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NONLINEAR DIFFRACTIVE OPTICS WITH CURVED PHASES: BEAM DISPERSION

61 pages
NONLINEAR DIFFRACTIVE OPTICS WITH CURVED PHASES: BEAM DISPERSION AND TRANSITION BETWEEN LIGHT AND SHADOW. E. Dumas Laboratoire de Mathematiques et Physique Theorique Parc de Grandmont 37200 Tours, FRANCE tel: 02-47-36-73-14 fax: 02-47-36-70-68 Abstract: We give asymptotic descriptions of smooth oscillating solu- tions of hyperbolic systems with variable coefficients, in the weakly nonlinear diffractive optics regime. The dependence of the coefficients of the system in the space-time variable (corresponding to propagation in a non-homogeneous medium) implies that the rays are not parallel lines –the same occurs with non-planar initial phases. Approximations are given by WKB asymptotics with 3-scales profiles and curved phases. The fastest scale concerns oscilla- tions, while the slowest one describes the modulation of the envelope, which is along rays for the oscillatory components. We consider two kinds of be- haviors at the intermediate scale: ‘weakly decaying' (Sobolev), giving the transverse evolution of a ‘ray packet', and ‘shock-type' profiles describing a region of rapid transition for the amplitude. Contents Introduction 2 Long time propagation in homogeneous media . . . . . . . . . . . . 2 Variable coefficients . . . . . . . . . . . . . . . .

  • wave transition

  • weakly nonlinear

  • linear wave

  • considers equations

  • schneider has

  • between light


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NONLINEARDIFFRACTIVEOPTICS
WITHCURVEDPHASES:BEAMDISPERSION
ANDTRANSITIONBETWEENLIGHTANDSHADOW.
E.Dumas
LaboratoiredeMathe´matiquesetPhysiqueThe´orique
ParcdeGrandmont
37200Tours,FRANCE
tel:02-47-36-73-14fax:02-47-36-70-68
dumas@gargan.math.univ-tours.fr
Abstract:
Wegiveasymptoticdescriptionsofsmoothoscillatingsolu-
tionsofhyperbolicsystemswithvariablecoefficients,intheweaklynonlinear
diffractiveopticsregime.Thedependenceofthecoefficientsofthesystemin
thespace-timevariable(correspondingtopropagationinanon-homogeneous
medium)impliesthattheraysare
not
parallellines–thesameoccurswith
non-planarinitialphases.ApproximationsaregivenbyWKBasymptotics
with3-scalesprofilesandcurvedphases.Thefastestscaleconcernsoscilla-
tions,whiletheslowestonedescribesthemodulationoftheenvelope,which
isalongraysfortheoscillatorycomponents.Weconsidertwokindsofbe-
haviorsattheintermediatescale:‘weaklydecaying’(Sobolev),givingthe
transverseevolutionofa‘raypacket’,and‘shock-type’profilesdescribinga
regionofrapidtransitionfortheamplitude.

Contents
Introduction2
Longtimepropagationinhomogeneousmedia............2
Variablecoefficients...........................4
Descriptionofthepaper........................6
1Dispersionofbeams9
1.1TheAnsatz............................10
1.2Firstequations..........................10
1.3Thesublinearitycondition....................13
1.3.1Functionspaces......................14
1.3.2Operators.........................16

1.3.3Profileequations.....................19
1.4Existenceofprofiles........................20
1.5Approximationofsolutions....................22
1.5.1Estimatesontheresidual................22
1.5.2Stability..........................24
1.6Diffractionfortheweaklycompressible,isentropic3-dEuler
equations.............................29
2Transitionbetweenlightandshadow(foroddnonlinearities)32
2.1Framework,notations.......................34
2.2Functionspaces,andthemeanoperator
M
..........36
2.3Formalderivationofprofileequations..............36
2.3.1Usingthemeanoperator
M
...............37
2.3.2Fastscaleanalysis....................38
2.3.3Analysisw.r.t.the(remaining)intermediatevariables.39
2.4Existenceofprofilesandapproximationofexactsolutions...41
2.4.1Solvingtheprofileequations...............41
2.4.2Stability..........................42
3Wavetransitionsforsystemsofconservationlaws44
3.1ThesystemandtheAnsatz...................46
3.2Theapproximatesolution....................46
3.3Stability..............................48
3.3.1Theconjugationoperator
V
...............49
ε3.3.2Thesingularsystem...................54
3.3.3ExampleofphasesforEulerequations.........58

Introduction
Longtimepropagationinhomogeneousmedia
Geometricopticsprovidesasymptoticapproximationsofwavesinthelimitof
zerowavelength.Theseapproximationsarevalidonlyforsomepropagation
distances(see[22]).Whenlookingatlongerpropagationscales,adiffractive
correctionisneeded.Thefirstrigorousworksinthiscontextareprobably
[7]and[8].Underoddnessassumptionsonthenonlinearities,theseauthors
giveanapproximationtothesolutionoftheinitialvalueproblemassociated
withanonlinearhyperbolicsystem
L
(
u,∂
)
u
=
F
(
u
),where
L
(
u,∂
)=

t
+
2

irtamcirtemmysera)u(jAeht,ereH.j∂)u(jAjP,uahomogeneousmedium.Theinitialdataoscillateatfrequency1

,andthe
approximationisprovided,ontheRayleighdistance(oforder1

),by:
(0.0.1)
ε
m
ε
n
a
n
εX,X,β

X,X
=(
T,Y
)

R
1+
d
,
XεNmn∈i.e.
:Thereis
t
?
suchthat,forall
ε

]0
,
1],theexactsolution
u
ε
issmooth
on[0
,t
?

]
×
R
d
,andadmitstheasymptoticexpansion(0.0.1)as
ε

0.
Theamplitude
ε
m
issmallerthantheoneofgeometricoptics(for
O
(1)
propagation),sothatdiffractionaffectstheprincipaltermoftheasymptotics
theoscillatingwaveis
β

x
=
j
β
j
x
j
.Theprofiles
a
n
(
X
˜
,X,θ
)aresmooth,
atthesametimeastheaccum
P
ulatedeffectsofnonlinearities.Thephaseof
periodicin
θ
(withmeanequaltozero).Theyaresolutionstoacoupledsys-
temoftransportequationattheintermediatescaleandSchro¨dingerequation
withslowtime.Thesystemisnonlinearforthefirstprofile:
(0.0.2a)
πa
0
=
a
0
,
(0.0.2b)
V
(

X
)
a
0
=0
,
(0.0.2c)
V
(

X
˜
)
a
0
+
R
(

Y
)

θ

1
a
0
+
π
[Φ(
a
0
)+Λ(
a
0
)

θ
a
0
]=0
.
Thenextprofilesaresolutionstosystemswiththesamestructure,butlinear.
Equation(0.0.2a)expressesthepolarizationof
a
0
,and
π
isa(matrix)pro-
jectorassociatedto
L
and
β
.Theoperator
V
(

X
)=

T
+
v.∂
Y
isthetransport
fieldalongrays,withgroupvelocity
v
.Thesetwoequationsaresimilartothe
onesofusualgeometricoptics.Finally,(0.0.2c)representstransversediffrac-
tion,atthetimescale
T
˜,viathescalaroperator
R
(

Y
)=
i,j
r
i,j

Y
i

Y
j
,
Pwhosecoefficientsarerelatedtothecurvatureofthecharacteristicvarietyof
L
.Thenonlineartermisthesameastheonearisingintheweaklynonlinear
geometricopticsequations.
Aqualitativedifferencebetweentheapproximatesolution(0.0.1)and
thegeometricoptic’onecomesfromEquation(0.0.2c),whichimpliesnon-
conservationofsupports:Eveniftheinitialdatahavecompactsupport,
a
0
(
εX,X,β

X/ε
)doesnot,whereasthegeometricopticsapproximationdoes,
becauseitistransportedalongrays.Thisexplainsthespatialdispersionof
alaserbeam,forexample.
ThiskindofasymptoticshasalsobeenstudiedbyJoly,Me´tivier,Rauchin
[19],whenrectificationeffectsarepresent,
i.e.
wheninteractionsofoscillating
3

nisetagaporpevawehttahtsnaemsiht:)x,t(setanidroocehtnotontubnognidnepedylbissopsec
modescangeneratenon-oscillatorywaves.In[21],D.Lannesconsidersthe
caseofdispersivesystems,withrectification.G.Schneiderhastreatedthe
caseofoneequation,inspacedimensionone,bymeansofnormalforms(see
[26]).In[6],T.Colinhasstudiedsystemswitha‘transparency’property,
allowingsolutionswithgreateramplitude;theprofilesarethensolutionsof
Davey-Stewartsonsystems(seealso[20]).Diffractionforpulses(
i.e.
when
theprofiles
a
n
(
X
˜
,X,θ
)havecompactsupportin
θ
)leadstoasomewhat
differentapproximation,withatypicalprofileequation2

T
˜

θ
a
n

Δ
Y
a
n
=

θ
f
(
a
n
);see[2],[1],and[3]foranapproachvia‘continuousspectra’.
Alltheseresultshavebeenobtainedinthegeneralframeworkof‘long
withrespecttoonelinearphase(Ansatz
ε
m
ε
n
u
n
(
εX,X,β

X/ε
)).
time’propagation(oforder1

whenthewa
P
velengthis
ε
),andoscillations
Variablecoefficients
Thepreviousresultsbreakdownassoonasoneconsidersequationswithvari-
n
(
εX
)
2
ablecoefficients,forexamplethefoll

owingwaveequati

onwithnon-constant
refractiveindex(seeExample0.1):
2

T
2

Δ
Y
u
ε
=0
.
cHere,weareinterestedinthecaseofcurvedphases,forwhichraysare
nolongerparallellines–butbeforefocusing(orcaustics):Ourstudyonly
concernssmooth(
C
1
)phases.Webeginwithachangeofscale,sothatthe
propagationoccursfortimesoftheorderone.Usingtheslowvariable
x
=
εX
(=
O
(1))insteadof
X
,theapproximatesolution(0.0.1)reads
(0.0.3)
ε
m
ε
n
a
n
x,x,β

x,
Xn

m
N
εε
2
√andsetting
ε
=

,
n
m/
2
X

n/
2
ax,

x,β

x.
n

m
N

Now,inthecaseofvariablecoefficientsystems,nonlinearphasesareinvolved,
aprioridefinedonaboundeddomainΩ(as
ε

0)only.That’swhyweuse
theAnsatz
(0.0.4)

m/
2

n/
2
ax,ψ

(
x
)

(
x
)
,
nXn

m
N

4

where
a
n
=
a
n
(
x,ω,θ
)
∈∩
s
H
s

×
R
p
×
T
q
).
ThisAnsatzwasintroducedbyJ.K.Hunterin[16],andbecomes
(0.0.5)
ε
m
ε
n
a
n
εX,ψ
(
εX
)

(
εX
)
Xn

m
N
εε
2
inthescalesof(0.0.1)(propagationondistancesoforder1

).Thephases
φ
and
ψ
dependslowlyon
X
,andthisregimeiscalled
weaklynonplanar
.
Example0.1.
Considerthelinearwaveequation
2n
(
εX
)

T
2

Δ
Y
u
ε
=0
,
2cwithrefractiveindex
n
smoothandbounded,andinitialdata

εm

η

Y



u
|
T
=0
=
εgY,ε
0=Tε


t
u
|
ε
=
ε
m

1
hY,η

Y.
Wechoose
h
=
nc
|
η
|

θ
g
(polarizeddata),and
g
∈∩
s
H
s
(
R
d
×
T
)with
gdθ
=
R0(purelyoscillatingprofiles).
Weareinterestedinthebehaviorof
u
ε
fortimesoftheorder1

.So
astoapplytheresultsofParagraph1.5,wechangevariables:
x
=
εX
,
v
ε
2
(
x
)=
u
ε
(
X
),andset
ε
2
=

.
2n
(
x
)

2

Δ
y
v

=0
t2c




y

v
|

t
=0
=

m/
2
g

,



y




t
v
|

t
=0
=

(
m

1)
/
2
h

,.
Theorems1.1and1.2giveanapproximationof
v

ontheconeΩ=
{
x
=
(
t,y
)

R
1+
d
/
0

t

t
?
,δt
+
|
y
|≤
ρ
}
:Thereare
V

,
V
app
∈∩
s
H
s

∩{
t

t

R
d
×
T
)(for
t<t
?
)suchthat:
0φψv

(
x
)=

m/
2
V

x,

,,

s

R
,
kV
app
−V

k
H
s




0
0
.
5

n
(
x
)

t
φ
=
|

y
φ
|
V
φ
(

x
)
ψ
µ
=0

Thephases
φ,ψ
0
=(
ψ
1
,..

.,ψ
d
)aredefinedby:
c,,φ
|
t
=0
=
η


µ
|
t
=0
=
y
µ
n
(
x
)
∂φ
where
V
φ
(
x,∂
x
)=


t

y
.∂
y
:
φ
satisfiesaneikonalequationasso-
c
|

y
φ
|
2ciatedwith
cn
2

t
2

Δ,andeach
ψ
µ
isannihilatedbytheassociatedtangent
transport,
i.e.
forthevectorfieldcorrespondingtothegroupvelocity.When
therefractiveindexreallydependson
x
=
εX
,noneofthese

phasesislinear.
Theapproximateprofile
V
app
isgivenby
V
app
=
v
0
+
v
1
+
v
2
.The
terms
v
n
(
x,ω,θ
)aredeterminedbyequations(1.3.2)to(1.3.4c)and(1.2.5d)
to(1.2.5g),whichhererestrictto(seeRemark1.4,
iii)
):
V
φ
(

x
)
v
n

D
(

ω
)

θ

1
v
n
=
f
n
,
with
D
(

ω
)=
2
(

t
ψ
µ
)
−|

y
ψ
µ
|

ω
µ
,and
f
n
afunctionof
v
n

1
,v
n

2
1
X
n
2222
c2µandtheirderivatives.
2
Comingbacktotheoriginalscalesandsetting
U
aεpp
=
V
app
,wegetan
approximationof
u
ε
fortimes
T

1

:
1∀
t<t
?
,

α

N
1+
d
,
onΩ
∩{
T

t/ε
}
,
ε0(
ε∂
)
α
u
ε

ε
m
U
aεpp
εX,ψ
(
εX
)

(
εX
)=
o
(
ε
m
)
.
L
εε
2


One
cannot
obtainsuchanapproximation(onΩ
t

)usingplanephases:
ψ
µ
(
εX
)

differsfromitslinearpart

x
ψ
µ
(0)

X
bya
O
(
ε
|
X
|
2
)=
O
(1

)
term.Becauseofdecayingof
U
aεpp
,replacing
ψ
µ
(
εX
)

byitslinearpart
wouldgeneratea
O
(
ε
m
)errorintheapproximatesolution.
Descriptionofthepaper
Themaindifficultiesintheanalysisarenonlinearitiesandvariablecoeffi-
cients.Nonlinearitiesallowinteractionsbetweenpropagatingmodes,and
thusphasemixing,andinducecouplingintheprofileequations.Theseequa-
tionsalsohavevariablecoefficientswhentheoriginalsystemdoes.Asolvable
systemdeterminingtheprofilescanbeobtainedonlywhentheoperatorsin-
volvedcommute.Thisisguaranteedbycoherenceassumptionsonthephases

6

(
cf.
Paragraph1.3.2andParagraph1.4).Theasymptoticsystemfinallyin-
heritspropertiesfromtheinitialone,whichprovideenergyestimates,in
functionspaceswithdifferentregularitiesfordifferentvariables;seePara-
graph1.4.
Ourstudydealswithquasilinearhyperbolicsystems,andthesamemeth-
odsapplytosemilinearsystems.Thepaperisorganizedasfollows:
1-
Inthefirstpart,welookattheweaklydecayingcase,whentheinitial
profiles
g
n
(
y,Y,θ
)
∈∩
s
H
s

0
×
R
p

1
×
T
).TheAnsatzisgiveninPara-
graph1.1.Weformallyderivethefirstprofileequationsbythemethodof
multiplescalesinParagraph1.2.
Weintroducean‘intermediatetime’
T
(
X
=(
T,Y
))inordertoanalyse
andsolvetheseequations.Inparticular,itisnatural

torequiretheprofile
u
1
tobesublinearwithrespectto
T
,sothattheterm
εu
1
isacorrectorof
u
0
(Paragraph1.3).
Thenextstepconsistsinlookingattheinteractionsbetweenwavescon-
stituting
u
0
,soastodeterminewhichoneshaveaninfluenceatleadingorder:
theothersareconsideredascorrectors.Thankstoourcoherenceassumptions
onthephases,theconditionsunderwhichtheprofilesaresublinearcanbede-
rivedviathetechniquesofJoly,Me´tivier,Rauch[19]orLannes[21].Thesize
ofthecorrectorscannotbemorespecified,becauseofinteractionsbetween
oscillatoryandnon-oscillatoryterms(rectificationphenomenon).Hence,the
asymptoticsisbasedonafirsttermandtwocorrectorsonly.
Weshowexistenceof
u
0
,
u
1
and
u
2
(Paragraph1.4),andthen,stability
ofexactsolutionsneartheapproximateone,viaasingularsystemmethod
andanadditionalcoherenceassumptioninvolvingbothfastandslowphases
(Paragraph1.5).
WeillustratetheseresultswiththeconcreteexampleofisentropicEuler
equations,exhibitingexplicitcoherentnonlinearphasesandprofileequations.
2-
Inthesecondpart,westudyoscillatingwaveswhoseamplitudeshave
arapidvariationacrosssomehypersurface:theydecaytozeroononeside,
whereasontheotherside,theybehavelikeusualslowlymodulatedoscillating
waves.Thisdescribestransitionbetweenlightandshadow(orsoundand
silence,inthecaseofsoundwaves).Insteadoftreatinga‘matchingproblem’
(suchasin[16]),weconstructwaveswithWKBasymptoticsintermsof
‘shock’profiles:theyadmitfinitelimitsat+

and
−∞
withrespecttoone
oftheintermediatevariables,
Y
1
.Theraysassociatedwith
φ
arethentangent
tothesurface
ψ
1
=0.Theprofiles
u
n
splitinto
u
n
=
χ
(
Y
1
)
a
n
(
Y
2
,...,Y
p
)+
b
n
(
Y
1
,...,Y
p
),where
χ
isafixed‘step’(functionwithfinitelimitsat+

7

and
−∞
).Theterm
a
n
givesthebehavio

r‘atinfinity’(w.r.t.
Y
1
),while
b
n
representsthetransitionlayer(ofwidth
ε
).
χa
k
χa
k
+
b
k
bkY
1
Y
1
Y
1

Figure1:Theprofiles’shape.
Theequationsdeterminingthe
a
n
areindependentofthe
b
n
.Inparticular,
inthemodelcasewhen
p
=1,
a
n
doesnotdependon
Y
,andsatisfiesthe
usualequationsofweaklynonlinearopticsin
{
ψ>
0
}
:seeRemark2.2.
shadowlight
ψ
1
=0
0=ψ1

obstacle
φ
-rays
Y
1
=
ψ
1
/

ε
Ωincidentrayscoordinatestretching
√εFigure2:From‘macroscopic’to‘microscopic’description.
InParagraphs2.1and2.2,wegivetheAnsatz.WeexplaininExample2.1
whyonlyoneintermediatephase
ψ
1
cangoverntherapidtransition,andthus,
whyrectificationeffectsmustbeavoidedateachstepoftheasymptotics.
Thesimplestwaytodosoconsistsinimposingastrongconditiononthe
nonlinearities:theirTaylorexpansion(atzero)includesoddpowersonly.
Wearethenabletoconstructinfinite-orderasymptotics,basedonpurely
oscillating(or‘zero-mean’)profiles.
WeconstructtheprofilesinParagraph2.4(underthesamecoherence
assumptionsasinSection1).Theysatisfyastrongerconditionthan
T
-
sublinearity(theyarebounded).
Finally,thankstotheinfiniteorderasymptotics,weprovestabilityofthe
exactsolutionsviaasmoothperturbationmethod(inthespiritofO.Gue`s,
[13]).

8

3-
Thethirdpartisdevotedtoanotherapproachconcerningthiskindof
rapidtransitions,wherewegetridofthepreviousoddnessassumptionon
nonlinearities.WeextendthesingularsystemtechniqueofPart1tothefunc-
tionspacesofPart2,usingsemi-classicalpseudo-differentialcalculus.This
methodsimplyrequiresnon-generationofprofilemeanvaluesatfirstorder
(whichisforexamplesatisfiedinthecaseofsystemsofconservationlaws),
butasaconsequence,weonlyobtainleading-orderasymptotics.Anaddi-
tionalgeometrical‘coherence-type’assumptionontheintermediatephase
ψ
isneededtoensureconverge.
Theseassumptionsaresatisfiedbytheexplicitexampleofacousticwaves
inParagraph3.3.3.
Remark0.1.
Ourprofilesdependonseveralintermediatephases,andonly
onerapidphase
φ
.Onemayprovethesameexistenceandstabilityproperties
formultiphaseasymptotics,andtreatinteractionsofdiffractedwaves(adding
acoherenceassumptiononthephases
φ
;see[10],[9]).

1Dispersionofbeams
Westudythesolutionsofaquasilinearsymmetrichyperbolicsystem
dd(1.0.6)
L
(
x,u,∂
)
u
=

t
u
+
A
j
(
x,u
)

j
u
=
A
j
(
x,u
)

j
u
=0
.
XXj
=1
j
=0
Wedenoteby
x
=(
t,y
)apointinΩ.Ωisaconnectedopensubsetof
R
1+
d
onwhichthematrices
A
j
satisfy:
Assumption1.1.
Thematrices
A
j
∈C


×
C
N
,
M
N
(
C
))
areHermitian,
and
A
0

I
.
WeareinterestedintheCauchyproblemassociatedto(1.0.6),forinitial
dataoftheform
00εgy,ψ

(
y
)

(
y
)
,
where
g
∈∩
s
H
s

×
R
p

1
×
T
)
.
εε

9

1.1TheAnsatz
Theprofilesare‘weaklydecaying’(
i.e.
H
s
)withrespecttotheintermediate
variable.WedenotebyΨthe(real)vec

torspacewithgenerators
ψ
.We
introduceanintermediatetime
T
=
t/ε
(thus,aphase
ψ
0

t
),soas
totreatCauchyproblems.Thephases
ψ
=(
ψ
0
,...,ψ
p

1
)are
R
-linearly
independent.Thesizeofcorrectorsisalsomeasuredby
T
(see1.3),sothat
weavoidill-posedsystemssuchasproposedin[16].
Remark1.1.
Inthesequel,thevectorspace
Ψ
willsatisfycoherenceas-
sumptions.Asexplainedin[18](p.56;seealso[17]),suchaspaceusually
containsatimelikephase
ψ
0
.Changingvariables,onecanusethis
ψ
0
as
timevariable.Butinthiscase,thematrix
A
0
(coefficientof

t
in(1.0.6))
thendependson
(
t,y
)
.Forthesakeofsimplicity,wesupposethat
ψ
0

t
.
Oneofthefeaturesemphasizedin[8]and[19]isrectification,
i.e.
the
possibilityofinteractionbetweenoscillatingandnon-oscillatingmodes(trav-
ellingatthesamespeed).ThisforcestheuseofanAnsatzwithonlyone
termandtwocorrectors,whichreads:
20(1.1.1)
u
ε

εε
n/
2
u
n
x,

t,ψ

(
x
)

(
x
)
,
Xn
=0
εεε
where
ψ
=(
t,ψ
0
)

Ψ
p
,with
u
n
=
u
n
(
x,X,θ
)=
u
n
(
x,T,Y,θ
)periodicw.r.t.
θ
andsmooth,and
u
n
(
x,T,.,θ
)
∈∩
s
H
s
(
R
p

1
).
Notation1.1.
Wedenoteby
Ψ
0
the(real)spacegeneratedbythephases
ψ
0
;
itisadimension
p

1
subspaceof
Ψ
,suchthat
Ψ=Ψ
0

t
R
.
1.2Firstequations
Oneformallygetsanasymptoticsolutionto(1.0.6)bypluggingtheAnsatz
intothesystemandinsistingthatthecoefficientsof
ε
0
,
ε
1
/
2
and
ε
inthe
residualallvanish.Thisyields:

(1.2.1)

(1.2.2)

L
1
(

)

θ
u
0
=0
,

L
1
(

)

θ
u
1
+
L
1
(

)

X
u
0
=0
,
01

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