  The coupling between negative and positive index medium waveguides is
investigated theoretically in this paper. A coupled mode theory is developed
for such a waveguide system and its validity is verified. Interesting phenomena
in the coupled waveguides are demonstrated, which occur in the case when the
negative index medium waveguide in isolation guides its mode backward. A new
type of coupled mode solution that varies exponentially with the coupling
length is found in the special case when the propagation constants of two
individual waveguides are nearly the same. A coupler operating in this case is
insensitive to the coupling length, and its coupling efficiency can reach 100%
as long as the coupling length is long enough. However, when the propagation
constants of the two individual waveguides differ greatly, the coupled mode
solution is still a periodic function of the coupling length, but the coupled
power is output backward. In addition, the modes in the composite waveguide
system are also studied using the coupled mode theory, and their fundamental
properties are revealed.
