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We propose a novel system of dual-wavelength micro-cavity based on the coupling between a photonic crystal membrane (PCM); operating at the Γ- point of the Brillouin zone, with a Fabry-Perot vertical cavity (FP). The optical coupling, which can be adjusted by the overlap between both optical modes, leads to the generation of two hybrid modes separated by a frequency difference which can be tuned using micro-opto-electromechanical structures. The proposed dual-wavelength micro-cavity is attractive for application where dual-mode behaviour is desirable as dual-lasing, frequency conversion. An analytical model, numerical (FDTD) and transfer matrix method investigations are presented.
©2011 Optical Society of America
1. Introduction
The photon mode confinement in photonic crystal membrane (PCM) at micrometer scale is of significant interest to control light-matter interaction [1–4] and to achieve different applications such as light-emitting diodes, photo-detectors, lasers [5,6] or frequency conversion devices [7,8]. Most of those applications are exploiting a band-gap defect or band edge photon mode confinement schemes to enhance photon-excitons interactions. Recently, an important extension of PCM applications using lossy modes above the light line has been proposed. The general aim of this proposal, so-called the 2.5D approach, is to exploit the third (vertical) direction by using a multi-layer to control the coupling between PCM resonant modes and free space modes [9]. Such an association can be used to achieve a variety of optical devices including filters, surface emitting lasers and all optical switches.
In this frame, we have proposed a new system of coupled optical micro-resonators to achieve a compact and tunable dual-wavelength device (section II). It is based on the association of a PCM resonator and a Fabry-Perot (FP) micro-cavity [10]. Indeed, it is well known that coupling between two tuned resonators leads to a splitting in two new resonant frequencies. Optical coupling has been demonstrated previously, theoretically and experimentally, with two vertical coupled cavities, leading to dual-wavelength lasing and second harmonic generation [7,8,11,12]. Although such applications have been reported successfully, the dual-λ micro-cavity proposed in this article will result in a more compact monolithic device with improved quantum efficiency, while offering new functionalities such as polarization control and tunability.
In this article, we investigate the coupling between a resonant PCM mode, at the Γ-point of the Brillouin zone, and a FP vertical mode. Optical coupling between both modes can be controlled by adjusting their field overlap. An analytical investigation using coupled mode theory (section III) provides design guidelines as well as the spectral properties of the micro-resonator. We finally report a design of a dual-wavelength micro-cavity, with resonance splitting in the THz range, using numerical 2D-FDTD [13] and transfer matrix method [9] simulations. For the former, infinite and finite lateral size dual-λ cavity simulations are presented.
2. Dual mode micro-cavity design
A schematic diagram of the proposed dual-λ cavity is shown in Fig. 1 . It consists of a PCM embedded in a vertical FP cavity. The PCM is a high index (nh) membrane with a periodic lateral pattern formed by a 2-D lattice-of holes or a 1-D lattice of slits surrounded by a lower index (nl) material.
Fig. 1 Schematic representations of a) the dual-λ cavity comprising a FP and a PCM membrane b) the dual-λ cavity system.
Let us first consider the PCM separately. Due to its finite height, we have to distinguish between photon modes below and above the so-called light line (or light cone). Full confinement in the wave-guiding slab membrane is achieved only for photon modes below the light line, while coupling with radiated modes is made possible for modes above the light line. The latter manifest themselves as resonant effect in the transmission and reflection of a beam propagating towards the PCM with an incident angle corresponding to the in-plane wave-vector of the PCM Bloch mode. It occurs, typically in a sub-wavelength grating when the first order diffracted mode corresponds to not only a freely propagating mode but also to a wave-guided mode. Both modes can therefore interfere as a function of wavelength and display diffraction efficiency variations [14]. The 2.5D micro-photonics concept aims at exploiting the third (« vertical ») direction by using a multi-layer approach to control such coupling between photonic lossy wave-guided modes and free space modes.
Let us now consider a PCM mode at frequency f0 located at the Γ-point of the Brillouin zone above the light line. Such pseudo-guided resonant Bloch mode can interact with optical beams, normal to the PCM. If a resonant “vertical” FP cavity mode is provided at the same frequency f0, optical coupling can occur between both modes, leading to a splitting into two “new” resonant frequencies at f1 and f2 as illustrated in Fig. 2 . The coupling strength depends on the overlap between the FP and PCM mode field profiles. This phenomenon can be seen as a photon-photon “strong coupling” interaction, by analogy with the well-known atom-photon or exciton-photon ones (Rabi oscillations) [2,3]. We draw the reader's attention to the fact that the dual-wavelength resonance is observable only at the strong coupling regime which will be later detailed in Section 3.
Such dual-λ micro-cavity can be realised by enclosing silicon (Si) or III-V hetero-structure (in prospect of developing active devices) patterned membrane within silica as low index medium, while the FP cavity mirrors could be composed by well known quarter-wave stack reflector (e.g Si/SiO2). In order to develop tunable dual-λ cavity, the lower index medium could be air gaps. In this latter case, electro-mechanical actuation can be used to tune the mode splitting Δf [9]. The proposed dual- micro-cavity is surface addressable, lending itself to direct butt coupling in optical fiber or by improved techniques based on the use of micro-lenses to enhance coupling efficiency. The dual-λ cavity can be also addressed in guided mode by a waveguide in the plane of the PCM.
3. Coupled mode theory modelling
A PCM can be accounted, under normal incidence, for a simple resonator embedded in the middle of a layer of thickness h0 (the PCM thickness) and optical index n0 (the “mean” refractive index of the PCM) [9]. The proposed dual-wavelength micro-cavity is therefore equivalent to a resonator inserted in a FP cavity as showed in Fig. 1(b). We assume the PCM, thus the resonator, supports only one mode in the frequency range of interest. Using coupled mode theory [15], we derive the Eq. of the resonant mode amplitude evolution Eq. (1):
where ω0 is the resonant frequency of the PCM mode, 1/τc its decay rate due to out of plane losses, 1/τ0 a time constant accounting for any other type of optical losses (absorption, lateral losses). C+i(C–i) is the incoming (outgoing) wave as defined in Fig. 1(b). K = √1/τc is the coupling factor and θ1(θ2) its dephasing associated with the forward (backward) propagating wave. Δθ = |θ1- θ2| is zero (π) for symmetric (anti-symmetric) PCM mode [9] with respect to the membrane middle horizontal plane. A0ejωt represents a monochromatic time dependent excitation source.In Eq. (1), we consider that the resonator (the PCM) exhibits only a zero order diffracted mode at normal incidence. Indeed high order diffracted modes from the PCM cannot couple with the FP cavity mode which resonance wavelength varies with the angle. Therefore, they will only contribute increasing undesirable losses. For that purpose, the PCM period should be always chosen smaller than the operating wavelength to assure a specular diffraction.
To go through the dual-λ micro-cavity’s intrinsic properties, 1/τ0 will be supposed negligible compared to 1/τc. To give a physical insight, and for the sake of simplicity, we start by considering r(λ), the reflection coefficient of the two mirrors enclosing the FP cavity, real, positive and constant. The PCM is located at the mid-thickness of the FP cavity (d1 = d2 = d). The relation between the in-coming (C+i) and out-going (C-i) fields at the PCM resonator is therefore simply given by Eq. (2) as their dephasing is caused only by the round-trip propagation of the wave through the layer of thickness di:
Then, from Eqs. (1) and (2) and for symmetric PCM mode, the output signal amplitude spectrum is found to take form: Figure 3 features a s(λ) normalized spectrum of λ0 and 1.5λ0 optically thick cavities and a PCM resonator of Qc = 5000 quality factor. It demonstrates that coupling occurs only when the FP cavity thickness is an integer number of the resonance wavelength (pλ0). In this case Ci and C-i constructively interfere at the centre of the cavity, resulting in an even FP mode profile with respect to the PCM middle plan. Conversely, for an anti-symmetric PCM mode scheme, we obtain dual-wavelength resonance only in the case of half-wavelength ((2p + 1/2)λ0) cavity. If we consider now a negative reflectivity, then the above observations are simply reversed between even and odd PCM mode schemes. These observations are consistent with the fact that the coupling strength is linked to the overlap integral between the FP and PCM field distributions.Fig. 3 Coupled mode theory: Calculated spectral response of the dual-mode cavity (Eq. (3)) for two different optical thicknesses d and Qc = ω0τc/2 = 5000.
An important feature of the dual-λ cavity is the splitting Δω. To determine it, let us consider a pλ0 optical length cavity and denote ω’ the new resonance frequencies. We assume that the photons lifetime in the resonator is much longer than their round- trip propagation time through the cavity FP alone, that is Qc = ω0τc/2>>pπ. We finally presume r≈1 and ε = (ω’- ω0)/ω0<<1. Hence, using the first order Taylor expansion of α, we derive from Eq. (3) the transmission maxima ω’(Eq. (4)). It shows that the transmission spectrum of the dual-λ cavity exhibits two symmetrically shifted peaks from ω0 as expected. It demonstrates besides that Δω depends on both the FP field intensity at the center of the cavity (through p parameter) and on the PCM quality factor (Qc).
It is worth noticing that the splitting Δω is only observable in the strong coupling regime i.e. when Δω/ω0 ≥ 1/QFP, where QFP = 2pπ/(1-R) is the FP cavity quality factor. Here R = r2 denotes the intensity reflection coefficient of the FP mirror. Therefore, mirrors with higher reflectivity are required for smaller targeted splitting. Hence for a given PCM of quality factor Qc and a dual-λ cavity system of pλ optical thickness (Fig. 1), there is a minimum reflectivity value R0 (Eq. (5)) for the mirrors (Fig. 1) to be met, below which a dual-λ output is impossible.At this step, we may distinguish two situations: Qc<QFP and Qc>QFP. In the first case, the splitting is still observable as the strong coupling regime condition (Eq. (5)) is always verified. In the second case, the strong coupling regime requires a minimum of reflectivity (R0) as defined in Eq. (5). For instance, for a cavity of optical thickness 2λ0 and Qc = 5.103, R0 is thus equal to ~0.929. Figure 4 shows the output signal amplitude spectrum derived from Eq. (3) for different values of R, where a dual mode spectrum is obtained only with R≥R0.Fig. 4 Calculated transmission spectra of s, given by Eq. (3) for different R values, for a 2λ0 thick cavity and Qc = ω0τc/2 = 5000.
For easy reading, we denote λ1 (resp: λ2) the lower resonance wavelength(resp: the upper). Under condition given by Eq. (5), λ1 and λ2, as well as their quality factor, variations are plotted in Fig. 5 as a function of the half-cavity thickness (d) and for Qc = 5.103. It shows that Δλ = |λ1- λ2| reaches a minimum, namely the anti-crossing point, for λ0 optically thick cavity.
Fig. 5 Hybrid modes resonance wavelengths (a) and quality factors (b) as a function of the half- cavity thickness, calculated with Eq. (3).
The quality factor of the hybrid modes is another essential feature. Far from the anti-crossing point, where the FP and PCM are detuned, the modes are purely FP and PCM. The PCM mode quality factor becomes therefore high as there is no FP mode to couple with. In our case, the Q factor reaches easily 20 times QFP with only 2% shift of the cavity thickness from its anti-crossing value (Fig. 5). This configuration is an interesting way to reduce the out of plane losses of a PCM modes operating above the light line and has been previously exploited for low threshold vertical emitting laser [5]. However, this increase of Q will be limited for real devices by in plane lateral losses. Notice that fabrication imperfections as interface roughnesses, the PCM sidewall deformations, can result in additional losses which can also affect the dual-λ cavity properties. Such additional losses can be taken into account in the model through the parameter τ0. They can destroy the dual-λ behavior in the worst case (1/2πτ0>Δf), otherwise, the Q factor of both modes will be limited compared to an ideal structure. Besides, it is a matter of technology to realize such dual-λ cavity with controlled geometrical parameters as the period, the filling factor, the thickness of the PCM and its position in the cavity that determine the coupling condition.
At the anti-crossing point, the two new resonant modes have the same quality factor which is twice QFP regardless of Qc. It indicates that the hybrid-modes are equally shared between the FP and the PCM parent modes. Photons need then twice more time to escape from the cavity compared to the FP cavity only.
4. Numerical validation
In this section, we focus on the design of device operating in the infrared region around 1.55 µm. We report here simulations of an optical passive structure using FDTD calculation and a phenomenological approach method developed in the reference [9], combining transfer matrix and coupled mode theory (TMM-CMT) analysis.
For numerical 2D-FDTD simulation, in order to determine the spectral properties (reflection and transmission), a plane wave time pulse is launched in the vertical direction, normal to the interface between the PCM and its environment. Two monitors record, along simulation time, the powers which are reflected and transmitted in the vertical direction and fast Fourier transforms are calculated at the simulation end. We first consider lateral infinite PCM structure with periodic boundary conditions along the horizontal axis, while perfectly matched layer (PML) absorbing media, of thickness 0.5 µm and reflectivity of 10−8, are added at the top and bottom of the simulation domain along the vertical direction.
The PCM consists of a 1-D lattice of air slits drilled in a high index membrane (nh = 3.17) enclosed in a lower index material (nl = 1.45) corresponding respectively to an InP membrane and a silica cladding medium. The PCM thickness is h = 0.57µm, its period 0.95µm and the filling factor 50%. Figure 6 illustrates the wavelength-dependent reflectance feature for TE polarization (that is, the E-field vector is along the slits axis). The resonance wavelength is λ0 = 1541.6 nm and the mode profile is symmetric with respect to the PCM middle plane (Fig. 6(b)). Using TMM-CMT method, we determine Qc as well as the mean refractive optical index of the PCM. We obtain Qc~4.104 with a mean refractive index n0~2.71. The PCM is thus a wavelength optically thick membrane. To fulfill the condition given by Eq. (5), we design a Bragg reflector composed of 4 quarter-wave stacks of n1 = 1.45 and n2 = 3.45 refractive index layers. For a plane wave at normal incidence, the intensity reflectivity is higher than 0.99 in the range [1.35µm - 1.97µm].
Fig. 6 a) PCM Reflectivity spectrum and b) the resonant mode field profile (the black line encloses InP region, the dot lines mark out the air slit zones).
The next step is to find the cavity thickness to adjust the coupling between the FP and the PCM modes. At this step, TMM-CMT method is faster and less memory consuming for exploratory design work compared to the numerical method.
We consider that the low index media 1&2 (Fig. 1), named spacer, consist in silica. To get maximum coupling, the spacer optical thickness is quarter-wavelength. Figure 7 shows the new resonant frequencies and their quality factor evolution as a function of the spacer thickness using TMM-CMT, where d0 denotes the spacer thickness at the anti-crossing point. The splitting is in the range [0.4THz, 1.2THz]. The minimum frequency difference (Δf) is obtained for 2.5 λ0 optically thick cavity at the anti-crossing point, with two modes (λ1 = 1539.6 nm and λ2 = 1542.9 nm) of same spectral purity (Q~8000).
Fig. 7 Hybrid modes resonance wavelengths (a) and their quality factor (b) evolution as function of the spacer thickness. Comparison between numerical 2D-FDTD and TMM-CMT methods.
In order to validate this simple TMM-CMT mode result, we perform a 2-D FDTD simulation of a lateral infinite dual-λ cavity. The FDTD simulation method is almost the same as described above for the PCM simulation. The new resonances and their quality factors evolution as a function of the spacer thickness are reported and compared to TMM-CMT results in Fig. 7. Both simulation methods show excellent agreement with a difference of 10% of silica spacer thickness (d0) at the anti-crossing point, which is due to the approximations of the TMM-CMT method. Let us point out that the variations in FDTD simulation come from numerical fluctuations when computing the resonance wavelength and the corresponding Q factor. They mainly appear for high Q resonant modes because of a limited spectral resolution.
The hybrid mode profiles, for d = d0, are shown in Fig. 8 . They are clearly linear combinations of PCM and FP modes.
Fig. 8 The TE polarisation electric field maps of dual micro-cavity modes (a: λ1 = 1.5396 µm, b:λ2 = 1.5429 µm).
Let us now consider the case of a realistic device with limited lateral dimensions. In this case, 1/τ0 (see section II) accounts for lateral losses. These additional losses can result in the dual-mode behavior destruction in the worst case or simply limit the Q factor of both modes compared to an infinite ideal structure. We perform numerical (FDTD) simulation for ~21 µm lateral sized structures. The resonant wavelengths and their quality factors evolution are plotted in Fig. 9 , as a function of the spacer thickness (d). The minimum splitting is about 0.6 THz and the quality factor of both modes is about Q~6000, at the anti-crossing point. We observe that the quality factor, for both new resonance frequencies, decreases due to lateral losses compared to previous lateral infinite dual-λ cavity simulations. However, they appear more important on the right of the anti-crossing point. For instance, the lateral losses are three times higher at B than at A; two symmetrical points with respect to the anti-crossing position (Fig. 9). As the drop of the Q factors is only due to lateral releases of photons, this difference should be related to higher in-plane kinetics of photons at point B than at point A.
Fig. 9 Hybrid modes (a) and the difference frequency (b) evolution as function of the Half-cavity thickness for 21µm lateral sized dual-mode cavity.
To check this point, we calculate the band diagram (ω(k)) of the dual-λ cavity using plane wave expansion (PWE) method. In order to model a single dual-λ cavity, we impose an “artificial” periodicity in the vertical direction. The structure is thus composed of periodic sequence of dual-λ cavities, separated by a sufficient amount of background (air) to avoid coupling between vertical adjacent structures, hence insuring modes localization in a single cavity. However, notice that this approach induces unfortunately not only bands folding but also the onset, in the band diagram, of some spurious modes, due to the artificial vertical periodicity. However, considerations about the symmetry and/or the field confinement allows for distinguishing the “real” modes of the dual-λ cavity.
Figure 10 shows the dispersion characteristic ω(k) of the PCM mode for parameters corresponding to A and B points, where the in-plane momentum component k// is linked to the PCM periodicity (a) along the horizontal axis. Note that the dispersion relation is plotted in a k// range which is coherent with the lateral size of our cavity. The kinetics of photons is given by the group velocity vg (vg = ∂ω/∂k), which illustrates how the energy flows along the propagation direction. The larger curvature of the PCM mode dispersion characteristic curvature at point B, unlike at point A, results therefore in higher lateral escape rate of photons at the former point.
Fig. 10 Dispersion characteristic of the PCM mode of the dual-λ cavity at point A (left) and B (right).
5. Conclusion
We have proposed an original system of surface addressable dual-wavelength optical resonator with adjustable resonant frequencies based on the coupling between a PCM wave-guided mode and a FP cavity mode. An analytical investigation of the coupling between both optical modes is presented, using coupled mode theory analysis. It appears that the dual mode behavior is related not only to the overlap between mode field profiles of individual component but also requires a minimum reflectivity of mirror enclosing the cavity. The hybrid modes resulting from this coupling have been found to be a linear combination of the FP and PCM modes and exhibit the same quality factor at the anti-crossing. A quasi analytical model based on TMM and CMT is shown to provide a complete understanding of the optical behavior, and is confirmed by ab initio FDTD calculations.
The proposed dual-λ resonant structures is very generic: it can be implemented in a variety of devices and applications where dual-mode resonance is desirable such as dual-λ lasing, second harmonic generation, frequency conversion or optical signal regeneration and multi-function (e.g. photo-detection/emission) systems.
Acknowledgments
This work is supported by the French National Research Agency (ANR) through the Nanotechnology program (BASTET project).
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