Experimental and theoretical analysis of the intensity of beams diffracted by three-dimensional photonic crystals

Expe imen al and heo e ical analysis o he in ensi y o beams di ac ed by h ee-dimensional pho onic c ys als 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, A gen ina Daniel Schinca Cen o de In es igaciones Óp icas (CIOp), (CIC-CONICET), La Pla a, A gen ina D o Ciencias Básicas, Facul ad de Ingenie ía, Uni e sidad Nacional de La Pla a, 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, Spain 共Recei ed 30 Ap il 2008; e ised manusc ip ecei ed 18 June 2008; published 4 Augus 2008兲 An analysis o he di ac ed beams eme ging om h ee-dimensional pho onic c ys als is he ein p esen ed. The wa e ec o s o nonspecula beams a e calcula ed o a iangula wo-dimensional la ice and he change in hei di ec ions as a unc ion o he wa eleng h is con i med expe imen ally o he case o ace-cen e ed- cubic colloidal c ys als illumina ed unde no mal incidence. A luc ua ing beha io o beam in ensi y as a unc ion o he wa eleng h o he inciden ligh is p edic ed o pe ec ly o de ed la ices. As i is he case o specula ly e lec ed and ballis ically ansmi ed beams, his modula ion a ises om mul ipole esonances o he sphe e ensemble ha a e smoo hed ou ia he di use ligh sca e ing p oduced by impe ec ions in he c ys alline s uc u e. When op ical ex inc ion is in oduced in o de o model he e ec o impe ec ions, i is possible o accu a ely ep oduce expe imen al obse a ions. DOI: 10.1103/PhysRe B.78.075102 PACS numbe 共s兲: 42.70.Qs, 41.20.Jb, 78.20.Bh, 78.40.⫺q Th ee-dimensional 共3D兲pho onic c ys als, which a e ma- e ials wi h a dielec ic unc ion ha ing 3D spa ial pe iodic- i y, ha e ecei ed much a en ion du ing he las decades mainly due o hei po en ial applica ions in op ical, in a ed, and mic owa e de ices.1,2This kind o ma e ial is he only one capable o a oiding ligh p opaga ion in all di ec ions when he dielec ic con as is high enough, so hey can p esen a comple e band gap in hei pho onic band s uc u e.3This p ope y has been used o mold he emission o op ically ac i e ma e ials and p oposed o se e al ech- nological applica ions ha a e s ill unde con inuous esea ch.4The ad en o ab ica ion echniques, ha ake ad- an age o he sel -assembling p ope ies o sphe ical colloi- dal pa icles in he mic ome e scale, has made i possible o obse e s op bands5and e en ull gaps6in he isible and nea in a ed spec a. Imp o emen s in hese echniques ha e led o high quali y colloidal c ys als wi h a low densi y o de ec s.7–9This has enabled he obse a ion o p e iously unde ec ed op ical e ec s in he so-called high ene gy ange, whe e he la ice cons an is equal o g ea e han he wa e- leng h. Fo his ange, in e es ing undamen al phenomena ha e been obse ed when ligh p opaga es h ough low dis- pe sion modes, such as he supe p ism e ec 10 o beam sel - ocusing.11 The op ical spec a ea u es obse ed in his ange, such as he appea ance o e lec ance peaks and ans- mi ance dips in he absence o any band gap, ha e gene a ed an in ense deba e on he physical mechanisms o igina ing hese ea u es.12–15 An in e es ing and almos unexplo ed phenomenon occu - ing in he high ene gy ange is he opening o di ac ion channels,13 ha is, a ini e numbe o di ac ed beams eme ge om he c ys al slab when he pho on ene gy is g ea e han a h eshold ene gy o di ac ion cu o . These di ac ed beams a e p opaga ing wa es ha can be p ojec ed on a sc een in o de o measu e hei in ensi ies. Howe e , up o da e, mos o he expe imen al and heo e ical analyses in he high ene gy ange a ailable in he li e a u e ha e been ocused on he in ensi ies o he specula ly e lec ed and o - wa dly 共o ballis ically兲 ansmi ed beams unde no mal incidence.12 In his con ex , he physical o igin o he ea- u es obse ed in he specula e lec ance and o wa d ans- mi ance spec a has ecen ly been explained by means o mul ipole esonances inside he sys em.16 Besides, he mod- eling o diso de ia ex inc ion has also shown ha sha p esonances a e smoo hed ou mo e d ama ically as a conse- quence o he di use ligh sca e ing.17,18 I has been ound ha he unin en ional impe ec ions in oduced in he ab i- ca ion p ocess19 p oduce di usely sca e ed ligh which e- mo es ene gy om he cohe en e lec ed and ansmi ed beams.16 This e ec has been simula ed by adding ex inc ion o he heo e ical model o a pe ec ly o de ed la ice o sphe es, ob aining i ings o he specula e lec ance and o - wa d ansmi ance expe imen al da a wi h a deg ee o accu- acy ha has no p eceden .17 Mul ipole esonances and ex- inc ion due o impe ec ions a e expec ed o ha e simila e ec s o beams di ac ed o no mal. Howe e , al hough hei obse a ion has been epo ed,13 no ac ual measu e- men s o di ac ed beam in ensi ies, no a heo e ical p edic- ion o hem, has been pe o med so a . In his pape , we p esen o he i s ime a comple e desc ip ion o nonspecula di ac ed beams, bo h heo e i- cally and expe imen ally. We measu ed he ela i e in ensi y o each di ac ed spo as he wa eleng h 共and he e o e he di ac ion angle兲is a ied in a lase beam emi ed om an PHYSICAL REVIEW B 78, 075102 共2008兲 1098-0121/2008/78共7兲/075102共5兲©2008 The Ame ican Physical Socie y075102-1 op ical pa ame ic oscilla o ha impinges pe pendicula ly on a sel -assembled h ee-dimensional colloidal c ys al. We use a Ko inga-Kohn-Ros oke 共KKR兲app oach o calcula e he expec ed op ical esponse o bo h pe ec and impe ec 共 eal兲la ices. The e ec o diso de is analyzed in de ail and modeled by adding an imagina y pa o he dielec ic con- s an o he sphe es. Wi h ex inc ion in oduced in his ash- ion, we a e able o ep oduce he main ea u es obse ed in ou measu emen s. We ind ha pe ec ly o de ed s uc u es p esen a luc ua ing spec al esponse whose op ical ea u es apidly smoo hens as ex inc ion is in oduced, al hough some ea u es a e obus agains diso de and can be obse ed in bo h pe ec and impe ec c ys als. Sel -assembled 3D colloidal c ys al ilms we e p epa ed using he p ocedu e desc ibed in Re . 20 by he me hod o e apo a ion-induced sel -assembly on o e ical o il ed subs a es. In pa icula , hey we e deposi ed on o la glass subs a es by e apo a ion o polys y ene sphe e suspensions 共IKERLAT, polydispe si y below 3%, densi y ␳ =1.1 g/cm3, e ac i e index n=1.59兲o pa icle olume ac ion com- p ised be ween 0.1% and 0.2%. E apo a ion empe a u e was kep cons an o a gi en la ice g ow h p ocess, and anged be ween 35 and 50 °C. As he suspension e apo a es, a c ys alline ilm is deposi ed on he subs a e a he con ac line wi h he suspension meniscus. The model s uc u e is he e o e a close-packed ace-cen e ed-cubic 共 cc兲la ice o sphe es o dielec ic cons an ␧s=2.53+i␧iembedded in a medium o ␧=1, which would co espond o la ex sphe es in ai . The c ys al-g ow h di ec ion is ypically he 关111兴,21 so we will ocus on he op ical p ope ies o di ac ed beams when ligh impinges in ha pa icula di ec ion. Hence, he su ace o he c ys al slab p esen s a iangula la ice s uc- u e and subsequen laye s a e o de ed in he known se- quence ABCABC..., he inciden ligh beam being no mal o he slab su ace 共no mal incidence兲. Pa icle diame e dwas measu ed om scanning elec on mic oscopy images o such ex e nal su ace. A e age cen e o cen e dis ance be ween neighbo ing sphe es was ound o be 752 nm. The ligh sou ce was a uneable op ical pa ame ic oscilla o 共OPO兲 pumped by he hi d ha monic 共355 nm兲o a Q-swi ched Nd:YAG 10 Hz epe i ion a e pulsed lase 共Con inuum Su e- li e II兲. As i is well known, nonlinea op ical equency mix- ing inside he be a-ba ium bo a e 共BBO兲c ys al o he OPO p oduces cohe en ou pu a wo equencies 共signal and idle 兲whose sum equals he pumping equency o a ce ain p opaga ion di ec ion 共phase ma ching condi ion兲. Fo 355 nm inpu wa eleng h, he signal ell in he isible ange while he idle ell in he nea in a ed 共NIR兲 ange. Since he phase ma ching di ec ion is equency dependen , he ou pu wa eleng h can be con inuously a ied by mic omechanical il o he BBO. In ou case, he IR idle ou pu was blocked using an IR-abso bing isible- ansmi ing il e . Di ac ed beam in ensi ies we e measu ed using a silicon pho odiode a ached o a XYZ⌰⌽ moun ha allowed us o accu a ely posi ion he de ec o in he co ec di ac ed beam di ec ion and angle. The pho odiode ou pu was ed in o a digi al os- cilloscope. Al hough no ime- esol ed measu emen s we e needed in his expe imen , he 50 ⍀inpu coupling was se- lec ed o a oid any possible signal sa u a ion. To minimize he e ec o sho - o-sho luc ua ion, in ensi y alues we e aken as he a e age o 32 sho s. Since e iciency o non- specula e lec ed beams was o be measu ed, a ac ion o he incoming beam was aken using a calib a ed beam spli - e and i s in ensi y measu ed wi h he same pho odiode. Di - ac ed beam in ensi ies o each selec ed wa eleng h we e no malized o he co esponding incoming in ensi y, and e- lec ance 共 ansmi ance兲was calcula ed in he usual way. The o e all e lec ance unce ain y was abou 5%. The in ensi ies o di ac ed beams we e calcula ed using he ec o KKR me hod22,23 in i s laye e sion.24 In his me hod, he c ys al slab is i s di ided in o laye s pa allel o a gi en c ys allog aphic plane, each laye con aining a wo- dimensional 共2D兲la ice o iden ical sphe es. Nex , a mul i- pole expansion in sphe ical wa es is used o calcula e he mul iple sca e ing be ween sphe es in a gi en laye . Finally, a plane-wa e expansion is used o accoun o he mul iple sca e ing be ween laye s. In all he calcula ions ha ollow, nume ical con e gence was ob ained using a cu o LMAX =9 in he sphe ical wa e expansion and 41 plane wa es.25 I he sphe e laye s a e pa allel o he xy plane, a se o 2D p imi i e la ice ec o s is a1=dx ˆand a2=d共x ˆ+冑3y ˆ兲/2, whe e dis he dis ance be ween la ice si es, equal o he sphe e diame e in a close-packed s uc u e. The p imi i e ecip ocal-la ice ec o s can be chosen as b1=4 ␲ y ˆ/共冑3d兲 and b2=2 ␲ 共y ˆ/冑3−x ˆ兲/d, hen any ecip ocal-la ice ec o can be exp essed as g=pb1+qb2, whe e 共p,q兲is a pai o in ege s. In no mal incidence, when an incoming ligh beam a els in he posi i e z-axis di ec ion, he wa e ec o o a di ac ed beam eme ging om he slab can be w i en as Kg ⫾=g⫾冑k2−兩g兩2z ˆ, whe e he +共–兲sign co esponds o a ansmi ed 共 e lec ed兲beam, k=2 ␲ /␭is he magni ude o he inciden wa e ec o and ␭is he wa eleng h. Each di - ac ed beam co esponds o a p opaga ing wa e i he z componen o Kg ⫾is pu ely eal. Then, di ac ion channels a e open when 兩g兩⬍kand we ha e a di ac ion cu o when- e e 兩g兩=k. Fo a iangula la ice, his condi ion educes o d ␭=1 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. In ai 共n=1兲, six di ac ion channels—co esponding o 共p,q兲=共1,0兲,共1,−1兲,共0,−1兲,共−1,0兲,共−1,1兲,共0,1兲—open when d/␭⬎2/冑3⬵1.155 o a/␭⬎2冑2/冑3⬵1.633, whe e a is he la ice cons an o he cc con en ional cubic cell. The wa e ec o s o hese di ac ed beams a e loca ed along a cone and he angle ␪ be ween Kg ⫾and he zaxis is gi en by sin共 ␪ 兲=2␭ 冑3d,共2兲 which gi es he angle ␪ o he i s six di ac ed beams as a unc ion o he wa eleng h. The specula ly e lec ed and o - wa dly ansmi ed beams co espond o channel 共0, 0兲which a e always open channels. This si ua ion is illus a ed in Fig. 1, whe e each e lec ed and ansmi ed beam is labeled ac- co ding o he pai o in ege s 共p,q兲. The ac ha he pho- onic c ys al is suppo ed on a glass subs a e 共 e ac i e in- dex n=1.53兲in oduces a i s di ac ion cu o o ansmi ed beams a a/␭=1.07. Al hough hose beams a e DORADO e al. PHYSICAL REVIEW B 78, 075102 共2008兲 075102-2 in e nally e lec ed a he glass-ai su ace and he e o e do no p opaga e in ai , hey can be expe imen ally obse ed by us a ing he o al e lec ion.13 The in ensi ies o di ac ed beams a e e e ed o he inciden -beam in ensi y. I we call R共p,q兲and T共p,q兲 o he e lec ance and ansmi ance coe i- cien s o di ac ion channel 共p,q兲, espec i ely, he o al e- lec ance and ansmi ance will be R=兺共p,q兲R共p,q兲and T =兺共p,q兲T共p,q兲, whe e he sums include he open channel 共0,0兲. The coe icien s R共p,q兲and T共p,q兲will also be e e ed o as e iciencies o channel 共p,q兲in e lec ion and ansmission, espec i ely. In he absence o ene gy losses, conse a ion o ene gy implies ha R+T=1. Since diso de emo es ene gy om he cohe en ly sca e ed beams, i wo ks as a so o loss mechanism and we ha e R+T⬍1 in an ac ual expe i- men . In Figs. 2共a兲and 2共b兲, pho og aphs o he di ac ion spo s can be obse ed on a sc een o wo di e en wa eleng hs, namely, ␭=484 nm and ␭=565 nm, espec i ely. These spo s a e p oduced by he six e lec ed beams ha a ise abo e he di ac ion cu o . The specula ly e lec ed beam 共0, 0兲passes h ough a hole pe o a ed in he sc een o allow he lase beam eaching he c ys al slab. Figu e 2共c兲shows he ansmi ed beams di ac ed om he slab and p ojec ed on a sc een also pa allel o he xy plane, as i is indica ed in Fig. 1. In his case he wa eleng h is ␭=539 nm and he spo p oduced by he o wa dly ansmi ed beam 共0, 0兲can be seen a he cen e o he pic u e. In hese pho og aphs, a uni o m backg ound due o di use ligh sca e ing can be obse ed. This ene gy is no cap u ed by he de ec o s mea- su ing he in ensi ies o di ac ion spo s and hen i can be conside ed as a kind o loss, which is simula ed by adding ex inc ion o he dielec ic cons an o he sphe es in he la - ice in ou heo e ical model. The exis ence o laye s ha ing a iangula la ice pa allel o he xy plane and piled up in he sequence ABCABC... along he zaxis implies a symme y ela ion o he e iciencies o di ac ed beams when he inciden ligh is pola ized. In all he expe imen s, he inciden wa e is linea ly pola ized along he xaxis, as i is indica ed by he inciden elec ic- ield ec o Einc in Fig. 1. Unde his condi ion, we ha e R共1,0兲,R共1,−1兲=R共0,1兲,R共0,−1兲=R共−1,1兲, and R共−1,0兲, hus he e a e essen ially ou di e en di ac ion spo s. A close look a Figs. 2共a兲–2共c兲allows us o obse e his symme y be ween spo s. Figu es 2共d兲–2共 兲show pho og aphs o di ac ion spo s ob ained by using a sc een pa allel o he yz plane o de- c easing alues o he wa eleng h, namely, 2共d兲␭=622 nm, 2共e兲␭=593 nm, and 2共 兲␭=512 nm. The wid h 共1mm兲o he subs a e on o which he sample 共a ound 5 ␮ m wide兲is deposi ed can be seen a he cen e o each pic u e and spo s o e lec ed 共 ansmi ed兲beams 共1,−1兲and 共0,−1兲a e on he igh 共le 兲o he sample. As we ha e men ioned, di ac- ion spo s o beams 共1,−1兲and 共0,−1兲ha e he same e i- ciencies as beams 共0, 1兲and 共−1,1兲, espec i ely. In Figs. 2共d兲–2共 兲we can see ha he spo s mo e away om he sample as he wa eleng h is dec eased, ollowing he change in he di ec ion o he di ac ed wa e ec o s Kg ⫾p edic ed by Eq. 共2兲. Subs i u ing he expe imen al alue a ained o he sphe e diame e d=752 nm in Eq. 共1兲yields a di ac ion cu o wa eleng h ␭c=651 nm. Hence, Fig. 2共d兲co esponds o a wa eleng h 共␭=622 nm兲close o he onse o di ac- ion spo s and Eq. 共2兲gi es ␪ nea 90°, which is ac ually wha we see in his igu e. Thus, looking a he di ac ed spo s on o a sc een pe pendicula o he c ys al p o ides a simple way o con i ming expe imen ally he di ec ional p ope ies o he wa e ec o s o di ac ed beams p edic ed by Eq. 共2兲. The calcula ed e lec ion e iciencies o di ac ed chan- nels a e plo ed in Fig. 3共a兲as unc ions o he wa eleng h o a sel -s anding c ys al 10 laye s wide wi hou ex inc ion 共␧i=0兲. The di ac ion cu o wa eleng h, ␭c, is indica ed by he e ical dashed line. No e ha di ac ion e iciencies o nonspecula beams a e all ze o o wa eleng hs g ea e han ␭c. Al hough he cu es p esen apid luc ua ions, wo main peaks can be app ecia ed o all he e iciencies, one close o ␭=530 nm and ano he nea ␭=475 nm, pa icula ly e i- den o he beam 共1,0兲. The angle ␪ o he di ac ed beams, gi en by Eq. 共2兲, is indica ed in he uppe ho izon al scale, FIG. 1. 共Colo online兲Scheme o he expe imen al se up used o measu e he e iciency o nonspecula e lec ed beams. T ansmi ed beams a e also indica ed. FIG. 2. 共Colo online兲共a兲–共b兲Di ac ion pa e ns o e lec ed beams p ojec ed on he sc een shown in Fig. 1共xy plane兲 o : 共a兲 ␭=484 nm and 共b兲␭=565 nm. 共c兲Di ac ion pa e n o ansmi - ed beams p ojec ed on a sc een pa allel o he xy plane o ␭ =539 nm. 共d兲–共 兲Di ac ion pa e ns o e lec ed and ansmi ed beams p ojec ed on a sc een pa allel o he yz plane o : 共d兲␭ =622 nm, 共e兲␭=593 nm, and 共 兲␭=512 nm. EXPERIMENTAL AND THEORETICAL ANALYSIS OF…PHYSICAL REVIEW B 78, 075102 共2008兲 075102-3 whe e we can see ha he main peak nea ␭=530 nm co e- sponds o ␪ ⬵55°. Since each sphe e in he la ice can be conside ed as a supe posi ion o elec ic and magne ic di- poles, quad upoles, oc upoles, e c., hese e iciency peaks o igina e om esonances inside he pho onic c ys al due o he in e ac ions be ween mul ipoles. In o he wo ds, he e is a na u al esonance mechanism inside he c ys al ha is ex- ci ed o only ce ain wa eleng hs. These esul s a e consis- en wi h p e ious obse a ions and calcula ions ha demon- s a ed ha he o igin o he op ical esponse o he e lec ed and ansmi ed 共0, 0兲beams, obse ed o a⬎␭ in 3D pho- onic c ys als, lies on mul ipola esonances o he sphe e ensemble.16 Hence i is no su p ising ha his mechanism also s ongly a ec s he di ac ion phenomena occu ing in hese la ices. In o de o ake in o accoun he e ec o dis- o de , he op ical spec a o di ac ed beams ha e been e- calcula ed adding an imagina y pa ␧i=0.04 o he dielec ic cons an o he sphe es, which is a alue ha has p o ided good i ing wi h measu emen s o specula e lec ance 共R共0,0兲兲and o wa d ansmi ance 共T共0,0兲兲. The esul s a e shown in Fig. 3共b兲, whe e a dec ease o he in ensi ies o he e lec ed beams is ob ained as well as a smoo hing o he cu es compa ed wi h Fig. 3共a兲. Calcula ions ha e been pe - o med o a glass-suppo ed slab since eal colloidal c ys als a e usually deposi ed on a glass subs a e 共 e ac i e index =1.53兲. Ne e heless, we mus poin ou ha he e lec ed beam e iciencies o he glass-suppo ed slab a e basically he same as he ones ob ained o he sel -s anding slab when ex inc ion is in oduced in he model. Fu he mo e, only he i s six laye s a e in ol ed in he op ical esponse o he slab because wa es ha e a ini e pene a ion dep h in o he slab.16,18 Figu e 4shows he measu ed 共solid lines兲and calcula ed 共dashed lines兲e iciencies o he e lec ed beams 共1, 0兲and 共1,−1兲, whe e a good co ela ion be ween heo y and expe i- men is ob ained. In his case, a shi o he peaks owa d highe wa eleng hs 共 ed shi 兲can be app ecia ed. To i he main peak posi ions o he expe imen al cu es we ha e used ␧s=2.45+0.03i. These esul s show ha , al hough many ine spec al ea u es a e smoo hed ou due o diso de e ec s, some esonances a e s ong enough o su i e he ex inc ion p ocess and can clea ly be app ecia ed in a eal expe imen . A he same ime, hey con i m ha ex inc ion due o impe - ec ions plays a c ucial ole in he op ical esponse o colloi- dal c ys al la ices and s ongly de e mines he esul o ac- ual measu emen s. In summa y, we ha e p edic ed and obse ed a luc ua ing op ical esponse o nonspecula beams di ac ed om pe - ec ly o de ed 3D pho onic c ys als, in good ag eemen wi h he beha io p e iously obse ed o e lec ed and ansmi - ed beams. Di ac ed beams in e lec ion and ansmission ha e been ecognized and labeled acco ding o hei associ- a ed ecip ocal-la ice ec o s, and he obse ed spec al de- pendence o he di ac ed beams in ensi y has been accu- a ely modeled using a KKR app oach. The e ec o diso de in he c ys alline s uc u e has been simula ed by adding ex- inc ion o he heo e ical model in o de o ep oduce he shape o he expe imen al cu es. Finally, we ha e ound some spec al ea u es in di ac ed beams a ising om s ong esonances ha a e obus agains diso de e ec s. ACKNOWLEDGMENTS 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.D. acknowledge suppo om Consejo Nacional de In es igaciones Cien í i- cas yTécnicas 共CONICET兲, Uni e sidad de Buenos Ai es 共UBA兲and Agencia Nacional de P omoción Cien í ica yTec- nológica 共BID 1728/OC-AR PICT-11-1785兲. D.S. aknowl- edges suppo om G an s No. PIP 5997 CONICET and No. PICT 26090 ANPCyT. H.M. is g a e ul o inancial suppo om he Spanish Minis y o Science and Educa ion unde G an No. MAT2005-03028. FIG. 3. 共Colo online兲Calcula ed e iciencies o nonspecula e lec ed beams o 共a兲a pe ec sel -s anding c ys al 10 laye s wide 共␧i=0兲and 共b兲 he same la ice a e in oducing ex inc ion 共␧i =0.04兲. FIG. 4. 共Colo online兲Measu ed 共solid lines兲and calcula ed 共dashed lines, ␧s=2.45+0.03i兲 e lec ion e iciencies o di ac ed channels 共1,0兲共black lines兲and 共1,−1兲共 ed lines兲. DORADO e al. PHYSICAL REVIEW B 78, 075102 共2008兲 075102-4 *[email p o ec ed] 1E. Yablono i ch, Phys. Re . Le . 58, 2059 共1987兲. 2K. Sakoda, Op ical P ope ies o Pho onic C ys als 共Sp inge - Ve lag, Be lin, 2001兲. 3K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Re . Le . 65, 3152 共1990兲. 4J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Pho onic C ys als 共P ince on Uni e si y P ess, P ince on, NJ, 1995兲. 5J. F. Be one, P. Jiang, K. S. Hwang, D. M. Mi leman, and V. L. Col in, Phys. Re . Le . 83, 300 共1999兲. 6A. Blanco, E. Chomski, S. G ab chak, M. Ibisa e, S. John, S. W. Leona d, C. Lopez, F. Mesegue , H. Míguez, J. P. Mondia, G. A. Ozin, O. Toade , and H. M. an D iel, Na u e 共London兲405, 437 共2000兲. 7H. Míguez, V. Ki ae , and G. Ozin, Appl. Phys. Le . 84, 1239 共2004兲. 8S. Wong, V. Ki ae , and G. A. Ozin, J. Am. Chem. Soc. 125, 15589 共2003兲. 9K. Wos yn, Y. Zhao, B. Yee, K. Clays, A. Pe soons, G. de Scha- e zen, and L. Hellemans, J. Chem. Phys. 118, 10752 共2003兲. 10 H. Kosaka, T. Kawashima, A. Tomi a, M. No omi, T. Tamamu a, T. Sa o, and S. Kawakami, Phys. Re . B 58, R10096 共1998兲. 11 A. Ma ínez, H. Míguez, A. G iol, and J. Ma í, Phys. Re . B 69, 165119 共2004兲. 12 J. F. Galis eo-López and C. López, Phys. Re . B 70, 035108 共2004兲. 13 F. 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. 14 X. Checou y, S. Enoch, C. López, and A. Blanco, Appl. Phys. Le . 90, 161131 共2007兲 15 A. Bales e i, L. C. And eani, and M. Agio, Phys. Re . E 74, 036603 共2006兲. 16 L. A. Do ado, R. A. Depine, G. Lozano, and H. Míguez, Op . Exp ess 15, 17754 共2007兲. 17 L. A. Do ado, R. A. Depine, and H. Míguez, Phys. Re . B 75, 241101共R兲共2007兲. 18 L. A. Do ado, R. A. Depine, G. Lozano, and H. Míguez, Phys. Re . B 76, 245103 共2007兲. 19 G. Lozano and H. Míguez, Appl. Phys. Le . 92, 091904 共2008兲. 20 G. Lozano and H. Míguez, Langmui 23, 9933 共2007兲. 21 P. Jiang, J. F. Be one, K. S. Hwang, and V. L. Col in, Chem. Ma e . 11, 2132 共1999兲. 22 A. Modinos, Physica A 141, 575 共1987兲. 23 K. Oh aka, J. Phys. C 13, 667 共1980兲. 24 N. S e anou, V. Yannopapas, and A. Modinos, Compu . Phys. Commun. 113,49共1998兲;132, 189 共2000兲. 25 N. S e anou, V. Ka a hanos, and A. Modinos, J. Phys.: Condens. Ma e 4, 7389 共1992兲. EXPERIMENTAL AND THEORETICAL ANALYSIS OF…PHYSICAL REVIEW B 78, 075102 共2008兲 075102-5