## Abstract

To derive a simple form of the multimode laser diode rate equations incorporating the band filling effect, the laser diode gain in the direct bandgap model is introduced into the conventional multimode laser diode rate equations. By numerically examining each modal gain under the gain-switching condition, it is found that both the differential gain coefficient and the carrier density at transparency show an approximately linear dependency on the oscillation frequency. As a result, it is possible to derive a simple form of the multimode laser diode rate equations with linearized gain, which can be used to simulate the behaviors of a gain-switched laser diode characterized by the band filling effect, in both the multimode and single-mode oscillation cases.

© 2011 OSA

## 1. Introduction

Rate equations have been widely used for analyzing the dynamic behavior of laser diodes [1–22]. For rigorous analysis of laser diode operation, the laser diode gain in the rate equations should be described with terms that take into account certain aspects of semiconductors, namely, the band structure and many-body effects [18,19]. In one such attempt, pioneering work by Osinski and Adams demonstrated that the asymmetric multimode power spectrum of a gain-switched pulse from an InGaAsP Fabry-Perot laser diode could be simulated by introducing a semiconductor gain term in the rate equations [20,21]. The asymmetry seemed to stem from the band filling effect in the laser diode, which is a phenomenon peculiar to a laser diode whereby the gain peak is blue-shifted with increasing carrier density. On the other hand, when a linear approximation of the relation between the laser diode gain and the carrier density holds, the gain term in the rate equations is reduced to a simple form *G*(*N – N*_{0}) (*G*: differential gain coefficient; *N*, *N*_{0}: carrier density and carrier density at transparency). This simple form of the linearized gain has been traditionally used in analyzing laser diode operation [1–17]. Even though the importance of introducing a realistic semiconductor gain has been noted, as mentioned above, the linearized gain form in the rate equations seems to remain valued by researchers in terms of its convenience. For GaAs-based laser diodes, at least, the numerical framework of the rate equations with the linearized gain provides foreseeable analytical solutions in steady-state or quasi steady-state analyses [8,9] and also provides many good estimations in analyzing both single-mode and multimode operation in the presence of optical feedback fields [4, 11, 13] or under large-amplitude modulation conditions [2, 5–7, 10, 12, 14, 16, 17]. Some typical aspects of laser diodes, namely, linewidth broadening [22, 23], spontaneous emission [2], gain saturation [6,7], the carrier-density-dependent carrier lifetime [24], and band filling [5,10], are taken into account by introducing phenomenologically-derived parameters into the rate equations. As for the band filling effect, however, although its expression is elegant, the modified gain is given as a polynomial with respect to the carrier density, which conflicts with the definition of the differential gain coefficient. In this paper, therefore, using the simple form of the linearized gain *G*(*N* – *N*_{0}), we derive consistently formulated multimode laser diode rate equations incorporating the band filling effect.

This paper consists of six sections. In Section 2, an example of the experimentally observed multimode power spectrum of the gain-switched pulse is shown with some explanations concerning its asymmetric shape. In Section 3, the dynamics of a gain-switched multimode laser diode are examined numerically with the direct bandgap model and the conventional differential gain model. The asymmetry in the multimode power spectrum is explained in terms of the contribution of the band filling effect. In Section 4, a simple form of multimode laser diode rate equations incorporating the band filling effect is derived by applying a linear approximation to the modal gain in the direct bandgap model. The feature of the linearized gain derived in this way is described by comparing it with the conventional differential gain. In Section 5, using the linearized multimode gain, the dynamics of a gain-switched single-mode laser diode are examined. In the Conclusion section, we summarize our paper with a discussion of the derivation of a simple form of the multimode laser diode rate equations.

## 2. Asymmetry in the power spectrum of a gain-switched pulse

Figure 1 shows an example of an experimentally observed multimode power spectrum of a gain-switched pulse from an 800 nm Fabry-Perot laser diode (Rohm, RLD-78PIT). The laser diode was dc-biased at nearly its threshold current and sinusoidally modulated with an amplitude of about four times the threshold current at a modulation frequency of 1 GHz. The power spectrum was observed with an optical spectrum analyzer with a resolution of 0.01 nm. Two spectral peaks seen in each mode [25], forming deformed trapezoidal shapes, are indicated by arrows in the magnified view. The ratio of the longer wavelength peak intensity to the shorter one (*SP _{l}/SP_{s}*) in the individual modes decreases gradually and continuously in the spectral range from the central mode at 802.5 nm to the shorter wavelength modes around 798 nm, through the peak ratio of nearly one in the 801 nm mode. In contrast, the peak ratio decreases continuously but slowly in the spectral range from the central mode to the longer wavelength modes around 805 nm, because of the existence of excessively high longer wavelength peaks. This leads to an asymmetric multimode power spectrum of a gain-switched pulse, as reported before [22, 26]. By introducing semiconductor gain into the multimode laser diode rate equations, the numerical simulation conducted by Osinski and Adams [20,21] demonstrated that the asymmetric spectral shape stemmed from the semiconductor gain aspect.

## 3. Dynamics of a gain-switched multimode laser diode

In a similar way to the pioneering work described above, we started by introducing laser diode gain terms into common multimode rate equations to find a better expression of the gain in analyzing laser diode operation. For simplicity, we used the laser diode modal gain *g _{n}* in the direct bandgap model [27,28] without taking into account many-body effects. We formulate the multimode rate equations by referring to the expressions described in [1–11] and partially modified them by introducing terms related to the spontaneous emission factor and the longitudinal mode spacing [12]:

*n*and

*j*indicate the longitudinal-mode numbers (

*n*,

*j*= 0 for the central mode). A minus sign on the mode number indicates that the wavelength of the mode of interest is longer than the central wavelength, and a plus sign indicates that it is shorter. The symbol

*i*appearing twice in Eq. (1) is the imaginary unit.

*E*is the slowly varying complex electric field of the

_{n}*n*-th mode, ${S}_{n}(={n}_{r}^{2}{\varepsilon}_{0}{\left|{E}_{n}\right|}^{2}/2\overline{h}{\omega}_{n}$,

*n*: refractive index,

_{r}*ɛ*

_{0}: dielectric constant of vacuum,

*h*(= 2

*πh̄*): Planck’s constant,

*ω*: the central angular frequency of

_{n}*n*-th mode) is the photon density of the

*n*-th mode,

*N*is the carrier density,

*I*is the injected current,

*T*

_{1}is the carrier density lifetime, and

*n*is the group refractive index. |

_{g}*M*|

^{2}is the mean square momentum matrix element,

*ρ*is the reduced density of states,

_{r}*E*and

_{a}*E*are the electronic energies of the valence band and the conduction band, respectively, and

_{b}*E*and

_{v}*E*are the valence band-edge energy and the conduction band-edge energy, respectively.

_{c}*F*and

_{v}*F*are the quasi-Fermi levels of the valence band and the conduction band, respectively.

_{c}*f*and

_{a}*f*are the electron occupation probabilities in the

_{b}*E*level and

_{a}*E*level, respectively.

_{b}*m*is the reduced effective mass,

_{r}*N*and

_{v}*N*are the effective densities of states of holes in the valence band and of electrons in the conduction band, respectively, under thermal equilibrium. Other parameters used in the rate equations are set to suitable values for an 800 nm AlGaAs Fabry-Perot laser diode, listed in Table 1. Note, however, that the following assumptions are made: Because the light hole effective mass

_{c}*m*is one-fifth to one-tenth of the heavy hole effective mass

_{pl}*m*, for simplicity, the contribution of the light holes is neglected in Eqs. (6), (8), (10), and (13). In Eq. (11), the relation

_{ph}*N*=

*n*=

_{e}*p*(

_{h}*n*: electron density,

_{e}*p*: hole density) is assumed because the injection level is high enough under the gain-switching condition. Because the values of the electron and hole effective masses suitable for 800 nm AlGaAs alloy in the active layer are not necessarily clear, they are set to known values for GaAs available in the literature [18,28]. The value of the bandgap is chosen in order that the central mode of the multimode spectrum is located at 800 nm under the given pumping condition. The contribution of Auger recombination is neglected because it is sufficiently small in shorter wavelength (∼ 800 nm) laser diodes. For simplicity, the group refractive index

_{h}*n*is set to the same constant value as the refractive index

_{g}*n*. These assumptions do not have a serious influence on the numerical results described below.

_{r}For comparison, the conventional differential gain model incorporating the band filling effect is also shown below [5,10]; the differential gain coefficient is used, and the gain has a parabolic spectrum (parameter values used in the gain expression are listed in Table 2):

*λ*and

_{n}*δλ*are the central wavelength of the

*n*-th mode and the wavelength difference between adjacent modes, respectively.

The injected current *I* is defined here as the sum of a dc current and a sinusoidally modulated current, given by

*I*and

_{dc}*I*are the dc-bias current and the amplitude of the modulated current, respectively, and

_{mw}*f*is the modulation frequency. These parameter values are set to

_{m}*I*= 0.95

_{dc}*I*(

_{th}*I*is the threshold current),

_{th}*I*= 2.0

_{mw}*I*, and

_{th}*f*= 1.0GHz throughout the simulation. When the value of

_{m}*I*is negative in Eq. (18), it must be replaced by zero according to the rectification property of diodes. Because the form of the above gain expressions is complicated, the threshold current

*I*was estimated to be 54 mA for both gain models by numerically calculating the I–L characteristics (injected dc current vs. light output power). The number of longitudinal modes assumed in the calculation is 41 (

_{th}*n*is varied from −20 to 20), and an initial small amplitude with a random phase is adopted for the optical field of each mode. The temporal evolutions of variables (

*E*(

_{n}*t*) and

*N*(

*t*)) in both of the above rate equations (the direct bandgap model and the conventional differential gain model) are then numerically evaluated by using the fourth-order Runge–Kutta method under the gain-switching condition of Eq. (18). Repeating the numerical integration produces a regular gain-switched pulse train within 30 cycles of the modulation period (1/

*f*). The Fourier transform of the temporal data of the optical fields thus obtained provides the power spectrum of the gain-switched pulse.

_{m}Figure 2 shows multimode power spectra of gain-switched pulses simulated when the laser diode gain is described with the direct bandgap model (a) and with the conventional differential gain model (b, c). The parameter *k* tied to the band filling effect is set to 0 nm in (b) (no band filling) and 20 nm in (c). By assuming an *α* parameter value of 5, all linewidths of the multimode spectra are broadened [22], corresponding to the experimental result in Fig. 1. The above-mentioned spectral peak ratio (*SP _{l}/SP_{s}*) in the individual modes gradually varies in all of the multimode spectra in Fig. 2, whereas symmetry of the peak ratio variation about the central mode is seen only in (b). The other two power spectra contain asymmetry, corresponding to the experimental result in Fig. 1. From these results, the asymmetry in the spectral peak variation stems from the band filling effect.

To examine this in detail, the gain spectra for these cases are depicted in Fig. 3. By varying the value of the carrier density from 1.25 × 10^{24}m^{−3} to 1.73 × 10^{24}m^{−3} in 0.04 × 10^{24}m^{−3} steps in Eq. (4) for (a) and in Eq. (14) for (b) and (c), thirteen gain spectra curves are plotted in each figure. In the case of Fig. 3(b), the gain spectra have a symmetric parabolic shape about the central mode at 800 nm, which leads to the symmetric power spectrum of the gain-switched pulse in Fig. 2(b). In Figs. 3(a) and 3(c), as a result of the decrease in the carrier density within the gain-switched pulse width, the gain remains in the longer wavelength region more than the shorter wavelength region. This phenomenon is based on the red shift of the gain peak with decreasing carrier density, which corresponds to the reverse process of the band filling effect. The residual gain produces the excessively high spectral peak in the individual modes in the longer wavelength region, causing the asymmetry in the power spectrum of the gain-switched pulse.

To see this phenomenon in the time domain, the peak intensity and the pulse width of the gain-switched pulse component in individual modes simulated in Fig. 2(a) (direct bandgap model) are plotted in Fig. 4. In Fig. 4, although the peak intensity varies significantly in the spectral range from mode numbers −10 to 10, through the maximum peak intensity at the central mode (*n*=0), the pulse width shortens monotonically, with a gentle slope in the corresponding spectral range. Because the mechanism of the gain-switching is the same as Q-switching, the gain-switched pulses acquire a fast leading edge and a slower trailing edge, and the speed of the leading edge and the peak intensity depend strongly on the laser gain [9, 30]. Therefore, it seems unreasonable that the gain-switched pulse components oscillating under greatly differing values of the modal gain acquire a nearly constant pulse width.

Some of the waveforms of those pulse components are shown in Fig. 5. To obtain a composite pulse of all pulse components of the 41 modes, those pulse components were summed under random phase conditions until interference beats disappeared. The composite pulse corresponds to the gain-switched pulse experimentally observed with a fast photodetector. It has a normal pulse shape, that is, a fast leading edge and a slower trailing edge, as shown by the curve labeled “all modes” in Fig. 5, which is approximately consistent with the pulse shape of the 0th mode pulse component. It is estimated from the corresponding temporal variation of the carrier density that the carrier density decreases linearly from 1.71 × 10^{24}m^{−3} to 1.58 × 10^{24}m^{−3} within the FWHM (full width at half maximum) of the composite pulse. In Fig. 5, a normal pulse shape of the gain-switched pulse components is seen at the −5th, −3rd, 0th, and 3rd modes, whereas an abnormal pulse shape, having a fast trailing edge and a slower leading edge, is seen at the −7th, 5th, and 9th modes. The reason is as follows: The speed of the leading edge of a gain-switched pulse component is determined by the initial gain in the individual modes. In this case, the top curve of the gain spectrum in Fig. 3(a) corresponds approximately to the initial gain. Then, stimulated emission for generating gain-switched pulses takes place dominantly near the central mode. As a result of this large consumption of the carrier density, the modal gain located around the edges of the gain spectrum decreases strongly. This makes pulse trailing edges faster sometimes than leading edges, as observed in the −7th, 5th, and 9th modes in Fig. 5. Consequently, the pulse widths of the pulse components in respective modes approach the pulse width of the central mode pulse component. In addition, the red shift of the gain peak with decreasing carrier density causes the pulse width to shorten monotonically with a gentle slope toward the shorter wavelength region. Thus, the result in Fig. 4 is reasonable. After generating gain-switched pulses, because more laser gain remains in the longer wavelength region due to the red shift of the gain peak, slower trailing edges are seen in the −3rd and −5th modes in Fig. 5. Because the gain-switched pulse certainly contains a frequency down-chirp within the pulse width [21, 31], the leading edge of the pulse forms the shorter wavelength peak (*SP _{s}*) in the individual modes and the trailing edge forms the longer wavelength peak (

*SP*). That is, the spectral peak ratio (

_{l}*SP*) in the individual modes in Fig. 2(a) corresponds to the speed ratio of the leading edge to the trailing edge of the corresponding pulse component. From this point of view, it is interesting that pulse shaping occurs gradually in the shorter wavelength region (0th–10th modes) in Fig. 2(a). As the mode number is decreased from 10 to 0, the leading edge and the trailing edge become faster and slower, respectively, via a symmetric pulse shape at the 3rd or 4th mode where the leading edge and the trailing edge have almost the same speed. The spectral peaks seen in the longer wavelength region (−1st – −5th modes) in Fig. 2(a) correspond to slower trailing edges of the pulse components caused by the red shift of the gain peak. Thus, the simultaneous interaction of the optical fields of all 41 modes with the carrier density (as only one variable) under the gain-switching condition produces the characteristic shape of the power spectrum of the gain-switched pulse.

_{l}/SP_{s}In addition to the above considerations, the pulse shape functions of those pulse components are estimated in detail by applying the TBP–CBP method (Time Bandwidth Product –Coherence-time Bandwidth Product) which was proposed to estimate shape functions of the power spectra of light emitting diodes [32]. To obtain TBP and CBP values, the pulse complex spectrum should be calculated by Fourier-transforming the complex optical pulse fields while assuming that the inside phase is constant. The power spectrum is obtained by calculating squared absolute values of the complex pulse spectrum. The FWHM of the power spectrum Δ*f* multiplied by the pulse width Δ*t* equals the TBP value. Then, the coherence function *C* is calculated by substituting the complex optical pulse field *E*(*t*) and its replica *E*(*t* + *τ*) into the following equation assuming the inside phase is constant:

*τ*is the time delay. The value of

*τ*is varied in Eq. (19) until the coherence function is obtained. The FWHM of the coherence function Δ

*τ*(called coherence time) multiplied by the FWHM of the power spectrum Δ

*f*equals the CBP value. The combination values (CBP, TBP) thus calculated characterize the pulse shape function in terms of the relative ratio of the average height of inflection points between the leading and trailing edges of the pulse to the pulse peak (called “wing height” in [32]) and the relative ratio of the average width between the pulse leading and trailing edges to the pulse width (called “wing width” in [32]). For simplicity, they are called “the position of the pulse waist” and “the ratio of the pulse edge width”, respectively. A small TBP value indicates a large ratio of the pulse edge width under a constant pulse width regardless of the CBP value, and a large CBP value indicates a high position of the pulse waist under a constant TBP value. Figure 6 plots the combination values (CBP, TBP) of the pulse components for the corresponding mode numbers in Fig. 2(a). To intuitively estimate the pulse shape functions and the degree of variation of those pulse shapes, the combination values of typical functions, namely, Gaussian, hyperbolic secant-squared, and Lorentzian, are also indicated in Fig. 6. Those combination values of the pulse components, located in the vicinity of the combination value of a sech

^{2}-shape function (0.78, 0.315), form a semielliptical-like locus. The apex of the semiellipse, composed of the combination values around the central mode (−2nd, −1st, and 0th modes), has a closest approach to the combination value of the sech

^{2}-shape function. As the absolute mode number becomes large, the shape function is found to approach a Lorentz-like shape. It is found in Fig. 6 that the combination values of the pulse component at the 9th mode include approximately the same TBP value as that at the −5th mode and the same CBP value as that at the −7th mode. This indicates the following: Comparing the shape function of the pulse component at the 9th mode with that at the −5th mode, both have approximately the same ratio of the pulse edge width, and the difference is that the former has a lower pulse waist position. Then, comparing the shape function of the pulse component at the 9th mode with that at the −7th mode, the former has a smaller ratio of the pulse edge width and a higher pulse waist position. These estimations correspond to the results in Fig. 5. Thus, a gain-switched pulse from a multimode laser diode contains a distinctive variation in the shape function of inside pulse components.

## 4. Derivation of multimode laser diode rate equations

As shown above, the direct bandgap model is found to be appropriate for describing the multimode laser diode gain. The good agreement between Figs. 2(a) and 2(c) implies that the conventional differential gain model is also appropriate for describing the multimode laser diode gain. However, the introduction of the term phenomenologically added in Eq. (14) to incorporate the band filling effect nullifies the definition of the differential gain coefficient, that is, *dg/dN* ≠ *G*_{0}. Because of this, despite a positive differential gain coefficient, the gain decreases with increasing carrier density at higher values in the wavelength region from 804 nm to 806 nm in Fig. 3(c), shown as a gain curve crossing. Therefore, we returned to the estimation of the differential gain coefficient with the direct bandgap model.

The variation of the modal gain as a function of the carrier density in the direct bandgap model is plotted under the gain-switching condition. Examples are shown in Fig. 7. It is found that a linear approximation is applicable to the relation between the gain and the carrier density, although each curve has slight saturation characteristics. As a simulation model that can be applied to the gain-switching operation of laser diodes, the linear approximation is employed in the carrier density interval between 1.58×10^{24}m^{−3} and 1.71×10^{24}m^{−3} where the composite gain-switched pulse mentioned above is generated. The modal differential gain coefficient (*G*_{0n}, where *n* is the mode number) is then estimated by Δ*g*/Δ*N*, as shown in the middle of Fig. 7. Here, Δ*g* and Δ*N* correspond to variations of the gain and the carrier density, respectively, within the FWHM of the composite pulse. For simplicity, Δ*N* is fixed to the above interval in estimating the differential gain coefficient of all the rest modes. Then the carrier density at transparency (*N*_{0n}) is estimated as the value of the carrier density when the linearized gain becomes zero, as is also shown in the middle of Fig. 7.

In the same way, the values of the differential gain coefficient and the carrier density at transparency are estimated for all 41 modes, as shown in Fig. 8. Those parameter values increase monotonically as a function of the mode number or the oscillation frequency. To our knowledge, the carrier density at transparency is treated as a constant for the conduction band of interest. The result in Fig. 8, however, makes it clear that the carrier density at transparency should be treated as a parameter that depends on the oscillation frequency.

By applying a linear approximation to the relation between those parameters and the mode number, the form of the laser diode gain expression is greatly simplified:

where, Here,*G*

_{0n}and

*N*

_{0n}are the differential gain coefficient of the

*n*-th mode and the carrier density at transparency of the

*n*-th mode, respectively;

*G*

_{00}and

*N*

_{00}are the corresponding values of the central mode; and

*δG*

_{0}and

*δN*

_{0}are the differences in differential gain coefficients and carrier densities at transparency between adjacent modes, respectively. The parameter values in Eqs. (21) and (22) are listed in Table 3.

Figure 9 shows the simulated power spectrum of the gain-switched pulse using the linearized laser diode gain derived in Eqs. (20)–(22). The same parameter values in Table 1 and the same pumping condition were used in the simulation, except for the gain term. The mode number *n* was varied from −20 to 20 in the same way as in the case of the direct bandgap model. The excellent agreement between Fig. 2(a) and Fig. 9 indicates that the linearized laser diode gain is equivalent to the gain in the direct bandgap model, and therefore, it incorporates the band filling effect despite its comparatively simple description.

To understand this, Eqs. (21) and (22) are substituted into Eq. (20). Then, the gain expression becomes a quadratic equation with respect to the mode number *n*:

*n*is proportional to the oscillation frequency when the central frequency is regarded as the origin of the coordinates, the laser diode gain

*g*is found to have a parabolic spectrum, which is the same as the conventional differential gain model. The modal gain becomes zero when the mode number is set to the values (

_{n}*N*−

*N*

_{00})/

*δN*

_{0}or −

*G*

_{00}/

*δG*

_{0}, which correspond to the angular frequencies

*ω*

_{0}+

*δω*(

*N*−

*N*

_{00})/

*δN*

_{0}and

*ω*

_{0}−

*δωG*

_{00}/

*δG*

_{0}, respectively. The latter lower angular frequency is independent of carrier density and therefore is plotted as a fixed point on the gain spectra when the carrier density is varied, meaning that the lower angular frequency corresponds to the band edge. In contrast, the gain spectra in the conventional differential gain model have no fixed angular frequency to become zero-gain because the values of the differential gain coefficient, the gain width, and the amount of blue shift are set independently of each other. This leads to a gain curve crossing, as shown in Fig. 3(c), in return for the convenience of the model. In the gain form of Eq. (23), the gain width Δ

*ω*defined under the condition

_{g}*g*= 0 and the gain peak

_{n}*g*show linear and quadratic dependencies on carrier density, respectively:

_{m}*N*is increased by Δ

*N*, the gain peak is shifted by

*δω*Δ

*N*/2

*δN*

_{0}toward a large value of the mode number

*n*(blue shift), maintaining constant modal differential gain coefficients (

*dg/dN*=

*G*

_{0n}). Thus, the laser diode gain

*g*consistent with the definition of the differential gain coefficient is formulated in Eqs. (20)–(22), and it implicitly incorporates the band filling effect.

_{n}Figure 10 plots the gain spectra using Eq. (23) to compare it with the result in Fig. 3(a). Because the linear approximation between the gain and the carrier density was applied to the higher carrier densities, the gain spectra in Fig. 3(a) and Fig. 10 show good agreement at higher carrier densities, although the discrepancy between them is increased at lower carrier densities. This is an essential problem in the use of the linearized gain model. If necessary, saturation effects in the laser diode gain should be introduced. On the other hand, a disagreement seen in the longer wavelength region above 804 nm stems from the linear approximation applied to the result in Fig. 8. However, this does not have a serious effect on the validity of the model. When using the same numerical method with the more realistic GHLBT-SME model that takes a bandtail into account [33], we confirmed that the calculated gain spectra showed good agreement with those in Fig. 10 in the longer wavelength region.

## 5. Dynamics of a gain-switched single-mode laser diode

In analyzing the dynamic behavior of a single-mode laser diode using the linearized laser diode gain, a set of the combination values (*G*_{0}, *N*_{0}) that satisfy the relations in Eqs. (20)–(22) should be chosen. In this case, the mode number *n* should be treated not as an integer but as a real number. Consequently, this model is the same as the conventional gain written in the form *G*_{0}(*N − N*_{0}). However, although only two parameter values are chosen, their values characterize the particular spectral position of the laser diode gain while taking into account the band structure of the semiconductor. Therefore, the parameter values chosen reflect the band filling effect.

In the single-mode oscillation case, the threshold current *I _{th}* is analytically calculated by assuming

*ɛ*,

*β*= 0, as follows:

*I*= 0.95

_{dc}*I*and

_{th}*I*= 2.0

_{mw}*I*, whose absolute values are approximately constant in the wavelength region below 804 nm. The modulation frequency is set to 1 GHz. The fourth term on the right-hand side of Eq. (1) concerning the frequency detuning was omitted throughout the simulation for the single-mode oscillation case. Figure 12 plots the peak intensity and the pulse width of gain-switched pulses simulated when different combination values (

_{th}*G*

_{0},

*N*

_{0}) are used. When a sufficient carrier density is supplied, the speed of the gain-switching becomes faster in the shorter wavelength region where the gain varies significantly in association with the band filling effect. However, because the amount of blue shift of the gain peak is limited under a given pumping condition, the speed of the gain-switching certainly has the fastest value at a particular spectral position in the shorter wavelength region, where the pulse must have the shortest pulse width. From Fig. 12, the particular spectral position seems to be located below 794 nm, and as the oscillation frequency becomes higher, the peak intensity increases and the pulse width decreases, concurrently showing saturation characteristics. The increase in the peak intensity seen in the longer wavelength region is based on the increase of the input energy accompanied with the increase of threshold current in the corresponding wavelength region.

Figure 13 shows examples of simulated gain-switched pulses from a single-mode laser diode. As simulated in the 20th mode in Fig. 13, a gain-switched pulse followed by a lower and broader second pulse is characteristically generated in the shorter wavelength region corresponding to the mode numbers higher than 13. In the single-mode oscillation case, all gain-switched pulses have a normal pulse shape; that is, the pulses acquire a fast leading edge and a slower trailing edge, regardless of the oscillation frequency.

A detailed analysis using the TBP–CBP method in Fig. 14, however, gives rise to a difference in shape functions of the gain-switched pulse simulated in Figs. 12 and 13. Increasing the mode number from −20 to 20, an inverted S-shaped locus of the combination values (CBP, TBP) is exhibited. The increase in both the CBP and TBP values at mode numbers from −20 to −12 implies that the long pulse tail caused by the red shift of the gain peak becomes shorter, accompanied by a rising position of the pulse waist, as the oscillation frequency is increased. As a result, the pulse oscillating in the −12th mode (shown in the middle-left panel of Fig. 13) has a shape function approximately the same as the sech^{2}-shape. Then, increasing the mode number from −12 to 13, both the CBP and TBP values decrease monotonically, which indicates that the pulse shape function varies from a nearly sech^{2}-shaped to slightly Lorentzian. Such variation of the pulse shape function makes the position of the pulse waist lower and the ratio of the pulse edge width larger. The increase in the CBP value at mode numbers from 13 to 20 implies that the position of the pulse waist becomes high inversely because a part of the trailing edge is separated from the main body of the pulse due to a speeding-up of the gain-switching. This also saturates the decreasing trend of the TBP value. High-speed gain-switching enhanced by the band filling effect is thus found to cause not only a shortening of the pulse width but also a distinctive variation of the pulse shape function in the single-mode oscillation case.

## 6. Conclusions

With the aim of deriving a simple form of the multimode laser diode rate equations incorporating the band filling effect, the laser diode gain in the direct bandgap model was introduced into the conventional multimode rate equations. Using this numerical framework, the behavior of a gain-switched multimode laser diode was examined in detail in both the time and frequency domains. From a series of related simulations, a distinctive shape of the power spectrum of the gain-switched pulse from a multimode laser diode was found to stem dominantly from the band filling effect, which was directly related to pulse shaping among pulse components of the respective modes. Such a numerical approach also clarified that the modal gain in each mode held approximately a linear relation with the carrier density under the gain-switching condition, making it possible to estimate the differential gain coefficient and the carrier density at transparency for all of the assumed oscillation modes. Then, both parameters were expressed as linear functions of the oscillation frequency (or the oscillation mode number). As a result, a simple form of the laser diode gain consistent with the definition of the differential gain coefficient was derived, as described in Eqs. (20)–(22). It was confirmed that the obtained multimode laser diode rate equations with the linearized gain could be used to simulate the behavior of a gain-switched laser diode characterized by the band filling effect in both the multimode and single-mode oscillation cases.

## Acknowledgments

This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, under Grant-in Aid for Scientific Research (C), No. 21560044.

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