NIM-comment-2002

Comment on \Wave Refraction in Negative-Index Materials: Always Positive andVery Inhomogeneous"Valanju, Walser and Valanju (VWV) [1] have shown moves away from k , !(k) can be expanded in a Taylor0that for a group consisting of two plane waves incident series to rst order in k k to a good approximation.0i(k r !(k )t0 0on the interface between a material of positive refractive This gives, E = E e g(r ctk =k ), where0 0 0R2 i(k k )R0index (PIM) and material of negative refractive index g(R) = d kf(k k )e . Inside the NIM,k and0(NIM), the group velocity refracts positively. Here we k in the argument of the exponent get replaced by k0 rshow that this is true only for the special two plane wave and k which are related to k and k by Snell’s law.r0 0case constructed by VWV, but for genericlocalized wave Then the wave packet once it enters the NIM is given bypackets, the group refraction is generically negative.0 i(k r !(k )tr0 r0E = E e g (r v t); (1)The sum of two plane waves of wavevector and fre- r r gr0quency (k ;! ) and (k ;! ), considered by VWV, can1 1 2 2 R2 iR[(k k )r k ]0 k ri(k r ! t)0 0 where g (R) = d kf(k k )e . Herer 0be written as 2e cos[(1=2)(k r !t)],k denotesk evaluated atk=k andv =r !(k )r0 r 0 gr k rwhere(k ;! )the averagewavevectorandfrequencyand r0 0evaluated at k =k . Thus, the refracted wave movesr r0(k;!)denotetheirdierences. Clearly,theargumentwith the group velocity v . Evaluation of Eq. (1) forgrof the cosine is constant ...