Propagation losses in photonic crystal waveguides: Effects of band tail absorption and waveguide dispersion

: Propagation losses in GaAs-based photonic crystal (PhC) waveguides are evaluated near the semiconductor band-edge by measuring the finesse of corresponding L n cavities. This approach yields simultaneously the propagation losses and the mode reflectivity at the terminations of the cavities. We demonstrate that the propagation losses are dominated by band tail absorption for shorter wavelengths and by fabrication disorder related scattering, near the photonic band edge, for longer wavelengths. Strategies for minimizing losses in such elongated cavities and waveguides are discussed, which is important for the monolithic integration of light sources with such optical elements. indexes up to 30 are observed. At energies below 1.286eV, the irregular values of the group indexes are a sign of Anderson localized modes which were not included in our analysis.


Introduction
Analyzing the sources of propagation loss in semiconductor nano-photonic waveguides is important for constructing compact optical elements, with applications in integrated quantum photonics. In such applications, single photons and other non-classical states of light are generated and routed on-chip, and the selection of proper waveguiding schemes is crucial for avoiding excessive photon loss. Whereas PhC defect cavities and waveguides can provide strong optical confinement, dispersion engineering and tailored light-matter interaction that may prove crucial for on chip quantum optic devices [1,2], they may introduce high scattering losses due to the fabricated surfaces defining the PhC [3]. Alternatively, ridge waveguides have been employed to achieve lower scattering and bend losses, for longer-haul on-chip light propagation [4]. For all semiconductor waveguide types, residual, below-gap optical absorption might be particularly detrimental when light sources such as semiconductor quantum dots (QDs), made of related heterostructures, are integrated on-chip.
Strategies for minimizing propagation losses in semiconductor PhC waveguides and associated devices should consider not only inherent optical material absorption but also the impact of fabrication induced disorder and waveguide dispersion effects [3]. Experimental values of the propagation losses, extracted using different techniques, have been reported for various materials, structures and wavelengths. In Si-based PhC waveguides, propagation losses as low as 4dB/cm at 1.5µm wavelength were measured [5], and the quadratic increase in losses with increasing group index n g due to slow light effects near the photonic bandedge was observed [6]. In GaAs PhC waveguides, propagation losses as low as 0.2 and 1.5dB/cm were reported for multimode W7 and W3 waveguides, respectively, at 1.5µm wavelength [7]. Typically, direct transmission measurements are used for extracting the propagation loss. Higher losses of 5-60dB/mm were reported for GaAs PhC W1 waveguides around 900nm wavelength by measuring the finesse of PhC cavities of different lengths [8].
In this work we analyze the wavelength dependence of the propagation losses in GaAsbased PhC waveguides by measuring the finesse of PhC L n cavities of increasing length. The propagation losses are extracted for the wavelength range of 900-960nm, compatible with the emission wavelengths of InGaAs/GaAs QDs that can be integrated with these PhC structures. Our method yields simultaneously the mode reflectivity at the edge of the cavities, which needs to be precisely determined for accurate measurement of the propagation losses. We measured losses as low as 17dB/mm at ~910-940nm, propagation losses that are acceptable for routing single photons across distances up to ~100µm. Moreover, we show that the increased losses at shorter wavelengths due to band-tail absorption and at longer wavelengths due to waveguide dispersion result in optimal wavelengths for which propagation losses are minimized.

Structure design and fabrication
The structures used in this study were L n PhC membrane cavities [9] incorporating (n-1)/2 pyramidal QDs [10] with n = 3,7,17,33 and 61 [11] (9 nominally identical structures for each length). The QD-PhC structures were fabricated on a GaAs/Al 0.7 Ga 0.3 As membrane wafer grown by molecular beam epitaxy on a (111)B GaAs substrate misoriented by 3° towards [211] . The site-controlled InGaAs/GaAs QDs were grown via metalorganic vapour phase epitaxy [12] over an array of inverted pyramids previously defined using electron beam lithography (EBL). The PhC pattern was aligned over the QDs using EBL with a ~20nm precision and etched in an inductive coupled plasma (ICP) system [13]. The 250nm thick GaAs membrane was suspended by wet etching of the sacrificial Al 0.7 Ga 0.3 As layer. The PhC structures were designed such that their 1D photonic band overlaps the QD emission spectra [7], with a bandedge at 966nm. The hole pattern was positioned on a triangular lattice with pitch a = 225nm and 60nm radius. The QDs were distributed uniformly along the cavities at distances of 0.45µm. At each end of these cavities, three holes were shifted outwards along x by 0.23a, 0.15a and 0.048a. This cavity design improves the theoretical M 0 mode Q-factor of an L 3 cavity up to 200 000 [14]. As an example, Fig. 1 shows the design for the L 33 cavity employed. To ensure a good spatial overlap of the QDs with cavity mode field patterns, all QDs were placed at the maxima of the in-plane electric field of the waveguide Bloch mode.

Finesse of Ln cavities
Propagation losses in optical waveguides can be extracted from measurements of the finesse of cavities formed by terminating the waveguides with reflectors to form a Fabry-Pérot (FP) resonator [15]. The finesse F of such cavities is related to the mirror reflectivity R (assumed identical at both terminations) and the distributed propagation loss coefficient α p by [16]: where L is the cavity length, Δλ is the free spectral range, δλ is the cavity mode spectral width, and Q is the cavity quality factor. In our case, the reflectivity is high (R≈1) the propagation losses low (Lα p <<1), and thus the inverse of F varies approximately linearly with L: We hence measured the finesse of the FP modes observed in the different L n cavities versus wavelength, and used expression (1) to extract the loss and reflectivity parameters. We note that when the propagation losses are comparable to the mirror losses per cavity length (i.e., when Lα p~l n(1/R)), R needs to be determined accurately enough in order to minimize the error in the extracted propagation loss [17]. The FP modes in the L n cavities were identified and characterized by measuring the low temperature photoluminescence (PL) spectra of the structures. The QDs were excited with a diffraction limited 1.5µm excitation spot a relatively high power (500µW), such that a large number of FP modes were excited by emission of the dots and their barriers [18]. Spectra measured with a 70µeV resolution for representative cavities of different lengths are shown in Fig. 2(a). On the background of the broadband emission of the highly excited QDs, the FP cavity modes are clearly visible. The measured Q-factors Q = λ/δλ, where δλ is the mode linewidth, were extracted with the aid of Lorentzian fits of the observed mode lineshapes, are displayed in Fig. 2(b). The measured FP mode spacing Δλ in the L 61 cavities yields the group index n g = λ 2 /(2LΔλ) from which the finesse of the FP mode is calculated as F = λQ/(2Ln g ), as shown in Fig. 2(c). The measured group index of one cavity is shown on Fig. 2(d) alongside the theoretical group index computed from 3D FDTD using the software meep [19]. Group indexes up to 30 are observed. At energies below 1.286eV, the irregular values of the group indexes are a sign of Anderson localized modes which were not included in our analysis.

Analysis of propagation losses
To obtain the propagation loss coefficient as a function of wavelength, the finesse data of Fig.  2(c) were aggregated in 5nm-wide wavelength bins, and the wavelength dependence of 1/F was fitted with expression (1). Notice that the fit is very close to the linear function of L predicted by expression (2) indicating that the waveguides are in the low cavity loss regime as shown on Fig. 3(a). We restricted our analysis to the 900-960nm range, far enough from the photonic band edge in order to avoid localized modes, which cannot be described by this Fabry-Pérot model. The extracted propagation loss coefficients α p and the transmission parameter 1-R are displayed versus wavelength in Fig. 3(b). The loss coefficient slowly decreases from 6mm −1 (26 dB/mm) at 910nm to 4mm −1 (17dB/mm) at 940nm, then increases more sharply to 17 mm −1 (74dB/mm) at 950-960nm, closer to the photonic band edge. This increase at longer wavelength is concomitant with the increase in the group index n g [4], suggesting effects of slow light on the propagation losses. The reflectivity parameter R varies between 98.2 and 99.6% in the 900-950nm wavelength range; these high reflectivities are consistent with computations of reflectivity for related cavity structures [20]. The propagation loss should increase towards the photonic band gap due to the higher group index n g . In an attempt to uncover dispersion effects beyond a simple linear dependence on n g , we plot in Fig. 3(c) the normalized propagation loss coefficient α p /n g versus wavelength. Clearly, the loss coefficient varies more rapidly than n g at longer wavelengths. A similar increase [6] obtained via transmission measurements was attributed to backscattering and modelled as a quadratic n g term [6]. On the shorter-wavelength side, the observed increasing loss can be explained by exponential absorption tails (e.g., Urbach tails due to lattice disorder [21]). We thus model the dependence of the propagation coefficient on photon energy E by: where the first term is the absorption component and the second one represents the scattering component. Here, E bg is the energy of the PhC band edge and E a is a characteristic energy of the absorption decay. The fitted parameters (see fit in Fig. 3(c)) are: α 1 = 16 ± 0.12mm −1 , α 2 = 0.04 ± 0.01mm −1 , and E a = 55 ± 21meV. The variation of the two terms of (3), using the fit parameters, are also shown in Fig. 3(c). Two processes contributing to the propagation loss are thus distinguished in this model: absorption losses dominating at higher photon energy near the semiconductor bandedge, and scattering processes dominating at lower energy near the photonic bandedge. The propagation losses could be further decreased at longer wavelengths by shifting the photonic bandedge to lower photon energies via proper PhC designs.
Uncovering the wavelength dependence of the different loss mechanisms in these PhC waveguides also provides insight into strategies for increasing Q-factors in PhC cavities, which is important for achieving high Purcell factors and strong coupling in QD-PhC integrated structures. Estimation of the energy-decay of the band tail absorption is useful for determining the red shift of the emitter wavelength needed for reducing the absorption effects. Besides suggesting the proper red-shift in photonic band edge for minimizing the slow light effects, the scattering contribution indicates the possibility of further increase in Q-factors by selecting higher order (blue shifted) cavity modes in order to stay away from the photonic band edge.

Conclusion
In summary we evaluated the wavelength dependence of the propagation loss and edge reflectivity in GaAs-based PhC waveguides of finite lengths. We measured propagation losses increasing from 17 to 26dB/mm in the 900-950 wavelength range to 74dB/mm in the 950-960nm range, and waveguide edge reflectivities R~98.4-99.6% in the 908-950nm wavelength range. Moreover, we showed how the increased losses at shorter wavelengths due to band-tail absorption and at longer wavelengths due to scattering result in optimal wavelengths for which propagation losses are minimized. In particular, we demonstrated that these propagation losses are quadratic in n g near the photonic band edge. These results are of interest for the design and optimization of integrated nanophotonic devices, e.g., semiconductor QDs embedded in PhC waveguides and cavities, in which propagation losses need to be minimized and cavity Q factors be maximized.

Funding
Swiss National Science Foundation.