The single mode Fabry-Perot (FP) semiconductor lasers are investigated systematically by a rigorous time-domain theoretical model based on the transfer matrix method. Static and high-speed dynamic performances under direct modulation and strong external optical feedbacks are simulated for both symmetric and asymmetric longitudinal structures of the lasers. Comparisons with the DFB and conventional FP lasers are made to confirm its effectiveness in achieving single-mode lasing with high spectrum purity under modulation and feedback conditions. Structural optimization is also carried out with respect to the key design parameters.
© 2011 OSA
Since the recent development of single mode Fabry-Perot (FP) lasers [1–4], much attention has been paid to this type of structure [5–7] for its extraordinary single mode lasing performance under various operation conditions . By etching a few shallow slots on the ridge of an FP laser, constructive interference of the cavity can be established and utilized to manipulate the gain threshold and lasing condition to suppress unwanted FP side modes. As demonstrated in a series of experiments [1,4,8], this laser showed excellent performance with characteristics, such as high side-mode suppression ratio (SMSR), high speed dynamic modulation response, and low-degree feedback sensitivity, etc. Therefore, the properly designed and re-growth-free single mode FP lasers can now serve as a promising low-cost alternative to the more expensive distributed feedback (DFB) lasers in applications such as the optical access networks. Following in Fig. 1 are the simplified 2D sketches of the slotted FP and DFB lasers, respectively, where the red-strap layer represents the active region.
The non-periodic nature of the longitudinal structure inherent in single-mode FP laser, however, presents some challenges to the modeling and design optimization for such devices. The conventional time-domain coupled mode theory  commonly used for the DFB lasers can no longer be applicable for this structures. More complicated quantum mechanical model [10,11] was applied only to the single-slot FP laser, without explicit consideration of the device temperature, which is critical for the operation of such devices under practical situations. For those reasons, comprehensive simulation of the dynamic performance of single-mode FP lasers under large signal direct modulation and/or strong optical feedback has not yet been reported to our best knowledge. Further, the necessity of using a time domain model to characterize the laser is for design optimization purposes. Although the powerful threshold gain approaches [1,4,7] can provide certain rules and guidance for single-mode FP laser designs, they cannot be readily extended to account for the above-threshold or dynamic characteristics, under high level external feedback conditions. To perform those optimizations with respect to the key design parameters, such as the strength and distribution of slots along the laser cavity, a large signal model is essential.
In this work, we develop a rigorous and straightforward time-domain model based on the transfer matrix method [12,13], to explicitly account for the thermal and feedback effects [14,15] on single-mode FP laser under both static and dynamic situations. Comprehensive simulation studies of the laser are carried out in a systematic manner. Design optimization with respect to the slot index contrast and number is performed. With the help of the comprehensive simulation method developed in this work, a more powerful design optimization procedure can be envisaged.
2. Model governing equations and simulation method
By combining the conventional transfer matrix method and the time domain evolution of electrical and optical fields, the time-domain transfer matrix method (TD-TMM) [12,13] have been developed as a cascade of elementary transfer matrices, i.e., the scattering and propagation matrices, to describe the time-space evolution of the counter-propagating waves along the laser cavity.
2.1 Governing equations for TD-TMM
As in Ref , we can write the forward and backward optical fields at the two boundaries of each structural section k at time t as followsEq. (1), the field at a future-step can be further written in terms of the previous-step fields and the matrix elements as12,13].
2.2 Additional effects considered in TD-TMM
a) Index change due to injection current: Due to the photon-electron interaction mechanism, the laser’s refractive index n of section k (as noted in subscript) will change, when current is injected into the active region, as
b) Wavelength shift due to injection current: The second effect aside from the index change under current injection is the lasing wavelength shift, due to alteration of the cavity resonance condition. This can be captured by two terms: one is the shift of reference wavelength as12] as
c) Finite gain profile: To consider the impact of finite bandwidth for the material gain profile on wavelength selections in simulation, as occurred in the actual situations, the flat gain profile used in Ref [12,13]. can be replaced by a transfer function to account for the material gain selection  as16].
d) Temperature effect: When the device temperature changes under different working conditions and external environments, refractive index of the material can be affected  as14]; is the temperature of the substrate viewed as a constant heat sink, a is the conversion coefficient and is the output power. At the same time, differential gain and transparent carrier density in the gain formula have to be modified to include the temperature factor as
e) Optical feedback interferences: To model the optical feedback’s interference effect on laser’s performance by the time-domain transfer matrix method, we can add one term to the field as a reflected portion of the previous-time output from the facet, with an extra phase shift and time delay  as
3. Simulation results
3.1 Single-mode FP laser design and TD-TMM verification
Single-mode FP lasers are designed by using the inverse scattering method [1,4] to guide synthesis of the shallowly etched slots onto the FP structure, which can enforce desired threshold gain target function and accurately matched phase condition at chosen wavelengths.
Briefly, considering slot perturbations to the lasing threshold equation, Eq. (18), as1,4]. for more details.
Based on this theory, we have reproduced both types of the single-mode FP structures, i.e., the symmetric and the asymmetric configurations, to be examined in the following section. However, to exam our TD-TMM approach, we first applied the method to a DFB structure  and obtained the static and dynamic performances as in Fig. 2 .
From the above L-I curve and large signal modulation by 0.5GHz square wave-forms, we can see that the TD-TMM can reproduce reference results with high accuracy. For certain increase of the environment temperature, lasing behavior in terms of the threshold current and L-I curve slope are significantly changed. The saturation of output power can also be observed, as indicated by the decrease of slope at higher injections in Fig. 2(a).
3.2 Single-mode FP laser simulations
After validation of the simulation method by way of the above DFB laser example, we apply it to simulations of single-mode FP lasers. For demonstration purposes, we examined two types of designs, namely, the symmetric and asymmetric structures, whose SMSR can both reach 50 dB, but with different manufacturing complexities.
a) Symmetric single-mode FP laser structure: From threshold analysis using the inverse scattering method of Ref [1,4], we obtained the slot distribution for longitudinal structure of a FP laser and calculated its threshold gain profile as shown in Fig. 3(a) (insert is for the distribution of 90 slots for this laser design). Then, we used the TD-TMM model to obtain its lasing spectrum as in Fig. 3(b).
Lasing wavelength and SMSR with respect to the injection current at room temperature are shown in Fig. 4(a) to indicate the wavelength shift at different pumping levels. Red shift can be seen above the 20mA threshold and stable while slight improvement of SMSR can also be observed. By varying injection current at different temperatures, we obtain the light-current relation as in Fig. 4(b), which demonstrates the T-dependent threshold current and L-I curve slope of the laser. At higher temperature, increases while the slope decreases rapidly.
Further, for small signal analysis, we calculate the optical response at room temperature for three different bias currents as in Fig. 5(a) , from which we can see that the relaxation oscillation frequency increases from 1GHz to around 5GHz at higher injection. To examine the large-signal modulation performances, we also calculate the output power and lasing wavelength shift, as shown in Fig. 5(b), for the modulation current switched between 30mA and 50mA at 10 Gbit/s.
To determine the impact of optical feedback on single-mode FP laser performance, we further include a high level backward reflection up to −25dB, i.e., , to the right facet of the laser. The output power, the lasing wavelength shift, and the overlapped spectrum taken at different simulation times are plotted in Fig. 6(a) . The reason to plot spectra on top of each other is to observe the possible mode-hopping during modulation, especially when injection current is switched between 0/1.
Comparing Fig. 6(a) with Fig. 3(b) and 5(b), we can see that the laser can maintain high single-mode spectrum under strong feedback and fast modulation condition, as experimentally described in Ref . Central lasing wavelength is also changing smoothly according to the injection modulation. For further comparison, a DFB laser with the same structural and material parameters, especially the same designed threshold gain difference and facet reflectivity, is calculated before and after feedback is applied, as in Fig. 6(b). From those graphs, we can see the different effects of the feedback on the two structures. It is observed that the DFB laser is sensitive to feedback mostly on its neighboring mode, i.e., mode hopping can occur to this vicinity wavelength. On the other hand, the feedback sensitivity of the single-mode FP laser is shared by all other suppressed FP modes covered by the material gain profile. This leads to the fact that total sensitivity to the external feedback is reduced, but at the price of possible mode hopping to wavelengths far away from the lasing mode.
b) Asymmetric single-mode FP laser structure: The asymmetric structure of the FP lasers is introduced for simplifying the manufacturing process as described in Ref [1,4]. In this work, we simulated the 15-slot asymmetric structure that has only one half of the FP cavity etched with slots, while the other half is un-etched and the facet is coated with highly reflective mirror (as insert of Fig. 7(a) ). This design can reduce the number of slots needed in the structure, while distributing them more evenly into the cavity, rather than putting high density of slots around the device center as in the symmetric design. This can also improve external efficiency for the useful output power as only one side of the laser is emitting light, with the other side used for sampling or monitoring.
Threshold gain of the structure is calculated as shown in Fig. 7(a), while the lasing spectrum in condition of dc-injection and zero optical feedback is shown in Fig. 7(b). From these plots we can observe the high SMSR single-mode lasing spectrum, even with very small designed threshold gain difference, due to the carefully engineered slot positions that enforce constructive phase condition to be satisfied simultaneously around the desired wavelength. It should be pointed out that, for the same threshold gain difference, a DFB laser can hardly operate in single-mode condition.
The same procedure is done to the asymmetric single-mode FP laser at high modulation speed and −25dB feedback level, as in Fig. 8(a) in which the output power, the lasing wavelength change, and the overlapped spectrum sampled at different time points during the simulation are displayed. For comparison, we also included calculations in Fig. 8(b) for the conventional FP laser that has the same structural and material parameters, but no slots etched.
From the above, we see that the asymmetric structure can achieve high SMSR single-mode condition, while maintaining structural simplicity and certain feedback immunity.
To further optimize the structure to produce more stable single-mode spectrum with higher SMSR, we can manipulate the total number of slots and their index-contrast as two adjusting parameters for the design. We calculated the asymmetric single-mode FP laser by either fixing the contrast of while changing slot number from 11 to 50, to plot threshold gain difference and lasing spectrum SMSR as in Fig. 9(a) and 9(b); or by using its 15-slot case but changing slot-index contrast from 0.001 to 0.01, to plot the same quantities as in Fig. 10(a) and 10(b).
From the plots, we can see that the threshold gain difference increases linearly with the number and/or depth of the slots along the cavity. However, more importantly, the effect of deeper slots will be leveled off at higher contrast, and their improvements over feedback will be reduced, as the more/deeper slots etched the more sensitive the structure will be to external perturbations. This also indicates that a recent development of single-mode FP laser  using the “pixel method” should prefer to adopt coarse meshing segments to reduce number of slots, considering the feedback perturbation effect on designed structures for use in the high speed, strong feedback optical communication networks. Therefore, the trade-off between slot number/depth and single-mode performance has to be balanced for an optimized single-mode laser.
We have analyzed the single-mode FP laser using time-domain traveling wave model and compared the device performance over their different (symmetric and asymmetric) configurations. From the simulations, we confirmed the effectiveness of the properly designed FP lasers in achieving single-mode lasing with high spectrum purity against high level of external feedbacks. We also carried out design optimization with respect to the key design parameters and revealed some interesting features about the dependence of threshold gain difference and SMSR on number and depth of slots.
Special thanks go to Dr. Stephen O'Brien and Prof. Liam Barry for their helpful comments for the feedback issue of this work. Thanks also go to Lin Han for his helpful discussions during the simulation.
References and links
1. S. O’Brien and E. P. O’Reilly, “Theory of improved spectral purity in index patterned Fabry-Perot lasers,” Appl. Phys. Lett. 86, 201101 (2005). [CrossRef]
2. J. S. Young, D. A. Kozlowski, J. M. C. England, and R. G. S. Plumb, “Spectral perturbation and mode suppression in 1.3μm Fabry-Perot lasers,” Electron. Lett. 31(4), 290–291 (1995). [CrossRef]
3. B. Corbett and D. McDonald, “Single longitudinal mode ridge waveguide 1.3μm Fabry-Perot laser by modal perturbation,” Electron. Lett. 31(25), 2181–2182 (1995). [CrossRef]
4. S. O’Brien, A. Amann, R. Fehse, S. Osborne, E. P. O’Reilly, and J. M. Rondinelli, “Spectral manipulation in Fabry-Perot lasers: perturbative inverse scattering approach,” J. Opt. Soc. Am. B 23, 1046–1056 (2006). [CrossRef]
5. Q. Y. Lu, W. H. Guo, R. Phelan, D. Byrne, J. F. Donegan, P. Lambkin, and B. Corbett, “Analysis of slot characteristics in slotted single-mode semiconductor lasers using the 2-D scattering matrix method,” IEEE Photon. Technol. Lett. 18(24), 2605–2607 (2006). [CrossRef]
6. D. C. Byrne, J. P. Engelstaedter, W. H. Guo, Y. Q. Lu, B. Corbett, B. Roycroft, J. O'Callaghan, F. H. Peters, and J. F. Donegan, “Discretely tunable semiconductor lasers suitable for photonic integration,” IEEE J. Sel. Top. Quantum Electron. 15(3), 482–487 (2009). [CrossRef]
7. G. Adolfsson, J. Bengtsson, and A. Larsson, “Spectral engineering of semiconductor Fabry-Perot laser cavities in the weakly and strongly perturbed regimes,” J. Opt. Soc. Am. B 27(1), 118–127 (2010). [CrossRef]
8. C. Herbert, D. Jones, A. Kaszubowska-Anandarajah, B. Kelly, M. Rensing, J. O'Carroll, R. Phelan, P. Anandarajah, P. Perry, L. P. Barry, and J. O'Gorman, “Discrete mode lasers for communication applications,” IET Optoelectron. 3(1), 1–17 (2009). [CrossRef]
9. L. M. Zhang, S. F. Yu, M. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side mode suppression in second-order DFB lasers using time-domain large signal traveling wave model,” IEEE J. Quantum Electron. 30(6), 1389–1395 (1994). [CrossRef]
10. D. D. Marcenac and J. E. Carroll, “Quantum-mechanical model for realistic Fabry-Perot lasers,” IEE Proc. J. 140, 157–171 (1993).
11. D. A. Kozlowski, J. S. Young, R. G. S. Plumb, and J. M. C. England, “Time-domain modeling of mode suppression in 1.3-μm Fabry-Perot lasers,” IEEE Photon. Technol. Lett. 8(6), 755–757 (1996). [CrossRef]
12. M. G. Davis and R. F. O’Dowd, “A new large-signal dynamic model for multielectrode DFB lasers based on the transfer matrix method,” IEEE Photon. Technol. Lett. 4(8), 838–840 (1992). [CrossRef]
13. O. A. Lavrova and D. J. Blumenthal, “Detailed transfer matrix method-based dynamic model for multisection widely tunable GCSR lasers,” J. Lightwave Technol. 18(9), 1274–1283 (2000). [CrossRef]
14. W. Li, X. Li, and W. P. Huang, “A traveling-wave model of laser diodes with consideration for thermal effects,” Opt. Quantum Electron. 36(8), 709–724 (2004). [CrossRef]
15. M. Homar, J. V. Moloney, and M. S. Miguel, “Traveling wave model of a multimode Fabry-Perot laser in free running and external cavity configurations,” IEEE J. Quantum Electron. 32(3), 553–566 (1996). [CrossRef]
16. D. J. Jones, L. M. Zhang, J. E. Carroll, and D. D. Marcenac, “Dynamics of monolithic passively mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 31(6), 1051–1058 (1995). [CrossRef]
17. Y. P. Xi, X. Li, and W. P. Huang, “Time-domain standing-wave approach based on cold cavity modes for simulation of DFB lasers,” IEEE J. Quantum Electron. 44(10), 931–937 (2008). [CrossRef]