Shape derivatives for diffraction by non-smooth periodic interfaces [Elektronische Ressource] / Norbert Kleemann. Betreuer: Dietmar Hömberg

Shapederivativesfordiffractionbynon-smoothperiodicinterfacesDISSERTATIONzurErlangungdesakademischenGradesDoktorderNaturwissenschaften(Dr.rer.nat.)eingereichtvonDipl.-Math.NorbertKleemanngeb. am31.08.1981inOrosházagenehmigteDissertationTECHNISCHE UNIVERSITÄT BERLINFakultätII-MathematikundNaturwissenschaftenPromotionsausschuss:Vorsitzender: Prof. Dr. WolfgangKönigGutachter: Prof. Dr. DietmarHömbergProf. Dr. JanSokolowskiDr.habil.AndreasRathsfeldTagderwissenschaftlichenAussprache: 19. August2011Berlin2011D83In der vorliegenden Arbeit werden konische Diffraktionsprobleme bei nichtglattenDiffraktionsgitternuntersucht. ZielistdieBerechnungvonFormableitungen,welchezurRekonstruktionderstreuendenStrukturgenutztwerdenkönnen. DazuwerdenzunächstA-priori-AbschätzungeningewichtetenSobolevräumenvomKondratiev-Typbewiesen.Anschließend werden Aussagen zu Existenz und Eindeutigkeit von Lösungen in diesenRäumen getroffen. Darauf aufbauend wird dann mit Hilfe der Theorie nichtlokaler Stö-rungenelliptischerRandwertproblemedieExistenzundEindeutigkeitvonFormableitun-gengezeigt. DieFormableitungenwerdenanschließendcharakterisiertalsLösungenvonDiffraktionsproblemenmitgleichemOperator,abermodifiziertenrechtenSeiten. DadieFormableitungen bei Anwesenheit von Ecken im Diffraktionsgitter eine niedrige Regu-laritätaufweisen,wirdzurnumerischenBerechnungeinAnsatzvorgeschlagen,derdarinbesteht, die Singularitäten an den Ecken mit Hilfe glatter Funktionen abzuschneiden.
Publié le : samedi 1 janvier 2011
Lecture(s) : 18
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Source : D-NB.INFO/1016533624/34
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Shapederivativesfordiffractionby
non-smoothperiodicinterfaces
DISSERTATION
zurErlangungdesakademischenGrades
DoktorderNaturwissenschaften(Dr.rer.nat.)
eingereichtvonDipl.-Math.NorbertKleemann
geb. am31.08.1981inOrosháza
genehmigteDissertation
TECHNISCHE UNIVERSITÄT BERLIN
FakultätII-MathematikundNaturwissenschaften
Promotionsausschuss:
Vorsitzender: Prof. Dr. WolfgangKönig
Gutachter: Prof. Dr. DietmarHömberg
Prof. Dr. JanSokolowski
Dr.habil.AndreasRathsfeld
TagderwissenschaftlichenAussprache: 19. August2011
Berlin2011
D83In der vorliegenden Arbeit werden konische Diffraktionsprobleme bei nichtglatten
Diffraktionsgitternuntersucht. ZielistdieBerechnungvonFormableitungen,welchezur
RekonstruktionderstreuendenStrukturgenutztwerdenkönnen. Dazuwerdenzunächst
A-priori-AbschätzungeningewichtetenSobolevräumenvomKondratiev-Typbewiesen.
Anschließend werden Aussagen zu Existenz und Eindeutigkeit von Lösungen in diesen
Räumen getroffen. Darauf aufbauend wird dann mit Hilfe der Theorie nichtlokaler Stö-
rungenelliptischerRandwertproblemedieExistenzundEindeutigkeitvonFormableitun-
gengezeigt. DieFormableitungenwerdenanschließendcharakterisiertalsLösungenvon
DiffraktionsproblemenmitgleichemOperator,abermodifiziertenrechtenSeiten. Dadie
Formableitungen bei Anwesenheit von Ecken im Diffraktionsgitter eine niedrige Regu-
laritätaufweisen,wirdzurnumerischenBerechnungeinAnsatzvorgeschlagen,derdarin
besteht, die Singularitäten an den Ecken mit Hilfe glatter Funktionen abzuschneiden.
Dann wird eine Randintegralformulierung für das modifizierte Problem, welches die
Formableitungen charakterisiert, hergeleitet. Abschließend werden für einige Beispiele
numerischeResultatevorgestellt.
Zusammenfassung„DasWissen,dasMachtist,kenntkeineSchranken,
wederinderVersklavungderKreaturnochinder
WillfährigkeitgegendieHerrenderWelt.”
THEODOR W. ADORNO, MAX HORKHEIMER
DialektikderAufklärung2.1 TheMaxwellsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Thequasi-periodicHelmholtzproblem . . . . . . . . . . . . . . . . . . . . 9
3.1 Aprioriestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Existenceanduniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Thevariationalformulation . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 TheFredholmproperty . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Fredholmindexandinvertibility . . . . . . . . . . . . . . . . . . . . 37
4.1 Existenceandregularityofmaterialandshapederivatives . . . . . . . . . 40
4.1.1 Smoothperturbationsoftheidentity . . . . . . . . . . . . . . . . . . 40
4.1.2 Perturbationswhichchangeanglesatcornerpoints . . . . . . . . . 45
4.2 Characterizationoftheshapederivative . . . . . . . . . . . . . . . . . . . . 49
5.1 Regularizationofcornersingularities . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 Construction of a smooth cut-off function and convergence of the
cut-offprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.2 Applicationtoboundaryvalueproblems . . . . . . . . . . . . . . . 65
5.2 Formulationwithintegralequations . . . . . . . . . . . . . . . . 66
5.2.1 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Boundaryintegralequationsfortheshapederivative . . . . . . . . 68
5.2.3 Equivalenceandsolvabilityoftheintegralequations . . . . . . . . 71
5.3 ComputedRayleighcoefficients . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Anotherpossiblenumericalapproach . . . . . . . . . . . . . . . . . . . . . 77
derivativesMaterialIntro4Computation15ConicalsolutionsshapofContentsy5rittables62e6ofry5p1ectiveductionList2guresdiractionListRegulaSummaeandshaperspand79derivativesof4085derivativesof3868We consider the scattering of a time-harmonic electromagnetic plane wave by a diffrac-
3tion grating inR . The simplest form of such a diffraction grating is a periodic interface
between two materials with different dielectric constants. More precisely, the grating
surface is a perturbation of the (x ,x )-plane, which is assumed to be periodic in the1 3
x -direction and invariant in the x -direction. Scattering by such gratings occurs in the31
micro-optics industry, where optical devices with certain features have to be designed,
see e.g. [1]. Additionally, it is important to solve the inverse problem of shape recon-
struction, i.e. to determine the grating structure from measured data of the diffracted
wave[20]. Mathematically,thiscanbeformulatedasaninverseproblemfortheMaxwell
equations. See the books of Petit [35] and Colton/Kress [12] for detailed explanations.
Undertheassumptionofaperiodicgratingilluminatedbyaplanewaveitispossibleto
2reduce the 3D Maxwell transmission problem to a system of Helmholtz equations inR
whicharecoupledbyconditionsontheinterface. Theinverseproblemcan
thenbesolvedbyaniterativeNewton-typemethod,whichmakesuseofcertainconcepts
of differentiability with respect to the domain. The theory of shape calculus and shape
optimization has been thoroughly investigated for example by Sokolowski and Zolesio
[44] and Simon [43]. Since inverse problems of this kind are typically ill-posed, iterative
methodsrequireregularization.
In the past, different settings and approaches have been discussed. Elschner, Schmidt
et al. focused on Eulerian derivatives of shape functionals. In their papers, these fuc-
tionals depend continuously on the Rayleigh coefficients of the scattered waves. The
Rayleigh coefficients themselves depend on the shape of the diffraction grating. These
investigations cover existence and uniqueness results for the direct problem in standard
Sobolev spaces [14] and gradient formulas for both classical TE/TM diffraction and bi-
narygratings[16]andforconicaldiffractionbygeneralnon-smoothstructures[15]. They
1alsoprovideanexistenceresultformaterialderivativesofsolutionswhichareH -regular
andstatementsaboutasymptoticexpansionsofthefieldcomponentsnearcornerpoints.
The formulas given in these papers involve solutions of direct and adjoint problems.
Therefore,twodifferentdiffractionproblemshavetobesolvedineachiterationstep.
Adifferentapproachfortheinverseproblemusestheshapederivativeofthesolution
operator F :G! u for a fixed incident wave, depending on the interfaceG. An iterative
methodisgivenforexamplebytheminimizationproblem

1 a
0 2 2min jjF (G )(G G ) u+F(G )jj + jjG G jj , (1.1)n n+1 n n n+1 n
2 2
0where F is the shape derivative of F, a is a regularization parameter, u is a given so-
5
1Intro7duction6
lution for which the scattering surface has to be determined, andG is an initial guess0
for the diffracting surface. In practical applications, u would be the measurement of
the diffracted wave. Potthast, Chandler-Wilde and Hohage and Schormann character-
izedshapederivativesofsolutionsofDirichletandNeumannboundaryvalueproblems
[36,37],transmissionproblemsforbounded,smoothdomains[22]andofDirichletprob-
lemsforunboundedroughsurfaces. Thesearesurfaceswhicharedescribedbycontiuous
non-periodic functions with Hölder continuous gradients [11]. The shape derivatives
are characterized as solutions of problems with the same operator and different right-
hand sides. These results were proven by representing the solution as single layer or
double layer potentials and taking the shape derivative of the resulting boundary in-
tegrals. Hettlich [21] obtained the same results for Dirichlet and Neumann problems,
and additionally for a transmission problem with a smooth interface, by means of weak
formulations of these problems. Kirsch [24] also employed this method for a Dirichlet
problem with a smooth periodic grating. This ansatz works if the shape derivative has
1H -regularity. However,iftheboundariesarenon-smooth,theshapederivatives,ifthey
1exist, are no longer in H . Bochniak and Cakoni [9] suggested a different approach for
non-smoothboundaries,usingnon-localperturbationtheoryandKondratiev’sweighted
Sobolev spaces [29, 25]. They showed shape differentiability of solutions for Dirichlet
and Neumann problems for domains with corners with the help of an a priori estimate.
Inthiswork, thisansatz isusedtoinvestigateshapederivativesof solutionsofasystem
of Helmholtz equations coupled by transmission conditions on a periodic, non-smooth
interface.
This thesis is structured in the following way. The second chapter recalls the coni-
caldiffractionproblemin threedimensions andits reductionto aHelmholtz problemin
a two-dimensional periodic cell. Chapter 3 introduces Kondratiev’s weighted Sobolev
spacesandshowsanaprioriestimateforconicaltransmissionproblemsinthesespaces.
This estimate can be sharpened if the solution is unique. In the second part of the third
section a uniqueness result is shown for absorbing materials, which makes use of a for-
merresultofElschner,Schmidtetal. [14]instandardSobolevspaces. Thefourthchapter
discussesexistence,uniquenessandthecharacterizationofshapederivativesofsolutions
totheconicaldiffractionproblem. Here,non-localperturbationtheoryandtheansatzof
Bochniak and Cakoni are used. We consider perturbations which preserve the opening
angles at corner points as well as perturbations which change the angles. Finally, the
shapederivativeischaracterizedasasolutionofaconicaldiffractionproblem. Morepre-
cisely, the solution operator is the same as for the original problem, only the right-hand
side is changed. For interfaces with corners, as opposed to smooth interfaces, the right-
hand sides of the transmission conditions involve terms which are concentrated on the
cornerpoints.
Althoughalloftheseresultsareformulatedforperiodicgratings,theapproachisgen-
eral and can be used also to show existence of shape derivatives and to provide their
characterization for transmission problems e.g. for bounded obstacles with corners. As
mentioned above, for Dirichlet and Neumann boundary conditions on bounded obsta-
clesthiswasdonein[9]. Infact,theinvestigationoftransmissionproblemsforbounded7
obstacles should be simpler than it is in the case of periodic gratings, because here the
radiationconditionleadstosometechnicaldifficulties,aswewillseeinChapter3.
The last chapter deals with numerical computation of shape derivatives. In order to
dealwiththesingularitiesatthecornerpoints,acut-offansatzisproposedwhichreduces
thetheoreticalresultsintheKondratievspacestothesettinginstandardSobolevspaces.
Wepresenttheideaofthisansatzandproveconvergence. Thenweuseresultsobtained
bySchmidt[39]togivearepresentationoftheshapederivativeasasolutionofasystem
of boundary integral equations, which can then be discretized using e.g. a collocation
method. Finally, examples for numerical computations of shape derivatives for some
simplegeometriesarepresented.Weconsideratime-harmonicincomingplanewavewithfrequencywilluminatingaperi-
3odicdiffractiongratinginR dividingtwomaterialswithdifferentdielectriccoefficients.
Thesurfacestructureisassumedtobeinfiniteandperiodicin x -directionandinvariant1
in x -direction. It is then determined by a profile curveG being the intersection of the3
interface with the (x ,x ) plane. In the conical diffraction case the angle between the in-1 2
coming wave direction and the (x ,x ) plane is allowed to be non-zero. If the direction1 2
oftheincomingwaveliesinthe (x ,x ) plane,wehaveTEdiffraction,TMdiffractionor1 2
acombinationthereof,dependingonthepolarization.
X_2
incoming
wave
π/2−φ−
X_1
X_3
Thediffractiongrating
Sincetheincomingwaveistime-harmonic,i.e. itadmitstheform

(i) (i) iax ibx +igx iwt iax ibx +igx iwt (i) (i) iwt1 2 3 1 2 3
E ,H = pe e ,qe e = E ,H e , (2.1)
3where p,q 2 R , k = (a, b,g) is the wave vector and k/jkj is the direction of the
incomingwave,weobtainthetime-harmonicMaxwellequationsfortheelectromagnetic
field (E,H), which is the sum of the incoming wave and the reflected wave above the
(i) (i) (r) (r)grating, i.e. (E ,H ) = (E ,H ) + (E ,H ). Below the grating it is equal to the1 1
8
22.1:diractionThesystemConicalell2.1MaxwFigure2.2.Thequasi-periodicHelmholtzproblem 9
(t) (t)transmittedwave (E ,H ) = (E ,H ). TheMaxwellsystemis2 2
rE = iwmH
(2.2)
rH = iw#E
withtransmissionconditions
n(E E ) = 01 2
(2.3)
n(H H ) = 021
on the interfaceGR, where n is the unit normal vector to GR, m is the magnetic
permeability and # is the dielectric coefficient. We will assume that m is constant, that
# = # abovethegratingand# = # belowthegrating,respectively. Here# > 0and#+ +
areconstant. Iftheincomingwaveisoftheform(2.1),then
igx3(E,H)(x ,x ,x ) = (E,H)(x ,x )e , (2.4)1 2 3 1 2
and the above Maxwell system can be reduced to a system of Helmholtz equations for
the third components E of E = (E ,E ,E ) and H of H = (H ,H ,H ) defined in the3 1 2 3 3 1 2 3
cross-section plane (x ,x ), which is described in the next subsection. For details, see1 2
[14],[15],[17]andthefollowingsection.
In this section we describe how in the given situation the Maxwell system reduces to a
systemofHelmholtzequationsintwodimesions. TheMaxwellequationsgive

i ¶ ¶ ¶ ¶
E = H igH ,igH H , H H ,3 2 1 3 2 1
w# ¶x ¶x ¶x ¶x2 1 1 2
(2.5)
1 ¶ ¶ ¶ ¶
H = E igE ,igE E , E E ,3 2 3 21 1iwm ¶x ¶x ¶x ¶x2 1 1 2
whichimmediatelyimplies
2i ¶ ig ¶ g
E = H + E + E .1 3 3 12 2
w#¶x w #m¶x w #m2 1
Thisis
2 2
w #m g i ¶ ig ¶
E = H + E ,1 3 32 2
w #m w#¶x w #m¶x2 1
Helmholtz2.2Theeriopdicroblemquasi-p2.2.Thequasi-periodicHelmholtzproblem 10
andlikewisefor E , H and H . Moreover,takingthederivativewithrespectto x ofthe2 1 2 1
secondcomponentsinthefirstandinthesecondlineof(2.5)gives
2
¶ ¶ ¶
E = ig E iwm H ,3 1 22
¶x ¶x
¶x 1 11
2
¶ ¶ ¶
E = ig E +iwm H .3 2 22
¶x ¶x ¶x2 22
When#isconstant,theMaxwellequationrH = iw#Eimplies
divE = 0.
Usingthistogetherwith(2.4)andthethirdcomponentofthefirstlineof(2.5),weget

¶ ¶ ¶ ¶ ¶ ¶
+ E = ig E +ig E iwm H H3 1 2 2 12 2
¶x ¶x ¶x ¶x
¶x ¶x 1 2 1 21 2
¶ (2.6)2= ig E w m#E3 3
¶x3
2= k E .3
Analogously,

¶ ¶ 2+ H = k H . (2.7)3 32 2
¶x ¶x1 2
p2 2 2 Letk := k g ,where k = w m# andk(x) := k if x2W ,andanalogouslyfor#.


Notethatthematerialconstantscanbecomplexvalued. Therefore,wechoosethesquare
p pif if/2rootof z = re withf2 [0,2p)tobe z = re . Thenwehave
¶ ¶2
k E = imw H +ig E ,1 3 3
¶x ¶x2 1
¶ ¶2
k H = i#w E +ig H ,1 3 3
¶x ¶x2 1 (2.8)
¶ ¶2
k E = imw H +ig E ,2 3 3
¶x ¶x1 2
¶ ¶2
k H = i#w E +ig H ,2 3 3
¶x ¶x1 2
from which follows that, under the assumption k = 0, the first two components of the
electric and the magnetic field are determined by the third one. Writing n = (n ,n ,0),1 2
thetransmissionconditions(2.3)implythatthejumpof
nE = (n E , n E ,n E n E ),2 3 1 3 1 2 2 1
nH = (n H , n H ,n H n H )2 3 1 3 1 2 2 1
6

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