Remarks on the projection operation

Admittedly, at the first glance the projection operation might appear redundant, since the CMT solution in the form (3.3) provides directly amplitudes for the basis fields that occur also in the external field representation (3.16), (3.17). Here perhaps further explanatory remarks are necessary.

The physical field around the exit planes of the CMT window can be seen as a superposition of the outgoing guided modes of the bus core with their constant amplitudes, and a remainder, that, when expanded in the modal basis associated with the straight waveguide, is orthogonal to the guided waves. The notions of ``vanishing interaction'' or ``decoupled'' fields, as used e.g. for the motivation of the assumptions underlying the abstract framework of Section 1.4, are to be concretized in precisely this way: The guided waves in the straight core are stationary, iff projections onto the mode profiles at growing propagation distances lead to constant amplitudes $A_q$, $B_q$ (apart from the phase changes according to the undisturbed propagation of the respective modes).

Now the coupled mode theory formalism is limited to the few non-orthogonal bend and straight modes included in the CMT ansatz, which are overlapping in the regions of the input and exit ports A and B of the coupler. Consequently, when the CMT procedures try to approximate both the guided and radiative part of the real field, the optimum approximations may well be superpositions with non-stationary coupled mode amplitudes $C_{\mbox{\scriptsize s}q}$ of the modes of the bus waveguide. Indeed, as observed in Sections 3.4.1 and 3.4.2, the projected amplitudes $\vert B_q\vert^2$ (or the related scattering matrix elements $\vert{\sf {S}}_{\mbox{\scriptsize s}q,wj}\vert^2$) become stationary, when viewed as a function of the exit port position $z_{\mbox{\scriptsize o}}$, while at the same time the associated CMT solution $\vert C_{\mbox{\scriptsize s}q}(z)\vert^2$ (or the elements $\vert{\sf {T}}_{\mbox{\scriptsize s}q,wj}\vert^2$ of the transfer matrix) exhibit an oscillatory behaviour. Still, in the sense of the projections, one can speak of ``non-interacting, decoupled'' fields. That justifies the limitation of the computational window to $z$-intervals where the elements of S (not necessarily of T) attain constant absolute values around the input and output planes.

In conclusion, it is at least partly misleading to stick to the familiar notion of ``mode evolutions'' computed by the CMT approach. If one abandons that viewpoint and regards the CMT procedures as just ``a'' method that generates an approximate field solution inside the computational window, then applying the projections to extract the external mode amplitudes appears perfectly reasonable.