With Leak Free Surface Waveguide Mode - Asymptotic Solution Analysis

In this paper, some analysis are provided for the asymptotic solution of the LeakyModes of the three-layer slab optical waveguides with free interface.The wave equation can be obtained from the Maxwell Equation. By FourierTransformation, the Helmholtz Equation is obtained.The original two dimension Helmholtz Equation is:Where (z,x) satisfies, {(z,x)|0＜z＜+∞,-∞＜x＜+∞}.In this paper, their interface is assumed free and at all levels and we study ofheterogeneous media.First, the original problem has a unbounded region, so the numerical method wecan't be used to solve it. So the boundary condition is introduced to obtain a boundedproblem.The problem we discussed has a curve interface, so it is quite complex. Therefore,in order to highlight our asymptotic method, the issue in line interface case is firststudied. On the one hand, in line interface circumstances, accurate characteristicequation can be more easily derived, so more energy can be made on the asymptoticmethod of research; On the other hand, in the curve case, the result of straight-linesituation can be use as a measure of the results of the curve case. In theory, when thedegradation of curve to linear, the case of curve should result consistent with thelinear case.Facing the curve interface, the traditional method can be used as the staircaseapproximation. But it will have big error if we use rough rule. And if the rule is verysmall, the volume of calculation is too much. So it is not an ideal method. Therefore,the local-level orthogonal coordinate transformation is adopted, the bending interfacewill be "straighten", thus enabling the transformation of the problems with linearinterface. Local orthogonal coordinate transformation designs the function: (?)= f(x, z),(?)=g(x, z)which satisfies the coordinate condition and boundary condition, then let u =WV, wehave the equation after transformation: V_?+αV_?+βV_?+γV=0.Then, By the variable separation method, characteristic equation can be obtained: aΦ″(?)+βΦ′(?)+(γ-λ²)Φ(?)=0,As our borders curve fluctuation is very small, that can be approximatedβ≈0, Thischaracteristic equation was similar transformated into:αΦ″(?)+(γ-λ²)Φ(?)≈0.Then, WKB method is used to deal with this characteristic equation. SetΦ(?)=A(?)e^iφ(?), incorporated into the equation, under the assumptionthe approximate solution of the characteristic equation can be obtained, theapproximate solution containing two unknown parameters, of course containeigenvalues. Since we have a four-tier division, the transformation is not the same asthe approximate solution containing the unknown parameters is not the same, Thiswill have eight unknown parameters. Eight equation is needed to determine them. Inaddition to six interface and boundary conditions, we also coupled with an interface.This condition is in fact very evident. After this eight equations being treated, we havea more complicated equation only with eigenvalue. Fortunately, the eigenvalueequation is very similar with the eigenvalue equation in the form of linear interface,We look straight interface approach for dealing with it. In order to make the finalasymptotic solution has a relatively good form, we do a certain approximation, butthese similar lead to a very small error, so it is feasible.After asymptotic solution is derived, we inspected it in the limit cases with linearcoherence, on the other hand, the error between asymptotic solution and accurate solution is inspected. Finally, we find that the former consistency is very good, thelatter error is very small when the fluctuation in the interface is very small. When theinterface more volatile, a big deviation than we desired, Perhaps this is deduced in thecourse of a series of similar treatment.