## Abstract

Vertical radiation loss and far-field pattern are investigated for microcylinder lasers by 3D FDTD simulation and experimentally. The numerical results show that an output waveguide connected to the microcylinder resonator can result in additional vertical radiation loss for high *Q* coupled modes and affect the far field pattern. The vertical radiation loss can be controlled by adjusting the up cladding layer thickness. Furthermore, two lobes of vertical far-field patterns are observed for a 15-μm-radius microcylinder laser connected with an output waveguide, which confirms the vertical radiation loss.

©2013 Optical Society of America

## 1. Introduction

Microdisk lasers have received a great deal of attention for two decades for their potential applications in photonic integrated circuits and optical interconnections, because of their ultralow threshold and small mode volume [1–3]. Semiconductor microdisk lasers supported by a pedestal have a strong vertical waveguide of air/semiconductor/air for whispering-gallery modes (WGMs). Compared to the microdisk cavity, microcylinder cavities with vertical semiconductor waveguide will have better thermal conductivity and current injection efficiency [4]. High-quality (*Q*) WGMs were observed in the microcylinder cavity with upper and lower distributed-Bragg reflectors [5]. For a microcylinder cavity fabricated from a common edge-emitting laser wafer, TE WGMs at wavelength around 1550 nm can have high *Q* factors as the radius of the microcylinder is larger than 5 μm, but TM WGMs can have high *Q* factors even the radius down to 1 μm [6,7]. Furthermore, mode *Q* factors can be adjusted for TE WGMs in a small microcylinder cavity based on the destructive interference effect of the leak waves [8, 9].

The directional emission of circular microcavity lasers is limited by their rotation symmetry. Deformed chaotic resonators [10, 11], coupled microcavities [12] and microdisks evanescently coupled to a bus waveguide [13] were applied to realize directional emission from WGMs based microlasers. Recently, connecting a waveguide to circular [14] and wave-chaotic [15] microcavities was proposed for realizing unidirectional emission, which allows a large fabrication tolerance compared to the evanescent wave coupling. Waveguide connected microcylinder lasers were realized continuous-wave electrically injected operations at room temperature [16,17]. However, the vertical far field patterns with multiple lobes are observed in a 15 μm radius circular microlaser connected with an output waveguide [18].

In this paper, the vertical radiation loss and far-field pattern are investigated for microcylinder lasers with an output waveguide using three-dimensional (3D) finite-difference time-domain (FDTD) technique. The high order radial WGMs involved in the high *Q* coupled mode will result in vertical radiation loss for the coupled mode in a microcylinder cavity with a large lateral size. Vertical-far-field patterns with two peaks are observed in a 15-μm-radius microcylinder laser with an output waveguide.

## 2. Numerical model

The perfect 3D microcylinder resonators, i.e., without an output waveguide, were simulated in a 2D space based on circular symmetry in [8,9]. Now we perform 3D FDTD simulations in real 3D space even for a perfect microcylinder resonator. Microcylinder resonator connected with an output waveguide in the radial direction as shown in Fig. 1(a) is simulated by 3D FDTD technique. The microcylinder in air consists of a center active layer with a thickness of *t _{g}*, upper and bottom InP cladding layers with thicknesses of

*t*and

_{u}*t*on the InP substrate. The refractive indices of air, the active layer and InP are set to be 1, 3.4 and 3.17. The coordinate system origin is chosen at the center of the active layer and the microcylinder. TE and TM WGMs are assigned based on the main electro-magnetic field components

_{b}*H*,

_{y}*E*,

_{x}*E*and

_{z}*E*,

_{y}*H*,

_{x}*H*, respectively. The spatial steps

_{z}*Δx*,

*Δy*and

*Δz*are 20, 30 and 30 nm, and the time step of 0.047 fs satisfies the Courant condition. A wide bandwidth exciting source is used to excite multiple modes, and the recorded FDTD output is then transformed into frequency-domain by the Padé approximation for calculating mode wavelengths and

*Q*factors [19]. Narrow-bandwidth exciting source centered at a mode wavelength

*λ*is used to simulate the mode-field distribution. Based on simulated near-field pattern

*U*(

*x*

_{0},

*y*

_{0}) at a

*z*

_{0}plane in air near the output waveguide, far-field pattern

*F*(

*x*,

*y*) at

*z*plane is calculated by Helmhots-Kirchohoff's diffraction formula:

*r*is the distance between (

*x*,

*y*,

*z*) and (

*x*

_{0},

*y*

_{0},

*z*

_{0}), and

*z*is 1 cm in the calculation.

## 3. Numerical results for TE and TM WGMs

The 2-μm-radius microcylinder resonator with 0.4-μm-width waveguide is simulated as *t _{g}* = 0.3 μm and

*t*= 2.1 μm. The

_{b}*Q*factors and wavelengths for high-

*Q*TE modes near 1550 nm versus the upper cladding layer thickness

*t*are plotted in Figs. 1(b) anti-symmetric and 1(c) symmetric modes relative to the output waveguide. The mode wavelengths increase from 1547 to 1554 nm as

_{u}*t*increases from 1.3 to 1.9 μm. The anti-symmetric mode

_{u}*Q*factor increases from 1.5 × 10

^{3}to 3.0 × 10

^{3}as

*t*increases from 1.3 to 1.4 μm, and then decreases with the further increase of

_{u}*t*. The symmetric mode

_{u}*Q*factor is 1.3 × 10

^{3}, 3.6 × 10

^{3}, and 1.1 × 10

^{3}at

*t*= 1.3, 1.45, and 1.9 μm, respectively. The distributions of |

_{u}*E*|

_{y}^{2}for the symmetric TE mode at the plane

*z*= 0 are plotted in Figs. 2(a)-2(c) at

*t*= 1.3, 1.45 and 1.9 μm, and the corresponding |

_{u}*H*|

_{y}^{2}are given in Figs. 2(e)-2(g). The main fields

*H*are confined well in the active layer, but the minor fields

_{y}*E*are mainly located in the upper cladding layer with a vertical radiation to the substrate, especially in Fig. 2(c). The mode field patterns in Figs. 2(d) and 2(h) are for TE WGM in the microcylinder without the output waveguide at

_{y}*t*= 1.9 μm, which shows vertical radiation loss. For the perfect microcylinder cavity, the mode wavelengths and

_{u}*Q*factors of TE WGMs are 1545.4, 1549.6 and 1554.0 nm, and 2.1 × 10

^{3}, 7.2 × 10

^{3}and 1.5 × 10

^{3}, at

*t*= 1.3, 1.45 and 1.9 μm, respectively.

_{u}The distributions of |*E _{y}*|

^{2}at 1549.7 nm for

*t*= 1.45 μm are plotted in Figs. 3(a) and 3(b) at the horizontal planes of

_{u}*y*= 1 and −1.5 μm, and those at 1554.0 nm for

*t*= 1.9 μm in Figs. 3(c) and 3(d) at

_{u}*y*= 1 and −1.5 μm. Based on the Fourier transformation as in [20], the mode field patterns in Figs. 3(a) and 3(c) consist of WGMs with azimuthal mode numbers

*v*= 21 and 17 at the percentages of 67.1% and 31.3%, and 67.2% and 27.8%, respectively. So the coupled mode field patterns are nearly square shapes due to the angular mode number difference

*Δv*of 4. However, four leakage modes with

*v*= 21, 17, 14 and 11 are obtained with the percentages of 6.8%, 32.9%, 47.8%, and 9.8% for the triangle-shaped field pattern in Fig. 3(b). The vertical radiation loss of the main mode with

*v*= 21 is almost canceled due to the destructive interference effect at

*t*= 1.45 μm, so the mode

_{u}*Q*factor reaches the maximum value. However, the field distribution in Fig. 3(d) has the percentages of 57.7% and 38.6% for the leakage modes with

*v*= 21 and 17, i.e., the two main modes are still the main leakage channels, so the corresponding coupled mode has a low

*Q*factor at

*t*= 1.9 μm. The distributions of |

_{u}*E*|

_{y}^{2}in the output waveguide at

*z*= 2.97 μm are plotted in Figs. 4(a)-4(c) at

*t*= 1.3, 1.45 and 1.9 μm, and corresponding |

_{u}*H*|

_{y}^{2}in Figs. 4(d)-4(f), for the microcylinder with a long output waveguide. The patterns of |

*H*|

_{y}^{2}are confined well in the active layer, while distributions of |

*E*|

_{y}^{2}are mainly confined in the upper cladding layer with obvious leakage into the substrate as in Fig. 2, because they come from the WGMs in the microcylinder cavity.

To calculate far-field pattern by Eq. (1), we choose the field distribution at *z*_{0} = 3.47 μm in air as the near field pattern *U*(*x*_{0}, *y*_{0}) for the microcylinder with a 0.5-μm-length output waveguide. The far-field intensity distributions for the main component *H _{y}* of the symmetric TE mode are plotted in Figs. 5(a)-5(c) at

*t*= 1.3, 1.45 and 1.9 μm, with the minus vertical angle in the substrate side. Due to the vertical radiation, the vertical far-field patterns broaden and the main peaks of the far-field patterns deviate from the normal direction. The far-field patterns of the anti-symmetric TE modes are plotted in Figs. 5(d)-5(f) at

_{u}*t*= 1.3, 1.4 and 1.9 μm, which show two lobes in the horizontal direction.

_{u}Finally, mode *Q* factors and wavelengths versus *t _{u}* are plotted in Figs. 6(a) for anti-symmetric TM modes at 1609 and 1754 nm and 6(b) for symmetric mode at 1754 nm. The mode

*Q*factors of the anti-symmetric modes reach the maximum values at

*t*= 1.35 and 1.65 μm, respectively, because different vertical propagating constants require different paths to cancel the vertical leakage waves [8,9]. The mode

_{u}*Q*factor of the anti-symmetric (symmetric) mode at 1754 nm firstly increases from 2.8 × 10

^{3}to 5.4 × 10

^{3}(4.0 × 10

^{3}to 6.6 × 10

^{3}) as

*t*increases from 1.1 to 1.35 (1.1 to 1.3) μm, and then decreases with the further increase of

_{u}*t*. The variations of mode wavelengths with

_{u}*t*for TM modes are much more slowly than that of TE modes in Fig. 1. The squared main field |

_{u}*E*|

_{y}^{2}at the plane of

*z*= 0 is confined well in the active layer as shown in Figs. 7(a)-7(c) at

*t*= 1.1, 1.3 and 1.8 μm, for the symmetric TM modes at 1754 nm. But the squared minor field |

_{u}*H*|

_{y}^{2}in Figs. 7(e)-7(g) mainly belongs to a higher order radial mode and leaks into the substrate greatly. Taking the mode analysis as for Fig. 3, we can assign the mode as the coupled mode between TM

_{12,3}and TM

_{19,1}. The mode field patterns in Figs. 7(d) and 7(h) are for a perfect microcylinder without the output waveguide at

*t*= 1.8 μm, and mode field patterns at

_{u}*t*= 1.1 and 1.3 μm are almost the same with near zero vertical leakage loss. The corresponding mode wavelengths and

_{u}*Q*factors are 1752.46, 1752.26 and 1752.75 nm and 4.6 × 10

^{5}, 3.5 × 10

^{5}and 4.6 × 10

^{5}for

*t*= 1.1, 1.3 and 1.8 μm in the perfect microcylinder.

_{u}## 4. Output characteristics of microcylinder lasers

AlGaInAs/InP microcylinder lasers with a 15-μm-radius connected with a 2-μm-wide output waveguide are fabricated in a similar technology process as in [16]. After cleaving over the output waveguide, microcylinder laser is bonded p-side up onto a heat sink and mounted on a thermoelectric cooler (TEC) to control temperature. The output powers coupled into a tapered single mode fiber (SMF) versus continuous-wave injection currents are plotted in Fig. 8(a) at the temperatures of 287, 292 and 298 K, with the threshold currents of 14, 17 and 21.6 mA, respectively. The lasing spectra measured at the currents of 35 and 70 mA are presented in Fig. 8(b), and the dominant mode wavelengths and the side mode suppression ratios (SMSR) versus the current are shown in Fig. 8(c). Single mode operations with the SMSRs of 31.5 and 34.0 dB are obtained at 35 and 70 mA, respectively. Lasing mode wavelength increases from 1535.9 to 1538.2 nm as the injection current rising from 12 to 50 mA, and the lasing mode jumps from 1538.2 to 1553.0 nm at 50 mA.

By mounting the microlaser at a platform rotating in a step of 1° horizontally and vertically, respectively, the horizontal and vertical far-field patterns are recorded by a photodetector placed at 15 cm away as in [17]. The horizontal and vertical far-field patterns at 35 and 70 mA are presented in Fig. 8(d). The lasing mode in the circular resonator is coupled to high order transverse mode in the output waveguide [17], so multi-peaks are observed in the horizontal far field pattern. In the vertical direction, an additional peak is observed in the substrate side due to the vertical leakage loss. The results indicate that the output waveguide can induce vertical radiation loss in a cylinder microcavity with a radius of 15 μm. The vertical radiation loss is expected to be zero in the perfect 15-μm-radius microcylinder [7].

## 5. Conclusions

We have investigated the vertical radiation loss and far-field patterns for microcylinder lasers connected with an output waveguide theoretically and experimentally. The high *Q* coupled modes in the circular resonator are related to two WGMs with different azimuthal and radial mode numbers, which result in an inevitable vertical radiation loss. The measured far field patterns confirm the existence of the vertical radiation loss.

## Acknowledgments

This work was supported by the High Technology Project of China under Grant 2012AA012202, and the National Nature Science Foundation of China under Grants 61235004, 61021003, 61106048, 61061160502 and 61006042.

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