Compact waveguide taper and waveguide crossing

A tapered waveguide, having:

• a narrow end;
• and
• a wide end,
• the tapered waveguide having:
• a taper angle, at a first point between the narrow end and the wide end, equal to ΩL1−x1, and
• a taper angle, at a second point between the narrow end and the wide end, equal to ΩL2−x2,
• wherein:
• ΩL1=ρ1/LB1 is an adiabatic taper angle at the first point,
• ρ1 is one half of the width of the waveguide at the first point,
• L
• B
• ⁢
• 2
• =
• 2
• ⁢
• π
• β
• 12
• -
• β
• 22
• is a beat length at the first point,
• ρ11 is a propagation constant, at the first point, of a fundamental mode,
• β21 is a propagation constant, at the first point, of a higher order mode with a greatest overlap with the fundamental mode,
• ΩL2=ρ2/LB2 is an adiabatic taper angle at the second point,
• ρ2 is one half of the width of the waveguide at the second point,
• L
• B
• ⁢
• 2
• =
• 2
• ⁢
• π
• β
• 12
• -
• β
• 22
• is a beat length at the second point,
• β12 is a propagation constant, at the second point, of a fundamental mode,
• β22 is a propagation constant, at the second point, of a higher order mode with a greatest overlap with the fundamental mode,
• ΩL1<ΩL2, and
• x1>x2.