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By employing two-dimensional InGaAsP photonic band-edge lasers, we have experimentally demonstrated that a random mixture of two different photonic crystals (PCs) possesses a new band structure that is intermediate to that of the two host PCs. The photonic band-edges shift monotonically, but with a strong bowing effect, as the mixed PC system is systematically transformed from one PC to the other. The experimental observations are in excellent agreement with finite-difference time-domain simulations and model calculations based on virtual-crystal approximation with compositional disorder effect included.
©2010 Optical Society of America
1. Introduction
As described by the Bloch theorem, a semiconductor having a periodically modulated electronic potential exhibits a complicated energy dispersion relation called band structure [1]. Electron behaviors and resultant physical properties of the material depend on the band structure and the energy of electrons. Interestingly, it was found that there exist photonic systems that correspond directly to semiconductors, namely, photonic crystals (PCs), a photonic structure whose dielectric constant varies periodically [2–4]. The underlying origin for the correspondence lies in the similarity between Maxwell’s wave equation governing the state of photons and Schrödinger’s equation that determines the electronic states [5]. So far, photonic analogies to the electronic phenomena associated with band structures have been identified, including the most prominent example of photonic Anderson localizations [6,7].
In the history of compound semiconductors, there was a topic of great scientific interest and also of tremendous technological impact: mixed semiconductors (or semiconductor alloys) [8–10]. Two or more semiconductors are randomly mixed together in order to create an entirely new semiconductor whose band structure is intermediate to those of the host materials. Such mixed semiconductors can be stacked together so as to produce novel functional photonic and electronic devices; this approach has earned a dedicated name of band-gap engineering [11]. Two outstanding examples of band-gap engineering include double-heterostructure laser diodes [12] and high-electron-mobility transistors [13]. Recently, we proposed that photonics should also have counterparts to the concepts of mixed semiconductors and band-gap engineering as well, which we call mixed photonic crystals (MPCs) and photonic band-gap (PBG) engineering, respectively. In fact, we have demonstrated the validity of the proposed concepts utilizing two different PC systems, colloidal PCs [14] and microwave PCs [15].
Although our proof-of-the-concept experiments provided clear evidence for the analogies, accurate identification of band-edge frequencies, which is crucial for any quantitative discussion on the phenomena, was difficult. Therefore, in this study, we employed a special type of PC devices, band-edge lasers (BELs), which enabled us to directly monitor the spectral positions of the photonic band-edges of MPCs. Here, it should be reminded that photonic band-edges are the extreme points and therefore represent the overall band structure of a PC. A significance of this study is to establish the universality of the physics of MPCs. Unlike our previous MPC studies [14,15] wherein photonic atoms composed of different materials (and therefore, of different dielectric constants) were stacked into three-dimensional MPC structures, in the present study, we prepared MPCs by structurally differentiating photonic atoms and arranging them into a two-dimensional (2D) PC, still confirming that the same physics holds.
BELs are in fact an ideal test-bed since they operate under the gain enhancement effect at the very band-edge points of high symmetry [16]. Therefore, band-edges can be accurately determined simply by observing sharp BEL emission peaks. This is a direct contrast to conventional cavity-based PC lasers, which utilize a localized defect mode inside the PBG and thus do not reveal any direct information on the photonic band structure of PC. In this study, we constructed membrane-type 2D BELs composed of two types of air holes with different radii. Photonic properties of such membrane-type BEL structures can be easily simulated using the finite-difference time-domain (FDTD) method and subsequently compared with experimental data.
2. Fabrication of mixed PC lasers
For the present experiment, we used an InGaAsP multiple-quantum-well (MQW) structure grown by the metal-organic chemical-vapor-deposition method, which exhibited MQW emission at wavelengths of λ ~1550 nm. We then employed electron-beam lithography, reactive ion etching, and selective wet chemical etching to fabricate an array of BELs in an air-bridge membrane structure [17,18]. Among the available 2D PC patterns and band-edge modes, we specifically utilized the Γ-point band-edge of the honeycomb lattice, which is characterized by a large optical gain and the surface-emitting nature [17–19]. Further, we randomly placed two types of air holes with different radii, r S and r L (r S < r L), at the atomic sites of the honeycomb lattice—Fig. 1(a) . We denote the MPC system as S 1– xLx (0 ≤ x ≤ 1), where S and L indicate the air holes with the radii r S and r L, respectively, while x represents the number ratio of the larger air holes (L). Figure 1(a) also shows a schematic dielectric-constant profile of the MPC. The shaded region denotes the difference profile between the two pure PCs, x = 0 and 1.
Fig. 1 (a) Schematic diagram of the 2D honeycomb-lattice MPC, S1– xLx, and its dielectric constant profile. The shaded regions indicate the index difference between two types of photonic atoms, S and L. (b) Photonic band diagram of the honeycomb-lattice PC for x = 0 (blue) and x = 1 (red). The insets show the magnetic field profiles for the monopole and dipole band-edge modes. (c) Scanning electron microscope images of a fabricated honeycomb-lattice MPC BEL structure. The two types of air holes with different radii are marked by the letters S and L as a guide for the eye. The radii of the air holes are r S = 0.380a and r L = 0.415a, where a = 450 nm is the distance between the closest air-holes.
Figure 1(b) shows the band diagrams calculated by the plane wave expansion (PWE) method for the two pure (x = 0 and 1) honeycomb-lattice PCs, which are composed of only one type of air holes, either S or L. The band diagrams are almost the same, except that the band structure for x = 1 is blue-shifted from that of x = 0, as the overall average dielectric constant is smaller for x = 1. By analogy with semiconductor alloys, we expect that the band structure of an MPC (0 < x <1) is somewhere in-between that of the two pure PCs. In this study, we investigated two Γ-point band-edge modes, monopole and dipole modes; also shown in Fig. 1(b) are their magnetic field profiles calculated by the FDTD method.
In order to operate the BELs within the gain range of MQW, we fixed the distance between the closest holes at a = 450 nm, but differentiated the air-hole radii by 6%–9%: (r S = 0.420a, r L = 0.450a) for the monopole mode and (r S = 0.380a, r L = 0.415a) for the dipole mode. The lattice constant of the corresponding honeycomb lattice is then ≈779 nm. However, the dipole-mode samples were still immersed in an index-matching fluid (n = 1.34; Cargille Labs, USA) so as to match the band-edges better with the MQW gain band. The thickness of the MQW membrane was nominally t = 200 nm, and we assume it to be 0.450a for model calculations. The lateral size of the fabricated sample was 13 × 11 μm2 with 566 air holes. Naturally, we selected x = 0, ⅓, ⅔, and 1 as four representative compositions of the honeycomb-lattice MPC system. Figure 1(c) shows scanning electron microscope (SEM) images of a fabricated x = ⅓ MPC. The difference in the air-hole diameter is about 27 nm, which is sufficiently large in comparison with the resolution of our electron-beam lithography equipment (~3 nm). In other words, intrinsic air-hole size fluctuations should be insignificant.
3. Experimental results
The fabricated MPC BELs were pumped using a micro-photoluminescence setup equipped with a 980-nm pulsed laser diode (20 ns pulse width, 1% duty cycle) as the excitation source. The pump spot size was ~10 μm in diameter, slightly smaller than an individual BEL pattern. Regardless of the mixing compositions, all MPC BELs lased at similar thresholds (P th ≈1 mW) in incident pump power. More importantly, the emission wavelength of the MPC BELs blue-shifted gradually and monotonically as the mixing composition x changed from 0 to 1. The lasing spectra of both dipole and monopole modes are shown in Fig. 2(a) . This is a direct proof that the concept of alloy crystals holds not only for semiconductors but also for PCs.
Fig. 2 (a) BEL lasing spectra for the dipole (left) and monopole (right) modes for a few representative composition ratios, x = 0, ⅓, ⅔, 1. (b) MPC BEL lasing spectra for the same composition ratio (x = ⅓) but different configurations.
On the other hand, there was a need to double-check that the MPC BEL lasing was not certain testing environmental coincidence originating from, for instance, a localized resonance in a specific MPC configuration, but rather a collective effect of randomly mixed photonic atoms. For this purpose, we repeated the experiment for three different MPC configurations of x = ⅓; the results obtained are shown in Fig. 2(b): For both the monopole and dipole Γ-point band-edge modes, all three configurations lased at practically identical spectral positions. The configuration-dependent spectral drifts were only δλ ≈2.2 nm for the monopole mode and δλ ≈2.0 nm for the dipole mode, far smaller than the lasing wavelength shifts themselves between x = 0 and ⅓ (Δλ ≈15 nm for the monopole mode and Δλ ≈30 nm for the dipole mode). Thus, we can conclude that the lasing wavelengths of MPC BELs (or band-edges of MPCs) are not configuration-specific, but solely dependent on the composition ratio.
4. Theoretical investigation
4.1 FDTD simulation
In parallel to the experimental investigation, we also performed theoretical simulations on the MPCs. Figure 3(a) illustrates the particular MPC configurations that were used in our BEL experiments and in fact produced the BEL emission spectra shown in Fig. 2(a). We simulated these MPC structures by using the FDTD method so as to obtain their modal spectra, which are shown in Fig. 3(b). In the frequency range of a/λ < 0.3, all the MPC spectra are dominated by one single peak near a/λ ~0.29, regardless of the composition ratios. The magnetic field profiles calculated at the peak frequencies, which are shown in Fig. 3(c), are characterized by a large spatial extension with a single dot-like field distribution in each of the honeycomb-lattice, indicating that the dominant modes are the monopole band-edge modes. It is interesting to note that the field distribution is partially localized at some places where air-holes of the same size happen to be clustered. Nevertheless, the overall field distribution spreads out across the entire simulation domain, much alike a homogeneous PC (x = 0 or 1). This implies that a band-edge mode of the MPC may contain partially localized components but is still an extended state across the structure. We also took the Fourier transformations of the field profiles in order to obtain their intensity distributions in the momentum space. Although calculated for different composition ratios, all the k-space intensity distributions in Fig. 3(d) are basically the same, exhibiting sharply defined dot patterns exactly at the Brillouin zone centers (the Γ points) of the reciprocal lattice of the honeycomb lattice. This implies that the optical modes established in the MPCs (x = ⅓ and ⅔) are of exactly the same nature as those established in the pure PCs (x = 0 and 1).
Fig. 3 (a) MPC configurations for representative composition ratios (x = 0, ⅓, ⅔, 1). (b) Modal spectra obtained by FDTD simulations on the corresponding MPC configurations given in (a). (c) Magnetic field distributions in real space, calculated at the peak frequencies observed in (b). (d) Intensity distributions in momentum space, obtained by Fourier transforming the field distributions shown in (c).
4.2 Virtual-crystal approximation theory with compositional disorder effect included
In Fig. 4 , the experimental and simulated band-edge frequencies are plotted together as a function of the composition ratio x. Except for an overall frequency shift of Δ(a/λ) ~0.005 between the experimental and simulated values, which we attribute to the structural discrepancies between the actual and model BEL structures, the FDTD simulations correctly reproduce the experimentally determined compositional dependence of the Γ-point band-edges, including the bowing effect.
Fig. 4 Experimentally determined lasing frequencies (solid squares) of the MPC BELs and theoretical Γ-point resonant frequencies (open circles) calculated by FDTD for the corresponding MPCs: (a) dipole mode and (b) monopole mode. Also shown are the band-edge frequencies calculated by PWE based on the virtual crystal models with and without the disorder effect. All frequencies are plotted as a function of the mixing composition x.
As seen from the experimental and FDTD data in Fig. 4, the alteration in band-edge frequency is not linear, but exhibits approximately quadratic dependence on the composition ratio x. The band-edge frequency of the MPC, S 1– xLx, can be expressed by the following formula:
In the equation, the constants a and b are determined by the band-edges of the two pure MPCs (x = 0 and 1), while c is a parameter that accounts for bowing. A similar dependence is already known in the case of semiconductor alloys [20,21]. Historically, theories to explain the alloy band gaps were highly demanded as a variety of sophisticated devices were developed based on semiconductor alloys. Among others, the virtual-crystal approximation (VCA) has played a pivotal role [22]. However, it failed to completely resolve the problem of the bowing phenomenon. Accordingly, a few theoretical elucidations were offered to account for the bowing effect [23–26]. In particular, the case considered by Lee et al. [24] that included the statistical analysis for compositional disorder into the VCA is directly relevant and applicable to our MPCs. By translating the results of Ref. 24, the dielectric constant of an MPC can be presented aswhere the first term from the VCA is given by Vegard’s law [27]. On the other hand, represents the term introduced by non-periodic compositional disorder. Statistical consideration of the difference between a real disordered crystal and a perfectly periodic virtual crystal results in being proportional to [24]. The spatial profile of the dielectric constant of an MPC composed of two types of photonic atoms S and L is then given byNote that the dielectric constant difference, , is already depicted schematically in Fig. 1(a). Further, the disorder parameter β calculated for the nearest neighbors is given by , in which N is the number of the nearest neighbors [24].At least from the theoretical standpoint, the bowing effect is a complicated problem when dealing with the properties of an alloy. In the case of semiconductors, it becomes even more complicated due to the effects of charge exchange contribution, volume deformation, and structural relaxation. However, since our MPCs have a well-defined crystal structure and a dielectric constant profile given by the abrupt step-function, the effects of disorder seem to be most dominant among the possible influences on the bowing parameter.
In Fig. 4, we show two theoretical curves for the Γ-point band-edges acquired by the PWE calculations, in which the dielectric constant profiles are given by the VCA theory with and without the disorder effect—Eq. (3). In particular, the VCA theory with the disorder effect is very satisfactory as it produces an upward bowing (c > 0), whereas that without the disorder effect results in an opposite bowing (c < 0), which is contradictory to the experimental observation. The bowing parameter c acquired through the experiments, FDTD simulations, and PWE model calculations are summarized in Table 1 .
Table 1. Bowing parameter c for the monopole and dipole modes at the Γ-point band-edge.
5. Conclusions
In summary, we have demonstrated that the concepts of mixed crystals and associated band-gap engineering are also valid in photonics. For this study, we employed BELs, as they are an ideal test-bed for the present purpose, offering sharp laser emission lines exactly at the photonic band-edge frequencies that may very well represent the overall band structure. Our MPC BELs were 2D honeycomb-lattice PC structures built in an InGaAsP MQW slab, wherein two types of air holes of different sizes were randomly distributed at the honeycomb-lattice sites. Similar to the case of semiconductor alloys, the band-edge frequencies of the MPCs exhibited systematic shifts with changes in the mixing composition. Further, the experimental data on the shifts of the Γ-point band-edge were adequately reproduced by the FDTD simulations for both monopole and dipole modes. In addition, we could explain the bowings in the band-edge shift in a quantitative manner by using the VCA theory with the disorder effect included. We believe that the observed results not only clarify the concept of mixed photonic crystals but also suggest a new paradigm for manipulating photonic band structures by using random PC systems.
Acknowledgments
This study was supported by the World Class University (WCU) project of the Ministry of Education, Science & Technology of Korea through Seoul National University (R31-2009-100320), and in part by the National Research Foundation of Korea through Inha University (2010-0001476).
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