Multifrequency spatial filtering: a general property of two-dimensional photonic crystals

PhotonicsandNanostructures–FundamentalsandApplications18(2016)1–9

Availableonlineatwww.sciencedirect.com

ScienceDirect

Multifrequency

spatial

filtering:

A

general

property

of

two-dimensional

photonic

crystals

A.E.

Serebryannikov

,

E.

Colak

,

A.

Petrov

,

P.V.

Usik

,

E.

Ozbay

a_{FacultyofPhysics,AdamMickiewiczUniversity,61-614Pozna´n,Poland} b_{NanotechnologyResearchCenter–NANOTAM,BilkentUniversity,06800Ankara,Turkey} c_{DepartmentofElectricalandElectronicsEngineering,AnkaraUniversity,Golbasi,06830Ankara,Turkey} d_{InstituteofOpticalandElectronicMaterials,HamburgUniversityofTechnology,21073Hamburg,Germany}

e_{ITMOUniversity,49KronverskiiAve.,197101St.Petersburg,Russia}

f_{InstituteofRadioAstronomy,NationalAcademyofSciencesofUkraine,61002Kharkiv,Ukraine}

Received23March2015;receivedinrevisedform5October2015;accepted19November2015 Availableonline28November2015

Abstract

Spatialfiltering,ananalogoffrequency-domainfilteringthatcanbeobtainedintheincidenceangledomainatafixedfrequency isstudiedinthetransmissionmodeforslabsoftwo-dimensionalrod-typephotoniccrystals.Inthepresentpaper,theemphasisis putonthedemonstrationofthepossibilitytoobtainvariousregimesofspatialfiltering,i.e.,band-stop,band-pass,andlow-pass filteringindifferentfrequencyrangesinonesimpleconfiguration.TheoperationisbasedontheuseofseveralFloquet–Bloch modeswithappropriatedispersionproperties,sothatsuchoneortwoco-existingmode(s)contributetotheformingofaproperfilter characteristicwithineachspecificfrequencyrange.Itisshownthathigh-efficiencytransmissionandsteepswitchingbetweenpass andstopbandscanbeobtainedintheangledomainforwiderangesofvariationoftheproblemparameters.Inparticular,byvarying therod-diameter-to-lattice-constantratio,oneattainslotsoffreedomintheengineeringofspatialfilterswithdesiredtransmission characteristics.

©2015ElsevierB.V.Allrightsreserved.

Keywords: Spatialfiltering;Photoniccrystal;Floquet–Blochmode;Transmission;Fabry–Perotresonance

1. Introduction

Spatial (angular) filters are important components requiredininformationprocessingandimage enhance-ment for various frequency ranges. They operate in the incidence angle domain at a fixed frequency, f,

and thus represent analogs of the conventional fre-quency filters that operate ata fixed incidence angle,

∗_{Correspondingauthor.}

E-mailaddress:[email protected](A.E.Serebryannikov).

θ. Spatialfilters are also considered fromthe spatial-frequencyfiltering perspective [1].Forexample, such filterswereemployedintheanalysisofthespatial spec-trum,enhancementoftheantennadirectivity,radardata processing, aerial imaging, and sorting the incoming radiationaccordingtothesourcelocation.Theknown theoreticalandexperimentalperformancesofthespatial filters includethosebased on anisotropic (anti-cutoff) media[2],multilayerstackscombinedwithaprism[3], resonantgratingsystems[4],metallicgridsoveraground plane [5], interference patterns [1], and axisymmet-ricmicrostructures[6].Variousphotoniccrystal(PhC)

http://dx.doi.org/10.1016/j.photonics.2015.11.001 1569-4410/©2015ElsevierB.V.Allrightsreserved.

structuresthatenableefficientspatialfilteringshouldbe mentioned,including thosebased ontwo-dimensional regular(defect-free)PhCsandone-andtwo-dimensional chirpedPhCs [7–11]. The co-existing spatial and fre-quencyfilteringhasalsobeenstudiedinconnectionwith thecontroloflaserradiationwiththeaidoftheresonant gratingbasedfilters[12].Thestate-of-the-artofspatial filteringhasbeenreviewedin[13].

Theknownmechanismsofspatialfilteringdifferin

thattheyuse[2,7–9,11]ordonotuse[1,4]thepeculiar

dispersionfeatures.Theformermaygivemorefreedom indesign,becausethe requiredfeatures canappearin widerangesoffrequencyand/orangleofincidence.For instance,low-passspatial filtering canbeobtained by usingvolumetric structureswithisotropic-type disper-sion,whichcorresponds totherefractiveindex values withintherangeof0<n<1[14].Inturn,high-passand band-passfilteringrequire structureswith anisotropic-typedispersion.Generally,thesetypesofspatialfiltering canbe attained byusing anti-cutoff media[2], which arealsoassociatedwithhyperbolicmetamaterials[15]

andPhCswiththedispersionfeaturesthatenable block-ing transmission in the vicinity of zero of tangential wavenumber [7–9]. The problem can appear when a nearlyperfecttransmissionwithintheentirewide angle-domainpass band isrequired [16,17]. High-pass/dual band-passspatial filteringthatfulfills thisrequirement has been demonstrated in the transmission mode for aslab of the two-dimensional dielectricPhC [7]. For operationinthereflectionmode,high-/band-passspatial filters haverecently been suggested, whichare based on two-dimensional PhCs andsingle-layer rod arrays over ametallic reflector[18]. Note that the reflection modehigh-/band-passspatialfiltersrequirethe redistri-butionof the incident-wave energy in favorof higher diffraction orders, i.e., this mode is connected with theblazingregime[19].Onthecontrary,the transmis-sionmode doesnot require thecontributionof higher orders,although blazing,if appears,can leadtosome advances in functionality. Regardless of the possible contributionofhigherorders,therichnessofdispersion features remains the mainargument infavor of using two-dimensionalPhCs.

Inthispaper, wedemonstrate thatarich varietyof thetypesofspatialfiltering,e.g.,low-pass, high-/band-passandbandstopfilteringcanberealizedinonesimple configurationthat represents afinite-thickness slab of atwo-dimensionalPhCcomposedofcirculardielectric rods.The crucialkey knobsofthe initialdesignstage thatisbasedonthedispersionanalysisincludethe fol-lowing:(i)engineeringsingleormultipleFloquet–Bloch waves by having bands with monotonous dispersion

whichare separated wellfromeach other,bandswith nonmonotonous dispersion, or co-existing bands; (ii) engineering the relative position of the modes within thefirstBrillouinZone(BZ)withrespecttothe equifre-quencydispersion contours (EFC)inair(forinstance, isotropic-typeEFCsof PhCnarrowerthaninairfavor low-passspatialfiltering;anisotropic-typeEFCswould favor high-/band-pass spatial filtering); and(iii) engi-neeringtheshapeofEFCs(e.g.,squareshapeofEFCsis expectedtobepreferabletokeepnearlythesame trans-missionefficiencywithintheentireband).Thesepoints are illuminatedbythe analysisof thepossible combi-nations of EFCs in airand PhC andrelated coupling scenarios.Then,weshowbyusingthesimulated trans-missionresultsthatsolelyaproperparameteradjustment canenabledifferenttypesofspatialfilteringinthe neigh-boringfrequencyranges.Torealizeadesiredresponse intheangle domainatafixedfrequency,eitherasole Floquet–Blochmodeortwosuchmodesareemployed. Thesharpfilterpropertiesareobtainedduetothe con-tributions from several effects, such as the shape of theEFCs,Fabry–Perottypeinterferences,andthe exci-tation of higher diffraction orders. The transmission resultshavebeenobtainedbyusingthe coupled-integral-equationtechnique,aflexibleiterativetechniquewhich enables acceleration in convergence by applying pre-conditioning[20].

2. Dispersionbasedanalysis

Thebasisofoperationofspatialfiltersisconnected withthedistributionoftheFloquet–Blochmodesinthe

entirewavevectorspace(notjustinthefirstBZ).Itiswell known that electromagneticwavesfollowinPhCsthe Floquet–Blochtheoremaselectronsinacrystal. There-fore, the distribution of the modes in the wavevector spacecanbereconstructedfromtheoneinthefirstBZ accordingtoarepeatedzonescheme,byfollowingthe symmetryofthelatticejustlikeintheelectroniccase. Thisapproachcansimplifybothanalysisanddesign sig-nificantly.Itisnoteworthythatanextensiveanalysisof refractioninPhCs,bytakingintoaccountthemode dis-tributionintheentirewavevectorspace,wasfirstdone byFoteinopoulouandSoukoulis[21].Forthepurposes ofspatialfilteringbyusingPhCs,theresultspresented in[21]areveryimportantsincetheyillustratethe possi-blebehavior(shapeandlocations)oftheEFCsforPhC intheentirewavevectorspace,whilstthesignof refrac-tionandhandednessarenotsoimportant,inthecontrast towhatmightbeimportantforotherapplications.The sameremains truefor otherpreviousstudiesof PhCs, e.g.,see[22,23].Comparingtothestudyofrefractionin

[21],thesituationwhentwobeamsaresimultaneously refractedatthesamevaluesof fandθisnotexpected tobeusefulforspatialfiltering.Instead,itisimportant forustoknowwhichshapesandlocationsofEFCsfor PhCsarepossible.

Werestrictourconsiderationtothecaseofa square-latticePhCandtheinterfacesoftheslabofthisPhCthat arealongthe−Xdirection.Inthebandregimeswith monotonousdispersion, oneFloquet–Blochwavemay coupletotheincidentandoutgoingwaves,leadingtoone transmissionband inθ-domain.However, inthe band regimes with two band solutions, two Floquet–Bloch modesmaybecoupledsimultaneously,resultingintwo transmissionbands.Alternatively,onebandwitha non-monotonousdispersioncanyieldtwotransmissionbands inthe θ-domain.Taking intoaccount the known rich-ness of dispersion types achievable using PhCs, it is possible to expect obtaining different types of spatial filteringwithintheneighboringfrequencyranges.Here, weassumethateitherasoleortwoFloquet–Blochmodes areused,butthenumberofthesimultaneouslyutilized modescanformallybearbitrary.

The above given description is depicted in Fig. 1

wheretypicalscenariosofspatialfilteringare schemati-callyshown.TheEFCshapesusedinFig.1areeitherthe sameorclosetotherealisticones,asfollowsfromthe numerous PhC studies [21–23]. Predictions regarding the existence andthe realizabletypes ofspatial filter-ingarebasedontheanalysisoftheshapeandlocation ofEFCs.Toensurethatadesiredcouplingregimeand relatedpassbandmayoccur,EFCsforPhCandthe sur-roundingairmustcoexistinthecorrespondingrangeof variation of the tangential wavevector component, kx,

whichisalongthevirtualinterface,atafixedfrequency. Forthesake ofdefiniteness, weassumethat the inter-facesofthe slabofPhCare alongthex-direction, see

Fig.2(a). Thus,each kx-value for thecircularEFC in

airunambiguouslycorrespondstoacertainvalueofθ, i.e.,kx=ksinθwherek=ω/c(ω=2πf)meanstheradius

ofthisEFC.Thedashedverticallinesrepresentthe con-structionlinesinthelimitingcase,i.e.,attheboundaries of kx-ranges, in whichcoupling of the incident wave

toaFloquet–Blochmodeisallowedbythe dispersion at kx<0. Generally,construction lines serve a

graphi-calrepresentationoftheconservationofthetangential component ofthewavevector,sothateach suchaline essentiallyindirectlyrepresentstheincidenceangle.If aconstructionlinecrossesboththeEFCforairandthe EFCforPhC,couplingispossibleatthechosenvalues ofθandf.

It is worth noticing the symmetry of EFCs with respect to kx=0, so that the passand stopbands can

(a) (b) (c) (d) (e) (f) (h) (g) (i) Γ X X X X X X X X X M M M M M M M M M x x y y y y y y y y y k k k k k k k k k x k k_x k_x x x x k k k_x k k k no coupling Γ Γ Γ Γ Γ Γ Γ Γ

Fig.1.(a)DifferentcombinationsofmutuallocationofEFCsforPhC (solidlines)andairhost(dottedlines–circles)intherepeatedzone diagramatfixedfrequency;componentsofwavevector,kxandky,are

assumedtovaryfrom−2π/ato2π/a(aislatticeconstant);triangles showboundariesofthefirstBZ;verticaldashedlines–construction linesattheboundariesofthekx-rangeswherecouplingispossible;

rectanglesschematicallyshowthelocationofthekx-rangesinwhich

couplingispossibleatkx<0;(a–c)caseoftwobandsolutionsorone

nonmonotonousbandsolutionthatlead(s)totwogroupsofEFCs includingthosearound-pointandM-pointofthefirstBZ,at dif-ferentvaluesoftheEFCradiusforair,k:(a)k=π/a,(b)k<π/a,and (c)k>π/a;(d–f)caseofonemonotonousbandsolutionthatleadsto onegroupofEFCs,whicharenarrowerthaninair,includinganEFC locatedaround-pointofthefirstBZ:(d)k=π/a,(e)k<π/a,and(f)

k>π/a;(g–i)caseofonebandsolutionthatleadstoonegroupofEFCs includingthataroundM-pointofthefirstBZ:(g)k=π/a,(h)k<π/a, and(i)k>π/a.

appearinthesamerangesofvariationof|kx|whenkx>0

andkx<0.Moreover,bothrangescansimultaneouslybe

employed.Inthiscase,morepassandstopbandscanbe obtainedinθ-domainatf=const(e.g., dualband-pass filteringinsteadofsingleband-passfiltering).However, inthispaper,considerationisrestrictedtothecasewhen sgnkx=const.Moreover,inthissectionweassumethat

higher diffraction orders (|m|>0), which may appear duetotheperiodicityoftheinterfacesthatisinherited fromthesquarelattice,arenotcoupledtoFloquet–Bloch modes.

InFig.1(a–c),the EFCsof thePhCconsistoftwo groups,includingthoselocatedaround-pointand M-point.Suchasituationcanappearduetotwomonotonous bandsolutions[7]oronenonmonotonousbandsolution

Fig.2.(a)Generalgeometryusedintransmissionstudies;(b)Mapoft0in(ka,θ)-planeatd/a=0.443,ε=9.61,andN=8;straightverticallines

–approximateboundariesofka-rangesthatcanbeusedforbandstop(BS)andlow-pass(LP)spatialfiltering,dashedcurve–thresholdlinefor thediffractionorderm=−1;largenumbers,2,3and4,indicatetransmissionareaduetothecorrespondingFloquet–Blochmode;(c–e)t0vskaat

d/a=0.443,ε=9.61,θ=47◦,and(c)N=6,(d)N=8,and(e)N=12.

relatedto the ratio of the sizes of EFCs for PhC and airhostalongthekx-axis.In fact,thedifferentradiiof

EFCsforairmeanheredifferentdispersion properties ofthePhCanddifferentscalingratiosforthekxandky

axesratherthandifferentpropertiesofthehostmedium. Onecanseethatbandstop,low-pass,anddualband-pass spatialfilteringcanbeobtainedaccordingtoFig.1(a), (b),and(c),respectively.Next,Fig.1(d–f)demonstrates thecasewhentheEFCsforPhCincludeanEFClocated around-point,butarenarrowerthantheEFCforair.It appearsinthecaseofasolemonotonousbandsolution. Now,low-passspatialfilteringcanonlyberealized,with therangeoftheallowedangles,whichisdeterminedby the ratio of the widths of EFCs for PhCand air,i.e., max|θ|=arcsin[maxk_xPhC/k](k_xPhC isx-component of thewavevectorinPhC).Finally,Fig.1(g–i)presentsthe casewhentheEFCsincludeanEFClocatedaround M-pointof the first BZ.This is also possible for asole bandsolution.Dependingonthe widthof EFC inair, onecanobtainhigh-passfiltering,perfectreflection,and band-passfiltering as shown in the plots (g), (h), (i), respectively.Toconclude,anEFCaround-pointis nec-essaryforlow-passfiltering,andanEFCaroundM-point isnecessaryforhigh-passfiltering.Ontheotherhand, dualband-passandbandstopfilteringcan beobtained whenEFCsarelocatedaround-pointandM-point; sin-gleband-passfilteringrequiresanEFCaroundM-point. Generally,involvinghigherBZsextendsthevarietyof theachievablescenarios.

3. Transmissionresultsanddiscussion

Geometryofthefinite-thicknessslabofPhCisshown inFig.2(a).ItcontainsNlayersofcirculardielectricrods withdiameterdandpermittivityε.Therodsareplacedin

asquarelatticewithconstanta.Thestructureis illumi-natedbys-polarizedplanewave(electricfieldisparallel totherodaxes)attheincidenceangleθ.Themapof zero-ordertransmittance,t0,ispresentedinFig.2(b)inthe

(ka,θ)-planefortheslabofPhC,whichhasbeenstudied theoreticallyandexperimentallyatmicrowave frequen-cies[8].Theareasofvanishingtransmissioncorrespond toincompletebandgaps,i.e.,thetransmissionisblocked forafiniteportionoftheentirerangeofθvariation.Such bandgapsareparticularlyappropriateforspatial filter-ing,atleastifasteepswitchingbetweenwidepassand stopbandsisrequired.Severalrangesthatare appropri-ate for spatial filteringat fixedfcan beseenbetween

ka=2.6andka=3.8.Theserangescorrespondtoasole nonmonotonous band solution or two band solutions. Thecouplingscenariosrealizedat2.6<ka<3.5are sim-ilartothoseillustratedbyFig.1(aandc).Intheareas of strong transmissionin Fig.2(b), thereare alternat-ingmountainsofnearlyperfecttransmissionandvalleys of lower transmission. In fact, they represent differ-entcasesoftheFabry–Perottyperesonances,including unusualones.Onlythosevalleysandmountains,which areshiftedtolargerkavalueswhileθisincreased, corre-spondtotheconventionalFabry–Perotresonancesand, thus,tothecasewhentheeffectiveindexofrefractionis

n>0[24].Onthecontrary,thoseshiftingtosmallerka

valuescorrespondtothecaseofn<0[25].Thesignof theeffectiveindexisconnectedtothecurvatureofthe EFCs.Bothcasesofn>0andn<0canbeutilizedforthe purposesofnarrowbandspatialfiltering.Forwideband bandstopandband-passfiltering,theFabry–Perot reso-nancesarerequired,whichwouldallowoneobtaining mountainsoft0≈1thatarenotshiftedwhilevaryingka

withinacertainθ-range.Thisshouldcorrespondtothe caseswhentheEFCcurvatureiszeroforacertainrange

ofkxand,thus,theconstantphaseconditioncanbe

ful-filledforaFabry–Perottransmissionresonancewithina desiredanglerange.Basically,itispossibleduetothe fact that ky=constfor thisrange.Thus,the closerthe

EFCshapetosquareoneis,theweakerthesensitivityof thespectrallocationofthemountainpeaktovariations inθcouldbe.Forexample,thissituationcanberealized for thethird lowestmodeandpartiallyfor thesecond lowestmodeat3.12<ka<3.5,arangeusedinour ear-lier studiesof spatialfiltering [7,8].In thiscase, dual band-passfilteringcanbeobtained.However,the prob-lemremainsregardingsimultaneouslylargevaluesoft0

forthebothlarge-andsmall-anglepassbands.Onthe otherhand,thesamekarangecanbeutilizedfor band-stopfilteringregimeregardlessoftheabove-mentioned problem.

Infact,theconditionofzerocurvatureofEFCs coin-cideswiththatrequiredforfiniteanglerangecollimation insidethePhC.Hence,efficientcollimationshould co-existwithspatialfilteringthatischaracterizedbyt0≈1

atthemountains. Strictlyspeaking,thesetwo regimes represent two sides of one phenomenon. In turn, the depth of the valleys anddistance between the neigh-boring mountain peaks depend onthe scenario of the evolutionofEFCsatvaryingka.Itcanbecharacterized, for instance,intermsof theeffectivegroupindexand effectiveimpedance.However,forthepurposesof engi-neeringspatialfilterswithsuitablecharacteristics,itis lessusefulthanthedirectanalysisoftransmission.Thus, themainattentionispaidtotheeffectexertedby vari-ation ind/aon thetopology andotherfeatures of the transmittancemapsinthe(ka,θ)-plane.

Let us mention two more interesting features observedinFig.2(b).First,inthevicinityofka=2.95 andθ=40◦,weobtainmultipleandverynarrow moun-tainpeaksoft0=1,whichappearforthesecondlowest

mode dueto the specifictransformation of the EFCs whilevaryingka.Thistypeofbehavior,whichis possi-bleforanonmonotonousbandsolution,hasbeenstudied in detail in [25]. Secondly, a semi-ring of t0≈1 at

3.08<ka<3.12 andθ>65◦,whichappearsduetothe secondandthirdlowestmodes,shouldbenoticed. More-over,oneshouldmentionlow-passfilteringthatoccurs at3.65<ka<3.8duetothefourthlowestmode,which corresponds to amonotonous band solution. Angular bandwidth isincreasedherewiththevalue ofkaand withthewidthof thenearly circularEFCsbeing sim-ilar tothoseinFig.1(d–f).Although low-passspatial filteringisknownasaneasilyobtainableregime,its co-existencewithothertypesoffilteringinotherfrequency ranges butin thesame structure canbe important for multifunctionaloperation.

It is noteworthy that t0 in Fig. 2(b) dramatically

decreases above the first-order threshold line, where redistributionoftheincident-waveenergyinfavorofthe diffractionorderm=−1takesplace.Asaresult, behav-ioroft0inthe(ka,θ)-planemaydifferfromthatpredicted

withtheaidofEFCanalysis,inwhichpossibleeffects ofhigherordersareignored.Infact,onemayconsider

θ-domainbehaviorat3.12<ka<3.5aseitherbandstop filteringordualband-passfiltering.Fig.2(c–e)presents dependencesoft0 onkaattheselectedvalueofθ,for

differentvaluesofN.Onecanseethatthetransmittance maxima are located within nearly the same ka-range,

whereasthedifferenceoccursinthenumberofthepeaks of t0=1 andvalleys anddistance between them.The

valueofNonlyaffects thedensity,i.e.,the numberof themountainpeaksandvalleysinthesamemanneras intheclassicalFabry–Perotresonators[24,25].

Inordertodemonstratethatthefeaturesobservedin

Fig.2(b) are quitegeneral,we varythe PhC parame-ters.ForthetransmittancemapshowninFig.3(a),we takealargervalueof N andsimultaneouslyasmaller valueofd/a.Thevalueofd/aaffectslocationand trans-missionpropertiesof the areasin(ka,θ)-plane, which areconnectedwithdifferentFloquet–Blochmodes.As expected,theyareshiftedtowardlargerka-valuesforall threemodesconsidered,owingtoasmallervalueofd/a.

Animportant observationinFig.3(a)concernsthe fact that now themountains of t0 for the third lowest

modetendtomerge,whilethevalleysbecome weaker pronounced, see also Fig. 3(d). This leads to rather largeareasofhightransmissioninthe(ka,θ)-plane,e.g., at3.5<ka<3.7and30◦<θ<60◦.The sameandeven strongerpronouncedeffectappearsforthefourth low-estmodeat3.85<ka<4.15,i.e.,inthelow-passspatial filteringregime.However,similarbutweakermerging thatappearsforthethirdlowestmodeismoreimportant, becausethismodecontributestobandstop/band-pass fil-tering.Itisworthnotingthestrongtransmissionowing tothe diffractionorderm=−1that takesabigpartof theincident-waveenergyintheareaabovethethreshold line(notshown).Asharpswitchingbetweentheorders

m=0andm=−1ispresentlyunderstudy.

In the contrast to Fig. 2(b), there is a range of band-pass spatial filtering at 3.58<ka<3.83 in

Fig.3(a), wheretransmissionvanishesatsmall values of θ,in line withthe scenariosthat are schematically shown in Fig. 1(g and i). The use of three neigh-boringka-ranges,e.g.,3.39<ka<3.58,3.58<ka<3.83 and3.85<ka<4.15,enablesbandstop/dualband-pass, band-pass,andlow-pass spatialfiltering,respectively. Thus,thevarietyofspatialfilteringregimesachievable in onesimple PhC based structure canbe quite rich.

Fig.3.(a)Mapoft0in(ka,θ)-planeatd/a=0.4,ε=9.61,andN=12;straightverticallines–approximateboundariesofka-rangeswhichcanbe

usedforbandstop(BS),band-pass(BP),andlow-pass(LP)spatialfiltering,dashedcurve–thresholdlineforthediffractionorderm=−1;large numbers,2,3and4,indicatetransmissionareaduetothecorrespondingFloquet–Blochmode;(b–d)t0vskaatd/a=0.4,ε=9.61,θ=40◦,and(b)

N=6,(c)N=8,and(d)N=12.

Thesecond and thirdof them appearduetothe third andfourthlowestmodes,respectively,whereasthefirst one owes to the second and third lowest modes. As inFig.2(b), bandstopfilteringcanalsobeobtained at 2.95<ka<3.08and3.17<ka<3.33duetothesecond mode corresponding to a nonmonotonous band solu-tion.Thisregimecorrespondstothecouplingscenarios shownintheschematicsinFig.1(aandc).Similarlyto

Fig.2(b),theproblemofkeepingt0highandconstant

simultaneouslywithinbothpassbandscancomplicate thepossibleobtainingofefficientwidebanddual band-passfilteringattheselectedvaluesofka,whichbelong to the range being appropriate for bandstop filtering.

Fig.3(b–d)demonstratestheeffectofNonthe depend-enciesoft0atfixedθ.Again,Fabry–Perottypebehavior

isevident.Notethatt0<1 atoneof thepeaksinthese

plotsbecauseoftheeffectoftheorderm=−1.

Next,letusfurtherdecreasethevalueofd/a.Fig.4

presentsthetransmissionresultsford/a=0.35,whereas

N is again the same as inFig. 2(b and d). As far as the roleoftheorderm=−1becomesmoreimportant, thetransmissionresultsarepresentedherefortheboth propagatingorders,i.e.,m=0andm=−1.Awider

ka-rangeofsingleband-passspatialfilteringcanbeobtained duetozero-ordertransmissionascomparedtoFig.3(a), seeFig.4(a).Itislocatednowat3.72<ka<4.2.Asin

Fig.3,thisregimemayoccurhereowingtothedispersion behaviorlikethatshownschematicallyinFig.1(gand i).The(ka,θ)-areaofhightransmissionduetotheorder

m=−1isseeninFig.4(b).Itcanbeproperlycombined withthepassbandsconnectedwiththeorderm=0for thesamemode.Comparingzero-ordertransmissionfor the second lowestmode inFigs. 2(b), 3(a), and4(a), onecansee thatnot onlytypesof spatial filteringbut

Fig.4.Mapsof(a)t0and(b)t−1in(ka,θ)-planeatd/a=0.35,ε=9.61,andN=8;straightverticallines–approximateboundariesofka-ranges

whichcanbeusedforbandstop(BS)andband-pass(BP)spatialfiltering,dashedcurve–thresholdlinefortheorderm=−1;largenumbers,2and 3,indicatetransmissionareasduetothecorrespondingFloquet–Blochmodes;notethatthedifferentscalesareusedinplots(a)and(b).

Fig.5. Mapsoft0in(ka,θ)-planeat(a)d/a=0.5,(b)d/a=0.55,ε=9.61,andN=8;straightverticallines–approximateboundariesofka-ranges

whichcanbeusedforbandstop(BS)andlow-pass(LP)spatialfiltering,dashedcurve–thresholdlinefortheorderm=−1;largenumbers,2and 3,indicatetransmissionareasduetothecorrespondingFloquet–Blochmodes.

alsobasiccharacteristics,e.g.,locationsoftransmittance mountainsandlocationsanddepthsofthevalleyscan easilybemodifiedjustbyvaryingd/a.

Zero-ordertransmissionbecomes weakerabovethe threshold line in Fig. 4(b), since the incident-wave energy is redistributed in favor of the order m=−1. Amongthe featuresobservedinFig.4(b), the follow-ing onesshould benoticed: themountains of t₋₁ that weaklydependonθatka=3.2,ka=3.35,andka=3.5;a ring-shapedmountainoft₋₁thatappearsnearka=3.65; andawide(ka,θ)-areawithgradualvariationsint₋₁that occursat3.8<ka<4.2and38◦<θ<73◦.Thefirstone canbeusedformultifrequencyband-passspatial filter-ing atlarge valuesof θ. In thiscase, deflection angle of theoutgoingbeam ofthe orderm=−1 variesfrom

φ−1=−77◦ atθ=64◦ to φ₋₁=−65◦ at θ=75◦ when

ka=3.35,andfromφ₋₁=−77◦atθ=55◦toφ₋₁=−56◦ at θ=75◦ when ka=3.5. The third one is especially appropriate for band-pass filtering when the incident wave has a wide frequency and/or angular spectrum. However,thedisadvantageoftheuseoftheorderm=−1 isthatthereflectionsarequitestrong.

Now, we increase d/a ratio compared to the case depicted in Fig. 2(b–e). The results for d/a=0.5 and

d/a=0.55 are presented in Fig. 5(a) and (b), respec-tively.Anarea,whichmightbeusedforhigh-/band-pass spatialfiltering,isnotobserved.Justtwotypesof filter-ing,i.e.,low-passandbandstopfiltering,areobserved in Fig. 5(a). These regimes require the same shapes and locations of EFCs as similar regimes in Figs. 3

and 4 do. Atthe sametime, a newimportant feature isobtained,whichmanifestsitselfinthattheindividual mountainsoft0≈1correspondingtothesecondlowest

modetendtomergeat2.8<ka<3.18andθ<35◦.For example,t0>0.85inthe(ka,θ)-planeat2.63<ka<3.08

andθ<15◦.Thisleadstotheappearanceofaverylarge area,whichisalmosttransparentfortheincidentwaves. Hence,thiscaseisparticularlyappropriatewhenspatial filteringwithastopband,whichislocatedbetweentwo passbandsofanearlyperfecttransmission,isrequired. Inparticular, itispossibletoovercome thedifficulties inachievingdualband-passfiltering,whichhavebeen mentioned in the analysis of Figs. 2(b) and3(a). For example,bandstopanddualband-passregimescanbe obtainedinFig.5(a)atka=2.9,wheret0≈1forθ<25◦

and53◦<θ<70◦whilet0≈0at27◦<θ<51◦.Besides,

alarge areawitht0≈0 occurs inthe (ka, θ)-plane at

3.24<ka<3.61 due to the band gap arising between thethirdlowestmodeandthefourthlowestmode.This resultsinall-angle separationofthe areasof bandstop (leftward)andlow-pass(rightward)spatialfiltering.At thesametime,twomentionedtypesofspatialfiltering canbeobtained inoneconfigurationatvery close fre-quenciesnearka=2.83duetoaverynarrowθ-dependent stopbandthatappearsbetweenthetransmissionareas, whichcorrespondtothesecondandthirdlowestmodes. However,thislow-passfilteringregimeenables flexibil-ityinneitherthespectrallocationnorthewidthofthe transmissionbandinθ-domain.

Fig.5(b)demonstratestheeffectexertedbyafurther increaseofd/aratio.Themainfeaturesarethesameas inFig.5(a).Thedifferencesincludetheshiftofthe trans-missionareasconnectedwiththesecondandthirdmodes towardsmallervaluesofka,wideningthestopband adja-centtotheloweredgeofthetransmissionareaconnected withthefourthmode,shiftoftheareaofthestrongEFC transformationforthesecondmodetowardlargervalues ofθandsmallervaluesofka,andthethirdlowestmode areabecomingfullyfreeofdiffraction.Thus,itisevident thatnotonlycombinationsofvariousregimesofspatial

filteringbutalsootherregimes,e.g.,theareaofstrong EFCtransformationstudiedin[25],canbeobtainedin slabsofPhCforawiderangeofd/avariation.

4. Conclusion

To summarize, various regimes of spatial filter-ing have been studied in simple photonic structures thatrepresentfinite-thicknessslabsoftwo-dimensional rod-type photonic crystals (PhCs). Spatial filtering withwidebands of nearly perfect transmissioninthe incidence-angledomaincanbeobtainedforawiderange of parameter variation. Strictly speaking, one always obtainsthisorthattypeofspatialfiltering,exceptforan exoticcasewhenthedispersionforthePhCis(nearly) the same as that for air. However, obtaining various combinationsofthespatialfilteringregimesinone con-figuration that utilize wide angle-domain pass bands withhighandnearlyconstanttransmittanceneedssome parameter adjustments. It has been shown that it can berealizedbypropervariationsinthe rod-diameter-to-latticeconstantratio,d/a,withoutchangingtheprincipal geometricalfeaturesoftheusedstructure.Someofthe filteringregimescanbestronglyaffectedbyblazing,i.e., redistributionoftheincident-waveenergyinfavorofa higher diffraction order. Small values of d/a are well suitableforband-passspatialfiltering.Inthiscase, sin-gleband-pass, bandstop,andlow-passfilteringcanbe obtained in the neighboring frequency ranges in one structure that opens a route to multifunctional appli-cations. In turn, large values of d/a are found to be appropriateforbandstop/dualband-passfiltering,with twohigh-transmittanceangle-domainbandsatlargerand smalleranglesthat areadjustedtothestopbands aris-ing at intermediate angles. Thus,the obtained results suggestasolutiontotheproblemoftwo simultaneous angle-domainbandsofnearlyperfecttransmission.Itis basedonthe useof twoFloquet–Blochmodes,oneof whichhasthenearlyperfecttransmissionwithinalarge areainthefrequency-angleplane.Anearlyperfect trans-missionandrelatedspatialfilteringcanbeachievedat multipleoperation frequencies,atleastsomeofwhich arenearlyequidistantfromtheirneighbors.Alongside thevarietyofdispersiontypescorrespondingto differ-entFloquet–Blochmodes,Fabry–Perotresonancesplay avery importantroleinthe richnessoftheachievable regimesofspatialfiltering.Ithascommonrequirements regarding dispersion and, thus, can co-exist with col-limation arising in a finite range of the angles. The presentedresultsmaybeusefulforbothtraditionaland newapplications.Atthenextsteps,hybridregimeswill bestudied,whichinclude butare notrestrictedtothe

combinationofcollimation,spatialfiltering,blazing,and controlofcouplingstrengthbycorrugationsplacedatthe interface(s).

Acknowledgements

ThisworkissupportedbytheprojectsDPT-HAMIT, ESF-EPIGRAT, and NATO-SET-181 as well as by TUBITAK under Project Nos. 107A004, 109A015, 109E301,and110T306.ThecontributionofA.E.S.has partially been supported by TUBITAK in the frame-work of the Visiting Researcher Programme and by NationalScienceCenterofPolandunderProject Mag-noWa DEC-2-12/07/E/ST3/00538. A.P. acknowledges financialsupportfromtheMinistryofEducationand Sci-enceofRussianFederationintheframeworkofstatetask 11.1227.2014/K andfromtheGovernment of Russian Federation,Grant074-U01.E.O.acknowledgespartial support from the Turkish Academy of Sciences. The authorsarethankfultotheanonymousreviewerwhose commentsallowedtoimprovethispaper.

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