WMM based coupled mode theory: Adiabatic directional couplers
M. Hammer,
MESA+ Research Institute, University of Twente
The table lists a collection of WMM application files that address
the problem of guided light propagation through a straight (port)
waveguide that is evanescently (vertically or horizontally) coupled to a
ring (cavity) waveguide segment:
Top view of the directional coupler configuration: A segment of a cylindrical waveguide core (WG2) with radius R, evanescently coupled to a straight waveguide (WG1). Letters A, B, a, b denote the ports of the device, p(z) is the local ydisplacement of the ring rim at position z.  Coupler cross section: The port waveguide with core width s, thickness v, and refractive index n_{g} is embedded in a medium with refractive index n_{b}, buried at a distance b below the surface x=0 of that medium. The cavity strip of width w and thickness t with refractive index n_{r} is placed on top of the surface, covered by a material with refractive index n_{a}. The horizontal position of the port waveguide is defined by the ycoordinate g (positive or negative values are to be considered) of the core center. p is the horizontal, zdependent displacement of the ring rim. 
Port waveguide, port.c:
Mode properties of an embedded channel waveguide with rectangular core. Program output: Propagation constants / effective mode indices (stderr stream), mode definition files (.mod), mode profile plots, horizontal and vertical mode profile cross sections (MATLAB mfiles).  
Ring waveguide, ring.c:
Mode properties of a rectangular raised strip waveguide. Program output: Propagation constants / effective mode indices (stderr stream), mode definition files (.mod), mode profile plots, horizontal and vertical mode profile cross sections (MATLAB mfiles).  
Ring waveguide, width dependence, ringdisp.c:
Propagation constants versus the width of a rectangular raised strip waveguide. Program output: Propagation constants (??pc??.xyf) and effective mode indices (??neff??.xyf) in twocolumn ASCII files. First column: the core width, second column: the propagation constant / effective mode index.  

Coupler simulation, cpl.c:
Coupled mode theory analysis of the guided wave propagation along the composite coupler structure. Program output: Properties of the basis modes: Propagation constants and effective indices (stderr), mode definition files (.mod, optional), mode profiles, translated to the core positions at z=0 (MATLAB mfiles). Properties of the supermodes evaluated for the composite refractive index profile at z=0: Propagation constants, effective mode indices, coupling length (stderr), mode profiles (MATLAB mfiles). The coupler scattering matrix with transfer coefficient tau and coupling constant kappa (stderr). Evolution of the amplitudes C_{1} (port) and C_{2} (ring) of the basis modes: For excitation in port A (C_{1}(Z) = 1, C_{2}(Z) = 0) the relative mode powers are stored in files pAB.xyf (twocolumn ASCII files representing pairs z, C_{1}(z)^{2}), pAb.xyf (C_{2}(z)^{2}), and pAB+pAb.xyf (C_{1}(z)^{2}+C_{2}(z)^{2}). For excitation in port a (C_{1}(Z) = 0, C_{2}(Z) = 1) the relative mode powers are stored in files paB.xyf (C_{1}(z)^{2}), pab.xyf (C_{2}(z)^{2}), and paB+pab.xyf (C_{1}(z)^{2}+C_{2}(z)^{2}). 
Coupler simulation, parameter dependences, cplsc.c:
CMT analysis of the coupler structure, scattering matrices versus the relative core position. Program output: Basis mode properties: Propagation constants and effective indices (stderr), mode definition files (.mod, optional). Transfer coefficients tau and coupling coefficients kappa in twocolumn ASCII files. First column: the geometrical parameter (the burying depth b in the original program file), second column: tau^{2} (tau.xyf), kappa^{2} (kappa.xyf), and tau^{2}+kappa^{2} (total.xyf). 
Some of the approximations and assumptions the coupled mode approach is based on are mentioned in Ref. [1]. Additionally, the recipes that were originally derived and implemented for composite, longitudinally invariant structures (see Ref. [3] for details) are here applied to a device with zdependent refractive index profile. Reasonable results can thus be expected only for an intermediate range of core distances (more precise: observations in [3]) and for relatively large ring radii (do not forget to adjust Z accordingly). The specific form of the couplers allows in a certain respect to check the consistency of the simulations:
Strictly speaking, the "power coupling" matrix should appear in the expressions that evaluate the modal power. This becomes relevant for local configurations with "strong coupling", with pronounced mode overlaps. Hence, in particular for intermediate positions (z around 0), the interpretation of the squares of the individual amplitudes of the basis modes C_{j}(z)^{2} (and of their sum) is not straightforward. Nevertheless, assuming "decoupled" waveguides at z=Z and at z=Z, the interpretation of the squared amplitudes as guided output powers is adequate.
For specific configurations, the relations may be pronouncedly violated, as a consequence of the approximations inherent in the present CMT version. However, in these cases we have observed the deviations bewtween the two offdiagonal elements of the coupler scattering matrix to be of the same order of magnitude as the violation of power conservation, where the average of the two coupling coefficients fits relatively well to the  almost coinciding  transmission coefficients, the diagonal elements of the scatter matrix. The programs cpl.c and cplsc.c therefore declare the average of the offdiagonal elements as the approximation of the coupling coefficient. Particular caution is advisable when using these results.
Technical remarks:
Microresonator simulations:
Relying completely on the properties of straight waveguides, these programs have to be complemented by results from a suitable bend mode solver when aiming at a 3D model for the microringresonators that are investigated in the NAIS project. This concerns the mode attenuation in the curved cavity waveguides in the first place. In a second place, available bend mode properties can prospectively help to improve a coupler design found by means of the WMM CMT programs: Given the mode profile and propagation constant of the straight basis field for the ring core, one could try to modify the core geometry and position, such that the bend mode calculated for the new core with the appropriate radius coincides as far as possible with the original straight field. This concerns the position of the field maximum, adjustable by a slight shift of the relative core positions, and the effective mode index, which can be matched e.g. by a change in the cavity core width.
References:
[1]
Coupled mode model for 3D directional couplers,
NAIS internal note, September 2002.
[2]
Standard ringresonator model,
NAIS internal note, April 2003.
[3]
M. Lohmeyer, N. Bahlmann, O. Zhuromskyy, and P. Hertel.
Radiatively coupled waveguide polarization splitter simulated by
wavematching based coupled mode theory.
Optical and Quantum Electronics 31, 877891 (1999).
M. Hammer 