Bend modes: A collection of 2-D example results
The bent slab waveguide structure is parametrized in terms of the refractive indices ns, nf, nc in the interior of the bend, in the core region, and of the cladding material, the bend radius R, measured from the origin to the core center, the core thickness d, and the angular frequency ω = k c = 2 π c/λ of the light, specified in terms of the vacuum speed of light c, vacuum wavenumber k, and vacuum wavelength λ. Data for the simulations on this page (see ): ns=1.6, nf=1.7, nc=1.6, d=1.0 µm, λ=1.3 µm.
Ansatz for the optical electric field
The single nonzero component Ey of the TE polarized fields
considered here is given in
in polar coordinates x=r cos φ,
z=r sin φ as
Ey(x, z, t) = Re E(r) exp(iωt -i(β-iα)Rφ),
where E(r) is the (complex) mode profile, corresponding to the angular phase propagation constant β and to the attenuation constant α.
Results, parameters for the fundamental bend modes
The table summarizes the results of (semi-) analytic evaluations of the bend mode properties in terms of Bessel functions, based on the routines of the MAPLE system:
|R / µm||∞||200||150||100||50||10|
|β / k||present||1.66034||1.66049||1.66061||1.66096||1.66303||1.69071|
|α / k||present||0||5.066 10-11||1.020 10-8||1.987 10-6||3.309 10-4||2.518 10-2|
|||5.08 10-11||1.02 10-8||2.66 10-5||3.31 10-4|
|Profiles||eps||Re Im |.| |.| arg||Re Im |.| |.| arg||Re Im |.| |.| arg||Re Im |.| |.| arg||Re Im |.| |.| arg||Re Im |.| |.| arg|
The data entries in the row 'Profiles' are linked to ASCII data files with the mode profiles sampled at 801 points around r=R: The columns specify the radius r and the real and imaginary part of the mode profile E, respectively, where the profile has been scaled to a maximum absolute value of one, with the phase adjusted such that E(R) is real and positive. Plots of these curves in encapsulated postscript format are connected to the entries in the 'eps' columns. The figures 'Re Im |.|' show the real part (dashed), the imaginary part (dash-dotted), and the absolute value (continuous); the files linked to '|.| arg' display the absolute value |E| and the phase argE of the mode profile.
The icons in the 'Fields' column lead to visualizations of the light propagation around these bends. The animated-gif pictures (1-1.5 Mb each) show a sequence of 30 snapshots of the time dependent physical (i.e. real) electric field, equally distributed over one time period. White and black regions indicate positive and negative values, the blue color marks the zero level.
Dependence of the propagation constant β and the attenuation constant α on the bend radius R: Figure (eps format, lines: the present results, circles: values from ) and data file (present results; entries: R [µm], β/k, and α/k in the first, second, and third column).
 C. Vassallo, Optical waveguide concepts, Elsevier, Amsterdam, 1991.