PCs waveguide discontinuities

Figure 1: The basic idea of the analysis of waveguide discontinuity is presented. The "defect" of waveguide structure is isolated by two
(or more) fictitious boundaries which will be used for field matching between our discontinuity structure and the eigenfields 
of connected waveguides. A reflection and transmission properties of the waveguide discontinuity are characterized by 
the transmission (T) and reflection coefficients (R) (which can be defined by means of the field amplitude or by means of the field power
for unit excitation (E)).   

Figure 2: The connection concept for a waveguide discontinuity analysis using MMP is described. At the beginning, the MMP
eigenvalue solver is used for eigenfield determination of PCs line defect waveguide at certain frequency. Having a such eigenfield
we are able to pack and store this information by using corresponding multipolar expansions. This expansions package is called
"connection". The waveguide discontinuity is introduced by a simple displacement of a single dielectric rod. At a given frequency 
it is necessary to compute eigensolution of the infinitely long waveguide. Based on this solution we will define three different 
connections with an appropriate positions and rotational angles in order to have a correct representation of the input, reflected, 
and transmitted electromagnetic power. As a result of field metching along the fictitous boundaries, using MMP apprach,
we are able to compute unknown complex amplitudes for a corresponding connections, i.e. corresponding reflection (R) and 
transmission (T) coefficients in the case of unit excitation (E).      


Figure 3: The infrared light propagation along the PCs waveguide defect structure (left hand side) and the time average Poynting vector
(right hand side) are presented. As one can see, our simple discontinuity in waveguide structure produces a very large reflection, R=82%,
and low transmission, T=18% (R and T are defined by means of power). The most important question of this connection approach 
for the waveguide discontinuity analysis is how far from discontinuity our fictitious boundaries should be placed. The distance between 
the discontinuity and the termination must be large enough in order to prevent that the evanescent field (produced by discontinuity) reach a port, i.e. fictitious boundary. Otherwise, this evanescent field will produce a significant field mismatching along the termination boundary, because of the fact that our connections consist only eigenfield informations. According to the analysis performed by
Esteban Moreno this distance should be at least one effective wavelength of our line defect waveguide at corresponding frequency. This effective wavelength
can be easily found by using dispersive relation of the line defect waveguide.

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