A finite element based fast eigensolver for three dimensional anisotropic photonic crystals

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Journal of Computational Physics

www.elsevier.com/locate/jcp

A finite element based fast eigensolver for three dimensional anisotropic photonic crystals

So-Hsiang Chou

^a,∗

, Tsung-Ming Huang

^b,∗

, Tiexiang Li

^c

, Jia-Wei Lin

^d

, Wen-Wei Lin

^d

aDepartmentofMathematicsandStatistics,BowlingGreenStateUniversity,OH,USA bDepartmentofMathematics,NationalTaiwanNormalUniversity,Taipei116,Taiwan cSchoolofMathematics,SoutheastUniversity,Nanjing211189,People’sRepublicofChina dDepartmentofAppliedMathematics,NationalChiaoTungUniversity,Hsinchu300,Taiwan

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received17December2017

Receivedinrevisedform3January2019 Accepted5February2019

Availableonline4March2019

Keywords:

Maxwell’sequations

Three-dimensionalanisotropicphotonic crystals

Face-centeredcubiclattice Finiteelementmethod

Null-spacefreeeigenvalueproblemandfast Fouriertransforms

ThestandardYee’sschemefortheMaxwelleigenvalueproblemplacesthediscreteelectric fieldvariable atthe midpointsoftheedges ofthegridcells.Itperformswell whenthe permittivityis ascalar field.However,when the permittivityis aHermitianfull tensor fielditwouldgenerateun-physicalcomplexeigenvaluesorfrequencies.Inthispaper,we proposea finiteelementmethod whichcanbe interpreted as amodifiedYee’sscheme toovercomethisdifficulty. ThisinterpretationenablesustocreateafastFFTeigensolver that cancomputeveryeffectivelythe bandstructureof theanisotropic photoniccrystal withSCandFCClattices.Furthermore,weovercometheusuallargenullspaceassociated with the Maxwelleigenvalue problem by derivinga null-space free discrete eigenvalue problemwhichinvolvesacrucialHermitianpositivedefinitelinearsystemtobesolvedin eachoftheiterationsteps.ItisdemonstratedthattheCGmethodwithoutpreconditioning convergesin37iterationsevenwhenthedimensionofamatrixisaslargeas5,184,000.

©^{2019ElsevierInc.Allrightsreserved.}

1. Introduction

InthispaperweareconcernedwithafastnumericaleigensolverfortheMaxwelleigenvalueproblem

∇ × ∇ × E = λ ε _E , λ = μω

²

,

(1a)

∇ · ε E =

^{0 (1b)}

posedon,whichisa primitivecellwithlattice vectorsa,=¹,2,3,ofa periodicstructure inR^{3 such asa photonic} crystal(PC). Here in (1), E=E(x) isthe electric field,

ε

is theperiodic permittivity complextensor field, and

μ

isthe constantpermeabilityscalarfield.Mathematically,thisproblemarisesnaturallyfromthesource-freeMaxwellequationsby assumingthat thetimedependentelectricandmagneticfieldsaretime-harmonic withafactorofe⁻ιωt (

ι

=√

−^1).Inan anisotropicmedium,thepermittivity

ε

maytakeaHermitianform[27–29]

*

Correspondingauthors.

E-mailaddresses:chou@bgsu.edu(S.-H. Chou),min@ntnu.edu.tw(T.-M. Huang),txli@seu.edu.cn(T. Li),jiawei.am05g@g2.nctu.edu.tw(J.-W. Lin), wwlin@math.nctu.edu.tw(W.-W. Lin).

https://doi.org/10.1016/j.jcp.2019.02.029

0021-9991/©^{2019ElsevierInc.Allrightsreserved.}

ε =

⎡

⎣ ε

11

ιε

12

ιε

13

− ιε

₁₂

ε

₂₂

ιε

₂₃

− ιε

13

− ιε

23

ε

33

⎤

⎦ .

(2)

Anisotropic PC material has many interesting and novel applications in practice, such as the hollow semiconductor nanorod [22], femtosecond laser pulses [38], photonic crystalfibers [2,6,37], among others. In 1993, it was shownthat the anisotropy ofthe material could split the degenerate band andnarrow the band gap ofa PC[41]. Laterin 1998, a largebandgapopeningbyusingananisotropicPCwas shownin[23,24].Thereafter,toexhibitlargebandgaptenability,a three-dimensional(3D)inverseopalPCstructureinfiltratedwithliquidcrystalswasproposedin[7].

Computational methods playan equallyimportantrole in the studyofthe anisotropic material.The spectral element method [30], the finite-element method (FEM) [3,4,10,11,35], the mixed finite element method [21,25,26], the finite- difference frequency-domain method(FDFD) [13,36], andtheplane wave expansion method[20] have been proposed to compute thebandstructureoftheanisotropicPC.Inthispaper,weareinterestedincreatingafiniteelementmethodthat resemblesthefinitedifferenceYee’sscheme[40] duetoitsrobustnessinsolvingtheMaxwell eigenvalueproblemsassoci- ated withisotropicmaterials. Yee’sschemeonlydiscretizes(1a) andleavesthedivergence-freecondition(1b).Thereason forthat isformerlythe exactsolutionof(1a) satisfies(1b). Indeed,thisstrategy workswell forisotropicmaterials. Thus, when itcomestothediscretizationof(1a)-(1b) orits manypossibleweak formulationsthroughfinitedifferenceorfinite element schemes, the traditionalwisdom is to usespatial discretizations that admit discrete analogueof the divergence freecondition.ThisclassofspatialdiscretizationsincludesYee’sscheme[14,16,40],theWhitneyform[5,39],thecovolume discretization[34],themimeticdiscretization[19] andtheedgeelement[32,33].Itshouldbenotedthatinallthesecases thepermittivity

ε

isascalarfield.Inthispaper,wewillfollowthisapproach,i.e.,wewilldiscretize(16a) insuchawaythat enablesustorigorouslyshowinTheorem5ofSec.5.2thatthediscretizedversion oftheweakdivergencefreecondition (16b) isautomaticallysatisfied,whichisbynomeansatrivialtask.Ontheotherhand,therearecertainlyotherapproaches, e.g., [4] thatkeep(16b). Theyhavethe advantageoftheavailabilityofageneralconvergencetheory of[3] foreigenvalue problemsposedongeneraldomains.However,forcuboiddomains ourapproachcan generatefasteigensolvers.Theavail- ability ofsuch solversis essential whenthesame largeMaxwell eigenvalue problemhas tobe solved alarge numberof timeswhenthematerialparametersintheconstitutiverelationsmustbevariedindesignproblems[17].

On a 3D PC,due to the divergence-free condition (1b), a large nullspace of the underlyingeigenvalue problem will be produced. The presenceof the huge nullspace is a numericallychallenging problem in solving the eigenvalueprob- lem.Moreover, howtoefficientlysolvetheassociatedlinearsystemineachstepoftheeigensolverisanotherchallenging problem.

Inthe3Dnon-anisotropiccases(isotropicorbi-isotropicmedium),wehavehadsuccesses[8,14] intacklingtheseobsta- clesthatwerepresentwhenusingYee’sscheme[40] inwhichE^{isassignedtotheedgecentersandthemagneticfield}H^to thefacecenters.Afterthediscretizationphase,ageneralizedeigenvalueproblem(GEP)withalargenullspaceisproduced.

Itisratherremarkablethatanull-spacefreemethodwithFFT(fastFouriertransform)-basedmatrix-vectormultiplications hasbeenfound[8,14] toexplicitlydeflatethishugenullspacesothattheGEPistransformedintoanull-spacefreestandard eigenvalueproblem(NFSEP).ThecoefficientmatrixoftheNFSEPisHermitianpositivedefinite(HPD)andwell-conditioned.

Therefore,theassociatedlinearsystemineachiterationoftheinvertLanczosmethodcanbeefficientlysolvedbythecon- jugategradient(CG)methodwithoutanypreconditioner.InthedispersivePCwithisotropicmedium(i.e.,

ε

isapiecewise nonlinear function of λ), a Newton-typeiterative method[18] withthe null-space free methodis proposed to solve the associatednonlineareigenvalueproblem.

The successes of the null-space free method in isotropic and bi-isotropic PCs are partially due to the fact that the permittivity

ε

isapiecewiseconstantscalarfield,andthereisnoconflictintheplacementofdiscretepermittivityvariable.

Inotherwords,itmesheswellwiththestandarddiscretizationbyYee’sscheme.However,fortheanisotropicmaterial,there is aconflictinthe placementofdiscretetensor permittivityvariable.Howtomodify Yee’sschemeso thatthe null-space freemethodcanbeappliedtoatensorfield

ε

isoneoftheissuesaddressedinthispaper.

In an anisotropic medium,the permittivity

ε

isgiveninthe tensorform (2) where

ε

i j, for1≤ⁱ,j≤^{3, are piecewise} constants or piecewisenonlinear functions. Moreover,in [27,28],the tensor

ε

isHPD. AFDTDmethod [36] for modeling electromagnetic scatteringis proposed todiscretizethe generalanisotropicobjects.The importantstructure andproperty of the tensor

ε

in (2) were not considered in this discretization.In this paper, we will exclusively consider the casein whichtheHermitianmatrixin(2) satisfiesthat

ε

i j arepiecewiseconstant,andproposeanewfiniteelementschemewhich preservestheHermitianandpositivedefiniteness (if

ε

has)inthediscretization.Themaincontributionsinthispaperare summarizedasfollows:

•^{Our new finite element method can be} interpreted asa modified finite difference Yee’s scheme. As a consequence, the discretizationofthedouble curloperator inourproposed methodisequivalent tothe corresponding operatorin Yee’s scheme so that the null-space free method[8,14] can be applied to solve the resultinggeneralized eigenvalue problem.Furthermore,awell-knownfeatureoftheclassicalfinitedifferenceYee’sschemeforisotropicmaterialsisthat thedivergencefreeconditionofthefieldisautomaticallysatisfiedbythefinitedifferencesolution.Inotherwords,one needs not todiscretize (1b) explicitly.We show inTheorem5 thatour methodpreservesthisfeatureforanisotropic materialsaswell.

• ^{Fortheright-handsideof(1a),ourmethodprovidesan}HPD-preservingcoefficientmatrixBε,i.e.,if

ε

isHPD,sois Bε. Furthermore,thematrixBεcanbefactorizedbyFFT-matrices,diagonalmatricesanda3×^{3 blockdiagonalmatrixwith} asuitablepermutation.Thespecialfactorizationof Bε resultsinminimaleffort ininverting Bε whilestillkeepingthe correctphysicsintact(realfrequencies).

• ^{Basedon theabove important}factorizationof Bε,the null-spacefree methodcanbe appliedto reduce theGEPinto a NFSEP with HPD coefficient matrix. The NFSEP can be solved by the invert Lanczos method and the CG method withoutpreconditioningcanbeusedtosolvethelinearsystemefficientlyateachiteration.Thecomputationalcostof thematrix-vector multiplicationin theCGmethodis almostequal tothat ofthe null-spacefree methodinisotropic PCs.

This paper is organized as follows. We introduce our proposed finite element method in Section 2. The equivalence betweenourmethodandYee’sschemeforthediscretizationofdouble-curloperatorisshowninSection3.Thediscretization of

ε

_E isintroduced inSection 4. In Section 5, we show that the coefficient matrix corresponding to

ε

is HPD, andthe divergencefreeconditionisautomaticallysatisfiedfortheapproximateelectricfield.Finally,thenumericalefficiencyofthe developedschemesisstudiedinSection6.

2. AfiniteelementmethodforMaxwelleigenvalueproblem

Inthissection, wepropose a weakformulationfortheMaxwell eigenvalueproblem(1) anda Galerkinfiniteelement methodtosolveit.

2.1. Weakformulationofproblem(1)

Denoteby(f,g)thecomplexinnerproductofC^{N-valuedfunctionsinL2}(),i.e.,

(

f

,

g

) =

N i=1

fi

(

x

)

gi

(

x

)

dx

=

g^∗

(

x

)

f

(

x

)

dx

.

(3)

Here“∗^{”denotestheconjugatetranspose.Let} beaprimitivecellwithlattice vectorsa,=¹,2,3.Itisknownthatthe electricfieldsatisfiesthequasi-periodiccondition

E(

^x

+

^a

) =

^eι2πk·a

E(

^x

), =

¹

,

2

,

3

,

(4) forallx∈R^{3,wheretheBlochwavevectorkliesinthefirstBrillouinzone}B^.

CorrespondingtoE^{,wecandefineaperiodicfunctionu,itsassociatedmodulatingamplitude,as}

u

(

x

) =

^{e−ι2πk·x}

E(

^x

),

(5)

whichsatisfiestheperiodiccondition

u

(

x

+

^a

) =

^u

(

x

), =

¹

,

2

,

3

.

(6)

ThusonecanalsostudytheMaxwelleigenvalueproblem(1) bytheequivalentsystemintermsofu:

∇

k

× ∇

k

×

^u

= λ ε

u

,

(7a)

∇

k

· ε

u

=

⁰

,

(7b)

wheretheshiftedgradientoperator∇k:= ∇ +

ι

2

π

k.

Remark1.This u-system(7) has the advantageof havingFourier analysisat ourdisposal when analyzingthe existence, stability,andconvergenceissues,see[11,35].However,inthispaperwearemoreconcernedwithcreatingfasteigensolvers and the E^{-system (1) will be used. We will have occasions of} interpreting some nice spectral-theory results from [35, Chapter4] backtotheE^-system.

Weneedtointroducesomecomplex-valuedfunctionspaces.Let H

(

curl

,) := {

^v

: → C

³

|

^v

, ∇ ×

^v

∈

^L2

()}

withthenorm

||

v

||

²_H(curl,)

:= ||

v

||

²_L2()

+ ||∇ ×

v

||

²_L2()

,

andlet

C^∞_per

() := { φ |

: φ : R

³

→ C , φ ∈

^C∞

( R

³

), φ (

x

+

^a

) = φ (

x

), =

¹

,

2

,

3

} ,

C^∞_per

() := {

v

|

:

v

: R

³

→ C

³

,

v

∈

C^∞

(R

³

),

v

(

x

+

a

) =

v

(

x

), =

1

,

2

,

3

}.

Definetheperiodicfunctionspaces

Hper

(

curl

,) :=

the closure ofC^∞_per

()

inH

(

curl

, ),

(8a)

H_per¹

() :=

the closure ofC^∞_per

()

inH¹

(),

(8b)

andtheirassociatedquasi-periodicfunctionspaces

H_qper

(

curl

, ) := {

^v

:

^{e−ι2πk·xv}

∈

^Hper

(

curl

, )},

^(8c)

H_qper¹

() := {φ :

^{e−ι2πk·x}

φ ∈

^H1per

()}.

^(8d)

Note that inview of(5), (6) and the Fourierseries representation,functions inthe above two spaces satisfy the quasi- periodicconditions(4).

Theorem1.ForanyE,W∈^C∞(R³)∩^Hqper(curl,)andφ∈^C∞(R³)∩^Hqper¹ (),wehave

(∇ × ∇ × E,W) = (∇ × E, ∇ × W),

(9a)

(∇( ε E), φ) = ( ε E, ∇φ).

(9b)

Proof. SinceE^andW^{aresmooth,Green’sformulaimplies}

(∇ × ∇ × E,W) = (∇ × E, ∇ × W) +

∂

ν × ∇ ×

E

· W

d A

,

(10)

where

ν

isthe outwardunit normalvector totheboundary. Forconvenience, wenamethe sixboundary surfacesofthe primitivecellas∂⁰_a1,∂¹_a1,∂⁰_a2,∂¹_a2,∂⁰_a3 and∂¹_a3,where

∂

^s_a1

= {

^sa1

+

^aa2

+

^ba3

|

^a

,

b

∈

[0

,

1]

},

∂

^s_a2

= {

^sa2

+

^aa3

+

^ba1

|

^a

,

b

∈

[0

,

1]

},

∂

^s_a3

= {

^sa3

+

^aa1

+

^ba2

|

^a

,

b

∈

[0

,

1]

},

fors=⁰,1.Thenthesurfaceintegraltermin(10) canbedividedas

3

=1

⎛

⎜ ⎜

⎝

∂⁰a

ν

_,0

× ∇ × E · W

d A

+

∂¹a

ν

_,1

× ∇ × E · W

d A

⎞

⎟ ⎟

⎠

where

ν

,0= −

ν

,1 are the outward unit normal vectors on ∂⁰_a and ∂¹_a, respectively. In particular, from the quasi- periodicityofE^andW^,wehave

∂⁰_a

ν

_,0

× ∇ × E (

x

) · W (

x

)

d A

=

∂⁰_a

ν

_,0

× ∇ × E (

x

+

^a

) · W (

x

+

^a

)

d A

=

∂¹a

− ν

_,1

× ∇ × E (

x

) · W (

x

)

d A

,

for=¹,2,3.Therefore,theresultin(9a) isproved.

Toshow(9b),weobservethat

(∇ · ε E, φ) = ( ε E, −∇φ) +

∂

ε E · ν φ

d A

.

(11)

Again,thesurfaceintegraltermin(11) canbedividedas

3

=¹

⎛

⎜ ⎜

⎝

∂⁰a

ε E · ν

,0

φ

d A

+

∂¹_a

ε E · ν

,1

φ

d A

⎞

⎟ ⎟

⎠ .

Fromtheperiodicityof

ε

,andthequasi-periodicityofE^andφ,wehave

∂⁰a

ε _E (

x

) · ν

_,0

φ (

x

)

d A

=

∂⁰a

ε (

x

+

^a

) E (

x

+

^a

) · ν

_,0

φ (

x

+

^a

)

d A

=

∂¹a

ε (

x

) E (

x

) · ( − ν

_,1

)φ (

x

)

d A

for=¹,2,3.Theresultin(9b) isobtained. 2

Inview ofTheorem1,asimpledensityargumentsuggeststhatweconsiderthefollowingweakformulationofproblem (1):Findapair(λ,E)withλ∈C^andE∈^Hqper(curl,)suchthat

(∇ × E,∇ × W) = λ( ε E,W), ∀ W ∈

Hqper

(

curl

, ),

(12a)

( ε E,∇φ) =

0

, ∀ φ ∈

Hqper

().

(12b) However, froma computationalelectromagnetics viewpoint,itis moreconvenienttowork withthefollowing twospaces whosedescriptionismoreaccessible:Forfunctionsu:R³→C^3andφ:R³→C^,wedefine

H_curl

() = {

^u

|

:

^u

∈

^H

(

curl

, ),

u

(

x

+

^a

) =

^eι2πk·au

(

x

), =

¹

,

2

,

3

},

⁽¹³⁾ Hgrad

() = {φ|

: φ ∈

H¹

(), φ (

x

+

a

) =

e^ι2πk·a

φ (

x

), =

1

,

2

,

3

}.

(14) Remark2.Alternatively,the quasi-periodic condition in (13) can be viewed asa wayto extendthe functionsoutsideof ,whiletheregularityconditionu∈^H(curl,),togetherwiththequasi-periodiccondition, impliesatangentialboundary (trace)conditionon∂:

u_T

(

x

+

^a

) =

^eι2πk·auT

(

x

), =

¹

,

2

,

3

,

(15)

whereuT=

ν

×uisthetangentialtraceofu.

Infact, H_grad()=^H1qper() andH_curl()=^Hqper(curl,).It isnothard toseethatthefirst relationholds.The proof ofthesecondrelationand(15) usessimilarargumentsinprovingtwocharacterizationsofH0(curl)areequivalent(cf.[31, Theorem3.33] or[12,Theorem2.12]).Seealso[1,Lemma2.1].Theanalogyisnotsurprising, ifwe viewthedifferenceof (15) asazero-tracecondition.

Insummary,weshallconsidertheweakformulationofproblem(1):Findapair(λ,E)^withλ∈C^andE∈^Hcurl()such that

(∇ × E,∇ × W) = λ( ε E,W), ∀ W ∈

Hcurl

(),

(16a)

( ε E,∇φ) =

0

, ∀ φ ∈

H_grad

(),

(16b) where

ε

:R³→C^{3×3 isatensorfieldofthegeneralform(not}necessarilyHermitian)

ε =

⎡

⎣ ε

11

ε

12

ε

13

ε

₂₁

ε

₂₂

ε

₂₃

ε

31

ε

32

ε

33

⎤

⎦ .

(17)

Remark3.DuetotheexistenceofanexactdeRhamsequence(cf.[35,p.58])andRemark1,itimpliesthat

∇

Hgrad

()

⊂

Hcurl

().

(18)

ThuswecansetW= ∇φin(16a) andseethat[9] whenλ=^{0,thedivergencefreeconditionis}automaticallysatisfied.Our discreteweakformulationbelowwillrespectthisproperty(seesubsection5.2):

∇

H^h_grad

()

⊂

^Hh_curl

().

(19)

Remark4.ItmaybecuriousthatwhythereisnoneedinintroducingaLagrangemultiplierintheaboveweakformulation.

See[35,Proposition4.20] forsuchacase.Usingthetechniquesin[35] via(5),wecanshowaresultsimilartotheProp.4.20:

Suppose that

ε

isHPD. Thenforeach k∈B^{,firstBrillouinzone,theeigenvalue problem(16) hasan increasing sequence} {λj}j∈N ofnon-negativerealnumberstendingtoinfinity. Eacheigenvaluehasfinitemultiplicity andzeroisan eigenvalue ifandonlyifk=^{0.Thedetailswillbereportedelsewhere.Asa}consequence,zeroeigenvalueisnotonlynonphysical,but mathematicallyspuriousforthek=^{0 case,whenadiscreteweak}formulationoftheMaxwelleigenvalueproblemproduces zeroeigenvalue.Anull-spacefreemethodwouldbecalledforinsuchacase.

2.2. Discretizationsofweakformulationsof(16)

Inthissubsection,weuseaGalerkinfiniteelementmethodbasedontheedgeelementstoapproximatetheexactfield E^{in(16a).Tothisend,let Hh}_curl()andH^h_grad()be3N-dimensionallinearsubspacesofH_curl()andN-dimensionallinear subspaceofHgrad(),respectively(tobedefinedin(23) and(24),respectively).ThenastraightforwardGalerkinmethodfor (16a)-(16b) is:Findapair(λ,E)withλ∈C^andE∈^H_curl^h ()suchthat

(∇ ×

E

,∇ ×

W

)

_h

= λ( ε

E

,

W

)

_h

, ∀

W

∈

H^h_curl

(),

(20a)

( ε

E

, ∇ φ)

_h

=

⁰

, ∀ φ ∈

^Hh_grad

().

(20b)

Here(f,g)hisadiscreteinnerproducttobespecifiedlater.

Tosimplifythediscussion,weonlygivethedetailswhenisasimplecubic(SC)primitivecellwiththelatticetransla- tionvectorsa=^axˆ,=¹,2,3,where^xˆisthe-thunitvectorinR^{3.Alongtheway,wealsogive sufficientremarksfor} theface-centeredcubic(FCC)latticesothatitcanbehandledsimilarly.Lethx,hy,andhzbetheequalgridspacingsalong thex,y,andzdirections,respectively.Gridpointsareindexedfrom0 toni(i=¹,2,3)withn1=^a/hx,n2=^a/hy,n3=^a/hz. Thusthenumberoftheedgesparalleltothex-, y- andz-axes,respectively,excludingthoseontheright,rear,andtopsur- faces,isjustN=ⁿ1n2n3.

From nowon,weusexm=^mhx andx_mˆ = ˆ^mhx,wheremˆ =^m+¹₂^.Furthermore,form= −¹,0,. . . ,n1,let ˆc_mˆ(x)denote thecharacteristicfunctionwhosevalueisoneover[^xm,xm+¹]^{andzeroelsewhere,}ˆl_i denotethegenericglobalhatfunction supportedover[^xi−¹,xi+¹]^,i.e.,ˆ_li(x)=δi,fori=¹,. . . ,n1−^{1.Hereandhereafter,}δidenotestheKroneckerdelta.Similar notationsareusedforthe yandzdirections.

Wenowdefinepiecewiseconstantfunctions

c_ˆi

(

x

) = ˆ

c_ˆi

(

x

),

c_ˆj

(

y

) = ˆ

c_ˆj

(

y

),

c_kˆ

(

z

) = ˆ

c_kˆ

(

z

),

(21a) fori=⁰,n1−^{1, j}=⁰,n2−^1,k=⁰,n3−^1,and

c_0ˆ

(

x

) = ˆ

c_0ˆ

(

x

) +

e^ι2πk·a1c

ˆ

_nˆ1

(

x

),

c_nˆ1−1

(

x

) = ˆ

c_nˆ1−1

(

x

) +

e^{−ι2πk·a1}c

ˆ

-1

(

x

),

(21b) c_0ˆ

(

y

) = ˆ

^c_0ˆ

(

y

) +

^eι2πk·a2c

ˆ

_nˆ2

(

y

),

c_nˆ2−1

(

y

) = ˆ

^cn_ˆ2−1

(

y

) +

^{e−ι2πk·a2}c

ˆ

-1

(

y

),

(21c) c_0ˆ

(

z

) = ˆ

^c_0ˆ

(

z

) +

^eι2πk·a3c

ˆ

n_ˆ3

(

z

),

c_nˆ3−1

(

z

) = ˆ

^cn_ˆ3−¹

(

z

) +

^{e−ι2πk·a3c}

ˆ

-1

(

z

),

(21d) aswellaspiecewiselinearfunctions,

li

(

x

) = ˆ

li

(

x

),

lj

(

y

) = ˆ

lj

(

y

),

l_k

(

z

) = ˆ

l_k

(

z

),

(22a) fori,j,k=^0,and

l₀

(

x

) =

⎧ ⎪

⎨

⎪ ⎩

−

h⁻_x¹

(

x

−

x1

),

ifx

∈ [

x0

,

x1

],

e^ι2πk·a1h⁻_x¹

(

x

−

xn1−¹

),

ifx

∈ [

xn1−¹

,

xn1

],

0

,

otherwise

,

(22b)

l₀

(

y

) =

⎧ ⎪

⎨

⎪ ⎩

−

^h−y¹

(

y

−

^y1

),

ify

∈ [

^y0

,

y₁

] ,

e^ι2πk·a2h⁻_y¹

(

y

−

^yn2−1

),

ify

∈ [

^yn2−1

,

y_n2

] ,

0

,

otherwise

,

(22c)

l0

(

z

) =

⎧ ⎪

⎨

⎪ ⎩

−

^h−_z¹

(

z

−

^z1

),

ifz

∈ [

^z0

,

z₁

],

e^ι2πk·a3h⁻_z¹

(

z

−

^zn3−1

),

ifz

∈ [

^zn3−1

,

z_n3

],

0

,

otherwise

.

(22d)

Based on the piecewise constant andlinear functionsin (21) and (22), we now define bases forthe 3N-dimensional subspace H_curl^h ()andN-dimensionalsubspaceH^h_grad()in(20),respectively,as

H^h_curl

() =

^span

{ φ

ⁱ₁^,j,k

_i

, φ

₂^r,s,t

_j

, φ

^,₃^m,n_k

} ⊂

^Hcurl

()

(23) and

H^h_grad

() =

^span

{φ

^i,j,k

} ⊂

^Hgrad

(),

(24)

where_i,_j,_{karethestandardunitvectorsin}R^3,and

φ

ⁱ₁^,j,k

(

x

,

y

,

z

) :=

^c_ˆi

(

x

)

l_j

(

y

)

l_k

(

z

),

(25a)

φ

₂^r,s,t

(

x

,

y

,

z

) :=

lr

(

x

)

c_ˆs

(

y

)

lt

(

z

),

(25b)

φ

₃^,m,n

(

x

,

y

,

z

) :=

^l

(

x

)

l_m

(

y

)

c_nˆ

(

z

),

(25c)

φ

^i,j,k

(

x

,

y

,

z

) :=

^li

(

x

)

l_j

(

y

)

l_k

(

z

)

(25d)

withthesup-indicesvaryinginthesetS describedas

S

= {(

ⁱ

,

j

,

k

) :

ⁱ

=

⁰

, . . . ,

n₁

−

¹

,

j

=

⁰

, . . . ,

n₂

−

¹

,

k

=

⁰

, . . . ,

n₃

−

¹

}.

⁽²⁶⁾

The components of an approximate electricfield E∈^H_curl^h () are piecewise constant in one direction andpiecewise linearintheothertworemainingdirections,i.e.,

E₁

(

x

,

y

,

z

) =

(i,j,k)∈^S

E₁

|

_ˆi,j,k^c_ˆi

(

x

)

l_j

(

y

)

l_k

(

z

) ≡

(i,j,k)∈^S

E₁

|

_ˆi,j,k

φ

ⁱ₁^,j,k

,

(27a) E₂

(

x

,

y

,

z

) =

(i,j,k)∈^S

E₂

|

_i,ˆj,k^li

(

x

)

c_ˆj

(

y

)

l_k

(

z

) ≡

(i,j,k)∈^S

E₂

|

_i,ˆj,k

φ

ⁱ₂^,j,k

,

(27b) E₃

(

x

,

y

,

z

) =

(i,j,k)∈^S

E₃

|

_i,j,ˆk^li

(

x

)

l_j

(

y

)

c_kˆ

(

z

) ≡

(i,j,k)∈^S

E₃

|

_i,j,ˆk

φ

ⁱ₃^,j,k

.

(27c) Here in (27a), a subindex (ˆi,j,k) indicates that a quantity is located atthe midpoint ofthat edge. Bydirect evaluation, we seethat E₁|_ˆi,j,k=^E1(x_i+1

2,y_j,z_k).Similarinterpretations holdfor E₂ and E₃.The E_i’s arethefamiliaredge elements closelyrelated tothe finitedifference Yee’s scheme[9].In consistent withthe quasi-periodicconditions (15), it iseasily checkedthat

Left-Right

E₂

|

_n1,ˆj,k

=

^eι2πk·a1E2

|

_0,ˆj,k

,

j

=

⁰

:

ⁿ2

−

¹

,

k

=

⁰

:

ⁿ3

−

¹

,

E3

|

_n

1,j,ˆk

=

e^ι2πk·a1E3

|

_0,j,ˆk

,

j

=

0

:

n2

−

1

,

k

=

0

:

n3

−

1

,

^(28a) Front-Rear

E1

|

_ˆi,n2,k

=

^eι2πk·a2E1

|

_ˆi,0,k

,

i

=

⁰

:

ⁿ1

−

¹

,

k

=

⁰

:

ⁿ3

−

¹

,

E₃

|

_i,n2,ˆk

=

^eι2πk·a2E3

|

_i,0,ˆk

,

i

=

⁰

:

ⁿ1

−

¹

,

k

=

⁰

:

ⁿ3

−

¹

,

^(28b) Top-Bottom

E₁

|

_ˆi,j,n3

=

^eι2πk·a3E1

|

_ˆi,j,0

,

i

=

⁰

:

ⁿ1

−

¹

,

j

=

⁰

:

ⁿ2

−

¹

,

E₂

|

_i,ˆj,n3

=

^eι2πk·a3E2

|

_i,ˆj,0

,

i

=

⁰

:

ⁿ1

−

¹

,

j

=

⁰

:

ⁿ2

−

¹

.

^(28c)

Forthederivationofthegeneralizedeigenvalueproblemfor(20a),thevectorizedformofthediscreteelectricfieldE in (27) canbewrittenas

e

=

e₁ e₂ e₃

(29) with

e₁

=

^vec

{

^E1

|

_ˆi,j,k

},

^e2

=

^vec

{

^E2

|

_i,ˆj,k

},

^e3

=

^vec

{

^E3

|

_i,j,kˆ

},

for(i,j,k)∈^S.Here

vec

{

^F

} :=

⎡

⎢ ⎢

⎢ ⎣

vec

(

F

(

0

:

ⁿ1

−

¹

,

0

:

ⁿ2

−

¹

,

0

))

vec

(

F

(

0

:

ⁿ1

−

¹

,

0

:

ⁿ2

−

¹

,

1

))

.. .

vec

(

F

(

0

:

ⁿ1

−

¹

,

0

:

ⁿ2

−

¹

,

n₃

−

¹

))

⎤

⎥ ⎥

⎥ ⎦

forgivenF∈C^n1×n2×n3.

IntermsofthebasisfunctionsofH_curl^h (),weexpectthediscreteweakformulation(20a) tohaveamatrixform

Ae

= λ

B_εe

,

(30)

whereeisasin(29),AandBεarematrixrepresentationsoftheleftandrighthandsidesof(20a),respectively.InSections3 and4,wewillusethetrapezoidalquadrature

(

f

(

x

),

g

(

x

)) =

xn

xm

f

(

x

)

g

(

x

)

dx

≈

^hx 2

⎡

⎣

f

(

x_m

)

g

(

x_m

) +

²

n−¹

i=^m+¹

f

(

x_i

)

g

(

x_i

) +

^f

(

x_n

)

g

(

x_n

)

⎤

⎦

≡ (

f

(

x

),

g

(

x

))

_h (31)

to approximatethe discretizationin(20a) with H^h_curl()in (23) sothat thematrix A in (30) isidenticalto thatof Yee’s schemeuptoafactorhxhyhz.

3. Trapezoidalquadratureapproximationfor(∇×^E,∇×^W)

ForagivenE∈^Hh_curl(),from(20a) and(23) itsufficestoconsidertheexpressions

(∇ ×

^E

,∇ × φ

₁^i,j,k

i

), (∇ ×

^E

,∇ × φ

₂^i,j,k

j

), (∇ ×

^E

, ∇ × φ

₃^i,j,kk

)

(32) with (i,j,k)∈^{S in(26) and} φ^i,k,j, =¹,2,3, in (25). In fact, the inner product (·,·) in(32) involves the evaluationof typicaltermslike(l,c),(c,l),(l,l)and(c,c).Fortechnicalderivation,insteadoftheexactintegrals(l_r(x),l_i(x)),(l_s(y),l_j(y)) and(lt(z),lk(z))in(32),weusethetrapezoidalquadrature(31) toobtain

(

l_r

(

x

),

l_i

(

x

))

h

= δ

rih_x

, (

l_s

(

y

),

l_j

(

y

))

h

= δ

sjh_y

, (

l_t

(

z

),

l_k

(

z

))

h

= δ

tkh_z (33) fortheirapproximations.

Let

c-1

(

x

) =

^eι2πk·a1cn_ˆ1−1

(

x

),

c-1

(

y

) =

^eι2πk·a2cn_ˆ2−1

(

y

),

c-1

(

z

) =

^eι2πk·a3cn_ˆ3−1

(

z

).

(34) Thenthefollowingrelationshold:

∂

xl_i

(

x

) =

^h−x¹

(

c_ˆi−1

(

x

) −

^c_ˆi

(

x

)),

(35a)

∂

yl_j

(

y

) =

^h−y¹

(

c_ˆj−1

(

y

) −

^c_ˆj

(

y

)),

(35b)

∂

_zl_k

(

z

) =

^h−z¹

(

c_kˆ−1

(

z

) −

^c_kˆ

(

z

)),

(35c)

and

∂

y

φ

₁^i,j,k

=

^h−y¹c_ˆi

(

c_ˆj−1

−

^c_ˆj

)

l_k

, ∂

z

φ

₁^i,j,k

=

^h−z¹c_ˆil_j

(

c_kˆ−1

−

^c_kˆ

),

(36a)

∂

x

φ

₂^i,j,k

=

^h−_x¹

(

c_ˆi−1

−

^c_ˆi

)

c_ˆjl_k

, ∂

z

φ

₂^i,j,k

=

^h−_z^1lic_ˆj

(

c_kˆ−1

−

^c_kˆ

),

(36b)

∂

x

φ

₃^i,j,k

=

^h−_x¹

(

c_ˆi−1

−

^c_ˆi

)

l_jc_kˆ

, ∂

y

φ

₃^i,j,k

=

^h−_y^1li

(

c_ˆj−1

−

^c_ˆj

)

c_kˆ

,

(36c) for(i,j,k)∈^S.

Beforeprovingthefollowingtheoremconcerningacertainconsistencybetweenouredgeelementschemeandthefinite differenceYee’sscheme,wedenotethedifferencematrixwiththequasi-periodicconditionas

Kn,a

=

0 In−¹ e^ι2πk·a 0

−

In

∈ C

^n×n

,

(37)

for=¹,2,3.

Theorem2.ForanSClattice,thevectorizedformsoftheinnerproductsdescribedin(32)takeonthematrixrepresentation

⎡

⎢ ⎣

vec

{ ( ∇ ×

^E

, ∇ × φ

₁^i,j,k

_i

)

_h

}

(i,j,k)∈S

vec

{(∇ ×

E

, ∇ × φ

₂^i,j,k

_j

)

_h

}

(i,j,k)∈S

vec

{(∇ ×

^E

, ∇ × φ

₃^i,j,kk

)

h

}

(i,j,k)∈^S

⎤

⎥ ⎦ = (

h_xh_yh_z

)

C^∗Ce

,

(38)