We present a comprehensive study for a new three-branch widely tunable semiconductor laser based on a self-imaging, lossless multi-mode interference (MMI) coupler. We have developed a general theoretical framework that is applicable to all types of interferometric lasers. Our analysis showed that the three-branch laser offers high side-mode suppression ratios (SMSRs) while maintaining a wide tuning range and a low threshold modal gain of the lasing mode. We also present the design rules for tuning over the dense-wavelength division multiplexing grid over the C-band.
© 2017 Optical Society of America
Semiconductor lasers emitting at a single wavelength with a high side-mode suppression ratio (SMSR) have many applications in wavelength-division multiplexing (WDM) and optical sensing systems [1,2]. Widely tunable lasers can be realized using interferometer-based cavities, such as cleaved-coupled cavity (C3), Y-branch, and V-cavity lasers [3–5], or cavities with wavelength-selective mirrors, such as sampled-grating distributed Bragg reflector (SGDBR), modulated grating Y (MGY) lasers, and micro-ring resonator (MRR) lasers [6–8]. SMSRs higher than 50 dB and tuning ranges wider than 35 nm have been reported in lasers based on sampled gratings [1,6] and microring resonators [8,9]. However, complex controls are required to access all wavelengths, and some wavelengths between the supermodes are sometimes hard to reach. Interferometer-based lasers benefit from the design simplicity and cost effectiveness, but often face a trade-off between the SMSR and tuning range. The V-cavity lasers reported recently showed a high SMSR and a wide tuning range simultaneously; however, the use of a truncated, non-self-imaging coupler introduces excessive optical loss and increases the threshold of the laser . In this paper, we propose a new, three-branch interferometric laser with an 1×3 self-imaging multi-mode interferometer (MMI) couplers. We developed a general theoretical framework to analyze all interferometric lasers and showed that the three-branch laser offers a superior SMSR and a wide tuning range at the same time, without excess optical loss.
2. Structure of three-branch MMI lasers
The schematic of a three-branch MMI laser is shown in Fig. 1. A self-imaging, lossless 1×3 MMI is used to connect the four cavity arms. Among the four arms, Arms 1 and 4 are used as gain sections; their carrier densities will be clamped at the threshold value and not contribute to index tuning. Arms 2 and 3 are tuning sections for coarse and fine wavelength tuning, respectively. Compared to the Y-branch lasers, the proposed three-branch MMI laser features an additional long arm (Arm 3) that acts as another optical interferometer that helps achieve a high SMSR and a wide tuning range. In the following sections, the SMSR and the tuning characteristics of the three-branch MMI laser are studied.
3. Side mode suppression ratio and cavity configuration
A high SMSR is one of the most important requirements for tunable laser sources in optical communication systems. The SMSR between two cavity modes is given by 
It is clear from Eq. (1) that a large Δ(αL), or equivalently, a large normalized threshold modal gain difference, is the key to achieve high SMSRs. Fabry-Pérot (FP) lasers, as an extreme example, have poor SMSRs because the envelope of the cavity transmission function is essentially flat so every cavity mode has the same threshold modal gain, as shown in Fig. 2(a). Increasing the mode selectivity can be accomplished by modulating the cavity transmission spectrum. Figures 2(b) and 2(c) show the calculated cavity transmission spectra of a two-branch Y laser, having cavity lengths L1 = 373.40 μm and L2 = 336.03 μm, and a SGDBR laser with two identical 1550-nm-centered SGDBR mirrors having 50 DBR pairs per burst, a burst period of 37.36 μm, and a sampling duty cycle of 32%. One can observe that adding an extra cavity to provide optical interference or introducing wavelength-selective mirrors such as sampled gratings readily increases mode selectivity, because only the modes that are able to form standing wave in both cavities in the former case, or to fulfill the phase-matching condition with the Bragg vector of the sampled grating in the latter case, have the lowest threshold modal gain.
In the proposed three-branch MMI laser, the additional long arm imposes an additional condition for the standing waves. That is, the lasing mode has to satisfy the round-trip condition in the additional arm, hence the SMSR can be greatly improved with a well-designed arm length. A quantitative analysis of the three-branch MMI laser is performed using the scattering parameter model shown in Fig. 3. In this model, the external excitation, the transmitted field corresponding to the excitation, and the fields entering and leaving the MMI structure from the j-th arm are labeled as Eex,j, Esc,j, aj, and bj, respectively. The scattering property of the MMI is described using the scattering matrix S, and the length, effective refractive index, and the field reflectivity of the end facet of the j-th arm are labeled as Lj, nj, and rj. For simplicity, the cavity arms are assumed to be single-mode waveguides and there is no coupling between adjacent waveguides. The field parameters are related by:Eq. (7) describe the resonant wavelength and the normalized threshold modal gain of the cavity modes.
The formulation in Eq. (5) is very general. It is applicable to C3, V-cavity, two-branch, and the proposed three-branch MMI lasers by plugging in their respective coupling ratios. The difference in the normalized threshold modal gain between two adjacent modes, ΔGth, as a function of the coupling ratio χ, is shown in Fig. 4(a). The arm lengths of the three-branch MMI laser are chosen for an optimized SMSR, as will be described in the following section. The coupling ratio χ is defined as the fractional power coupled out from the main cavity, as illustrated in Figs. 4(b)–4(e). Furthermore, it is assumed that the coupler in the three-branch laser is symmetric about the center output port so that the out-coupled power is equally distributed in Arms 2 and 4. Figure 4(a) shows that the threshold modal gain difference is very low over a wide range of χ for C3 and two-branch lasers. The threshold modal gain difference is much higher for V-cavity and three-branch lasers, and peaks at χ =~ 0.03 for V-cavity lasers and ~ 0.67 for three-branch MMI lasers. It is clear that, with a short-to-long arm length ratio of 0.95, the threshold modal gain difference decays much slower on either side of the peak value for three-branch MMI lasers compared to V-cavity lasers. Furthermore, the fact that threshold modal gain difference peaks at χ ~ 0.67 in three-branch MMI lasers confirms that the 1×3 MMI is operating the self-imaging mode, where the power exiting the MMI is evenly distributed into three output arms. Consequently, the use of a self-imaging MMI not only gives a high SMSR tolerant to fluctuations in χ due to fabrication errors but eliminates the excessive scattering loss present in the V-cavity lasers.
4. Three-branch MMI laser cavity design
As shown in Fig. 4(a), to obtain the highest SMSR possible, a MMI coupler with a coupling ratio χ =~ 0.67 is preferred. An elegant way to realize such a coupling ratio is to make the power leaving the MMI equally split into the three output ports . The scattering matrix of the 1×3 MMI of this type is given by12].
As residual intra-cavity reflections may cause undesired tuning behavior or degrade SMSR in any kind of tunable lasers, tapered waveguides may be employed to minimize the reflections at the input and output ports of the MMI coupler. Simulations using eigenmode expansion (EME) and finite-difference time-domain (FDTD) methods have shown that a single-pass optical transmission greater than 98% can be achieved for a 1×3 MMI coupler with tapered waveguides. The impact of residue intra-cavity reflections need to be further investigated in future experimental efforts.
For a standing wave pattern to exist in the 1-to-j-th-arm optical path when lasing at a specific wavelength λ0, the total phase shift after a round-trip has to be an integer multiple of 2π. Using Eqs. (7) and (8) and assuming all the end facets do not introduce any parasitic phase shift, the arm lengths satisfy:
When the laser is designed to match the 100-GHz dense wavelength-division multiplexing (DWDM) channel spacing defined by the International Telecommunication Union (ITU) , one of the integers, say M, is fixed. Provided an effective refractive index of 3.22 for InP-based waveguides, one finds M = 1932 for λ0 = 1550 nm. Since Vernier effect is used to coarsely tune the lasing wavelength, the length of Arm 4, corresponding to the integer P, is readily found. Assuming a ratio between the two optical paths associated with Arms 2 and 4 of x, P is found to be ⌊xM⌋, the largest integer less than xM.
Figures 5(a) and 5(b) show the normalized interference patterns, effectively the envelope of the cavity transmission, for the cases of x = 0.95 and 1.2, without the addition of Arm 3. The interference pattern is defined as
Adding Arm 3 introduces additional interference between arms. Due to the larger length difference between Arms 3 and either Arm 2 or 4, the interference pattern is sharp and dense. In this case, the pattern of such a “fine interferometer” can be found by setting j = 3, 4 in Eq. (13), as shown in the left panel of Fig. 5(c). The overall interference pattern between the three arms can be found again from Eq. (13) by setting j = 2, 3, 4. The fine interferometer modulates the coarse interference pattern and suppresses not only the modes directly next to the 1550-nm main mode but those whose transmission was originally comparable to the main mode. As a result, the cavity appears to be single-mode across the entire C-band. The length of Arm 3, however, has to be properly chosen such that the valleys of the fine interference pattern match the peaks of the coarse interference pattern. Consequently, the ratio of the length differences between Arms 2 and 3, and Arms 2 and 4, has to be an odd multiple of 1/2:
Figures 6(a) and 6(b) show the interference pattern and the cavity transmission for designs with m = 7 and m = 8, respectively. In the case of m = 7, the modes at ~ 1534 and ~ 1566 nm are effectively pressed down by wisely positioning the fine interference pattern. Figure 6(c) plots the threshold modal gain difference between the 1550-nm main mode and the modes having similar transmission in the ±5-nm neighboring range, ΔGth,neighboring, and those in the entire C-band, ΔGth,global. Although any integer m provides a nonzero ΔGth,neighboring, the odd m’s actually provide additional modal discrimination ΔGth,global across the C-band. A configuration with m = 7 turns out to be the optimized design given the parameters mentioned previously.
5. MMI laser wavelength tuning characteristics
When currents are applied to the tuning arms of the three-branch MMI laser, the effective refractive indices of the arms decrease due to the plasma effect and shift the interference pattern. In the following simulation, the lasing wavelength is defined by the mode having the lowest normalized threshold modal gain in the entire C-band, which can be found from Eq. (7) with changing the effective refractive indices of the tuning arms, as shown in Fig. 1. The lengths of the cavity arms are determined by the three integers: M = 1932, N = 2291, and P = 1835, and the length of a 15-μm-wide MMI is LMMI = 188 μm based upon . The refractive index change due to the thermal-optic effect is not taken into account.
Figures 7(a)–7(d) show the change in wavelength and threshold modal gain of the lasing mode for the cases of coarse and fine tuning, respectively. The discrete hopping of the lasing wavelength is due to the shift of the interference pattern relative to the cavity transmission peaks, as will be explained in the following. There are two types of hopping in the case of coarse tuning, with a wider spectral spacing of 13.6 nm and a narrower one of 3.4 nm. As n2 decreases, the entire coarse interference pattern blue shifts and defines the lasing wavelength by aligning its peak to the mode that has the highest transmission. For example, as shown in Fig. 7(e), the coarse interferometer peak aligns with the 1563.6-nm mode in the Arms 3–4 pair transmission when Δn2 = −0.05% and aligns with the 1546.6-nm mode when Δn2 = −0.1%. The hopping can also be explained by the change in the threshold modal gain of the lasing mode, which appears to be almost periodic for the coarse tuning case. As the interferometer peaks becomes misaligned from a certain mode the threshold modal gain starts to increase. Hopping occurs when another mode turns to have a lower threshold. It is thus clear that while the narrower hopping, 3.4 nm, is determined by the spectral spacing between two consecutive peaks of the Arms 3–4 pair transmission, the wider hopping, 13.6 nm, is associated with the separation between the main mode and the mode having the highest transmission near the edge of the C-band. In fact, the wider hopping spacing will be an integer multiple of the narrower one, and in this specific design the ratio between them is 4. The same principle applies to the case of fine tuning, except for the fact that the Arms 2–4 pair transmission is now much broader and slowly-varying. Consequently, as it blue shifts, the fine interference pattern picks the lasing mode following the FSR of the main cavity, which is designed to match the ITU channel spacing of 0.8 nm, as shown in Fig. 7(f). The wider hopping is again associated with the 3.4-nm spacing between two consecutive peaks in the fine interference pattern, and the greater variation in the threshold modal gain is due to the sharper nature of the fine interferometer.
An interesting feature of the three-branch MMI laser is that, the fine tuning process can be used to fill up the gaps resulted from the hopping of the coarse tuning. In practice, a set of coarsely-spaced lasing wavelength can be obtained by first sweeping the coarse tuning current with the fine tuning arm rest. With this coarse wavelength grid in mind, one can easily access all the channels in the second current sweep for the fine tuning arm with the coarse tuning arm pre-biased at a certain desired state. The simulated fine tuning of the three-branch MMI laser is given in Fig. 8(a) following the biasing methodology described above, and a demonstration of a quasi-continuous wavelength coverage over the entire C-band is shown in Fig. 8(b). The bias configurations are listed in Table 1.
In this paper, a compact, widely-tunable three-branch semiconductor laser based upon a self-imaging lossless MMI is proposed, and the design rules for a high SMSR and tuning characteristics are studied in detail. The three-branch design not only significantly improves the low SMSRs associated with two-branch Y lasers but separates the coarse and fine tuning from the gain section since the carrier densities in these two arms do not clamp at threshold. Tapered waveguides can be included to minimize the reflections at the input and output ports of the 1×3 MMI, and bending can be applied to Arms 2 and 4 for electrical isolation from Arm 3. The three-branch MMI laser exhibits two-dimensional wavelength tuning characteristics, where the fine tuning fills up the discrete spectral hopping gaps resulted from the coarse tuning process. While both the proposed three-branch MMI laser and the microring-based laser use Vernier effect to achieve wide tuning ranges, the main difference is in the wavelength tuning map. The FSRs of the microrings are generally much larger than the ITU channel spacing, so it requires shifting both microring modes simultaneously to achieve fine tuning. In contrast, in the proposed three-branch MMI laser, one of the cavity arms is designed to have an FSR matching the ITU channel spacing so the control of fine tuning along the ITU grid is simpler. The V-cavity laser has the same benefit in fine tuning control  but the three-branch MMI laser should have lower cavity loss. The tuning speed will be on the order of a nanosecond for carrier injection [8, 14], and a microsecond for thermal tuning . It is expected that the proposed three-branch MMI laser, with the simplest coupler design and maximum utilization of the device length, will contribute as a cost-effective optical light source solution for telecommunications in the C-band.
DARPA DODOS program (grant number HR0011-15-C-0057), Berkeley Sensor and Actuator Center (BSAC), HPE.
The authors thank helpful discussions with Dr. Robert Lutwak at Defense Advanced Research Project Agency (DARPA) DODOS program, Drs. Mike Tan, Marco Fiorentino, Wayne Sorin, Stanley Cheung, and Sagi Mathai at the Hewlett Packard Enterprise (HPE).
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