Page 345 - Introduction to Information Optics

330 6. Interconnection with Optics

Fig. 6.30. Output profiles from the 45" surface-normal microcoupler. (a) z = 100 pm (experiment),
(b) z = 1 mm (experiment), (c) z - 5mm (experiment), (a') z = 100/<m (theory), (h') z — I mm
(theory), (c') z = 5mm (theory).

where U(£, r\) is the complex amplitude of the excitation at point (£, ?j) on the
45° microcoupler, U(x, y) the complex amplitude of the observed field at point
(x, y), k the magnitude of the wave vector, and z the distance from the upper
surface of the 45° microcoupler to the point of observation. However, direct
integration based on Fresnel approximation fails for a small z due to the fast
oscillations of the Fresnel factor. A modified convolution approach was used
to calculate the U(x, y) and output profiles. Theoretical output profiles at
z — 100 /im, z = 1 mm, and z = 5 mm from the 45° microcoupler are shown in
Fig. 6.30(a'), (b'), and (c'), respectively. In these calculations, the input to the
microcoupler was assumed to be the fundamental mode of the waveguide. This
is a reasonable assumption, since the most of the energy in a multimode
waveguide remains confined to the fundamental propagating mode. Note that
the input is in the TEM 00 mode. Compared to Fig. 6.30(c'), the side lobes in
Fig. 6.30(c) are not clear due to the poor contrast of the image. As z becomes
larger, the output light diverges faster in the direction that corresponds to the
smaller dimension of the 45° microcoupler. There is a good agreement between
the theoretically simulated and experimentally observed output profiles.
Figure 6.31 shows a SEM micrograph of the mirror coupler integrated to
an array of channel waveguides. The measured output coupling efficiency is