The Modal Transmission

The FDTD method is not a modal numerical method, therefore it is difficult to get information on the modal features of the calculated transmission. On the other side, it is as well difficult calcu-lating the transmission of a single bend with modal expansion methods. In the FDTD method, the transmission is calculated detecting the outgoing power along a line perpendicular to the waveg-uide axis. The power is averaged along that line and divided by the incident power. No modal information is preserved when the power is averaged. In order to retain full information on the modal composition of the outgoing power, one should project the detected fields on the guided modes of the straight waveguides, which implies the knowledge of the profile of all guided modes for all frequencies. This procedure is time consuming and not easy to implement. However, one can avoid to calculate the modal transmission for all modes as long as the main interest concerns the transmission into the fundamental mode. Assume that the incident power is mainly in the fundamental guided mode, that is true for frequencies not in the mini-stop band region. The bend couples the fundamental guided mode to the other modes of the waveguide. After the bend, each mode propagates independently along the straight channel. At the exit one can thus know which modes were excited by the bend. It is important to know how much power is preserved into the fundamental mode and consider the others as lost. For this piece of information it is not necessary to calculate the profile of every mode. The outgoing fields are projected onto their even/odd com-ponents with respect to the waveguide axis and the even/odd Poynting vectors are subsequently calculated. In practice, the FDTD source and detector are swapped with respect to each others:

the source is place in front of the tilted section and the detector is in front of the waveguide section whose axis is aligned with the x direction. The trick facilitates the decomposition into even and odd fields. Even/odd transmission is the first step to estimate the modal mixing.

The second step is to extract the power of the fundamental guided mode from the even transmis-sion. This part is not yet optimized. At the moment it is only possible to show contour plots of the Poynting vector where the x−y plane represents the frequency and the distance from the waveguide axis. The profile of the Poynting vector provides to some extent details on the modal

composition of the even transmission: a Gaussian profile suggests high transmission into the fun-damental guided mode. The projection onto the funfun-damental guided mode is feasible and will be done in the near future. In summary, with the even/odd decomposition the conversion of the fundamental mode into even and odd modes is calculated. The spectra are normalized with respect to the incident power. Secondly, with the contour plots, one roughly knows how much power goes into the fundamental mode. Here, the Poynting vector, corresponding to spatially even modes, is weighted with a gaussian function exp(−(ω−ω_◦)²/σ²), to account for the spectral profile of the incident pulse, and it is normalized to unity.

0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

a/λ

0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

a/λ

Figure 3.18 Modal transmission for a three-holes-moved bend (a) and a ten-holes-moved bend (b). The black line refer to the total transmission. The red (blue) line corresponds to the trans-mission into spatially even (odd) modes with respect to the waveguide axis. Parameters: ²= 10.5,f = 35%, and ²⁰⁰ = 0.1.

Following the above directions, the transmission into even and odd modes has been calculated for the bend designs discussed in the previous section. Two representative cases are shown in Fig. 3.18, where the total transmission (black curves) is also shown⁴. First of all, there are transmission peaks where the contribution of odd modes is important, specially for the case of p = 3. These peaks, which are apparently suitable for wave propagation if one looks only at the total transmission,

4The spectra are slightly different form those of Figs. 3.16 and 3.17, because they have not been stretched.

must be excluded, because a relevant portion of the power is “lost” into odd modes. In general, the bends with larger p exhibit less coupling with odd modes, as confirmed by a comparison between Fig. 3.18a and Fig. 3.18b. Moreover, for any design, it turns out that the odd transmission is zero for a narrow frequency window close to a/λ = 0.24. This is not because the bend does not couple the fundamental to the odd modes, but because the W3 waveguide has a mini-stop band in the first odd mode, as displayed in Fig. 3.14a. Therefore, the propagating field cannot have a component into odd modes and transmission may occur into even modes only. The fact that the even transmission is maximum close to this mini-stop band has to be attributed to resonant scattering, rather to the above cited mini-stop band.

Given that the best even transmission is obtained for a/λ ' 0.24, for knowing how much power remains to the fundamental mode, one has to analyze the distribution of the Poynting vector along the linear detector. Fig. 3.19a and Fig. 3.19b show contour and surface plots of the

Poynting-(a)p= 3 (b)p= 10

Figure 3.19 Poynting vector as a function of frequency and distance from the waveguide axis for the transmission into spatially even modes. y=1 corresponds to y = 3√

3/2a. (a) three-holes-moved bend; (b) ten-holes-moved bend. Param-eters as in Fig. 3.18.

vector profile for the bend designs corresponding top= 3 andp= 10, respectively. Thexaxis refers

to frequency, whereas they axis represents the distance from the waveguide axis. Both designs are characterized by a main transmission contribution of the fundamental mode fora/λ∼0.24, where the higher-order even mode has a small group velocity. Looking at the whole band gap, notice that the are regions where the transmission goes primarily into higher-order modes.

In conclusion, analyzing the modal transmission provides a better understanding of the mode mixing occurring at the bend transition. In some cases, the presence of mini-stop bands may reduce the number of modes, which necessarily results in a minor mode scrambling, without changing the mode-coupling coefficients associated to a certain bend design. A particularly favorable situation is represented by the mini-stop band between the odd modes at a/λ ' 0.235, which overlaps to a resonant scattering condition for even modes. There, only two even modes remain for guiding light to and from the bend, namely the fundamental and the next higher-order mode. If one could eliminate also this higher-order mode, single-mode transmission would be achieved. In other words, once identified the frequency region where the waveguide supports the smallest number of guided modes, including the fundamental mode, the idea is to work on the bend design to eliminate only those mode-coupling coefficients, which are essential for attaining single-mode transmission.

However, even if a few coupling coefficients are involved, it is difficult to know how they depend on the bend geometry.