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A single-line-defect low-loss photonic crystal waveguide based on a perforated GaAs membrane in an aluminium-free material system is demonstrated. The GaInP lattice is matched to GaAs as the cladding/sacrificial layer. Fabry-Perot resonances are analyzed to obtain the group velocity dispersion for a 1-mm long guide. The losses are deduced to be close to 5 dB/cm, taking into account the wavelength dependent reflectivity of the guide extremities. In this framework, side-coupled nanocavities are also investigated. Feasibility of low-loss photonic-crystal-based devices combined with a reliable industrial material systems is thus demonstrated.
©2006 Optical Society of America
1. Introduction
The recent years have witnessed huge advances in the performances of photonic crystal (PhC) waveguides and cavities based on perforated membranes. The so-called “W1” waveguide, a single line defect in a triangular lattice PhC array, is a model system benchmarked by several authors [1–4] in its basic form or through various geometrical variants. Lowest propagation losses are in the dB/cm range [4], considering operation under the light line to inhibit Bloch mode leakage.
The ultimate limits of the performances of these waveguides are dictated by imperfections, mostly details of the hole shape such as verticality and micro-roughness of walls and diameter fluctuations. E-beam lithography generally guarantees a nm-scale accuracy of the hole center location, making this latter issue manageable. The scope of this paper is to propose a method to determine the propagation loss and the facet reflectivities of a long PhC waveguide as well as the Q factors of a side-coupled cavity, by analyzing the Fabry-Perot type fringes that arise at the cleaved ends of the waveguide. The idea, as detailed below, is that the fringe spacing first provides the group index ng at the local frequency. Then, by exploiting an accurate computation of the facet reflectivity, the propagation losses can be deduced from the contrast.
A search for optimal operation implies the selection of a proper material. As far as optoelectronic systems are concerned, III-V materials offer a superior potential for further functions requiring emission or detection or tuning through carrier injection. While GaAs is a reliable semiconductor, the traditional AlGaAs lattice-matched system has the disadvantage of being too prone to Al oxidation. We recently developed a membrane technology on GaInP lattice-matched to GaAs. This technology includes e-beam lithography, high quality ICP etching, and highly selective dissolution. The GaInP/GaAs system, due to its large compositional difference between anions, naturally offers a high etching selectivity. Additionally, GaInP is not subject to oxidation.
2. Fabrication of the PhC structure
Electron-beam lithography on a Leica EBPG 5000+ machine with a 2.5-nm grid using positive resist followed by Reactive Ion Etching (RIE) was performed in order to define a hexagonal lattice of holes in a silica hard mask. The lattice period a was 418 nm, with hole radius r = 0.287 a. An Al-free heterostructure consisting of a 2-μm thick sacrificial bottom layer of GaInP (lattice-matched to GaAs), followed by a 300-nm thick layer (0.72 a) of GaAs, was used as the starting material. Transfer of the patterns into the semiconductor was achieved by high-density plasma etching (ICP) with a method similar to the one described in [5]. The underlying sacrificial layer was removed by HCl-based selective wet etching. The resulting structure is an air-clad, 300-nm thick GaAs suspended membrane, as shown in Fig. 1.
3. Waveguides
1-mm long straight-line-defect PhC waveguides composed of a single missing row of holes (W1) along the ΓK direction were obtained by cleaving both sides to enable end-fire coupling (Fig. 1). The surface quality of both the membrane and the cleaved facets is remarkable. The proper choice of geometric parameters leads to a single-mode transmission window ranging between 1525 and 1638 nm, below which the transmission drops drastically due to the intrinsic optical leakage of modes situated in the light cone, and the upper limit corresponding to the cut-off of the W1 waveguided mode, as shown in Fig. 2(b).
Fig. 1. Scanning electron microscope (SEM) images of W1 waveguide cleaved facets. Top (a) and bottom-surface (b). Holes cylindrical shapes are seen by transparency through the GaAs membrane and exhibit very vertical and smooth surfaces (c). The radius was measured by SEM to be equal to r = 0.2895 a.
Cleavage takes place preferably along the hole rows. This fact, in addition to the crystalline aspect of GaAs, leads to a very reproducible facet cleaving process, which is a convenient asset for the Fabry-Perot method described here.
Fig. 2. (a) Transmission spectrum of the W1 waveguide; (b) associated dispersion diagram scaled to (a); (c) detail of the transmission spectrum with fully resolved Fabry-Perot fringes.
3.1. Measurement set-up
The setup is based on an end-fire method with lock-in detection. The output of a narrow linewidth (Δv < 1 pm) tuneable diode laser (Tunics MC), with controlled TE polarisation, is modulated by a chopper and then end-fire coupled into the PhC waveguide using a ×63, 0.95 N.A. microscope objective. A ×40, 0.65 N.A. objective is employed to collect the transmitted signal onto a photodiode. Its photocurrent is fed to the lock-in amplifier. The wavelength is swept in steps of 5-10 pm between 1500 and 1630 nm, resulting in a transmission spectrum like that shown in Fig. 2(a).
We readily observe a sharp change in the transmission regime below and above the light cone at 1525 nm. The mode cut-off frequency cannot be determined because of the limited wavelength range of the laser. Figure 2(c) shows neat and evenly spaced fringes indicating a pure single-mode behaviour.
3.2. Analysis of FP fringes
Several techniques are employed to obtain information about the group index, such as time resolved near field imaging [6], phase-shift [7, 8], interference [9], time-of-flight [10], and Fabry-Perot fringe analysis [1]. This latter method requires a simpler experimental set-up, however, data analysis is more involved.
In the view of deducing the effective group index as a function of the wavelength, the transmission spectrum was analysed by local Fourier transforms in a window of a certain spectral width, i. e. the number of fringes per wavelength unit was determined. This window has been slid across the spectrum. The result can be seen in Fig. 3. Furthermore, the spectral width of the window was adapted (corresponding to the different colours) in order to fit correctly throughout the transmission range. This is also an explicit way to compensate for numerical truncation effects when tracking fringe spacing variations. Detailed analysis reveals that the actual dispersion curve is given by the lower envelope of the set of resulting points.
Fig. 3. Effective group index as a function of the wavelength; measurement (colours), PWE simulation (black).
The minimum value of the group index is 4.5, a value substantially larger than the 1550 nm GaAs bulk group index (~ 3.6), in spite of the large overlap of the waveguide mode with air. In our case, this aspect is related to the particular nature of the photon propagation in such structures, that is coherent reflection rather than total internal reflection. A group velocity of vg = c/11 in the 1-mm long guide can be obtained. The measured dispersion curve is in very good agreement with a 3D calculation using the MPB code [11] (dashed black line).
3.3. Reflectivity and loss analysis
The examination of the fringe contrast C = (T max - T min)/(T max + T min), where T max/min are adjacent maxima/minima in the transmission spectrum, is a common and convenient way to determine the losses, once the sample length and facet reflectivities are known. For this, we determine the quantity R˜ [12], where
The local fringe contrast is extracted using a method of sliding windows as described above. The propagation losses, α, are then found through the expression
Here, L is the sample length. R is the reflectivity of the waveguide facets, which has to be determined either using the cutback method (measuring the transmission of identical waveguides of different lengths) or, as in our case, through 3D FDTD calculations. The common belief is that coupling to low-group velocity modes is usually poor, and correspondingly, both reflectivity and loss increase when approaching high group index regions [7, 13]. In an analysis that we shall publish elsewhere we detail 3D FDTD calculations of the reflectivity.
In Fig. 4, we compare the waveguide facet reflectivity R obtained by calculation to the quantity RϜ, which has been measured in dependence of the wavelength and which is a function of both the waveguide loss and the reflectivity R. The distance between the two curves is a measure for the propagation loss, as indicated by Eq. (2).
Fig. 4. Comparison between the calculated reflectivity R (solid line) and the parameter RϜ as a function of the wavelength and the frequency.
Figure 5 shows the propagation losses α in the waveguide as a function of the excitation wavelength as they can be deduced from the calculated reflectivity and the measured data of Fig. 4. The minimum value is situated at about (4.5 ± 1) dB/cm at λ = 1570 nm. This loss level corresponds to worldwide state-of-the-art values [4].
Approaching longer wavelengths and thus regions of low group velocity, the propagation loss shows the well-known dependence on vg. Physically, the mechanisms induced by imperfections, mostly deviations from the perfect hole shape like diameter fluctuations and micro-roughness, lead to either out of plane scattering (in air) or backscattering [14]. This fact has important consequences when it comes to make good use of “slow light”, in the flatter region of the dispersion near the band edge (k > 0.4 π/a). Nanosecond-range management of delay and dispersion are indeed highly desirable features in modern devices [7]. More generally, there are several identified benefits to the enhanced light-matter interaction arising from the low group velocity, e.g. for modulation and nonlinearities [15, 16].
4. Cavities
For cavity studies, waveguides were widened by 0.1√3 a and donor-type cavities based on aligned missing holes were added with a separation of N rows from the waveguide (N = 5). Figure 6 shows a micrograph of a particular realisation. As in Noda’s design [17], the outer holes are slightly shifted off the cavity center (by δx = 0.15 a) to provide larger reflectivity accompanied by a “gentler” field confinement [17].
4.1. Measurement
In order to observe the cavity resonance, we collected its vertically emitted light by an objective coupled to either a photodiode or an InGaAs focal array. The movie in Fig. 7 displays the cavity lighting at ω 0 = 1625.9 nm as well as the photodiode signal when upsweeping the laser wavelength. The resonance has a FWHM of 55 pm, hence a directly measured quality factor of Q dm ≈ 30 000 is found, as indicated in Fig. 8(a). A zoom on the resonant region for both the vertically collected signal and the transmitted intensity is depicted in Figs. 8(b) and 8(c). The transmission response exhibits the resultant dip on the rising edge of a FP resonance.
Fig. 7. Movie of the cavity lighting at resonance (a); corresponding waveguide transmission (b) and cavity vertical emission (c) (2.2 MB).
Fig. 8. Spectra in the vicinity of the cavity resonance. (a) Directly measured (dm) quality factor; fitted curve (red) to the experimental data (blue), both for the vertical irradiance of the cavity (b) and the waveguide transmission (c).
4.2. Extraction of the parameters
We have worked out as comprehensively as possible the system parameters from these spectra: the coupling quality factor, Q in, the intrinsic one, Q v, and the facet reflectivities, r. This task has never been explicitly done to the best of our knowledge. It is interesting to see by the case we investigated in some depth that a very good fit can be obtained not only on the cavity signal but also on the waveguide transmission, as shown in Figs. 8(b) and 8(c). Assessing accurate cavity quality factors is a delicate task. Even with the fits shown, an uncertainty of about ± 5% remains on the actual bare (intrinsic) cavity Q v. It is also clear that the exact reflectivity value of the waveguide facets plays a role on all spectra when performing the fit and induces asymmetric behaviours of the transmission spectra since it modifies the extraction and the coupling between the waveguide and the cavity. We therefore believe that there is a genuine added value in outlining the details of the current study, that becomes the more relevant in the present state of PhC research: high Q cavity designs are now well mastered, but the exploration of limiting factors, either for simple cavity-guide systems or more complex systems, will increasingly demand a precise evaluation of Q values and their evolution.
As mentioned above, the particular appearance of the transmission function at the cavity resonance in Fig. 8(b) suggests that the coupling mechanism between the waveguide and the cavity is quite complex. This is attributed to the intensity variations induced by the Fabry-Perot resonator constituted by the waveguide's end facets. Therefore, the irradiance extracted by the cavity is also varying and it is mandatory to consider the spectral position of the resonance with respect to the fringe system since this will alter its spectral emission form [18]. A model using the T-matrix formalism, which describes the two resonators simultaneously, is used to account for this [19]. For an incident wave at frequency ω, we define the matrices describing the cavity coupled to the waveguide, T c, the propagation in the waveguide of length l, T p, and the reflecting waveguide facets, T r, as follows:
where γaccounts for the waveguide-cavity-coupling, Γ0 is the intrinsic linewidth of the cavity, φ is the additional phase that the guided wave acquires while travelling from one waveguide facet to the cavity, and t = (1 - r 2)1/2. The total transfer matrix is then determined by T tot = T r·T p·T c·T p·T r.The transmission of the coupled system, T, and the cavity vertical emission, E, are found through the relations T = 1/|t 22|2 and E = 1 - T - R, with R = |t 12/t 22|2 and the matrix elements 4. We are then able, through the error minimization between measured and simulated signals, to extract the quality factors of the cavity correctly. These are given by Q v = ω 0/Γ0 and Q in = ω 0/γ. The loaded quality factor given by the FWHM of the fit curve (51 pm) is Q load ≈ 32 000 [Fig. 8(b)]. This is actually in a good agreement with the measured value in Fig. 8(a). Simultaneously, we found Q in ≈ 400 000 and Q v ≈ 44 000 for the coupling between waveguide and cavity (without facets contribution) and the intrinsic quality factors, respectively. These values are quite close to the ones obtained using 3D FDTD calculations.
5. Conclusion
In this paper, we demonstrated the utilization of the Fabry-Perot method to accurately evaluate waveguide and cavity performances by detailed analysis of both their transmission and vertical emission spectra. Our side-coupled waveguide-cavity system is based on an Al-free PhC-membrane structure. By analyzing the spacing and the contrast of the Fabry-Perot fringes combined with the modelling of the facet reflectivity, both the group index and the propagation loss could be deduced. In the slow light regime, a group velocity of c/11 has been obtained. Furthermore, a minimum loss value of 4.5 dB/cm has been measured in our 1-mm long guides, which is comparable to the worldwide state-of-the-art value. The utilization of an S-matrix formalism describing the guide-cavity system allows us to correctly take into account the contribution of the facet reflectivity in the calculation of the Q factors of the cavity, which are in a good agreement both with the measured and the 3D FDTD calculated ones.
Acknowledgments
E. W. acknowledges the allocation of the EU Descartes Prize, given to the IST-QuComm Project in 2004, as a postdoctoral grant.
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