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2612 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 A Comprehensive Circuit-Level Model of Vertical-Cavity Surface-Emitting Lasers P. V. Mena, Member, IEEE, Member, OSA, J. J. Morikuni, Member, IEEE, S.-M. Kang, Fellow, IEEE, A. V. Harton, Member, IEEE, and K. W. Wyatt Abstract— The increasing interest in vertical-cavity surfaceemitting lasers (VCSEL’s) requires the corresponding development of circuit-level VCSEL models for use in the design and simulation of optoelectronic applications. Unfortunately, existing models lack either the computational efﬁciency or the comprehensiveness warranted by circuit-level simulation. Thus, in this paper we present a comprehensive circuit-level model that accounts for the thermal and spatial dependence of a VCSEL’s behavior. The model is based on multimode rate equations and empirical expressions for the thermal dependence of the activelayer gain and carrier leakage, thereby facilitating the simulation of VCSEL’s in the context of an optoelectronic system. To conﬁrm that our model is valid, we present sample simulations that demonstrate its ability to replicate typical dc, small-signal, and transient operation, including temperature-dependent lightcurrent (LI) curves and modulation responses, multimode behavior, and diffusive turn-off transients. Furthermore, we verify our model against experimental data from four devices reported in the literature. As the results will show, we obtained excellent agreement between simulation and experiment. Index Terms—Circuit-level models, multimode rate equations, spatial hole burning, thermal modeling, vertical-cavity surfaceemitting lasers (VCSEL’s). I. INTRODUCTION RECENT years have witnessed the increased popularity of vertical-cavity surface-emitting lasers (VCSEL’s), which offer a variety of advantages compared to edge-emitting semiconductor lasers. In a typical VCSEL, an optical cavity is formed along the device’s growth direction, with distributed Bragg reﬂectors (DBR’s) typically forming the cavity mirrors. The many advantages of VCSEL’s can be related to this simple design. First, because the cavity length is typically very short, the correspondingly large mode spacing limits the optical output to a single longitudinal mode [1]. Second, a VCSEL’s planarity allows symmetric transverse cross sections, thereby resulting in circular output beams [1]–[2]. This feature, a signiﬁcant improvement over the elliptical beams exhibited by edge-emitters [2], is particularly attractive since it improves Manuscript received June 2, 1999. P. V. Mena was with the Department of Electrical and Computer Engineering, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. He is now with Motorola, Inc., Schaumburg, IL 60196 USA. S.-M. Kang is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. J. J. Morikuni Motorola, Inc., Schaumburg, IL 60196 USA. He is now with Tellabs Operations, Inc., Bolingbrook, IL 60440 USA. A. V. Harton and K. W. Wyatt are with the Motorola, Inc., Schaumburg, IL 60196 USA. Publisher Item Identiﬁer S 0733-8724(99)09662-0. coupling of the optical output to a ﬁber [3]. Planarity also results in other important advantages, including support for on-wafer probe testing, two-dimensional (2-D) integration of VCSEL arrays [1], and the ability to limit device area to a particular spot size [3]. Finally, because of their small volume, VCSEL’s should ultimately have relatively high modulation bandwidths [4]. As a consequence of their advantages, VCSEL’s have been studied as elements of a variety of systems, including multichannel optical links [5], smart pixel systems [6]–[7], optoelectronic switches [8]–[9], WDM applications [10], optical storage [11], and laser printing [12]. The effective design of optoelectronic systems incorporating VCSEL’s requires the availability of a VCSEL model that satisﬁes the various requirements of circuit and system design and simulation. First, the model must be able to accurately replicate the operating characteristics of actual devices [13]. The model can then be reliably used to simulate a VCSEL’s interaction with other elements in a particular design, such as the transistors in a laser driver. Second, the model should be compact and numerically efﬁcient [13], and not computationally intensive like a multidimensional device-level model. A typical system will incorporate a large number of photonic and electronic components, such as one-dimensional (1-D) and 2D VCSEL arrays [5]–[6]. Furthermore, system design often requires a large number of simulations for design optimization and veriﬁcation. For example, the design of drive circuitry for a VCSEL may require many iterations to determine optimal transistor topology and sizing. In these cases, numerically efﬁcient models are essential. This situation is analogous to that from IC design, where the use of computationally intensive models would make the efﬁcient development of million- or even hundred-transistor designs a practical impossibility. Initially, it would seem that existing circuit-level models of edge-emitting semiconductor lasers, such as those described in [14]–[17], could be used to describe VCSEL’s. Generally, however, these models neglect physical effects which are intrinsic to a VCSEL’s behavior. First, due to their poor heat dissipation and the large resistance introduced by their DBR’s [18], typical VCSEL’s undergo relatively severe heating, and consequently can exhibit strong thermally dependent behavior, including thermal lensing, temperature-dependent threshold current, and output power rollover [19]. Second, spatial effects related to the transverse variation of the active-layer carrier and mode proﬁles play an important role in a VCSEL’s behavior. For example, multitransverse mode operation is possible [20]. Furthermore, carrier diffusion and spatial hole 0733–8724/99$10.00 © 1999 IEEE MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2613 burning can limit a VCSEL’s performance by contributing to mode competition [21] and secondary pulsations in the turn-off transient [22]. Obviously, a circuit-level VCSEL model must account for these mechanisms. Unfortunately, while the literature has presented a variety of VCSEL models, to the best of our knowledge none have adequately addressed all of the above issues simultaneously. Many approaches have relied on detailed multidimensional analysis to provide a description of the interplay between optical, electrical, and thermal behavior in VCSEL’s [23]–[27]. For example, Scott et al. [23] modeled the thermal gain, leakage, and spatial-dependence of VCSEL’s via quantum-well gain calculations and ﬁnite-element analysis of the VCSEL active layer. As we noted previously, however, while these models are comprehensive, their computationally intensive nature makes them unsuitable for circuit- and system-level design and simulation. Simpler approaches, however, have resulted in models which generally not only provide an incomplete picture of a VCSEL’s behavior, but also are rarely implemented in the context of circuit-level simulation. Yu et al. [28], for example, presented a rate-equation-based model which accounts for spatial hole burning, as well as the variation of gain with temperature. However, the model is limited to a single mode and neglects thermally dependent carrier leakage out of the active layer. Morozov et al. [29] and Dellunde et al. [30], meanwhile, both implemented rate-equation-based VCSEL models which account for multimode behavior, but neglect thermal effects. While the simpler models of [28]–[30] lend themselves to implementation in standard computeraided design environments, the authors did not choose to do so, further limiting the usefulness of their models in optoelectronic system design and simulation. On the other hand, circuit-level models which do exist are incomplete. In [31]–[33], circuit models are presented which only describe a VCSEL’s electrical characteristics, while Su et al. describe a model in [34] which is limited to the static simulation of a VCSEL’s thermal behavior. Clearly, then, despite these prior efforts, there continues to exist a need for a comprehensive model which can be used in the design and simulation of optoelectronic systems. In this paper, we present a comprehensive circuit-level VCSEL model that addresses the above concerns. First, the model accounts for the physical effects which are critical to a VCSEL’s operation. These include the thermal dependence of the active layer’s gain [35], thermal leakage of carriers out of the active layer at elevated temperatures and carrier densities [23], and spatial characteristics such as the transverse proﬁle of the optical modes [20], the resulting spatial hole burning of the transverse carrier proﬁle, and lateral diffusion of carriers in the active region [23]. Second, the model is based on rate equations that permit a compact and numerically efﬁcient implementation which lends itself readily to circuitlevel simulation. As a result, we have implemented the model in Analogy’s Saber [36], an industry-standard circuit- and system-level simulator. Our discussion begins in Section II, where we describe in detail the various components of the model as well as its ﬁnal Saber implementation. We then review in Section III various options for modeling the transverse optical mode proﬁles. Next, in Section IV we demonstrate the utility of the model for simulating single- and multimode VCSEL’s, and in Section V we validate the model against experimental data from four devices presented in the literature. Finally, we present conclusions in Section VI. II. MODEL DEVELOPMENT Our comprehensive circuit-level VCSEL model provides an accurate description of a VCSEL’s terminal characteristics, namely, the generation of an optical output in response to an input current, via a computationally efﬁcient description of the internal device physics. In particular, the model accounts for thermal and spatial effects without resorting to the numerically intensive calculations of a multidimensional device model. In the following subsections, we present a detailed description of the model’s main features, which we brieﬂy summarize here for clarity. First, we discuss the foundation of the model, a set of spatially dependent multimode rate equations which are used to account for the spatial dependence of a VCSEL’s behavior. The ﬁrst equation models the lateral distribution of carriers within the active layer, while the remaining equations each account for the photons in a particular transverse mode. Second, we present simple empirical expressions for the thermally dependent active-layer gain and carrier leakage. These empirical expressions provide a simple alternative to more detailed quantum-mechanical calculations. Next, we present a methodology by which the spatially dependent equations are converted into more numerically efﬁcient spatially independent ones that implicitly account for a VCSEL’s spatial dependence. Obviously, full spatial calculations would make our model unsuitable for circuit-level simulation. However, the spatially independent rate equations do not suffer from this deﬁciency. We then present a thermal rate equation that models the device temperature as a function of dissipated heat, as well as expressions that model a VCSEL’s output power and electrical characteristics. Finally, we discuss the model’s implementation in Saber. A. Spatially Dependent Operation While thermal behavior is certainly a signiﬁcant component of VCSEL operation, spatial effects can play an important role as well. Consequently, our comprehensive model must be capable of accounting for them. First, as many researchers have observed, while their short cavity length allows VCSEL’s to have a single longitudinal mode, multitransverse-mode operation can still occur [37]. This multimode behavior has been observed in both index- and gain-guided devices [20], [37]–[41]. The speciﬁc forms of the various transverse mode proﬁles are an important component of a VCSEL’s spatially dependent behavior. Because the optical modes are not uniform in the transverse direction, they burn holes in the transverse carrier distribution where their intensity is largest. This spatial hole burning (SHB) allows different modes to compete with one another, as described in [38]. If the VCSEL begins lasing in its fundamental transverse mode, eventually it will burn a hole in the center of the carrier proﬁle. The 2614 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 corresponding increase in carrier number outside of this hole allows additional modes, whose proﬁles overlap these carriers, to begin lasing. This interplay between the modes and carriers also can play a role in self-focusing and thermal-lensing effects, which alter the mode proﬁles and their impact on device performance. For example, the spatial hole burning can cause a transverse variation in the active-region index proﬁle, resulting in self-focusing that can accelerate the onset of multimode effects in weakly index-guided devices [41]. From the above discussion, it is clear that in addition to the transverse mode proﬁles, we must also properly take into account the transverse spatial dependence of the active region’s carrier distribution. This nonuniform carrier proﬁle, induced not only by SHB, but also by nonuniform current injection and poor carrier conﬁnement [42], can result in lateral carrier diffusion. This diffusion can be an important mechanism in VCSEL’s. First of all, as has been well documented in edge-emitters, diffusion can act as a damping mechanism during high-speed modulation [43]–[45]. Second, diffusion can signiﬁcantly alter a laser’s transient operation. Ikegami [46] reported the presence of a tail in a DH injection laser’s turnoff transient, while Chinone et al. [47] presented results on the impact of lateral effects on the turn-on behavior of semiconductor lasers. Similar phenomena have been investigated in VCSEL’s. For example, the study of mode competition during transient operation [21] has revealed that diffusion can be an important contributor to the transient evolution of each mode’s output. Finally, carrier diffusion can lead to the onset of secondary pulsations and optical bumps during the turnoff transient. This phenomenon has already been observed in VCSEL’s [48] as well as edge-emitters [49]. As discussed in [22], if we assume fundamental mode operation, then as the current through the VCSEL is increased, a spatial hole is burned into the carrier proﬁle. When the VCSEL is eventually turned off, carriers begin to diffuse back into the hole. This serves to delay the turn-off, and in some cases results in an increase of the output power [50] in the form of secondary pulsations or optical bumps. Typical attempts to model all of the above behavior have involved detailed multidimensional analysis, ranging from the beam propagation method (BPM) for determining the transverse mode structure [51] to ﬁnite-element and ﬁnitedifference analysis for modeling the transverse distribution of the active layer carriers [23]–[24], [52]. However, because such approaches are unsuitable for circuit-level simulation, we have adopted an alternative that avoids explicit spatial calculations. The technique involves the use of assumed solutions for the carrier and mode proﬁles within spatially dependent rate equations. As we shall show later, by substituting into these rate equations an orthogonal series expansion for the carrier proﬁle and an assumed functional form for the mode proﬁle, we can generate a set of spatially independent rate equations that need only be solved in the time domain. This approach has been used successfully by various authors for VCSEL’s [28], [30], [53] as well as edge emitters [43], [45], [54]. From a physical standpoint, spatially dependent rate equations provide an excellent foundation for our VCSEL model. The use of rate equations is a well-accepted technique for describing the general operating characteristics of lasers. Generally, these equations describe the rates of change of the carrier and photon populations in the active layer. Thus, by making these quantities spatially dependent, as well as introducing a diffusion term into the carrier rate equation, we can account for the complex spatial interaction between the carrier and mode proﬁles in a VCSEL, such as SHB. Furthermore, we can introduce a different rate equation for each mode in the device, thereby allowing us to model multimode behavior. Finally, by making the gain thermally dependent and introducing a current term to describe the thermal carrier leakage, we can account for a VCSEL’s thermal behavior. The spatially dependent rate equations that form the foundation of our model are based largely on those presented by Moriki et al. in [53]. The ﬁrst equation, which describes the carrier number in the active region, is (1) where is the spatially dependent injection current; and are the total photon number and normalized transverse mode proﬁle in the th transverse mode, respectively; is the device temperature; is the gain of the th mode; is the thermal leakage current; is the current-injection efﬁciency; is the carrier lifetime; is the effective carrier diffusion length; and is the electron charge. In the above equation, the carrier number is the carrier density scaled by the effective active-layer volume which we shall discuss in a later subsection. Equation (1) is similar to (7) from [53], with the most noticeable differences being the addition of thermal gain and leakage. As we can see, it accounts for multiple transverse modes by including a stimulated emission term for each one. Furthermore, the general spatial vector is used to account for arbitrary coordinate systems; as we shall show later, in cylindrical coordinates it reduces to the radius Diffusion is included via the Laplacian of the carrier density. Finally, the leakage current is included as an offset to the injection current, thereby accounting for the reduction in injection efﬁciency as the leakage increases. The remaining rate equations describe the total photon number in each mode The equation for mode is (2) where and are the spontaneous-emission coupling coefﬁcient and photon lifetime, respectively, for the th mode, and is the total volume of the active region. As we can see, the time rate of change of the total photon number depends on the total photon loss, as well as coupled spontaneous and stimulated emission. The integrations over the active volume account for all of the contributing emission events occurring within the active region. Note that we neglect the MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2615 dependence of the spontaneous emission on the mode proﬁles [54], assuming instead that it is accounted for in In addition to supplying a means for describing a VCSEL’s spatially dependent behavior, (1) and (2) provide a framework in which the thermally dependent gain and active-layer carrier leakage can be modeled. In the above equations, the gain is accounted for via while the leakage is modeled via the current By using simple empirical temperaturedependent expressions for these two terms, we can avoid more computationally expensive approaches based on detailed physical descriptions. The next two subsections describe such expressions. B. Thermally Dependent Gain Because of its common usage in other rate-equation-based laser models, we have elected to model the gain as a linear function of carrier density [17], [45]. Consequently, the full gain term is based on the expression where is the gain constant (proportional to the differential gain), and is the carrier transparency number. While this expression indicates the gain’s dependence on the active layer carriers, it does not yet account for the gain’s thermal dependence, which, as we noted earlier, is a critical component of a VCSEL’s behavior. This thermal dependence plays a particularly important role in the threshold current’s depen- dence on temperature [19], as well as the thermal variation that one might expect in the small-signal relaxation oscillation frequency In the latter case, analogous behavior has been experimentally observed in the 3-dB bandwidth of edge emitters [55], and should be evident in VCSEL’s as well. The effect is due to the fact that is a linear function of the differential gain [56]. The thermal dependence of the gain can also contribute to thermal rollover of a VCSEL’s light-current (LI) curves [19]. The gain’s temperature dependence can be explained as follows. As its temperature increases, a VCSEL’s gain spec- trum broadens and its peak location shifts to longer wave- lengths. The device’s emission wavelength also increases with temperature, though considerably less than the gain peak [35]. Consequently, depending on the initial location of the gain peak relative to the wavelength, the laser gain will either decrease or increase with temperature as the gain peak and wavelength become more or less mismatched [35]. This nonmonotonic thermal dependence of the gain contributes to the thermal variation of the threshold current. In fact, an optimal value of temperature typically exists in which a minimum threshold is achieved [19], [35]. By making and functions of temperature, the gain’s temperature dependence can be included in a simple manner without resorting to complex calculations. This approach has been used extensively throughout the literature for VCSEL’s as well as edge emitters. In the case of edge emitters, the gain and transparency number have been modeled as linear functions of temperature [57], exponential functions of temperature such as where is a characteristic temperature [58], and exponential functions of temperature in conjunction with a description of the gain spectrum in which the peak location varies linearly with temperature [59]. In VCSEL’s, similar thermal expressions have also been used. Hasnain et al. [19] modeled a gain-guided VCSEL’s differential gain as a combination of a temperature-dependent gain-spectrum and an inverse function of temperature, and the transparency density as a linear function of temperature. Other researchers have modeled a VCSEL’s gain using a logarithmic expression, with the gain constant and transparency number described as polynomial functions of temperature [23], [60]. While there clearly exists ample precedent for modeling a VCSEL’s gain constant and transparency number as simple analytical functions of temperature, some of the approaches listed above assume a monotonic dependence of gain on temperature, which we know is not the case in VCSEL’s. Furthermore, it is not clear which expressions will work well across a large cross section of device designs. In order to gain some insight into what kind of expressions are needed, it is instructive to examine two prior approaches related to VCSEL’s. First, as we mentioned before, Hasnain et al. [19] modeled a gain-guided VCSEL’s transparency density as a linear function of temperature, and the differential gain as the product of an inverse function of temperature and a temperature-dependent gain spectrum: (3) where is a ﬁtting constant, is the temperature- dependent peak-gain wavelength, and is the gain spectrum’s temperature-dependent full width at half maximum (FWHM). Note that (3) does not contain the leading term of that was spuriously included by the authors, making the spectral term of (3) equivalent to that presented in [61]–[62]. If we assume a square-root dependence on temperature for the FWHM, and a linear dependence on temperature for both and [19], then we can reduce (3) into the simple form where are constants. Second, in [23], Scott et al. model a VCSEL’s gain using detailed quantum-mechanical gain calculations. They then ﬁt plots of the gain versus carrier density to a logarithmic function of the form where is a constant and and are polynomial functions of temperature. The authors use two separate polynomials to model at temperatures both above and below 430 K. By linearizing their gain expression about the transparency density, we can obtain an equivalent linear version of the gain, where the transparency density and the differential gain These results are similar to those of the Hasnain model discussed above. First, the transparency density is again a polynomial function of temperature, in this case a quadratic. Furthermore, the differential gain is described by the ratio of polynomial functions of temperature. Motivated by the above observations, we elected to model the carrier transparency number using a quadratic function of temperature, and the gain constant using the ratio of two quadratics, Essentially, the approach is a more general implementation of the equations utilized by Hasnain et al., which by default 2616 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 Fig. 1. Comparison of differential gain as determined by the simple ratio of two quadratic functions of temperature (dotted lines) to more detailed calculations (solid lines) from (a) the model of Scott et al. [23], and (b) our own simple quantum-mechanical gain calculations [63]. can be generated by setting and the quadratic terms in the transparency number expression equal to zero. Similarly, our quadratic expression for the transparency number auto- matically covers the expression used by Scott et al. However, to conﬁrm that our model could be used to describe their differential gain constant as well, we used our new expression to ﬁt their equation As Fig. 1 demonstrates, our equation does an excellent job of ﬁtting the data generated by Scott’s model. To further demonstrate the utility of our approach, we also ﬁt our gain expression to the results of our own simple quantum-mechanical gain calculations [63]. For simplicity, we assumed a single set of transitions between the conduction and valence bands, we neglected the effects of strain, and we calculated the electron and heavy-hole carrier densities using well accepted analytical expressions [64]. Furthermore, we based our calculations on a device with a 10-nm In0.2Ga0.8As QW, GaAs barrier layers, a 990-nm emission wavelength at room temperature, and a 0.084-nm/K linear variation of this wavelength with temperature [19], [23]. The differential gain was then determined by linearizing the gain at the transparency density. As we see in Fig. 1, again our simple ratio of two quadratics provides an excellent match to the more detailed quantum-mechanical calculations. As the above results demonstrate, simple expressions based on polynomial functions of temperature can be used to model the thermal dependence of both the gain constant and trans- parency number in our model. Thus, the complete set of expressions that describe the gain are (4) (5) (6) where is a temperature-independent gain constant; is a temperature-independent transparency number; and are ﬁtting constants; and is the gain saturation factor of mode due to mode Based on the above discussion, (5) should be able to model the nonmonotonic thermal dependence of the gain constant. Also, as noted above, (6) provides a general formulation in which the carrier transparency number’s thermal dependence can be modeled. However, for some devices, such as the gain-guided VCSEL of [19], a simple linear relationship is sufﬁcient, in which case and can be set equal to zero. Two additional features of (4) are worth noting. First, because of the small spacing between a VCSEL’s transverse modes [20], [65], we use the same material gain for each mode. Second, we have included a gain saturation term in the denominator, as suggested by Channin [66]. In this case, the saturation term assumes contributions from all of the transverse modes. A similar approach has been used to model the interaction of longitudinal modes in semiconductor lasers [56]. While in principle the gain saturation should also be spatially dependent [41], [67], for simplicity we choose to neglect this dependence [22]. C. Active-Layer Carrier Leakage While the gain is the most well-recognized thermally depen- dent mechanism in VCSEL’s, thermal leakage of carriers out of the active region can also have a severe impact on device performance [23]. As the device temperature increases, the bandgap of a VCSEL’s active layer shrinks. Furthermore, the active-layer carrier number increases due to a relative increase of the quasi-Fermi levels. Eventually, the large number of car- riers and the high temperature no longer allow the active layer to adequately conﬁne carriers, and leakage current becomes a dominant inﬂuence on the VCSEL’s operation [23]. The increase of the carrier number due to spatial hole burning can further accelerate the increase in the leakage [23]. Because the leakage acts to reduce the laser’s overall efﬁciency [23], we have included it in our model via the subtracted current in (1). Obviously, this leakage current should be modeled as a function of both carrier number and temperature [23], preferably via a simple analytical expression. An obvious choice would be the well-known formulation of thermionic emission [68]. In this case, the leakage current density is proportional to where is a constant that characterizes the emission’s exponential temperature dependence. A similar expression can be derived for heterojunction leakage if we assume that it is proportional to the carrier density immediately outside of the active region [69]. The resulting expression does not include a square-root temperature dependence, however. Alternatively, Scott et al. assumed that the carrier leakage could be modeled using an approximate homojunction-diode relationship proportional to where is the bandgap of the conﬁnement layers surrounding the active region, and is the active region’s quasi-Fermi-level separation [23]. Carrier number can be introduced into this expression if we crudely approximate it using expressions for bulk material [70]. In this case, the leakage is proportional to While these expressions appear promising, in reality they over-predict the carrier-number dependence at lower temper- MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2617 2 0 0 Fig. 2. Leakage current as calculated by (7) at temperatures from 250 to 400 2 and K using Ilo a3 = 6:13 = 10 A, 109: a0 = 3574; a1 = 2:25 10 5; a2 = 10 7; atures. This deﬁciency is a direct result of the carrier and temperature dependencies being independent of one another, whereas in reality, one would expect the carrier dependence to become more sensitive at higher temperatures. Thus, instead of relying on the above expressions, we chose an alternate approach based on the work of Scott et al. in [23]. As mentioned above, they modeled the leakage as a function of the quasi-Fermi-level separation Furthermore, to simplify matters, they performed detailed calculations of as a function of carrier density and then used a curve-ﬁt to model the carrier and temperature dependence of analytically using where is the carrier density and are constants. Examination of their results revealed that could be neglected with little effect on the end result. If we use their expression in terms of carrier number and substitute it into the homojunction diode equation, we obtain the following formula for the thermal leakage current as a function of carrier number and temperature: (7) Fig. 2 illustrates plots of (7) at temperatures of 250–400 K using values of based on data in [23]. As we can see, at 250 K the leakage is negligible for low values of carrier number, and increases dramatically after some threshold value. As the temperature increases, the leakage becomes much more sensitive to carrier number. It should be noted that for extremely high values of carrier number, the leakage of (7) actually decreases with temperature. However, as we shall see later in the parameter extraction from experimental device data, this regime of operation is typically not encountered in practice. Consequently, we elected to use (7) in our compre- hensive VCSEL model. D. Spatially Independent Rate Equations Obviously, because they require full spatial calculations, the spatially dependent rate equations (1)–(2) are unsuitable for a circuit-level model. However, as we noted earlier, we can remove their explicit spatial dependence, making them more appropriate for circuit-level simulation. The technique involves the use of assumed solutions for the carrier and mode proﬁles. Furuya et al. [43] presented one of the original implementations of this approach and applied it to edge emitters. The ﬁrst step in this technique is to describe the laser operation via spatially dependent rate equations such as (1)–(2). A functional form of the mode proﬁles is then assumed in advance, with the actual photon number of each mode left as the only unknowns. An orthogonal-series expansion of the carrier proﬁle with time-dependent expansion coefﬁcients is then substituted into the rate equations. Based on the orthogonality condition, a series of integrations can be performed on the spatial rate equations to produce a set of spatially independent differential equations for the photon numbers and expansion coefﬁcients, thereby providing an implicit description of a device’s spatial dependence. In the case of [43], sinusoidal mode proﬁles and 1-D Fourier-series expansions were used. Similar equations have been used by other authors [45], [54]. Not surprisingly, because of the power of this approach, it has also been used in VCSEL’s. Moriki et al. [53] used a Bessel-series expansion in cylindrical coordinates to determine the single transverse mode condition of buried-heterostructure VCSEL’s, while Dellunde et al. [30] used the same approach to model the statistics of a VCSEL’s turn-on transient. Similarly, Yu et al. [28] used a two-term Bessel-series expansion to model the impact of diffusion and SHB on a single-mode laser. Due to the success of this approach, we have chosen it for our own model. For many VCSEL’s, a cylindrical coordinate system is the most appropriate choice for modeling the device geometry [30], [53]. By neglecting azimuthal variations in the carrier and mode proﬁles [30], we performed our analysis in terms of the radial coordinate, replacing with Furthermore, we assumed that integrations over the active-layer volume could be converted into integrations over a radius with azimuthal and longitudinal contributions lumped into the gain constant and spontaneous emission coupling coefﬁcient. Based on these simpliﬁcations, we proceeded to convert our spatially dependent rate equations into spatially independent ones. First, we identiﬁed a suitable series expansion for the carrier proﬁle. In cylindrical coordinates the appropriate choice is a Bessel series. Thus, we modeled the carrier number using [30], [53]: (8) where and are the time-dependent expansion coefﬁ- cients, is an effective active-layer radius (which corre- sponds to the effective active-layer volume and is the th root of the ﬁrst-order Bessel function with As illustrated in Fig. 3, the two-term version of (8) [28] looks very much like that of a two-term Fourier-series expansion [45]; in this case, one can interpret as a measure of the average carrier number, and as the spatial hole produced by the VCSEL’s fundamental transverse mode. As Fig. 3 suggests, the above expansion is calculated over an effective active-layer radius This essentially 2618 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 shortly, its spatial dependence is eliminated from the carrier rate equation via the application of the orthogonal series expansion of the carrier distribution. Meanwhile, we modeled the spatially dependent current as (10) where is the total current ﬂowing through the device, and is the normalized current distribution deﬁned such that (11) Here we assumed that the current is conﬁned to the radius Substituting (8) and (10) into (1)–(2), we next integrated out the spatial dependence, making use of the following relations: (12) (13) Fig. 3. Plot of a two-term Bessel-series expansion of the radial carrier proﬁle over the effective active-layer radius W WT ; illustrated in comparison to a slice of the active layer. assumes that the slope of the carrier proﬁle is zero at [30], which corresponds, roughly, to the radius of the current injection. This assumption can be understood as follows. First, if we set then we are simply forcing the carrier proﬁle to remain ﬂat at the active layer boundaries. When however, corresponds to an effective active-layer boundary deﬁned by the current injection proﬁle, where the majority of the current is near the center of the VCSEL [30]. In this case, we assume that the carrier distribution peaks near this boundary, and that this peak does not appreciably move with changing operating conditions. Because the carrier-proﬁle expansion is only valid within the width the accuracy of the model should begin to decrease as becomes increasingly smaller than Typically, though, we can approximate even in devices where the carriers can laterally diffuse away from the current conﬁnement region. In this case, the rel- ative widths of the injection-current, carrier, and mode proﬁles need to be adjusted through parameter extraction to compen- sate for the absence of more detailed geometrical information. Next, we modeled the normalized photon distribution using an arbitrary mode shape based on the particular device under consideration. Various alternatives are discussed in Section III. In general, we deﬁned the distribution such that it satisﬁes the following normalization condition: where (12) is the orthogonality condition of the Bessel-series expansion and (13) is the diffusion term from (1) in terms of that same expansion. First, we scale (1) by and integrate over thereby producing the rate equation for Next, we scale (1) by and again integrate over this time yielding the rate equations for each term in the Bessel-series expansion. Finally, we carry out the integrations in (2) for each transverse mode. Note that this approach is equivalent to the one taken in [28], [30], and [53], where spatially dependent rate equations were transformed using a Bessel-series expansion for the carrier proﬁle. In performing the above integrations, it should be clear that the leakage current as deﬁned in (7) does not lend itself well to eliminating the spatial dependence from (1)–(2). In fact, the use of (7) in conjunction with a series expansion for the carrier number means that the resulting overlap integrals would no longer be constant, but instead functions of the carrier-proﬁle expansion coefﬁcients, which are themselves functions of the model’s operating conditions. In this case, we would have to solve the overlap integrals during simulation, an unacceptable proposition. One solution would be to linearize (7). However, because the leakage itself is highly nonlinear, this approach is also untenable. Instead, we opted to replace in (7) with , which, as we noted, is the constant component of the carrier proﬁle’s series expansion and in general accounts for the average carrier number in the active region. Thus, (14) (9) It should be noted that the mode proﬁle does not need to be described via a series expansion, since, as we shall see This equation allows us to model the nonlinear leakage without requiring the calculation of the overlap integrals during simulation. Though it appears that any spatial dependence has been eliminated from the leakage, in reality implicitly accounts for it, since effects such as SHB act to increase the total carrier MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2619 number in order to compensate for the loss of carriers in the spatial holes. The spatially independent rate equations which result from the above analysis are Because of the use of a Bessel-series expansion, in almost all cases numerical integration is necessary to calculate the above integrals. While this would appear to make the simulations more complex, in fact it does not. Because the integrals are functions of model parameters and not operating conditions, they need only be calculated once for a given device. E. Complete Model As the above results demonstrate, the spatial and thermal dependence of a VCSEL’s operation can be modeled via the spatially independent rate equations (15)–(17), the gain terms (15) (5)–(6), the leakage expression (14), and (18)–(23) for the constants and However, in order for our circuit-level model to be complete, we still require expressions for a VCSEL’s optical output power, device temperature, and current-voltage characteristics. First, we related the photon number in each mode to the corresponding optical output power using (24) (16) (17) These equations are very similar to the equations from [53], with the major exception being the inclusion of terms for the thermal gain and leakage. As we can see, we have managed to replace explicit spatial dependencies with constants. Diffusive effects are modeled via the parameters where (18) The integrated spontaneous emission in (2) is modeled through the parameters where (19a) (19b) Meanwhile, the current distribution is accounted for through (20) Finally, the overlap of the gain and mode proﬁles is accounted for by the overlap integral values and where where is the output-power coupling coefﬁcient of mode Next, we identiﬁed a suitable expression for the VCSEL temperature While it is certainly possible to adopt detailed numerical representations of the VCSEL temperature proﬁle as a function of the heat dissipation throughout the device [71], a much simpler method is to describe the temperature via a thermal rate equation that accounts for the transient temperature increase as a result of heat dissipation [28], [72]. Following this approach, we used (25) where is the VCSEL’s thermal impedance (which relates the change in device temperature to the power dissipated as heat), is a thermal time constant (which is necessary to account for the nonzero response time of the device temper- ature, observed to be on the order of 1 s [19]), is the ambient temperature, is the total current ﬂowing through the device, and is the device voltage. We have assumed that any input power not carried in the multimode output powers is dissipated as heat in the device. Finally, while the VCSEL’s electrical characteristics could be modeled in great detail based on the complex VCSEL device structure, for simplicity we modeled them using the following empirical function of current and temperature: (26) where we assumed that the current corresponding to the device (21) voltage is the cavity injection current . By introducing a capacitor or other parasitic components in parallel with this voltage, we can account for the complete electrical characteristics of the VCSEL. Fig. 4 illustrates this arrangement, where (22) the currents and are clearly identiﬁed and a parasitic shunting capacitance is used to account for parasitic effects. (23) The advantage of this technique is that the speciﬁc functional form of (26) can be determined on a device-by-device 2620 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 Fig. 4. Complete VCSEL model, including elements for modeling current-voltage characteristics, parasitics (e.g., shunting capacitance), and the intrinsic VCSEL behavior. basis. If we use experimental IV data to help determine all of the other model parameters ﬁrst, then the exact form of (26) can be determined at the very end of parameter extraction for a speciﬁc device. This simpliﬁed approach not only allows the voltage’s current and temperature dependence to be accurately modeled, but also permits the optical and electrical device characteristics to be largely decoupled from one another, thereby simplifying the extraction of model parameter values from experimental data. Equations (24)–(26), along with (5) and (6) and (14)–(23), form the complete theoretical basis of our comprehensive circuit-level VCSEL model. As we noted earlier, one of our main goals was to simulate VCSEL’s in conjunction with elec- tronics and other optoelectronic devices, thereby facilitating the design and simulation of optoelectronic applications. Thus, we have implemented these equations in Analogy’s Saber. A particular advantage of Saber is that, while it sup- ports circuit-level netlisting like SPICE, it also allows device behavior to be described explicitly in terms of differential equations using the behavioral modeling language MAST. This capability dramatically simpliﬁes the implementation of circuit-level models, making our approach particularly attractive for use in optoelectronic system design. Because of this simplicity, we have implemented two separate versions of our model. The ﬁrst describes a single-mode VCSEL with a two-term carrier-proﬁle expansion, and the second describes a two-mode VCSEL with a three-term expansion. Of course, because the model equations support an arbitrary number of modes and expansion terms, there is no reason that other implementations cannot be adopted that include more than two modes or three terms in the carrier-proﬁle expansion. An important feature of our Saber implementations is the use of variable transformations such as those described in [14] and [17] in order to improve the convergence properties of the model during simulation. Because of the nonlinear character and multiple solution regimes of the rate equations, such transformations help the simulator converge to a correct numerical solution [14]. In this case, we transformed and into the variables and respectively, using (27) (28) (29) where and are arbitrary constants. Equations (27)–(28) ensure a nonnegative solution for and [14], while (29) scales to improve convergence. It is worth noting that our full model bears a number of similarities to the efforts of other researchers. First, Moriki et al. [53] originally suggested the use of a Bessel-series expansion to describe VCSEL’s, but neglected thermal effects and parasitics. Dellunde et al. [30] also used a Bessel-series expansion to generate spatially independent multimode rate equations, but also failed to include thermal behavior or parasitic effects. In [28], rate equations similar to the ones presented here are used, with the carrier proﬁle modeled as a two-term Bessel-series expansion. Furthermore, like our approach, device temperature is modeled via a thermal rate equation and thermal gain is accounted for through polynomial functions of that temperature. However, despite these features, the authors fail to include thermally dependent leakage current, and limit their analysis to a single mode. They also do not model parasitic effects in the device. Clearly, our model dramatically improves upon these earlier approaches, eliminating many of their major deﬁciencies. Furthermore, this is the ﬁrst time that this type of comprehensive model is being implemented in a circuit-level simulation environment. In later sections, we shall demonstrate that our model can indeed be used to simulate the complex behavior of VCSEL’s. However, we will ﬁrst discuss various analytical forms for the mode proﬁles III. MODELING OF MODE PROFILES In the comprehensive circuit-level VCSEL model described above, each mode is described by a total photon number and a normalized transverse mode proﬁle . In order to generate the spatially independent rate equations through which our model is implemented, the speciﬁc form of must be chosen in advance for a given VCSEL. While the exact choice of will most certainly vary from device to device, general analytical representations exist whose parameters can be adjusted to yield a suitable ﬁnal expression for . Below, we identify some examples based on the cylindrical coordinate system used in our model. One possibility is to model the various mode proﬁles as Laguerre–Gaussian (LG) modes [40], analogous to the Hermite–Gaussian solutions used in rectangular coordinates. These solutions are based on Laguerre polynomials. Though in general these polynomials should have complex arguments [40], [73], we can still obtain a useful approximation of the transverse mode shapes by neglecting any complex parameters and using the standard Laguerre–Gaussian function [73] (30) Note that we have replaced the mode-index with where is the radial index and is the azimuthal index. Also, having assumed cylindrical symmetry [28], we ignored any explicit reference to angular dependencies. In (30), are the generalized Laguerre polynomials, is the normalization coefﬁcient deﬁned by (9), and is the characteristic width of the family of LG modes. Fig. 5 illustrates three of the lowest-order LG modes, with the normalization constants set equal to 1. These distribu- MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2621 Fig. 5. Illustration of the 00, 10, and 11 Laguerre–Gaussian modes, with their normalization constants pl set equal to 1. Fig. 6. modes n2 = Illustration for a cavity 3:4: of the radial dependence with Wm = 3 m, o of = the LP01, 850 nm, nL1P11=, a3n:d5;LaPn2d1 tions correspond to the 00, 10, and 11 generalized Laguerre polynomials and respectively. These kinds of solutions have previously been used to represent the measured mode proﬁles in actual devices, namely gain-guided VCSEL’s [40], [74]. In cases where low-order donut modes are required, an alternative representation for the mode proﬁles can be used, namely LP modes. These solutions have been used by many researchers to describe the transverse mode proﬁles of VCSEL’s. They have been observed in actual devices [38], as well as used extensively to model mode competition, mode- partition noise, and gain-switching in VCSEL’s [21]–[22], [30], [75]–[77]. Neglecting angular dependencies, we can model the LP modes using [21], [30]: (31) where, because of the absence of any angular dependence, we have dropped the mode-index In (31), is the normal- ization constant, and are the Bessel functions of the ﬁrst and second kind, respectively, and is a characteristic mode width. The parameters and are eigenvalue solutions of the optical waveguide for which (31) is a solution. To be exact, (31) actually represents approximate solutions for a weakly-index-guided cylindrical waveguide [21], [30]. As discussed in [78], for a waveguide with core index cladding index core radius and free-space wave number the solutions for the transverse mode proﬁles can be approximately determined in terms of the propagation constant We can deﬁne and in terms of via the expressions and The th-mode solution from (31) can then be determined based on the eigenvalue equation (32) Finally, can be calculated using (9). Fig. 6 depicts the LP01, LP11, and LP21 modes for a cavity with m, nm (corresponding to m and Similar results are obtained for even smaller index steps such as 0.01. As we can see, the LP01 mode is very similar to a Gaussian, while the LP11 and LP21 modes both correspond to donut-shaped mode proﬁles. While the above expressions for implement ﬁxed mode shapes, in many VCSEL’s the proﬁles change as a function of bias. For example, self-focusing in some devices can cause the fundamental mode to shrink [41], and thermal lensing in gain- guided VCSEL’s can result in a similar effect [19]. While more exact representations would account for this bias dependence, we have elected to keep our mode proﬁles ﬁxed, as would be the case in index-guided devices. The discrepancy that may occur in gain-guided or weakly-index-guided devices can then be accounted for during parameter extraction. Now that we have identiﬁed various alternatives for the mode proﬁles we can use our VCSEL model to simulate various interesting features of VCSEL operation. In the next section, we will use the LP01 and LP11 modes to demonstrate the capabilities of our approach through a set of example simulations, while in the following section we will use a Gaussian mode proﬁle to ﬁt our model to various experimental devices reported in the literature. IV. EXAMPLE SIMULATIONS OF VCSEL BEHAVIOR In order to demonstrate the ability of our comprehensive VCSEL model to replicate actual VCSEL operation, we adopted two approaches. First, we chose a representative set of model parameters and performed example simulations of typical VCSEL behavior. Second, we validated the model against experimental data for four devices presented in the literature. In this section, we discuss the former, namely the generation of sample simulations for single- and two-mode VCSEL’s. As the following results will demonstrate, our model is indeed capable of simulating a wide range of VCSEL characteristics, including thermally dependent threshold current, output-power 2622 TABLE I REPRESENTATIVE SET OF MODEL PARAMETERS USED IN GENERATING SAMPLE SIMULATIONS OF SINGLE- AND TWO-MODE VCSEL’s JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 G T N T Fig. 7. Plots of ( ) and t( ) based on the representative set of model parameters in Table I. rollover, small-signal behavior as a function of temperature, transient phenomena, and multimode operation. The model parameters used in our example simulations are summarized in Table I. In generating many of these parameters, a two-term Bessel-series expansion was used for the carrier number of the single-mode VCSEL, while a three-term expansion was used for the two-mode device. The additional term is included in the latter case to account for the more complicated spatial behavior resulting from multimode operation. As can be seen, the parameters for the two-mode device consist of the single-mode parameters in addition to parameters accounting for the extra mode and the additional term in the series expansion. Thus, those parameters in Table I corresponding to and can be ignored in the case of the single-mode VCSEL. Many of the model parameters, such as the photon life- times and spontaneous-emission coupling coefﬁcients, take on typical values. Others, in particular the spatial, gain, and leakage parameters, warrant additional discussion. First, to generate the spatial parameters of the model (e.g., and we represented the optical mode proﬁles of the single- and two-mode VCSEL’s as LP modes, where LP01 models mode 0, and, in the case of the two-mode device, LP11 models mode 1. Speciﬁcally, we used the proﬁles of Fig. 6 as a representative set of distributions, with the modes approximated to be zero for We also assumed that the carriers are conﬁned to a radius thereby resulting in and therefore Furthermore, we assumed a uniform injection current over the width as a result of which It is worth noting that, in general, there is no need to include an explicit value for the mode width It is easy to show that for a ﬁxed ratio the overlap-integral values are independent of the actual device geometry and instead are functions of In other words, the overlap integrals capture the shape of the mode relative to the active region dimensions. The only parameters which do explicitly account for the VCSEL geometry are which are functions of However, as Table I shows, we elected to use directly as one of the model parameters. Thus, all spatial and diffusive effects can be accounted for via and Also of interest in Table I are the parameters relating to the gain and active-layer carrier leakage. In both cases, we chose parameter values which resulted in reasonable descriptions of the corresponding mechanisms. Fig. 7 depicts plots of the gain constant and transparency number while Fig. 8 illustrates the leakage current As we can see, the gain peaks at approximately room temperature, while the transparency number is an increasing function of temperature. The leakage current, meanwhile, exhibits a form similar to that of Fig. 2. Finally, not included in Table I are parameters related to electrical characteristics, including a functional form for the device voltage and any parasitics. For simplicity, we have modeled the voltage as a series combination of a 100- resistor and a diode with saturation current A and thermal voltage mV. In other words, (33) Meanwhile, we included parasitics via a 1-fF shunting capacitance, in effect neglecting their role in our simulations. MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2623 Fig. 8. Plots of Il(N0; T ) based on the representative set of model parameters in Table I at temperatures from 250 to 400 K. A. Single-Mode VCSEL Simulations Using the model parameters discussed above, we ﬁrst generated example simulations of the single-mode device under dc, small-signal, and transient conditions. These results allowed us to verify that our model is capable of replicating the basic thermal and spatial behavior typical of VCSEL’s. Of initial interest were the single-mode LI characteristics at various ambient temperatures. Thus, we began by simulating these curves at temperatures of 25, 50, and 65 C. Fig. 9 illustrates the resulting family of LI curves at these temperatures. As we can see, our model is able to capture the temperaturedependent threshold current and output-power rollover of VCSEL’s. The device performance becomes progressively worse as the ambient temperature increases, with the threshold current gradually growing, and the thermal rollover becoming increasingly severe. In the curves of Fig. 9, while the thermal variation of the gain certainly contributes to the observed behavior, leakage current plays a dominant role in rolling over the output power at elevated currents [23]. To demonstrate this fact, we forced the leakage to zero and ran a second set of simulations. The results are also shown in Fig. 9. As we can see, the elimination of the leakage removes the rollover; however, the threshold current still shifts with temperature, obviously due to the thermal dependence of the gain. In the past, researchers have attributed the LI-curve rollover to the gain alone [19]. While parameters could be chosen to duplicate this behavior, we feel that in many cases, such as the one shown here, the gain will be largely responsible for shifting the threshold at lower temperatures, while the leakage will be dominant at higher temperatures and currents [23]. For example, for a device from [23], the threshold current is seen to vary by only a few mA over an approximate 60 C increase in ambient temperature, while the device begins to rollover after an approximate current increase of 10 mA or less. For a worst-case turn-on voltage of 5.0 V and a thermal impedance of roughly 1 C/mW, this corresponds to a 50- C change in temperature. Obviously, as the authors point out, different mechanisms, namely the thermal gain and leakage, must be affecting the threshold current and rollover in different ways. Fig. 9. Simulated LI curves for the single-mode VCSEL with leakage (at ambient temperatures of 25, 50, and 65 C), and without leakage (at ambient temperatures of 25, 80, and 150 C). Fig. 10. Additional dc simulation results for the single-mode VCSEL (with leakage) at 25 C showing the carrier numbers N0 and N1; and leakage current Il versus the input current. To further demonstrate the importance of the leakage cur- rent, Fig. 10 illustrates the variation of and the leakage current for the 25 C LI curve. As we can see, as the current increases and the device begins to lase, a hole is burned into the carrier proﬁle, corresponding to the initial increase of Consequently, also begins to increase in order to maintain an above-threshold modal gain. Eventually, the output power rolls over and both and begin to decrease again. However, as Fig. 10 shows, by this time they have contributed to a thermal leakage current which ultimately shuts off the device completely. We next ran simulations of the single-mode VCSEL under small-signal conditions. Fig. 11 depicts ac transfer functions at 25 C and bias currents of 2, 5, 10, and 20 mA. The curves were normalized at a low-frequency value of 10 MHz. For the three biases below the rollover point (2, 5, and 10 mA), the transfer functions’ resonance frequencies increase with bias, with the magnitudes of the peaks eventually decreasing. This result is analogous to what one would expect in regular edge- emitting lasers. For bias currents beyond the rollover point, 2624 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 Fig. 11. Normalized small-signal transfer functions of the single-mode VCSEL at an ambient temperature of 25 C and input bias currents of 2, 5, 10, and 20 mA. Fig. 12. Small-signal transfer functions of the single-mode VCSEL at an output-power bias of 0.5 mW and ambient temperatures of 25 and 50 C. such as that for the 20-mA curve, we see that the resonance frequency begins to decrease; in other words, it rolls over as well, much like the rollover of an LI curve. Not surprisingly, similar results have been observed experimentally [79]. We sought further insight into the thermal effects on small- signal modulation by keeping the bias ﬁxed while varying the temperature. Fig. 12 illustrates the resulting transfer function for an output-power bias point of 0.5 mW and ambient temperatures of 25 and 50 C. As we can see, the resonance frequency decreases with temperature. As we brieﬂy discussed in Section II, this result is related to the gain constant de- creasing over the temperature range 25 to 50 C; because the relaxation oscillation frequency is a function of the gain, the associated resonance frequency should decrease for a ﬁxed bias power. As noted earlier, analogous behavior has been ex- perimentally observed in edge-emitters’ 3-dB bandwidth [55]. Thermal effects can also impact the small-signal modulation via the thermal time constant. Typical values of this time constant are on the order of a few s [19]; for frequencies greater than the thermal cutoff frequency the Fig. 13. Small-signal transfer function of the single-mode VCSEL, illustrating a low-frequency shift due to thermal effects. modeled temperature will not be able to change in response to the modulation. The thermal mechanism in the model that is most affected by this result is the leakage, which is a highly sensitive function of temperature. Because the temperature cannot be modulated at frequencies above in many cases the leakage remains constant. Hence, the VCSEL should respond as if there is no small-signal modulation of the leakage (or any other thermally varying behavior), resulting in an altered value of the simulated modulation response. Furthermore, the response should shift to its dc value at frequencies below Fig. 13 illustrates this behavior in a simulated ac response at a bias current of 10 mA and a 25 C ambient temperature. Because of the effect of the thermal time constant, we had previously normalized the results of Fig. 11 at 10 MHz to avoid unnecessary confusion in comparing the data at different biases. Obviously, however, the low-frequency effect can be quite important. While this low-frequency shift has been observed experimentally [80], to the best of our knowledge it has not been addressed in the literature. Thus, additional research is necessary for a better understanding of whether or not the experimental results correspond to the mechanisms at work in our model, or to some other effect. To complete our single-mode simulations, we simulated the VCSEL’s transient response to a square-pulse-train input current whose low and high levels were 1.5 and 9.0 mA, respectively. As Fig. 14 shows, the behavior described in Section II is clearly replicated by our model, with an optical bump occurring in the output-power’s turn-off transient as a result of the interplay between the optical mode and carrier proﬁles due to SHB. While this result provided an initial veriﬁcation of our model’s spatial capabilities, we obtained further conﬁrmation through the two-mode VCSEL simulations of the next subsection. B. Two-Mode VCSEL Simulations Using the complete set of model parameters from Table I, we next simulated the two-mode VCSEL under dc and transient conditions. While much of the behavior exhibited by MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2625 Fig. 14. Simulated transient response of the single-mode VCSEL, demonstrating an optical bump in the turn-off transient. Fig. 16. Simulated transient response of the two-mode VCSEL, illustrating an additional turn-on delay for mode 1 and an optical bump in the turn-off transient of mode 0. Fig. 15. Simulated LI characteristics for the two-mode VCSEL at ambient temperatures of 25 and 40 C. In the case of the 25 C curves, results are shown both with and without SHB. Note that mode 1 essentially does not lase in the latter case. the single-mode device is to be expected in this laser as well, there are other features worth noting that are speciﬁc to a multimode VCSEL and further elucidate the ability of our model to describe spatially dependent behavior. First, we considered the two-mode device’s LI characteristics. Fig. 15 illustrates LI curves for both modes at ambient temperatures of 25 and 40 C. As we can see, the second mode (mode 1, or LP11) has a higher threshold as compared to the fundamental (mode 0, or LP01). This result is largely due to the fact that mode 1 achieves threshold after SHB allows its modal gain to reach threshold itself. Once the LP11 mode begins to lase, there is a kink in the LP01 mode’s output power, as would be expected since the two modes share the available laser gain. Such a kink has also been observed in nonthermal simulation results presented in the literature [21]. To better understand the role of spatial effects in arbitrating the competition of the two modes, we ran an additional dc simulation at 25 C while neglecting the spatial dependence of the carrier proﬁle. In other words, and were ﬁxed at zero. Fig. 15 includes the resulting simulation data as compared to the earlier results. Two observations are immediately obvious. First, without SHB, the LP11 mode essentially never lases, further conﬁrming the importance of spatial effects in a VCSEL’s operation. Because no hole is burned in the carrier proﬁle, the LP11 mode’s modal gain can never achieve threshold. Second, the overall output power of the LP01 mode is clearly increased when SHB is removed. This corroborates the role SHB plays in reducing the efﬁciency with which a VCSEL converts current into photons [23]. As a ﬁnal simulation, we generated the transient response of the two-mode VCSEL to a pulse input at 25 C, where the lowand high-level currents were 1.5 and 8 mA, respectively. The output power of each mode is illustrated in Fig. 16. Again, the importance of SHB can be seen in the additional delay necessary before the LP11 mode can lase, as has been discussed elsewhere [30]. This delay is most likely due to the time necessary for the LP11 mode’s modal gain to reach threshold. In addition, despite the presence of a second mode, secondary pulsations can still be seen in the turn-off transient of the LP01 mode, similar to the results seen in Fig. 14. V. COMPARISON WITH EXPERIMENT While the results presented in the previous section demonstrate that our comprehensive VCSEL model can be used to simulate typical VCSEL behavior, they do not indicate whether or not the model can be used to replicate the operating characteristics of real devices. Because this latter capability is critical to the modeling of VCSEL’s in the context of optoelectronic-system design and simulation, it was necessary to verify our model against experimental data. To accomplish this task, we identiﬁed four experimental devices from the literature and ﬁt our model to the reported operating characteristics. In generating the model parameters of each device, most of the numerical optimization was carried out using CFSQP from the University of Maryland [81]. As the results will show, we obtained excellent agreement between simulation and experiment. However, the results also highlight the need for more comprehensive device characterization than what has been previously presented in the literature. 2626 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 It should be noted that because none of the devices’ experi- mental data reported any multimode information, we generated model parameters for a single-mode VCSEL. Consequently, we used a two-term Bessel-series expansion to describe the carrier proﬁle. Furthermore, for simplicity, we assumed Also, we used the lowest-order Laguerre–Gaussian mode i.e., a Gaussian proﬁle, to describe the fundamental lasing mode. Finally, as discussed in the previous section, it sufﬁces to consider only the ratio when calculating the overlap integrals for a given mode proﬁle. Thus, instead of calculating and from and we used and during parameter optimization to account for spatial effects in the model. A. Index-Guided InGaAs VCSEL The ﬁrst device is an index-guided, vertically-contacted VCSEL discussed in [23]. The device has a 100- m2 area and is composed of GaAs-AlAs DBR mirrors, three In0.2Ga0.8As quantum wells, and Al0.2Ga0.8As conﬁnement layers. Lateral carrier conﬁnement is provided through an etched-mesa design. In addition to presenting measured LI characteristics at ambient temperatures of 25, 45, 65, and 85 C, the authors also provided an analytical estimate for the thermal impedance and a formula for the device voltage as a polynomial function of current and temperature. Utilizing these additional equations in conjunction with the measured LI curves, we determined the measured IV data at 25, 45, 65, and 85 C. Using the LI and IV data, we determined model parameters for this device. They are listed in Table II and are valid over the range of operating conditions in the experimental data. Furthermore, we obtained the following empirical expression for the voltage : TABLE II PARAMETERS RESULTING FROM THE FITTING OF THE VCSEL MODEL TO EXPERIMENTAL DATA FROM FOUR DEVICES. (a) INDEX-GUIDED InGaAs VCSEL [23]. (b) SELECTIVELY OXIDIZED AlGaInP VCSEL [82] AND [83]. (c) BOTTOM-EMITTING AlGaAs VCSEL [85]. (d) THIN-OXIDE-APERTURED VCSEL [86] (34) where and Figs. 17 and 18 compare the resulting simulated light-current-voltage (LIV) data with the measured curves, demonstrating a good match between the two. The biggest discrepancy is found in the 25 C LI data, where the simulation exhibits a lower output power than that in the measured characteristic. However, the error is still less than 10%. The IV data also exhibits good agreement, with the error increasing below threshold. It is worth noting that the level of agreement shown here in the LI data is superior even to the results of detailed multidimensional analysis, such as those presented in [23] for the same device. We extracted the model parameters needed to ﬁt the LIV data, resulting in the values listed in Table II, valid over the range of operating conditions in the experimental data. Furthermore, the following expression was determined for the voltage : B. Selectively Oxidized AlGaInP VCSEL The second device is an AlGaInP-based 683-nm selectivelyoxidized VCSEL with a 3 m 3 m area, reported in [82]–[83] by Crawford et al. This device consists of compressively-strained InGaP quantum wells, AlGaInP barrier and cladding layers, and AlGaAs graded DBR’s. The authors provided both LI and IV curves at ambient temperatures of 25, 40, 60, 80, and 85 C. (35) Using these parameters, we were again able to obtain excellent agreement between the simulated and experimental curves, as shown in Figs. 19 and 20. The largest discrepancies can be seen in the reasonably small error in the 25- C LI curve at rollover, as well as the below-threshold IV data. MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2627 Fig. 17. Comparison of simulated (lines) and experimental (points) LI data Fig. 20. Comparison of simulated (lines) and experimental (points) IV data for the index-guided InGaAs VCSEL of [23]. for the selectively oxidized AlGaInP VCSEL of [82] and [83]. over the complete range of reported ambient temperatures. In contrast, the present model provides a much improved level of agreement between simulation and experiment, especially at the higher temperatures, thereby illustrating the importance of including descriptions for the thermal gain, leakage, and spatial behavior in our model Fig. 18. Comparison of simulated (lines) and experimental (points) IV data for the index-guided InGaAs VCSEL of [23]. C. Bottom-Emitting AlGaAs VCSEL The next device is the 863-nm bottom-emitting AlGaAs VCSEL presented by Ohiso et al. in [85]. This 16- m diameter device was grown on an Al0.1Ga0.9As substrate and consists of a Si-doped Al0.15Ga0.85As-AlAs, GaAs-Al0.2Ga0.8As -type DBR, six quantum wells, and a C-doped Al0.15Ga0.85AsAl0.5Ga0.5As-AlAs -type DBR. The authors presented a family of LI curves over a 20–130 C range of ambient temperatures, as well as a room-temperature IV characteristic. Unlike the ﬁrst two devices presented in this section, the experimental data for this VCSEL depict the complete thermal rollover of the LI characteristics, which allowed us to validate our model across the full lasing regime of the device for each of the reported ambient temperatures. Through numerical optimization, we obtained the parameters listed in Table II, again valid over the range of operating conditions in the experimental data. We also used the following expression to model the voltage Fig. 19. Comparison of simulated (lines) and experimental (points) LI data for the selectively oxidized AlGaInP VCSEL of [82] and [83]. We previously had analyzed this device using a simpler model that we developed [84], which neglected explicit thermal and spatial dependencies. Though this model provided a good semi-empirical description of the VCSEL’s thermal LI characteristics, it failed to adequately replicate the curves (36) The use of these parameters resulted in excellent agreement between simulation and experiment, as conﬁrmed by Fig. 21. Most notably, the rollover in the LI data is correctly captured by the model. A potential limitation of the result is the absence of IV data at different ambient temperatures. However, based on the results from the ﬁrst two devices presented above, we are conﬁdent that such data would not signiﬁcantly change the accuracy of the match between simulation and experiment. D. Thin-Oxide-Apertured VCSEL The ﬁnal device is the VCSEL reported by Thibeault et al. in [86], a 3.1- m diameter, thin-oxide-apertured de- 2628 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999 Fig. 21. Comparison of simulated (lines) and experimental (points) LIV data for the bottom-emitting AlGaAs VCSEL of [85]. Fig. 22. Comparison of simulated (lines) and experimental (points) LIV data for the thin-oxide-apertured VCSEL of [86] at an ambient temperature of 23 C. vice composed of an Al0.9Ga0.1As-GaAs -type DBR, three In0.17Ga0.83As-GaAs quantum wells, an Al0.3Ga0.7As cavity, and an AlAs-GaAs -type DBR. The authors reported a single LI curve at a temperature of 23 C, and the corresponding wall-plug efﬁciency, from which we were able to determine IV data. They also reported modulation responses at an ambient temperature of 22 C and bias currents of 0.5, 0.7, 1.0, 1.3, and 2.1 mA. Despite the lack of LIV data at additional ambient temperatures, the 23 C curve does exhibit thermal rollover. More importantly, however, is the fact that the data allowed us to analyze the merits of our model under nondc conditions. In the ﬁrst three devices, we were verifying the static capabilities of our model. In actual practice, however, it is critical that our model be able to capture dynamic device behavior as well. As we shall see shortly, it can indeed accomplish this task. After optimizing the model parameters, we obtained the values listed in Table II, valid over the range of operating conditions present in the experimental data. We also generated the following expression for the voltage (37) Furthermore, as suggested in [86], we accounted for parasitics via a shunting capacitance, for which we determined a value of 248.85 fF. Based on these results, Fig. 22 shows a comparison between the simulated and experimental LIV data. As one might expect for such a limited set of dc data, the agreement is very good, with the simulation reasonably matching the thermal rollover at 6 mA. Fig. 23 illustrates a comparison of the simulated and experimental data. Analogous to the sample simulation results for the single-mode VCSEL presented earlier, the simulated data discussed here also exhibited a low-frequency shift of roughly 5–10% due to the thermal time constant. However, in all likelihood the experimental data was normalized at a low-frequency value not equal to dc [80]. Thus, we normalized our simulation results at a low nondc frequency of 10 MHz, effectively removing Fig. 23. Comparison of simulated (lines) and experimental (points) S21 data for the thin-oxide-apertured VCSEL of [86] at an ambient temperature of 22 C. the thermal shift. As we can see, the overall agreement between simulation and experiment is quite good. It should be noted, though, that the reported data suggested a dip at frequencies just below resonance, such as that depicted in Fig. 11. However, because it was difﬁcult to resolve to which curve this dip corresponded, we excluded it from our parameter optimization. As we shall discuss shortly, the absence of adequate comprehensive VCSEL characterization in the literature did not allow us to validate our model across a broader range of operating conditions, namely, dc- and small-signal modulation at different ambient temperatures. However, even the limited dc and small-signal data presented here provides evidence that our model should be capable of replicating the general operating characteristics of VCSEL’s. E. Discussion The results of this section demonstrate that our comprehensive VCSEL model can be used to replicate the dc and nondc behavior of actual devices. However, it is important MENA et al.: COMPREHENSIVE CIRCUIT-LEVEL MODEL OF VCSEL’S 2629 G T Fig. 24. Plots of ( ) using the model parameters of (a) the index-guided InGaAs VCSEL, (b) the selectively-oxidized AlGaInP VCSEL, (c) the bot- tom-emitting AlGaAs VCSEL, and (d) the thin-oxide-apertured VCSEL. to recognize that because of the relatively large number of parameters involved in matching simulation to experiment, in conjunction with the relatively limited amount of experimental data that is typically available, it can be difﬁcult to ensure that all of the parameter values correctly conform to the corresponding phy