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 light–matter interaction time with respect to that of the number of the active medium layers in the optical amplification. [DOI: 10.2971/jeos.2007.07010]


INTRODUCTION
The propagation of light through photonic band gap (PBG) structures [1] has been the focus of several investigations in the recent past.Their capability to control the flow of the light makes them very attractive for a lot of applications in numerous different fields like integrated optics, telecommunication and sensing.Since it was demonstrated that the photonic crystals (PCs) can slow down the propagating pulses, thanks to their unique dispersive properties [2,3], the interest versus these structures is increased a lot.In fact slow light is so attractive for photonic devices because it leads to more efficient linear and nonlinear interaction [4]; in particular the long interaction time between radiation field and the matter caused by the small group velocity near the edge of the photonic band gap can enhance the gain of an active PC [5,6].The stopping of the light can also be achieved by introducing a defect in the periodic lattice; this provokes a transmission resonance inside the band gap, that results in a high-Q field localization in the well bounded zone of the defect [7].In this way, higher is the lifetime of the photon inside the microcavity region higher will be the efficiency of any non-linear interaction as self-phase modulation [4], second harmonic generation [8] or optical amplification [6].Recently, it has been demonstrated guiding and bending of electromagnetic waves along a periodic arrangement of defects inside a three-dimensional photonic crystal at microwave frequencies [9]; in the coupled-cavity structures, photons hop from one evanescent defect mode to the neighbouring one due to overlapping between the tightly confined modes at each defect site [9]- [11].Furthermore it has been numerically and experimentally shown that by placing more than one defect in cascade the light is guided through the coupled microcavity very slowly [12].This phenomenon could be suitable to design an optical compact amplifier if an active medium is introduced in the structure.
In this paper we investigate the propagation of the slow modes in a coupled cavity (CC) realized in a one-dimensional SiO 2 -TiO 2 photonic crystal [13]; in particular we estimate the photon lifetime and the field intensity in such a structure.By erbium doping the SiO 2 layers we calculate the transmission coefficient focusing on a comparison between the performance of a CC structure and a photonic crystal microcavity (PCM).The comparison is made by considering two localised state resonances characterized by the same quality factor.

NUMERICAL ANALYSIS
The analysis of the CC structure and the PCM is performed by means of a transfer-matrix-method-based code [13,14] while the simulations of the active CC and PCM are performed by considering a finite difference time domain (FDTD) approach [15].The 1D PC investigated is a quarter-wave Bragg stack consisting of alternated layers of SiO 2 and TiO 2 ; the refractive indices of the silica and titania layers are n s = 1.453 and n t = 2.304 at the wavelength λ = 1.55 µm, respectively.The Bragg wavelength has been fixed to λ B = 1.532 µm, where the erbium emission cross section is centered.The introduction of a defect half-wave long inside the periodic structure makes arise a resonance at the center of the band gap; then we consider a PCM constituted by 20 periods, 10 on the right and 10 on the left of the central SiO 2 local defect (see Figure 1a).The localized state depicted in Figure 1b is characterized by a cold-cavity quality factor Q ≈ 15320.By adding another defect, two resonances appear at wavelengths shifted of the same quantity with respect to λ B = 1.532 µm.The two localized states move towards λ B as the separation between the defects increases and consequently the coupling between them decreases; Figure 2 shows that for a separation Λ = 10 periods, we obtain a single frequency tuned at λ B characterized by a quality factor Q ≈ 510, that is about six times smaller than that of the single cavity shown in Figure 1b.Thus, a third defect is introduced: Figure 3a depicts the designed 3-CC-PC structure in order to have a central resonance characterized by a quality factor Q ≈ 15320 as for the PCM.In this case the frequency transmission spectrum of the structure shows three different resonances inside the bandgap, one of which is centered at λ B (see Figure 3b).By looking at Figure 3c, it is possible to observe that the two localized states match each other with a very high accuracy, allowing a rea-sonable comparison between the two investigated structures.By considering a mode-matching technique the field intensity profile for a plane wave tuned at λ B is obtained and plotted in Figure 4a for the 3-CC-PC and in Figure 4b for the single microcavity.While in the PCM the field is mainly concentrated in the defect, in the 3-CC-PC the field is localized overall in the lateral defect showing a node in the central cavity; moreover the intensity value reached by the field in the PCM is about two and half times the value reached in the 3-CC-PC; this means that although the localized states are characterized by the same quality factor, the light spends more time in the defect of the PCM with respect to the 3-CC-PC.
The photon lifetime of each defect in the 3-CC structure has been evaluated, by considering each one as a Fabry-Perot cavity; this allows us to foresee the amplification performance of such a structure compared to the single microcavity constituted by a defect half-wave long embedded in a 20 periods one-dimensional photonic crystal.The photon lifetime τ c of a Fabry-Perot cavity can be calculated as follows: where L c is the cavity length, n is the cavity refractive index, c is the speed of light in vacuum, R 1 and R 2 are the reflectivity of the two mirrors aside the defect.Figure 5a  fects, τ 1 and τ 3 respectively, are bigger than that of the central defect τ 2 verifying the results of the field intensity calculation; moreover the light stays in each of the three defects of the CC for a time that is more than three times smaller than that of the PCM.This results in a weaker interaction with the silica and so when the cavity is doped with erbium ions, the light interacts with the active medium more in the PCM than in 3-CC-PC.
The analysis of the erbium doped CC amplifier has been performed by means of a home made computer code which implements a model [17] based on the auxiliary differential equation finite-difference time domain (ADE-FDTD) scheme [15] which solves together the Maxwell equations, the electron oscillator (EO) equation and the erbium rate equations that give the energy states population time variation during the propagation of pump and input signals.In particular the model considers the absorption and emission erbium cross sections by solving the EO equation (the ADE) that takes into account the field effect on the medium during its propagation [18].For self-consistency, we briefly describe the implemented model referring the interested readers to the paper [17] for further details.The laser system is a four-level energy state one and the related rate equation model (Eqs.2-6) considers the following energy state transitions: the ground state absorption (GSA), that occurs at the pump wavelength λ p = 980 nm, the stimulated emission (SE) that occurs at the input signal wave- .4) (5) while the population density on different energy levels N i are related to the total dopant concentration by the conservation equation: The rate equations are linked to the propagating electric field e(t) and the macro-polarization vector p(t), by considering the EO equation: A Lorentzian susceptibility lineshape fits the experimentally measured erbium absorption and emission cross sections.The transmittance has been evaluated by calculating the FFT of the input and output electromagnetic field components and evaluating the ratio between the Poynting vector along the propagation direction at the output of the structure and the Poynting vector at the input of the structure.Figure 5b depicts the transmission coefficient of the 3-CC-PC and of the PCM at λ B ; the PCM shows higher amplification with respect to the 3-CC-PC.This is not an obvious result because if it is true that the light stays into the PCM more time than in each of the three coupled cavity of the CC structure it is also true that in the 3-CC-PC there are more erbium doped layers.In an erbium doped waveguide amplifier longer is the structure higher is the gain so by increasing the number of erbium doped layers an increase of the transmission coefficient is expected.The 3-CC structure is longer than the PCM, having more doped defects; but, observing Figure 5b, one can see that the contribution to the gain due to the structure length isn't enough to fill up the gap in photon lifetimes between the two compared structures.It comes out that the light confinement in an active layer is more important than the total quantity of the active layers in a one-dimensional photonic crystal.

CONCLUSION
This paper reports a comparison between a microcavity and 3coupled cavity in a one-dimensional SiO 2 -TiO 2 photonic crystal for the amplification application.The estimation of the photon lifetime in each defect of both structures allows us to foresee the amplification performance of these devices when the silica layers are doped with erbium.In fact the optical amplification results to be dependent on the photon lifetime more strongly than on the number of active layers.For this reason photonic crystal microcavity configuration can be chosen to design the amplifier since it is more compact than the 3-coupled cavity amplifier to a parity of gain.

D
FIG. 1 a) 1D-PCM.b) Localized state resonance of the structure in Figure 1a; it is tuned at λ B = 1.532 µm and is characterized by a quality factor Q ≈ 15320.

FIG. 2
FIG. 2 Localized state resonances of a 2-coupled cavity-photonic crystal.By increasing the spacing between the two coupled cavities the two resonances move towards λ B until reaching one single resonance tuned at λ B .The intercavity distance is Λ and quoted per integer number of periods.
FIG. 3 a) 3-CC-photonic crystal with an intercavity distance of one period.b) Localized state resonance of the structure in Figure 3a; the central resonance is tuned at λ B .c) Comparison between the resonance in Figure 1b and the central resonance in Figure 3b; the two localized states are charecterized by almost the same quality factor.
FIG. 4 a) Field intensity for a signal, tuned at λ B , which through propagates through the 3-CC.b) Field intensity for a signal, tuned at λ B , which propagates through the PCM.
length λ a = 1532 nm, where the peak of stimulated emission (SE) takes place, and the radiative and non-radiative transitions that are governed by the fluorescence times τ 43 = τ 32 = 1 • 10 −9 s and τ 21 = 7.1 • 10 −3 s.Moreover it includes the concentration quenching effects such as the up-conversion characterized by the coefficient rates C 3 = C up = 5 • 10 −23 m 3 /s and the cross-relaxation characterized by the coefficient rate C 14 = 3.5 • 10 −23 m 3 /s [19]

For
the FDTD simulations, a longitudinal step ∆z = 5 • 10 −9 m and a time step ∆t = 1.5 • 10 −17 s, values which verify the Courant limit, have been considered.Each structure under analysis has been excited by a sinusoidal pump signal considering different power values and an input signal having a Gaussian lineshape centered around the erbium emission wavelength λ = 1.532 µm.The erbium ion concentration for both structures is N t = 2.2 • 10 26 [ ions/m 3 ] and the pump rate W p = 11.42 • 10 4 .

FIG. 5
FIG. 5 a) Calculated photon lifetime of the 3-CC in Figure 3a and of the PCM depicted in Figure 1a, as a function of the wavelength.In particular, referring to the 3-CC structure, t 1 and t 3 are the photon lifetimes of the lateral defects and t 2 the photon lifetime of the central defect.t pcm is the photon lifetime of the PCM.b) Transmission coefficient around λ B of the 3-CC-PC and the PCM doped with erbium.