## 1. Introduction

It is one of the primary interests in recent nano- and micro-photonics to achieve a strong confinement of light in a small region, because it finds a variety of applications in optical physics and engineering where it is exploited in low-threshold lasers [1], nonlinear optical devices [2], and cavity quantum electrodynamics devices [3]. Extensive efforts have therefore been devoted to developing a cavity that can confine light efficiently—a high-quality optical resonator. The quality of resonators is described here by the photon lifetime

In order to achieve high-quality optical resonators, the two directions seem to have been explored so far: one is the use of the extended waves and another is the use of the localized waves. The photonic crystals (PCs) may be the first candidate high-quality resonators, the *Q* factors for which have been found to be increased by the slowed-down light (the extended waves, or the Bloch waves in this case) near the photonic band edge [5-7]. The typical example for the exploitation of the localized waves can be found in the defect mode that is localized around a disorder in the PC [8-13], which provides more pronounced light-confinement than the band edge modes in the PCs. Although the defect itself generally occupies a very small region, this confinement requires the presence of a large periodic medium around it in order for the defect mode to be sufficiently isolated from its environment. Light can also be localized in the central part of a three-dimensional (3D) fractal structure (Menger sponge) made up of cubes that need not have high

In the context mentioned above, we describe in Sec. 3 an entirely different type of resonator: a closed chain array made up of dielectric microstructures arranged periodically in the background material (e.g., the air). We call it a photonic atoll (PA) resonator because it resembles an atoll in the ocean. This PA resonator is thought to have a prominent function to confine light very strongly for the following reasons: (1) the multiple scattering of light by the periodic quasi-one-dimensional (q1D) array causes a slowing down of extended light-waves and (2) the closed optical path forces a photon once trapped in the array to keep circulating in the loop, both of which would undoubtedly increase the photon lifetime. Factor (1) is the same as the factor responsible for the lifetime enhancement at the band edge of the PCs (see the preceding paragraph) while factor (2) reminds us of the analogy to the ring accelerator for elementary particles. Because of the features mentioned above, this PA structure could also be called a distributed feedback ring-resonator. The above concept was previously [20] applied to the PA resonator of the two-dimensional (2D) circular array consisting of the fifty rods. This resonator was actually found to create an extremely high radiative ^{15} and the resultant very long lifetime of the order of one second for visible light at the modes near the photonic band edges created in this q1D closed PC. The idea of PA was conceived during the investigation of circulating modes in a two-dimensional PC [6, 21] and a microdisk [22], so we believe that the investigation of it and its structure effects will also help us understand the behavior of light in those structures.

Since we have confirmed that the PA structure has a potential to achieve very long lifetimes, our next step of research is to investigate what kind of PA shapes would provide the most efficient optical resonator (Sec. 4). This is because the PA has the degree of freedom that permits it to have an arbitrary loop form: note that the first work (Sec. 3) has focused on a circular PA. In the process of investigations to pursue the optimum PA structure that maximizes the

Finally, we describe in Sec. 5 the laser actions in the PA resonators with extremely high

## 2. Theory

### 2.1. Multiple scattering of light

The analytic multiple-scattering theory is used here to evaluate the light confinement effects. Since the general theory is described in the reports [6, 7], here we briefly outline the framework of the calculation. We consider a 2D array consisting of a finite number

where

where

### 2.2. Modes and lifetimes

To determine the photon lifetime in the photonic atoll, we assume real dielectric constants (i.e., no optical gain) and a complex photon frequency

Here, we refer to the physical meaning of the above method for determining the photon lifetimes. The imaginary part

### 2.3. Threshold amplitude-gain for laser oscillation

In the calculation of lasing thresholds [6] in the photonic atoll, we assume that every rod has the same optical amplitude gain

where the photon frequency *unlasing modes*, which do not laser-oscillate even under very high

## 3. Modes and lifetimes in photonic atolls

The schematic photonic-atoll structure is shown by the inset in Fig. 1. It consists of periodically arranged 50 GaAs rods (with the dielectric constant

### 3.1. Mode distributions

Figure 1 shows the distribution of optical modes and

Let us call these bands #1, #2… etc. from lower to higher frequencies. The #2 and #4 bands are very narrow, but the others are so wide they can be regarded as real bands. These narrow bands are not localized modes, however, because this structure does not contain disorders causing light localization. Actually, these modes have extended (unlocalized) distributions of the light intensity [see Fig. 3(b)]. Although these

The significance of the above results can be better understood by comparing them with the results obtained with other resonators. The ^{4} [8, 9], which are still lower than the present

### 3.2. Filling-factor effects

The finding of bands and band gaps has given impetus to the study of filling-factor effects, as is often carried out in the ordinary PCs [26]. Figure 2 shows the variation of the positions of bands (shaded areas) and band gaps (blank areas) as a function of the *f* value. Here, we focused on bands and band gaps created by lasing modes (solid lines in Fig. 1), because they are verified later to be generated along the rod loop (see Fig. 3). The vertical broken line corresponds to Fig. 1. Let us scan the results from low to high *f* splits band #3 to create a new narrow band #2. This new band remains narrow until *f* reaches its maximum value. A similar phenomenon occurs in band #5, which splits to form a narrow band #4 around

### 3.3. Light intensity distributions

In order to clarify what occurs for these modes, we next investigate the field intensity distributions in the photonic atoll. Note again that no gain is assumed for this calculation. We first select several lasing modes from band #1, because the modes in the first band with relatively long wavelengths are expected to provide a variety of clues to the understanding of the fundamental processes of light localization. Figure 3(a) shows the light-intensity distributions for the four lower lasing modes in band #1 (indicated by arrows in Fig. 1). In the colored figures, the intensity increases in the order blue, white, yellow, red, and black. The numerals in the figure are the mode frequency values and from left to right in Fig. 3(a), they respectively correspond to

Several examples for the modes with very high ^{11} and the last one (^{5}. Note that this intense light is obtained in the array with no optical gain (no gain is assumed here!). It is entirely due to the extremely long photon lifetimes attained by the use of photonic-atoll resonators. Although light is focused on the rod array, its intensity distributions are not easy to construe. Let us focus on the bright regions, which are the loops of light waves. While the contour of bright regions for the mode in band #1 (

Shown in Fig. 3(c) are for the modes that never lase even if they have very high gains (the modes shown by dotted lines in Fig. 1). Light for these modes is clearly confined in the inner region of the atoll but not along the rod array. We see an increase in the number of loops and nodes as the mode frequency increases, indicating that they are formed by light trapped in the inner region and reflected at the rod array of the atoll. The observed

## 4. Shape effects of photonic atolls

In this section, we assume the PA that consists of 20 GaAs rods (with the dielectric constant *e* from 0 to 0.968. For all the PAs studied here, however, the filling factor *d*: rod radius,

### 4.1. Splitting of degenerate modes

Prior to showing the detailed properties of the PAs, we first present the basic results for the optical modes created in the PA. Figure 4 shows the angular frequency *n*=0, 1, 2, …, 10 for low to high *n*=1, 2, …, 9. Note that no twin modes are created for *n*=0 and 10.

The similar studies have been carried out for elliptical PAs with a variety of eccentricities. The results are summarized in Fig. 5, which shows the variation of the mode frequency as a function of the eccentricity *e* (from 0 to 0.968). As clearly shown in Fig. 5, each of modes 1-9 is found to split into two with the increasing eccentricity. Those modes that are increasing and decreasing, respectively, with the growing eccentricity are denoted by open and close circles. While we succeeded in locating almost all split-modes very precisely, we failed in isolating several higher modes (open circles) split for mode 1 at *e* values (*e* values studied here. However, noteworthy here is that mode 0 slightly increases while mode 10 decreases to a certain extent, according as the eccentricity grows. This fact again suggests the difference in the behavior between the near-

Figure 6 shows the mode separation *e* value for each curve is given in the caption of Fig. 6. As mentioned before, modes 0 and 10 have no split modes. What is intriguing here is that the separation

### 4.2. Light intensity distributions

The results mentioned in Sec. 4.1 have prompted us to investigate the light field distributions for these modes. Hereafter, we focus on the PA with *n*H and *n*L, respectively, for mode *n*.

Figure 7 shows the light intensity distributions for (a) mode 2L with

Next, we display the results for the modes near the band edge as a matter of convenience for explanation. Figure 8 shows the light intensity distributions for (a) mode 9L with

Finally, we briefly mention the behavior of the modes in the middle of the band. Figure 9 shows the light intensity distributions for (a) mode 5L with

### 4.3. Discussion

Let us discuss the creation of eigen modes and their splitting by the structural modification of the PA resonator. For this purpose, we simplify the discussion by regarding the closed q1D chain as a closed pure-1D system with the position variable *x* along the circumference. The periodic boundary condition can be applied to this system exactly for its closed structure, and the Bloch theorem for the *L*-periodicity of the PA. We thus obtain the optical modes specified by the wave number *n* ranges over –9,…, –1, 0, +1,…, +9, and +10, creating 20 modes. Note that *n*= –10 is excluded since it is identical to *n*= +10 in the reciprocal space. The modes with *n* are thus known to be degenerate with those with –*n* for *n*=1, 2,…, 9 and their wave functions are the complex conjugate of each other, which propagate in the opposite directions with the wave numbers

When we look at Figs. 4-6 together with the above considerations, it is not unusual for modes 0 and 10 to remain single under any perturbations given to the structure because of their nondegeneracy. As for the other modes (*n*=1, 2,…, 9), it is reasonable to consider that their degeneracy is lifted by the modification of the PA structure. If we apply the group theory to this phenomenon straightforwardly, it may be said that the degeneracy lifting is caused by the reduction of the rotational symmetry in the whole PA structure. Although this is an elementary but important interpretation for these degeneracy-lifting phenomena, we here refer to another perspective. Under the assumption to regard the PA as a very long closed 1D structure—it is actually possible as mentioned before, the waves propagating in the opposite directions (*N*: we obtained 0.0097, 0.0060, and 0.0036 for *N*=10, 20, and 50, respectively, for, e.g., mode 1 at *n*-dependence of *n*, light is loosely bound around the PA because of their shorter lifetime, which ought to render its intensity distributions more sensitive to the PA modification. The modes near the band edge, on the other hand, have the lifetime that is barely retained very long in a fixed (symmetric) PA structure. This implies that their life is vulnerable even to a slight perturbation to the structure and hence its abrupt reduction may cause marked splittings of degenerate modes.

As mentioned on the light field distributions in Sec. 4.2, the structurally deformed PAs have a variety of optical responses. In particular, the band-edge modes (e.g., modes 9H and 9L) exhibit a strong anisotropy of excitation. Moreover, it should be emphasized that this anisotropy is very sensitive to the modification of the structure, i.e., it occurs even under a slight modification of the structure. This implies that optical excitations can be controlled by the mechanical deformation of the structure, which could have a potential to be exploited as high-function devices such as opto-mechanical devices [28]. We therefore believe that the present results will find a number of valuable applications as very high-

## 5. Laser oscillations

Because of the scaling rule that holds in our calculation in a similar manner to in the PCs, *d* nor *L*). We here use the

We studied the characteristics of a photonic atoll as a laser oscillator by assuming that every rod has the same optical amplitude gain

## 6. Conclusion

We have theoretically demonstrated that very high *Q* factors and resultant very long photon lifetimes can be achieved by using the closed periodic array of microstructures, which we call a photonic-atoll (PA) resonator. Although other possible losses of light remain to be considered before this structure is put to practical use, the results we obtained suggest that it would be an excellent structure for confining light. In particular, the fact that it does not require a large size to achieve a strong light confinement will prove a great advantage over other ways of light confinement when it is incorporated into optical integrated circuits. Through the investigation for the PAs with a variety of elliptical forms, we found that the photon lifetime is maximized for the symmetric (or circular) form of the resonator. This structure deformation was also shown to give rise to the degeneracy lifting for eigen modes: even a slight deformation created pronounced splitting widths especially for the near-