1. Introduction

Nonlinear period-one (P1) dynamics of semiconductor lasers have actively attracted attention due to unique properties like eliminating high-speed electronics, nearly single-sideband spectrum, and wideband microwave frequency tunability [1–7]. Novel applications have been demonstrated using P1 dynamics, such as photonic microwave generation [1–3,7,8], radio-over-fiber transmission [9,10], self-mixing interferometry sensing [6,11,12], carrier recovery and Doppler-free coherent detection for OFDM-RoF systems [13,14], microwave time delay [15], and frequency-modulated waveform generation [16–19]. P1 dynamics generates an optical wave with intensity modulation at a microwave frequency. It can be obtained when a stable emission semiconductor laser experiences a Hopf-bifurcation through undamping the relaxation oscillation by perturbations such as optical injection from an additional laser and optical feedback from an external cavity [2,20–22]. P1 microwave oscillations from optical injection have instabilities induced by intrinsic spontaneous emission noise. These instabilities can degrade the microwave linewidths to even broader than the linewidth of the free-running laser [23], thereby degrade the performance in photonic microwave generations [1,24,25], reduce the data rate in radio-over-fiber transmissions [10], and limit the accuracy in sensing applications [6,11]. Optical feedback is of particular interest because it is the most straightforward approach by only involving a laser with a mirror [2,26]. A semiconductor laser under optical feedback can generate various nonlinear dynamics such as P1, period-doubling, quasi-periodic, and chaotic states. Different from optical injection, an optical feedback scheme contains two characteristic time scales that governs the dynamics, namely the laser relaxation oscillation time and the external cavity round-trip time. The former characterizes energy exchange speed between carriers and photons, thus regulates the oscillation frequency of P1 dynamics. The latter corresponds to external cavity modes, which could be helpful to stabilize P1 dynamics. Once P1 frequency components are locked to these modes, the phase noise originating from intrinsic spontaneous emission will be effectively suppressed [16,27–30]. Then the corresponding P1 oscillation will be stabilized by reducing the microwave linewidth down to be much smaller than the linewidth of the free-running laser. However, other external cavity modes, which are not occupied by P1 frequencies, are usually pronounced by residual noise peaks near the P1 microwave frequency, hence compromise the stability [24,29,31].

Various approaches towards P1 stabilization in optical feedback schemes have been reported, where P1 stabilization was achieved by reducing microwave linewidth and suppressing residual noise peaks. For instance, approaches using short-cavity optical feedback were numerically and experimentally demonstrated [32–36]. In these approaches, the cavity round-trip time was set to be shorter than the laser relaxation oscillation time. Then these external cavity modes were utilized for P1 microwave generation rather than forming residual noise peaks [34]. This mechanism was later extended to P1 stabilization using short-cavity optical feedback with optical injection [37]. However, compared to long-cavity configurations, short-cavity feedback may compromise the reduction of microwave linewidth [36]. As for long-cavity feedback, an approach using mirror feedback was utilized for generating narrow linewidth P1 oscillations with residual side peaks suppressed by another optoelectronic feedback loop through Vernier effect [29]. The mechanism was also reported in a scheme using dual-loop optical feedback with optical injection [24,31]. Besides broadband feedback, narrowband feedback using optical filter can generate laser dynamics as well. Fischer $et~al.$ pioneered the generation of intensity and frequency oscillations using optical feedback with a Fabry-Perot filter [38,39]. Recently it was extended to P1 stabilization in an optical injection scheme [40]. Though optical feedback from a fiber Bragg grating (FBG) has been demonstrated for chaos generation and optimization [41–44], its effect towards P1 generation and stabilization is still worth investigating.

In this paper, we investigate and analyze the generation and stabilization of P1 oscillations in a semiconductor laser with optical feedback from a narrowband FBG. The Bragg frequency is aligned to the laser free-running frequency, and the grating bandwidth is smaller than the laser relaxation oscillation frequency. It can generate P1 oscillations even under relatively strong feedback by suppressing the coherence collapse. Moreover, a uniform FBG typically has a comb-filtered reflection. Hence it can limit the external cavity modes by each lobe. The generation of P1 dynamics is investigated in the feedback parameter space of feedback strength and delay time. The stability of P1 oscillations, in the form of microwave linewidth, phase noise variance, and side-peak suppression ratio (SPSR), is investigated under the effects of feedback parameters as well as grating bandwidth. The SPSR is defined as the ratio of the microwave power at P1 frequency over that at the strongest noise side peak in the intensity power spectrum. Following this introduction, the simulation model is presented in Section 2. The numerical results are described in Section 3. A conclusion follows them in Section 4.

2. Model

In the proposed scheme, a continuous-wave emission from a single-mode semiconductor laser is externally reflected by an FBG, then is fed back into the laser. Here, a narrowband uniform FBG is adopted. It provides comb-filtered reflection with a narrow main lobe surrounded by several side lobes. Hence the reflection is totally different from that of the conventional mirror feedback. The laser is described by the normalized charge carrier density $\tilde {n}(t)$ and the normalized intracavity optical field amplitude $a(t)$ in referencing to the free-running optical frequency. Denoting the impulse response of the FBG reflection by $r(t)$, the field amplitude coupling back to the laser is proportional to a convolution, $r(t) \ast a(t-\tau _\textrm {RT})$, where $\tau _\textrm {RT}$ is the round-trip feedback delay time between the laser output facet and the FBG input end. The rate equations that govern the laser dynamics are [41,45,46]

The last term in Eq. (1) is the Langevin fluctuating force of $f_\textrm {sp}~=~f_\textrm {1}~+~\textrm {i}f_\textrm {2}$ which accounts for the spontaneous emission noise of the laser [24,48]. The real part $f_\textrm {1}$ and imaginary part $f_\textrm {2}$ are mutually independent. The Langevin force has an effective delta-function self-correlation in time [24]:

In fact, the rate-equation model in Eqs. (1)$-$(2) is a generic model for arbitrary feedback impulse response $r(t)$, which is suitable for any physical reflections. For mirror feedback, the impulse response $r(t)$ can be set as $\delta (t)$, where the rate equations in Eqs. (1)$-$(2) are reduced to the conventional Lang-Kobayashi model [45]. For FBG feedback, the impulse response $r(t)$ is obtained from an inverse Fourier transform of the FBG reflection frequency response $r(\Omega )$ which is described as follows [41,49]:

3. Results and analysis

3.1 Dynamical mapping

Figure 1 shows the mapping of the dynamical states of the laser in the feedback parameter space ($\xi _\textrm {f}$, $\tau _\textrm {RT}$) without the influence of Langevin noise. The FBG has a bandwidth corresponding to $f_\textrm {BW}/f_\textrm {r}$ = 0.5. The stable states (white), P1 states (red), quasi-periodic states (gray), period-doubled states (yellow), and chaotic states (black) are identified according to the intensity time series obtained from the simulations [41]. The laser remains stable over a large region on the map. This is because the narrowband FBG does not reflect enough power at optical sidebands at laser relaxation oscillation frequency $f_\textrm {r}$. Thus the relaxation oscillations remain damped for a range of small $\xi _\textrm {f}$ [10]. Further increasing the feedback strength can nonetheless undamp the relaxation oscillations, which results in regions of P1 oscillation states on the map. P1 oscillation states are found even when $\tau _\textrm {RT}= 0$ for $\xi _\textrm {f}>0.05$ because the feedback light still experiences some delay within the FBG. Repeated regions of P1 states are found across a wide range of $\tau _\textrm {RT}$ at relatively strong feedback strength of $0.16<\xi _\textrm {f}<0.2$. Interestingly, each of these P1 regions is found to have microwave oscillations corresponding to a fixed order of external cavity mode, which will be discussed later in Fig. 4. Qualitatively, FBG feedback significantly expands the regions of P1 states towards stronger $\xi _\textrm {f}$ where chaotic states are usually excited in conventional mirror feedback [41]. Despite being occupied by regions of P1 oscillation states, small regions of quasi-periodic states and chaotic states are also found on the map, thus quasi-periodic routes to chaos are identified, as for most semiconductor lasers under feedbacks [2,26,50].

To detailed investigate the expanding of P1 regions, intensity bifurcation diagrams as a function of the feedback strength $\xi _\textrm {f}$ at $\tau _\textrm {RT}$ = 0.65 ns from FBG and mirror feedback are compared in Fig. 2 without the influence of Langevin noise. As for FBG feedback in Fig. 2(a), when $\xi _\textrm {f}$ is relatively weak, the laser remains in stable state. By increasing $\xi _\textrm {f}$ to above 0.0296, it is driven in to P1 state through a Hopf-bifurcation. Further increasing $\xi _\textrm {f}$ to exceeding 0.0328, the laser is driven out of P1 state. Interestingly, when $0.073 \leq \xi _\textrm {f} \leq 0.093$ and $0.172 \leq \xi _\textrm {f} \leq 0.193$, another two regions of P1 state appear following Hopf-bifurcations. As for mirror feedback in Fig. 2(b), the laser is operating in P1 state only within a narrow range of $0.0115 \leq \xi _\textrm {f} \leq 0.012$. Further increasing $\xi _\textrm {f}$ quickly drives the laser into quasi-periodic state and finally chaotic state. Obviously, FBG feedback significantly expands P1 operation range over $\xi _\textrm {f}$, which allows much better flexibility in P1 generation.

 

Fig. 1. Mapping of the dynamical states when $f_\textrm {BW}/ f_\textrm {r}$ is $0.5$. Regions of stable states (white), P1 states (red), quasi-periodic states (gray), period-doubled states (yellow), and chaotic states (black) are identified.

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Fig. 2. Bifurcation of a semiconductor laser under optical feedback from (a) an FBG and (b) a mirror. The feedback delay time $\tau _\textrm {RT}$ is 0.65 ns.

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3.2 P1 microwave oscillation

Figure 3 illustrates P1 (i) optical spectrum and (ii) power spectrum from the Fourier transform of the field amplitude $a(t)$ and corresponding intensity $|a(t)|^{2}$. FBG feedback with bandwidths correspond to $f_\textrm {BW}/f_\textrm {r}$ = 0.5, 0.5, and 0.62 are respectively presented by Figs. 3(b)$-$(d), while mirror feedback is presented by Fig. 3(a) for a comparison. The feedback delay time is $\tau _\textrm {RT}$ = 0.65 ns. The feedback strength are $\xi _\textrm {f}$ = 0.0119, 0.0321, 0.188, and 0.188 for Figs. 3(a)$-$(d), respectively. Black curves in column (i) represent optical spectrum of P1 oscillations while green curves represent reflectivity spectrum of reflectors.

 

Fig. 3. Numerical results of (i) optical spectra and (ii) power spectra from (a) mirror feedback and (b)$-$(d) FBG feedback. The feedback delay time is $\tau _\textrm {RT}$ = 0.65 ns. The feedback strength are $\xi _\textrm {f}$ = 0.0119, 0.0321, 0.188, and 0.188 in (a)$-$(d), respectively. The bandwidth of FBG corresponds to ($f_\textrm {BW}/f_\textrm {r}$, $\tau _\textrm {g}$) = (0.5, 57 ps), (0.5, 57 ps), and (0.62, 48 ps) in (b)$-$(d), respectively.

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Starting with Fig. 3(a) where mirror feedback is adopted for a comparison, the feedback strength of $\xi _\textrm {f}$ = 0.0119 is adopted for generating P1 oscillation. In Fig. 3(a-i), the reflectivity spectrum in green is a flat line as the mirror has no frequency-selectivity. The P1 optical spectrum in black consists of a primary frequency of the center mode, two adjacent sidebands originating from the undamping of relaxation oscillation, and other frequency components from four-wave mixing of the former three frequencies. The primary frequency has a negative offset because of optical feedback induced red-shift of laser cavity resonances [5]. The strongest adjacent relaxation oscillation sideband, which is usually at the negative frequency offset due to the effect of red-shift as well, is named as the secondary frequency in this work for convenience. P1 oscillation with a microwave frequency of $f_{0}$ = 10.29 GHz is mainly from the beating of primary and secondary frequencies [5]. P1 frequencies in mirror feedback are found to be bounded around $f_\textrm {r}$. Though $f_\textrm {r}$ could be changed by varying the laser threshold through proper control of the feedback parameters, in practice the allowed feedback for P1 oscillation is usually weak, and thus the change in $f_\textrm {r}$ is relatively limited [28]. Besides, the optical spectrum in Fig. 3(a-i) contains periodic features by $1/\tau _\textrm {RT}$ denoting the external cavity modes. These modes are also pronounced in the corresponding power spectrum in Fig. 3(a-ii) by periodic residual noise peaks at frequencies about $1/\tau _\textrm {RT}$ and its multiples. The P1 microwave frequency $f_{0}$, marked by red arrow in Fig. 3(a-ii), is equal to the multiple of $1/\tau _\textrm {RT}$, which shows a frequency locking between $f_{0}$ and the external cavity mode. The SPSR between $f_{0}$ and the strongest side peak is about 25 dB, which agrees well with the experimental results in [29]. Moreover, the inset of Fig. 3(a-ii) shows that the P1 microwave has an FWHM linewidth of $\Delta f$ = 2.4 MHz based on a Lorentzian fit. The P1 microwave linewidth is significantly smaller than the 10-MHz free-running optical linewidth due to the frequency locking.

In Fig. 3(b), the mirror is replaced by an FBG with a bandwidth corresponding to $f_\textrm {BW}/f_\textrm {r}$ = 0.5 at $\xi _\textrm {f}$ = 0.0321. In Fig. 3(b-i), the reflectivity spectrum in green has a comb-filtered shape with a relatively strong main lobe surrounded by several side lobes roughly separated by $f_\textrm {l}$ = 5 GHz. P1 oscillation is achieved with its primary frequency in FBG main lobe and other frequency components in the side lobes. The beating of primary and secondary frequencies results in a P1 microwave frequency of $f_{0}$ = 11.69 GHz. Interestingly, the external cavity modes are effectively suppressed due to the comb-filtered effect. In the corresponding power spectrum in Fig. 3(b-ii), the noise peak at frequency about $1/(\tau _\textrm {RT} + \tau _\textrm {g})$ is clear but those at the multiples are suppressed. $\tau _\textrm {g}$ = 57 ps represents the group delay at Bragg resonance, which is calculated using $\textrm {d}\theta _\textrm {FBG}/\textrm {d}\Omega$ at $\Omega$ = 0, where $\theta _\textrm {FBG}$ represents the phase of the FBG reflection frequency response $r(\Omega )$ [49]. The SPSR of P1 microwave signal is increased to about 38 dB, and the inset shows a microwave linewidth of $\Delta f$ = 2.9 MHz, which is comparable with that in mirror feedback. Note that higher harmonics are in phase with the P1 frequency thus do not degrade the stability.

In Fig. 3(c), the feedback strength is increased to $\xi _\textrm {f}$ = 0.188 while other configurations are the same with Fig. 3(b). In Fig. 3(c-i), the optical spectrum shows that the primary frequency is still in FBG main lobe, while the secondary frequency is pushed away to the lower frequency side lobe, hence results in an increased P1 microwave frequency of $f_{0}$ = 22.48 GHz. Compared to mirror feedback in Fig. 3(a), FBG feedback can generate much higher P1 microwave frequency due to sustaining much stronger feedback power. In Fig. 3(c-ii), the noise peak at frequency about $1/(\tau _\textrm {RT} + \tau _\textrm {g})$ is significantly suppressed. The SPSR of P1 microwave signal is improved to about 49 dB, and the microwave linewidth $\Delta f$ is reduced to 400 kHz in the inset.

Finally, in Fig. 3(d), the FBG bandwidth is increased to $f_\textrm {BW}/f_\textrm {r}$ = 0.62 while other configurations are the same with Fig. 3(c). In Fig. 3(d-i), a similar comb-filtered reflectivity spectrum in green is observed except for a small $f_\textrm {BW}$ increment. The optical spectrum, which corresponds to $f_{0}$ = 22.46 GHz, has frequencies roughly the same as that in Fig. 3(c-i). In Fig. 3(d-ii), the suppression on the noise peak at frequency about $1/(\tau _\textrm {RT} + \tau _\textrm {g})$ is similar to that in Fig. 3(c-ii), though $\tau _\textrm {g}$ is 48 ps instead. Improvements are observed as well, as the SPSR is further improved to about 54 dB, and the microwave linewidth $\Delta f$ is further reduced to about 110 kHz in the inset. In short, compared to mirror feedback, FBG feedback reduces the microwave linewidth by up to more than an order of magnitude and improves the SPSR by up to more than two orders of magnitude than mirror feedback. Besides the enhancements on P1 stability, FBG feedback also improves the microwave power, as the power difference between the primary and secondary frequencies is significantly smaller than that of mirror feedback.

Figure 4 further shows how the P1 microwave frequency $f_{0}$ changes with feedback parameters when using FBG feedback with $f_\textrm {BW}/f_\textrm {r}$ = 0.5. Relatively strong feedbacks are adopted to make sure that the laser is operating in the repeated P1 regions mentioned in Fig. 1. Closed symbols show the data as a function of $\tau _\textrm {RT}$ at $\xi _\textrm {f}$ = 0.188, while open symbols show that as a function of $\xi _\textrm {f}$ at $\tau _\textrm {RT}$ = 0.65 ns. Note that the laser noise is not considered in the numerical calculation here in order to better investigate the intrinsic variation of microwave frequency. As for the dependence on $\tau _\textrm {RT}$, the microwave frequency $f_{0}$ is found to red-shift continuously with the delay time $\tau _\textrm {RT}$ and then followed by an abrupt blue-shift after a certain delay time range. The feedback cavity sets external cavity modes for the laser system to satisfy, and thus the P1 oscillations can be locked to the closest mode. Increasing $\tau _\textrm {RT}$ results in the red-shift of external cavity modes, hence leads to the red-shift of $f_{0}$. However, the narrowband FBG provides a filtering effect on the external cavity modes around the P1 frequency as shown in Figs. 3(b)$-$(d), thus P1 frequency hopping to the adjacent higher frequency mode would be achieved if $\tau _\textrm {RT}$ is increased across a value when two adjacent modes have approximately identical magnitude response. The frequency hopping leads to an abrupt blue-shift of $f_{0}$. For instance, when increasing $\tau _\textrm {RT}$ from 0.46 ns to 0.48 ns where the laser is operating in one of the repeated P1 regions, in Fig. 4(a) the microwave frequency $f_{0}$ is observed to red-shift continuously, and in Fig. 4(b) the corresponding frequency ratio $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$ roughly equals to an integer revealing frequency locking between $f_{0}$ and the closest external cavity mode [29,30,51]. Further increasing the delay time $\tau _\textrm {RT}$, in Fig. 4(a) an abrupt blue-shift of $f_{0}$ is observed and the corresponding frequency difference could approach the reciprocal of $\tau _\textrm {RT}$ if a considerable long delay is adopted. In Fig. 4(b), the corresponding frequency ratio shows stair-like evolution revealing mode hopping between adjacent external cavity modes. The continuously red-shift followed by an abrupt blue-shift of $f_{0}$ repeats periodically by about 43 ps which roughly equals to the reciprocal of P1 frequency. As for the dependence on $\xi _\textrm {f}$, the P1 frequency $f_{0}$ is slightly increased with $\xi _\textrm {f}$ in Fig. 4(a), and the corresponding $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$ is approaching to an integer ratio revealing an improvement on the frequency locking in Fig. 4(b). Similar evolutions have been reported in most mirror feedback schemes [27–29,51,52].

 

Fig. 4. FBG feedback generated P1 (a) microwave frequency $f_{0}$ and corresponding (b) ratio $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$. The data as a function of $\tau _\textrm {RT}$ at $\xi _\textrm {f}$ = 0.188 are shown by closed symbols and as a function of $\xi _\textrm {f}$ at $\tau _\textrm {RT}$ = 0.65 ns are shown by open symbols. The FBG has $f_\textrm {BW}/f_\textrm {r}$ = 0.5 and $\tau _\textrm {g}$ = 57 ps.

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Though P1 frequency $f_{0}$ is found to be bounded around 10 GHz in mirror feedback, by replacing the mirror with a narrowband FBG, $f_{0}$ can be increased by enhancing the feedback strength together with adjusting the delay time. The increment of $f_{0}$ is enabled due to much stronger feedback power can be sustained in FBG feedback than in mirror feedback. However, the highest $f_{0}$ is found to be about 24 GHz where $\xi _{f}$ is approaching 0.2, as too much feedback power tends to drive the laser out of P1 regime. $f_{0}$ may be further increased by enhancing the laser power through changing the bias current, but nonetheless, it may have an upper limit related to the maximum frequency response bandwidth set by the $K$ factor as in most case of semiconductor lasers [47,53].

In Fig. 4 at relatively short delay, there exist gaps between two adjacent periods of microwave frequency, which correspond to the stable states between two adjacent P1 regions as shown in Fig. 1. While at relatively long delay, these stable state gaps are found to vanish, as the adjacent P1 regions are well overlapped as shown in Fig. 1. To detailed investigate the effect of laser noise on P1 frequency characteristics when the repeated P1 regions are well overlapped, Figs. 5(a) and (b) respectively plot the microwave frequency $f_{0}$ and frequency ratio $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$ as a function of $\tau _\textrm {RT}$ at long delay. Using the same FBG and $\xi _\textrm {f}$ with Fig. 4, the data by open symbols are recorded without laser noise, while that by closed symbols are recorded under the conditions that not only the laser noise is included but also multiple independent noise realizations are conducted [51]. Similar to the phenomenon reported in [51], when the laser noise is taken into account, the abrupt blue-shift of $f_{0}$ is observed not only at one specific $\tau _\textrm {RT}$ but also over a limited range of delay time around it. Within such a delay time range, two $f_{0}$ values are possible for each $\tau _\textrm {RT}$, revealing the effect of laser-noise-induced frequency hopping.

 

Fig. 5. FBG feedback generated P1 (a) microwave frequency $f_{0}$ and corresponding (b) ratio $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$ as $\tau _\textrm {RT}$ varies at long delay. Closed and open symbols respectively show the data with and without laser noise. $\xi _\textrm {f}$ = 0.188. The FBG has $f_\textrm {BW}/f_\textrm {r}$ = 0.5 and $\tau _\textrm {g}$ = 57 ps.

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3.3 P1 stability versus ($\xi _\textrm {f}$, $\tau _\textrm {RT}$)

As pointed out in previous discussions, P1 microwave frequency $f_{0}$ is more sensitive to the feedback delay time rather than to the feedback strength. Due to interactions between $f_{0}$ and external cavity modes, frequency locking and mode hopping can be observed alternatively by increasing the feedback delay time. To study the stability of P1 oscillations, Figs. 6 and 7 examine the microwave linewidth $\Delta f$, signal SPSR, and phase noise variance by varying feedback parameters. The phase noise variance is estimated by integrating the noise single sideband of the power spectrum centered at $f_{0}$, with normalization to the microwave power, over an offset from 10 MHz to 500 MHz [24,51]. This integration range does not include external cavity modes, thus eliminating the contribution of the noise side peaks.

 

Fig. 6. (a) Microwave linewidth $\Delta f$, (b) SPSR, and (c) phase noise variance as a function of $\tau _\textrm {RT}$. Closed and open symbols respectively correspond to FBG and mirror feedback. The bandwidth of FBG corresponds to $f_\textrm {BW}/f_\textrm {r}$ = 0.5. Dashed line is the linewidth of the free-running laser.

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Fig. 7. (a) Microwave linewidth $\Delta f$, (b) SPSR, and (c) phase noise variance as a function of $\xi _\textrm {f}$. Closed and open symbols respectively represent FBG feedback and mirror feedback. The bandwidth of FBG corresponds to $f_\textrm {BW}/f_\textrm {r}$ = 0.5.

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The dependence of microwave linewidth $\Delta f$, signal SPSR, and phase noise variance on $\tau _\textrm {RT}$ are respectively shown in Figs. 6(a)$-$(c). Closed symbols represent data from FBG feedback with $\xi _\textrm {f}$ = 0.188, which correspond to microwave frequencies when varying $\tau _\textrm {RT}$ in Fig. 4(a). And open symbols correspond to mirror feedback with $\xi _\textrm {f}$ = 0.0118 for a comparison. The dashed line denotes the linewidth of the free-running laser. In Fig. 6(a), for FBG feedback, when $\tau _\textrm {RT}$ varies from 0.46 ns to 0.48 ns where $f_{0}$ continuously red-shift while the frequency ratio $f_{0}(\tau _\textrm {RT} + \tau _\textrm {g})$ dose not change much in Fig. 4, a local minimum of $\Delta f$ is observed at around the middle of the delay time range. As addressed above, the constant frequency ratio reveals a frequency locking to the closest external cavity mode. The local minimum $\Delta f$ is attained when this external cavity mode reaches its strongest magnitude response by the filtering effect, denoting the optimal frequency locking. When this external cavity mode shifts away as $\tau _\textrm {RT}$ varies, the $\Delta f$ starts to broaden until reaching a local maximum right before the mode hopping occurs. Globally, there are a bunch of local minimum linewidth that are roughly spaced apart by about 43 ps corresponding to the reciprocal of $f_{0}$ [28]. Besides the apparent periodicity, the local minimum linewidth becomes smaller at longer feedback delay time. Similar evolution of $\Delta f$ has been observed in an optically injected semiconductor laser with optical feedback stabilization [51]. For mirror feedback shown in open symbols, a similar evolution is observed, except for a longer gap time of about 97 ps due to lower P1 microwave frequencies [28]. Independent of the grating or the mirror, the linewidth $\Delta f$ can be reduced to much smaller than the linewidth of the free-running laser. It is clear from Fig. 6(a) that the FBG is better than the mirror for linewidth reduction by varying $\tau _\textrm {RT}$. The improvement of the $\Delta f$ is further conformed by the evolution of phase noise variance in Fig. 6(c). In Fig. 6(b), for FBG feedback, the evolution of signal SPSR roughly exhibits a reverse trend to that of linewidth $\Delta f$ in Fig. 6(a). As expected, each local maximum SPSR is achieved at $\tau _\textrm {RT}$ roughly corresponding to the local minimum linewidth due to optimal locking between $f_{0}$ and the closest external cavity mode. Though the local maximum SPSR becomes smaller at longer $\tau _\textrm {RT}$ due to denser external cavity modes, Fig. 6(b) shows that the FBG is better than the mirror for SPSR improvement by varying $\tau _\textrm {RT}$.

In addition to the feedback delay time, the stability of P1 oscillations is also investigated by varying the feedback strength $\xi _\textrm {f}$. The dependence of microwave linewidth $\Delta f$, signal SPSR, and phase noise variance on $\xi _\textrm {f}$ are respectively shown in Figs. 7(a)$-$(c). Closed symbols represent data from FBG feedback correspond to microwave frequencies when varying $\xi _\textrm {f}$ in Fig. 4(a). And open symbols represent results from mirror feedback within the P1 region in Fig. 2(b) for a comparison. $\tau _\textrm {RT}$ = 0.65 ns roughly corresponds to the local optimal locking in both FBG and mirror feedback in Fig. 6. In Fig. 7(a), independent of the grating or the mirror, the linewidth $\Delta f$ is reduced by increasing $\xi _\textrm {f}$, as the effect of frequency locking is improved shown by Fig. 4(b). It is clear that the FBG is better than the mirror for linewidth reduction by varying $\xi _\textrm {f}$. The improvement of the $\Delta f$ is further conformed by the results of phase noise variance in Fig. 7(c). In Fig. 7(b), for FBG feedback, the evolution of signal SPSR roughly exhibits a reverse trend to that of linewidth $\Delta f$ in Fig. 7(a). By contrast, for mirror feedback, it has a limited variation by varying $\xi _\textrm {f}$. Figure 7(b) shows that the FBG is better than the mirror for SPSR improvement by varying $\xi _\textrm {f}$.

Generally, the FBG feedback can generate P1 oscillations with better stability than the mirror feedback. On the one hand, P1 oscillations from the FBG feedback can sustain much more feedback to achieve better linewidth and phase noise reduction. On the other hand, the external cavity modes can be limited by the comb-filtered effect of FBG, hence the signal SPSR of P1 oscillations is significantly improved.

3.4 P1 stability versus $f_\textrm {BW}$

Similar to mirror feedback, FBG feedback can optimize the stability of P1 oscillations by proper control of the feedback parameters ($\xi _\textrm {f}$, $\tau _\textrm {RT}$). More than mirror feedback, FBG feedback can further improve the stability by varying the grating bandwidth $f_\textrm {BW}$ through adjusting the grating coupling coefficient $\kappa$. Figures 8(a)$-$(c) respectively investigate the dependence of linewidth $\Delta f$, signal SPSR, and phase noise variance on the normalized bandwidth $f_\textrm {BW}/f_\textrm {r}$ by closed symbols. The open symbols in Fig. 8(a) examine the corresponding microwave frequency $f_{0}$. When $f_\textrm {BW}/f_\textrm {r}$ < 1, the grating bandwidth is smaller than the laser relaxation resonance frequency, which denotes an effective suppression of the coherence collapse. The data are recorded when P1 states are achieved under a relatively strong feedback of $\xi _\textrm {f}$ = 0.188. Circles and squares respectively represent results using the feedback delay time of $\tau _\textrm {RT}$ = 0.65 ns and 1.30 ns. Independent of $\tau _\textrm {RT}$, the open symbols in Fig. 8(a) show that adjusting the grating bandwidth does not contribute much to the microwave frequency. When using $\tau _\textrm {RT}$ = 0.65 ns, in Fig. 8(a), the linewidth $\Delta f$ is reduced from 2.8 MHz to a level of about 110 kHz by increasing $f_\textrm {BW}/f_\textrm {r}$. Such a dependence of $\Delta f$ on grating bandwidth is related to the properly enhanced feedback power from the FBG side lobes. Since increasing the grating bandwidth in this simulation is conducted by increasing the grating coupling coefficient $\kappa$, which is accompanied by an increment of the reflectivity of side lobes, and thus enhances the power reflected from the side lobes [49]. The enhancement of the $\Delta f$ is further conformed by examining the phase noise variance in Fig. 8(c). In Fig. 8(b), the signal SPSR is found to have a reverse trend to that of $\Delta f$. It can be improved from 32 dB to a level of about 54 dB by increasing $f_\textrm {BW}/f_\textrm {r}$, which reveals an excellent suppression of the external cavity modes. Though increasing feedback delay time is an effective method to reduce the linewidth $\Delta f$, a degradation of SPSR is observed in Fig. 8(b) by using a longer delay time of $\tau _\textrm {RT}$ = 1.30 ns. Because the external cavity modes become much denser as $\tau _\textrm {RT}$ increases. Nonetheless, squares in Fig. 8 show that the linewidth can be reduced down to 20 kHz with a relatively high SPSR of about 49 dB when $f_\textrm {BW}/f_\textrm {r}$ = 0.59. While further decreasing or increasing the main lobe bandwidth quickly drives the laser out of P1 regime, adjusting the side lobe bandwidth via changing the grating length could be a potential degree of freedom for further optimizing P1 oscillations in the future work.

 

Fig. 8. (a) Microwave linewidth $\Delta f$, (b) SPSR, and (c) phase noise variance as a function of $f_\textrm {BW}/f_\textrm {r}$ in closed symbols. For reference, the microwave frequencies $f_{0}$ are shown in open symbols. Circles and squares respectively represent $\tau _\textrm {RT}$ = 0.65 ns and 1.30 ns. The feedback strength is $\xi _\textrm {f}$ = 0.188.

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4. Conclusion

In conclusion, a semiconductor laser under comb-filtered optical feedback from a narrowband FBG is numerically investigated for generating and stabilizing P1 oscillations. By FBG feedback, the laser can oscillate in P1 state across a Hopf-bifurcation through undamping the relaxation oscillation. Because grating bandwidth is smaller than the laser relaxation oscillation frequency, the FBG feedback effectively suppresses the coherence collapse. Hence it stabilizes P1 operations even under relatively strong feedback where chaotic dynamics are induced in mirror feedback. The FBG feedback effectively expands P1 operation range over $\xi _\textrm {f}$, which not only allows better flexibility in P1 generation but also further reduces P1 microwave linewidth and phase noise by using stronger feedback. Besides, a uniform FBG provides comb-filtered reflection with the main lobe surrounded by several side lobes. Then it can limit the external cavity modes by each lobe, hence significantly improve the SPSR of P1 microwave. The effects of stabilization are enhanced by properly increasing grating bandwidth. Similar to mirror feedback, a longer feedback delay time leads to a better microwave linewidth and phase noise reduction. Nevertheless, FBG feedback reduces the microwave linewidth by up to more than an order of magnitude and improves the SPSR by up to more than two orders of magnitude than mirror feedback for the same delay time.