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@article{lu2002comment,

title = {Comment on},

author = {WT Lu and JB Sokoloff and S Sridhar},

year  = {2002},

date = {2002-01-01},

journal = {arXiv preprint cond-mat/0207689},

abstract = {Valanju, Walser and Valanju (VWV) [1] have shown

that for a group consisting of two plane waves incident

on the interface between a material of positive refractive

index (PIM) and material of negative refractive index

(NIM), the group velocity refracts positively. Here we

show that this is true only for the special two plane wave

case constructed by VWV, but for generic localized wave

packets, the group refraction is generically negative.

The sum of two plane waves of wavevector and frequency (k1, ω1) and (k2, ω2), considered by VWV, can

be written as 2e

i(k0·r−ω0t)

cos[(1/2)(∆k · r − ∆ωt)],

where (k0, ω0) the average wavevector and frequency and

(∆k, ∆ω) denote their differences. Clearly, the argument

of the cosine is constant along planes, which propagate

in time along the direction of their normal, ∆k. We

have carried out numerical simulations of wave packets

incident on the PIM-NIM interface and for the 2-wave

case arrive at conclusions similar to VWV. For arbitrary

number of incident plane waves whose k vectors are all

parallel, the group refraction is again positive. Note that

in all these special cases the packet remains nonzero on

infinite planes.

Here we show that for any wave packet that is spatially

localized, the group refraction is generically negative.

For 3 (or more) waves whose wave vectors not aligned,

the group refraction will be negative. For example, consider three wave vectors in PIM in the x-z-plane, whose

magnitudes are, k − δk, k, k − δk and whose angles with

the z-axis are, θ − δθ, θ, θ + δθ, respectively. The dispersion k

2 = (ω

2 − ω

2

p

)(ω

2 − ω

2

b

)/c2

(ω

2 − ω

2

0

) were used for

NIM. The results are shown in Fig. 1. The wave packet

refracts negatively, in obvious contrast to VWV. As we

have seen, two plane waves result in a wave packet-like

structure which is constant along planes; the addition of

a third wave breaks the planes into localized wave packets

which refract negatively.

A packet constructed from a finite number of plane

waves will always give a collection of propagating wave

pulses, as seen in Fig. 1. A wave packet localized in one

region of space, as occurs in all experimental situations,

can only be constructed from a continuous distribution of

wavevectors. Consider such a wave packet incident from

outside the NIM, E = E0

R

d

2kf(k − k0)e

i(k·r−ω(k)t)

,

where ω(k) = ck. If f(k − k0) drops off rapidly as k

moves away from k0, ω(k) can be expanded in a Taylor

series to first order in k − k0 to a good approximation.

This gives, E = E0e

i(k0·r−ω(k0)t

g(r − ctk0/k0), where

g(R) = R

d

2kf(k−k0)e

i(k−k0)·R. Inside the NIM, k and

k0 in the argument of the exponent get replaced by kr

and kr0 which are related to k and k0 by Snell’s law.

Then the wave packet once it enters the NIM is given by

Er = E

′

0

e

i(kr0·r−ω(kr0)t

gr(r − vgrt), (1)

where gr(R) = R

d

2kf(k − k0)e

iR·[(k−k0)·∇kkr]

. Here

kr0 denotes kr evaluated at k = k0 and vgr = ∇kr ω(kr)

evaluated at kr = kr0. Thus, the refracted wave moves

with the group velocity vgr. Evaluation of Eq. (1) for

a Gaussian wave packet shows that the incident packet

gets distorted but the maximum of the packet moves at

vgr. For the case of an isotropic medium, considered by

VWV [1], the group velocity is anti-parallel to the wave

vector in the medium. Hence, the group velocity will be

refracted the same way as the wavevector is, contrary to

the claim of VWV [1].

Thus VWV’s statement that the “Group Refraction is

always positive” is true only for the very special (and

rare) wave packets constructed by them and is incorrect

for more general wave packets that are spatially localized.

This work was supported by the National Science

Foundation, the Air Force Research Laboratories and the

Department of Energy.

W. T. Lu, J. B. Sokoloff and S. Sridhar

Department of Physics, Northeastern University, 360

Huntington Avenue, Boston, MA 02115.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}