 ## Coupled mode theory

Codirectional light propagation in

Commonly, the term `mode coupling' adresses one of at least three different means of power transfer. These include coupling modes of distinct waveguides by evanescent fields, coupling modes in the same waveguide by longitudinally homogeneous perturbations, and co- and contradirectional coupling by longitudinally inhomogeneous, usually periodical perturbations. While one of the example files on magnetooptic polarization conversion adresses the second topic, this section is concerned with the first version, i.e. with codirectional light propagation in closely spaced adjacent waveguides. See the paper on radiatively coupled polarization splitters or the theory chapter in the survey for details on the employed formulation of coupled mode theory. However, some remarks on the notion of supermodes in the current framework may be helpful:

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.cpp, 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.cpp.

#### Initialize CMT:

The calculations that establish the coupled mode theory model are performed by the procedure

```    CMTsetup```,

defined in wmmcmt.h and wmmcmt.cpp. 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.cpp:

 `sigma` A symmetrical matrix of normalized overlap integrals (`lscalprod`s) 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.cpp 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 `translate`d 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.

#### Supermodes:

While the former procedure computes the supermode propagation constants and the amplitude fractions, an explicit assembly of the mode profiles needs a call to

```    CMTsupermodes```.

wmmcmt.h and wmmcmt.cpp 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_Mode`s; 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.cpp 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_Mode`s. 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.

#### CMT propagation:

The sources wmmcmt.h and wmmcmt.cpp 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:

 `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` `CMTprop2` 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.

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.

The cmt.cpp example applies the `WMM_ModeArray.prop` procedure to write a m-script for displaying a few coupling lengths. The MATLAB output looks like this: #### Power transmission:

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.cpp supply the following functions for this purpose:

 `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` `CMTpower2` 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`).
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). 