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Abstract
In this work we perform a theoretical and simulation analysis of the behavior of an integrated four section distributed Bragg reflector semiconductor laser under optical injection and Q-switching operation. An electro-absorption modulator is introduced into the laser cavity to control the total losses and perform Q-switching. The simulations are done using a rate equation model. Q-switching operation produces very short and high power pulses. This, together with the use of optical injection, allows obtaining flat and broad optical frequency combs with up to 2100 optical lines within 10 dB (642 lines within 3 dB) at a repetition frequency of 100 MHz. The high chirp of the pulses is responsible for the broad spectra of these combs in comparison with gain switched combs, and the device structure allows fabrication in commercial foundries using standard building blocks.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical frequency combs (OFCs) are coherent light sources consisting of discrete equally spaced lines [1–3]. They have applications in multiple fields, including frequency metrology [4], high-speed optical communications [5,6], optical arbitrary waveform generation [7], ranging [8,9], spectroscopy [10,11] and remote gas-sensing. Taking advantage of the usual benefits of semiconductor sources (high efficiency, low cost, and small footprint), OFCs have been generated from semiconductor lasers by different techniques, the most common being mode-locking [12], micro-ring resonators [13,14], electro-optical modulation (EOM) [7–11] and gain-switching (GS) [15–22]. In the first two cases, the repetition frequency is provided by the cavity and cannot be tuned (only slightly), but they provide very wide combs (up to several nm) with repetition frequencies in the GHz range. In the case of EOM, an individual or a series of intensity and/or phase EOMs are used to achieve flat and broad combs at repetition frequencies in the MHz or GHz ranges. GS is a very simple and versatile technique based on the direct modulation of the laser by a combination of a bias and an alternating current that switches the laser emission on and off, producing short pulses and high quality OFCs when the phase coherence between consecutive pulses is maintained, either by avoiding the complete extinction of the pulse, by applying an external optical injection (OI) that provides the phase reference [16,17] or by using an optical feedback scheme [20]. This technique provides flat combs with typical maximum widths around 100 GHz at repetition frequencies in the GHz range [18], and slightly wider when driving the laser at 100 MHz with short electrical pulses [19,22]. Recent theoretical investigations predict much wider OFCs using gain-switched semiconductor nanolasers [21]. High repetition frequencies (GHz range) are especially suited for telecom applications, while low repetition frequencies (around 100 MHz) are used for spectroscopic and ranging applications.
An additional advantage of the OFCs generated by semiconductor lasers is that they can be fabricated as a component of a photonic integrated circuit (PIC) [23], incorporating additional functions and even enabling dual comb systems [24].
Q-switching is a very well-known technique for producing high peak power pulses from solid-state or fiber lasers [25]. It consists of actively or passively modulating the losses of the laser resonator, in such a way that the pump energy accumulated during the off time (high losses) is released with high peak power during a short on time (low losses). Q-switching has been applied to generate short optical pulses from semiconductor lasers since the beginning of laser diode research [26]. The technique as well as the main theoretical and experimental results during the 80s and 90s were reviewed in [27]. The goal was either to obtain short pulses with high peak power, or to modulate the laser for telecom applications. He et al [28] proposed Q-modulation as an advantageous alternative to direct modulation and the same group studied the Q-modulation in detail with a rate-equation model (REM) [29], but as far as we know, this modulation approach was not experimentally demonstrated later.
In this work we propose active pulsed Q-switching with OI as a technique to generate broad OFCs at low repetition frequencies. As we will analyze using detailed simulations this procedure provides very broad OFC due to the short and strongly chirped pulses emitted during the on-time. This technique is similar to the generation of OFCs by pulsed GS and OI, with the advantage of providing shorter pulses and wider OFCs. We analyze the behavior of a distributed Bragg reflector (DBR) laser with an intracavity electro-absorption modulator (EAM) and we select realistic design parameters based on PICs fabricated in open access platforms. The role of the driving parameters in the quality of the OFCs generated at 100 MHz is discussed, and we predict very broad combs (up to 200 GHz) by selecting the best driving conditions.
This paper is organized as follows: section 2 covers the fundamentals of the generation of short optical pulses and the corresponding OFCs by Q-switching semiconductor lasers; section 3 details the rate equation model used to perform the simulations, as well as the physical parameters of the laser considered in the simulations; section 4 explains the principles followed for the laser design in order to achieve the generation of broad OFCs; section 5 describes and discusses the results and finally the main conclusions are summarized in section 6.
2. Fundamentals
The fundamentals of this technique can be understood with the help of Fig. 1. We have considered a DBR laser integrated with an intracavity EAM in single frequency operation. Short voltage pulses are applied to the EAM, changing its optical losses and making it nearly transparent (with low losses) for short periods of time with duration and absorbing (high losses) for the rest of the repetition period ${T_r}$. This results in a relatively high value of the photon lifetime ${\tau _{ph}}$ during the on-periods and a low value during the off-periods, as it is schematically shown in Fig. 1 (a). The bias current is kept constant at a value below the threshold current corresponding to high losses in the EAM (${I_{thOFF}}$), but well above the threshold current with low losses (${I_{thON}}$). During the off-time, since there is no (or very little) stimulated recombination, there is a high accumulation of carriers, reaching a steady state carrier density much higher than the threshold carrier density under low loss conditions, marked in Fig. 1 (b). When the losses are switched from a high to a low value (low ${\tau _{ph}}$ to high ${\tau _{ph}}$), at the point marked with A in Fig. 1, the gain is much higher than the losses, and the number of photons in the cavity increases very rapidly with a time constant related to ${\tau _{ph}}$, as can be seen in Fig. 1(c)). The increasing number of photons reduces the carrier concentration, so that if the pulse were long enough, relaxation oscillations would be produced, as will be illustrated later. However, if the losses are switched back to a high value (low ${\tau _{ph}}$) before the second relaxation oscillation spike occurs, since the gain is lower than the losses, the number of photons decreases drastically, and a short and powerful optical pulse is generated. Along the off-time, the carrier density increases according to the carrier lifetime, building up a high carrier density again, until the next switching of the losses. If the laser emission were completely extinguished during the period between successive pulses, the next pulse would be generated from spontaneous emission, and then the phase correlation would be lost, resulting in a noisy optical spectrum, similar to what has been observed in GS lasers [16,17]. However, if the laser is externally injected by a Master Laser (ML) the coherence between pulses will be preserved, giving rise to an OFC, as illustrated in the inset of Fig. 1(c).
Fig. 1. Principle of pulse generation in semiconductor lasers by Q-switching. Evolution of photon lifetime (a), carrier density (b) and photon density (c) as a function of time. The losses of the cavity are modulated through the EAM between a low value during a short time and a high value during the rest of the repetition period. The laser is driven with a constant bias below the threshold current in the high losses condition. When the losses are switched to a low value, the accumulated carriers rapidly recombine by stimulated emission giving rise to a short optical pulse.
The active Q-switching process in a semiconductor laser is very similar to the well-known GS technique, but there are some important differences that should be taken into account. In GS, the optical pulse begins to grow when the gain is slightly higher than the losses, whereas in Q-switching the gain suddenly becomes much higher than the losses, giving rise to a faster pulse growth and consequently to a shorter pulse. In addition, the fast growth of the pulse leads to a fast depletion of carriers, which induces a large index change and a large dynamic chirp during the optical emission, and this in turn results in a broad OFC. However, this advantage requires a high initial carrier concentration, and this implies a relatively long time between pulses, which is limited by the relatively long carrier lifetime. Then, OFC generation by Q-switching is appropriate for low repetition frequencies (up to hundreds of MHz), but not for the GHz range, as there would be no time between pulses to accumulate a high carrier concentration. It is also clear that switching using electrical pulses is more appropriate than using sinusoidal signals, similar to what happens in GS at low repetition frequencies [19].
3. Simulation model
The generation of OFCs by Q-switching under optical injection has been simulated by a rate equation model similar to that described in [17] for GS. This model, together with the model parameters extracted from experimental characterization of the relative intensity noise and power-current characteristics, provided excellent agreement between simulations and OFCs generated by GS with sinusoidal excitation in a wide range of operating conditions [17]. Here we explain the main features of the model, especially those changes introduced to simulate the Q-switching, while further details can be found in [17]. The following set of equations is used to describe the dynamics of the photon density (S(t)), carrier density (N(t)) and optical phase (Φ(t)):
Equations (1)–(5) were solved using a numerical Heun’s algorithm for stochastic differential equation (SDE) systems implemented in Matlab. The simulation parameters related to the material properties are those reported in [17], that were extracted from experimental measurements in a high-speed 1.5 µm laser, and are summarized in Table 1. Other parameters related to the geometry and losses were defined following a trade-off design explained in the next section.
Table 1. Material parameters
The quality of the OFCs was characterized by means of the 10 dB spectral width (δf10), the carrier-to-noise ratio (CNR) as defined in [16], and the spectral flatness. This parameter, defined according to [31], is obtained by calculating the ratio between the geometric mean and the arithmetic mean of the intensity of all the tones of the comb within δf10. It provides a figure of merit for the uniformity of the line intensities.
4. Laser design
We have selected for the design a four section DBR laser similar to those fabricated in open access foundries [32,33] but with an additional intracavity EAM section. It consists of a rear DBR, a semiconductor optical amplifier (SOA), the EAM section, and a front DBR with lengths ${L_{RDBR}}$, ${L_{SOA}}$, ${L_{EAM}}$ and ${L_{FDBR}}$, respectively, as shown in Fig. 2. Making use of the effective length approximation [34] the laser can be analyzed as a Fabry-Perot laser with two mirrors of effective reflectivities $R_F^{\prime}$ and $R_R^{\prime}$ placed at effective lengths ${L_{effR}}$ and ${L_{effF}}$, respectively. Then the total cavity length ${L_T}$ is given by the addition of the SOA and EAM section length, the two DBR effective lengths and three times the length of the electrical isolation region (${L_{ISO}}$) between sections. A simple analysis of the round-trip condition shows that the total cavity loss coefficient ${\alpha _T}$ can be expressed as:
Fig. 2. Schematic of the designed four-section laser. RDBR: Rear DBR; ISO: Electrical isolation region; SOA: Semiconductor Optical Amplifier; EAM: Electro-Absorption Modulator; FDBR: Front DBR.
We have used a standard value for the coupling coefficient of the DBRs, $\kappa = 50\; \textrm{c}{\textrm{m}^{ - 1}}$, and a value similar to that of a previous experimental design [35] for the SOA length, ${L_{SOA}} = 500\;\mathrm{\mu}\textrm{m}$. The estimated active volume for this SOA is $7,65\cdot {10^{ - 18}}\; {\textrm{m}^3}$. We have also used standard values for the internal loss coefficient of the active and passive sections, ${\alpha _{in}} = 25\; \textrm{c}{\textrm{m}^{ - 1}}$ [36], the insertion losses, ${T_{ON}} ={-} \; 2.3\; \textrm{dB}$, and the extinction ratio of the EAMs, $ER = 35\; \textrm{dB}$ for an EAM length of 250 µm [37]. Then, our design degrees of freedom were the DBR lengths, which were optimized to provide a low value of m together with a low ${I_{thON}}$. The selected lengths were ${L_{RDBR}} = 250\;\mathrm{\mu}\textrm{m}$ and ${L_{FDBR}} = 100\;\mathrm{\mu}\textrm{m}$. The resulting design has a total cavity length of 971 $\mathrm{\mu}\textrm{m}$ (taking the isolation section length as $30\;\mathrm{\mu}\textrm{m}$), ${\tau _{phON}} = 3.46\; \textrm{ps}$, $m = 0.29$., and ${I_{thON}}$ = 9.2 mA.
Figure 3 illustrates the dependence of ${I_{thOFF}}$ calculated by solving expressions (1) and (2) without optical injection in the steady state, using previous parameters and changing ${\tau _{ph}}$ to the correspondent value for the given m. For m = 0.29 (ER = 35 dB), it is possible to use a bias current up to 130 mA without reaching ${I_{thOFF}}$, thus maximizing the carrier density in the off-state.
Fig. 3. Threshold current in the OFF-state ${I_{thOFF}}$ as a function of the parameter $m = {\raise0.7ex\hbox{${{\tau _{phOFF}}}$} \!\mathord{/ {\vphantom {{{\tau_{phOFF}}} {{\tau_{phON}}}}}}\!\lower0.7ex\hbox{${{\tau _{phON}}}$}}$.
5. Results and discussion
We have simulated the OFCs generated by Q-switching the previously described laser at a repetition frequency of 100 MHz. The simulation time step has been set to 34.6 fs. The optical injection frequency has been adjusted in each case to ensure injection locking and thus a good quality of the OFCs. The injected power has been set to $5\; \textrm{dBm}$ with a coupling coefficient of ${k_c} = 3.5\cdot {10^{10}}\; {s^{ - 1}}$, which makes an effective injected power of ${\textrm{P}_{\textrm{inj}}} = 4.62\; \textrm{dBm}$, although the dependence of the injection conditions on the OFC characteristics has not been analyzed in detail. The optical spectra is obtained by applying a Fast Fourier Transform (FFT) to the simulated complex optical field. Figure 4 shows the simulated OFCs for different bias currents and a pulse width ${w_{pulse}} = 30\; \textrm{ps}$. This width has been selected because it provides the widest OFCs, as it will be explained later. For a low bias current ${I_{bias}} = 20\; \textrm{mA}$, an OFC with $\delta {f_{10}} = 70\; \textrm{GHz}$, CNR = 27 dB and a flatness value of 0.8 is obtained (Fig. 4(a)). The strong peak comes from the optical injection, and the reference frequency is that of the laser emission ${I_{bias}} = {I_{thON}}$. As ${I_{bias}}\; $ is increased (Fig. 4(b)-(d)), the width of the combs increases while the CNR and flatness remain similar. The widest comb is obtained at the highest bias, 120 mA (Fig. 4(d)). For this OFC, $\delta {f_{10}}$ = 202 GHz, CNR = 27 dB and the flatness is 0.77, and the optical power per line is -29.82 dBm. The inset of Fig. 4(d)) shows a zoomed spectrum for this case, where the comb lines, separated by the repetition frequency (100 MHz), are clearly visible.
Fig. 4. Simulated optical spectra for ${w_{pulse}} = 30\; ps$ and different bias and detuning (see text for other operation conditions): (a) ${I_{bias}} = $ 20 mA, $\delta \upsilon = $ 40 GHz; (b) ${I_{bias}} = $ 50 mA, $\delta \upsilon = $ 90 GHz; (c) ${I_{bias}} = $ 80 mA, $\delta \upsilon = $ 140 GHz; (d) ${I_{bias}} = $ 120 mA, $\delta \upsilon = $ 180 GHz, including a zoomed region on the right.
The evolution of the comb width $\delta {f_{10}}$ as a function of the bias current has been plotted in Fig. 5, where an almost linear dependence can be seen. The reasons for this increase can be well understood with the help of Fig. 1(b)), where the carrier dynamics has been plotted: the higher ${I_{bias}}$, the higher the value of the carrier density at the end of the off-state, and consequently, the higher the net gain (modal gain minus losses) at the moment of switching. During the on-state, the higher the net gain, the faster the growth of the photon density and consequently, the decrease of the carrier density. This results in a large carrier variation during the pulse and therefore a highly chirped pulse. This high chirp is the origin of a broader optical spectrum compared to the case of GS, where the carrier variation during the pulse is not as abrupt.
Fig. 5. OFC width $\delta {f_{10}}$ as a function of bias for ${w_{pulse}} = 30\; ps$ (see text for other operation conditions).
Further understanding of the characteristics of the OFCs generated by Q-switching can be gained by comparing the pulses and OFCs with those generated by GS under similar operating conditions. We have solved the rate equations for GS with pulsed electrical injection, using the same laser parameters and similar driving conditions, with the only exception of the pulse width which has to be wider in GS (we used 100 ps instead of 30 ps). These equations are those of section 3 but with variable current and constant losses. Figure 6(a)) illustrates the evolution of the pulse intensity and chirp for the case of the OFC of Fig. 4(d)), in comparison with the pulse intensity and chirp obtained by GS (Fig. 6(b)). The Q-switching pulse grows much faster than the GS pulse and reaches a much higher peak power. In addition, the chirp variation during the pulse is greater in the case of Q-switching, and this is the origin of the broader combs. The spectrum corresponding to the GS case is plotted in Fig. 6(c)); it features a $\delta {f_{10}}$ around 94 GHz, a CNR around 27 dB and a flatness of 0.77. The detuning in this case is $\delta \nu = $ 71 GHz. A comparison of this spectrum with the corresponding Q-switching OFC (Fig. 4(d)) shows that the Q-switching OFC is much broader at 10 dB, and even more at 20 dB (202 GHz vs 113 GHz), but the flatness is better for the GS comb. Some oscillations at the high frequency side of the comb envelope are clearly seen in the Q-switching case, but not in the GS comb. We attribute these oscillations to the fast decay of the optical pulse in the case of Q-switching (see Fig. 6(a)), which resembles a rectangular pulse whose Fourier transform is characterized by lobes, the frequency spacing of which is inversely related to the pulse width.
Fig. 6. Comparison between simulated optical pulses and chirp generated by Q-switching (a) and Gain-switching (b) using similar operating conditions. (c) Simulated OFC corresponding to (b).
The evolution of the OFCs as a function of the pulse width is shown in Fig. 7, where we have plotted the optical spectra at the maximum bias of 120 mA for ${w_{pulse}}$ = 10, 50, 80 and 100 ps and $\delta \upsilon = $ 180 GHz. The corresponding evolution of $\delta {f_{10}}$ is shown in Fig. 8, where a maximum at 30 ps and a linear decrease of $\delta {f_{10}}$ with increasing ${w_{pulse}}$ can be observed. The average frequency spacing between the previously commented lobes on the high frequency side of the spectra has also been plotted in Fig. 8. A clear decrease of the lobe frequency spacing with the pulse width can be seen, supporting their previously commented origin. The changes in the shape of the spectra and the evolution of $\delta {f_{10}}$ can be well understood with the help of Fig. 9, where the temporal evolution of the pulse power and the carrier density has been plotted for the case of maximum width (100 ps). It is clear that when the excitation pulse width is 10 ps the optical pulse has no time to reach its maximum, and then the width is limited to 150 GHz. However, for ${w_{pulse}}$ = 30 ps the width is maximum (Fig. 4(a)), as this value corresponds to the minimum of the carrier density during the first relaxation oscillation. Longer excitation pulses result in narrower spectra with less flatness due to the appearance of the lobes in the spectral envelope.
Fig. 7. Simulated optical spectra for ${I_{bias}} = 120\; mA$ and different ${w_{pulse}}$ (see text for other operation conditions): (a) 10 ps; (b) 50 ps; (c) 80 ps; (d) 100 ps.
Fig. 8. OFC width $\delta {f_{10}}$ and average frequency distance between lobes flobes as a function of the pulse width for ${I_{bias}} = 120\; mA$.
6. Conclusion
We propose the use of Q-switching as a technique to generate flat and broad OFCs from semiconductor lasers. This technique can be easily applied using a modulation element such as an EAM inside the cavity of a single-frequency laser. We have demonstrated the feasibility of the technique through numerical simulations using a rate equation model. We have designed a realistic DBR laser using typical parameters of lasers and EAMs fabricated in open access foundries at 1550 nm. An optimized design of this laser enables the generation of broad OFCs under Q-switching operation with external optical injection. We have analyzed the quality of the OFCs as a function of the operating conditions at a repetition frequency of 100 MHz in terms of optical width, CNR and flatness. We have shown that the optical width of the OFCs increases with the bias current due to the accumulation of carriers in the off-state, which is responsible for a very fast transition to the on-state and a strong chirp during the pulse emission. A maximum width of 200 GHz is achieved at the maximum bias current avoiding lasing with the EAM in the off-state. This maximum OFC width is achieved by using 30 ps pulses to drive the EAM in the on-state, as shorter driving pulses do not allow the maximum peak power to be reached and longer pulses excite the second spike of the relaxation oscillations and decrease the OFC width.
In summary, this technique is shown to be a simple and promising method for obtaining broad and flat OFCs from semiconductor lasers at frequencies in the tens to hundreds of MHz range, which can be applied to high resolution spectroscopy and distance measurements.
Funding
Ministerio de Ciencia e Innovación (PID2021-1234590B-C21, TED2021-131957B-100).
Acknowledgments
This work has been supported by the grants TED2021-131957B-100 and PID2021-1234590B-C21 funded by MCIN/ AEI /10.13039/501100011033 and by the European Union NextGenerationEU/PRTR and ERDF “A way of making Europe”.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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