1. Introduction

Ever since the concept of photonic crystals (PCs) emerged [1,2], we have witnessed a series of photonic analogues to the electronic phenomena. Photonic bandgap (PBG), which was first demonstrated in the microwave range [3], is a rather straightforward analogue to the electronic bandgap that is associated with a semiconductor. A photonic defect state within the PBG, which corresponds to an impurity state inside the electronic bandgap, was then produced by intentionally introducing a structural imperfection [4]. This translational symmetry breaking has led to the recent development of PC-based cavity with extreme Q-value [5]. The correspondence between photonics and electronics was culminated in the recent experimental demonstration of the Anderson localization of photons [6]. As a new found entity of the correspondence between photonics and electronics, here we report on the mixed photonic crystals (MPCs), the photonic version of semiconductor alloys, and the associated phenomenon of PBG engineering. The history of semiconductor alloys dates back to as early as the 1950s, during which it was realized that the solid mixtures of binary compound semiconductors could be used to control some of the key material parameters. Their usage in the fabrication of functional heterostructure devices, such as laser diodes and high-speed transistors, led to the Nobel Prize in 2000 [7]. In significant contrast, however, the photonic analogy to the mixed semiconductor system has been left unexplored thus far, despite its potential impact on the future nanophotonic structures and devices of high functionality.

 

Fig. 1. Atomic configuration of the diamond structure in the conventional cubic cell (a), and in an extended scheme after reoriented such that the [111] crystal direction points upward (b). In a-b, two groups of atoms, (0,0,0) and (¼,¼,¼) in their relative coordinates, are distinguished by color contrast. (c) Diamond lattice photonic crystal composed of hard spheres of equal size, with the nearest neighbor spheres in contact with each other. Different crystal planes of the diamond lattice photonic crystal: (111) (d) and (-110) (e). In (c)-(e), colors are used to distinguish among different unit layers that are stacked vertically in the actual samples.

Download Full Size | PDF

Previously, we proposed the existence of PBG engineering in optical frequency range, based on the experimental observation that the L-point bandgap could be tuned when equisized sub-micron spheres of silica and polystyrene were randomly mixed and self-assembled to form an MPC [8]. Although it may be considered as an initial proof of the concept, neither mixing composition ratio nor the randomness of the colloidal MPCs could be identified or confirmed. Further, the total amount of PBG shift was only comparable to the resonance bandwidth. These experimental incompletenesses prevented us from deducing any definitive conclusion about the general properties of the MPCs. In this study, we employ an MPC system that operates at microwave frequencies. Since the feature sizes in that case are subsequently in the range of ~cm, we can directly dictate the details of the MPC structures (such as mixing composition ratio and randomness in mixing), thereby eliminating the structural unassurances that were experienced in our previous experiment with the colloidal MPCs. However, it should be recalled that the physical properties underlying the microwave MPCs should be equally valid for MPCs in optical frequencies, as the Maxwell equations are scale-invariant [9]. In fact, the microwave structures have been actively utilized to demonstrate certain of the significant photonic properties and functionalities [1, 3, 4, 10-14], which include the first demonstration of invisibility cloak based on metamaterials [14].

In this study we experimentally discovered that the PBG edges of the MPCs shift monotonically with negligible bowing effect as the mixing composition ratio is varied. The observed PBG shift is in excellent agreement with the band structure calculation results based on the virtual crystal approximation (VCA), which was originally developed to model the electronic structure of alloy crystals [15].

2. Sample preparation

Our microwave MPC system is composed of silica (SiO₂) and alumina (Al₂O₃) spheres nominally with equal diameter (φ=5 mm), which we abbreviate of this system as So_1-xAo_x(0≤x≤1). As in its GaAlAs electronic counterpart, the lattice-match between the two types of constituent spheres is still important since the crystalline quality of the resultant close-packed MPCs is directly affected by it. The dielectric constants of SiO₂ and Al₂O₃ in microwave frequencies are ~5.0 and ~9.0, respectively. The dielectric spheres are then arranged into a diamond lattice structure, which offers a relatively large PBG for a given index contrast [16]. The diamond structure is essentially a face-centered-cubic (fcc), but with a pair of basis atoms at the sites of (0,0,0) and (¼,¼,¼), as shown in Figs. 1(a)–1(b). The diamond lattice is not the most densely packed crystal structure; thus it cannot be produced by simple stacking of hard spheres unless an appropriate mechanical support is provided. The strategy we employed in constructing the diamond lattice PC involved preparing unit plates parallel to the (111) plane and then stacking them in the [111] direction in a semi-self-assembly manner —Figs. 1(c)–1(e).

The preparation of the unit plates began with thin template plates, which were composed of a highly porous sponge material of a dielectric constant of ~1.2 and formed the skeleton of the MPC when completed. The two-dimensional hexagonal lattice array of air-holes was then punched through the template plate at predetermined lattice sites. Then a pair of either silica or alumina spheres was inserted into each air-hole. These two spheres were glued together with a small amount of epoxy; the pair served as the two basis atoms at (0,0,0) and (¼,¼,¼). The air-holes in each template plate were filled with both silica and alumina sphere pairs with an appropriate number ratio, depending on the composition x of the MPC, with their sites chosen randomly. Filling up all the air-holes with sphere pairs completed the unit plate, which was the basic building unit for the formation of the diamond lattice MPC. The unit plates prepared as such were then stacked on top of each other to form a diamond lattice MPC. Stacking of the unit plates was performed in the fcc arrangement sequence (ABCABC…), in which three unit plates constituted one period in the stacking sequence. The stacking resulted in a semi-self-assembled diamond lattice MPC. It should be noted that the resultant PC is the most densely-packed diamond lattice PC since the nearest dielectric spheres are physically in contact with each other —Fig. 1(c).

Figures 2(a)–2(c) show the photographs of the actual MPCs constructed as such for a few representative composition ratios: x=0, 0.5, and 1. With the exception of the pure crystals (x=0 or 1), the stacked unit plates are unidentical; pairs of silica and alumina spheres are randomly distributed, conforming to the given composition of x. One notable advantage of our stacked-but-separable layer scheme is that a variety of alternative MPCs for a given x can be obtained by simply rearranging the stacking order and/or the orientation of the unit plates—Fig. 2(d). This allows us to construct a set of MPCs even with a limited number of the unit plates, which enables the application of a statistical approach in studying the properties of the MPCs.

 

Fig. 2. Photographic images of the completed diamond lattice photonic crystal alloys with alumina sphere compositions of x=0 (a), x=0.5 (B), and x=1 (C). Each plate in the stack contains 504 holes that are filled with pairs of either silica or alumina spheres. Insets are the amplified images showing the detailed sphere arrangements for the corresponding photonic crystals. (d) Anatomy of the x=0.5 mixed photonic crystal, which explicitly shows that individual unit plates have different sphere configurations.

Download Full Size | PDF

3. Transmission spectra

A prerequisite for this experiment is to determine the appropriate number of unit plates required to constitute a MPC. We examined the transmission spectra as a function of the number of stacked unit plates, N, for a few different compositions. Transmission spectra were measured along the [111] crystal direction, which is perpendicular to the stacked plates. An Anritsu network analyzer (37247D) was used in conjunction with two antenna horns for collimated probe beam. Figure 3(a) shows the measured transmission spectra of the alumina PC (x=1) when N=10, 12, and 14. It distinctly demonstrates that the transmission stopband (or the L-point PBG) develops as N increases. We found that N≥14 was sufficient for the complete development of the L-point PBG for unmixed PCs, whereas a larger number of unit plates were required for mixed ones. To ensure unambiguous identification of the PBG, we stuck to N=24 throughout the experiment regardless of the mixing composition. The transmission stopband (or the L-point PBG) was identified in the following manner: (1) Measure the transmission spectra while increasing the number of unit plates in the stack. (2) Monitor the transmission stopband development. (3) Identify as the bandedges the two Fabry-Perot peaks on both ends of the stopband whose responses to the increase in the stacked unit plate number are least sensitive (see the arrows marked in Fig. 3(a)).

Next we proceeded with two constituent PCs: PCs composed of either silica or alumina spheres only. Shown in Fig. 3(b) are the measured transmission spectra for x=0 and 1, together with the corresponding band structures calculated by the three-dimensional plane-wave-expansion (PWE) method for comparison [16]. The silica and alumina PCs exhibited bandgaps across the frequency ranges of 13.4–15.9 GHz and 10.9–13.8 GHz, respectively. The concurrence between the measured transmission stopbands and the calculated L-point PBGs is high for both PCs, supporting that the design and implementation of the diamond lattice PC was properly accomplished.

 

Fig. 3. (a). Evolution of transmission spectrum of the alumina photonic crystal (x=1). As the number of the unit plates increases, the transmission stopband becomes distinct. Well-defined bandedges, which are indicated by two arrows, develop themselves on both sides of the transmission stopband. (b) Comparison between the measured transmission stopband and the calculated L-point bandgap for the alumina photonic crystal (x=1; top two panels) and for the silica photonic crystal (x=0; bottom two panels)

Download Full Size | PDF

With reasonable confidence on our diamond lattice PC structure, we finally approached the MPCs. The transmission spectra were measured for the MPCs by employing various mixing compositions and configurations. Figure 4(a) displays the measured transmission spectra of the So_1-xAo_x MPCs for x=0, 0.1, 0.3, 0.5, 0.7, 0.9, and 1, which cover the entire mixing composition range of 0≤x≤1. The L-point bandedges determined from Fig. 4(a) are plotted in Fig. 4(b) as a function of the mixing composition ratio x. Analogous to the electronic bandgap shift in the GaAlAs mixed compound semiconductor system, the L-point PBG redshifts monotonically as the alumina composition x increases. To theoretically account for the PBG shift, we applied the VCA in a linear interpolation scheme, which was quite successful in accounting for the electronic bandgap variation of mixed semiconductors [17]. We assumed that the MPC is a virtual PC: a homogeneous diamond lattice PC composed of identical dielectric spheres of a single effective medium whose dielectric constant varies linearly with composition in a manner similar to the Vegard’s law [18]: ${\epsilon }_{\mathit{eff}}\left(x\right)={\epsilon }_{{\mathit{SiO}}_2}·\left(1-x\right)+{\epsilon }_{{\mathit{Al}}_2{O}_3}·x$ . For a given x, the band structure calculation using the PWE method and the bandedges identification thereof were quite straightforward. Calculational results, which exhibited a very small bowing effect, are also shown in Fig. 4(b) for direct comparison with the experimentally determined values. Despite the simplicity of the linear effective medium assumption, the calculation results are in excellent agreement with the measured data for the entire range of mixing compositions.

 

Fig. 4. (a). Transmission spectra of the mixed photonic crystals for the full range of alumina sphere mixing composition: From the top, x=1, 0.9, 0.7, 0.5, 0.3, 0.1, and 0. (b) Photonic bandedge positions as a function of alumina sphere composition x. Squares are experimental results deduced from (a), whereas the lines are calculated using the virtual crystal approximation theory. The theoretical lines, especially the dielectric bandedge line, show slight bowing effect. (c) Transmission spectra for an ensemble of the x=0.5 mixed photonic crystals. Multiple transmission spectra are intentionally drawn on top of each other to make directly visible the intrinsic fluctuation associated with the ensemble (of the same composition but different mixing configurations).

Download Full Size | PDF

4. Statistical consideration

We are now in a position to answer a fundamental question regarding the statistical nature of the MPCs. For a given mixing composition x (≠0, 1), there exist many possible MPC structures in different atomic configurations—an ensemble. Furthermore, for the MPCs to have any physical implication, the ensemble should exhibit identical PBG properties that are immune to the configurational variations. In Fig. 4(c) we intentionally overlap the transmission spectra measured for an ensemble of 16 MPCs of x=0.5, each having its own unique sphere arrangement. In fact, the measured transmission spectra overlapped satisfactorily with a well defined common stopband. The fluctuation in the bandedge position was Δf≈250 MHz (or Δf/f<0.02) in frequency domain. This observation is consistent with the fundamental assumption of statistical physics: The physical properties of a many-particle system are sharply defined and given by the most probable configuration of the corresponding ensemble [19]. Obviously, the most probable configuration for the MPC system is a random mixture for which the system entropy becomes maximum. Therefore, any configuration should be a representative MPC with a well defined PBG as long as the mixing is reasonably random.

Although a thorough theoretical consideration at a more fundamental level is needed to explain the physical origin and the limitations of the PBG engineering in the MPC system, the following simple argument could be an intuitive physical rationale for the observed phenomena. Regarding the origin of a PBG formation, it is known that the otherwise degenerate electromagnetic modes at the band crossing possess different energies in the presence of periodic dielectric contrast of a PC due to distinct spatial mode profiles and that this causes the opening of a PBG [9]. The total electromagnetic field energy of a mode in a PC can be represented by the total energy distributed over the entire PC. In this context, the energy in the proposed MPC can be reasonably approximated to that stored in a homogeneous PC whose dielectric constant is given by the averaged value. Therefore, our experimentally determined PBG shift was in reasonable agreement with the calculation results based on the VCA and the linear effective medium assumption.

5. Conclusion

The MPCs and photonic bandgap engineering that we have demonstrated in this study should only be considered a continuation of answering the fundamental questions posed: how far the analogy between electronics and photonics could go and what would be the limitation of this analogy. In the meanwhile, the proposed microwave diamond lattice photonic crystal structure in the stackable layer format offers a versatile testbed in which the answers to these fundamental questions could be efficiently sought. Nevertheless, one may hypothesize that, when photonic crystals operating in the optical frequency could be self-assembled in a layer-by-layer format, the resultant MPCs might realize highly functional photonic heterostructures. This could usher in another technologically prosperous era much like semiconductor alloys and heterostructures.