Photonics and Nanostructures - Fundamentals and Applications
Far-field emission profiles from L3 photonic crystal cavity modes
Highlights
► The experimental characterization of the far-field emission profiles of the modes of a L3 photonic crystal cavity. ► A detailed study of the effect of disorder and fabrication imperfections on the far-field profiles. ► A simple Fabry–Perot model which allows us to get a clear physical picture of the link between relevant features of the far-field profiles and parameters of the near-fields.
Introduction
Photonic crystals (PhC) offer unprecedented control over electromagnetic field confinement in all three spatial directions [1]. In particular, two-dimensional PhC nanocavities in a planar waveguide have already found applications in different fields such as nanolasers, nonlinear optics and quantum information processing [2], [3]. Similar to any electromagnetic resonator, PhC nanocavity modes are essentially characterized by two figures of merit: the cavity quality factor, Q, and the effective confinement volume of each mode, Vmode [4]. The quality factor is proportional to the photon lifetime in the cavity which depends on the cavity losses to the external world. The mode volume is a quantitative measure of the spatial confinement of the electromagnetic mode. In most applications, it is crucial to maximize the Q/Vmode ratio. For example, the Purcell factor, which measures the enhancement of the spontaneous emission rates for atoms resonant with a cavity is directly proportional to this figure of merit. In PhC nanocavities the mode is strongly confined to a very small volume, on the order of (λ/n)3, where λ is the mode wavelength. In a planar membrane nanocavity, in-plane confinement is provided by spatial localization of a structural defect in a perfectly periodic PhC with a photonic band-gap, while out-of-plane confinement is given by total internal reflection between the slab and the air cladding (assuming a suspended membrane as a planar waveguide). Very high quality factors, in the range 104–106 [5], [6], [7] have been demonstrated in the literature. In particular, the L3-type cavity, consisting of three missing holes in a triangular lattice, was the first PhC cavity to show quality factors larger than 104 [5].
The spectral mode structure for L3 cavities has been thoroughly investigated by Chalcraft et al. [8] who compared the calculated resonant energies, quality factors and emission polarizations for the lowest order modes with experimental data. Most experiments coupling single quantum dot emitters to a nanocavity exploit the fundamental (i.e. lowest energy) cavity mode [9], [10]. However, higher order modes can still be important for, e.g., efficient pumping in nanocavity lasers [11], selective excitation of quantum dots embedded within the cavity [12], [13], or mutually coupling quantum dots in different spatial positions [14]. Several groups have studied the near-field emission profiles of photonic crystal nanocavities [15], [16], even with polarization-resolving imaging [17].
In this paper we report an experimental and theoretical investigation of the spatial far-field profile of the out-of-plane emission for the two lowest order modes of L3-type PhC nanocavities. We believe the characterization of the out-of-plane far-field emission for PhCs is important for two main reasons. First, for single-photon source applications the emitted radiation needs to be efficiently collected into a fiber, and simultaneous optimization of far-field emission for multiple nanocavity resonances could be useful. In addition, in the case of cavity-QED experiments in the “one-dimensional atom” approximation, a perfect mode-matching is needed to get a large enough interference between the input light field and the field radiated by the atom [18], [19], [20]. Recently, quite some work has been done to get a beam-like vertical emission from PhC nanocavities [21], [22], [23], [24] for the fundamental mode. Here we extend previous work by experimentally analyzing the far-field emission properties of both the fundamental and the second-order mode, finding good agreement with numerical simulations. We introduce a simple model, based on a one-dimensional Fabry–Perot resonator, to estimate the essential far-field characteristics of a given near-field mode profile and link them to the relevant device parameters. We believe that such a model can be useful for fast parameter optimization, while full-scale numerical simulations can provide an accurate but time-consuming description of the electromagnetic field in the structure. Finally, we will discuss the effect of fabrication imperfections on the far-field cavity emission profiles. As we will show, measurements of far-field profiles are relatively easy to perform and they can provide insightful information about the parameters and the quality of the cavities under examination.
The paper is organized as follows: in Section 2, we present experimental measurements of the far-field profiles and a comparison with theoretical far-fields extracted from finite-difference-time-domain (FDTD) simulations; in Section 3, we introduce a simple model, based on a Fabry–Perot resonator, which is sufficient to give indications of what the actual far-field profile looks like for a given near-field and to link far-field properties to actual device parameters.
Section snippets
Theoretical modeling and experimental data
Our sample consists of a 180 nm GaAs membrane grown by molecular beam epitaxy on top of a 0.92 μm Al0.7Ga0.3As sacrificial layer on a GaAs substrate. An In0.4Ga0.6As quantum dot layer is grown at the center of the GaAs membrane by depositing 10 periods of 0.55 Å thick InAs and 1.2 Å thick In0.13Ga0.87As. The L3 PhC cavities were fabricated on the sample using standard electron beam lithography and reactive ion etching techniques [25], [26]. The lattice constant of the triangular hole lattice is a =
A simple Fabry–Perot model
In general, 3D FDTD simulations can provide accurate modeling of near-field and far-field properties of PhC cavity modes. However, they give little physical insight on how the detected features of such far-field profiles can be related to specific device parameters. In this section, we will show that the experimental data can be reproduced by a simple model, elaborated from the proposal of Sauvan et al. [33].
For a line of N ≥ 3 missing holes, the PhC nanocavity can be described quite accurately
Conclusions
In conclusion, we have presented an extensive characterization of far-field emission profiles from L3-type photonic crystal nanocavities, introducing a simple imaging technique as an efficient tool to give a two-dimensional mapping of the emitted intensity. The measurements have been directly compared to theoretically modeled far-field projections from the 3D FDTD near-field cavity modes profiles, and we believe these results to be useful to PhC cavity designs for specific purposes. The effect
Acknowledgments
The authors acknowledge helpful discussions with Morten Bakker. This work was supported by NSF NIRT Grant No. 0304678, Marie Curie EXT-CT-2006-042580 and FOM\NWO Grant No. 09PR2721-2. A portion of this work was done in the UCSB nanofabrication facility, part of the NSF funded NNIN network.
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- 1
Current address: Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario M5S 3G4, Canada.
- 2
Current address: Department of ECE, IREAP, University of Maryland, College Park, MD 20742, USA.