1 Introduction

In recent years, optical frequency comb (OFC) generation has gained considerable attention from researchers because of its wide-ranging applications such as optical arbitrary waveform generation [1], radio-photonics [2], optical communications [3, 4], etc. The essential characteristics required for most of these applications are generating a large number of comb lines with a high degree of flatness together with good linewidth characteristics [5]. This is regardless of the other different comb characteristics needed depending on the application being targeted [6].

Generating OFCs using semiconductor laser is based on three main strategies; namely, traditional mode-locked lasers (MLLs) [7], electro-optic modulators (EOMs) [8] and gain switching (GS) laser [9,10,11,12,13,14]. Although traditional MLLs can generate high-stability OFCs, they are relatively complicated and costly [7], and do not provide a wavelength tunability due to the fixed comb line spacing, which is decided by the laser cavity length (or the inverse of the round trip time for light propagating in the cavity) [10]. Adjusting the comb line spacing can be obtained using external modulations [15]. In this case, multiple external modulators are required to improve the flatness of the comb, which results in increases the cost and insertion loss [15, 16]. Another method that has recently gained attention is frequency-modulated (FM) mode-locking, which is favorable for many single-section semiconductor lasers with fast gain dynamics [17,18,19,20,21]. In this method, the spatial hole burning effect initially enhances lasing on multiple longitudinal modes with non-equidistant spacing. The four-wave mixing effect can lock the phases of these modes together and equalize the spacing [17]. This approach is preferred for applications requiring high average optical power and quasi-continuous-wave intensity [19]. Although the wavelength portfolio of FM combs is expanding, there are several practical relevance wavelength ranges that have yet to be explored [19]. EOMs method for generating OFCs provide a relatively large number of optical comb lines with easily adjustable spacing, but they suffer from significant insertion loss from the modulators, and need extra optical components that increase the complexity and cost of the transmitter [6, 10]. On the other hand, compared with previous techniques, generating OFCs by GS technique based on direct modulation of semiconductor laser has advantages of simple implementation, low cost, stability, low losses, and flexibility in tuning the frequency spacing of the comb lines (frequency repetition rate) by varying the modulation frequency [6, 10, 11]. Therefore, such a technique has become a focus of research for more interesting applications, such as radio-over-fiber (RoF) technology [22], multi-wavelength sources [23] and multicarrier transmission systems [24]. Moreover, it has been proved as achievable dual-comb spectroscopy by OFCs generated by GS technique [25]. However, generating OFCs by GS technique suffers from a limited bandwidth and large phase noise [26].

Several studies were concerned with enhancing the characteristics of the gain-switched OFCs and improving its spectral quality using external optical injection method [12, 13, 27, 28]. However, this method increases the total cost and needs considerable care to avoid unstable operation induced by external effects like temperature, vibration, polarization, etc. In literature, to the best of our knowledge, only Anandarajah et al. [10] reported on possibility of generation of an OFC without need for external injection via gain switching. The study was concerned with modulation frequencies around the relaxation oscillation frequency of the laser. However, there were insufficient details on influence of the gain-switching conditions and intrinsic nonlinear properties of the laser on the generated OFC [10].

The GS technique based on direct modulation of semiconductor laser is typically accomplished by modulating the semiconductor laser by a sinusoidal RF current in conjunction directly with the bias current to switch on and off the laser gain [11, 13]. Semiconductor lasers modulated with such pulsed pump current, often exhibit damped oscillations (commonly referred to as relaxation oscillations) whose pulse width is significantly shorter than the electrical pulse. These oscillations result from the coupling between electrons and photons through stimulated emission [29, 30]. The idea then is to pick up the first intensity spike of these oscillations, at the beginning of the laser operation, without exciting the next ones by switching off the RF current period [31, 32]. Therefore, single pulses can be generated simply by modulating the semiconductor laser biased below or near the threshold with a large RF current amplitude (i.e., strong modulation depth) [31, 33], to ensure the signal clipping. Ahmed et al. [34] showed that modulating the laser under deep modulation with frequencies around the relaxation oscillation frequency induces pulsed signals with period-doubling.

The characteristics of OFCs generated by GS the laser diode mainly depend on both the amplitude and frequency of the RF current and the bias current [11, 13]. Another intrinsic property of the laser that may control OFCs is the nonlinear gain suppression because it affects the dynamic response of the laser [35]. Gain suppression was proved to suppress the carrier-photon resonance and increase the damping rate of relaxation oscillations and is hence affected to the generation of OFCs [36,37,38]. Nonlinear gain suppression may be attributed to different physical mechanisms, in particular, spectral hole burning [39], nonlinear absorption [40], carrier diffusion [41] and dynamic carrier heating effects [42, 43].

In this paper we introduce a comprehensive simulation study on the generation of OFCs induced by GS technique basing direct modulation of semiconductor laser by a sinusoidal current signal as a potential comb generation technique. We investigate the generation of OFCs and bring together details on their temporal and spectral characteristics. We present insight into influence of gain suppression on laser dynamics in terms of transient relaxation oscillations, corresponding phase portraits, and small-signal modulation response. Also, we examine impact of gain-switching conditions on the generated OFCs; namely, the bias current and the modulation depth at the important frequency of carrier-photon resonance. The results show that when the laser is biased near the threshold value, short optical pulses and large number of comb lines with low-power flatness are generated at low degree of gain suppression and/or strong modulation depth. Moreover, the influence of gain suppression is more robust when the laser is biased far above threshold value.

2 Theoretical model and simulation method

The dynamics of single-mode semiconductor laser including generation of OFCs are simulated basing on the following pair of rate equations of the photon number S(t) and injected electron number N(t) in a single field mode [44]

$$\frac{{{\text{d}}S}}{{{\text{d}}t}} = [G - G_{{{\text{th}}}} ]S + \frac{a\xi }{V}N$$
(1)
$$\frac{{{\text{d}}N}}{{{\text{d}}t}} = \frac{I(t)}{e} - AS - \frac{N}{{\tau_{s} }},$$
(2)

where G is the gain per unit time and described as [45]:

$$G = A - BS$$
(3)

The coefficient \(A = {{a\xi } \mathord{\left/ {\vphantom {{a\xi } V}} \right. \kern-\nulldelimiterspace} V}(N - N_{g} )\) is the linear gain term, while B is the coefficient of nonlinear gain suppression and is given by

$$B = \frac{9}{2}\frac{\pi c}{{\varepsilon_{o} n_{a}^{2} \hbar \lambda_{o} }}\left( {\frac{{\xi \tau_{{{\text{in}}}} }}{V}} \right)^{2} a\left| {R_{{{\text{cv}}}} } \right|^{2} (N - N_{s} )$$
(4)

Ng and Ns are the electron numbers at transparency and that character B, respectively. In Eq. (1), Gth is the threshold gain level and is given in terms of the material loss κ, power reflectivities of the front and back facets (Rf and Rb, respectively), and active region length L as [46]

$$G_{{{\text{th}}}} = \frac{c}{{n_{a} }}\left\{ {k + \frac{1}{2L}\ln \left( {\frac{1}{{R_{{\text{f}}} R_{{\text{b}}} }}} \right)} \right\} \approx \frac{a\xi }{V}\left( {N_{{{\text{th}}}} - N_{g} } \right),$$
(5)

where Nth is the electron number at the threshold level. The other parameters in the above equations are defined with their typical values in Table 1.

Table 1 Definitions and typical values of the FP-InGaAsP laser parameters used in calculations
Full size table

The direct modulation of SL is described in the rate equations by representing the injection current I(t) in Eq. (2) by the following sinusoidal form.

$$I(t) = I_{{\text{b}}} \left( {1 + m\sin (2\pi f_{{\text{m}}} t} \right),$$
(6)

where Ib is the bias current, m is the modulation depth and fm is the modulation frequency.

Rate Eqs. (1) and (2) are integrated numerically by the 4th-order Runge–Kutta algorithm [47]. The integration time step is set Δt = 5 ps and the integration is taken over a period of T = 300 ns. The time step is short enough to provide a good resolution of the temporal trajectories of S(t). Over a finite period T, the frequency content of the temporal trajectory of S(t) is determined by the fast Fourier transformation (FFT) as [48]

$$S_{f} = \frac{1}{T}\left| {\int\limits_{0}^{T} {S(t)e^{ - j2\pi f\tau } d\tau } } \right|^{2} \cong \sqrt {\frac{\Delta t}{T}} \left| {{\text{FFT}}(S(t))} \right|,$$
(7)

where f is the Fourier frequency. This frequency spectrum Sf is used to reveal the generated frequency components of the oscillating mode (fundamental, harmonic and sub-harmonic frequencies), which are the comb lines, and to determine their power flatness (power differences between comb lines).

The small-signal modulation response Hm(fm) at a given modulation frequency fm with a small sinusoidal perturbation ΔI = mIb around Ib is defines as the transfer function from input current modulation to optical output power [49]. Therefore, Hm(fm) is evaluated as the ratio of the modulated photon number Sm(fm) to the corresponding non-modulated value Sm(0) and is given by [50]

$$H\left( {f_{{\text{m}}} } \right) = \frac{{S_{{\text{m}}} (f_{{\text{m}}} )}}{{S_{{\text{m}}} (0)}} = \frac{{f_{{\text{r}}}^{2} }}{{\frac{j}{\pi }\Gamma_{{\text{r}}} f_{{\text{m}}} + f_{{\text{r}}}^{2} - f_{{\text{m}}}^{2} }},$$
(8)

where fr and Γr are the relaxation oscillation frequency and the average damping rate of the laser in the transient regime, respectively, and are determined approximately from the small-signal approach as [50]

$$f_{{\text{r}}} \approx \frac{1}{2\pi }\sqrt {\left( {\frac{a\xi }{V}} \right)\left[ {\frac{{a\xi \tau_{s} }}{eV}\left( {I_{{\text{b}}} - I_{{\text{g}}} } \right) + BS_{{\text{b}}} } \right]S_{{\text{b}}} } ,$$
(9)
$$\Gamma_{{\text{r}}} \approx \frac{1}{2}\left[ {\frac{a\xi }{V} + B} \right]S_{{\text{b}}} ,$$
(10)

where Ig = eNg/τs is the transparency current, and Sb is the bias component of S(t) and is determined by \(S_{{\text{b}}} = {{\left( {I_{{\text{b}}} - I_{{{\text{th}}}} } \right)} \mathord{\left/ {\vphantom {{\left( {I_{{\text{b}}} - I_{{{\text{th}}}} } \right)} {eG_{{{\text{th}}}} }}} \right. \kern-\nulldelimiterspace} {eG_{{{\text{th}}}} }}\) with Ith is the threshold current Ith = eNth/τs.

3 Results and discussion

3.1 Influence of gain suppression on laser dynamics

First, we examine the influence of nonlinear gain suppression on the transient characteristics of free running SL by varying the coefficient of gain suppression B relative to its initial value Bo = 18,885 C2m s−2 as calculated via Eq. (4) and using the parameters in Table 1. The modulation depth in this case is canceled (m = 0) and hence, the laser is injected only by the dc bias current Ib. Figure 1a, b show typical dynamics of the photon number S(t) during the laser transients under three different values of gain suppression; namely, without gain suppression (B = 0), with its initial value (B = Bo), and with high degree (3Bo), when the laser is biased near and far above the threshold value, Ib = 1.2Ith and 3Ith, respectively. Figure 1a indicates that the stimulated emissions are negligible up to 0.65 ns (turn-on delay times). The photon number S(t) and electron N(t) display the transient spikes through the known relaxation damped oscillations before they reach their steady-state values [37, 51]. The figure shows that the relaxation oscillation frequencies are nearly constant at fr ~ 2.7 GHz. The influence of gain suppression appears clearly in the damping rates Γr of relaxation oscillations; the increase in B works to strength the damping rates, which agree with the results in [50, 52, 53]. This may be attributed to the nonlinear dependence of the gain on the electron density as reported by Zhao et al. [53] when they derived expressions for the carrier distribution functions accounting for the spectral hole burning. This enhancement in the damping rates shrinks the time instants at which oscillations disappear (setting time of relaxation oscillations). As a numerical example, the oscillations disappear at 6 ns without gain suppression (B = 0), while they disappear at 3.5 and 2 ns when B increases to Bo and 3Bo, respectively. The decrease in these setting times is considered as a desired condition for enhancing the performance of optical communication systems that operate at high bit rates [51]. More details are illustrated in the inset of the Fig. 1a; it is seen that the increase in gain suppression induces a slight increase in the pulse width of the relaxation oscillations. For example, the full width half maximum (FWHM) of the first spike is about 70 ps when B = 0 and it broadens to 72 and 85 ps when B increases to Bo and 3Bo, respectively. These effects are seen clearly in the second spikes.

Fig. 1
figure 1

Transients of photon numbers S(t) with different gain suppression degrees of B = 0 (black lines), B = Bo (red lines) and B = 3Bo (blue lines), when bias current Ib is a 1.2Ith and b 3Ith

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When the bias current is increased to far above threshold (Ib = 3Ith), the turn-on delay time is reduced to 0.13 ns, whereas the relaxation oscillation frequency is still constant but increases to 8.9 GHz [see Eq. (9)] with shorter pulses as shown in Fig. 1b. The figure reveals also that the increase in B induces a significant increase in the damping rates. This significant increase in the damping rates at high bias current is in agreement with the findings of Zhao et al. [53] who attributed this to the increase in the nonlinearity of the gain-electron density dependence at high bias levels [also see Eq. (10)]. The inset of Fig. 1(b) illustrates also that the FWHM pulse width of the first spike is about 18.5 ps when B = 0 and is broadened to 24 and 37 ps when B is increased to Bo and 3Bo, respectively. Comparing Fig. 1a, b indicates that the influence of B is more robust at high bias currents, which agrees with the findings in Refs. [35, 38, 50]. Moreover, it can be found that the influence of B on the relaxation oscillation frequencies fr is almost negligible in both bias levels in consistence with the findings of Ahmed et al. [50] and Zhao et al. [53].

The transient characteristics of SL can also be understood by looking at the phase-space trajectory of the dynamic variables S(t) and N(t). Figure 2 plots the corresponding phase portrait of the transient characteristics of Fig. 1 to elucidate the effect of B on the correlations among electron and photon numbers. The direction of time evolution is indicated by the arrow in each figure. Figure 2a–f correspond to the condition of Ib = 1.2Ith and 3Ith, respectively. For each bias level, B is varied as in Fig. 1 (B = 0, Bo and 3Bo). The figures illustrate that all trajectories are often characterized by a spiral flattened motion towards a fixed point (a stable focus), which is the steady-state value. The intensity of oscillations reduces with the time variation, which is the reason why the phase-space trajectory is converged. The increase in B induces a decrease in the number of loops and the fixed points (focuses) tend to become nodes. This is due to the fact that the electron and photon numbers become more and more out of phase with the increase in B and induces an increase in the damping rates of relaxation oscillations. Our results related to the trajectory shape agree with those plotted by Lingnau et al. [54] when they investigated effect of electron scattering lifetimes on the damping rates of quantum dot lasers. Comparing Fig. 2a–f indicate that there is a decrease in the number of loops resulting from raising Ib from 1.2 to 3Ith, which is an indicator for increasing the damping rates as a function of Ib as confirmed by the results of Fig. 1.

Fig. 2
figure 2

Phase-space trajectory of (N-S) under different B degrees of 0, Bo and 3Bo, when a–c Ib = 1.2Ith (upper row) and d–f 3Ith (lower row)

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Figure 3a, b plot the small-signal intensity modulation responses |Hm(fm)| of semiconductor laser as a function of B when Ib = 1.2Ith and 3Ith, respectively. The modulation depth in this case is very week (m = 0.1). Regardless the influence of B, the figures exhibit clear response peaks at modulation frequency fm approximately equal to the relaxation oscillation frequency fr. This indicates a complete synchronized phase between the electron and photon numbers (resonance case). Figure 3a shows that the response peak appears at fm(peak) ≈ fr = 2.7 GHz when Ib = 1.2Ith, while it appears at fm(peak) ≈ fr = 8.9 GHz when Ib = 3Ith as seen in Fig. 3b. Increasing the gain suppression degree works to reduce the spectrum of |Hm(fm)|. The reduction in |Hm(fm)| is pronounced around the peaks and is more robust at high bias level (3Ith) than at low (1.2Ith), as seen in Fig. 3a, b, respectively. This confirms the fact that the influence of B on the laser characteristics is significant at high bias levels as mentioned in the above discussions. Figure 4 plots variation of the response peak as a function of B/Bo when Ib = 1.2 and 3Ith. The figure shows that when B increases from 0 to 3Bo, the response peak is slightly reduced with 0.24 dB (from 0.47 to 0.23 dB) when Ib = 1.2Ith, whereas it is significantly reduced with 0.6 dB when Ib = 3Ith. The figure indicates also that the influence of B is significant when it increases beyond Bo.

Fig.3
figure 3

Modulation response under different gain suppression degrees (B = 0, Bo, 2Bo and 3Bo) when a Ib = 1.2Ith and b Ib = 3Ith

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Fig.4
figure 4

Variation of the response peak as a function of gain suppression degree when Ib = 1.2Ith (black line) and 3Ith (red line)

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3.2 Influence of gain suppression on the generated OFC characteristics

In this subsection, we investigate in details the suggestion of generation of OFCs by large-signal direct modulation of the investigated semiconductor laser and dependence of their temporal and spectral characteristics on gain suppression. The cavity length of the used laser is constant and the modulation frequency fm is set to be the relaxation oscillation frequency fr because this high-frequency regime is rich with nonlinearities. This condition along with strong modulation depth ensures depletion of carrier density when the modulation current goes under the threshold current, which accumulates the lasing photons in the cavity and then emitted when the current goes above threshold (gain-switching of the laser) [11, 13]. The investigations are carried out under the two investigated bias currents of Ib = 1.2Ith and 3Ith.

Figure 5 displays evolution of the temporal and spectral characteristics of the laser outputs which look like OFCs generated under using three degrees of gain suppression of B = 0, Bo and 3Bo. The modulation parameters are set to be Ib = 1.2Ith and modulation frequency fm = fr = 2.7 GHz with strong modulation depth m = 1. These modulation parameters ensure clipping the signal with a large quantity of population pulsations [32] and hence induce period-doubling pulses [34]. Figure 5a, c, e (left-hand side column) display the temporal trajectories of the photon numbers S(t) normalized to its time-averaged value (Sav) (on the left axis), and the associated injected electron numbers N(t) normalized to its threshold value (Nth) (on the right axis). Figure 5b, d, f (right-hand side column) display the corresponding FFT power spectra Sf. Figure 5a shows that when B = 0, the laser emission appears as intense pulses, each one induces a significant drop in the electron population below the threshold value. Therefore, there is no sufficient population for the production of lasing pulse in the successive period. Thus, one optical pulse appears during two successive periods of the modulation current; a character of period-doubling pulsation. On the other hand, the FFT power spectra reveal several harmonic products as seen in Fig. 5b. The power differences between the fundamental signal (fm) and these products are commonly called distortions effects. The figure indicates that there are strong sub-harmonics at fm/2 appear together with the higher-order harmonics that appear at multiples of fm. As a result, 10 comb lines are generated with a power flatness of 2.9 dB and a spacing of 1.35 GHz (fm/2) with spectral width of 12.15 GHz within 2.9 dB of the spectral envelope peak, which in good agreement with the results reported in [11]. These harmonics and sub-harmonics result when the laser resonance is pulled towards this frequency [55, 56]. They may be attributed to lower laser damping rate obtained when gain suppression effect is ignored (B = 0). The relation between damping rate and distortion effects was investigated in the experiments of Hemery et al. [57] even at week modulation depth.

Fig. 5
figure 5

Temporal and spectral characteristics of laser output when Ib = 1.2Ith and m = 1 with fm = fr = 2.7 GHz when a, b B = 0, c, d Bo and e, f 3Bo. The left-hand side column (a, c, e) plots the temporal trajectories of S(t) normalized to average values (black lines on left axis), and the temporal trajectories of N(t) normalized to its threshold values (red lines on right axis). The right-hand side column (b, d, f) plots the corresponding FFT power spectrum Sf

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With the increase of gain suppression to B = Bo, the harmonic distortion is little released with the appearance of other weaker pulses in the repetition of the half-harmonic fr/2, as illustrated in Fig. 5c. That is non-equal two pulses repeats every two periods as a character of distorted period-doubling pulsation. The weaker pulse is generated due to a slight increase in the electron numbers over its threshold value as shown in the right axis of Fig. 5c. Figure 5d confirms the slight decrease in the distortion effects, which seems to be due to an increase in the damping rate. The laser still generates 10 comb lines with fm/2 spacing (1.35 GHz), but the power flatness is increased to 3.6 dB.

The further increase of gain suppression to 3Bo induces generation of period-1 pulsation, the pulses have the same height and are generated in repetition of the modulation frequency fm = fr, as shown in Fig. 5e. In this case, the number of electrons exceeds the threshold value and equals for each period as shown in the right axis of the figure. This condition is often a desired operation for optical communication systems [57]. The corresponding power spectrum is illustrated in Fig. 5f, which indicates that the strong degree of gain suppression results in releasing the power spectrum from sub-harmonics. The spectrum is characterized only with the higher harmonics at multiples of fm. As a result, only five comb lines are generated with a spacing of fm = 2.7 GHz, and power flatness as large as 3.85 dB. It is worth noting that the increase in B induces a decrease in the pulse height (photon number) and a significant increase in the pulse width due to the reduction in population pulsation. As a numerical example, the FWHM pulse width is 67 ps when B = 0 and spreads to 80 and 119 ps when B increases to Bo and 3Bo, respectively.

Figure 6 displays the evolution of the temporal and spectral characteristics of the OFCs generated at three degrees of gain suppression (B = 0, Bo and 3Bo) when Ib = 3Ith and fm = fr = 8.9 GHz with strong modulation depth m = 2.8. The increase in the bias current requires an increase in the modulation depth to ensure the signal clipping and hence gain-switching laser. The temporal trajectories of both S(t) and the associated N(t) are plotted in Fig. 6a, c, e (left-hand side column), whereas the corresponding FFT power spectra are plotted in Fig. 6b, d, f (right-hand side column). Although the temporal and spectral characteristics of the generated OFCs are almost similar to those of Fig. 5, the comb lines generated under these modulation conditions are with higher power flatness and at larger spaces, in addition to generated wider pulses. The frequency spacing of the comb lines (frequency repetition rate) therefore can be controlled by the modulation frequency fm; the frequency spacing increases by increasing the modulation frequency. Figure 6b displays 10 comb lines with a power flatness of 3.98 dB and a spacing of 4.45 GHz (fm/2) when B = 0. The maximum spectral width of the generated OFC, in this case, reaches 40 GHz within 3.98 dB of the spectral envelope peak, which in good agreement with the results reported in [11]. When B increases to Bo, the laser still generate 10 comb lines with a spacing of 4.45 GHz, but the power flatness increased to 4.75 dB as seen in Fig. 6d. At high degree of gain suppression (3Bo), the sub-harmonics disappear and only the harmonic distortions remain as seen in Fig. 6f. Therefore, there are five comb lines separated with fm = 8.9 GHz with higher power flatness of 4.88 dB. Regarding pulse widths, it is noticed a clear spread in the pulse widths with increase of B as illustrated in Fig. 6a, c, e. As a numerical example, the FWHM pulse width is about 85 ps at B = 0 and spreads to 141 and 146 ps when B increases to Bo and 3Bo, respectively.

Fig. 6
figure 6

Temporal and spectral characteristics of laser output when Ib = 3Ith and m = 2.8 with fm = fr = 8.9 GHz when a, b B = 0, c, d Bo and e, f 3Bo. The left-hand side column (a, c, e) plots the temporal trajectories of S(t) normalized to average values (black lines on left axis), and temporal trajectories of N(t) normalized to its threshold values (red lines on right axis). The right-hand side column (b, d, f) plots the corresponding FFT power spectrum Sf

Full size image

Influence of gain suppression on the signal distortion and the generated combs is investigated in more details in Fig. 7a, b, which plot the number of the generated comb lines and their power flatness under three modulation depths for gain switching of m = 0.5, 1 and 2.8 when Ib = 1.2 and 3Ith, respectively. The number of comb lines is located below each point in the figures. Both figures indicate increase of power flatness with the increase of B. This power flatness decreases with the increase of the modulation depth. In Fig. 7a, when B increases from 0 to 3Bo the number of comb lines is ten and decreases to 4 when m = 0.5 and to 5 when m = 1.0. On the other hand the case of Ib = 3Ith in Fig. 7b, the number is almost unchanged with the increase of B. That is, it could be concluded in general that larger number of comb lines with lower-power flatness is generated at lower degree of gain suppression and/or stronger modulation depth m. Comparing Fig. 7a, b reveals that when the laser is biased at Ib = 1.2Ith, the power flatness is lower and more acceptable (< 5 dB) than when biased at Ib = 3Ith. Moreover, the influence of gain suppression on the power flatness is weaker at Ib = 1.2Ith than at 3Ith. As a numerical example, when B increases from 0 to 3Bo at m = 1, the power flatness increases from 2.93 to 3.65 dB when Ib = 1.2Ith, whereas it increases from 4.7 to 13 dB when Ib = 3Ith. On the other hand, it can be noticed that the number of comb lines increases significantly when m increases from 1 to 2.8 at Ib = 3Ith (large enough to avoid the total demise of stimulated emissions), which agree with the explanations reported in [11]. It can be concluded from the above discussions that the optimum conditions of generating OFCs (10 comb lines with power flatness of 2.1 dB) are at B = 0 and m = 2.8 when Ib = 1.2Ith.

Fig. 7
figure 7

Variation of numbers and power flatness of the generated comb lines as a function of B normalized to Bo under three distinct modulation depths of m = 0.5 (black line), 1 (red line) and 2.8 (blue line) when a Ib = 1.2 and b 3Ith

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Comprehensive details on influence of gain suppression on the width of the generated pulses are illustrated in Fig. 8a, b using the same parameters of Fig. 7a, b, respectively. The figures indicate that there is a pronounced broadening of the generated pulse width associated with increase of B and/or a decrease of m. Comparing Fig. 8a, b reveals that the pulses generated when Ib = 1.2Ith are narrower than those generated when Ib = 3Ith. Moreover, the influence of gain suppression on the generated pulse width when Ib = 1.2Ith is weaker than on those generated when Ib = 3Ith. As a numerical example, when B increases from 0 to 3Bo at m = 1, the FWHM of the pulses increases from 67 to 119 ps (increase of 52 ps) at Ib = 1.2Ith, whereas it increases from 169 to 291 ps (increase of 122 ps) when Ib rises to 3Ith. This effect is more significant under week modulation depth of m = 0.5 (low populations) than under strong depth of m = 2.8 (high populations). The idea of broadening the emitted pulses with reduction of population pulsation agrees with the experimental results in [32].

Fig. 8
figure 8

Variation of the generated pule width as a function of B under three distinct modulation depths of m = 0.5 (black line), 1 (red line) and 2.8 (blue line) when a Ib = 1.2 and b 3Ith

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4 Conclusions

We showed that it is possible to produce a route to OFCs of semiconductor laser by gain-switching method using the cost-effective technique of direct sinusoidal current modulation of semiconductor laser without need for external optical injection. We examined in details influence of nonlinear gain suppression on the temporal and frequency characteristics of the generated pulses and OFCs. The investigations were in terms of the relaxation oscillations, phase portrait, and small-signal modulation response for the non-modulated laser, and the temporal trajectories of electron and photon numbers and the FFT spectra for the generated OFCs. Based on the results obtained in the present study, conclusions can be itemized as follows:

  1. (1)

    The damping rate of the laser relaxation oscillations increases with the increase of gain suppression coefficient B and/or Ib.

  2. (2)

    The modulation response is reduced with the increase of B especially around the response peak. This effect is more robust at higher biasing current; when B increases from 0 to 3Bo, the reduction is 0.6 dB when Ib = 3Ith, whereas it is 0.24 dB when Ib = 1.2Ith.

  3. (3)

    The optimum parameters to generate a large number of comb lines with low-power flatness are low gain suppression along with stronger modulation depth m when the laser is biased near the threshold. Ten comb lines with power flatness of 2.1 dB are predicted at B = 0 and m = 2.8 when Ib = 1.2Ith. Increasing B to 3Bo and decreasing m to 0.5 results in significant reduction in the number of comb lines to 5 with a higher flatness of 4.99 dB. This effect is more significant when Ib = 3Ith with high ranges of power flatness.

  4. (4)

    The frequency spacing between comb lines (frequency repetition rate) increases by increasing the modulation frequency; the comb spacing increases from 1.35 to 4.45 GHz when the modulation frequency increases from 2.7 to 8.9 GHz, respectively.

  5. (5)

    Generation of short pulses results when gain suppression is low and/or the modulation depth is strong when the laser is biased near the threshold. Pulses with 60 ps width are generated at B = 0 and m = 2.8 when Ib = 1.2Ith. Increasing B to 3Bo and decreasing m to 0.5 induce a significant broadening of pulses to 132 ps. This effect is more significant when Ib = 3Ith with higher widsth.