IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 487 Extraction of VCSEL Rate-Equation Parameters for Low-Bias System Simulation Matt Bruensteiner and George C. Papen Abstract—Laser diode rate-equation parameters are extracted for simulation of on–off keyed digital communication links with below-threshold laser prebiases. The extraction procedure uses measurements of the current–voltage–light characteristic, the ac small-signal response above threshold, and the turn-on delay due to an isolated pulse. Characteristics of the extracted models are discussed in detail. The predictive capability of the extracted models is demonstrated by good agreement between modeled and measured transient response pulse shapes for a prebias current of 0.3 times the threshold current. Index Terms— Characterization, low prebias operation, semiconductor laser, system modeling. I. INTRODUCTION VERTICAL-CAVITY surface-emitting lasers (VCSEL’s) are becoming the preferred transmitters in short-distance optical communication systems such as local-area networks (LAN’s), where a 1–2-Gb/s signal must be carried 0–3 km over multimode ﬁber. The multimode output of VCSEL’s reduces modal noise, and their low threshold may allow drive circuits integrated with standard logic families. Low-power systems provide the additional beneﬁt of achieving eye safety without interlocks. However, the maximum output power of a VCSEL driven by a standard logic power supply, or constrained by eye safety, is limited to at most a few milliwatts. Near- or below-threshold biases are required to maximize the extinction ratio, and biases well below threshold are required to minimize power consumption. Robust circuit models for VCSEL’s, which include large-signal below-threshold transient behavior, are therefore vital for integrating VCSEL’s with standard system design tools. The well-known laser diode rate equations have been implemented as a SPICE subcircuit by many authors, and the simplest of these [1]–[4] are well suited to system simulation. More complex SPICE models [5] might be applied to system simulation but are more commonly applied to the design of the laser itself. The available models vary in the level of detail introduced, particularly in the gain carrier density dependence, and the carrier recombination rate. Parameter extraction to reproduce ac and dc characteristics of laser diodes has been demonstrated for the simplest forms of the rate equation parameters using ac response measurements of edge-emitting Manuscript received December 1, 1998; revised April 6, 1999. This work was supported by an external research grant from Hewlett-Packard Laboratories. The authors are with the Department of Electrical and Computer Engineering, University of Illinois, Urbana, IL 61801 USA. Publisher Item Identiﬁer S 1077-260X(99)06137-7. laser diodes [4], [6]–[8] and etched-undercut [9] and oxideaperture [10] VCSEL’s. Of these studies, only [6] reported the large-signal system simulation behavior of the extracted model, using a prebias current of 0.98 times the threshold current . It has also been shown that the rate equations may be recast in terms of parameters which may be extracted from measurement of the dc – curve, and the intensity modulation (AM) and optical frequency modulation (FM) responses [11], but this procedure failed to predict the laser turn-on delay for below-threshold prebias. Measurements of the threshold current and AM and FM modulation responses have been used to extract system simulation models of distributed feedback (DFB) quantum-well (QW) edge-emitting lasers [12]; however, low-prebias conditions were not considered. For below-threshold operation, models should predict the turn-on delay of the laser in response to input pulses because the operation of a digital link with a laser diode biased below threshold may be limited by pattern-dependent jitter [13]. Pattern-dependent jitter arises because the turnon delay in response to a step current pulse applied to a laser diode biased below threshold depends on the gain region carrier density at the time the current pulse arrives. For a digital data signal, the carrier density depends on the number of 0’s preceding an arriving 1, because the carrier density has been decaying during the 0’s with an approximate exponential dependence whose time constant is typically a few nanoseconds, or several bit periods for multigigabit/second links. Thus, for a particular prebias and pulse magnitude is also the peak-to-peak amplitude of pattern-dependent jitter for an uncoded on–off keyed digital system with equivalent prebias and modulation currents. The below-threshold dynamic behavior has been investigated to determine properties of the laser materials. Turn-on time measurements have been used to obtain the recombination rate and to distinguish between recombination mechanisms [14]. However, to our knowledge, measurements have not been used in a procedure to extract complete laser models sufﬁcient for system simulation. In this paper, we demonstrate an enhanced parameter extraction procedure for system simulation of multimode oxide VCSEL’s biased below threshold. The extracted models are optimized to reproduce the measured AM response at one bias point as well as the dc current–light–voltage ( – – ) characteristic, and the laser diode turn-on time in response to an isolated square pulse with a below-threshold prebias. To our knowledge, no previous study of rate-equation parameter extraction for system simulation has accurately predicted the 1077–260X/99$10.00 © 1999 IEEE 488 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 transient turn-on pulse shape for prebiases below . We demonstrate prediction of the transient pulse shape and eye pattern shapes for prebias below . TABLE I SUMMARY OF MODEL VARIATIONS II. LASER DIODE RATE-EQUATION MODEL Many characteristics of laser diode behavior can be described by a set of rate equations, which may be written as (1) (2) where is the carrier density, is the injection current, and is the normalized photon density , with being the photon number. This deﬁnition provides symmetry in the rate-equation stimulated emission terms. Further, is the current injection efﬁciency, is the electron charge, is the number of QW’s, is the volume of a single QW, is the carrier recombination rate, is the carrier conﬁnement factor, is the velocity of light in the cavity, is the gain function, including gain compression, is the coupled spontaneous emission, and is the photon lifetime. The output power is thus , where is the output coupling efﬁciency, and is the photon energy. We use the SPICE implementation of [3] which optionally includes logarithmic or linear gain dependence, and monomolecular, bimolecular, and Auger recombination. For system simulation purposes, we do not require that the extracted parameter values exactly reﬂect the true physical character of the materials or device, but only that the model reﬂect the observable behavior of the device. We use the physical origins of the model to provide insight into what are reasonable values for the parameters and what are the consequences of modifying the parameters. We have neglected chirp due to our interest in multimode ﬁber systems. Although measurement of the chirp (FM) and intensity (AM) modulation response can be combined to obtain the photon lifetime [11], this has been shown to be unnecessary for obtaining rate-equation parameters sufﬁcient for system simulation [12]. The recombination rate may be written . When and are set to zero, we obtain the linearized form often used when simple rate equations are desired. The coupled spontaneous emission may be written as or when the linearized recombination rate is used. The gain function may be written as (3) where is the gain suppression factor. The gain conﬁnement factor appears in the gain suppression term because the photon density is equal to where is the volume of the optical mode. The gain carrier dependence may be written as (4) The gain function may be linearized about the transparency carrier density, yielding , where Fig. 1. The simulated laser diode and parasitic network. Parasitic parameter values were optimized to reproduce S11 prior to extracting cavity parameters. is a linearization constant. This linearized form is equivalent to the form frequently used in the literature. We will consider three model variations which deﬁne and , as shown in Table I. We will use the term model to indicate either a set of equations such as (1) and (2) which describe laser diodes in general, or to indicate a speciﬁc set of parameters which describe a particular device. Typically, when microwave frequency measurements are made, signals are brought to the laser diode by a trans- mission line, and the source impedance, cavity impedance, and parasitic effects due to the chip and package geometry can signiﬁcantly affect the measurement result. The cavity impedance may be obtained by relating the carrier density to the voltage across the cavity by the relationship [15] , where is the equilibrium carrier density, is a diode ideality factor, and is the thermal voltage. The space charge capacitance may also be important in modeling the device behavior well below threshold. Parasitic elements have been accounted for by ﬁtting the reﬂection ( ) response of an assumed parasitic network to the measured behavior [6], [16], by measurement of a relative intensity noise (RIN) spectrum rather than [8], or by a simple response subtraction technique [17]. Parasitics have also been treated by including an additional pole frequency in modeling the forward ac response [7], [10]. A typical parasitic network model (Fig. 1) includes a se- ries inductor representing the wirebond, a shunting capacitor representing the contact capacitance, and a series resistor representing the contact resistance and the Bragg mirror stacks. The source conductance may also be included in the parasitic model so that an ac simulation obtains the transmission - parameter of the packaged laser diode as one half of the complex ratio of the output optical power to input current. The parasitic network is represented in ac analysis by its hybrid equations, which express the input current and cavity voltage as functions of the input voltage and cavity current. In transient analysis, a state variable and a differential equation are introduced for each independent inductor or capacitor in this network. BRUENSTEINER AND PAPEN: EXTRACTION OF VCSEL RATE-EQUATION PARAMETERS FOR SIMULATION 489 By linearizing the rate equations (1) and (2) and the parasitic network hybrid equations, and inverting the combined matrix equation, we obtain the well-known modulation transfer function (5) where is the transfer function of the parasitic network, and we have identiﬁed the resonant frequency Fig. 2. Experimental setup for turn-on delay measurement. The position of the optical rising edge was observed as the bias was adjusted from well above threshold to well below threshold. (6) and the damping coefﬁcient (7) Here we have deﬁned , and primed functions indicate differentiation with respect to the argument. Well above threshold, the ﬁrst term dominates in (6) and the ﬁrst two terms dominate in (7), thus and have approximate proportionality . Near threshold, the ﬁnal two terms of (7) dominate. A well-known technique obtains best-ﬁt and values for the ac response at a number of biases and uses the relationship between and implied by (6) and (7) to obtain rate-equation parameters. It has been shown, however, that for single-frequency laser diodes operated well above threshold it is sufﬁcient for system simulation purposes to obtain rate-equation parameters that produce the required as well as and at a single bias point [12]. Since this extraction procedure is straightforward and readily im- plemented, we investigate if a similar procedure can be used with a multimode oxide VCSEL operating near and below threshold. III. MEASUREMENT The device tested was a 5- m oxide-aperture VCSEL [18] operating at 850 nm, with mA. The active region consisted of ﬁve 80-A˚ GaAs QW’s in an Al Ga As seper- ate conﬁnement heterostructure (SCH) region extending over 1100 A˚ . The device emits in multiple lasing modes; in model- ing, we apply the supermode approximation, representing by the sum of photons in all modes that can achieve zero round-trip loss, and averaging and over the modes. We note also that the SCH regions extend only 150 A˚ beyond the outermost QW’s, minimizing the effect of SCH transport, and allowing us to choose a model which does not separately account for carriers in the SCH region. The laser was mounted on a microstrip test ﬁxture. The dc – characteristic was measured using a calibrated broad-area photodetector (Newport 818-SL) and a picoamme- ter (Keithley 485). The input voltage was measured simul- taneously using a multimeter. The current controller (ILX LDC-3722) had a measured accuracy of 0.01 mA for bias currents less than 2.5 mA. The turn-on delay was measured using a low-repetition-rate pulsed source (Avtech AVM-2C), with the output attenuated by a 10-dB attenuator. The signal was split using a 26.5- GHz resistive splitter, to permit observation of the input pulse magnitude on the oscilloscope. A prebias current was added through a 100-MHz–26.5-GHz bias tee. By choosing a pulse repetition rate of 10–100 kHz, we assured that the laser state when the pulse arrived was the equilibrium state due to the prebias current. The laser diode light output was collected by a 9- m butt-coupled ﬁber, and the optical signal was detected by an ampliﬁed 6-GHz (electrical 3 dB bandwidth) detector (New Focus 1531). The output was observed using a digitizing oscilloscope (Tektronix 11801) with a 20-GHz sampling head (Tektronix SD-24). This setup is shown in Fig. 2. As the prebias current was adjusted, the output signal was observed, and the turn-on time was measured by comparing the crossing time at the 20% level of the rising edge to the crossing time when the prebias was . The laser’s small-signal ac response was measured with an HP 8510B vector network analyzer. The high-speed detector described above was used. A standard ﬁve-term error correc- tion procedure (1-path 2-port procedure) was used to eliminate the effect of nonidealities in the network analyzer, which included a bias network, or the coaxial cables connecting the analyzer to the test ﬁxture and the detector. The response of the detector was measured using a calibrated 1500-nm signal at modulation frequencies to 10 GHz, and this response was deconvolved from the measurements of the test device. The dispersion of the approximately 2-m–9- m core ﬁber was negligible. Optical feedback effects attributed to reﬂections from the far end of this cable were observed in the data; however, they had little effect on the extracted parameters. The ac response was measured at a bias of 2.5 mA. The response was normalized at 1 GHz, which was below the res- onant frequency. The maximum frequency of the measurement was 8 GHz, at which frequency the response had reached the noise ﬂoor. Correction of the measurements to provide the current-to-power signal transfer function was not required because the simulation models included the effects of the parasitic network and source impedance. IV. PARAMETER EXTRACTION Parameter values for the parasitic network were obtained by optimization to reproduce the measured above-threshold characteristic, using a commercial RF and microwave 490 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 TABLE II PARASITIC PARAMETER EXTRACTED VALUES TABLE III CAVITY PARAMETER STARTING GUESSES AND EXTRACTED VALUES CAD tool (HP MDS). The extracted values are given in Table II. The parasitic network parameters were held ﬁxed as the cavity parameters were extracted using the CFSQP nonlinear minimization routine [4], [19]. An interface program written for this routine speciﬁed the parameter space over which minimization should be performed, rescaled the param- eters to improve the numerical properties of the problem, and calculated the error function to be minimized. The parameter space could be speciﬁed as an arbitrary subset of the rate- equation parameters, with arbitrary upper and lower bounds for each. Parameters could be scaled linearly or logarithmi- cally. The error function minimized by the routine was the weighted sum of squared errors in the modeled reproduction of the dc voltage and optical power output at the biases measured, the normalized response for the measured bias current, and the turn-on delay for the measured prebias currents and pulse magnitudes. We compared the CFSQP procedure with a simulated annealing routine (ASA [20]) and a Levenberg–Marquardt method (MINPACK). The simulated annealing routine required substantially more function eval- uations than CFSQP, and the Levenberg–Marquardt routine did not contain provisions for constrained problems and thus frequently returned physically unreasonable results. For extraction purposes, simulation was performed by func- tions built in to the extraction code. The dc characteristics of the model were calculated by the solution of (1) and (2) and the parasitic network hybrid equations with time derivatives set to zero, using a binary search method for robustness. The ac response was calculated by inversion of the network matrix at each frequency. The simulated ac response is normalized at 1 GHz, to reproduce the measured response. The turn-on delay was calculated using a root-ﬁnding differential equation solver (LSODAR in ODEPACK), applied to (1) and (2) and two additional equations representing the second-order parasitic network model. Although direct integration of the differential equations to obtain the turn-on time might be expected to be very inefﬁcient, the time required for the robust but inefﬁcient dc solution method was found to dominate the calculation time. A ﬁtting operation in 5–10 parameters and requiring about 20 dc solutions per error function evaluation consumed roughly 5–20 min on an HP 700 series workstation. The choice of starting guesses for the cavity parameters can signiﬁcantly affect the performance of the ﬁtting routine. Our approach was not to make the most accurate ﬁrst principles estimates possible, but to choose physically reasonable values at which variation of the value signiﬁcantly affected the simulation result—values at which the modeled behavior was sensitive to the values and, thus, at which the ﬁtting routine would have signiﬁcant information about the inﬂuence of each parameter. We used typical values from the literature [21], [22] for (240 000 m ), (2.6 10 cm ), and (2 ps). A typical value for the carrier lifetime (2.5 ns) was used, and (133 10 s ), (51.28 10 m s ), and (19.72 10 m s ) were chosen by allotting equal portions of the recombination current to each process at transparency. The modal velocity was estimated from the index of GaAs and the known effective index for modes of a similar ﬁeld diameter in step-index waveguides. The well volume (9.0 10 m ) was initially estimated from the area of the aperture and the length of the QW; however, more rapid convergence was achieved by increasing the value by a factor of approximately 1.5. The lateral conﬁnement factor was taken from the ratio of the well length to cavity length, and the lateral conﬁnement factor from a typical value from the literature (0.8) [22], giving an initial estimate for of 0.05. The gain suppression factor was estimated to be 10 m . Typically, these starting values were used to optimize a restricted set of parameters to reproduce the – – charac- teristics only. This result was used as the starting point for a further optimization of an expanded set of parameters and including the ac response measurement. Finally, the turn-on delay measurement was included, and all of the parameters were optimized. The extracted parameters for the base model (Model 1) were used to produce two simpliﬁed models. In one (Model 2), the gain function was linearized about the transparency car- rier density, and, in the other (Model 3), the recombination rate was linearized about the threshold carrier density. The relationships distinguishing these model forms are summarized in Table I. For Models 2 and 3, the parameter extraction procedure was repeated, using the parameters extracted for Model 1, or their linearized approximations, for the starting guesses for the new models. The extracted cavity parameter values for the three models are presented in Table III. The – and – curves resulting from the three models are compared to the measurement in Fig. 3. Here, and in the ac and responses, the characteristics of the three models were very similar. The above-threshold – and – characteristics agree well because, if the threshold current is reproduced accurately, the parameters , , and may be adjusted to bring agreement in the dc characteristics without BRUENSTEINER AND PAPEN: EXTRACTION OF VCSEL RATE-EQUATION PARAMETERS FOR SIMULATION 491 (a) Fig. 4. Measured and simulated S11 over the frequency range 49 MHz and 10 GHz. (b) Fig. 3. Measured and simulated (a) I–L and (b) I–V responses. Results for the three models are indistinguishable. affecting the modeled ac and turn-on delay characteristics. The below threshold – measurement was not ﬁt well due to the signiﬁcant contribution of side modes that never lase and are, therefore, not part of the supermode. In single-mode lasers, the lasing mode’s below threshold – relationship can be measured using a monochromator [11], [23]. For the modeled multimode device, this procedure would re- quire a measurement which accepts exactly those modes contributing to the supermode and excludes noncontributing sidemodes. The main features of the below-threshold – characteristic are determined by and the recombination parameters , , and . However, the measurements we use obtain information about these parameters without resorting to a spectrally resolved measurement via their effect on the AM response near threshold, where may contribute signiﬁcantly to and, in the turn-on delay measurement, where , , and contribute directly, and contributes via stimulated emission. Because nonlasing sidemodes are neglected, our models necessarily overestimate the extinction ratio for lasers biased below threshold; however, this does not signiﬁcantly affect the modeled signal-to-noise ratio (SNR) when the extinction ratio is large. In Fig. 4, the modeled of the packaged device is compared to the measurement. The modeled response agrees well with the measurement. Deviations occur at fre- quencies above 7 GHz, where high-order parasitic effects become important. Good reproduction of the response is critical because, when designing interfacing circuitry to the VCSEL, the packaged device’s input impedance may be as im- Fig. 5. Measured and simulated jS21j for the three laser diode models. The differences in the models are described in Table I. Results for the three models are indistinguishable. portant in limiting system performance as the device’s modulation transfer characteristic. Signiﬁcant variability was observed between the measured of different devices. This variability was similar in magnitude to the variability over the measured frequency range for a single device and is attributed to variation in contacting and device placement on the test ﬁxture. The reproduced ac response is accurate to within 3 dB for frequencies less than 7 GHz, as shown in Fig. 5. The resonant peak height and full-width at half-maximum (FWHM) response are reproduced well; however, the resonant frequency is displaced by approximately 500 MHz. From the slope of the modeled response tail, we infer that some parasitic effects are neglected in the model. The optimization procedure, in order to reproduce the response optimally over all frequencies, has obtained parameters which shift the characteristic frequencies of all poles and zeros in the ac response. The modeled and measured turn-on delays are presented in Fig. 6. The three models were able to reproduce the turn-on 492 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 Fig. 6. Measured and simulated turn-on time in response to 14- and 20-mA pulses, for the three laser diode models. Only the response for the 14-mA pulse was used in optimization. Results for the three models are indistinguishable. Fig. 8. Transient responses of all three models to the same input as in Fig. 7. Fig. 7. Modeled transient behavior using extracted parameters for Model 1 and measured data. The signiﬁcant turn-on delay is predicted by the model. time within 20 ps for prebias currents as low as 0.2 mA for the 14-mA input pulse magnitude. After optimizing the models as described above, the modeled turn-on delay for a pulse magnitude of 20 mA was compared to the measured response, and agreement was found to within 31 ps for mA, that is, when ps. Fig. 6 also indicates the ability of the models to reproduce the pattern-dependent jitter amplitude for particular prebias and modulation currents. The good agreement in the turn-on delay measurements indicate that, despite the exclusion of many effects, which precludes reproducing the ac response with high accuracy, these models are appropriate for system simulation. To test this conjecture, MDS was used to simulate a turn-on transient using Model 1, and the results are presented in Fig. 7. The input pulse consisted of a 0.5-mA constant prebias component and a pair of 5.5-mA square pulses with duration 0.5 ns, rise and fall times of approximately 60 ps, and separated by 0.75 ns. This transient was chosen to reproduce the conditions that lead to pattern-dependent jitter, with the initial pulse beginning when the VCSEL carrier density is in an equilibrium state, while the second pulse begins when the carrier density has had only 0.75 ns to decay from the threshold density toward the equilibrium value. In system design, the overshoot magnitude must be known to prevent saturation of the detector preampliﬁer. For the test signal, the overshoot magnitude was predicted with approximately 33% error. We found that, for some choices of subsets of the measurement data input and weighting coefﬁcients for the optimization, the extraction routine obtained models which signiﬁcantly underestimated the height of the resonant peak in the ac response, and, in this case, the transient overshoot magnitude was also signiﬁcantly underestimated. The modeled transient response shows very good agreement in the pulsewidth, indicating accurate reproduction of turn-on delay. The simulation was repeated with the same input using Models 2 and 3, and the results for all three models are shown in Fig. 8. These results also agree well with the measured response, indicating that simpliﬁed gain or recombination models can be sufﬁcient even for simulation of below-threshold dynamic behavior. In order to demonstrate the utility of the model for system design, we compare the eye patterns generated by simulation and measurement of a simple link. The link is driven by a 1 Gb/s pseudorandom bit stream (PRBS). The signal is coupled to the laser through a bias-tee. A 250-mV 50- signal with a 5.0-mA bias current and a 500-mV signal with a 3.0- mA bias are considered. These signals correspond to prebias currents of approximately 3.0 and 0.2 mA and signal currents of approximately 3.9 and 6.7 mA, respectively. The optical signal is coupled through 0.5-m 9- m core ﬁber to the 6-GHz photodetector. The receiver output is coupled through a 1.2- GHz low-pass ﬁlter (LPF) such as might be used in a real link to improve the SNR. The 128-bit waveform is captured, and the bit intervals are overlayed on each other to generate the eye patterns of Fig. 9(a). The link is simulated using the extracted Model 1 with no further optimization to improve the reproduction of the eye pattern. The measured PRBS generator output is used as input to the simulation. The photodetector response is approximately ﬂat for frequencies where the LPF response is signiﬁcant. The LPF is approximated by a 1.2-GHz ﬁfth-order Butterworth ﬁlter which provides a good estimate of the frequency response magnitude to 10 GHz. The resulting 128-ns output waveform is broken into overlapping 2-ns segments and overlayed to produce the eye patterns of Fig. 9(b). BRUENSTEINER AND PAPEN: EXTRACTION OF VCSEL RATE-EQUATION PARAMETERS FOR SIMULATION 493 (a) (b) Fig. 9. Eye patterns generated by (a) measurement and (b) simulation for two signals. A 50- signal is coupled through a bias-tee to the laser. The received signal is ac coupled and low-pass ﬁltered. The model predicts important trends in the eye pattern shape such as overshoot, ring, and the point of crossing of the rising and falling edges. Deviation between the jitter apparent in the measured and simulated eyes is attributed to the difference in the signal magnitudes of 3.9 and 6.7 mA from that used to extract the model parameters, 14 mA. The enhanced signal variance in the measurement, which also accounts for a portion of the jitter discrepancy, is attributed to effects excluded from the model, such as optical feedback and junction heating, which may be signiﬁcant over the 10–100-ns time scale of the 1-Gb/s PRBS. V. CONCLUSION This paper has demonstrated parameter extraction for rateequation simulation of multimode VCSEL’s by a SPICE-like simulator. Operation near and below threshold was emphasized by the choice of input data to the extraction routine. Three laser diode models, distinguished by the forms of the recombination rate and gain functions, were considered. Rateequation parameters were extracted for each model, using a method which simultaneously considers the dc – and – characteristics, the ac modulation response at multiple biases, and the turn-on delay for below-threshold prebias. The extracted models were seen to reproduce with good accuracy the above-threshold – – response, the threshold current, and the ac response at one bias. All three models were able to reproduce the measured characteristics with similar accuracy. All three models were able to predict the transient response to a pair of isolated pulses with a prebias of , and thus we see that the simplest rate-equation models, not including nonlinear recombination or gain functions, are adequate for system simulations of this kind. Finally, Model 1 was used to generate the eye pattern for this laser used in a simple link, demonstrating the ability to predict trends in the eye pattern shape for signals outside the range of impulses used to characterize the below-threshold dy- namics. Work on parameter extraction procedures for models incorporating thermal effects, which reduce the ability of the present model to predict the eye pattern, is ongoing. This paper has demonstrated the ability to produce models which reproduce the large-signal turn-on delay for biases below threshold, and to predict large-signal transient pulse shapes for a prebias of . Trends in the eye pattern shape are predicted for signal magnitudes differing from those used in parameter extraction and prebiases above and below threshold. This represents a substantial improvement over previously reported parameter extraction procedures for short- haul low-power system simulation. We expect the extracted models will be valuable for designing low-prebias systems. 494 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999 ACKNOWLEDGMENT The authors would like to thank P. Mena for valuable discussions and for providing the SPICE and C-code models described in [4] and K. Choquette of Sandia National Laboratories for providing VCSEL’s. REFERENCES [1] R. S. 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