are given in (4.23–4.24), for t ∈ { e, o } . These boundary conditionsand the closed system of stochastic differential equations described at the end of theprevious section define the random amplitudes (5.30). We now introduce the forward scat-tering approximation, where the coupling between forward and backward going modescan be neglected. It follows from the following facts, under the assumption that thepower spectral density (cid:98) R ( κ ) has compact support or fast decay so that (cid:98) R ( κ ) = (cid:90) ∞−∞ R ( z ) e iκz dz ≈ , if | κ | ≥ k. (5.33)1. In the limit ε →
0, the coupling between the forward and backward guided modes epends on the coefficients (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,l (cid:48) (0) C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:3) cos (cid:2) ( β t (cid:48) ,l (cid:48) + β t,j ) z (cid:3) dz, (5.34)while the coupling among the forward-guided modes depends on (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,l (cid:48) (0) C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:3) cos (cid:2) ( β t (cid:48) ,l (cid:48) − β t,j ) z (cid:3) dz. (5.35)Recalling definitions (5.2) and (5.6–5.9) and that β t,j ∈ ( k, nk ), we conclude that thecoefficients (5.34), which are proportional to (cid:98) R ( β t (cid:48) ,l (cid:48) + β t,j ), are negligible under theassumption (5.33). Therefore, we can neglect the coupling between the forward andbackward guided modes. Nevertheless, the forward guided modes are coupled amongthemselves, because the coefficients (5.35), which are proportional to (cid:98) R ( β t (cid:48) ,l (cid:48) − β t,j ),are not negligible.2. The coupling between forward guided and backward radiation modes dependsin the limit ε → (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,γ (cid:48) (0) C t,j,t (cid:48) ,γ (cid:48) ( z ) (cid:3) cos (cid:2) ( (cid:112) γ (cid:48) + β t,j ) z (cid:3) dz, (5.36)for γ (cid:48) ∈ (0 , k ). These are proportional to (cid:98) R ( β t,j + √ γ (cid:48) ) and since β t,j ∈ ( k, nk ), thistype of coupling is negligible under the assumption (5.33).3. The coupling between any radiation modes is negligible in our regime ε → z = L , we can set b ( ε ) t,j ( z ) ≈ , b ( ε ) t,γ ( z ) ≈ , j = 1 , . . . , N t , γ ∈ (0 , k ) , t ∈ { e, o } . The expression (5.3) of the wavefield simplifies to p (cid:16) zε , x (cid:17) = (cid:88) t ∈{ e,o } (cid:34) N t (cid:88) j =1 a ( ε ) t,j ( z ) (cid:112) β t,j e iβ t,j zε φ t,j ( x ) + (cid:90) k a ( ε ) t,γ ( z ) γ / e i √ γ zε φ t,γ ( x ) dγ (cid:35) + o (1) , (5.37)and the forward guided mode amplitudes a ( ε ) t,j and a ( ε ) t,γ satisfy ∂ z a ( ε ) t,j ( z ) = i ∆ k ε (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,j,t (cid:48) ,l (cid:48) ( zε ) (cid:112) β t (cid:48) ,l (cid:48) β t,j a ( ε ) t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) − β t,j ) zε + i ∆ k ε (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,j,t (cid:48) ,γ (cid:48) ( zε ) √ γ (cid:48) (cid:112) β t,j a ( ε ) t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) − β t,j ) zε dγ (cid:48) + i ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,j,t (cid:48) ,γ (cid:48) ( zε ) (cid:112) β t,j q ( ε ) t (cid:48) ,γ (cid:48) ( z ) e − iβ t,j zε dγ (cid:48) , (5.38) nd ∂ z a ( ε ) t,γ ( z ) = i ∆ k ε (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 C t,γ,t (cid:48) ,l (cid:48) ( zε ) (cid:112) β t (cid:48) ,l (cid:48) √ γ a ( ε ) t (cid:48) ,l (cid:48) ( z ) e i ( β t (cid:48) ,l (cid:48) −√ γ ) zε + i ∆ k ε (cid:88) t (cid:48) ∈{ e,o } (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( zε ) √ γ (cid:48) γ a ( ε ) t (cid:48) ,γ (cid:48) ( z ) e i ( √ γ (cid:48) −√ γ ) zε dγ (cid:48) + i ∆ k (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ C t,γ,t (cid:48) ,γ (cid:48) ( zε ) √ γ q ( ε ) t (cid:48) ,γ (cid:48) ( z ) e − i √ γ zε dγ (cid:48) , (5.39)where q ( ε ) t,γ ( z ) = ∆ k (cid:112) | γ | (cid:88) t (cid:48) ∈{ e,o } (cid:90) L/ε (cid:34) N t (cid:48) (cid:88) l (cid:48) =1 C t,γ,t (cid:48) ,l (cid:48) ( z (cid:48) ) (cid:112) β t (cid:48) ,l (cid:48) a ( ε ) t (cid:48) ,l (cid:48) ( z ) e iβ t (cid:48) ,l (cid:48) z (cid:48) + (cid:90) k C t,γ,t (cid:48) ,γ (cid:48) ( z (cid:48) ) √ γ (cid:48) a ( ε ) t (cid:48) ,γ (cid:48) ( z ) e i √ γ (cid:48) z (cid:48) dγ (cid:48) (cid:35) e − √ | γ | (cid:12)(cid:12) zε − z (cid:48) (cid:12)(cid:12) dz (cid:48) . (5.40) The next theorem gives the ε → Theorem 5.1.
Suppose that the wavenumbers β t,j are distinct, for ≤ j ≤ N t and t ∈ { e, o } . Then, the process (cid:0) ( a ( ε ) t,j ( z )) N t j =1 , ( a ( ε ) t,γ ( z )) γ ∈ (0 ,k ) , t ∈ { e, o } (cid:1) convergesin distribution in C (cid:0) [0 , L ] , C N × L (0 , k ) (cid:1) , where C N × L (0 , k ) is equipped with theweak topology, to the Markov process (cid:0) ( a t,j ( z )) N t j =1 , ( a t,γ ( z )) γ ∈ (0 ,k ) , t ∈ { e, o } (cid:1) . Theinfinitesimal generator of this limit process is L = L + L + L , where {L j } ≤ j ≤ arethe differential operators: L = 12 (cid:88) t,t (cid:48) ∈{ e,o } N t (cid:88) j =1 N t (cid:48) (cid:88) l (cid:48) =1 Γ ct,j,t (cid:48) ,l (cid:48) (cid:0) a t,j a t,j ∂ a t (cid:48) ,l (cid:48) ∂ a t (cid:48) ,l (cid:48) + a t (cid:48) ,l (cid:48) a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t,j − a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) − a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) (cid:1) ( t,j ) (cid:54) =( t (cid:48) ,l (cid:48) ) + 12 (cid:88) t,t (cid:48) ∈{ e,o } N t (cid:88) j =1 N t (cid:48) (cid:88) l (cid:48) =1 Γ t,j,t (cid:48) ,l (cid:48) (cid:0) a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) + a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) − a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) − a t,j a t (cid:48) ,l (cid:48) ∂ a t,j ∂ a t (cid:48) ,l (cid:48) (cid:1) + 12 (cid:88) t ∈{ e,o } N t (cid:88) j =1 (cid:0) Γ ct,j,t,j − Γ t,j,t,j (cid:1)(cid:0) a t,j ∂ a t,j + a t,j ∂ a t,j (cid:1) + i (cid:88) t ∈{ e,o } N t (cid:88) j =1 Γ st,j,t,j (cid:0) a t,j ∂ a t,j − a t,j ∂ a t,j (cid:1) , (5.41) L = − (cid:88) t ∈{ e,o } N t (cid:88) j =1 (Λ ct,j + i Λ st,j ) a t,j ∂ a t,j + (Λ ct,j − i Λ st,j ) a t,j ∂ a t,j , (5.42) L = i (cid:88) t ∈{ e,o } N t (cid:88) j =1 (cid:0) κ t,j + κ ev t,j (cid:1)(cid:0) a t,j ∂ a t,j − a t,j ∂ a t,j (cid:1) . (5.43) n these definitions we use the classical complex derivative: ζ = ζ r + iζ i (cid:32) ∂ ζ = 12 ( ∂ ζ r − i∂ ζ i ) , ∂ ζ = 12 ( ∂ ζ r + i∂ ζ i ) , and the coefficients of the operators (5.41) – (5.43) are defined for indexes j = 1 , . . . , N t ,l (cid:48) = 1 , . . . , N t (cid:48) and t, t (cid:48) ∈ { e, o } , as follows:For all ( t, j ) (cid:54) = ( t (cid:48) , l (cid:48) ) : Γ ct,j,t (cid:48) ,l (cid:48) = ∆ k β t,j β t (cid:48) ,l (cid:48) (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,l (cid:48) (0) C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:3) cos (cid:2) ( β t (cid:48) ,l (cid:48) − β t,j ) z (cid:3) dz, Γ st,j,t (cid:48) ,l (cid:48) = ∆ k β t,j β t (cid:48) ,l (cid:48) (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,l (cid:48) (0) C t,j,t (cid:48) ,l (cid:48) ( z ) (cid:3) sin (cid:2) ( β t (cid:48) ,l (cid:48) − β t,j ) z (cid:3) dz. For all ( t, j ) , ( t (cid:48) , l (cid:48) ) : Γ t,j,t (cid:48) ,l (cid:48) = ∆ k β t,j β t (cid:48) ,l (cid:48) (cid:90) ∞ E (cid:2) C t,j,t,j (0) C t (cid:48) ,l (cid:48) ,t (cid:48) ,l (cid:48) ( z ) (cid:3) dz. For all ( t, j ) : Γ ct,j,t,j = − N t (cid:88) l =1 ,l (cid:54) = j Γ ct,j,t,l − (cid:88) t (cid:48) (cid:54) = t N t (cid:48) (cid:88) l (cid:48) =1 Γ ct,j,t (cid:48) ,l (cid:48) , Γ st,j,t,j = − N t (cid:88) l =1 ,l (cid:54) = j Γ st,j,t,l − (cid:88) t (cid:48) (cid:54) = t N t (cid:48) (cid:88) l (cid:48) =1 Γ st,j,t (cid:48) ,l (cid:48) , Λ ct,j = (cid:88) t (cid:48) ∈{ e,o } (cid:90) k ∆ k √ γβ t,j (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,γ (0) C t,j,t (cid:48) ,γ ( z ) (cid:3) cos (cid:2) ( √ γ − β t,j ) z (cid:3) dzdγ, Λ st,j = (cid:88) t (cid:48) ∈{ e,o } (cid:90) k ∆ k √ γβ t,j (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,γ (0) C t,j,t (cid:48) ,γ ( z ) (cid:3) sin (cid:2) ( √ γ − β t,j ) z (cid:3) dzdγ,κ ev t,j = (cid:88) t (cid:48) ∈{ e,o } (cid:90) −∞ ∆ k (cid:112) | γ | β t,j (cid:90) ∞ E (cid:2) C t,j,t (cid:48) ,γ (0) C t,j,t (cid:48) ,γ ( z ) (cid:3) cos( β t,j z ) e − √ | γ | z dzdγ,κ t,j = ∆ k D β t,j R (0) (cid:34) ∂ x ( φ t,j ) (cid:16) d D (cid:17) − ∂ x ( φ t,j ) (cid:16) d (cid:17)(cid:35) . The proof of this diffusion limit theorem is based on a martingale approach usingthe perturbed test function method. It is an extension of the diffusion approximationtheorem in [23, 10] that is carried out in [15] and [16, Sections 3, 4.1]. Here we repro-duce the result but we simplify its statement. The rigorous statement [16, Theorem4.1] has two steps: The first step deals with the convergence of truncated processes,obtained using the resolution of the identity Π R \ ( − γ (cid:63) ,γ (cid:63) ) defined in section (4.3). Thisremoves the vicinity of γ = 0, in order to take the diffusion limit ε →
0. The secondstep deals with the convergence of the truncated processes themselves, in the limit γ (cid:63) → Remark 5.1.
Note that:1. The convergence result in Theorem 5.1 holds in the weak topology. . The infinitesimal generator L does not involve the derivatives ∂ a t,γ . Therefore (cid:16) ( a ( ε ) t,j ( z )) N t j =1 , t ∈ { e, o } (cid:17) converges in distribution in C ([0 , L ] , C N ) to the Markovprocess (cid:16) ( a t,j ( z )) N t j =1 , t ∈ { e, o } (cid:17) with generator L . Of course, the weak and strongtopologies are the same in C N .3. If the generator L is applied to a test function that depends only on the modepowers (cid:16) ( P t,j = | a t,j | ) N t j =1 , t ∈ { e, o } (cid:17) , then the result is a function that depends onlyon (cid:16) ( P t,j ) N t j =1 , t ∈ { e, o } (cid:17) . Thus, the mode powers (cid:16) ( P t,j ( z )) N t j =1 , t ∈ { e, o } (cid:17) define aMarkov process, with infinitesimal generator L P = (cid:88) t,t (cid:48) ∈{ e,o } N t (cid:88) j =1 N t (cid:48) (cid:88) l (cid:48) =1 Γ ct,j,t (cid:48) ,l (cid:48) (cid:16) P t (cid:48) ,l (cid:48) P t,j (cid:0) ∂ P tj − ∂ P t (cid:48) ,l (cid:48) (cid:1) ∂ P t,j + (cid:0) P t (cid:48) ,l (cid:48) − P t,j (cid:1) ∂ P t,j (cid:17) − (cid:88) t ∈{ e,o } N t (cid:88) j =1 Λ ct,j P t,j ∂ P t,j . (5.44)
4. The radiation mode amplitudes remain constant on L (0 , k ) , equipped withthe weak topology, as ε → . However, this does not describe the power transportedby the radiation modes (cid:88) t ∈{ e,o } (cid:90) k | a t,γ | dγ , because the convergence does not hold inthe strong topology of L (0 , k ) . The third point in Remark 5.1 implies that the mean mode powers satisfy theclosed system of equations d E [ P t,j ( z )] dz = (cid:88) t (cid:48) ∈{ e,o } N t (cid:48) (cid:88) l (cid:48) =1 Γ ct,j,t (cid:48) ,l (cid:48) (cid:0) E [ P t (cid:48) ,l (cid:48) ( z )] − E [ P t,j ( z )] (cid:1) − Λ ct,j E [ P t,j ( z )] , (5.45)with initial conditions P t,j (0) = | a (0) t,j | , j = 1 , . . . , N t , t ∈ { e, o } . (5.46)These equations (5.45) can be found in the literature [22, 24]. Here we derived themfrom first principles, taking into account the coupling between all types of modes. Remark 5.2.
The three terms {L j } ≤ j ≤ of the the limit infinitesimal generator L account for the mode coupling as follows:1. The operator L accounts for coupling of the guided modes, modeled by thecoefficients { Γ ct,j,t (cid:48) ,j (cid:48) } ≤ j ≤ N t , ≤ j (cid:48) ≤ N t (cid:48) ,t,t (cid:48) ∈{ e,o } . This coupling results in the power ex-change between these modes.2. The operator L is due to coupling between the guided and radiation modes,which causes power leakage from the guided modes to the radiation ones (effectivediffusion), modeled by the coefficients { Λ ct,j } ≤ j ≤ N t ,t ∈{ e,o } .3. The operator L accounts for coupling between the guided and evanescentmodes, which gives the term proportional to { κ ev t,j } ≤ j ≤ N t ,t ∈{ e,o } . The other term,proportional to { κ t,j } ≤ j ≤ N t ,t ∈{ e,o } , is due to the neglected terms discussed at the endof section 5.1. Both contributions give only additional phase terms on the guide modesi.e., an effective dispersion. . Two weakly coupled single mode random waveguides. Using the re-sults of the previous section and the decay in range (like ( z/ε ) − = ε /z ) of thecontribution of the radiation and evanescent modes, we obtain that p (cid:16) zε , x (cid:17) = (cid:88) t ∈{ e,o } N t (cid:88) j =1 a t,j ( z ) (cid:112) β t,j exp (cid:16) iβ t,j zε (cid:17) φ t,j ( x ) + o (1) , z > , (6.1)where the amplitudes { a t,j ( z ) } ≤ j ≤ N t ,t ∈{ e,o } are described in Theorem 5.1 and theresidual o (1) tends to zero as ε → N e = N o = 1, where we can applyLemma 4.1, under the assumptions (3.1) and (3.11), and we can use the expansion(4.29) of the eigenfunctions. The goal is to derive the results stated in section 3.2, inthe three coupling regimes defined by the scaling relations (3.22–3.24). In the regime defined by the scaling relation(3.22), the expression (6.1) becomes p (cid:16) zε , x (cid:17) = φ ( | x | ) √ β e iβ zε (cid:104) (0 , ∞ ) ( x ) u + ( z ) + 1 ( −∞ , ( x ) u − ( z ) (cid:105) + o (1) , (6.2)with φ defined in (3.4) and β defined in Lemma 4.1. It corresponds to a single guidedwave in each waveguide, with transverse profile defined by φ ( | x | ) and range dependentrandom amplitude u ± ( z ) = a e ( z ) exp (cid:16) i ∆ β e zε (cid:17) ± a o ( z ) exp (cid:16) i ∆ β o zε (cid:17) . (6.3)This is the expression of the wave field given in section 3.2, where we simplified thenotation as a t, ( z ) (cid:32) a t ( z ) and introduced∆ β t = β t, − β = O (cid:0) e − ηd (cid:1) , t ∈ { e, o } , (6.4)satisfying 1 (cid:29) | ∆ β t | (cid:29) ε , by equations (3.22) and (4.28).The statistics of the amplitudes (6.3) follows from Theorem 5.1, in the case N e = N o = 1. The coefficients in the infinitesimal generator L are described in the nextcorollary, proved in Appendix B. Corollary 6.1.
The effective coefficients in the operator L are given by Γ ce, ,o, = Γ co, ,e, = − Γ ce, ,e, = − Γ co, ,o, = Γ , (6.5) and Γ t, ,t (cid:48) , = Γ , Γ st, ,t, = 0 , t, t (cid:48) ∈ { e, o } , (6.6) where Γ has the expression (3.31) . The effective coefficients in the operator L are Λ ct, = Λ , Λ st, = Θ , t ∈ { e, o } , (6.7) where Λ is given by (3.30) and Θ = ∆ k D β (cid:0) η + D (cid:1) cos (cid:16) ξD (cid:17) (cid:90) k dγπη γ √ γ (cid:34) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) (cid:35) × (cid:90) ∞ dz R ( z ) sin (cid:2) ( √ γ − β ) z (cid:3) . (6.8) he effective coeffients in the operator L are κ ev t, = κ ev , κ t, = κ, t ∈ { e, o } , (6.9) where κ ev = ∆ k D β (cid:0) η + D (cid:1) cos (cid:16) ξD (cid:17) (cid:90) −∞ dγπη γ (cid:112) | γ | (cid:34) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) (cid:35) × (cid:90) ∞ dz R ( z ) cos (cid:0) βz (cid:1) exp( − (cid:112) | γ | z ) . (6.10) and κ = − ∆ k D ξβ (cid:0) η + D (cid:1) R (0) sin( ξD ) . (6.11)Using Theorem 5.1 and Corollary 6.1, we obtain the joint moments of the modeamplitudes a e ( z ) and a o ( z ): The mean amplitudes are given by E [ a t ( z )] = a (0) t e − (Γ+Λ / z − i ( κ ev + κ +Θ / z , (6.12)with a (0) t defined in (3.10) for t ∈ { e, o } . The mean powers and field covariance are E [ | a e ( z ) | ] = ( | a (0) e | + | a (0) o | )2 e − Λ z + ( | a (0) e | − | a (0) o | )2 e − (2Γ+Λ) z , (6.13) E [ | a o ( z ) | ] = ( | a (0) e | + | a (0) o | )2 e − Λ z − ( | a (0) e | − | a (0) o | )2 e − (2Γ+Λ) z , (6.14) E [ a o ( z ) a e ( z )] =( a (0) o a (0) e ) e − (Γ+Λ) z , (6.15)and the fourth-order moments are E [ | a e ( z ) | ] E [ | a o ( z ) | ]2 E [ | a e ( z ) | | a o ( z ) | ] = ( | a (0) e | + | a (0) o | ) )3 e − z − ( | a (0) e | − | a (0) o | )2 e − (2Λ+2Γ) z − − ( | a (0) e | + | a (0) o | − | a (0) e | | a (0) o | )6 e − (2Λ+6Γ) z − − . (6.16)We conclude from equations (6.13–6.14) and (6.16) that E (cid:2) | a e ( z ) | + | a o ( z ) | (cid:3) =( | a (0) e | + | a (0) o | ) e − Λ z , (6.17)Var (cid:0) | a e ( z ) | + | a o ( z ) | (cid:1) =0 . (6.18)This implies the results (3.28) and (3.33), which model the effective leakage of powerfrom the guided modes to the radiation modes. Equations (6.13–6.15) also give theexpression (3.35) of the imbalance of power between the waveguides. Remark 6.1.
These results show that of all the coefficients in Corollary 6.1, Γ and Λ are the important ones. The coefficient Γ determines the coupling between the wo guided modes, whereas Λ determines the power leakage to the radiation modes.Note from (3.30) – (3.31) that Γ depends on power spectral density (cid:98) R , the Fouriertransform of the covariance R , evaluated at zero wavenumber, while Λ depends on (cid:98) R evaluated at wavenumbers larger than β − k , where we recall that β ∈ ( k, nk ) . Thisresult is consistent with the observation in [21] that radiation losses should only dependon the power spectral density evaluated at wavenumbers between β − k and β + k . It is possible to encounter situations where Γ > β − k , thenΛ = 0. Otherwise, if (cid:98) R (0) = 0 (this happens for instance when the random processes ν j are derivatives of stationary processes), then Γ = 0. However, in general, both Γand Λ are positive. In the regime defined by the scaling relation(3.23), the wave field has the same expression (6.2), but the amplitudes u ± ( z ) havedifferent statistics. They are obtained as the ε → u ( ε ) ± ( z ) = a ( ε ) e, ( z ) e iβ (cid:48) θz ± a ( ε ) o, ( z ) e − iβ (cid:48) θz , (6.19)where we used the expression of β e, and β o, given in equations (4.28) and setexp( − ηd ) = ε θ with θ ∈ [0 , ∞ ). To obtain this limit, we let α ( ε ) e ( z ) = a ( ε ) e, ( z ) e iβ (cid:48) θz , α ( ε ) o ( z ) = a ( ε ) o, ( z ) e − iβ (cid:48) θz , (6.20)and then use equations (5.38)–(5.39) to derive a closed system of stochastic differentialequations satisfied by the process (cid:0) α ( ε ) t ( z ) , (cid:0) a ( ε ) t,γ ( z ) (cid:1) γ ∈ (0 ,k ) , t ∈ { e, o } (cid:1) . The limit ofthis process is given in the next theorem, with proof outlined in Appendix C. As wasthe case in Theorem 5.1, the infinitesimal generator does not involve the radiationmode amplitudes, so we describe directly the limit of (cid:0) ( α ( ε ) t ( z ) , t ∈ { e, o } (cid:1) . Theorem 6.2.
In the limit ε → , the process (cid:0) α ( ε ) e ( z ) , α ( ε ) o ( z ) (cid:1) converges indistribution in C (cid:0) [0 , L ] , C (cid:1) to the Markov process α ( z ) = ( α e ( z ) , α o ( z )) with in-finitesimal generator L α ,θ = L α + L α + L α + θ L α , (6.21) where θ is defined in (3.38) , and the operators are defined by L α = Γ2 (cid:16) | α o | + | α e | )( ∂ α o ∂ α o + ∂ α e ∂ α e ) + 2( α o α e + α o α e )( ∂ α o ∂ α e + ∂ α e ∂ α o ) − ( α o + α e )( ∂ α o + ∂ α e ) − ( α o + α e )( ∂ α o + ∂ α e ) − α o α e ∂ α o ∂ α e − α o α e ∂ α o ∂ α e − α o ∂ α o + α e ∂ α e ) − α o ∂ α o + α e ∂ α e ) (cid:17) , (6.22) and L α = − (cid:88) t ∈{ e,o } (Λ + i Θ) α t ∂ α t + (Λ − i Θ) α t ∂ α t , (6.23) L α = i (cid:0) κ + κ ev (cid:1) (cid:88) t ∈{ e,o } (cid:0) α t ∂ α t − α t ∂ α t (cid:1) , (6.24) L α = iβ (cid:48) (cid:0) α e ∂ α e − α e ∂ α e − α o ∂ α o + α o ∂ α o (cid:1) , (6.25) ith coefficients given in Corollary 6.1. The result of Theorem 6.2 can be reformulated as stated in the following corollary,by the change of variables: u ( ε ) ± ( z ) = α ( ε ) e ( z ) ± α ( ε ) o ( z ) . Corollary 6.3.
In the limit ε → , the process ( u ( ε )+ ( z ) , u ( ε ) − ( z )) convergesin distribution in C ([0 , L ] , C ) to the Markov process u ( z ) = ( u + ( z ) , u − ( z )) withinfinitesimal generator L u ,θ = L u + + L u − + θ L u + ,u − , defined by L u =Γ (cid:0) | u | ∂ u ∂ u − u ∂ u − u ∂ u − u∂ u − u∂ u (cid:1) − Λ + i Θ2 u∂ u − Λ − i Θ2 u∂ u + i ( κ + κ ev ) (cid:0) u∂ u − u∂ u (cid:1) , (6.26) L u + ,u − = iβ (cid:48) (cid:0) u + ∂ u − + u − ∂ u + − u + ∂ u − − u − ∂ u + (cid:1) . (6.27)Note that for any test function F , the generator gives L u ,θ F ( | u + | + | u − | ) = − Λ( | u + | + | u − | ) F (cid:48) ( | u + | + | u − | ) . (6.28)This shows that | u + ( z ) | + | u − ( z ) | = 2( | α o ( z ) | + | α e ( z ) | ) is deterministic andsatisfies (3.33), as stated in section 3.2.2. The total power transported by the twoguided modes decays as exp( − Λ z ) with probability one, due to an effective leakagetowards the radiation modes: | u + ( z ) | + | u − ( z ) | = 2( | a (0) o | + | a (0) e | ) exp( − Λ z ) . (6.29)To determine the imbalance of power between the two guided modes P ( z ) = | u + ( z ) | − | u − ( z ) | | u + ( z ) | + | u − ( z ) | , (6.30)we apply the infinitesimal generator L u ,θ to the two functions( u + , u + , u − , u − ) (cid:55)→ | u + | − | u − | , ( u + , u + , u − , u − ) (cid:55)→ u + u − , and we get L u ,θ ( | u + | − | u − | ) = − Λ( | u + | − | u − | ) − θβ (cid:48) Im( u + u − ) , (6.31) L u ,θ ( u + u − ) = ( − − Λ)( u + u − ) + iθβ (cid:48) ( | u + | − | u − | ) . (6.32)Denoting I ( z ) = 2Im( u + u − ( z )) / ( | u + ( z ) | + | u − ( z ) | ), we find that ∂ z E [ P ( z )] = − θβ (cid:48) E [ I ( z )] ,∂ z E [ I ( z )] = − E [ I ( z )] + 2 θβ (cid:48) E [ P ( z )] , which gives the harmonic oscillator equation (3.37) satisfied by E [ P ( z )]. .3. Very weak coupling regime. When the separation d between the waveg-uides is so large that exp( − ηd ) (cid:28) ε , we obtain from (4.28) that β t, = β + o ( ε ) , t ∈ { e, o } . (6.33)The ε → β (cid:48) = 0 in Theorem 6.2. The expression (6.2) of the wave fieldstill holds, but the mode amplitudes have different statistics, as described in the nextcorollary: Corollary 6.4.
In the limit ε → , the process ( u ( ε )+ ( z ) , u ( ε ) − ( z )) converges indistribution in C ([0 , L ] , C ) to the process ( u + ( z ) , u − ( z )) where u + ( z ) and u − ( z ) areindependent and identically distributed Markov processes with infinitesimal generator L u defined by (6.26). Note that, for any test function F , we have L u F ( | u | ) = − Λ | u | F (cid:48) ( | u | ) . This shows that the processes ( | u + ( z ) | , | u − ( z ) | ) are deterministic and exponentiallydecaying with the rate Λ. We conclude that ( | u ( ε )+ | ( z ) , | u ( ε ) − | ( z )) converges in prob-ability to the deterministic function ( | u + ( z ) | , | u − ( z ) | ) that decays exponentially asexp( − Λ z ). Therefore, there is no mode coupling in this regime, except between guidedand radiation modes, which results in effective leakage. In particular, the imbalanceof power between the two waveguides is constant, as stated in section 3.2.3.
7. Summary.
In this paper we introduced an analysis of wave propagation in adirectional coupler consisting of two parallel step-index waveguides. The waveguideeffect is due to a medium of high index of refraction separated from a uniform back-ground by randomly fluctuating interfaces. The fluctuations occur on a length scale(correlation length) that is similar to the wavelength and have small amplitude mod-eled by a dimensionless parameter ε satisfying 0 < ε (cid:28)
1. The analysis is based onthe decomposition of the wave field in a complete set of guided, radiation and evanes-cent modes. It accounts for the interaction of all these modes and derives a closedsystem of stochastic differential equations for the guided and radiation mode ampli-tudes. These equations are driven by the random fluctuations of the interfaces andmodel the net scattering effect that becomes significant at distances of propagationof the order of ε − . We analyze them in the asymptotic limit ε →
0, under the as-sumption that the covariance function of the fluctuations is smooth. This allows us touse the forward scattering approximation and characterize the limit mode amplitudesas a Markovian process with infinitesimal generator that we calculate explicitly. Theanalysis applies to waveguides that support an arbitrary number of guided modes.Since many directional couplers use single guided mode waveguides, we studied in de-tail this case and obtained a detailed quantification of the transfer of power for threedifferent regimes, where the coupling between the guided waves is stronger or weaker,depending on how far apart the waveguides are. In all regimes there is a self-averagingpower leakage from the guided to the radiation modes. This is the only coupling ifthe waveguides are very far apart. Otherwise, the guided modes are coupled andthe random boundary fluctuations induce a blurring of the periodic transfer of powerthat would occur in the absence of the random fluctuations. We quantified this blur-ring and showed that at sufficiently long distances of propagation the power becomesevenly distributed among the waveguides, independent of the initial condition definedby the wave source. cknowledgements. This research is supported in part by the Air Force Officeof Scientific Research under award number FA9550-18-1-0131 and in part by the NSFgrant DMS1510429.
Appendix A. Proof of Lemma 4.1.
The solution β of (4.26) can be obtainedas follows: First, note that cos ( q ) /q is a monotone, strictly decreasing function in theinterval q ∈ (0 , π/ q (cid:38) cos qq = ∞ , lim q (cid:37) π/ cos qq = 0 . Thus, there is a unique solution q (cid:63) ofcos qq = 2 kD √ n − , which gives tan q (cid:63) = (cid:104) − (cid:16) q (cid:63) kD √ n − (cid:17) (cid:105) / q (cid:63) kD √ n − . Substituting q (cid:63) = D (cid:112) n k − β into this equation we obtain that (4.26) is satisfied,with β = (cid:114) k n − (cid:16) q (cid:63) D (cid:17) . Obviously, β < kn and the condition β > k is consistentwith q (cid:63) < kD (cid:112) n − < π , where the last inequality is by the assumption (3.1).A similar construction shows that (4.27) has no solution in the interval ( k, nk ).Therefore, the right-hand sides of equations (4.6) and (4.10) vanish at the single root β ∈ ( k, nk ) of (4.26).If d is large, then the left-hand sides of equations (4.6) and (4.10) are small, andby the implicit function theorem, they have unique solutions, close to β calculatedabove. Consequently, N e = N o = 1. Appendix B. Proof of Corollary 6.1.
Theorem 5.1 states thatΓ ce, ,o, = ∆ k β e β o (cid:90) ∞ E (cid:2) C e, ,o, (0) C e, ,o, ( z ) (cid:3) cos (cid:2) ( β e − β o ) z (cid:3) dz, (B.1)where β e ≈ β o ≈ β and the expectation follows by definitions (5.11), (3.4) andequation (4.29), E (cid:2) C e, ,o, (0) C e, ,o, ( z ) (cid:3) ≈ R ( z ) D (cid:104) φ e, (cid:16) d (cid:17) φ o, (cid:16) d (cid:17) + φ e, (cid:16) d D (cid:17) φ o, (cid:16) d D (cid:17)(cid:105) , ≈ D (cid:16) η + D (cid:17) R ( z ) cos (cid:16) ξ D (cid:17) . Here the approximation is up to an error of the order of exp( − ηd ) (cid:28)
1, which weneglect in this appendix. Substituting in (B.1) we get the results (6.5) and (3.31).Proceeding similarly, we find Γ t, ,t (cid:48) , = Γ , for all t, t (cid:48) ∈ { e, o } . he expression of Λ ct, in Theorem 5.1 isΛ ct, = (cid:88) t (cid:48) ∈{ e,o } (cid:90) k dγ ∆ k √ γβ t, (cid:90) ∞ dz E (cid:2) C t, ,t (cid:48) ,γ (0) C t, ,t (cid:48) ,γ ( z ) (cid:3) cos (cid:0) ( √ γ − β t, ) z (cid:1) , (B.2)where the expectation is calculated using the definition (5.7), E (cid:2) C t, ,t (cid:48) ,γ (0) C t, ,t (cid:48) ,γ ( z ) (cid:3) = 2 R ( z ) D (cid:104) φ t, (cid:16) d (cid:17) φ t (cid:48) ,γ (cid:16) d (cid:17) + φ t, (cid:16) d D (cid:17) φ t (cid:48) ,γ (cid:16) d D (cid:17)(cid:105) . (B.3)As explained above, φ t ≈ φ for t ∈ { e, o } , and definition (3.4) gives that φ (cid:16) d (cid:17) = φ (cid:16) d D (cid:17) = (cid:16) η + D (cid:17) − cos (cid:16) ξ D (cid:17) . We also have from the expressions of φ e,γ and φ o,γ in section 4.2 that φ e,γ (cid:16) d (cid:17) + φ e,γ (cid:16) d D (cid:17) = A e,γ (cid:110) ξ γ η γ cos ( η γ d (cid:2) − sin( ξ γ D ) sin( η γ d ξ γ η γ cos( ξ γ D ) cos( η γ d (cid:3) (cid:111) , with A e,γ given by (4.15). Substituting in (B.3), we obtain E (cid:2) C t, ,e,γ (0) C t, ,e,γ ( z ) (cid:3) = D πη γ (cid:16) η + D (cid:17) R ( z ) cos (cid:0) ξD (cid:1) × cos( η γ d ) (cid:104) ξ γ η γ − sin ( ξ γ D )(1 + ξ γ η γ ) (cid:105) − sin( η γ d ) ξ γ η γ sin(2 ξ γ D ) + 2 ξ γ η γ + sin ( ξ γ D )(1 − ξ γ η γ )cos( η γ d ) sin ( ξ γ D ) (cid:16) ξ γ η γ − (cid:17) + sin( η γ d ) ξ γ η γ (cid:16) ξ γ η γ − (cid:17) sin(2 ξ γ D ) + 2 ξ γ η γ + sin ( ξ γ D )(1 − ξ γ η γ ) . As d is large, we take the weak limit d → + ∞ in this expression, seen as a functionof γ , and we get E (cid:2) C t, ,e,γ (0) C t, ,e,γ ( z ) (cid:3) = D πη γ (cid:16) η + D (cid:17) R ( z ) cos (cid:16) ξD (cid:17)(cid:34) ξ γ η γ + 2 sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) ξ γ η γ + sin ( ξ γ D ) (cid:0) − ξ γ η γ (cid:1) (cid:35) by the following calculation: for any α , α , α , α , α , α , with α > (cid:112) α + α > π (cid:90) π α + α cos( s ) + α sin( s ) α + α cos( s ) + α sin( s ) ds = α + α α + α α α + α (cid:0)(cid:112) α − α − α − α (cid:1)(cid:112) α − α − α . This gives Λ ce, = Λ, as stated in Corollary 6.1. We can deal with φ o,γ (cid:16) d (cid:17) + φ o,γ (cid:16) d D (cid:17) = A o,γ (cid:110) ξ γ η γ sin (cid:16) η γ d (cid:17) + (cid:104) sin( ξ γ D ) cos( η γ d ξ γ η γ cos( ξ γ D ) sin (cid:16) η γ d (cid:17)(cid:105) (cid:111) nd E (cid:2) C t, ,o,γ (0) C t, ,o,γ ( z ) (cid:3) in the same way, which gives Λ co, = Λ. The coefficientsΛ st, , κ t, , t ∈ { e, o } are obtained in a similar way. Appendix C. Proof of Theorem 6.2.
The result follows from an extendedversion of Theorem 5.1, when all β t,j are equal, as was considered in [19]. We obtain L α ,θ = (cid:88) t ,t ,t ,t ∈{ e,o } ∆ k β (cid:90) ∞ E (cid:2) C t , ,t , ( z ) C t , ,t , (0) (cid:3) dz (cid:0) α t ∂ α t α t ∂ α t − α t ∂ α t α t ∂ α t (cid:1) − (cid:88) t ,t ,t (cid:48) ∈{ e,o } (cid:90) k ∆ k β √ γ (cid:48) (cid:90) ∞ E (cid:2) C t (cid:48) ,γ (cid:48) ,t , ( z ) C t , ,t (cid:48) ,γ (cid:48) (0) (cid:3) e i ( β −√ γ (cid:48) ) z dzdγ (cid:48) α t ∂ α t + i ( κ + κ ev ) (cid:0) α e ∂ α e + α o ∂ α o (cid:1) + iθβ (cid:48) (cid:0) α e ∂ α e − α o ∂ α o (cid:1) + c.c., where c.c. stands for complexe conjugate. The statement of Theorem 6.2 follows fromthis expression, by using the definition (5.11) of the random coefficients C t , ,t , ( z )and C t (cid:48) ,γ (cid:48) ,t , ( z ) and the symmetry of the eigenfunctions i.e., that φ e, is even and φ o, is odd. REFERENCES[1] R. Alonso and L. Borcea,
Electromagnetic wave propagation in random waveguides , SIAM Mul-tiscale Modeling & Simulations, , 847–889, 2015. 1[2] R. Alonso, L. Borcea and J. Garnier, Wave propagation in waveguides with random boundaries ,Communications in Mathematical Sciences, , 233–267, 2012. 1[3] M. D. Bedford and G. A. Kennedy, Modeling microwave propagation in natural caves passages ,IEEE Transactions on Antennas and Propagation, , 6463–6471, 2014. 1[4] W. Bogaerts, P. De Heyn P, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes,P. Dumon, P. Bienstman, D. Van Thourhout and R. Baets, Silicon microring resonators ,Laser & Photonics Reviews , 47–73, 2012. 1[5] L. Borcea and J. Garnier, Paraxial coupling of propagating modes in three-dimensional waveg-uides with random boundaries , SIAM Multiscale Modeling & Simulation, , 832–878, 2014.1[6] L. Borcea and J. Garnier, Pulse reflection in a random waveguide with a turning point , SIAMMultiscale Modeling & Simulations, , 1472–1501, 2017. 1[7] L. Borcea, J. Garnier and D. Wood, Transport of power in random waveguides with turningpoints , Communications in Mathematical Sciences , 2327–2371 (2017). 1[8] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations , R. E. KriegerPublishing Company, Malabar, FL, 1984. 10, 14[9] R. E. Collin,
Field theory of guided waves (2nd edition) , IEEE Press, Piscataway, 1990. 1[10] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna,
Wave Propagation and Time Reversalin Randomly Layered Media , Springer, New York, 2007. 24[11] J. Garnier,
Light propagation in square law media with random imperfections , Wave Motion , 1–19, 2000. 1[12] J. Garnier, The role of evanescent modes in randomly perturbed single-mode waveguides , Dis-crete and Continuous Dynamical Systems-Series B , 455–472, 2007. 1[13] J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides ,SIAM J. Appl. Math. , 1718–1739, 2007. 1[14] C. Gomez, Time-reversal superresolution in random waveguides , SIAM Multiscale Model. Simul. , 1348–1386, 2009. 1[15] C. Gomez, Wave propagation in shallow-water acoustic random waveguides , Commun. Math.Sci. , 81–125, 2011. 1, 20, 24[16] C. Gomez, Wave propagation in shallow-water acoustic random waveguides , arxiv:0911.5646v1.24[17] C. Gomez,
Wave Propagation in Underwater Acoustic Waveguides with Rough Boundaries ,Communications in Mathematical Sciences, , 2005–2052, 2015. 1[18] W.-P. Huang, Coupled-mode theory for optical waveguides: an overview, J. Opt. Soc. Am. A , 963–983, 1994. 1 3319] W. Kohler and G. Papanicolaou, Wave Propagation in Randomly Inhomogeneous Ocean , Lec-ture Notes in Physics, J. B. Keller and J. S. Papadakis, eds., Wave Propagation and Under-water Acoustics, Springer-Verlag, Berlin, 70, 1977. 1, 33[20] R. Magnanini and F. Santosa,
Wave propagation in a 2-d optical waveguide , SIAM J. Appl.Math. , 1237–1252, 2000. 4, 5, 10, 14, 16[21] D. Marcuse, Radiation losses of dielectric waveguides in terms of the power spectrum of thewall distortion function , Bell System Technical Journal , 3233–3242, 1969. 28[22] D. Marcuse, Theory of Dielectric Optical Waveguides , Academic Press, New York, 1974. 1, 2,4, 25[23] G. Papanicolaou and W. Kohler,
Asymptotic theory of mixing stochastic ordinary differentialequations , Commun. Pure Appl. Math. , 641–668, 1974. 3, 24[24] I. Papakonstantinou, R. James and D. R. Selviah, Radiation- and bound-mode propagationin rectangular, multimode dielectric, channel waveguides with sidewall roughness , IEEE J.Lightwave Technology , 4151–4163, 2009. 25[25] E. Perrey-Debain and I. D. Abrahams, A diffusion analysis approach to TE mode propagationin randomly perturbed optical waveguides , SIAM J. Appl. Math. , 523–543, 2007. 1[26] H. E. Rowe, Electromagnetic Propagation in Multi-mode Random media , Wiley, New York,1999. 1[27] R. Syms and J. Cozens,
Optical Guided Waves and Devices , McGraw-Hill, London, 1992. 1, 2,7[28] C. Wilcox,
Spectral analysis of the Pekeris operator in the theory of acoustic wave propagationin shallow water , Arch. Rational Mech. Anal.60