Interplay between crystal-size and disorder effects in the high-energy optical response of photonic crystal slabs

In e play be ween c ys al-size and diso de e ec s in he high-ene gy op ical esponse o pho onic c ys al slabs Luis A. Do ado and Rica do A. Depine* G upo de Elec omagne ismo Aplicado, Depa amen o de Física, Facul ad de Ciencias Exac as y Na u ales, Uni e sidad de Buenos Ai es, Buenos Ai es C1428EGA, A gen ina Gab iel Lozano and He nán Míguez Ins i u o de Ciencia de Ma e iales de Se illa, Consejo Supe io de In es igaciones Cien í icas, Se illa 41092, Spain 共Recei ed 20 June 2007; e ised manusc ip ecei ed 13 Augus 2007; published 5 Decembe 2007兲 Expe imen al e lec ance spec a ha e been ob ained o colloidal c ys als whose wid hs anged om one o se e al sphe e monolaye s, and hei ea u es in he highe o de band ene gy ange ha e been ep oduced heo e ically. In o de o i he measu ed da a, op ical ex inc ion has been in oduced in he heo e ical model, which accoun s o s uc u e impe ec ions and diso de , he main sou ces o losses in an ac ual measu emen . A complex spec um in he high equency egion is obse ed e en o one o de ed monolaye , being his peak s uc u e g adually modi ied as mo e laye s a e piled up. This allowed us o iden i y which peaks a e eminis- cen o he op ical e lec ance ea u es o a single close-packed laye and which a e he esul o building up a h ee dimensional pe iodici y. A clea co ela ion be ween he amoun o ex inc ion in oduced in he i ing and he slab wid h has been ound, which demons a es ha wide eal c ys als p oduce less di usely sca e ed ligh . A he same ime, we ind ha he op ical esponse o hinne c ys als is mo e obus agains he in oduc ion o ex inc ion han ha o hicke ones, o which he e ec is d ama ic. DOI: 10.1103/PhysRe B.76.245103 PACS numbe 共s兲: 78.20.Bh, 78.40.⫺q, 42.70.Qs, 41.20.Jb Imp o emen s in he p ocesses o ab ica ion o sel - assembled h ee dimensional 共3D兲pho onic c ys als 共PhCs兲, which a e ma e ials wi h a spa ially pe iodic dielec ic unc- ion in all h ee dimensions, ha e made possible he obse - a ion o high quali y op ical spec a in highe o de band equencies.1–3Also, he appea ance o localized modes in- side band gaps in he pho onic band s uc u e due o he addi ion o local de ec s is a well-known e ec , which is analogous o he doping o semiconduc o s.4Poin , line, o plane de ec s a e c ea ed wi hin he c ys al by locally adding o emo ing ma e ial in a con olled manne ,5,6so hey a e di e en om in insic de ec s, which a e unin en ional dis- up ions o he spa ial pe iodici y o he dielec ic unc ion ha a ise du ing he ab ica ion p ocess. Despi e new ab i- ca ion echniques ha ha e alle ia ed he e ec o diso de in he so-called low-ene gy ange, whe e he la ice cons an is less han he wa eleng h o ligh , e y ecen ly, i has been demons a ed ha ex inc ion due o in insic de ec s in 3D PhCs s ongly a ec s he shape o he expe imen al spec a in he high-ene gy ange, whe e he la ice cons an is g ea e han he wa eleng h.7Also, calcula ions o he pho onic band s uc u e, pe o med conside ing ex inc ion, ha e shown a clea co ela ion be ween he beha io o he imagina y pa o he wa e ec o and he op ical ea u es obse ed in he spec a. This means ha diso de , while ha ing a small in- luence on he measu ed spec a in he low-ene gy ange, ampli ies i s e ec s a highe ene gies and, consequen ly, u - he imp o emen o he ab ica ion echniques is equi ed o achie e op ical quali y a hose pho on ene gies.8 In o de o pe o m a alid compa ison be ween he op i- cal spec a o ini e c ys al slabs and he pho onic band s uc- u e o an in ini e c ys al, PhC slabs o se e al laye s we e analyzed in a p e ious wo k; an excellen ag eemen be ween he expe imen al and he calcula ed op ical esponse a high ene gies has been ound. He e, we p esen an analysis o he beha io o he op ical spec a ea u es in he high-ene gy ange as he wid h o he c ys al slab is g adually inc eased. We compa e he expe imen al and heo e ical e olu ions o he specula e lec ance spec a as we inc ease he c ys al hickness om 1 o 18 sphe e laye s. We ound complex spec a in he high equency egion e en o he ini ial monolaye s uc u e,9being his peak s uc u e g adually modi ied as mo e laye s a e piled up. This allowed us o iden i y which peaks a e eminiscen o he op ical e lec- ance ea u es o a single close-packed laye and which a e he esul o building up a h ee dimensional pe iodici y. A he same ime, we ind a clea co ela ion be ween he c ys al size and he amoun o ex inc ion needed o a ain he bes i . The in e play o c ys al size and diso de e ec s on he high- ene gy op ical esponse is ho oughly analyzed, inding a much mo e d ama ic e ec o he o me on he e lec ance spec a o la ge c ys als. Pho onic c ys al ilms we e made by e apo a ion induced sel -assembly10 on glass subs a es o polys y ene sphe e 共IKERLAT, a e age diame e o 700 nm, polydispe si y be- low 3%, densi y ␳ =1.1 g/cm3, e ac i e index n=1.59兲sus- pended in wa e wi h pa icle olume ac ion anging om 0.05% o 0.20% and e apo a ed a empe a u es anging om 30 o 60 °C. Se e al independen measu emen s o he high-ene gy op ical esponse o simila s uc u es can be ound in he li e a u e.1,11,12 Thus, he echnique is su i- cien ly ma u e and eliable o p o ide ep oducible esul s in di e en labo a o ies. The ypical c ys al g ow h di ec ion is he 关111兴, and we will ocus on he op ical p ope ies when ligh impinges in ha pa icula di ec ion since hey a e bes known.13 Thus, in his case, a monolaye consis s o a ian- gula la ice o close-packed sphe es and he slab wid h is inc eased by piling monolaye s up in he sequence PHYSICAL REVIEW B 76, 245103 共2007兲 1098-0121/2007/76共24兲/245103共4兲©2007 The Ame ican Physical Socie y245103-1 ABCABC¯. As i is usual, he pho on ene gy is exp essed in educed uni s a/␭, whe e ais he la ice cons an o he classical cubic cell, whose alue is a=冑2 ␾ in an cc close- packed s uc u e, ␾ being he sphe e diame e and ␭ he wa eleng h o he inciden ligh . Re lec ance measu emen s we e pe o med using a Fou- ie ans o m in a ed spec opho ome e 共BRUKER IFS- 66兲a ached o a mic oscope. A 4⫻objec i e wi h a nume i- cal ape u e o 0.1 共ligh cone angle o ±5.7°兲was used o i adia e he la ices and o collec he e lec ed ligh a qua- sino mal incidence wi h espec o i s su ace. A spa ial il e was used o selec i ely de ec ligh om squa e spo s o 1000⫻1000 ␮ m2. Simula ed e lec ance spec a we e calcula ed by means o he ec o Ko inga-Kohn-Ros oke me hod.14,15 To ensu e con e gence in he high-ene gy ange, we used 41 wo di- mensional 共2D兲 ecip ocal la ice ec o s in he plane wa e expansions and sphe ical wa es wi h angula momen um up o lmax=9. These alues a e he minimum ones ha p o ide good con e gence in ha ange. Lowe alues o lmax and less ecip ocal la ice ec o s s ill gi e a co ec esul o lowe -ene gy bands, bu i is no su icien o p o ide us ul in o ma ion on he egion we a e in e es ed in. Measu e- men s we e pe o med wi h he PhC slab deposi ed on a glass subs a e o e ac i e index 1.53, so his subs a e is also included in he heo e ical model. The incoming me- dium is ai 共 e ac i e index=1兲. The sphe es a e embedded in ai and ha e a complex dielec ic cons an ␧s=2.5+i␧i, which co esponds o la ex sphe es 共 e ac i e index=冑2.5 ⬵1.58兲. An imagina y pa ␧iis added o he dielec ic con- s an o he sphe es in o de o ake in o accoun all possible sou ces o losses in ac ual measu emen s. Then, he alue o ␧iis chosen o i he expe imen al da a. Ene gy losses due o powe dissipa ion a e negligible in la ex sphe es, so s uc u e impe ec ions and diso de a e e- sponsible o he expe imen al de ia ion o he spec um shape wi h espec o he heo e ical op ical esponse wi hou ex inc ion.7In he high-ene gy ange, he a e age size o hese s uc u al impe ec ions, measu ed in wa eleng hs, be- comes mo e impo an han ha in he low-ene gy bands. Fo his eason, hese in insic de ec s16 cause ligh o be sca - e ed di usely, emo ing pa o he ene gy om he specu- la ly e lec ed o ballis ically ansmi ed beam. In his wo k, we will ocus on he analysis o specula e lec ance esul s, since hey can be eadily a ained by mic ospec oscopy om single domains o he c ys alline ilm using a low nume ical ape u e objec i e, ensu ing he hickness uni o mi y o he es ed spo and he no mal incidence o he beam. This is a much mo e complica ed ask o be pe o med in ansmission mode, since he inciden beam is ypically ocused wi h a la ge nume ical ape u e objec i e. In Figs. 1and 2, he measu ed and calcula ed specula e lec ances 共o R0,0兲can be seen as a unc ion o he la ice cons an a/␭ o di e en c ys al slab wid hs. Ve y good ag eemen be ween he expe imen al and heo e ical e lec- ance spec a has been ob ained by in oducing ex inc ion in he heo e ical model, as we ha e demons a ed o much hicke c ys als be o e. Figu e 1shows he expe imen al and heo e ical specula e lec i i y spec a o PhC slabs o a ew laye s 共N=numbe o laye s兲. The alue o ␧iin he calcula ed spec a was chosen o ob ain he bes i . No e ha all oscilla ions p esen a simila heigh wi h espec o he backg ound in bo h he expe imen al 共black solid lines兲and he simula ed 共 ed dashed lines兲spec a, which was ac ually he c i e ion ollowed o choose he bes i . Di e ences in he baseline a e due o he ac ha only one in e ace o he glass subs a e is conside ed o he calcula ions in o de o a oid he p esence o e y sho equency lobes in he spec- a. These a ise om he in e e ence be ween beams e- lec ed a he uppe and lowe aces o he hick subs a e 共1mm兲unde conside a ion. Also, some dispe sion in he eal pa o he dielec ic cons an was added, which accoun s o a small e ac i e index inc ease o ma e ials as we mo e owa d highe ene gies. In Fig. 1共a兲, wo main peaks o he monolaye 共N=1兲can FIG. 1. 共Colo online兲Measu ed 共solid lines兲and calcula ed 共dashed lines兲specula e lec ance spec a o glass-suppo ed slabs o Nlaye s composed o sphe es o dielec ic cons an ␧s=2.5+i␧i in ai . 共a兲N=1, ␧i=0.30. 共b兲N=2, ␧i=0.25. 共c兲N=3, ␧i=0.15. 共d兲 N=4, ␧i=0.13. FIG. 2. 共Colo online兲Measu ed 共solid lines兲and calcula ed 共dashed lines兲specula e lec ance spec a o glass-suppo ed slabs o Nlaye s composed o sphe es o dielec ic cons an ␧s=2.5+i␧i in ai . 共a兲N=6, ␧i=0.08. 共b兲N=13, ␧i=0.06. 共c兲N=18, ␧i=0.06. DORADO e al. PHYSICAL REVIEW B 76, 245103 共2007兲 245103-2 be seen: one nea a/␭=0.55 and he o he a a/␭=1.0. These peaks mo e g adually owa d highe ene gies as he slab wid h is inc eased, as Figs. 1共a兲–1共c兲show. Also, seconda y peaks ela ed o he Fab y-Pe o oscilla ions appea be ween hese main peaks o slabs o N=2, 3, and 4 laye s. Since all ou measu emen s we e aken a om he edges o he c ys- al, in which di e en ypes o s acking sequences ha e been epo ed,17 he ai ness o he i ings we a ain conside ing only he ABCABC¯sequence indica es ha an cc la ice is buil as mo e monolaye s a e piled up. In Fig. 2, he e lec i i y spec a o PhC slabs o N=6, 13, and 18 laye s can be app ecia ed. The main peak in he low-ene gy band ends o i s inal posi ion, which is a a/␭ =0.61, as can be p edic ed om he pho onic band s uc u e o he sphe e cc la ice unde conside a ion, because he e is a pseudoband gap in he ⌫Ldi ec ion cen e ed a his pa - icula ene gy.7The i s peak in he high-ene gy egion has mo ed o a/␭=1.07, and a second double degene a ed peak nea a/␭=1.2 is p esen . I is in e es ing o no e ha his second peak becomes double degene a ed in slabs wi h a leas six laye s. In o he wo ds, we mus comple e a leas wo ABC sequences in he slab in o de o ob ain he peak a a/␭=1.2. Figu e 2also shows ha his peak is di icul o see in he measu ed da a o N=6 and 13 laye s, bu Fig. 2共c兲 o N=18 laye s con i ms i s exis ence. The e is a second peak s uc u e in he ange o 1.5⬍a/␭⬍1.8, which is composed o wo peaks o simila ampli udes. They can be obse ed in Figs. 1共c兲,1共d兲, and 2, so he numbe o laye s needed mus be Nⱖ3, which co esponds o one o mo e comple e se- quences o ABC. The g adual appea ance o hese high- ene gy peaks as he numbe o sphe e laye s is inc eased can be p edic ed om he pho onic band s uc u e calcula ed wi h ex inc ion.7Howe e , o a/␭⬎1.63, di ac ed spo s emo e pa o he ene gy om he specula ly e lec ed beam. Then, abo e ha ene gy, bo h diso de and di ac ion con ibu e o he educ ion o in ensi y o he specula ly e- lec ed ligh . The a io be ween he powe ca ied by each di ac ed mode and he inciden powe is gi en by a e lec ion coe i- cien Rp,q, whe e he pai o in ege s 共p,q兲indica e he o de o di ac ion, and 共p,q兲=共0,0兲co esponds o he specula ly e lec ed ligh , he quan i y ha has been measu ed o his wo k. The onse o di ac ion in ai a a/␭=1.63 can be p edic ed by equi ing he conse a ion o he angen ial componen o he wa e ec o s o a 2D iangula la ice, ha is, a ␭艌冑2 n冑p2+共2q+p兲2 3,共1兲 whe e nis he e ac i e index o he di ac ion medium 共n=1 o ai 兲. This same exp ession explains he appea - ance o di ac ed modes only wi hin he glass subs a e 共n=1.53兲and no in he incoming medium a a/␭=1.07 o he suppo ed la ice.7In Fig. 3, we compa e he calcula ed spec um o R0,0 and ha o 兺共p,q兲⫽共0,0兲Rp,q共 he sum o he in ensi ies o all di ac ed modes in he homogeneous inci- den medium o e ac i e index n=1兲 o he slab composed o 18 laye s. The o al e lec i i y R=兺共p,q兲Rp,qis also plo ed in Fig. 3共a兲, whe e he in ensi y educ ion o R0,0 compa ed o Rcan be app ecia ed o pho on ene gies a/␭⬎1.63. The impac o diso de in he op ical esponse o he glass-suppo ed PhC slab can be obse ed in Fig. 4, whe e he specula e lec ance spec a o N=1 and 5 laye s is shown o di e en alues o he imagina y dielec ic con- s an ␧i. The a enua ion o he e lec ance alues compa ed o he case ␧i=0.0001, which co esponds o a slab wi h al- mos no diso de , can clea ly be seen in bo h cases. Howe e , his e ec is much mo e d ama ic in he case o he i e laye la ices, whose maximum a a/␭=1.15 is dec eased by a ac- o abo e 10, while o he monolaye , he co esponding maximum a a/␭=1.0 is educed by ac o smalle han 2. This is a heo e ical con i ma ion ha he hinne he c ys als, FIG. 3. 共Colo online兲共a兲Calcula ed specula 共solid line兲and o al 共dashed line兲 e lec ance spec a, R0,0 and R, o a glass- suppo ed slab o 18 laye s and sphe es o dielec ic cons an ␧s =2.4964+0.06iin ai . 共b兲Re lec ed di ac ed ligh , 兺共p,q兲⫽共0,0兲Rp,q, spec a. The e ical dashed line indica es he onse o di ac ion. FIG. 4. 共Colo online兲Calcula ed specula e lec ance spec a o 共a兲a glass-suppo ed monolaye 共N=1兲and 共b兲a glass- suppo ed slab o N=5 laye s, bo h wi h sphe es o dielec ic con- s an ␧s=2.5+i␧iin ai o di e en alues o ␧i. INTERPLAY BETWEEN CRYSTAL-SIZE AND DISORDER…PHYSICAL REVIEW B 76, 245103 共2007兲 245103-3 he mo e obus i s op ical esponse agains he in oduc ion o ex inc ion. In he expe imen al spec a o Figs. 1and 2, we can ap- p ecia e a g adual inc ease o he e lec ance ampli udes as he numbe o laye s is inc eased. Also, he imagina y pa ␧i o he dielec ic cons an o he sphe es dec eases as he numbe o laye s Nis inc eased in he calcula ed spec a. This endency is plo ed in Fig. 5, whe e ␧ican be seen as a unc ion o N. Hence, he e exis s a clea co ela ion be ween he ex inc ion in oduced and he wid h o he slab, which indica es ha di usely sca e ed ligh inc eases o hinne c ys als. In he case o c ys als o a ew laye s, besides he e ec o diso de , in e ac ions wi h he glass subs a e p o- duce an addi ional a enua ion o he e lec i i y alues. The e o e, he inc ease o ␧ias he c ys al wid h is dec eased means ha mo e di usely sca e ed ligh is p oduced, he combina ion o diso de and glass-c ys al in e ac ion being esponsible o his e ec . Ne e heless, o c ys als wi h six o mo e laye s, ␧ibecomes p ac ically a cons an , as Fig. 5 shows, which means ha any su ace e ec and glass-c ys al in e ac ion a e negligible, he alue o ␧ibeing a measu e o he deg ee o diso de in hese cases. Thus, we ound a me hod o quan i ying diso de in a eal c ys al because we can use a single pa ame e such as ␧i, which is de e mined by i ing he heo e ical model o he expe imen al da a, as a measu e o he s uc u al quali y o he PhC slab. A simila me hod has al eady been p oposed o he same ype o PhC ilms bu based on he i ing o he low-ene gy op ical e- sponse o he la ices.18 In e es ingly, he same end o he ex inc ion needed o i he expe imen al cu e e sus he numbe o close-packed laye s in he ilm is ound. In conclusion, we ha e ealized an in eg al app oach o he op ical esponse o 3D PhC slabs as a unc ion o he slab wid h in he highe o de band ene gy ange. By analyzing he g adual modi ica ion o ha esponse om one o se e al monolaye s, we could iden i y which peaks a e eminiscen o he op ical e lec ance ea u es o a single close-packed laye and which a e he esul o building up a h ee dimen- sional pe iodici y. Fu he mo e, we ha e ound a clea co e- la ion be ween he ex inc ion in oduced in ou heo e ical model and he wid h o he eal slab, which indica es ha wide c ys al slabs p oduce less di usely sca e ed ligh and p o ides a me hod o quan i ying diso de in hese cases. This wo k has been ealized in he amewo k o a join Spanish-A gen inian coope a ion p ojec CSIC-CONICET 共G an No. 2005AR0070兲. R.A.D. and L.A.D. acknowledge suppo om Consejo Nacional de In es igaciones Cien í i- cas y Técnicas 共CONICET兲, Uni e sidad de Buenos Ai es 共UBA兲, and Agencia Nacional de P omoción Cien í ica y Tecnológica 共ANPCYT-BID 802/OC-AR03-14099兲. H.M. is g a e ul o inancial suppo om he Ramón A eces Foun- da ion and he Spanish Minis y o Science and Educa ion unde G an No. MAT2005-03028. *[email p o ec ed] 1H. Míguez, V. Ki ae , and G. Ozin, Appl. Phys. Le . 84, 1239 共2004兲. 2F. Ga cía-San ama ía, J. F. Galis eo-López, P. V. B aun, and C. López, Phys. Re . B 71, 195112 共2005兲. 3K. Wos yn, Y. Zhao, B. Yee, K. Clays, A. 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