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Abstract
The conventional direct parameter extraction method generally suffers from cumbersome due to redundant experiments. An efficient and systematical parameter extracting solution is proposed based on an equivalent circuit model of distributed feedback (DFB) lasers. The successfully built circuit model includes the necessary intrinsic parameters in the rate equations and the extrinsic parameters to provide a better approximation of the actual laser. This method is experimentally verified through a DFB laser chip measurement of electronic and optical performance under the same conditions. Finally, the nine intrinsic parameters in the rate equations and five extrinsic parameters in the model are efficiently extracted using this technique from a set of experimental characteristics of a DFB laser chip. Modeled and measured results for the laser output characteristics exhibit good agreement when the extracted parameters are used. The method is versatile for other semiconductor lasers that can be modeled using rate equations. Comparison with simulation results of published laser models further validates the reliability of the presented model and extraction method.
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1. Introduction
Increasing demand for long-distance, high-capacity fiber-optic communication systems has stimulated a boom in semiconductor laser design technology [1–3]. An important method for estimating the system performance is the modelling of lasers on the basis of the rate equation theory [4]. Compared with the traditional Fabry-Perot (FP) laser, the semiconductor distributed feedback (DFB) laser has the advantages of narrow linewidth, single longitudinal mode output, and good stability. Recently, it has become the primary light source for long-distance fiber-optic communication system and is attracting increasing research attention [5–8]. Furthermore, extracting the parameters of the rate equations has been proven to be an essential step in achieving accurate model construction and consistency between simulated and experimental results [9]. Gao categorizes semiconductor laser model parameter extraction techniques into three types, namely direct extraction method, semi-analytical method, and numerical optimization method [10]. He also concludes that the first two methods are more suitable for parameter extraction than the third, which is highly dependent on initial values and may yield results that are not physically meaningful. Although Cartledge and Bjerkan's work on parameter extraction for DFB lasers is excellent, further improvements can be made due to insufficient analysis of parasitic parameters or the complexity of fiber experiments [9,11]. And in the case of too many parameters, the method to extract the exact value of single parameter is not applicable [12].
In recent years, there has been a growing interest in the extraction of laser parameters and the construction of a laser equivalent circuit model [13–16]. The construction of a laser equivalent circuit model visualizes the structure of a complex laser device and facilitates the analysis of the laser [17–19]. Furthermore, circuit modelling can provide accurate extraction of the intrinsic laser parameters without the need for intensive numerical calculations. Pioneering work has explored the application of the equivalent circuit model in laser parameter extraction. Minoglou et al. developed a compact nonlinear equivalent circuit model for a laser input and extracted the required parameters based on current-light-voltage and small-signal S11 measurements [20]. Gao et al. derived a laser diode (LD) intrinsic small-signal equivalent circuit model from the linear rate equations and extracted LD parasitic intrinsic as well as rate-equation model parameters [21]. Bacon et al. considered the electrical access and the optical cavity of the VCSEL chip as two independent networks for parasitic removal and then extracted the intrinsic parameters [22]. Overall, a model that accurately reflects the characteristics of the laser is an important tool for parameter extraction, while the determination of the laser parameters has a direct impact on the accuracy of the model.
The goal of laser parameter extraction based on a laser model is to determine a physically reasonable set of parameter solutions for accurate reproduction of laser characteristics and simulation of fiber-optic communication system behavior. Researchers always use two methods to combine the equivalent circuit model with parameter extraction. One is to derive the components from laser structure or material property analysis to build the physical structure model. The parameters extracted by this method have real physical significance, but may be subject to large errors [21,23]. The other is to vary the rate equations to obtain discrete components with capacitive or resistive magnitudes and then build a model based on Kirchhoff's laws. Based on this model, the researchers first obtain some parameter values that are considered as capacitance or resistance, which makes the extraction of the internal parameters not efficient enough. However, the parameter extraction studies of other research groups based on equivalent circuit models are mainly concerned with VCSEL lasers [24] or Schottky diodes [25]. Our group is devoted to the parameter extraction study of DFB lasers based on the equivalent circuit model.
For the purpose to overcome the limitation of previous literature, this paper proposes an efficient technique to easily extract laser parameters according to the laser equivalent circuit model built with Advanced Design System (ADS). The procedure combines the traditional direct extraction method with model building to extract the laser parameters in a relatively simple way and then to construct a model closer to the actual laser. Our research team proposed an improved equivalent circuit model of DFB laser on the basis of rate equations in PSpice [19,26]. Following the previous work, a model is constructed in ADS for efficient parameter extraction using the model optimization function and more accurate output at high frequencies. Prior to model building, several easily measurable parameters are extracted using the direct extraction method. Subsequently, the photon lifetime and the remaining parameters that are shown to be correlated with the photon lifetime are defined as variables to be written into the constructed model. Using the experimental data as a target, the photon lifetime is scanned within a reasonable range obtained by the calculation to determine the remaining parameters. A resistance-capacitance circuit is included in the model to simulate the effects of parasitic parameters on the laser, bringing the simulated results closer to the measured data. To illustrate the above technique, the parameters of a DFB laser chip are extracted from the measured small-signal characteristics and added to the model built. The measured and modelled results for the small-signal intensity modulation frequency response (hereafter referred to as the frequency response) of the DFB laser exhibit good agreement, demonstrating the accuracy of the parameter extraction method. To further validate the model, the large signal impulse response of the semiconductor laser is also simulated in the paper. The output characteristics of the published Pspice model are compare with the results of the model presented here, and they show good agreement. Altogether, the extraction technique used in this paper is more efficient and faster without additional fiber testing. Additionally, the equivalent model can provide guidance for optimization and improvement of fiber-optic communication systems in the simulation phase. Various sensor types have been developed for broader applications [27,28]. Considering the close connection between lasers and fiber sensors, the present method may also have potential applications for the improvement of the fiber sensors [29–31].
2. Theory
2.1 Single mode rate equations
The rate equation theory is a microscopic image-only theory dealing with the interaction between light and substance, which can be used to describe the internal photoelectric conversion law of semiconductor lasers. The single mode rate equations for semiconductor lasers are given in the following form [11,16].
The optical output power $P(t )$ as a function of the photon density $S(t )$ is given by
where $S(t )$ is the photon density, $N(t )$ is the carrier density, $I(t )$ is the injection current, $\Gamma $ is the optical confinement coefficient, ${g_0}$ is the gain slope, ${N_0}$ is the transparency carrier density, $\varepsilon $ is the gain saturation coefficient, ${\tau _p}$ is the photon lifetime, ${\tau _n}$ is the carrier lifetime, $\beta $ is the spontaneous emission coefficient, ${V_a}$ is the active layer volume, $\eta $ is the total quantum efficiency, h is the Planck constant, and v is the optical frequency. Here the number of parameters that we want to determine is nine (${V_a}$, $\Gamma $, ${g_0}$, $\varepsilon $, ${N_0}$, $\beta $, ${\tau _p}$, ${\tau _n}$ and $\eta $).2.2 Parameter extraction procedure
The time derivatives of the photon and carrier densities are equal to zero when the laser is operating in the DC steady state. And it is referred to as the threshold condition when the gain counterbalances the loss in (1), i.e.,
Here ${N_{th}}$ is the carrier density at threshold. Then $X(t) = N(t)/{N_{th}}$ is defined and substituted into (1)-(2) to obtain the following (5)-(6), which contain only directly measurable transition parameters.
Thus, the nine parameters to be extracted can be determined from the transition parameters by measuring the external characteristics of the laser.
The following expression is derived from the combination of (1)-(4)
Equation (8) is effective near the threshold, where the photon density is very low, i.e., $\varepsilon S$ is small enough to be neglected. It is also clear from [9] and [21] that ${\tau _c}\beta /{\tau _n} \ll 1$, which allows the following approximation
For currents well above the threshold, the spontaneous radiation ${I_s}$ from the laser can be ignored according to laser theory. Thus, (10) is derived from (9) as
This form means that the power is linearly related to the injected current for values well above the threshold. Consequently, the parameter F can be extracted from (10) by fitting the linear part of the power versus current curve, while ${I_s}$ can be obtained from (9) with points close to the threshold.
Unlike other externally modulated lasers, semiconductor lasers can be modulated directly by adjusting the injection current. In small-signal modulation, as the laser injection current varies with the modulation signal, the carrier concentration in the laser cavity changes, causing a variation in the optical output power. The frequency response of a small-signal modulated laser can be determined by assuming a harmonic current superimposed on a constant bias current above the threshold [32]. Derived from (1) and (2), a well-known expression for the small-signal intensity modulation frequency response of a laser [11] is
Here ${f_r}$ is the relaxation oscillation frequency and $\gamma $ is the damping coefficient. And the relationships between them are shown in the following equations.
Here K is the K-factor.
Traditionally, Eq. (11) is considered sufficient to extract the values of ${f_r}$ and $\gamma $, but the study done by [16] showed the deficiency of (11) under high current conditions. Therefore, a more accurate frequency response by taking into account the gain compression effect of a laser at high currents is expressed as
The subtraction method is often used to eliminate the effect of the parasitic parameter in the measured frequency response, which is generally considered independent of the bias current [33], and the resulting ${H_{21}}$ is expressed as
In consequence, equations (15a) and (15b) are obtained, which give more precise relationships between B, ${\tau _n}$ and K for determining their values.
2.3 Model construction
In order to extract the remaining parameters without conducting more fiber experiments, we built an equivalent circuit model to simulate the external characteristics of a laser. ADS has powerful electronic design automation features that greatly improve the convenience, speed, and accuracy of circuit and system design. It is also one of the most widely used microwave/RF circuit and communication system simulation software in domestic universities and research institutes [34]. The following model in Fig. 1 is constructed on the basis of rate equations using the symbolically-defined device (SDD) in ADS [35].
As shown in Fig. 1, port 1 is the injection current, port 2 is the output light intensity, and port 3 is the carrier density. The current and voltage values of port 1 are set to be equal and the current value of port 2 is assigned to m. In addition, ${i_1}$, ${i_2}$, and ${i_3}$ are the currents of port 1, port 2 and port 3, respectively. Besides, the current and voltage relationships of the three ports are represented by F[p, w], where p is the port index and w is the weighting factor. Then, the DC characteristics and small-signal characteristics of the laser can be simulated by adding external circuits to the model. And each parameter in the model is defined separately, facilitating the determination or modification of any rate equation parameter value directly. Furthermore, the parametric optimization function of ADS allows us to build a model that better fits the actual output more quickly, and thus to obtain more accurate parameters.
The rate equation used in Fig. 1 undergoes the following transformations with the aim of improving the convergence of the model [36].
where ${N_e}$ is the equilibrium minority carrier density, ${V_L}$ is the voltage across the model, $\xi $ is an empirical constant, k is the Boltzmann constant, T is the temperature, ${S_n}$ is the normalization constant, m is a new variable, and $\delta $ is a small constant to improve the convergence. Then (1)-(3) are transformed intoFurther analysis of the conventional method shows that the parameters not be determined in the previous part can be represented by the parameters already obtained and the photon lifetime. The model constructed contains only ${\tau _p}$ as a variable after writing the transformed parameters into it. Therefore, a set of physically reasonable combinations of laser parameters is obtained by determining the value of ${\tau _p}$ in the model. Finally, parasitic parameters are added into the model and then the outputs are adjusted to match the measured results. Since SDD is a powerful tool for pure equation models, it is used for modeling many nonlinear devices. Therefore, for other semiconductor lasers that can be characterized by rate equations, this method can also be applied to their modeling and parameter extraction.
3. Measurement and discussion
The expressions for the output optical power, threshold current and frequency response obtained in the theoretical part contain the parameters of the rate equation that need to be extracted. And the turn-on voltage as well as the differential resistance of the laser can be obtained from the V-I characteristic, which can be used for parasitic parameter analysis. Thus, accurate measurement and analysis of the P-I-V and frequency response characteristics of the laser is an important step for parameter extraction. The device used in the following experiment is a 1330 nm single-mode DFB laser. It is an electrically pumped laser that provides energy for the stimulated emission by injecting current into the active region. And the laser lases only when the injection current is greater than the threshold. For all measurements described below, the laser chip was mounted on the chip carrier with a temperature control (TEC) stage underneath (25 °C).
3.1 Direct parameter extraction
Figure 2 is a photograph of the experimental device for the frequency response test system. Figure 3 shows the setup for measuring the frequency response. The DFB laser was powered by KEYTHLEY 2450 Source Meter and the frequency response was measured by KEYSIGHT N5227A PNA Microwave Network Analyzer. The DC signal generated by the 2450 Source Meter and the RF signal generated by the N5227A PNA Microwave Network Analyzer were added using the Marki Bias Tee and fed to the DFB laser through the RF probe. The output power was collected by a single-mode tapered fiber and added to the Lightwave Component Analyzer, which then feeds the converted signal to the network analyzer. The network analyzer generated a swept signal with a frequency increase from 100 MHz to 25 GHz with a sweep interval of 31.125 MHz.
Fig. 2. Experimental device for frequency response test system and the laser chip. (a) System setup; (b) Top view of the DFB laser chip.
The frequency response of the laser was then measured with the network analyzer at multiple bias currents. The bias current is the stabilized DC current injected into the laser to keep the laser in the proper excitation state. In addition, the output power and voltage versus current (P-I-V) characteristics of the laser are measured through a test set with an optical sensor, and the injection current is increased from 0 mA to 120 mA. The frequency response can directly reflect the high-speed characteristics of the DFB laser from the frequency domain. A typical frequency response curve is relatively flat at low frequencies, then gradually rises to the peak at the relaxation oscillation frequency, after which the curve declines rapidly. Additionally, the light output from the laser is in the form of optical energy, whose output power is a physical quantity that measures the amount of laser energy output per unit of time. And the voltage versus current (V-I) characteristic is the measured forward voltage of the laser at different currents, which can reflect the advantages and disadvantages of the structural characteristics of the laser.
As shown in Fig. 4, the P-I results of the laser show good linearity above the threshold until distortion occurs near the top of the curve. This means that with the increase of the injected current, the thermal effect of the laser is enhanced and the luminous efficiency gradually becomes lower. For the V-I curve in Fig. 4, the change of voltage with current is basically linear in the initial stage, and then the curve becomes slower when the turn-on voltage of the laser is reached, which is 0.68 V read from the plot.
The linear fit of Fig. 4 by (10) gives the threshold current Ith = 7.5 mA and F = 2.315 A/W. Additionally, the parameter ${I_s}$ for the laser is calculated to be 5.333 × 10−6 A by fitting the data around the threshold current to (9). Besides, the frequency response of the laser is measured at different bias currents over the range from 20 mA to 70 mA and the results are normalized in Fig. 5. The relaxation oscillation frequency and the bandwidth of the laser increase with the bias current in Fig. 5, which is consistent with the reported results. However, factors such as the increasing damping effect as high currents cause the frequency response curves to flatten out and even lead to a backward shift of ${f_r}$. In addition, the output power in Fig. 3 is strongly affected above bias currents of 70 mA. It is therefore appropriate to select the 20 mA−70 mA current range for analysis in this paper.
Since the data at 20 mA is closest to the threshold current in this paper, it was chosen as the baseline data for the subtraction method proposed by Morton [33]. To eliminate the influence of inherent parasitic parameters that are less affected by the bias currents, the curves in Fig. 6 are derived by subtracting the 20 mA bias frequency response from the curves in Fig. 5. Using the Levenberg-Marquardt algorithm [37], the relative frequency response curves in Fig. 6 are fitted based on (14) to obtain the corresponding ${f_r}$ and $\gamma $ for each current. The results in Fig. 6 demonstrate a good agreement with errors all below 0.01 from 100 MHz−25 GHz, which also verifies the validity of (14). The fitting process is repeated several times in order to find the corresponding ${f_r}$ and $\gamma $ for each current in Fig. 5. Moreover, averaging of the experimental results helps to reduce the error between different bias currents.
Fig. 6. Comparison of the relative frequency response and fitted results. The symbols demonstrate the frequency response for currents above 20 mA minus the frequency response for 20 mA bias current and the lines show the fitted results.
Figure 7(a) shows the fit of the relationship between bias current versus M using (15a). Simultaneously, Fig. 7(b) depicts the match of the extracted values of $\gamma $ versus M to the fitted curve. As shown in Fig. 7(a), the relationship between the bias current and the relaxation oscillation frequency is not simply linear. This result stems from the more pronounced gain compression effect of semiconductor lasers at high currents. And the parameter M shows a good linear correlation with $\gamma $, which is consistent with (15b). However, the measurement uncertainty also leads to the poor overlap of the fitting results at high biases, which is in accordance with the results in [16].
Fig. 7. Comparison of the parameter results obtained with the fitted values at different biases. (a) The results of bias current versus M. (b) The results of M versus $\gamma $.
The values of B, K, and ${\tau _n}$ are determined by fitting Fig. 7 to (15) and recorded in Table 1. Meanwhile, the active layer volume ${V_\textrm{a}}$ and the optical confinement coefficient $\Gamma $ are calculated to be 54.4 um3 and 0.15, respectively, based on the structure of the laser. Under these conditions, the values of ${g_0}$ and $\eta $ can be determined as shown in Table 2 by substituting all the parameters obtained above into (7).
Table 1. Parameter Values Obtained Using Traditional Method
Table 2. Values and Expressions of the Extracted Intrinsic Parameters
Previous studies [11,16] give the orders of magnitude for the parameters in the rate equations, which are in general agreement with the results shown in Table 1 and Table 2, demonstrating the reliability of the values obtained. The slope efficiency of the laser is 43.2%, calculated from the reciprocal of F in Table 1. Furthermore, the remaining parameters that have not yet been extracted are derived from the quantitative relationships between the parameters described in the theory part. And they are then summarized in Table 2 and expressed in forms related to ${\tau _p}$ as shown below.
The results in Table 2 indicate that the other parameters would be determined subsequently once ${\tau _p}$ is known. Therefore, the next section moves on to discuss the extraction procedure using the equivalent circuit model for ascertaining those parameters.
3.2 Model-based parameter extraction
The model proposed in Section II is initially validated before being used for parameter extraction. More specifically, the output characteristics of a laser are simulated and analyzed using the intrinsic and parasitic parameters from [17]. With biases of 50 mA and 60 mA, the frequency responses of the model are performed in Fig. 8. As can be seen in Fig. 8, the outputs are in good agreement with the measured results in the reference, which proves the reliability of the model.
Next, the parameters determined from Table 1 are written directly into the model, while the other parameters are set in the forms associated with ${\tau _p}$ according to Table 2. Thus, the model can be determined by specifying the value of the unique unknown ${\tau _p}$. Take the relaxation oscillation frequency and 3-dB bandwidth of the actual data as the optimization target, and then adjust the values of ${\tau _p}$ in the model to make the output match the experimental data. Perform the same simulation with different bias currents, as the averaging of the simulation results will help to reduce the error between the different curves and grant more accurate parameters for the particular simulation. Finally, the frequency response profile corresponding to the target conditions has been obtained with the appropriate combination of parameters.
The scan range of ${\tau _p}$ is obtained from Table 2, with reference to the values in [18]. Under a bias current of 40 mA, the modelled results have a relaxation frequency of 14.075 GHz and a 3-dB bandwidth of 17.623 GHz, with an error of 0.031 GHz and 0.28 GHz, respectively, from the measured curve. Compared to the method in [11], some of the optical experiments are omitted here, simplifying the extraction procedure. Meanwhile, the corresponding estimated parameter sets based on the circuit model are recorded in Table 3 with the extraction results using the method in [16]. Closer inspection of Table 3 shows that the extraction results of ${\tau _p}$ and related parameters using the present method are close to those calculated using formulae. Apart from this, an equivalent circuit model of the corresponding laser is established during the extraction of parameters, which is useful for more analysis of the laser characteristics.
Table 3. Comparison of Intrinsic Parameter Values Extracted by Calculation and Using the Presented Method
In this way, a set of parameters for the laser rate equations can be obtained within reasonable bounds using the laser equivalent circuit model without fiber measurements. The laser parameters are also extracted by calculation and the results are recorded in Table 3. As can be seen from Table 3, the values of ${\tau _p}$ and other related parameters have small difference between the calculated and model-based results, indicating the efficiency of the method in this paper. However, the model at this point is an ideal case that does not truly represent the characteristics of the actual laser.
3.3 Model optimization and validation
The frequency response subtraction used in the parameter extraction section eliminates the effects of parasitic parameters that are mandatory to be considered when building the model. Therefore, a model is constructed here that more completely reflects the actual laser output characteristics by taking into account more external chip parameters. In Fig. 9, the parasitic parameters of the laser package are equated as a resistance-capacitance (R-C) circuit [38]. Specifically, Cp and Rs are the shunt capacitance and series resistance introduced by the contact between the chip and the electrodes, respectively, Ca is the shunt capacitance introduced when the chip electrodes contact the carrier, the La and Ra are the inductance and resistance presented by the gold wire connecting the chip to the external circuit. Here the diode is used to improve convergence, while the non-linear CSRC1 device is used to transform the output. Supplementally, the thermal effects caused by temperature variations are not considered due to the constant 25 °C experimental environment, which will be the subject of future work.
For the purpose to make the parasitic network act more closely to the actual situation, we first relate the measured data to the theoretical analysis. From the I-V curve of the laser, the turn-on voltage of 0.68 V can be read directly, while the resistance of the entire device is calculated as 7.68 Ω from the linear part of the curve above the threshold. And the calculated values will provide a guide for the further determination of the parasitic parameters.
Prior to the optimization of the model, the values of each component in the R-C circuit are set to an initial value within a reasonable range. Choosing appropriate initial values allows the optimization to reach the optimal solution faster and avoid getting stuck in a local minimum instead of a global solution. Furthermore, the relaxation oscillation frequency and the 3-dB bandwidth of the laser are selected as the targets for optimization. Finally, the parasitic network is obtained by scanning each parameter to narrow the gap between the modelled and measured data. For each solution, the corresponding I-V curve is plotted in order to filter out the combination of parameters that matches the overall device impedance. In this way, the values of the parasitic parameters that reproduce the actual results of the laser are obtained, as shown in Table 4. And the simulation results of the optimized model including the parasitic circuit show a good agreement with the measured data, as can be seen in Fig. 10. Besides, increasing the bias currents leads to an addition of the 3-dB bandwidth, which ties well with the measured results. To test the model performance, the transient simulation is performed with a pulse signal (−10 mA to 10 mA, rise and fall time are both 0.1 ns, time delay 1 ns) in 3 ns and compared with the published Pspice model results. It can be seen that there are relaxation oscillations on both the rising and falling edges of the pulse responses, which is an inherent property of the laser. Additionally, the relaxation amplitude and time decrease with increasing bias current due to the gain saturation effect of the laser. As can be seen in Fig. 11, the response results of the models have good conformity, providing further validation of the model in this paper.
Fig. 10. Comparison of the simulated frequency response using the optimized model with the measurements.
Fig. 11. Comparison of the impulse response using extracted parameters for the presented model and Pspice model. Symbols: Pspice model, lines: model in this paper.
Table 4. Parameter Values Obtained with the Equivalent Circuit Model
The modelling process shows that each element of the parasitic circuit has a significant effect on the performance of the frequency response. In particular, the modification of La in our model causes a change in the relaxation oscillation frequency, while almost every parasitic parameter has an impact on the amplitude of the frequency response. This indicates that the model construction is expected to provide guidance for laser performance optimization and subsequently improve the behavior of the fiber-optic communication system.
4. Conclusion
The technique described in this paper is capable of efficiently and systematically extracting both the intrinsic and extrinsic parameters of the DFB laser. It differs from the conventional extraction method in that it compounds the direct method with the construction of the laser equivalent circuit model. With the model, it is not necessary to rely on the additional experiments to determine the values of the overall rate equation parameters. And the inclusion of parasitic circuits in the model also brings the output much closer to the measured data. The value of ${\tau _p}$ is successfully obtained by optimizing the model and is then used to directly extract the other four parameters which are associated with ${\tau _p}$. To verify the feasibility of the method, the extraction procedure uses experimental results from a DFB laser chip. In conclusion, the proposed technique extracts a physically reasonable set of parameters with in a simpler and more efficient way than the traditional method. The validity of the constructed model and the accuracy of the extraction method have been demonstrated by the good agreement between the modelled results using the extracted parameters and the measurements. Furthermore, there is a small gap between the large-signal simulation results using the Pspice model from a published work and the model presented in this paper. And the method can be further refined by considering the effects of temperature and other factors. Importantly, the extraction technique can be very promising for predicting the behavior of DFB lasers during the simulation stage and is expected to provide guidance for improving fiber-optic communication systems.
Funding
Natural Science Foundation of Shandong Province (ZR2017MF070, ZR2020MF014, ZR2022QF065); National Natural Science Foundation of China (No. 62201324).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this manuscript may be obtained from the authors upon reasonable request.
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