NONLINEAR DIFFRACTIVE OPTICS WITH CURVED PHASES: BEAM DISPERSION

Description

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: [phone] fax: [phone] 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

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

NONLINEARDIFFRACTIVEOPTICS
WITHCURVEDPHASES:BEAMDISPERSION
ANDTRANSITIONBETWEENLIGHTANDSHADOW.
E.Dumas
LaboratoiredeMathe´matiquesetPhysiqueThe´orique
ParcdeGrandmont
37200Tours,FRANCE
tel:[phone]fax:[phone]
[e-mail]
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
/ε
,andthe
approximationisprovided,ontheRayleighdistance(oforder1
/ε
),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
?
/ε
]
×
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
/ε
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
√