Physics Letters B 760 (2016) 149–152
Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Wave-scattering
from
a
gently
curved
surface
Giuseppe Bimonte
a,
b,
∗
aDipartimento di Fisica E. Pancini, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy bINFN Sezione di Napoli, I-80126 Napoli, Italy
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Article history:Received16May2016
Receivedinrevisedform21June2016 Accepted24June2016
Availableonline29June2016 Editor:M.Cvetiˇc
We studywavescatteringfromagentlycurvedsurface.Weshowthat therecursiverelations,implied byshiftinvariance,amongthecoefficientsoftheperturbativeseriesforthescatteringamplitudeallow to perform an infinite resummationof the perturbative seriesto all orders in the amplitude of the corrugation.Theresummedseriesprovidesaderivativeexpansionofthescatteringamplitudeinpowers ofderivatives ofthe heightprofile,whichis expectedtobecome exactin thelimitof quasi-specular scattering.Wediscusstherelationofourresultswiththeso-calledsmall-slopeapproximationintroduced sometimeagobyVoronovich.
©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Theproblemofwave scatteringatroughinterfaceshasalways attractedmuch attention, due to its importance in diverseareas ofphysicsrangingfromoptics,toacoustics,communications, geo-physics, etc. Its difficulty is universally acknowledged, and even withpresentlyavailablecomputersitrepresentsaformidable chal-lenge in realistic situations. Despiteimpressive progress madein numerical methods, approximate analytical approaches are still much valuable as the only ones capable of providing an insight intogeneralphysicalfeaturesofwavescattering. Itisnot surpris-ing then that a number of diverseapproximation schemes, with differentranges ofvalidity, have been developed over the years. Forareview,weaddressthereadertothemonograph[1]ortothe recentreview [2].Historically,the first andperhaps mostwidely knownapproximatetheory of scatteringbyrough surfaces isthe
smallperturbationmethod (SPM) originally developed by Rayleigh tostudyscatteringofsoundwavesbysinusoidallycorrugated sur-faces of small amplitude, and later generalized by several other authorstoelectromagneticscattering.Anotherclassicapproximate schemeistheKirchhoffapproximation (KA),alsoknownasthe tan-gentplane approximation(TPA), whichrepresents a valid approx-imationfor locallysmooth surfaces withlarge radii ofcurvature. A schemeaimingatreconcilingSPMandKAwasproposedtwenty years ago by Voronovich [1,3], i.e. the small-slopeapproximation
(SSA).It consistsof an ingeniousansatz for the scattering
ampli-*
Correspondenceto:DipartimentodiFisicaE.Pancini,UniversitàdiNapoli Fed-ericoII,ComplessoUniversitariodiMonteS.Angelo,ViaCintia,I-80126Napoli,Italy.E-mail address:bimonte@na.infn.it.
tude(SA),thatmanagestocapturetheleadingcurvaturecorrection to theKA.The ansatz involves unknowncoefficients thatare de-termined aposteriori by matching theSSA withthe SPM intheir commonregionofvalidity. Numericalinvestigationsrevealedthat withinitsdomainofvaliditytheSSAisindeedremarkablyaccurate
[2].Severalauthorshaveattemptedtoproviderigorous mathemat-icalderivationsoftheSSA,utilizingtheextinctiontheorem[4,5]or theMeecham–Lysanovmethod[6].
Inrecentyearsithasbeenshown[7–9]that curvature correc-tions to the Casimir interaction between two gently curved sur-faces can be estimatedusing a derivative expansion (DE) of the Casimir-energy functional, in powers of derivativesof the height profiles ofthesurfaces.TheDE hasbeenlaterusedto study cur-vatureeffects intheCasimir–Polder interactionofa particlewith a gentlycurvedsurface [10,11].The samemethodhasbeenused veryrecentlytoestimatetheshiftsoftherotationallevelsofa di-atomicmoleculeduetoitsvanderWaalsinteractionwithacurved dielectricsurface[12].Animportantinsightintothenatureofthe DE wasachievedin[13] (see also[10]) whereit wasshownthat the derivative expansion amounts to an infinite resummation of theperturbative seriesfortheCasimirenergytoallorders inthe amplitude of the corrugation, forsmall in-plane momenta of the electromagneticfield.Thesemethodscanindeedbeadaptedtothe problemof wave scattering by a rough surface. In this letterwe demonstratethat theinfiniterecursiverelationsamongthe coeffi-cientsoftheperturbativeseriesfortheSA,engenderedbyitsexact invarianceunderverticalandhorizontal shiftsofthesurface,allow toperformaninfinite resummationoftheperturbativeseries,toall ordersintheamplitudeofthecorrugation.Theresummation pro-cedure resultsintoa DEoftheSAinpowersofderivativesofthe
http://dx.doi.org/10.1016/j.physletb.2016.06.058
0370-2693/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
150 G. Bimonte / Physics Letters B 760 (2016) 149–152
height profile,whichis expectedto becomeexact inthe limit of quasi-specularscattering.WeshowthatourDEoftheSAisindeed equivalenttoVoronovich’sSSAansatz,thusprovidingaformal jus-tificationforit.
2. PerturbativeexpansionoftheSA
We consider a surface
separating two media of different (optical or acoustical) properties. A cartesian coordinate system
(
x,
y,
z)
is chosen such that the z-axis is directed upwards in the direction going from medium 2 towards medium 1, while(
x,
y)
span a reference plane orthogonal to the z-axis. It is as-sumedthatthesurfacecanberepresentedby a(single-valued)
smooth height profile of equation z
=
h(
x)
=
h(
x,
y)
. A down-wardpropagating (electromagnetic oracoustic)wave E(αin0)(
x,
z)
=
E(α00)
/
√
q0 exp
[
i(
k0·
x−
q0z)
]
with wave vector K0= (
k0,
−
q0)
, wavenumber K0=
2π/λ
01 is incident on the surface. The indexα
0takesvaluesinadiscretesetconsistingofoneortwoelements, dependingonwhetheranacousticorelectromagneticwaveis con-sidered.ThescatteredfieldE(sc)(
r,
z)
atpointsaboveandfarfromcanbeexpressedthroughtheSASαα0
(
k,
k0)
asasuperpositionsofupwardspropagatingwaveswithwavevectorK
= (
k,
q)
as:E(αsc)
(
x,
z)
=
d2k 4π
2 1√
qei(k·x+qz)Sαα0(
k,
k0)
E (0) α0.
(1)TheSAsatisfiestwogeneralproperties.Thefirstoneisreciprocity:
Sαα0
(
k,
k0)
=
Sα0α(
−
k0,
−
k) ,
(2)whichfollowsfrommicroscopicreversibility[14].Thesecond gen-eralpropertysatisfiedbytheSAisshiftinvariance,whichamounts tothe following transformationproperty oftheSA undera hori-zontalandverticaldisplacementofthesurface
:
Sαα0
(
k,
k0)
|h
(x−a)−b=
ei[(k0−k)·a−(q0+q)b]S
αα0
(
k,
k0)
|h
(x).
(3)According to the SPM, we postulate that for sufficiently small heightprofilesh theSAcanbeexpandedasapowerseriesinthe heightprofile: Sαα0
(
k,
k0)
=
n≥0 1 n!
d2x1· · ·
d2xn×
G(ααn)0(
x1, . . . ,
xn;k,
k0)
h(
x1) . . .
h(
xn) ,
(4)wherethe kernels G(μνn)
(
x1,
. . . ,
xn;
k,
k0)
are symmetric functions of(
x1. . .
xn)
.In momentumspace, the perturbative expansion oftheSAreads: Sαα0
(
k,
k0)
=
n≥0 1 n!
d2k1(
2π
)
2· · ·
d2kn(
2π
)
2× ¯
G(ααn)0(
k1, . . . ,
kn;k,
k0) ˜
h(
k1) . . . ˜
h(
kn) .
(5)Shiftinvariance under a horizontal translation of the profile h
(
x)
implies that the kernels G
˜
(ααn)0(
k1,
. . . ,
kn;
k,
k0)
must be of theform:
¯
G(ααn)0
(
k1, . . . ,
kn;k,
k0)
= (
2π
)
2δ
(2)(
k1+ . . .
kn+
k0−
k)
× ˜
G(ααn)0(
k1, . . . ,
kn;k,
k0) ,
(6)where G
˜
(ααn)0(
k1,
· · · ,
kn;
k,
k0)
are symmetricfunctionsof the in-plane momentak1,
. . . ,
k2,whichare definedonly onthe hyper-planeP
(n)≡ {
k1+ . . .
kn+
k0−
k=
0}
.Ofcourse1 WefollowthenormalizationofwavesadoptedbyVoronovich[3].
¯
G(αα0)0
(
k,
k0)
= (
2π
)
2δ
(2)(
k0
−
k)
Rαα0(
k0) ,
(7)where Rαα0
(
k0)
arethefamiliarreflectioncoefficientsforaplanarsurface.ByinsertingEq.(6)intoEq.(5),theperturbativeseriescan berewrittenas: Sαα0
(
k,
k0)
=
n≥0 1 n!
d2k 1(
2π
)
2· · ·
d2k n(
2π
)
2 h˜
(
k1) . . . ˜
h(
kn)
× (
2π
)
2δ
(2)(
k1+ . . .
kn+
k0−
k) ˜
G(ααn)0(
k1, . . . ,
kn;k,
k0) .
(8)Nextweshow thattheperturbativekernels havetosatisfy an in-finite set of relations, asa result ofthe shift invariance under a
vertical shift of theprofile. Toderive theserelations we note the identitythatfollowsfromEq.(3):
ei(q0+q)b S
αα0
(
k,
k0)
|h
(x)−b=
Sαα0(
k,
k0)
|h
(x).
(9)Upontaking p derivativesofbothsidesoftheaboverelationwith respecttotheshiftb,weobtaintheidentities:
dp d bp
ei(q0+q)bS αα0(
k,
k0)
|h
(x)−b b=0≡
A(ααp)0(
k,
k0)
|h
(x)=
0 (10)that have to be satisfiedfor any profile h.By making useof the perturbative expansion of the SA Eq. (4) into the l.h.s. of the above identities,oneobtains thefollowingexpansion forthe ker-nels A(ααp)0
(
k,
k0)
|
h(x): A(ααp)0(
k,
k0)
|h
(x)=
n≥0 1 n!
d2x1· · ·
d2xn×
Aαα(p,0n)(
x1, . . . ,
xn;k,
k0)
h(
x1) . . .
h(
xn)
=
0,
(11) with: A(ααp,n0)(
x1, . . . ,
xn;k,
k0)
=
p k=0(
−
1)
k p!
k!(
p−
k)
!
[
i(
q0+
q)
]
p−k×
d2xn+1. . .
d2xn+kG(ααn+0k)(
x1, . . . ,
xn+k;k,
k0) .
(12)SincetheidentitiesinEq.(10)mustbesatisfiedforarbitrary pro-filesh,itfollowsthat allthekernels Aαα(p,0n)
(
x1,
. . . ,
xn;
k,
k0)
mustvanishidentically:
A(ααp,n0)
(
x1, . . . ,
xn;k,
k0)
=
0,
∀
n≥
0,
p>
0.
(13) Whenexpressedinmomentumspace,theseconditionsread:(
2π
)
2δ
(2)(
k1+ . . .
kn+
k0−
k)
p k=0(
−
1)
k p!
k!(
p−
k)
!
× [
i(
q0+
q)
]
p−kG˜
(ααn+0k)(
k1, . . . ,
kn,
0, . . . ,
0)
=
0.
(14)Foranyfixedn,theaboverelationscanbesolvediteratively lead-ingto:
˜
G(ααn+0m)
(
k1, . . . ,
kn,
0, . . . ,
0;
k,
k0)
= [
i(
q0+
q)
]
mG˜
αα(n)0(
k1, . . . ,
kn;k,
k0) .
(15)Werecallthat theseidentitiesholdon
P
(n).Equation(15)consti-tutes a very importantresult, andin next Section we show that withtheir helpit ispossibletoperform aninfinite re-summation
G. Bimonte / Physics Letters B 760 (2016) 149–152 151
3. Resummationoftheperturbativeseries
Nowwe considera surface
whoselocal radiusofcurvature is everywhere large compared to the wavelength of the incom-ingwave. For sucha gently curvedprofile, theFouriertransform
˜
h
(
k)
of theheight profileissignificantly differentfromzeroonly for small in-plane wave vectors k. For quasi specular-scattering, i.e. for kk0, it is therefore legitimate to Taylor-expand the kernels G˜
(ααn)0(
k1,
. . . ,
kn;
k,
k0)
around k1= · · · =
kn=
0.Exis-tence of this Taylor expansion is not guaranteed a priori, but hasto be checkedcaseby case. Withrespect to thisquestion, it is important to observe that by virtue of the iterative relations Eq.(15) existenceofthe Taylorexpansionofordern for the ker-nel G
˜
(ααn)0(
k1,
. . . ,
kn;
k,
k0)
automatically implies existence of theTaylorexpansion tothesameordern forall higherorderkernels
˜
G(ααn+0m)
(
k1,
. . . ,
kn+m;
k,
k0)
, m=
1,
2,
. . .
. Let usassume that theperturbativekernelscanbeTaylor-expandedtosecondorderinthe wavevectors2:
˜
G(n)(
k1, . . . ,
kn;k,
k0)
=
A(n)(
k,
k0)
+
Bμ(n)(
k,
k0)(
k1+ · · · +
kn)
μ+
C(μνn)(
k,
k0)
n i=1 kiμkνi−
Dμν(n)(
k,
k0)
i<j kμi kνj+
o(
k2) ,
(16) where Greek indices take values(
1,
2)
, and for brevity we sup-pressed all polarization indices.Since then-point Green function is defined only on the hyperplanesP
(n), we are free to add to˜
G(n)
(
k1
,
· · · ,
kn;
k,
k0)
anyfunction f(n)(
k1,
· · · ,
kn;
k,
k0)
vanish-ingonP
(n)oftheformf(n)
(
k1, . . . ,
kn;k,
k0)
= (
k1+ . . .
kn+
k0−
k)
μ×
gμ(n)(
k1, . . . ,
kn;k,
k0)
(17) whereg(μn)(
k1,
· · · ,
kn;
k,
k0)
arearbitrarysmoothsymmetric func-tionsofk1,
· · · ,
kn.Letustakegμ(n)
= ˆ
Aμ(n)(
k,
k0)
+ ˆ
Bμν(n)(
k,
k0)(
k1+ · · · +
kn)
ν.
(18)It iseasy toverify that the coefficients A
ˆ
μ(n),
Bˆ
(μν cann) always bechosensuchthataddition of f(n)
(
k1
,
· · · ,
kn;
k,
k0)
toG˜
(n)(
k1,
· · · ,
kn;
k,
k0)
removes from Eq. (16) the terms proportional to B(μn)and C(μν .n) Without loss ofgenerality, the second order Taylor ex-pansion of the kernels G
˜
(n)(
k1
,
· · · ,
kn;
k,
k0)
around k1= · · · =
kn=
0 canthusbeassumedtobeoftheform:˜
G(n)(
k1,
· · · ,
kn;k,
k0)
=
A(n)(
k,
k0)
−
D(μνn)(
k,
k0)
i<j kμikνj+
o(
k2) .
(19)Aftertheabove small-k expansion issubstituted intothe pertur-bativeseriesEq.(8),onefinds:
2 Byanexplicitperturbativecomputation tosecond orderinthe height pro-file, it is possible to verify that the perturbative kernels G˜αα(1)0(k1;k,k0) and
˜
G(αα2)0(k1,k2;k,k0)doadmitasecond-orderTaylor-expansionasperEq.(16),for
bothascalarwavesatisfyingDirichletorNeumannboundaryconditionsonthe sur-face, aswellasfor thescatteringofanelectromagneticwavebyadielectric surface.Explicitformulaefortherelevantperturbativekernelsinthe electromag-neticcaseareprovidedin[3].
S
(
k,
k0)
=
n≥0 1 n!
d2k 1(
2π
)
2· · ·
d2k n(
2π
)
2h˜
(
k1) . . . ˜
h(
kn)
× (
2π
)
2δ
(2)(
k1+ . . .
kn+
k0−
k)
× [
A(n)(
k,
k0)
−
D(μνn)(
k,
k0)
i<j kμi kνj]
= (
2π
)
2δ
(2)(
k0−
k)
R(
k0)
+
A(1)(
k,
k0) ˜
h(
k−
k0)
+
n≥2 d2x ei(k0−k)·x 1 n!
A (n)(
k,
k 0)
hn(
x)
+
hn−2(
x)
2(
n−
2)
!
D (n) μν(
k,
k0)∂
μh(
x)∂
νh(
x)
.
(20)However, theidentities Eq.(15) satisfiedbythe perturbative ker-nelsforn
=
0,
1,
2 implyatonce:A(m)
(
k0,
k0)
= (
2 i q0)
mR(
k0) ,
(21)A(m+1)
(
k,
k0)
= [
i(
q0+
q)
]
mA(1)(
k,
k0) ,
(22)A(m+2)
(
k,
k0)
= [
i(
q0+
q)
]
mA(2)(
k,
k0) ,
(23)D(μνm+2)
(
k,
k0)
= [
i(
q0+
q)
]
m Dμν(2)(
k,
k0) .
(24) Bymaking use intoEq. (20)ofthe above identities,it iseasy to performtheinfinitesumsinther.h.s.ofEq.(20):S
(
k,
k0)
= (
2π
)
2δ
(2)(
k0−
k)
R
(
k0)
−
A(1)(
k0,
k0)
2iq0+
d2x ei(k0−k)·xA(1)
(
k,
k0)
i(
q0+
q)
+
1 2D (2) μν(
k,
k0)∂
μh(
x)∂
νh(
x)
n≥0 1 n
!
[
i(
q+
q0)
h(
x)
]
n⎫
⎬
⎭
=
d2x ei[(k0−k)·x+(q+q0)h(x)]A(1)
(
k,
k 0)
i(
q0+
q)
+
1 2D (2) μν(
k,
k0) ∂
μh(
x)∂
νh(
x)
.
(25)The above formula forthe SA representsour main result. As we see,thecontributionproportionaltoA(1)
(
k,
k0)
reproducesthe KA, whilethesecondtermoforder(
∇
h)
2 providestheleading curva-ture correction to theKA. Bearing inmind Eq.(19),we seethat the coefficients A(1)(
k,
k0)
and Dμν(2)(
k,
k0)
occurring in this for-mula can be extracted from the second order Taylor expansion oftheperturbativekernelsG˜
(αα1)0(
k1;
k,
k0)
andG˜
(2)
αα0
(
k1,
k2;
k,
k0)
,respectively.Wenow provethatourEq.(25)isindeedequivalent (toorder
(
∇
h)
2)totheSSAansatzmadebyVoronovich[3]twenty yearsago.4. ComparisonwithVoronovich’ssmall-slopeapproximation
In Ref. [3]Voronovich postulated the following ansatz for the SSAvalidtoorder
(
∇
h)
2:S
(
k,
k0)
=
2√
qq0 q+
q0 d2x ei[(k0−k)·x+(q+q0)h(x)]×
B11(
k,
k0)
−
i 4 d2k1M11(
k,
k0;
k1) ˜
h(
k1)
eik1·x,
(26) where152 G. Bimonte / Physics Letters B 760 (2016) 149–152
M11
(
k,
k0;
k1)
=
B112(
k,
k0;
k−
k1)
+
B112(
k,
k0;
k0+
k1)
+
2(
q+
q0)
B11(
k,
k0) ,
(27) andforbrevitywesuppressedallpolarizationindices.The pertur-bativekernelsoccurringintheaboveEquationsarerelatedtoours bythefollowingrelations:˜
G(1)(
k−
k0;
k,
k0)
=
A(1)(
k,
k0)
=
2i√
qq0B11(
k,
k0) ,
(28) and˜
G(n+1)(
k−
k1,
k2, . . . ,
kn,
kn−
k0;
k,
k0)
= (
n+
1)
!
√
qq0(
Bn+1)
11(
k,
k0;
k1, . . . ,
kn) ,
n=
1,
2, . . .
(29)Inview of Eq.(28),itis clearthat thefirsttermbetweenthe squarebracketsinEq.(26)reproducesthefirsttermbetweenthe square bracketsofEq.(25).Byan explicitcomputation, itis pos-sibleto verifythatthesecond termsbetweenthesquare brackets ofEqs.(26)and(25)coincideaswell.InviewofEq.(29),and us-ing thesecond orderTaylorexpansionof G
˜
(2)(
k1,
k2;
k,
k0)
given inEq.(19),wefind:√
qq0(
B112(
k,
k0;
k−
k1)
+
B112(
k,
k0;
k0+
k1))
=
1 2[ ˜
G (2)(
k 1,
k−
k0−
k1;
k,
k0)
+ ˜
G(2)(
k−
k0−
k1,
k1;
k,
k0)
]
= ˜
G(2)(
k1,
k−
k0−
k1;
k,
k0)
=
A(2)(
k,
k0)
−
D(μν2)(
k,
k0)
kμ1(
k−
k0−
k1)
ν.
(30) Therefore√
qq0M11(
k,
k0;
k1)
=
A(2)(
k,
k0)
−
i(
q+
q0)
A(1)(
k,
k0)
−
D(μν2)(
k,
k0)
kμ1(
k−
k0−
k1)
ν= −D
(2) μν(
k,
k0)
kμ1(
k−
k0−
k1)
ν,
(31) whereinthelast passagewe usedtherecursive relationEq.(22)tocancelthefirsttwotermsintheintermediateexpression.Then:
√
qq0 d2k1M11(
k,
k0;
k1) ˜
h(
k1)
eik1·x= −D
(2) μν(
k,
k0)
d2k1eik1·xkμ1(
k−
k0−
k1)
νh˜
(
k1)
eik1·x=
i D(μν2)(
k,
k0)(
k−
k0+
i∂)
ν∂
μh(
x) .
(32)Usingtheaboveformula,weseethatthetermofEq.(26)involving thek1 integralcanberecastas:
−
i√
qq 0 2(
q+
q0)
d2x ei[(k0−k)·x+(q+q0)h(x)]×
d2k1M11(
k,
k0;
k1) ˜
h(
k1)
eik1·x=
D (2) μν(
k,
k0)
2(
q+
q0)
d2x ei[(k0−k)·x+(q+q0)h(x)]× (
k−
k0+
i∂)
ν∂
μh(
x)
=
1 2D (2) μν(
k,
k0)
d2x ei[(k0−k)·x+(q+q0)h(x)]∂
μh(
x)∂
νh(
x) ,
(33)where in the last passage we performedan integration by parts on the term involving second derivatives of h. Comparison with Eq.(25)showsthattheabovetermcoincideswiththesecondterm onther.h.s. ofEq.(25).
5. Conclusions
Wehaveshownthat theSAdescribingscatteringofawave by agentlycurvedsurfaceadmitsaderivativeexpansioninpowersof derivatives ofthe height profile. Thisderivative expansion of the SA hasbeenderived herebyperforming an infiniteresummation of the perturbative seriesfor the SA, to all orders inthe ampli-tude of the corrugation. Based on this formal derivation it can be expectedthat the derivative expansionis asymptotically exact in thelimit of quasi-specularscattering. In theleading order the derivative expansion coincides with the classic KA, while in the next orderitprovides theleadingcurvaturecorrection tothe KA. We have also shown that the derivative expansion is equivalent totheorder
(
∇
h)
2 totheSSAansatz,proposed sometimeago by Voronovich to describe wave scattering by a rough surface. The resummation of the perturbative series performed here to order(
∇
h)
2,can beeasily generalizedtohigherorders in∇
h,provided onlythattheperturbativekernelsforthescatteringamplitude ad-mitaTaylorexpansionofsufficientlyhighorderforsmallin-plane wavevectors.Acknowledgements
TheauthorthanksT. Emig,N. Graham,M. Kruger,R.L. Jaffeand M. Kardar for valuable discussions while the manuscript was in preparation.
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