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We report on far-field measurements of L3 photonic crystal (PhC) cavities with high quality beaming. This is achieved by means of the so-called “band folding” technique, in which a modulation of the radius of specific holes surrounding the cavity is introduced. Far-field patterns are measured from photoluminescence of quantum wells embedded in the PhC. A very good agreement between experimental results and simulated radiation patterns has been found. Laser effect is demonstrated in the beaming cavity with a threshold comparable to the regular one. In addition, free-space input coupling to this cavity has been achieved. In order to fully analyze the coupling efficiency, we generalize the approach developed in S. Fan, et al., [J. Opt. Soc. Am. A 20, 569 (2003)], relaxing the hypothesis of mirror symmetry. The obtained coupling efficiencies are about 15% with quality factors (Q) exceeding 104. These results further validate the “folding” technique on L3 cavities for nanocavity realization with efficient free-space coupling and high Q factors.
© 2012 Optical Society of America
1. Introduction
Photonic crystal (PhC) nanocavities have been intensively investigated during the last decade due to their capabilities of achieving tight light confinement and low optical losses simultaneously. Among the different geometries, the cavity proposed by Noda et al. [1], namely a L3 cavity (three holes missing) with shifted end-holes, has been successfully implemented in several applications including cavity quantum electrodynamics and single photon sources [2], laser emission [3] and nonlinear devices [4, 5]. Unfortunately, the reduced modal volume of small cavities leads to a very strong diffraction. As a result, the radiation patterns of high Q nanocavities are highly non-directional, and free-space input/output light coupling becomes inefficient. As an alternative road, near field coupling schemes have been recently developed, such as evanescent coupling using tapered optical fibers [6, 7]. In spite of this, optimizing input/output free-space optical coupling is a key element for many applications in nanophotonics such as efficient single-photon nanosources and parallel optical injection in nonlinear coupled cavity networks, among others.
In order to overcome the poor free-space coupling, diverse nanocavity designs that optimize the radiation patterns have been investigated in the last years. Romer et al. have proposed an optimized photonic crystal H1 cavity for spatial and spectral enhancement of the spontaneous emission [8]. Kim et al. have studied possible strategies towards a far-field optimization for hexapolar modes in H1 cavities as well [9]. Experimentally, a characterization of the far-field of a modified hexapole mode with optimized beaming has been carried out in [10].
Recently, an elegant path, the so-called “band-folding” method has been developed by Tran et al., which allows control over the beaming from cavity modes minimizing the off-plane leakage [11]. This is based on the modulation of the hole size at twice the period of the underlying PhC, which considerably increases the coupling efficiency in the vertical direction [11, 12]. Using this approach, different geometries and designs such as 1D, H0, hexapolar and heterostructure cavities have been investigated [13]. Very recently, laser emission of a H0 PhC nanocavity, modified to enhance vertical emission, was demonstrated [14]. In particular, the Ln cavity (n holes missing in the PhC periodicity in the Γ – K direction of a triangular lattice) with shifted end-holes has been widely used [12, 13]. In this work we have studied L3 PhC cavities with directive far-field profiles using the band folding method. Radiation patterns from “folded” and “unfolded” cavities are systematically measured and compared.
This paper is organized as follows: in Section 2 we describe the PhC samples. In Section 3 we report on far-field measurements from folded L3 nanocavities; a systematic comparison with unfolded cavities is carried out, both experimentally and numerically. Laser operation of the folded L3 nanocavity is demonstrated and compared to the unfolded one. In Section 4, input coupling experiments are performed, and Fano resonances are obtained. A model of optical Fano resonances, derived in the Appendix, is used in order to account for optical coupling efficiency. Finally, the connection between the far-field profiles and the coupling efficiency is discussed in Section 5, and conclusions are given in Section 6.
2. Photonic crystal samples
The basic cavities used in this work are L3 cavities, i.e. three holes missing in the Γ – K direction of a triangular lattice. Two different hexagonal lattice periods have been realized, a = 437nm and a = 450nm, with target hole radius r0 = 0.266a. The two holes closing the cavity are shifted apart by s = 0.16a, and their radius is reduced to r1 = r0 − 0.06a, in order to increase the theoretical Q-factor up to 105.
The “band folding” technique has been implemented as follows. The basic L3 cavities have been modified introducing a modulation of hole radius with period 2a. The radius of modified holes is chosen to be r2 = r0 − 0.02a. Thereafter “unfolded” and “folded” cavity will refer to the basic and the modulated L3 nanocavities, respectively. Note that, as in Ref. [11], only the nearest neighbors are modified. 3D-FDTD simulations indicate that further increasing the number of periods in the modulation does not improve the cavity performance.
The fabricated samples are InP-based cavities on a suspended membrane, with resonant modes at around 1.5 − 1.55μm. The InP membrane (265 nm-thick), grown by Metalorganic Vapour Phase Epitaxy (MOCVD), incorporates four central layers of InGaAs/InGaAsP quantum wells (QWs). The QW luminescence at 300 K is centered at 1.49μm with a spectral broadening of 86 nm. A SiO2 sacrificial layer underneath the InP is bonded onto a Si substrate through a benzocyclobutene (BCB) layer. A 1 μm-air spacer, obtained after wet etching the sacrificial layer, lies between the InP membrane and the substrate.
Scanning Electron Microscopy (SEM) images of the unfolded and folded cavities are shown in Figs. 1(a) and 1(d), respectively. As it has been shown in [11, 13], this additional periodicity at twice the pitch of the original lattice folds the M points onto Γ points in reciprocal space. This enables directional leakage of modes that were located below the light line in the non-modified configuration. The modulation of hole-radius used in this work has been chosen such that it ensures the best compromise between the quality factor, the coupling efficiency and the expected far-field profile [12].
Fig. 1 SEM images of the unfolded (a) and folded (d) L3 nanocavities. Circles filled in red denote the shifted and shrunk-end holes to boost the Q-factor, those filled in green denote the size modulation at twice the period of the original lattice to achieve the band-folding effect. Measured sizes are a = 437nm, s = 66nm, r0 = 122nm, r1 = 97nm and r2 = 114nm. (b) and (e) are the spectral emissions of the unfolded (9μW-pump power) and folded (16.3μW-pump power) nanocavities, respectively. The output power (black) and spectral width (red) versus input power are presented in (c) for the unfolded nanocavity and (f) for the folded one, showing laser emission in both cases.
3. Far-field measurements
We have first characterized photoluminescence (PL) spectra of our active cavities with an experimental setup as the one described in [15]. This allows us to obtain spectrally resolved PL, as well as spatially resolved PL both in the near and far-field. Unfolded and folded cavities with a = 437nm (for the other parameters, see caption of Fig. 1) have been investigated. The sample is surface pumped with a 100 fs-pulse duration, 80 MHz-repetition rate Ti:Sa laser source emitting at 810 nm. A high N.A. objective (N.A=0.95, 160x, 0.2 mm-working distance) was used, which focuses the pump beam down to a 1.5μm-diameter spot, and the emission is collected through the same microscope objective and sent to a spectrometer and to an InGaAs camera (“Sensors Unlimited”, SU320). Pumping powers are set close to ∼ 10μW, measured after the objective. A resonant mode centered at 1504 nm for the unfolded cavity, and one centered at 1511 nm for folded cavity are observed [Figs. 1(b) and 1(e), respectively]. Such a red-shift of the resonant modes of modulated cavities with respect to the basic ones is systematically observed and it is in good agreement with the theoretical prediction as we will discuss at the end of this section; this is attributed to an increased effective refractive index of the cavity mode due to the smaller size of surrounding holes. We point out that the Q factors of the cavities cannot be obtained from a simple measurement of the FWHM of the resonant PL, since the PL peaks are broadened essentially due to frequency chirp as a result of the pulsed pump. Q-factor measurements of folded cavities are carried out in Section 4 using a resonant probe: Q-factors of about 104 are obtained, which shows that the hole size modulation does not substantially degrade the optical quality of the nanocavities. For comparison, intrinsic Q-factors of about 4000 have been previously measured in similar unfolded cavities using a tapered-assisted coupling method [4].
In Figs. 1(c) and 1(f) the output intensity and FWHM are presented as a function of the incident pump power, for the unfolded and folded cavities, respectively. They both show laser emission. The threshold, determined as the pump power at the minimum of the FWHM of the laser spectrum, is Pth = 3.7μW for the unfolded nanocavity [Fig. 1(c)], and Pth = 6.5μW [Fig. 1(f)] for the folded one. As a result of the relatively high Q-factor of the modified cavity, laser behavior is still observed for this cavity with a higher but same order of magnitude threshold compared to the unfolded cavity.
The far-field measurement is the most direct method to investigate the beaming properties of the folded cavity approach. To the best of our knowledge, such measurements have never been performed in folded Ln cavities. In the following, far-field images are measured to experimentally study the beaming properties of the folded L3 cavity.
Figure 2(a) shows the experimental radiation pattern of the unfolded L3 cavity. This is achieved using the technique described in detail in [15]: the Fourier plane is imaged on the InGaAs camera by adding a f1-focal length lens (f1=250 mm) at a distance f1 from the back focal plane of the objective. Pump intensity is set above the laser threshold (P ∼ 20.4μW), for both unfolded and folded cavities. The emission pattern of the unfolded cavity is not directional: quite strong lobes are observed at large emission angles (∼ 70° in ky-direction). Figure 2(b) shows the emission diagram of the folded cavity. Note that the far-field is nearly circular, slightly elongated in the kx-direction, and centered at k = 0. Figure 2(b) clearly shows that most of electromagnetic energy is directed along the sample normal within a 30°-cone. Beam sizes in k-space (to 1/e2 of the intensity) are δkx ≈ 0.57(2π/λ), θx ∼ 35°, and δky ≈ 0.43(2π/λ), θy ∼ 25°. The beaming improvement of folded L3 cavities is thus experimentally demonstrated.
Fig. 2 Experimental and simulated far-field of the unfolded (a and c) and folded (b and d) L3 nanocavity. The white line corresponds to the light line (emission at 90°) while the dashed white line corresponds to N.A.=0.95, i.e. the maximum angle collected in our set up (∼ 72°). Short dashed line corresponds to 30° (N.A.=0.5).
In order to obtain numerical far-field images, 3D-FDTD simulations have been performed [16]. For this we considered an isolated PhC membrane with a refractive index of n = 3.3. Boundary conditions have been set to perfect matched layers (PML), and mirror symmetries have been exploited to reduce calculation time. Grid resolution is set to a/20 in x and z directions and in y direction. Mode frequencies and Q-factors are computed using the harmonic inversion algorithm Harminv [17]. For a = 437nm, the resonant modes are centered at λ0 = 1515nm with Q0 ∼ 9.1 × 104 for the unfolded cavity, and at λf = 1525nm with Qf ∼ 4.4 × 104 for the folded one. Resonances of the modified cavities are red-shifted by ∼ 10nm with respect to the original ones, in agreement with the experimental observation (∼ 7nm-red shift). Emission profiles in k-space have been calculated from the Fourier transform of the electromagnetic field distribution on a monitor located at 0.38a from the slab surface in z-direction. Figures 2(c) and 2(d) show the calculated angular emission from the unfolded and folded L3 cavities, respectively. The vertical emission of the cavity is totally modified as a result of the band-folding procedure. Calculated beam sizes are δkx ≈ 0.5(2π/λ), θx ∼ 30°, and δky ≈ 0.3(2π/λ), θy ∼ 17°. We remark a good agreement between experimental and numerical far-field images. This result shows that output coupling becomes efficient with moderate N.A. collection optics. In the following section we show that input coupling can also be improved using folded cavities.
4. Input light coupling measurements
In this Section we show that the high quality beaming obtained from folded cavities gives rise to efficient input light coupling. In order to reduce the absorption penalty, absorption losses from the QW must be minimized. For this, a L3 cavity with a = 450nm (a = 437nm in the previous Section) is chosen, which gives a resonant mode red detuned from the maximum of QW absorption of about 55 nm. The cavities are probed using a 120 fs-pulse duration, 80 MHz-repetition rate Optical Parameter Oscillator (OPO) emitting at 1.55μm. The beam, perpendicularly incident with respect to the PhC surface, is spatially filtered with a 100μm-pinhole and two lenses (10 cm-focal length) and collimated in order to optimize the beam quality. The probe power impinging the sample is 1.24μW. A beam diameter of D = 3mm, where D = 2w and w the beam waist @1/e2 of the intensity, has been measured before the microscope objective. The beam waist on the sample, i.e. after the microscope objective, is w = (0.7 ± 0.1)μm, which is close to the diffraction limit. As shown in Ref. [18], the coupling conditions are optimized for a tightly focused beam over the cavity. The reflectivity is obtained as R(ω) = Pr(ω)/P0(ω), where Pr(ω) is the reflected signal and P0(ω) the incoming power, which is obtained from reflection onto a high reflectivity (>99%) dielectric mirror.
4.1. Fano resonance
Figure 3 (black line) shows the probe reflectivity spectrum exhibiting a strongly asymmetric profile. Such a lineshape, known as Fano resonance, is typical of interference processes in which a discrete mode, given in this case by the nanocavity, interacts with a quasi-continuum of radiative modes. In our case, the latter comes from multiple reflections in the vertical direction associated to the PhC membrane, the air-gap and substrate underneath. It has been shown that Fano resonances of this kind can be fitted with the following generic formula for Fano processes [12, 18]:
where A0 and F0 are constants, Γ is the linewidth and q a dimensionless parameter related to the asymmetry of the resonance. As far as reflectivity measurements are concerned, F(ω) = R(ω) → r2 for both ω ≪ ω0 and ω ≫ ω0, where r2 is the background reflectivity. Hence, A0 + F0 = r2, and only two free parameters, F0 and q, define the contrast and shape of the resonance for a given Γ. Equation (1) then becomes Note that Eq. (2) remains invariant under the transformation q → −1/q and F0 → −F0q2. Hence, two different F0, one positive and the other one negative, give the same lineshape. We take F0 > 0 from now on without loss of generality. Three regimes can thus be distinguished: i) for |q| ≫ 1, the resonant scattering dominates over the direct scattering, and the resonance becomes a symmetric Lorentzian; ii) for |q| ≪ 1, the situation is reversed and the resonance becomes a reversed Lorentzian; and iii) for |q| ∼ 1 the amplitudes of both resonant and direct scattering are similar and the resonance is strongly asymmetric. Figure 3(a) depicts a fit of the measured reflectivity using Eq. (2). The obtained q parameter is q ≈ 0.9, showing that the resonant scattering intensity is close to the reflected intensity on the sample (direct scattering). Let us define the contrast efficiency as the amplitude of the reflectivity variation close to resonance, i.e. ηC = max{F(ω)} − min{F(ω)}. In terms of Eq. (2), this becomesFig. 3 (a) Experimental (black line) data and fit with Fano model [Eq. (2) in the text, red line], of the reflectivity spectrum. Fitted parameters are r2 = 0.1804, F0 = 0.06373, λ0 = 1546.22nm, Γ = 1.0585 × 1011 Hz and q = 0.9. The contrast efficiency is ηC ≈ 12%. (b) Zoom of (a), with an additional fit: Coupled Mode Theory (CMT) Fano model with mirror symmetry (Eq. (8), green line). Fitted parameters are r = −0.4254, τc = 1.29 × 10−10 s, τ = 1.83 × 10−11 s and λ0 = 1546.23nm. The corresponding coupling efficiency is η ≈ 14%. Inset: schematic of coupling channels (CMT model).
The magnitude in Eq. (3) slightly differs from the resonant scattering efficiency defined in Ref. [12], ηRS = F0q2 (F0 is normalized to the incoming power), which accounts for the coupled light at ω = ω0. From the fit of the experimental data with Eq. (2) [Fig. 3(a), red line], we obtain ηC ≈ 0.12 and Q ≈ 11500.
While Eqs. (1) and (3) are universal in the sense of describing generic Fano processes, an optical insight is still missing. In particular, a relation between the contrast efficiency and the coupling efficiency is worthy. In the following we will derive a model from coupled mode theory accounting for Fano resonances.
4.2. Optical coupling efficiency
Fan. et al. [19] have derived a formula of a Fano lineshape as a result of the interaction of a discrete optical resonance and a quasi-continuum of leaky modes. Here we consider a generalization of R(ω) in [19] in which we relax the hypothesis of mirror symmetry. Indeed, the presence of a substrate underneath the PhC membrane breaks the vertical symmetry. In addition, we analyze the general case in which the coupling efficiency can be less than unity. Our study is carried out in the framework of the Coupled Mode Theory (CMT) formalism. We first define E⃗0(r⃗) the optical injection mode. The coupling rates to the reflection and transmission ports are given by the spatial overlap of the resonantly scattered fields with E⃗0(r⃗). The transmission and reflection ports will be called mode-matched radiative ports. We consider coupling times τ1 and τ2 of the cavity field to reflection and transmission ports, respectively [see Fig. 3(b), inset]. The coupling time to mode-matched radiative ports is τc. The Q-factor is Q = ωτ/2, with τ defined as
where τ0 accounts for losses, i.e. radiative coupling to mismatched modes (τmis), and other absorption losses (τa). Moreover, it can be shown that (see Appendix): We define the asymmetry parameter . The coupling efficiency can be written as The magnitude η is the probability that a photon in the cavity is coupled to either the reflection or the transmission mode-matched ports. In the Appendix A.1 we derive the following expression for the reflectivity of a probe beam: where r′ and t′ are given in Eqs. (19), and r is the (field) reflectivity of the direct process. We will show at the end of this section that a mapping exists from parameters in the generic Fano model [Eq. (2)] to the parameters in the CMT model given by Eq. (7). Before we proceed, let us discuss analytical approximations of the optical coupling efficiency given in Eq. (6), and its connection to ηC. We first consider a system with mirror symmetry, f = 1. In this case, the positive root of Eq. (17) into Eqs. (19) gives r′ = r and . Equation (7) then yields In the limit η → 1 (i.e. τc = τ), Eq. (8) becomes R(ω) given in Ref. [19] for the symmetric solution of the CMT equations. A relation between these parameters and those of Section 4.1 can be found in the Appendix A.2. It can be easily shown that the extrema of Eq. (8), ω±, can be written in terms of q± as ω± = ω0 − q∓/τ, and q+ = −1/q−, where q± are given in Eqs. (20). Using Eq. (3) and Eqs. (20), the amplitude of the reflectivity variation close to resonance, ΔR = R(ω+) − R(ω−), can be then related to the coupling efficiency as: Note that the contrast efficiency of Section 4.1 is ηC = |ΔR|. Thus, Eq. (9) gives a general relation between ηC and η. First, we point out that the root q−, which can be either positive or negative depending on the sign of r, leads to F0 > 0, whereas q+ leads to F0 < 0. Since we assumed F0 > 0 in Sec. 4.1, we take q = q−. The relation between ηC and η can be now analytically studied in three regimes: i) for r → 0, q → −η/rt then |q| → ∞; Eq. (9) yields and the resonance is a Lorentzian peak; ii) for |r| → 1, q → −rt/(2 − η) → 0 and Eq. (9) gives , the resonance is a Lorentzian dip; iii) for |q| ≈ 1, Eq. (9) gives η → |ΔR/2rt|. These expressions show that ηC ≠ η in general. In particular, for strongly asymmetric lineshapes [case (iii)], closely approximates the optical coupling efficiency in terms of resonance contrast and reflection coefficients. Finally, we remark that the resonant efficiency ηRS fails to be accurate in the case (ii). In this regime (|r| → 1, q → 0), ηRS = F0q2 ∼ r2(1 − r2)η/(2 − η) → 0 regardless the coupling efficiency η.A fit of the experimental data using Eq. (8) [Fig. 3(b)] gives η ≈ 0.14 and Q ≈ 11160. The efficiency can also be approximated using Eq. (10): from ηC = 0.12 and |r| = 0.425 as obtained from the fit in Sec. 4.1, Eq. (10) gives η ≈ 0.156, in good agreement with the fitted efficiency. When comparing the generic Fano model with the CMT symmetric model, we observe that the quality of the fit of the former is slightly better compared to that of the latter. This is due to the fact that, for a given linewidth and background reflectivity, the CMT model with mirror symmetry has only one free parameter η, while the generic Fano model has two free parameters, F0 and q. The CMT model without symmetry restrictions [Eq. (7)] provides an additional degree of freedom contained in the asymmetry parameter f.
Parameters of the generic Fano model can indeed be mapped to the ones from the non-symmetric CMT model. In the Appendix A.2 we derive such a mapping from the generic Fano model [Eq. (2)] to the CMT model without symmetry restrictions [Eq. (7)]. From the fitted parameters of Eq. (2) [see caption of Fig. 3(a)], Eqs. (21) and (22) give t′ = 0.9638 and τ1 = 1.3472×10−10 s, respectively; these allow to compute the asymmetry parameter [Eq. (23)], f = 1.079, and the coupling efficiency [Eq. (24)], η = 0.15. The fact that the obtained asymmetry parameter is close to unity validates the analysis for a nearly symmetric system carried out before.
5. Discussion
In order to relate the coupling efficiency η obtained in the previous section to the far-field emission features of Section 3, let us first compute the following overlap integral in k-space:
where the E⃗cav(k⃗) is the far-field of the cavity mode, and the integration is performed over the surface S defined as the light cone (k = |k⃗| ≤ ω/c = 2π/λ). The overlap integral ηop can be related to the coupling time τc in the following way, where we have defined the cavity coupling time to radiative modes τrad as 1/τrad = 1/τc + 1/τmis, and τmis has been defined in Eq. (4). Equations (4), (7) and (12) lead to the following relations: Now, we estimate ηop by computing the overlap integral [Eq. (11)] from: i) the fully vectorial far-field of the simulated single cavity, and ii) approximating the far-field of the injected beam E⃗0(k⃗) as the radiation of an ideal dipole oriented in the x-direction, E⃗0(k⃗) = E⃗dip(k⃗) for k ≤ (2π/λ) × (N.A.), and E⃗0(k⃗) = 0 otherwise. As a result, we obtain ηop ≈ 0.2. Using Q = 11000 and η = 0.15 from the previous sections, and ηop = 0.2, Eqs. (13) give Qmis ≈ 18300, Qrad ≈ 14700 and Qa ≈ 44000. Taking into account that the theoretical Q-factor for the folded cavity is Q(theo) ≈ 44000, we obtain Qrad/Q(theo) ∼ 1/3, which is a typical factor coming from fabrication imperfections in our technological processes. We can now predict the degree of improvement of coupling efficiency when improving mode-matching. Indeed, replacing the N.A.=0.95 by a N.A.=0.5 microscope objective, a numerical evaluation of Eq. (11) gives ηop ∼ 0.6, and hence η = ηopQ/Qrad ∼ 0.75ηop ∼ 0.45. Therefore, the coupling efficiency (η), as obtained from the contrast of the Fano lineshape, is intrinsically limited to 75% due to the material absorption, and is lowered to 15% in our coupling conditions due to imperfect spatial mode-matching of the injection beam profile to the cavity mode.6. Conclusions
We have investigated the emission properties of L3 nanocavities optimized for efficient beaming perpendicular to the PhC periodicity. By implementing a far-field monitoring of the spatial emission profile, we have systematically compared optimized and standard L3 nanocavities and demonstrated the beaming effect, i.e. most of electromagnetic energy is directed along the sample normal within a 30°-cone. We further demonstrated laser operation with such an increased directionality. The laser power threshold is increased by a factor <2, which demonstrates the non-degradation of the nanocavity Q-factor. Such a redirection of nanolaser emission is crucial for several applications and particularly those needing the optimization of the coupling to standard optical fibers.
In many domains of applications of PhC, such as nonlinear optics, a key issue is to efficiently couple light into nanocavity modes. Thanks to the high quality beaming, we demonstrated a high reflection contrast at resonance. In order to study the coupling efficiency and the nanocavity Q-factor, we generalized the coupled mode theory approach developed in [19] to take into account both the absence of mirror symmetry due to the substrate and the presence of losses. Relaxing symmetry restrictions allows us to obtain a Fano resonance formula with an extra degree of freedom (the asymmetry parameter f), that can be mapped to a generic Fano-resonance model. The analysis of the experimental results in the framework of this model allows inferring a coupling efficiency of 15%, which is reduced both due to material absorption and imperfect mode matching. Our theory allows to predict coupling efficiencies as large as 50% by simply adjusting the numerical aperture of the microscope objective to the nanocavity mode.
A. Appendix
A.1. Derivation of Fano resonance from coupled mode theory
In the framework of the Coupled Mode Theory (CMT) formalism, we consider a cavity coupled to two nonidentical ports: the reflection port (port 1) and the transmission port (port 2) with coupling times τ1 and τ2, respectively. Due to the presence of a substrate, mirror symmetry considerations cannot be rigorously applied. We then extend the work of Fan et al. [19] for the case of a non-symmetric system. We begin our analysis by considering a conservative system, for which the cavity lifetime τ is equal to the coupling time τc, the latter related to τ1 and τ2 through energy conservation [see (Eq. (5))]. The scattering matrix is
where C is the scattering matrix of the direct process, and the output coupling coefficients are . As shown in [19], time reversal symmetry leads to the condition Now, considering a direct scattering process of the form the following expression can be obtained from Eqs. (14) to (16): where Δβ± = β1 − β2 and is herein called the asymmetry parameter. Note that for a system with mirror symmetry (f = 1), Δβ+ = 0 and Δβ− = π which correspond to symmetric and antisymmetric modes, respectively. The squared root in Eq. (17) is real for . We now assume that the optical injection is given by an incoming field coupled to port 1 (s1+) and then compute the reflected field, given by the outgoing field s1−, where we have definedThe reflectivity can be readily obtained as R = |s1−/s1+|2, and it is explicitly given in Eq. (7). Finally we introduce losses in the model. We simply add a loss cavity lifetime τ0 [Eq. (4)] accounting for radiative losses —i.e. mismatched modes— and absorption. Such a loss term leads to a coupling efficiency η = τ/τc which is less than unity in general.
A.2. Relation between generic Fano model and CMT Fano model
In comparing the generic Fano resonance model [Eq. (2)] with the Fano model from CMT with mirror symmetry [Eq. (8)], the following relations can be found:
For given reflectivity (r) and linewidth (Γ), the CMT model with mirror symmetry has only one free parameter, η. Since the generic Fano model —as applied to reflectivity signals— has two independent parameters, namely F0 and q, the parameters of the former can only be mapped to a subset of parameters of the latter. The inclusion of parameter f in the CMT equations when considering the general case without mirror symmetry introduces an additional degree of freedom in the CMT model.Indeed, a mapping exists from the generic Fano model to the CMT model without symmetry restrictions. Given F0 and q, the transmission T′ = t′2 can be found as the positive root of a second order polynomial,
The coupling time is subsequently found as where τ = 2/Γ, and . The asymmetry parameter then reads Finally, the coupling efficiency becomes Note that the sign of q determines the sing of r [with the convention F0 > 0, sgn(r) = −sgn(q)].Acknowledgments
These results are within the scope of C’Nano IdF and were supported by the Région Ile-de-France. C’Nano IdF is a CNRS, CEA, MESR and Région Ile-de-France Nanosciences Competence Center.
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16. Commercial FDTD software from Lumerical Solutions Inc. has been used for the 3D-FDTD simulations.
17. http://ab-initio.mit.edu/wiki/index.php/Harminv
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