Quasi ormal Modes desriptio of trasmissio properties for Photoi Bad Gap strutures. A. Settimi (1-), S. Severii (3), B. J. Hoeders (4) (1) FILAS (Fiaziaria Laziale di Sviluppo) via A. Farese 3, 19 Roma, Italy () IGV (Istituto azioale di Geofisia e Vulaologia) via di Viga Murata 65, 143 Roma, Italy (3) Cetro Iterforze Studi per le Appliazioi Militari via della Bigattiera 1, 561 Sa Piero a Grado (Pi), Italy (4) Researh group theory of odesed matter i the Istitute for theoretial physis ad Zerie Istitute for advaed materials Uiversity of Groige ijeborg 4, L 9747 AG Groige, etherlads Abstrat I this paper, we use the Quasi ormal Modes (QM) approah for disussig the trasmissio properties of double-side opeed optial avities: i partiular, this approah is speified for oe dimesioal (1D) Photoi Bad Gap (PBG) strutures. Moreover, we ojeture that the desity of the modes (DOM) is a dyamial variable whih has the flexibility of varyig with respet to the boudary oditios as well as the iitial oditios; i fat, the e.m. field geerated by two moohromati outer-propagatig pump waves leads to iterferee effets iside a quarter-wave (QW) symmetri 1D-PBG struture. Fially, here, for the first time, a large umber of theoretial assumptios o QM metris for a ope avity, ever disussed i literature, are proved, ad a simple ad diret method to alulate the QM orm for a 1D-PBG struture is reported. PACS umbers: 4.5.Bs;.6.Lj; 78..Ci; 4.55.Tv 1
1. Itrodutio. Photoi rystals a be viewed as partiular optial avities havig the properties of presetig allowed ad forbidde bads for the eletromageti radiatio travellig iside, at optial frequeies. For these motivatios these strutures are also amed Photoi Bad Gap (PBG) [1]. I these strutures, dispersive properties are usually evaluated assumig ifiite periodi oditios []. The fiite dimesios of PBG strutures oeptually modify the alulatio ad the ature of the dispersive properties: this is maily due to the existee of a eergy flow ito ad out of the rystal. A pheomeologial approah to the dispersive properties of oe-dimesioal (1D) PBG strutures has bee preseted i. [3]. The appliatio of the effetive-medium approah is disussed, ad the aalogy with a simple Fabry-Perot struture is developed by Sipe et al. i. [4]. The problem of the field desriptio iside a ope avity was disussed by several authors. I partiular, Leug at al. [5] itrodued the desriptio of the eletromageti field i a oe side ope miroavities i terms of Quasi ormal Modes (QMs). Miroavities are mesosopi ope systems. Sie they are ope, ad theore leay, i.e. ooservative, the resoae field eige-futios are QMs with omplex eige-frequeies. They play importat roles i may optial proesses. The aalogy with ormal modes of oservative systems is emphasized. I partiular, i ertai ases, the QMs form a omplete set, ad muh of the usual formalism a be arried through. Miroavities are also ope system, i whih there has to be some output ouplig. O aout of the output ouplig, the eletromageti field ad eergy i the miroavity aloe would be otiuously lost to the outside. Thus, i physial terms, the miroavity is a o oservative system, while, from a mathematial stadpoit, the operators whih appear would be o-hermitia (or ot self-adjoit). This the leads to iterestig halleges i attemptig to geeralize the familiar tools of quatum mehais ad mathematial physis to suh a o hermitia settig. The issues whih arise, ad the framewor developed for addressig them, are geeri to may other situatios ivolvig ope systems. Theoretial results were obtaied i. [6] whih developed a Hilbert-Shmidt type of theory, leadig to a biliear expasio i terms of the atural modes (QMs) of the resolvet erel oeted with the itegral equatios of eletromageti ad potetial satterig theory (see erees therei [6]). The iitial value problem osidered by. [6] ould be solved usig this Hilbert-Shmidt type of theory taig the temporal Laplae trasform of the Maxwell equatios. Appliative results were obtaied by Bertolotti [7] who disussed the liear properties of oe dimesioal resoat avities by usig the matrix ad the ray method (see erees therei [7]).
Fabry-Perot, photoi rystals ad 1D avities i PBG strutures are osidered. QMs for the desriptio of the eletromageti field i ope avities are itrodued ad some appliatios are give. Theoretial ad appliative results were obtaied by. [8] whih developed a satterig theory for fiite 1D-PBG strutures i terms of the atural modes (QMs) of the satterer. This theory geeralizes the lassial Hilbert-Shmidt type of biliear expasios of the field propagator to a biliear expasio ito atural modes (QMs) (see erees therei [8]). It is show that the Sturm-Liouville type of expasios for dispersive media differs osiderably from those for odispersive media, they are e.g. overomplete. Colusive theoretial ad appliative results were obtaied by Masimovi [9] who used QMs to haraterize trasmissio resoaes i 1D optial defet avities ad the related field approximatios. Ref. [9] speializes to resoaes iside the badgap of the periodi multilayer mirrors whih elose the defet avities. Usig a field model with the most relevat QMs, a variatioal priiple permits to represet the field ad the spetral trasmissio lose to resoaes. I s. [1], for the first time, the QM approah was used ad exteded to the desriptio of the salar field behaviour i double-side opeed optial avities, i partiular 1D-PBG strutures. The validity of the approah is disussed by provig the QM ompleteess, disussig the omplex frequeies distributio, as well as the orrespodig field distributios, ad reoverig the behaviour of the desity of modes (DOM). I. [11], the eletromageti field iside a optial ope avity was aalyzed i the framewor of the QM theory. The role of the omplex frequeies i the trasmissio oeffiiet ad their li with the DOM is larified. A appliatio to a quarter-wave (QW) symmetri 1D-PBG struture is disussed to illustrate the usefuless ad the meaig of the results. I. [1], by usig the QM formalism i a seod quatizatio sheme, the problem of the outer-propagatio of eletromageti fields iside optial ope avities was studied. The lis betwee QM operators ad aoial destrutio ad reatio operators desribig the exteral free field, as well as the field orrelatio futios, are foud ad disussed. A appliatio of the theory is performed for ope avities whose rative idex satisfies symmetri properties. I this paper, we use the QM approah for disussig the trasmissio properties of doubleside opeed optial avities: i partiular, this approah is speified for 1D-PBG strutures. Moreover, we ojeture that the DOM is a dyamial variable whih has the flexibility of varyig with respet to the boudary oditios as well as the iitial oditios; i fat, the e.m. field geerated by two moohromati outer-propagatig pump waves leads to iterferee effets iside a quarter-wave (QW) symmetri 1D-PBG struture. Fially, here, for the first time, a large 3
umber of theoretial assumptios o QM metris for a ope avity, ever disussed i literature, are proved, ad a simple ad diret method to alulate the QM orm for a 1D-PBG struture is reported. This paper is orgaized as follows. I setio, the QM approah is itrodued. I setio 3, a large umber of theoretial assumptios o QM metris for a ope avity, ever disussed i literature, are proved. I setio 4, the trasmissio oeffiiet for a ope avity is alulated as a superpositio of QMs. I setio 5, the QM approah is applied to a 1D-PBG struture ad a trasmissio oeffiiet formula is obtaied for a QW symmetri 1D-PBG struture. I setio 6, the trasmissio resoaes are ompared with the QM frequeies ad the trasmissio modes at the resoaes are alulated as super-positios of the QM futios. I setio 7, it has bee show that the e.m. field geerated by two moohromati outer-propagatig pump waves leads to iterferee effets iside a QW symmetri 1D-PBG struture, suh that a physial iterpretatio of the DOM is proposed. I setio 8, a fial disussio is reported ad a ompariso is proposed about all the obtaied results ad their assoiated theoretial improvemets with respet to similar topis preseted i literature (Masimovi [9]). The Appedix desribes a simple ad diret method to alulate the QM orm for a 1D-PBG struture.. Quasi ormal Mode (QM) approah. With eree to fig. 1, osider a ope avity as a regio of legth material of a give rative idex L, filled with a, whih is elosed i a ifiite homogeeous exteral spae. The avity iludes also the termial surfaes, so it is represeted as C = [, L] ad the rest of uiverse as U = (,) ( L, ). The rative idex satisfies [5]: x ( ) the disotiuity oditios, i.e. x ( ) presets a step at x = ad x = L, i this hypothesis a atural demaratio of a fiite regio is provided; the o tail oditios, i.e. x ( ) = for x < ad x > L, i this hypothesis outgoig waves are ot sattered ba. The e.m. field Ext (, ) i the ope avity satisfies the equatio [13] x ρ( x) E( x, t) =, (.1) t where ρ ( x) = [ x ( )/ ], beig the speed of light i vauum. If there is o exteral pumpig, the e.m. field satisfies suitable outgoig waves oditios [5][1] 4
Ext (, ) = ρ Ext (, ) for x<, (.) x t Ext (, ) = ρ Ext (, ) for x> L, (.3) x t where ρ = ( / ), beig the outside rative idex. I fat, o the left side of the same avity, i.e. x <, the e.m. field is travellig i the egative sese of the x-axis, i.e. Ext (, ) = Ex [ + ( ) t, ] so eq. (.) holds as oe a easily prove. O the right side of the avity, i.e. x > L, the e.m. field is travellig i the positive sese of the x-axis, i.e. Ext (, ) = Ex [ ( ) t], so eq. (.3) holds as well. To tae the avity leaages ito aout, the Laplae trasform of the e.m. field is osidered [14], Ex (, ) = Ext (, )exp( itdt ), (.4) where is a omplex frequey. The e.m. field has to satisfy the Sommerfeld radiative oditio [13]: lim Ex (, ) =. (.5) x ± Sie s= i = exp( iπ ) with C [14], eq. (.4) defies a trasformatio whih loos lie a Fourier trasform with a omplex frequey, but it is a π -rotated Laplae trasform. The π -rotated Laplae trasform of the e. m. field overges to a aalytial futio Ex (, ) oly over the half-plae of overgee Im >. I fat, if the Laplae trasform (.4) is applied to the outgoig waves oditios (.), it follows ad, solvig the last equatio (.7): Ex (, ) = i ρ Ex (, ) for x <, (.6) x Ex (, ) = i ρ Ex (, ) for x > L, (.7) x Ex (, ) exp( i ρ x) = exp( ire ρ x)exp( Im ρ x) for x > L. (.8) The Sommerfeld radiative oditio (.5) a be satisfied oly if Im >. The trasformed Gree futio G ( x, x, ) a be defied by [13] x ; (.9) + ρ( x) Gxx (,, ) = δ( x x ) it is a e.m. field so, over the half-plae of overgee Im >, it satisfies the Sommerfeld radiative oditios [13]: exp( i ρ x) for x Gxx (,, ) exp( i ρ x) for x. (.1) 5
Two auxiliary futios g± ( x, ) a be defied by [5] ρ( x) g ( x, ) + ± = ; (.11) x they are ot defied as e.m. fields, beause, over the half-plae of overgee Im >, they satisfy oly the asymptoti oditios [5][1]: g+ ( x, ) exp( i ρ x) for x g ( x, ) exp( i ρ x) for x. (.1) However, the trasformed Gree futio Gxx (,, ) a be alulated i terms of the auxiliary futios g± ( x, ). I fat, it a be show that [5] the Wrosia W( x, ) assoiated to the two auxiliary futios g± ( x, ) is x-idepedet, ad for the trasformed Gree futio [5]: W( x, ) = g ( x, ) g ( x, ) g ( x, ) g ( x, ) = W( ), (.13) + + g ( x, ) g ( x +, ) for x < x W ( ) Gxx (,, ) =. (.14) g+ ( x, ) g ( x, ) for x < x W ( ) I what follows it is proved that, just beause of the asymptoti oditios (.1), the auxiliary futios g± ( x, ) are liearly idepedet over the half-plae of overgee Im >, ad so the Laplae trasformed Gree futio Gxx (,, ) is aalyti over Im >. The asymptoti oditios establish that, oly over Im >, the auxiliary futio g+ ( x, ) ats as a e.m. field for large x, beause it is expoetially deayig. I fat, from eq. (.1): g ( x, ) exp( i x) e ire x)exp( Im for x. The, still over + = ρ = xp( ρ ρx) Im >, the other auxiliary futio g ( x, ) i geeral does ot at as a e.m. field for large x, so it is expoetially ireasig. I fat, aordig to eq. (.1): g ( x, ) ρx) = A( )exp( i + B( )e xp( i ρ x) for x, with B( ), ad so g ( x, ) B( )exp( i ρ x) = B( ) exp( ire ρx)exp( Im ρx), for x. It follows that the auxiliary futios g± ( x, ) are liearly idepedet over Im >, beause the Wrosia W ( ) is ot ull; i fat, from eq. (.13): W ( ) = lim[ g ( x, ) g ( x, ) g ( x, ) g ( x, )] = i ρ B( ) over Im >. Thus, the x + + 6
trasformed Gree futio G ( x, x, ) is aalyti over Im >, where G ( x, x, ) does ot diverge; i fat, from eq. (.14): G ( x, x, ) 1/ W( ), with W ( ) over Im >. For aalytial otiuatio [14], the trasformed Gree futio G ( x, x, ) a be exteded also over the lower omplex half-plae Im < Gxx (,, ), over the lower omplex half-plae Im <. I other words, there exists a ifiite set {, } = ± 1, ±,, with egative imagiary parts Im, for W ( ) = g ( x, ) = ( ) g ( x, ) = f( x, ), (.16) + where ( ) is a suitable omplex ostat. The above futios f ( x, ) = f( x) are erred as Quasi-ormal-Mode (QM) eige-futios [15]. The ouples [, f ( x )] are erred as Quasi- uder the disotiuity ad o tail oditios, the wave futios f ( ) x form a d dx + ρ ( x) f. Aordig to. [15], it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio of omplex frequeies, whih the Wrosia (.13) is ull [5]: <. (.15) The poles of the trasformed Gree futio are erred to as Quasi-ormal-Mode (QM) eigefrequeies [15]. The defiitio of the QM eige-frequeies implies that the auxiliary futios g ( x, ) beome liearly depedet whe they are alulated at the QM frequeies { ± }, =, ±1, ±, ; so, the auxiliary futios i the QM s are suh that [5] ormal-modes beause: they are haraterized by omplex frequeies, so they are the ot-statioary modes of a ope avity [5] (they are observed i the frequey domai as resoaes of fiite width or i the time domai as damped osillatios); orthogoal basis oly iside the ope avity [5] (it is possible to desribe the QMs i a maer parallel to the ormal modes of a losed avity). Applyig the QM oditio (.16) to the equatio for the auxiliary futios (.11), it follows that the QMs [, f ( x)] satisfy the equatio [5]: ( x ) =. (.17) Moreover, applyig the QM oditio (.16) to the asymptoti oditios for the auxiliary futios (.1), it follows that, oly for log distaes from the ope avity, the QMs do ot represet e.m. fields beause they satisfy the QM asymptoti oditios [5][1], 7
f ( x) = exp( ± i ρ x) for x ±, (.18) while, iside the same avity ad ear its termial surfaes from outside, the QMs represet ot statioary modes, i fat the asymptoti oditios (.18) imply the formal QM outgoig waves oditios: f ( x ) = i ρ f (), (.19) x x= f ( x) = i ρ f ( L ). (.) x x= L The oditios (.19)-(.) are alled as formal beause they are erred to the QMs whih do ot represet e.m. fields for log distaes from the ope avity, ad as outgoig waves beause they are formally idetial to the real outgoig waves oditios for the e.m. field i proximity to the surfaes of the avity. I fat, eqs. (.19)-(.) for the QMs a be derived if ad oly if the requiremet of outgoig waves holds for the e.m. field. A ope system is ot oservative beause eergy a esape to the outside. As a result, the time-evolutio operator is ot Hermitia i the usual sese ad the eigefutios (fatorized solutios i spae ad time) are o loger ormal modes but quasi-ormal modes (QMs) whose frequeies are omplex. QM aalysis has bee a powerful tool for ivestigatig ope systems. Previous studies have bee mostly system speifi, ad use a few QMs to provide approximate desriptios. I s. [5], the authors review developmets whih lead to a uifyig treatmet. The formulatio leads to a mathematial struture i lose aalogy to that i oservative, Hermitia systems. Hee may of the mathematial tools for the latter a be trasribed. Emphasis is plaed o those ases i whih the QMs form a omplete set ad thus give a exat desriptio of the dyamis. More expliitly tha s. [5], we osider a Laplae trasform of the e.m. field, to tae the avity leaages ito aout, ad remar that, oly i the omplex domai defied by a π -rotated Laplae trasform, the QMs a be defied as the poles of the trasformed Gree futio, with egative imagiary part. Our iter follows three steps: 1. the π -rotated Laplae trasform of the e. m. field [see eqs. (.4)-(.5)] overges to a aalytial futio Ex (, ) oly over the half-plae of overgee Im >.. just beause of the asymptoti oditios [see eq. (.1)], the auxiliary futios g ( x, ) [see eq. (.11)] are liearly idepedet over the half-plae of overgee ± Im >, ad so the Laplae trasformed Gree futio G ( x, x, ) [see eq. (.9)] is aalyti over Im >. 8
3. with respet s. [5], we remar that: the trasformed Gree futio G ( x, x, ) [see eq. (.14)] a be exteded also over the lower omplex half-plae Im <, for aalytial otiuatio [14]; ad, it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio G ( x, x, ) [see eqs. (.15)-(.16)], over the lower omplex half-plae Im <, aordig to. [15]. 3. QM metris. The QM orm is defied as [5] f f dw =, (3.1) d = ad, with the method proposed for a oe side ope avity [5], oe a prove, for a both side ope avity, usig the QM oditio (.18) ad the eqs. (.11)-(.1), that [5][1]: L = ρ + ρ f f ( x) f ( x) dx i [ f () + f ( L)]. (3.) Several remars about this geeralized orm are i order: it ivolves f ( x ) rather the f (x) ad it is i geeral omplex; it ivolves the two termial surfae terms i ρ f () ad i ρ f L. ( ) If the QM futio f ( x ) is ormalized, aordig to f ( x) = f( x), (3.3) f f the: f f =. (3.4) The QM ier produt is defied as [5] [1] L ˆ ˆ m = m + m + ρ m + m + f f i f ( x) f ( x) f ( x) f ( x) dx i f () f () f ( L) f ( L), (3.5) if the QM ojugate mometum f ˆ ( x ) is itrodued, aordig to: ˆ f ( x) = iρ( x) f ( x). (3.6) Oe a prove that: the ier produt (3.5) is i aordae with the QM orm (3.) [5]; the QMs form a orthogoal basis iside the ope avity [5], i.e. [1] 9
f f = + ), (3.7) m δ, m( m beig δ m, the Kroeer delta, i.e. δ, m 1, =, = m m. Let us itrodue the overlappig itegral of the th QM L I = f ( x) ρ( x) dx, (3.8) th whih is lied to the statistial weight i the desity of modes (DOM) of the QM [1]. If the ope avity is haraterized by very slight leaages, Im << Re, (3.9) th the the overlappig itegral of the QM overges to [1]: Proof: L eq.(3.3) I = f ( x) ρ( x) dx = = f f L I 1. (3.1) eq.(3.) f ( x) ρ( x) dx = L L f ( x) ρ( x) dx f ( x) ρ( x) dx = 1 L L ρ f ( ) ( ) [ () ( )] ( ) ( ) x ρ x dx+ i f f x x dx + f L ρ ; (3.11) if Im << Re, the f ( x) f ( x), so eq. (3.1) holds. Istead, if the ope avity is haraterized by a some leaage, the the overlappig itegral of the th QM a be alulated as [1]: ρ I f () f ( L). (3.1) = Im + Proof: L ˆ ˆ m = m + m + ρ m + m f f i f ( x) f ( x) f ( x) f ( x) dx i f () f () f ( L) f ( L) L m m m m = ( + ) f ( x) ρ( x) f ( x) dx+ i ρ f () f () + f ( L) f ( L) = = δ ( + ) m, m so 1 eq.(3.7) eq.(3.6) =, (3.13)
L ρ f x x f x dx i f f f L f L ( ) ρ( ) ( ) = m δ, m [ () m () ( ) m ( )] + + m ; (3.14) if m=, the =, f ( x) = f ( x) ad more δ, =, so eq. (3.14) a be redued to eqs. (3.8) ad (3.1). idex Fially, if the ope avity is haraterized by very slight leaages [eq. (3.9)] ad its rative x ( ) satisfies the symmetry property the the QM orm a be approximated i modulus: L ( / x) = L ( / + x), (3.15) f f ρ. (3.16) Im Proof of eq. (3.16): so ρ ρ eq.(3.1) I f () f ( L) f () f ( L) 1, (3.17) eq.(3.3) = Im + = + f f Im f f f f L f L Hip: f () 1 = ρ () + ( ) = ρ 1 + ( ) Im Im ; (3.18) if the ope avity is symmetri [eq. (3.15) holds], the f ( L) = ( 1) f () = ( 1), so eq. (3.18) a be redued to eq. (3.16). Physial iterpretatio of eq. (3.16) : The modulus of QM orm is expressed oly i terms of the QM frequeies; The modulus of the QM orm is high ( f f >> ρ ) whe the leaages of the ope avity are very slight ( Im << Re ); The QM theory a be applied to ope avities ad is based o the outgoig waves oditios [eqs. (.)-(.3)] whih formalize some leaages ( Im < ), so the QM theory a ot ilude the oservative ase, whe the avities are losed ad are ot haraterized by ay leaages ( Im = ). 4. Calulatio of trasmissio oeffiiet. With eree to fig. 1.b., let us ow osider a ope avity of legth L, filled with a rative idex x ( ), i the presee of a pump iomig from the left side. The avity iludes 11
also the surfaes, so it is represeted as C = [, L] ad the rest of uiverse as U = (,) ( L, ). The rative idex satisfies the disotiuity ad o tail oditios [5], as speified above. Uder these oditios, the QMs form a omplete basis oly iside the avity, ad the e.m. field a be alulated as a superpositio of QMs [5] Ext (, ) = a( t) f ( x), for x L, (4.1) where f ( x ) are the ormalized QM futios [see eq. (3.3)]. The superpositio oeffiiets a ( t ) satisfy the dyami equatio [5] i a () t + ia() t = f () b() t, (4.) ρ where b( t ) is the drivig fore bt () = ρ xep( xt,). x= The left-pump E ( P x, t ) satisfies the iomig wave oditio [5]: Eah QM is drive by the drivig fore b( t ) ad at the same time deays beause of Im <. The ouplig to the fore is determied by the surfae value of the QM wave futio (). x P x= t P bt () = ρ E ( xt,) = ρ E (,) t. (4.3) The avity is i a steady state, so the Fourier trasform with a real frequey f Ex (, ) = Ext (, )exp( itdt ) a be applied to equatios (4.1)-(4.3), ad it follows Ex (, ) = a ( ) f ( x) f () b ( ) a ( ) =, (4.4) ρ b ( ) = iρe P(, ) so f () f ( x) Ex (, ) = E P(, ) i ρ. (4.5) ( ) With eree to fig. 1.b., the e.m. field is otiuous at avity surfaes x = ad x = L, so E + (, ) = E (, ) ad EL (, ) = EL ( +, ), ad the e.m. field E (, ) at surfae x = is the superpositio of the iomig pump E ( E (, ) P, ) ad the leted field R, so E (, ) = E (, ) + E (, ), while the e.m. field EL (, ) at the surfae x = L is oly the P R trasmitted field E ( L, ), so EL (, ) = E ( L, ). T T 1
It follows that the trasmissio oeffiiet t( ) for a ope avity of legth L a be defied as the ratio betwee the trasmitted field EL (, ) at the surfae x = L ad the iomig pump E (, ) at surfae x = [13]: EL (, ) t( ) = E (, ). (4.6) P The trasmissio oeffiiet is obtaied as superpositio of QMs isertig eq. (4.5) i eq. (4.6): f () f ( L) t( ) = i ρ ( ). (4.7) P Applyig the QM ompleteess oditio [5] [19] trasmissio oeffiiet simplifies as: f ( x) f ( x ) = for x, x L, the t( ) = i f () f ( L) ρ. (4.8) Isertig the ormalized QM futios f ( x ) [see eq.(3.3)], with f () = 1, equatio (4.8) beomes: f( L) t( ) = i ρ f f. (4.9) For a symmetri avity, suh that f ( L) = ( 1) f () = ( 1), fially [11] where γ = f f /i ρ. ( 1) t( ) =, (4.1) γ I aordae with. [11], the trasmissio oeffiiet t( ) of a ope avity a be alulated as a superpositio of suitable futios, with QM orms f f as weightig oeffiiets ad QM frequeies as parameters. 5. Oe dimesioal Photoi Bad Gap (1D-PBG) strutures. With eree to fig., let us ow osider a symmetri oe dimesioal (1D) Photoi Bad Gap (PBG) struture whih osists of periods plus oe layer; every period is omposed of two layers respetively with legths h ad l ad with rative idies ad, while the added h layer is with parameters h ad. h l 13
The symmetri 1D-PBG struture osists of + 1 layers with a total legth L= ( h+ l) + h. If the two layers exteral to the symmetri 1D-PBG struture are osidered, the 1D-spae x a be divided ito + 3 layers; they are L = [ x, x 1], =,1,, + 1, +, with x =, x1 =, x = L, x 3 =+. + + The rative idex x ( ) taes a ostat value i every layer L, =,1,, + 1, +, i.e. + for x L, L + x ( ) = h for x L, = 1,3,, 1, + 1. (5.1) l for x L, =,4,, For the symmetri 1D-PBG struture with rative idex (5.1), the QM orm f f a be obtaied i terms of the frequey ad of the A ( ) +, B ( ) + oeffiiets for the g ( x, ) auxiliary futio i the L + layer o the right side of the 1D-PBG struture (see Appedix): db+ f f = i A+ ( ). (5.) d = As proved i [1], for a quarter wave (QW) symmetri 1D-PBG struture with periods ad as eree frequey, there are + 1 families of QMs, i.e. F Q M family of QMs osists of ifiite QM frequeies, i.e. m,, F, [, ] ; the QM m =, ± 1, ±,, whih have the same imagiary part, i.e. Im, = Im,,, ad are lied with a step some rages, i.e. S = { m < Re < ( m+ 1) } m m =, i.e. Rem, = Re, + m, m. It follows that, if the omplex plae is divided ito m QM, m, eah of the QM family drops oly F { } oe QM frequey over the rage S m, i.e. = (Re + m, Im ), ; there are + 1 QM m,, frequeies over the rage S m ad they a be erred as m, =, + m, [, ]. Thus, there are +1 QM frequeies over the basi rage, i.e. S = { < Re < } ; they orrespod to ].,, [, The QM orm (5.) beomes: m, db + fm, fm, = i A+ ( m, ) d = m,. (5.3) The oeffiiets A + ( ), B ( ) + are of the type exp( iδ ) where δ = π ( ) [1]. If oeffiiets A ( ) +, B ( ) + are alulated at QM frequeies m, =, + m, where 14
=, it follows A+ (, m) = A + (,) ad ( db+ / d) = = ( db+ / d) =, so from eq. (5.3): = m, m, m ( i/ ) =,,,, m,, f f γ γ ( + m ). (5.4) Thus, for the F QM family, the orm is periodi with a step Γ ( ). QM γ m, = γ,, The trasmissio oeffiiet (4.1) beomes t( ) ad, taig ito aout eq. (5.4), = t( ) = ( 1) + m ( 1), + m γ m, (5.5) = m= m,,, m ( 1) = γ, m=, m 1, (5.6) it overges to: π t( ) = ( 1) γ, =, π s[ (,)]. (5.7) 6. QMs ad trasmissio peas of a symmetri 1D-PBG struture. 6.a. QM frequeies ad trasmissio resoaes. The trasmissio oeffiiet of a QW symmetri 1D-PBG struture, with periods ad as eree frequey, is alulated as a superpositio of + 1 QM-oseats, whih orrespod to the umber of QMs i [, ) rage. I fig. 3.a., the trasmissio oeffiiet predited by QM theory ad the by umeri methods existig i the literature [16] are plotted for a QW symmetri 1D-PBG struture, where the eree wavelegth is λ = 1µ m, the umber of periods is = 6 ad the values of rative idies are h =, l =1.5. I figure 3.b., the two trasmissio oeffiiets are plotted for the same struture of fig. 3.a. but with h = 3, =. I figure 3.., the two trasmissio oeffiiets are l plotted for the same struture as i figure 3.b., with a ireased umber of periods = 7. I aordae with. [11], we ote the good agreemet betwee the trasmissio oeffiiet predited by the QM theory ad the oe obtaied by some umeri methods [16]. I the tables 1, the same examples of QW symmetri 1D-PBG strutures are osidered: (a) λ = 1µ m, = 6, =, = 1,5, (b) λ = 1µ m, = 6, = 3, =, () λ = 1µ m, = 7, h l 15 h l
= 3, =. For eah of the three examples, the low ad high frequey bad-edges are h l desribed i terms of their resoaes ( / ) ad phases t (../ ); besides, there is Bad Edge BE oe QM lose to every sigle bad-edge: the real part of the QM-frequey Re( / ) QM is reported together with the relative shift from the bad-edge resoae (Re QM B. E. ) / B. E.. From tables 1.a. ad 1.b. ( = 6 ), the low ad high frequey bad-edges are haraterized by egative phases, whih sum is early ( π ), while, from table 1.. ( = 7 ), their respetive phases are positive with a sum π approximately. I fat, a QW symmetri 1D-PBG struture, with periods ad re resoaes i the [, ) f as eree frequey, presets a trasmissio spetrum with +1 trasmissio pea rage, i.e. t( pea ) = exp( iϕ ), =,1,,. Oly by umerial alulatios (MATLAB) o eq. (5.7), oe a sueed i provig that: ϕ 1 1) ( 1) + ϕ + ϕ(+ π. (6.1), = 1,, pea So, the lowest resoae i the rage [, ) has always a phase ϕ. The phases ϕ ad ϕ of the low ad high frequey bad-edges ad are suh that +1 pea pea + 1 ϕ + ϕ + 1 ( 1 1 ) + π is a odd umber or ( π ) the bad-edge [11]. I fat, a QW symmetri 1D-PBG struture, with periods ad as + 1 QMoseats (5.7), etred to the + 1 QM frequeies i [, re ) rage, i.e. QM, =,1, ; so, for a QW symmetri 1D-PBG struture with periods, the phases of the two bad-edges are suh that their sum is early π whe approximately if is a eve umber. Agai from tables 1, there is oe QM lose to every sigle bad-edge, with a relative shift from eree frequey, presets a trasmissio oeffiiet whih is a superpositio of,. The QM-oseats are ot sharp futios, so, whe they are superposed, f th a aliasig ours; the pea of the QM oseat a feel the tails of the ( 1) th ad ( + 1 ) th QM-oseats, so the th resoae of the trasmissio spetrum results effetively shifted from th pea the QM frequey i.e. = Re,, =,1,,. Comparig values i the tables 1.a. ( = =.5 ) ad table 1.b. ( = 1), the relative shift h l (Re QM ) / B E. dereases with the rative idex step, while, omparig values i B. E.. tables 1.b. ( = 6 ) ad 1.. ( = 7 ), the same shift ireases with the umber of periods. I aordae with. [11], some umerial simulatios prove that, the more the 1D-PBG struture 16
presets a large umber of periods ( L >> λ ) with a high rative idex step ( >> ), the th th more the QM desribes the trasmissio pea i the sese that pea. Re, omes ear to 6.b. QM futios ad trasmissio modes. Let us osider oe moohromati pump E ( x, ), whih is iomig from the left side with uit amplitude ad zero iitial phase E p ( x, ) p ( x) = exp( i x), (6.) beig is the rative idex of the uiverse. The e.m. field Ex (, ), iside the QW symmetri 1D-PBG struture, a be alulated as a superpositio of QMs [see setios 3 ad 4]: Ex (, ) E ( x) = f () f ( x) f () f ( x) f ( x) QMs = p () i ρ = i ρ = i ρ = ( ) f f γ f ( x) + m f ( x) QW symmetri 1 D-PBG struture m,,, m, = = = m= γm,, m γ, m, m th pea to the QM of the [, rage, i.e. Re, p = = f ( ) x ) with =,1,, the the = (6.3) If the pump (6.), iomig from the left side, is tued at the trasmissio pea whih is lose trasmissio mode E ( x) a be approximated as a superpositio oly of the domiat terms where a pea E ( x) = a ( ) f ( x) = = m=, m, m pea a m, ( ) fm, ( m= m= = x) + a ( ) f ( m= a ( ) f ( x) pea m, m, pea m, m, x), (6.4) pea, / γ, ( ) =,,1, 1, m {, ± 1, ±, } m = + =. (6.5) m, pea, pea I fat, if the superpositio (6.4)-(6.5) is alulated at the frequey Re, the eah, domiat term is haraterized by a oeffiiet 17 a ) m, ( pea with a deomiator iludig the
pea pea addedum, ( Re, ) iim, iim, pea pea =, so a m, ( ) 1, 1 Im,, pea beig =,1, + 1, while eah egleted term is haraterized by a oeffiiet a m, ( ) with a deomiator iludig the addedum >>, so pea pea,, a < 1 << 1 a m( ), ( ), beig. pea pea pea pe a,,, m Thus, the e.m. mode pea E ( x) th trasmissio pea lose to the QM of the [, ) rage, beig =,1,, a be approximated as the superpositio of the QM futios f m, ( x) whih belog to the th QMfamily, beig {, 1,, } m = ± ± iside the QW symmetri 1D-PBG struture, tued at the ; moreover, the weigh-oeffiiets of the superpositio pea a m, ( ) pea th are alulated i the trasmissio resoae ad deped from the QMfamily. By umerial alulatios (MATLAB) oe o eqs. (6.4)-(6.5) oe a prove that, whe the trasmissio pea is loser to the QM mode (6.4) by the I order approximatio [9] pea Re E x a, pea ( ),( ) f,, it is suffiiet to alulate the trasmissio ( x), (6.6) where [9] / γ a =,1, + 1, (6.7) pea,,,( ) =, pea, otherwise, by the II order approximatio, more ied, E ( x) a ( ) f ( x) + pea,, + a ( ) f ( x) + a ( pe, 1 pea, 1, 1 a +, + 1, (6.8) ) f ( x) where / γ, =. (6.9) pea,, a, ± 1( ) =,1, + 1 pea (, ± ) pea I aordae with. [9], i the hypothesis that Re,, it follows pea pea pea pea pea, << (, ± ) < < (, + m ) < a a, ± 1 a m pea beig m : i fat, the e.m. mode (6.4)-(6.5) tued at the trasmissio pea, for the I th order, a be approximated by eq. (6.6), orrespodig to the QM of the [, ) rage (the pea pea, so, ( ) >> ( ) > >, ( ) >, resoae is lose eough to the QM frequey Re, ) (see. [9]); the, for the II order, by eq. (6.8), iludig the two adjaet ( ±1) th QMs (if the resoae is far from the QM 18 pea
frequeies Re =, 1 ±, but the ( ) pea ±, a, futio tued at feels the tails of the a ( ), ± 1 futios tued i Re, ± 1); fially, the ext orders a be egleted (beause the resoae pea is too far from the QM frequeies Re m,, = + m, so the a, ( ) futio tued at pea does ot feel the tails of the a m, ( ) futios tued i Re m,, beig m ). By umerial alulatios (MATLAB) o eqs. (6.6) ad (6.8) oe a prove that, if the QM is eve loser to the trasmissio pea of a QW symmetri 1D-PBG struture, i.e. pea Re,, whih is haraterized by a large umber of periods, i.e. L >> λ, ad a high rative idex step, i.e. approximatio: >>, the the I order approximatio (6.6)-(6.7) a be redued to the QM E ( ) ( x) f, x. (6.1) I fat, if pea Re,, the eq. (6.7) a be approximated as / γ / γ a = =,1, + 1; (6.11) ( pea ),,,,, ) Im, pe a ( Re, i, iim, if L >> λ, >>, the the 1D-PBG struture is haraterized by very slight leaages [see eq. (3.9)], ad, sie the rative idex satisfies suitable symmetry properties [see eq. (3.15)], the approximated QM orm (3.16) a be used f, f,, γ, =, =, 1, + 1 (6.1) ρ Im ad, i modulus, the weigh-oeffiiet (6.11) overges to:, a (, pea ) 1 γ, Im,, 1, =, 1, + 1. (6.13) Thus, i aordae with. [11], some umerial simulatios prove that the more the 1D-PBG struture presets a large umber of periods with a high rative idex step, the more the QMs desribe the trasmissio peas i the sese that the QM futios approximate the e.m. modes i the trasmissio resoaes. I fig. 4, with eree to a QW symmetri 1D-PBG struture ( λ = 1µm, = 6, = 3, = ), the e.m. mode itesities (a) I edge at the low frequey bad edge (.8 I II bad edge Ibad I bad e dge / = ) ad (b) at the high frequey bad edge ( = II bad edge / 1 itesity for a iomig pump I pump, are plotted as futios of the dimesioless spae x / L, where L is the legth of the 1D-PBG struture. It is lear the shiftig betwee the QM 19 h.178 ), i uits of the l
approximatio (6.1) (the e.m. mode itesity I Ibad edge is due to QM lose to the bad-edge) ad the first order approximatio (6.6)-(6.7) [more ied tha (6.1), I bad edge is alulated as the produt of the QM, lose to the frequey bad-edge, for a weigh oeffiiet, whih taes ito aout the shift betwee the bad-edge resoae ad the QM frequey] or the seod order approximatio (6.8)-(6.9) [eve more ied, but almost superimposed to (6.6)-(6.7), bad edge alulated as the sum of the first order otributio due to the QM, lose to the frequey badedge, ad the seod order otributios of the two adjaet QMs, with the same imagiary part, so belogig to the same QM family]. I is 7. Quarter wave (QW) symmetri 1D-PBG strutures exited by two outer-propagatig field pumps. Let us osider a QW symmetri 1D-PBG struture with periods ad as eree frequey. If the uiverse iludes the termial surfaes of the 1D-PBG, the avity is represeted as C = [, L] ad the rest of uiverse as U = (,) ( L, ). The 1D-PBG presets a rative idex x ( ) whih satisfies the symmetry properties L ( / x) =L ( / + x). The avity is haraterized by a trasmissio spetrum with + 1 resoaes i the [, ) rage [1], i.e. t( ) = exp[ iϕ ], r ( ) =, =,1, [17]. th Let us osider oe pump, omig from the left side, whih is tued at the resoae of the 1D-PBG struture, with uit amplitude ad zero iitial phase ( ) ( ) x), (7.1) p x = exp( i ( beig the rative idex of the uiverse. It exites i the avity a e.m. mode E ) ( x) whih satisfies the equatio [13] ( ) de ( ( x) E ) + ( x) = dx, (7.) ad the boudary oditios are [13]: ( ) ( ) ( r ) ( ) ( ) E ( ) () = p () + p () = p () = 1. (7.3) ( ) ( ) E ( L) = t( ) p () = t( ) Let us osider aother pump, omig from the right side, tued at the 1D-PBG struture with uit amplitude ad ostat phase-differee ϕ, i.e. th resoae of the
p ( x) = exp[ i ( )( x L)]exp[ i ϕ]. (7.4) ( ) If aloe, this pump exites i the avity a e.m. mode to (7.) [13] de dx ( E ) ( x) ( ) ( ) + ( x) E ( ) x = whih satisfies a equatio similar, (7.5) beause of the symmetry properties of the rative idex, with boudary oditios whih are differet from (7.3) [13]: ( ) ( ) ( ) ( ) ( ) E ( L) = p ( L) + r ( ) p ( L) = p ( L) = exp[ i ϕ] ( ) ( ). (7.6) E () = t( ) p ( L) = t( )exp[ i ϕ] It is easy to verify that the e.m. mode ( E ) ( x), whih is exited by the pump (7.4) ad solves the equatio (7.5) with the boudary oditios (7.6), is related to the e.m. mode ( E ) ( x), whih is exited by the pump (7.1) ad solves equatio (7.) with the boudary oditios (7.3), by the li: E ( x) = [ E ( x)] t( )exp[ i ϕ]. (7.7) ( ) ( ) The two outer-propagatig pumps, whih are tued at the struture, exite a e.m. mode ( E ) ( x) ad E ( ) ( x ), i.e.[13] E ( x) th resoae of the 1D-PBG iside the avity, whih is the liear super-positio of E x E x E x E x E x t i. (7.8) ( ) ( ) ( ) ( ) ( ) = ( ) + ( ) = ( ) + [ ( )] ( )exp[ ϕ] It is easy to alulate from (7.8) the e.m. mode itesity iside the avity [13] I ( x) = E ( x)[ E ( x)] = ( ) = I ( x) + Re{[ E ( x)] t ( )exp[ i ϕ]} = = + + =, (7.9) ( ) ( ) I ( x){1 os[ φ ( x) ϕ ϕ]} ϕ ϕ = + ( ) ( ) 4 I ( x)os [ φ ( x) ] if the e.m. mode exited by the pump (7.1) is represeted by E ( ) ( x) = I ( ) ( x)exp[ iφ ( ) ( x)]. Eq. (7.9) shows that the e.m. field geerated by two moohromati outer-propagatig pump waves leads to iterferee effets iside a QW symmetri 1D-PBG struture. 7.1. Desity of modes (DOM). l = I fig. 5, with eree to a symmetri QW 1D-PBG struture ( λ = 1µ m, = 6, = 3, ), the e.m. mode itesity exited iside the ope avity by oe field pump (- - -) omig 1 h
from the left side [I order approximatio (6.6)] ad by two outer-propagatig field pumps i phase ( ) [see eq. (7.9)] are ompared whe eah of the two field pumps are tued at (a) the low frequey bad-edge ( I bad edge / =.8 ) or (b) the high frequey bad-edge ( II bad edge / = 1.178 ). The e.m. mode itesities I Ibad edge ad III bad edge, i uits of the itesity for the two pumps L I pump, are plotted as futios of the dimesioless spae x / L, where is the legth of the 1D-PBG struture. If the two outer-propagatig field pumps i phase are tued at the low frequey bad-edge, there is a ostrutive iterferee ad the field distributio i that bad-edge is remiiset of the distributio exited by oe field pump i the same trasmissio pea; while, if the two pumps are tued at the high frequey bad-edge, there is a destrutive iterferee ad almost o e.m. field peetratio ours i the struture. Some umerial simulatios o eq. (7.9) prove that, whe a QW symmetri 1D-PBG struture with periods is exited by two outer-propagatig field pumps with a phase-differee ϕ, if is a eve umber, the e.m. mode itesities i the low (high) frequey bad-edge ireases i stregth whe the two field pumps are i phase ϕ = (out of phase ϕ = π ) ad almost flags whe the two pumps are out phase ϕ = π (i phase ϕ = ). Moreover, if is a odd umber, the two field distributios i the low ad high frequey bad-edges exhage their physial respose with respet to the phase-differee of the two pumps. The duality betwee the field distributios i the two bad-edges, for whih the oe ireases i stregth whe the other almost flags, a be explaied reallig that the trasmissio phases of the two bad edges are suh that their sum is π ( π ) whe is a odd (eve) umber. Theore, if a 1D-PBG struture is exited by two outer-propagatig field pumps whih are tued at oe trasmissio resoae, o e.m. mode a be produed beause it depeds ot oly o the boudary oditios (the two field pumps) but also o the iitial oditios (the phasedifferee of the two pumps) [1]. We ojeture that the desity of modes is a dyami variable whih has the flexibility of varyig with respet to the boudary oditios as well as the iitial oditios, i aordae with. [1]. 8. A fial disussio. The Quasi ormal Mode (QM) theory osiders the realisti situatio i whih a optial avity is ope from both sides ad is elosed i a ifiite homogeeous exteral spae; the la of eergy oservatio for optial ope avities gives resoae field eige-futios with omplex
eige-frequeies. The time-spae evolutio operator for the avity is ot hermitia: the modes of the field are quasi-ormal, i.e. they form a orthogoal basis oly iside the avity. The behavior of the eletromageti field i the optial domai, iside oe-dimesioal photoi rystals, is aalyzed by usig a extesio of the QM theory. Oe-dimesioal (1D) Photoi Bad Gap (PBG) strutures are partiular optial avities, with both sides ope to the exteral eviromet, with a stratified material iside. These 1D-PBG strutures are fiite i spae ad, whe we wor with eletromageti pulses of a spatial extesio loger tha the legth of the avity, the 1D-PBG aot be studied as a ifiite struture: rather we have to osider the boudary oditios at the two eds of the avity. I the preset paper, the QM approah has bee applied to a double-side opeed optial avity, ad speified for a 1D-PBG struture. All the osideratios made are exhaustively demostrated ad all the subsequetly results agree with the oes preseted i literature (see s. [1] ad [11]). I this paper (se. ), suitable outgoig wave oditios for a ope avity without exteral pumpig are formalized [see eqs. (.)-(.3)]. More expliitly tha s. [5], here we have osidered a Laplae trasform of the e.m. field, to tae the avity leaages ito aout, ad we have remared that, oly i the omplex domai defied by a π -rotated Laplae trasform, the QMs a be defied as the poles of the trasformed Gree futio, with egative imagiary part. Our iter has followed three steps: 1. the π -rotated Laplae trasform of the e. m. field [see eqs. (.4)-(.5)] overges to a aalytial futio Ex (, ) oly over the half-plae of overgee Im >.. just beause of the asymptoti oditios [see eq. (.1)], the auxiliary futios g ( x, ) [see eq. (.11)] are liearly idepedet over the half-plae of overgee ± Im >, ad so the Laplae trasformed Gree futio G ( x, x, ) [see eq. (.9)] is aalyti over Im >. 3. Differetly from s. [5], here we have remared that: the trasformed Gree futio Gxx (,, ) [see eq. (.14)] a be exteded also over the lower omplex half-plae Im <, for aalytial otiuatio [14]; ad, it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio G ( x, x, ) [see eqs. (.15)-(.16)], over the lower omplex half-plae Im <, aordig to. [15]. Moreover (se. 3), if the ope avity is haraterized by very slight leaages ad its rative idex satisfies suitable symmetry properties, more tha s. [5], here we have proved that the modulus of the QM orm i modulus a be expressed oly i terms of the QM frequeies [see eq. (3.16)] 3
ad, more tha s. [1] ad [11], here we have proposed its physial iterpretatio: the QM orm is high whe the leaages of the ope avity are very slight; the QM theory a be applied to ope avities ad is based o the outgoig waves oditios whih formalize some leaages (Im < ), so the QM theory a ot ilude the oservative ase, whe the avities are losed ad are ot haraterized by ay leaages ( Im = ). The paper (se. 4) has give a proof that the trasmissio oeffiiet t( ) a be alulated as a superpositio of suitable futios, with QM orms f f as weightig oeffiiets ad QM frequeies as parameters [see eq.(4.1)], thesis oly postulated i. [11]. The QM approah has bee applied to a 1D-PBG struture (se. 5): more tha. [11], here the trasmissio oeffiiet of a quarter wave (QW) symmetri 1D-PBG struture, with periods ad as eree frequey, has bee alulated as a superpositio of +1 QM-oseats, whih orrespod to the umber of QMs i [, ) rage [see eq. (5.7)]; differetly from the assertio of. [8] for whih the frequey depedee of t( ) ould suggest at first glae that the positios of maxima of T( ) = t( ) are realized whe the real frequey oiides with the real part of the QM frequeies, here we have dedued that this fat is true oly i the further hypothesis of real parts of QM frequeies well separated o the real frequey axis, whereas, o the otrary, overlappig of the tails of two ear QM-oseat otributios to T ( ) lead to a sigifiatly displayed positio of the maxima of T ( ), a sort of aliasig proess (see figs. 4.a-. ad tab. 1.a-.). I se. 6., the trasmissio resoaes have bee ompared with the QM frequeies (se. 6.a.) ad the trasmissio modes i the resoaes have bee alulated as super-positios of the QM futios (se. 6.b.): differetly from Masimovi paper [9], here we affirm expliitly that a proper field represetatio for the trasmittae problem a be established i terms of QMs, ad the boudary oditios are ot violated [see eqs. (6.)-(6.3)]; eve if the fields are etirely differet whe osidered o the whole real axis, the e.m. field a be represeted as a superpositio of QMs iside the ope avity ad o the limitig surfaes too. I the preset paper (se. 7), it has bee show that the e.m. field geerated by two moohromati outer-propagatig pump waves leads to iterferee effets iside a QW symmetri 1D-PBG struture [see eq. (7.9)]. 4
Eve if this paper is hroologially subsequet to. [11], ayway it is oeived as a trait d uio betwee s. [11] ad [1], i fat the paper forestalls oeptually, aordig to lassi eletrodyamis, what. [1] develops i terms of quatum mehais. We have show that if a 1D-PBG struture is exited by two outer-propagatig pumps whih are tued at oe trasmissio resoae, o e.m. mode a be produed beause it depeds ot oly o the boudary oditios (the two pumps) but also o the iitial oditios (the phase-differee of the two pumps) [see fig. 5.a-b.]; i aordae with. [1], we have ojetured (se 7.1) that the desity of the modes (DOM) is a dyamial variable whih has the flexibility to adjust with respet to the boudary oditios as well the iitial oditios. I Appedix, a diret method has bee desribed to obtai the QM orm for a QW symmetri 1D-PBG struture: the QM orm f f a be obtaied i terms of the frequey ad of the A + ( ), B ( ) + oeffiiets for the g ( x, ) auxiliary futio i the L + layer o the right side of the 1D-PBG struture. 8.1. A ompariso with Masimovi [9]. Let us propose a ritial ompariso betwee Masimovi ad our methods: Masimovi speializes. [9] to fiite periodi strutures whih possess trasmissio properties with a badgap, i.e. with a regio of frequeies of very low trasmissio. He hooses a field model for the trasmittae problem ad taes the relevat QMs as those with the real part of their omplex frequey i the give frequey rage. To determie the deompositio oeffiiets i the field model, a variatioal form of the trasmittae problem is employed. The trasmittae problem orrespods to the equatio ad atural boudary oditios, arisig form the oditio of statioarity of a futioal FExt [ (, )]. If FExt [ (, )] beomes statioary, i.e. if the first variatio of FExt [ (, )] vaishes for arbitrary variatios of the e.m. field Ext (, ), the Ext (, ) satisfies the atural boudary oditios. The optimal oeffiiets a be obtaied as solutios of a system of liear equatios. By solvig the system for eah value of the real frequey, the deompositio oeffiiets i the field represetatio for trasmittae problem are obtaied. This eables approximatio of the spetral trasmittae ad letae ad the related field profile. Masimovi applies a method whih is based o a mathematial priiple, ivolvig arduous alulatios. Istead, we have proposed, for a ope avity, a physial method whih starts from the boudary oditios at the surfaes of the avity (se. 4). Moreover, 5
we have provided, for a symmetri 1D-PBG struture (se. 5), a ew algorithm to alulate the QM orm by a derivative [see eq. (5.)], more diret tha the itegral of. [9]; ad, we have proved that a QW symmetri 1D-PBG struture with periods is haraterized by +1 families of QM frequeies, periodially distributed o the omplex plae, suh that, for eah family, the QM orm is periodial with a proper step [see eq.(5.4)]: as a result, the alulatio of the trasmissio oeffiiet is eve more simplified with respet to. [9]. A ommo assumptio made i the literature [5][7][1][18] is that the spetral trasmissio for sigle resoae situatio, as desribed, is of a Loretia lie shape. We have falle ito lie of this assumptio (se. 6.b): i fat, the e.m. field, iside a QW symmetri 1D-PBG struture, has bee alulated as a superpositio of QMs; so, eah sigle resoae is desribed by a Loretia lie shape [see eqs. (6.3), (6.4)-(6.5) ad ommets below]. The method of. [9] justifies aalytially this ommo assumptio. Masimovi osiders the otributio of a sigle QM i the field model. Here (se. 6.b.), we have osidered the otributios of the adjaet QMs, by more ad more ied approximatios iludig suessive orders: i fat, some umerial simulatios have proved that (see figs. 4.a-b.), whe the trasmissio pea is lose to the QM, it is suffiiet to alulate the trasmissio mode by the I order approximatio [see eqs. (6.6)-(6.7)], otherwise, by the II order approximatio, more ied [see eqs. (6.8)-(6.9) ad ommets below]; so, i our opiio, the otributio of a sigle QM is suffiiet for the field model oly i the limit of arrow resoaes [see eqs. (6.1), (6.11)-(6.13) ad ommets below]. Appedix: QM orm for 1D-PBG strutures. With eree to fig., let us ow osider a symmetri oe dimesioal photoi bad gap (1D-PBG) struture whih osists of period plus oe layer; every period is omposed of two layers respetively with legths h ad l ad with rative idies ad h is with parameters h ad. h l, while the added layer The symmetri 1D-PBG struture osists of + 1 layers with a total legth L= ( h+ l) + h. If the two layers exteral to the symmetri 1D-PBG struture are osidered, the 1D-spae x a be divided ito + 3 layers; the two exteral layers, L = (,) ad L + = ( L, + ), ad the 1D- PBG layers, the odd oes L = 1 [( 1)( h+ l), ( 1)( h+ l) + h], oes L = [( 1) ( h+ l) + h, ( h+ l)], = 1,,,. 6 = 1,,, +1 ad the eve
The rative idex x ( ) taes a ostat value i every layer L, =,1,, + 1, +, i.e. for x L, L + x ( ) = h for x L, = 1,3,, 1, + 1. (A.1) l for x L, =,4,, At first, the alulatio of quasi ormal modes (QMs) is stated for the 1D-PBG struture with rative idex (A.1). The homogeeous equatio is solved for the auxiliary futios (.11), with the "asymptoti oditios" (.1). The auxiliary futio g ( x, ) is i x i x g ( x, ) = Ae + Be ϑ( x) + + 1 ih x ih x + A 1e + B 1e ϑ[ x ( 1)( h+ l)] ϑ[( 1)( h+ l) + h x] + = 1, (A.) il x il x + Ae + Be ϑ[ x ( 1)( h+ l) h] ϑ[ ( h+ l) x] + = 1 i x i x + A + e + B + e ϑ( x L) where ϑ ( x) is the uit step futio; while the auxiliary futio g+ ( x, ) has a similar expressio to (A.), where some oeffiiets C ( ) ad D ( ) replae A ( ) ad B ( ) with =, 1,, + 1, +. The auxiliary futio g ( x, ) is a right to left wave for x <, so A ( ) =, while the auxiliary futio g ( x, ) + is a left to right wave for x > L, so D + ( ) =. If the otiuity oditios are imposed at the 1D-PBG surfaes for the auxiliary futio g ( x, ) ad its spatial derivative ( x, ), it follows g x 7
1 1 A 1 1 1 A 1 h = B1 1 B 1 h 1 i l [( 1)(h+ l) + h] i l [( 1)(h+ l) + h] e e i h [( 1)(h + l) + h] ih [( 1)(h + l) + h] A 1 l e e A 1 = B i l [( 1)(h+ l) + h] 1 i l [( 1)(h+ l) + h] i h [( 1)(h+ l) + h] i h [( 1)(h+ l) + h] B 1 e e e h e h l i L 1 i L e e ih L ih L A+ 1 e e A+ 1 = B + i L 1 i L ih L ih L B + 1 e e e h e h ih (h+ l) 1 ih (h+ l) e e il (h+ l) il (h+ l) A+ 1 1 h e e A = B+ 1 ih (h+ l) 1 ih (h+ l) il (h+ l) il (h+ l) B e e e l e l h (A.3) while the otiuity oditios for g+ ( x, ) are similar to (A.3), but the oeffiiets C ( ) ad D ( ) replae A ( ) ad B ( ) with =,1,, + 1, +. The B ( ) oeffiiet is fixed hoosig a ormalizatio oditio ad all the [ A ( ), B ( )] ouples with = 1,, + are determied applyig the otiuity oditios (A.3). Similarly for the C the [ C ( ), D ( )] ouples with =,1,, + 1. + ( ) oeffiiet ad The QM frequeies a be alulated, if suitable oditios are imposed to the g ( x, ) oeffiiets. At QM frequeies =, where Im <, the auxiliary futios g± ( x, ) are liearly depedet g ( x, ) g + ( x, ) ; they are right to left waves for x <, so C ( ) =, ad left to right waves for x > L, so: B A + ( ) =. (A.4) ( ) = C ( ) + + Fially, the QM orm a be alulated for the stratified medium with rative idex (A.1). The Wrosia W( x, ) of the auxiliary futios g± ( x, ) is obtaied from eq. (.13), usig eq. (A.) ± 8