## Abstract

Defected circular resonators laterally confined by a metal layer with a flat side as an emitting window are numerically investigated based on the boundary element method for realizing unidirectional emission microlasers. The results indicate that Fabry-Pérot (FP) modes become high *Q* confined modes in the defected circular resonator with a metallic layer. The mode coupling between the FP mode and chaotic-like mode can result in high *Q* confined mode for unidirectional emission with a narrow far field pattern.

©2013 Optical Society of America

## 1. Introduction

Semiconductor microdisk lasers [1] are suitable for realizing ultra-low-threshold operation due to the ultra-small volumes and high *Q* factors. However, the output power and directional emission of microdisk lasers are greatly limited by the rotational symmetry of the circular cavities, and the high efficiency unidirectional emission from the microlaser is a key issue for practical applications. Many works are focused to realize the directional emission semiconductor microlasers, such as deformed circular resonators [2–5], circular resonators evanescently coupled to a bus waveguide [6,7], and microresonators directly connected to a bus waveguide [8–10]. Circular resonators connected with a bus waveguide were proposed to realize directional emission based on coupled mode between two whispering-gallery modes (WGMs) with equilateral-polygonal shape mode pattern [8]. Limaçon cavities based on the dipolar smooth deformation of the circle were proposed for realizing high-*Q* modes with unidirectional light emission [11]. In addition, the directional emission microlasers were expected in dielectric deformed cavities due to mode coupling or dynamic tunneling from an isotropic high *Q* WGM to a directional emitting low *Q* mode [11, 12].

The classical and quantum chaos of a particle in a circular billiard with a straight cut was studied in [13], with the billiard as integrable or nonintegrable at various cut size. In addition, the characteristics of deformed dielectric circular cavities were studied, and the Far-field emission patterns were compared numerically and experimentally [14]. Recently, a defected circular resonator laser confined laterally by a metallic layer with a flat side was fabricated and realized wide-angle far-field emission with a low threshold current of 2 mA [15]. The high reflectivity provided by a metallic layer can support additional high Q modes in the deformed resonators confined by the metallic layer. Metallic confinement can be used to realize lasing in nanocavities [16]. In this paper, we investigate the mode properties for defected microresonator confined by a metallic layer with a flat side based on the boundary element method (BEM) [17, 18]. The results indicate that the well-designed defected cavities can have high *Q* confined modes for unidirectional emission with a narrow far-field pattern.

## 2. Defected microdisk with laterally confined metallic layer

The defected circular microresonator in the air, with a flat side as an emission window and the other parts of the perimeter of the resonator confined by a metallic layer, as shown in Fig. 1(a)
, is simulated by the BEM method. The refractive indices are *n*_{1} = 3.2 and *n*_{2} = 0.18 + 10.2i for the resonator and the metallic layer Au at the wavelength of 1.55 μm. The metallic layer thickness is *t* = *R*_{2} - *R*_{1}, where *R*_{1} and *R*_{2} are the radius of the circular resonator and the external boundary of the metallic layer, respectively, the distance between the center of the microcircular and the flat side is *d*, and the length of the flat side is *l*.

Defected microcircular resonator with *R*_{1} = 4 μm and *t* = 100 nm is simulated. First we simulate light transmission inside the defected microresonator with *d* = 3.98, 3.7 and 3 μm based on ray motional dynamics, and plot the ray trajectories of 200 reflections in Figs. 1(b)-1(d) for an initial light ray with an incident angle of 60°. In Fig. 1(b), the defected degree is very small and the light ray trajectory is still similar to a WGM in the circular resonator. In Fig. 1(c), the light ray trajectory displays equilateral-polygonal shape, as the coupled mode between two WGMs [8]. The light ray trajectory in Fig. 1(d) is near a Fabry-Pérot mode of a two-bounce orbit oscillating upper and down.

## 3. Scattering spectra and mode characteristics for defected microresonator

To investigate the mode characteristics in detail, we solve the Maxwell equation numerically for transverse magnetic (TM) modes in the defected microresonator by using BEM [17]. We first calculate the scattering spectra of the resonator under the exciting of an incident plane wave, by integrating the scattering intensity around the resonator. The obtained scattering spectra are plotted in Fig. 2
for the defected circular microresonators with *R*_{1} = 4 μm at *d* = 3.9 μm (a) and 3.1μm (b), respectively. The numerical results indicate that the number of high *Q* confined modes decrease with the increase of the flat side length. So the flat side length can be used for suppression higher order modes.

Since the whole structure has vertical symmetry, all modes can be assigned as symmetric and anti-symmetric modes relative to horizontal axis. At *d* = 3.9 μm, we can obtain symmetric and anti-symmetric confined modes with the mode wavelengths of 1608.6 nm and 1608.3 nm, and the mode *Q* factors of 1500 and 8600. A small wavelength interval between the symmetry and the anti-symmetric modes comes from the perturbation of the short flat side. The anti-symmetric mode has higher *Q* factor than the symmetric mode, because the symmetric mode has higher radiation loss. The squared mode electric field patterns and the far-field patterns (FFPs) are plotted in Fig. 3(a)
for the symmetric mode and (b) for antisymmetric mode, respectively. The FFP of the symmetric mode is very wide with the full width at half maximum (FWHM) of 48° and the FFP of the anti-symmetric mode has two petals. For the defected microresonator with *d* = 3.1 μm, corresponding to the length of the flat side *l* = 5.06 μm, we have two high *Q* modes, as shown in Fig. 2(b), with the mode wavelengths of 1634 nm and 1583.2 nm for the symmetric and anti-symmetric modes, respectively, and the corresponding mode *Q* factors of 3279 and 8135. The squared mode electric field patterns and FFPs are plotted in Fig. 3(c) for the symmetric mode and (d) antisymmetric mode at *d* = 3.1 μm. The mode intensity patterns in Fig. 3(c) and 3(d) are similar to that of the Fabry-Pérot modes. The FFP of the symmetry mode in Fig. 3(c) has a very narrow main petal and two small side petals. The radial mode numbers, i.e., the number of the maximum in the radial direction, are 15 and 16 for modes in Figs. 3(a) and 3(c), respectively. The large mode wavelength intervals between the symmetric and anti-symmetric modes make it possible to control the modes independently. The obtained mode *Q* factors ranged from 3279 to 8600 are in the same order as that in a 300-μm-length Fabry-Pérot resonator with cleaved mirrors. In fact, the high *Q* mode with a large radial order number has mode light ray nearly vertically impinging on the metallic layer, and experiences a large reflectivity and a small absorption loss of the metallic layer.

## 4. Symmetric mode quality factor and FFP at different flat length

In the following section, we focus on the characteristics of the symmetric mode. The mode wavelengths versus the flat side length are plotted in Fig. 4
for the symmetry modes classified as branches 1’ and 2’. The results show that the mode wavelength of the branch 1’ decreases from 1636.6 to 1634.3 nm as the flat side length increases from 4.956 to 5.018 μm, then takes near constant value around 1634 nm as the length increases from 5.02 to 5.16 μm, and decreases from 1633.8 to 1632.8 nm as the length increases from 5.16 to 5.19 μm again. Mode wavelength of the branch 2’ varies with the length *l* similar to that of the branch 1’. Furthermore, we can confirm mode coupling from the anti-crossing of the mode wavelengths around *l* = 5.16 μm for the branches 1’ and 2’.

The mode quality factor and directionality factor, defined as the ratio of the integrated intensity of the FFP among ± 45° to the total integrated intensity, are plotted as functions of the flat side length in Fig. 5(a)
and 5(b) for the branches 1’ and 2’. The modes have high *Q* factors as the mode wavelengths varied slowly with the length *l*. The mode *Q* factors are larger than 3000 and 9000 for the branches 1’ and 2’, respectively, as 5.05 μm < *l* < 5.13 μm and 5.20 μm < *l* < 5.32 μm. Furthermore, the mode *Q* factors reduce as the mode coupling happens. In addition, the directionality factor rises from 0.377 to 0.838 as the flat side length increases from 4.956 to 5.104 μm in Fig. 5(a). For the branch 2’, mode *Q* factors are 10969 and 9837 and the directionality factors are 0.26 and 0.67 at *l* = 5.23 and 5.268 μm.

The mode intensity patterns and the FFPs are plotted in Fig. 6(a)
-6(c) for the branch 1’ at *l* = 4.96, 5.1 and 5.19 μm, and in Fig. 6(d) for the branch 2’ at *l* = 5.268 μm. The mode intensity pattern in Fig. 6(a) at *l* = 4.96μm is still similar to that of WGMs, and the mode pattern at *l =* 5.19 μm is one of chaotic-like modes as in non-integral system [13], which has very good unidirectional emission, and those in Fig. 6(b) at *l* = 5.1 and Fig. 6(d) at *l* = 5.268 μm like Fabry-Pérot modes. The Fabry-Pérot type modes are less affected by the opening of the flat side, so the corresponding mode wavelengths vary slowly with the length *l* as shown in Fig. 4, and the corresponding mode *Q* factors take peak values. In contrast, the WGMs and chaotic-like modes have lower mode *Q* factors than the Fabry-Pérot type modes and suffer more influence from the flat side. As the flat side length changes, the mode coupling between chaotic-like mode and FP mode can result in unidirectional emission from the opening window of the flat side. By precision control of the flat side length, we can realize a narrow far-field emission from the defected microcircular resonator. At the optimal flat side length of 5.1 μm, we have a mode *Q* factor of 3895 and the corresponding FWHM of the FFP of 6° for branch 1’. For the mode of branch 2’ at *l* = 5.268μm, the mode *Q* factor is 9837 and the directionality factor is 0.67.

To further investigate the properties of the confined modes, we also calculate the Husimi projection [19] onto the Poincaré surface of section (SOS). Figure 7
shows Husimi projections of two mode fields in Fig. 6 at *l* = 5.1 and 5.19 μm. The vertical axis is sinχ, where χ is the incident angle of the mode light ray, and the horizontal axis *θ* is the angle at the boundary in the polar coordinate, with the two vertical lines corresponding to the angles of two ends of the flat side. The two horizontal lines mark the critical angle χ_{c} = ± asin(1/*n*_{1}) for the total internal reflection. The outgoing mode light rays are located in the center rectangle region formed by the four lines, due to the metallic confinement and the total internal reflection. The high *Q* Fabry-Pérot-like mode as shown in Fig. 7(a) mainly locates at the polar position *θ* = ± π/2 with low incident angle, and has very weak distribution at the emission window of the center rectangle region. But the low *Q* mode in Fig. 7(b) has a large proportion in the emission window with a good directional emission.

Without metallic confinement, the Fabry-Pérot-like modes cannot have high mode *Q* factors in the defected circular microresonator. With the help of the metallic confinement, the mode coupling between Fabry-Pérot-like and chaotic-like modes can result in high unidirectional emission. The highest mode *Q* factor about 10^{4} is in the same order of *Q* factor determined by the material absorption loss of the semiconductor laser materials and is suitable to realize high output efficiency, although is far less than the *Q* factors of 10^{7} in Limaçon cavities with a radius of 3.75μm [11]. Furthermore, the confined mode with higher *Q* factor and good directional emission can be expected in a bigger defected resonator based on the mode coupling for potential high power emission.

## 5. Conclusion

In conclusion, we investigate the mode characteristics for defected circular microresonators confined by a metallic layer with a flat side as output window by the boundary element method. The metallic layer confinement can make the Fabry-Pérot-like modes becoming high *Q* confined modes in the defected microcircular resonator. In this non-integral system, the Fabry-Pérot-like mode and chaotic mode suffer different influence from the cut size, and the mode coupling between the high *Q* Fabry-Pérot-like mode and low *Q* chaotic-like mode result in high *Q* confined mode for directionality emission with narrow far-field pattern.

## Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grants 60838003, 61006042, 61106048, and 61061160502, Science Fund for Creative Research Group under Grant 61021003, and High Technology Development Project under Grant 2012AA012202.

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