Introduction
With the small size and high mode Q factor, microresonator lasers are potential light sources for photonic integrated circuits and optical interconnects [ 1– 19]. In recent decades, varieties of microlasers have been investigated and demonstrated, such as circular microlasers [ 1– 6], square microlasers [ 7– 12] and other equilateral polygon microlasers [ 13– 18]. Through connecting an output waveguide to the microresonators, single mode microlasers with unidirectional emission have been demonstrated [ 5, 6, 10– 14]. Furthermore, single mode lasing without mode hopping was realized through adjusting the position and width of the output waveguide [ 12]. High-Q coupled modes can be obtained for the perfect rectangular microresonators with proper length and width [ 19].
In this paper, the mode characteristics are investigated for the rectangular microresonators using the two-dimensional (2D) finite-difference time-domain (FDTD) technique. An output waveguide is connected to the midpoint of the long side for unidirectional emission and mode selection. High-Q modes with large mode interval suitable for wide and continuous wavelength tuning can be obtained via proper design of the rectangle aspect ratio and the output waveguide width. This paper is organized as follows. In Section 2, we give the numerical model for the simulation of rectangular microresonators. In Section 3, we investigate the influences of the aspect ratio of the rectangular microresonator on the mode characteristics through fixing the area of the microresonator. In Section 4, the influences of the output waveguide width on the mode characteristics are investigated. In Section 5, the characteristics of square microlaser as a special case for the rectangular microlasers are reported, and finally the conclusion is summarized in Section 6.
Numerical model
The mode characteristics for the rectangular microresonator, with an output waveguide connected to the midpoint of the long side, are numerically investigated by 2D FDTD technique. The 2D schematic diagram of the rectangular microresonator is presented in Fig. 1, where a and b are the width and length of the microresonator, respectively, and w is the width of the output waveguide. As shown in Fig. 1, the rectangular microresonator is laterally confined by benzocyclobutene (BCB), and a 200-nm silicon nitride (SiNx) is sandwiched between them as the adhesion layer. The refractive indices of AlGaInAs/InP laser wafer, SiNx and BCB layers are taken to be 3.2, 2 and 1.54, respectively. The spatial steps Dx and Dy are set to be 20 nm and the time step is fixed to be 0.0467 fs based on the Courant limit. A 50-cell perfectly matched layer (PML) absorbing boundary condition is applied to terminate the simulation area. A cosine impulse modulated by a Gaussian function P(t) = exp[ - (t-t0)2/tw2]cos(2pf0t) is used as the exciting source, with the pulse center t0 = 6.6 fs, the pulse half width tw = 19.8 fs, and the pulse center frequency f0 = 193.5 THz. Symmetric and antisymmetric excitation sources are used to simulate symmetric and antisymmetric modes relative to the middle line of the output waveguide, respectively. The variations of Hz are recorded as the FDTD outputs for the transverse-electric (TE) modes, and then the outputs are transformed from the time domain to frequency domain using Padé approximation [ 20]. Through fitting the resonance peak with a Lorentzian function, the mode wavelengths and Q factors can be obtained.
Influences of rectangle aspect ratio on the mode characteristics
Rectangular microresonators with S = 100 mm2
The mode characteristics are investigated for the rectangular microresonators with the area S = 100 mm2 and the width of the output waveguide w = 1 mm. The mode Q factors for the symmetric and antisymmetric modes, with the highest mode Q factors of the modes with the wavelengths ranging from 1450 to 1650 nm, are presented in Fig. 2(a) for the rectangular microresonators with b/a ranging from 1 (square microresonator) to 2.4 at a step of 0.1. The mode Q factors for the symmetric and antisymmetric modes first change slightly when b/a is less than 1.6 and then increase rapidly as 1.6<b/a<1.8. As b/a = 1.8, the mode Q factor for the symmetric mode is almost four times of that at b/a = 1 while the mode Q factor for the antisymmetric mode double that at b/a = 1. Moreover, the mode Q factor for the symmetric mode is much higher than the antisymmetric mode as b/a = 1.8. As b/a continues to increase, the mode Q factors decrease gradually except for the kink at b/a= 2 for the symmetric mode. Furthermore, the mode field distributions for the symmetric modes are calculated and plotted in Figs. 2(b)−2(e) at b/a= 1, 1.4, 1.8 and 2.1, respectively, where the field distributions in the output waveguide are magnified by 5 times.
Fig.2 (a) Mode Q factors for the symmetric and antisymmetric modes, with the highest mode Q factors of the modes with the wavelengths ranging from 1450 to 1650 nm for the rectangular microresonator with b/a ranging from 1 to 2.4 at a step of 0.1 as S = 100 mm2 and w = 1 mm, and the mode field distributions for the corresponding symmetric modes in the rectangular microresonators at b/a of (b) 1, (c) 1.4, (d) 1.8 and (e) 2.1, respectively. The field distributions in the output waveguide are magnified by 5 times and the proportional scales of (b)−(e) are different |
Based on the mode field distributions and the simulation results in Ref. [ 19], the high-Q modes in the rectangular microresonators are coupled modes from two original modes TE(p1,q1) and TE(p2,q2) with the same symmetries and close wavelengths. The mode order number for the coupled modes in the rectangular microresonator can be assigned by the wave nodes number along the long side. Due to the strong mode field distribution in the midpoint of the long side, the fundamental modes are suppressed by the midpoint-connected output waveguide [ 11]. Therefore, the first-order transverse mode has the highest mode Q factor in the rectangular microresonators with an output waveguide connected to the midpoint of the long side if it can be obtained. For the first-order transverse coupled modes, the mode numbers of the original modes TE(p1,q1) and TEp2,q2) have the relation of
Based on the mode analysis in Ref. [ 12], the propagation constants in the x and y directions kx and ky for the modes TE(p,q) in the rectangular microresonators can be estimated from
For the high-Q coupled modes, the incident angles of the original modes are close to 45°, which indicates kx is close to ky. Therefore, the mode numbers of the original modes TE(p1,q1) and TE(p2,q2) with close wavelengths have the relation of
Since the original modes have the same symmetries, |q1 – q2| will be an even number. To obtain the high-Q first order transverse mode, a and b have to satisfy the following relation
The mode field distributions in Figs. 2(b), 2(d) and 2(e) show that the symmetric modes with the highest mode Q factors in the rectangular microresonators with b/a = 1, 1.8 and 2.1 are the first-order transverse modes, which are in well agreement with the above numerical analysis. At b/a = 1.4, Eq. (5) is not satisfied and thus the high Q mode is the higher-order mode, as shown in Fig. 2(c).
The mode intensity spectra for the symmetric and antisymmetric modes are presented in Figs. 3(a) and 3(b) for the square microresonator and the rectangular microresonator with b/a = 1.8, respectively. For the square microresonator with w = 1 mm, two high-Q symmetric modes are observed at the wavelengths of 1452.80 and 1553.53 nm. They are TEo,(30,34) and TEo,(28,32) modes with the mode Q factors of 9.93 × 103 and 7.51 × 103, respectively, corresponding to the first-order modes. The mode Q factors for the antisymmetric first-order modes TEo,(29,33) and TEo,(27,31) are 7.93 × 103 and 5.89 × 103, respectively, which are close to the adjacent symmetric first-order modes.
For the rectangular microresonator with w = 1 mm as b/a = 1.8, five high-Q symmetric modes are observed at the mode wavelengths of 1473.13, 1507.92, 1544.83, 1583.11 and 1623.64 nm, which are marked as C1, C2, C3, C4 and C5. The corresponding mode Q factors for modes C1, C2, C3, C4 and C5 are 2.90 × 104, 3.09 × 103, 9.17 × 103, 4.83 × 103 and 2.28 × 103, respectively, with the mode Q factor for mode C1 much higher than the other four modes. For the antisymmetric modes, six high-Q modes are observed at the wavelengths of 1456.10, 1490.61, 1526.03, 1563.56, 1603.15 and 1644.56 nm. They are marked as D1, D2, D3, D4, D5 and D6, with the corresponding mode Q factors of 4.68 × 103, 4.03 × 103, 9.45 × 103, 2.75 × 103, 1.43 × 104 and 4.18 × 103, respectively. The mode Q factor for mode D5 is larger than those of modes D1, D2, D3, D4 and D6 but smaller than the symmetric mode C1. Furthermore, the mode field distributions for modes C1, C3, D3 and D5 are calculated and plotted in Figs. 4(a)−4(d), respectively, which are magnified by 5 times in the output waveguide. Through applying Fourier transform on the mode field distributions, the mode numbers of the original modes can be obtained [ 11] and they are summarized in Table 1 for modes C1, C2, C3, C4, C5, D1, D2, D3, D4, D5 and D6. The mode numbers (p1,q1) and (p2,q2) for the original modes TE(p1,q1) and TE(p2,q2) have the relation of |q1 – q2| = 4, which indicates that modes C1, C2, C3, C4, C5, D1, D2, D3, D4, D5 and D6 are the first-order modes. Different from the square microresonator, the differences between the mode Q factors for the first-order modes in the rectangular microresonator is large, and this may be induced by different kx/ky of the original mode. For mode C1, kx/ky of the original modes TE(23,34) and TE(21,38) are 1.21 and 1.00, which indicates that the incident angles of the modes are close to 45°. Therefore, the mode Q factor for mode C1 is much higher than the other first-order transverse modes, which makes the rectangular microresonators suitable for tunable lasers.
Tab.1 Mode Q factors and original modes for the high-Q coupled modes in the rectangular microresonator with b/a = 1.8 and w = 1 mm |
| modes | Q factor | original modes | modes | Q factor | original modes |
|---|---|---|---|---|---|
| C1 | 2.90 × 104 | TE(23,34), TE(21,38) | D1 | 4.68 × 103 | TE(26,31), TE(24,35) |
| C2 | 3.09 × 103 | TE(22,34), TE(20,38) | D2 | 4.03 × 103 | TE(25,31), TE(23,35) |
| C3 | 9.17 × 103 | TE(22,32), TE(20,36) | D3 | 9.45 × 103 | TE(22,33), TE(20,37) |
| C4 | 4.83 × 103 | TE(23,28), TE(21,32) | D4 | 2.75 × 103 | TE(23,29), TE(21,33) |
| C5 | 2.28 × 103 | TE(23,26), TE(21,30) | D5 | 1.43 × 104 | TE(22,29), TE(20,33) |
| D6 | 4.18 × 103 | TE(23,25), TE(21,29) |
Rectangular microresonators with S = 144 mm2
For the rectangular microresonators with S = 144 mm2 and w = 1 mm, the mode Q factors versus b/a are presented in Fig. 5(a) for the symmetric and antisymmetric modes, with the highest mode Q factors of the modes with the wavelengths ranging from 1450 to 1650 nm, as b/a increases from 1 to 2.4 at a step of 0.1. The mode Q factors for the symmetric and antisymmetric modes decrease as b/a<1.6. For the symmetric modes, the mode Q factors increase rapidly as 1.6<b/a<2.1 except for the sharp kinks at b/a = 1.9 and 2. When b/a = 2.1, the mode Q factor for the symmetric mode is 9.26 × 104, which is almost five times of that at b/a = 1, and it is much higher than symmetric mode with the second highest mode Q factor of 4.22 × 104 and the corresponding antisymmetric modes. The large differences between the mode Q factors for different modes make the rectangular with b/a = 2.1 and w = 1 mm suitable for the tunable lasers. Furthermore, the mode filed distributions are presented Figs. 5(b)−5(e) for the symmetric modes with the highest mode Q factors in the rectangular microresonators with b/a = 1, 1.4, 1.8 and 2.1, respectively, which are magnified by 5 times in the output waveguide. The four symmetric modes at b/a = 1, 1.4, 1.8 and 2.1 are marked as modes H, I, J, K, respectively. The mode field distributions show that modes H, J and K are the first-order modes while mode I is the third-order transverse mode, which agrees well with the numerical analysis in Section 3.1.
Fig.5 Mode Q factors for the symmetric and antisymmetric modes (a), with the highest mode Q factors of the modes with the wavelengths ranging from 1450 to 1650 nm for the rectangular microresonator with b/a ranging from 1 to 2.4 at a step of 0.1 as S = 144 mm2 and w = 1 mm. and the mode field distributions for the corresponding symmetric modes in the rectangular microresonators at b/a of (b) 1, (c) 1.4, (d) 1.8 and (e) 2.1, respectively. The field distributions in the output waveguide are magnified by 5 times and the proportional scales of (b)−(e) are different |
Influences of the output waveguide width on the mode Q factors
The influences of the output waveguide are investigated for the square microresonator and the rectangular microresonator with b/a = 1.8 as S = 100 μm2. The mode Q factors versus the output waveguide width w are presented in Fig. 6(a) for the symmetric modes TEo,(30,34) and TEo,(28,32) and antisymmetric modes TEo,(29,33) and TEo,(27,31) in the square microresonator. The mode Q factors for the symmetric and antisymmetric modes decrease with w and they are close to each other under different output waveguide widths. As shown in Fig. 6(b), the mode Q factor for the high-Q symmetric mode C1 decreases rapidly from 2.90 × 104 to 7.13 × 103 due to the increase of the coupling loss caused by the output waveguide as w increases from 1 to 1.3 mm. When w> 1.3 mm, the mode Q factors for modes C1, C3, D3 and D5 are close to each other and the small mode interval between these modes makes the rectangular unsuitable for the tunable lasers.
Output characteristics of square microlasers
The output characteristics are tested for the square microlasers laterally confined by BCB. For a square microlaser with the side length of 17.8 mm and the output waveguide width of 1.8 mm, the output power is measured by butt-coupling a multi-mode fiber to the cleaved facet of the output waveguide and presented in Fig. 7(a). The threshold current is about 4.5 mA at 291 K and the corresponding threshold current density is 1.42 kA/cm2. From the I-V curve around the threshold current, a series resistor of 24.2 W is estimated.
Fig.7 (a) Output power and applied voltage versus CW injection current at 291 K, (b) lasing spectra at different currents at 291 K, and (c) lasing wavelength and SMSR versus the current at 291 K, for the square microlaser with the side length of 17.8 mm and the output waveguide width of 1.8 mm |
The lasing spectra are measured and plotted in Fig. 7(b) at the injection currents of 5, 15, 25, 35, 45, 55 and 59 mA, respectively, where the adjacent lasing spectra are relatively shifted by 10 dB for clarity. As I = 5 mA, three evident peaks are observed at the wavelengths of 1518.39, 1531.22 and 1544.31 nm at 291 K with the wavelength intervals of 12.83 and 13.09 nm, respectively, which can be fitted by the longitudinal mode interval λ2/23/2/a/ng with the group index ng of 3.59. The dominant lasing mode is mode A with the side mode suppression ration (SMSR) of 18 dB at 5 mA, and it jumps 27.1 nm over two longitudinal mode intervals to mode C with an additional thermal-induced wavelength redshift of 1.1 nm as I increases to 15 mA. This indicates that mode C has a higher mode Q factor than mode B and the mode interval between the adjacent high-Q modes for the square microresonator is twice of the longitudinal mode interval, which agrees well with the simulation results in Fig. 3(a). As the injection current continues to increase, the dominant lasing mode keeps constant at mode C because of the high mode Q factor and large mode interval for the high-Q modes. Therefore, the lasing wavelength can be tuned continuously through tuning the injection current. The mode wavelengths and SMSR versus the continuous wave (CW) injection currents are summarized in Fig. 7(c) at 291 K. As shown in Figs. 7(b) and 7(c), the mode wavelength can be continuously tuned from 1545.4 to 1554.51 nm with the SMSR larger than 26 dB as I increases from 15 to 59 mA, and the corresponding wavelength tuning range is 9.11 nm. The influences of the injection current on the lasing wavelength can be mainly contributed to three mechanisms, which are carrier-induced variation of refractive index, thermal expansion of the device and the temperature-induced variation of refractive index. Since the carrier density of the microlaser almost keeps constant as the injection current is larger than the threshold current, the carrier-induced variation of refractive index is small and it has slight influence on the lasing wavelength. The lasing wavelength shift caused by the thermal expansion and the temperature-induced variation of refractive index on the lasing wavelength is almost linear to the temperature of the microlaser [21]. Under a pulsed current with a pulse duty of 1% and a pulse width of 10 ns, the mode wavelength redshift rate of 0.114 nm/K is obtained by varying the thermoelectric cooler (TEC) temperature. When the injection current and the TEC temperature are fixed at certain values, the temperature of the microlaser almost keeps constant, and thus the refractive index seldom changes. Therefore, the lasing wavelength almost keeps constant at certain injection current and the wavelength tuning can be repeated.
Conclusions
In conclusion, the mode characteristics are investigated for the rectangular microresonators with an output waveguide connected to the midpoint of the long side. For a rectangular microresonator with fixed area, the mode Q factors can be greatly enhanced through choosing the rectangular aspect ratio. As the aspect ratio is around 2, mode Q factors are much higher than the corresponding square microresonator for the rectangular microresonator with a 1-mm-width output waveguide. Furthermore, the influences of the output waveguide width are investigated. When the width of the output waveguide is less than 1.2 mm, the rectangular microresonators are suitable for tunable lasers due to the large difference between the adjacent modes with the same transverse mode number and the large mode interval. The square microresonators with the mode interval for the high-Q modes twice of the longitudinal mode interval can also be utilized for the tunable lasers. For a square microlaser with the side length of 17.8 mm and the output waveguide width of 1.8 mm, single-mode operation is achieved with a continuous tuning range of 9.1 nm. With the compact size, the microlaser can provide compact tunable light source for the photonic integrated circuits and optical interconnects.