Quasi Normal Modes desriptio of trasmissio properties for Photoi Bad Gap strutures. A. Settimi (1), S. Severii (), B. J. Hoeders (3) (1) INGV (Istituto Nazioale di Geofisia e Vulaologia) via di Viga Murata 65, 143 Roma, Italy () Cetro Iterforze Studi per le Appliazioi Militari via della Bigattiera 1, 561 Sa Piero a Grado (Pi), Italy (3) Researh group theory of odesed matter i the Istitute for theoretial physis ad Zerie Istitute for advaed materials Uiversity of Groige Nijeborg 4, NL 9747 AG Groige, Netherlads 1
Abstrat I this paper, we use the Quasi Normal Modes (QNM) 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 QNM metris for a ope avity, ever disussed i literature, are proved, ad a simple ad diret method to alulate the QNM orm for a 1D-PBG struture is reported. OCIS umbers: 6.11 (Eletromageti optis) 6.574 (Resoae) 1.7 (Trasmissio) 3.598 (Photoi rystals)
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) [1-4]. I these strutures, dispersive properties are usually evaluated assumig ifiite periodi oditios [5]. 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 [6]. The appliatio of the effetivemedium approah is disussed, ad the aalogy with a simple Fabry-Perot struture is developed by Sipe et al. [7]. The problem of the field desriptio iside a ope avity was disussed by several authors. I partiular, Leug et al. i s. [8-15] itrodued the desriptio of the eletromageti field i a oe side ope miroavities i terms of Quasi Normal Modes (QNMs). Miroavities are mesosopi ope systems. Sie they are ope, ad theore leay, i.e. ooservative, the resoae field eige-futios are QNMs 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 QNMs 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 3
issues whih arise, ad the framewor developed for addressig them, are geeri to may other situatios ivolvig ope systems. Theoretial results were obtaied i s. [16-19] whih developed a Hilbert-Shmidt type of theory, leadig to a biliear expasio i terms of the atural modes (QNMs) of the resolvet erel oeted with the itegral equatios of eletromageti ad potetial satterig theory (see erees therei [16-19]). The iitial value problem osidered by s. [16-19] 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 [] who disussed the liear properties of oe dimesioal resoat avities by usig the matrix ad the ray method (see erees therei []). Fabry-Perot, photoi rystals ad 1D avities i PBG strutures are osidered. QNMs for the desriptio of the eletromageti field i ope avities are itrodued ad some appliatios are give. Theoretial ad appliative results were obtaied by. [1] whih developed a satterig theory for fiite 1D-PBG strutures i terms of the atural modes (QNMs) 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 (QNMs) (see erees therei [1]). 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 et al. [] who used QNMs to haraterize trasmissio resoaes i 1D optial defet avities ad the related field approximatios. Ref. [] speializes to resoaes iside the badgap of the periodi multilayer mirrors whih elose the defet avities. Usig a field model with the most relevat QNMs, a variatioal priiple permits to represet the field ad the spetral trasmissio lose to resoaes. I s. [3,4], for the first time, the QNM approah was used ad exteded to the desriptio of the salar field behaviour i double-side opeed optial avities, i partiular 1D- 4
PBG strutures. The validity of the approah is disussed by provig the QNM ompleteess, disussig the omplex frequeies distributio, as well as the orrespodig field distributios, ad reoverig the behaviour of the desity of modes (DOM). I. [5], the eletromageti field iside a optial ope avity was aalyzed i the framewor of the QNM 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. [6], by usig the QNM formalism i a seod quatizatio sheme, the problem of the outer-propagatio of eletromageti fields iside optial ope avities was studied. The lis betwee QNM 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 QNM 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 umber of theoretial assumptios o QNM metris for a ope avity, ever disussed i literature, are proved, ad a simple ad diret method to alulate the QNM orm for a 1D-PBG struture is reported. This paper is orgaized as follows. I setio, the QNM approah is itrodued. I setio 3, a large umber of theoretial assumptios o QNM 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 QNMs. I setio 5, the QNM 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, 5
the trasmissio resoaes are ompared with the QNM frequeies ad the trasmissio modes at the resoaes are alulated as super-positios of the QNM 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 []. The Appedix desribes a simple ad diret method to alulate the QNM orm for a 1D-PBG struture.. Quasi Normal Mode (QNM) approah. With eree to fig. 1, osider a ope avity as a regio of legth L, filled with a material of a give rative idex ( x ), 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 [8-15]: 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 E( x, t ) i the ope avity satisfies the equatio [7] x ρ( x) E( x, t) = t, (.1) 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 [8-15][3,4] 6
E( x, t) = ρ E( x, t) for x <, (.) x t E( x, t) = ρ E( x, t) 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. E( x, t) = E[ x + ( ) 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. E( x, t) = E[ x ( ) t], so eq. (.3) holds as well. To tae the avity leaages ito aout, the Laplae trasform of the e.m. field is osidered [8], = Eɶ ( x, ) E( x, t) exp( it) dt, (.4) where is a omplex frequey. The e.m. field has to satisfy the Sommerfeld radiative oditio [7]: lim Eɶ ( x, ) =. (.5) x ± Sie s = i = exp( iπ ) with C [8], 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 Eɶ ( x, ) oly over the half-plae of overgee Im >. I fat, if the Laplae trasform (.4) is applied to the outgoig waves oditios (.), it follows ɶ(, ) = ρ ɶ (, ) for <, (.6) E x i E x x x ɶ = ρ ɶ >, (.7) E ( x, ) i E x ( x, ) for x L ad, solvig the last equatio (.7): Eɶ ( x, ) exp( i ρ x) = exp( i Re ρ x)exp( Im ρ x) for x > L. (.8) The Sommerfeld radiative oditio (.5) a be satisfied oly if Im >. 7
The trasformed Gree futio Gɶ ( x, x, ) a be defied by [7,9] x ɶ ; (.9) + ρ( x) G( x, x, ) = δ ( x x ) it is a e.m. field so, over the half-plae of overgee Im >, it satisfies the Sommerfeld radiative oditios [7,9]: exp( i ρ x) for x G( x, x, ) exp( i ρ x) for x ɶ. (.1) Two auxiliary futios g± ( x, ) a be defied by [8-15] ρ( x) g ( x, ) + ± = x ; (.11) they are ot defied as e.m. fields, beause, over the half-plae of overgee Im >, they satisfy oly the asymptoti oditios [8-15][3,4]: g+ ( x, ) exp( i ρ x) for x. (.1) g ( x, ) exp( i ρ x) for x However, the trasformed Gree futio Gɶ ( x, x, ) a be alulated i terms of the auxiliary futios g± ( x, ). I fat, it a be show that [8-15] the Wrosia W ( x, ) assoiated to the two auxiliary futios g± ( x, ) is x-idepedet, W ( x, ) = g ( x, ) g ( x, ) g ( x, ) g ( x, ) = W ( ), (.13) + + ad for the trasformed Gree futio [8-15]: g ( x, ) g ( x +, ) for x < x W ( ) Gɶ ( x, x, ) =. (.14) g+ ( x, ) g ( x, ) for x < x W ( ) 8
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 Gɶ ( x, x, ) 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 i x i x x + (, ) = exp( ρ ) = exp( Re ρ ) exp( Im ρ ) for x. The, still over 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, ) A( ) exp( i ρ x) B( )exp( i ρ x) = + for x, with ( ) B, ad so g x B i ρ x B i ρ x ρ x (, ) ( )exp( ) = ( ) exp( Re )exp(im ), 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 + + trasformed Gree futio Gɶ ( x, x, ) is aalyti over Im >, where Gɶ ( x, x, ) does ot diverge; i fat, from eq. (.14): Gɶ ( x, x, ) 1/ W ( ), with W ( ) over Im >. For aalytial otiuatio [8], the trasformed Gree futio Gɶ ( x, x, ) a be exteded also over the lower omplex half-plae Im <. Aordig to. [3], it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio Gɶ ( x, x, ), over the lower omplex half-plae Im <. I other words, there exists a ifiite set of omplex frequeies, = {, ± 1, ±, } whih the Wrosia (.13) is ull [8-15]: Z, with egative imagiary parts Im <, for W ( ) =. (.15) 9
The poles of the trasformed Gree futio are erred to as Quasi-Normal-Mode (QNM) eigefrequeies [3]. The defiitio of the QNM eige-frequeies implies that the auxiliary futios g± ( x, ) beome liearly depedet whe they are alulated at the QNM frequeies { }, Z =, ± 1, ±, ; so, the auxiliary futios i the QNM s are suh that [8-15] g ( x, ) = ( ) g ( x, ) = f ( x, ), (.16) + where ( ) is a suitable omplex ostat. The above futios f ( x, ) = f( x) are erred as Quasi-Normal-Mode (QNM) eige-futios [3]. The ouples [, f ( x)] are erred as Quasi- Normal-Modes beause: they are haraterized by omplex frequeies, so they are the ot-statioary modes of a ope avity [8-15] (they are observed i the frequey domai as resoaes of fiite width or i the time domai as damped osillatios); uder the disotiuity ad o tail oditios, the wave futios f ( x ) form a orthogoal basis oly iside the ope avity [8-15] (it is possible to desribe the QNMs i a maer parallel to the ormal modes of a losed avity). Applyig the QNM oditio (.16) to the equatio for the auxiliary futios (.11), it follows that the QNMs [, f ( x)] satisfy the equatio [8-15]: d dx + ( ) ( ) ρ x f x =. (.17) Moreover, applyig the QNM oditio (.16) to the asymptoti oditios for the auxiliary futios (.1), it follows that, oly for log distaes from the ope avity, the QNMs do ot represet e.m. fields beause they satisfy the QNM asymptoti oditios [8-15][3,4], f ( x) = exp( ± i ρ x) for x ±, (.18) 1
while, iside the same avity ad ear its termial surfaes from outside, the QNMs represet ot statioary modes, i fat the asymptoti oditios (.18) imply the formal QNM 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 QNMs 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 QNMs 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 (QNMs) whose frequeies are omplex. QNM aalysis has bee a powerful tool for ivestigatig ope systems. Previous studies have bee mostly system speifi, ad use a few QNMs to provide approximate desriptios. I s. [8-15], 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 QNMs form a omplete set ad thus give a exat desriptio of the dyamis. More expliitly tha s. [8-15], 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 QNMs a be defied as the poles of the trasformed Gree futio, with egative imagiary part. Our iter follows three steps: 11
1. the π -rotated Laplae trasform of the e. m. field [see eqs. (.4)-(.5)] overges to a aalytial futio Eɶ ( x, ) 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 Gɶ ( x, x, ) [see eq. (.9)] is aalyti over Im >. 3. with respet s. [8-15], we remar that: the trasformed Gree futio Gɶ ( x, x, ) [see eq. (.14)] a be exteded also over the lower omplex half-plae Im <, for aalytial otiuatio [8]; ad, it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio Gɶ ( x, x, ) [see eqs. (.15)-(.16)], over the lower omplex half-plae Im <, aordig to [8-15]. 3. QNM metris. The QNM orm is defied as [8-15] f f dw =, (3.1) d = ad, with the method proposed for a oe side ope avity [8-15], oe a prove, for a both side ope avity, usig the QNM oditio (.18) ad the eqs. (.11)-(.1), that [8-15][3,4]: 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 If the QNM futio f ( x ) is ormalized, aordig to 1 i ρ f () ad i ρ f L. ( )
f N ( x) = f( x), (3.3) f f the: f N f N =. (3.4) The QNM ier produt is defied as [8-15][3,4] L N N N ˆ N ˆ N N N N N N 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 QNM ojugate mometum f ˆ ( x ) is itrodued, aordig to: ˆ N N f ( x) = i ρ( x) f ( x). (3.6) Oe a prove that: the ier produt (3.5) is i aordae with the QNM orm (3.) [8-15]; the QNMs form a orthogoal basis iside the ope avity [8-14], i.e. [3,4] N N f fm = δ, m( + m), (3.7) beig δ, m the Kroeer delta, i.e. δ, m 1, =, = m m. Let us itrodue the overlappig itegral of the th QNM L N = ( ) ρ( ), (3.8) I f x x dx whih is lied to the statistial weight i the desity of modes (DOM) of the If the ope avity is haraterized by very slight leaages, th QNM [3,4]. Im << Re, (3.9) the the overlappig itegral of the th QNM overges to [3,4]: I 1. (3.1) 13
Proof: L eq.(3.3) N I = f ( x) ρ( x) dx = = f f L 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 QNM a be alulated as [3,4]: ρ I f f L. (3.1) N N = () ( ) Im + Proof: L N N N ˆ N ˆ N N N N N N 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 N N N N N N m m m m = ( + ) f ( x) ρ( x) f ( x) dx + i ρ f () f () + f ( L) f ( L) = = δ ( + ), m m eq.(3.7) eq.(3.6), (3.13) so L ρ f x x f x dx i f f f L f L N N N N N N ( ) 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). Fially, if the ope avity is haraterized by very slight leaages [eq. (3.9)] ad its rative idex ( x ) satisfies the symmetry property ( L / x) = ( L / + x), (3.15) 14
the the QNM orm a be approximated i modulus: f f ρ. (3.16) Im Proof of eq. (3.16): ρ ρ eq.(3.1) I = f () f ( L) f () f ( L) + = + 1, (3.17) eq.(3.3) N N Im f f Im so 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 QNM orm is expressed oly i terms of the QNM frequeies; The modulus of the QNM orm is high ( f f >> ρ ) whe the leaages of the ope avity are very slight ( Im << Re ); The QNM theory a be applied to ope avities ad is based o the outgoig waves oditios [eqs. (.)-(.3)] whih formalize some leaages ( Im < ), so the QNM 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 15
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 [8-15], as speified above. Uder these oditios, the QNMs form a omplete basis oly iside the avity, ad the e.m. field a be alulated as a superpositio of QNMs [8-15] N E( x, t) = a( t) f ( x), for x L, (4.1) N where f ( x ) are the ormalized QNM futios [see eq. (3.3)]. The superpositio oeffiiets a( t ) satisfy the dyami equatio [8-15] i aɺ ( t) + i a ( t) = f () b( t), (4.) N ρ where b( t ) is the drivig fore b( t) = ρ xep ( x, t). x= The left-pump EP ( x, t ) satisfies the iomig wave oditio [8-15]: b ( t ) = ρ E ( x, t ) = ρ E (, t ). (4.3) x P x= t P Eah QNM is drive by the drivig fore b( t ) ad at the same time deays beause of Im <. N The ouplig to the fore is determied by the surfae value of the QNM wave futio f (). The avity is i a steady state, so the Fourier trasform with a real frequey = Eɶ ( x, ) E( x, t)exp( it) dt a be applied to equatios (4.1)-(4.3), ad it follows N Eɶ ( x, ) = aɶ ( ) f ( x) N f () bɶ ( ) aɶ ( ) =, (4.4) ρ bɶ ( ) = iρeɶ P(, ) so N N f () f ( x) Eɶ ( x, ) = Eɶ P(, ) i ρ. (4.5) ( ) 16
With eree to fig. 1.b., the e.m. field is otiuous at avity surfaes x = ad x = L, so E ɶ + (, ) = E ɶ (, ) ad E ɶ( L, ) E ( + = ɶ L, ), ad the e.m. field Eɶ (, ) at surfae x = is the superpositio of the iomig pump E ɶ (, ) ad the leted field Eɶ (, ), so P R P E ɶ (, ) = E ɶ (, ) + E ɶ (, ), while the e.m. field Eɶ ( L, ) at the surfae x = L is oly the trasmitted field E ɶ ( L, ), so E ɶ( L, ) = E ɶ ( L, ). T T It follows that the trasmissio oeffiiet t( ) for a ope avity of legth L a be defied as the ratio betwee the trasmitted field Eɶ ( L, ) at the surfae x surfae x = [7]: = L ad the iomig pump Eɶ (, ) at E( L, ) t( ) = ɶ E (, ). (4.6) ɶ P The trasmissio oeffiiet is obtaied as superpositio of QNMs isertig eq. (4.5) i eq. (4.6): R P N N f () f ( L) t( ) = i ρ ( ). (4.7) Applyig the QNM ompleteess oditio [8-15] N N f ( x) f ( x ) = for x, x L, the trasmissio oeffiiet simplifies as: t( ) = i N N f () f ( L) ρ. (4.8) N Isertig the ormalized QNM 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 [5] ( 1) t( ) =, (4.1) γ 17
where γ = f f / i ρ. I aordae with. [5], the trasmissio oeffiiet t( ) of a ope avity a be alulated as a superpositio of suitable futios, with QNM orms f f as weightig oeffiiets ad QNM 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 N periods plus oe layer; every period is omposed of two layers respetively with legths h ad l ad with rative idies h ad l, while the added layer is with parameters h ad h. The symmetri 1D-PBG struture osists of N + 1 layers with a total legth L = N( h + l) + h. If the two layers exteral to the symmetri 1D-PBG struture are osidered, the 1D-spae x a be divided ito N + 3 layers; they are L = [ x, x + 1], =,1,,N + 1, N +, with x =, x =, x = L, x = +. 1 N + N + 3 The rative idex ( x ) taes a ostat value i every layer L, =,1,, N + 1, N +, i.e. for x L, L N + ( x) = h for x L, = 1,3,,N 1, N + 1. (5.1) l for x L, =,4,,N For the symmetri 1D-PBG struture with rative idex (5.1), the QNM orm f f a be obtaied i terms of the frequey ad of the A ( ) N +, B ( ) N + oeffiiets for the g ( x, ) auxiliary futio i the L N + layer o the right side of the 1D-PBG struture (see Appedix): 18
db N + f f = i A N + ( ). (5.) d = As proved i. [3], for a quarter wave (QW) symmetri 1D-PBG struture with N periods ad as eree frequey, there are N + 1 families of QNMs, i.e. F QNM, [, N] ; the F family of QNMs osists of ifiite QNM frequeies, i.e. m = { ± ± } QNM,, Z, 1,,, m whih have the same imagiary part, i.e. Im, = Im,, Z, ad are lied with a step m m =, i.e. Re, m = Re, + m, m Z. It follows that, if the omplex plae is divided ito S = m < Re < ( m + 1), m Z, eah of the QNM family some rages, i.e. { } m QNM F drops oly oe QNM frequey over the rage S, i.e., = (Re, + m, Im,) ; there are N + 1 QNM m m frequeies over the rage S m ad they a be erred as, m =, + m, [, N ]. Thus, there N + QNM frequeies over the basi rage, i.e. S { } are 1, [, ], N. The QNM orm (5.) beomes: = < Re < ; they orrespod to, m db N + f, m f, m = i A N + (, m) d =, m. (5.3) The oeffiiets A ( ) N +, B ( ) N + are of the type exp( iδ ) where δ = π ( ) [3]. If oeffiiets A ( ) N +, B ( ) N + are alulated at QNM frequeies, m =, + m, where =, it follows A N + (, m) = A N + (,) ad ( db / d) = ( db / d), so N + =, m N + =, from eq. (5.3): γ f f γ = = ( + m ). (5.4), m, m,, m, i( / ), Thus, for the QNM F QNM family, the, m γ orm is periodi with a step Γ = γ, (,). The trasmissio oeffiiet (4.1) beomes 19
t( ) = ( 1) + m N + m,, (5.5) = m= γ, m, m ad, taig ito aout eq. (5.4), N, m 1 t( ) = ( 1) ( 1) γ m, (5.6) =, m=, it overges to: t π π. (5.7) N, ( ) = ( 1) s[ (,)] = γ, 6. QNMs ad trasmissio peas of a symmetri 1D-PBG struture. 6.a. QNM frequeies ad trasmissio resoaes. The trasmissio oeffiiet of a QW symmetri 1D-PBG struture, with N periods ad as eree frequey, is alulated as a superpositio of N + 1 QNM-oseats, whih orrespod to the umber of QNMs i [, ) rage. I fig. 3.a., the trasmissio oeffiiet predited by QNM theory ad the by umeri methods existig i the literature [31-33] are plotted for a QW symmetri 1D-PBG struture, where the eree wavelegth is λ = 1µ m, the umber of periods is N = 6 ad the values of rative idies are =, = 1.5. I figure 3.b., the two trasmissio oeffiiets are plotted for the same h l struture of fig. 3.a. but with = 3, =. I figure 3.., the two trasmissio oeffiiets are h l plotted for the same struture as i figure 3.b., with a ireased umber of periods N = 7. I aordae with. [5], we ote the good agreemet betwee the trasmissio oeffiiet predited by the QNM theory ad the oe obtaied by some umeri methods [31-33].
I the tables 1, the same examples of QW symmetri 1D-PBG strutures are osidered: (a) λ = 1µ m, N = 6, =, = 1,5, (b) λ = 1µ m, N = 6, = 3, =, () λ = 1µ m, N = 7, 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 oe QNM lose to every sigle bad-edge: the real part of the QNM-frequey Re( / ) is reported together with the relative shift from the bad-edge resoae (Re..) /... h l B E QNM QNM B E B E From tables 1.a. ad 1.b. ( N = 6 ), the low ad high frequey bad-edges are haraterized by egative phases, whih sum is early ( π ), while, from table 1.. ( N = 7 ), their respetive phases are positive with a sum π approximately. I fat, a QW symmetri 1D-PBG struture, with N periods ad as eree frequey, presets a trasmissio spetrum with N+1 trasmissio pea pea resoaes i the [, ) rage, i.e. t( ) = exp( iϕ ), =,1,,N. Oly by umerial alulatios (MATLAB) o eq. (5.7), oe a sueed i provig that: ϕ ϕ + ϕ π = + 1 (N + 1) ( 1), 1,, N. (6.1) pea So, the lowest resoae i the rage [, ) has always a phase ϕ. The phases ϕ N ad ϕ N + 1 of the low ad high frequey bad-edges pea N ad pea N + 1 are suh that 1 1 ( 1) N + ϕn ϕn + π + ; so, for a QW symmetri 1D-PBG struture with N periods, the phases of the two bad-edges are suh that their sum is early π whe N is a odd umber or ( π ) approximately if N is a eve umber. Agai from tables 1, there is oe QNM lose to every sigle bad-edge, with a relative shift from the bad-edge [5]. I fat, a QW symmetri 1D-PBG struture, with N periods ad as eree frequey, presets a trasmissio oeffiiet whih is a superpositio of N + 1 QNMoseats (5.7), etred to the N + 1 QNM frequeies i [, ) rage, i.e. 1
QNM, =,1,,N. The QNM-oseats are ot sharp futios, so, whe they are superposed, a aliasig ours; the pea of the th QNM oseat a feel the tails of the ( 1) th ad ( + 1) th QNM-oseats, so the th resoae of the trasmissio spetrum results effetively shifted from the th QNM frequey i.e., pea = Re, =,1,,N. Comparig values i the tables 1.a. ( = =.5 ) ad table 1.b. ( = 1), the relative shift h l (Re ) / dereases with the rative idex step, while, omparig values i QNM B. E. B. E. tables 1.b. ( N = 6 ) ad 1.. ( N = 7 ), the same shift ireases with the umber of periods N. I aordae with. [5], some umerial simulatios prove that, the more the 1D-PBG struture presets a large umber of periods ( L >> λ ) with a high rative idex step ( >> ), the more the th QNM desribes the th trasmissio pea i the sese that Re, omes ear to pea. 6.b. QNM futios ad trasmissio modes. Let us osider oe moohromati pump Eɶ ( x, ), whih is iomig from the left side with uit amplitude ad zero iitial phase beig is the rative idex of the uiverse. p Eɶ (, ) ( ) exp( ) p x pɶ x = i x, (6.) The e.m. field Eɶ ( x, ), iside the QW symmetri 1D-PBG struture, a be alulated as a superpositio of QNMs [see setios 3 ad 4]:
Eɶ ( x, ) Eɶ ( x) = f () f ( x) f () f ( x) f ( x) f ( x) = pɶ () i ρ = i ρ = i ρ = = QNMs N N N N ( ) f f γ f ( x) + m f ( x) QW symmetri 1D-PBG struture N N, m,,, m = = = m= γ, m, m γ, m, m = = (6.3) If the pump (6.), iomig from the left side, is tued at the trasmissio pea whih is lose to the th QNM of the [, ) pea rage, i.e. Re, with =,1, N, the the trasmissio mode Eɶ ( x) a be approximated as a superpositio oly of the domiat terms N pea ɶ, m, m = m= Eɶ ( x) = a ( ) f ( x) = pea pea ɶ, m, m ɶ, m, m m= m= = a ( ) f ( x) + a ( ) f ( x), (6.4) m= aɶ ( ) f ( x) pea, m, m where / γ aɶ = = N + m Z = ± ±. (6.5) { } pea,,, m( ),,1, 1,, 1,, pea, m pea I fat, if the superpositio (6.4)-(6.5) is alulated at the frequey Re,, the eah pea domiat term is haraterized by a oeffiiet aɶ, ( ) with a deomiator iludig the m pea pea addedum, ( Re, ) i Im, i Im, =, so, ( pea pea aɶ m ) 1, 1 Im,, pea beig =,1, N + 1, while eah egleted term is haraterized by a oeffiiet aɶ, ( ) with pea pea a deomiator iludig the addedum, >>,, so m aɶ ( ) < 1 << 1 aɶ ( ), beig. pea pea pea pea, m,,, m Thus, the e.m. mode Eɶ ( x) iside the QW symmetri 1D-PBG struture, tued at the pea trasmissio pea lose to the th QNM of the [, ) 3 rage, beig =,1, N, a be approximated as the superpositio of the QNM futios f, ( x ) whih belog to the th QNM- m
family, beig m Z = {, ± 1, ±, } ; moreover, the weigh-oeffiiets of the superpositio pea pea aɶ, m( ) are alulated i the trasmissio resoae ad deped from the th QNMfamily. By umerial alulatios (MATLAB) oe o eqs. (6.4)-(6.5) oe a prove that, whe the pea trasmissio pea is loser to the QNM Re,, it is suffiiet to alulate the trasmissio mode (6.4) by the I order approximatio [] Eɶ ( x) aɶ ( ) f ( x), (6.6) pea,, where [] / γ aɶ N, (6.7) pea,,,( ) =,,1, 1 pea = +, otherwise, by the II order approximatio, more ied, pea Eɶ ( x) aɶ,( ) f,( x) +, (6.8) + aɶ ( ) f ( x) + aɶ ( ) f ( x) pea pea, 1, 1, + 1, + 1 where / γ aɶ N. (6.9) pea,,, 1( ) =,,1, 1 ± pea (, ± ) = + pea I aordae with. [], i the hypothesis that Re,, it follows pea pea pea, << (, ± ) < < (, + m ) <, so pea pea pea aɶ,( ) >> aɶ, ± 1 ( ) > > aɶ, ( ) >, m beig m : i fat, the e.m. mode (6.4)-(6.5) tued at the trasmissio pea pea, for the I order, a be approximated by eq. (6.6), orrespodig to the th QNM of the [, ) rage (the pea resoae is lose eough to the QNM frequey Re, ) []; the, for the II order, by eq. (6.8), iludig the two adjaet ( ± 1) th pea QNMs (if the resoae is far from the QNM frequeies Re =, 1 ±, but the ( ) pea ±, aɶ, futio tued at feels the tails of the aɶ ( ), ± 1 4
futios tued i Re, ± 1 ); fially, the ext orders a be egleted (beause the resoae is too far from the QNM frequeies Re, m =, + m, so the aɶ ( ), futio tued at does ot feel the tails of the aɶ, m( ) futios tued i Re, m, beig m ). pea pea By umerial alulatios (MATLAB) o eqs. (6.6) ad (6.8) oe a prove that, if the QNM is pea eve loser to the trasmissio pea of a QW symmetri 1D-PBG struture, i.e. Re,, whih is haraterized by a large umber of periods, i.e. L >> λ, ad a high rative idex step, i.e. >>, the the I order approximatio (6.6)-(6.7) a be redued to the QNM approximatio: Eɶ ( x) f ( x). (6.1), pea I fat, if Re,, the eq. (6.7) a be approximated as / γ / γ aɶ = = N + ; (6.11) pea,,,,,( ),,1, 1 pea ( Re,) i Im, i Im, 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 QNM orm (3.16) a be used γ, = f, f,,,,1, N 1 ρ Im = + (6.1), ad, i modulus, the weigh-oeffiiet (6.11) overges to: 1 aɶ N. (6.13) pea,,( ) 1,,1, 1 γ, Im = +, Thus, i aordae with. [5], 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 QNMs desribe the trasmissio peas i the sese that the QNM futios approximate the e.m. modes i the trasmissio resoaes. 5
I fig. 4, with eree to a QW symmetri 1D-PBG struture ( λ = 1µ m, N = 6, = 3, = ), the e.m. mode itesities (a) I I bad edge at the low frequey bad edge ( I bad edge / =.8 ) h l ad (b) I at the high frequey bad edge ( / = 1.178 ), i uits of the II bad edge II bad edge 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 QNM approximatio (6.1) (the e.m. mode itesity I I bad edge is due to QNM lose to the bad-edge) ad the first order approximatio (6.6)-(6.7) [more ied tha (6.1), Ibad edge is alulated as the produt of the QNM, lose to the frequey bad-edge, for a weigh oeffiiet, whih taes ito aout the shift betwee the bad-edge resoae ad the QNM frequey] or the seod order approximatio (6.8)-(6.9) [eve more ied, but almost superimposed to (6.6)-(6.7), Ibad edge is alulated as the sum of the first order otributio due to the QNM, lose to the frequey badedge, ad the seod order otributios of the two adjaet QNMs, with the same imagiary part, so belogig to the same QNM family]. 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 N 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 N + 1 resoaes i the [, ) rage [3,4], i.e. t( ) = exp[ iϕ ], r ( ) =, =,1, N [34]. 6
Let us osider oe pump, omig from the left side, whih is tued at the the 1D-PBG struture, with uit amplitude ad zero iitial phase th resoae of ( ) pɶ ( x) = exp( i x), (7.1) beig the rative idex of the uiverse. It exites i the avity a e.m. mode satisfies the equatio [7] ( E ɶ ) ( x) whih d Eɶ dx ad the boudary oditios are [7]: ( ) ( ) + ( x) E ( ) x = ɶ, (7.) ( ) ( ) ( ) ( ) ( ) E ɶ () = pɶ () + r ( ) pɶ () = pɶ () = 1. (7.3) ( ) ( ) Eɶ ( L) = t( ) pɶ () = 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 pɶ ( x) = exp[ i ( )( x L)]exp[ i ϕ]. (7.4) ( ) If aloe, this pump exites i the avity a e.m. mode ( E ɶ ) ( x) whih satisfies a equatio similar to (7.) [7] d Eɶ dx ( ) ( ) + ( x) E ( ) x = ɶ, (7.5) beause of the symmetry properties of the rative idex, with boudary oditios whih are differet from (7.3) [7]: ( ) ( ) ( ) ( ) ( ) E ɶ ( L) = pɶ ( L) + r ( ) pɶ ( L) = pɶ ( L) = exp[ i ϕ]. (7.6) ( ) ( ) Eɶ () = t( ) pɶ ( L) = t( )exp[ i ϕ] It is easy to verify that the e.m. mode ( E ɶ ) ( 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 ɶ ) ( x), whih is exited by the pump (7.1) ad solves equatio (7.) with the boudary oditios (7.3), by the li: 7
Eɶ ( x) = [ Eɶ ( x)] t( )exp[ i ϕ]. (7.7) ( ) ( ) The two outer-propagatig pumps, whih are tued at the th resoae of the 1D-PBG struture, exite a e.m. mode Eɶ ( x) iside the avity, whih is the liear super-positio of ( E ɶ ) ( x) ad ( E ɶ ) ( x), i.e. [7] Eɶ x Eɶ x Eɶ x Eɶ x Eɶ x t i. (7.8) ( ) ( ) ( ) ( ) ( ) = ( ) + ( ) = ( ) + [ ( )] ( ) exp[ ϕ] It is easy to alulate from (7.8) the e.m. mode itesity iside the avity [7] Iɶ ( x) = Eɶ ( x)[ Eɶ ( x)] = ( ) = Iɶ ( x) + Re{[ Eɶ ( x)] t ( ) exp[ i ϕ]} = = Iɶ x + x + =, (7.9) ( ) ( ) ( ){1 os[ φ ( ) ϕ ϕ ]} ϕ ϕ = Iɶ x x + ( ) ( ) 4 ( )os [ φ ( ) ] if the e.m. mode exited by the pump (7.1) is represeted by Eɶ ( ) ( ) ( x I ) ( x) exp[ i ( = ɶ φ ) ( 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). I fig. 5, with eree to a symmetri QW 1D-PBG struture ( λ = 1µ m, N = 6, = 3, l = ), the e.m. mode itesity exited iside the ope avity by oe field pump (- - -) omig 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 I bad edge ad I II bad edge, i uits of the itesity for the two pumps I pump, are plotted as futios of the dimesioless spae x / L, where L is the legth of the 1D-PBG struture. If the two outer-propagatig field pumps i phase are 8 h
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 N periods is exited by two outer-propagatig field pumps with a phase-differee ϕ, if N 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 N 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 N 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) [6]. 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. [6]. 8. A fial disussio. The Quasi Normal Mode (QNM) 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 9
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 QNM 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 QNM 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 [3-5]. I this paper (se. ), suitable outgoig wave oditios for a ope avity without exteral pumpig are formalized [see eqs. (.)-(.3)]. More expliitly tha s. [8-15], 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 QNMs 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 Eɶ ( x, ) 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 Gɶ ( x, x, ) [see eq. (.9)] is aalyti over Im >. 3
3. Differetly from s. [8-15], here we have remared that: the trasformed Gree futio Gɶ ( x, x, ) [see eq. (.14)] a be exteded also over the lower omplex half-plae Im <, for aalytial otiuatio [8]; ad, it is always possible to defie a ifiite set of frequeies whih are the poles of the trasformed Gree futio Gɶ ( x, x, ) [see eqs. (.15)-(.16)], over the lower omplex half-plae Im <, aordig to. [3]. Moreover (se. 3), if the ope avity is haraterized by very slight leaages ad its rative idex satisfies suitable symmetry properties, more tha s. [8-15], here we have proved that the modulus of the QNM orm i modulus a be expressed oly i terms of the QNM frequeies [see eq. (3.16)] ad, more tha s. [3-5], here we have proposed its physial iterpretatio: the QNM orm is high whe the leaages of the ope avity are very slight; the QNM theory a be applied to ope avities ad is based o the outgoig waves oditios whih formalize some leaages ( Im < ), so the QNM 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 QNM orms f f as weightig oeffiiets ad QNM frequeies as parameters [see eq.(4.1)], thesis oly postulated i. [5]. The QNM approah has bee applied to a 1D-PBG struture (se. 5): more tha. [5], here the trasmissio oeffiiet of a quarter wave (QW) symmetri 1D-PBG struture, with N periods ad as eree frequey, has bee alulated as a superpositio of N + 1 QNM-oseats, whih orrespod to the umber of QNMs i [, ) rage [see eq. (5.7)]; differetly from the assertio of. [1] 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 QNM frequeies, here we have dedued that this fat is true oly i the further hypothesis of real parts of QNM frequeies well separated o the real frequey axis, whereas, 31
o the otrary, overlappig of the tails of two ear QNM-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 QNM frequeies (se. 6.a.) ad the trasmissio modes i the resoaes have bee alulated as super-positios of the QNM futios (se. 6.b.): differetly from. [], here we affirm expliitly that a proper field represetatio for the trasmittae problem a be established i terms of QNMs, 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 QNMs 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)]. Eve if this paper is hroologially subsequet to. [5], ayway it is oeived as a trait d uio betwee s. [5] ad [6], i fat the paper forestalls oeptually, aordig to lassi eletrodyamis, what. [6] 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. [6], 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 QNM orm for a QW symmetri 1D-PBG struture: the QNM orm f f a be obtaied i terms of the 3
frequey ad of the A ( ) N +, B ( ) N + oeffiiets for the g ( x, ) auxiliary futio i the L N + layer o the right side of the 1D-PBG struture. 8.1. A ompariso with Masimovi et al.. []. Let us propose a ritial ompariso betwee Masimovi ad our methods: Masimovi speializes. [] to fiite periodi strutures whih possess trasmissio properties with a badgap, i.e. with a regio of frequeies of very low trasmissio. Ref. [] hooses a field model for the trasmittae problem ad taes the relevat QNMs 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 F[ E( x, t )]. If F[ E( x, t )] beomes statioary, i.e. if the first variatio of F[ E( x, t )] vaishes for arbitrary variatios of the e.m. field E( x, t ), the E( x, t ) 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, we have provided, for a symmetri 1D-PBG struture (se. 5), a ew algorithm to alulate the QNM orm by a derivative [see eq. (5.)], more diret tha the itegral of. []; ad, we have proved that a QW symmetri 1D-PBG struture with N periods is haraterized by 33
N + 1 families of QNM frequeies, periodially distributed o the omplex plae, suh that, for eah family, the QNM 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. []. A ommo assumptio made i the literature [8-15][][3,4][35] 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 QNMs; 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. [] justifies aalytially this ommo assumptio. Masimovi osiders the otributio of a sigle QNM i the field model. Here (se. 6.b.), we have osidered the otributios of the adjaet QNMs, 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 QNM, 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 QNM 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: QNM 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 N period plus oe layer; every period is omposed of two 34
layers respetively with legths h ad l ad with rative idies h ad l, while the added layer is with parameters h ad h. The symmetri 1D-PBG struture osists of N + 1 layers with a total legth L = N( h + l) + h. If the two layers exteral to the symmetri 1D-PBG struture are osidered, the 1D-spae x a be divided ito N + 3 layers; the two exteral layers, L = (, ) ad L N + = ( L, + ), ad the 1D- PBG layers, the odd oes L = 1 [( 1)( h + l), ( 1)( h + l) + h], = 1,,, N + 1 ad the eve oes L = [( 1)( h + l) + h, ( h + l)], = 1,,, N. The rative idex ( x ) taes a ostat value i every layer L, =,1,, N + 1, N +, i.e. for x L, L N + ( x) = h for x L, = 1,3,,N 1, N + 1. (A.1) l for x L, =,4,,N At first, the alulatio of quasi ormal modes (QNMs) 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) + N + 1 ih x ih x + A 1e + B 1e ϑ [ x ( 1)( h + l)] ϑ[( 1)( h + l) + h x] + = 1, (A.) N il x il x + A e + Be ϑ [ x ( 1)( h + l) h] ϑ[ ( h + l) x] + = 1 i x i x + A N + e + B N + 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,, N + 1, N +. The auxiliary futio g ( x, ) is a right to left wave for x <, so 35