codirectional coupling

Chris Spencer1

1Affiliation not available

1. Introduction

Coupled-mode theory is concerned with coupling spatialmodes of differing polarizations, distribution, or both.To understand codirectional coupling it is useful to havean understand of background material that builds tocodirectional coupling. First, consider coupling normalmodes in a single waveguide that is affected by a per-turbation. Such case is single-waveguide mode coupling.The perturbation in question is spatially dependent andis represented as ∆P (r), a perturbing polarization. Con-sider the following Maxwell’s equations

∇× E = iωµ0H

∇×H = −iωεE − iω∆P

Consider two sets of fields (E1, H1) and (E2, H2), theysatisfy the Lorentz reciprocity theorem give by ∇ ·(E1 ×H∗2 + E∗2 ×H1) = −iω (E1 ·∆P ∗2 − E∗2 ·∆P1) For∆P1 = ∆P and ∆P2 = 0 and integrating over the resultfor the cross section of the waveguide in question, we get∑ν

∂

∂zAν(z)ei(βν−βµ)z = iωe−iβνz

∫ ∞−∞

∫ ∞−∞

E∗µ·∆Pdxdy

Evoking orthonormality, we can get the coupled-modeequation

±∂Aν∂z

= iωe−iβµz∫ ∞−∞

∫ ∞−∞

E∗ν ·∆Pdxdy

The plus sign indicates forward propagating modes whenBν > 0 and the minus sign indicates a backward propa-gating mode with Bν < 0Many applications are concerned with the coupling be-tween two modes. This coupling between two modes canbe within the same waveguide or can be coupled betweentwo parallel waveguides. For a system where we are inter-ested in coupling two modes for either the parallel waveg-uides case or within the same waveguide, the two modesare described by two amplitudes A and B. The coupledequations are given by ±∂A∂z = iκaaA + iκabBe

i(βb−βa)z

and ±∂B∂z = iκbbB + iκbaAei(βa−βb)z

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