Recently discovered laser frequency combs represent a novel versatile tool with a wide range of applications such as high-capacity optical communications [1], RF photonics [2], optical metrology [3], and broadband molecular spectroscopy [4,5]. Among them, the dual frequency comb (DFC) spectroscopy is seen as an emerging spectroscopic tool providing high-resolution and high-sensitivity broadband spectroscopy [6]. The state-of-the-art approaches for DFC generation usually rely on femtosecond erbium-doped fiber lasers [7], mode-locked erbium fiber ring lasers [8], dissipative Kerr solitons in optical microresonators [9], parametric processes in nonlinear crystals [10], or architectures involving one or several electro-optic modulators [11–14]. However, such schemes are usually power consuming and/or not suitable for photonic integration, and although providing significant capabilities, the performance, complexity, and cost of these setups usually exceed the requirements of most practical applications [14]. Simpler schemes in which electro-optic modulators are replaced with gain-switched (GS) laser diodes (LDs) have also been proposed [15,16]. Solutions relying on fiber lasers can provide DFCs with extremely high comb spans (CSs) with power per teeth pair(s) (t.p.) of the order of 20 pW [17] up to 4 nW [18]. On the other hand, capabilities of GS LD high-resolution DFC spectroscopy have been proven on much lower spans, of the order of 35 GHz [16] or 70 GHz, with power per line of the order of 100 nW [15]. Schemes proposed in [15,16] generate DFCs via mixing of two separate combs obtained by two independent GS LDs. The complexity and power consumption of the schemes is increased by introducing the third, master laser, which provides mutual coherence of the generated combs by injection-locking two slave LDs. Both schemes include an acousto-optic modulator, making them difficult for photonic integration.

In this paper, we theoretically investigate a simple scheme for DFC generation based on one LD, by designing the adequate temporal form of the driving current pulse for the LD large signal modulation. Apart from providing simplicity and power efficiency by reducing the number of active components, such a scheme could provide easy photonic integration and comb line mutual coherence.

To fully describe the LD dynamics in the regime of large signal modulation, we employ an extended rate equation model accounting for the carrier transport and parasitic effects [19]. This model provides more realistic small and large signal modulation response of the LD, and it can account for the nonlinear optical phase temporal response caused by strong current modulation, which is essential for the shaping of the comb profile [19]. The model describes the dynamics of a multiple quantum well distributed feedback (DFB) laser and comprises four differential equations following the dynamics of the carrier density in the barrier states (${n _{\rm b}}$), carrier density in the bound states of the well region (${n _{\rm w}}$), photon density ($S$), and the evolution of the optical phase ($\phi$):

In the equations above, ${\eta _{{\rm inj}}}$ is the injection efficiency, while $I(t)$ stands for the driving electrical current entering the LD. ${V_{{\rm tot}}}$ is the total volume of the separate confinement heterostructure (SCH) and the active region, and ${V _{\rm w}}$ is the volume of the wells. ${\tau _{\rm b}}$ and ${\tau _{\rm w}}$ are the carrier recombination lifetimes in the barrier and well region, respectively. The effective carrier diffusion across the SCH region and capture time by the wells is modeled by the capture time ${\tau _{{\rm {bw}}}}$, while the escape time ${\tau _{{\rm wb}}}$ is the thermionic emission and carrier diffusion time from the well to the barrier states. Furthermore, parameter ${v_{\rm g}}$ stands for the group velocity of light, $\Omega$ for differential gain, ${n_0}$ for transparency carrier density, $\Gamma$ for optical confinement factor, ${\tau _{\rm p}}$ for photon lifetime, $\varepsilon$ for nonlinear gain suppression coefficient, and ${R_{{\rm sp}}}$ for spontaneous emission rate. Finally, $\alpha$ stands for the linewidth enhancement factor. Parameters of the model are fitted to the Gooch & Housego AA0701 series, 1.55 µm high modulation bandwidth DFB LD with values given in [19]. Although inclusion of the stochastic terms in the rate equation system provides a more general and sophisticated model of GS dynamics [20], we neglect the noise terms due to the fact that we exploit large signal modulation, which keeps the laser above the threshold level, suppressing the contribution of the spontaneous and carrier noise.

The LD is driven by the electrical current $I(t)$ in the form

The output optical frequency comb is obtained after solving the rate equation system (1–4) for the given electrical current waveform and taking the Fourier transform of the LD complex electrical field, expressed as $E(t) = \sqrt {S(t)} {e^{i\phi (t)}}$ [19]. Equations are solved by implicit backward differentiation formulas of the fourth order, adapting the time step size so that the estimated error in the solution is within allowed ${10^{- 8}}$ relative error. The time span of the simulation is set to cover around 2000 electrical current pulses, while computational time is of the order of 60 s. Finally, the output optical power $P$ is $P(t) = {\eta _0}{V _{\rm w}} \hbar {\omega _0}S(t)/({\tau _{\rm p}}\Gamma)$ [19], where ${\eta _0}$ stands for the optical efficiency and ${\omega _0}$ for the lasing central angular frequency.

We investigate DFCs generated directly from a single LD, utilizing two current waveforms, and analyze the influence of electrical current parameters and linewidth enhancement factor $\alpha$ on the shape, span, and optical power of the output combs. For both current waveforms, resulting DFCs are composed of two combs with repetition rates corresponding to ${f_1}$ and ${f_2}$. In our simulations, ${f_2} = {f_1} + \delta f$, where $\delta f \ll {f_1}$. Two combs overlap at the fundamental lasing frequency (central line), while series of t.p. (heterodynes), with each t.p. composed of two comb lines, appear on both sides of the central line. The first line of each t.p., measured from the central line in the direction of both positive and negative relative frequencies, appears at multiples of ${f_1}$, while the second line of each t.p. appears at multiples of ${f_2}$. In the case of ${\rm sinc}$-shaped pulses, lines appear at even multiples of ${f_1}$ and ${f_2}$, while in the case of ${{\rm sinc}^2}$-shaped pulses, lines appear at integer multiples of the two frequencies. We define the position of each t.p., with respect to the central line, as the relative frequency corresponding to the frequency in the middle of a pair, meaning that t.p. will appear at even or integer multiples of ${\pm}({f_1} + {f_2})/2$, for ${\rm sinc}$ and ${{\rm sinc}^2}$ pulses, respectively. In such an arrangement, frequency separation of two lines in a pair (heterodyne frequency) increases as the relative position of a t.p. from the laser dominant mode increases. Heterodyne frequencies correspond to even or integer multiples of $\delta f$, with each heterodyne appearing on both sides of the dominant mode.

In the case of $p = 1$ [Eq. (5)], we set modulation frequencies to ${f_1} = 100\;{\rm MHz}$ and ${f_2} = {f_1} + \delta f = 100 + 0.25\;{\rm MHz}$. The resulting output DFC consists of series of t.p. appearing at ${\pm}2n \times ({f_1} + {f_2})/2 = \pm n \times 200\;{\rm MHz}$, relative to the central lasing frequency, where $n$ denotes the ordinal of the t.p. with respect to the central comb line, i.e., t.p. appear at ${\pm}200,400,600,\ldots\;{\rm MHz}$. Heterodyne frequency of the $n$th t.p. is equal to $2n \times ({f_2} - {f_1}) = 2n \times \delta f = n \times 0.5\;{\rm MHz}$. In our simulations, we use odd values of $N$, since in this case, the current waveform consists of positive current spikes providing better shaped DFCs.

In Fig. 1, we show DFCs and corresponding comb envelopes, representing the loci of the comb lines maxima. They are obtained for $\alpha = 3$, two values of bias current ${I_{\rm b}}$, modulation amplitude $\Delta I$, and $N$, providing for two different electrical current bandwidths $B = {{Nf}_2}$. We define the CS as the frequency region comprising t.p., or DFC envelope, whose optical power lies in the range of 20 dB below maximal power, excluding the power of the central mode. In Fig. 1(a), we show DFC and corresponding envelope (solid line) for moderate bias current (${I_{\rm b}} = 6{I_{{\rm th}}} \approx 66\;{\rm mA}$), small modulation amplitude ($\Delta I = 2{I_{{\rm th}}}$), and $N = 201$, i.e., electrical current bandwidth of $B \approx 20\;{\rm GHz}$. The insets provide a clearer visualization of the comb lines arrangements. The right inset shows a zoom of the 51st and 52nd t.p. on the positive spectrum side, with heterodyne frequencies corresponding to 25.5 and 26 MHz, respectively, while the left inset shows a zoom of the 11th and 12th t.p. on the negative spectrum side, with heterodyne frequencies corresponding to 5.5 and 6 MHz, respectively. Shaded rectangle outlines the 20 dB CS of 48 GHz, i.e., 240 t.p. with optical power per t.p. ranging from 60 nW down to 100 times lower power of 0.6 nW. Since a 20 dB span will be discussed for all other obtained DFCs, in further text only the maximal power per pair will be stated. In the 40 GHz span (from ${-}20$ to 20 MHz) the resulting DFC is ultraflat, with comb flatness of the order of 1 dB. In the case of moderate $\Delta I = 6{I_{{\rm th}}}$, we depict only the spectrum envelope [dashed line in Fig. 1(a)]. Larger amplitudes of modulation raise the maximal power per t.p. to 413 nW, on the expense of somewhat disturbed line flatness in the ${\pm}20\;{\rm GHz}$ region, which is around 3 dB. The 20 dB criterion outlines the CS of 55 GHz (275 t.p.).

 

Fig. 1. DFCs for $\alpha = 3$ and two amplitudes of modulation, defined with respect to the threshold current ${I_{{\rm th}}}$, $\Delta I = 2{I_{{\rm th}}}$ (solid lines) and $6{I_{{\rm th}}}$ (dashed lines), for electrical current bandwidth (a), (c) $B \approx 20\;{\rm GHz}$ and (b) $B \approx 40\;{\rm GHz}$ and for two bias currents. Shaded rectangles denote dual frequency comb 20 dB span expressed in GHz and as number of teeth pairs (t.p.). Insets in (a) show zooms of teeth pairs for $\Delta I = 2{I_{{\rm th}}}$.

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In Fig. 1(b), we present envelopes of DFCs obtained for the same bias current and amplitudes of modulation, but for larger $N = 401$, i.e., larger electrical current bandwidth ($B \approx 40\;{\rm GHz}$). Larger $B$ lowers the maximal power in t.p. in comparison with spectra presented in Fig. 1(a), but contributes to larger 20 dB CS. For smaller amplitudes of modulation ($\Delta I = 2{I_{{\rm th}}}$, solid line), CS equals 78 GHz (390 t.p.), with maximal power per t.p. of 17 nW. Higher amplitudes of modulation ($\Delta I = 6{I_{{\rm th}}}$, dashed line) raise both the power and the span, resulting in ${\rm CS} = 84\; {\rm GHz}$ (420 t.p.), with maximal power per t.p. of 114 nW, denoted with a red shaded rectangle [Fig. 1(b)]. In both cases, the spectrum has high flatness (${\lt}3\;{\rm dB}$) in the ${\pm}20\;{\rm GHz}$ span. The total optical power of both spectra in Figs. 1(a) and 1(b) is around 5.7 mW, out of which the central line is taking 97%–99%, while the rest is distributed into DFC t.p. To compare different current forms and resulting spectra, we define the spectrum power efficiency (SE) as the ratio of power distributed into t.p. with respect to the total power, giving that power efficiency of the spectra presented in Figs. 1(a) and 1(b) is of the order of 1%–3%.

Although higher bias currents raise the total optical power in the spectrum, SE is decreased and goes even below 1%, resulting in less power per t.p. and smaller CS in comparison with the spectra obtained for low bias currents. Figure 1(c) shows DFC envelopes obtained for ${I_{\rm b}} = 10{I_{{\rm th}}}$, under different modulation amplitudes, for $B \approx 20\;{\rm GHz}$. Small amplitude modulation ($\Delta I = 2{I_{{\rm th}}}$, solid line) provides CS of 40 GHz (200 t.p.) with 3 dB of flatness and maximal optical power per t.p. of 50 nW. Larger amplitudes of modulation ($\Delta I = 6{I_{{\rm th}}}$, dashed line) raise the power per t.p. and slightly raise the 20 dB margin span, while degrading the line flatness, resulting in CS of 50 GHz, i.e., 250 t.p. with maximal power per t.p. of 375 nW. Electrical currents with higher bandwidth ($B \approx 40\;{\rm GHz}$) lead to the CS of $\approx 80\;{\rm GHz}$, similar to the spectra obtained for smaller bias current in Fig. 1(b), although with lower power per t.p.

Finally, a LD with larger $\alpha$, although causing more pronounced phase noise, leads to more pronounced nonlinearities in the optical phase response, providing stronger comb lines. For $\alpha = 6.5$, moderate bias current (${I_{\rm b}} = 6{I_{{\rm th}}}$), small amplitude modulation [$\Delta I = 2{I_{{\rm th}}}$, solid line in Fig. 2(a)], and $B \approx 20\;{\rm GHz}$, we obtain CS of 57 GHz (285 t.p.), with maximal power per t.p. of 390 nW. As for a small amplitude of modulation, the ${\pm}20\;{\rm GHz}$ span exhibits high flatness ${\lt}3\;{\rm dB}$. Larger amplitudes of modulation [$\Delta I = 6{I_{{\rm th}}}$, dashed line in Fig. 2(a)] provide higher CS equal to 73 GHz (365 t.p.) with significantly higher maximal power per t.p. of 2.6 µW, on the expense of distorted flatness in the ${\pm}20\;{\rm GHz}$ region. In this case, SE is around 6%. Further raising of the electrical current bandwidth will contribute to the CS increase, at the expense of lower SE and lower maximal power per t.p. In the case of $B \approx 40\;{\rm GHz}$, small $\Delta I = 2{I_{{\rm th}}}$ [solid line in Fig. 2(b)] provides CS of 78 GHz (390 t.p.), with maximal power per t.p. of 103 nW, while a higher span is obtained for large amplitude modulation [$\Delta I = 6{I_{{\rm th}}}$, dashed line in Fig. 2(b)], yielding 103 GHz (515 t.p.) with maximal power per t.p. of 656 nW. For larger $B$, SE is reduced to the order of 1%.

 

Fig. 2. DFC envelopes for $\alpha = 6.5$ and bias current ${I_{\rm b}} = 6{I_{{\rm th}}}$, under two amplitudes of modulation: $\Delta I = 2{I_{{\rm th}}}$ (solid line) and $6{I_{{\rm th}}}$ (dashed line), for (a) $B \approx 20$ (b) and $B \approx 40\;{\rm GHz}$.

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Although the approach based on the ${\rm sinc}$-shaped current waveform can provide a broad spectrum with regions of high line flatness, it is not particularly efficient in converting the power to DFC lines, since it provides SE of the order of only a few percent (in our simulations, up to 6%). Larger efficiencies could be obtained for lower bias currents and higher modulation amplitudes. However, for lower bias currents, optical power in t.p. drops below nW.

In the case of $p = 2$ [Eq. (5)], we set modulation frequencies to ${f_1} = 200\;{\rm MHz}$ and ${f_2} = 200 + 0.5\;{\rm MHz}$ to obtain the same comb teeth configuration as above, i.e., t.p. appear at ${\pm}n \times ({f_1} + {f_2})/2 = \pm n \times 200\;{\rm MHz}$, with heterodyne frequency of the $n$th t.p. equal to $n \times ({f_2} - {f_1}) = n \times 0.5\;{\rm MHz}$. We analyze two values, $N = 100$ and $N = 200$, setting electrical current bandwidth to $B \approx 20\;{\rm GHz}$ and $B \approx 40\;{\rm GHz}$, respectively.

In Fig. 3(a), we present obtained DFC envelopes under small bias current ${I_{\rm b}} = 2.5{I_{{\rm th}}}$ and a moderate value of $\alpha = 4.25$, for $B \approx 20\;{\rm GHz}$. Our simulations prove that for the presented current waveform, low bias currents in combination with moderate to large modulation amplitudes lead to an increase in SE, which is around 25% for the presented DFCs, which significantly raises the optical power per t.p. in the 20 dB margin span in comparison to the current waveform above. For modulation amplitude of $\Delta I = 7{I_{{\rm th}}}$ [solid line, Fig. 3(a)], we obtain CS of 55 GHz (275 t.p.) with maximal power per t.p. of 4.7 µW, while larger $\Delta I = 14{I_{{\rm th}}}$ [dashed line, Fig. 3(a)] results in ${\rm CS} = 71\;{\rm GHz}$ (355 t.p.) with slightly smaller maximal power per t.p. of 4.2 µW. Due to low bias current, the total power of the optical spectrum is lower in comparison with Figs. 1(a) and 1(b) and Fig. 2 and equals around 2 mW. However, SE is significantly higher and equals 20%–25%. Span increase can be obtained for larger $B \approx 40\;{\rm GHz}$ [Fig. 3(b)], at the expense of decreasing SE and lowering the power per t.p. In the case of large amplitude $\Delta I = 14{I_{{\rm th}}}$ [dashed line, Fig. 3(b)], the CS is increased to 116 GHz (580 t.p.) with maximal power per t.p. of 1.3 µW and ${\rm SE} = 15\%$.

 

Fig. 3. DFC envelopes for bias currents ${I_{\rm b}} = 1.5{I_{{\rm th}}}$, $2.5{I_{{\rm th}}}$, and $\Delta I = 7{I_{{\rm th}}}$ (solid line) and $14{I_{{\rm th}}}$ (dashed and dotted lines). (a) Envelopes for $\alpha = 4.25$ all for $B \approx 20\;{\rm GHz}$. (b) Envelopes for $\alpha = 4.25$, ${I_{\rm b}} = 2.5{I_{{\rm th}}}$, $B \approx 40\;{\rm GHz}$ (dashed line) and for $\alpha = 6$, ${I_{\rm b}} = 1.5{I_{{\rm th}}}$, $B \approx 20\;{\rm GHz}$ (dotted line), all for $\Delta I = 14{I_{{\rm th}}}$. Insets show zooms of teeth pairs for (a) $\alpha = 4.25$ and (b) 6.

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Finally, larger values of $\alpha$ in combination with very low bias currents and high amplitudes of modulation can significantly increase the SE and consequently raise the power per t.p. Our simulations prove that higher values of $\alpha$ require low current bandwidths to circumvent pronounced peaks and dips in the DFC envelope. For $\alpha = 6$, with ${I_{\rm b}} = 1.5{I_{{\rm th}}}$, $\Delta I = 14{I_{{\rm th}}}$, and $B \approx 20\;{\rm GHz}$ [dotted line in Fig. 3(b)], we obtain CS of 103 GHz (515 t.p.) with maximal power per t.p. of 8.8 µW. In this case, SE is around 52%, which is significantly higher in comparison to the spectra with lower $\alpha$.

The increase in $\alpha$ leads to an increase in comb linewidth. However, this linewidth is still sufficiently small in comparison with the majority of heterodyne frequencies, preventing in this way a decay of the t.p. coherence level (c.f. insets in Fig. 3). The major reason for spectrum distortions and asymmetry is strong and nonlinear phase variation, which under some operating conditions, usually large magnitudes of modulation current, may lead to sidelobe occurrence [c.f. Fig. 1(a) and Fig. 3]. The increase in $\alpha$ leads to larger phase variation, which consequently may cause more pronounced sidelobes and spectrum flatness degradation as shown in Fig. 3. For more thorough insight into the temporal laser dynamics, see Supplement 1, Visualization 1, and Visualization 2.

We present a simple dual comb generation scheme based on a single directly modulated LD, driven by superposition of two electrical current waveforms providing for large signal modulation. In the case of current form based on the superposition of two trains consisting of ${\rm sinc}$-shaped current pulses, our simulations predict spectra with t.p. spanning up to 103 GHz in a 20 dB margin, with maximal power per pair of 656 nW, or up to 2.6 µW with a 73 GHz span. In addition, we show that smaller spans of 40 GHz can be obtained with a high level of comb line flatness, of the order of 1 dB, with power per pair of 60 nW. The spectral power efficiency in the case of ${\rm sinc}$ pulses is rather low and goes up to 6%. In the case of ${{\rm sinc}^2}$ pulses, our simulations predict spectra with t.p. spanning up to 116 GHz in a 20 dB margin, with maximal power per t.p. of 1.3 µW, or up to 8.8 µW with a 103 GHz span. In the case of ${{\rm sinc}^2}$ pulses, SE goes up to 52%. For both current forms, a higher linewidth enhancement factor $\alpha$ increases power per t.p. at the expense of spectrum envelope smoothness distortion.