Valanju, Walser, Valanju Reply to Pendry-Smith Comment on "Wave Refraction in Negative-Index Media: Always Positive and Inhomogeneous"

1 Pages·0.15 MB·English

by  P. M. Valanju

1
Valanju et. al. (VWV) reply: The central argu- inhomogeneous waves in their Figures 2 and 3. Vese-
ment of Pendry and Smith (PS) [1] is: “VWV correctly lago considered only monochromatic refraction, which
calculate positively refracted modulation fronts at a vac- by definition contains no modulation to misalign with
uum/NIM interface, but wrongly interpret their normal ~v . Neither PS’s Eq.2norVeselago’smonochromaticray
p
as the direction of group velocity ~v . Both the phase ve- diagrams represent inhomogeneous waves with θ 6= π
g pg
locity ~v and ~v refract negatively so the angle between because they neglect dispersion.
p g
them is θpg = π. The propagation of the front is crab- AlthoughallequationsinRef.[3] arecorrect,the inad-
wise, and antiparallel to ~vp, as required by Veselago.” vertentuseof“wave”for“wavefront”inthetitle,andthe
However, PS do not show how θpg = π, or Veselago’s association of ~vg with energy flux in the first paragraph
strictly monochromatic refraction [2], can yield “crab- of Ref.[3], may have caused the PS Comment. However,
wise”waves,whicharenothingbuttherigorouslyderived neither the central result (wave inhomogenization), nor
inhomogeneous waves of VWV [3] with θpg 6=π. any equations of Ref.[3], are disproved by PS. The key
PS’s use of Eq.2 to imply θpg =π is incorrect because issue here is wave inhomogenization that invalidates ray
3 it neglects the non-collinearity of refracted ~k(ω) at dif- diagrams,notthedirectionof~vg. Causalityimpliesthat,
0 ferent ω due to dispersion. PS do not derive the value
0
of θ , and their assertion, that ~v must be either par-
2 pg g
allel or antiparallel to ~k in an isotropic NIM, is wrong
n because there is no unique ~k(ω) after refraction at even
a
J an isotropic, but dispersive, NIM. The direction of~k(ω)
3 can not be both independent of ω as in their Eq.2, and
yet depend on ω as required to generate inhomogeneous

1 wavesintheirFigures2and3. Infact,thesefiguresagree
v
with the wave inhomogeneity first shown in VWV, but
0
not with PS’s Eq.2 or Veselago’s rays which imply un-
3 FIG. 1: Intensity of wave with three ω components, each
10 pPhSysaiscsaolc,ianteega~vtgivweliythrethfreacPtoedynhtionmgovgeecnteooruP~s,wbauvtefVroWntVs. fwroitnhtstw(osotlriadn)svsteirlsler±efkra(ωct).pTorsaitnisvveelyrseinlyNchIoMp.peTdhmeyodaurleatnioont
/030 cswhoanovwseeseqduatetnhataltyPtIP~hMe)-rNNeIIfMrMacditniistnperedfraisffcioeernbencetrcea(au±tse)esd~viingrheacontmidoon~gvspe.n(eaonuds nd(doeacramshyaealdn)t,oomit.eha,leoθu~vpspgly(6=dfaoπsttt.,edTe)vheoenrrfPeosrouyilnntifitninngigtien(shPi~momaolroIg~vmeEn[)neopdu(iωrse)cw]t.aiovness
at Fig.3 of PS is underived: collinear ~k’s in Eq.2 can-
m not give wave inhomogeneity (θ 6= π). In contrast, by
pg
- addingtransverse±kcomponentstoeachk(ω)inEq.1of afterrefraction,~vg differsfrom~vp andenergyvelocity~vE,
d Ref.[3], we derive our Fig.1. It shows that static, trans- causing anomalously rapid, dispersive wave decay [4].
n versespatialfeaturesdonotqualitativelychangeVWV’s Insummary,PS’sneglectofdispersionattheinterface
o results. The wave modulation fronts are still oriented in (by assuming a unique ~k(ω)) makes their Eq.2 wrong,
c
: thepositivedirection,thewavesarestillinhomogeneous, irrelevant, and inconsistent with their underived Figures
v and hence decay rapidly. Static effects of transverse ±k 2and3. Instead,theirfiguressupportVWV’sderivation
i
X are unaffected by causality because they carry no time- of wave inhomogenization by oblique refraction in NIM.
varying signal. Hence they were correctly omitted from
r
a VWV to present with clarity the first theory of causal
P.M. Valanju, R.M. Walser, and A.P. Valanju
signal refraction by NIM.
The University of Texas at Austin.
PS’s conclusion, that Veselago was correct in saying
that~v is antiparallelto~v (θ =π), disagreeswith the
g p pg
[1] J.B.PendryandD.R.Smith,precedingComment,Phys. Rev.Lett. 88, 187401 (2002).
Rev.Lett. [4] P.M.Valanju,R.M.Walser,andA.P.Valanju,submitted
[2] V. G. Veselago, Sov. Phys.Usp. 10, 509 (1968). to Phys.Rev. Lett.(July 2002).
[3] P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys.