Multi-dimensional photon confinement in optical cavities and waveguides provides efficient means for tailoring light-matter interaction via the control of the photon density of states, polarization and wavefunction [1,2]. Three-dimensional (3D) confinement in photonic-crystal (PhC) nano-cavities was instrumental in attaining substantial modifications of the spontaneous emission in semiconductor nanostructures via the Purcell effect [3], and in enhancing the efficiency and photon indistinguishability of quantum dot (QD) single-photon emitters [4,5]. It also allowed for the generation of QD polaritons [6], as well as for the construction of ultra-low-threshold microcavity lasers featuring very high spontaneous-emission coupling factors [7,8]. The particularity of one-dimensional (1D), and quasi-1D photonic structures lies not only in the provision of singular density of states, but also in the possibility for photon transport within the 1D photonic bands. This makes them attractive for realizing both on-chip generation [2] and manipulation of single and entangled photons, useful for applications in quantum information communications [9,10] and processing [11]. Unlike fully confined 0D photonic cavities, however, 1D systems are highly sensitive to disorder in the dielectric-constant distribution, which can induce photon localization [12,13]. Such localization can significantly alter the characteristics of light-matter interaction and photon transport in quasi-1D photonic systems –featuring either regular [14] or coupled-cavity waveguides [15]– due to the modifications in the spectral and spatial distributions of the photon wavefunction, with crucial implications on single-photon sources and photon-transfer lines. The experimental investigation of the formation and perturbation of 1D photonic bands in real structures thus deserves special attention.

Semiconductor PhC membrane L_N cavities, consisting of line defects formed by N missing holes in a 2D PhC hexagonal lattice, have been employed in many important experiments in nano-photonics and quantum physics, in particular on single-photon sources [5], single-photon transfer [14], and PhC bandgap microlasers [7] [8]. More recently, long (10-20 unit cells) 1D-like L_N cavities have been identified as good candidates for producing extremely high spontaneous-emission enhancement factors, realizing “on chip” single-photon guns [2]. A PhC waveguide-based single-photon source has also been demonstrated experimentally [17], however a lack of site and photonic-wavefunction control makes it difficult to conclude whether the device performance was based on quasi-1D modes or rather on localized photonic states.

In the present work, we present evidence for the transition from fully confined photonic cavities to a 1D PhC waveguide in a series of L_N cavities of increasing length (N = 3 to 35). By expressing the L_N cavity modes as a linear combination of the 1D Bloch eigenstates of the corresponding W1 waveguide, we show that in an ideal case the spectra of the confined modes approach the 1D dispersion starting from N~35. However, fabrication-induced disorder limits the extension of the photon wavefunction to shorter lengths. Using finite-difference (FD) computations and spatially resolved micro-photoluminescence (PL) measurements, we directly probe this localization and its dependence on the photon mode index.

The photonic band structure [Fig. 1(b) ] of the W1 PhC waveguide shown schematically in Fig. 1(a) was evaluated numerically by means of two-dimensional FD calculations, in the effective-index approximation and with periodic boundary conditions. The calculated dispersion for a typical GaAs-air structure [see Fig. 1(b)] shows two bands of odd and even parity [16] confined within the TE₀ PhC bandgap. As the L_N cavities [see Fig. 1(c)] tend to the W1 waveguide for N→∞, it is intuitive to expect their spectral features to bear a connection to the W1 dispersion properties at any finite N. Focusing on the lowest-order cavity states –the ones most interesting for applications [2,14]– the evolution of the spectrum of an L_N cavity [see example in Fig. 1(c)] with increasing N is depicted in Fig. 2(a) [lattice parameters being the same as for Fig. 1(b)]. Clearly, the density of states grows as N is increased, with same-order modes (M₀, M₁, M₂ …) progressively red-shifting. This red shift is a signature of the reduced photon confinement along the x-direction, and is summarized in Fig. 2(b) for the first 5 lowest-order modes. For different modes the red shift saturates (becoming negligibly small) above a certain cavity length. For M₀, this takes place at N~35, which can be taken as a signature of the onset of the 1D photonic band formation.

 

Fig. 1 (Color online) (a) Sketch of a GaAs-air 2D PhC W1 waveguide, (b) Band diagram computed using effective index n_eff=3.223 (as estimated from 3D FDTD) and corrected for the infrared dispersion of n_GaAs. The PhC continuum and radiative (above the light line) leaky modes are shaded in gray. The band parity notation follows the full 3D field distributions [16] (verified by 3D FDTD analysis). The odd band (red symbols) is well-defined up to the anticrossing point with the upper PhC bands. (c) SEM top-view of a fabricated L₁₁ cavity.

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Fig. 2 (Color online) Simulations and experiments for L_N cavity modes with increasing cavity length N. (a) Calculated spectra (insets show the E_y field-component distributions). The “mode weight” is related to the out-of-plane cavity losses [18], and it is calculated as the inverse of the Fourier transform of a given field integrated within the light-cone area normalized to that integrated over the full domain. (b) Calculated mode frequencies vs. N for the first 5 lowest-order modes (continuous lines are guides to the eye; inset shows a magnified view near saturation); (c) Measured micro-PL spectra. For L₃ and L₆ cavities r/a ≈ 0.24 while for L_11,21,35 r/a ≈ 0.265 was chosen in order to overlap with the QWR emission. The resulting spectral offset (Δλ ≈ 8 nm) is taken into account in the figure; (d) Same as (b), but with measured values superimposed on calculated trends [the 13-nm offset is taken into account (see text)].

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Experimentally, the mode spectra of L_N cavities with N=3,6,11,21 and 35 were characterized using GaAs membrane structures incorporating InGaAs/GaAs site-controlled (lateral alignment precision ~40nm) V-groove QWRs serving as an internal light source (ILS) (see [19] for fabrication details). A scanning-electron-microscope (SEM) view of an L₁₁ sample is shown in Fig. 1(c). Post-processing by digital etching [20] was performed to enhance the cavity Q-factor and the QWRs were designed to have their peak PL spectrum near ~890 nm at 10K. The micro-PL spectra show a series of cavity modes superimposed on the QWR background emission [Fig. 2(c)]. The measured Q-values ranged from 3000 (L₃, M₀ state) up to ~7500 (in L₂₁, L₃₅). The measured mode wavelengths differ from the calculated ones by ≈ 13nm, mainly due to a difference of ~2 nm in the lattice constant (yielding Δλ~+8 nm) and to fluctuations in the hole diameter along z (reduced diameter at the half-slab level giving Δλ~+5 nm) as compared with the design target. However, since such a small offset does not involve an appreciable spectrum stretching, the experimental spectra and the N-dependence match the computations quite well.

In order to relate the spectra of the L_N cavities to those of a W1 waveguide and gain more insight into the formation of 1D photonic bands, it is possible to expand the m-th mode of the L_N cavity, ${\overrightarrow{E}}_{\text{N}}^m\left(x,y\right),$ in terms of the Bloch states of the W1 structure, ${\overrightarrow{E}}^{\text{w}{k}_x}\left(x,y\right)e^{i{k}_{x}x}:$

Here, V is the domain volume, and the sum was centered on the minimum of the W1 band, k_x=π/a. Since we are focusing on the odd cavity modes, only Bloch modes belonging to the corresponding odd band of the waveguide [see Fig. 1(b)] were taken into account. For the cavity modes studied in the present work, the $c_{\text{w}{k}_x}^{m,\text{N}}$ coefficients can be written as [21]

$E_y^{m,N}\left(x\right)$ is the y-component of ${\overrightarrow{E}}_{\text{N}}^m\left(x,y\right)$ integrated over y, G=$\pm \raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$a$}\right.\cdot n$ (for n=0,1,…,∞) is the one-dimensional reciprocal lattice vector, and the $W_G^{\text{w}{k}_x}$ term only depends on the W1 dielectric constant and eigenstates. In principle, Eqs. (1-2) are valid for any L_N cavity mode possessing the right parity (i.e., odd for reflections about the xz plane); only for very small values of N (approximately N<3) the cavity modes become too localized to be expressed in terms of a small number of “extended” photonic crystal states, such as the waveguide eigenmodes. In these cases, the sum in Eq. (1) has to be modified to include a larger number of PhC bands. The $c_{\text{w}{k}_x}^{m,\text{N}}$ coefficients provide the necessary connection between the m-th mode of the L_N cavity –via the mode’s 1D Fourier transform, $\text{FT}\left\{E_y^{m,N}\left(x\right)\right\}$– and the eigenmodes of the corresponding PhC W1 waveguide –via the $W_G^{\text{w}{k}_x}$ term. It should be noted that the latter term is completely independent of the cavity length, so that the evolution of the cavity modes with increasing N is fully determined by $\text{FT}\left\{E_y^{m,N}\left(x\right)\right\}.$ This evolution is quite evident if we superimpose $\text{FT}\left\{E_y^{m,N}\left(x\right)\right\}$ to the waveguide dispersion for the different L_N cavities, as shown in Fig. 3 . Even for the smallest values of N, all cavity modes localize around particular k_x-values, and with increasing cavity length their distributions converge to discrete values of the W1 band. Hence, the cavity modes shift rapidly in energy down to a certain level [compare also to Figs. 2(b) and 2(d)], which, for the ground state (M₀) is near the minimum of the dispersion curve. Starting from this point (i.e., for N~35 in the ideal case), any additional shift of M₀ is negligible, and the state practically becomes 1D.

 

Fig. 3 (Color online) Comparison of the W1 dispersion (black line) with the 1D Fourier transform $\text{FT}\left\{E_y^{m,N}\left(x\right)\right\}$ (colormap) of the confined modes of the L₃, L₁₁, L₂₁ and L₃₅ cavities (for the latter, only the first 16 modes are shown). PhC bands are shown as grey-shaded regions, and the light line is in green.

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Though, in general, the experiment confirms these tendencies, relatively long cavities (N= 21, 35) show evidence for PhC structural disorder in the measured spectra (for example, compare Fig. 2(c), and Fig. 2(a) for L₃₅). In order to get further insight, we performed 2D-FD numerical computations based on a realistic dielectric constant imported directly from SEM images of the fabricated samples [see Fig. 4(a) ]. Notwithstanding a good PhC quality (a preliminary statistical analysis suggests a relative standard deviation smaller than 3% for both the hole position and radius), in the fabricated L₃₅ cavity the computed near-field patterns are clearly localized in different sections [Fig. 4(b)]. Analyzing the field patterns along the lines followed for the ideal cavities, we observe that such spatial localization manifests itself through a broadening of the mode field distributions in k_x-space, and hence in a blueshift [see Fig. 4(c)] arising from the random-disorder potential. For the localized modes the effective cavity length is thus shorter, e.g., for the M₀ mode of the fabricated L₃₅ the effective cavity length is actually similar to that of an ideal L₁₁. Since the different modes are localized at different sections of the cavity, it is possible to probe the localization directly using spatially resolved micro-PL spectroscopy of these PhC structures. Figure 5 shows the measured micro-PL spectra of an L₃₅ cavity excited with a ~1.5µm wide laser spot at several positions along its axis. The different modes in each spectrum are identified based on their spectral position withthe aid of the calculated spectra (corresponding to the field distributions shown in Fig. 4). As evidenced by the variation of the integrated micro-PL intensity of the first 6 cavity modes as a function of the position of the laser spot within the cavity –displayed in Fig. 5(a)– the relative mode intensities depend strongly on the excitation position, reflecting the localization of the different modes in different sections of the long cavity as predicted by the model calculations. In particular, the fundamental mode M₀ is excited most efficiently when the excitation spot is located at the center of the cavity, whereas modes M₁ and M₂ are best excited when the laser spot is positioned at either extreme end of the cavity. The observed variation in the excitation efficiency of the different modes with the position of the laser spot is qualitatively consistent with the FD-calculated mode patterns [Fig. 4(b)]. It is important to note that a significant localization [bringing also disorder in the mode spectral positions, see Fig. 2(c)] arises only for the lowest-order modes [e.g., from M₀ to M₃ in Fig. 4(c)]. On the other hand, the higher-order ones (starting from ~M₄-M₅ for the measured L₃₅) are less disorder-sensitive. This observation reflects the fact that the high-order cavity modes stem mainly from the index-guided W1 states (linear part of the dispersion band, see Fig. 1(b) and Ref [22].), which are inherently insensitive to the fine details of the dielectric constant, whereas the photonic gap-guided modes (close to the minimum of the W1 dispersion band) “sense” even slight fabrication-induced PhC-lattice periodicity imperfections [22]. Obviously, the mode localization is of crucial importance for applications requiring near-ideal 1D photonic bands, e.g., QD coupling with photonic wires [17] or 1D polaritons [23]. Furthermore, localization is detrimental for applications based upon on-chip photon transfer [14], and particularly in the case of a III-V PhC waveguide on Si (where the waveguide dispersion is limited mostly to the gap-guided modes due to the presence of the bonding layer [24]).

 

Fig. 4 (Color online) (a) Top-view SEM image of the measured L₃₅ cavity (close-up of the cavity termination). (b) Mode patterns (E_y component) obtained from 2D-FD computations using the SEM image shown in panel (a) (lowest-order modes shown). (c) Same as Fig. 3, for the first 14 modes of the measured L₃₅ cavity. The zoomed part shows the lowest-order states compared to an ideal L₃₅-cavity structure (both with a=196 nm).

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Fig. 5 (Color online) (a) Micro-PL intensity profile of the first 6 cavity modes across the measured L₃₅ cavity. (b) Representative micro-PL spectra excited by a ~1.5µm wide laser spot at different positions in the cavity. The spectra are vertically shifted for clarity. Excitation conditions: pulsed (3 ps, 78 MHz), 700 nm, 12.5 μW. T=50K. (c) Top-view SEM image of the measured L₃₅ cavity [this image was used to compute the modes shown in Fig. 4(a)].

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In summary, we followed the evolution into a 1D photonic band structure of the fully confined cavity modes of PhC membrane L_N cavities of increasing length, both experimentally and in the ideal case. The complete band formation, generally speaking, depends on the spectral width of each cavity mode, which needs to be greater than the spectral mode separation in order to allow for propagation along the axis of the photonic wire. In practice, the observed saturation in the variation of the eigenmode frequencies with increasing N suggests the onset of the formation of a 1D photonic band already for N~35. In addition, we presented direct evidence of disorder-induced photonic mode localization along the cavities, with characteristic localization patterns depending on the mode index. Such weak localization effects limit the formation of fully-extended quasi-1D photon states with predictable mode patterns, and need to be removed in case efficient QD-cavity coupling or photon transfer along the 1D photonic structure is desired, e.g., for applications in on-chip single-photon generation and transfer.