Microwave Frequency Comb Optimization for FMCW Generation Using Period-One Dynamics in Semiconductor Lasers Subject to Dual-Loop Optical Feedback

Abstract

Microwave frequency comb (MFC) optimization for frequency-modulated continuous-wave (FMCW) generation by period-one (P1) dynamics with dual-loop optical feedback are numerically investigated. The linewidth, the side peak suppression (SPS) ratio, and the comb contrast are adopted to quantitatively evaluate the optimization performance, which directly influence the phase stability, spectral purity and repeatability of the MFC. The results show that intensity modulation of the optical injection can generate a sweepable FMCW signal after photodetection via the optical beat effect. When optical feedback loops are introduced, the single-loop configuration can reduce the phase noise of the FMCW signal whereas a dual-loop configuration exploits the Vernier effect to achieve further linewidth reduction and wide tolerance to the feedback strength. Finally, for both the SPS ratio and comb contrast, the dual-loop configuration achieves a higher SPS ratio and maintains high contrast across a wide range of optical feedback loop delays, which outperforms the loop time tolerance of the single-loop configuration.

1. Introduction

Frequency-modulated continuous-wave (FMCW) is essential for high-resolution detection in diverse applications, from automotive radar to coherent distributed sensing [1,2,3]. To meet the rigorous performance specifications in these scenarios, the FMCW signals are expected to possess excellent linearity, minimal phase noise, and adaptability over wide spectral ranges [4,5,6]. However, conventional electronic oscillators or basic photonic architecture generation solutions face intrinsic trade-offs among the system complexity, frequency tunability, and spectral stability [7]. Therefore, addressing these challenges requires advances in generation techniques that transcend the bottlenecks of purely electronic or simplified photonic approaches.

For the electronic synthesis of FMCW signals, a voltage-controlled oscillator (VCO) [8], and a direct digital synthesizer (DDS) [9] are commonly employed. However, the nonlinear characteristics of varactors in VCOs and the finite sampling rates of DDSs fundamentally constrain the achievable sweep bandwidth and linearity [10]. For optoelectronic methodologies such as optoelectronic oscillators (OEOs), there are essential pathways to address the phase noise issues [11,12]. Nevertheless, the frequency tunability is restricted by the bandwidth limitations of electronic components and filters within the feedback loop [13]. In addition, many advanced FMCW generation schemes also rely on an underlying optical frequency comb [14,15]. To overcome the constraints of the above-mentioned approaches, photonic schemes for FMCW generation have been developed, which mainly include frequency-to-time (FTM) mapping [16,17], the optical heterodyne [18,19] and period-one (P1) dynamics [20,21,22,23,24,25,26]. FTM performs spectral manipulation of an optical pulse followed by time–domain conversion in a dispersive medium; however, the static filter characteristics result in limited tunability [16]. The optical heterodyne methods, which are based on mixing a continuous-wave laser with either a tuned optical source or distinctively chirped pulses, can provide broad frequency tunability, but the phase instabilities degrade the quality of the converted electrical FMCW signals [18]. Recently, based on the P1 nonlinear dynamics, an interesting photonic technique for generating FMCW signals has been proposed [20,21,22,23,24,25,26]. Nevertheless, electronic and simpler photonic sources are still suitable for applications with relaxed performance requirements and a small form factor. Some critical needs remain for achieving exceptional spectral purity for the demanding scenarios. To achieve excellent FMCW signals, P1 dynamics-based systems often require advanced stabilization techniques. Although this can increase system complexity, size, weight, or power requirements, it is a necessary trade-off for achieving superior phase noise performance. Recent work has focused on optimizing the quality of MFCs in terms of narrowing the signal linewidth in optically injected lasers. Early experiments, such as the work by Zhuang et al., utilized a single optoelectronic feedback loop and achieved a signal linewidth of 1.9 MHz [20]. Tseng et al. demonstrated that stable FMCW signals with reconfigurable linear and triangular sweeps could be generated by phase-locking [21]. To achieve narrow linewidths in MFC generated from optically injected lasers, dual-loop optoelectronic feedback is adopted, which suppresses the linewidth to 500 Hz [22].

The aforementioned approaches primarily focus on optoelectronic feedback, whereas all-optical optimization configurations remain scarce, except for a single-loop optical feedback [10,24]. This work focuses on a numerical exploration of the processes involved in generating and stabilizing the FMCW signals with P1 dynamics in a semiconductor laser under dual-loop optical feedback. A quantitative analysis of the MFC is presented in terms of its linewidth, comb contrast, and sideband suppression ratio with a single-loop optical feedback configuration included for comparative discussion.

2. Theoretical Model

Figure 1 shows the schematic illustration of MFC generation and stabilization via dual-loop feedback. A master laser’s (ML) continuous-wave output is injected into a slave laser (SL) after being modulated by a Mach-Zehnder modulator (MZM) and routed by an optical circulator (OC). With suitable injection conditions, the P1 dynamics can first be obtained in the SL without external modulation. Then, a sinusoidal modulation signal is introduced by the MZM, which results in a frequency-modulated microwave signal generation. However, the spectral properties of the MFC generally need further optimization in terms of the linewidth, the comb contrast, as well as the side peak suppression ratio. Consequently, in this paper, dual optical feedback loops are adopted to stabilize the spectral characteristics of the MFC. In the feedback loops, the optical feedback strength and delay time are controlled with a variable attenuator (VA) and a variable optical delay line (VDL), respectively. Finally, the stabilized microwave frequency comb can be detected and further analyzed after optical-to-electrical conversion.

Accordingly, to theoretically characterize the system, the model in [27] is utilized, which extends the Lang–Kobayashi model by incorporating normalized variables, refined carrier dynamics, and nonlinear gain saturation. The specific rate equations, adapted to include the modulated injection and dual-loop optical feedback, are described as:

$$\begin{gathered} \begin{array}{c}{\displaystyle \frac{da}{dt}}={\displaystyle \frac{1-ib}{2}}\left[{\displaystyle \frac{{\gamma }_c{\gamma }_n}{{\gamma }_s\tilde{J}}}\tilde{n}-{\gamma }_p\left({\left|a\right|}^2-1\right)\right]a+\left(1+m\mathit{cos}\left(2\pi f_{m}t\right)\right){\xi }_i{\gamma }_c{e}^{-{i2\pi f}_{i}t}+f\end{array} \\ \begin{array}{c}+{\xi }_f{\gamma }_{c}a\left(t-{\tau }_1\right)+{\xi }_f{\gamma }_{c}a\left(t-{\tau }_2\right)\end{array} \end{gathered}$$

(1)

$$\begin{array}{c}{\displaystyle \frac{d\tilde{n}}{dt}}=-\left({\gamma }_s+{\gamma }_n{\left|a\right|}^2\right)\tilde{n}-{\gamma }_s\tilde{J}\left(1-{\displaystyle \frac{{\gamma }_p}{{\gamma }_c}}{\left|a\right|}^2\right)\left({\left|a\right|}^2-1\right)\end{array}$$

(2)

where the normalized complex field amplitude is denoted by a, the normalized carrier density is ñ, the linewidth enhancement factor is b, γ_c represents the cavity decay rate, γ_s is the spontaneous carrier relaxation rate, γ_n represents the differential carrier relaxation rate, γ_p stands for the nonlinear carrier relaxation rate, J̃ indicates the normalized bias current above threshold, ξ_i signifies the optical injection strength, the frequency detuning f_i refers to the difference between the injection optical frequency and the free-running optical frequency of the SL, and ξ_f is the optical feedback strengths. The times of τ₁ and τ₂ correspond to the optical feedback delay time of loop 1 and loop 2, respectively. For the modulation term, it is expressed as mV_π/2 cos (2πf_mt), where m denotes the modulation index, V_π/2 represents the quarter-wave voltage of the MZM, and f_m is the modulation frequency [20,27]. It should be pointed out that the MZM is represented by a simplified sinusoidal modulation term and captures the essential modulation effect responsible for generating sidebands spaced by the modulation frequency. Our previous work discussed dual-drive MZMs for FMCW generation [26], while an improved MZM model would substantially increase the model complexity without leading to essential differences in the main physical conclusions of this study, specifically regarding the focus of the MFC optimization under dual-loop optical feedback. For the fiber couplers, the beam splitter model is adopted. The splitting ratio of the couplers in the feedback loops is set to 50:50 to obtain equal feedback strengths for both loops, and each is half of that in the single-loop case for simplicity. The stochastic character of spontaneous emission noise in the SL is quantitatively represented through a Langevin stochastic term f incorporated into Equation (1), and a systematic estimation of uncertainty could also be considered for the noise estimation [28]. The fluctuation term exhibits zero cross-correlation between its real and imaginary parts, with both spatial and temporal statistical independence. The ergodic hypothesis holds under this formulation, as evidenced by the vanishing cross-temporal correlations and the characteristic statistical moments defined by:

$$\begin{array}{c}<f\left(t\right)f^{*}\left(t^{\prime }\right)>={\displaystyle \frac{4\pi \Delta v}{1+b^2}}\delta \left(t-t^{\prime }\right)\end{array}$$

(3)

$$\begin{array}{c}<f\left(t\right)f\left(t^{\prime }\right)>=0\end{array}$$

(4)

$$\begin{array}{c}<f\left(t\right)>=0\end{array}$$

(5)

where, Δν is the full width at half-maximum (FWHM) of the solitary SL output, which is set to 100 MHz [27]. In our numerical simulation, the parameters are set as follows [20]: γ_c = 5.36 × 10¹¹ s⁻¹, γ_s = 5.96 × 10⁹ s⁻¹, γ_n = 7.53 × 10⁹ s⁻¹, γ_p = 1.91 × 10¹⁰ s⁻¹, b = 3.2, J̃ = 1.222, and V_π/2 = 1 V. The numerical solution to Equations (1) and (2) is obtained using a second-order Runge–Kutta method with a fixed 2 ps step, yielding a spectral resolution of 7.45 kHz.

3. Results and Discussions

In this section, the results of microwave frequency comb (MFC) optimization for FMCW generation through period-one (P1) dynamics with dual-loop optical feedback are numerically studied. As shown in Figure 2a, the P1 oscillation is first obtained in the slave laser (SL) with an injection strength ξ_i = 0.17 and frequency detuning of f_i = 14.2 GHz. The optical spectrum exhibits sidebands separated by the fundamental oscillation frequency f₀ = 23.79 GHz. Following optical to electrical conversion, the beating between the regenerated optical injection component at f_i and the red-shifted cavity component at f_r produces the microwave signal shown in Figure 2b. To generate the FMCW signal, intensity modulation is introduced to the output of the master laser with a modulation index m = 0.2 and modulation frequency f_m = 20 MHz, and the modulation effect can be observed in the inset of Figure 2c. As a result, in Figure 2c, the discrete sidebands of the pure P1 oscillation transform into a comb-like structure, where multiple optical lines spaced by f_m emerge around each one. It can be found that the higher order of the P1 sidebands has a broader bandwidth than the first sideband at the frequency of f_r, which results from the accumulation of modulation-induced frequency spreading. Specifically, the spectral components around f₀ arise from the fundamental beating between the injected signal and the red-shifted cavity mode, while those around 2f₀ are produced by nonlinear mixing and harmonic interactions among the injection, cavity resonance, and side peaks. Consequently, in Figure 2d, after optical-to-electrical conversion, a FMCW signal with a bandwidth of around 4 GHz is achieved (gray curve). The mechanism originates from the beating of the frequency-modulated optical frequency components at the cavity resonance of the SL and the regeneration injection component. Thus, a continuous spectral envelope in place of the discrete peaks can be observed in the unmodulated P1 case. Meanwhile, the generated FMCW signal with dual-loop feedback is presented (black curve). Obviously, the power as well as the comb contrast of all approximately 180 frequency components get enhanced due to dual-loop feedback stabilization, and the other characteristics will be discussed in detail later.

To characterize the instantaneous frequency of the generated FMCW signal, we perform a short-time Fourier transform analysis. For this analysis, a Gaussian window with a length of 2048 samples and a shape parameter of 0.4 is utilized. The instantaneous frequency of the P1 oscillation varies over cycles of 1/f_m = 50 ns, which reveals the frequency modulation effects, and the sweep range Δf is about 4 GHz shown in Figure 3. Accordingly, the fastest sweep rate of the instantaneous frequency is πΔf f_m = 0.25 GHz/ns [20]. Furthermore, to evaluate the tunability of the MFC bandwidth, the dependency of the bandwidth on the modulation index m is analyzed. To prevent the SL from entering other dynamic states, the optical injection strength ξ_i is fixed at 0.17, and the detuning frequencies are set as f_i/2, f_i, and 3f_i/2, respectively. As depicted in Figure 3b, for m = 0, no modulation is introduced; the instantaneous frequencies of the microwaves remain constant but are still larger than the detuning frequencies due to the antiguidance effect-induced cavity resonance red-shift [7]. The sweep range undergoes a nearly monotonic expansion driven by the rising modulation index, which also determines the corresponding lower and upper frequency limits. For a fixed modulation index, increasing the detuning frequency results in a narrower instantaneous frequency sweep range. This is due to the reduced red-shift cavity resonance, where modulation-induced frequency deviation is partially suppressed and has been observed in similar experimental work [10].

The following content will focus on the MFC optimization. the 3-dB linewidth of the spectral components is calculated by Lorentzian fitting. The inset of Figure 4 shows the fitting results for a single peak located around the central frequency of the spectrum. For consistency, the injection parameters are fixed at (ξ_i, f_i) = (0.17, 14.2 GHz) unless otherwise stated, and the case of single-loop optical feedback is studied for comparison. As shown in Figure 4a, the modulated optical injection without the optical feedback scheme is analyzed first. The linewidth of the MFC is 7.33 MHz. Additionally, the frequency spacing between adjacent frequency components is about 20 MHz, which matches the modulation frequency f_m. The broad linewidth arises from relatively strong phase noise, which could degrade range resolution and accuracy in FMCW radar systems [29]. To achieve phase noise suppression, a single optical feedback loop is employed. The feedback strength ξ_f is set to 0.005, and the optical delay time τ equals the reciprocal of the modulation frequency f_m to obtain the best stabilization performance [20]. Consequently, in Figure 4b, the linewidth is narrowed to 173.70 kHz, and the comb contrast, which will be quantitatively discussed later, is apparently improved. Furthermore, for the dual-loop optical feedback configuration, the feedback strengths in both loops are set equal, each being half of that used in the single-loop case for simplicity, and the optical delay times τ₁ and τ₂ are 50 ns and 75 ns, respectively. Obviously, in Figure 4c, the linewidth further decreases to 12.76 kHz, which is lower than that obtained with the single-loop optical feedback configuration and indicates that dual-loop optical feedback offers superior linewidth narrowing performance.

Subsequently, the influence of optical feedback strength on the MFC linewidth is explored. For a fair comparison, the feedback strengths for the single- and the dual-loop configurations are the same, and the optical delay times are kept consistent with those in Figure 4. To avoid the SL entering unstable or unwanted dynamical regimes induced by strong optical feedback strength, the feedback strength is restricted to 0.01. In Figure 5, the average MFC linewidth, calculated across all frequency components within the bandwidth of the FMCW signal, is plotted against a range of feedback strengths for both single-loop and dual-loop configurations. As shown in Figure 5, the MFC linewidth variations under both configurations exhibit similar trends. Both trends show a decreasing tendency as the feedback strength increases gradually, though some fluctuations occur during the process. This phenomenon can be attributed to nonlinear fluctuations induced by increasing feedback strength [24]. Furthermore, the dual-loop feedback curve exhibits superior linewidth-narrowing performance compared to the single-loop curve. At the feedback strength of 0.001, the dual-loop linewidth is about 23% smaller than that of the single-loop configuration, decreasing to 45 kHz from around 60 kHz in the single-loop configuration. This enhanced performance can be attributed to the suitable optical delay difference induced Vernier interaction between the two external-cavity modes, which minimizes the sensitivity to phase variations [30].

Apart from the linewidth of the MFC, the side peak suppression (SPS) ratio as well as the comb contrast of each frequency component are also critical parameters for optimizing MFC performance. The magnitude of the SPS ratio, which corresponds to the precision and resolution level of FMCW signals in radar systems, is described as the ratio of the power at the central frequency of an individual frequency component of the MFC to its maximum side-peak power [30]. Similarly, the comb contrast, which indicates the repeatability of FMCW signals [24], is the ratio of the central frequency power of an individual frequency component of the MFC to the average noise floor of the MFC [31]. Figure 6 illustrates the zoom-in power spectra of an individual spectral component near the central frequency of the MFC with single-loop optical feedback and dual-loop optical feedback. The optical injection and feedback parameters remain identical to those in Figure 4. For the single-loop optical feedback case, the SPS ratio is about 3.5 dB, and the comb contrast is 52.2 dB. For the dual-loop configuration, in Figure 6b, the comb contrast is 51.5 dB, which is slightly reduced compared with the single-loop case, while the SPS ratio is significantly improved to 23.1 dB. To the previous work [27], it can be expected that the dual loops induced optical feedback delay time τ₁ and τ₂ play crucial roles in the SPS ratio of the MFC, and this issue will be addressed next.

Figure 7 shows the averaged SPS ratio of all spectral components of the MFC as a function of the optical feedback delay time. For the dual-loop optical feedback configuration, the optical feedback delay time τ₁ is fixed at 50 ns, while τ₂ varies from 5 ns to 100 ns, and the other operation parameters are identical to those in Figure 4. The averaged SPS ratio exhibits fluctuations as τ₂ varies. The maxima occur when the relative delay difference between the two loops yields strong destructive interference among residual side peaks [32]. However, when τ₂ is close to τ₁ or the delays become nearly commensurate, the SPS ratio drops, resulting from the reinforcement of external-cavity modes by both optical feedback loops. As a comparison, for the single-loop configuration, the averaged SPS ratio fluctuates within a narrow range of 5 dB, which indicates that under moderate optical feedback strength, the external cavity modes introduced by single-loop feedback are evident and inevitably result in SPS ratio degradation compared with the dual-loop optical feedback configuration. The achieved SPS of approximately 20 dB in the dual-loop configuration is mainly limited by factors such as residual external-cavity modes and spontaneous emission noise. Further improvements to the SPS performance could be realized by introducing photonic filter feedback [30] or the implementation of a hybrid approach, which combines all-optical feedback with optoelectronic stabilization.

The subsequent analysis is to elucidate the relationship between the feedback strength and the SPS ratio of the MFC. In Figure 8, the single-loop configuration exhibits a monotonic degradation in SPS ratio with increasing feedback strength, dropping from approximately 20 dB to below 5 dB when feedback strength is larger than 0.006. In contrast, the dual-loop configuration maintains a higher SPS ratio than the single-loop case, remaining above 20 dB for feedback strength as large as 0.008. Although the dual-loop curve shows some fluctuations, the mechanism may result from the optical phase variation induced by the feedback strength [33], it sustains superior suppression performance compared with a single-loop optical feedback configuration.

Finally, the dependency of the MFC’s averaged comb contrast on the optical feedback delay time is examined. For single-loop optical feedback, the optical feedback delay τ varies from 5 ns to 105 ns. In dual-loop optical feedback, the optical feedback delay time τ₁ is fixed at 50 ns, while τ₂ varies from 5 ns to 105 ns. As shown in Figure 9, for the single-loop configuration, the black curve remains low except for two clear peaks at 50 ns and 100 ns. One peak reaches 52 dB at 50 ns, and the other is near 50 dB at 100 ns, corresponding to 1/f_m and twice of it, respectively. The results agree with the previous single-loop optical feedback stabilization work, which proposed that the optimal optical feedback delay for comb contrast should be set at the reciprocal modulation frequency f_m [20]. As a comparison, for dual-loop optical feedback, the red curve exhibits a consistently high comb contrast across a wide range of τ₂, whose values are mostly above 47 dB with small variations. The suppression of deep contrast dips observed in the single-loop case indicates that the dual-loop configuration effectively mitigates the sensitivity of comb contrast to delay time variations due to the flatness gain and suppressed phase variations in the vicinity of the modulation frequency [34]. The dual-loop configuration, therefore, enables robust comb contrast performance without the need for precise matching of τ₂ to the reciprocal modulation frequency, offering a more tolerant and stable operational regime for MFC stabilization.

4. Conclusions

In summary, microwave frequency comb (MFC) optimization for FMCW generation through period one dynamics with dual-loop optical feedback is theoretically investigated. The results show that intensity modulation of the optical injection results in MFCs and yields an FMCW generation. However, the generated MFCs generally need to be further stabilized in terms of the linewidth, the side peak suppression ratio, and the comb contrast. For linewidth narrowness, although single-loop feedback with a delay near the modulation period could suppress the phase noise by locking the MFC, dual-loop optical feedback, which is based on the Vernier interaction between external cavity modes, can further reduce the linewidth of each individual spectral component of the MFC. For SPS ratio and comb contrast enhancement, the dual loop configuration maintains the SPS ratio and comb contrast above 20 dB and 47 dB, respectively, over a wide operational range of optical feedback, demonstrating superior stabilization performance and greater parameter tolerance compared with the single-loop feedback configuration. Though the primary aim of this work is to provide comprehensive theoretical analysis and numerical simulations of an all-optical stabilization approach using dual-loop feedback, it should be noticed that some idealized operating conditions are assumed. However, in practice, environmental fluctuations such as temperature variations, polarization drift in the MZM can significantly influence system stability and spectral performance. In addition, practical implementations will inevitably require active devices such as temperature and current controllers, which introduce additional noise for FMCW generation. Therefore, a dedicated experimental demonstration will be carried out based on the numerical simulations, and the results will be reported elsewhere in due course. This work will help to realize low phase noise, high SPS ratio, and comb contrast of MFC for FMCW generation and deliver an all-optical stabilization approach for photonic FMCW sources applicable to coherent radar, LiDAR, and other microwave photonic systems.