forchiral media Electromagnetic field behavior of 3D Maxwell’s equations Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Electromagnetic field behavior of 3D Maxwell’s equations for chiral media

Tsung-Ming Huang

^a

, Tiexiang Li

^b

^,∗ , Ruey-Lin Chern

^c

, Wen-Wei Lin

^d

aDepartmentofMathematics,NationalTaiwanNormalUniversity,Taipei116,Taiwan bSchoolofMathematics,SoutheastUniversity,Nanjing211189,People’sRepublicofChina cInstituteofAppliedMechanics,NationalTaiwanUniversity,Taipei106,Taiwan

dDepartmentofAppliedMathematics,NationalChiaoTungUniversity,Hsinchu300,Taiwan

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received13July2018

Receivedinrevisedform19October2018 Accepted17November2018

Availableonline28November2018 Keywords:

Maxwell’sequations

Three-dimensionalchiralmedium Null-spacefreeeigenvalueproblem Shift-invertresidualArnoldimethod Anticrossingeigencurves

Resonancemode

Thisarticle focuses onnumerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell’s equationswithmagnetoelectriccouplingeffects whichmodel3Dreciprocalchiralmedia.

Itischallengingtosolvesuchalarge-scaleGEPefficiently.Wecombinethenull-spacefree methodwiththeinexactshift-invertresidualArnoldimethodandMINRESlinearsolverto solvetheGEPwithamatrixdimensionaslargeas5,308,416.Theeigenstructureisheavily determinedbythe chiralityparameter

γ

.We showthatalltheeigenvaluesare realand finiteforasmallchirality

γ

.Foracriticalvalue

γ

₌

γ

^∗,theGEPhas2×^{2 Jordanblocks} atinfinity eigenvalues. Numerical results demonstratethat when

γ

increasesfrom

γ

^∗, the 2×^{2 Jordan blockwill first splitinto acomplex conjugate eigenpair, thenrapidly} collidewiththerealaxisandbifurcateintopositive(resonance)andnegativeeigenvalues withmodulus smallerthan theother existingpositive eigenvalues. Theresonance band alsoexhibitsananticrossinginteraction.Moreover,theelectricandmagneticfieldsofthe resonance modes are localized inside the structure,with only aslight amountof field leakingintothebackground(dielectric)material.

©^{2018ElsevierInc.Allrightsreserved.}

1. Introduction

Mathematically, the propagationofelectromagneticfields inbianisotropicmedia is modeled bythe three-dimensional (3D)frequencydomainsource-freeMaxwell’sequationswiththeconstitutiverelations

∇ ×

^E

(

x

) = ιω

B

(

x

), ∇ ·

^B

(

x

) =

⁰

,

(1.1a)

∇ ×

H

(

x

) = − ιω

D

(

x

), ∇ ·

D

(

x

) =

0

,

(1.1b)

where

ω

isthefrequency, E

,

H

,

D andBaretheelectric,themagneticfields,thedielectricdisplacementandthemagnetic induction,respectively,atthepositionx∈R^3.Bianisotropicmaterialsareimportantclassesofcomplexmediaofwhichthe couplingeffectsbetweenelectricandmagneticfieldscanbedescribedbytheconstitutiverelations

*

Correspondingauthor.

E-mailaddresses:[email protected](T.-M. Huang),[email protected](T. Li),[email protected](R.-L. Chern),[email protected](W.-W. Lin).

https://doi.org/10.1016/j.jcp.2018.11.026

0021-9991/©^{2018ElsevierInc.Allrightsreserved.}

B

= ˜ μ

H

+ ˜ ζ

E

,

D

= ˜ ε

E

+ ˜ ξ

H

,

(1.2) where

μ

˜ is thepermeability,

ε

˜ is the permittivity,

ζ

˜ and

ξ

˜ are magnetoelectric parameters. Fordetaileddescriptions of bianisotropicmaterialswithrespectto(1.2),wereferto[1–6,22,24,25] andreferencestherein.

Ifthe photoniccrystal ismade ofcomplex media that contain magnetoelectriccouplings in(1.2), that is,the electric (magnetic)polarizationbeinginduced bythemagnetic(electric) field,theeigensystemdistinctly differsfromtheclassical one:thesingle curl,aswellasthedoublecurloperator, appearsinthewave equations[7,21].Thisfeaturecorrespondsto thesymmetrybreakinginthesystemandintroduceschiralityintheeigensystem.Itisexpectedthatcircularlyorelliptically polarizedwaveswillserveaseigenwaves inthesystem.Tosolvetheband structuresforthistypeofphotoniccrystals,in particular,inthreedimensions,extracarehastobetakenonthechoiceoftherangespacewiththesinglecurloperator.

Themagnetoelectriccouplingcoefficients

μ

˜

, ε

˜

, ζ

˜ and

ξ

˜ in(1.2) areusuallytensormatricesinvariousforms[19,23].In particular,abianisotropicmediumisalsocalledabiisotropicmedium,if

μ

˜

, ε

˜

, ζ

˜ and

ξ

˜ arescalardyadics,orequivalently,

˜

μ = μ

0

˜

_I

, ε ˜ = ε (

x

) ˜

_I

, ζ ˜ = ζ (

x

) ˜

_I

, ξ ˜ = ξ(

x

) ˜

_I

,

(1.3) where˜I istheidentitydyadics,

μ

0≡^{1,specially,the}permittivity

ε (

x

)

andthereciprocalchiralmedium(Pasteurmedium)

ζ (

x

), ξ(

x

)

in(1.3) whichweareinterestedinthispaper,aretypesofbiisotropicmediawith

ε (

x

) =

ε

_i

,

x

∈

^material

,

ε

0

,

otherwise

,

^(1.4a)

ζ (

x

) =

− ιγ ,

x

∈

^material

,

0

,

otherwise

, ξ(

x

) =

ιγ ,

x

∈

^material

,

0

,

otherwise

,

^(1.4b)

and

ε

0

>

0

, ε

i

>

0

, γ

≥^0.

TheMaxwell’sequations(1.1) togetherwith(1.2) canberewrittenas

0

− ι ∇×

ι _∇×

0 H E

= ω μ ˜ ζ ^˜

ξ ˜ ε ˜

H E

.

(1.5)

BasedontheBloch Theorem[18,p. 167],theeigenvectors E and H onagivencrystallattice,satisfyingthequasi-periodic conditions

E

(

x

+

^al

) =

^eι2πk·alE

(

x

),

H

(

x

+

^al

) =

^eι2πk·alH

(

x

),

(1.6) are ofinterest, where2

π

k isthe Bloch wave vector inthe firstBrillouinzone B ^andal,l=¹

,

2

,

3 arethe lattice trans- lationvectors(e.g.[17,p. 34]).Using Yee’sfinitedifferencescheme[26] on (1.5) satisfyingthe source-freeconditionsand thequasi-periodicconditions(1.6),thediscretizedMaxwell’sequationswithbiisotropicmedia (1.4) resultinageneralized eigenvalueproblem(GEP)intheform

0

− ι

C

ι

C^H 0 h e

= ω

μ

_d

ζ

_d

ξ

d

ε

d

h e

,

(1.7)

whereh

,

e∈C^3n,

μ

d

, ε

d

, ξ

d

, ζ

d,andC∈C^{3n×3n withC havingthespecial structurewhich caneasily be treatedwiththe} fastFouriertransform(FFT)toacceleratethenumericalsimulation[8,11] (seeSection2fordetails).

Inthepresenceofthechiralityparameter

γ

in(1.4b),thedegeneracybetweenthefirsttwobandshasbeenlifted,that is,thetwo bandsare nolongerdegenerate [8,Figures2and5],asa resultofthesymmetry breakingintheconstitutive relation.As

γ

increases,a largerdiscrepancyisfound betweenthetwobands.Foranytwo Hermitianmatrices A and B, wesaythatthematrixpair

(

A

,

B

)

orthematrixpencil A−

ω

B ispositive definiteifB ispositivedefinite.Alleigenvalues of positive definite matrix pair

(

A

,

B

)

are real. For photonic crystals made of isotropic chiral media (1.4) with a small chirality

γ

, namely, the weakly-coupled case, the matrix pair in (1.7) is positive definite for which efficient numerical algorithms andband structureshavebeenwell-studied by Chernetal. [8].A critical condition occurswhenthe chirality parameter isequalto thesquare rootofthe permittivity

ε

i in(1.4a), wherethe constitutivematrix isnolonger positive definite[8].Inthissituation,theright-handsidecoefficientmatrix

μ

d

ζ

_d

ξ

d

ε

d

in(1.7) becomessingular.Whenthechirality parameterexceedsthecriticalvalue,thematrixpairsin(1.7) mayintroduceverydifferentandcomplicatedeigenstructures.

AsshowninSection5,sucheigenstructuresleadtothefollowingtwonewinterestingphysicalphenomena.

• ^{Thebandstructurechangesso}drasticallythatalargenumberofresonancemodesemergefromlowerfrequenciesdue tothe bifurcation ofeigenvalues,pushing theoriginal modesto higherfrequencies.The resonancemodestend tobe dispersionless,that is,insensitivetothechangeofwavevectors, andarerepresentedby flatbands.Inparticular,each oftheresonancebandsexhibitsananticrossing(avoidedcrossing)interactionwiththeoriginalone.

•^A distinguished feature ofthe resonance mode is that the electromagnetic fields are highly concentrated inside the chiralmaterial,withonlyaslightamountoffieldsleakingintothebackground(dielectric)material.Inahomogeneous chiral medium characterizedby the dielectricconstant

ε

andchiralityparameter

γ

,thereare two effective refractive indices√

ε

+

γ

and√

ε

−

γ

[7].Foralarger

γ

suchthat

γ >

√

ε

,√

ε

+

γ

becomesevenlargerandthechiralmedium behaves asa waveguide whenit issurrounded by themedium witha lower refractiveindex. In thissituation,most fields willbe concentrated inthe region witha higherrefractive index, leading to theso-calledwaveguidemode.On the otherhand,√

ε

₋

γ

becomes negative andthechiralmedium behaveslike anegative indexmaterial.When itis connectedto thebackgroundwithapositive refractiveindex, thefieldswillbe concentratedattheinterface, leading to the so-calledsurfacemode. Inthe presentproblem, the emergingresonance modesare basically a combinationof waveguide modes and surface modes, with the fields highly localized in the structure and slightly smeared to the background.This isconsidered aunique feature oftheeigenmodes inthe chiralphotoniccrystalswhen thechirality parametergoesbeyondthecriticalvalue.

However, howtosolve theGEP(1.7) withthe chiralityparametergreater thanthecriticalvalue, namely,thestrongly- coupled case, efficientlyisstill open. The difficulty isthat thematrix pairin(1.7) is nolonger apositive definite matrix pairwhichmayintroduceaverydifferentandcomplicatedeigenstructure.Numericalresultsbynewlydevelopedalgorithms (seeSection4)showthatthematrixpairof(1.7) cancreatesomenewstatewhoseenergy(frequency)issmallertothatof theoriginalgroundstate.

Inthispaper,wemakethefollowingcontributiononthe3DMaxwell’sequationswithstronglycoupledreciprocalchiral media:

•^{Foracriticalvalue}

γ

=

γ

^∗,thematrixpairin(1.7) becomespositivesemidefinitesuchthatnullspacesofbothmatrices in (1.7) genericallyhavenonon-trivialintersection(thisproperty alwaysholds inourpractical applications),andhas 2×^{2 Jordanblocksat}

ω

= ∞^.

•^For

γ

=

γ

^∗₊0⁺,thematrixpairin(1.7) createslotsofcomplexconjugateeigenvaluepairs near±

ι

_∞whichrapidly collidewiththerealaxisat

γ

≡

γ

¹

> γ

^∗andbifurcateintopositive(resonance)andnegativeeigenvalueswithmodulus smallerthantheother existingpositiveeigenvalues.Thenewlycreatedpositiveeigenmodepushestheoriginalmodes tohigherfrequencies.

•^Weusetheshift-invertresidualArnoldi(SIRA)method[16,20] combinedwithMINRES[10] andtheFFT-basedscheme [11] tofind a few smallest positive eigenvalues of (1.7) for

γ > γ

^∗ andshow that the associated fields are highly localizedinthestructureandslightlysmearedtothebackgroundmaterial.Numericalexperimentsalsoshowthatthe resonancebandandtheoriginalbandexhibitananticrossinginteraction.

This paperis outlined asfollows.In Section 2,we briefly introduce the matrixrepresentation ofthe discretizationof Maxwell’s equations and the associated singular value decomposition. In Section 3, we analyze and discuss the eigen- structure behavior ofdiscretized Maxwell’sequations.A null-spacefree method andshift-invert residualArnoldi method are introduced in Section 4 to solve the GEP (1.7). Numerical results are demonstrated in Section 5 to show the itera- tion numbersoftheeigensolver,collidingeigenvalues, anticrossingeigencurves andcondensationsofeigenvectors.Finally, a concludingremarkisgiveninSection6.

Notation.Bold letters denote vectors;

ι

=√

−^{1 is the imaginary unit; I}n is the identity matrix of size n; I⁽ⁱ⁾∈R^3n×3n denotes the diagonal matrix with the j-th diagonal entrybeing 1 for the corresponding j-th discrete point inside the material and zerootherwise; I⁽⁰⁾=^I3n−^I(i);I⁽⁰⁾ andI⁽ⁱ⁾ denote the matricesconsistingof all nonzerocolumns of I⁽⁰⁾ and I⁽ⁱ⁾, respectively. Formatrices A and B, A and A^H are the transpose and conjugatetranspose, respectively; N

(

A

)

and R

(

A

)

are thenullspaceandtherangespaceof A,respectively; A⊗^{B and A}⊕^B=^diag

(

A

,

B

)

are,respectively, the KroneckerproductandthedirectsumofA andB(withsuitablesizes).

2. DiscretizationofMaxwell’sequations

From crystallography,itiswell-known thatcrystalstructurescan beclassifiedas14Bravaislattices. Becauseofvarious lattices, the matrix C in (1.7) of the discretized single-curl operator on the electric field may have different forms. To lookthroughthedetailsofthediscretizationprocessofYee’sschemeforMaxwell’sequations(1.5) withthree-dimensional photoniccrystals,wereferthereaderto[14].Forconvenience,inthispaper,weonlyconsiderthefacecenteredcubic(FCC) latticewith

a₁

= √

^a

2

[

¹

,

0

,

0

] ,

a₂

= √

^a 2

1 2

,

√

3 2

,

0

,

a₃

= √

^a 2

1 2

,

²

2

√

3

,

2 3

,

(2.1)

whereaisthelatticeconstant.Letn1,n2andn3denotethenumbersofgridpointsinthex1-,x2- andx3-axis,respectively.

Set

δ

1=_n^a

1

√2,

δ

2=_n^a√3

22√

2,

δ

3=_n^a

3

√3,theassociatedmeshlengths,andn=ⁿ1n₂n₃.Thentheresulting3n×^{3nmatrixC is} oftheform[11,12]

C

=

⎡

⎣

⁰

−

^C3 C₂ C₃ 0

−

^C1

−

C2 C1 0

⎤

⎦ ,

(2.2)

where

C1

=

_δ¹₁^In₂n₃

⊗

^K1,n₁

(

e^ι2πk·a1

),

(2.3a)

C₂

=

_δ¹₂^In3

⊗

^Kn1,n2

(

e^ι2πk·a2J₁

),

(2.3b)

C3

=

_δ¹₃^Kn₁n₂,n₃

(

e^ι2πk·a3J2

)

(2.3c)

with2

π

kbeingBlochwavevectorsasin(1.6),

J1

=

0 e^{−ι2πk·a1}In₁/2

I_n1/2 0

,

(2.4a)

J₂

=

0 e^{−ι2πk·a2}In₂/3

⊗

In1

I_2n2/3

⊗

^J1 0

,

(2.4b)

and

K_m1,m2

(

X

) =

⎡

⎢ ⎢

⎢ ⎣

−

^Im₁ Im₁

. . . . . .

−

^Im₁ Im₁

X

−

^Im1

⎤

⎥ ⎥

⎥ ⎦ ∈ C

^m1m2×m1m2

.

(2.4c)

It has been shownin [11] that the matricesC1, C2, and C3 can be diagonalized by a common unitary matrix asin Theorem2.1.

Theorem2.1(EigendecompositionsofCi’s[11]).ThematricesC1,C2andC3in(2.3)aresimultaneouslydiagonalizablebytheunitary matrixT=^√_n1¹_n2n3[^T1

,

T2

,

· · ·

,

Tn1]^withTi= [^Ti,1

,

· · ·

,

Ti,n2]^andTi,j= [^zi,j,1⊗^yi,j⊗^xi

,

· · ·

,

zi,j,n3⊗^yi,j⊗^xi]^,intheforms

T^HC1T

=

n₁

⊗

^In₂n₃

≡

1

,

(2.5a)

T^HC₂T

= ⊕

ⁿ_i=¹₁

(

i,n2

⊗

^In3

) ≡

2

,

(2.5b)

T^HC3T

= ( ⊕

ⁿ_i=¹₁

⊕

ⁿ_j=²₁

_i,j,n3

) ≡

3

,

(2.5c)

where

n1

= δ

1−¹diag

e^θ1

−

¹

,· · · ,

e^θn1

−

¹

,

(2.5d)

i,n2

= δ

2−1diag

e^θi,1

−

¹

,· · · ,

e^θi,n2

−

¹

,

(2.5e)

i,j,n3

= δ

3−1diag

e^θi,j,1

−

¹

,· · · ,

e^θi,j,n3

−

¹

,

(2.5f)

and

θ

i

=

^ι2π(i_n⁺₁^k·a1)

,

xi

=

1

,

e^θi

, · · · ,

e^(n1−1)θi

,

^(2.5g)

θ

i,j

=

^ι2π(j−_n²ⁱ₂^+k·a2)with

a2

=

^a2

−

¹₂^a1

,

y_i,j

=

1

,

e^θi,j

, · · · ,

e^{(n2−1)θi,j}

,

(2.5h)

θ

_i,j,k

=

^{ι2π(k−13(}_nⁱ⁺₃ ^{j)+k·a3)with}

a3

=

^a3

−

¹₃

(

a1

+

^a2

),

z_i,j,k

=

1

,

e^θi,j,k

,· · · ,

e^{(n3−1)θi,j,k}

,

(2.5i)

fori=¹

,

· · ·

,

n1,j=¹

,

· · ·

,

n2,k=¹

,

· · ·

,

n3.

ThefollowingimportantsingularvaluedecompositionofC isderivedin[8].

Theorem2.2([8]).Thereexistunitarymatrices

Q

=

Qr Q0

≡ (

I3

⊗

^T

)

1

2

0

≡ (

I3

⊗

^T

)

⎡

⎣

_1,1

_1,2

_1,0

2,1

2,2

2,0

3,1

3,2

3,0

⎤

⎦ ,

(2.6a)

P

=

Pr P0

= (

I₃

⊗

^T

)

− ¯

2

¯

1

¯

0

,

(2.6b)

whereQ_r

,

P_r∈C^3n×2nand

i,j∈C^n×n,i=¹

,

2

,

3

,

j=⁰

,

1

,

2,arediagonalsuchthatC hasasingularvaluedecomposition C

=

^{P diag}

(

¹q^/2

,

¹q^/2

,

0

)

Q^H

=

^Pr

rQ_r^H

,

r

=

^diag

(

¹q^/2

,

¹q^/2

)

(2.7) with

q=

₁^H

1+

^H₂

2+

₃^H

3.

We nowconsiderthediscretizationoftherighthandsidein(1.7).Thediagonalmatrices

μ

d,

ε

d,

ξ

_d and

ζ

_d in(1.7) can bedeterminedbytheshapeofthemedium.From(1.3) and(1.4) wehave

μ

_d

₌

I3n

, ε

_d

₌ ε

0I⁽⁰⁾

+ ε

iI⁽ⁱ⁾

,

(2.8a)

ζ

_d

= − ιγ

I⁽ⁱ⁾

, ξ

_d

= ιγ

I⁽ⁱ⁾

,

(2.8b)

where

ε

i

, ε

0 are the permittivities inside andoutside the medium, respectively,

γ >

0 is the chirality, I⁽⁰⁾ and I⁽ⁱ⁾ are definedinNotationsofSection1.

3. Eigenstructurebehavior ofdiscretizedMaxwell’sequations

Wenowstudytheeigenstructurebehavior oftheGEPin(1.7).Itiseasilyseenthatequation(1.7) canberewrittenas

I_3n 0

ξ

d

μ

⁻_d¹ I3n

0

− ι

C

ι

C^H

ι ξ

d

μ

⁻_d¹C

− ι

C^H

μ

⁻_d¹

ζ

d

− ω

μ

d 0 0

ε

_d

₋ ξ

_d

μ

_d⁻¹

ζ

_d

I_3n

μ

⁻_d¹

ζ

_d 0 I3n

h e

=

⁰

.

(3.1)

Togetherwiththechoicesof

μ

d

, ε

d

, ξ

d

, ζ

dasin(2.8),wehavethefollowingmatrixpencilinstead.

0

− ι

C

ι

C^H

− γ [

I⁽ⁱ⁾C

+

C^HI⁽ⁱ⁾

]

− ω

I_3n 0

0

ε

0I⁽⁰⁾

+ ( ε

_i

₋ γ

²

)

I⁽ⁱ⁾

≡

^Aγ

− ω

B_γ

.

(3.2)

Furthermore,itiseasilyseenthat

( ω ,

h

e

)

isaneigenpairof(1.7) ifandonlyif

( ω ,

h−

ιγ

I⁽ⁱ⁾e e

)

isaneigenpairof(3.2).

With

ω

=^{0 itholdsthath}=

ι ( γ

I⁽ⁱ⁾−

ω

⁻¹C

)

e.Notethat Aγ

,

Bγ in(3.2) areHermitianwithAγ beingindefinite;

(

Aγ

,

Bγ

)

isregular(i.e.,det

(

Aγ−

ω

Bγ

)

≡^{0) if}

γ

₌^√

ε

i;For

γ <

√

ε

i,thematrixpair

(

Aγ

,

Bγ

)

withBγ

>

0 being positivedefinite hasallrealeigenvalues[8];For

γ >

√

ε

i, Bγ isindefiniteand

(

Aγ

,

Bγ

)

mayhavecomplexeigenvalues.

Wenowstudytheeigenstructureofthecriticalcasewhen

γ

=

γ

^∗₌^√

ε

i.Forthiscase,from(3.2) wehave A_γ∗

=

0

− ι

C

ι

C^H

− γ

^∗

_[

I⁽ⁱ⁾C

+

^CHI(i)

]

,

B_γ∗

=

I_3n 0 0

ε

0I⁽⁰⁾

.

(3.3)

ItiseasilycheckedthatN

(

Bγ^∗

)

=R

0

I⁽ⁱ⁾

,whereI⁽ⁱ⁾isdefinedinNotationsofSection1.FromTheorem2.2,itfollows thatN

(

C

)

=R((I3⊗^T

)

0

)

andN

(

C^H

)

=R((I3⊗^T

)

¯0

)

.Thus,N

(

Aγ∗

)

=R(Nγ∗

)

,where

N_γ∗

=

− ιγ

^∗I⁽ⁱ⁾

(

I₃

⊗

^T

)

0

(

I₃

⊗

^T

) ¯

0

(

I₃

⊗

^T

)

₀ 0

.

(3.4)

Theorem3.1.ItholdsgenericallythatN

(

Aγ^∗

)

∩N

(

Bγ^∗

)

= {⁰}^.

Proof. Letx∈N

(

Aγ^∗

)

∩N

(

Bγ^∗

)

.Thenthereare z1∈C²ⁿ

,

z2∈C^{ni withn}i beingthenumberofcolumnsofI⁽ⁱ⁾suchthat x=^Nγ^∗z1=

0

I⁽ⁱ⁾

z2.Itfollowsthat

N

˜

_γ∗z

≡

N_γ∗

−

⁰I⁽ⁱ⁾

z₁ z₂

=

^{0 (3.5)}

withN˜γ^∗∈C^6n×(2n+ni).N˜γ^∗ isgenericallyoffullcolumnrank.Thisimpliesthatz=^0,andthusx=^0. 2

Theorem3.2.Supposethatthen_i×ⁿisubmatrixI⁽ⁱ⁾

(

C+^CH

)

I⁽ⁱ⁾inAγ^∗hasnullityⁿˆi≥^{1.ThenthematrixpencilA}γ^∗−

ω

Bγ^∗in (3.2)hasatleastnˆi2×^{2Jordanblocksat}

ω

= ∞^.

Proof. FromtheassumptionthatI⁽ⁱ⁾

(

C+^CH

)

I⁽ⁱ⁾in Aγ^∗ correspondingtothen_i×ⁿizerodiagonalblockof

ε

0I⁽⁰⁾in Bγ^∗ hasanullspaceofdimensionⁿˆi.BecauseofN

(

Aγ^∗

)

∩N

(

Bγ^∗

)

= {⁰}^{asinTheorem3.1,fromTheorem4.1of[9], A}γ^∗−

ω

Bγ^∗ hasatleastnˆi2×^{2 Jordanblocksat}

ω

= ∞^. 2

Remark3.1.In fact, the condition of the singularity ofI⁽ⁱ⁾

(

C+^CH

)

I⁽ⁱ⁾ in Theorem 3.2 heavily depends on shapes of materialsin(1.4).ThisconditionalwaysholdsforournumericalexamplesinSection5.3.

Theorem3.3.For

γ

⁺=

γ

^∗₊

η

as

η

→^{0+,itholdsthatA}γ⁺−

ω

Bγ⁺hasatleastonecomplexconjugateeigenvaluepairsofthe forms

ω

_±

( γ

⁺

) :=

¹ 1

+ η ^±

√ ι

η (

1

+ η ) ,

as

η →

⁰⁺

.

(3.6)

Proof. FromTheorem3.2, Aγ∗−

ω

Bγ∗ hasatleastone 2×^{2 Jordanformat}

ω

_{= ∞}.Therefore, Aγ⁺−

ω

Bγ⁺ musthavea canonicalsub-blockoftheform

0 1 1 0

−

ω

1

η

η

₋

η

,as

η

→^0+.Here,forconvenience,wewrite

η

≡O(

η ) >

0 with“O^” denotingthebigO.Thus,wehave

ηω

²+

(

1−

ηω )

²=^{0 solvingwhichweget(3.6).} 2

Discussion1.

(i)Let[

(

h−

ιγ

I⁽ⁱ⁾e

) ,

e ] ^betheeigenvectorofAγ−

ω

Bγin(3.2)correspondingto

ω

.Thenitholdsthat

c

(

e

) − ωγ

b

(

e

) − ω

²

_[ ε

0a0

(

e

) + ( ε

_i

₋ γ

²

)

a_i

(

e

) ] =

⁰

,

(3.7) where

c

(

e

) =

^eHCHCe

≥

⁰

,

b

(

e

) =

^eH

I⁽ⁱ⁾C

+

^CHI(i)

e

,

(3.8a)

a₀

(

e

) =

^eHI(0)e

≥

⁰

,

a_i

(

e

) =

^eHI(i)e

=

^eHe

−

^a0

(

e

) ≥

⁰

.

(3.8b) Thenwehave

ω

_±

( γ ) = γ

b

(

e

) ± √ (

e

)

−

2

[ ε

0a0

(

e

) + ( ε

i

− γ

²

)

ai

(

e

)]

^(3.9)

with

(

e

) = γ

²b

(

e

)

²

+

^4c

(

e

) [ ε

0a0

(

e

) + ( ε

_i

₋ γ

²

)

a_i

(

e

) ]

= γ

²b

(

e

)

²

+

^4c

(

e

) [ ( ε

0

+ γ

²

− ε

i

)

a₀

(

e

) + ( ε

i

− γ

²

)

e^He

] .

(3.10) FromTheorem3.3,when

γ

⁺=

γ

^∗+

η

(as

η

→^0+)with

(

e

) <

0,itmustholdthata₀

(

e

)

≈^0andb

(

e

)

≈^0.Thus,e(0)≡ I⁽⁰⁾e≈^0,i.e.,at

γ

=

γ

⁺,theelectricfieldE

(

x

)

almostvanishwhenxisoutsidethematerial.

(ii)Weincrease

γ

⁺to

γ

⁰sothattwocomplexconjugateeigenvalues

ω

_±

( γ )

collideontherealaxisat

γ

=

γ

⁰with

(

e

)

=⁰ andcreate(bifurcateinto)twonewrealeigenvalues

ω

_±

( γ

¹

)

with

γ

¹

> γ

⁰inwhich

ω

₊

( γ

¹

) >

0isthesmallestoneamong otherexistingpositiverealeigenvalues.Inthiscase,from(3.9)and(3.8),weseethata0

(

e

)

≈O

( η )

andb

(

e

)

≈O

(

1

)

withsmall

η >

0.Thisimpliesthatat

γ

=

γ

¹,theelectricfieldE

(

x

)

isalsoflatwhenxisoutsidethematerial(seenumericalexperimentsin Section5.5).

(iii) Interchangingtherolesof

μ

dand

ε

daswellashandein(1.7),similarlyto(3.1)and(3.2),wehavethematrixpencil

− γ _[

I⁽ⁱ⁾C

˜

^H

+ ˜

^{C I(i)}

] − ι

C

˜ ι

C

^˜

^H 0

− ω

ε

0I⁽⁰⁾

+ ( ε

_i

₋ γ

²

)

I⁽ⁱ⁾ 0

0 I_3n

(3.11)

withtheeigenpair

( ω ,

h

˜

e−

ιγ

I⁽ⁱ⁾_h˜

)

,whereC˜=

ε

¹_d^/2C

ε

_d^−1/2,h˜=

ε

⁻_d^1/2hande˜=

ε

_d^1/2e.Asin(3.7)–(3.10),italsoholdsthat

ω

_±

( γ ) = − γ

b

^˜ (

_h

˜ ) + ( ˜

_h

˜ )

−

2

[ ε

0a

˜

0

(

_h

˜ ) + ( ε

i

− γ

²

)˜

ai

(

_h

˜ )]

^(3.12)

with

(

˜ _h˜

)

=

γ

²b^˜

(

_h˜

)

²+⁴˜c

(

_h˜

)

[

ε

0a˜0

(

_h˜

)

+

( ε

i−

γ

²

)˜

ai

(

_h˜

)

]^,where

˜

a0

(

_h

˜ ) = ˜

^hHI(0)_h

˜ ≥

⁰

,

(3.13a)

˜

ai

(

_h

˜ ) = ˜

^hHI(i)_h

˜ ≥

⁰

,

(3.13b)

b

˜ (

_h

˜ ) = ˜

h^H

[

I⁽ⁱ⁾C

˜

^H

+ ˜

C I⁽ⁱ⁾

]˜

h

∈ R,

(3.13c)

˜

c

(

_h

˜ ) = ˜

h^HC

˜

^HC

˜

_h

˜ ≥

0

.

(3.13d)

Asin(ii),wecanalsoconcludethatfor

ω

₊

( γ

¹

)

with

γ

¹

> γ

^∗ thesmallesteigenvalueamongtheotherexistingpositivereal eigenvalues,theassociatedmagneticfieldH

(

x

)

isflatwhenxisoutsidethematerial.

4. Null-spacefreemethodwithshift-invertresidualArnoldimethod

Since the6n×^{6nHermitianmatrix Aγ in(3.2) hasahugenullspacewithnullity2n,itwould}significantly affectand slowdown theconvergenceofsmallestpositiveeigenvalues. Toremedy thisdrawback,we developan efficientnumerical algorithmforthecomputationofafewsmallestpositiveeigenpairsof(1.7) byapplyingtheSIRAtothe4n×^4nnull-space freeGEP(NFGEP)derivedin[8].

Theorem4.1([8]).If

γ

=

γ

^∗,thentheGEPin(1.7)canbereducedtoa4n×^{4n NFGEP}

A_ry_r

= ω

ι

0

⁻_r¹

−

⁻_r¹ 0

y_r

≡ ω

B_ry_r

,

(4.1)

and

h e

= ι

−

^I3n

−ζ

d

ξ

d

ε

d

₋1

diag

(

P_r

,

Q_r

)

y_r

,

where

A_r

:=

A_r

( γ ) ≡

^diag

(

P_r^H

,

Q_r^H

)

ζ

d

−

I3n

I_3n 0

⁻¹ 0 0 I_3n

ξ

d I3n

−

^I3n 0

diag

(

P_r

,

Q_r

)

(4.2)

with

:=

( γ )

≡

ε

_d−

ξ

_d

ζ

_dbeingHermitian.

Form(3.7)–(3.10) itfollowsthatfor

γ > γ

^∗,eachcomplexeigenvalueof

(

Aγ

,

Bγ

)

in(3.2) appearsinacomplexconjugate pair.Inthefollowingtheorem,wefurthershowthattheeigenvaluesofthenull-spacefreematrixpair

(

Ar

,

Br

)

in(4.1) have thecommonpropertyasthecomplexHamiltonianmatrix.

Theorem4.2.For

γ > γ

^∗,if

ω

isacomplexeigenvalueof

(

Ar

,

Br

)

asin(4.1),then

ω

¯isalsoaneigenvalue.

Proof. Lety=

0

⁻_r^1/2

⁻r^1/2 0

yr.Thentheequation(4.1) canberewrittenas

H^y

≡

¹_r^/2 0 0

−

¹r^/2

A_r

0

¹_r^/2

¹r^/2 0

y

= ιω

y

.

(4.3)

With J=

0 I

−^{I 0}

,itiseasytocheckthat

H^J

= −

^J

0

¹_r^/2

_r^1/2 0

A_r

_r^1/2 0 0

−

¹r^/2

= −

^JH^H

.

So,H^isacomplexHamiltonianmatrix.Itfollowsthatif

λ

=

ιω

isaneigenvalueofH^,then−¯

λ

=

ι ω

¯ isalsoaneigenvalue ofH^{.Therefore,eachcomplexeigenvalueappearsinacomplexconjugatepair.} 2