Coupled mode equation

Evaporated PDA(12-8) film was used as a nonlinear optical medium in a layered guided wave directional coupler. The directional coupling phenomenon happens in two adjacent waveguide by periodical energy transfer. The theory of linear directional coupler was exactly established [11]. It can be reduced to coupled mode equations ... 

In this study we suppose nonlinear organic material shows optical Kerr effect as n = n0+n2lEl2 and n2 = X<3)/(2n0). Moreover for simplification, we suppose the waveguides allow single mode propagations and TE polarization. After appropriate handling we get the following nonlinear coupled mode equations [ 12] ... 

Now we can insert expressions for the modal fields and normalization constant into coupled mode equation (13.15)... 

By inserting Eqs. 12 in Eqs. 11 one obtains the coupled mode equations for guided... 

Marcatili, E. A. J., Improved coupled-mode equations for dielectric guides, IEEE J. Quantum Electron., QE-22, 988 (1986). 

The coupled mode theory is often used to describe the polarization characteristics, or modes coupling in optical fibre, and the reflection spectra of FBG sensors. When the optical response of an optical fibre is analysed based on the coupled mode theory, the electric field is normalized as j j E ( E (dxdy = 1. Based on the perturbation approach, the slowly varying amplitudes c are determined by the following coupled mode equations ... 

Coupled mode equations 27-2 Weakly guiding fibers 27-3 Example Weak power transfer... 

Nevertheless, the fields of the perturbed fiber at position z can be described by a superposition of the fields of the complete set of bound and radiation modes of the unperturbed fiber. An individual mode of the set does not satisfy Maxwell s equations for the perturbed fiber, and hence the perturbed fields must generally be distributed between all modes of the set. This distribution varies with position along the fiber and is described by a set of coupled mode equations, which determine the amplitude of every mode [1-3],... 

The set of coupled mode equations satisfied by the modal amplitudes can be derived directly from the scalar wave equation for the perturbed fiber, as we show in Section 33-11, or may be regarded as the weak-guidance limit of the corresponding equations for arbitrary profile fibers, derived in Section 31-11. The latter approach leads to... 

The fiber is assumed nonabsorbing so that the are real, and 4 is the infinite cross-section. In subsequent sections we examine those special situations where an analytical solution of the coupled mode equations can be found. 

When only a small fraction of the total power of the perturbed fiber is transferred between modes, the coupled mode equations can be solved iteratively. The solution of the first iteration is then identical to the induced-current solution of Chapter 22. To demonstrate this equivalence, we assume that only the /th forward-propagating mode of the unperturbed fiber is excited at z = 0 of the perturbed fiber. To lowest order we ignore coupling to all other modes, whence the solution of Eq. (27-3a) is... 

With the help of Table 13-2, page 292, it is readily verified that this is the weak-guidance limit of Eq. (21-2), and, consequently, the first iteration of the coupled mode equations is identical to the induced-current solution. 

Under certain resonance conditions, a small perturbation of the fiber leads to the transfer of a large proportion of total power between only two modes of the complete set of modes of the unperturbed fiber, e.g. resonant coupling on a sinusoidally deformed fiber as discussed below. The coupled mode equations describe arbitrary power transfer exactly, whereas the induced-current method of Chapter 22 is inaccurate, since it assumes only a slight transfer of total mode power. If we ignore the weak power transfer to all other modes, the set of coupled mode equations reduces to just two equations for the resonant modes. 

The corrections to the slow-scale solution of the coupled mode equations, due to fast-scale variations in r, can be included by iterating Eq. (27-17). To determine the first correction, we set... 

In Section 27-1 we showed that in applying the coupled mode equations there is an intrinsic restriction to weakly guiding waveguides. This restriction does not occur in the application of the coupled local-mode equations. A general solution of the coupled local-mode equations is derived for weak power transfer in the following section, and the radiation modes are discussed later in the chapter. 

The inclusion of the local radiation modes in Eq. (28-1) follows by analogy with the coupled mode equations derived in Section 31-11. For example, the amplitude and phase dependence bj (z, Q) of the j th forward-propagating local radiation mode satisfies the coupled local-mode equation... 

To solve Eq. (29-2) we need to know . An expression could be obtained by expanding over the complete set of Pj for the bound and radiation modes of the first fiber, and this would, of course, lead to the infinite set of coupled mode equations derived in Section 33-11. The disadvantage of this description is that each mode of the first fiber by itself is a poor approximation to the field within the second fiber. Consequently large numbers of modes are required for accuracy, and. the set of coupled equations is then intractable. 

The sets of coupled, integro-differential equations of Eqs. (31-50) and (31-51) are an exact restatement of Maxwell s equations for waveguides with continuous profiles. This restriction follows because we have used the expansions of Eq. (31-46). However, the coupled mode equations also apply to discontinuous profiles if we adopt the smoothing procedure described in Section 31-10. The coupled mode equations can be solved by perturbation analysis, provided n = n,for reasons given in Section 31-10. 

The coupled mode equations of the previous section can be derived intuitively. This also provides insight into the physical mechanism of the coupling process. Consider a differential section of the perturbed waveguide of length dz, as shown in Fig. 31-2, and its effect on the k th forward-propagating bound mode. The z dependence of the fields, hi(z) of Eq. (31-45), is expressible as... 

For convenience we only include bound modes in our derivation radiation modes are readily incorporated by analogy with the coupled mode equations of Section 31-11. We express the total transverse field of the perturbed waveguide as an expansion over the complete set of forward- and backward-propagating local modes... 

Relationship with coupled mode equations for arbitrary waveguides... 

---