Codirectional light propagation in
Suppose a composite longitudinally homogeneous waveguide is to be simulated. Its permittivity profile shall be such that the propagating light E, H may be reasonably assumed a superposition of a number of known guided mode profiles, Em, Hm with longitudinally varying amplitudes Cm:
The factors Pm normalize the members of the superposition.
Given the basic modes (propagation constants and profiles) and the total underlying structure, coupled mode theory predicts the following dependence of the coupled mode amplitudes on the propagation distance:
This can be interpreted as follows: A fraction ams of the basic mode number m participates in supermode number s (a superposition of basic modes), which propagates strictly harmonically with propagation constant betas. The supermodes are normalized by factors Ps and excited with amplitudes As. If one defines the supermode profiles Es, Hs by
and with the two equations above inserted into the ansatz at the beginning, the total guided field can be rearranged as:
Here the interpretation of the superposed fields as supermodes becomes evident.
The application cmt.c, a simulation of two parallel raised strip waveguides, shall serve to illustrate the WMM procedures related to coupled mode theory. All functions are encoded in wmmcmt.h and wmmcmt.c.
The calculations that establish the coupled mode theory model are performed by the procedure
defined in wmmcmt.h and wmmcmt.c. Arguments are the basic modes, collected in a WMM_ModeArray, and the combined permittivity profile, a Waveguide object. On completion, the procedure sets the following reference arguments, double matrices or vectors as defined in matrix.h, matrix.c:
sigma A symmetrical matrix of normalized overlap integrals (lscalprods) of the basic modes. The matrix is to be reused by some of the following procedures. smpc The supermode propagation constants, collected in a double vector. By construction, the number of supermodes equals the number umode.num of basic modes (all indices range from 0 to umode.num-1 !). smav The amplitude fractions ams, as introduced above. The column smav(?, s) corresponds to the amplitude vector a?s for supermode number s.
The example cmt.c investigates a symmetrical coupler segment composed of two separate waveguides:
The simulation starts with computing the single mode (for fixed polarization) of one isolated waveguide. The mode is then duplicated and the two copies translated to the positions of the coupler ports, with the two port modes combined into a mode array. This WMM_ModeArray and the Waveguide analogon of the structure shown above constitute the input to CMTsetup. The function responds with the line
* WMM-CMT setup (2) - OK.
in the stderr output. Afterwards, the supermode propagation constants are accessible by the two entries of smpc, and with them the coupling length as the reciprocal of their difference.
While the former procedure computes the supermode propagation constants and the amplitude fractions, an explicit assembly of the mode profiles needs a call to
wmmcmt.h and wmmcmt.c contain the code. The function fills its last reference argument, a WMM_ModeArray, with the supermode information. The supplied Waveguide serves as the wg member of the new WMM_Modes; the propagation constants are taken from the vector smpc as output from CMTsetup. Symbolic superpostions of all trial functions of the basic modes constitute the mode profile of the new supermodes.
For the cmt.c example, the stderr output of CMTsupermodes
* WMM-CMT supermode-composition (2)
* -> sm(0): beta: 9.8337, neff: 2.03461, npcB: 0.324939
* -> sm(1): beta: 9.81473, neff: 2.03068, npcB: 0.309558
informs about the supermodes propagation constants. While the coupled mode approximation naturally enters the new mode object (the wave equation relating fields and the propagation constant holds only approximately), from the programmers viewpoint the two entries of the mode array are ordinary WMM_Modes. Hence all visualization routines may be applied. Surface plots of the dominant component show a symmetric and an antisymmetric TM mode, as expected:
(Here the raw MATLAB output obviously requires some polishing.)
Internally, the supermodes are represented as common WMM_Modes, i.e. symbolically. Each supermode object requires the same amount of memory as the underlying basic modes together. Analogously, any operation on the supermodes needs the time for performing the operation separately on each of the basic modes (e.g. for a field evaluation) or a higher order of this duration (e.g. in case of a field overlap). Therefore, if possible, one should avoid constructing the supermodes explicitely. The next sections show that many useful information about the light propagation can be obtained without the supermodes at hand, i.e. on the level of the coupled mode theory field representation in terms of basic modes with z-dependent coefficients Cm.
The sources wmmcmt.h and wmmcmt.c define the following functions for evaluating the coupled mode field. All functions assume the basic modes to be normalized, i.e. as output from the mode solver:
At present, the above functions refer to the local intensity, the longitudinal component of the time averaged Poynting vector only. If other field components are of interest, the mode interference and interference visualization capabilities of the supermode WMM_ModeArray may be employed. This applies to other input configurations as well. As long as only few calculations are required (cf. the remarks at the end of the previous section), that is the most convenient way. In particular, the MATLAB output code is accessible directly in this way.
CMTsz Given the set of basic modes, the index of the input mode (assuming an input power of 1 W), the output of CMTsetup, and a point specified by three coordinates (x, y, z), the function returns the local intensity of the interference field at that position. CMTprop
For a fixed x-elevation, the functions evaluate the local intensity of the coupled mode field on a rectangular regular mesh in the y-z-plane. The last reference argument returns a Dmatrix of intensity values. CMTprop assumes one of the basic coupled modes to be excited at the beginning z=0 of the coupling segment with a unit amplitude. CMTprop2 computes the coupled mode field, if power is inserted at the same time in two of the basic modes with arbitrary complex amplitudes.
The cmt.c example applies the WMM_ModeArray.prop procedure to write a m-script for displaying a few coupling lengths. The MATLAB output looks like this:
Referring again to the three-segment-coupler configuration, as discussed in the framework of mode interference and visualization, a common configuration consists of a multi-waveguide coupler segment of finite length between two input and output segments, where the modes of the input/output waveguides enter the coupled mode description of the coupling segment. See e.g. the paper on radiatively coupled waveguides as an example. Once the coupled mode theory has been established for the coupling segment, the power throughput from one or more of the coupled modes at the input z=0 to other members of the coupled mode set at the output can be computed very efficiently.
The files wmmcmt.h and wmmcmt.c supply the following functions for this purpose:
Note that in the current framework the coupled modes are not necessarily orthogonal. Therefore the output of CMTamp and CMTpower is not directly the (square of) the relevant z-dependent coefficient Cm (overlap integrals from two waveguide junctions enter).
CMTamp Computes the relative (complex) output amplitude for a specified mode, given an input in another one of the coupled modes. Besides the indices of the relevant modes, the function takes the output of CMTsetup and the device length as arguments. CMTpower
As before, but for the relative output power, in the case that one of the coupled modes is excited with unit input power (CMTpower), or for two of the basic modes excited with arbitrary complex amplitudes (CMTpower2).