Efficient and accurate parameter extraction for quantum-well DFB lasers: a comprehensive approach integrating multiple mathematical models

Changpeng Li; Feng Gao; Jiewen Chi; Shengtong Sang; Chuanning Niu; Jia Zhao

doi:10.1364/OE.516299

1. Introduction

Semiconductor lasers play a pivotal role in optical communications [1–3], medical diagnostics [4,5], and sensing systems [6–8]. In particular, quantum-well distributed feedback (DFB) lasers have attracted extensive research attention because of their low threshold current, stable single-mode operation, narrow linewidth, and high-speed modulation capability [9–12]. The efficient and precise simulation of DFB lasers has important significance in predicting and guiding DFB device design and optimization. However, considering the difference in fabrication techniques, processes, and equipment, the laser output characteristics can vary significantly even with the same device design [13]. Extracting objective parameters from the measured laser performance is a critical step in validating theoretical models, identifying manufacturing defects, and guiding design optimization.

Extensive research efforts have been dedicated to modeling simulation and parameter extraction of semiconductor lasers based on the rate equation [14–18]. In 2023, Ding et al. [19] demonstrated a parameter extraction solution based on rate equations by an equivalent-circuit DFB laser model. The rate equation has proven to be a quick and simple laser model. However, as a zero-dimensional model that does not involve grating information, the rate equation is not suitable for describing the wave coupling due to the gratings in DFB lasers. Considering the non-uniform distribution of the optical field and carrier density along the DFB laser cavity, accurate parameter extraction of DFB lasers usually employs complex models such as the one-dimensional traveling-wave model (TWM) [20–23]. In 2020, Ma et al. [24] proposed an inverse design method to find the target parameters of semiconductor lasers based on a deep-learning neural network algorithm and TWM simulations. In 2022, Chi et al. [25] proposed a method for extracting almost all DFB laser parameters by fitting the output characteristics using the TWM. However, it can be highly time-consuming and computationally expensive to extract most of parameters by fitting several output characteristics of the DFB laser simultaneously.

In this paper, we propose an efficient and accurate method for parameter extraction of quantum-well DFB lasers that combines the rate equation model, finite element method (FEM), transmission matrix method (TMM), and TWM. The companion Fabry-Perot (FP) lasers are fabricated on the same wafer as the target DFB laser to extract the common parameters involved with the material and structural properties. By deriving analytical expressions from rate equations, the common parameters are efficiently and accurately extracted from the light–current (LI) curves, amplitude modulation (AM) responses, and chirp characteristics of the companion FP lasers. By solving the optical mode using the FEM, the coupling situation of multiple parameters during the extraction process is solved by fitting the far field. The grating-related parameters are extracted individually by fitting the spectrum of the target DFB laser using the TMM. By integrating multiple mathematical models, the proposed method of extracting DFB laser intrinsic parameters can greatly improve computational accuracy and reduce the possibility of multiple solutions. The output characteristics of the DFB laser based on extracted parameters were finally simulated by the TWM, and showed close agreement with the measured results.

2. Parameter extraction theory and process

DFB laser output characteristics are influenced by a combination of structural and material parameters, including cavity length, active region volume, optical confinement factor, effective index, current injection efficiency, internal loss, differential gain coefficient, transparent carrier density, nonlinear gain saturation coefficient, carrier lifetime, linewidth enhancement factor, grating period, and grating coupling coefficient. The DFB laser shares some common parameters with the FP lasers on the same wafer, including fundamental material properties, some physical structural parameters, and basic process parameters. Because FP laser parameters can be quickly and easily extracted using the rate equation, fabricating companion FP lasers during the production of the DFB laser provides a straightforward and efficient method for extracting the common parameters.

2.1 Parameter extraction through far field

In practical fabrication processes, the actual geometric dimensions and material refractive index of the laser may differ slightly from the design values because of factors such as process tolerances, equipment precision, and changes in material properties. These deviations affect the transverse mode field of the laser, resulting in changes in the far-field distribution. By fitting the experimentally measured far-field distribution, the actual material refractive index and cross-section waveguide structural parameters are extracted, and parameters such as the active region volume, effective index, and optical field confinement factor can be further obtained. This process is the foundation for extracting the other parameters and a crucial step in decoupling interdependent parameters.

On the basis of the designed cross-sectional structure, we solve the transverse mode field by the FEM [26,27] and obtain the far-field distribution by a Fourier transform. An error function between the predicted far field of the model and the experimental measurement is then constructed at a series of sampling points. By adjusting the structural parameters and material refractive index to minimize the error function, the active region volume, effective index, and optical field confinement factor of the laser are determined. The adaptive differential evolution algorithm, a common and efficient global optimization algorithm [28,29], is used to minimize the error function.

The fitting process can be simplified because the geometric parameters obtained through cutting the laser are generally accurate, and the only potential inaccuracies arise primarily from the determination of the refractive index for each layer. Considering the effect of the etched grating on the transverse mode field, it is necessary to extract these parameters from the companion FP lasers and DFB laser, respectively.

2.2 Parameter extraction using companion FP lasers

Although complex models such as the TWM are more accurate and comprehensive, the rate simulation model is usually sufficient for parameter extraction of FP lasers because they do not involve distributed feedback structures or complex optical phenomena such as spatial hole burning. Considered the total photon density and carrier density variation with time, the standard rate equations are given as

(1)$$\frac{{dN(t)}}{{dt}} = \frac{{\eta I(t)}}{{eV}} - \frac{{N(t)}}{{{\tau _n}}} - \frac{{{v_g}{g_N}\ln [{N(t)/{N_{tr}}} ]}}{{1 + \varepsilon S(t)}}S(t), $$

(2)$$\frac{{dS(t)}}{{dt}} = \left\{ {\frac{{{v_g}\Gamma {g_N}\ln [{N(t)/{N_{tr}}} ]}}{{1 + \varepsilon S(t)}} - \frac{1}{{{\tau_p}}}} \right\}S(t) + {R_{sp}}, $$

(3)$$\frac{{d\Phi (t)}}{{dt}} = \frac{{{\alpha _H}}}{2}\left\{ {{v_g}\Gamma {g_N}\ln [{N(t)/{N_{tr}}} ]- \frac{1}{{{\tau_p}}}} \right\}, $$

where N and P are the density of the carrier and photon, respectively, $\eta $ is the current injection efficiency, I is the injected current, V is the volume of the active region, e is the electron charge, ${\tau _n}$ is the carrier lifetime, ${v_g}$ is the group velocity, ${g_N}$ is the differential gain coefficient, ${N_{tr}}$ is the carrier density at transparency, $\varepsilon $ is the nonlinear gain saturation coefficient, $\mathrm{\Gamma }$ is the optical confinement factor, ${\tau _p} = 1/\{{{v_g}[{\alpha + \textrm{ln}({1/{R_f}/{R_b}} )/({2L} )} ]} \}$ is the photon lifetime, $\alpha $ is the internal loss, L is the cavity length, ${R_f}$ and ${R_b}$ are the reflectivity of the front and back facet, respectively, ${R_{sp}}$ is the spontaneous emission rate, $\mathrm{\Phi }$ is the phase of the optical field, and ${\alpha _H}$ is the linewidth enhancement factor.

In Eqs. (1)–(3), ${v_g}$ can be calculated from the mode spacing observed in the measured FP laser spectrum, and $\Gamma $ and V can be extracted as described in Section 2.1. In addition, the equations have seven unknown parameters: $\eta $, ${g_N}$, ${N_{tr}}$, $\varepsilon $, ${\tau _n}$, ${\tau _p}$ and ${\alpha _H}$, which can be extracted by analyzing the steady-state characteristics, small-signal modulation response, and chirp characteristics of the FP lasers.

First, by setting the time derivative in the equation to zero and neglecting the spontaneous emission noise, the steady-state equations of the laser can be obtained. When the laser operates at the threshold, the gain and loss are balanced, and the following equations are derived:

(4)$$\frac{{{v_g}\Gamma {g_N}\ln ({N_{th}}/{N_{tr}})}}{{1 + \varepsilon S}} = \frac{1}{{{\tau _p}}}, $$

(5)$${I_{th}} = \frac{{eV}}{{\eta {\tau _n}}}{N_{tr}}{e^{\frac{1}{{{v_g}\Gamma {g_N}{\tau _p}}}}}. $$

The output power of the laser can be expressed as

(6)$$P(t) = \frac{{F{v_g}hv\ln [{1/({R_f}{R_b})} ]VS(t)}}{{2\Gamma L}}, $$

where P is the output power of the front facet, $F = ({1 - {R_f}} )/\left[ {1 - {R_f} + ({1 - {R_b}} )\left( {\sqrt {{R_f}/{R_b}} } \right)} \right]$ represents the ratio of output power from the front facet to the total power, h is Planck’s constant, and v is the optical frequency corresponding to the reference wavelength. The slope efficiency of laser can be derived as

(7)$$\frac{{{P_f}}}{{{I_f} - {I_{th}}}} = \frac{{F\eta hv{v_g}\ln [{1/({R_f}{R_b})} ]{\tau _p}}}{{2eL}}. $$

Using Eqs. (1) and (2), the normalized small-signal modulation response of quantum well lasers can be derived as

(8)$${S_{21}} = \frac{{\omega _r^2}}{{\omega _r^2 - {\omega ^2} + \frac{{j\omega {\omega _r}}}{Q}}}, $$

where the relaxation oscillation frequency ${\omega _r}$ and quality factor Q are given by

(9)$${\omega _r} = \sqrt {\frac{{\eta {v_g}\Gamma {g_N}({I - {I_{th}}} )}}{{eV{N_{tr}}{e^{\frac{1}{{{v_g}\Gamma {g_N}{\tau _p}}}}}}}}, $$

(10)$$\frac{1}{Q} = \frac{{\varepsilon {\omega _r}{N_{tr}}{e^{\frac{1}{{{v_g}\Gamma {g_N}{\tau _p}}}}}}}{{{v_g}{g_N}}} + {\tau _p}{\omega _r} + \frac{1}{{{\tau _n}{\omega _r}}}. $$

The effects of spontaneous radiation and gain compression are neglected in the derivation. Therefore, the measurements used for parameter extraction should be taken when the laser is operating above threshold and the optical power does not reach an extremely high level.

During current modulation of the active region, the variation of carrier density leads to a change in the refractive index, which shifts the laser resonance mode in the frequency domain, i.e., frequency chirp. Using Eqs. (2), (3), and (6), the frequency chirp is expressed in terms of the output power as

(11)$$\Delta v = \frac{{{\alpha _H}}}{{4\pi }}\left( {\frac{d}{{dt}}\ln P(t) + kP(t)} \right), $$

where $k = \frac{{2\Gamma \varepsilon L}}{{FhvV{\tau _p}{v_g}ln[{1/({{R_f}{R_b}} )} ]}}$ is the adiabatic chirp coefficient. As the modulation of frequency is proportional to the linewidth enhancement factor, the parameter can be extracted by frequency chirp.

From Eqs. (5)–(11), the unknown parameters can be extracted by the method represented by the flowchart in Fig. 1. The known parameters that have been confirmed or extracted must be set before extraction, including L, V, ${R_f}$, ${R_b}$, ${v_g}$ and $\mathrm{\Gamma }$. From Eq. (7), $\eta $ and ${\tau _p}$ are first extracted from the measured LI curves of the FP lasers with at least two different cavity lengths. Then, we preset a reasonable initial value of ${\tau _n}$ for extraction of parameters from the AM response. The detailed method is to extract ${g_N}$ and ${N_{tr}}$ from the AM response at the same bias current and obtain $\varepsilon $ at a different bias current from Eqs. (5), (9), and (10), and then update ${\tau _n}$ by Eq. (10) according to the calculated ${g_N}$, ${N_{tr}}$, and $\varepsilon $ until convergence. Finally, from the measured frequency chirp and output power, ${\alpha _H}$ is calculated by Eq. (11).

Fig. 1. Flowchart for parameter extraction from the FP laser output characteristics.

Download Full Size | PDF

2.3 Parameter extraction and validation using the DFB laser

After the aforementioned extraction process, the grating parameters of the DFB laser remain to be extracted. The grating period can also be accurately measured by cutting the laser, and the grating coupling coefficient can be obtained by fitting the simulated spectrum to the measured spectrum. For different DFB lasers, the grating phases of the cavity facet are considered to be different and serve as initial variables during the spectrum fitting process. The spectrum is calculated using the TMM [30,31], and the fitting approach is also based on the adaptive differential evolution algorithm. The main mode, side modes and stopband width of the laser are considered as key features during the spectrum fitting, and are given higher weights when constructing the error function.

At this point, all parameters of the DFB laser are extracted. The accuracy of parameter extraction is well checked by simulating the output characteristics of the DFB laser on the basis of the extracted parameters through the TWM and comparing them with the measured results.

The governing equations of the TWM can be expressed as follows:

(12)$$\left( {\frac{1}{{{v_g}}}\frac{\partial }{{\partial t}} + \frac{\partial }{{\partial z}}} \right){e^f}(z,t) = [{G(z,t) - j\delta } ]{e^f}(z,t) + j\kappa {e^b}(z,t) + {\widetilde s^f}(z,t), $$

(13)$$\left( {\frac{1}{{{v_g}}}\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial z}}} \right){e^b}(z,t) = [{G(z,t) - j\delta } ]{e^b}(z,t) + j\kappa {e^f}(z,t) + {\widetilde s^b}(z,t), $$

(14)$$\frac{{dN(z,t)}}{{dt}} = \frac{{\eta I}}{{eV}} - \frac{{N(z,t)}}{{{\tau _n}}} - \frac{{{v_g}\Gamma {g_N}\ln [N(z,t)/{N_{tr}}]}}{{1 + \varepsilon S(z,t)}}S(z,t), $$

where ${e^f}$ and ${e^b}$ are the forward and backward traveling optical fields, ${\tilde{s}^f}$ and ${\tilde{s}^b}$ denote the spontaneous emission noises, and $\kappa $ is the coupling coefficient due to the gratings. G and $\delta $ are the modal gain and the detuning factor, respectively, which are calculated by

(15)$$G(z,t) = \frac{1}{2}\left\{ {\frac{{\Gamma {g_N}[{N(z,t)/{N_{tr}}} ]}}{{1 + \varepsilon S(z,t)}} - \alpha } \right\}, $$

(16)$$\delta = \frac{{2\pi }}{{{\lambda _0}}}\left\{ {{n_{eff0}} - \frac{{{\lambda_0}}}{{4\pi }}{\alpha_H}\Gamma {g_N}\ln [N(z,t)/{N_{tr}}]} \right\} - \frac{\pi }{\Lambda }, $$

where ${\lambda _0}$ is the reference wavelength, ${n_{eff0}}$ is the effective refractive index without injection, and $\Lambda $ is the grating period. Because the rate equation model can be derived from the TWM, it is guaranteed that all the above-extracted parameters are applicable.

3. Extraction implementation and validation

The DFB lasers for parameter extraction were grown on an InP substrate using low-pressure metal-organic chemical vapor deposition (LP-MOCVD). The active region consists of 11 layers of AlGaInAs compression strained quantum well embedded in AlGaInAs SCH layers. The first order corrugations were formed using holographic lithography and wet etching. The companion FP lasers were fabricated on the same wafer without etching the grating layer material. For device fabrication, a 1.6-µm-wide ridge waveguide was formed through photolithography and wet etching. Finally, the wafer was cleaved into bars for facet coating with a chosen cavity length. The actual geometric dimensions of the DFB laser and the companion FP lasers, including cavity length, ridge width, active region thickness, and grating period, were precisely measured with an electron microscope after cutting the lasers. The reflectivity of the laser facets is determined by simultaneously coating a reference sample as well as the measured lasers, and measuring the surface reflectivity of the reference sample. Figure 2(a) shows the laser far-field and LI curves measurement system, and the fiber-coupled platform illustrated in Fig. 2(b) is utilized for the spectrum, small-signal response and large-signal response measurements. The measured laser was mounted on a heat-conducting carrier with a thermoelectric cooler (TEC) to maintain a constant operating temperature of 25 °C.

Fig. 2. (a) Laser far-field and LI curves measurement system. (b) Fiber-coupled platform for the spectrum, small-signal response and large-signal response measurements.

Download Full Size | PDF

The far-field distribution in the horizontal and vertical directions was first measured around the threshold current of the laser to mitigate the effect of injected carriers on the refractive index. The scanning arms are controlled by an electrical control system to rotate horizontally and vertically around the emission point of the laser and record the light intensity at different angles. Because of the low output power of the laser operating near the threshold, the measurement results are susceptible to noise. Therefore, the measured results are processed by the smoothing function included in the test software, which facilitates fitting with the simulation results. The refractive index of each layer is calculated by self-developed software according to the information including temperature (room temperature) and material components. By fitting the calculated far-field distribution with the measured result, the effective refractive index and the optical field confinement factor of the companion FP lasers and the DFB laser were extracted, respectively. The extracted effective refractive index and optical confinement factor were 3.231 and 8.3% for the FP laser and 3.227 and 8.0% for the DFB laser. Figure 3 shows the measured far-field distribution in the horizontal and vertical directions as well as the calculated result by the extracted parameters.

Fig. 3. Measurement and calculated result of far-field distributions of the FP lasers in the (a) horizontal and (b) vertical directions, and of the DFB laser in the (c) horizontal and (d) vertical directions.

Download Full Size | PDF

We then extracted the common parameters on the basis of the output characteristics of the FP laser. Figure 4(a) shows the measured LI curves of the FP lasers with cavity lengths of 168 µm and 214 µm. The threshold currents and slope efficiencies for the 168-µm lasers were 7.2 mA and 0.26 mW/mA, respectively, and those for the 214-µm lasers were 8.3 mA and 0.24 mW/mA, respectively. According to Eq. (7), the injection efficiency and internal loss of the lasers were calculated to be 0.73 and 13.8 cm⁻¹. Then, the AM responses of the 168-µm laser were measured using a KEYSIGHT N5227A PNA network analyzer and an N4373D lightwave component analyzer. The RF signal generated by the network analyzer is superimposed on the DC signal by a Bias Tee and fed to the DFB laser through a GSG RF probe. The output power is collected via a lensed fiber and fed into the lightwave component analyzer. The measured results of AM response are shown in Fig. 4(b). The relaxation oscillation frequencies and quality factors were obtained as 8.5 GHz and 2.66 at an operating current of 20 mA, and 11.0 GHz and 2.07 at 30 mA. Following the steps outlined in Section 2, the laser’s differential gain coefficient, transparent carrier density, nonlinear gain saturation coefficient, and carrier lifetime were determined, as listed in Table 1.

Fig. 4. (a) LI curves of the FP lasers with cavity lengths of 168 µm and 214 µm; (b) AM response of the FP laser at the operating current of 20 mA and 30 mA.

Download Full Size | PDF

Table 1. Extracted parameters of the FP lasers

View Table | View all tables in this article

The linewidth enhancement factor can be extracted from the spectrum of the FP laser under digital amplitude modulation because the frequencies for bits ‘0’ and ‘1’ are different according to Eq. (11), which resulted in a split of the carrier in the spectrum [32,33]. A 100-MHz square wave signal generated by the arbitrary waveform generator is amplified by an RF amplifier and superimposed with a DC signal via a Bias Tee. The optical power collected by the fiber is fed to the KEYSIGHT DSAZ594A real-time oscilloscope and the YOKOGAWA optical spectrum analyzer via a 9:1 fiber coupler. Figure 5(a) shows the spectrum under the large-signal modulation, which exhibits two separated peaks corresponding to bits ‘0’ and ‘1’. The measured output power is shown in Fig. 5(b). The linewidth enhancement factor was finally extracted as 3.28.

Fig. 5. (a) Measured spectrum and (b) output power of the FP laser under the large-signal modulation at 100 MHz.

Download Full Size | PDF

Table 1 lists all the extracted parameters of the FP lasers. The output characteristics of the FP lasers were simulated by the rate equation model and compared with the measured values, as shown in Fig. 6. Figure 6(d) shows the close agreement between the simulated result and the measurement of the large-signal modulation at 5 GHz, proving the accuracy of the FP laser parameter extraction.

Fig. 6. Comparison of the calculated result and the measured result of the FP laser: (a) LI curve with cavity length of 168 µm, (b) LI curve with cavity length of 214 µm, (c) AM response, and (d) output power during the large-signal modulation at 5 GHz.

Download Full Size | PDF

The grating coupling coefficient of the DFB laser was extracted by fitting the spectrum of the DFB laser. Figure 7 shows the spectrum of the DFB laser, which was measured near the threshold current to mitigate the impact of spatial hole burning. By fitting the calculated spectrum by the TMM with the measurement, the grating coupling coefficient was obtained to be 161 cm⁻¹.

Fig. 7. Measured lasing spectrum of the DFB laser and corresponding simulation result using the extracted parameters.

Download Full Size | PDF

Table 2 presents all the extracted parameters of the DFB laser. Using the extracted parameters, the output characteristics of the DFB laser were simulated through the TWM to evaluate the accuracy of the parameter extraction. Figure 8 shows a comparison of the calculated and measured results of the LI curves, AM response, and large-signal modulation. The final simulation results agree closely with the experimental measurements, which proves the excellent feasibility and accuracy of the proposed method.

Fig. 8. (a) LI curve, (b) AM response, and (c) output power during the large-signal modulation at 5 GHz of the DFB laser.

Download Full Size | PDF

Table 2. Extracted parameters of the DFB laser

View Table | View all tables in this article

4. Conclusion

This study proposed an efficient and accurate parameter extraction method for quantum-well DFB lasers. Using multiple mathematical models, all the intrinsic parameters of the DFB laser were accurately extracted. Companion FP lasers were fabricated on the same wafer as the DFB lasers and provided an efficient means to extract their common parameters. Benefiting from the simple optical properties of FP lasers without spatial-hole burning, the common parameters were efficiently extracted from the LI curves, AM responses, and chirp characteristics of the companion FP lasers using the analytical expression derived from the rate equation. The effective refractive index, active region dimensions, and optical field confinement factor were obtained by fitting the far field and cutting the laser, which is a crucial step for decoupling the interdependent parameters. The grating coupling coefficient of the DFB laser was complemented by fitting the spectrum of the DFB lasers with the TMM. Compared to directly extracting all parameters by fitting the output characteristics of the DFB laser through the TWM, the proposed method can significantly improve computational efficiency and reduce the possibility of multiple solutions. The feasibility and accuracy of the extraction method are finally confirmed through the congruence between the TWM simulation results using the extracted parameters and measurements. The accuracy of the extracted results can be further improved by considering the effect of factors such as operating temperature. It is believed that the comprehensive parameter extraction can be adopted as an efficient and accurate method for identifying manufacturing defects and guiding design optimization.

Funding

Key Research and Development Program of Shandong Province, China (2023CXGC010107).

Acknowledgments

We thank Professor Xun Li for technical assistance and guidance.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. K. Kikuchi, “Fundamentals of coherent optical fiber communications,” J. Lightwave Technol. 34(1), 157–179 (2016). [CrossRef]

2. Y. Shi, Y. Zhang, Y. Wan, et al., “Silicon photonics for high-capacity data communications,” Photonics Res. 10(9), A106–A134 (2022). [CrossRef]

3. W. Jin, Q.-F. Yang, L. Chang, et al., “Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high- Q microresonators,” Nat. Photonics 15(5), 346–353 (2021). [CrossRef]

4. J. Lepselter and M. Elman, “Biological and clinical aspects in laser hair removal,” J. Dermatol. Treat. 15(2), 72–83 (2004). [CrossRef]

5. Y. Li, Z. Kang, and L. Hu, “New application of semiconductor laser in medical field,” Laser J. 31(6), 73–75 (2010).

6. A. Schmohl, A. Miklós, and P. Hess, “Detection of ammonia by photoacoustic spectroscopy with semiconductor lasers,” Appl. Opt. 41(9), 1815–1823 (2002). [CrossRef]

7. Y.-H. Lai, M.-G. Suh, Y.-K. Lu, et al., “Earth rotation measured by a chip-scale ring laser gyroscope,” Nat. Photonics 14(6), 345–349 (2020). [CrossRef]

8. S. Gao, S. Xiang, Z. Song, et al., “Hardware implementation of ultra-fast obstacle avoidance based on a single photonic spiking neuron,” Laser Photonics Rev. 17(12), 2300424 (2023). [CrossRef]

9. M. Okai, M. Suzuki, and T. Taniwatari, “Strained multi-quantum-well corrugation-pitch-modulated distributed feedback laser with ultranarrow (3.6 kHz) spectral linewidth,” Electron. Lett. 29(19), 1696–1697 (1993). [CrossRef]

10. K. Nakahara, Y. Wakayama, T. Kitatani, et al., “Direct modulation at 56 and 50 Gb/s of 1.3-µm InGaAlAs ridge-shaped-BH DFB lasers,” IEEE Photonics Technol. Lett. 27(5), 534–536 (2015). [CrossRef]

11. J. Chi, X. Li, C. Niu, et al., “Enhanced modulation bandwidth by delayed push-pull modulated DFB lasers,” Micromachines 14(3), 633 (2023). [CrossRef]

12. S. Liu, H. Wu, Y. Shi, et al., “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photonics Technol. Lett. 31(10), 751–754 (2019). [CrossRef]

13. X. Li, Optoelectronic Devices: Design, Modeling, and Simulation (Cambridge University, 2009).

14. H. Yasaka, K. Takahata, and M. Naganuma, “Measurement of gain saturation coefficients in strained-layer multiple quantum-well distributed feedback lasers,” IEEE J. Quantum Electron. 28(5), 1294–1304 (1992). [CrossRef]

15. L. Bjerkan, A. Royset, L. Hafskjaer, et al., “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996). [CrossRef]

16. J. C. Cartledge and R. C. Srinivasan, “Extraction of DFB laser rate equation parameters for system simulation purposes,” J. Lightwave Technol. 15(5), 852–860 (1997). [CrossRef]

17. P. Pérez, A. Valle, I. Noriega, et al., “Measurement of the intrinsic parameters of single-mode VCSELs,” J. Lightwave Technol. 32(8), 1601–1607 (2014). [CrossRef]

18. S. Xiang, Y. Shi, X. Guo, et al., “Hardware-algorithm collaborative computing with photonic spiking neuron chip based on integrated Fabry–Perot laser with saturable absorber,” Optica 10(2), 162–171 (2023). [CrossRef]

19. Q. Ding, Q. Yang, J. Li, et al., “Efficient and systematic parameter extraction based on rate equations by DFB equivalent circuit model,” Opt. Express 31(24), 40604–40619 (2023). [CrossRef]

20. L. M. Zhang, S. F. Yu, M. C. Nowell, et al., “Dynamics analysis of radiation and side mode suppression in a 2nd-order DFB laser using time-domain laser signal traveling wave model,” IEEE J. Quantum Electron. 30(6), 1389–1395 (1994). [CrossRef]

21. C. F. Tsang, D. D. Marcenac, E. Carroll, et al., “Comparison between ‘power matrix model (PMM)’ and ‘time domain model (TDM)’ in modeling large signal responses of DFB lasers,” IEE Proc.-Optoelectron. 141(2), 89–96 (1994). [CrossRef]

22. W. Li, W.-P. Huang, X. Li, et al., “Multiwavelength gain-coupled DFB laser cascade: design modeling and simulation,” IEEE J. Quantum Electron. 36(10), 1110–1116 (2000). [CrossRef]

23. B. S. Kim, Y. Chung, and J. S. Lee, “An efficient split-step time-domain dynamic modeling of DFB/DBR laser diodes,” IEEE J. Quantum Electron. 36(7), 787–794 (2000). [CrossRef]

24. Z. Ma and Y. Li, “Parameter extraction and inverse design of semiconductor lasers based on the deep learning and particle swarm optimization method,” Opt. Express 28(15), 21971 (2020). [CrossRef]

25. J. Chi, C. Niu, and J. Zhao, “Parameter extraction for quantum well DFB lasers based on 1D traveling wave model,” IEEE Photonics J. 14(5), 1–8 (2022). [CrossRef]

26. F. Assous, P. Degond, E. Heintze, et al., “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109(2), 222–237 (1993). [CrossRef]

27. M. F. Wong, O. Picon, and V. F. Hanna, “A finite element method based on Whitney forms to solve Maxwell equations in the time domain,” IEEE Trans. Magn. 31(3), 1618–1621 (1995). [CrossRef]

28. R. Storn and K. Price, “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim. 11(4), 341–359 (1997). [CrossRef]

29. A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009). [CrossRef]

30. G. Björk and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5(1), 140–146 (1987). [CrossRef]

31. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26(16), 3474–3478 (1987). [CrossRef]

32. A. Villafranca, J. Lasobras, and I. Garces, “Precise characterization of the frequency chirp in directly modulated DFB lasers,” in Proceedings of Spanish Conference on Electron Devices (2007), pp. 173–176.

33. C. del Río Campos, P. R. Horche, and A. Martin-Minguez, “Analysis of linewidth and extinction ratio in directly modulated lasers for performance optimization in 10 Gbit/s CWDM systems,” Opt. Commun. 283(15), 3058–3066 (2010). [CrossRef]

Parameters	Symbol	Value	Unit
Facet reflectivity	$R_{f}, R_{b}$	0.5,0.5
Laser length	$L_{1}, L_{2}$	168,214	$μ m$
Active region volume	$V_{1}, V_{2}$	14.78,18.83	$μ m^{3}$
Optical confinement factor	$Γ$	0.082
Effective index without injection	$n_{e f f 0}$	3.231
Current injection efficiency	$η$	0.733
Internal loss	$α$	13.84	$c m^{- 1}$
Differential gain coefficient	$g_{N}$	1094	$c m^{- 1}$
Transparent carrier density	$N_{t r}$	0.57	$10^{18} c m^{- 3}$
Nonlinear gain saturation coefficient	$ε$	3.59	$10^{- 17} c m^{3}$
Carrier lifetime	$τ_{n}$	0.47	$ns$
Linewidth enhancement factor	$α_{H}$	3.28

Parameters	Symbol	Value	Unit
Facet reflectivity	$R_{f}, R_{b}$	0.96,0.003
Laser length	$L$	180	$μ m$
Active region volume	$V$	15.84	$μ m^{3}$
Optical confinement factor	$Γ$	0.08
Effective index without injection	$n_{e f f 0}$	3.227
Current injection efficiency	$η$	0.733
Internal loss	$α$	13.84	$c m^{- 1}$
Differential gain coefficient	$g_{N}$	1094	$c m^{- 1}$
Transparent carrier density	$N_{t r}$	0.57	$10^{18} c m^{- 3}$
Nonlinear gain saturation coefficient	$ε$	3.59	$10^{- 17} c m^{3}$
Carrier lifetime	$τ_{n}$	0.47	$ns$
Linewidth enhancement factor	$α_{H}$	3.28
Grating period	$Λ$	202.9	$nm$
Grating coupling coefficient	$κ$	161	$c m^{- 1}$

Parameters	Symbol	Value	Unit
Facet reflectivity	$R_{f}, R_{b}$	0.5,0.5
Laser length	$L_{1}, L_{2}$	168,214	$μ m$
Active region volume	$V_{1}, V_{2}$	14.78,18.83	$μ m^{3}$
Optical confinement factor	$Γ$	0.082
Effective index without injection	$n_{e f f 0}$	3.231
Current injection efficiency	$η$	0.733
Internal loss	$α$	13.84	$c m^{- 1}$
Differential gain coefficient	$g_{N}$	1094	$c m^{- 1}$
Transparent carrier density	$N_{t r}$	0.57	$10^{18} c m^{- 3}$
Nonlinear gain saturation coefficient	$ε$	3.59	$10^{- 17} c m^{3}$
Carrier lifetime	$τ_{n}$	0.47	$ns$
Linewidth enhancement factor	$α_{H}$	3.28

Parameters	Symbol	Value	Unit
Facet reflectivity	$R_{f}, R_{b}$	0.96,0.003
Laser length	$L$	180	$μ m$
Active region volume	$V$	15.84	$μ m^{3}$
Optical confinement factor	$Γ$	0.08
Effective index without injection	$n_{e f f 0}$	3.227
Current injection efficiency	$η$	0.733
Internal loss	$α$	13.84	$c m^{- 1}$
Differential gain coefficient	$g_{N}$	1094	$c m^{- 1}$
Transparent carrier density	$N_{t r}$	0.57	$10^{18} c m^{- 3}$
Nonlinear gain saturation coefficient	$ε$	3.59	$10^{- 17} c m^{3}$
Carrier lifetime	$τ_{n}$	0.47	$ns$
Linewidth enhancement factor	$α_{H}$	3.28
Grating period	$Λ$	202.9	$nm$
Grating coupling coefficient	$κ$	161	$c m^{- 1}$

Efficient and accurate parameter extraction for quantum-well DFB lasers: a comprehensive approach integrating multiple mathematical models

Abstract

1. Introduction

2. Parameter extraction theory and process

2.1 Parameter extraction through far field

2.2 Parameter extraction using companion FP lasers

2.3 Parameter extraction and validation using the DFB laser

3. Extraction implementation and validation

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (16)

Optics Express