The propagation of light in one-dimensional SiO2-TiO2 coupled cavity photonic crystal is investigated. In particular the potential application in light amplification is proposed considering the small group velocity that characterizes the propagation at the edge of the resonance band due to the defects. Then, by means of a transfer-matrix method and a mode matching method code, an estimation of the photon lifetime and of the field intensity in a three-coupled cavity-photonic crystal is reported comparing it with those pertaining to a microcavity photonic crystal. This calculation allows us to underline the role of the light–matter interaction time with respect to that of the number of the active medium layers in the optical amplification

Biallo, D.

D'Orazio, Antonella

Petruzzelli, Vincenzo

English

D. Biallo

A. D’Orazio

V. Petruzzelli

EDP Sciences OAI-PMH repository (1.2.0)

Active microcavity and coupled cavities in one-dimensional photonic crystal

The propagation of light in one-dimensional SiO2-TiO2 coupled cavity photonic crystal is investigated. In particular the potential application in light amplification is proposed considering the small group velocity that characterizes the propagation at the edge of the resonance band due to the defects. Then, by means of a transfer-matrix method and a mode matching method code, an estimation of the photon lifetime and of the field intensity in a three-coupled cavity-photonic crystal  is reported comparing it with those pertaining to a microcavity photonic crystal. This calculation allows us to underline the role of the lightâ€“matter interaction time with respect to that of the number of the active medium layers in the optical amplification

D'Orazio, A.

Petruzzelli, V.

JEOS:RP - Journal of the European Optical Society Rapid publications

J O U R N A L  O FTORH E  E U R O P E A N  P T I C A L  S O C I E T YA PID  PU B LIC AT IO N SJournal of the European Optical Society - Rapid Publications 2, 07010 (2007) www.jeos.orgActive microcavity and coupled cavities inone-dimensional photonic crystalD. Biallo Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari,Via Orabona 4, 70124 Bari, ItalyA. D’Oraziodorazio@poliba.itDipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari,Via Orabona 4, 70124 Bari, ItalyV. Petruzzelli Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari,Via Orabona 4, 70124 Bari, ItalyThe propagation of light in one-dimensional SiO2-TiO2 coupled cavity photonic crystal is investigated. In particular the potential applicationin light amplification is proposed considering the small group velocity that characterizes the propagation at the edge of the resonance banddue to the defects. Then, by means of a transfer-matrix method and a mode matching method code, an estimation of the photon lifetimeand of the field intensity in a three-coupled cavity-photonic crystal is reported comparing it with those pertaining to a microcavity photoniccrystal. This calculation allows us to underline the role of the light–matter interaction time with respect to that of the number of the activemedium layers in the optical amplification. [DOI: 10.2971/jeos.2007.07010]Keywords: photonic crystal, coupled-cavity, erbium1 INTRODUCTIONThe propagation of light through photonic band gap (PBG)structures [1] has been the focus of several investigations inthe recent past. Their capability to control the flow of the lightmakes them very attractive for a lot of applications in numer-ous different fields like integrated optics, telecommunicationand sensing. Since it was demonstrated that the photonic crys-tals (PCs) can slow down the propagating pulses, thanks totheir unique dispersive properties [2, 3], the interest versusthese structures is increased a lot. In fact slow light is so attrac-tive for photonic devices because it leads to more efficient lin-ear and nonlinear interaction [4]; in particular the long interac-tion time between radiation field and the matter caused by thesmall group velocity near the edge of the photonic band gapcan enhance the gain of an active PC [5, 6]. The stopping of thelight can also be achieved by introducing a defect in the peri-odic lattice; this provokes a transmission resonance inside theband gap, that results in a high-Q field localization in the wellbounded zone of the defect [7]. In this way, higher is the life-time of the photon inside themicrocavity region higher will bethe efficiency of any non-linear interaction as self-phase mod-ulation [4], second harmonic generation [8] or optical ampli-fication [6]. Recently, it has been demonstrated guiding andbending of electromagnetic waves along a periodic arrange-ment of defects inside a three-dimensional photonic crystal atmicrowave frequencies [9]; in the coupled-cavity structures,photons hop from one evanescent defect mode to the neigh-bouring one due to overlapping between the tightly confinedmodes at each defect site [9]-[11]. Furthermore it has beennumerically and experimentally shown that by placing morethan one defect in cascade the light is guided through the cou-pled microcavity very slowly [12]. This phenomenon couldbe suitable to design an optical compact amplifier if an activemedium is introduced in the structure.In this paper we investigate the propagation of the slowmodes in a coupled cavity (CC) realized in a one-dimensionalSiO2-TiO2 photonic crystal [13]; in particular we estimate thephoton lifetime and the field intensity in such a structure.By erbium doping the SiO2 layers we calculate the transmis-sion coefficient focusing on a comparison between the perfor-mance of a CC structure and a photonic crystal microcavity(PCM). The comparison is made by considering two localisedstate resonances characterized by the same quality factor.2 NUMERICAL ANALYSISThe analysis of the CC structure and the PCM is performed bymeans of a transfer-matrix-method-based code [13, 14] whilethe simulations of the active CC and PCM are performed byconsidering a finite difference time domain (FDTD) approach[15]. The 1D PC investigated is a quarter-wave Bragg stackconsisting of alternated layers of SiO2 and TiO2; the refrac-tive indices of the silica and titania layers are ns = 1.453 andnt = 2.304 at the wavelength λ = 1.55 µm, respectively. TheBragg wavelength has been fixed to λB = 1.532 µm, wherethe erbium emission cross section is centered. The introduc-tion of a defect half-wave long inside the periodic structuremakes arise a resonance at the center of the band gap; then weconsider a PCM constituted by 20 periods, 10 on the right and10 on the left of the central SiO2 local defect (see Figure 1a).The localized state depicted in Figure 1b is characterized by acold-cavity quality factor Q ≈ 15320.Received January 13, 2007; published March 14, 2007 ISSN 1990-2573Journal of the European Optical Society - Rapid Publications 2, 07010 (2007) D. Biallo, et. al.abFIG. 1 a) 1D-PCM. b) Localized state resonance of the structure in Figure 1a; it is tunedat λB = 1.532 µm and is characterized by a quality factor Q ≈ 15320.By adding another defect, two resonances appear at wave-lengths shifted of the same quantity with respect to λB =1.532 µm. The two localized states move towards λB as theseparation between the defects increases and consequentlythe coupling between them decreases; Figure 2 shows that fora separation Λ = 10 periods, we obtain a single frequencytuned at λB characterized by a quality factor Q ≈ 510, that isabout six times smaller than that of the single cavity shown inFigure 1b. FIG. 2 Localized state resonances of a 2-coupled cavity-photonic crystal. By increasingthe spacing between the two coupled cavities the two resonances move towards λBuntil reaching one single resonance tuned at λB. The intercavity distance is Λ andquoted per integer number of periods.Thus, a third defect is introduced: Figure 3a depicts the de-signed 3-CC-PC structure in order to have a central resonancecharacterized by a quality factor Q ≈ 15320 as for the PCM.In this case the frequency transmission spectrum of the struc-ture shows three different resonances inside the bandgap, oneof which is centered at λB (see Figure 3b). By looking at Fig-ure 3c, it is possible to observe that the two localized statesmatch each other with a very high accuracy, allowing a rea-sonable comparison between the two investigated structures.abcFIG. 3 a) 3-CC-photonic crystal with an intercavity distance of one period. b) Localizedstate resonance of the structure in Figure 3a; the central resonance is tuned at λB.c) Comparison between the resonance in Figure 1b and the central resonance inFigure 3b; the two localized states are charecterized by almost the same quality factor.By considering a mode-matching technique the field intensityprofile for a plane wave tuned at λB is obtained and plotted inFigure 4a for the 3-CC-PC and in Figure 4b for the single mi-crocavity. While in the PCM the field is mainly concentratedin the defect, in the 3-CC-PC the field is localized overall inthe lateral defect showing a node in the central cavity; more-over the intensity value reached by the field in the PCM isabout two and half times the value reached in the 3-CC-PC;this means that although the localized states are characterizedby the same quality factor, the light spends more time in thedefect of the PCM with respect to the 3-CC-PC.The photon lifetime of each defect in the 3-CC structure hasbeen evaluated, by considering each one as a Fabry-Perot cav-ity; this allows us to foresee the amplification performance ofsuch a structure compared to the single microcavity consti-tuted by a defect half-wave long embedded in a 20 periodsone-dimensional photonic crystal. The photon lifetime τc of aFabry-Perot cavity can be calculated as follows:τc = − nLcc ln√R1R2(1)where Lc is the cavity length, n is the cavity refractive index,c is the speed of light in vacuum, R1 and R2 are the reflectiv-ity of the two mirrors aside the defect. Figure 5a depicts theevaluated photon lifetime of the three defects in the investi-gated 3-CC-PC and of the PCM. It is possible to note that, incorrespondence of λB, the photon lifetimes of the lateral de-07010- 2Journal of the European Optical Society - Rapid Publications 2, 07010 (2007) D. Biallo, et. al.abFIG. 4 a) Field intensity for a signal, tuned at λB, which through propagates throughthe 3-CC. b) Field intensity for a signal, tuned at λB, which propagates through thePCM.fects, τ1 and τ3 respectively, are bigger than that of the centraldefect τ2 verifying the results of the field intensity calculation;moreover the light stays in each of the three defects of the CCfor a time that is more than three times smaller than that ofthe PCM. This results in a weaker interaction with the silicaand so when the cavity is doped with erbium ions, the lightinteracts with the active medium more in the PCM than in 3-CC-PC.The analysis of the erbium doped CC amplifier has been per-formed by means of a home made computer code which im-plements amodel [17] based on the auxiliary differential equa-tion finite-difference time domain (ADE-FDTD) scheme [15]which solves together the Maxwell equations, the electron os-cillator (EO) equation and the erbium rate equations that givethe energy states population time variation during the prop-agation of pump and input signals. In particular the modelconsiders the absorption and emission erbium cross sectionsby solving the EO equation (the ADE) that takes into accountthe field effect on the medium during its propagation [18]. Forself-consistency, we briefly describe the implemented modelreferring the interested readers to the paper [17] for furtherdetails. The laser system is a four-level energy state one andthe related rate equation model (Eqs. 2-6) considers the fol-lowing energy state transitions: the ground state absorption(GSA), that occurs at the pump wavelength λp = 980 nm, thestimulated emission (SE) that occurs at the input signal wave-length λa = 1532 nm, where the peak of stimulated emission(SE) takes place, and the radiative and non-radiative transi-tions that are governed by the fluorescence times τ43 = τ32 =1 · 10−9 s and τ21 = 7.1 · 10−3 s. Moreover it includes the con-centration quenching effects such as the up-conversion char-acterized by the coefficient rates C3 = Cup = 5 · 10−23m3/sand the cross-relaxation characterized by the coefficient rateC14 = 3.5 · 10−23m3/s [19].dN4dt= −N4τ43+ CupN22 + C3N23 − C14N1N4 (2)dN3dt= WpN1 − N3τ32 +N4τ43− 2C3N23 (3)dN2dt=N3τ32+e(t)hνsdp(t)dt− N2τ21+ 2C14N1N4 − 2 · CupN22 (4)dN1dt= −WpN1− e(t)hνsdp(t)dt+N2τ21−C14N1N4+CupN22 +C3N23(5)while the population density on different energy levels Ni arerelated to the total dopant concentration by the conservationequation:NT = N1 + N2 + N3 + N4 (6)The rate equations are linked to the propagating electric fielde(t) and the macro-polarization vector p(t), by consideringthe EO equation:d2p (t)dt2+ ∆ωadp (t)dt+ω2ap (t) = k∆N12(t)e (t) (7)A Lorentzian susceptibility lineshape fits the experimentallymeasured erbium absorption and emission cross sections. Thetransmittance has been evaluated by calculating the FFT of theinput and output electromagnetic field components and eval-uating the ratio between the Poynting vector along the prop-agation direction at the output of the structure and the Poynt-ing vector at the input of the structure.For the FDTD simulations, a longitudinal step ∆z = 5 · 10−9mand a time step ∆t = 1.5 · 10−17 s, values which verify theCourant limit, have been considered. Each structure underanalysis has been excited by a sinusoidal pump signal con-sidering different power values and an input signal havinga Gaussian lineshape centered around the erbium emissionwavelength λ = 1.532 µm. The erbium ion concentration forboth structures is Nt = 2.2 · 1026[ ions/m3] and the pump rateWp = 11.42 · 104.Figure 5b depicts the transmission coefficient of the 3-CC-PCand of the PCM at λB; the PCM shows higher amplificationwith respect to the 3-CC-PC. This is not an obvious result be-cause if it is true that the light stays into the PCM more timethan in each of the three coupled cavity of the CC structure itis also true that in the 3-CC-PC there are more erbium dopedlayers. In an erbium doped waveguide amplifier longer is thestructure higher is the gain so by increasing the number of er-bium doped layers an increase of the transmission coefficientis expected. The 3-CC structure is longer than the PCM, hav-ing more doped defects; but, observing Figure 5b, one can seethat the contribution to the gain due to the structure lengthisn’t enough to fill up the gap in photon lifetimes between the07010- 3Journal of the European Optical Society - Rapid Publications 2, 07010 (2007) D. Biallo, et. al.two compared structures. It comes out that the light confine-ment in an active layer is more important than the total quan-tity of the active layers in a one-dimensional photonic crystal.abFIG. 5 a) Calculated photon lifetime of the 3-CC in Figure 3a and of the PCM depicted inFigure 1a, as a function of the wavelength. In particular, referring to the 3-CC structure,t1 and t3 are the photon lifetimes of the lateral defects and t2 the photon lifetime ofthe central defect. tpcm is the photon lifetime of the PCM. b) Transmission coefficientaround λB of the 3-CC-PC and the PCM doped with erbium.3 CONCLUSIONThis paper reports a comparison between amicrocavity and 3-coupled cavity in a one-dimensional SiO2-TiO2 photonic crys-tal for the amplification application. The estimation of thephoton lifetime in each defect of both structures allows us toforesee the amplification performance of these devices whenthe silica layers are doped with erbium. In fact the opticalamplification results to be dependent on the photon lifetimemore strongly than on the number of active layers. For thisreason photonic crystal microcavity configuration can be cho-sen to design the amplifier since it is more compact than the3-coupled cavity amplifier to a parity of gain.References[1] J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystal: Mold-ing the Flow of the Light (Princeton University Press, 1995).[2] M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M.J. Bloemer, C. M. Bowden, I. Nefedov “Dispersive properties offinite, one-dimensional photonic band gap structures: Applicationsto nonlinear quadratic interactions” Phys. Rev. E 60, 4891–4898(1999).[3] J. P. Dowling, M. Scalora, M. Bloemer, C. M. Bowden, “The photonicband edge lasers: a new approach to gain enhancement” J. Appl.Phys. 75, 1896–1899 (1994).[4] M. Soljacic, J. D. Joannopoulos, “Enhancement of nonlinear effectsusing photonic crystals” Nat. Mater. 3, 211–219 (2004).[5] M. Botey, M. Maymò, D. Biallo, J. Martorell, “Photon lifetime atthe inner band edges of a 3-D photonic crystal” Laser Phys. 14,643–647 (2004).[6] D. Biallo, A. D’Orazio, V. Petruzzelli, “Enhanced light extraction inEr3+ doped SiO2-TiO2microcavity embedded in one-dimensionalphotonic crystal” J. Non-Cryst. Solids 352, 3823–3828 (2006).[7] D. J. Ripin, K. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen,J. D. Joannopoulos, E. P. Ippen, A. Kolodziejski, “One-dimensionalphotonic bandgap microcavities for strong optical confinementin GaAs and GaAs/AlxOy semiconductor waveguides” J. LightwaveTechnol. 17, 2152–2160 (1999).[8] M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden,R. Viswanathan, J. W. Haus, “Pulsed second-harmonic generationin nonlinear, one-dimensional, periodic structures” Phys. Rev. A56, 3166–3174 (1997).[9] M. Bayindir, B. Temelkuran, E. Ozbay, “Tight-Binding Descriptionof the Coupled Defect Modes in Three-Dimensional Photonic Crys-tals” Phys. Rev. Lett. 84, 2140–2143 (2000).[10] M. Bayindir, S. Tanriseven, E. Ozbay, “Propagation of light throughlocalized couplet-cavity modes in one-dimensional photonic band-gap structures” Appl. Phys. A 72, 117–119 (2001).[11] M. Bayindir, C. Kural, E.Ozbay, “Coupled optical microcavities inone-dimensional photonic bandgap structures” J. Opt. A: PureAppl. Op. 3, S184–S189 (2001).[12] H. Altug, J. Vuckovic, “Experimental demonstra-tion of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays”http://arxiv.org/ftp/physics/papers/0406/0406109.pdf.[13] A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Con-trol of Laser Radiation (John Wiley & Sons, 1984).[14] J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressionsfor the electromagnetic mode density in finite, one-dimensional,photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).[15] A. Taflove, S. C. Hagness, Computational Electrodynamics: theFinite-Difference Time-Domain Method (Artech House, Boston-London, 2000).[16] J. M. Liu, Photonic Devices (Cambridge University Press, 2005).[17] D. Biallo, A. D’Orazio, M. De Sario, V. Petruzzelli, F. Prudenzano,“Time domain analysis of optical amplification in Er3+ doped SiO2-TiO2 planar waveguide” Opt. Express 13, 4683–4692, (2005).[18] A. E. Siegman, Lasers, (University Science Book, 1986).[19] A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano,A. Chiasera, M. Montagna, C. Tosello, M. Ferrari, “Design of Er3+Doped SiO2-TiO2 Planar Waveguide Amplifier,” J. Non-Cryst. Solids322, 278–283 (2003).07010- 4

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Active microcavity and coupled cavities in one-dimensional photonic crystal

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JEOS:RP - Journal of the European Optical Society Rapid publications