We propose a simple structure for manipulating resonant conditions in random structures, in which a “defect” region where no scatterer is set is deliberately made in the structure. By employing a two-dimensional finite-difference time-domain method including rate equations, we examine the resonant and lasing properties observed at the defect region by changing the filling factor of scatterers. From the numerical results, we confirm that a distinct localized spot at the defect can be realized by determining an optimal filling factor and scatterer size and selecting the appropriate defect size.
©2009 Optical Society of America
Since light-matter interactions can be greatly enhanced within micro- to nanoscale cavity structures due to increased photon lifetime, drastic improvements in the performance of various optical applications can be expected. Although several types of small cavity structures have been investigated intensively for such purposes, we have been especially interested in the potential for photon localization within a wavelength-scale-disordered structure (random structure) [1–17]. Because photon diffusion constants are reduced drastically in the random structures by the interference effect of multiply scattered light, they can also be used as microcavities for the realization of extremely low threshold lasers, efficient nonlinear optical devices, and highly efficient functional materials. In addition, because a random structure has the advantages of easy fabrication, such as simple nanoparticle assembly or utilizing self-generated surface roughness, we expect that random structures can be used for easily fabricated, low-cost applications. However, although randomness is a key feature of random structures, it makes it difficult to induce or prepare the intended modes within the structure. Since high spectral and spatial overlap between photons and materials doped in a random structure is necessary for efficient light-matter interactions (e.g., for decreasing the threshold of nonlinear phenomena), technological developments in manipulating resonant properties are essential for applications.
Several approaches such as temperature control , reducing structure size [12-14], limiting excitation area [7, 18] have been reported thus far. For instance, a single microlaser in a 1.7-μm-sized ZnO random structure was demonstrated experimentally [12, 13], and by a reduction in the excitation area, the existence of only a few localized modes that felt the strongest gain near the excitation spot was confirmed numerically to initiate laser oscillation [7, 18]. Furthermore, Topolancik et al. demonstrated that they introduced the shape disorder of airholes in photonic crystal waveguides and experimentally confirmed that the strong localization was able to be achieved in the waveguide [19, 20]. In their studies, although the disorder also played an important role for the optical localization, their structure was still highly ordered structure rather than the structures considered here and the localized modes were induced due to the coherent interference of scattered waves from periodic airholes. Among various types of proposals for mode control, we are interested in the numerical analysis of random structures reported by several groups [14, 18, 21-24], in which resonant peaks or transmittance dips dominated specific spectral regions, the so-called frequency windows. In particular, Miyazaki et al.  and Rockstuhl et al.  reported that the transmittance of size-monodispersed random structures exhibited a sharp dip at certain frequency bands, similar to a photonic band gap. This type of structure could be used as a kind of a mirror, owing to the modal coupling between the Mie resonances of neighboring scatterers. From these studies, we have conceived the idea that if we made a defect region in a random structure by mixing polymer nanoparticles with scatterers as defects, we would manipulate resonant conditions at the defect regions and utilize the defect as a photoreactive site.
In this paper, we propose a simple method for controlling the spatial and spectral resonant properties in a random structure, by which a defect region where no scatterer is set is deliberately made in the structure. By adjusting the filling factor of scatterers in a random structure to fill interspaces and making a defect region with a specific size, we attempt to control resonant conditions at the defect region and utilize it as a reactive site for photophysical phenomena. By employing a two-dimensional finite-difference time-domain (2D-FDTD) method combined with rate equations [18, 21-22, 25], we numerically analyze the spatial, spectral, and lasing properties at the defect region in random structures with different filling factors, in order to confirm the validity of our proposed method. From the results of resonant spectra and intensity distributions, we found the existence of an optimal filling factor, optimal scatterer size, and appropriate defect size for realizing the photon confinement in the defect region. In addition, from the results of lasing intensity distributions of the same random structures used for the analysis of resonant properties, we confirm that even when multimode lasing occurs due to the high pumping rate, a strong lasing spot is induced primarily at the defect region of a random structure with the optimal filling factor.
2. Simulation and analysis
The numerical models we used were composed of randomly distributed dielectric circular scatterers (diameter: 400 nm, refractive index: 2.6, surface filling factor: 40–60%) and a single defect region made at the center of structures (diameter 600 nm), where no scatterer was set. The surrounding medium was assumed to be air (refractive index 1.0). The size of the entire calculation area and the dispersion area of the scatterers were 50 × 50 and 15 × 15 μm2, respectively. For making random structure with a defect region, we first fixed scatterer size (diameter 400 nm) and calculated the frequency windows of surrounding random structures. Then, the defect size was determined to be larger than the scatter size and to roughly form a half-wavelength resonator at frequencies in the frequency window by taking into account of the average refractive index of surrounding structures. Then, we calculated resonant and lasing properties by changing filling factors in the following section. Note that, when making random structures, we allowed to contacting scatterers, but prohibited overlapping of scatterers. Filling factors of scatterers were controlled by changing the number of scatterers randomly dispersed in the dispersion area.
Figure 1 shows typical spatial distributions of scatterers (solid circles) with surface filling factors of (a) 40, (b) 50, and (c) 60%, respectively. Many interspaces occur in the structure with a filling factor of 40%, while the number of interspaces decreases with increasing filling factor, and the defect region at the center of the structure becomes clear at 60%.
By employing a 2D-FDTD method, the structures were analyzed numerically. Only the electric field parallel to the cylinder’s axis was considered. A time step and cell size of 7.0 × 10-17 s and 50 nm, respectively, were chosen. Mur’s second-order absorbing boundary condition was used, and each calculation was performed for 530000 steps (37 ps). Light sources were set at individual cells in the surrounding medium of the dispersion area in the structures. To obtain intensity distributions, the light sources were excited homogeneously by cosine waves at a given frequency until a steady state was achieved. For the calculation of resonant spectra, a short Gaussian pulses (duration time about 10-15 s) were launched at individual light sources and electric field at the center within the defect was recorded during a whole calculation time (37 ps) and resonant spectra were calculated from Fourier transform of the signal. For the calculation of resonant intensity distributions and spectra, random structures were assumed to be passive systems (without gain or absorption).
For the analysis of laser oscillations, the FDTD method combined with rate equations and a polarization equation for 4-level atoms was employed [18, 21-22, 25]. The gain medium, including rate equations and the polarization equation, was assumed to be set in the interspaces of random structures (black areas in Fig. 1). In the rate equations, the decay rates of individual states and the total density of atoms were assumed to be r 32 = 1012 s-1, r 21 = 109 s-1, r 10 = 1011 s-1, and N = 3.31 × 1024 m-3. The levels represented by subscripts 0 and 3 indicate the ground and excited states of atoms, respectively, and those represented by 1 and 2 denote the lower and upper laser levels, respectively. In the polarization equation, the gain spectrum was assumed to have a center frequency of ω a = 280 THz and a bandwidth of Δω a = 16 THz. The pumping rate was sufficiently large (r p = 109 s-1) to simultaneously excite many lasing modes appearing in each structure. The electromagnetic field was calculated by numerically solving Maxwell’s equations and was absorbed or stimulated by the rate equations depending on the pumping rate. From the population inversion of the rate equations, polarization was calculated and changes in the electromagnetic field were induced via the polarization term of Maxwell’s equations. By repeating the calculation until laser oscillations achieved steady state, we calculated the lasing properties. From the steady state, we obtained the lasing emission intensity distributions and also their lasing emission spectra by Fourier transformation of the recorded time series data of electromagnetic field at the defect region.
3. Results and discussion
To confirm the optical properties of surrounding random structures with individual filling factors, we first calculated the transmitted intensity spectra of surrounding random structures (15× 15 μm2), which were different distributions of scatterers from Fig. 1 and without a defect region. For the calculation of the transmitted intensity spectra in Fig. 2, different from the procedure described in the previous section, a point light source was set at the center of structures (within a defect region) and 40 detection points set around the edge of the structures. Then, by launching a short Gaussian pulse at the light source, the transmitted electric fields were recorded and averaged all of the transmitted intensity spectra calculated at individual detection points. The results are shown in Fig. 2. The filling factors vary from 40 to 60% (from top to bottom), and the dotted curve indicates the spectrum profile of the incident pulse. Individual spectra are offset for clarity. The transmitted intensities clearly exhibited several sharp dips (frequency windows) at similar frequencies for all structures with different filling factors. The visibilities of the frequency windows at 40 and 60% were lower, and their transmittances became higher compared with those at 50%. This tendency was similar to the numerical results reported in previous papers [23, 24]. Using this feature, we considered that the structures surrounding a defect region work as a mirror or a filter with random reflection or transmission properties.
Figure 3 shows the resonant spectra calculated at the defect region in random structures with filling factors of (a) 40, (b) 50, (c) 50, and (d) 60%. The spectra in (a), (c), and (d) were the calculated results from the numerical models shown in Fig. 1. Spectrum (b) was the result of a different distribution of scatterers with the same filling factor of 50% (not shown in Fig. 1). From the results at 40 and 50%, we found that resonant peaks randomly appeared in the spectra, and the distribution of these peaks coincided well with the frequency windows in the transmitted intensity spectra of surrounding random structures (see Fig. 2). By repeating the calculations for different distributions of scatterers with the same filling factors, we confirmed that a similar tendency was observed, and the most intense resonant peaks appeared within the frequency windows. At 50%, in particular, although resonant frequencies in each different distribution of scatterers with the same filling factors of 50% randomly appeared, the results exhibiting single or a few intense resonant peaks in each structure were noteworthy. This tendency was also confirmed for several different distributions of scatterers with the same filling factor [see Figs. 3 (b) and (c)]. However, comparing the resonant spectrum at 60% with those at 40 and 50%, it was difficult to find clear resonant peaks, and for the most part, a broad spectrum was observed. This suggests that most of the energy of the incident pulse immediately dissipated from the defect region, and only leaky modes were induced at 60%. This was also confirmed by calculating the quality factors of individual resonant peaks appeared in Fig. 3. The quality factors were estimated from their full width at half maximum of resonant peaks and the typical values were about 5 × 103, 1 × 104, and 5 × 102 for the structures with the filling factors of 40, 50, and 60%, respectively. These results suggest that the quality factors would strongly depend on the filling factors and observed modes at 60% would be more leaky rather than those at 40 and 50%.
To confirm the appearance of a localized spot bound in the defect region for each filling factor, the intensity distributions at the resonant frequencies (arrows in Fig. 3) were calculated (Fig. 4). We found that the intensity spots occurred in the defect regions located at the centers of individual structures (arrows in Fig. 4). However, at 40 and 60%, other intense spots induced at locations other than the defect region were mainly observed. In particular, at 60%, we found that distinct intensity distributions mainly appeared in higher index regions where connected scatterers occur, not in interspaces or the defect region, and spread throughout the structure, unlike the cases at 40 and 50%, as seen in Fig. 4(c). Since these spread modes would provide leak paths for energy dissipation from the defect region, we considered that the leakiness of these modes explains why only a few weak resonant peaks or a broad resonant spectrum was observed at 60%. However, when the filling factor was 50%, we found that intense spots appeared mainly in the defect region and the probability that the modes appeared besides the defect was low, compared with the cases at 40 and 60% [Fig. 4(b)]. By repeating the calculations for different distributions of scatterers with the same filling factor of 50%, we confirmed that some weak spots induced in interspaces were occasionally observed also in the case of 50%, because the distributions of scatterers were randomly generated. However, an intense and distinct intensity spot at the defect was always confirmed. Thus, the above numerical results suggested that a filling factor of about 50% was optimal for our numerical model, and a certain degree of mode control could be achieved in space (defect region) and frequency (frequency window) by utilizing our simple method.
To examine the role of a defect region and filling factor, we produced histograms of the distances from the center of one scatterer to the center of its nearest neighbor for individual filling factors. The results are shown in Fig. 5. The black vertical line indicates the size of the defect region that we set in the calculation. It seems abnormal that the frequency of distances less than 400 nm between proximal scatterers was not 0, although the scatterers had a diameter of 400 nm. Since the shapes of scatterers were not precisely circular, the distances between proximal scatterers occasionally became less than 400 nm in our numerical model.
At 40%, since many interspaces were larger than the defect compared with the cases of the higher filling factors, we considered that photons could also be localized in such interspaces, resulting in the observation of many bright intensity spots in the distribution. On the other hand, with an increase in the filling factor, the frequency of interspaces larger than the defect became smaller, and the frequency of connected scatterers simultaneously became higher. As a result, we found that the defect region becomes a specific point in the structure at high filling factors, as seen in Fig. 1(c). However, at 60%, since the filling factor of scatterers was very high and many scatterers contacted each other, the intensity distributions bound in the higher index region (bound in the connected scatterers), not in interspaces or a defect region, would be dominant. Certainly, in the distribution, fringes where the photons propagate through connected scatterers were confirmed. Therefore, a filling factor of around 50% was optimal for realizing photon confinement at the defect under the given numerical conditions. Similar results were also observed in different distributions with individual filling factors. Thus, the numerical results could be explained from the balance between the filling factor and the defect size in individual structures. However, it is necessary to note the point that even if their filling factor is kept constant, since the interspaces between scatterers become larger when the size of scatterers becomes larger, the optimal defect size should also be selected appropriately depending not only on the filling factor but also on the scatterer size.
Finally, to discuss the influence of the defect region on nonlinear phenomena, we calculated the spatial distributions of lasing emission intensities in the same structures used in the above calculations by employing a 2D-FDTD method combined with rate equations [18, 21-22, 25]. Figure 6 shows the results and arrows indicate the locations of individual defect regions. The pumping rate was 109 s-1, which was sufficiently high to simultaneously excite many lasing modes in the structure. From the results, we found that at 40%, several intense spots were also found in other interspaces, in addition to the defect region, and at 60%, mainly lasing modes bound in the connected scatterers were induced. In both cases, laser oscillations in interspaces or connected scatterers were dominantly induced rather than those in defect regions. In contrast, at 50%, an intense lasing spot was observed at the defect region, and unlike the result of Fig. 4, several bright spots were also found in the structure because the gain was so high that various modes with lower quality factors also appeared in the distribution. However, even under such conditions, the numerical result with the filling factor of 50% indicated that the lasing was distinctly induced in the defect, rather than any other intespaces or higher index regions. These results suggest the possibility that the resonant and lasing properties in random structures can be moderately controlled and limited in space and frequency domains by use of the defect region.
To achieve the control of resonant and lasing properties of random structures, we proposed a method simply by making a defect region in the structure. From the analysis performed by the use of a 2D-FDTD method combined with rate equations, we found that resonant and lasing properties could be moderately limited in space and frequency domains by selecting the size and filling factor of scatterers and an appropriate defect size according to the frequency window of a surrounding random structure. In addition, the numerical results suggested that by use of this method, laser oscillation could be dominantly induced and limited at the defect. This suggests the possibility that intended resonant and lasing properties could be realized even in “random” structures. Considering the potential for various applications and the significant attention paid to random structures, even though further numerical studies and experimental verifications are required, we believe that the proposed method for manipulating randomly appearing resonant and lasing properties in space and frequency domains can provide a novel viewpoint for the technological applications of photon manipulation within random structures.
This work was supported by PRESTO from the Japan Science and Technology Agency and partly by KAKENHI on Priority Area “Strong Photon-Molecule Coupling Fields” from MEXT.
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