We propose a simple structure for manipulating resonant conditions in random structures, which is composed of a waveguide structure as a defect region embedded in a random structure. Using the two-dimensional finite-difference time-domain method, we examine the resonant properties of localized modes bound in the waveguide. From the results, we confirm that long-lived modes are strongly confined in the waveguide only when the resonant frequency matches the frequency windows in the transmitted intensity spectrum of the surrounding random structure.
© 2009 OSA
Wavelength-scale-disordered structures composed of strong scattering nanoparticles (random structures) have attracted attention as unique microcavity structures because of their potential for photon localization due to the interference of multiply scattered light [1–21]. Utilizing nanoparticle assembly, self-generated surface roughness, or biological tissues, random structures have suggested a potential for the realization of easily fabricated and low-cost applications, such as low-threshold nonlinear optical devices. Although spectral and spatial overlaps between long-lived modes and materials doped in the random structure are required to achieve highly efficient light-matter interactions within the structures, their resonant frequencies and positions are randomly determined by local conditions of the structure. Therefore, it is difficult to induce intended long-lived modes in random structures. In addition, since such long-lived modes would typically exist deep inside the structure because of leakage loss from the surface, it is also difficult to access these long-lived modes from the outside or collect their output because of multiple scattering. Therefore, technological developments for manipulating intended modes, including their location and frequency as well as input-output characteristics within the structure, would be indispensable.
Among several proposals for mode control [12–27], we have numerically demonstrated a simple method to control the resonant properties by deliberately making a point-like defect region, in which no scatterers are arranged, within a random structure . On the basis of previous studies by Miyazaki et al.  and Rockstuhl et al. , in which their numerical analysis suggested that the transmittance of an ensemble of scatterers exhibited a sharp dip at certain frequency bands (frequency windows) due to modal coupling between the Mie resonances of neighboring scatterers, we have numerically demonstrated that the resonant conditions can be manipulated in the defect region within a random structure by optimizing the structure parameters (filling factor and sizes of scatterers and size of defect) . On the other hand, improving the excitation efficiency of long-lived modes typically existing deep inside a random structure has been achieved experimentally by multiphoton excitation [9,10]. Since the influence of multiple light scattering on the excitation light can be much weaker than that on emission, according to Rayleigh’s λ−4 law, the excitation light can penetrate deeply into a random structure, while up-converted emission can be strongly confined in small volumes. Using multiphoton excitation, we successfully demonstrated the up-conversion lasing of a random medium composed of Tm3+-doped glass powder and TiO2 nanoparticles with an extremely low threshold of several kW/cm2 . However, although multiphoton excitation certainly makes it possible to efficiently excite internal long-lived modes within which strong light-matter interactions can be achieved, it is also difficult to take the outputs from these modes to the outside of the structure, as photons from these modes were scattered and emitted in random directions.
To address this issue, we have conceived that a waveguide structure can be used as a defect region within a random structure, similar to photonic crystal waveguides, and is expected to improve both the input-output characteristics and the control of resonant conditions of long-lived modes. The results reported by Topolancik et al. suggested useful information regarding such a waveguide structure [28,29]. They experimentally demonstrated that the shape disorder of airholes in photonic crystal waveguides played an important role in the realization of strong localization in the waveguide due to the coherent interference of scattered waves from periodic airholes. However, their structure was still a highly ordered structure, unlike that considered here. Another idea for guiding random laser output using a waveguide were also experimentally demonstrated by Watanabe et al.  and Yuen et al . In Ref. 17, a dual-layered waveguide dye laser was demonstrated, in which a waveguide layer was over coated on a random active layer, whereas in Ref. 18, random laser action was also observed from a ridge waveguide formed on a disordered ZnO thin film. In both cases, they suggested that lasing characteristics were improved by using the waveguide structures. However, these proposed structures are markedly different from our proposed structure. Among several proposals using waveguide structures, Miyazaki et al. also proposed a defect waveguide structure within uniformly distributed scatterers , which was very similar to the structure we propose in this paper. From the numerical results, they suggested that the electric field could be almost completely confined within the waveguide. However, in their numerical proposal, in order to confine photons within the waveguide, a wavelength-scaled periodic sidewall for the defect waveguide was used. Therefore, for making a wavelength-scaled periodic sidewall along the waveguide region, a nanofabrication technique would be necessary and, therefore, the advantage of the easy fabrication of random structures would be spoiled.
In this paper, to improve the input-output characteristics of long-lived modes within random structures as well as the control of the resonant conditions, we propose a simple structure composed of a slab waveguide surrounded by randomly distributed scatterers as a defect region, instead of using a periodic sidewall. By employing a two-dimensional finite-difference time-domain (2D-FDTD) method [22–25], we numerically analyze the resonant properties of long-lived modes bound in the waveguide within the random structure. From the analysis, we found that long-lived modes strongly bound in the waveguide could be realized, while waves at off-resonant frequencies were rapidly dissipated from the structure. Thus, we expect that the proposed method would provide a useful technique for accessing or manipulating long-lived modes in the random structure and would also facilitate application of such modes in integrated photonic devices or circuits using random structures.
2. Simulation and analysis
Figure 1 shows a typical spatial distribution of scatterers (solid circles) and a waveguide (gray line). The numerical model was composed of a waveguide as a defect region (width 900 nm, length 52.5 µm, refractive index 1.5) embedded in the center of randomly distributed dielectric circular scatterers (diameter 400 nm, refractive index 2.6, surface filling factor 50%). The dispersion area of the scatterers was 5.9 × 62.5 µm2. When setting up the distribution of scatterers, we allowed contact between scatterers, but did not allow them to overlap each other or the waveguide. The surrounding medium was assumed to be air (refractive index 1.0). The size of the entire calculation area was set to 30 × 100 µm2. The filling factor and size of scatterers were referenced from our previous paper , in which we confirmed that a localized intensity spot was clearly induced only in a point-like defect region at a 50% filling factor under the conditions for scatterers described here.
Using a 2D-FDTD method [22–25], we calculated resonant spectra and intensity distributions of the numerical model. We considered only the electric field parallel to the cylinder’s axis. The time step and cell size were set at 7.0 × 10−17 s and 50 nm, respectively. Each calculation was performed for 106 steps (70 ps), and Mur’s second-order absorbing boundary condition was used. Light sources for exciting a fundamental waveguide mode in the waveguide were set at the left edge of the waveguide and generated light waves were propagated only to the right along the waveguide. For creating intensity distributions, the light sources were excited by cosine waves at a given frequency with optimal phases for exciting a fundamental waveguide mode. To obtain resonant spectra, a short Gaussian pulse (duration time about 3.5 × 10−15 s, center frequency about 280 THz, and spectral width about 370 THz) was launched from the light source, and the electric fields of the reflected light waves were recorded at the left side of the light source during the entire calculation time (70 ps). Then, because we wanted to examine only long-lived modes, only about the last 1.3 × 105 steps of the recorded signals (from 63 to 70 ps) were Fourier-transformed.
3. Results and discussion
We calculated the reflected intensity spectrum from the waveguide in the random structure [curve (a) in Fig. 2 ]. In addition, to investigate the influence of the frequency windows of the surrounding random structure, we calculated the transmitted intensity spectra of random structures with a different distribution of scatterers from Fig. 1 and no waveguide [curve (b) in Fig. 2]. This calculation was different from the procedure described in the previous section, a plane electric field was made to impinge for about 10−15 s on a structure with a thickness of 10 µm and filling factor of 50%, and we recorded the transmitted electric fields at 10 detection points on the opposite side. In addition, the reflected intensity spectrum of the waveguide shown in Fig. 1, but without the surrounding random structure, was also calculated for reference [curve (c) in Fig. 2].
From the resonant spectrum indicated by curve (a) in Fig. 2, we found sharp resonant peaks randomly appearing in the spectrum. Their resonant frequencies coincided well with the frequency windows in the transmitted intensity of the surrounding random structure [curve (b) in Fig. 2], which shows three sharp dips around 170, 290, and 400 THz. By repeating the calculations for ten different distributions of scatterers with the same filling factor and waveguide structure, a similar tendency was also observed, and the resonant peaks of long-lived modes appeared only within the frequency windows. The circle with error bars in Fig. 2 indicates the average resonant frequency and its deviation (286 ± 6 THz); we confirmed that the resonant peak positions of long-lived modes coincided well with the frequency windows. Indeed, when we Fourier-transformed the recorded electric fields for every 10 ps, we confirmed that spectral peaks at frequencies outside the frequency windows completely disappeared within 10 ps. This result suggested that these peaks were leaky modes, which should be immediately scattered out from the waveguide and also from the random structure. In addition, from the calculation of the reflected intensity spectrum of the waveguide alone [curve (c) in Fig. 2], we clearly found that there was no distinct resonant peak, only weak ripple structures due to interference between waves propagating back and forth in the waveguide. From these results, the observed sharp resonant peaks originated from the surrounding scatterers, not from interference in the waveguide. Therefore, we considered that the surrounding scatterers could work as filters or mirrors with specific frequency bands and give random feedback depending on the frequency windows. Thus, the data suggests that the observed resonant peaks in the frequency windows could be long-lived modes surviving only within the waveguide.
Note that it seems strange that no resonant peaks of long-lived modes were observed in the first frequency window around 170 THz [curve (a) in Fig. 2], since the first frequency window exhibits higher visibility than the second and third widows. This could be simply explained by the insufficient thickness of the surrounding random structure. The thickness of the random structure surrounding the waveguide was 2.5 µm, which is comparable to the wavelength of the first frequency window, while the thickness of the random structure used to calculate the frequency windows was 10 µm for the clarity of frequency windows. Since spatial intensity distributions at lower frequency would be broader than those at higher frequency because of the diffraction limit, leakage from the boundary of the structure at the resonant frequency within the first frequency window would become larger than that at the second and third frequency windows. Indeed, although resonant peaks in the first frequency window could also be observed when only the first 10 ps of the recorded signal was Fourier-transformed, these peaks rapidly disappeared within the next 10 ps, suggesting that these modes were short-lived (leaky) modes. In addition, we also note that when the thickness of the surrounding random structure was increased to 5 µm, we could also observe the resonant peaks of long-lived modes. Thus, the insufficient thickness of the surrounding random structure was the reason no long-lived modes were observed at the first frequency window.
To confirm the intensity distribution of long-lived modes appearing in Fig. 2, we calculated these distributions at on- and off-resonant frequencies. Figure 3 shows the results at (a) on- and (b) off-resonant frequencies in the spectrum of Fig. 2 (281 and 325 THz, respectively). Each distribution was normalized by each maximum, and the maximum intensity of the image at the off-resonant frequency was about 103 times smaller than that at the on-resonant frequency. At the on-resonant frequency (281 THz), we clearly found that the intensity distribution was strongly bound in the waveguide, and no distinct mode was observed in the surrounding structure. However, when the frequency was set at the off-resonant condition (325 THz), the intensity distribution was spread over the structure, in contrast with the on-resonant case, and immediately dissipated from the structure. As seen in Fig. 3(b), a light wave incident on the waveguide at the off-resonant frequency was strongly dissipated by surrounding scatterers and leaked from the structure, resulting in steep intensity decay along the waveguide. By repeating the calculations at different resonant frequencies, we confirmed that the intensity distributions exhibited different resonant mode distributions from the result shown in Fig. 3, but the individual distributions were strongly bound in the waveguide and showed similar tendencies.
In our previous paper , we found that resonant peaks coincided well with the frequency window and that a filling factor of 50% was optimal for filling out other interspaces to induce other localized spots in the surrounding random structure at the resonant frequency. From these facts, we considered that when the frequency matches one of the frequency windows, most light waves cannot penetrate into the surrounding random structure and thus are reflected back to the waveguide. Therefore, the surrounding scatterers could work as filters or mirrors with specific frequency bands and give random feedback depending on the frequency windows. However, waves at frequencies outside of the windows were scattered out from the waveguide and leaked through the surrounding random structure, thus inducing only leaky modes. As a result, we concluded that long-lived modes could survive only in the waveguide structure, not within the surrounding random structure. Thus, the numerical results of the resonant spectra and the intensity distributions suggest that a certain degree of localized mode control can be achieved in space (waveguide) and frequency (frequency window) regions using our proposed structure.
In the above discussion, the existence of the long-lived mode was attributed solely to the surrounding random structure. However, the details of the mechanism for inducing the long-lived modes are still unclear and it might be more complicated. In Fig. 3(a), the intensity pattern in the waveguide suggests that the fundamental mode of the waveguide is coupled to a higher order mode. This situation looks similar to the discussions of a ministop band in a photonic-crystal waveguide, due to the anticrossing between the fundamental and higher order modes of the waveguide caused by the surrounding photonic crystal . Therefore, we should note the possibility that the long-lived modes are induced inside the waveguide because of the ministop band as in the photonic crystal waveguide. However, the mechanism of the long-lived modes is still under investigation and remains as our future work.
To manipulate long-lived modes in random structures, we proposed a simple structure composed of a waveguide embedded in a random structure. From numerical analysis using a 2D-FDTD method, we found that resonant peaks of long-lived modes in the spectra appeared only when the resonant frequency matched the frequency windows of the surrounding random structure. In addition, the intensity distributions indicated that photons at the resonant frequencies could be strongly confined within the waveguide, while those at off-resonant frequencies rapidly dissipated from the waveguide and the random structure. These results suggest that intended long-lived modes could be realized in a waveguide structure embedded in a random structure, and therefore, we have believed the possibility that the long-lived modes could be easily accessed from outside of the random structure via the waveguide. Although further numerical studies and experimental verifications are necessary, we expect that the potential of the proposed structure to improve the input-output characteristics as well as to manipulate the resonant conditions of long-lived modes within a waveguide can open novel possibilities for technological applications, such as integrated photonic devices or circuits using random structures.
This work was supported by the PRESTO program of the Japan Science and Technology Agency and partly by a KAKENHI grant in the Priority Area “Strong Photon-Molecule Coupling Fields” from MEXT.
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